1. Introduction

Sea ice is an important component of the global climate system, providing a dynamic and thermodynamic coupling between the polar oceans and the lower atmosphere. Simple free drift theories to model the dynamics of sea ice were first proposed by Nansen (1902) and Rossby and Montgomery (1935). These were essentially point mass models, which neglected the resistance of the ice floes to compression, with the result that unphysically thick ice was produced in regions of net mean convergence (Semtner 1976). A particularly important feature of the Hibler (1979) sea ice model is that the pack is treated as a two-dimensional continuum with large-scale material properties to represent the net effect of these floe interactions. The similarity of the highly fractured mosaic of ice floes with granular materials suggested that plasticity might be appropriate. A two-dimensional viscous–plastic rheology (Hibler 1979) was therefore developed from an elliptical yield curve (Coon et al. 1974) and associated flow rule. This has since become the classic sea ice model and has been used extensively in ice and coupled ice–ocean studies. In this paper the viscous-plastic sea ice model (Hibler 1979) is shown to change type as it switches from uniaxial convergence to uniaxial divergence, and the implications of this for the linear stability of the system are investigated.

2. Governing equations

The viscous–plastic sea ice model (Hibler 1979) consists of two continuity equations and one vector momentum balance for the ice concentration A, the ice thickness per unit pack area h, and the two-dimensional in-plane velocity v. The ice concentration and thickness continuity equations are

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where d/dt = ∂/∂t + v · ∇ is the two-dimensional Lagrangian time derivative, div and ∇ are the two-dimensional divergence and gradient operators, and S^h and S^A are thermodynamic source terms. These equations are derived from mass balance and kinematic conditions and neglect the effects of sea ice ridging (Gray and Morland 1994; Gray and Killworth 1996), which occurs during convergence. The momentum balance is

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where m = ρh is the ice mass per unit pack area, ρ is the density of ice, N is the two-dimensional depth-integrated stress tensor, k is the unit vector normal to the plane of motion, f is the Coriolis parameter, τ_a is the wind stress, τ_w is a dynamic ocean drag, and H is the sea surface height.

The wind stress and the ocean drag are modeled with quadratic drag laws

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where ρ_a, ρ_w are the densities of air and water, and c_a and c_w are the quadratic drag coefficients for air and water, respectively. The geostrophic wind, U_a, and water, U_w, velocities are rotated by atmospheric and water turning angles ϕ_a and ϕ_w, respectively.

A viscous–plastic rheology is used to describe the large-scale material properties of the sea ice, in which the depth-integrated stress

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where D = (∇v + ∇v^T)/2 is the two-dimensional strain-rate tensor, 1 is the unit tensor, ζ and μ are bulk and shear viscosities, and P is the ice strength. In the plastic flow regime the viscosities are

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where e is the ratio of the minor and major axes of the elliptical yield curve. The normal flow rule implies that

η²e⁻²γ^21/2(8)

which is a function of the strain-rate invariants

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The ice strength P is a function of the sea ice thickness and concentration

PPhCA(10)

where P* = 5 × 10³ N m⁻² and C = 20 are constants. At low strain rates the plastic flow regime produces arbitrarily large viscosities. To circumvent this a viscous creep state for low strain rates was introduced in which the viscosities attain maximum values ζ_max = 2.5 × 10⁸ P and μ_max = ζ_maxe⁻². All the theoretical results presented in this paper assume that the strain rate is sufficiently large to lie outside the viscous creep domain.

3. Uniaxial flow

Let O(xy) be a rectangular Cartesian coordinate system in the two-dimensional sea ice plane and let the velocity have components u and υ along the x and y axes, respectively. The viscous–plastic sea ice model is now reduced to the case of uniaxial flow parallel to the x axis by assuming that all variables are independent of the coordinate y and the cross-flow velocity component υ is identically zero,

yυ(11)

The uniaxial velocity u is therefore independent of y, and the strain rate components reduce to D_xx = η, D_yy = 0, and D_xy = 0, where η = u_x. The functions γ² = η²/4 and Δ = (1 + e⁻²)^1/2|η|. Substituting these into the constitutive relation (6), it follows that in uniaxial plastic flow the depth-integrated stresses

N_xxψP,N_xy(12)

where the piecewise linear function

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and the constants

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The normal stress is illustrated as a function of η in Fig. 1. For simplicity uniaxial divergence above the strain rate η_m will be referred to in this paper as divergence, uniaxial convergence below the strain rate −η_m will be referred to as convergence, and strain rates in the range −η_m ⩽ η ⩽ η_m will be referred to as in the viscous creep state. It follows from (13) that ψ is equal to a positive constant ψ^d in divergent flow and is equal to a negative constant ψ^c in convergent flow. These two constant states are joined by a linear transition at low strain rates. The sign reversal of the constant ψ between diverging and converging flow is crucial to the analysis that follows.

In uniaxial flow the ice concentration equation (1), the ice thickness equation (2), and the x component of the momentum balance (3) reduce to

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where R^x is the residual driving force in the x direction. All derivatives of A, h, and u with respect to time and space are indicated by the subscripts t and x, respectively. Following notational convention the spatial derivative of tensor components are indicated by a preceding comma. It follows from (12) that, provided |η| ≥ η_m, the x derivative of the in-plane stress N_xx is

N_xx,xψP_AA_xP_hh_x(19)

where P_A and P_h are the partial derivatives of the ice strength P with respect to A and h, respectively. All terms involving derivatives of the dependent variables A, h, and u have, therefore, been written on the left-hand side of the equations in (16)–(18). It follows that in uniaxial flow the viscous–plastic model (Hibler 1979) reduces to a quasi-linear first-order system of partial differential equations.

4. Classification

Quasi-linear systems of partial differential equations can be classified as being of either hyperbolic or elliptic type (Courant and Hilbert 1962). Physically one would expect a time-dependent problem of this form to give rise to a hyperbolic system in which the characteristic curves are well defined. However, it shall be shown here that the type changes as the ice switches from convergence to divergence.

The system of viscous–plastic sea ice dynamics equations (16)–(18) can be written in the matrix form (22) by defining the dependent variable vector s = (A, h, u)^T. It follows that the residual vector r = (S^A, S^h, R^x)^T and the coefficient matrices are

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The characteristic determinant of the system is

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After expansion of the determinant the eigenvalue problem, Q = 0 reduces to

uτρhuτ²ψAP_AhP_h(27)

and the characteristic directions of the system are given by the roots

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where

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For a strength-hardening material in uniaxial divergent flow there is one real root (28) and a pair of complex conjugate roots (29) and (30) since ψ = ψ^d > 0. This situation does not satisfy the definition of either totally hyperbolic or elliptic systems. Courant and Hilbert (1962, p. 176) do not formally classify these intermediate types and note that such systems “do not seem to occur in problems of mathematical physics.” Whitham (1974, section 14.2, p. 489) investigated a similar smaller system with two characteristics that are either both real or both imaginary implying either hyperbolic or elliptic type. He notes that “the elliptic case turns out not to be uncommon” although it “leads to ill-posed problems in the wave propagation context” and unstable periodic wave trains. In the viscous-plastic case the system possesses both hyperbolic and elliptic properties in divergence since there is one characteristic direction along each particle trajectory and relative to this Lagrangian framework there is elliptic behavior.

To illuminate this further, the viscous–plastic sea ice model is now linearized about a state of uniform flow u₀ with constant ice concentration A₀ and thickness h₀. Differentiating (18) with respect to time and adding u₀ times (18) differentiated with respect to x allows the x derivatives of (16) and (17) to be substituted to obtain a single second-order partial differential equation,

û_ttu₀û_xtu²₀ψλ²₀û_xxrû, û_xû_t(36)

for small velocity perturbations û, where λ²₀ = (AP_A + hP_h)₀/(ρh₀) and the residual r is a function of lower-order derivatives. Quasi-linear second-order partial differential equations of the form,

aû_ttbû_xtcû_xxd,(37)

are classified (e.g., Sneddon 1957; Meister 1996) as

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For Eq. (36) the discriminant ac − b² = ψλ²₀ and the linearized viscous–plastic model is hyperbolic in convergent flow and elliptic in divergent flow since ψ changes sign. Here the type definition is clear cut, and the ambiguous intermediate type classification in divergent flow with the full quasi-linear first-order system is avoided.

5. Positive feedback, instability, and ill-posedness

The loss of hyperbolicity is synonymous with the ill-posedness of the system, as it is the imaginary wave speeds (29) and (30) that cause the growth rate of the linear instability to increase without bound with increasing wavenumber k. The equations can be integrated until the point when the wave speeds become imaginary and then the predictive capability is lost. The subsequent solution can only be computed when additional higher-order terms are introduced into the theory to make it well posed. It is therefore of importance to understand why the equations are ill posed and to decide whether this represents a model of a physical process taking place in real sea ice packs.

The ill-posedness of the system arises because of a subtle positive feedback loop in the coupled system of equations, which becomes stronger and stronger as the wavenumber is increased. Some understanding of the feedback mechanism can be obtained by considering the effect of neglecting the accelerative terms in the momentum balance (18). If the advection uu_x is neglected, the linearized partial differential equation for û reduces to

û_ttu₀û_xtψλ²₀û_xxrû, û_xû_t(42)

The discriminant ac − b² = ψλ²₀ − u²₀/4, which is negative in convergence, and the system is hyperbolic as before. However, in divergence, when ψ^d is positive, the system is

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The convective term therefore assists the positive feedback mechanism but is not the root cause. Consider now what happens when the time derivative u_t is neglected in (18) instead. The linearized partial differential equation reduces to

u₀û_xtu²₀ψλ²₀û_xxrû, û_xû_t(44)

and the discriminant is strictly negative:

acb²u²₀(45)

It follows that the system is hyperbolic in both convergence and divergence, and the positive feedback mechanism is destroyed. This can also be seen by substituting normal modes into (44) to give

ωu₀ψλ²₀u₀ik(46)

and Re(ω) = 0, implying that the instability is removed. In addition, neglecting the stress divergence term in (18) is equivalent to setting ψ = 0 in (36) and (40) and the equations are parabolic, stable, and the feedback is again eliminated.

It is clear that the positive feedback arises from the coupling of the local acceleration to the strength hardening model through the ice concentration and thickness equations. In a divergent motion the acceleration and pressure gradient set up a transient dynamic resistance to the motion, whose strength is proportional to P. Relative to the ice surrounding a weak area

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The positive feedback loop is therefore complete, and a small weakness of the ice will grow. It is therefore expected, that as the ice diverges, weaker areas will become ever weaker, eventually fragmenting the ice cover into regions of thick ice separated by regions with no ice or very little ice. Indeed as the wavelength of the weakness becomes infinitesimally small, its amplitude will grow infinitely quickly, and this will produce dynamic discontinuities within the ice, which are sensitive to the smallest perturbation.

6. Numerical simulations

Consider a sea ice motion in which there is no freezing or melting, S^A = S^h = 0, and there is no net driving force R^x = 0. The system of equations, (16)–(18), in conservative form reduces to

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where the piecewise linear function ψ is defined in (13) and the ice strength P is defined in (10).

The continuity equations (48) and (49) have a similar structure that allows a useful reduction for special initial conditions. If the initial thickness is proportional to the initial concentration h(x, 0) = h*A(x, 0), where h* is the constant of proportionality, then h(x, t) = h*A(x, t) for all time. With this special initial condition the governing equations reduce to

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where the constant of proportionality cancels out and the results are therefore valid for arbitrary initial ice thicknesses (providing they are similar).

It is advantageous to nondimensionalize the equations, and the following scalings are introduced:

xLx̃,tLut̃,uuũ,(53)

where u* = (ψ^dP/ρ)^1/2 and tildes are used to indicate nondimensional variables. Dropping the tildes for simplicity, the final conservative system of nondimensional equations is

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where

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the constant Ψ = ψ^c/ψ^d < 0 and η_m = 0.001.

Initially a finite symmetric sea ice concentration is prescribed in the region |x| ⩽ 1 with an antisymmetric velocity field to ensure that |u_x| > η_m. Outside this region there is no ice. The initial conditions are

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Boundaries lie at x = ±3 at which the boundary conditions are

Atut(59)

These boundary conditions are sufficient to model the flow provided the moving ice edge does not reach x = ±3.

A total variational diminishing Lax–Friedrich monotone upwind scheme for conservation laws (Yee 1989;Toth and Odstrcil 1996) was used to numerically solve the reduced system (54)–(56) subject to the initial value and boundary conditions (57)–(59). This is a high-resolution shock capturing method that has the advantage that it does not require the use of a Riemann solver and can be applied to any system of conservation laws. A good review of the method is presented by Toth and Odstrcil (1996).

The results of the simulation using an equal grid spacing Δx = 10⁻² nondimensional units and a time step Δt = 10⁻⁴ nondimensional units are shown in Fig. 2 at six time intervals. The initial concentration is ice free, |x| > 1, and then rises rapidly so that there is a large central region in which the ice concentration is almost constant. At t = 0.1, some slight concentration ripples can be seen. Weak areas, where the concentration is less than in the surrounding ice, develop and grow by the positive feedback mechanism (47). This sends a series of fingers through the ice concentration field, with adjacent regions of high and low concentration ice, that grow in amplitude with increasing time. Eventually the fingers penetrate through the ice sheet, fragmenting it into numerous discrete blocks. The linear instability mechanism selects only the divergent part of the wave train since ψ = ψ^c < 0 implies linear stability in convergence and the perturbation will decay. The fingers therefore only grow downward and are naturally bounded below by the ice-free state, consistent with the global energy statements of Schulkes (1995) and Dukowicz (1997). The linear stability analysis simply shows that there are large deviations from the uniform state, which is demonstrated numerically here.

The numerical grid and time step resolve the solution to a finite resolution and necessarily filter out the high wavenumber and high-frequency response. However, the growth rate of the linear instability becomes stronger with increasing wavenumber (40), and it is expected that, as the spatial resolution and time step are refined, more of these high growth rate modes will be resolved. Figure 3 shows the results of the simulation on a refined grid with Δx = 2.5 × 10⁻³ and Δt = 5 × 10⁻⁵ at the same time intervals as before. Once again the fingers grow and fragment the ice. The fourfold increase in the resolution also increases the number of fingers by approximately a factor of 4, and by t = 0.5 units they have penetrated farther through the ice than before. The small wavelength and high-frequency perturbations introduced by the numerical discretization and rounding error are amplified by the instability and dominate the numerical solution. This has two important consequences. First, numerical results are not unique as they are dependent on the numerical grid and time step size, and second, the symmetry of a symmetric problem, such as this, is destroyed by the asymmetric rounding and discretization error.

7. Rate-dependent models

The numerical simulations show that it is highly undesirable to have a mixed type ill-posed theory, as it looses its predictive power, and a new theory is required. In this section and the next two new well-posed theories are derived using different approaches. The first is a regularization procedure, which introduces higher-order terms to control the ill-posedness, and the second moves the elliptical yield curve in principal stress space so that the positive feedback mechanism is removed. The two approaches raise a basic physical question about the nature of the sea ice pack: Does the positive feedback mechanism (47) represent realistic behavior or not? The answer is unclear at present. The pack is implicitly assumed to be a highly fractured continuum, so the instability cannot represent the break up of floes in tension. However, it might play an important role in the opening of polynyas in coastal regions.

The first theory, presented in this section, retains the positive feedback mechanism (47) by introducing a rate-dependent constitutive relation, instead of rate-independent plasticity, to regularize the problem. Consider the modified constitutive relation

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with viscosities

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where Δ_M > 0 is constant. This constitutive relation is asymptotically similar to the Hibler (1979) law, given in (6) and (7), for Δ ≫ Δ_M. In fact, the two are identical when Δ_M = 0. However, the linear stability properties for Δ_M > 0 are quite different. To see this (60) is reduced to the case of uniaxial flow in which η = u_x, γ² = η²/4, Δ = E|η|, and E = (1 + e⁻²)^1/2. The in-plane stress

N^r_xxPψ^r(62)

where

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and the positive constant η_M = Δ_M/E. In contrast to ψ, defined in (13), ψ^r is a continuously varying function of η, as shown in the dash–dot graph in Fig. 1. Its derivative with respect to η is

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and this therefore contributes to the stress gradient

N^r_xx,xP_xψ^rPψ^r_ηu_xx(65)

If N^r_xx,x is substituted into the momentum balance (18), a nonlinear diffusion term is introduced, which will tend to smooth discontinuities away.

As |η| becomes large, relative to η_M, the diffusion coefficient becomes small and we may linearize the system around a state with a small diffusion coefficient ε₀ = P₀(ψ^r_η)₀/(ρh₀), which is strictly positive. Combining the linearized equations as before yields the third-order equation

û_ttu₀û_xtu²₀ψ^r₀λ²₀)û_xx0û_xxtu₀û_xxxrû,û_xû_t(66)

where the small parameter ε₀ multiplies the highest derivative terms, and we expect singular behavior. Introducing normal modes the dispersion relation

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which has a root with a positive real part in divergence

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which is illustrated in Fig. 4. The system is therefore linearly unstable. However, there is an upper bound,

ω⁺ω_mψ^rd₀λ²₀₀(69)

and the system is well posed by (41). Note that, if ε₀ → 0, the system tends to a rate-independent state and the upper bound ω_m → ∞, implying ill-posedness, which is consistent with the previous results. Nonlinear diffusion therefore introduces smoothing into the positive feedback mechanism that suppresses the ill-posedness but does not prevent the linear instability from growing. It follows that the ice will still fragment, but the formation of dynamic discontinuities is prevented.

Introducing rate-dependence into the constitutive relation does not always remove the ill-posedness. Consider the alternative function

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which is illustrated in Fig. 1 with the dashed line. Its derivative,

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is negative for |η| large enough. It follows that this model leads to a diffusion coefficient ε₀ = P₀(ψ^Rd_η)₀/(ρh₀), which is negative, implying antidiffusion. The growth rate (68) is unbounded in both convergence and divergence and the system is always ill posed, as illustrated in Fig. 4.

8. Shift models

The rate-dependent model (60) introduced higher-order derivatives to control the ill-posedness, but the positive feedback mechanism (47) was retained implying that the linear instability still occurs. As it is unclear whether or not this instability actually represents physical behavior in real sea ice packs, it is of interest to see how it can be eliminated completely from the theory.

The root cause of the positive feedback is that tensile stresses develop during divergent motions. A simple translation of the yield curve in principal stress space is sufficient to ensure that the tensile stress, positive feedback, and ill-posedness are eliminated without including higher-order derivatives. Consider the modified constitutive relation

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where s is the shift parameter. This is a generalization of the constitutive relation (6). The position of the center of the elliptical yield curve is simply moved along the line of equal principal stresses, N₁ = N₂, by changing the value of the shift parameter as shown in Fig. 5. The Hibler (1979) model is recovered by setting s = 1. In uniaxial flow the in-plane stress

N^s_xxψ^sP,(73)

where the piecewise linear function

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and the constants are

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The classification procedure proceeds in exactly the same way as before, and the characteristic directions of the system are again given by (28)–(30). Assuming that a strength-hardening law is used, the characteristics are real provided ψ^s is negative. It follows that in convergence

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and in divergence

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Hyperbolicity is therefore guaranteed in both uniaxial convergence and uniaxial divergence provided

sE.(83)

It also follows from the dispersion relation (40) that the system is linearly stable and well posed since Re(ω) = 0 for ψ^s ⩽ 0. The acceleration and strength-hardening relation now couple to produce a dynamic acceleration with a strength proportional to P. It follows that in a region of slightly weaker ice

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and the system equilibrates itself. That is, there is a negative feedback loop instead of the positive one (47). A repositioning of the yield curve to avoid the tensile stress regime is sufficient to eliminate the positive feedback mechanism and hence the linear instability.

To demonstrate the behavior of the shifted ellipse model numerical computations for the case, s = 2E + 1, are presented using the nondimensional variables introduced in (53). This implies that ψ̃^sd = −1 in this case. The results for Δx = 10⁻² and Δt = 10⁻⁴ nondimensional units are shown in Fig. 6 at the same time steps as before. The ice now spreads out laterally, driven by the initial velocity and its own internal pressure. The ice concentration decreases slowly in response to the divergent velocity field. No instabilities or fingers are generated, the symmetry is preserved, and the solution does not change with further grid refinement.

9. Conclusions

In uniaxial flow the viscous–plastic sea ice model (Hibler 1979) reduces to a quasi-linear system of three first-order partial differential equations, which are hyperbolic in convergence and have two imaginary wave speeds in divergence, implying mixed elliptic/hyperbolic properties. The mixed problem cannot be solved as an elliptic boundary value problem as this would violate causality. Instead, it has to be treated as an initial boundary value problem, which has wavelike solutions. These are linearly unstable (Gray and Killworth 1995) and, since the growth rate of a small disturbance increases without bound with increasing wavenumber, the equations are ill posed.

The source of the problem is a subtle unchecked positive feedback loop (47) between the acceleration terms and the strength-hardening law, which sets up a transient dynamic resistance to a divergent motion. Areas of weak ice have less transient resistance and diverge more than the surrounding ice, which further reduces their strength, and the weakness grows. As the wavelength of this instability tends to zero, its growth rate tends to infinity, implying that the problem is ill posed. Numerical simulations show that fingers are generated by the instability, which propagate through the ice and eventually fragment it. The numerical results are not unique. As the grid size and time step are refined, the smallest wavelength and highest-frequency responses are amplified by the instability and dominate the numerical solution. That is, the fingers have a shorter wavelength and grow faster as the resolution increases as shown in Figs. 2 and 3.

This demonstrates that it is highly undesirable to have a mixed type ill-posed theory as the systems predictive capability is lost. A new well-posed theory is therefore required, and two new models have been proposed. The first model retains the positive feedback mechanism (47) by introducing rate-dependent higher-order derivatives to regularize the theory and suppress the unbounded grow rate of the instability. The second eliminates the positive feedback loop, and hence the instability, by moving the elliptical yield curve into the third quadrant of principal stress space. This theory is totally hyperbolic and a numerical simulation demonstrates that it remains stable as the ice diverges (see Fig. 6). These two approaches raise a fundamental question as to the physical properties of real sea ice packs: Does the positive feedback mechanism (47) exist or not, and, if so what physical process does it represent? The answer is unclear at present.

Many sea ice simulations use the Hibler (1979) code to produce numerical results. Usually the equations are uncoupled from one another and solved sequentially with updates once a day. Such long time steps are sufficient for the ice to be very close to steady state and the transient resistance, and hence the positive feedback loop, in divergence, are not resolved. The results that are generated are not solutions of the underlying system of partial differential equations that claim to be solved since the high-frequency and small wavelength signals, which have a large effect, are filtered out. This paper demonstrates that as both grid sizes and time steps are reduced in size, the numerical results do not converge to a solution.