Physical Oceanography, Vol.
NONLINEAR RADIAL OSCILLATIONS OF AN ISOLATED EDDY IN THE OCEAN
WITH REGARD FOR THE EXTERNAL ACTION
S. F. Dotsenko
and A. Rubino
Within the framework of an reduced-gravity model of the ocean dynamics, we find a class of
exact analytic solutions of the problem of description of nonlinear axisymmetric oscillations of a
subsurface eddy under the action of stationary radial mass forces. The radial projection of the
velocity of oscillations of this sort is a linear function and the azimuthal velocity, the thickness of
the eddy, and the mass forces are polynomials as functions of the radial coordinate with time-de-
pendent coefficients. The method used to find the analytic solution is based on the exact replace-
ment of the original mathematical model by a system of ordinary differential and algebraic equa-
tions. A new class of motions of the eddy appears as a result of nonlinear interaction between
the lowest mode of oscillations and the geostrophic circulation inside the eddy.
Eddies formed in the ocean are characterized by well-pronounced variability as functions of time participat-
ing in horizontal displacements and oscillations about the center of mass . The theoretical investigations of
the dynamics of eddy formations are carried out by numerous researches by using various numerical and analytic
methods. However, only for a small number of model situations, it is possible to find exact (explicit or implicit)
analytic solutions of the problems on nonlinear oscillations of eddies. Investigations of this sort are carried out
mainly within the framework of the so-called reduced-gravity model of two-layer ocean whose mathematical
description formally coincides with the system of equations for long barotropic waves. This fact makes it pos-
sible to obtain analytic solutions of the problems of oscillations of oceanic eddies.
The analytic solution of the problem of axisymmetric oscillations of a rotating liquid in basins of parabo-
loidal form expressed via the elementary functions was obtained in [2, 3]. For motions of this type (seiches), the
projections of the horizontal velocity are linear functions of the distance from the axis of symmetry of the model
and the displacements of the free surface are quadratic functions of the indicated distance. In , it is shown that
oscillations of this kind form a fairly general class of oscillations of dynamic formations in the ocean and atmo-
sphere described by shallow-water-type systems of equations.
The exact analytic solutions of the problem of description of horizontal oscillations of an eddy without
changes in its geometric shape and radial inertial oscillations of the fields about the center of mass of the eddy
were found for the first time in [1, 5]. The method used to find analytic solutions is based, as in the problems of
oscillations of liquid in a paraboloidal basin, on the exact substitution of the original mathematical model by a
system of nonlinear ordinary differential equations and finding of its analytic solutions. The procedures of find-
ing analytic solutions of a similar structure used to describe nonlinear oscillations of elliptic eddies can be found
A class of analytic solutions of the reduced-gravity model describing nonlinear inertial oscillations of cir-
cular eddies was found in . As earlier, the radial velocity of motion of the liquid in the eddy is a linear func-
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol.
University of Venice, Italy.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
15–25, January–February, 2005. Original article submitted September
14 0928-5105/05/1501–0014 © 2005 Springer Science+Business Media, Inc.