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Vorticity Coordinates, Transformed Primitive Equations, and a Canonical Form for Balanced Models

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  • 1 Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
  • | 2 Department of Applied Mathematics and Theoretical Physics, Centre for Atmospheric Science, Cambridge, United Kingdom
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Abstract

A potential pseudodensity principle is derived for the quasi-static primitive equations on the sphere. An important step in the derivation of this principle is the introduction of “vorticity coordinates”—that is, new coordinates whose Jacobian with respect to the original spherical coordinates is the dimensionless absolute isentropic vorticity. The vorticity coordinates are closely related to Clebsch variables and are the primitive equation generalizations of the geostrophic coordinates used in semigeostrophic theory. The vorticity coordinates can be used to transform the primitive equations into a canonical form. This form is mathematically similar to the geostrophic relation. There is flexibility in the choice of the potential function appearing in the canonical momentum equations. This flexibility can be used to force the vorticity coordinates to move with some desired velocity, which results in an associated simplification of the material derivative operator. The end result is analogous to the way ageostrophic motions become implicit when geostrophic coordinates are used in semigeostrophic theory.

Abstract

A potential pseudodensity principle is derived for the quasi-static primitive equations on the sphere. An important step in the derivation of this principle is the introduction of “vorticity coordinates”—that is, new coordinates whose Jacobian with respect to the original spherical coordinates is the dimensionless absolute isentropic vorticity. The vorticity coordinates are closely related to Clebsch variables and are the primitive equation generalizations of the geostrophic coordinates used in semigeostrophic theory. The vorticity coordinates can be used to transform the primitive equations into a canonical form. This form is mathematically similar to the geostrophic relation. There is flexibility in the choice of the potential function appearing in the canonical momentum equations. This flexibility can be used to force the vorticity coordinates to move with some desired velocity, which results in an associated simplification of the material derivative operator. The end result is analogous to the way ageostrophic motions become implicit when geostrophic coordinates are used in semigeostrophic theory.

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