Arakawa–Lamb Scheme in Application to Stratified Incompressible Fluid in the Absence of Friction

S. G. Demyshev

Marine Hydrophysical Institute of RAS, Sevastopol, Russian Federation

e-mail: demyshev@gmail.com

Abstract

Purpose. The study is aimed at generalizing the Arakawa–Lamb scheme for discrete equations of the horizontal components of three-dimensional absolute vorticity of an ideal fluid and analyzing its features.

Methods and Results. To derive the finite-difference three-dimensional equations of absolute vorticity, a grid containing more unknowns than equations is applied, that permits obtaining the discrete motion equations which, in their turn, yield the equation for absolute vorticity. The resulting expression is presented in the form of three terms reflecting different features of the discrete equations. The first term provides the fulfillment of the energy conservation law for discrete statement, the second term represents the presence of two quadratic invariants for a divergence-free flow, the addition of the third term results in the Arakawa–Lamb scheme under the shallow water approximation. It follows from the presented expression that the second and third terms, which have no analogues in the continuous statement, can be interpreted as a zero approximation with the second order of accuracy. Thus, selection of these expressions makes it possible to construct the schemes with the required features of the conservation laws.

Conclusions. The presented form of the discrete equation for three-dimensional absolute vorticity enables the construction of schemes with the desired features. The difference equations for the horizontal components of absolute vorticity are derived, which possess two quadratic invariants.

Keywords

Arakawa-Lamb scheme, discrete equations of model, sea dynamics, kinetic energy, absolute vortex, quadratic invariants

Acknowledgements

The study was carried out as part of the state assignment of FSBSI FRC MHI under theme FNNN- 2021-0004.

Original russian text

Original Russian Text © S. G. Demyshev, 2025, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 41, Iss. 3, pp. 346–357 (2025)

For citation

Demyshev, S.G., 2025. Arakawa–Lamb Scheme in Application to Stratified Incompressible Fluid in the Absence of Friction. Physical Oceanography, 32(3), pp. 361-371.

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