Physical Oceanography, Vol. 16, No. 4, 2006
MATHEMATICAL MODELING OF MARINE SYSTEMS
SPECTRAL BAROTROPIC MODEL FOR THE PREDICTION OF SYNOPTIC
CURRENTS IN THE OPEN PART OF THE OCEAN
I. I. Karpatovich
We consider a quasigeostrophic spectral model used for the prediction of synoptic currents in the
barotropic ocean. The spectral method is based on the expansion of the current function in a
double series in cosines. An algorithm of numerical evaluation of the nonlinear term in the equa-
tion of potential eddy is described in detail.
The meteorological and oceanological problems are, as a rule, solved by using the finite-difference and
spectral methods of integration of the corresponding equations. The principal advantage of the finite-difference
method is a high rate of numerical calculations and its serious drawback is connected significant errors in the
evaluation of the phase and group velocities. At the same time, the spectral method is characterized by a much
higher accuracy of determination of the space derivatives. Its disadvantages are mainly connected with the diffi-
culties encountered in the numerical realization of boundary conditions and low rates of numerical calculations.
In the oceanological studies, the spectral method is used, as a rule, in the model problems with periodic boundary
conditions and expansion of the sought functions in series in complex trigonometric functions. The series in
these functions are rapidly convergent and, in addition, admit the realization of fast algorithms of calculations.
However, the periodic boundary conditions do not reflect the actual processes running in the ocean with the re-
quired degree of completeness. Therefore, in what follows, we show that the spectral technique of integration
can also be used in the case of nonperiodic boundary conditions if the sought functions are expanded in series in
Among the existing spectral models of the dynamics of the ocean, we especially mention the models de-
scribed in [1, 2]. The main difficulty encountered in the application of the spectral method in bounded regions is
connected with the numerical realization of boundary conditions. In the present work, this problem is solved by
introducing tangential velocities on the boundary of the region of integration.
Statement of the Problem
The equation of a quasigeostrophic potential eddy (with regard for the friction forces and the action of the
wind) has the following form:
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
62–69, July–August, 2006. Original article submitted September 8,
2003; revision submitted April 21, 2005.
0928-5105/06/1604–0247 © 2006 Springer Science+Business Media, Inc. 247