Physical Oceanography, Vol. 18, No. 6, 2008
THERMOHYDRODYNAMICS OF THE OCEAN
NONLINEAR RADIAL OSCILLATIONS OF A FLUID IN A PARABOLIC
BASIN WITH REGARD FOR THE EXTERNAL ACTION
S. F. Dotsenko
and A. Rubino
Within the framework of the theory of long waves, we find a class of exact analytic solutions of
the problem of description of nonlinear axially symmetric oscillations of a fluid in a parabolic
basin with regard for the action of stationary radial bulk forces. The radial projection of the
velocity of these oscillations (seiches) is a linear function of the radial coordinate, whereas the
azimuthal velocity and displacements of the free surface of the fluid are polynomials in the radial
coordinate with time-dependent coefficients. The method of finding solutions is based on the
exact replacement of the original problem by a system of ordinary differential and algebraic
equations. The action of the bulk forces may result either in the increase in the frequency of os-
cillations of the fluid and in the decrease in this frequency and affect the motion of the water
edge, the characteristics of waves, and the velocity field.
The exact analytic solutions can be found only for a small number of nonlinear wave problems, including
the investigation of Gerstner waves  and the problems of dam destruction  and run-up of waves on a plane
beach . The application of the model of long waves in which the distribution of pressure in a moving fluid is
assumed to be hydrostatic makes it possible to enlarge the list of nonlinear problems for which it is possible to
find exact analytic solutions. Within the framework of this approach, the problems of axially symmetric oscilla-
tions of fluids in basins of parabolic shape [4–7] and fluctuations of eddy formations in the ocean [8–14] were
solved. The investigations of the fluctuations of eddy formations were initiated by the problems of oceanology.
In the present work, we get a new class of exact analytic solutions of the nonlinear system of equations of
long waves. These solutions describe the axially symmetric periodic oscillations of an ideal homogeneous fluid
in a rotating basin in the form of a paraboloid of revolution with regard for the action of horizontal axially sym-
metric bulk forces. The general form of solutions of these sort was proposed for the problem of nonlinear inerti-
al oscillations of circular eddies in . Later, these results were extended to the case of oscillations of eddies
under the action of external forces . In [13, 14], the radial velocity is a linear function of the radial coordi-
nate, whereas the azimuthal velocity and the thickness of the eddy are polynomials of different degrees in the ra-
dial coordinate with time-dependent coefficients. The solution of the linear problem of seiches in a parabolic ro-
tating basin is presented in . Various aspects of the plane linear problem of free and forced oscillations of flu-
ids in basins with parabolic distributions of depth are studied in [15, 16].
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol., Ukraine.
Ca' Foscari University, Venice, Italy.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
3–13, November–December, 2008. Original article submitted June 7,
0928-5105/08/1806–0297 © 2008 Springer Science+Business Media, Inc. 297