Earth Surf Dynam 1 29ndash43 2013wwwearth-surf-dynamnet1292013doi105194esurf-1-29-2013copy Author(s) 2013 CC Attribution 30 License

Earth Surface Dynamics

Open A

ccess

The role of hydrological transience inpeatland pattern formation

P J Morris1 A J Baird2 and L R Belyea3

1Soil Research Centre Department of Geography and Environmental Science University of ReadingReading RG6 6DW UK

2School of Geography University of Leeds Leeds LS2 9JT UK3School of Geography Queen Mary University of London 327 Mile End Road London E1 4NS UK

Correspondence toP J Morris (pjmorrisreadingacuk pauljohnmorrisgmailcom)

Received 7 May 2013 ndash Published in Earth Surf Dynam Discuss 6 June 2013Revised 18 September 2013 ndash Accepted 30 September 2013 ndash Published 16 October 2013

Abstract The sloping flanks of peatlands are commonly patterned with non-random contour-parallel stripesof distinct micro-habitats such as hummocks lawns and hollows Patterning seems to be governed by feedbacksamong peatland hydrological processes plant micro-succession plant litter production and peat decomposi-tion An improved understanding of peatland patterning may provide important insights into broader aspectsof the long-term development of peatlands and their likely response to future climate change

We recreated a cellular simulation model from the literature as well as three subtle variants of the modelto explore the controls on peatland patterning Our models each consist of three submodels which simulatepeatland water tables in a gridded landscape micro-habitat dynamics in response to water-table depths andchanges in peat hydraulic properties

We found that the strength and nature of simulated patterning was highly dependent on the degree to whichwater tables had reached a steady state in response to hydrological inputs Contrary to previous studies wefound that under a true steady state the models predict largely unpatterned landscapes that cycle rapidly be-tween contrasting dry and wet states dominated by hummocks and hollows respectively Realistic patterningonly developed when simulated water tables were still transient

Literal interpretation of the degree of hydrological transience required for patterning suggests that the modelshould be discarded however the transient water tables appear to have inadvertently replicated an ecolog-ical memory effect that may be important to peatland patterning Recently buried peat layers may remainhydrologically active despite no longer reflecting current vegetation patterns thereby highlighting the poten-tial importance of three-dimensional structural complexity in peatlands to understanding the two-dimensionalsurface-patterning phenomenon

The models were highly sensitive to the assumed values of peat hydraulic properties which we take to in-dicate that the models are missing an important negative feedback between peat decomposition and changesin peat hydraulic properties Understanding peatland patterning likely requires the unification of cellular land-scape models such as ours with cohort-based models of long-term peatland development

1 Introduction

11 Background

The surface of northern peatlands often comprises a patch-work of distinct small-scale (lt10 m known as scale-level 1or SL1 ndash Baird et al 2009) micro-habitats each with a char-

acteristic vegetation type micro-topographic relief water-table and soil-moisture regimes soil hydraulic propertiesand soil biogeochemical regime (Ivanov 1981 Alm et al1997 Belyea and Clymo 2001 Rydin and Jeglum 2006)These micro-habitats commonly aggregate into larger spatial

Published by Copernicus Publications on behalf of the European Geosciences Union

30 P J Morris et al The role of hydrological transience in peatland pattern formation

structures at horizontal scales of tens to hundreds of metres(SL2) often forming landscapes composed of strongly direc-tional non-random patterns that may be linear and contourparallel polygonal or maze like (eg Aber et al 2002 Ep-pinga et al 2008 Korpela et al 2009) Baird et al (2009)demonstrated that the frequency distribution of water-tabledepths from across a peatland landscape depends not juston the proportion of the landscape covered by differentmicro-habitats but also on the pattern In consequence pat-tern may play an important role in determining peatlandndashatmosphere fluxes of greenhouse carbon gases (carbon diox-ide and methane) which vary with water-table depth (Bubieret al 1993 1995 Roulet et al 2007) Additionally it seemslikely that the development and maintenance of peatland pat-terning is governed by the same mechanisms that control peataccumulation decomposition and the development of soilhydraulic properties (Belyea and Clymo 2001 Nungesser2003 Eppinga et al 2009) An improved understanding ofthe mechanisms that control patterning may therefore revealfundamental rules that govern broader aspects of peatlandecosystem and soil development at the landscape scale

Observational (eg Foster and Fritz 1987 Belyea andClymo 2001 Comas et al 2005) and modelling (egNungesser 2003 Swanson 2007) studies have identified avariety of aspatial or one-dimensional (vertical only) feed-back mechanisms that may help to explain directionlessclumping of SL1 units into larger features However un-derstanding the highly directional nature of patterning seenin many peatlands clearly requires an explicit considera-tion of spatial interactions This problem lends itself natu-rally to investigation using 2- or 3-dimensional simulationmodels in which directional transfers of water energy andnutrients and their effects on pattern can be explored di-rectly Interplay between long- and short-range processes isa recurring theme in many apparently successful models ofpatterned landscapes including peatlands (Rietkerk et al2004a b Eppinga et al 2009) and other landscape typessuch as marshes (van de Koppel and Crain 2006) and dry-lands (Lefever and Lejeune 1997)

One current hypothesis on peatland pattern formation theponding mechanism has been explored using cellular land-scape models in a number of previous studies The pondingmechanism consists of a pair of competing feedbacks be-tween water-table depth peatland micro-habitat successionand peat hydraulic properties Areas with deeper water tablesare assumed to be more likely to support hummock vegeta-tion whereas areas with shallower water tables are assumedto be more likely to support hollow vegetation (cf Rydinand Jeglum 2006) Furthermore hummock vegetation is as-sumed to produce near-surface peat that is less permeablethan that produced in hollows (cf Ivanov 1981) Contrast-ing micro-habitat states (hummock hollow) between adja-cent SL1 units allow a positive feedback that reinforces localdifferences in water tables Ponding occurs upslope of hum-mocks leading to hollows there while areas downslope of

Figure 1 Aerial photograph showing contour-parallel striped pat-terning on a peatland complex in the James Bay lowlands OntarioCanada The directions of slope and regional water flow are fromthe top right to the bottom left of the picture Horizontal distancebetween tops of successive ridges is approximately 5 to 10 m Imagebelongs to Brian Branfireun reproduced here with kind permission

hummocks are deprived of water encouraging the develop-ment of hummocks Conversely rapid drainage occurs up-slope of hollows leading to hummocks there while areasdownslope of hollows are supplied readily with water en-couraging the development of hollows This short-range pos-itive feedback competes against a long-range negative feed-back whereby water flow between the modelrsquos boundariestends to homogenise local variations in water tables Swan-son and Grigal (1988) Couwenberg (2005) and Couwen-berg and Joosten (2005) found that when implemented in anumerical cellular landscape in which shallow groundwaterflow is predominantly downslope the ponding mechanismgenerated realistic-looking contour-parallel stripes of hum-mocks and hollows at SL2 (we henceforth refer to these threepapers and the versions of the ponding model that they de-scribe collectively as SGCJ) However Eppinga et al (2009)came to different conclusions when using their own cellularmodel to explore the individual and combined effects of theponding mechanism and two other feedbacks They foundthat contour-parallel striped patterning such as that shown inFig 1 developed under several combinations of these feed-backs but not when the ponding mechanism alone was usedthat is when using the ponding model in isolation they wereunable to find a combination of parameters that generatedanything other than a homogeneous unpatterned landscapeseemingly in disagreement with the earlier studies Conse-quently it is difficult to judge whether the ponding mecha-nism alone still represents a viable theory of peatland pat-terning

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 31

Table 1 Glossary of algebraic terms including default values where appropriate for each of the four model versions

Symbol Description Dimensions Units Model 1 Model 2 Model 3 Model 4

B peatland surface height above arbitrary datum L m 022 to 420 m 022 to 420 m 048 to 417 m 048 to 417 m∆te hydrological submodel runtime (steady-state criterion) T h 50 50 50 17 520 (2 yr)d thickness of flow L m ndash auxiliary variable auxiliary variable auxiliary variableH water-table height above arbitrary datum L m auxiliary variable auxiliary variable auxiliary variable auxiliary variableK hydraulic conductivity L Tminus1 m sminus1 ndash ndash ndash ndashKave depth-averaged hydraulic conductivity L Tminus1 m sminus1 ndash ndash ndash ndashKdeep hydraulic conductivity of deep peat L Tminus1 m sminus1 ndash 0 125times10minus5 125times10minus5

Khol hydraulic conductivity of hollow peat L Tminus1 m sminus1 ndash 10times10minus3 10times10minus3 10times10minus3

Khum hydraulic conductivity of hummock peat L Tminus1 m sminus1 ndash 50times10minus5 50times10minus5 50times10minus5

p probability of hummock formation ndash ndash Eq (3) Fig 3 Eq (3) Fig 3 Eq (3) Fig 3 Eq (3) Fig 3Q hummock turnover rate ndash ndash state variable state variable state variable state variableR relative variance of hummocks per row ndash ndash state variable state variable state variable state variableS proportion of model landscape occupied by hummocks ndash ndash state variable state variable state variable state variableθ peat drainable porosity ndash ndash 03 03 03 03T peat transmissivity L2 Tminus1 m2 sminus1 ndash ndash ndash ndashThol transmissivity of hollow cells L2 Tminus1 m2 sminus1 20times10minus4 ndash ndash ndashThum transmissivity of hummock cells L2 Tminus1 m2 sminus1 10times10minus5 ndash ndash ndashU net rainfall rate (precip minus evapotran) L Tminus1 mm yrminus1 0 0 400 (constant) 400 (time series)x spatial index (across-slope direction) L m ndash ndash ndash ndashy spatial index (along-slope direction) L m ndash ndash ndash ndashZ water-table depth below peat surface L m auxiliary variable auxiliary variable auxiliary variable auxiliary variableZfinal final water-table depth at end of developmental step L m state variable state variable state variable ndashZmean mean water-table depth during second half of dev step L m ndash ndash ndash state variable

12 Aim and objectives

We recreated the SGCJ version of the ponding model to ex-plore three characteristics that we suspected may have indi-vidually or collectively led to a Type-1 error in the SGCJstudies (ie causing a model to predict patterning despite theponding mechanism being incapable of generating patterningin reality)

i Numerical implementation of shallow groundwaterflow the transmissivityT [dimensions of L2 Tminus1] (seeTable 1 for a glossary of algebraic terms) of any gridsquare in the original SGCJ models depends entirely onwhether that square is currently a hummock or a hol-low Only two values ofT are possible one canonicalvalue for hummocksThum and one for hollowsThol Amore realistic scheme would have been to assign canon-ical values of a more intrinsic property of peat such assaturated hydraulic conductivityKhum andKhol [L Tminus1]to hummocks and hollows respectively and to calcu-late transmissivity as the product ofK and the thicknessof flow in that square thereby allowing for continuousvariation inT across the model landscape We extendedthe hydrological submodel from the SGCJ models inthis way in order to remove any unrealistic constraintsthat the simplified numerical implementation of shallowgroundwater flow may have placed upon overall modelbehaviour We also wished to explore the effects of theabsolute values ofT or K (as appropriate) upon modelbehaviour

ii Conceptual hydrogeological setting the original SGCJmodels considered only a shallow layer of near-surfacepeat and assumed that deeper peat was impermeableto groundwater flow The top few decimetres of peat

are usually the most permeable (eg Fraser et al 2001Clymo 2004) and are therefore prone to the most rapidsubsurface flow although deeper peat is rarely truly im-permeable Indeed a number of studies have indicatedthat drainage through deep peat layers may play an im-portant role in peatland development and the ability ofthese ecosystems to self-organise (eg Ingram 1982Belyea and Baird 2006 Morris et al 2011) The as-sumption of impermeable peat below the uppermost fewdecimetres may have prevented the SGCJ models fromrepresenting a potentially important hydrological inter-action between surficial hydraulic structures and deeperpeat layers Furthermore in the original SGCJ modelsrainfall was not included the only inputs of water tothe models were from shallow groundwater flow andorsurface runon In order to address these conceptual is-sues we experimented with the effects of incorporatinga permeable lower layer and driving the model usingsimulated rainfall

iii Dependence on hydrological transience the originalSGCJ authors reported that strong striped patterning oc-curred in their models under ldquosteady-staterdquo hydrolog-ical conditions whereby micro-habitat transitions onlytook place once the simulated water-table map had equi-librated with with the current distribution of hummocksand hollows (and so the spatial arrangement of transmis-sivity) (see Sect 2 for full model description) The cri-terion used to determine steady state considers the pro-portion of cells in which the rate of water-table change[L Tminus1] is less than a threshold rate Steady-state con-ditions are deemed to occur when the proportion ofcells below the threshold is greater than a proportion setby the model user However preliminary experiments

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

32 P J Morris et al The role of hydrological transience in peatland pattern formation

(not reported here in full) with our own version of theponding model indicated that the simulations reportedin the earlier SGCJ studies may not have attained truehydrological steady state and that curiously pattern-ing only developed under intermediate steady-state cri-teria where water tables were still transient We choseto experiment with the effects of hydrological tran-sience upon model patterning either by manipulatingthe water-table steady-state criterion or by driving themodel with a real time series of rainfall data

2 Models and methods

21 Overview

We began with a model that was as similar as possible tothat employed by Couwenberg (2005) as far as the origi-nal model description allows As well as this replica model(henceforth Model 1) we created three additional models(Models 2 3 and 4) with slightly altered routines We de-signed Model 2 so as to address the numerical implementa-tion issue identified in objective (i) while we designed Mod-els 3 and 4 so as to address the conceptual issues identifiedin objective (ii) (see Sect 12 above and Table 2) Each ofthe four models may be thought to consist of three submod-els that simulate shallow saturated groundwater movementsand the spatial distribution of water-table depths at SL1 (hy-drological submodel) switches in micro-habitat type (eco-logical submodel) and changes in peat soil hydraulic prop-erties (soil properties submodel) Model time progresses indevelopmental steps during each developmental step the hy-drological submodel is run until a predetermined steady-statecriterion is met The output from the hydrological submodelis a map of water-table depths within the model landscapeThe ecological submodel then simulates a new micro-habitatmap within the model landscape on the basis of the water-table map such that every cell in the model landscape isassigned one of two binary micro-habitat types hummockor hollow Finally the soil properties submodel uses the newmicro-habitat map to reassign the spatial distribution of near-surface peat hydraulic properties The new map of peat hy-draulic properties is then used as an input to the hydrologicalmodel at the beginning of the next developmental step Aswith the SGCJ studies our models do not simulate explic-itly the processes of peat formation or decomposition andtherefore do not incorporate a peat mass balance Rather themodels represent plant community succession as a shiftingmosaic of SL1 tiles superimposed onto a static peat land-form

22 Hydrological submodel

The hydrological submodel uses a modification of the Di-giBog model of peatland saturated hydrology (Baird et al2012) We took the original DigiBog Fortran 95 code and

added routines to represent the ecological and hydrophys-ical submodels In the current study we deactivated Digi-Bogrsquos peat accumulation decomposition and hydrophysicalsubroutines described by Morris et al (2012) We refer thereader to Baird et al (2012) for a comprehensive descrip-tion of DigiBogrsquos governing equations and numerical imple-mentation although a few points are pertinent here DigiBogrepresents a peatland as a grid of vertical peat columns inplan each column is equivalent to one square grid cell inthe SGCJ models although DigiBog also allows for ver-tical variation in peat hydraulic properties Horizontal sat-urated groundwater flow occurs between adjacent columns(four-square neighbourhood) at a rate equal to the productof the harmonic mean of inter-cell transmissivity (the prod-uct of depth-averaged hydraulic conductivity and the thick-ness of flow) and inter-cell hydraulic gradient according tothe Boussinesq equation (see also McWhorter and Sunada1977)

partHpartt=part

partx

(Kave(d)θ

dpartHpartx

)+part

party

(Kave(d)θ

dpartHparty

)+

Uθ (1)

whereH is water-table height [L] above an arbitrary datumt is simulated time [T]x andy are across-slope and along-slope horizontal spatial dimensions respectively [L]Kave isdepth-averaged hydraulic conductivity [L Tminus1] θ is drainableporosity of the peat [dimensionless] assumed constant at 03d is thickness of flow [L] defined as the height of the watertable H minus the height of the impermeable base layerand U is net rainfall [L Tminus1] BecauseU is added directlyto the saturated zone it may be thought of as a black-boxrepresentation of total precipitation minus the sum of sur-face runoff interception evapotranspiration and unsaturatedwater-storage change

The hydrological model is run for a predetermined lengthof simulated time in order to allow the water-table map tochange in response to the boundary conditions and the cur-rent distribution of hydraulic properties In Models 1 to 3 thelength of time that the hydrological model is allowed to run∆te [T] before an ecological transition takes place (Sect 23)determines how close to a genuine steady state the simulatedwater tables are Short equilibration times lead to highly tran-sient water tables that are still changing rapidly and are farfrom being in equilibrium with the hydrological inputs orboundary conditions the opposite is true of long runtimes InModels 1 2 and 3 we varied∆te between 1 h (highly transientwater tables) and 10 000 h (approximately 417 days highlysteady water tables) to examine the effect of the steady-statecriterion upon model behaviour We performed only a singlesimulation with Model 4 with∆te = 17520 h (equal to twoyears of simulated time) and introduced hydrological tran-sience by driving the model using a daily rainfall time seriesderived from observed data (Sect 25)

We performed a simple test on Models 1 2 and 3 in orderto verify that increasing∆te led to increasing hydrologicalsteadiness We ran each model for a spin-up period of 25

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 33

Table 2 Summary of the four models and the objectives addressed by each

Model Transmissivity Deep peat layer Net rainfallU Up-slope boundary Aquifer Objectivesscheme (precip minus evapotran) condition shape addressed

1 binary impermeable off constant headZ = 0025 m constant slope iii2 continuous impermeable off constant headZ = 0025 m constant slope i iii3 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (constant) impermeable (no-flow) hemi-elliptical i ii iii4 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (time series) impermeable (no-flow) hemi-elliptical iii

Figure 2 Time series of daily-average absolute water-table changeper grid square during five-year test periods with Models 1 2 and3 See main text for full details Note the logarithmic scales on bothaxes

developmental steps with all default parameter values (seeTable 1) and∆te = 50 h We then allowed the hydrologicalsubmodel to continue for a further five years of simulatedtime without any further changes in the spatial distributionsof micro-habitats or peat properties For each simulated dayof the five-year test we calculated the daily sum of absolutewater-table movements in each model grid square and thenaveraged these absolute changes for all grid squares In allthree models the average rate of water-table change declinedrapidly from initial values of between 14 and 19 mm dayminus1

to between 003 and 014 mm dayminus1 by the end of the five-year test period (Fig 2) The tests indicate that water-tablebehaviour for all three models converges on steady state withincreasing∆te

The depth of the water table below the surface at any pointin model time is given simply by

Z = BminusH (2)

whereZ is water-table depth [L] andB is the height of thepeatland surface relative to the arbitrary datum [L] We referto the value ofZ at the end of a developmental step asZfinaland the time-averaged value ofZ during the second half of adevelopmental step asZmean These two metrics are used asinputs to the ecological submodel (Sect 23)

Figure 3 Graphical representation of the linear probability func-tion used by the ecological submodel to assign hummock and hol-low states to each model cell based on water-table depth

23 Ecological submodel

In Models 1 2 and 3 the probabilityp [dimensionless]of any model cell being designated as a hummock duringa given developmental step is a linear function of water-table depthZfinal at the end of the previous developmentalstep When the water table is at the surface in any cell (iewhenZfinal = 000 m) p= 0 (ie that cell is necessarily des-ignated as a hollow for the following developmental step)The value ofp increases linearly with increasing water-tabledepth up toZfinal = 005 m WhenZfinal is equal to or greaterthan 005 m the cell is automatically designated a hummock(ie p= 1) The relevant equations (see also Fig 3) are

p= 00 for Z le 000m

p= 20Z for 000mlt Z lt 005m

p= 10 for Z ge 005m (3)

whereZ is equal to either final water-table depth at the endof a developmental stepZfinal (Models 1 2 and 3) or time-averaged water-table depth during the second half of a two-year developmental stepZmean(Model 4) (see below)

We refer to the upper 005 m of the soil profile as the tran-sition zone All cells that are not designated as hummocksare automatically designated as hollows Swanson and Gri-gal (1988) provide a full description of micro-habitat tran-sitions as simulated in our Models 1 2 and 3 In Model 4micro-habitat transitions are dealt with slightly differentlyRather thanZfinal Model 4 assigns hummocks and hollows

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

34 P J Morris et al The role of hydrological transience in peatland pattern formation

based on each grid squarersquos time-averaged water-table depthZmean [L] during the second half of the previous 2 yr devel-opmental step The first 365 days of each developmental stepare used to allow Model 4rsquos simulated water tables to adjustto the newly updated distribution of peat hydraulic proper-ties but this adjustment period is not incorporated intoZmeanThis measure was intended to ensure thatZmean in Model4 contains no meaningful artefact of water-table geometriesfrom earlier developmental steps Along with a variable rain-fall time series (see Sect 25 below) the use ofZmean pro-vided a means of introducing transience to water-table be-haviour in an arguably more realistic manner than the shortequilibration times used in Models 1 to 3

24 Soil properties submodel

Hummocks are assumed to produce peat that is less perme-able than that produced by hollows DigiBogrsquos implemen-tation of the Boussinesq equation uses depth-averaged satu-rated hydraulic conductivityKave and the thickness of flowd to calculate saturated groundwater flux between adjacentcolumns (Eq 1) However the original SGCJ models simplyassigned a single value of transmissivityThum to hummocksand another higher valueThol to hollows Model 1 uses thesimple treatment of canonical transmissivities as per the orig-inal SGCJ models thus in the case of Model 1 Eq (1) sim-plifies to

partHpartt=part

partx

(Tθ

partHpartx

)+part

party

(Tθ

partHparty

)+

Uθ (4)

We assumed default transmissivity values ofThol = 20times10minus4 m2 sminus1 andThum= 10times10minus5 m2 sminus1 thereby preservingthe Thol to Thum ratio of 20 1 that led to strong pattern-ing in the original SGCJ models Models 2 3 and 4 use amore sophisticated and realistic treatment allowed by Digi-Bog whereby hummocks and hollows are assigned canonicalvalues of hydraulic conductivityKhum andKhol respectivelyT for each cell is recalculated during each iteration of the hy-drological submodel as the product ofK and the thickness offlow d (height of water table above the modelrsquos impermeablebase layer) (cf Freeze and Cherry 1979) In this wayT isable to vary in a continuous manner based on water-table po-sition which is more realistic than the simple binary-T treat-ment of the original SGCJ models (Eq 1) For Models 2 3and 4 we assumed default values ofKhol = 10times10minus3 m sminus1

and Khum= 50times10minus5 m sminus1 thereby giving aKhol to Khum

ratio of 20 1 In models 1 2 and 3 we also manipulated thedefault values ofT or K by factors of between 005 and 20in line with objective (i) In Model 1 we variedThum between50times10minus7 and 20times10minus4 m2 sminus1 andThol between 10times10minus5

and 40times10minus3 m2 sminus1 whilst always maintaining aThol toThum ratio of 20 1 Similarly in Models 2 and 3 we variedKhum between 25times10minus6 and 10times10minus3 m sminus1 andKhol be-tween 5times10minus5 and 2times10minus2 m sminus1 whilst maintaining aKhol

to Khum ratio of 20 1

25 Model spatial and temporal domains boundaryconditions

We implemented all simulations in a 70 (across-slopex di-rection)times200 (along-slopey direction) grid of 1 mtimes1 mgrid squares In Models 1 and 2 the simulated peat aquiferoverlaid a sloping impermeable base (replicating the assump-tion of an impermeable lower peat layer in the SGCJ mod-els) Both the peat surface and the impermeable base had aconstant slope of 1 50 the permeable upper peat had a uni-form thickness of 02 m In Models 3 and 4 we assumed thethick lower peat layer is also permeable with its own hy-draulic conductivityKdeep meaning thatK for each cell isdepth-averaged to account for the vertical transition inK be-tween the upper and lower layers Baird et al (2012) pro-vide a full description of DigiBogrsquos calculation ofKave andinter-cell T We used the groundwater mound equation (In-gram 1982) to calculate the dimensions of a deep peat layerthat is hemi-elliptical in cross section has a uniformK of125times10minus5 m sminus1 is 200 m from central axis to the margin isunderlain by a flat impermeable base and receives a net wa-ter input (from an implied upper peat layer) of 155 mm yrminus1The resulting deep peat layer was 397 m thick along its cen-tral axis curving elliptically down to 028 m at the peatlandrsquosmargin Within DigiBog this deep layer was overlain by asurficial more permeable layer which like Models 1 and 2was 02 m thick

We allowed all simulations to run for 100 developmentalsteps the initial condition consisted of randomly generatedwater tables between 00 and 005 m below the peat surfacein each cell The initial micro-habitat and soil-propertiesmaps were based on this random initial water-table map inthe usual manner described above

In all simulations we set the lateral (along-slope) bound-aries to impermeable (Neumann zero-flux condition) in or-der to simplify the representation of the simulated peatlandwithin its local hydrogeological setting In the simulationswith an impermeable lower peat layer (Models 1 and 2) boththe upslope and downslope boundaries were set to constantwater-table (Dirichlet) conditions of 0025 m below the sur-face such that the overall hydraulic gradient along the modelwas equal to the topographic gradient and the gradient of theimpermeable peat layer As with the original SGCJ modelsModels 1 and 2 received no rainfall all inputs of water inthese models came from the upper boundary condition rep-resenting shallow groundwater influx andor surface runonIn this way Models 1 and 2 might be thought to be more rep-resentative of throughflow fens than of raised bogs In Mod-els 3 and 4 we replaced the constant-head condition at theup-slope boundary with a no-flow (Neumann zero flow) con-dition to represent the drainage divide along the crest of araised bog in a manner similar to Morris et al (2012) (seealso Ingram 1982) but retained the constant water-table con-dition at the lower boundary to represent shallow drainageoutflow to a lag stream for example Water-table levels in

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 35

0

10

20

30

40

50

60

70

80

1 61 121 181 241 301 361

Dai

ly R

ainf

all (

mm

day

-1)

Day of Year

ObservedScaled

Figure 4 365-day time series of daily observed rainfall and itsconversion to net rainfallU after scaling by a factor of 0245 Seemain text for full details

Model 3 were maintained by a constant simulated net rainfallrate ofU = 400 mm yrminus1 (U is assumed to be net of evapo-transpiration interception and runoff hence its low value)We drove Model 4 using a daily time series ofU We took365 days of daily rainfall data from close to Malham TarnMoss a raised bog in North Yorkshire United Kingdom forthe calendar year 2011 The rain gauge recorded precipitationon 294 days of that year with an annual total of 1633 mm ofprecipitation Maximum daily precipitation for the year was786 mm on 10 August We multiplied each daily rainfall to-tal in the time series by approximately 0245 so as to give anannual total of 400 mm of net precipitation equal to the con-stantU assumed in Model 3 whilst maintaining a plausibletemporal variation (Fig 4) During each 2 yr developmentalstep in Model 4 we cycled twice through the 365-day rain-fall series By using the same year of rainfall data repeatedlyin this way we were able to induce water-table transiencewithout the potentially complicating effects of inter-annualvariability in rainfall patterns The thick peat deposit repre-sented in Models 3 and 4 and the fact that their groundwa-ter mounds were maintained by simulated net rainfall meansthat they might be thought to be most representative of om-brotrophic raised bogs

26 Analysis

Previous studies (eg Andreasen et al 2001 Eppinga etal 2009) have demonstrated that the human eye is a pow-erful tool for assessing pattern strength in landscape mod-els and we used a visual appraisal of the two-dimensionalmicro-habitat maps produced by the four models as our pri-mary means of estimating simulated pattern strength In ad-dition we calculated the relative varianceR [dimensionless]of hummocks per across-slope row of cells following themethod of Swanson and Grigal (1988) as an objective andreproducible metric of pattern strength The value ofR in-

creases with pattern strength values less than 2 indicate un-ordered landscapes without discernible patterning values ofR greater than 4 represent highly ordered landscapes withclear strong patterning We also calculated the proportionS[dimensionless] of model cells occupied by hummocks dur-ing each developmental step Finally we calculated modelturnover rateQ [dimensionless] defined as the proportionof model cells that undergo a transition from hummock tohollow or vice versa in each developmental step

3 Results

31 Hydrological transience

In Models 1 2 and 3 the strength of across-slope stripedpatterning at SL2 initially increased strongly with increas-ing hydrological steadiness from an apparently entirely un-ordered random mixture of hummocks and hollows at SL1when∆te = 1 h to peak ldquostrengthsrdquo at around∆te = 100 h (forModels 1 and 2) or 50 h (Model 3) This increase in pattern-ing strength is evident from both a visual appraisal of thefinal micro-habitat maps (Fig 5) and an increase in the val-ues of relative varianceR with increasing∆te (Fig 6a b c)For values of∆te greater than 50 h (Models 1 and 2) and 20 h(Model 3) an increase in the spatial scale of the simulatedSL2 stripes is apparent with increasing∆te (ie the stripesbecame broader in they (along-slope) direction) Increasing∆te also led to SL2 stripes whose upslope and downslopeedges were increasingly straight and sharply defined (Fig 5)For values of∆te greater than 100 h (Models 1 and 2) or 50 h(Model 3) the simulated SL2 stripes had broadened to thepoint of appearing to be ldquoover-developedrdquo such that theyno longer resembled realistic peatland striped patterning (cfFig 1)

As well as governing the spatial configuration of simulatedpatterning the value of∆te also affected the temporal dynam-ics of Models 1 2 and 3 For low values of∆te turnover ratesof model cells were high when∆te = 1 h approximately halfof all cells would undergo a transition from designation as ahummock to a hollow (or vice versa) during each develop-mental step providing further indication of a random land-scape Increasing∆te led to a reduction in turnover rates inModels 1 and 2 as SL2 structures began to stabilise to a min-imum of approximately 40 of cells per developmental stepwhen∆te = 100 h (Fig 6d e) Increasing∆te had little ef-fect on turnover rates in Model 3 until∆te = 1000 or 10 000 h(Fig 6f) The temporal trends in SL1 turnover rate and pro-portional hummock coverage also indicate what appears tobe a limit cycle in the behaviour of all models under highlystringent hydrological steady-state criteria Particularly when∆te = 17520 (Model 4 only) or 10 000 h and to a lesser ex-tent when∆te = 1000 h the models exhibited a cyclical be-haviour in time whereby the simulated landscape would al-ternate on an approximately regular cycle of between fourand ten developmental steps between two contrasting states

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

36 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 5 Final micro-habitat maps after 100 developmental steps showing effects of increasing∆te on patterning in typical simulationswith Model 1 (top panel) Model 2 (middle panel) and Model 3 (bottom panel) The value of∆te assumed for each simulation is indicatedimmediately above the map Light pixels represent hummocks dark pixels represent hollows Low values ofy represent upslope locations(y= 0 is the upslope boundary) high values ofy represent downslope locations (y= 200 is the downslope boundary) as such groundwaterflow is generally down the page All maps have the same horizontal (x y) scale as that shown for the upper-leftmost map

a dry landscape with deeper water tables dominated by hum-mocks and a wet landscape with water tables near the bogsurface dominated by hollows (Fig 6) The contrast be-tween the wet and dry states was particularly pronouncedin Model 2 which cycled between approximately 15 and95 hummock coverage when∆te = 10000 h (Fig 6h) Thesingle simulation with Model 4 (∆te = 17520 h) behaved ina very similar manner to Models 1 2 and 3 when∆te =10000 h predicting nearly homogeneous unpatterned land-scapes (Fig 7) that cycled rapidly between being dominatedby hummocks and hollows (Fig 6f i)

In all simulations that generated striped SL2 pattern-ing those SL2 units migrated consistently downslope (not

shown) in the same manner as that previously reported in thepublished accounts of the SGCJ models

32 Simplified calculation of transmissivity

Any artefacts introduced to model behaviour by the sim-ple binary treatment of peat permeability employed in theoriginal SGCJ models (see objective i above) would havebeen evident as differences in behaviour between Model 1(simple binary transmissivity scheme based solely on hum-mockhollow designation) and Model 2 (continuous treat-ment of transmissivity as product of peat hydraulic conduc-tivity and thickness of flow) Models 1 and 2 behaved in avery similar manner to one another in terms of their response

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P J Morris et al The role of hydrological transience in peatland pattern formation 37

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

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10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

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08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

a) b) c)

d) e) f)

g) h) i)

Model 1 Model 2 Models 3 and 4

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

30 P J Morris et al The role of hydrological transience in peatland pattern formation

structures at horizontal scales of tens to hundreds of metres(SL2) often forming landscapes composed of strongly direc-tional non-random patterns that may be linear and contourparallel polygonal or maze like (eg Aber et al 2002 Ep-pinga et al 2008 Korpela et al 2009) Baird et al (2009)demonstrated that the frequency distribution of water-tabledepths from across a peatland landscape depends not juston the proportion of the landscape covered by differentmicro-habitats but also on the pattern In consequence pat-tern may play an important role in determining peatlandndashatmosphere fluxes of greenhouse carbon gases (carbon diox-ide and methane) which vary with water-table depth (Bubieret al 1993 1995 Roulet et al 2007) Additionally it seemslikely that the development and maintenance of peatland pat-terning is governed by the same mechanisms that control peataccumulation decomposition and the development of soilhydraulic properties (Belyea and Clymo 2001 Nungesser2003 Eppinga et al 2009) An improved understanding ofthe mechanisms that control patterning may therefore revealfundamental rules that govern broader aspects of peatlandecosystem and soil development at the landscape scale

Observational (eg Foster and Fritz 1987 Belyea andClymo 2001 Comas et al 2005) and modelling (egNungesser 2003 Swanson 2007) studies have identified avariety of aspatial or one-dimensional (vertical only) feed-back mechanisms that may help to explain directionlessclumping of SL1 units into larger features However un-derstanding the highly directional nature of patterning seenin many peatlands clearly requires an explicit considera-tion of spatial interactions This problem lends itself natu-rally to investigation using 2- or 3-dimensional simulationmodels in which directional transfers of water energy andnutrients and their effects on pattern can be explored di-rectly Interplay between long- and short-range processes isa recurring theme in many apparently successful models ofpatterned landscapes including peatlands (Rietkerk et al2004a b Eppinga et al 2009) and other landscape typessuch as marshes (van de Koppel and Crain 2006) and dry-lands (Lefever and Lejeune 1997)

One current hypothesis on peatland pattern formation theponding mechanism has been explored using cellular land-scape models in a number of previous studies The pondingmechanism consists of a pair of competing feedbacks be-tween water-table depth peatland micro-habitat successionand peat hydraulic properties Areas with deeper water tablesare assumed to be more likely to support hummock vegeta-tion whereas areas with shallower water tables are assumedto be more likely to support hollow vegetation (cf Rydinand Jeglum 2006) Furthermore hummock vegetation is as-sumed to produce near-surface peat that is less permeablethan that produced in hollows (cf Ivanov 1981) Contrast-ing micro-habitat states (hummock hollow) between adja-cent SL1 units allow a positive feedback that reinforces localdifferences in water tables Ponding occurs upslope of hum-mocks leading to hollows there while areas downslope of

Figure 1 Aerial photograph showing contour-parallel striped pat-terning on a peatland complex in the James Bay lowlands OntarioCanada The directions of slope and regional water flow are fromthe top right to the bottom left of the picture Horizontal distancebetween tops of successive ridges is approximately 5 to 10 m Imagebelongs to Brian Branfireun reproduced here with kind permission

hummocks are deprived of water encouraging the develop-ment of hummocks Conversely rapid drainage occurs up-slope of hollows leading to hummocks there while areasdownslope of hollows are supplied readily with water en-couraging the development of hollows This short-range pos-itive feedback competes against a long-range negative feed-back whereby water flow between the modelrsquos boundariestends to homogenise local variations in water tables Swan-son and Grigal (1988) Couwenberg (2005) and Couwen-berg and Joosten (2005) found that when implemented in anumerical cellular landscape in which shallow groundwaterflow is predominantly downslope the ponding mechanismgenerated realistic-looking contour-parallel stripes of hum-mocks and hollows at SL2 (we henceforth refer to these threepapers and the versions of the ponding model that they de-scribe collectively as SGCJ) However Eppinga et al (2009)came to different conclusions when using their own cellularmodel to explore the individual and combined effects of theponding mechanism and two other feedbacks They foundthat contour-parallel striped patterning such as that shown inFig 1 developed under several combinations of these feed-backs but not when the ponding mechanism alone was usedthat is when using the ponding model in isolation they wereunable to find a combination of parameters that generatedanything other than a homogeneous unpatterned landscapeseemingly in disagreement with the earlier studies Conse-quently it is difficult to judge whether the ponding mecha-nism alone still represents a viable theory of peatland pat-terning

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 31

Table 1 Glossary of algebraic terms including default values where appropriate for each of the four model versions

Symbol Description Dimensions Units Model 1 Model 2 Model 3 Model 4

B peatland surface height above arbitrary datum L m 022 to 420 m 022 to 420 m 048 to 417 m 048 to 417 m∆te hydrological submodel runtime (steady-state criterion) T h 50 50 50 17 520 (2 yr)d thickness of flow L m ndash auxiliary variable auxiliary variable auxiliary variableH water-table height above arbitrary datum L m auxiliary variable auxiliary variable auxiliary variable auxiliary variableK hydraulic conductivity L Tminus1 m sminus1 ndash ndash ndash ndashKave depth-averaged hydraulic conductivity L Tminus1 m sminus1 ndash ndash ndash ndashKdeep hydraulic conductivity of deep peat L Tminus1 m sminus1 ndash 0 125times10minus5 125times10minus5

Khol hydraulic conductivity of hollow peat L Tminus1 m sminus1 ndash 10times10minus3 10times10minus3 10times10minus3

Khum hydraulic conductivity of hummock peat L Tminus1 m sminus1 ndash 50times10minus5 50times10minus5 50times10minus5

p probability of hummock formation ndash ndash Eq (3) Fig 3 Eq (3) Fig 3 Eq (3) Fig 3 Eq (3) Fig 3Q hummock turnover rate ndash ndash state variable state variable state variable state variableR relative variance of hummocks per row ndash ndash state variable state variable state variable state variableS proportion of model landscape occupied by hummocks ndash ndash state variable state variable state variable state variableθ peat drainable porosity ndash ndash 03 03 03 03T peat transmissivity L2 Tminus1 m2 sminus1 ndash ndash ndash ndashThol transmissivity of hollow cells L2 Tminus1 m2 sminus1 20times10minus4 ndash ndash ndashThum transmissivity of hummock cells L2 Tminus1 m2 sminus1 10times10minus5 ndash ndash ndashU net rainfall rate (precip minus evapotran) L Tminus1 mm yrminus1 0 0 400 (constant) 400 (time series)x spatial index (across-slope direction) L m ndash ndash ndash ndashy spatial index (along-slope direction) L m ndash ndash ndash ndashZ water-table depth below peat surface L m auxiliary variable auxiliary variable auxiliary variable auxiliary variableZfinal final water-table depth at end of developmental step L m state variable state variable state variable ndashZmean mean water-table depth during second half of dev step L m ndash ndash ndash state variable

12 Aim and objectives

We recreated the SGCJ version of the ponding model to ex-plore three characteristics that we suspected may have indi-vidually or collectively led to a Type-1 error in the SGCJstudies (ie causing a model to predict patterning despite theponding mechanism being incapable of generating patterningin reality)

i Numerical implementation of shallow groundwaterflow the transmissivityT [dimensions of L2 Tminus1] (seeTable 1 for a glossary of algebraic terms) of any gridsquare in the original SGCJ models depends entirely onwhether that square is currently a hummock or a hol-low Only two values ofT are possible one canonicalvalue for hummocksThum and one for hollowsThol Amore realistic scheme would have been to assign canon-ical values of a more intrinsic property of peat such assaturated hydraulic conductivityKhum andKhol [L Tminus1]to hummocks and hollows respectively and to calcu-late transmissivity as the product ofK and the thicknessof flow in that square thereby allowing for continuousvariation inT across the model landscape We extendedthe hydrological submodel from the SGCJ models inthis way in order to remove any unrealistic constraintsthat the simplified numerical implementation of shallowgroundwater flow may have placed upon overall modelbehaviour We also wished to explore the effects of theabsolute values ofT or K (as appropriate) upon modelbehaviour

ii Conceptual hydrogeological setting the original SGCJmodels considered only a shallow layer of near-surfacepeat and assumed that deeper peat was impermeableto groundwater flow The top few decimetres of peat

are usually the most permeable (eg Fraser et al 2001Clymo 2004) and are therefore prone to the most rapidsubsurface flow although deeper peat is rarely truly im-permeable Indeed a number of studies have indicatedthat drainage through deep peat layers may play an im-portant role in peatland development and the ability ofthese ecosystems to self-organise (eg Ingram 1982Belyea and Baird 2006 Morris et al 2011) The as-sumption of impermeable peat below the uppermost fewdecimetres may have prevented the SGCJ models fromrepresenting a potentially important hydrological inter-action between surficial hydraulic structures and deeperpeat layers Furthermore in the original SGCJ modelsrainfall was not included the only inputs of water tothe models were from shallow groundwater flow andorsurface runon In order to address these conceptual is-sues we experimented with the effects of incorporatinga permeable lower layer and driving the model usingsimulated rainfall

iii Dependence on hydrological transience the originalSGCJ authors reported that strong striped patterning oc-curred in their models under ldquosteady-staterdquo hydrolog-ical conditions whereby micro-habitat transitions onlytook place once the simulated water-table map had equi-librated with with the current distribution of hummocksand hollows (and so the spatial arrangement of transmis-sivity) (see Sect 2 for full model description) The cri-terion used to determine steady state considers the pro-portion of cells in which the rate of water-table change[L Tminus1] is less than a threshold rate Steady-state con-ditions are deemed to occur when the proportion ofcells below the threshold is greater than a proportion setby the model user However preliminary experiments

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

32 P J Morris et al The role of hydrological transience in peatland pattern formation

(not reported here in full) with our own version of theponding model indicated that the simulations reportedin the earlier SGCJ studies may not have attained truehydrological steady state and that curiously pattern-ing only developed under intermediate steady-state cri-teria where water tables were still transient We choseto experiment with the effects of hydrological tran-sience upon model patterning either by manipulatingthe water-table steady-state criterion or by driving themodel with a real time series of rainfall data

2 Models and methods

21 Overview

We began with a model that was as similar as possible tothat employed by Couwenberg (2005) as far as the origi-nal model description allows As well as this replica model(henceforth Model 1) we created three additional models(Models 2 3 and 4) with slightly altered routines We de-signed Model 2 so as to address the numerical implementa-tion issue identified in objective (i) while we designed Mod-els 3 and 4 so as to address the conceptual issues identifiedin objective (ii) (see Sect 12 above and Table 2) Each ofthe four models may be thought to consist of three submod-els that simulate shallow saturated groundwater movementsand the spatial distribution of water-table depths at SL1 (hy-drological submodel) switches in micro-habitat type (eco-logical submodel) and changes in peat soil hydraulic prop-erties (soil properties submodel) Model time progresses indevelopmental steps during each developmental step the hy-drological submodel is run until a predetermined steady-statecriterion is met The output from the hydrological submodelis a map of water-table depths within the model landscapeThe ecological submodel then simulates a new micro-habitatmap within the model landscape on the basis of the water-table map such that every cell in the model landscape isassigned one of two binary micro-habitat types hummockor hollow Finally the soil properties submodel uses the newmicro-habitat map to reassign the spatial distribution of near-surface peat hydraulic properties The new map of peat hy-draulic properties is then used as an input to the hydrologicalmodel at the beginning of the next developmental step Aswith the SGCJ studies our models do not simulate explic-itly the processes of peat formation or decomposition andtherefore do not incorporate a peat mass balance Rather themodels represent plant community succession as a shiftingmosaic of SL1 tiles superimposed onto a static peat land-form

22 Hydrological submodel

The hydrological submodel uses a modification of the Di-giBog model of peatland saturated hydrology (Baird et al2012) We took the original DigiBog Fortran 95 code and

added routines to represent the ecological and hydrophys-ical submodels In the current study we deactivated Digi-Bogrsquos peat accumulation decomposition and hydrophysicalsubroutines described by Morris et al (2012) We refer thereader to Baird et al (2012) for a comprehensive descrip-tion of DigiBogrsquos governing equations and numerical imple-mentation although a few points are pertinent here DigiBogrepresents a peatland as a grid of vertical peat columns inplan each column is equivalent to one square grid cell inthe SGCJ models although DigiBog also allows for ver-tical variation in peat hydraulic properties Horizontal sat-urated groundwater flow occurs between adjacent columns(four-square neighbourhood) at a rate equal to the productof the harmonic mean of inter-cell transmissivity (the prod-uct of depth-averaged hydraulic conductivity and the thick-ness of flow) and inter-cell hydraulic gradient according tothe Boussinesq equation (see also McWhorter and Sunada1977)

partHpartt=part

partx

(Kave(d)θ

dpartHpartx

)+part

party

(Kave(d)θ

dpartHparty

)+

Uθ (1)

whereH is water-table height [L] above an arbitrary datumt is simulated time [T]x andy are across-slope and along-slope horizontal spatial dimensions respectively [L]Kave isdepth-averaged hydraulic conductivity [L Tminus1] θ is drainableporosity of the peat [dimensionless] assumed constant at 03d is thickness of flow [L] defined as the height of the watertable H minus the height of the impermeable base layerand U is net rainfall [L Tminus1] BecauseU is added directlyto the saturated zone it may be thought of as a black-boxrepresentation of total precipitation minus the sum of sur-face runoff interception evapotranspiration and unsaturatedwater-storage change

The hydrological model is run for a predetermined lengthof simulated time in order to allow the water-table map tochange in response to the boundary conditions and the cur-rent distribution of hydraulic properties In Models 1 to 3 thelength of time that the hydrological model is allowed to run∆te [T] before an ecological transition takes place (Sect 23)determines how close to a genuine steady state the simulatedwater tables are Short equilibration times lead to highly tran-sient water tables that are still changing rapidly and are farfrom being in equilibrium with the hydrological inputs orboundary conditions the opposite is true of long runtimes InModels 1 2 and 3 we varied∆te between 1 h (highly transientwater tables) and 10 000 h (approximately 417 days highlysteady water tables) to examine the effect of the steady-statecriterion upon model behaviour We performed only a singlesimulation with Model 4 with∆te = 17520 h (equal to twoyears of simulated time) and introduced hydrological tran-sience by driving the model using a daily rainfall time seriesderived from observed data (Sect 25)

We performed a simple test on Models 1 2 and 3 in orderto verify that increasing∆te led to increasing hydrologicalsteadiness We ran each model for a spin-up period of 25

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 33

Table 2 Summary of the four models and the objectives addressed by each

Model Transmissivity Deep peat layer Net rainfallU Up-slope boundary Aquifer Objectivesscheme (precip minus evapotran) condition shape addressed

1 binary impermeable off constant headZ = 0025 m constant slope iii2 continuous impermeable off constant headZ = 0025 m constant slope i iii3 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (constant) impermeable (no-flow) hemi-elliptical i ii iii4 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (time series) impermeable (no-flow) hemi-elliptical iii

Figure 2 Time series of daily-average absolute water-table changeper grid square during five-year test periods with Models 1 2 and3 See main text for full details Note the logarithmic scales on bothaxes

developmental steps with all default parameter values (seeTable 1) and∆te = 50 h We then allowed the hydrologicalsubmodel to continue for a further five years of simulatedtime without any further changes in the spatial distributionsof micro-habitats or peat properties For each simulated dayof the five-year test we calculated the daily sum of absolutewater-table movements in each model grid square and thenaveraged these absolute changes for all grid squares In allthree models the average rate of water-table change declinedrapidly from initial values of between 14 and 19 mm dayminus1

to between 003 and 014 mm dayminus1 by the end of the five-year test period (Fig 2) The tests indicate that water-tablebehaviour for all three models converges on steady state withincreasing∆te

The depth of the water table below the surface at any pointin model time is given simply by

Z = BminusH (2)

whereZ is water-table depth [L] andB is the height of thepeatland surface relative to the arbitrary datum [L] We referto the value ofZ at the end of a developmental step asZfinaland the time-averaged value ofZ during the second half of adevelopmental step asZmean These two metrics are used asinputs to the ecological submodel (Sect 23)

Figure 3 Graphical representation of the linear probability func-tion used by the ecological submodel to assign hummock and hol-low states to each model cell based on water-table depth

23 Ecological submodel

In Models 1 2 and 3 the probabilityp [dimensionless]of any model cell being designated as a hummock duringa given developmental step is a linear function of water-table depthZfinal at the end of the previous developmentalstep When the water table is at the surface in any cell (iewhenZfinal = 000 m) p= 0 (ie that cell is necessarily des-ignated as a hollow for the following developmental step)The value ofp increases linearly with increasing water-tabledepth up toZfinal = 005 m WhenZfinal is equal to or greaterthan 005 m the cell is automatically designated a hummock(ie p= 1) The relevant equations (see also Fig 3) are

p= 00 for Z le 000m

p= 20Z for 000mlt Z lt 005m

p= 10 for Z ge 005m (3)

whereZ is equal to either final water-table depth at the endof a developmental stepZfinal (Models 1 2 and 3) or time-averaged water-table depth during the second half of a two-year developmental stepZmean(Model 4) (see below)

We refer to the upper 005 m of the soil profile as the tran-sition zone All cells that are not designated as hummocksare automatically designated as hollows Swanson and Gri-gal (1988) provide a full description of micro-habitat tran-sitions as simulated in our Models 1 2 and 3 In Model 4micro-habitat transitions are dealt with slightly differentlyRather thanZfinal Model 4 assigns hummocks and hollows

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

34 P J Morris et al The role of hydrological transience in peatland pattern formation

based on each grid squarersquos time-averaged water-table depthZmean [L] during the second half of the previous 2 yr devel-opmental step The first 365 days of each developmental stepare used to allow Model 4rsquos simulated water tables to adjustto the newly updated distribution of peat hydraulic proper-ties but this adjustment period is not incorporated intoZmeanThis measure was intended to ensure thatZmean in Model4 contains no meaningful artefact of water-table geometriesfrom earlier developmental steps Along with a variable rain-fall time series (see Sect 25 below) the use ofZmean pro-vided a means of introducing transience to water-table be-haviour in an arguably more realistic manner than the shortequilibration times used in Models 1 to 3

24 Soil properties submodel

Hummocks are assumed to produce peat that is less perme-able than that produced by hollows DigiBogrsquos implemen-tation of the Boussinesq equation uses depth-averaged satu-rated hydraulic conductivityKave and the thickness of flowd to calculate saturated groundwater flux between adjacentcolumns (Eq 1) However the original SGCJ models simplyassigned a single value of transmissivityThum to hummocksand another higher valueThol to hollows Model 1 uses thesimple treatment of canonical transmissivities as per the orig-inal SGCJ models thus in the case of Model 1 Eq (1) sim-plifies to

partHpartt=part

partx

(Tθ

partHpartx

)+part

party

(Tθ

partHparty

)+

Uθ (4)

We assumed default transmissivity values ofThol = 20times10minus4 m2 sminus1 andThum= 10times10minus5 m2 sminus1 thereby preservingthe Thol to Thum ratio of 20 1 that led to strong pattern-ing in the original SGCJ models Models 2 3 and 4 use amore sophisticated and realistic treatment allowed by Digi-Bog whereby hummocks and hollows are assigned canonicalvalues of hydraulic conductivityKhum andKhol respectivelyT for each cell is recalculated during each iteration of the hy-drological submodel as the product ofK and the thickness offlow d (height of water table above the modelrsquos impermeablebase layer) (cf Freeze and Cherry 1979) In this wayT isable to vary in a continuous manner based on water-table po-sition which is more realistic than the simple binary-T treat-ment of the original SGCJ models (Eq 1) For Models 2 3and 4 we assumed default values ofKhol = 10times10minus3 m sminus1

and Khum= 50times10minus5 m sminus1 thereby giving aKhol to Khum

ratio of 20 1 In models 1 2 and 3 we also manipulated thedefault values ofT or K by factors of between 005 and 20in line with objective (i) In Model 1 we variedThum between50times10minus7 and 20times10minus4 m2 sminus1 andThol between 10times10minus5

and 40times10minus3 m2 sminus1 whilst always maintaining aThol toThum ratio of 20 1 Similarly in Models 2 and 3 we variedKhum between 25times10minus6 and 10times10minus3 m sminus1 andKhol be-tween 5times10minus5 and 2times10minus2 m sminus1 whilst maintaining aKhol

to Khum ratio of 20 1

25 Model spatial and temporal domains boundaryconditions

We implemented all simulations in a 70 (across-slopex di-rection)times200 (along-slopey direction) grid of 1 mtimes1 mgrid squares In Models 1 and 2 the simulated peat aquiferoverlaid a sloping impermeable base (replicating the assump-tion of an impermeable lower peat layer in the SGCJ mod-els) Both the peat surface and the impermeable base had aconstant slope of 1 50 the permeable upper peat had a uni-form thickness of 02 m In Models 3 and 4 we assumed thethick lower peat layer is also permeable with its own hy-draulic conductivityKdeep meaning thatK for each cell isdepth-averaged to account for the vertical transition inK be-tween the upper and lower layers Baird et al (2012) pro-vide a full description of DigiBogrsquos calculation ofKave andinter-cell T We used the groundwater mound equation (In-gram 1982) to calculate the dimensions of a deep peat layerthat is hemi-elliptical in cross section has a uniformK of125times10minus5 m sminus1 is 200 m from central axis to the margin isunderlain by a flat impermeable base and receives a net wa-ter input (from an implied upper peat layer) of 155 mm yrminus1The resulting deep peat layer was 397 m thick along its cen-tral axis curving elliptically down to 028 m at the peatlandrsquosmargin Within DigiBog this deep layer was overlain by asurficial more permeable layer which like Models 1 and 2was 02 m thick

We allowed all simulations to run for 100 developmentalsteps the initial condition consisted of randomly generatedwater tables between 00 and 005 m below the peat surfacein each cell The initial micro-habitat and soil-propertiesmaps were based on this random initial water-table map inthe usual manner described above

In all simulations we set the lateral (along-slope) bound-aries to impermeable (Neumann zero-flux condition) in or-der to simplify the representation of the simulated peatlandwithin its local hydrogeological setting In the simulationswith an impermeable lower peat layer (Models 1 and 2) boththe upslope and downslope boundaries were set to constantwater-table (Dirichlet) conditions of 0025 m below the sur-face such that the overall hydraulic gradient along the modelwas equal to the topographic gradient and the gradient of theimpermeable peat layer As with the original SGCJ modelsModels 1 and 2 received no rainfall all inputs of water inthese models came from the upper boundary condition rep-resenting shallow groundwater influx andor surface runonIn this way Models 1 and 2 might be thought to be more rep-resentative of throughflow fens than of raised bogs In Mod-els 3 and 4 we replaced the constant-head condition at theup-slope boundary with a no-flow (Neumann zero flow) con-dition to represent the drainage divide along the crest of araised bog in a manner similar to Morris et al (2012) (seealso Ingram 1982) but retained the constant water-table con-dition at the lower boundary to represent shallow drainageoutflow to a lag stream for example Water-table levels in

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 35

0

10

20

30

40

50

60

70

80

1 61 121 181 241 301 361

Dai

ly R

ainf

all (

mm

day

-1)

Day of Year

ObservedScaled

Figure 4 365-day time series of daily observed rainfall and itsconversion to net rainfallU after scaling by a factor of 0245 Seemain text for full details

Model 3 were maintained by a constant simulated net rainfallrate ofU = 400 mm yrminus1 (U is assumed to be net of evapo-transpiration interception and runoff hence its low value)We drove Model 4 using a daily time series ofU We took365 days of daily rainfall data from close to Malham TarnMoss a raised bog in North Yorkshire United Kingdom forthe calendar year 2011 The rain gauge recorded precipitationon 294 days of that year with an annual total of 1633 mm ofprecipitation Maximum daily precipitation for the year was786 mm on 10 August We multiplied each daily rainfall to-tal in the time series by approximately 0245 so as to give anannual total of 400 mm of net precipitation equal to the con-stantU assumed in Model 3 whilst maintaining a plausibletemporal variation (Fig 4) During each 2 yr developmentalstep in Model 4 we cycled twice through the 365-day rain-fall series By using the same year of rainfall data repeatedlyin this way we were able to induce water-table transiencewithout the potentially complicating effects of inter-annualvariability in rainfall patterns The thick peat deposit repre-sented in Models 3 and 4 and the fact that their groundwa-ter mounds were maintained by simulated net rainfall meansthat they might be thought to be most representative of om-brotrophic raised bogs

26 Analysis

Previous studies (eg Andreasen et al 2001 Eppinga etal 2009) have demonstrated that the human eye is a pow-erful tool for assessing pattern strength in landscape mod-els and we used a visual appraisal of the two-dimensionalmicro-habitat maps produced by the four models as our pri-mary means of estimating simulated pattern strength In ad-dition we calculated the relative varianceR [dimensionless]of hummocks per across-slope row of cells following themethod of Swanson and Grigal (1988) as an objective andreproducible metric of pattern strength The value ofR in-

creases with pattern strength values less than 2 indicate un-ordered landscapes without discernible patterning values ofR greater than 4 represent highly ordered landscapes withclear strong patterning We also calculated the proportionS[dimensionless] of model cells occupied by hummocks dur-ing each developmental step Finally we calculated modelturnover rateQ [dimensionless] defined as the proportionof model cells that undergo a transition from hummock tohollow or vice versa in each developmental step

3 Results

31 Hydrological transience

In Models 1 2 and 3 the strength of across-slope stripedpatterning at SL2 initially increased strongly with increas-ing hydrological steadiness from an apparently entirely un-ordered random mixture of hummocks and hollows at SL1when∆te = 1 h to peak ldquostrengthsrdquo at around∆te = 100 h (forModels 1 and 2) or 50 h (Model 3) This increase in pattern-ing strength is evident from both a visual appraisal of thefinal micro-habitat maps (Fig 5) and an increase in the val-ues of relative varianceR with increasing∆te (Fig 6a b c)For values of∆te greater than 50 h (Models 1 and 2) and 20 h(Model 3) an increase in the spatial scale of the simulatedSL2 stripes is apparent with increasing∆te (ie the stripesbecame broader in they (along-slope) direction) Increasing∆te also led to SL2 stripes whose upslope and downslopeedges were increasingly straight and sharply defined (Fig 5)For values of∆te greater than 100 h (Models 1 and 2) or 50 h(Model 3) the simulated SL2 stripes had broadened to thepoint of appearing to be ldquoover-developedrdquo such that theyno longer resembled realistic peatland striped patterning (cfFig 1)

As well as governing the spatial configuration of simulatedpatterning the value of∆te also affected the temporal dynam-ics of Models 1 2 and 3 For low values of∆te turnover ratesof model cells were high when∆te = 1 h approximately halfof all cells would undergo a transition from designation as ahummock to a hollow (or vice versa) during each develop-mental step providing further indication of a random land-scape Increasing∆te led to a reduction in turnover rates inModels 1 and 2 as SL2 structures began to stabilise to a min-imum of approximately 40 of cells per developmental stepwhen∆te = 100 h (Fig 6d e) Increasing∆te had little ef-fect on turnover rates in Model 3 until∆te = 1000 or 10 000 h(Fig 6f) The temporal trends in SL1 turnover rate and pro-portional hummock coverage also indicate what appears tobe a limit cycle in the behaviour of all models under highlystringent hydrological steady-state criteria Particularly when∆te = 17520 (Model 4 only) or 10 000 h and to a lesser ex-tent when∆te = 1000 h the models exhibited a cyclical be-haviour in time whereby the simulated landscape would al-ternate on an approximately regular cycle of between fourand ten developmental steps between two contrasting states

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

36 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 5 Final micro-habitat maps after 100 developmental steps showing effects of increasing∆te on patterning in typical simulationswith Model 1 (top panel) Model 2 (middle panel) and Model 3 (bottom panel) The value of∆te assumed for each simulation is indicatedimmediately above the map Light pixels represent hummocks dark pixels represent hollows Low values ofy represent upslope locations(y= 0 is the upslope boundary) high values ofy represent downslope locations (y= 200 is the downslope boundary) as such groundwaterflow is generally down the page All maps have the same horizontal (x y) scale as that shown for the upper-leftmost map

a dry landscape with deeper water tables dominated by hum-mocks and a wet landscape with water tables near the bogsurface dominated by hollows (Fig 6) The contrast be-tween the wet and dry states was particularly pronouncedin Model 2 which cycled between approximately 15 and95 hummock coverage when∆te = 10000 h (Fig 6h) Thesingle simulation with Model 4 (∆te = 17520 h) behaved ina very similar manner to Models 1 2 and 3 when∆te =10000 h predicting nearly homogeneous unpatterned land-scapes (Fig 7) that cycled rapidly between being dominatedby hummocks and hollows (Fig 6f i)

In all simulations that generated striped SL2 pattern-ing those SL2 units migrated consistently downslope (not

shown) in the same manner as that previously reported in thepublished accounts of the SGCJ models

32 Simplified calculation of transmissivity

Any artefacts introduced to model behaviour by the sim-ple binary treatment of peat permeability employed in theoriginal SGCJ models (see objective i above) would havebeen evident as differences in behaviour between Model 1(simple binary transmissivity scheme based solely on hum-mockhollow designation) and Model 2 (continuous treat-ment of transmissivity as product of peat hydraulic conduc-tivity and thickness of flow) Models 1 and 2 behaved in avery similar manner to one another in terms of their response

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 37

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

a) b) c)

d) e) f)

g) h) i)

Model 1 Model 2 Models 3 and 4

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

P J Morris et al The role of hydrological transience in peatland pattern formation 31

Table 1 Glossary of algebraic terms including default values where appropriate for each of the four model versions

Symbol Description Dimensions Units Model 1 Model 2 Model 3 Model 4

B peatland surface height above arbitrary datum L m 022 to 420 m 022 to 420 m 048 to 417 m 048 to 417 m∆te hydrological submodel runtime (steady-state criterion) T h 50 50 50 17 520 (2 yr)d thickness of flow L m ndash auxiliary variable auxiliary variable auxiliary variableH water-table height above arbitrary datum L m auxiliary variable auxiliary variable auxiliary variable auxiliary variableK hydraulic conductivity L Tminus1 m sminus1 ndash ndash ndash ndashKave depth-averaged hydraulic conductivity L Tminus1 m sminus1 ndash ndash ndash ndashKdeep hydraulic conductivity of deep peat L Tminus1 m sminus1 ndash 0 125times10minus5 125times10minus5

Khol hydraulic conductivity of hollow peat L Tminus1 m sminus1 ndash 10times10minus3 10times10minus3 10times10minus3

Khum hydraulic conductivity of hummock peat L Tminus1 m sminus1 ndash 50times10minus5 50times10minus5 50times10minus5

p probability of hummock formation ndash ndash Eq (3) Fig 3 Eq (3) Fig 3 Eq (3) Fig 3 Eq (3) Fig 3Q hummock turnover rate ndash ndash state variable state variable state variable state variableR relative variance of hummocks per row ndash ndash state variable state variable state variable state variableS proportion of model landscape occupied by hummocks ndash ndash state variable state variable state variable state variableθ peat drainable porosity ndash ndash 03 03 03 03T peat transmissivity L2 Tminus1 m2 sminus1 ndash ndash ndash ndashThol transmissivity of hollow cells L2 Tminus1 m2 sminus1 20times10minus4 ndash ndash ndashThum transmissivity of hummock cells L2 Tminus1 m2 sminus1 10times10minus5 ndash ndash ndashU net rainfall rate (precip minus evapotran) L Tminus1 mm yrminus1 0 0 400 (constant) 400 (time series)x spatial index (across-slope direction) L m ndash ndash ndash ndashy spatial index (along-slope direction) L m ndash ndash ndash ndashZ water-table depth below peat surface L m auxiliary variable auxiliary variable auxiliary variable auxiliary variableZfinal final water-table depth at end of developmental step L m state variable state variable state variable ndashZmean mean water-table depth during second half of dev step L m ndash ndash ndash state variable

12 Aim and objectives

We recreated the SGCJ version of the ponding model to ex-plore three characteristics that we suspected may have indi-vidually or collectively led to a Type-1 error in the SGCJstudies (ie causing a model to predict patterning despite theponding mechanism being incapable of generating patterningin reality)

i Numerical implementation of shallow groundwaterflow the transmissivityT [dimensions of L2 Tminus1] (seeTable 1 for a glossary of algebraic terms) of any gridsquare in the original SGCJ models depends entirely onwhether that square is currently a hummock or a hol-low Only two values ofT are possible one canonicalvalue for hummocksThum and one for hollowsThol Amore realistic scheme would have been to assign canon-ical values of a more intrinsic property of peat such assaturated hydraulic conductivityKhum andKhol [L Tminus1]to hummocks and hollows respectively and to calcu-late transmissivity as the product ofK and the thicknessof flow in that square thereby allowing for continuousvariation inT across the model landscape We extendedthe hydrological submodel from the SGCJ models inthis way in order to remove any unrealistic constraintsthat the simplified numerical implementation of shallowgroundwater flow may have placed upon overall modelbehaviour We also wished to explore the effects of theabsolute values ofT or K (as appropriate) upon modelbehaviour

ii Conceptual hydrogeological setting the original SGCJmodels considered only a shallow layer of near-surfacepeat and assumed that deeper peat was impermeableto groundwater flow The top few decimetres of peat

are usually the most permeable (eg Fraser et al 2001Clymo 2004) and are therefore prone to the most rapidsubsurface flow although deeper peat is rarely truly im-permeable Indeed a number of studies have indicatedthat drainage through deep peat layers may play an im-portant role in peatland development and the ability ofthese ecosystems to self-organise (eg Ingram 1982Belyea and Baird 2006 Morris et al 2011) The as-sumption of impermeable peat below the uppermost fewdecimetres may have prevented the SGCJ models fromrepresenting a potentially important hydrological inter-action between surficial hydraulic structures and deeperpeat layers Furthermore in the original SGCJ modelsrainfall was not included the only inputs of water tothe models were from shallow groundwater flow andorsurface runon In order to address these conceptual is-sues we experimented with the effects of incorporatinga permeable lower layer and driving the model usingsimulated rainfall

iii Dependence on hydrological transience the originalSGCJ authors reported that strong striped patterning oc-curred in their models under ldquosteady-staterdquo hydrolog-ical conditions whereby micro-habitat transitions onlytook place once the simulated water-table map had equi-librated with with the current distribution of hummocksand hollows (and so the spatial arrangement of transmis-sivity) (see Sect 2 for full model description) The cri-terion used to determine steady state considers the pro-portion of cells in which the rate of water-table change[L Tminus1] is less than a threshold rate Steady-state con-ditions are deemed to occur when the proportion ofcells below the threshold is greater than a proportion setby the model user However preliminary experiments

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

32 P J Morris et al The role of hydrological transience in peatland pattern formation

(not reported here in full) with our own version of theponding model indicated that the simulations reportedin the earlier SGCJ studies may not have attained truehydrological steady state and that curiously pattern-ing only developed under intermediate steady-state cri-teria where water tables were still transient We choseto experiment with the effects of hydrological tran-sience upon model patterning either by manipulatingthe water-table steady-state criterion or by driving themodel with a real time series of rainfall data

2 Models and methods

21 Overview

We began with a model that was as similar as possible tothat employed by Couwenberg (2005) as far as the origi-nal model description allows As well as this replica model(henceforth Model 1) we created three additional models(Models 2 3 and 4) with slightly altered routines We de-signed Model 2 so as to address the numerical implementa-tion issue identified in objective (i) while we designed Mod-els 3 and 4 so as to address the conceptual issues identifiedin objective (ii) (see Sect 12 above and Table 2) Each ofthe four models may be thought to consist of three submod-els that simulate shallow saturated groundwater movementsand the spatial distribution of water-table depths at SL1 (hy-drological submodel) switches in micro-habitat type (eco-logical submodel) and changes in peat soil hydraulic prop-erties (soil properties submodel) Model time progresses indevelopmental steps during each developmental step the hy-drological submodel is run until a predetermined steady-statecriterion is met The output from the hydrological submodelis a map of water-table depths within the model landscapeThe ecological submodel then simulates a new micro-habitatmap within the model landscape on the basis of the water-table map such that every cell in the model landscape isassigned one of two binary micro-habitat types hummockor hollow Finally the soil properties submodel uses the newmicro-habitat map to reassign the spatial distribution of near-surface peat hydraulic properties The new map of peat hy-draulic properties is then used as an input to the hydrologicalmodel at the beginning of the next developmental step Aswith the SGCJ studies our models do not simulate explic-itly the processes of peat formation or decomposition andtherefore do not incorporate a peat mass balance Rather themodels represent plant community succession as a shiftingmosaic of SL1 tiles superimposed onto a static peat land-form

22 Hydrological submodel

The hydrological submodel uses a modification of the Di-giBog model of peatland saturated hydrology (Baird et al2012) We took the original DigiBog Fortran 95 code and

added routines to represent the ecological and hydrophys-ical submodels In the current study we deactivated Digi-Bogrsquos peat accumulation decomposition and hydrophysicalsubroutines described by Morris et al (2012) We refer thereader to Baird et al (2012) for a comprehensive descrip-tion of DigiBogrsquos governing equations and numerical imple-mentation although a few points are pertinent here DigiBogrepresents a peatland as a grid of vertical peat columns inplan each column is equivalent to one square grid cell inthe SGCJ models although DigiBog also allows for ver-tical variation in peat hydraulic properties Horizontal sat-urated groundwater flow occurs between adjacent columns(four-square neighbourhood) at a rate equal to the productof the harmonic mean of inter-cell transmissivity (the prod-uct of depth-averaged hydraulic conductivity and the thick-ness of flow) and inter-cell hydraulic gradient according tothe Boussinesq equation (see also McWhorter and Sunada1977)

partHpartt=part

partx

(Kave(d)θ

dpartHpartx

)+part

party

(Kave(d)θ

dpartHparty

)+

Uθ (1)

whereH is water-table height [L] above an arbitrary datumt is simulated time [T]x andy are across-slope and along-slope horizontal spatial dimensions respectively [L]Kave isdepth-averaged hydraulic conductivity [L Tminus1] θ is drainableporosity of the peat [dimensionless] assumed constant at 03d is thickness of flow [L] defined as the height of the watertable H minus the height of the impermeable base layerand U is net rainfall [L Tminus1] BecauseU is added directlyto the saturated zone it may be thought of as a black-boxrepresentation of total precipitation minus the sum of sur-face runoff interception evapotranspiration and unsaturatedwater-storage change

The hydrological model is run for a predetermined lengthof simulated time in order to allow the water-table map tochange in response to the boundary conditions and the cur-rent distribution of hydraulic properties In Models 1 to 3 thelength of time that the hydrological model is allowed to run∆te [T] before an ecological transition takes place (Sect 23)determines how close to a genuine steady state the simulatedwater tables are Short equilibration times lead to highly tran-sient water tables that are still changing rapidly and are farfrom being in equilibrium with the hydrological inputs orboundary conditions the opposite is true of long runtimes InModels 1 2 and 3 we varied∆te between 1 h (highly transientwater tables) and 10 000 h (approximately 417 days highlysteady water tables) to examine the effect of the steady-statecriterion upon model behaviour We performed only a singlesimulation with Model 4 with∆te = 17520 h (equal to twoyears of simulated time) and introduced hydrological tran-sience by driving the model using a daily rainfall time seriesderived from observed data (Sect 25)

We performed a simple test on Models 1 2 and 3 in orderto verify that increasing∆te led to increasing hydrologicalsteadiness We ran each model for a spin-up period of 25

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 33

Table 2 Summary of the four models and the objectives addressed by each

Model Transmissivity Deep peat layer Net rainfallU Up-slope boundary Aquifer Objectivesscheme (precip minus evapotran) condition shape addressed

1 binary impermeable off constant headZ = 0025 m constant slope iii2 continuous impermeable off constant headZ = 0025 m constant slope i iii3 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (constant) impermeable (no-flow) hemi-elliptical i ii iii4 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (time series) impermeable (no-flow) hemi-elliptical iii

Figure 2 Time series of daily-average absolute water-table changeper grid square during five-year test periods with Models 1 2 and3 See main text for full details Note the logarithmic scales on bothaxes

developmental steps with all default parameter values (seeTable 1) and∆te = 50 h We then allowed the hydrologicalsubmodel to continue for a further five years of simulatedtime without any further changes in the spatial distributionsof micro-habitats or peat properties For each simulated dayof the five-year test we calculated the daily sum of absolutewater-table movements in each model grid square and thenaveraged these absolute changes for all grid squares In allthree models the average rate of water-table change declinedrapidly from initial values of between 14 and 19 mm dayminus1

to between 003 and 014 mm dayminus1 by the end of the five-year test period (Fig 2) The tests indicate that water-tablebehaviour for all three models converges on steady state withincreasing∆te

The depth of the water table below the surface at any pointin model time is given simply by

Z = BminusH (2)

whereZ is water-table depth [L] andB is the height of thepeatland surface relative to the arbitrary datum [L] We referto the value ofZ at the end of a developmental step asZfinaland the time-averaged value ofZ during the second half of adevelopmental step asZmean These two metrics are used asinputs to the ecological submodel (Sect 23)

Figure 3 Graphical representation of the linear probability func-tion used by the ecological submodel to assign hummock and hol-low states to each model cell based on water-table depth

23 Ecological submodel

In Models 1 2 and 3 the probabilityp [dimensionless]of any model cell being designated as a hummock duringa given developmental step is a linear function of water-table depthZfinal at the end of the previous developmentalstep When the water table is at the surface in any cell (iewhenZfinal = 000 m) p= 0 (ie that cell is necessarily des-ignated as a hollow for the following developmental step)The value ofp increases linearly with increasing water-tabledepth up toZfinal = 005 m WhenZfinal is equal to or greaterthan 005 m the cell is automatically designated a hummock(ie p= 1) The relevant equations (see also Fig 3) are

p= 00 for Z le 000m

p= 20Z for 000mlt Z lt 005m

p= 10 for Z ge 005m (3)

whereZ is equal to either final water-table depth at the endof a developmental stepZfinal (Models 1 2 and 3) or time-averaged water-table depth during the second half of a two-year developmental stepZmean(Model 4) (see below)

We refer to the upper 005 m of the soil profile as the tran-sition zone All cells that are not designated as hummocksare automatically designated as hollows Swanson and Gri-gal (1988) provide a full description of micro-habitat tran-sitions as simulated in our Models 1 2 and 3 In Model 4micro-habitat transitions are dealt with slightly differentlyRather thanZfinal Model 4 assigns hummocks and hollows

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

34 P J Morris et al The role of hydrological transience in peatland pattern formation

based on each grid squarersquos time-averaged water-table depthZmean [L] during the second half of the previous 2 yr devel-opmental step The first 365 days of each developmental stepare used to allow Model 4rsquos simulated water tables to adjustto the newly updated distribution of peat hydraulic proper-ties but this adjustment period is not incorporated intoZmeanThis measure was intended to ensure thatZmean in Model4 contains no meaningful artefact of water-table geometriesfrom earlier developmental steps Along with a variable rain-fall time series (see Sect 25 below) the use ofZmean pro-vided a means of introducing transience to water-table be-haviour in an arguably more realistic manner than the shortequilibration times used in Models 1 to 3

24 Soil properties submodel

Hummocks are assumed to produce peat that is less perme-able than that produced by hollows DigiBogrsquos implemen-tation of the Boussinesq equation uses depth-averaged satu-rated hydraulic conductivityKave and the thickness of flowd to calculate saturated groundwater flux between adjacentcolumns (Eq 1) However the original SGCJ models simplyassigned a single value of transmissivityThum to hummocksand another higher valueThol to hollows Model 1 uses thesimple treatment of canonical transmissivities as per the orig-inal SGCJ models thus in the case of Model 1 Eq (1) sim-plifies to

partHpartt=part

partx

(Tθ

partHpartx

)+part

party

(Tθ

partHparty

)+

Uθ (4)

We assumed default transmissivity values ofThol = 20times10minus4 m2 sminus1 andThum= 10times10minus5 m2 sminus1 thereby preservingthe Thol to Thum ratio of 20 1 that led to strong pattern-ing in the original SGCJ models Models 2 3 and 4 use amore sophisticated and realistic treatment allowed by Digi-Bog whereby hummocks and hollows are assigned canonicalvalues of hydraulic conductivityKhum andKhol respectivelyT for each cell is recalculated during each iteration of the hy-drological submodel as the product ofK and the thickness offlow d (height of water table above the modelrsquos impermeablebase layer) (cf Freeze and Cherry 1979) In this wayT isable to vary in a continuous manner based on water-table po-sition which is more realistic than the simple binary-T treat-ment of the original SGCJ models (Eq 1) For Models 2 3and 4 we assumed default values ofKhol = 10times10minus3 m sminus1

and Khum= 50times10minus5 m sminus1 thereby giving aKhol to Khum

ratio of 20 1 In models 1 2 and 3 we also manipulated thedefault values ofT or K by factors of between 005 and 20in line with objective (i) In Model 1 we variedThum between50times10minus7 and 20times10minus4 m2 sminus1 andThol between 10times10minus5

and 40times10minus3 m2 sminus1 whilst always maintaining aThol toThum ratio of 20 1 Similarly in Models 2 and 3 we variedKhum between 25times10minus6 and 10times10minus3 m sminus1 andKhol be-tween 5times10minus5 and 2times10minus2 m sminus1 whilst maintaining aKhol

to Khum ratio of 20 1

25 Model spatial and temporal domains boundaryconditions

We implemented all simulations in a 70 (across-slopex di-rection)times200 (along-slopey direction) grid of 1 mtimes1 mgrid squares In Models 1 and 2 the simulated peat aquiferoverlaid a sloping impermeable base (replicating the assump-tion of an impermeable lower peat layer in the SGCJ mod-els) Both the peat surface and the impermeable base had aconstant slope of 1 50 the permeable upper peat had a uni-form thickness of 02 m In Models 3 and 4 we assumed thethick lower peat layer is also permeable with its own hy-draulic conductivityKdeep meaning thatK for each cell isdepth-averaged to account for the vertical transition inK be-tween the upper and lower layers Baird et al (2012) pro-vide a full description of DigiBogrsquos calculation ofKave andinter-cell T We used the groundwater mound equation (In-gram 1982) to calculate the dimensions of a deep peat layerthat is hemi-elliptical in cross section has a uniformK of125times10minus5 m sminus1 is 200 m from central axis to the margin isunderlain by a flat impermeable base and receives a net wa-ter input (from an implied upper peat layer) of 155 mm yrminus1The resulting deep peat layer was 397 m thick along its cen-tral axis curving elliptically down to 028 m at the peatlandrsquosmargin Within DigiBog this deep layer was overlain by asurficial more permeable layer which like Models 1 and 2was 02 m thick

We allowed all simulations to run for 100 developmentalsteps the initial condition consisted of randomly generatedwater tables between 00 and 005 m below the peat surfacein each cell The initial micro-habitat and soil-propertiesmaps were based on this random initial water-table map inthe usual manner described above

In all simulations we set the lateral (along-slope) bound-aries to impermeable (Neumann zero-flux condition) in or-der to simplify the representation of the simulated peatlandwithin its local hydrogeological setting In the simulationswith an impermeable lower peat layer (Models 1 and 2) boththe upslope and downslope boundaries were set to constantwater-table (Dirichlet) conditions of 0025 m below the sur-face such that the overall hydraulic gradient along the modelwas equal to the topographic gradient and the gradient of theimpermeable peat layer As with the original SGCJ modelsModels 1 and 2 received no rainfall all inputs of water inthese models came from the upper boundary condition rep-resenting shallow groundwater influx andor surface runonIn this way Models 1 and 2 might be thought to be more rep-resentative of throughflow fens than of raised bogs In Mod-els 3 and 4 we replaced the constant-head condition at theup-slope boundary with a no-flow (Neumann zero flow) con-dition to represent the drainage divide along the crest of araised bog in a manner similar to Morris et al (2012) (seealso Ingram 1982) but retained the constant water-table con-dition at the lower boundary to represent shallow drainageoutflow to a lag stream for example Water-table levels in

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 35

0

10

20

30

40

50

60

70

80

1 61 121 181 241 301 361

Dai

ly R

ainf

all (

mm

day

-1)

Day of Year

ObservedScaled

Figure 4 365-day time series of daily observed rainfall and itsconversion to net rainfallU after scaling by a factor of 0245 Seemain text for full details

Model 3 were maintained by a constant simulated net rainfallrate ofU = 400 mm yrminus1 (U is assumed to be net of evapo-transpiration interception and runoff hence its low value)We drove Model 4 using a daily time series ofU We took365 days of daily rainfall data from close to Malham TarnMoss a raised bog in North Yorkshire United Kingdom forthe calendar year 2011 The rain gauge recorded precipitationon 294 days of that year with an annual total of 1633 mm ofprecipitation Maximum daily precipitation for the year was786 mm on 10 August We multiplied each daily rainfall to-tal in the time series by approximately 0245 so as to give anannual total of 400 mm of net precipitation equal to the con-stantU assumed in Model 3 whilst maintaining a plausibletemporal variation (Fig 4) During each 2 yr developmentalstep in Model 4 we cycled twice through the 365-day rain-fall series By using the same year of rainfall data repeatedlyin this way we were able to induce water-table transiencewithout the potentially complicating effects of inter-annualvariability in rainfall patterns The thick peat deposit repre-sented in Models 3 and 4 and the fact that their groundwa-ter mounds were maintained by simulated net rainfall meansthat they might be thought to be most representative of om-brotrophic raised bogs

26 Analysis

Previous studies (eg Andreasen et al 2001 Eppinga etal 2009) have demonstrated that the human eye is a pow-erful tool for assessing pattern strength in landscape mod-els and we used a visual appraisal of the two-dimensionalmicro-habitat maps produced by the four models as our pri-mary means of estimating simulated pattern strength In ad-dition we calculated the relative varianceR [dimensionless]of hummocks per across-slope row of cells following themethod of Swanson and Grigal (1988) as an objective andreproducible metric of pattern strength The value ofR in-

creases with pattern strength values less than 2 indicate un-ordered landscapes without discernible patterning values ofR greater than 4 represent highly ordered landscapes withclear strong patterning We also calculated the proportionS[dimensionless] of model cells occupied by hummocks dur-ing each developmental step Finally we calculated modelturnover rateQ [dimensionless] defined as the proportionof model cells that undergo a transition from hummock tohollow or vice versa in each developmental step

3 Results

31 Hydrological transience

In Models 1 2 and 3 the strength of across-slope stripedpatterning at SL2 initially increased strongly with increas-ing hydrological steadiness from an apparently entirely un-ordered random mixture of hummocks and hollows at SL1when∆te = 1 h to peak ldquostrengthsrdquo at around∆te = 100 h (forModels 1 and 2) or 50 h (Model 3) This increase in pattern-ing strength is evident from both a visual appraisal of thefinal micro-habitat maps (Fig 5) and an increase in the val-ues of relative varianceR with increasing∆te (Fig 6a b c)For values of∆te greater than 50 h (Models 1 and 2) and 20 h(Model 3) an increase in the spatial scale of the simulatedSL2 stripes is apparent with increasing∆te (ie the stripesbecame broader in they (along-slope) direction) Increasing∆te also led to SL2 stripes whose upslope and downslopeedges were increasingly straight and sharply defined (Fig 5)For values of∆te greater than 100 h (Models 1 and 2) or 50 h(Model 3) the simulated SL2 stripes had broadened to thepoint of appearing to be ldquoover-developedrdquo such that theyno longer resembled realistic peatland striped patterning (cfFig 1)

As well as governing the spatial configuration of simulatedpatterning the value of∆te also affected the temporal dynam-ics of Models 1 2 and 3 For low values of∆te turnover ratesof model cells were high when∆te = 1 h approximately halfof all cells would undergo a transition from designation as ahummock to a hollow (or vice versa) during each develop-mental step providing further indication of a random land-scape Increasing∆te led to a reduction in turnover rates inModels 1 and 2 as SL2 structures began to stabilise to a min-imum of approximately 40 of cells per developmental stepwhen∆te = 100 h (Fig 6d e) Increasing∆te had little ef-fect on turnover rates in Model 3 until∆te = 1000 or 10 000 h(Fig 6f) The temporal trends in SL1 turnover rate and pro-portional hummock coverage also indicate what appears tobe a limit cycle in the behaviour of all models under highlystringent hydrological steady-state criteria Particularly when∆te = 17520 (Model 4 only) or 10 000 h and to a lesser ex-tent when∆te = 1000 h the models exhibited a cyclical be-haviour in time whereby the simulated landscape would al-ternate on an approximately regular cycle of between fourand ten developmental steps between two contrasting states

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

36 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 5 Final micro-habitat maps after 100 developmental steps showing effects of increasing∆te on patterning in typical simulationswith Model 1 (top panel) Model 2 (middle panel) and Model 3 (bottom panel) The value of∆te assumed for each simulation is indicatedimmediately above the map Light pixels represent hummocks dark pixels represent hollows Low values ofy represent upslope locations(y= 0 is the upslope boundary) high values ofy represent downslope locations (y= 200 is the downslope boundary) as such groundwaterflow is generally down the page All maps have the same horizontal (x y) scale as that shown for the upper-leftmost map

a dry landscape with deeper water tables dominated by hum-mocks and a wet landscape with water tables near the bogsurface dominated by hollows (Fig 6) The contrast be-tween the wet and dry states was particularly pronouncedin Model 2 which cycled between approximately 15 and95 hummock coverage when∆te = 10000 h (Fig 6h) Thesingle simulation with Model 4 (∆te = 17520 h) behaved ina very similar manner to Models 1 2 and 3 when∆te =10000 h predicting nearly homogeneous unpatterned land-scapes (Fig 7) that cycled rapidly between being dominatedby hummocks and hollows (Fig 6f i)

In all simulations that generated striped SL2 pattern-ing those SL2 units migrated consistently downslope (not

shown) in the same manner as that previously reported in thepublished accounts of the SGCJ models

32 Simplified calculation of transmissivity

Any artefacts introduced to model behaviour by the sim-ple binary treatment of peat permeability employed in theoriginal SGCJ models (see objective i above) would havebeen evident as differences in behaviour between Model 1(simple binary transmissivity scheme based solely on hum-mockhollow designation) and Model 2 (continuous treat-ment of transmissivity as product of peat hydraulic conduc-tivity and thickness of flow) Models 1 and 2 behaved in avery similar manner to one another in terms of their response

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 37

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

a) b) c)

d) e) f)

g) h) i)

Model 1 Model 2 Models 3 and 4

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

32 P J Morris et al The role of hydrological transience in peatland pattern formation

(not reported here in full) with our own version of theponding model indicated that the simulations reportedin the earlier SGCJ studies may not have attained truehydrological steady state and that curiously pattern-ing only developed under intermediate steady-state cri-teria where water tables were still transient We choseto experiment with the effects of hydrological tran-sience upon model patterning either by manipulatingthe water-table steady-state criterion or by driving themodel with a real time series of rainfall data

2 Models and methods

21 Overview

We began with a model that was as similar as possible tothat employed by Couwenberg (2005) as far as the origi-nal model description allows As well as this replica model(henceforth Model 1) we created three additional models(Models 2 3 and 4) with slightly altered routines We de-signed Model 2 so as to address the numerical implementa-tion issue identified in objective (i) while we designed Mod-els 3 and 4 so as to address the conceptual issues identifiedin objective (ii) (see Sect 12 above and Table 2) Each ofthe four models may be thought to consist of three submod-els that simulate shallow saturated groundwater movementsand the spatial distribution of water-table depths at SL1 (hy-drological submodel) switches in micro-habitat type (eco-logical submodel) and changes in peat soil hydraulic prop-erties (soil properties submodel) Model time progresses indevelopmental steps during each developmental step the hy-drological submodel is run until a predetermined steady-statecriterion is met The output from the hydrological submodelis a map of water-table depths within the model landscapeThe ecological submodel then simulates a new micro-habitatmap within the model landscape on the basis of the water-table map such that every cell in the model landscape isassigned one of two binary micro-habitat types hummockor hollow Finally the soil properties submodel uses the newmicro-habitat map to reassign the spatial distribution of near-surface peat hydraulic properties The new map of peat hy-draulic properties is then used as an input to the hydrologicalmodel at the beginning of the next developmental step Aswith the SGCJ studies our models do not simulate explic-itly the processes of peat formation or decomposition andtherefore do not incorporate a peat mass balance Rather themodels represent plant community succession as a shiftingmosaic of SL1 tiles superimposed onto a static peat land-form

22 Hydrological submodel

The hydrological submodel uses a modification of the Di-giBog model of peatland saturated hydrology (Baird et al2012) We took the original DigiBog Fortran 95 code and

added routines to represent the ecological and hydrophys-ical submodels In the current study we deactivated Digi-Bogrsquos peat accumulation decomposition and hydrophysicalsubroutines described by Morris et al (2012) We refer thereader to Baird et al (2012) for a comprehensive descrip-tion of DigiBogrsquos governing equations and numerical imple-mentation although a few points are pertinent here DigiBogrepresents a peatland as a grid of vertical peat columns inplan each column is equivalent to one square grid cell inthe SGCJ models although DigiBog also allows for ver-tical variation in peat hydraulic properties Horizontal sat-urated groundwater flow occurs between adjacent columns(four-square neighbourhood) at a rate equal to the productof the harmonic mean of inter-cell transmissivity (the prod-uct of depth-averaged hydraulic conductivity and the thick-ness of flow) and inter-cell hydraulic gradient according tothe Boussinesq equation (see also McWhorter and Sunada1977)

partHpartt=part

partx

(Kave(d)θ

dpartHpartx

)+part

party

(Kave(d)θ

dpartHparty

)+

Uθ (1)

whereH is water-table height [L] above an arbitrary datumt is simulated time [T]x andy are across-slope and along-slope horizontal spatial dimensions respectively [L]Kave isdepth-averaged hydraulic conductivity [L Tminus1] θ is drainableporosity of the peat [dimensionless] assumed constant at 03d is thickness of flow [L] defined as the height of the watertable H minus the height of the impermeable base layerand U is net rainfall [L Tminus1] BecauseU is added directlyto the saturated zone it may be thought of as a black-boxrepresentation of total precipitation minus the sum of sur-face runoff interception evapotranspiration and unsaturatedwater-storage change

The hydrological model is run for a predetermined lengthof simulated time in order to allow the water-table map tochange in response to the boundary conditions and the cur-rent distribution of hydraulic properties In Models 1 to 3 thelength of time that the hydrological model is allowed to run∆te [T] before an ecological transition takes place (Sect 23)determines how close to a genuine steady state the simulatedwater tables are Short equilibration times lead to highly tran-sient water tables that are still changing rapidly and are farfrom being in equilibrium with the hydrological inputs orboundary conditions the opposite is true of long runtimes InModels 1 2 and 3 we varied∆te between 1 h (highly transientwater tables) and 10 000 h (approximately 417 days highlysteady water tables) to examine the effect of the steady-statecriterion upon model behaviour We performed only a singlesimulation with Model 4 with∆te = 17520 h (equal to twoyears of simulated time) and introduced hydrological tran-sience by driving the model using a daily rainfall time seriesderived from observed data (Sect 25)

We performed a simple test on Models 1 2 and 3 in orderto verify that increasing∆te led to increasing hydrologicalsteadiness We ran each model for a spin-up period of 25

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 33

Table 2 Summary of the four models and the objectives addressed by each

Model Transmissivity Deep peat layer Net rainfallU Up-slope boundary Aquifer Objectivesscheme (precip minus evapotran) condition shape addressed

1 binary impermeable off constant headZ = 0025 m constant slope iii2 continuous impermeable off constant headZ = 0025 m constant slope i iii3 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (constant) impermeable (no-flow) hemi-elliptical i ii iii4 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (time series) impermeable (no-flow) hemi-elliptical iii

Figure 2 Time series of daily-average absolute water-table changeper grid square during five-year test periods with Models 1 2 and3 See main text for full details Note the logarithmic scales on bothaxes

developmental steps with all default parameter values (seeTable 1) and∆te = 50 h We then allowed the hydrologicalsubmodel to continue for a further five years of simulatedtime without any further changes in the spatial distributionsof micro-habitats or peat properties For each simulated dayof the five-year test we calculated the daily sum of absolutewater-table movements in each model grid square and thenaveraged these absolute changes for all grid squares In allthree models the average rate of water-table change declinedrapidly from initial values of between 14 and 19 mm dayminus1

to between 003 and 014 mm dayminus1 by the end of the five-year test period (Fig 2) The tests indicate that water-tablebehaviour for all three models converges on steady state withincreasing∆te

The depth of the water table below the surface at any pointin model time is given simply by

Z = BminusH (2)

whereZ is water-table depth [L] andB is the height of thepeatland surface relative to the arbitrary datum [L] We referto the value ofZ at the end of a developmental step asZfinaland the time-averaged value ofZ during the second half of adevelopmental step asZmean These two metrics are used asinputs to the ecological submodel (Sect 23)

Figure 3 Graphical representation of the linear probability func-tion used by the ecological submodel to assign hummock and hol-low states to each model cell based on water-table depth

23 Ecological submodel

In Models 1 2 and 3 the probabilityp [dimensionless]of any model cell being designated as a hummock duringa given developmental step is a linear function of water-table depthZfinal at the end of the previous developmentalstep When the water table is at the surface in any cell (iewhenZfinal = 000 m) p= 0 (ie that cell is necessarily des-ignated as a hollow for the following developmental step)The value ofp increases linearly with increasing water-tabledepth up toZfinal = 005 m WhenZfinal is equal to or greaterthan 005 m the cell is automatically designated a hummock(ie p= 1) The relevant equations (see also Fig 3) are

p= 00 for Z le 000m

p= 20Z for 000mlt Z lt 005m

p= 10 for Z ge 005m (3)

whereZ is equal to either final water-table depth at the endof a developmental stepZfinal (Models 1 2 and 3) or time-averaged water-table depth during the second half of a two-year developmental stepZmean(Model 4) (see below)

We refer to the upper 005 m of the soil profile as the tran-sition zone All cells that are not designated as hummocksare automatically designated as hollows Swanson and Gri-gal (1988) provide a full description of micro-habitat tran-sitions as simulated in our Models 1 2 and 3 In Model 4micro-habitat transitions are dealt with slightly differentlyRather thanZfinal Model 4 assigns hummocks and hollows

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

34 P J Morris et al The role of hydrological transience in peatland pattern formation

based on each grid squarersquos time-averaged water-table depthZmean [L] during the second half of the previous 2 yr devel-opmental step The first 365 days of each developmental stepare used to allow Model 4rsquos simulated water tables to adjustto the newly updated distribution of peat hydraulic proper-ties but this adjustment period is not incorporated intoZmeanThis measure was intended to ensure thatZmean in Model4 contains no meaningful artefact of water-table geometriesfrom earlier developmental steps Along with a variable rain-fall time series (see Sect 25 below) the use ofZmean pro-vided a means of introducing transience to water-table be-haviour in an arguably more realistic manner than the shortequilibration times used in Models 1 to 3

24 Soil properties submodel

Hummocks are assumed to produce peat that is less perme-able than that produced by hollows DigiBogrsquos implemen-tation of the Boussinesq equation uses depth-averaged satu-rated hydraulic conductivityKave and the thickness of flowd to calculate saturated groundwater flux between adjacentcolumns (Eq 1) However the original SGCJ models simplyassigned a single value of transmissivityThum to hummocksand another higher valueThol to hollows Model 1 uses thesimple treatment of canonical transmissivities as per the orig-inal SGCJ models thus in the case of Model 1 Eq (1) sim-plifies to

partHpartt=part

partx

(Tθ

partHpartx

)+part

party

(Tθ

partHparty

)+

Uθ (4)

We assumed default transmissivity values ofThol = 20times10minus4 m2 sminus1 andThum= 10times10minus5 m2 sminus1 thereby preservingthe Thol to Thum ratio of 20 1 that led to strong pattern-ing in the original SGCJ models Models 2 3 and 4 use amore sophisticated and realistic treatment allowed by Digi-Bog whereby hummocks and hollows are assigned canonicalvalues of hydraulic conductivityKhum andKhol respectivelyT for each cell is recalculated during each iteration of the hy-drological submodel as the product ofK and the thickness offlow d (height of water table above the modelrsquos impermeablebase layer) (cf Freeze and Cherry 1979) In this wayT isable to vary in a continuous manner based on water-table po-sition which is more realistic than the simple binary-T treat-ment of the original SGCJ models (Eq 1) For Models 2 3and 4 we assumed default values ofKhol = 10times10minus3 m sminus1

and Khum= 50times10minus5 m sminus1 thereby giving aKhol to Khum

ratio of 20 1 In models 1 2 and 3 we also manipulated thedefault values ofT or K by factors of between 005 and 20in line with objective (i) In Model 1 we variedThum between50times10minus7 and 20times10minus4 m2 sminus1 andThol between 10times10minus5

and 40times10minus3 m2 sminus1 whilst always maintaining aThol toThum ratio of 20 1 Similarly in Models 2 and 3 we variedKhum between 25times10minus6 and 10times10minus3 m sminus1 andKhol be-tween 5times10minus5 and 2times10minus2 m sminus1 whilst maintaining aKhol

to Khum ratio of 20 1

25 Model spatial and temporal domains boundaryconditions

We implemented all simulations in a 70 (across-slopex di-rection)times200 (along-slopey direction) grid of 1 mtimes1 mgrid squares In Models 1 and 2 the simulated peat aquiferoverlaid a sloping impermeable base (replicating the assump-tion of an impermeable lower peat layer in the SGCJ mod-els) Both the peat surface and the impermeable base had aconstant slope of 1 50 the permeable upper peat had a uni-form thickness of 02 m In Models 3 and 4 we assumed thethick lower peat layer is also permeable with its own hy-draulic conductivityKdeep meaning thatK for each cell isdepth-averaged to account for the vertical transition inK be-tween the upper and lower layers Baird et al (2012) pro-vide a full description of DigiBogrsquos calculation ofKave andinter-cell T We used the groundwater mound equation (In-gram 1982) to calculate the dimensions of a deep peat layerthat is hemi-elliptical in cross section has a uniformK of125times10minus5 m sminus1 is 200 m from central axis to the margin isunderlain by a flat impermeable base and receives a net wa-ter input (from an implied upper peat layer) of 155 mm yrminus1The resulting deep peat layer was 397 m thick along its cen-tral axis curving elliptically down to 028 m at the peatlandrsquosmargin Within DigiBog this deep layer was overlain by asurficial more permeable layer which like Models 1 and 2was 02 m thick

We allowed all simulations to run for 100 developmentalsteps the initial condition consisted of randomly generatedwater tables between 00 and 005 m below the peat surfacein each cell The initial micro-habitat and soil-propertiesmaps were based on this random initial water-table map inthe usual manner described above

In all simulations we set the lateral (along-slope) bound-aries to impermeable (Neumann zero-flux condition) in or-der to simplify the representation of the simulated peatlandwithin its local hydrogeological setting In the simulationswith an impermeable lower peat layer (Models 1 and 2) boththe upslope and downslope boundaries were set to constantwater-table (Dirichlet) conditions of 0025 m below the sur-face such that the overall hydraulic gradient along the modelwas equal to the topographic gradient and the gradient of theimpermeable peat layer As with the original SGCJ modelsModels 1 and 2 received no rainfall all inputs of water inthese models came from the upper boundary condition rep-resenting shallow groundwater influx andor surface runonIn this way Models 1 and 2 might be thought to be more rep-resentative of throughflow fens than of raised bogs In Mod-els 3 and 4 we replaced the constant-head condition at theup-slope boundary with a no-flow (Neumann zero flow) con-dition to represent the drainage divide along the crest of araised bog in a manner similar to Morris et al (2012) (seealso Ingram 1982) but retained the constant water-table con-dition at the lower boundary to represent shallow drainageoutflow to a lag stream for example Water-table levels in

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 35

0

10

20

30

40

50

60

70

80

1 61 121 181 241 301 361

Dai

ly R

ainf

all (

mm

day

-1)

Day of Year

ObservedScaled

Figure 4 365-day time series of daily observed rainfall and itsconversion to net rainfallU after scaling by a factor of 0245 Seemain text for full details

Model 3 were maintained by a constant simulated net rainfallrate ofU = 400 mm yrminus1 (U is assumed to be net of evapo-transpiration interception and runoff hence its low value)We drove Model 4 using a daily time series ofU We took365 days of daily rainfall data from close to Malham TarnMoss a raised bog in North Yorkshire United Kingdom forthe calendar year 2011 The rain gauge recorded precipitationon 294 days of that year with an annual total of 1633 mm ofprecipitation Maximum daily precipitation for the year was786 mm on 10 August We multiplied each daily rainfall to-tal in the time series by approximately 0245 so as to give anannual total of 400 mm of net precipitation equal to the con-stantU assumed in Model 3 whilst maintaining a plausibletemporal variation (Fig 4) During each 2 yr developmentalstep in Model 4 we cycled twice through the 365-day rain-fall series By using the same year of rainfall data repeatedlyin this way we were able to induce water-table transiencewithout the potentially complicating effects of inter-annualvariability in rainfall patterns The thick peat deposit repre-sented in Models 3 and 4 and the fact that their groundwa-ter mounds were maintained by simulated net rainfall meansthat they might be thought to be most representative of om-brotrophic raised bogs

26 Analysis

Previous studies (eg Andreasen et al 2001 Eppinga etal 2009) have demonstrated that the human eye is a pow-erful tool for assessing pattern strength in landscape mod-els and we used a visual appraisal of the two-dimensionalmicro-habitat maps produced by the four models as our pri-mary means of estimating simulated pattern strength In ad-dition we calculated the relative varianceR [dimensionless]of hummocks per across-slope row of cells following themethod of Swanson and Grigal (1988) as an objective andreproducible metric of pattern strength The value ofR in-

creases with pattern strength values less than 2 indicate un-ordered landscapes without discernible patterning values ofR greater than 4 represent highly ordered landscapes withclear strong patterning We also calculated the proportionS[dimensionless] of model cells occupied by hummocks dur-ing each developmental step Finally we calculated modelturnover rateQ [dimensionless] defined as the proportionof model cells that undergo a transition from hummock tohollow or vice versa in each developmental step

3 Results

31 Hydrological transience

In Models 1 2 and 3 the strength of across-slope stripedpatterning at SL2 initially increased strongly with increas-ing hydrological steadiness from an apparently entirely un-ordered random mixture of hummocks and hollows at SL1when∆te = 1 h to peak ldquostrengthsrdquo at around∆te = 100 h (forModels 1 and 2) or 50 h (Model 3) This increase in pattern-ing strength is evident from both a visual appraisal of thefinal micro-habitat maps (Fig 5) and an increase in the val-ues of relative varianceR with increasing∆te (Fig 6a b c)For values of∆te greater than 50 h (Models 1 and 2) and 20 h(Model 3) an increase in the spatial scale of the simulatedSL2 stripes is apparent with increasing∆te (ie the stripesbecame broader in they (along-slope) direction) Increasing∆te also led to SL2 stripes whose upslope and downslopeedges were increasingly straight and sharply defined (Fig 5)For values of∆te greater than 100 h (Models 1 and 2) or 50 h(Model 3) the simulated SL2 stripes had broadened to thepoint of appearing to be ldquoover-developedrdquo such that theyno longer resembled realistic peatland striped patterning (cfFig 1)

As well as governing the spatial configuration of simulatedpatterning the value of∆te also affected the temporal dynam-ics of Models 1 2 and 3 For low values of∆te turnover ratesof model cells were high when∆te = 1 h approximately halfof all cells would undergo a transition from designation as ahummock to a hollow (or vice versa) during each develop-mental step providing further indication of a random land-scape Increasing∆te led to a reduction in turnover rates inModels 1 and 2 as SL2 structures began to stabilise to a min-imum of approximately 40 of cells per developmental stepwhen∆te = 100 h (Fig 6d e) Increasing∆te had little ef-fect on turnover rates in Model 3 until∆te = 1000 or 10 000 h(Fig 6f) The temporal trends in SL1 turnover rate and pro-portional hummock coverage also indicate what appears tobe a limit cycle in the behaviour of all models under highlystringent hydrological steady-state criteria Particularly when∆te = 17520 (Model 4 only) or 10 000 h and to a lesser ex-tent when∆te = 1000 h the models exhibited a cyclical be-haviour in time whereby the simulated landscape would al-ternate on an approximately regular cycle of between fourand ten developmental steps between two contrasting states

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

36 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 5 Final micro-habitat maps after 100 developmental steps showing effects of increasing∆te on patterning in typical simulationswith Model 1 (top panel) Model 2 (middle panel) and Model 3 (bottom panel) The value of∆te assumed for each simulation is indicatedimmediately above the map Light pixels represent hummocks dark pixels represent hollows Low values ofy represent upslope locations(y= 0 is the upslope boundary) high values ofy represent downslope locations (y= 200 is the downslope boundary) as such groundwaterflow is generally down the page All maps have the same horizontal (x y) scale as that shown for the upper-leftmost map

a dry landscape with deeper water tables dominated by hum-mocks and a wet landscape with water tables near the bogsurface dominated by hollows (Fig 6) The contrast be-tween the wet and dry states was particularly pronouncedin Model 2 which cycled between approximately 15 and95 hummock coverage when∆te = 10000 h (Fig 6h) Thesingle simulation with Model 4 (∆te = 17520 h) behaved ina very similar manner to Models 1 2 and 3 when∆te =10000 h predicting nearly homogeneous unpatterned land-scapes (Fig 7) that cycled rapidly between being dominatedby hummocks and hollows (Fig 6f i)

In all simulations that generated striped SL2 pattern-ing those SL2 units migrated consistently downslope (not

shown) in the same manner as that previously reported in thepublished accounts of the SGCJ models

32 Simplified calculation of transmissivity

Any artefacts introduced to model behaviour by the sim-ple binary treatment of peat permeability employed in theoriginal SGCJ models (see objective i above) would havebeen evident as differences in behaviour between Model 1(simple binary transmissivity scheme based solely on hum-mockhollow designation) and Model 2 (continuous treat-ment of transmissivity as product of peat hydraulic conduc-tivity and thickness of flow) Models 1 and 2 behaved in avery similar manner to one another in terms of their response

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 37

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

a) b) c)

d) e) f)

g) h) i)

Model 1 Model 2 Models 3 and 4

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

P J Morris et al The role of hydrological transience in peatland pattern formation 33

Table 2 Summary of the four models and the objectives addressed by each

Model Transmissivity Deep peat layer Net rainfallU Up-slope boundary Aquifer Objectivesscheme (precip minus evapotran) condition shape addressed

1 binary impermeable off constant headZ = 0025 m constant slope iii2 continuous impermeable off constant headZ = 0025 m constant slope i iii3 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (constant) impermeable (no-flow) hemi-elliptical i ii iii4 continuous Kdeep= 125times10minus5 m sminus1 400 mm yrminus1 (time series) impermeable (no-flow) hemi-elliptical iii

Figure 2 Time series of daily-average absolute water-table changeper grid square during five-year test periods with Models 1 2 and3 See main text for full details Note the logarithmic scales on bothaxes

developmental steps with all default parameter values (seeTable 1) and∆te = 50 h We then allowed the hydrologicalsubmodel to continue for a further five years of simulatedtime without any further changes in the spatial distributionsof micro-habitats or peat properties For each simulated dayof the five-year test we calculated the daily sum of absolutewater-table movements in each model grid square and thenaveraged these absolute changes for all grid squares In allthree models the average rate of water-table change declinedrapidly from initial values of between 14 and 19 mm dayminus1

to between 003 and 014 mm dayminus1 by the end of the five-year test period (Fig 2) The tests indicate that water-tablebehaviour for all three models converges on steady state withincreasing∆te

The depth of the water table below the surface at any pointin model time is given simply by

Z = BminusH (2)

whereZ is water-table depth [L] andB is the height of thepeatland surface relative to the arbitrary datum [L] We referto the value ofZ at the end of a developmental step asZfinaland the time-averaged value ofZ during the second half of adevelopmental step asZmean These two metrics are used asinputs to the ecological submodel (Sect 23)

Figure 3 Graphical representation of the linear probability func-tion used by the ecological submodel to assign hummock and hol-low states to each model cell based on water-table depth

23 Ecological submodel

In Models 1 2 and 3 the probabilityp [dimensionless]of any model cell being designated as a hummock duringa given developmental step is a linear function of water-table depthZfinal at the end of the previous developmentalstep When the water table is at the surface in any cell (iewhenZfinal = 000 m) p= 0 (ie that cell is necessarily des-ignated as a hollow for the following developmental step)The value ofp increases linearly with increasing water-tabledepth up toZfinal = 005 m WhenZfinal is equal to or greaterthan 005 m the cell is automatically designated a hummock(ie p= 1) The relevant equations (see also Fig 3) are

p= 00 for Z le 000m

p= 20Z for 000mlt Z lt 005m

p= 10 for Z ge 005m (3)

whereZ is equal to either final water-table depth at the endof a developmental stepZfinal (Models 1 2 and 3) or time-averaged water-table depth during the second half of a two-year developmental stepZmean(Model 4) (see below)

We refer to the upper 005 m of the soil profile as the tran-sition zone All cells that are not designated as hummocksare automatically designated as hollows Swanson and Gri-gal (1988) provide a full description of micro-habitat tran-sitions as simulated in our Models 1 2 and 3 In Model 4micro-habitat transitions are dealt with slightly differentlyRather thanZfinal Model 4 assigns hummocks and hollows

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

34 P J Morris et al The role of hydrological transience in peatland pattern formation

based on each grid squarersquos time-averaged water-table depthZmean [L] during the second half of the previous 2 yr devel-opmental step The first 365 days of each developmental stepare used to allow Model 4rsquos simulated water tables to adjustto the newly updated distribution of peat hydraulic proper-ties but this adjustment period is not incorporated intoZmeanThis measure was intended to ensure thatZmean in Model4 contains no meaningful artefact of water-table geometriesfrom earlier developmental steps Along with a variable rain-fall time series (see Sect 25 below) the use ofZmean pro-vided a means of introducing transience to water-table be-haviour in an arguably more realistic manner than the shortequilibration times used in Models 1 to 3

24 Soil properties submodel

Hummocks are assumed to produce peat that is less perme-able than that produced by hollows DigiBogrsquos implemen-tation of the Boussinesq equation uses depth-averaged satu-rated hydraulic conductivityKave and the thickness of flowd to calculate saturated groundwater flux between adjacentcolumns (Eq 1) However the original SGCJ models simplyassigned a single value of transmissivityThum to hummocksand another higher valueThol to hollows Model 1 uses thesimple treatment of canonical transmissivities as per the orig-inal SGCJ models thus in the case of Model 1 Eq (1) sim-plifies to

partHpartt=part

partx

(Tθ

partHpartx

)+part

party

(Tθ

partHparty

)+

Uθ (4)

We assumed default transmissivity values ofThol = 20times10minus4 m2 sminus1 andThum= 10times10minus5 m2 sminus1 thereby preservingthe Thol to Thum ratio of 20 1 that led to strong pattern-ing in the original SGCJ models Models 2 3 and 4 use amore sophisticated and realistic treatment allowed by Digi-Bog whereby hummocks and hollows are assigned canonicalvalues of hydraulic conductivityKhum andKhol respectivelyT for each cell is recalculated during each iteration of the hy-drological submodel as the product ofK and the thickness offlow d (height of water table above the modelrsquos impermeablebase layer) (cf Freeze and Cherry 1979) In this wayT isable to vary in a continuous manner based on water-table po-sition which is more realistic than the simple binary-T treat-ment of the original SGCJ models (Eq 1) For Models 2 3and 4 we assumed default values ofKhol = 10times10minus3 m sminus1

and Khum= 50times10minus5 m sminus1 thereby giving aKhol to Khum

ratio of 20 1 In models 1 2 and 3 we also manipulated thedefault values ofT or K by factors of between 005 and 20in line with objective (i) In Model 1 we variedThum between50times10minus7 and 20times10minus4 m2 sminus1 andThol between 10times10minus5

and 40times10minus3 m2 sminus1 whilst always maintaining aThol toThum ratio of 20 1 Similarly in Models 2 and 3 we variedKhum between 25times10minus6 and 10times10minus3 m sminus1 andKhol be-tween 5times10minus5 and 2times10minus2 m sminus1 whilst maintaining aKhol

to Khum ratio of 20 1

25 Model spatial and temporal domains boundaryconditions

We implemented all simulations in a 70 (across-slopex di-rection)times200 (along-slopey direction) grid of 1 mtimes1 mgrid squares In Models 1 and 2 the simulated peat aquiferoverlaid a sloping impermeable base (replicating the assump-tion of an impermeable lower peat layer in the SGCJ mod-els) Both the peat surface and the impermeable base had aconstant slope of 1 50 the permeable upper peat had a uni-form thickness of 02 m In Models 3 and 4 we assumed thethick lower peat layer is also permeable with its own hy-draulic conductivityKdeep meaning thatK for each cell isdepth-averaged to account for the vertical transition inK be-tween the upper and lower layers Baird et al (2012) pro-vide a full description of DigiBogrsquos calculation ofKave andinter-cell T We used the groundwater mound equation (In-gram 1982) to calculate the dimensions of a deep peat layerthat is hemi-elliptical in cross section has a uniformK of125times10minus5 m sminus1 is 200 m from central axis to the margin isunderlain by a flat impermeable base and receives a net wa-ter input (from an implied upper peat layer) of 155 mm yrminus1The resulting deep peat layer was 397 m thick along its cen-tral axis curving elliptically down to 028 m at the peatlandrsquosmargin Within DigiBog this deep layer was overlain by asurficial more permeable layer which like Models 1 and 2was 02 m thick

We allowed all simulations to run for 100 developmentalsteps the initial condition consisted of randomly generatedwater tables between 00 and 005 m below the peat surfacein each cell The initial micro-habitat and soil-propertiesmaps were based on this random initial water-table map inthe usual manner described above

In all simulations we set the lateral (along-slope) bound-aries to impermeable (Neumann zero-flux condition) in or-der to simplify the representation of the simulated peatlandwithin its local hydrogeological setting In the simulationswith an impermeable lower peat layer (Models 1 and 2) boththe upslope and downslope boundaries were set to constantwater-table (Dirichlet) conditions of 0025 m below the sur-face such that the overall hydraulic gradient along the modelwas equal to the topographic gradient and the gradient of theimpermeable peat layer As with the original SGCJ modelsModels 1 and 2 received no rainfall all inputs of water inthese models came from the upper boundary condition rep-resenting shallow groundwater influx andor surface runonIn this way Models 1 and 2 might be thought to be more rep-resentative of throughflow fens than of raised bogs In Mod-els 3 and 4 we replaced the constant-head condition at theup-slope boundary with a no-flow (Neumann zero flow) con-dition to represent the drainage divide along the crest of araised bog in a manner similar to Morris et al (2012) (seealso Ingram 1982) but retained the constant water-table con-dition at the lower boundary to represent shallow drainageoutflow to a lag stream for example Water-table levels in

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 35

0

10

20

30

40

50

60

70

80

1 61 121 181 241 301 361

Dai

ly R

ainf

all (

mm

day

-1)

Day of Year

ObservedScaled

Figure 4 365-day time series of daily observed rainfall and itsconversion to net rainfallU after scaling by a factor of 0245 Seemain text for full details

Model 3 were maintained by a constant simulated net rainfallrate ofU = 400 mm yrminus1 (U is assumed to be net of evapo-transpiration interception and runoff hence its low value)We drove Model 4 using a daily time series ofU We took365 days of daily rainfall data from close to Malham TarnMoss a raised bog in North Yorkshire United Kingdom forthe calendar year 2011 The rain gauge recorded precipitationon 294 days of that year with an annual total of 1633 mm ofprecipitation Maximum daily precipitation for the year was786 mm on 10 August We multiplied each daily rainfall to-tal in the time series by approximately 0245 so as to give anannual total of 400 mm of net precipitation equal to the con-stantU assumed in Model 3 whilst maintaining a plausibletemporal variation (Fig 4) During each 2 yr developmentalstep in Model 4 we cycled twice through the 365-day rain-fall series By using the same year of rainfall data repeatedlyin this way we were able to induce water-table transiencewithout the potentially complicating effects of inter-annualvariability in rainfall patterns The thick peat deposit repre-sented in Models 3 and 4 and the fact that their groundwa-ter mounds were maintained by simulated net rainfall meansthat they might be thought to be most representative of om-brotrophic raised bogs

26 Analysis

Previous studies (eg Andreasen et al 2001 Eppinga etal 2009) have demonstrated that the human eye is a pow-erful tool for assessing pattern strength in landscape mod-els and we used a visual appraisal of the two-dimensionalmicro-habitat maps produced by the four models as our pri-mary means of estimating simulated pattern strength In ad-dition we calculated the relative varianceR [dimensionless]of hummocks per across-slope row of cells following themethod of Swanson and Grigal (1988) as an objective andreproducible metric of pattern strength The value ofR in-

creases with pattern strength values less than 2 indicate un-ordered landscapes without discernible patterning values ofR greater than 4 represent highly ordered landscapes withclear strong patterning We also calculated the proportionS[dimensionless] of model cells occupied by hummocks dur-ing each developmental step Finally we calculated modelturnover rateQ [dimensionless] defined as the proportionof model cells that undergo a transition from hummock tohollow or vice versa in each developmental step

3 Results

31 Hydrological transience

In Models 1 2 and 3 the strength of across-slope stripedpatterning at SL2 initially increased strongly with increas-ing hydrological steadiness from an apparently entirely un-ordered random mixture of hummocks and hollows at SL1when∆te = 1 h to peak ldquostrengthsrdquo at around∆te = 100 h (forModels 1 and 2) or 50 h (Model 3) This increase in pattern-ing strength is evident from both a visual appraisal of thefinal micro-habitat maps (Fig 5) and an increase in the val-ues of relative varianceR with increasing∆te (Fig 6a b c)For values of∆te greater than 50 h (Models 1 and 2) and 20 h(Model 3) an increase in the spatial scale of the simulatedSL2 stripes is apparent with increasing∆te (ie the stripesbecame broader in they (along-slope) direction) Increasing∆te also led to SL2 stripes whose upslope and downslopeedges were increasingly straight and sharply defined (Fig 5)For values of∆te greater than 100 h (Models 1 and 2) or 50 h(Model 3) the simulated SL2 stripes had broadened to thepoint of appearing to be ldquoover-developedrdquo such that theyno longer resembled realistic peatland striped patterning (cfFig 1)

As well as governing the spatial configuration of simulatedpatterning the value of∆te also affected the temporal dynam-ics of Models 1 2 and 3 For low values of∆te turnover ratesof model cells were high when∆te = 1 h approximately halfof all cells would undergo a transition from designation as ahummock to a hollow (or vice versa) during each develop-mental step providing further indication of a random land-scape Increasing∆te led to a reduction in turnover rates inModels 1 and 2 as SL2 structures began to stabilise to a min-imum of approximately 40 of cells per developmental stepwhen∆te = 100 h (Fig 6d e) Increasing∆te had little ef-fect on turnover rates in Model 3 until∆te = 1000 or 10 000 h(Fig 6f) The temporal trends in SL1 turnover rate and pro-portional hummock coverage also indicate what appears tobe a limit cycle in the behaviour of all models under highlystringent hydrological steady-state criteria Particularly when∆te = 17520 (Model 4 only) or 10 000 h and to a lesser ex-tent when∆te = 1000 h the models exhibited a cyclical be-haviour in time whereby the simulated landscape would al-ternate on an approximately regular cycle of between fourand ten developmental steps between two contrasting states

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

36 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 5 Final micro-habitat maps after 100 developmental steps showing effects of increasing∆te on patterning in typical simulationswith Model 1 (top panel) Model 2 (middle panel) and Model 3 (bottom panel) The value of∆te assumed for each simulation is indicatedimmediately above the map Light pixels represent hummocks dark pixels represent hollows Low values ofy represent upslope locations(y= 0 is the upslope boundary) high values ofy represent downslope locations (y= 200 is the downslope boundary) as such groundwaterflow is generally down the page All maps have the same horizontal (x y) scale as that shown for the upper-leftmost map

a dry landscape with deeper water tables dominated by hum-mocks and a wet landscape with water tables near the bogsurface dominated by hollows (Fig 6) The contrast be-tween the wet and dry states was particularly pronouncedin Model 2 which cycled between approximately 15 and95 hummock coverage when∆te = 10000 h (Fig 6h) Thesingle simulation with Model 4 (∆te = 17520 h) behaved ina very similar manner to Models 1 2 and 3 when∆te =10000 h predicting nearly homogeneous unpatterned land-scapes (Fig 7) that cycled rapidly between being dominatedby hummocks and hollows (Fig 6f i)

In all simulations that generated striped SL2 pattern-ing those SL2 units migrated consistently downslope (not

shown) in the same manner as that previously reported in thepublished accounts of the SGCJ models

32 Simplified calculation of transmissivity

Any artefacts introduced to model behaviour by the sim-ple binary treatment of peat permeability employed in theoriginal SGCJ models (see objective i above) would havebeen evident as differences in behaviour between Model 1(simple binary transmissivity scheme based solely on hum-mockhollow designation) and Model 2 (continuous treat-ment of transmissivity as product of peat hydraulic conduc-tivity and thickness of flow) Models 1 and 2 behaved in avery similar manner to one another in terms of their response

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 37

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

a) b) c)

d) e) f)

g) h) i)

Model 1 Model 2 Models 3 and 4

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

34 P J Morris et al The role of hydrological transience in peatland pattern formation

based on each grid squarersquos time-averaged water-table depthZmean [L] during the second half of the previous 2 yr devel-opmental step The first 365 days of each developmental stepare used to allow Model 4rsquos simulated water tables to adjustto the newly updated distribution of peat hydraulic proper-ties but this adjustment period is not incorporated intoZmeanThis measure was intended to ensure thatZmean in Model4 contains no meaningful artefact of water-table geometriesfrom earlier developmental steps Along with a variable rain-fall time series (see Sect 25 below) the use ofZmean pro-vided a means of introducing transience to water-table be-haviour in an arguably more realistic manner than the shortequilibration times used in Models 1 to 3

24 Soil properties submodel

Hummocks are assumed to produce peat that is less perme-able than that produced by hollows DigiBogrsquos implemen-tation of the Boussinesq equation uses depth-averaged satu-rated hydraulic conductivityKave and the thickness of flowd to calculate saturated groundwater flux between adjacentcolumns (Eq 1) However the original SGCJ models simplyassigned a single value of transmissivityThum to hummocksand another higher valueThol to hollows Model 1 uses thesimple treatment of canonical transmissivities as per the orig-inal SGCJ models thus in the case of Model 1 Eq (1) sim-plifies to

partHpartt=part

partx

(Tθ

partHpartx

)+part

party

(Tθ

partHparty

)+

Uθ (4)

We assumed default transmissivity values ofThol = 20times10minus4 m2 sminus1 andThum= 10times10minus5 m2 sminus1 thereby preservingthe Thol to Thum ratio of 20 1 that led to strong pattern-ing in the original SGCJ models Models 2 3 and 4 use amore sophisticated and realistic treatment allowed by Digi-Bog whereby hummocks and hollows are assigned canonicalvalues of hydraulic conductivityKhum andKhol respectivelyT for each cell is recalculated during each iteration of the hy-drological submodel as the product ofK and the thickness offlow d (height of water table above the modelrsquos impermeablebase layer) (cf Freeze and Cherry 1979) In this wayT isable to vary in a continuous manner based on water-table po-sition which is more realistic than the simple binary-T treat-ment of the original SGCJ models (Eq 1) For Models 2 3and 4 we assumed default values ofKhol = 10times10minus3 m sminus1

and Khum= 50times10minus5 m sminus1 thereby giving aKhol to Khum

ratio of 20 1 In models 1 2 and 3 we also manipulated thedefault values ofT or K by factors of between 005 and 20in line with objective (i) In Model 1 we variedThum between50times10minus7 and 20times10minus4 m2 sminus1 andThol between 10times10minus5

and 40times10minus3 m2 sminus1 whilst always maintaining aThol toThum ratio of 20 1 Similarly in Models 2 and 3 we variedKhum between 25times10minus6 and 10times10minus3 m sminus1 andKhol be-tween 5times10minus5 and 2times10minus2 m sminus1 whilst maintaining aKhol

to Khum ratio of 20 1

25 Model spatial and temporal domains boundaryconditions

We implemented all simulations in a 70 (across-slopex di-rection)times200 (along-slopey direction) grid of 1 mtimes1 mgrid squares In Models 1 and 2 the simulated peat aquiferoverlaid a sloping impermeable base (replicating the assump-tion of an impermeable lower peat layer in the SGCJ mod-els) Both the peat surface and the impermeable base had aconstant slope of 1 50 the permeable upper peat had a uni-form thickness of 02 m In Models 3 and 4 we assumed thethick lower peat layer is also permeable with its own hy-draulic conductivityKdeep meaning thatK for each cell isdepth-averaged to account for the vertical transition inK be-tween the upper and lower layers Baird et al (2012) pro-vide a full description of DigiBogrsquos calculation ofKave andinter-cell T We used the groundwater mound equation (In-gram 1982) to calculate the dimensions of a deep peat layerthat is hemi-elliptical in cross section has a uniformK of125times10minus5 m sminus1 is 200 m from central axis to the margin isunderlain by a flat impermeable base and receives a net wa-ter input (from an implied upper peat layer) of 155 mm yrminus1The resulting deep peat layer was 397 m thick along its cen-tral axis curving elliptically down to 028 m at the peatlandrsquosmargin Within DigiBog this deep layer was overlain by asurficial more permeable layer which like Models 1 and 2was 02 m thick

We allowed all simulations to run for 100 developmentalsteps the initial condition consisted of randomly generatedwater tables between 00 and 005 m below the peat surfacein each cell The initial micro-habitat and soil-propertiesmaps were based on this random initial water-table map inthe usual manner described above

In all simulations we set the lateral (along-slope) bound-aries to impermeable (Neumann zero-flux condition) in or-der to simplify the representation of the simulated peatlandwithin its local hydrogeological setting In the simulationswith an impermeable lower peat layer (Models 1 and 2) boththe upslope and downslope boundaries were set to constantwater-table (Dirichlet) conditions of 0025 m below the sur-face such that the overall hydraulic gradient along the modelwas equal to the topographic gradient and the gradient of theimpermeable peat layer As with the original SGCJ modelsModels 1 and 2 received no rainfall all inputs of water inthese models came from the upper boundary condition rep-resenting shallow groundwater influx andor surface runonIn this way Models 1 and 2 might be thought to be more rep-resentative of throughflow fens than of raised bogs In Mod-els 3 and 4 we replaced the constant-head condition at theup-slope boundary with a no-flow (Neumann zero flow) con-dition to represent the drainage divide along the crest of araised bog in a manner similar to Morris et al (2012) (seealso Ingram 1982) but retained the constant water-table con-dition at the lower boundary to represent shallow drainageoutflow to a lag stream for example Water-table levels in

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 35

0

10

20

30

40

50

60

70

80

1 61 121 181 241 301 361

Dai

ly R

ainf

all (

mm

day

-1)

Day of Year

ObservedScaled

Figure 4 365-day time series of daily observed rainfall and itsconversion to net rainfallU after scaling by a factor of 0245 Seemain text for full details

Model 3 were maintained by a constant simulated net rainfallrate ofU = 400 mm yrminus1 (U is assumed to be net of evapo-transpiration interception and runoff hence its low value)We drove Model 4 using a daily time series ofU We took365 days of daily rainfall data from close to Malham TarnMoss a raised bog in North Yorkshire United Kingdom forthe calendar year 2011 The rain gauge recorded precipitationon 294 days of that year with an annual total of 1633 mm ofprecipitation Maximum daily precipitation for the year was786 mm on 10 August We multiplied each daily rainfall to-tal in the time series by approximately 0245 so as to give anannual total of 400 mm of net precipitation equal to the con-stantU assumed in Model 3 whilst maintaining a plausibletemporal variation (Fig 4) During each 2 yr developmentalstep in Model 4 we cycled twice through the 365-day rain-fall series By using the same year of rainfall data repeatedlyin this way we were able to induce water-table transiencewithout the potentially complicating effects of inter-annualvariability in rainfall patterns The thick peat deposit repre-sented in Models 3 and 4 and the fact that their groundwa-ter mounds were maintained by simulated net rainfall meansthat they might be thought to be most representative of om-brotrophic raised bogs

26 Analysis

Previous studies (eg Andreasen et al 2001 Eppinga etal 2009) have demonstrated that the human eye is a pow-erful tool for assessing pattern strength in landscape mod-els and we used a visual appraisal of the two-dimensionalmicro-habitat maps produced by the four models as our pri-mary means of estimating simulated pattern strength In ad-dition we calculated the relative varianceR [dimensionless]of hummocks per across-slope row of cells following themethod of Swanson and Grigal (1988) as an objective andreproducible metric of pattern strength The value ofR in-

creases with pattern strength values less than 2 indicate un-ordered landscapes without discernible patterning values ofR greater than 4 represent highly ordered landscapes withclear strong patterning We also calculated the proportionS[dimensionless] of model cells occupied by hummocks dur-ing each developmental step Finally we calculated modelturnover rateQ [dimensionless] defined as the proportionof model cells that undergo a transition from hummock tohollow or vice versa in each developmental step

3 Results

31 Hydrological transience

In Models 1 2 and 3 the strength of across-slope stripedpatterning at SL2 initially increased strongly with increas-ing hydrological steadiness from an apparently entirely un-ordered random mixture of hummocks and hollows at SL1when∆te = 1 h to peak ldquostrengthsrdquo at around∆te = 100 h (forModels 1 and 2) or 50 h (Model 3) This increase in pattern-ing strength is evident from both a visual appraisal of thefinal micro-habitat maps (Fig 5) and an increase in the val-ues of relative varianceR with increasing∆te (Fig 6a b c)For values of∆te greater than 50 h (Models 1 and 2) and 20 h(Model 3) an increase in the spatial scale of the simulatedSL2 stripes is apparent with increasing∆te (ie the stripesbecame broader in they (along-slope) direction) Increasing∆te also led to SL2 stripes whose upslope and downslopeedges were increasingly straight and sharply defined (Fig 5)For values of∆te greater than 100 h (Models 1 and 2) or 50 h(Model 3) the simulated SL2 stripes had broadened to thepoint of appearing to be ldquoover-developedrdquo such that theyno longer resembled realistic peatland striped patterning (cfFig 1)

As well as governing the spatial configuration of simulatedpatterning the value of∆te also affected the temporal dynam-ics of Models 1 2 and 3 For low values of∆te turnover ratesof model cells were high when∆te = 1 h approximately halfof all cells would undergo a transition from designation as ahummock to a hollow (or vice versa) during each develop-mental step providing further indication of a random land-scape Increasing∆te led to a reduction in turnover rates inModels 1 and 2 as SL2 structures began to stabilise to a min-imum of approximately 40 of cells per developmental stepwhen∆te = 100 h (Fig 6d e) Increasing∆te had little ef-fect on turnover rates in Model 3 until∆te = 1000 or 10 000 h(Fig 6f) The temporal trends in SL1 turnover rate and pro-portional hummock coverage also indicate what appears tobe a limit cycle in the behaviour of all models under highlystringent hydrological steady-state criteria Particularly when∆te = 17520 (Model 4 only) or 10 000 h and to a lesser ex-tent when∆te = 1000 h the models exhibited a cyclical be-haviour in time whereby the simulated landscape would al-ternate on an approximately regular cycle of between fourand ten developmental steps between two contrasting states

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

36 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 5 Final micro-habitat maps after 100 developmental steps showing effects of increasing∆te on patterning in typical simulationswith Model 1 (top panel) Model 2 (middle panel) and Model 3 (bottom panel) The value of∆te assumed for each simulation is indicatedimmediately above the map Light pixels represent hummocks dark pixels represent hollows Low values ofy represent upslope locations(y= 0 is the upslope boundary) high values ofy represent downslope locations (y= 200 is the downslope boundary) as such groundwaterflow is generally down the page All maps have the same horizontal (x y) scale as that shown for the upper-leftmost map

a dry landscape with deeper water tables dominated by hum-mocks and a wet landscape with water tables near the bogsurface dominated by hollows (Fig 6) The contrast be-tween the wet and dry states was particularly pronouncedin Model 2 which cycled between approximately 15 and95 hummock coverage when∆te = 10000 h (Fig 6h) Thesingle simulation with Model 4 (∆te = 17520 h) behaved ina very similar manner to Models 1 2 and 3 when∆te =10000 h predicting nearly homogeneous unpatterned land-scapes (Fig 7) that cycled rapidly between being dominatedby hummocks and hollows (Fig 6f i)

In all simulations that generated striped SL2 pattern-ing those SL2 units migrated consistently downslope (not

shown) in the same manner as that previously reported in thepublished accounts of the SGCJ models

32 Simplified calculation of transmissivity

Any artefacts introduced to model behaviour by the sim-ple binary treatment of peat permeability employed in theoriginal SGCJ models (see objective i above) would havebeen evident as differences in behaviour between Model 1(simple binary transmissivity scheme based solely on hum-mockhollow designation) and Model 2 (continuous treat-ment of transmissivity as product of peat hydraulic conduc-tivity and thickness of flow) Models 1 and 2 behaved in avery similar manner to one another in terms of their response

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P J Morris et al The role of hydrological transience in peatland pattern formation 37

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

a) b) c)

d) e) f)

g) h) i)

Model 1 Model 2 Models 3 and 4

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

P J Morris et al The role of hydrological transience in peatland pattern formation 35

0

10

20

30

40

50

60

70

80

1 61 121 181 241 301 361

Dai

ly R

ainf

all (

mm

day

-1)

Day of Year

ObservedScaled

Figure 4 365-day time series of daily observed rainfall and itsconversion to net rainfallU after scaling by a factor of 0245 Seemain text for full details

Model 3 were maintained by a constant simulated net rainfallrate ofU = 400 mm yrminus1 (U is assumed to be net of evapo-transpiration interception and runoff hence its low value)We drove Model 4 using a daily time series ofU We took365 days of daily rainfall data from close to Malham TarnMoss a raised bog in North Yorkshire United Kingdom forthe calendar year 2011 The rain gauge recorded precipitationon 294 days of that year with an annual total of 1633 mm ofprecipitation Maximum daily precipitation for the year was786 mm on 10 August We multiplied each daily rainfall to-tal in the time series by approximately 0245 so as to give anannual total of 400 mm of net precipitation equal to the con-stantU assumed in Model 3 whilst maintaining a plausibletemporal variation (Fig 4) During each 2 yr developmentalstep in Model 4 we cycled twice through the 365-day rain-fall series By using the same year of rainfall data repeatedlyin this way we were able to induce water-table transiencewithout the potentially complicating effects of inter-annualvariability in rainfall patterns The thick peat deposit repre-sented in Models 3 and 4 and the fact that their groundwa-ter mounds were maintained by simulated net rainfall meansthat they might be thought to be most representative of om-brotrophic raised bogs

26 Analysis

Previous studies (eg Andreasen et al 2001 Eppinga etal 2009) have demonstrated that the human eye is a pow-erful tool for assessing pattern strength in landscape mod-els and we used a visual appraisal of the two-dimensionalmicro-habitat maps produced by the four models as our pri-mary means of estimating simulated pattern strength In ad-dition we calculated the relative varianceR [dimensionless]of hummocks per across-slope row of cells following themethod of Swanson and Grigal (1988) as an objective andreproducible metric of pattern strength The value ofR in-

creases with pattern strength values less than 2 indicate un-ordered landscapes without discernible patterning values ofR greater than 4 represent highly ordered landscapes withclear strong patterning We also calculated the proportionS[dimensionless] of model cells occupied by hummocks dur-ing each developmental step Finally we calculated modelturnover rateQ [dimensionless] defined as the proportionof model cells that undergo a transition from hummock tohollow or vice versa in each developmental step

3 Results

31 Hydrological transience

In Models 1 2 and 3 the strength of across-slope stripedpatterning at SL2 initially increased strongly with increas-ing hydrological steadiness from an apparently entirely un-ordered random mixture of hummocks and hollows at SL1when∆te = 1 h to peak ldquostrengthsrdquo at around∆te = 100 h (forModels 1 and 2) or 50 h (Model 3) This increase in pattern-ing strength is evident from both a visual appraisal of thefinal micro-habitat maps (Fig 5) and an increase in the val-ues of relative varianceR with increasing∆te (Fig 6a b c)For values of∆te greater than 50 h (Models 1 and 2) and 20 h(Model 3) an increase in the spatial scale of the simulatedSL2 stripes is apparent with increasing∆te (ie the stripesbecame broader in they (along-slope) direction) Increasing∆te also led to SL2 stripes whose upslope and downslopeedges were increasingly straight and sharply defined (Fig 5)For values of∆te greater than 100 h (Models 1 and 2) or 50 h(Model 3) the simulated SL2 stripes had broadened to thepoint of appearing to be ldquoover-developedrdquo such that theyno longer resembled realistic peatland striped patterning (cfFig 1)

As well as governing the spatial configuration of simulatedpatterning the value of∆te also affected the temporal dynam-ics of Models 1 2 and 3 For low values of∆te turnover ratesof model cells were high when∆te = 1 h approximately halfof all cells would undergo a transition from designation as ahummock to a hollow (or vice versa) during each develop-mental step providing further indication of a random land-scape Increasing∆te led to a reduction in turnover rates inModels 1 and 2 as SL2 structures began to stabilise to a min-imum of approximately 40 of cells per developmental stepwhen∆te = 100 h (Fig 6d e) Increasing∆te had little ef-fect on turnover rates in Model 3 until∆te = 1000 or 10 000 h(Fig 6f) The temporal trends in SL1 turnover rate and pro-portional hummock coverage also indicate what appears tobe a limit cycle in the behaviour of all models under highlystringent hydrological steady-state criteria Particularly when∆te = 17520 (Model 4 only) or 10 000 h and to a lesser ex-tent when∆te = 1000 h the models exhibited a cyclical be-haviour in time whereby the simulated landscape would al-ternate on an approximately regular cycle of between fourand ten developmental steps between two contrasting states

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

36 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 5 Final micro-habitat maps after 100 developmental steps showing effects of increasing∆te on patterning in typical simulationswith Model 1 (top panel) Model 2 (middle panel) and Model 3 (bottom panel) The value of∆te assumed for each simulation is indicatedimmediately above the map Light pixels represent hummocks dark pixels represent hollows Low values ofy represent upslope locations(y= 0 is the upslope boundary) high values ofy represent downslope locations (y= 200 is the downslope boundary) as such groundwaterflow is generally down the page All maps have the same horizontal (x y) scale as that shown for the upper-leftmost map

a dry landscape with deeper water tables dominated by hum-mocks and a wet landscape with water tables near the bogsurface dominated by hollows (Fig 6) The contrast be-tween the wet and dry states was particularly pronouncedin Model 2 which cycled between approximately 15 and95 hummock coverage when∆te = 10000 h (Fig 6h) Thesingle simulation with Model 4 (∆te = 17520 h) behaved ina very similar manner to Models 1 2 and 3 when∆te =10000 h predicting nearly homogeneous unpatterned land-scapes (Fig 7) that cycled rapidly between being dominatedby hummocks and hollows (Fig 6f i)

In all simulations that generated striped SL2 pattern-ing those SL2 units migrated consistently downslope (not

shown) in the same manner as that previously reported in thepublished accounts of the SGCJ models

32 Simplified calculation of transmissivity

Any artefacts introduced to model behaviour by the sim-ple binary treatment of peat permeability employed in theoriginal SGCJ models (see objective i above) would havebeen evident as differences in behaviour between Model 1(simple binary transmissivity scheme based solely on hum-mockhollow designation) and Model 2 (continuous treat-ment of transmissivity as product of peat hydraulic conduc-tivity and thickness of flow) Models 1 and 2 behaved in avery similar manner to one another in terms of their response

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 37

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

a) b) c)

d) e) f)

g) h) i)

Model 1 Model 2 Models 3 and 4

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

Hu

mm

Pro

po

rtio

n S

(-)

Developmental Step

00

02

04

06

08

10

0 10 20 30 40 50 60 70 80 90 100

SL1

Tu

rno

ver

Rat

e Q

(-)

Developmental Step

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Rel

ativ

e V

aria

nce

R (

-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

36 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 5 Final micro-habitat maps after 100 developmental steps showing effects of increasing∆te on patterning in typical simulationswith Model 1 (top panel) Model 2 (middle panel) and Model 3 (bottom panel) The value of∆te assumed for each simulation is indicatedimmediately above the map Light pixels represent hummocks dark pixels represent hollows Low values ofy represent upslope locations(y= 0 is the upslope boundary) high values ofy represent downslope locations (y= 200 is the downslope boundary) as such groundwaterflow is generally down the page All maps have the same horizontal (x y) scale as that shown for the upper-leftmost map

a dry landscape with deeper water tables dominated by hum-mocks and a wet landscape with water tables near the bogsurface dominated by hollows (Fig 6) The contrast be-tween the wet and dry states was particularly pronouncedin Model 2 which cycled between approximately 15 and95 hummock coverage when∆te = 10000 h (Fig 6h) Thesingle simulation with Model 4 (∆te = 17520 h) behaved ina very similar manner to Models 1 2 and 3 when∆te =10000 h predicting nearly homogeneous unpatterned land-scapes (Fig 7) that cycled rapidly between being dominatedby hummocks and hollows (Fig 6f i)

In all simulations that generated striped SL2 pattern-ing those SL2 units migrated consistently downslope (not

shown) in the same manner as that previously reported in thepublished accounts of the SGCJ models

32 Simplified calculation of transmissivity

Any artefacts introduced to model behaviour by the sim-ple binary treatment of peat permeability employed in theoriginal SGCJ models (see objective i above) would havebeen evident as differences in behaviour between Model 1(simple binary transmissivity scheme based solely on hum-mockhollow designation) and Model 2 (continuous treat-ment of transmissivity as product of peat hydraulic conduc-tivity and thickness of flow) Models 1 and 2 behaved in avery similar manner to one another in terms of their response

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 37

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rno

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(-)

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SL1

Tu

rno

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(-)

Developmental Step

a) b) c)

d) e) f)

g) h) i)

Model 1 Model 2 Models 3 and 4

00

02

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10

0 10 20 30 40 50 60 70 80 90 100

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mm

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rno

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60

0 10 20 30 40 50 60 70 80 90 100

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ativ

e V

aria

nce

R (

-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

P J Morris et al The role of hydrological transience in peatland pattern formation 37

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d) e) f)

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Model 1 Model 2 Models 3 and 4

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0 10 20 30 40 50 60 70 80 90 100

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aria

nce

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-)

Developmental Step

Δte

ndash 1 hr ndash 10 hr ndash 100 hr ndash 1000 hr ndash 10000 hr ndash 17520 hr (Model 4 only)

Figure 6 Influence of hydrological equilibration time∆te over temporal development of relative variance of hummocks per model rowR(top row) SL1 turnover rateQ (middle row) and proportion of model landscape occupied by hummocksS (bottom row) in Models 1 (leftcolumn) 2 (middle column) 3 and 4 (right column)

to both changing values of∆te and changing absolute val-ues of peat permeability Both models developed realisticlooking contour-parallel SL2 stripes over the entire modeldomain for intermediate values of∆te In both models thestriped patterning was relatively weak and discontinuous for∆te = 15 and 20 h the patterns were stronger and continuouswhen∆te was between 35 and 100 h the SL2 stripes becameunrealistically broad at∆te = 1000 h and were mainly ab-sent when∆te = 10000 h (Fig 5) Temporal patterns of sum-mary metrics (SL1 turnover rate hummock proportion rel-ative variance of hummocks per row) were qualitatively andquantitatively similar between models 1 and 2 and again re-sponded similarly to changes in∆te (Fig 6) and the ratio ofThum to Thol or Khum to Khol (not shown)

33 Hydrogeological setting

Any artefacts introduced to model behaviour by the simplify-ing assumptions made in the original SGCJ models about thehydrogeological setting of the simulated peatland (see ob-jective ii above) would have been evident as differences in

behaviour between Model 2 (impermeable deep peat layerconstant-head upslope boundary condition zero rainfall ad-dition constant slope of peatland surface) and Model 3 (per-meable deep peat layer no-flow upslope boundary con-stant net rainfall rate of 400 mm yrminus1 hemi-elliptical aquifershape) Model 2 simulations that generated patterning did soover the entire model domain although the same was not trueof Model 3 Striped patterning at SL2 developed in Model 3only in the downslope portion of the model domain Further-more the area of the model domain that exhibited pattern-ing extended upslope with increasing values of∆te For in-stance when∆te = 10 h Model 3 only produced striped pat-terning within 60 m or so of the downslope boundary butwhen ∆te = 100 h the patterned area covered the majorityof the model domain and extended as far as approximately140 m from the downslope boundary (Fig 5) Clear con-tinuous striped patterning that extended all the way acrossthe across-slope direction (x direction) of the model domaindeveloped in Model 3 at much lower values of∆te than inModel 2 Continuous closely spaced SL2 stripes were ev-ident in the downslope area of Model 3 when∆te = 20 h

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

38 P J Morris et al The role of hydrological transience in peatland pattern formation

Figure 7 Final micro-habitat map after 100 developmental stepsfrom Model 4 with∆te = 17520 h (2 yr) Light pixels representhummocks dark pixels represent hollows Low values ofy repre-sent upslope locations (y= 0 is the upslope boundary) high val-ues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the page

although sharply defined continuous SL2 stripes in Model 2did not develop for values of∆te less than 35 h AdditionallyModel 3 behaved differently from Model 2 in terms of its re-sponses to changes in the absolute values of peat hydraulicconductivity (see below)

34 Manipulation of peat properties

Modest changes in the absolute values ofT or K broughtabout large changes in the nature and strength of simulatedpatterning in Models 1 2 and 3 although the nature of theseresponses varied between models In Models 1 and 2 thelowest values ofT or K led to largely unpatterned landscapesdominated by apparently random distributions of hummocksand hollows with either a complete absence of patterning(times005 treatment) or very weak discontinuous patterning(times01 treatment) (Fig 8) In both Models 1 and 2 patternstrength increased with increasing values ofT or K up to thedefault combination For combinations ofT and K greaterthan the default values Models 1 and 2 both predicted an in-crease in the spatial scale of SL2 stripes in a manner similarto the effect of increasing∆te for thetimes10 andtimes20 treatmentsModel 1 became almost a uniform landscape of hummockswith only small SL2 groupings of hollows The behaviour ofModel 3 was quite different Only the default and thetimes02treatments generated any kind of contour-parallel stripes allother treatments produced near-uniform unpatterned land-scapes composed almost entirely of either hollows (for thetimes005 andtimes01 treatments) or hummocks (for thetimes5 times10andtimes20 treatments)

Figure 8 Final micro-habitat maps after 100 developmental stepsshowing effects of different combinations ofThum andThol (in thecase of Model 1 top panel) orKhum and Khol (Model 2 middlepanel Model 3 bottom panel) on pattern geometries in typicalmodel runs The values ofT or K used in each simulation are in-dicated immediately above the maps and are expressed as a factorof the default values See main text for full explanation Light pixelsrepresent hummocks dark pixels represent hollows Low values ofy represent upslope locations (y= 0 is the upslope boundary) highvalues ofy represent downslope locations (y= 200 is the downslopeboundary) as such groundwater flow is generally down the pageAll maps have the same horizontal (x y) scale as that shown for theupper-leftmost map

4 Discussion

41 Hydrological transience and ecological memory

The striking similarity between the patterns generated by ourModels 1 2 and 3 at intermediate values of∆te and thestriped patterns reported by the previous SGCJ authors aswell as the absence of patterning in our models under morestringent steady-state hydrological conditions are highlysuggestive that the simulations reported in the previous SGCJ

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

P J Morris et al The role of hydrological transience in peatland pattern formation 39

studies had not attained genuine steady-state hydrologicalconditions The dependence of realistic patterning in ourmodels on intermediate equilibration times can be reconciledwith the currently popular theory that landscape patterningcommonly arises from interplay between long- and short-range processes (eg Rietkerk and van de Koppel 2008see also Turing 1952) The strength of the long-range neg-ative feedback (which acts to homogenise local differencesin water-table position in response to distant hydrologicalboundary conditions) appears to increase as the hydrologicalequilibration time is lengthened When equilibration timesare very short each grid squarersquos water-table behaviour is in-fluenced only by its immediate neighbours and long-rangeinfluences are unable to propagate across the model domainConversely the longest equilibration times cause the sim-ulated water-table map to be dominated by its long-rangeboundary conditions which under steady state eliminatemuch of the short-range effects of contrasting peat hydraulicproperties in neighbouring SL1 and SL2 units Without ad-ditional feedbacks such as those explored by Eppinga etal (2009) it is only at intermediate values of∆te that themodel strikes the balance of long- and short-range feedbacksseemingly required for patterning

If the absolute values of∆te are taken literally then themodel predicts that realistic patterning only occurs if micro-succession at SL1 operates on unrealistically short timescalesof hours to days rather than years Moreover Model 4 inwhich we introduced water-table transience not through shortequilibration times but through a daily rainfall series did notgenerate patterning This indicates clearly that it is the shortequilibration times used in Models 1 2 and 3 rather thansimply non-steady water tables (which are also present inModel 4) that caused patterning in our simulations It is per-haps initially tempting to discard the model in light of itsreliance on hydrological transience

However an alternative and more abstract interpretationof the modelsrsquo dynamics allows one to conceptualise thelength of a developmental step simply as a measure of hydro-logical steadiness under which succession takes place Withthis in mind we can think of succession as occurring overa period of years during which the degree of hydrologicalsteadiness is determined by∆te Therefore∆te is no longera literal time period during which plant community succes-sion occurs but merely a means of representing hydrologicalsteadiness For lower values of∆te the water tables acrossthe modelsrsquo domains are not in full equilibrium with the cur-rent distribution of hummocks and hollows As such thosewater-table maps reflect the distribution of hummocks andhollows during not only the current developmental step butalso partly the previous developmental step It may be thatthe formation of peatland patterning relies on some ecologi-cal memory effect (cf Peterson 2002) such as the influenceof recently buried peat that no longer reflects current posi-tions of hummocks and hollows but which remains hydro-logically important due to its shallow depth Despite Digi-

Bog allowing for 3-dimensional variation in peat propertiesthe ponding model is in essence a 2-dimensional model inso-far as the new micro-habitat map during each developmentalstep supersedes the previous map entirely the modelled sys-tem retains no memory of previous SL1 units or their associ-ated soil properties As such our models neglect some of thestructural complexity that peatlands exhibit in three spatialdimensions (eg Barber 1981) Our simulations with non-conservative hydrological steady-state criteria (particularly∆te = 50 and 100 h) implicitly contain a type of ecologicalmemory effect because the influence of previous patterns ofsurface vegetation and shallow soil properties are expressedin the modelrsquos hydrological behaviour this memory effectdecreases in strength with increasingly stringent hydrologi-cal steady-state criteria Our results indicate that this mem-ory effect is important to the formation of patterning in theponding model and should be investigated directly in futurestudies perhaps via the use of cohort-based peat accumula-tion models (eg Frolking et al 2010 Morris et al 20112012)

42 Peat hydraulic properties

Models 1 2 and 3 were all highly sensitive to the abso-lute values ofT or K and patterning only occurred withina narrow region of parameter space Even in thetimes005 andtimes20 treatments the values ofT or K were well within re-ported ranges the fact that these parameterisations failed toproduce patterning suggests that a full suite of relevant pro-cesses and feedbacks has not been represented Particularlyin Model 3 in which model water-table levels are maintainedby the simulated addition of rainfall even modest changes inthe absolute values ofKhol and Khum caused simulations toldquorun awayrdquo to either wet or dry end-member states In sim-ulations with higher values of hydraulic conductivity Model3 drained rapidly to the downslope boundary causing wa-ter tables to fall below the transition zone and giving riseto uniformly dry simulated landscapes dominated by hum-mocks Conversely the simulations with lower values of hy-draulic conductivity caused Model 3 to drain so slowly thatwater tables rose to the surface of all columns leading to uni-formly wet landscapes dominated by hollows The same ef-fect was not evident in Models 1 and 2 because the constant-head condition at the upslope boundary maintained water-table levels within the transition zone Nonetheless manipu-lating T andK in Models 1 and 2 respectively still resultedin landscapes devoid of realistic patterning by strengtheningthe long-range negative feedback relative to the short-rangepositive feedback

The high sensitivity to the absolute values ofT andK may be taken to suggest that our models are lackingat least one important negative feedback namely that be-tween peat decomposition and changes in peat hydraulicproperties For example saturated hydraulic conductivity isknown to decrease strongly with increasing time-integrated

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

40 P J Morris et al The role of hydrological transience in peatland pattern formation

decomposition of peat (eg Boelter 1969 1972 Grover andBaldock 2013) Areas with deep water tables are prone tomore rapid decomposition and more rapid collapse of porespaces The resultant reduction in hydraulic conductivity inturn causes peat to retain water more readily and causesdecay rates to fall stabilising system behaviour (see Be-lyea 2009 and Morris et al 2012) This concept is par-tially represented in the ponding model by lower values ofT or K in hummocks (where peat spends more time underoxic conditions and hence peat at the water table is moredegraded and less permeable) than in hollows (where peathas a shorter residence time under oxic conditions produc-ing better-preserved and more permeable peat at the depth ofthe water table) However in the ponding model only twoKvalues are possibleKhum and Khol meaning that hydraulicconductivity cannot vary as a continuous function of decom-position the ponding modelrsquos representation of this relation-ship may therefore be overly constrained Modelling studiesby Morris et al (2011) and Swindles et al (2012) have in-dicated that a continuous relationship between peat decom-position and hydraulic conductivity may be highly impor-tant to the ability of peatlands to self-organise and to main-tain homeostatic water-table behaviour The representation ofthis negative feedback within models of peatland patterningsuch as ours has the potential to stabilise model behaviourby allowing simulated water tables and peat permeabilityto self-organise thereby reducing model sensitivity to smallchanges in soil hydraulic parameter values This would re-quire the expansion of the ponding model so as to includeroutines that describe litter production and decompositionand continuous changes in peat hydraulic conductivity Theinclusion of these processes in patterning models would alsohelp to address the question raised above of ecological mem-ory effects and suggests the need to unify models of peatlandsurface patterning such as those considered here and cohortmodels of long-term peatland development (eg Frolking etal 2010 Morris et al 2011 2012)

43 Improved model hydrology

Our alterations to the ponding modelrsquos hydrological basiscompared to previously published versions had little impacton model behaviour evidenced by the fact that Models 1 2and 3 all behaved in qualitatively similar manners includingtheir response to hydrological transience The previous SGCJauthors reported that patterning is stronger and forms morereadily as the slope angle of the model domain and asso-ciated hydraulic gradients are increased The curved cross-sectional shape of the bog in Model 3 means that slope an-gle near the downslope boundary is greater than the uniform1 50 slope in Models 1 and 2 This appears to have allowedweak striping to develop at the downslope end of Model 3even under hydrological conditions that were so transientas to prevent patterning in Models 1 and 2 Increasing hy-drological steadiness in Model 3 then allowed patterning to

spread upslope onto increasingly shallow slopes The differ-ences in behaviour between Models 2 and 3 can therefore beexplained by the curved aquifer shape reflecting the slopeangle effect reported by previous authors In particular thenature of SL2 patterning in Model 3 (Fig 5) is strikinglysimilar to that seen in the simulations with domed aquifersreported by Couwenberg and Joosen (2005) We are thereforeleft to deduce that the permeable deep peat layer in Model 3had little independent effect and that the generation of pat-terning by the ponding mechanism is not dependent on eitherhydrological interaction with or isolation from deeper peatlayers

The simple binary treatment of hummock and hollowtransmissivity used in the earlier SGCJ studies also appearsto have produced no artefact in the behaviour of those models(or in our Model 1) compared to our more realistic Boussi-nesq treatment (in Models 2 and 3) Nonetheless it is im-portant to recognise that neither the simplified treatment oftransmissivity nor the assumption of an impermeable deeppeat layer despite both being questionable assumptions inthemselves were responsible for the modelrsquos reliance on thehydrological steady-state criterion and the absolute values ofpeat hydraulic properties nor the prediction of downslopemigration of SL2 stripes As such we are confident that thesebehaviours (reliance on hydrological transience high sensi-tivity to absolute values of transmissivity downslope patternmigration) are genuine facets of the ponding model and arenot artefacts of numerical implementation

44 Pattern migration

The downslope movement of SL2 stripes in the pondingmodel is an emergent behaviour that should be treated as atestable hypothesis against which the ponding model couldin part be tested (cf Grimm et al 2005) We are awareof very little direct evidence as to the long-term stationarityor otherwise of peatland patterning likely because the longtimescales involved preclude direct observation Koutaniemi(1999) found that SL2 units in a Finnish aapa mire expandedand contracted on a seasonal basis due to frost heave but hesaw no evidence of consistent downslope migration of any ofthese features over a 21 yr period Foster and Wright (1990)analysed the age of pool-bottom sediments from a Swedishraised bog They also found no evidence of migration overthousands of years Rather they concluded that the age ofpools was determined largely by the timing of the lateral ex-pansion of the bog and that once formed the pools remainedstationary However Kettridge et al (2012) used data froma combination of ground-penetrating radar and soil coresto investigate the below-ground physical properties of peatsoil in a raised bog in Wales They found subsurface layersthat dipped consistently towards the bogrsquos margins whichthey interpreted as evidence of down-slope migration of SL1or SL2 units during the bogrsquos development More observa-tional work is clearly required to ascertain whether or not

Earth Surf Dynam 1 29ndash43 2013 wwwearth-surf-dynamnet1292013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

Aber J S Aaviskoo K Karofeld E and Aber S W Patterns inEstonian bogs as depicted in color kite aerial photographs Suo53 1ndash15 2002

Alm J Talanov A Saarnio S Silvola J Ikkonen E AaltonenH Nykaumlnen H and Martikainen P J Reconstruction of thecarbon balance for microsites in a boreal oligotrophic pine fenFinland Oecologia 110 423ndash431 doi101007s0044200501771997

Andreasen J K OrsquoNeill R V Noss R and Slosser N CConsiderations for the development of a terrestrial index ofecological integrity Ecol Indic 1 21ndash35 doi101016S1470-160X(01)00007-3 2001

Baird A J Belyea L R and Morris P J Upscaling of peatland-atmosphere fluxes of methane small-scale heterogeneity in pro-cess rates and the pitfalls of ldquobucket-and-slabrdquo models in Car-bon Cycling in Northern Peatlands Geophysical Monograph Se-ries 184 edited by Baird A J Belyea L R Comas XReeve A and Slater L American Geophysical Union Wash-ington DC 37ndash53 doi1010292008GM000826 2009

Baird A J Morris P J and Belyea L R The DigiBog peatlanddevelopment model 1 Rationale conceptual model and hydro-logical basis Ecohydrology 5 242ndash255 doi101002eco2302012

Barber K E Peat Stratigraphy and Climate Change A Palaeoe-cological Test of the Theory of Cyclic Peat Bog RegenerationBalkema Rotterdam Netherlands 1981

Belyea L R Nonlinear dynamics of peatlands and potential feed-backs on the climate system in Carbon Cycling in North-ern Peatlands Geophysical Monograph Series 184 editedby Baird A J Belyea L R Comas X Reeve A andSlater L American Geophysical Union Washington DC 5ndash18doi1010292008GM000829 2009

Belyea L R and Baird A J Beyond ldquothe limits topeat bog growthrdquo Cross-scale feedback in peatland de-velopment Ecol Monogr 76 299ndash322 doi1018900012-9615(2006)076[0299BTLTPB]20CO2 2006

Belyea L R and Clymo R S Feedback control of the rateof peat formation P Roy Soc Lond B 268 1315ndash1321doi101098rspb20011665 2001

Boelter D H Physical properties of peats as related to degree ofdecomposition Soil Sci Soc Am Pro 33 606ndash609 1969

Boelter D H Methods for analysing the hydrological character-istics of organic soils in marsh-ridden areas in Proceedings ofthe IAHS Symposium on the Hydrology of Marsh Ridden AreasMinsk IAHSUNESCO Paris 161ndash169 1972

wwwearth-surf-dynamnet1292013 Earth Surf Dynam 1 29ndash43 2013

P J Morris et al The role of hydrological transience in peatland pattern formation 41

the findings of Kettridge et al (2012) hold in the generalcase but it appears that the ponding modelrsquos prediction ofthe downslope migration of SL2 stripes cannot be used tofalsify the model at this stage

5 Summary and conclusions

Our numerical experiments suggest strongly that simulationsreported in previous studies had not attained true hydrologi-cal steady-state conditions under true steady state patterningdoes not occur Realistic striped patterns only form whenmicro-habitat transitions occur in response to highly tran-sient water-table patterns The equilibration times requiredby the hydrological submodel to attain the level of transiencenecessary for patterning are of the order of hours to daysthese timescales are largely meaningless in terms of vege-tation dynamics The modelsrsquo reliance on the hydrologicalsteady-state criterion indicates that ecological memory mayplay a role in pattern formation Although ecological mem-ory is not represented explicitly (or deliberately) in our mod-els the transient water tables appear to have inadvertentlyreplicated such an effect

We increased incrementally the sophistication of the mod-elsrsquo numerical implementation and the realism of their con-ceptual bases chiefly by representing a permeable deep peatlayer improving the modelsrsquo calculation of transmissivityand introducing a plausible temporal variation in rainfall Indoing so we removed a number of simplifying assumptionsmade in previous studies However these improvements hadlittle effect on the modelsrsquo behaviours indicating that thesensitivity to hydrological steady state is a genuine facet ofthe models and is not an artefact of numerical implementa-tion

The models appear to be unrealistically sensitive to the ab-solute values of the parameters used to represent peat per-meability We interpret this sensitivity as indicating that themodels are missing a negative feedback between peat decom-position and hydraulic conductivity which previous studieshave shown is important to the ability of peatlands to self-organise

Peatland structures and processes exhibit complexity inthree spatial dimensions A logical next step for pattern-ing research would be to combine 2-dimensional (horizon-tal only) cellular landscape models such as those presentedhere with (mostly 1-dimensional vertical only) peatland de-velopment models that provide a more detailed and realis-tic representation of peat formation decomposition and dy-namic changes in soil hydraulic properties This would allowa direct process-based exploration of both the mechanism in-volved in ecological memory and a continuous relationshipbetween peat decomposition and hydraulic properties

Acknowledgements This research was funded in part by aQueen Mary University of London PhD studentship awarded toPaul Morris We are grateful to Brian Branfireun (University ofWestern Ontario) for permission to use the photograph in Fig 1and to The Field Studies Council at Malham in particular RobinSutton for providing us with the data from Malham Tarn WeatherStation We are grateful to the associate editor for helpful dialogueand to three referees for insightful and constructuive comments onan earlier version of the manuscript

Edited by F Metivier

References

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