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List of Papers
The following papers, referred to in the text by their Roman numerals, areincluded in this thesis.
PAPER I: Ridged sea ice characteristics in the Arctic from a coupledmulticategory sea ice modelMårtensson, S., H. E. M. Meier, P. Pemberton, and J. Haapala(2012), J. Geophys. Res., 117, C00D15, DOI:10.1029/2010JC006936
PAPER II: Long-term characteristics of simulated ice deformation inthe Baltic Sea (1962-2007)Löptien, U., S. Mårtensson, H. E. M. Meier, and A. Höglund(2013), J. Geophys. Res.: Oceans, 118, DOI:10.1002/jgrc.20089
PAPER III: Modeling the impact of reduced sea ice cover in future cli-mate on the Baltic Sea biogeochemistryEilola, K., S. Mårtensson, and H. E. M. Meier (2013), Geophys.Res. Lett., 40, DOI:10.1029/2012GL054375
PAPER IV: Note on methods for assessing ice compression in continuumscale geophysical models.Mårtensson, S., U Löptien, H. E. M. Meier, Manuscript.
Reprints were made with permission from the publishers.
Author’s contribution
The original idea to Paper I came from a Swedish Research Council projectproposal written by Markus Meier and Peter Lundberg. The paper developedthrough discussions between mainly me and Markus Meier. I developed themodel and set it up for the Arctic Ocean. Jari Haapala provided his originalHELMI code and Markus Meier supported the model development with hisknowledge about the ocean model RCO. Further, I performed all the analysisand produced all figures except Figure 12 which shows the freshwater analysisperformed by Per Pemberton. I wrote most parts of the text. The introductionwas co-written by me and Jari Haapala, the model description except the newHELMI extension was written by Markus Meier and Per Pemberton providedinput for the fresh water discussion. Markus Meier helped with formulationsand proof reading. The original idea to Paper II developed from discussionsbetween me and Ulrike Löptien and also to some extent Markus Meier as partof an EU project in that SMHI was a partner. Ulrike Löptien had the idea toapply a regression model to explain ridging. Anders Höglund ported the RCO-HELMI code from the Arctic Ocean to the Baltic Sea. I finalized the modelparameter tuning and setup and evaluated the model. Further, I analyzed theseasonal cycle and wrote a first draft of the paper. Ulrike Löptien performedthe statistical analysis and wrote the final version of the paper. My main task inPaper III was to set up the model and to analyze the sea ice results in the sce-nario simulations. The original idea to Paper III came from Markus Meier whoperformed also the model simulations. Kari Eilola performed the analysis ofthe biogeochemical cycling and wrote the text with the help by me and MarkusMeier. Paper IV was completely my idea following up the discussions withinan EU project in that I participated. I did all model experiments, analysis andfigures and wrote all the text. Ulrike Löptien and Markus Meier helped withsupervision and text editing.
Papers not included in this thesis:Warm winds from the Pacific caused extensive Arctic sea-ice melt in sum-mer 2007Graversen R. G., T. Mauritsen, S. Drijfhout, M. Tjernström, S Mårtensson(2011), Climate Dynamics, 36, doi:10.1007/s00382-010-0809-z
Contents
List of Papers v
Author’s contribution vii
1 Historical context 11
2 Sea ice and Climate 132.1 Projections for the future . . . . . . . . . . . . . . . . . . . . 152.2 Sea ice as a habitat . . . . . . . . . . . . . . . . . . . . . . . 17
3 The development of modern sea ice modelling 193.1 What is sea ice and what are we actually trying to model? . . . 203.2 Anatonomy of a basin scale sea ice model . . . . . . . . . . . 22
3.2.1 Discretisation . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Sea ice rheology . . . . . . . . . . . . . . . . . . . . 233.2.3 Physics and parameterisations . . . . . . . . . . . . . 243.2.4 The Thermodynamics . . . . . . . . . . . . . . . . . 26
4 Summary of papers 294.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Sammanfattning xxxiii
Acknowledgements xxxv
References xxxvii
1. Historical context
Throughout history, sea ice has been both an asset and an obstacle for thepeople living in the north. It has sparked the interest of politicians, adventurersand scientists. Sea ice is something most people rarely have to deal with andit might be a somewhat abstract subject. Therefore, some historical events arehighlighted below to give context to the more technical discussion that willfollow.
• In 1658, the Swedes surprised the Danes by crossing the Belts by footover the ice, something that is very rarely possible. This action was ahuge military success that lead to the treaty of Roskilde later that year.A treaty that defined the southern Swedish border of today.
• In 1893 the Fram expedition lead by Fridtiof Nansen set sail towards theNorth Pole. A dramatic journey where Nansen used both boat and dogsleds, yet failed to reach the pole.
• In 1912, RMS Titanic sank on its maiden journey after colliding with alarge iceberg in the Atlantic ocean.
• In 1958 the U.S. submarine U.S.S. Nautilus was the first submarine toreach the North Pole and traverse across the entire Arctic under the ice.Later, submarines armed with nuclear armed ballistic missiles played animportant role in the defence strategies of both the U.S. and the formerU.S.S.R. Sea ice thickness, ridges and leads all affected the operationsof submarines. Knowledge of the ice thickness was thus paramount.
• The summer ice minimum in 2007 reached a much lower value than ex-pected, shattering all previous records. In some ways this minimum hasbeen the canary in the coal mine for the accelerated warming in the Arc-tic that has raised both environmental as well as geopolitical questions.The Arctic is a very sensitive biotope and beneath the ice, large quan-tities of natural resources are expected to be precent. Further, if the icemelts, shipping in the Arctic will increase utilising shorter trade routesthat will open up. The summer minima of 2007 was exceeded as seen inFigure 1.1.
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Figure 1.1: August ice extent and trend from the National Snow and Ice DataCenter. Minima usually occur in mid September.
Important questions are not only where the ice edge will be but also theamount of drift, where new ice is formed and how the ice deforms creating,among others, pressure ridges.
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2. Sea ice and Climate
Sea ice has a strong impact on both the ocean and the atmosphere. It limits heatfluxes between ocean and air, allowing for a shallow atmospheric boundarylayer over the ice. For example Guest et al. [10] who did measurements inthe marginal ice zones of the Nordic Seas found the spring median height ofthe atmospheric boundary layer to be 155 meters over the pack ice and 1240meters over the open ocean for off-ice winds. For on-ice winds, the median iceboundary layer height was measured to 160 meters and to 1490 meters overopen water. Stratiform clouds usually form at temperature inversions such asthose found at the boundary layer top. In the Arctic, low level clouds formed atthe boundary layer top is common. As clouds are a very effective absorber andtransmitter of long wave (heat) radiation, the cloud height is important for thethermodynamic balance. As the temperature generally decreases with heightdue to the adiabatic cooling (cooling due to slow expansion), changes in cloudheight alters the emission of long wave radiation.
The moisture balance is also highly dependent on the ice state, as shown inPaper I of this theses. We showed that by introducing an ice thickness distribu-tion, the net precipitation minus evaporation changed by 7%. This differenceis attributed to an increased frequency of leads.
Polynyas, which are areas of open water within the ice, are created eitherthrough melting of ice or through divergent advection of the ice. In the lattercase and under freezing conditions, the strong temperature difference betweenthe cold air and warm water causes large heat losses of up to several hundredwatts per square meter. Further, as ice is created, brine rejection causes theformation of very dense and cold saline water. As this dense water sinks,significant ventilation of deep and intermediate water masses occurs, [37]. Theventilation of deep water is mainly confined to the shelf areas. At the shelf,persistent off-shore winds may maintain a somewhat consistent polynya whichgives the deep water time to form. The dense water then flows of the shelf intothe deep, ventilating the deep water masses. There are several areas where offshore winds, or warm currents and a high mixing of the upper water columnleads to frequently recurring polynyas. Some examples in the Arctic are theKara Sea, Barents Sea and Chukchi Sea polynyas, [22].
However, formation of deep water masses is also dependent on the initialsalinity of the surface water. Maus [23] showed that advection of ice in the
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previous summer, not winter polynyas, is the most likely factor for dense waterformation in the Barents Sea.
During winter, polynyas and leads account for 50% of the atmosphere-ocean heat exchange in the Arctic [24]. Naturally, ice production rates arereflected in the heat exchange figures making a proper representation of leadsand polynyas a key process in sea ice modelling. Hence, leads and polynyasare important both for the ice volume but also in terms of the oceanic andatmospheric heat budgets [36].
The momentum flux from the atmosphere to the ocean is damped in thepresence of sea ice. As the atmosphere is hindered from acting directly on thesea surface, stress has to be transferred by the ice motion. While the Arcticice cover is quite mobile, stress transfer is affected by the internal ice stresswhich may dominate the force balance. In this sense, sea ice may also affectupwelling as shown by e.g. Pickart et al. [31].
Ice and snow albedo is a key factor in the radiative budged where the fa-mous ice albedo feedback mechanism amplifies changes in the downward ra-diation flux. It is not only important in a warming climate, but also in colderclimates. In a warmer climate the ice albedo feedback accelerates the melt-ing and additional heat is stored in the upper level of the ocean. In the fall,this heat has to be returned to the atmosphere (or space) before ice can form,thus delaying the onset of ice formation. The resulting shorter freezing seasonbefore next spring in turns means a thinner ice cover in the following springmelt season. The ice albedo feedback is not dependent on heat transportedinto the system as it is enough that a significant open water fraction is formedearlier than usual during the melt season. An increase in open water forma-tion could in addition to anomalous warming also be caused by an increasein storm activity. Other factors such as melt ponds may also contribute to thealbedo feedback. At present, both a warming and a change in the atmosphericflow has been observed by e.g. [27; 34]. Likewise, during past cold periods,the ice cover over sea and land helped shift the radiative balance to a muchcolder one.
Both the albedo feedback and the ice dynamics are dependent on the icethickness distribution. Generally speaking, ice of uniform thickness is lesssusceptible of ice albedo feedbacks during spring. For non-uniform ice ofsimilar volume, the thinner ice melts more rapidly changing the albedo ofthe area allowing for a higher absorption of incoming solar radiation and thealbedo-feedback is established. The ice dynamics is also much dependent onthe thinnest ice as it is the thin ice that largely controls the ice strength. Thisallows for the ice cover to remain relatively mobile even when the ice volumeincreases, as long as at least some deformation occurs. For a thinning ice cover,this process is a negative feedback, [15].
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Figure 2.1: Figure showing the basic principle of the ice-albedo effect due to aspread in the ice thickness distribution. On the left hand side, the melting of seaice from winter conditions at the top to summer conditions depicted in the bottomleft reveals open water which absorbs incoming solar radiation to a higher degree,thus changing the energy balance. To the right, the same amount of reduction insea ice thickness depicted in a single category model where the ice is representedby a single average value. Here, the ice albedo feedback is limited to melt ponds.Note though that ice motion, which may cause open water to form, is ignored inthis sketch.
2.1 Projections for the future
Observed trends in the Arctic raise the question if and when the Arctic willbe ice free. Current climate models indicate ice free summers in the Arctic assoon as 2030 according to Wang and Overland [44] who based their estimateon projections from CMIP5 models and observations. Later dates are givenby Mheel et al. [25] who estimate the Arctic to become ice free in 2040-2060according to simulations using the two models CESM1(CAM5) and CCSM4driven by the RCP8.5 emission scenario. The RCP8.5 is a high emission sce-
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nario where 8.5 stands for an increase in radiative forcing by 8.5 watts persquare meter by the year 2100 as compared to pre-industrial (19 century) val-ues. For the earliest estimates, the specific emission scenario is not that criticalin the timing for when the Arctic will become ice-free as no drastic changein green house gas emission is expected to happen in the near future. Thespread in estimates between different climate projections is noted by for ex-ample [28; 38] who also notes that most CMIP5 are too conservative in theirestimates.
One reason for the conservative estimates might be due to deficiencies inthe ice kinematics as noted by Rampal et al. [32]. They show that the Interna-tional Panel on Climate Change - Assessment Report 4 (IPCC-AR4) models donot reproduce the observed trends in ice kinematics and ice thickness. Neitherdo they have a correct seasonal cycle in the ice kinematics. In the models, icevelocities correlate with wind speeds suggesting a poor connection betweenthe ice state and the ice kinematics. This conclusion comes from the fact thatmodelled ice speed maxima occurs during late fall and winter in contrast to ob-served maxima during summer and early fall. Surprisingly, Rampal et al. [32]found no indication that more advanced ice models performed significantlybetter than quite simple ones in the IPCC-AR4 model ensemble. The IPCC-AR4 dates back to 2007. Todays models are better at capturing current trends,[18; 38], yet, large uncertainties in future projections remain as noted above.The improvement in these newer models, at least for the advanced ones, aremostly found in implementation details such as better melt pond parametri-sation rather than changes in basic principles. A counter example to Rampalet al. [32] can be found in Zhang et al. [46] who showed a marked increase inice drift speeds during the period 2007-2011 compared to 1979-2006 in theirsimulation, connecting the ice state with the ice kinematics. An explanationto the findings in Rampal et al. [32] might be found in the atmospheric dragcoefficient. Lüpkes et al. [21] found that drag coefficients traditionally used instate of the art models differ with a factor of 0.5-1.2 drag coefficients whereform drag from both the ice edge as well as melt pond edges has been ac-counted for. Most interestingly is that the maximum drag is achieved at 50%ice concentrations in these new parameterisation.
Together, these studies stresses the complex nature of sea ice modelling,especially in the Arctic with its multi-year ice.
Despite sure signs of a warmer Arctic in the future, local climate in nearbyareas may not follow the trend. For example Petoukhov and Semenov [30]showed that a decrease of winter ice concentrations in the Barents and KaraSeas may in some situations result in cold winters over Eurasia. Also Francisand Vavrus [7] noted a link between the Arctic amplification and an increasein extreme weather due to more persistent weather patterns. Where Arctic
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warming is partly the direct result of local changes in for example surfacealbedo, the climate in sub polar areas are more dependant on the atmosphericcirculation.
Contrary to the Arctic, in Antarctica the sea ice extent show a positivetrend. This can be explained by mainly two things. The presence of the Antarc-tic continent at the south pole prevents an effective ice albedo feedback. Fur-ther, the circumpolar wind pattern around the Antarctic continent isolates thearea from warm northern winds, while in the Arctic, the sub polar land massdistribution give rise to Rossby waves which promotes transportation of heatnorthwards. The Antarctic circumpolar wind is driven by the strong temper-ature gradient between the cold pole and the warmer air masses to the north.This gradient has increased due to the Antarctic ozone hole which cools theupper parts of the atmosphere, strengthening the polar vortex. [40; 43]
2.2 Sea ice as a habitat
Sea ice is a habitat for several species of plankton and algae. These speciesform a sympagic under-ice food web. Estimates of the primary productionfrom this food web in the Arctic ranges between 7-20% for the entire Arcticand up to 40-50% locally, [1; 5]. In the Antarctic the sea ice ecosystem ac-counts for about 4% of the total primary production. Diatoms are the majorcontributors for photosynthetic production and may account for over 90% ofthe total, [1].
These sympagic faunas serve important roles for the overall food web inboth the Arctic and the Antarctic. In the Antarctic, the Antarctic krill is aspecies which life cycle depends on the presence of these algae during wintertime. The krill in turn is a key food species for larger animals such seals, birds,penguins and others. Antarctic krill is even trawled to some extent.
In the Arctic, the zooplankton Calanus glacialis serves a similar role, whoalso during parts of the year depends on ice algae as food source. This zoo-plankton, being rich in fats, is in turn food for the Arctic cod, birds and whales.The cod is staple food for many animals higher up in the food chain.
Sea ice thickness, brine content and structure are factors in the growth ofthese life forms. Sea ice dynamics can play a major role here in its effect onboth ice thickness and brine content. The thickness of the ice regulates avail-able light for photosynthesis and brine content affects the chemical conditionsfor biological activity. While a high salinity is detrimental for productivity,brine pockets is the habitat for many diatoms, revealing a complex interplaybetween positive and negative growth factors.
In a warming climate, primary production in the Arctic is projected to in-crease. However, how the food chain will adapt is less known. More sun-
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light will be available for photosynthesis as the ice retreats. It is also possiblethat more nutrients will be disposed into the Arctic via river runoff as climatechange inland, [8]. At the same time, species now dependent on sea ice algaeor sea ice will need to adapt or see their habitats shrink significantly. Also,with higher temperatures, Atlantic species will migrate northwards.
Even today, relatively minor warm spells has been observed to affect thepolar bear population negatively. The polar bear is dependent on the ice as it isthe habitat for many of its preys. Another example of a species affected by therecent warming is the black guillemot, [26]. These two examples highlightsthe fragility of the Arctic environment.
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3. The development of modernsea ice modelling
Sea ice modelling as we know it today came about during the 60ies and the70ies. It was probably a combination of several factors that made this develop-ment possible. The increased political interest in the Arctic due to the cold waris one, improved observations due to satellites another and the development ofpowerful computer facilities a third key factor.
Despite the contribution from satellites, observations of sea ice is still acostly activity, often requiring air borne measurements or special rigs attachedto the ice. Modelling has therefore been a cheap alternative for increasingour understanding of the ice. Cheap being a relative term as basin scale icemodels are run on supercomputers, often requiring weeks of wall-time for eachsimulation.
Much of the basic work of describing the ice in ways suitable for discreti-sation was done in the 60ies and 70ies. Especially during the 70ies through theArctic Ice Dynamics Joint EXperiment (AIDJEX) (1970-1978) [41] the seaice dynamics we use today was developed. It was during AIDJEX the reali-sation that ice can be described as a material with plastic properties emerged,in contrast to previous unsatisfactory attempts to describe the ice as a viscousmaterial. Sea ice modelling therefore has in some sense closer connections tomechanical engineering than fluid dynamics.
The question remained though how to regularise the equations for efficienttime-integration on computers. Different approaches included both elastic-plastic and viscous-plastic solutions. In the end the viscous-plastic solutionsuggested by Hibler [13] was favoured due to its computational efficiency.He introduced a constitutive law relating strain rates and ice strength to theice stress which can be solved using implicit numerical schemes that allowfor long time steps. This solution has for a long time been more or less thestandard solution to sea ice rheology and it was not before the late 90ies thatan alternative became popular [16; 17]. An alternative that more or less hasthe same rheological properties but which is faster on highly parallel machines.Similarly, the thermodynamics used in RCO was developed during the 70ies,c.f Semtner [35]. Other contemporary models use thermodynamics of similardesign. Most improvements in the thermodynamics has been done in how the
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surface is handled, particularly in how ice and snow albedo are described. Thenotion of describing the ice thickness not as a grid box average, but as a icethickness distribution function also dates back to at least the AIDJEX, [33],but wasn’t implemented in most models until recently.
Improvements has been limited largely by two factors, (1) quality andquantity of observations and (2) available computational power. Today, thequality of the observations have become good enough to expose basic prob-lems in our models, namely the viscous-plastic rheology. Observations haveshown that the models struggle to reproduce the real ice motion in detail [20]and produce incorrect scaling properties compared to observations [9]. An-other problem is the isotopic assumption that was based on an infinitesimalelement being 100km [3]. Todays ice models resolve much smaller lengthscales, calling for a reevaluation of the isotropic assumption.
3.1 What is sea ice and what are we actually trying tomodel?
Sea ice is often thought of as large ice floes floating around or huge areasof the polar seas covered with meter thick ice. Both of these views are trueand in addition there are also various forms of smaller ice floes, deformed ice,packed ice and other small scale features. These different features of sea icearea characterised by length scales in the range between meters to nauticalmiles. From a modelling perspective this poses a problem as any model thatcan resolve ridges or other features on scales of a meter becomes impracticalfor applications at the basin scale. The solution to the problem is to identifylarge scale properties that can be modelled, leaving the small scale properties tobe parameterised. If we describe ice on really large scales, say 100km, sea iceis very two dimensional. That is, sea ice is very thin compared to its horizontaldimensions. Further, like gravel on a road, there is no point in talking aboutindividual stones, but the average size and the size distribution are meaningfulquestions. Thus, standard assumptions on sea ice is are:
1. Ice thickness is random in space and time, defined by a thickness distri-bution function.
2. Structures in the ice, such as leads, have no preferred direction, that is,isotropy is assumed.
3. Ice motion follows a plastic flow law.
4. Failure strength is a function of the thickness distribution and concen-tration
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5. Sub-grid scale features and structures are described by their statisticalproperties.
6. sea ice is a continuum.
The practical implications of these assumptions are maybe not evident,hence I will here elaborate on them following the listed order above. Assump-tion 1) means that ice thickness is either described by an average thickness orby a discrete thickness distribution function where each thickness is coupledwith an ice fraction. Assumption 2) means that at any point, the ice strength isnot a function of the direction. This assumption is not true on smaller scaleswhere for example a long crack in the ice breaks the isotropy. Referring toassumption 3), a plastic material is a material that resist deformation up toa threshold, called the yield stress. Above this threshold it is incapable ofresisting any excess forces leading to a ‘catastrophic’ failure. That is, the de-formation is irreversible and abrupt. To understand this intuitively, think ofthe coloured clay kids play with. Clay is a plastic material. Note here thatdeformation in this context only means non uniform motion like for exampleconvergence. It does not imply creation of pressure ridges, though it may hap-pen. Assumption 4) This simply means that an area with much open water hasa very low ice strength, and that thin ice is weaker than thicker ice. For modelswith a sub-grid scale thickness distribution, it is possible to use the work doneby the change in potential energy and frictional losses during deformation asa measure of the ice strength. Independent of the chosen approach the resultare the same, that is, thinner ice is weaker than thicker ice. Assumption 5)means that all sub-grid scale features are described in a statistical framework.So for example ridging, which is a sub-grid scale process is described as aempirically based function that modifies the ice thickness distribution given aconvergent motion and a dense enough ice field. The continuum assumptionno. 6) makes it possible to apply standard derivatives without having to worryabout the presence of leads or cracks in the ice. This assumption is also thecause of some confusion when talking about plasticity and deformation. It isthe continuum that is deformed, not any individual ice floe. Hence, sea ice asrepresented in models is a quite abstract construction.
Since ice isn’t truly two dimensional, a vertical model need to be employedto take care of heat exchange, thermodynamic growth etc. Consequently, thevertical dimension is disconnected to the horizontal ones. A way to think aboutthe division is that the model is divided into a dynamical (horizontal) part anda thermodynamic (vertical) part. Both parts operate on the ice thickness butrepresent completely different physics. Since the thermodynamics handle heatloss from the ocean, this part of the model framework includes the formationof new ice even this affects the horizontal structure.
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This makes for a fairly simple model with clear separation between thethermodynamics, dynamics and the sub-grid-scale physics.
In the next chapter, the three parts of a state-of-the-art sea ice model (ther-modynamics, dynamics and the sub-grid scale physics) will be described in de-tail. But before that, the discretization on a fixed numerical grid and the trans-formation from continuous derivates to finite differences will be explained. Asthese topics are the basic methods in ice, ocean and even atmospheric models,an introduction might be useful for the discussion.
3.2 Anatonomy of a basin scale sea ice model
In the following, a brief description of the constituent parts of a sea ice modelfor climate applications are presented. This introduction is not intended to be acomplete description. Instead the aim is to present the basics for non-experts.
3.2.1 Discretisation
The equations that describe ice motion are a set of partial derivative equations(PDEs) describing Newton’s second law. They are quite easy to understand, aswhat they basically are stating is how much something moves if it is pushedon a little. However, finding an analytical solution is only very rarely possible.Instead, the set of PDEs are exchanged for their finite difference counterparts.That is (somewhat simplified)
F = mdvdt−> F = m
∆v∆t
(3.1)
Thus, if we define a numerical grid on where the ice state is known at each gridpoint, it will be possible, by application of the finite difference equations, tocompute what the state of the ice at these grid points will be some short timestep after the initial state. Do this repeatedly and you get a time series showingthe evolution of the ice.
The most important item when choosing the numerical grid is the distancebetween the grid points or the so-called spatial resolution. The grid spacingneeds to be a minimum of half the distance of the size of the feature one wouldlike to model, preferable a little bit less. This limitation is also known in sam-ple theory as the Nyquist rate. This is reason why the CD-standard sets thesampling rate to 44.1kHz for accurate sound reproduction all the way up to thehearing threshold at around 20kHz. In the Arctic, we have used a resolutionof about 0.25 degrees and in the Baltic a resolution of about 2 nautical miles.Both resolutions can be regarded as state-of-the-art values. Another reason towhy a high resolution grid is used is the minimisation of computational errors
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introduced by the finite differences. A finite difference can be seen as a trun-cated tailor series. This is in contrast to the exact form of the derivate ∂/∂ t,which is equivalent to an infinite Tailor series. In other words, there is an er-ror introduced when switching from derivatives to finite differences. An errorwhich is dependent on the grid resolution.
There are three common grids used for solving equations based upon finitedifferences. The Arakawa A, B and C grid. The A grid is the naive implemen-tation while the B and C grid are both used in ocean models. The B and Cgrids are both so-called staggered grids where mainly the velocities are shiftedin space increase the effective resolution of the grid.
While the finite difference models are common in oceanography, in otherfields other methods are more common. In mechanics, the finite elementmethod is popular and in meteorology a spectral representation is often used.
3.2.2 Sea ice rheology
Simply put, the purpose of the rheology is to update the ice velocites.As mentioned earlier, sea ice on large scales behaves like a plastic material
[3]. This might be confusing as on the small scales that we are used to, iceis a little bit elastic, but mostly brittle. The aggregate of ice cracking, crush-ing, ridging and all these things we may intuitively understand is neverthelessapproximately a plastic behaviour.
Plastic behaviour can be described through a constitutive law [13].
σi, j = 2ηεi, j +(ζ −η)εi, jδi, j−P2
δi, j (3.2)
For numerical reasons, the equation needs to be regularised. That is, a divisionby zero has to be avoided. The trick is to set lower limits on the viscosities η
and ζ resulting in the common viscous-plastic rheology where for non criticalstresses, the ice undergoes a slow viscous creep. This rheology has an ellipticyield curve where the ice resists compression more than it do shearing and thetensile strength is zero.
An alternative regularisation technique is to scale the numerator when thestrain rates are approaching zero [12]. This is equivalent to shrinking the yieldcurve (strength) for very small velocities. The result is the same in both caseswhere completely stationary ice does not exist in the model. This techniqueallow unphysical motions for velocities that are sufficiently small that theireffect is negligible.
The viscous-plastic rheology was extended with an elastic part by [17] forbetter performance on parallel computers. This rheology is referred to as theelastic-viscous-plastic (EVP) rheology. Technically, the introduced elasticity
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turns the numerical problem from implicit to explicit. In the damped case, theelastic solution converges to the viscous-plastic solution and by sub cyclingthe rheology in between model time steps, the rheology behaves like viscous-plastic. In this rheology, the elastic waves are completely unphysical and onlya mean to solve the viscous-plastic rheology more efficiently. In Hunke [16]the EVP rheology was updated to better represent the plastic flow, fixing issuesprecent in both the original EVP and VP rheologies. This type of viscousityis not related to the viscousity in early viscous-plastic rheologies, e.g., Coonet al. [3]. In RCO, we use the EVP rheology.
While popular and successful the standard VP rheology is not without crit-icism. For example, it can be argued that ice has a stronger shear strength if atthe same time it is undergoing compression. This would lead to a differentlyshaped yield curve (e.g. tear drop shaped instead of elliptic). For exampleWilchinsky et al. [45] included not only frictional work in deriving the yieldcurve and ice strength, but also the thickness distribution. In their simula-tions, utilising a modified version of the CICE model, they achieved a betterthickness distribution where the common overestimation of ice thickness in theBeaufort sea and under estimation in the central Arctic is much reduced.
An assumption that has been more and more challenged is the isotropicnature of the VP-rheology. Sea ice is clearly not isotropic at small scales (e.g.leads, pressure ridges). The isotropic assumption was made when the resolu-tion of ice models where on the order of 100km, or about 1-2 degrees. Todaymodels run at around a quarter degree resolution in the Arctic and are get-ting close to length scales where anisotropy matters. Work on incorporatinganisotropic rheologies is ongoing, e.g. Tsamados et al. [39]
Moreover, recent observations (e.g., Kwok et al. [20]) have shown that theVP rheology has some deficiencies in modelled ice velocities. Girard et al. [9]showed a poor representation of the scaling laws characterizing ice deforma-tion and that for example an Elasto-brittle framework performs better in thisregard. However, Girard et al. [9] used a finite element approach which mightnot be practical for implementation in current climate models.
3.2.3 Physics and parameterisations
As briefly touched on earlier, there are important sub-grid scale processes thatdeals with the ice floes interaction. The most important ones are the mechan-ical redistribution of ice that affect the thickness distribution function, the icestrength and the closing and opening of leads.
The opening and closing of leads are very simply handled through the iceconcentration. Divergent motion creates a proportionally sized area of openwater. For convergent motion, there is a mixture of closing of leads and defor-
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mation of the ice floes.In RCO, only convergence leads to deformation, while in reality shear may
also lead to ridging in conjunction to lead openings depending on the orienta-tion of the ice floes.
When ice floes deform, a number of different forms of ice types can beformed, such as ridged, rafted and rubble ice. Depending on the resultingice, the ice thickness distribution is modified accordingly. These processes arereferred to as mechanical redistribution of the ice thickness.
Rafting creates ice of the combined thickness of the original ice floes whileridges are roughly triangular in shape. According to observations, ridged icethickness is dependent on the undeformed ice that undergoes ridging accordingto
hr = 17.64√
hu (3.3)
where, hr is the ridged ice thickness and hu is the undeformed ice floe thickness[19].
Further, in a multi-category ice model one has to consider which part of theice is undergoing deformation. We have used the assumption that the thinnest15%, by area, participates in the deformation [33]. In models with only one icethickness, deformation is handled by simply reducing the ice area while pre-serving the volume. To the authors best knowledge HELMI differs comparedto most multi-category model. It has explicit categories for deformed ice, sothat when ice floes are deformed, it is not only the thickness distribution thatis affected, but also the distribution of undeformed to deformed ice, [11].
In our model, we assume that ice thinner than a threshold of 17cm rafts[29] (8 cm in the Baltic Sea, pers. comm. Jari Haapala) and ice thicker thanthis threshold ridges with a smooth but rapid transition between the two modes.How thick ice can be and still raft in reality depends on the strength of the icesheet which is a function of several factors, including salinity and thereforeage.
During deformation, kinetic energy is transformed into potential energyby the work against gravity when ridged or rafted ice is formed. There is alsowork done by the sliding friction between the ice floes. This energetics is oftenused in multi-category ice models to assess the strength of the ice cover as awhole and is defined as
P = C fCd
∫h2dh (3.4)
where for single category models the strength is usually formulated as (sometimes called two layer models, one ice and one water)
P = P∗he−(C(1−c) (3.5)
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P (h) = P ⇥he�20(1�c)
P (h) = CfCp
� �
0
(�r + �u)h2dh
Figure 3.1: Sketch showing difference between ice the ice strength formulationin the old version of RCO (left box) with a single ice class versus the multi-category version (right box) in which the ice strength is based on the energeticsand has a strong dependance on the thin ice in the area.
where C f , C and P∗ are tunable constants, Cd is the apparent density of icefloating in water and c is the fraction of ice within the grid cell.
These formulations of the ice strength is based on isotropic behaviour, thatis, sliding friction, the breaking of the ice floe into smaller pieces et cetera isproportional to the potential energy formation. On smaller scales, the princi-ples are the same, but due to the importance of geometry on small scales, nosimple rules like the ones above can be formulated.
In summary, these are the most important sub-grid processes incorporatedinto the model. In addition to these, there are some minor important processesthat can also be handled here like for example flooding of sea water due to tooheavy snow.
3.2.4 The Thermodynamics
The purpose of the thermodynamics is to freeze and melt ice, and to presentfluxes of heat, salt and water to the air and ocean.
The thermodynamics is typically of column type, evaluated independentlyof the surrounding ice. In these equations the ice is not represented by averagesin the same sense as in the rheology, rather the average ice floe thickness isconsidered. In multi-category ice models, every category is represented byone floe thickness each. In addition, fluxes for the open water fraction hasto be computed or be provided from an underlying ocean model. In the caseof a model with five ice categories, there would be five plus one columns forwhich to solve the thermodynamic equations, one for each category of iceplus the open water. The results are then summed together, weighted by eachcategory’s fraction of the total area, to give the total in and outgoing fluxes.
RCO utilises the two-layer thermodynamics of Semtner [35]. There the iceand snow column is represented by three discrete layers, two for the ice andone for the snow as depicted in Figure 3.2. For reasons of numerical stability,
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Figure 3.2: The layered thermodynamic column model of Semtner [35]: Thethick ice model is depicted to the left and thin ice model to the right. T denotestemperature, Q fluxes and h the thickness.
thin ice is handled via a single layer model. The temperature at each layer iscomputed by the use of one dimensional diffusion:
(ρci,s)∂T∂ t
= ki,s∂ 2T∂ z2 (3.6)
The most difficult interface to get right is the surface. Absorbed incom-ing radiation is dependent on the albedo or whiteness of the surface. This inturn depends on whether snow is present, if the snow is dry or wet, if thereare melt ponds and so on. Most commonly is to let the albedo be a functionof the ice state, where temperature and snow thickness are the most impor-tant parameters. Limitations of this approach have been documented by forexample Curry et al. [4]. However, with CCSM4 [14] a big step towards aphysics based albedo treatment has been taken by employing a radiative trans-fer scheme rather than an empirically based treatment. The use of a radiativescheme is especially important for long climatology runs where for examplethe effect of an increased presence of black carbon can be modelled.
Sea ice, in contrast with lake ice, is salty. Even though salt is rejected whenwater freezes to ice, followed by a rapid desalination during the first two weeksor so, first year ice still has a salinity of around 0.5% where it for very old ice(10 years) drops to around 0.3%. Young ice has a fairly uniform salinity profilewhere for old ice the ice is much more fresh at the top than at the bottom. Not
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only does this rejection of the salt affect the salinity of the ocean, it also affectthe thermal properties of the ice [42].
Most models have a fixed salinity or salinity profile representing the av-erage state, while some models (e.g. LIM3) models the time evolution of thesalinity profile.
The salt in the ice is concentrated to brine pockets. Freezing of these pock-ets releases heat into the ice and melting consumes heat. Whether the salinityprofile of the ice is variable or not, this process is commonly included into thethermodynamics. See for example Bitz and Lipscomb [2] for more details.
While vertical growth is fairly straight forward, ice formation in open wateris based on several complex processes. In the real world ice is not formed witha uniform thickness over large areas. Rather, it grows out horizontally frompre-existing ice and/or clump together in areas of open water. Therefore, itis common to add new ice as ice with some minimum thickness, which maydepend on the pre-existing ice in the grid-cell. In RCO we have used 2.5 cmas minimum thickness for new ice in the Baltic and for example CICE version3.14, which is more focused on the Arctic, uses 5 cm.
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4. Summary of papers
4.1 Paper I
The Arctic is experiencing a tremendous change in climate at the moment. Onekey question is what will happen to the thick multi-year ridged ice that consti-tutes the bulk of the summer ice cover. A loss of this ice would mean a muchdecreased summer ice extent. In this paper we investigate the climatology ofdeformed as well as undeformed ice in the Arctic 1980-2002. We also lookat some differences of ice concentration and freshwater budged between RCOand the then newly implemented RCO-HELMI multi-category model. Themost interesting finding is that of the ridge ice volume trend. In the centralparts of the Arctic, we find an increase in mean ridged ice thickness of about4-6 cm per year during 1980-2002. Observations of an increased ice drift speedcorroborates well with these findings, as a more mobile ice field may undergomore deformation. From these results, the effect of an increasingly youngerice on the ice cover is mitigated by an increase in ridging.
4.2 Paper II
The North Atlantic Oscillation (NAO) is an often used index in climate stud-ies. In this paper we first try to link the NAO index to ridged ice fraction in theBaltic Sea but can only explain at most 20-25% of the internal variability lo-cally. A positive NAO index is statistically linked to both higher temperaturesand stronger westerly winds. The first being a detriment to ice formation whilethe latter is positive factor for ridging. These adverse effects explain why theNAO gives poor results for ridge ice prediction. However, through multivariatelinear regression reconstruction of annually averaged ridged ice we achieve lo-cal correlations of up to 0.8. The parameters used in the regression are winteraveraged wind extremes, surface air temperature, and sea surface temperature(SST). For large parts of the basin, the atmospheric parameters are sufficient,while in the eastern part of the Bothnian Bay and southern Gulf of Finland theinclusion of the SST to reconstruct the bulk of the ridged ice fraction is re-quired. Because of the high skill in reconstruction we can conclude that thesethree routinely measured parameters are the main drivers for ridged ice forma-
29
tion. In addition to the reconstruction, we examine the average seasonal cycleof ridged, rafted and undeformed ice in the different basins. Most noteworthyis the continuation of growth of the deformed ice types well beyond the peakin the ice cover. This is an effect of the fact that deformed ice fraction growsthrough mechanical deformation, rather than due to thermodynamics as therest of the ice pack does.
4.3 Paper III
Climate scenarios all indicate a warmer future. In this paper we investigatethe impact of a warmer climate on the Baltic sea ice cover and the effectsof the changes has on the biological activity. Through an ensemble of fourdifferent climate scenarios forcing the RCO-SCOBI model we find that theBaltic is mostly ice free during spring in the future scenarios, as compared totoday where the Bothnian Bay, Bothnian Sea and the Gulf of Finland all havesignificant quantities of sea ice cover in spring. The shrinking ice cover leadsto an increase in wave height, increased spring irradiance and an increasedresuspension of organic matter in areas that today is sheltered by the sea icecover. Of the changes in the biogeochemical cycling, we found that only aspring bloom that starts and ends considerably earlier can be attributed to thedecreased sea ice cover. Other changes cannot be unambiguously attributed tothe change in ice cover as the impacts of changing water temperature, salinity,horizontal nutrient transport, and external nutrient supply might be importantas well.
4.4 Paper IV
Shipping is and will continue to be an important channel for the transportationof goods. In areas with sea ice presence, the sea ice may be a significant un-certainty factor, leading to increased costs and complications for the shippingindustry. The ice is therefore routinely charted by various ice services in coun-tries affected by the ice. Ice concentration, thickness and type is well charted.What remains as an uncertainty factor is the sea ice dynamics, specificallyice compression. Ice compression leads to higher resistance when movingthrough the ice as ice is pushed against the ship hull. In severe cases, shipsmay get stuck and even experience hull damage. Ship captains therefore oftenuse established channels through the ice, opting for a longer route rather thanrisk incidents in the ice. Assessing ice compression with a continuum scaleice model is still a young field of research. In this paper, we investigate theproperties of mainly two different ice parameters, the internal ice force in the
30
equation of motion, and the internal ice stress. The difficulty lies in transform-ing information from the model-scale to something relevant at the ship scale.The paper identifies some specific situations where the internal ice force givesa poor understanding of the ice state.
31
32
Sammanfattning
Syftet med den här avhandlingen är att öka förståelsen kring havsisens dyna-mik och dess roll för klimatet. Havsis är en viktig komponent i klimatsystemetdå utbytet av värme och vatten kraftigt påverkas av närvaron av havsis. Fler-talet vitt skilda arter så som diatomer, isbjörnar och människor lever nära, påeller i havsis och påverkas därför markant av dess utbredning och dynamik.Genom den storskaliga strömningen tvärs Arktis, genom bildandet av isval-lar samt öppningar i isen påverkar isens dynamiska egenskaper både klimatetoch det lokala vädret. Vi har valt att fokusera på bildandet av isvallar av fler-talet anledningar. Direkta observationer av isens interna tillstånd är svåra ochomständiga att utföra och därför extremt ovanliga. Vidare är observationer avhavsis normalt sett antingen begränsade i sin geografiska utbredning eller i sinomfattning. Vallad is kan betraktas som minnet från händelser av hög internbelastning i isen vilket på så sätt ger oss en viss inblick i isens interna tillstånd.Den tjockaste isen kan bara skapas mekaniskt genom isvallning och vallad is ärdärmed en viktig faktor för ett istäckes tjocklek och dess tjockleksdistribution.Vallad is är även av direkt intresse för båttrafiken genom istäckta farvatten dåisvallar kan bilda barriärer som hindrar fartygens framfart samt att komprime-rande is kan innebära stora risker för fartygen som befinner sig i istäcket.
För att kunna simulera dessa egenskaper hos havisen på ett adekvat sätthar “Rossby Centre Ocean model” (RCO) utökats med en ismodul av multi-kategorityp kallad “HELsinki Multi category Ice model” (HELMI). HELMIen explicit hantering av vallad och hopskjuten is, har en isstyrka baserat påenergiomvandlingen vid deformation samt, i egenskap av att vara en multi-kategorimodell, en representation av istjockleksfördelningen inom en enskildgridruta.
I papper I undersökte vi havsisens egenskaper i Arktis under perioden1980-2002. Vi fann att volymen vallad is i centrala Arktis har ökat trots endalande trend i havsisens utbredning. I Papper II tar vi upp möjligheterna attestimera mängden vallad is via en statistisk modell. I papper III undersöker vihur ett reducerat istäcke i framtida scenarion påverkar den biologin i Öster-sjön. Slutgiltligen tar papper IV upp problematiken kring hur iskompressionkan modelleras med denna typ av modell.
Acknowledgements
First and foremost, I would like to thank my supervisor Markus Meier forgiving me this opportunity and for the support during this overly long journey.Thanks goes also to my co-supervisor Kristofer Döös for getting me into theoceanography scene to begin with. I would also like to thank Ulrike Löptien forthe good cooperation and for cheering me on. A shout out goes to the collegesat SMHI for the lunch discussions and the occasional parties and beers thatmade my short stint in Norrköping a pleasant one. I guess my former officemate is now part of this group so, hi Heiner! At MISU, past and precent, Iwould like to thank a whole bunch of people but in particularly and in thisparticular order! Anders for asking so many questions about programming,forcing me to learn how to google ;), Jonas for proving that all oceanographersfrom skåne are endlessly cheerful people, Marie for calling me out on being atotal klugscheisser (sorry), Qiong and Gao for teaching me to say silly thingsin Chinese and teaching me some proper tea culture. Susanne said she wouldbe upset lest I mention her... :D There are so many people I’m forgetting rightnow, but it’s 04.13 and I still need to produce the ‘spikblad’ before the printshop wakes up so... (what why is he writing this here?? why am I reading myown thoughts, this is weird) And of course, I have to thank Mia for turning meinto a hipster-farmer >.<
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