On the dynamical Rayleigh–Taylor instability of 2D inviscid geophysical fluids with geostrophic balance

We investigate the nonlinear instability of a steady-state solution to the two-dimensional nonhomogeneous incompressible Euler equations. The solution has a smooth, steady density profile and a background shear, which is derived from the balance of a uniform gravitational field, Coriolis forcing, and pressure. The density profile is characterized by an increasing heavier density with height, which is referred to as the Rayleigh–Taylor instability. First, we establish the linear instability of the solution by demonstrating the existence of a positive eigenvalue for the corresponding linear problem through the classical variational method. Second, we establish an existence theorem of local solutions to the original nonlinear equations. By utilizing this theorem, we then substantiate the instability in a nonlinear sense. By doing so, we extend the mathematical understanding beyond the original Rayleigh–Taylor instability to include the influence of Coriolis forcing and geostrophic balance. © 2024 Elsevier B.V.