In MITgcm we relax the hydrostatic balance approximation without incurring a prohibitive increase in computational cost. The vertical component of velocity is predicted rather than diagnosed.

MITgcm supports both hydrostatic, quasi-hydrostatic and non-hydrostatic algorithms.

Inclusion of the non-hydrostatic terms means that MITgcm can properly represent processes that involve overturning of fluid parcels leading to static instability. Examples of such processes include deep convection and Kelvin-Helmholtz instability processes that both appear in this simulation of a dense gravity plume.

Look here for more details on the non-hydrostatic capability of MITgcm.

The papers describing the non-hydrostatic algorithm are here:

Marshall, J., C. Hill, L. Perelman, and A. Adcroft, (1997)
Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling, 
J. Geophysical Res., 102(C3), 5733-5752.