  The stochastic Arctic sea ice model described as a single periodic
non-autonomous stochastic ordinary differential equation (ODE) is useful in
explaining the seasonal variability of Arctic sea ice. However, to be nearer to
realistic approximations we consider the inclusion of long-term forcing
implying the effect of slowly-varying ocean or atmospheric low-frequencies. In
this research, we rely on the equivalent Fokker-Planck equation instead of the
stochastic ODE owing to the advantages of the Fokker-Planck equation in dealing
with higher moments calculations. We include simple long-term forcing into the
Fokker-Planck equation and then seek approximate stochastic solutions. The
formalism based on the Fokker-Planck equation with a singular perturbation
method is flexible with regard to accommodating further complexity that arises
due to the inclusion of long-term forcing. These solutions are then applied to
the stochastic Arctic sea ice model with long-term forcing. Strong seasonality
in the Arctic sea ice model combined with long-term forcing, changes the
seasonal variability depending on the phase of the long-term forcing. The
change includes the shift of mean and the increase or decrease of variance and
skewness. Stochastic realisations show that the change of the statistical
moments due to long-term forcing is realised by unusual fluctuations
particularly concentrated at a specific time of a year.
