SOLITARY WAVES OF VORTICES

We study the propagation of solitary waves of vortices within a spheri cal shell which constitutes the uppermost layer of a solid planet. Thi s solid-liquid configuration rotates with constant angular velocity ab out an axis which is fixed with respect to the solid surface. The flui d within the shell is inviscid, incompressible, and of constant densit y. The motion imparted by the planetary rotation upon 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, we establish that the stream function of a solitary wave of vo rtices must satisfy a third-order partial differential equation. We ob tain solutions to this wave equation by imposing the condition that th e vertical component of vorticity be functionally related to the strea m function. We find that this dependence must necessarily be of the ex ponential type and that the solution to the wave equation then reduces to a quadrature depending on some arbitrary parameters. We prove that we can always choose the values of these parameters in order to appro ximate the integral in question by means of an analytic function: we r each a representation of the stream function of a solitary wave of vor tices in terms of hyperbolic functions of time and position.