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Editorial Type:
Article
Article Type:
Research Article

Effects of the Earth’s Curvature on the Dynamics of Isolated Objects. Part I: The Disk

P. Ripa1
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  • 1 CICESE, Ensenada, Mexico
Print Publication:
01 Aug 2000
DOI:
https://doi.org/10.1175/1520-0485(2000)030<2072:EOTESC>2.0.CO;2
Page(s):
2072–2087
Restricted access

Abstract

A disk over the frictionless surface of the earth shows an interaction between the center of mass and internal motions. At low energies, the former is an “inertial oscillation” superimposed to a uniform zonal drift c and the latter is a rotation with variable vertical angular velocity ω (as measured by a terrestrial observer).

The dynamics is understood best in a stereographic frame following the secular drift. The center of mass has a circular but not uniform motion; its meridional displacement induces the variations of the orbital and internal rotation rates. On the other hand, the temporal mean of the Coriolis forces due to both rotations produces the secular drift.

In spherical terrestrial coordinates geometric distortion complicates the description. For instance, the zonal velocity of the center of mass U is not equal to the average zonal component of the particle velocities 〈u〉, as a result of the earth’s curvature. The drift c and the temporal means U and 〈u〉 are all three different. In addition, ω differs from the local vertical angular velocity σ (as measured by an observer following the disk). The classical“β plane” approximation predicts correctly the value of c but makes order-one errors in everything else (e.g., it makes U = 〈u〉 = c and ω = σ).

The results of this paper set up the basis to study curvature effects on an isolated vortex. This, more difficult, problem is discussed in Part II.

Corresponding author address: Dr. Pedro Ripa, CICESE, P.O. Box 434844, San Diego, CA 92143-4844.

Email: ripa@cicese.mx

Abstract

A disk over the frictionless surface of the earth shows an interaction between the center of mass and internal motions. At low energies, the former is an “inertial oscillation” superimposed to a uniform zonal drift c and the latter is a rotation with variable vertical angular velocity ω (as measured by a terrestrial observer).

The dynamics is understood best in a stereographic frame following the secular drift. The center of mass has a circular but not uniform motion; its meridional displacement induces the variations of the orbital and internal rotation rates. On the other hand, the temporal mean of the Coriolis forces due to both rotations produces the secular drift.

In spherical terrestrial coordinates geometric distortion complicates the description. For instance, the zonal velocity of the center of mass U is not equal to the average zonal component of the particle velocities 〈u〉, as a result of the earth’s curvature. The drift c and the temporal means U and 〈u〉 are all three different. In addition, ω differs from the local vertical angular velocity σ (as measured by an observer following the disk). The classical“β plane” approximation predicts correctly the value of c but makes order-one errors in everything else (e.g., it makes U = 〈u〉 = c and ω = σ).

The results of this paper set up the basis to study curvature effects on an isolated vortex. This, more difficult, problem is discussed in Part II.

Corresponding author address: Dr. Pedro Ripa, CICESE, P.O. Box 434844, San Diego, CA 92143-4844.

Email: ripa@cicese.mx

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