The viscous-plastic model

The implemented version is based on Reference HiblerHibler’s (1979) viscous-plastic model. The momentum balance for sea ice, with the inertial term neglected, is described by the vector equation

where m is the ice mass per unit area, u the horizontal ice velocity, f the Coriolis parameter, k a unit vector normal to the surface, T_a and T_w forces due to non-linear air and water stresses, g the acceleration due to gravity and H the surface dynamic height. F is the internal force described as the divergence of the stress tensor σ with sea ice considered as a non-linear-viscous compressible fluid obeying the constitutive law

where ζ and η are non-linear bulk and shear viscosities depending on strain rate , δ_ij is the Kronecker symbol, and the pressure term P is a function of ice-thickness characteristics and the strain rate. The viscosities are related to the strain rates such that the stress state lies on an elliptical plastic yield curve. In the case of plastic flow, ice strength is parameterized as

where P* is a strength parameter (N m⁻²), h is the mean ice thickness (m), C is a dimensionless strength parameter and A is ice compactness. P* essentially determines the magnitude of ice strength, whereas C controls the dependence of ice strength on compactness. For very small strain rates, strength is modified assuming replacement closure (Reference IpIp, 1993: Reference Gerdes and KöberleHarder, 1996).

The compressible newtonian fluid model

The compressible Newtonian fluid model represents a linear-viscous medium. In this case, the bulk and shear viscosities of Equation (2) depend on the ice characteristics but not on the strain rate.The viscosities are prescribed as

with Δ₀ = 10⁻⁷ s⁻¹. The pressure term in Equation (2) is set to zero. With this rheology, sea ice has the same resistance to dilation (opening of leads) as to compressive deformation, in contrast to the viscous-plastic model, which enables opening with little or no stress.

The cavitating fluid model

In this scheme (after Reference Flato and HiblerFlato and Hibler, 1992), pack ice is assumed to have plastic behavior in the case of compressive deformation. The fundamental assumption is that this idealized medium has no shear or tensile strength. This makes the model simple to formulate and to implement. The initial ice-force term of Equation (1) can now be expressed simply as

where P is internal ice pressure. P is set to zero if the velocity field is divergent, and P is equal to compressive Strength if there is a convergent component. The compressive strength parameterization is the same as in the viscous-plastic model (Equation (3)).

Ice-drift scheme with step-function stoppage

The simplest model within the model hierarchy starts with the free-drill solution of Equation (1), i.e. the internal ice forces are set to zero. In a subsequent correction step all velocity components are set to zero if (a) the ice thickness exceeds the critical ice thickness h _max, and (b) the ice would be advected from thinner to thicker ice conditions. This correction to free drift excessively damps the velocity field, but prevents ice buildup near the coasts from becoming unrealistically large.

Computational grid and forcing fields

The model is run on a rotated spherical grid covering the whole Arctic, with a resolution of 1° (110 km) and a daily time-step.

Atmospheric forcing data are derived from European Centre for Medium-Range Weather Forecasts (ECMWF) analyses for the 7 years 1986–92. 10 m wind is applied as 24 hour mean value in accordance with the model time-step. 2m air and dew-point temperatures are linearly interpolated between 7 year monthly means to remove the artificial temperature shifts due to changes in the ECMWF model over the 7 year period (Reference Gerdes and KöberleHarder, 1996). Precipitation (Reference Vowinckel, Orvig and OrvigVowinckel and Orvig, 1970) and cloud cover (Reference Ebert and CurryEbert and Curry, 1993) are prescribed as spatially constant climatological monthly means. Geostrophic ocean currents are described as annual means from an ocean model (Reference Gerdes and KöberleGerdes and Köberle, 1995), and sea-surface tilt is calculated by geostrophy. An annual average heal flux from the deep ocean into the fixed mixed layer is taken from the coupled ice-ocean model of Reference Hibler, Zhang and PeltierHibler and Zhang (1993).

The model starts with an ice-free ocean. After one spin-up cycle of 7 years of forcing data, the model results are taken from a second 7 year cycle when the model has reached a cyclostationary state.

The models are implemented on different computer systems; therefore a quantitative intercomparison of CPU time consumption has not yet been made.

Drift speed statistics

Buoy data monitored by the Internadonal Arctic Buoy Program (IABP) (e.g. Reference Colony and RigorColony and Rigor, 1991) are used as verification data. Over 50 000 daily buoy velocities were recorded in the period 1986–92. Model velocities were interpolated onto the buoy positions to facilitate comparison with obserrvations.

The effects of ice interactions are best reflected in drift-Speed statistics (Reference Lemke, Hibler, Flato, Harder and KreyscherLemke and others, 1997). Speed distributions of the models and the observations are calculated for four seasons (January-March, April-June, July-September, October-December) and five separate regions covering the whole Arctic, each containing about 10 000 daily buoy velocities. The error of the drift characteristics is measured by the root mean square (rms) of speed distribution differences between simulations and observations. An example of rms distribution error is shown in Figure 1 for the viscous-plastic model in the period January-March.

Fig. 1.

Observed (thick line) and simulated (thin line) daily speed distributions for buoy data in the period January-March. Error measurement is based on the differences of the individual histogram bars. The root mean square (rms) of these differences is 0.9%.

All models are run with different ratios of air and water drag coefficients c_a/c_w to obtain realistic average daily-drift speed. The ice model with step-function stoppage slows the velocity field so much that it does not make sense to compensate this artificial error with an unrealistically high ratio of drag coefficients. Therefore, the ratio here is set to the value of the viscous-plastic model. Table 1 shows the ratios of drag coefficients used for the different models.

Table 1.

Parameter configurations for optimized ice-drift statistics

Subsequently, the ice-strength parameter P* (h _max for the ice-drift stoppage scheme) is optimized with regard to the rms speed-distribution error. The results (Fig. 2) reflect that the viscous-plastic rheology predicts the most realistic drift statistics, followed by the compressible Newtonian fluid model. The cavitating fluid simulation, with no shear forces, differs significantly because it does not provide ice-drift stoppage near coasts. The ice-drift scheme with step-function stoppage damps the ice drift in an unrealistic way, reflected in the highest error of the drift statistics (Reference Lemke, Hibler, Flato, Harder and KreyscherLemke and others, 1997).

Fig. 2.

Rms speed distribution errors (%) for four different dynamic schemes (based on drift-speed statistics calculated for five separate regions and four seasons).

Fram strait ice export

The previous section presented the quality of drift characteristics predicted by the different dynamic schemes. This raises the issue of how important these differences are for the ice export forcing the thermohaline circulation in coupled sea-ice ocean models.

Figure 3a compares the ice thickness at the North Pole for the different models to observations. Seven mean ice-thickness observations at the North Pole from 50 km submarine transects in the period 1986–92 (Reference McLaren, Bourke, Walsh, Weaver, Johannessen, Muench and OverlandMcLaren and others, 1994) are used as verification data. The best prediction of North Pole ice thickness is obtained with the viscous-plastic model. In contrast, the cavitating-fluid model underestimates and the two other schemes overestimate the ice thickness at the North Pole.

Fig. 3.

(a) Ice thickness at the North Pole (m), (b) averaged daily-drift speed (cm s⁻¹) in Fram Strait region, and (c) mean ice export through Fram Strait (Sv) for the different models. The horizontal lines indicate the observations.

Figure 3b shows the ice-drift speed in Fram Strait region. The two plastic rheologies slightly underestimate ice-drift speed. The other two models predict drift velocities that are much too low. Ice thickness in the central Arctic and the ice-drift speed in Fram Strait determine the ice export through Fram Strait as displayed in Figure 3c . In agreement with the observed Fram Strait outflow of 0.1–0.16 Sv given by Reference Aagaard and CarmackAagaard and Clarmack (1989), the viscous-plastic and the Cavitating-fluid model both predict a Fram Strait ice export of 0.11 Sv. The other dynamic schemes yield signifi-cantly smaller Fram Strait outflows, and also predict markedly different seasonal cycles of ice export (Fig. 4).

Fig. 4.

Time series of mean seasonal cycles of ice export through Fram Strait (Sv) for 1986–92. Results for four different dynamics schemes are shown.