An analysis of the vertical velocity field using the full generalized {omega} equation ($\omega$-equation) in a single mesoscale baroclinic oceanic gyre is carried out. The evolution of the gyre over 20 inertial periods is simulated using a new three-dimensional numerical model which directly integrates the horizontal ageostrophic vorticity, explicitly conserves the potential vorticity (PV) via contour advection on isopycnal surfaces, and inverts the nonlinear PV definition via the solution of a three-dimensional Monge–Ampère equation. In this framework the $\omega$-equation comes simply from the horizontal divergence of the horizontal ageostrophic vorticity prognostic equation. The ageostrophic vorticity is written as the Laplacian of a vector potential $\varphib$, from which both the velocity and the density fields are recovered, respectively, from the curl and divergence of $\varphib$. A new initialization technique based on the slow, progressive growth of the PV field during an initial time interval is used to avoid the generation of internal gravity waves during the initialization of the gyre. This method generates a nearly balanced baroclinic gyre for which the influence of internal gravity waves in the mesoscale vertical velocity field is negligible.

The numerical fields obtained are then used to carry out a first numerical analysis of the $\omega$-equation. The analysis shows that, for moderately high Rossby numbers, the local and the advective rates of change of the differential ageostrophic vertical vorticity $(\zeta'_z)$ are of the same order of magnitude as the three largest terms in the $\omega$-equation. There is, however, a large cancellation between these two terms, resulting in the approximate material conservation of $\zeta'_z$. This might explain the ‘over-applicability’ of the quasi-geostrophic (QG) $\omega$-equation for Rossby numbers larger than 0.1. The QG vertical velocity is only 22% smaller than the total vertical velocity for the case studied (having a Rossby number of −0.5).