STABILITY OF SEA ICE DYNAMICS MODELS

To learn if a model is stable, all parts of the model must be studied; its momentum equation and constitutive laws. Thus, model stability requires more than a constitutive law that dissipates energy during all deformations. To study the model behavior, small perturbations to the model solutions are analyzed. The perturbation equations are solved using normal modes of motion. Models are stable if the modes decay and unstable if modes grow. A growing mode can generate spurious motions from small initial perturbations. Behavior of dynamic and quasi-steady, isotropic elastic-plastic (EP) and viscous-plastic(VP) models are described. The method can be generalized to study anisotropic elastic-plastic and viscous-plastic models or elastic-viscous-plastic (EVP) models. INTRODUCTION Little attention has been given to model stability. Gray (1999) showed that the Hibler (1979) VP model is unstable during uniaxial opening when the ice condition is described by the standard two-component approximation to the thickness distribution. His approach requires that the system remain hyperbolic, which is similar to one approach used by Schreyer (2001). Here we assume that a model is stable and problems are well posed if small perturbations to the forcing or initial conditions give rise to small perturbations to the solution, and that these perturbations do not grow with time. Growth of the perturbations changes the solution a finite amount over a finite time interval. Chaos (Lorenz, 1993) can be defined in terms of this sensitive dependence to initial conditions. While unstable models might have acceptable behavior for some time interval, solutions that are initially similar eventually diverge, so that we eventually lose predictability. We derive equations governing linear perturbations to the model equations. Solutions to the perturbation equations are expressed in terms of normal modes, or eigenvectors. They are the fundamental solutions that can be combined linearly into general solutions. Finally, we determine the characteristic equation to determine conditions under which non-zero solutions can exist even when the forcing is zero. 1 IceCasting, Inc., San Rafael, CA USA 94901. http://www.icecasting. com. ICE DYNAMICS MODEL The system of equations has momentum balance, a strength law, and a constitutive law relating stress and stretching. Momentum Balance Inertia, Coriolis, and tilt accelerations balance the applied air and water stress and divergence of the internal stress σ τ b v ⋅ ∇ + = + dt d m (1) where the body force ( ) v b contains the Coriolis and water drag forces that depend on ice velocity v, and τ contains air drag and sea surface tilt forces that are assumed independent of ice velocity. Here m is mass per unit area and σ is internal ice stress resultant in excess of hydrostatic equilibrium (stress for short). The equations are expressed in a Lagrangian description with dt d indicating material rate and ∇ the spatial gradient. For daily averages, the inertial forcing is negligible relative to other forces in the momentum balance; if discarded (Pritchard, 2001), a quasi-steady model results. We consider τ to be the external forcing. Water stress w τ is assumed a quadratic function of ice velocity v B v τ ⋅ = w w w w C | | ρ (2) where water density is ρw, drag coefficient is Cw, and rotation operator Bw turns the water stress by angle β through the mixed layer. We have ignored ocean current here. Strength Ice conditions are usually described by thickness distribution (Thorndike et al., 1975) or by a two-level approximation using compactness and mean thickness (Hibler, 1979). Strength in the former is defined by equating energy dissipated by stress during deformation to known small scale ridging sinks. Strength in the latter is defined empirically. Here we ignore the ice condition formulation and introduce a form that should be useful for either description * * t p p dt dp + − = D : H (3) where H is a hardening modulus, p D is plastic stretching, and * t p describes external thermal effects. The operator : is the double inner product of the second-order tensors. Constitutive Law We consider two different types of constitutive laws, one that can describe EP behavior, and the other that can describe VP behavior. Our purpose is to describe an analytical technique that can be applied to specific laws in future studies. Elastic-plastic law Incremental EP laws describe stress rate as a function of stretching (or strain rate) * * t p ep p dt d M D : M σ + = . (4) where the modulus Mep is a fourth-order tensor. It has the same units as stress. The operator : is the double inner product of the fourth and second-order tensors. The thermal modulus is a second-order tensor that describes changes to the stress rate due to changes in strength caused by thermal growth or decay. The EP modulus σ : H σ : M : σ M : σ σ : M M M ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂     ∂ ∂ ⊗     ∂ ∂