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- Author or Editor: Peter R. Gent x
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Abstract
It is shown that the very frequently used form of the viscous, diabatic shallow-water equations are energetically inconsistent compared to the primitive equations. An energetically consistent form of the shallow-water equations is then given and justified in terms of isopycnal coordinates. Examples are given of the energetically inconsistent shallow-water equations used in low-order dynamical systems and simplified coupled models of tropical air–sea interaction and the E1 Niño–Southern Oscillation phenomena.
Abstract
It is shown that the very frequently used form of the viscous, diabatic shallow-water equations are energetically inconsistent compared to the primitive equations. An energetically consistent form of the shallow-water equations is then given and justified in terms of isopycnal coordinates. Examples are given of the energetically inconsistent shallow-water equations used in low-order dynamical systems and simplified coupled models of tropical air–sea interaction and the E1 Niño–Southern Oscillation phenomena.
Abstract
The low-order, nine-component, primitive equation model of Lorenz (1980) is used as the basis for a comparative study of the quality of several intermediate models. All the models are intermediate between the primitive equations and quasi-geostrophy and will not support gravity-wave oscillations; this reduces to three the number of independent components in each. Strange attractors, stable limit cycles, and stable and unstable fixed points are found in the models. They are used to make a quantitative intercomparison of model performance as the forcing strength, or equivalently the Rossby number, is varied. The models can be ranked from best to worst at small Rossby number as follows: the primitive equations, the balance equations, hypogeostrophy, geostrophic momentum approximation, the linear balance equations, and quasi-geostrophy. At intermediate Rossby number the only change in this ranking is the demotion of hypogeostrophy to the position of worst. Caveats about the low-order model, and hence the generality of the conclusions, are also discussed.
Abstract
The low-order, nine-component, primitive equation model of Lorenz (1980) is used as the basis for a comparative study of the quality of several intermediate models. All the models are intermediate between the primitive equations and quasi-geostrophy and will not support gravity-wave oscillations; this reduces to three the number of independent components in each. Strange attractors, stable limit cycles, and stable and unstable fixed points are found in the models. They are used to make a quantitative intercomparison of model performance as the forcing strength, or equivalently the Rossby number, is varied. The models can be ranked from best to worst at small Rossby number as follows: the primitive equations, the balance equations, hypogeostrophy, geostrophic momentum approximation, the linear balance equations, and quasi-geostrophy. At intermediate Rossby number the only change in this ranking is the demotion of hypogeostrophy to the position of worst. Caveats about the low-order model, and hence the generality of the conclusions, are also discussed.
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.
Abstract
Strongly curved fronts occur when the ratios of cross-front to alongfront horizontal velocities and scales are comparably small for an order one distance in the alongfront direction. This means that all three velocity terms in the continuity equation contribute at leading order. The balance equations are shown to be formally accurate through two orders in powers of the small parameter in this situation; however, the semigeostrophic equations are accurate only at leading order. Both approximate models are accurate through two orders of magnitude if the front is more weakly curved.
Abstract
Strongly curved fronts occur when the ratios of cross-front to alongfront horizontal velocities and scales are comparably small for an order one distance in the alongfront direction. This means that all three velocity terms in the continuity equation contribute at leading order. The balance equations are shown to be formally accurate through two orders in powers of the small parameter in this situation; however, the semigeostrophic equations are accurate only at leading order. Both approximate models are accurate through two orders of magnitude if the front is more weakly curved.
