  The Cotter-Holm Slice Model (CHSM) was introduced to study the behavior of
whether and specifically the formulation of atmospheric fronts, whose
prediction is fundamental in meteorology. Considered herein is the influence of
stochastic forcing on the Incompressible Slice Model (ISM) in a smooth 2D
bounded domain, which can be derived by adapting the Lagrangian function in
Hamilton's principle for CHSM to the Euler-Boussinesq Eady incompressible case.
First, we establish the existence and uniqueness of local pathwise solution
(probability strong solution) to the ISM perturbed by nonlinear multiplicative
stochastic forcing in Banach spaces $W^{k,p}(D)$ with $k>1+1/p$ and $p\geq 2$.
The solution is obtained by introducing suitable cut-off operators applied to
the $W^{1,\infty}$-norm of the velocity and temperature fields, using the
stochastic compactness method and the Yamada-Watanabe type argument based on
the Gy\"{o}ngy-Krylov characterization of convergence in probability. Then,
when the ISM is perturbed by linear multiplicative stochastic forcing and the
potential temperature does not vary linearly on the $y$-direction, we prove
that the associated Cauchy problem admits a unique global-in-time pathwise
solution with high probability, provided that the initial data is sufficiently
small or the diffusion parameter is large enough. The results partially answer
the problems left open in Alonso-Or{\'a}n et al. (Physica D 392:99--118, 2019,
pp. 117).
