Mathematical model of sea dynamics in a σ-coordinate system

The aim of this paper is to formulate a mathematical σ-model of thermohaline sea dynamics and its numerical solution. The novelty of the work is taking account of the nonhydrostatic effect, establishing the law of conservation for a complete nonlinear problem, and the generalization (for the nonhydrostatic case) of the numerical algorithm for solving the problem. The algorithm is based on the method of splitting with respect to physical processes. We describe nonhydrostatic effects at a separate splitting stage and introduce a new function describing the deviation of pressure from the hydrostatic one, which is calculated at an additional splitting stage. The elimination of this stage from the chain of split systems automatically leads to a special model case describing hydrostatic dynamics. Further, main attention is given to the barotropic dynamics problem. We formulate two finite difference algorithms of its solution: the first one by solving a linear hyperbolic system in terms of (u,v,ζ ), the second algorithm by reducing it to the equation ζ .