Purpose. The study is purposed at deriving a finite-difference equation of potential vorticity for a three-dimensional baroclinic fluid with regard for diffusion and viscosity in a quasi-static approximation. Its terms are calculated and analyzed in numerical modeling of the Black Sea circulation for two periods – winter and summer 2011.

Methods and Results. A finite-difference equation for the potential vorticity of a stratified incompressible fluid is obtained for a system of discrete equations of sea dynamics in the hydrostatic approximation allowing for viscosity, diffusion, river inflow, water exchange through the straits and atmospheric forcing. It is shown that the main contribution to the potential vorticity is made by its vertical component. The horizontal components are predominant in the areas of river inflow and water exchange through the straits. The vertical component of potential vorticity, except for the river inflow zones, is conditioned by the value and structure of an absolute eddy. The main contribution in the sea upper layer of the coastal region, its northwestern part and along the Anatolian coast is made by the advection of potential vorticity. At the lower horizons, its highest values are observed in the coastal strip, at that its character is more pronounced near the southern coast of the sea.

Conclusions. Analysis of the potential vorticity equation has shown that the value of the advective terms is conditioned by the divergence of the product of nonlinear terms in the motion equations and density gradient. The main conclusion consists in the following: locally, the sum of vertical and horizontal advection of potential vorticity is two orders of magnitude less than each of them separately.