Finite-Difference Approximation of the Potential Vorticity Equation for a Stratified Incompressible Fluid and an Example of its Application for Modeling the Black Sea Circulation. Part I. Finite-Difference Equation of Potential Vorticity of Ideal Fluid

S. G. Demyshev

Marine Hydrophysical Institute of RAS, Sevastopol, Russian Federation

e-mail: demyshev@gmail.com

Abstract

Purpose. The study is purposed at deriving the discrete equations of absolute and potential vorticity for a three-dimensional stratified incompressible fluid as an exact consequence of the finite-difference equations of sea dynamics in the field of a potential mass force in the adiabatic approximation provided that viscosity and diffusion are absent. The properties of two-dimensional projections of the absolute vorticity equation onto coordinate planes and the three-dimensional potential vorticity equation are analyzed.

Methods and Results. In order to determine the discrete analogues of absolute and potential vorticity, an additional grid is introduced, where the finite-difference equations for the components both of absolute and potential vorticity are written down. Two-dimensional analogues of the three-dimensional equation of absolute vorticity on the planes (x, y), (y, z) and (x, z) are obtained; they possess the feature of preserving vorticity, energy and enstrophy (square of vorticity). A discrete equation for potential vorticity of a stratified incompressible fluid is derived from the finite-difference system of three-dimensional equations of sea dynamics in the adiabatic approximation at the absence of viscosity and diffusion.

Conclusions. In the case of a linear equation of state, the discrete equations of absolute vorticity and potential vorticity which are the exact consequence of finite-difference formulation are obtained. The equation of potential vorticity is of a divergent form, and two-dimensional analogues of the absolute vorticity equation on the planes (x, y), (y, z) and (x, z) have two quadratic invariants that provide preservation of the average wave number.

Keywords

discrete equation, dynamics of sea, kinetic energy, vortex, potential vorticity, Ertel invariant

Acknowledgements

The study was carried out with financial support of the Russian Science Foundation grant 23-27-00141.

Original russian text

Original Russian Text © S. G. Demyshev, 2024, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 40, Iss. 2, pp. 165–179 (2024)

For citation

Demyshev, S.G., 2024. Finite-Difference Approximation of the Potential Vorticity Equation for a Stratified Incompressible Fluid and an Example of its Application for Modeling the Black Sea Circulation. Part I. Finite-Difference Equation of Potential Vorticity of Ideal Fluid. Physical Oceanography, 31(2), pp. 149-160.

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