Stochastic dynamics of Arctic sea ice Part II: Multiplicative noise

We analyze the numerical solutions of a stochastic Arctic sea ice model with multiplicative noise over a wide range of external heat-fluxes, $\Delta F_0$, which correspond to greenhouse gas forcing. When the noise is multiplicative, the noise-magnitude depends on the state-variable, and this will influence the statistical moments in a manner that differs from the additive case, which we analyzed in Part I of this study. The state variable describing the deterministic backbone of our model is the energy, $E(t)$, contained in the ice or the ocean and for a thorough comparison and contrast we choose the simplest form of multiplicative noise $\sigma E(t) \xi(t)$, where $\sigma$ is the noise amplitude and $\xi(t)$ is the noise process. The case of constant additive noise (CA) we write as $\sigma\overline{E_S}\xi(t)$, in which $\overline{E_S}$ is the seasonally averaged value of the periodic deterministic steady-state solution $E_S(t)$, or the deterministic seasonal cycle. We then treat the case of seasonally-varying additive noise (SVA), $\sigma E_S(t) \xi(t)$, as well as two types of multiplicative noise that depend on the form of stochastic calculus (It\^{o} or Stratonovich) used to interpret the noise itself. The comparison of these four cases reveals the stochastic anatomy of the system over the entire range of the $\Delta F_0$ from the perennial ice states to near the ice-free state.