## 1. Introduction

[2] Sea ice models typically consist of a thermodynamic component, which computes vertical heat conduction, growth, and melting, and a dynamic component, which determines horizontal motion. Both thermodynamic and dynamic properties depend on the ice thickness, which can vary from less than a centimeter to several meters. In order to better represent thickness-dependent processes, many sea ice models in recent years have introduced an ice thickness distribution (ITD). The evolution of the ITD can be described following *Thorndike et al.* [1975] as

where *h* is the ice thickness, *g*(*h*, ** x**,

*t*) is a probability distribution function for thickness,

**u**= (

*u*,

*v*) is the horizontal velocity,

*f*=

*dh*/

*dt*is the thermodynamic growth rate, and

*ψ*is a function to be specified. The first term on the RHS describes horizontal transport; the second term describes thermodynamic transport in thickness space

*h*; and the last term describes mechanical redistribution in thickness space.

[3] The velocity **u** is computed from a momentum equation that includes the effects of winds, ocean currents, sea surface tilt, the Coriolis force, and internal ice stress. Much research has focused on parameterizing the internal stress in a realistic way. Following *Hibler* [1979] (henceforth *H79*), many sea ice models treat the ice pack as a viscous plastic material that has strength under convergence and shearing, but offers little or no resistance to divergence. Various solution methods have been developed. One class of methods [e.g., *Zhang and Hibler*, 1997; *Zhang and Rothrock*, 2000] solves the momentum equation implicitly over the entire ice pack. Another method, the elastic viscous plastic (EVP) scheme of *Hunke and Dukowicz* [1997], introduces an elastic term as a numerical artifice so that the solution can be computed explicitly. Both approaches have been shown to give stable, accurate solutions.

[4] Much less effort has focused on *ψ*, which determines how ice is redistributed among thickness categories by mechanical processes such as rafting and pressure ridging. (In this paper we will generally refer to all such mechanical processes as “ridging.”) Current ridging schemes are largely heuristic and are difficult to verify empirically. Our starting point for this paper is the ridging scheme used in the Community Ice Code (CICE), the Los Alamos sea ice model [*Hunke and Lipscomb*, 2004]. This scheme is based on the work of *Rothrock* [1975] (henceforth *R75*), *Thorndike et al.* [1975] (henceforth *T75*), *Hibler* [1980], and *Flato and Hibler* [1995], who used a combination of observations, mathematical reasoning, and physical intuition to develop ridging parameterizations suitable for multicategory sea ice models. We refer to this scheme as the standard ridging scheme.

[5] In some models, including CICE, the ridging scheme is closely connected to the dynamics through the ice strength. Following *R75*, these models assume that the ice strength (defined as the compressive stress below which the ice is rigid and at which it fails) is a function of the energy dissipated during ridge creation. The energy dissipation rate depends in turn on the participation of various thickness categories in ridging and on the thickness of the resulting ridges. For example, a model that generates thick ridges will have stronger ice and larger stresses than a model that builds thinner ridges. Larger stresses result in smaller velocities and strain rates. To date there has been little study of the relation between ridging and dynamics in sea ice models.

[6] Global climate models typically have a horizontal resolution of 1° or more for sea ice and the underlying ocean. There is increasing interest, however, in running sea ice and ocean models at scales of the order of 10 km (about 0.1°) or less. These small scales are necessary to resolve mesoscale eddies in the ocean [*Smith et al.*, 2000] and detailed features of sea ice motion [e.g., *Maslowski and Lipscomb*, 2003]. In ocean and atmosphere models, the time step Δ*t* and grid cell size Δ*x* must satisfy a Courant-Friedrichs-Lewy (CFL) condition of the form Δ*x* ≤ max(∣**u**∣Δ*t*), where ∣**u**∣ is the greatest physical speed allowed by the model equations (for example, the speed of gravity waves). When the time step exceeds the CFL limit at a given spatial resolution, the model becomes inaccurate and often unstable. We show in this paper that sea ice models have an analogous limit, which depends on the time and spatial scales at which the ice strength changes. This limit is especially acute for models that use the *R75* ice strength formulation.

[7] Section 2 summarizes the dynamics and ridging schemes used in CICE, and section 3 describes model behavior when the stability limit is violated. We explain the source of the instability in section 4, using a one-dimensional test case to simplify the analysis. In section 5 we show that model stability can be improved by changing the ridging participation function and the thickness distribution of ridges. Although our motivation is primarily numerical, these changes can also be justified on physical grounds. Section 6 concludes the analysis and discusses some remaining uncertainties.