## Abstract

Results of a variety of numerical simulations are presented and the accuracy of quasigeostrophic (QG) and semigeostrophic (SG) vertical velocity estimates of the total vertical velocity is analyzed. The authors examine the dependence of the results on the potential vorticity (PV) anomaly of the flow, its time evolution, and the amount of numerical diffusion. The SG *ω* equation is solved in a novel way in the original physical coordinates rather than in geostrophic coordinates. A three-dimensional numerical model is used that *explicitly* conserves the PV on isopycnal surfaces through a contour-advective semi-Lagrangian (CASL) algorithm. The numerical simulations consist initially of one or two horizontal cylinders of anomalous PV: a shear zone that induces two or three counterflowing jets. These jets destabilize and break into cyclones or anticyclones. This is accompanied by enhanced vertical motion, which exhibits a dominantly balanced quadrupole pattern in horizontal cross sections, depending on the ellipticity of the gyres, together with weak second-order inertia–gravity waves. For flows containing only negative PV anomalies the magnitude of both the QG and SG vertical velocities are smaller than the magnitude of the total vertical velocity, while the opposite occurs for flows containing only positive PV anomalies. The reason for this behavior is that the QG *ω* equation misses a term proportional to the Laplacian of the horizontal velocity. A new, more accurate, *ω* equation is proposed to recover the vertical velocity when both experimental density and horizontal velocity data are available. The SG solution is nearly always more accurate than the QG solution, particularly for the largest vertical velocity values and when the flow has single-signed PV anomalies. For flows containing both positive and negative PV anomalies, for example, mushroomlike eddies, the QG vertical velocity is a better approximation to the total vertical velocity than the SG solution. The reason for this anomalous behavior lies in one additional assumption concerning the conservation of volume that is usually adopted to derive the SG *ω* equation.

*Corresponding author address:* Dr. Álvaro Viúdez, Institut de Ciències del Mar, P. Marítim 37, 08003-Barcelona, Spain. Email: aviudez@cmima.csic.es