effects of rheology and ice thickness distribution in a ...ice being more readily melted in the...

2656 VOLUME 29J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

q 1999 American Meteorological Society

Effects of Rheology and Ice Thickness Distribution in a Dynamic–Thermodynamic SeaIce Model

T. E. ARBETTER, J. A. CURRY, AND J. A. MASLANIK

Program in Atmospheric and Oceanic Sciences, Department of Aerospace Engineering Sciences, University of Colorado,Boulder, Colorado

(Manuscript received 12 September 1997, in final form 21 December 1998)

ABSTRACT

Realistic treatment of sea ice processes in general circulation models is needed to simulate properly globalclimate and climate change scenarios. As new sea ice treatments become available, it is necessary to evaluatethem in terms of their accuracy and computational time. Here, several dynamic ice models are compared usingboth a 2-category and 28-category ice thickness distribution. Simulations are conducted under normal windforcing, as well as under increased and decreased wind speeds. It is found that the lack of a shear strengthparameterization in the cavitating fluid rheology produces significantly different results in both ice thicknessand ice velocity than those produced by an elliptical rheology. Furthermore, use of a 28-category ice thicknessdistribution amplifies differences in the responses of the various models. While the choice of dynamic modelis governed by requirements of accuracy and implementation, it appears that, in terms of both parameterizationof physical properties and computational time, the elliptical rheology is well-suited for inclusion in a GCM.

1. Introduction

Arctic sea ice plays an integral role in global climatechange. Partial or complete disappearance of the icecover is hypothesized to bring about profound changesin both regional and global climate. This has been seenin doubled CO2 experiments (e.g., Washington andMeehl 1986; Rind et al. 1995), which indicate that CO2-induced warming will be enhanced by the thinning andretreat of sea ice in the Arctic. The reason for this in-creased response is due in part to the sea ice–albedofeedback, in which the disappearance of ice and snowdecreases the surface albedo and thereby increases theamount of incident solar radiation absorbed at the sur-face. This in turn brings about further melt of ice andsnow. Also important is the export of sea ice into theGreenland–Iceland–Norwegian (GIN) Sea throughFram Strait. Decadal-scale fluctuations in ice export arehypothesized to be responsible for events such as theGreat Salinity Anomaly (e.g., Mysak et al. 1990; Mysakand Power 1992; Hakkinen 1993). A substantial or com-plete disappearance of the ice could result in an anom-alously fresh mixed layer in the GIN Sea, suppressingor even shutting down thermohaline circulation in thisregion, and affecting climate farther south via the so-

Corresponding author address: Dr. Todd E. Arbetter, Departmentof Aerospace Engineering Sciences, P.O. Box 429, Boulder, CO80309-0429.E-mail: [email protected]

called ‘‘conveyor-belt’’ oceanic circulation (Broecker etal. 1985; Mysak and Power 1992).

The importance of sea ice has been recognized by theclimate modeling community (Houghton et al. 1996),and improved parameterizations of its thermodynamicand dynamic processes are being included in generalcirculation models (GCMs). Inclusion of sea ice inGCMs requires accurate and computationally efficientnumerical representations of the main sea ice processes.It is necessary not only to reproduce present-day seaice conditions correctly, but also to correctly predicthow the sea ice will respond to the perturbations as-sociated with climate change.

Sea ice was treated relatively crudely in early GCMs.Ice thermodynamics were included, but ice dynamicswere not (e.g., Washington and Meehl 1984). Manabeand Stouffer (1980, 1988) used a simple sea ice dy-namics model in which the ice drifted freely with theocean currents if it was less than a certain thickness (4m). Thicker ice remained stationary. More recently,GCMs have begun to incorporate sophisticated sea icedynamics parameterizations such as the viscous-plasticrheology (Hibler 1979; e.g., Hibler and Bryan 1987;Fichefet and Morales Maqueda 1997) and the cavitatingfluid rheology (Flato and Hibler 1992; e.g., Pollard andThompson 1994; Weatherly et al. 1997). These rheol-ogies, along with the recently developed elastic–vis-cous–plastic formulation (Hunke and Dukowicz 1997),are described in section 2.

Ice dynamics plays an important role in prediction of

OCTOBER 1999 2657A R B E T T E R E T A L .

climate change. The choice of ice dynamics parame-terization affects modeled ice thickness and transport.Several previous studies can be used to guide our un-derstanding of these effects. Hibler (1979) investigatedthe importance of parameterization of physical pro-cesses using a model with an elliptical (viscous–plastic)ice rheology and a 2-category ice thickness distributionin which ice is separated into categories of thick ice andopen water (which included ice of up to 50 cm thick-ness). Studies were performed in which the prescribedcompressive strength of ice was increased, the shearstrength was increased, strength parameterization wasaltered, and open water was not allowed. Results in-dicated that the shear strength parameterization wasmost important in determining the spatial ice thicknessvariations and ice export. Meanwhile, the open waterparameterization was important in determining meanthickness and growth characteristics. However, the ef-fects of the open water parameterization were compar-atively small.

Using a multicategory dynamic–thermodynamic seaice model, Hibler (1980) found that the inclusion ofridging substantially affects the heat exchange charac-teristics of the Arctic, as thin ice formed by open waterproduction is transferred by ridging to thicker ice cat-egories, creating open water and allowing more thin iceto form thermodynamically. The ice cover was foundto be sensitive to both thermodynamic parameterization(in the form of surface albedo) as well as ice strengthparameterization (in the form of frictional losses due toridging). Flato and Hibler (1995), using a model whichincluded separate distributions for level and ridged iceas well as a parameterization for the evolution of snowcover, found that the parameterization of the ridgingprocess—how much thin ice participates in the ridgingprocess and the thickness of the ridges formed—gov-erned both the ice circulation and the amount of energydissipated by ridging during shear deformation. Factorssuch as shear strength parameterization also had effectsbut were not as dominant. Both Hibler (1980) and Flatoand Hibler (1995) found the use of an ice thicknessdistribution produced thicker ice than occurred with the2-category model.

Using a fully explicit time-stepping numericalscheme, which guarantees energetic consistency, Ip etal. (1991) compared the results of elliptical, square, cav-itating fluid, and Mohr–Coulomb rheologies. It wasfound that the ice drift fields produced by the cavitatingfluid and Mohr–Coulomb treatments were faster thanthose produced by the elliptical and square rheologies,as was the outflow through Fram Strait. This was attri-buted to the latter two rheologies allowing a greateramount of shear stress. While spatial ice coverage pat-terns were similar for all four rheologies, the total icevolume was larger for the elliptical and square rheol-ogies, attributed to a larger Fram Strait outflow for thecavitating fluid and Mohr–Coulomb cases.

The effects of compressibility and shear strength were

also investigated by Flato and Hibler (1992). Comparinga velocity field produced by an incompressible cavitat-ing fluid model with one produced by free drift, theeffects of rheology were apparent, as regions of con-vergence in the free drift case were removed and theflow was steered along the coast. Allowing some com-pressibility of the ice resulted in some areas of con-vergent flow. Comparison of the compressible cavitatingfluid results with those of an elliptical (viscous–plastic)rheology and a Mohr–Coulomb rheology, both of whichcontain some shear strength parameterization, showedthat the lack of shear strength in the cavitating fluidparameterization yielded dramatically increased circu-lation compared to the other two. The Mohr–Coulombresults were intermediate. Spatial ice thickness patternswere similar, although the cavitating fluid case producedthinner ice. This was due in part to increased Fram Straitoutflow in the cavitating fluid case, as well as the thinnerice being more readily melted in the summer months.The effects of rheology on ice growth and heat fluxwere found to be small.

An extensive series of sensitivity tests was performedon a 2-category dynamic–thermodynamic sea ice modelby Holland et al. (1993). Among these was a comparisonof viscous–plastic rheology with cavitating fluid. Whilethey did not find significant differences in the annualcycles of ice thickness and ice-covered area, the kineticenergy fields associated with the cavitating fluid rhe-ology were 20% higher, consistent with a generally fast-er velocity field. They also performed tests on the winddrag parameterization, finding that while a decrease re-sulted in little change in ice covered area and thickness,an increase in wind drag increased the ice thickness dueto enhanced open water production. However, a morecomplicated treatment of wind drag as a function of icecompactness decreased the ice thickness slightly. Morerecently, Fichefet and Morales Maqueda (1997) com-pared the viscous–plastic and cavitating fluid rheologiesin a global sea ice–upper ocean model. They found littledifference in the seasonal cycles of ice areal coverage,but did notice the cavitating fluid simulation showed asmall decrease in ice volume throughout the year, par-ticularly in the Southern Hemisphere. This was due toincreased ice transport. The ice velocity fields producedby the cavitating fluid rheology were considerably fasterthan those of the viscous–plastic rheology. Also, North-ern Hemisphere features such as the Beaufort gyre andTranspolar Drift Stream had altered trajectories with thecavitating fluid model, resulting in it having a differentspatial distribution of ice thickness, most notably in theNorthern Hemisphere. Cavitating fluid ice was generallythicker than viscous–plastic ice in the Canadian basin,while in the Eurasian basin and GIN Sea it was thinner.

The study presented here further defines the effectsof ice rheology on simulations of Arctic sea ice coverand considers the role of ice thickness distribution inthese simulations. Three numerical formulations of icerheology and two ice thickness distributions are imple-


FIG. 1. Yield curves for the elliptical (viscous–plastic, elastic–viscous–plastic) and linear (cavitating fluid) rheologies in principalstress space. For very small strain rates, the viscous–plastic stressstate moves inside the yield curve to the smaller (shaded) ellipse,approximating a linear viscous fluid. After Hibler (1979), Flato andHibler (1992).

mented. All models are subjected to the same forcingfields. However, the resulting baseline conditions of thevarious models are not identical. While the models couldbe tuned to the same baseline conditions, that there arevaried responses to the same set of forcings demon-strates the model sensitivity to physical parameteriza-tions. Sensitivity studies are then performed in whicheach model is subjected to changes in wind speed, andthe model responses are intercompared and comparedto the baseline runs.

2. Governing equations and model description

The dynamics of sea ice is governed by a momentumbalance:

]um 5 2mf̂ k 3 u 1 t 1 t 2 mĝ=H 1 F, (1)a w]t

where, as in Flato and Hibler (1992, 1995), the nonlinearmomentum advection term is neglected. Here, m is theice mass per unit area, u is the ice velocity, f̂ is theCoriolis parameter, k is a unit vector normal to the sur-face, H is the sea surface dynamic height, ĝ the accel-eration due to gravity, and F is the force due to variationin internal ice stress; t a and t w are forces due to the airand water stresses given by

t 5 r C |U |(U cosf 1 k 3 U sinf) (2)a a a g g g

t 5 r C |U 2 u|[(U 2 u) cosuw w w w w

1 k 3 (U 2 u) sinu]. (3)w

Here, Ug is the geostrophic wind, Uw is the ocean cur-rent, Ca and Cw the air and water drag coefficients, raand rw the air and water densities, and f and u turningangles at the ice–air and ice–ocean interfaces. The rhe-ologies considered in this paper differ in their solutionfor F. These methods are summarized in the next threesubsections; the reader is encouraged to refer to theoriginal papers for a more detailed description. Section2d briefly describes the 2-category formulation of themodels, while section 2e describes the ice thicknessdistribution (28 category) formulation. The thermody-namics used for all models in this study are describedin Hibler (1980, appendix B).

a. Viscous-plastic model

Hibler (1979) presented a rheology in which sea iceis modeled as a nonlinear viscous compressible fluid.Under normal deformation rates the ice acts in a rigid–plastic manner, while under very small deformationrates it acts as a linear viscous fluid. The forces due tointernal stress are given by

F 5 = · s, (4)

where s is the two-dimensional stress tensor given by

pdi js 5 2h«̇ 1 (z 2 h)«̇ d 2 . (5)i j i j kk i j 2

Here, is the strain rate, p is the hydrostatic ice pressure«̇(defined below), and z and h are nonlinear bulk andshear viscosities, determined by

pz 5 (6)

2D

zh 5 , (7)

2e

where2 2 22 22 2 22D 5 Ï(«̇ 1 «̇ )(1 1 e ) 1 4e «̇ 1 2«̇ «̇ (1 2 e ).11 22 12 11 22

(8)

These viscosities are defined such that the stress statelies on an elliptical yield curve (where e is the ratio ofmajor to minor axes of the ellipse) for plastic flow (seeFig. 1). For viscous flow, which occurs at very smallstrain rates, maximum limits are imposed on z and hsuch that the stress state lies on a smaller, concentricellipse.

b. Cavitating fluid model

The cavitating fluid rheology introduced by Flato andHibler (1992) provides a computationally simpler rhe-ology than viscous–plastic, in which the pack ice doesnot resist divergence or shear but does resist conver-gence. The solution for F becomes simply

F 5 2=pstress, (9)

where pstress is the internal ice pressure, equal to the


magnitude of the principal stresses. [Note the change innotation from Flato and Hibler (1992); pmax (their no-tation) refers to p here.] Graphically, this yield curvecan be seen in Fig. 1. Computationally, the solution isachieved by starting with free drift, then correcting thevelocity fields in a manner reflecting the compressivestrength of the pack ice:

= · (u 1 ũ) $ 0 if p 1 p̃ 5 0stress stress

= · (u 1 ũ) 5 0 if 0 , p 1 p̃ , pstress stress

= · (u 1 ũ) # 0 if p 1 p̃ 5 p, (10)stress stresswhere the tilde indicates the corrected field. The firstterm represents divergence, the third term representsconvergence, while the second term represents the in-compressibility of the pack ice if it is subject to com-pressive pressure less than its compressive strength p.The iterative solution method for the cavitating fluidrheology is detailed in Flato and Hibler (1992).

c. Elastic–viscous–plastic model

The elastic–viscous–plastic model presented by Hun-ke and Dukowicz (1997) incorporates a numerical so-lution of an elliptical rheology in which an elastic con-tribution is added to the viscous–plastic equation. Thiselastic component is introduced for numerical efficiencyand is not intended to be physically realistic. Rewriting(5), we obtain

1 h 2 z ps 1 s d 1 d 5 «̇ , (11)i j kk i j i j i j2h 4hz 4z

which represents the viscous–plastic contribution to thestrain rate. Recognizing that an ice model need onlysimulate viscous–plastic behavior at timescales on theorder of the wind forcing (days), an elastic componentis added. This component, rather than a linear viscousapproximation (as in Hibler 1979), regularizes the smallstrain rate case. Hunke and Dukowicz (1997) argue thatthis component also provides more accuracy under tran-sient behavior while reducing to the original viscous–plastic behavior for longer timescales. The elastic con-tribution is represented by

]s1 i j5 «̇ , (12)i jE ]t

where E is a term analogous to Young’s modulus, de-fined to be a function of compactness and thickness.Summing (11) and (12) yields

]s1 1 h 2 z pij1 s 1 s d 1 d 5 «̇ . (13)i j kk i j i j i jE ]t 2h 4hz 4z

The momentum equation is integrated over an effectiveelastic–viscous–plastic time step (subcycle) of lengthDte 5 Dt/N, where Dt is the model time step and N isan integer greater than 1, with the stress tensor s updatedevery subcycle while the viscosities, compressive

strength, and Young’s modulus are held constant overa model time step. In the steady-state limit of (13), theelastic contribution disappears and the viscous–plasticsolution [(5), (11)] is recovered. As in the viscous–plas-tic formulation, (6)–(8) are used to determine the bulkand shear viscosities. Further details on the solutionmethod can be found in Hunke and Dukowicz (1997).

d. Two-category formulation

A 2-category (thick ice and open water) thicknessdistribution is used by Hibler (1979) using continuityequations for thickness,

]h ](uh) ](yh)5 2 2 1 S , (14)h]t ]x ]y

and for area,

]A ](uA) ](yA)5 2 2 1 S (15)A]t ]x ]y

in which h is the mean thickness of the ice and A is thefraction of ice-covered area, defined to be between zeroand unity; u and y are directional components of theice velocity vector u, and Sh and Sa are thermodynamicsource terms defined in Hibler (1979). The internal icestrength is given by

p 5 p*h exp[2C(1 2 A)], (16)

where p* and C are empirical constants given in Flatoand Hibler (1992).

e. Twenty-eight-category ice thickness distribution

Following Thorndike et al. (1975) and Hibler (1980),Flato and Hibler (1995) introduce continuity equationsfor level and ridged ice:

]g ]l 5 2= · (ug ) 2 ( f g ) 1 C 1 F (17)l l l l l]t ]h

]g ]r 5 2= · (ug ) 2 ( f g ) 1 C 1 F , (18)r r r r r]t ]h

where gl and gr are the level and ridged ice thicknessdistributions, and f l and f r are the level and ridged icethermodynamic growth rates, defined to be identical ( f r5 f l 5 f ); c l and cr, the level and ridged ice redistri-bution functions, and Fl and Fr, the thermodynamicsource terms, are defined in Flato and Hibler (1995). Adetailed description of the theory and numerical solutionof a two-dimensional dynamic–thermodynamic icethickness distribution model can by obtained by con-sulting Hibler (1980) and Flato and Hibler (1995).

The constraint proposed by Rothrock (1975), namelythat the potential energy of the displaced water per unitarea is equal to the sum of the energy of the ice andthe energy required to displace the water, implies


`

2p 5 C C (v 1 v )h dh, (19)f p E l r0

where Cp is a constant depending on the densities of iceand water and Cf is the ratio of total energy losses topotential energy change, defined by Flato and Hibler(1995) to be 17; vl and vr are ridging modes detailedin Flato and Hibler (1995). A method for numericallyevaluating (19) is given in Hibler (1980, appendix C)and is implemented here.

The total normalized energy dissipation rate can beobtained from (5):

121p s «̇ 5 (D 2 «̇ ). (20)i j i j kk2

Similarly, it can be derived from (13) if the elastic con-tribution is neglected. The right side of (20) can beseparated into components representing energy dissi-pated by shear and by convergence, respectively:

121p s «̇ 5 (D 2 |«̇ |) 2 min(«̇ , 0). (21)i j i j kk kk2

The energy dissipated by ridging is taken to be

1M 5 C (D 2 |«̇ |) 2 min(«̇ , 0), (22)s kk kk2

where M is the normalized mechanical energy dissi-pation rate due to ridge creation; CS is the fraction ofthe shearing component of mechanical energy dissipatedby ridging, and is set to 0.5 for the viscous–plastic andelastic–viscous–plastic formulations. For the cavitatingfluid rheology, the ratio of major to minor axes e of the‘‘elliptical’’ yield curve goes to infinity, and thus (8)reduces to

D 5 ||«̇kk (23)

(i.e, the positive root of the square of the sum of theprincipal strains), and the first term on the right-handside of (21) and (22) is identically zero. Thus, analogousto the large-scale rheology, no energy is dissipated byshear ridging in the cavitating fluid rheology; (21) and(22) are identical, and all energy dissipated is due toconvergence (G. Flato 1997, personal communication).

3. Model implementation and the baseline runs

The models used here are identical with the exceptionof ice rheology and ice thickness distribution. The ther-modynamics are essentially the same as those describedin Hibler (1980, appendix B). The model domain is an80-km rectangular Cartesian grid, resolving the ArcticOcean and adjacent seas including the GIN Sea southto Iceland as well as portions of the Bering Sea. Theannual and monthly averages presented here considerthe Arctic basin as defined in Gloersen et al. (1992).Forcing fields are derived from National Centers of En-vironmental Prediction reanalysis fields for 1992 at 6-

h intervals for surface air temperature, specific humid-ity, downwelling shortwave and longwave radiation, andgeostrophic winds computed from sea level pressure. Alatitudinally varying Coriolis parameter is used. Themodels are integrated for 30 years using 6-h time steps.The numerical solver described in Zhang and Hibler(1997) is used for the viscous–plastic rheology, with 1,5, and 15 iterations implemented in the solver to com-pare approximations to a plastic response. Modificationsare also made to the cavitating fluid rheology to accom-modate the high frequency (6 h) of the forcing by usinga tighter convergence condition. This is necessary inorder to maintain numerical stability for the cavitatingfluid solver (G. Flato 1997, personal communication).The elastic–viscous–plastic numerical model is imple-mented using its standard implementation of N 5 100subcycles. While it is possible to increase or decreasethe effect of the elastic waves by decreasing or increas-ing the value of N, this is not investigated here. The icethickness bins for the 28-category ice thickness distri-bution are given in Flato and Hibler (1995). Since therewas little natural variability in the runs once equilibriumwas reached, baseline averages are obtained in all casesby considering year 30 of the normal wind scenario.

Recently, the Sea Ice Model Intercomparison Project(SIMIP) (Lemke et al. 1997; see also http://www.ifm.uni-kiel.de/me/research/Projekte/SIMIP/sim-ip.html) has compiled requirements for a numericalmodel to be considered a valid implementation of aplastic rheology. Namely, the scheme should asymp-totically approach the exact solution for the stress statewith a sufficient number of iterations on the numericalsolver. As noted in Zhang and Hibler (1997), the line-arization of the bulk and shear viscosities in their so-lution procedure yields only an approximation to plasticflow. Thus, it is necessary to repeat the solution pro-cedure a number of times for each time step. Zhang andHibler (1997) recommend 15 iterations as close to trueplastic flow; thus, we choose viscous–plastic with 15iterations as our reference solution for this study. Figure2 shows the normalized stress states (s/p) for a typicalday in winter once equilibrium has been reached. It isseen that, for both 2- and 28-category formulations,nearly all of the stress states lie on or within the ellipticalyield curve.

a. Two-category simulations

The annual averages of thickness, areal coverage, andtotal kinetic energy per unit mass, as well as the Marchand September monthly means from the baseline two-layer simulations are given in Table 1. It is seen thatthe values for the viscous–plastic rheologies increaseasymptotically with increased number of iterations usedin the numerical solver. Values of total kinetic energyper unit mass, indicative of the overall motion of theice, decrease asymptotically with the number of itera-tions on the plastic solver for the annual averages. For


FIG. 2. Normalized stress states for a typical winter day for the viscous–plastic model with 15 iterations onthe numerical solver: (a) 225% wind speed case, 2 categories (left) and 28 categories (right); (b) normal windspeed case, 2 categories (left) and 28 categories (right); (c) 125% wind speed case, 2 categories (left) and 28categories (right).


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(1it

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Ann

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Mar

Sep

2.45

3.06

1.60

2.53

3.18

1.63

2.49

3.01

1.74

89.8

99.3

72.7

88.1

99.0

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92.4

99.6

79.2

0.39

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2533

3.52

69

0.74

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7864

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200.

2742

1.81

72

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(5it

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2.48

3.08

1.64

2.56

3.20

1.68

2.49

3.02

1.74

90.0

99.3

73.2

88.4

99.0

69.1

92.2

99.6

78.9

0.37

821.

2642

3.57

82

0.73

472.

9441

5.89

85

0.15

790.

3000

1.82

15

Vis

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stic

(15

iter

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Ann

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Mar

Sep

2.49

3.08

1.65

2.56

3.20

1.69

2.59

3.03

1.75

90.0

99.3

73.5

88.4

99.0

69.3

92.2

99.6

78.8

0.37

471.

2755

3.54

22

0.72

972.

9302

5.84

09

0.15

520.

3142

1.81

11

Ela

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–vis

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–pla

stic

Ann

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Mar

Sep

2.43

3.03

1.59

2.51

3.15

1.63

2.43

2.97

1.67

89.9

99.3

73.1

88.3

99.0

68.8

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99.6

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0.40

641.

4554

3.22

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353.

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4.50

33

0.17

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2.20

2.78

1.35

2.10

2.74

1.19

2.43

2.95

1.68

90.4

99.4

74.4

88.7

99.2

68.5

92.1

99.5

78.6

0.59

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6241

4.14

02

0.94

944.

1951

6.81

13

0.31

391.

4079

2.14

33

the March values (high areal coverage), the speed of theflow increases slightly, while for September (low arealcoverage) there is no pattern. However, these resultssuggest that a better approximation of a plastic solutionmay be important.

The elastic–viscous–plastic model gives ice thicknessslightly thinner (about 5 cm) than the viscous–plasticformulation, while areal ice coverage is about the same.The total kinetic energy values are about 10% higherfor the annual and March averages, although the Sep-tember values are actually lower than those of the vis-cous–plastic results. This results from the slightly dif-ferent treatment of the small strain rate regime. Stressstate plots (not shown, see e.g., Hunke and Zhang 1999)indicate that the solution method is not obeying theelliptical yield curve rule. The cavitating fluid rheologyaverage thicknesses are notably thinner than any of theelliptical yield curve results (viscous–plastic, elastic–viscous–plastic), while there is slightly more ice-cov-ered area. We note also that the spatial thickness patternof the cavitating fluid rheology differs from that of theelliptical formulations (not shown, see e.g., Flato andHibler 1992). Furthermore, kinetic energy values forcavitating fluid are higher than any of the elliptical yieldcurve cases. This is primarily due to the lack of shearstrength in the cavitating fluid rheology.

b. Ice thickness distribution simulations

The average thicknesses and areal coverages and totalkinetic energy per unit mass for the 28-category sim-ulations are given in Table 2. While the tendencies seenin the 2-category simulations are repeated here, the ad-dition of a multiple layer ice thickness distribution notonly substantially increases the average ice thicknesses,but also enhances the differences between the differentrheologies. This is most obvious in comparing the cav-itating fluid rheology with the elliptical yield curve re-sults. The increased difference between the elastic–vis-cous–plastic and viscous–plastic formulations is not in-significant, however. Average areal coverages are alsoincreased, most significantly in September. Kinetic en-ergy per unit mass values decrease for all cases, mostnotably in March, due to the substantial increase in icethickness (mass).

The thickness differences are examined spatially inFig. 3. Here, the spatial thickness pattern of the viscous–plastic 15-iteration run for March is shown. The othersolutions are subtracted from the reference, and theirthickness anomalies are shown. The most obvious dif-ferences are again with the cavitating fluid model, whereit is readily seen that the ice is thinner over most of themodel domain. The viscous–plastic solution with oneiteration has thicker ice along the Canadian coast, whilea slightly thinner area in the region north of Fram Strait.The viscous–plastic 5-iteration solution is nearly iden-tical to the 15-iteration reference solution. The elastic–viscous–plastic solution has thinner ice over most of the


TA

BL

E2.

Sum

mar

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28-c

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ory

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stic

(1it

erat

ion)

Ann

ual

Mar

Sep

3.65

4.27

2.76

3.71

4.42

2.70

4.21

4.59

3.67

92.3

99.8

85.0

89.4

99.6

77.7

98.2

100.

096

.7

0.35

820.

5514

3.30

04

0.71

802.

0370

5.41

28

0.08

560.

0469

1.38

74

Vis

cous

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erat

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)A

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lM

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3.74

4.35

2.87

3.81

4.51

2.82

3.77

4.24

3.12

92.7

99.8

85.7

89.7

99.6

78.6

96.8

99.9

95.2

0.34

580.

6067

3.23

68

0.69

861.

9664

5.39

33

0.13

100.

1027

1.55

95

Vis

cous

–pla

stic

(15

iter

atio

ns)

Ann

ual

Mar

Sep

3.75

4.36

2.89

3.81

4.51

2.82

3.70

4.19

3.01

92.7

99.8

85.8

89.8

99.6

78.8

96.5

99.9

94.0

0.34

500.

6291

3.16

25

0.68

881.

9987

5.31

33

0.13

780.

1267

1.60

69

Ela

stic

–vis

cous

–pla

stic

Ann

ual

Mar

Sep

3.59

4.21

2.71

3.69

4.40

2.69

3.51

4.02

2.80

92.3

99.7

84.8

89.4

99.6

78.0

95.7

99.9

92.9

0.39

720.

8651

3.21

98

0.75

722.

5032

4.85

60

0.17

360.

2384

1.71

46

Cav

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arS

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3.14

3.65

2.43

2.78

3.35

1.96

3.70

4.14

3.09

95.7

99.8

91.7

93.9

99.8

86.9

97.1

99.8

95.6

0.48

512.

1958

3.29

73

0.82

323.

6723

5.72

05

0.24

071.

1064

1.62

56

Arctic Basin, particularly in the Beaufort Sea and in thearea surrounding Svalbard, including Fram Strait.

Figure 4 shows the mean ice drift for the month ofMarch. While all formulations show the same generalpattern, the cavitating fluid model shows faster drift,particularly in the Beaufort Sea. This increased ice speedis typical of cavitating fluid and has been observed be-fore (e.g., Ip et al. 1991; Flato and Hibler 1992; Hollandet al. 1993; Fichefet and Morales Maqueda 1997). Alsoof note is a region of eastward transport north of thecoast of Greenland that is much slower in the otherrheologies. This is due to the lack of shear strength inthe cavitating fluid rheology. Ice flow is unimpeded,even in areas of thick ice where shear strength wouldotherwise inhibit the flow. As will be seen later, theoutflow of ice through Fram Strait is also affected bythe rheology. The viscous–plastic models show increas-ing ice drift with increased iterations on the numericalsolver, while the elastic–viscous–plastic solution resem-bles the viscous–plastic cases, with slightly faster driftin the Beaufort Sea.

As the 28-category ice thickness distribution explic-itly includes ridged ice, the fraction of ridged ice-cov-ered area can be determined, as seen in Fig. 5. Again,the most obvious differences are between cavitating flu-id and the elliptical models. The cavitating fluid resultsshow substantially less ridged ice, with most of it con-centrated along the Canadian archipelago and the northcoast of Alaska. As explained in section 2e, the onlyenergy dissipated by the ridging process is due to con-vergence. The ridging parameterization is driven by netconvergence into a grid cell (Hibler 1980), and thickerice is more incompressible, resisting this convergence.Thus, with no shear contribution to ridging, there is lessridged ice than in the elliptical yield curve results inwhich shear deformation contributes to the productionof ridged ice. The elliptical yield curve results varyslightly but are in general agreement as to the locationand amount of ridged ice.

c. Computational time requirements

It is worthwhile to consider the central processingunit (CPU) time required for each model, as this is aconsideration for inclusion in GCMs. Table 3 summa-rizes the total CPU time required for a 20-yr simulation.It is seen that the addition of a 28-category ice thicknessdistribution substantially increases the CPU time re-quired for all rheologies. While Zhang and Hibler (1997)recommend 10 to 15 iterations to more fully approxi-mate a plastic solution, a substantial amount of CPUtime is required for increased iterations. The use of aviscous–plastic rheology with only one iteration on thenumerical solver was fastest, but both Zhang and Hibler(1997) and Hunke and Dukowicz (1997) indicate thatone iteration is not sufficient to approximate a plasticsolution. The cavitating fluid rheology requires nearlytwice as much CPU time as the elastic–viscous–plastic


FIG. 3. (a) Baseline March ice thickness results (m) for viscous-plastic, 15 iterations. (b–e) Thickness anomaly (m)from viscous-plastic, 15 iterations, for (b) viscous-plastic, 1 iteration, (c) viscous-plastic, 5 iterations, (d) elastic-viscous-plastic, (e) cavitating fluid. Negative anomalies are gray shaded.


FIG. 4. Baseline March ice drift results for (a) viscous-plastic, 15 iterations, (b) viscous-plastic, 1 iteration, (c) viscous-plastic, 5 iterations, (d) elastic-viscous-plastic, (e) cavitating fluid.


FIG. 5. Baseline March fraction of ridged-ice-covered area for (a) viscous-plastic, 15 iterations, (b) viscous-plastic, 1iteration, (c) viscous-plastic, 5 iterations, (d) elastic-viscous-plastic, (e) cavitating fluid.


TABLE 3. CPU time required for 20-yr spinup runs.*

Rheology2-category

(h)28-category

(h)

Viscous–plastic, 1 iterationViscous–plastic, 5 iterationsViscous–plastic, 15 iterationsCavitating fluidElastic–viscous–plastic

3.6612.1525.7021.2712.07

27.3337.9753.9450.5038.58

* Runs were performed on a Silicon Graphics Power Challenge XLR10000.

rheology for the two-layer case, and is comparable tothe viscous–plastic rheology with 15 iterations. Thisarises because of the more strict convergence criterionused. As mentioned in Zhang and Hibler (1997), theirtechnique can also be used to solve the cavitating fluidequations. Such an implementation would be much fast-er than any of the viscous–plastic cases (G. Flato 1997,personal communication). However, this more efficientscheme was not used with the cavitating fluid rheologyin this study. We note also that improved numericalsolution techniques are constantly under development,some of which could substantially improve upon theCPU times presented here. We do not investigate thesemethods, as our study is focused primarily on comparingthe sensitivity of the various ice dynamics formulations.

4. The effects of a momentum flux perturbation

In order to investigate the sensitivity of the sea icemodels as a function of rheology, each was subjectedto a momentum flux perturbation. Cases were run for30 years, the first 20 under baseline forcing. For the last10 years, the winds were either increased or decreasedby 25%. All other forcing remained the same.

a. Two-category simulations

The responses of the 2-category simulations to per-turbations in momentum flux are shown in Table 1. Themost drastic differences are seen in the response of thecavitating fluid model compared to the elliptical yieldcurve models. While an increase in wind speed bringsabout an increase in ice thickness in the other models,it decreases the average thickness in the cavitating fluidcase. Increasing the wind speed increases open waterproduction. This is seen as a decrease in ice coveredarea within the Arctic basin in all models. With in-creased open water (leads), more ice production andhence a greater mean ice thickness should result, as seenin the elliptical yield curve models. However, the in-creased wind speed also brings about an increase in icedrift and ice transport, as indicated by the kinetic energyper unit mass.

The model responses to a decrease in wind speed arenot as straightforward. Annual averages show an in-crease in mean thickness for all models except the elas-

tic–viscous–plastic model. The cavitating fluid respondswith an increase in mean thickness for March and Sep-tember, while the elliptical yield curve rheologies allhave a decrease in March mean thickness but an increasein September. The reason for this is that the decreasedwind speed means decreased open water productioncompared to a baseline. This can be seen by increasesin ice-covered areas for all the models, and the sub-stantial decrease (greater than 75%) in March kineticenergy. It is also suggested by Fig. 2, in which morestress states lie within the ellipse, where the viscousflow regularization for small strain rates is in effect. Inthe winter, there are fewer leads and thus less new icegrowth, and the mean thickness drops. However, less ofthis ice is advected out of the region, leaving more ofit at the beginning of the summer melt season. Thus,more ice persists throughout the summer, increasing themean thickness compared to the baseline. The range ofresponse of the viscous–plastic models with varying it-erations on the numerical solver is interesting. It willbe more relevant in the next subsection, in which the28-category ice thickness distribution proves to be im-portant.

b. Ice thickness distribution simulations

The results for the 28-category model simulations aregiven in Table 2. The cavitating fluid response seen inthe 2-category models is repeated here. Thus, the in-creased thermodynamic resolution provided by the ad-ditional thickness categories does not affect the cavi-tating fluid rheology’s response to perturbations in mo-mentum flux. Meanwhile, the elliptical curve model re-sponses are also analogous to the 2-categorysimulations. The spatial distribution of thickness anom-alies for increased winds compared to the baseline isshown in Fig. 6. As in the 2-category cases, the responseis dependent on the shape of the yield curve. It is clearfrom the figure that the decrease in thickness for thecavitating fluid rheology is nearly basinwide. The el-liptical yield curve models, however, show an increasein thickness along the Canadian Archipelago and thenorth coast of Alaska. When the ice drift (Fig. 4) andridged ice location (Fig. 5) are taken into account, theregions of increased thickness correspond with regionsof convergence and high concentrations of ridged ice.The patterns for the various viscous–plastic results andthe elastic–viscous–plastic results are generally similar.Slight differences in the spatial location and magnitudeof ice thickness anomalies can be seen throughout thebasin, however.

The response of the models to a decrease in windspeed is more interesting. The viscous–plastic resultsfor average ice thickness are strongly dependent on thenumber of iterations on the numerical solver. With onlyone iteration, the annual average thickness is 4.21 m,with 98.2% of the Arctic basin covered with sea ice,and a total kinetic energy of 0.0856 J kg21. Increasing


FIG. 6. March thickness anomaly (m), 125% increased winds minus baseline for (a) viscous–plastic, 15 iterations,(b) viscous-plastic, 1 iteration, (c) viscous-plastic, 5 iterations, (d) elastic–viscous-plastic, (e) cavitating fluid. Negativeanomalies are gray-shaded.


TABLE 4. Annual Fram Strait ice export results, 28-category models.

Rheology

Export (Sv)

Baseline125%winds

225%winds

Mean thickness in Strait (m)

Baseline125%winds

225%winds

Mean u velocity (cm s21)

Baseline125%winds

225%winds

Viscous–plastic, 1 iterationsViscous–plastic, 5 iterationsViscous–plastic, 15 iterationsElastic–viscous–plasticCavitating fluid

0.1030.0960.0980.1070.061

0.1540.1420.1420.1490.071

0.0430.0530.0560.0670.049

4.454.464.504.242.62

5.345.275.274.882.37

3.103.483.553.482.98

3.923.653.684.213.80

4.954.604.645.144.86

2.522.662.703.232.74

to five iterations, the average thickness drops nearly one-half meter, to 3.77 m, 96.8% ice-covered area, and0.1310 J kg21 kinetic energy. A further increase in it-erations decreases the thickness to 3.70 m, 96.5% icecovered area, and 0.1378 J kg21. Corresponding behav-ior is seen in the March and September values, partic-ularly in March. While not as important in the 2-cate-gory cases, the number of iterations on the viscous–plastic solver becomes crucial when a 28-category icethickness distribution is used. The elastic–viscous–plas-tic results compare well with those of the viscous–plas-tic models for spatial patterns of thickness and area,although the ice motion in the elastic–viscous–plasticcases is somewhat faster. We again note that the stressstate plots (not shown) indicate that the elastic–viscous–plastic model is not adhering to the elliptical yield curverule.

c. Fram Strait ice export

Ice export through Fram Strait is the main connectionbetween the Arctic Basin and the North Atlantic Ocean,and thereby the global ocean. It is important, then, tobe able to model the Fram Strait export correctly, inboth regular and climate change scenarios. Table 4 sum-marizes the annual export for the 28-category models.Except for the cavitating fluid model, these results arecomparable to the climatological value of 0.097 Sv (Sv[ 106 m3 s21), derived from Aagard and Carmack(1989). There is considerably more seasonal variabilityin the cavitating fluid ice export than in the ellipticalmodels. The baseline March values (not shown) for theviscous–plastic models are typically 50% higher thanthe annual averages, and about 75% higher for the elas-tic–viscous–plastic, while the cavitating fluid March iceexport is nearly 150% higher than its annual average.While all models show an increase in ice export withincreased winds, the mean ice thickness in Fram Straitfor the cavitating fluid model decreases. Thus, the in-crease in export there is due to thinner ice that is movingfaster. Increases in the elliptical model ice export aredue to both increased ice thickness and ice velocity. Theviscous–plastic results depend on the number of itera-tions on the numerical solver, while the elastic–viscous–plastic results show generally slightly thinner, faster-moving ice than the viscous–plastic results.

5. Summary and conclusions

An intercomparison of three sea ice dynamic modelsand two ice thickness distributions has been presented.By assessing not only the differences among the baselineresults but also the responses of the various models toa momentum flux perturbation, insight is given into theirphysical behavior. This is useful for appraising the util-ity of the various models in global climate and climatechange studies. It is seen that inclusion of a 28-categoryice thickness distribution that explicitly calculates ridg-ing and open water production substantially increasesice thickness throughout the year as well as summerareal coverage. Thus, new ice growth and exchange ofheat and moisture between the atmosphere and ocean isaffected.

The response of the cavitating fluid model, whichlacks a parameterization for shear resistance, differsgreatly from those of the viscous–plastic and elastic–viscous–plastic formulations, both of which contain ashear strength parameterization. In the 2-category for-mulation, the responses of the elliptical models are near-ly identical, and the number of iterations on the viscous–plastic solver has a small, although nonnegligible, ef-fect. The addition of a 28-category ice thickness dis-tribution, however, accentuates the differences betweenthe elliptical rheologies, as well as the importance ofthe number of iterations on the viscous–plastic numer-ical solver. Furthermore, it is shown that, in particular,the Fram Strait ice export is sensitive to the type ofdynamic model used.

The cavitating fluid model was originally developed foruse with smoothed forcing in long-term, crude-resolutionclimate applications (Flato and Hibler 1992). It lent itselfwell to implementation in GCMs, necessitated by theirrequirement for computational speed and efficiency. Cur-rently, several GCMs use the cavitating fluid rheology(e.g., Weatherly et al. 1998). Our results emphasize theimportance of including shear strength in the parameter-ization of sea ice, and given the ongoing improvementsin numerical solvers, these more physically realistic rhe-ologies now lend themselves well to GCM use.

While widely accepted among the sea ice modelingcommunity, we would like to emphasize that the ellip-tical yield curve for sea ice is not necessarily the mostcorrect dynamic parameterization available (e.g., Hibler


and Schulson 1997). However, it does reproduce wellmany of the observed physical characteristics of sea icemost important for large-scale modeling, and thus ap-pears adequate for climate model simulations. Here, wehave compared a new elliptical numerical method, theelastic–viscous–plastic model, with several formula-tions of the viscous–plastic model. The elastic–viscous–plastic model gives results generally comparable withthe viscous–plastic model. In terms of accuracy, it hasbeen argued by many (e.g., Zhang and Hibler 1997;Kreyscher et al. 1997) that a viscous–plastic model, es-pecially with a sufficient number of iterations on thenumerical solver to approximate a plastic solution, isthe most accurate of the dynamic ice models in wideuse, at least when monthly and longer averages are con-sidered. A companion study (Arbetter and Meier 1998unpublished manuscript) compares modeled ice motionsfor various implementations of an elliptical yield curvewith daily observations derived from satellite and buoydata.

Comparison of stress states of the two models (e.g.,Zhang and Hibler 1997; Hunke and Zhang 1999) in-dicate that the viscous–plastic model obeys the ellipticalyield curve constraint much more consistently than theelastic–viscous–plastic model; indeed, the best that theelastic-viscous-plastic model can possibly do is emulatethe viscous–plastic model with one iteration (E. Hunke1997, personal communication). However, Hunke andZhang (1999) suggest that the elastic–viscous–plasticmodel is better suited than the viscous–plastic modelfor implementation on multiprocessor machines. Thus,the choice of which numerical model to use is somewhatdependent on its application. Nevertheless, the resultsof this study indicate that a elliptical yield curve rhe-ology is well-suited for inclusion in a GCM.

Acknowledgments. We are grateful to G. Flato for hissuggestions and help during the development of ourmodel. We also thank J. Zhang and E. Hunke for pro-viding their subroutines. This work was supported byGrants NSF OPP-9614492 and NSF OPP-9423546.

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effects of rheology and ice thickness distribution in a ...ice being more readily melted in the...

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