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, $\mathrm{\Delta }{F}_0$, 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 $\mathrm{\Delta }{F}_0$ from the perennial ice states to near the ice-free state.