Intermediate Models of Planetary Circulations in the Atmosphere and Ocean

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  • 1 Center for Atmospheric Research, Boulder, CO 80307
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Abstract

Large-scale extratropical motions (with dimensions comparable to, or somewhat smaller than, the planetary radius) in the atmosphere and ocean exhibit a more restricted range of phenomena than are admissible in the primitive equations for fluid motions, and there have been many previous proposals for simpler, more phenomenologically limited models of these motions. The oldest and most successful of these is the quasi-geostrophic model. An extensive discussion is made of models intermediate between the quasi-geostrophic and primitive ones, some of which have been previously proposed [e.g., the balance equations (BE), where tendencies in the equation for the divergent component of velocity are neglected, or the geostrophic momentum approximation (GM), where ageostrophic accelerations are neglected relative to geostrophic ones] and some of which are derived here. Virtues of these models are assessed in the dual measure of nearly geostrophic momentum balance (i.e., small Rossby number) and approximate frontal structure (i.e., larger along-axis velocities and length scales than their cross-axis counterparts), since one or both of these circumstances is usually characteristic of planetary motions. Consideration is also given to various coordinate transformations, since they can yield simpler expressions for the governing differential equations of the intermediate models. In particular, a new set of coordinates is proposed, isentropic geostrophic coordinates,(IGC), which has the advantage of making implicit the advections due to ageostrophic horizontal and vertical velocities under various approximations. A generalization of quasi-geostrophy is made. named hypo-geostrophy (HG), which is an asymptotic approximation of one higher order accuracy in Rossby number. The governing equations are simplest in IGC for both HG and GM; we name the latter in these coordinates isentropic semi-geostrophy (ISG), in analogy to Hoskins’ (1975) semi-geostrophy (SG). HG, GM and BE are, in our opinion, the three most valuable intermediate models for future consideration. HG and BE are superior to GM asymptotically in small Rossby number, but HG in IGC and GM are superior to HG in other coordinates and BE in frontal asymptotics. GM has global (not asymptotic) integral invariants of energy and enstrophy, which HG lacks, and this may assure physically better solutions in weakly asymptotic situations. BE has one global (energy) and one asymptotic (enstrophy) invariant. BE has difficulties of solution existence and uniqueness. Further progress in the search for intermediate models requires obtaining an extensive set of solutions for these models for comparison with quasi-geostrophic and primitive equation solutions.

Abstract

Large-scale extratropical motions (with dimensions comparable to, or somewhat smaller than, the planetary radius) in the atmosphere and ocean exhibit a more restricted range of phenomena than are admissible in the primitive equations for fluid motions, and there have been many previous proposals for simpler, more phenomenologically limited models of these motions. The oldest and most successful of these is the quasi-geostrophic model. An extensive discussion is made of models intermediate between the quasi-geostrophic and primitive ones, some of which have been previously proposed [e.g., the balance equations (BE), where tendencies in the equation for the divergent component of velocity are neglected, or the geostrophic momentum approximation (GM), where ageostrophic accelerations are neglected relative to geostrophic ones] and some of which are derived here. Virtues of these models are assessed in the dual measure of nearly geostrophic momentum balance (i.e., small Rossby number) and approximate frontal structure (i.e., larger along-axis velocities and length scales than their cross-axis counterparts), since one or both of these circumstances is usually characteristic of planetary motions. Consideration is also given to various coordinate transformations, since they can yield simpler expressions for the governing differential equations of the intermediate models. In particular, a new set of coordinates is proposed, isentropic geostrophic coordinates,(IGC), which has the advantage of making implicit the advections due to ageostrophic horizontal and vertical velocities under various approximations. A generalization of quasi-geostrophy is made. named hypo-geostrophy (HG), which is an asymptotic approximation of one higher order accuracy in Rossby number. The governing equations are simplest in IGC for both HG and GM; we name the latter in these coordinates isentropic semi-geostrophy (ISG), in analogy to Hoskins’ (1975) semi-geostrophy (SG). HG, GM and BE are, in our opinion, the three most valuable intermediate models for future consideration. HG and BE are superior to GM asymptotically in small Rossby number, but HG in IGC and GM are superior to HG in other coordinates and BE in frontal asymptotics. GM has global (not asymptotic) integral invariants of energy and enstrophy, which HG lacks, and this may assure physically better solutions in weakly asymptotic situations. BE has one global (energy) and one asymptotic (enstrophy) invariant. BE has difficulties of solution existence and uniqueness. Further progress in the search for intermediate models requires obtaining an extensive set of solutions for these models for comparison with quasi-geostrophic and primitive equation solutions.

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