Ice in the Environment: Proceedings of the 16th IAHR International Symposium on IceDunedin, New Zealand, 2nd–6th December 2002International Association of Hydraulic Engineering and Research

MULTIPLE EQUILIBRIUM ARCTIC ICE COVERSTATES INDUCED BY ICE MECHANICS

William D. Hibler III1 and Jennifer K. Hutchings1

ABSTRACTIce mechanics induced multiple equilibrium states of the arctic ice cover are investigated.A non-dimensional analysis of reduction of flow through narrow passages, culminatingwith ice arching arising from uniaxial plastic compressive strength, is performed. Byadding growth rates to this analysis it is shown that two stable equilibrium states of agiven ice cover under appropriate fixed thermodynamic and wind forcing are possibledepending on when the ice flow is restrained. The potential of such mechanical effectsfor inducing multiple equilibrium states for the Arctic Basin ice cover is investigatedusing realistic mean monthly winds and an idealized thermodynamic model. The analysisindicates three possible states, two of which are found to be stable. Integrations fromfixed thickness initial conditions demonstrate the existence of these states under moderateclimate cooling. The ramifications of this phenomenon to numerical investigations ofclimate employing dynamic thermodynamic sea ice models is discussed.

INTRODUCTIONCirculation of the Arctic ice cover is affected by ice mechanics especially in the vicinityof narrow passages such as the Fram Strait. Annually, the mass budget of the Arcticconsists of a net growth of about 1 m of ice balanced by an equivalent amount exiting theArctic Basin. Numerical investigations of the Arctic ice cover indicate that nonlinear icemechanics formulations substantially affect the flow of ice both locally through the Framstrait (e.g. Ip et al., 1991) and concomitantly throughout the Arctic Basin. However,the highly nonlinear nature of ice interaction still causes the outflow of ice to fluctuatein response to local wind fields (Hibler and Walsh, 1982) so that empirical correlationstudies (Vigne, 2001) suggest the local wind field controls ice export. This belief existsdespite the fact that satellite based observational studies (Kwok and Rothrock, 1999) showthat regression coefficients between outflow and local winds vary seasonally.

Satellite observations of narrow passages such as the Bering Strait (e.g. Sodhi, 1977)indicate ice motion can be totally stopped by the formation of static arches. A notableexample is the Nares Strait north of Baffin Bay where ice flow ceases every year around

1International Arctic Research Center, University of Alaska:Fairbanks, 903 Koyukuk Dr., Fairbanks,Alaska 99775-7320, USA

November. Stoppage of flow through this and other narrow passages in the CanadianArchipelago (Melling et al., 2001) plays an important role in the formation of the “North-water Polyna” in Baffin Bay. Analytical (Sodhi, 1977) and numerical analysis (Ip, 1993)of flow through narrow passages indicate that plastic rheologies with uniaxial compres-sive strengths can lead to total stoppage of flow. For a given wind or body forcing theseanalyses require the ice strength divided by the channel width to reach a critical valuefor total stoppage. It is of note that when scaled to the Fram Strait, results from non-dimensional numerical studies (Ip, 1993) of ice arching in narrow passages suggest theFram strait is not far from the “arching” limit, which is qualitatively apparent in directsimulations (Hibler, 1980).

Since equilibrium ice thickness of the Arctic basin depends nonlinearly upon residencetime, ice growth and ice outflow, the Arctic basin is thought to have a capability for mul-tiple equilibrium states. However, this has not yet been demonstrated. The restrictionof ice flow through narrow passages possibly causing static ’arching’ is the key compo-nent leading to the multi-equilibrium states. The existence of such multiple equilibriumstates is examined both with an idealized channel model and a fully coupled dynamicthermodynamic Arctic Basin model. Multiple equilibrium states can be investigated as afunction of climatic warming or cooling in a manner similar to Budyko–Sellers models ofglobal climate. The coupled model is used to assess the existence and stability of differentequilibrium states. In addition several hundred year simulations with different initial con-ditions demonstrate the actual existence and characteristics of the two stable equilibriumstates.

CHARACTERISTIC OF PLASTIC FLOW THROUGH NARROW PASSAGESAs first noted by Richmond and Gardner (1962) in a coulombic granular based anal-ysis, arching (Fig. 1a) is a statically indeterminate problem in that, for example, dif-ferent shapes of arches will have different strengths and braking limits. Subsequently,Pritchard et al. (1979) utilized a thick walled concentric cylinder analysis to analyze iceflow through the Bering strait and emphasized the need for uniaxial compressive strengthfor a stationary arch to form. Independent of the particular analysis, however, it is wellknown that a stationary free surface in stopped flow requires, by continuity of stresses atthe surface (Fig. 1a), the presence of uniaxial compressive strength (e.g. Fung, 1997).

Analysis of plastic flow through narrow passages requires not only analyzing station-ary stoppage, but understanding the behavior of the flow as the arching limit is ap-proached. With the appropriate formulation this flow can be numerically analyzed ina non-dimensional manner (Ip, 1993). Consider a tapered channel Fig. 1b, with parame-ters: length of the opening λ, ice strength P , and wind stress ρacau2

g. For this system thex momentum equation in the absence of coriolis force is given by

0 = ρacau2g − ρacwu2 +

(

∂σxx∂x

+∂σxy

∂y

)

. (1)

Expressing x and y in terms of λ and stress in terms of P , we have after dividing by thewind stress the dimensionless equation

1 = β − γ(

∂σx′x′

∂x′+∂σx′y′

∂y′

)

(2)

(a) (b)

3 02 01 000 .0

0 .2

0 .4

0 .6

0 .8

0 .0

0 .1

0 .2ice velocity outflow

thickness (m)

aver

age

ice

velo

city

(m

/s)

outf

low

(m/h

r)

(c)

Figure 1: (a) top panel: Cartoon of a typical ice arch, bottom panel: boundary conditionsat the free surface. (b) Numerical grid and (c) constant thickness ice outflow and velocitycharacteristics from plastic equilibrium solution.

where σx′x′ , σx′y′ , x and y are dimensionless. β and γ are dimensionless parameters:

β =ρwcwu

2

ρacau2g

, γ =P

lρacau2g

. (3)

Consequently in dimensionless form the solution for β which is a measure of the icevelocity, should only depend on γ unless the geometry of the boundaries change.

To examine the character of the flow we consider a fifty by fifty, 40 km resolution grid asshown in Fig. 1b. The shaded bands show regions where zero ice strength is assumed sothe ice can freely flow into or away from these bands. A constant stress of τ = .4 Nm−2

in the direction of the arrow is used for the body force (i.e. the term ρacau2g in Eqn. (1)).

To simplify scaling a linear water drag (i.e. ρwcwu) is used with ρwcw = 0.56. For plas-tic rheology the modified Coulombic yield curve (Fig. 1b inset) of Hibler and Schulson(2000) is utilized. This rheology has the requisite uniaxial compressive stress needed forstatic arching, which for a coulombic rheology requires some tensile stress. Solution ofthe equations of motion is carried out using the finite differences and relaxation proce-dure of Hibler (1979). Ice strength is taken to scale linearly with thickness according toP = 4×104h. Figure 1c shows a dimensional plot of mean ice velocity and ice export as afunction of ice thickness, or equivalently ice strength. Since linear water drag is used thesame mean velocity curve will apply in non-dimensional form (verified by simulation)with the abscissa being γ = P/λτ and the ordinate being β = ρwcwu/τ. The basic char-acter of the solutions is a gradual decrease of the ice velocity as the strength increases orthe opening span (λ) decreases. At some point the velocity stops and an effectively staticsolution with an arch is obtained. The system is close to motionless with higher velocitiesexisting in an arch shaped region near the outflow opening. With no ice interaction theice velocity is a constant fraction of the wind speed.

The ice outflow is particularly relevant to the multiple equilibrium problem. Since ice

(a) (b)

Figure 2: Characteristic thickness patterns for (a) high and (b) low growth equilibriumsimulations. Simulations initialized with h = 4 m uniformally.

strength scales with thickness the outflow peaks for intermediate values of the flow whereboth the velocity and thickness are significant. For thin ice the outflow scales with h, asvelocity is affected little by small thickness changes. For thicker ice the outflow becomesnegligible as an ice arch forms. This occurs even though there is some mass in the velocity“cirque” formed near the downstream opening in the channel. For outflow it can be shownthat the appropriate non-dimensional y axis value, call it α, scales as α ≈ (P/τ2)∆, where∆ is the outflow in dimensional form.

If ice growth occurs the outflow can be largely stopped or freely flowing depending onthe rate of growth. This is illustrated in Fig. 2 where thickness characteristics are shownfor growth rates inversely dependent upon thickness according to

f = α[1/(h + 0.4) − 0.005],∂h

∂t+ ∇ · (hu) = f (4)

with α a variable and f in units of meters/hour. With high growth rates (α = 5/2.25) theice gradually forms a strong enough arching formation (Fig. 2a) to stop the outflow. Inthe case of weaker growth (α = 2.5) a narrow channel (Fig. 2b) is created through theice pack and ice flows rapidly enough to maintain a thin ice channel. Such features arecharacteristic of the ice thickness patterns observed near the Bering Strait (L. Shapiro,personal communication). The transition to total stoppage occurs rapidly with growthrate. Since both states are forced with the same wind, the physical notion is that thereis potential for multiple equilibrium states with the same wind forcing dependent on theinitial conditions. In particular, once the ice has stopped the growth rates could be reducedand the ice allowed to further thicken. Examples of such multiple equilibrium steadystates are shown in Fig. 3. The thicker states were obtained from integrations with largergrowth rates for 72 hours, lowering the growth rate after 72 hours. Once the growth isstopped, there is very small outflow. Hence it is possible to obtain both thick and thinsolutions for a very wide range of growth rates.

(a) (b)

Figure 3: Different equilibrium thicknesses for identical forcing for idealized geometryof Fig. 1b. (a) Equilibrium thickness and (b) outflow versus growth parameter.

MULTIPLE EQUILIBRIUM STATES FOR AN ARCTIC BASIN DYNAMIC THER-MODYNAMIC MODELTo investigate these concepts in a full Arctic Basin simulation, we consider an idealizedaverage model of the Arctic basin composed of a viscous plastic dynamic thermodynamicsea ice model together with a simplified ice thickness distribution and idealized heat bud-get thermodynamics. For the momentum balance, we use a steady state solution of theviscous plastic momentum balance (e.g., Hibler, 1979; Hibler and Walsh, 1982) with theice stress tensor σ characterized by a viscous plastic rheology and an elliptical yield curve.Following Hibler and Schulson (2000) the pressure term in the rheology depends on thestrain rate so that the rheology is energy dissipative when taken to plastic flow. The icestrength P ∗ is related to the mean ice thickness h by P ∗ = (4 × 104)h. The mean icethickness evolves according to ∇ · vh + f (h) = 0 where f (h) is the ice growth rate asa function of thickness. The ice is forced with climatological monthly mean geostrophicwind calculated from NCAR/NCEP reanalysis mean sea level pressure.

To quantitatively test the effect of mechanics on sensitivity to climatic change an ideal-ized thermodynamic model (Thorndike 1992) is used. Growth and melt seasons are equallength running from October to March and April to September. Climate is described bydownwelling long wave radiation (flwc = 180 Wm−2) in the cold season and long andshort wave radiation during the warm season (flww = 270 Wm−2 and fsw = 200 Wm−2).The heat supplied from the ocean is idealized as 2 Wm−2 (Maykut and Untersteiner,1971). Ice growth and decay are obtained from a stefan like thermodynamic model withlinear temperature profile and fixed conductivity. Heat capacity effects are included ascooling and warming times between seasons up to several weeks in length. Climaticchange is introduced via a uniform perturbation δ to the long wave radiation fluxes flwcand flww.

To examine the potential for multiple equilibrium states, a series of two year simulationsinitialized with different constant thicknesses were carried out with δ = −11 Wm−2. Thesecond year mean annual outflow and growth are shown in Fig. 4a. To the degree thatsecond year results are indicative of quasi-equilibrium outflow and thickness, intersec-tions of annual growth and outflow versus initial thickness represent potential solutions.There are three possible solutions (Fig. 4a). Solutions (1) and (3) are stable, and hencereachable in the numerical simulations, which can be shown by a local stability analysis.

(a) (b)

Figure 4: (a) Second year outflow and Basin averaged ice growth as a function of initialice thickness for δ = −11 Wm−2. (b) Basin averaged ice thickness as a function of timefor δ = −9.5 Wm−2 with thin and thick initial conditions.

In particular, the rate of change of ice thickness is given by

dh

dt= G(h) − O(h) (5)

whereG(h) is the growth rate andO(h) is the outflow rate. Considering ho to be a solutionof this equation and h1 = h − ho a small thickness perturbation relative to ho; expandingG and O in a Taylor series expansion about ho, the rate of change of the perturbation tolowest order is

dh1

dt= [G′(h) − O′(h)]h1 (6)

where G′(h) = dG/dh and O′(h) = dO/dh. Clearly the small perturbation will growunless [G′(h)−O′(h)] < 0, a condition which, by inspection of Fig. 4a, is met for solutions(1) and (3) but not for solution (2). As solution (2) is not stable, as one proceeds from acold climate to a warm climate there will be a rapid jump to a lower thickness state whichthen changes less rapidly with warming.

To assess that multiple equilibrium solutions actually exist several hundred year simula-tions are carried out, initializing with very thick ice and very thin ice. The equilibriumtime scales for δ = −9.5 Wm−2 are shown in Fig. 4b. The thick solution converges moreslowly than the thin solution as thicker ice changes through thermodynamic response withvery little outflow. The two solutions are independent of initial conditions provided theinitial perturbation is not large enough to cause branching to the second stable solution.Results below show the outflow is greatly reduced in the thicker solution, although notaltogether stopped.

The range of states over which multiple solutions can occur is shown in Fig. 5, showingresults from several hundred year simulations with differing δ. The thickness jump be-tween the two solutions is significant with the ice mass increasing about 20 % in the thicksolution where outflow is very low. The range over which equilibrium states occur is rel-

atively small in keeping with the very high degree of sensitivity, of the idealized initialflow problem (Fig. 4a), to changes in growth.

(a) (b)

Figure 5: Equilibrium states as a function of climatic cooling parameter δ. States obtainedby intializing with thick ice are shown as crosses and a solid line and thin states withtriangles and dashed line.

(a) (b)

Figure 6: Seasonal outflow and growth characteristics of equilibrium solutions.

Independent of the range of the multiple solutions, the main effect of the nonlinear me-chanics is to induce a rapid effectively discontinuous jump in the thickness characteristics.This jump is due to the fact that the intermediate equilibrium state (Fig. 4a) is not stableso the equilibrium solution has to move from the thick to thin solution over a wide rangeof thicknesses. In actual climatic change this would of course occur more gradually be-cause of the the time needed to come to pure equilibrium. However, the fact that suchmultiple states exist indicates rapid changes in the ice state can occur in response to smallperturbations.

The seasonal outflow and growth characteristics of the two characteristic equilibriumstates are shown in Fig. 6. The main feature of the thick solution is a greatly reduced

(a) thick (b) thin (c) current

6

6

6

7

7

7

7

7

(d) thick

34

4 5

55

6

6

6

77

7

7

7

(e) thin

2

2

3

3

(f) current

Figure 7: Thickness (dark 6 to 7 m, light 1 to 3 m) and ice velocity during winter for dif-ferent equilibrium solutions (initialised with thick and thin ice δ = −9.5; current climateδ = 0). Scale arrow represents 0.001 ms−1 in [a] and [b] and 0.1 ms−1 in [c]

outflow, with outflow occurring mainly during the melt season with little correlation towind stress in the Fram Strait region. In the standard case, for example, because of theweaker strengths the outflow tends to correlate strongly with the wind causing the outflowto peak in late winter even though the strength is largest then. However, with the thicksolution, and to a lesser degree, the thin solution, the outflow is seasonal with much lessoutflow in the growth period even though the winds may be stronger then. In the thicksolution the outflow begins during the melt period and then gradually increase as the iceweakens.

While there are differences in seasonal behavior the dominant effect in the thick solutionis a greatly reduced outflow, with the thick case’s annual average outflow being 16 % ofthe thin solution and 12 % of the standard simulation. Overall it is clear that as the icestrengthens there is less correlation with the local wind forcing. It should be noted thatalthough the outflow is small compared to the growth rates, it is of course comparable tothe difference between the growth and melt rate.

Examination of the thickness and velocity states (some shown in Fig. 7) of the two equi-librium solutions shows it is largely changes in the ice velocity that account for the largeoutflow. In the thick solution the velocity is suppressed except in summer. This suppres-sion takes the form of reduced gyre motion and a slightly broader gyre. In the thin and

control solution there is a more concentrated and robust gyre which can recirculate theice. This occurs in conjunction with a transpolar drift like feature moving ice out of thebasin somewhat reminiscent of the channel like flow shown above (Fig. 2) in the idealizedarching case with low growth rates.

The thickness features show a similar spatial pattern in the two cold climate cases withthinner ice in the center of the Basin and near the Beaufort Sea and thicker ice up againstthe archipelago. In the thick case the thickness pattern does not change drastically; ratherthe thickness in the outflow region thickens and becomes greater. The thickening in theWestern Arctic and the outflow region appears to account for most of the thickness change.The current climate case has thicker ice against the archipelago, much like the thicknesspatterns observed in the last 20 years.

CONCLUDING REMARKSThis paper examined the role of nonlinear mechanics on controlling outflow through theArctic Basin such that multiple equilibrium states of the Arctic ice cover exist with iden-tical forcing. While this concept has long been suspected, this is the first explicit demon-stration of multiple equilibrium states in a numerical dynamic thermodynamic sea icemodel. The fact that multiple equilibrium states exist in coupled numerical models hasimportant ramifications for the use of such models in coupled ice ocean simulations andin broader climate applications. For example it is implicitly assumed the response ofdynamic thermodynamic sea ice models yields a well defined climate state for specifiedforcing, emphasized in the early development of the coupled viscous plastic models (Hi-bler, 1979; Flato and Hibler, 1995). While multiple equilibrium states may be outside therange of present climate conditions, their existence may affect the response of the modelto major changes in the wind fields related to interdecadel variability. Changes of thewinds between high and low arctic oscillation conditions might for example be strongenough to induce shifts between different equilibrium states if the wind were to persistlong enough. Consequently simple interpretations of the overall circulation of the arcticice cover based on correlations with winds is unlikely to be realistic.

This study has of course been idealized, but we expect the physical principles to be trans-ferable to more complete thickness distribution models together with thickness strengthcoupling. Analysis of such improved model simulations is currently in progress. Clearlyin this work, the detailed nonlinearity of plastic flow through narrow channels is criti-cal. Hence this paper presented some non-dimensional characteristics of such flow andthe arching mechanism as it applies in conventional viscous plastic and elastic viscousplastic sea ice dynamics models. These non-dimensional scaling characteristics can beused in other applications and to serve as a guide to determine when multiple equilibriumstates may be expected, as in Fig. 4a. This general analysis could be useful in numericalinvestigations of climate.

An important point is the existence of two stable and one unstable solution. Even thoughthe range of overlapping states may be small there is a tendency for sudden large changesin thickness characteristics as the climate variable changes. Whether this has happenedin the last several decades is an interesting question that impacts our understanding ofclimate warming and ice thinning. The analysis framework presented here will be used

to address this issue. It is clear analysis of arctic ice thicknesses on thermodynamic con-siderations alone with outflow corrections induced by local winds must be consideredsuspect.

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