ITERATED GEOSTROPHIC INTERMEDIATE MODELS

Intermediate models contain physics between that in the primitive equa tions and that in the quasigeostrophic equations and are capable of re presenting subinertial frequency motion over O(1) topographic variatio ns typical of the continental slope while filtering out high-frequency gravity-inertial waves. We present here a formulation for stratified flow of a set of new intermediate models, termed iterated geostrophic (IG) models, derived under the assumption that the Rossby number epsil on is small. We consider the motion of a rotating, continuously strati fied fluid governed by the hydrostatic, Boussinesq, adiabatic primitiv e equations (PE) with a spatially variable Coriolis parameter and with weak biharmonic momentum diffusion. The IG models utilize the pressur e field as the basic variable [as in the quasigeostrophic (QG) approxi mation], are capable of providing solutions of formally increasing acc uracy in powers of epsilon in a systematic manner, and are straightfor ward to solve numerically. The IG models are obtained by iteration, at a fixed time t = t0, of the momentum and thermodynamic equations usin g the known pressure field phi(x, t). The iteration procedure produces a sequence of estimates of increasing accuracy for the velocity compo nents and for the time derivative of the pressure field phi(t)(x, t0). The formulation is asymptotic in the sense that, given the pressure f ield, at each iteration the velocity components and phi(t)(x, t0) are formally determined to a higher order of accuracy in powers of epsilon . The order of the IG model is specified by the predetermined fixed nu mber of iterations N. Thus, a set of models is produced depending on t he choice for N, and the different models are denoted by IGN. The valu e of phi(t)(x, t0) obtained from iteration N is used with a time diffe rence scheme to advance the pressure field in time, and the process ma y be repeated. Energy and potential enstrophy conservation in the IG m odels are asymptotic. In the following companion paper (Allen and Newb erger), the accuracies of several intermediate models, including IG2 a nd IG3, are investigated by a comparison of numerical finite-differenc e solutions to those of the primitive equations. For moderate Rossby n umber flows, it is found that IG2 gives approximate solutions of reaso nable accuracy, with errors substantially smaller than those obtained from QG and several other intermediate models. The IG3 model is found to give extremely accurate approximate solutions for flows with Rossby numbers that range from moderately small to O(1).