Purpose. The study is purposed at deriving the discrete equations of absolute and potential vorticity for a three-dimensional stratified incompressible fluid as an exact consequence of the finite-difference equations of sea dynamics in the field of a potential mass force in the adiabatic approximation provided that viscosity and diffusion are absent. The properties of two-dimensional projections of the absolute vorticity equation onto coordinate planes and the three-dimensional potential vorticity equation are analyzed.

Methods and results. In order to determine the discrete analogues of absolute and potential vorticity, an additional grid is introduced, where the finite-difference equations for the components both of absolute and potential vorticity are written down. Two-dimensional analogues of the three-dimensional equation of absolute vorticity on the planes (x, y), (y, z) and (x, z) are obtained; they possess the feature of preserving vorticity, energy and enstrophy (square of vorticity). A discrete equation for potential vorticity of a stratified incompressible fluid is derived from the finite-difference system of three-dimensional equations of sea dynamics in the adiabatic approximation at the absence of viscosity and diffusion.

Conclusions. In the case of a linear equation of state, the discrete equations of absolute vorticity and potential vorticity which are the exact consequence of finite-difference formulation are obtained. The equation of potential vorticity is of a divergent form, and two-dimensional analogues of the absolute vorticity equation on the planes (x, y), (y, z) and (x, z) have two quadratic invariants that provide preservation of the average wave number.