Andrew Lane

Development of a Lagrangian sediment model to reproducethe bathymetric evolution of the Mersey Estuary

Received: 10 February 2005 / Accepted: 24 May 2005 / Published online: 25 August 2005� Springer-Verlag 2005

Abstract The development and application of aLagrangian particle-tracking model to simulate sedi-ment transport in the Mersey Estuary (UK) is described.Each of the particles (up to a million in total) is advectedhorizontally by the 3-D tidal currents. Related verticalmovements are: (1) downwards by settlement at a pre-scribed velocity ws and (2) both upwards and down-wards by a distance related to the vertical eddydiffusivity. In a novel departure from traditional prac-tice, where initial distributions of surficial sediments arespecified, all particles are introduced at the seawardboundary of the model. Provenance studies indicatesurficial sediments are overwhelmingly of marine origin.For the predominant fine sediments, ‘cyclical conver-gence’ in suspended sediment concentrations is approx-imated after about two spring-neap tidal cycles.Comparisons are shown between the suspended sedi-ment concentrations and net deposition rates computedby this model against observed values and earlier com-putations utilising both 1-D and 3-D Eulerian models.While all of these results are in broad agreement, theflexibility of the Lagrangian approach for simulatingflocculation, consolidation and mixed sediments illus-trates its future potential.

1 Introduction

A series of modelling and observational studies of theMersey estuary (Lane et al. 1997; Lane 2004) haveprovided insight into both the tidal and sedimentdynamics of the system. Fine resolution (120-m rectan-gular grid) 3-D models accurately reproduce both the

magnitude and cross-sectional variations in tidal currentconstituents. As commonly experienced, correspondingreproduction of suspended sediment concentrationsproves more difficult. However, by appropriate ‘cali-bration’ of erosion rates and selection of settling veloc-ities, reasonable reproduction of the magnitudes andcharacteristics of observed time series is achieved.

In the Mersey, observed net accretion by marinesediments has continued over the last century at a fairlysteady rate of approximately one million cubic metresper year, i.e., around 3% (by volume) of the net ebb/flood tidal exchange. Earlier studies indicated that thisaccretion was directly linked to the phenomenon of‘delayed settlement’ and hence was sensitive to effectivesettling velocities (by flocculation, etc., Manning andDyer 2002). In an effort to reproduce this accretion andexplore the related sensitivities, a random-walkLagrangian particle model has been formulated. Thedevelopment, application and evaluation of this modelare described in this paper.

Section 2 summarises the background of previousstudies, and Sect. 3 describes the formulation of therandom-walk model. Section 4 illustrates its applicationand the sensitivity of net accretion calculations in themodel to settling velocities, and finally, Sect. 5 outlinesfuture plans for further developments of this modellingapproach.

2 Background: observations and model studies

2.1 The Mersey Estuary

Close to the mouth of the estuary, the Mersey Narrows(Fig. 1) is approximately 1.5 km wide with a mean depthof 20 m, and tidal currents through this section can ex-ceed 2 m s�1. This deep entrance channel, important forshipping, extends outside of the estuary and is kept inplace by training walls. The inner estuary is muchbroader than at the mouth, and consists of mud and

Responsible Editor: Paulo Salles

A. LaneProudman Oceanographic Laboratory,Joseph Proudman Building, 6 Brownlow Street,Liverpool, L3 5DA UKE-mail: A.Lane@pol.ac.uk

Ocean Dynamics (2005) 55: 541–548DOI 10.1007/s10236-005-0011-8

sandbanks with highly mobile low water channels.Freshwater flow into the Mersey Estuary, Qf, variesfrom 25 to 200 m3 s�1, with a mean flow ratio (Qf

·12.42 h/volume between high and low water) ofapproximately 0.01. Flow ratios of less than 0.1 usuallyindicate well-mixed conditions, though in certain sec-tions during part of the tidal cycle, the Mersey is onlypartially mixed. Historically, the Mersey has been seri-ously polluted by industrial discharges and adjacent seadumping. An ongoing comprehensive programme hasimproved water quality in the river.

2.2 Tidal currents

Prandle et al. (1990) described earlier attempts at mon-itoring currents in this estuary using electromagneticcurrent meters mounted on a floating buoy. In a sub-sequent exercise (Lane et al. 1997), acoustic Dopplercurrent profiler (ADCP), electromagnetic and mechani-

cal current meters were mounted on low-profile framesacross a section of the estuary (shown as AB in Figs. 1and 5). Additional ADCP transects across the Narrowswere made continuously over a 15-day spring-neap cy-cle. These towed ADCP data provided detailed 3-Dspatial patterns of current ellipse distributions for themajor tidal constituents. Analyses of these data pre-sented difficulties in their transformation to a geographicreference frame, relative to which is also an irregularsampling frequency. Whole-estuary residual currentswere obtained from a 3-D numerical model (Fig. 1) witha 120-m rectangular grid and 10 vertical (sigma-coordi-nate) layers.

2.3 Cross-sectional distribution

Figure 2 shows how the M2 constituent of tidal currentsdominates the time series at the monitoring transect (ABin Fig. 1); it is almost rectilinear with maximum ampli-tude of 1.5 m s�1. The N2 constituent has approximatelyhalf the amplitude of the S2 constituent which, in turn, isabout one-third of the magnitude of the M2 amplitude(N2 and S2 not shown). In the vertical profiles of the

Fig. 1 Residual currents in theMersey Estuary near the bedfrom the high resolution 3-Dmodel (in cm s�1)

Fig. 2 Observed and computed M2 current amplitudes (m s�1)across the Mersey Narrows in July 1992

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axial component of the above current ellipses along thetransect line, the largest M2 currents occur at the surfacein the centre of the channel, decreasing with distancetowards the solid boundaries. A simplified model basedon a theory by Prandle (1982) is able to reproduce thesalient characteristics, indicating that most of the verti-cal and transverse variability in the tidal current distri-bution represents a localised response to depthvariations.

2.4 Suspended sediments

Figure 3 shows suspended sediment time series frommodels (Lane 2004) at the MIDAS (Mersey InshoreData Acquisition Station) location in the Narrows.Observations of suspended sediment (Fig. 4) weremade by the Environment Agency (NW) at the samelocation; analyses of these data are described byPrandle et al. (1990). These simulated time series of

suspended sediments for fall velocities, ws, of0.005 m s�1 (fine sand) and 0.0005 m s�1 (silt) arecomparable with observations in a recent study by Hillet al. (2003). The latter study suggested a median fallvelocity ws=0.0003 m s�1.

3 Formulation of the random-walk model

3.1 Model description

Random walk particle movements are used to replicatesolutions of the Eulerian advection-diffusion equation:

@C@t þu@C

@xþ v@C@y �ws

@C@z ¼ @

@z Kz@C@z

� �þsource

change inconcentration

advection settling diffusion

where C is the suspended sediment concentration, u and vare orthogonal velocity components, ws is the fallvelocity, and Kz is the vertical eddy diffusivity coefficient.

Fig. 3 Suspended sediment inthe Mersey Narrows from a 1-Dand b 3-D models, depth-averaged concentrations forsettling velocitiesws=0.005 m s�1 (upper) andws=0.0005 m s�1 (lower)

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3.2 Erosion

A simple erosion source cUp is assumed, with coefficientc. The power p, to which the tidal velocityU ¼ Ri¼1;n Ui cos xit is raised, is selected as 2 here,where Ui is the tidal current amplitude and xi is the tidalfrequency. The potential erosion in each time step Dt issummed until it exceeds a prescribed mass, M of anindividual particle. The height of release of each particlecorresponds to a normal distribution where the standarddeviation is l (diffusive path length—see diffusion, be-low). A value of c=0.0003 kg m�2 s2 was used follow-ing the specifications adopted from the Eulerian models(results shown in Fig. 3).

3.3 Settlement

Deposition occurs when the height of the particle abovethe bed calculated in a discrete advective settlement step�wsDt is less than zero.

3.4 Diffusion

Particles are displaced upwards or downwards randomlyby a length l = �(2 Kz Dt) (Fischer et al. 1979). Kz isapproximated by f UD where f is a bed friction coeffi-cient and D is the water depth. Movements ‘above’ thesurface are reflected downwards. Movements ‘below’ thebed are reflected upwards, but reduced by a ‘bounce’coefficient b (assumed as b=1 here). Deposition solelyby advection at the rate �ws C0 is consistent with tra-ditional ‘engineering’ approaches. However, sedimenta-tion of neutrally buoyant (ws=0) particles can occur.The model incorporates this bounce coefficient to rep-resent trapping via diffusion bed contact where required.

3.5 Numerical solution

The model simulation extends over N time steps for NC

tidal cycles, starting with no particles, either deposited

or in suspension. All particles are initially introducedon the flood tide at the seaward boundary, where thecondition dC/dx=0 is applied. N is chosen so thatDt=P/N, where P is the tidal period, produces a ran-dom walk length l (see section on diffusion) less than0.1 D. The condition l<0.1 D provides resolution forthe diffusive excursions (analogous to Dx=k/N usedfor horizontal wave resolution) and prevents the par-ticle from moving between the bed and the surface toorapidly. The solution involves calculations, for succes-sive time steps, of the height above the bed, z of eachparticle following:

1. a vertical advective movement �wsDt (downwards)2. a diffusive displacement l (up or down),3. horizontal advection,4. additional new particles are released into suspension

by accumulation of the erosion potential; likewise,particles may be ‘lost’ by settlement following theadvective movement in (1).

4 Model applications

Figure 5 illustrates an instantaneous distribution ofparticles calculated from the 3-D random-walk model,assuming unlimited supplies originating exclusively fromthe mouth of the estuary. Related tidal cycles of sedi-ment movement across a section in the Narrows areshown in Fig. 6; these show sediment import and export.Bathymetric evolution reflects the long-term temporalintegration of spatial gradients in relative erosion–deposition budgets.

Figures 7 and 8 show time series over two spring-neap cycles (starting from the initial introduction ofsediments) of:

1. cross-sectionally averaged suspended sediment con-centration (landwards from the mouth),

2. net sediment inflow and outflow (at the mouth)and

Fig. 4 Observed suspendedsediment concentrations in theMersey Narrows at surface(above), and mid-depth (below)

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Fig. 5 ‘Active’ particlesrepresent suspended sedimentsin the 3-D random-walk model

Fig. 6 The top panel shows thetidal elevation, the second panelshows the number of particlescrossing the Narrows(representing suspendedsediment transport during thetidal cycle). The lower diagramsshow the spatial variation ofparticles crossing the section(Fig. 5), moving upstream (in)and downstream (out)

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3. net-accumulated deposition for sediment fall veloci-ties, ws of 0.005 m s�1 (black lines) and 0.0005 m s�1

(grey lines) respectively.

Table 1 summarises these results and comparesthem with earlier observational and modelling resultsdescribed previously. Figure 7 (grey lines,ws=0.0005 m s�1) indicates how the SPM time serieschanges is predominantly semi-diurnal (linked toadvection) at the mouth. This follows from the sub-stantial drying out of the estuary at low water; onspring tides, the volume at high water is approxi-mately five times that at low water. More localisedresuspension introduces the familiar quarter-diurnalcomponent. Peak concentrations occur some three ti-dal cycles after maximum spring tides close to themouth and at up to seven cycles further upstream.Prandle (1997) calculated this phase lag as tan�1 (x/a),where x is the 15-day spring-neap cycle and 0.693/a isthe ‘half-life’ of SPM in suspension. This suggestshalf-lives ranging from about 1 day at the mouth to30 days further upstream (the latter value reflecting

the influence of axial gradients in concentration notconsidered in the original theory). Prandle (1997) usesan exponential decay model to represent sedimentdiffusion and deposition, in which the tidal and depth-averaged deposition rate is Ce�at. Although the pres-ent model formulation is independent of such theories,calculating the half-life is useful to characterise thedifferent dynamic and sediment regimes within theestuary.

Careful examination of numerical values for theseresults indicates the generation of a turbidity maximumupstream of the mouth on spring tides. The resultsindicate cyclical convergence, over a spring-neap cycle, isapproached within the two cycles shown, forws=0.0005 m s�1. By contrast, for ws=0.005 m s�1

(black line), Fig. 7 shows much reduced concentrationslargely confined to the seaward region, and a slower‘adjustment’ rate requiring longer simulation to intro-duce such coarse sediments further upstream. The timeseries is predominantly quarter-diurnal and peak con-centrations coincide with peak tides—implying a muchshorter half-life in suspension as discussed above.

Fig. 7 Suspended sedimentconcentrations at 12 positionsupstream along the Mersey(location 1 is at the mouth) forsettling velocitiesws=0.0005 m s�1 (grey lines)and ws=0.005 m s�1 (blacklines) coarser sediment

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Figure 8(a, b) shows related time-series of net inflowand outflow of SPM at the mouth, with the difference,representing net retention, highlighted. For

ws=0.0005 m s�1, the mean tidal exchange of sedimentsis 200,000 tonnes per tide, of which approximately 6% isretained amounting to 8.5 million tonnes per year. For

Fig. 8 a Cumulative inflow,outflow at the mouth of theMersey and b net depositedsediment

Table 1 Comparison of Lagrangian model results with observations and Eulerian modela values

Sediment settlingvelocity, ws (m s�1)

Suspended sedimentconcentration in theNarrows position2 (mg l�1)

Net sedimentdeposition(million tonnesyear�1)

Net tidal flux(m3 s�1)

Model

Mean Maximum Minimum Spring Neap

ws=0.005 25 67 0 1.8 46,500 2,300 Lagrangian125 550 0 1–D Eulerian150 300 0 3–D Eulerian

ws=0.0005 213 442 0 4.9 306,000 8,800 Lagrangian250 700 0 1–D Eulerian300 600 0 3–D Eulerian300 1,100 0 2.3 a, c 200,000b 60,000b Observed (surface)500 1,500 0 Observed (mid–depth)

Bathymetric records

a Lane (2004)b Prandle et al. (1990)c Thomas et al. (2002)

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ws=0.005 m s�1, the mean exchange is 40,000 tonnes ofwhich approximately 25% is retained, i.e., 7.0 mil-lion tonnes per year.

From Table 1, these concentrations, tidal fluxes andnet deposition rates are broadly consistent with bothearlier modelling studies and observations. Recognisingthat the concentrations associated with the finer sedi-ment are most likely to reflect conditions in the estuary,it is unlikely that the marine source of sediments entirelysatisfies the condition of unlimited supply assumed inthe model.

5 Conclusions and further studies

The Lagrangian particle approach to sediment model-ling in estuaries has been illustrated here and shown tobe effective in reproducing both the character andquantities of suspended concentrations, net tidal fluxesand long-term deposition found from earlier modellingand observational studies. Moreover, the differing nat-ure of the estuarine responses to the two sediment typesconsidered (ws=0.005 m s�1, d �70 lm silt/sand; andws=0.0005 m s�1, d �22 lm silt/mud) is consistent withunderlying theories.

An attractive feature of the model is its essentialsimplicity coupled with the potential for extensive elab-oration to accommodate effects such as flocculation orconsolidation and bioturbation, and biological andchemical labelling. The model is also well-suited fordiagnostic analyses to determine the contribution ofindividual mechanisms in the overall sediment budget.By introducing all sediments at the open boundary, both

the sensitivity to changes in this supply and the rates ofadjustment (i.e., subsequent distribution within theestuary) are evident—essential factors in consideringlikely geomorphological adjustments which might resultfrom Global Climate Change or localised ‘interven-tions’.

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