Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid

A second order numerical method for the primitive equations(PEs) of large-scale oceanic flow formulated in mean vorticity is proposed andanalyzed, and the full convergence in L² is established. In the reformulation ofthe PEs, the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative ofthe horizontal velocity. The total velocity field (both horizontal and vertical)is statically determined by differential equations at each fixed horizontal point.The standard centered difference approximation is applied to the prognosticequations and the determination of numerical values for the total velocity fieldis implemented by FFT-based solvers. Stability of such solvers are establishedand the convergence analysis for the whole scheme is provided in detail.