1. Introduction

A model of sea ice dynamics predicts the movement of the ice pack based on winds, ocean currents, and a model of the material strength of the ice. Nonuniform motion of the ice is responsible for the thickness and extent of the ice pack, which in turn influences the exchange of energy between the atmosphere and polar oceans. The dynamic characteristics of sea ice thereby play an essential role in climate-related processes of the ocean and atmosphere.

Many models have been developed to describe the ice dynamics. Some early studies focused on free drift descriptions with no ice interaction (Felzenbaum 1961; Bryan et al. 1975; Manabe et al. 1979; Parkinson and Washington 1979); others included more complex sea ice rheologies, treating the ice as a Newtonian viscous fluid (Campbell 1965), a linear viscous fluid (Hibler 1974; Hibler and Tucker 1979), or a plastic material. The Arctic Ice Dynamics Joint Experiment (AIDJEX) in the 1970s proposed an elastic–plastic rheology for the sea ice pack (Coon et al. 1974), and several other nonlinear plastic rheologies have been studied since then (e.g., Pritchard et al. 1977; Flato and Hibler 1992; Ip et al. 1991). A nonlinear viscous–plastic (VP) rheology proposed by Hibler (1979) has become the standard sea ice dynamics model and the basis for many recent sea ice studies.

The VP model suffers from numerical difficulties related to the enormous range of effective viscosities present in the model and requires large computational resources that become particularly cumbersome when the model is coupled to an ocean or atmophere model (Hibler and Bryan 1987; Oberhuber 1993a,b). To avoid the stringent time step restriction for stability of an explicit numerical scheme in regions where the ice is relatively rigid, the model equations are typically solved with implicit methods such as successive overrelaxation (Hibler 1979) and line relaxation (Oberhuber 1993a; Holland et al. 1993). However, these methods suffer from poor convergence characteristics as the mesh resolution is increased. Attempts to overcome the inherent problems of the model have included improved numerical methods as well as simplifications of the model itself. As part of this paper, we present a more efficient implicit numerical method for solving the VP model equations that uses preconditioned conjugate gradients.

Simpler versions of the VP model, such as free drift descriptions with no ice interaction and cavitating fluid models in which the ice has no resistance to shear forces (Nikiforov et al. 1967; Flato and Hibler 1989, 1992), are more tractable numerically, but the model behavior is sensitive to these simplifications (Holland et al. 1993). Likewise, simulations with more complicated rheologies than the standard elliptical yield curve (Hibler 1979), such as teardrop (Coon et al. 1974), sine wave lens (Bratchie 1984), Mohr–Coulomb, and square (Ip et al. 1991) shapes, show that the rheology can have a significant effect on long-term simulations of ice drift (Ip et al. 1991). Since an ice model need only simulate a visco–plastic material at timescales on the order of those imposed by wind forcing (days), we also present a modification of the model, the addition of elastic behavior, that realizes significant gains in numerical efficiency, reduces to the original VP model behavior at long timescales, and is more accurate for transients. Our model avoids the complexities of the early elastic–plastic models (Pritchard 1975; Colony and Pritchard 1975) because the elastic-like behavior is not intended to be physically realistic and is introduced for numerical expediency.

The VP model also suffers from inaccuracies in calculating transient behavior. For example, given daily time steps, the VP model behavior is acceptable only for surface stresses that vary on the order of a week or more. Hibler (1979) states that several time steps are needed between changes in the forcing (he uses 8-day averaged winds with a 1-day time step), and more recently, Stössel et al. (1994) have noted that the sea ice components of some ice–ocean coupled models are slow to converge, especially under daily forcing. The VP numerical model does produce correct transient behavior if the time step is taken sufficiently small, on the order of minutes for 1-day forcing timescales. Our implementation of the elastic–viscous–plastic (EVP) model is more accurate in resolving transients, even using relatively large time steps, and therefore will produce more accurate ice behavior.

The VP ice dynamics model is not well suited to parallel architectures. Implicit methods required for larger time steps typically entail a great deal of communication between processors, making parallel computation less attractive. Therefore, explicit models are generally preferable for parallel implementations. Ip et al. (1991) optimized the VP model for multiprocessor computers using an explicit, Euler time-stepping scheme, but stability requirements of the numerical method severely limited the time step. The new EVP model presented in this paper permits a fully explicit implementation with an acceptably long time step. Its efficiency is compared with three methods of solving the viscous–plastic equations: the preconditioned conjugate gradient method and two relaxation schemes (Hibler 1979; Zhang and Hibler 1996).

The present work is part of an effort to develop a computationally efficient sea ice component for a fully coupled atmosphere–ice–ocean global climate model. The sea ice model, which also includes thermodynamic and transport components, is designed to be compatible with the Parallel Ocean Program (POP), an ocean circulation model developed at Los Alamos National Laboratory for use on massively parallel computers (Smith et al. 1992; Dukowicz et al. 1993, 1994).

2. The ice dynamics model

a. Viscous–plastic model equations

Pack ice typically consists of rigid plates, which may drift freely in areas of relatively open water or be closely packed together in regions of high ice concentration. Although individual ice floes range from tens of meters to several kilometers across, the ice pack is considered to be a highly fractured two-dimensional continuum, to make modeling it tractable (Pritchard 1975; Rothrock 1975b; Hibler 1980; Gray and Morland 1994).

The force balance per unit area in the ice pack is given by a two-dimensional momentum equation (Hibler 1979), obtained by integrating the 3D equation through the thickness of the ice in the vertical direction:

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where τ_a = (τ_ai, τ_aj) and τ_w = (τ_wi, τ_wj) are wind and ocean stresses, respectively, assumed to be of the form

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The strength of the ice is represented by the internal stress tensor σ_ij. Definitions of the other variables and constants are given in Tables 1 and 2.

There has been a great deal of disagreement about the relative importance of the various terms in (1) (Parkinson and Washington 1979). The primary components are the air and water stresses, Coriolis force, and ice interaction effects (Hibler 1986); the most predominant of these is wind stress (Coon 1980). Rothrock (1975a) demonstrated through scale analyses that the acceleration term is three orders of magnitude smaller than the stress terms. In contrast to Hibler (1979) and following Oberhuber (1993a), we neglect nonlinear advection, which is at least an order of magnitude smaller than the acceleration term. The ice interaction term is essential in balancing the stresses in much of the ice field (Hibler 1979; Parkinson and Washington 1979; Coon 1980; Hibler 1986), and although they are smaller in magnitude, current and tilt effects are significant over long periods of time (Hibler 1986; Warn-Varnas et al. 1991).

The momentum equation must be consistent for any combination of ice and open water in a grid cell. Our particular model differentiates between thick and thin ice and tracks ice concentration with compactness, c, the fractional area of the cell covered with thick ice. When c = 0, there is no thick ice (H = 0), and there may be either thin ice (h > 0) or open water (h = 0). The mass m in (1) is the total mass of ice and snow per unit area, corresponding to ∫^H₀ ρ dz:

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Thin ice is assumed to have no strength, so that the internal stress tensor is nonzero only for thick ice. Since the surface stress terms τ_ai and τ_wi apply over the entire area, we see that thin ice in a cell that does not contain thick ice essentially exists in free drift, given by the momentum equation without ice interaction. In the special case when there is only open water, m = 0 and the “ice” velocity is that of the interface between atmosphere and ocean, calculated with τ_a + τ_w = 0.

The viscous–plastic rheology proposed by Hibler (1979) is given by a constitutive law that relates the internal ice stress σ_ij and the rates of strain ϵ̇_ij through an internal ice pressure P and nonlinear bulk and shear viscosities, ζ and η, such that the principal components of stress lie on an elliptical yield curve with the ratio of major to minor axes e equal to 2. The constitutive law is given by

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where

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Alternatively, this can be rewritten in the form

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which will be useful to us later. This rheology allows the ice pack to diverge with little or no stress, yet resist compression and shearing motion under convergent conditions.

The pressure P, a measure of ice strength, depends on both thickness and compactness: P = P*cHe exp[−c*(1 − c)], where P* and c* are constants given in Table 1. This definition of P is equivalent to the standard formulation of Hibler (1979), because cH is approximately the same as his “equivalent ice thickness” h.

The viscosities increase with pressure and with decreasing strain rates:

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These parameters represent an idealized visco–plastic material whose effective viscosities become infinite in the limit of zero strain rate. Hibler (1979) chose to regularize this behavior by bounding the viscosities when the rates of strain are small and the ice pack moves as an essentially rigid solid; the limiting viscosities are set to large, constant values so that the ice pack is treated as a linear viscous fluid undergoing very slow creep. The maximum value for ζ is 2.5 × 10⁸ P g s⁻¹; η is similarly bounded through equations (7) and (8). He also set minimum values to provide against nonlinear instabilities, with ζ_min = 4 × 10¹¹ g s⁻¹. For a sufficiently small value of c, ζ_max < ζ_min, in which case ζ = ζ_min. For a general account of constitutive laws for sea ice, see Hibler (1986).

Equations (1), (2), (5), and (6) may be combined as follows:

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where c′ = ρ_wC_w|U_w − u|.

b. Motivation for alternative methods

Most sea ice models, starting with the models developed for the AIDJEX project, agree on a visco–plastic rheology at normal levels of strain rate, differing perhaps in the shape of the yield curve. The ideal visco–plastic rheology, however, becomes singular as the strain rate approaches zero. The AIDJEX model (Coon et al. 1974; Pritchard 1975) regularized this behavior by adopting a rheology that converts to that of an elastic material at small strain rates (an “elastic–plastic” rheology). It is important to realize that such an elastic rheology is physically realistic for ice only at a laboratory scale, but at geophysical scales there is no reason to prefer the elastic regularization to any other closure that ensures that the ice pack behaves as a rigid slab in the singular, small strain rate regime. Unfortunately, the AIDJEX model took the limiting elastic behavior quite literally (Pritchard 1975; Colony and Pritchard 1975), needlessly introducing severe theoretical and numerical complexities. Hibler (1979), on the other hand, realized that what was really needed was a simple regularization that gave sufficiently “rigid” behavior in the singular regime. He introduced a regularization (i.e., a “viscous–plastic” rheology), as described previously, in which the nonlinear viscosity of the visco–plastic rheology was bounded at a very high value such that the limiting behavior was really a very slow creep. This regularization, although simple, has its own severe numerical difficulties, which will be discussed shortly. An elastic formulation, on the other hand, has certain advantages from the numerical point of view when viewed merely as a regularization, as explained below. Thus, ironically, we are led to reintroduce a model that has some resemblance to the original AIDJEX model, but in which the regularization is a simplified elastic model whose parameters are chosen for numerical, rather than physical reasons.

The difficulty in solving (10) and (11) is primarily associated with the presence of shear strength (η ≠ 0). The case η = 0 corresponds to the much simpler and easier to solve cavitating fluid model (e.g., Flato and Hibler 1992). This difficulty may be illustrated for the case of divergence-free velocity (∇·u = 0) and constant η and m. Setting the pressure, surface stresses, Coriolis, and tilt terms equal to R, assumed known, the equations decouple to give

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a simple parabolic equation. The one-dimensional stability condition for an explicit discretization of (12) is

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Given the maximum value of viscosity allowed in the VP model, the time step is on the order of a second for a mesh spacing of about 100 km (Ip et al. 1991) and a hundredth of a second at a resolution of about 10 km, which we anticipate in our application. This consideration led to the adoption of semi-implicit discretization schemes so that the equations could be integrated with a much less stringent time step. The solution methods currently in use are typically iterative relaxation methods (Hibler 1979; Oberhuber 1993a) whose rates of convergence scale asymptotically as (1 − αΔx²)^k for simple test problems, where α is a positive constant and k is the number of iterations (Elman 1994). Furthermore, iterative methods are usually recursive and therefore difficult to adapt to parallel machines. There are some iterative methods, such as the conjugate gradient method, for example, whose convergence rate is linear with resolution (Elman 1994) and which can be used successfully on parallel machines (Smith et al. 1992). Use of a preconditioner further improves this method, but good preconditioners usable on parallel machines are hard to find. We will consider the use of the preconditioned conjugate gradient method later as one of the two methods studied in this paper.

Now consider a schematic hyperbolic equation of the form

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We have written (14) so that it models a damped elastic-wave mechanism that converges to the same steady-state solution as (12); E is a parameter analogous to Young’s modulus. This equation is meant to represent, albeit very crudely, an elastic–plastic model of the sea ice in the regime where elastic waves dominate. The one-dimensional stability bound for an explicit discretization of (14) is

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and it is possible to arrange the stability restriction due to damping to be subsumed in (15). Thus, one might expect such an explicit scheme to converge to the steady state with a convergence rate proportional to (1 − αΔx)^k, similar to that of optimum methods for parabolic equations, and much better than that of standard iterative methods. This is the rationale for considering an elastic wave adjustment process as the regularization method for the visco–plastic model in the small strain rate regime. The parameter E controls the value of the strain rate at which the regularization kicks in.

c. The elastic formulation

To construct such a model, it is usual to separate the strain rate into the sum of plastic and elastic contributions (Reuss 1930). The plastic part has already been given by (6), and the elastic part is approximated by

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where E, as before, corresponds to Young’s modulus. Note that this is not a physically realistic elastic equation, but one that is simplified for use as a regularization. Consistent with (1), we have neglected nonlinear advection terms. Adding the elastic and plastic contributions, we obtain

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Note that the VP rheology (6) is obtained as the steady-state limit of (17) or alternatively in the limit E → ∞, while in the limit η, ζ → ∞ we recover the elastic equation (16). Thus, the elastic equation (16) controls the behavior in the limit of infinite viscosity, and therefore represents a regularization of the visco–plastic rheology. We also retain the limiting viscosities of Hibler (1979) in the elastic–viscous–plastic model. Although this is unnecessary, it allows more direct comparison with the VP model behavior. Equations (1) and (17) constitute the EVP model. These prognostic equations for the velocity and stress components, u_i and σ_ij respectively, are discretized explicitly, as described in the following section. The characteristics of this discretization are analyzed in section 4, where we obtain the appropriate choice of E and Δt to permit efficient integration while maintaining viscous–plastic balance at slow timescales.

3. Numerical formulations

In this section we outline our numerical techniques for both the preconditioned conjugate gradient method and the explicit elastic–viscous–plastic method. The spatial discretization is specialized for a generalized orthogonal B-grid as in Smith et al. (1996) or Murray (1996), and each logically rectangular grid cell is divided into four triangles, as illustrated in Fig. 1. All of the thermodynamic and transport variables are given at the center of the cell, velocity is defined at the corners, and the stress tensor is constant across each triangle. We assume contravariant velocity components (velocity components aligned along grid lines). Here, σ_ij may take on four different values within a grid cell. This tends to avoid the grid decoupling problems associated with the B-grid. Note that the rates of strain ϵ̇_ij, and therefore the viscosities η and ζ, are also defined in each triangle. A land mask M_h is specified in the cell centers, with 0 representing land and 1 representing oceanic cells. A corresponding mask M_u for velocity and other corner quantities is given by M_u(i, j) = min{M_h(l), l = (i, j), (i + 1, j), (i, j + 1), (i + 1, j + 1)}.

The velocity component equations [see (1), (5), (17), or (10), (11)] are coupled through the strain rate ϵ̇_ij, the viscosities, and the ocean stress τ_w. We lag the viscosities and c′ to obtain a linear system, but leave the equations otherwise coupled.

a. Conjugate gradient solution of the viscous–plastic model

Equations (10) and (11) are discretized semi-implicitly in time as follows: if n indicates the previous time step, then

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Here

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All coefficients, including ζ and η, are evaluated at time level n. The viscous–plastic time step, Δt_υ, is typically on the order of hours.

At time level n + 1, spatial discretization of (18) and (19) produces a system of simultaneous equations that must be solved iteratively for the values of uⁿ⁺¹ and υⁿ⁺¹ at each grid point. The viscous–plastic rheology operator ∂σ_ij/∂x_j arises from a variational principle with the functional

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where η and ζ are assumed constant for the purpose of the variation in u and υ, and we have temporarily ignored the pressure term. Formulas for ∂u_i/∂x_j are provided in appendix A. We discretize I, then take its variation with respect to u and υ discretely to obtain the second-order derivative terms in (10) and (11). This procedure ensures that the discrete operator involved in computing ∂σ/∂x is both symmetric and negative definite, thereby mimicking the corresponding properties of the continuum operator. Besides providing better fidelity, these properties are highly desirable from the numerical point of view. A symmetric, negative definite operator has only negative real eigenvalues, which means that it is dissipative, ensuring nonlinear stability due to its dissipation of the total kinetic energy (Dukowicz 1997). Furthermore, symmetry is a prerequisite for some highly effective iterative solution methods, such as the conjugate gradient method to be discussed later. There is no guarantee, particularly on a nonuniform grid, that a conventional spatial discretization such as that used by Hibler will have these properties. However, on a uniform grid, the variational discretization reduces to the standard centered difference formula.

The coefficients of all “n + 1 terms” in (18) and (19) translate into a banded matrix, which may be represented by the symmetric operator

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where

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The resulting matrix equation is solved iteratively with a preconditioned conjugate gradient method (Elman 1994; Ashby et al. 1990). The conjugate gradient method, along with other similar methods in its family, is particularly attractive because it is optimal among all linear iterative methods in reducing the error in a particular error norm. Because of this, it has received much attention in the literature and is being widely used, although not in the ice modeling community. The preconditioning matrix is given by

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where A′ is the tridiagonal matrix extracted from the coefficients of A^TA, which couples the u-velocity components along a line of constant j, and C′ is the corresponding tridiagonal matrix extracted from C^TC, which couples υ-velocity components along a line of constant i.

Success of the method hinges on symmetry of the iterating and preconditioning matrices; for this reason we lag the terms ±βu during the solution of (18) and (19). This treatment of the Coriolis term, which restricts the time step to about 2 hours for accuracy, might be remedied by applying a predictor–corrector method to these terms as in Zhang and Hibler (1996). This and other improvements to the VP time stepping scheme are reserved for future work.

We have employed a simple linearized Backward-Euler time discretization scheme for (18) and (19). Other methods for dealing with the nonlinearity, such as those employed by Hibler (1979) and Zhang and Hibler (1996), are somewhat more accurate but have their own difficulties. The numerical method of Hibler (1979), which we will refer to as “H79,” iteratively solves the system (10) and (11) at each time step with successive overrelaxation, utilizing a predictor–corrector method to march the equations in time. Specifically, predicted velocities at time level n + 1/2 are used to compute the coefficients of the linearized terms (namely, ζ, η, α, and β) before advancing to the next time level. Hibler and Ackley (1983) found a splitting problem with this procedure in cases of small nonlinear viscosities (free drift), which was corrected by a modified averaging procedure.

As with the predictor–corrector scheme, problems also arise in methods that use numerical spatial splitting and, in particular, in those methods that do not treat the entire strain rate tensor implicitly. For example, Zhang and Hibler (1996) also use successive overrelaxation to solve (10) and (11), along with a predictor–corrector time discretization scheme similar to that of Hibler (1979). In this case, however, the cross derivative terms are treated at time level n instead of n + 1, and the equations decouple. Then the equations for u_ij are solved iteratively along an entire row (i.e., constant j) before continuing to the next row, and the equations for υ_ij are solved similarly along columns. We will refer to this method as “ZH96.” Stössel et al. (1994) found that treating the diagonal part of the strain rate tensor implicitly and the off-diagonal terms explicitly produced anomalous ice drifts of 6 cm s⁻¹. For the conjugate gradient method described above, the strain rate tensor remains unsplit.

b. The elastic–viscous–plastic model

Discretization in time of the momentum equation (1) is analogous to that of (18) and (19), except that the stress tensor is determined prognostically, and both (1) and (17) are subcycled with an effective EVP time step of length Δt_e = Δt_ζ/N for some integer N > 1 and time interval Δt_ζ. That is, N smaller time steps are taken with (1) and (17), holding η and ζ constant, for each time interval [t_n, t_n + Δt_ζ]. Typically, Δt_ζ = Δt_υ, so that Δt_υ is often both the viscous–plastic implicit time step and the interval at which viscosity is updated in the EVP model. Subcycling maintains the time scale on which the viscous–plastic material characteristics are changing, ensuring that the VP and EVP formulations are equivalent in the limit Δt_e → 0, as will be seen later.

Denoting the subcycling with the index k, we time step (17) as follows, holding the viscosities constant at time level n:

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(Since they both depend on the thickness variables, E and P also change on the Δt_ζ time scale, as will be seen.) This is a simultaneous equation for the three distinct stress tensor components, σ₁₁, σ₁₂, and σ₂₂, which may be inverted directly. Incidentally, we found that computing σ₁₁ and σ₂₂ from formulas that have the same form is important for maintaining symmetry of the numerical solutions in the x and y directions, even at some computational expense.

Given the updated stress tensor σ^k+1_ij, the momentum equation (1) is marched as follows:

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where c′ = ρ_wC_w|U_w − u^k|. This equation may be solved for the velocity components as follows:

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where α = m/Δt_e + c′cosθ and β, τ_x and τ_y are defined in section 3a.

The proper spatial discretization of σ_ij is determined analogously to the variational principle method of (20). Given formulas for ∂u_i/∂x_j provided in appendix A, we demand that in each triangle

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Taking the variation of (23) with respect to u_i yields formulas for the spatial derivatives of σ_ij. This is equivalent to the formalism used in the conjugate gradient solution of the VP model. Since Eq. (23) is another form of the variational formulation for deriving the spatial discretization, it also results in a symmetric, negative definite discrete operator, which ensures a decreasing total kinetic energy. Note that this spatial discretization is different from that in Hibler (1979).

4. Heuristic analysis of the elastic–viscous–plastic model

a. Simplified model description

The sea ice model equations are strongly nonlinear and very difficult to analyze. In this section we will consider a simplified one-dimensional version of the equations describing the EVP model in order to better understand the behavior of the model and as an aid in the selection of parameters. The simplified model assumes all spatial variation and motion occurs only in the x direction, all coefficients are constant, all forcing is absorbed into a single term τ, the constant term P/4ζ is absorbed into σ = σ₁₁, and σ₁₂ = σ₂₂ = 0. The model therefore is not an exact representation of the EVP model but is sufficiently similar to be useful for a heuristic analysis. The resulting equations are

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where ζ is taken to be an effective constant viscosity. The VP model is recovered in the limit E → ∞:

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which will be considered as the reference for later comparisons. Conversely, in the limit ζ → ∞, (24) and (25) reduce to a purely elastic model, which supports undamped elastic waves,

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where c_e = E/m is the elastic wave speed. It is convenient to introduce a viscous timescale

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and an elastic timescale,

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As discussed in section 2, T_υ is on the order of a hundredth of a second for resolutions of 10 km. In contrast, as we will see shortly, we may be allowed to choose T_e to be several orders of magnitude larger.

Equations (24)–(26) may be discretized analogously to the full set of equations given in section 3. Since the equations are linear with constant coefficients, a von Neumann stability analysis may be performed; it is outlined in appendix B for the one-dimensional EVP case described in section 5. A two-dimensional stability analysis for the complete set of equations, analogous to that given in appendix B and assuming Δx = Δy, is summarized in Fig. 2 in terms of the timescales (28) and (29). It is remarkable that, provided Δt ≤ T_e2, the numerical scheme is stable irrespective of the value of the viscous timescale T_υ. Had (26) been discretized explicitly, the stability limit would have been Δt ≤ T_υ/2, implying a prohibitively small value of the time step. We will thus be able to integrate the EVP model with a time step

t_eT_e2(30)

which is much larger than the shortest viscous timescale T_υ, without resorting to implicit discretization. The time discretization of the EVP model therefore subsumes the viscous stability limit as mentioned in section 2.

In what follows it is essential to understand the effect of the time discretization and so, in the interest of simplicity, we will consider a continuous spatial representation, keeping in mind that only wavenumbers k which satisfy k²Δx² ≤ 1 are meaningful on a grid. The time discretization of the implicit VP model is

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where uⁿ is the value of u at time level n, and the corresponding discretization of the EVP model is

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where Δt_e is given by (30), and

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is the number of steps, or subcycles, that the EVP model takes for each step of the VP model. One of the objectives in this section is to estimate a suitable value for N, or in other words, to define an EVP time step Δt_e such that min(T_υ) ≪ Δt_e ≪ Δt_υ. In this section, Δt_ζ = Δt_υ.

b. Forced response

Since (31)–(33) are linear with constant coefficients, it is possible to do a rather complete analysis. However, it is sufficient for our purposes, since the ice is forced primarily by time-varying winds, to focus on the amplitude response to periodic forcing for solutions in the form of plane waves, that is, (u, τ) = (û, τ̂)e^i(kx−ωt), where ω is the angular frequency of the forcing. We characterize the response by a nondimensional parameter F,

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utilizing the damping factor

δe^−iωΔt_υe^−iωNΔt_eδ^N_e

Here, δ_e = e^−iωΔt_e is the damping factor for an elastic time step Δt_e. For convenience, define χ = k²Δx²Δt_υ/T_υ. Substituting plane wave solutions into the “exact” equation (26) and discretizations (31)–(33), we obtain the following response in the three cases:Exact:

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Viscous–plastic:

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Elastic–viscous–plastic:

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Assuming ωΔt_e ≪ 1, the EVP response parameter becomes

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We can deduce by inspection that the response in the viscous–plastic case is accurate (i.e., approximates the exact response) whenever δ ∼ 1, χ ≪ 1, and therefore the conditions for accuracy are

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In the elastic–viscous–plastic case, there are two possibilities:

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and

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These are consistent with the assumption made to obtain (34).

c. Choosing appropriate parameters

Conditions (35)–(40) may be more easily understood graphically. Figure 3 illustrates the domains of accuracy for the two models, using k²Δx²Δt_e/T_υ as the ordinate and ωΔt_e as the abscissa. In general, the EVP domain is larger than the VP domain: if 1/N < T_υ/T_e, then the VP domain is entirely contained within the EVP domain.

It is reasonable to choose parameters so that the accuracies of the two models are approximately balanced, or equivalently, that the domains of accuracy of the EVP model and the VP model are as common as possible. It is therefore desirable to have

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In view of (30), this is equivalent to

t²_eT_υt_υ2(42)

and therefore E may be determined by means of (29):

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It is interesting that (42) implies that a suitable subcycling time step Δt_e is proportional to the harmonic average of Δt_υ and the viscous timescale. This result highlights the benefits of the EVP model: the EVP time step may be orders of magnitude larger than the explicit VP time step. Since there is an entire range of viscous timescales associated with the very large range of effective viscosity coefficients, it may be appropriate to choose some intermediate value of the viscous timescale to use in estimating E. The larger the value of E one chooses, the larger the role of the elastic versus the visco–plastic strain rates and the shorter the time step Δt_e must be.

The parameter E cannot be considered a constant since then the EVP model would have dynamical effects even under free-drift conditions (cH → 0) when the ice rheology should play no role. To avoid this problem, it is sufficient to assume that E has the form E = cHE* for some constant E*. Given suitable values of T_υ and Δt_υ(Δt_ζ), we calculate Δt_e by (42), then calculate E for the two-dimensional problem as

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where 0 < E_○ < 1. Thus, E → ∞ as Δt_e → 0, and the VP rheology (6) is obtained from the EVP formulation (17). This behavior is illustrated in Fig. 8, discussed in the following section.

5. A one-dimensional test problem

In this section we further compare the behavior of the elastic–viscous–plastic and viscous-plastic models for an essentially 1D test problem, but one which now includes nonlinear effects. Consider the more complete one-dimensional form of (1) and (17):

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where, as in section 4, we have lumped all forcing into τ and assumed that all motion and spatial variation occurs only in the x direction.

a. Steady state

We now consider the associated steady-state problem, which we can solve analytically with constant τ and boundary conditions u = 0 on the domain 0 ≤ x ≤ L. At steady state, the stress tensor components are obtained from (45)–(47):

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Noting that η = ζ/4 for e = 2 in (49), we have

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Equation (48) states that

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Combining (52) and (53), assuming P is constant, we have

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Recalling (7) and (9) and noting that the strain rate has only one component, ϵ̇₁₁ = ∂u/∂x, we obtain Δ = 5/4|∂u/∂x|. Therefore, the viscosity ζ can have one of three possible values: ζ_min, ζ_max, and

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Now, ∂σ₁₁/∂x cannot simultaneously be both a constant, as required by (53), and a delta function, as implied by (52) and (55); hence the solution must be composed of segments characterized by ζ_min and ζ_max. Each segment is of the form

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where c₁, c₂ are constants. There will be three segments: two boundary segments characterized by ζ_min and one in the middle characterized by ζ_max. We therefore have a total of six undetermined constants, plus the location of the interior break points. Boundary conditions and continuity of the solution at the two interior break points provide four constraints. Because of the lack of slope continuity at the break points, integral moments of (54) provide the additional constraints:

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where n = 0, 1, 2, ··· . This closes the system.

This solution is confirmed by numerical results, to be shown shortly. We now present a series of simulations that explore and compare the behavior of the EVP and VP models. Unless otherwise noted, parameter values for the simulations in this section are those given in Table 3. The 1D solution was obtained with a simple numerical code that integrates (45)–(48) to steady state.

As predicted by our analysis, the steady-state solution is composed of line segments, illustrated in Fig. 4 (labeled “1D”). Figure 4 also presents corresponding numerical solutions of this problem from the 2D models. Due to the imposed land mask, the numerical solutions remain fundamentally two-dimensional, as illustrated in Fig. 5, and therefore not exactly comparable to the 1D solution. Implementing Neumann or periodic boundary conditions in the SOR viscous–plastic codes in order to make the solution more one-dimensional would have been time consuming and not necessary for our purposes. The four 2D models produce remarkably similar steady-state solutions.

The 2D equations were solved on a 40 × 100 grid of square cells (Δx = Δy = 12.7 km), and the cross sections shown are centrally located in the y direction (j = 50). The integration began with a uniform ice field at rest, no-slip conditions were maintained along all four boundaries, and all of the forcing terms were replaced by a single stress τ = (τ, 0). The CPU times shown in Table 4 represent the time used for the dynamics calculation alone; for each case, 31 s were spent in other sections of the calculation and are not included in the table. These calculations were performed by a CRAY Y-MP8/8128 supercomputer. The models were integrated for 2700 simulated hours, taking 450 time steps with Δt_ζ = Δt_υ = 21 600 s. The EVP dynamics were subcycled 72 times for each viscous–plastic time step, thus taking an effective EVP time step of length Δt_e = 300 s. The EVP numerical model is nearly 40 times more efficient than the original VP code on this test problem.

The elastic and conjugate gradient solutions shown in Figs. 4 and 5 are at steady state; the others are not. Since the corresponding EVP solution is essentially identical to Fig. 5, it is not shown. Doubling the size of the domain from 40 × 100 to 40 × 200, keeping the resolution the same, reduces the magnitude of the EVP steady-state velocity to about 38 cm s⁻¹, closely approximating the 1D numerical solution.

b. Transient behavior

We investigate the transient behavior of the EVP and VP models using two one-dimensional numerical codes, the 1D EVP code mentioned earlier and its viscous–plastic counterpart, which uses tridiagonal matrix inversion to implicitly solve the equation

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Both 1D models are discretized as described in section 3 for the 2D models.

Although the steady-state solutions of the VP and EVP models are the same, their transient behavior differs for typical values of Δt_υ and Δt_ζ. First, consider the VP model transient behavior for different values of Δt_υ, shown in Fig. 6. The very slow response of the VP model for large Δt_υ is due to the linearization used in the rheology operator. If the viscosity ζ is held at time t_n during integration to time t_n+1 = t_n + Δt_υ, the linearized ice rheology operator in the viscous–plastic case (57) takes the form

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Steady state is reached when ζⁿ⁺¹ = ζⁿ. Convergence of this “outer” iteration determines the effective time response. The adjustment process is described fully for this test problem in appendix C, where we determine the steady-state solution analytically and estimate the effective transition time to steady state, about 35 days for Δt_υ = 6 h. Decreasing Δt_υ lessens the time lag between ζⁿ and ∂uⁿ⁺¹/∂x; time steps on the order of a minute produce the “true inertial limited response” (Hibler 1979; appendix B), illustrated in Fig. 6. That is, in order to respond accurately to an impulsively applied τ, the viscous–plastic numerical model must be integrated with a time step of 60 s or less. We refer to this solution (obtained with Δt_υ = 60 s) as the reference solution.

Subcycling the EVP dynamics overcomes this difficulty somewhat. In this case, the ice rheology term has the form

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where k = 1, 2, ··· , N denotes the subcycling. The improved estimates of ∂u/∂x during the VP time step improve the adjustment of the solution. When Δt_ζ = Δt_υ exceeds the stability limit, as it often does, the EVP results generally lie within an envelope bounded by the viscous–plastic solutions for Δt_υ → 0 and Δt_υ → ∞, as indicated in Fig. 7.

Without subcycling, Δt_e = Δt_ζ and the elastic waves do not damp out within the viscous–plastic time step. The EVP results are then quite energetic for larger time steps, as illustrated in Fig. 8. As the time step approaches zero, however, the solutions converge to the reference solution. Furthermore, the two models produce identical results when Δt_υ and Δt_ζ are much shorter than the viscous–plastic stability limit, regardless of subcycling.

Poor adjustment of the VP model has been noticed previously. Hibler (1979) remarks that the viscous–plastic rheology is slow to converge to steady state and requires several time steps with constant forcing to respond accurately. Similarly, Flato and Hibler (1992) note that even the cavitating fluid model should be subcycled several times without changing the forcing. However, many numerical simulations that utilize the viscous–plastic rheology, including numerous sensitivity studies, use 1-day time steps with daily varying winds (e.g., Hibler and Walsh 1982; Hibler and Ackley 1983; Walsh et al. 1985; Ip et al. 1991; Riedlinger and Preller 1991; Chapman et al. 1994). These wind stresses may vary significantly on timescales of a day or so. For example, the wind stress imposed in this example is less than 0.1 dyn cm⁻². Since the initial change in wind stress occurs over the first time step (6 h), this is equivalent to a change in the applied wind stress of 0.4 dyn cm⁻² per day. The physical wind stress may vary as much as 5 dyn cm⁻² per day (Coon 1980), an order of magnitude larger. Not surprisingly, we observe that when integrated with 1-day time steps, the VP numerical model exhibits a weak response to strongly varying winds. The improved transient behavior of the EVP model enhances its ability to capture the response of the ice to such variations in the stress. We will explore the models’ responses to more realistic, time-dependent forcing in the next section.

Both the viscous–plastic transition to steady state and the magnitude of u at steady state depend on ice concentration, as shown in Fig. 9, since the maximum viscosity ζ_max varies with compactness as cexp[c*(1 − c)] through the pressure P. Because of this exponential dependence on c, P and ζ_max are about two orders of magnitude less for ice concentrations of 0.8 than for 0.9, and therefore the ice rheology is immaterial for c < 0.8, and one cannot distinguish between elastic and viscous–plastic models.

All of the calculations reported here were done with c = 0.9. Holland et al. (1993) point out that shear stress becomes significant for ice concentrations greater than about 0.9. Furthermore, while open water typically exists year round throughout the Arctic, both Arctic and Antarctic ice concentrations are predominantly greater than 90% during the winter (Stössel and Claussen 1993; Gloersen et al. 1992).

6. A two-dimensional problem

As a further test, we compare the results of the numerical formulations on a geometrically simple 2D problem in which the geostrophic ocean current and wind stress terms have physically realistic magnitudes. Table 5 contains the parameter values used here. A circular ocean current is used that has an amplitude on the order of 10 cm s⁻¹:

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where 0 ≤ x ≤ L_x and 0 ≤ y ≤ L_y. The ocean drag terms are computed as in (2). The wind stress is also specified analytically, but is based on Arctic data for the month of January 1986 provided by the Naval Research Laboratory. Fourier analysis of data in the Greenland Sea (9.2°W, 75.5°N) indicates that the characteristic timescale of the wind forcing is generally between 1 and 5 days. Based on these data, we allow the wind stress to vary 33% from a divergent stress field whose average amplitude is 3 dyn cm⁻², with a period T = 4 days:

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Coriolis and ocean tilting effects have been omitted. Note that the time variation of this forcing occurs only in its magnitude. Although directional variation is not included, this (relatively quiescent) wind stress varies sufficiently to illustrate the difficulties one encounters with the VP model.

The model equations were integrated for 25 simulated days from rest with a time step Δt_υ = 1 day, on a 40 × 40 grid of square cells (Δx = Δy = 12.7 km). Such a large time step is not feasible for the EVP dynamics model; for this case, Δt_ζ = 6 h and Δt_e = 216 s. These time steps were chosen to illustrate the VP model’s inaccuracy for conditions under which it is often used, and the improvement offered by the EVP formulation. Strictly speaking, results from the various codes are comparable only for very small Δt_υ and Δt_ζ, although we observe in Figs. 10 and 11 that the values of Δt_υ and Δt_ζ used here are sufficiently small to produce comparable results.

These results differ slightly from a time-accurate reference solution, which we define as that produced by the conjugate gradient method with a time step of 60 s. In Fig. 12, we present the differences of domain-averaged kinetic energies per unit mass for each of the methods with that of the reference solution. This comparison indicates that while all of the methods reach a quasi-steady state, the EVP model is much more accurate during the initial “spin up” from rest, and suggests that the EVP model will behave significantly better under the severe wind forcing conditions observed in the polar regions. For example, Arctic winds have been observed to change as much as 350% in a three-day period (Reynolds 1984), and the ice edge may move 35 km day⁻¹ under gale conditions (Roed and O’Brien 1983). In general, geostrophic winds are responsible for 60%–80% of the daily ice variance (Serreze et al. 1989). On these timescales, it is essential that a numerical model for ice dynamics respond accurately to the imposed forcing.

Furthermore, the magnitude of the differences between the viscous–plastic model solutions and the reference case in Fig. 12 indicate that the VP models are slow to respond to more typical forcing variations. The kinetic energy of the H79 solution is better than the conjugate gradient solution by about a factor of 2 due to effectively two iterations of the linearization being taken in the predictor–corrector method used for the time stepping. Incorporating a predictor–corrector method into the time discretization of the conjugate gradient numerical model would improve its accuracy to that of the H79 model, but degrade its efficiency. Regardless, neither VP model is as accurate as the EVP model.

The CPU times given in Table 6 represent the time used for the dynamics calculation alone; the 4 s spent performing I/O for each case is not included in the table. Implementing a two-step time discretization scheme for the conjugate gradient VP numerical model would improve its forcing response to roughly the level of the H79 code and slow it down by approximately a factor of 2. Note that for the Δt_υ = 60 s calculation, the conjugate gradient dynamics used 1379 s CPU. We have not made the corresponding calculation with the H79 method, but based on the figures in Table 6, the H79 model would have taken about 12 times longer, or 4.5 h, to perform this calculation. Thus, the standard VP model would require several CPU hours to reach the level of accuracy obtained with Δt_υ = 60 s, which the EVP model simulates fairly well in only 41 CPU seconds using Δt_ζ = 6 h and Δt_e = 216 s.

7. Summary

Despite its physical and computational problems, the nonlinear viscous–plastic rheology proposed by Hibler (1979) is the most widely accepted model for sea ice dynamics. In the model’s physical description, the ice viscosity suffers a severe singularity: treated as a viscous fluid, rigid sea ice has infinite viscosity. Hibler regularized this problem by setting a maximum viscosity bound, thereby allowing the ice to creep slowly rather than being completely rigid. Even so, the viscosity ranges over many orders of magnitude, and integrating the implicit VP numerical model requires large computational resources, particularly for high resolution grids on parallel architectures. Using smaller maximum viscosity values increases the model’s computational efficiency but produces less accurate results. Our explicit elastic–viscous–plastic model utilizes an elastic mechanism in regions of rigid ice to significantly increase the computational efficiency of the VP numerical model. For comparison purposes, we have chosen to retain the maximum viscosity bound for the results presented here. In this paper we also present a fast, though still implicit, conjugate gradient method for solving the VP equations. Although the conjugate gradient method’s efficiency is comparable to the EVP method’s on serial machines, the explicit EVP model will be substantially more efficient on parallel computers.

Furthermore, due to its semi-implicit treatment of the ice rheology, the standard numerical formulation of the VP model has very poor time response for time steps typically used by researchers in the field, which are often as long as a day. Our investigation of a simplified, one-dimensional version of the VP model indicates that the viscous–plastic model behavior is acceptable only for wind stresses that vary slowly. However, for wind stresses that vary significantly on timescales of a day, the viscous–plastic model response is weak.

This computational pathology may be resolved by improving the numerical method or by changing the physical parameterization in the model. The EVP model represents a combination of these approaches: its (albeit nonphysical) elastic waves enable the use of an efficient, explicit numerical method. We observe improved transient behavior of the solutions, enhancing the model’s ability to capture the ice response to variations in the imposed stress. However, because the EVP model is based on the same linearized viscous–plastic rheology as the VP model, it may inherit similar problems in some parameter regimes.

We have shown that a large range of the elastic wave parameter E exists for which the EVP numerical method is both stable and efficient. In particular, this allows the elastic time step to be orders of magnitude larger than the viscous–plastic timescale in areas of rigid ice. Several considerations must be weighed when choosing the model parameters. The timescale of the external forcing places an upper bound on Δt_υ or Δt_ζ. The choice of the subcycling time step Δt_e is based on considerations of efficiency and accuracy; some guidelines for choosing Δt_e are given in section 4. The parameter values used in this paper, namely for Δt_ζ, E, and Δt_e, are representive of suitable values that improve both the numerical efficiency and accuracy of the viscous–plastic ice model. A more complete parameter sensitivity study will be reported later.

Other numerical concerns involve maintaining operator symmetry and energy dissipation properties in the discretization of the stress tensor, which arises from a variational principle. Dividing the grid cells into four triangles for spatial discretizations results in higher resolution of the stress tensor and viscosity fields than of thickness and velocity. These numerical improvements, along with the formulation of the EVP model, have resulted in a fast, efficient model of sea ice dynamics well suited to climate studies on parallel machines.

We have coupled the EVP dynamics model to thermodynamic and transport components and will be testing this ice model with daily atmospheric fluxes and validating it with remotely sensed and in situ observations. More complete descriptions of the thermodynamics and transport components and results from the validation of the complete sea ice model are forthcoming.