### Abstract:

Intermediate models contain physics between that in the primitive equations and that in the quasigeostrophic equations and are capable of representing subinertial frequency motion over O(1) topographic variations 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 ϵ is small. We consider the emotion of a rotating, continuously stratified fluid governed by the hydrostatic, Boussinesq, adiabatic primitive equations (PE) with a spatially variable Coriolis parameter and with weak biharmonic momentum diffusion. The IG models utilize the pressure field as the basic variable [as in the quasigeostrophic (QG) approximation], are capable of providing solutions of formally increasing accuracy in powers of ϵ in a systematic manner, and are straightforward to solve numerically. The 1G models are obtained by iteration, at a fixed time t = t₀, of the momentum and thermodynamic equations using the known pressure field ϕ(x, t). The iteration procedure products a sequence of estimates of increasing accuracy for the velocity components and for the time derivative of the pressure field ϕt(x, t₀). The formulation is asymptotic in the sense that, given the pressure field, at each iteration the velocity components and ϕ₁(x, t₀) are formally determined to a higher order of accuracy in powers of ϵ. The order of the IG model is specified by the predetermined fixed number of iterations N. Thus, a set of models is produced depending on the choice for N, and the different models are denoted by IGN. The value of ϕ₁(x, t₀) obtained from iteration N is used with a time difference scheme to advance the pressure field in time, and the process may be repeated. Energy and potential enstrophy conservation in the IG models are asymptotic. In the following companion paper (Allen and Newberger), the accuracies of several intermediate models, including IG2 and IG3, are investigated by a comparison of numerical finite-difference solutions to those of the primitive equations. For moderate Rossby number flows, it is found that IG2 gives approximate solutions of reasonable 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).