# Super-parameterization of Lagrangian sea ice dynamics using the Boltzmann equation

A. Davis(1), D. Giannakis(1), G. Stadler(1), and S. Stechmann(2)

(1) *New York University*

(2) *University of Wisconsin*

## Abstract

We devise a superparameterized sea ice (SPICE) model that captures dynamics at multiple spatial and temporal scales. Arctic sea ice contains many ice floes—chunks of ice—whose macro-scale behavior is driven by oceanic/atmospheric currents and floe-floe interaction. There is no characteristic floe size; data suggest the floe size distribution follows a power law. Therefore, accurately modeling sea ice dynamics requires a multi-scale approach. Our two-tiered model couples basin-scale conservation equations with small-scale discrete element particle methods. The basin-scale sea ice dynamics are primarily driven by external forces (wind and ocean stresses), which advect ice floes. Small-scale floe dynamics—deformation, fracture, and collisions—lead to emergent behavior such as lead formation and ridging. Unlike many other sea ice models, we do not average quantities of interest (e.g., mass/momentum) over a representative volume element; we explicitly model small-scale dynamics. We formulate a mathematical modeling framework that rigorously couples a macro-scale PDE with small scale particle methods.

Our framework constructs a time dependent probability distribution over floe position and velocity. In theory, the particle density function evolves according to the Boltzmann equation. In practice, numerically solving the Boltzmann equation is computationally intractable. The SPICE model decomposes the density function into a mass density that models how ice is distributed in the spatial domain and a velocity density that models the small-scale variation in velocity at a given location. We show that the mass density and macro-scale quantities of interest (e.g., expected velocity) evolve according to a conservation equation that only depends on the macro-scale spatial coordinate. However, the flux term depends on expectations with respect to the velocity density at each point. We, therefore, use particle methods to simulate the conditional density at key points, using macro-scale variables to define auxiliary particle forces so that the small-scale particle methods are independent.