Solitary waves of vortices
The author studies the propagation of solitary waves of vortices within a spherical shell which constitutes the uppermost layer of a solid planet. This solid-liquid configuration rotates with constant angular velocity about a fixed axis. The fluid within the shell is inviscid, incompressible, and of a constant density. The motion imparted by the planetary rotation to this fluid mass is governed by the Laplace tidal equation from which the potential of the extraplanetary forces has been deleted. Consistent with this ocean model, the stream function of a solitary wave of vortices must satisfy a third-order partial differential equation. The author obtains solutions to this wave equation by imposing the condition that the vertical component of vorticity is functionally related to the stream function. This dependence must necessarily be of the exponential type; this solution then can be reduced to a quadrature depending on some arbitrary parameters. It is shown that one can always choose these parameters in order to approximate the integral in question by means of an analytic function; thus a representation of the stream function of a solitary wave of vortices in terms of hyperbolic functions of time and position is reached.
third-order partial differential equation
scientific article; zbMATH DE number 866170