Nonlinear Invariants of a Discrete System of the Sea Dynamics Equations in a Quasi-Static Approximation

Purpose. The study is purposed at obtaining the approximations providing the presence of discrete non-linear invariants for the difference system of the sea dynamics equations in the absence of external forces, friction and diffusion, and at analyzing the features of the resulting schemes at the example of calculating the Black Sea circulation for 2011. Methods and Results. The method of undetermined coefficients at which the new unknowns are introduced is applied, that makes it possible to satisfy the additional conditions. The schemes providing simultaneous preservation of temperature in the first and the K-th (K > 1) degrees and salinity in the first and the L-th (L > 1) degrees, were obtained. The approximations of temperature and salinity found on the box faces with a polynomial dependence of density on temperature and salinity lead to a divergent form of the density advection equation. This form provides fulfilling the law of conservation both of the total energy and the sum of kinetic and dynamic potential energy in a discrete formulation. Based on the analysis of circulation in the Black Sea in 2011, it is shown that at increase of the degree of invariants, the following effects take place: the gradients in the temperature field in the frontal zones as well as the processes of the saltier water upwelling in the sea center and the fresher water downwelling along its periphery are intensified, and the intensity of small-scale features in the vertical velocity field decreases. Conclusions. A discrete dynamical model in a quasi-static approximation was obtained. It has a number of nonlinear invariants corresponding to the continuous problem. The results of calculating the Black Sea circulation for real conditions in 2011 showed that presence of the degree invariants exceeding two made it possible to specify the circulation features on small scales.