Rayleigh-Taylor instability for nonhomogeneous incompressible geophysical fluid with partial viscosity

In this article, we investigate the Rayleigh-Taylor instability in a system of two-dimensional nonhomogeneous incompressible fluid equations with Coriolis force and partial viscosity. First, we employ variational methods to construct linear unstable solutions to the corresponding linearized equations of the system. Second, we utilize the classical Osgood lemma to derive nonlinear energy estimates for the perturbed equations. The local existence of solutions to the perturbed equations is established by using the semi-Galerkin method and the expanding domain method. Finally, we prove the nonlinear instability by combining the properties of the linear unstable solutions and the nonlinear energy estimates.