Exact solution and instability for geophysical waves at arbitrary latitude

We present an exact solution to the nonlinear governing equations in the $ \beta $-plane approximation for geophysical waves propagating at arbitrary latitude on a zonal current. Such an exact solution is explicit in the Lagrangian framework and represents three-dimensional, nonlinear oceanic wave-current interactions. Based on the short-wavelength instability approach, we prove criteria for the hydrodynamical instability of such waves.