On the superposition of solutions of the (3 + 1) dimensional Charney–Obukhov equation for the ocean

Abstract

New exact solutions describing Rossby waves and vortices in ocean propagating along the zonal direction at a constant velocity are found for the (3 + 1)-dimensional nonlinear Charney–Obukhov equation. These solutions are a partial superposition of previously discovered exact solutions of the Charney–Obukhov equation. Partial superposition is found for that part of the solution of the Charney–Obukhov equation, which complements the zonal stream. The presence of such a superposition in the solutions of a nonlinear equation with two nonlinear boundary conditions is a remarkable property of the Charney–Obukhov equation for the ocean and allows one to simulate a wide class of fluid flows based on exact solutions. As an example, we discuss solutions that include superposition of trigonometric functions and the functions of Bessel in a horizontal plane, and superposition of spherically symmetrical solutions in the vertical coordinate. Visualization of the solutions found shows that, depending on the values of the parameters included in the solutions, they can describe both a flow with a large number of vortices and a periodic structure with alternating high and low pressure fronts.