“Inertial” Oscillations and the β-Plane Approximation(s)

P. Ripa Centro de Investigación Científica y de Educación Superior de Ensenada, Ensenada, México

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Abstract

The general solution for the motion of a particle in the frictionless surface of a rotating planet is reviewed and a physical explanation of asymptotic solutions is provided. In the rotating frame at low energies there is a well-known quasi-circular oscillation superimposed to a weak westward drift; the latter is shown to be due to three different contributions, in the relative proportion of 1:tan2ϑ0:− tan2ϑ0, where ϑ0 is the mean latitude. The first contribution is due to the “β effect,” that is, the variation of the Coriolis parameter with latitude. The other two contributions are geometric effects, due to the tendency to move along a great circle and the change of the distance to the rotation axis with the latitude. The mean zonal velocity is produced by the first two contributions, and therefore is underestimated by the classical β-plane approximation [by a factor of (1 +tan2 ϑ0)−1 = cos2ϑ0] because of the lack of geometric effects in such a system. Correct first-order approximations are derived and found to belong to a one-parameter family, whose optimum element is obtained.

The key to develop a consistent approximation, with the right conservation laws, is to redefine three geometric coefficients by means of an expansion in a meridional coordinate, up to a fixed order. Making the expansion directly in the equations of motion, as done by other authors, leads to undesirable consequences for the conservation laws. This is true not only for particle dynamics but also for fields, as illustrated with the shallow-water equations. Correct approximations developed for this system are found to have the same integrals of motion as the exact one (angular momentum, energy, volume, and the potential vorticity of any fluid element). In the quasigeostrophic approximation, the geometric corrections cancel out in the potential vorticity law, and therefore the classical β plane gives the right prognostic equation, even though the (diagnostic) momentum equations are incorrect.

Corresponding author address: Dr. Pedro Ripa, CICESE, Km 107 Carretera Tijuana-Ensenada, (22800) Ensenada, BC, Mexico.

Email: ripa@cicese.mx

Abstract

The general solution for the motion of a particle in the frictionless surface of a rotating planet is reviewed and a physical explanation of asymptotic solutions is provided. In the rotating frame at low energies there is a well-known quasi-circular oscillation superimposed to a weak westward drift; the latter is shown to be due to three different contributions, in the relative proportion of 1:tan2ϑ0:− tan2ϑ0, where ϑ0 is the mean latitude. The first contribution is due to the “β effect,” that is, the variation of the Coriolis parameter with latitude. The other two contributions are geometric effects, due to the tendency to move along a great circle and the change of the distance to the rotation axis with the latitude. The mean zonal velocity is produced by the first two contributions, and therefore is underestimated by the classical β-plane approximation [by a factor of (1 +tan2 ϑ0)−1 = cos2ϑ0] because of the lack of geometric effects in such a system. Correct first-order approximations are derived and found to belong to a one-parameter family, whose optimum element is obtained.

The key to develop a consistent approximation, with the right conservation laws, is to redefine three geometric coefficients by means of an expansion in a meridional coordinate, up to a fixed order. Making the expansion directly in the equations of motion, as done by other authors, leads to undesirable consequences for the conservation laws. This is true not only for particle dynamics but also for fields, as illustrated with the shallow-water equations. Correct approximations developed for this system are found to have the same integrals of motion as the exact one (angular momentum, energy, volume, and the potential vorticity of any fluid element). In the quasigeostrophic approximation, the geometric corrections cancel out in the potential vorticity law, and therefore the classical β plane gives the right prognostic equation, even though the (diagnostic) momentum equations are incorrect.

Corresponding author address: Dr. Pedro Ripa, CICESE, Km 107 Carretera Tijuana-Ensenada, (22800) Ensenada, BC, Mexico.

Email: ripa@cicese.mx

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