Angular Momentum of β-Plane Flows

Joseph Egger Meteorologisches Institut, Universität München, Munich, Germany

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Abstract

The prognostic equations for the total angular momentum vector M are derived for f- and β-plane geometries and compared to those of spherical models. It is shown that the omission of the centrifugal effects and the corresponding adjustment of gravity in atmospheric f- (β-) plane models imply that a torque is exerted in analogy to the spherical case where this torque is caused by the nonspherical shape of the earth. For hydrostatic flow on the f- (β-) plane, it is only for the vertical component Mz of angular momentum that a prognostic equation can be derived. If the traditional approximation is introduced, Mz becomes a conserved quantity on the f plane in the absence of orographic and frictional torques while the corresponding component z on the sphere is not conserved. The prognostic equation for Mz on the β plane is an approximation to that on the sphere at least for nondivergent flow. The f- (β-) plane equations for the horizontal components of M deviate substantially from those valid on the sphere in the nonhydrostatic case.

Numerical integrations of the shallow water equations are performed in order to illustrate these points. The total angular momentum is evaluated for localized flow structures. It is found that the β-plane model captures the most important characteristics of the corresponding changes of z on the sphere at least for short times and for initially geostrophic flows. Moreover, z is reasonably well conserved for isolated flow structures of small scale as suited for the f plane.

Corresponding author address: Dr. Joseph Egger, Meteorologisches Institut, Universität München, Theresienstr. 37, 80333 München, Germany. Email: j.egger@lrz.uni-muenchen.de

Abstract

The prognostic equations for the total angular momentum vector M are derived for f- and β-plane geometries and compared to those of spherical models. It is shown that the omission of the centrifugal effects and the corresponding adjustment of gravity in atmospheric f- (β-) plane models imply that a torque is exerted in analogy to the spherical case where this torque is caused by the nonspherical shape of the earth. For hydrostatic flow on the f- (β-) plane, it is only for the vertical component Mz of angular momentum that a prognostic equation can be derived. If the traditional approximation is introduced, Mz becomes a conserved quantity on the f plane in the absence of orographic and frictional torques while the corresponding component z on the sphere is not conserved. The prognostic equation for Mz on the β plane is an approximation to that on the sphere at least for nondivergent flow. The f- (β-) plane equations for the horizontal components of M deviate substantially from those valid on the sphere in the nonhydrostatic case.

Numerical integrations of the shallow water equations are performed in order to illustrate these points. The total angular momentum is evaluated for localized flow structures. It is found that the β-plane model captures the most important characteristics of the corresponding changes of z on the sphere at least for short times and for initially geostrophic flows. Moreover, z is reasonably well conserved for isolated flow structures of small scale as suited for the f plane.

Corresponding author address: Dr. Joseph Egger, Meteorologisches Institut, Universität München, Theresienstr. 37, 80333 München, Germany. Email: j.egger@lrz.uni-muenchen.de

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