• Ball, F., 1963: Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech.,17, 240–256.

  • Benilov, E. S., 1996: Beta-induced translation of strong isolated eddies. J. Phys. Oceanogr.,26, 2223–2229.

  • Cushman-Roisin, B., 1982: Motion of a free particle on a beta-plane. Geophys. Astrophys. Fluid Dyn.,22, 85–102.

  • Goldstein, H., 1981: Classical Mechanics. Addison-Wesley, 672 pp.

  • Graef, F., 1998: On the westward translation of isolated eddies. J. Phys. Oceanogr.,28, 740–745.

  • Killworth, P. D., 1983: On the motion of isolated lenses on the beta-plane. J. Phys. Oceanogr.,13, 368–376.

  • McDonald, N., 1998: The time-dependent behaviour of a spinning disc on a rotating planet: A model for geophysical vortex motion. Geophys. Astrophys. Fluid Dyn.,87, 253–272.

  • Nof, D., 1981: On the β-induced movement of isolated baroclinic eddies. J. Phys. Oceanogr.,11, 1662–1672.

  • Nycander, J., 1996: Analogy between the drift of planetary vortices and the precession of a spinning body. Plasma Phys. Rep.,22, 771–774.

  • Paldor, N., and P. D. Killworth, 1988: Inertial trajectories on a rotating earth. J. Atmos. Sci.,45, 4013–4019.

  • Ripa, P., 1997: “Inertial” oscillations and the β-plane approximation(s). J. Phys. Oceanogr.,27, 633–647.

  • ——, 2000: Effects of the earth’s curvature on the dynamics of isolated objects. Part II: The uniformly translating vortex. J. Phys. Oceanogr., in press.

  • Stommel, H. M., and D. W. Moore, 1989: An Introduction to the Coriolis Force. Columbia University Press, 297 pp.

  • White, A., 1989: A relationship between energy and angular momentum conservation in dynamical models. J. Atmos. Sci.,46, 1855–1860.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 1 1 1
PDF Downloads 1 1 1

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

View More View Less
  • 1 CICESE, Ensenada, Mexico
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

Save