A Class of Exact Solutions for Two–Dimensional Equations of Geophysical Hydrodynamics with Two Coriolis Parameters

Abstract

The article proposes a class of exact solutions of the Navier–Stokes equations for a rotating viscous incompressible fluid. This class allows us to describe steady shear inhomogeneous (i.e., depending on several coordinates of the selected Cartesian system) flows. Rotation is characterized by two Coriolis parameters, which in a rotating coordinate system leads to the fact that even for shear flows the vertical velocity is nonzero. The inclusion of the second Coriolis parameter also clarifies the well-known hydrostatic condition for rotating fluid flows, used in the traditional approximation of Coriolis acceleration. The class of exact solutions allows us to generalize Ekman’s classical exact solution. It is known that the Ekman flow assumes a uniform velocity distribution and neglect of the second Coriolis parameter, which does not allow us to describe the equatorial counterflows. In this paper, this gap in theoretical research is partially filled. It was shown that the reduction of the basic system of equations, consisting of the Navier-Stokes equations and the incompressibility equation, for this class leads to an overdetermined system of differential equations. The solvability condition for this system is obtained. It is shown that the constructed nontrivial exact solutions in the general case belong to the class of quasipolynomials. However, taking into account the compatibility condition, which determines the solvability of the considered overdetermined system, leads to the fact that the spatial accelerations characterizing the inhomogeneity of the distribution of the flow velocity field turn out to be constant. The article also provides exact solutions for all components of the pressure field.

Keywords

• layered flows
• shear flows
• exact solutions
• coriolis parameter
• overdetermined system
• compatibility conditions