Theory of the Sea Ice Thickness Distribution

We use concepts from statistical physics to transform the original evolution equation for the sea ice thickness distribution $g(h)$ from Thorndike et al. into a Fokker-Planck-like conservation law. The steady solution is $g(h)=\mathcal{N}(q)h^q{e}^{-h/H}$, where $q$ and $H$ are expressible in terms of moments over the transition probabilities between thickness categories. The solution exhibits the functional form used in observational fits and shows that for $h\ll 1$, $g(h)$ is controlled by both thermodynamics and mechanics, whereas for $h\gg 1$ only mechanics controls $g(h)$. Finally, we derive the underlying Langevin equation governing the dynamics of the ice thickness $h$, from which we predict the observed $g(h)$. The genericity of our approach provides a framework for studying the geophysical-scale structure of the ice pack using methods of broad relevance in statistical mechanics.

Figure 1

Comparison of our theory with satellite measurements for February through March (F-M) of (a) 2008 and (b) 2004. Circles are the distribution functions from ICESat [9] and lines are the fits using Eq. (10). In (a), $q=1.849$ and $H=0.783\mathrm{m}$, and in (b), $q=1.848$ and $H=0.910\mathrm{m}$.

Figure 2

Solution to the Langevin equation (11) for F-M 2008. Here, $\epsilon =0.046$, $k_1=0.048$, and $k_2=0.025$. The ensemble size used is $N_{\mathrm{en}}={10}^5$ and the time step for the Euler-Maruyama scheme [10] is $\mathrm{\Delta }t={10}^{-5}$. The total nondimensional integration time is $T=140$. Panels (a) and (b) show the thickness distribution and four realizations from the ensemble, respectively. In (a), the circles represent $g(h)$ from the Langevin equation and the solid curve is the fit using Eq. (10), which gives $q=1.838$ and $H=0.777\mathrm{m}$.