ON THE USE OF THE BOUSSINESQ EQUATIONS, THE REDUCED SYSTEM, AND THE PRIMITIVE EQUATIONS FOR THE COMPUTATION OF GEOPHYSICAL FLOWS

Motivated in part by the mathematical problems associated with the app lication of open boundary conditions to the hydrostatic primitive equa tions (PE), Browning et al. (1990, Dyn. Atmos. Oceans, 14: 303-332) pr oposed the use of the reduced system (RS) of equations to replace PE f or oceanographic problems. The RS are essentially the Boussinesq equat ions (BO) with the non-hydrostatic terms in the vertical momentum equa tion multiplied by a constant delta(2) much greater than 1. This artif icially alters the physics (e.g, changing the internal-inertial wave p roperties) to facilitate numerical integration, but the changes are as sumed to have negligible effects on the dynamics of interest. We asses s the accuracy and utility of the RS (following the guidelines for the choice of delta) by comparing numerical finite difference solutions o f RS, PE and BO for initial-value problems involving three-dimensional instability of an ocean front and atmospheric frontal development in a two-dimensional Eady wave. Both explicit (BO, PE) and semi-implicit (BOSI, PESI) time-difference schemes are used for the Boussinesq and p rimitive equations. For RS, the same explicit scheme as for BO is used where delta much greater than 1 allows larger time steps than with th e other explicit models. It is found that relative to BO solutions, th e errors for RS are small but increase rapidly and monotonically with increasing delta (over a range consistent with the guidelines) and are greater than the errors for the other models. The use of BOSI allows time steps at least as large as those for RS and results in smaller er rors than RS, For these problems, BOSI is the preferable model to repl ace PE.