Computational aspects of calculating vertical velocity in a wind circulation model

Qualitative calculation of the vertical component of the velocity does not lose its relevance even today. Basically, the results obtained are compared by calculations using other models and with a~priori information about the dynamics of waters in the studied area. More accurate testing of calculation methods can be carried out if there is an accurate analytical solution to the problem. For these purposes, a~three-dimensional analytical solution to the problem of wind circulation in a rectangular reservoir with a flat bottom is used for a given wind effect. Analytical expressions for the barotropic and additional three-dimensional components of the velocity field are obtained. The expression for the vertical component of the velocity field in this paper is used for comparison with the values calculated from the continuity equation. The vertical velocity was calculated in various ways: by standard integration of the vertical continuity equation and by the run-through method. In addition, in this work, when integrating the equation for the current function, a family of difference discretizations obtained for solving a similar class of problems is used. As the results of numerical experiments presented in the tables have shown, even with a sufficiently accurate solution of the problem for the current function, the calculation of the horizontal components of the total flow can lead to significant errors if the specifics of the problem are not taken into account. The algorithms used give a fairly good accuracy of reproducing the solution of the equation for the current function. On a fine grid, the error is hundredths of a percent of the norm used. The technique used in this work allows, within the framework of a single approach, to solve the problem for the current function and calculate the derivatives of this solution, which guarantees the accuracy of determining the horizontal components of the total flow. The paper shows that the use of the run-through method in calculating vertical velocity significantly improves the calculation results. The results of the work can be used in modeling dynamic processes in the sea.