Physical Oceanography, Vol. 15, No. 5, 2005
MATHEMATICAL MODELING OF MARINE SYSTEM
NUMERICAL EXPERIMENTS AIMED AT THE COMPARISON OF TWO
FINITE-DIFFERENCE SCHEMES FOR THE EQUATIONS OF MOTION
IN A DISCRETE MODEL OF HYDRODYNAMICS OF THE BLACK SEA
S. G. Demyshev
By using the box method and the method of indefinite coefficients, we perform the analysis of fi-
nite-difference schemes with two invariants for the equations of motion in the barotropic appro-
ximation. We establish a functional dependence, which makes it possible to construct difference
schemes with the properties of antisymmetry of the solution and conservation of energy and/or
enstrophy. Two approximations of nonlinear terms in the equations of motion are analyzed. The
first of these approximations guarantees the conservation of two quadratic invariants for the bar-
otropic divergent motion. In the second approximation, the same is true for the barotropic non-
divergent motion. The results of prognostic experiments
of model time) demonstrate that
the first scheme gives a more accurate quantitative description of the well-known physical fea-
tures of large-scale circulation in the Black Sea.
For the adequate reconstruction of circulation in the sea, it is necessary to be able to give correct description
of the energy exchange between motions on different space scales. As shown in the theory of differential equa-
tions [1, 2], there exists a one-to-one correspondence between the conservation laws and the properties of solu-
tions of hydrodynamic-type systems. Hence, it is possible to assume that the best approximation of the proper-
ties of solutions of differential problems can be attained in the case where the conservation laws are obeyed in
the numerical model (to within the rounding errors).
For the equations of advection and heat and/or salt diffusion, an approximation guaranteeing the conserva-
tion of the first and higher moments was proposed by Lorenz . For the nonlinear equations of motion in the
barotropic approximation, difference analogs guaranteeing the validity of the laws of conservation of the total
energy and enstrophy (conservative schemes) were obtained. A finite-difference scheme with two quadratic in-
variants (energy and potential enstrophy) was proposed for the first time for the analysis of the motion of a baro-
tropic two-dimensional fluid in . In the case of nondivergent flow with constant density, a similar scheme was
obtained for the equations of motion written in the Gromeka–Lamb form on a
grid . Later, this scheme was
successfully used in the atmospheric models of weather forecasting .
The aim of the present work is to analyze different types of conservative schemes for the equations of mo-
tion and perform calibration numerical experiments based on a three-dimensional model of hydrodynamics of the
Black Sea .
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
47–59, September–October, 2005. Original article submitted March 25,
2004; revision submitted July 12, 2004.
0928-5105/05/1505–0299 © 2005 Springer Science+Business Media, Inc. 299