Verification of the Cabaret schemes for the shallow water equations based on the conservation of angular momentum

Abstract

To solve the problems of large-scale dynamics of the atmosphere and the ocean, a system of single-layer two-dimensional shallow water equations on a sphere is often used [1, 2]. Such a system is the basis for solving complete baroclinic systems of prognostic equations and is used to assess the accuracy and efficiency of computational algorithms for solving direct and inverse problems. Important issues here are the choice of a suitable form of writing differential equations and effective algorithms for their numerical solution.

In [3], instead of using the equation for the conservation of momentum in a spherical coordinate system, the authors proposed a technique based on the use of the equation for the conservation of angular momentum in a Cartesian coordinate system. This technique allows us to get rid of the degeneracy of the coordinate system near the poles, while preserving only two equations of motion. The semi-discrete finite-volume scheme written in [3] should be supplemented with a method for determining flows on the edges of a grid. The authors used the Cabaret technique [4] for this purpose, which ensures time reversibility on flows where the characteristics of one family do not intersect.

This paper presents the original shallow-water equations in an integral form, a semi-discrete finite-volume scheme, and a brief description of the closure of this scheme using the Cabaret technique. The main attention is paid to the verification of the scheme on a series of traditional tests.