This paper establishes the large deviation principle (LDP) for the two-dimensional stochastic primitive equations with horizontal viscosity, a model for anisotropic geophysical flows under small noise. Using the weak convergence method, we first prove an LDP in the space Formula: see text. The main novelty lies in demonstrating that the solutions possess enhanced pathwise regularity, belonging almost surely to Formula: see text. This stronger regularity allows us to subsequently strengthen the LDP to the finer topology of Formula: see text, providing a more precise description of the asymptotic behavior. The analysis requires handling the intrinsic anisotropy of the system, particularly in the convection term, which introduces technical challenges distinct from the anisotropic Navier-Stokes cases.
Sun et al. (Fri,) studied this question.