The method of solution employed in the HPE, QH and NH models is summarized in soln_strat. Under all dynamics, a 2-d elliptic equation is first solved to find the surface pressure and the hydrostatic pressure at any level computed from the weight of fluid above. Under HPE and QH dynamics, the horizontal momentum equations are then stepped forward and ṙ found from continuity. Under NH dynamics a 3-d elliptic equation must be solved for the non-hydrostatic pressure before stepping forward the horizontal momentum equations; ṙ is found by stepping forward the vertical momentum equation.
There is no penalty in implementing QH over HPE except, of course, some complication that goes with the inclusion of cos φ Coriolis terms and the relaxation of the shallow atmosphere approximation. But this leads to negligible increase in computation. In NH, in contrast, one additional elliptic equation -a three-dimensional one - must be inverted for pnh. However the ‘overhead’ of the NH model is essentially negligible in the hydrostatic limit (see detailed discussion in Marshall et al. (1997) marshall:97a resulting in a non-hydrostatic algorithm that, in the hydrostatic limit, is as computationally economic as the HPEs.
Basic solution strategy in MITgcm. HPE and QH forms diagnose the vertical velocity, in NH a prognostic equation for the vertical velocity is integrated.