Effects of the Earth’s Curvature on the Dynamics of Isolated Objects. Part II: The Uniformly Translating Vortex

P. Ripa CICESE, Ensenada, México

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Abstract

Zonally propagating solutions of the primitive equations for an isolated volume of fluid are considered. In a moving stereographic projection (from the antipode of the center of mass) geometric distortion enters at O(R−2), with R the radius of the earth, whereas planet curvature effects are O(R−1). The imbalance between the centrifugal force and the poleward gravitational force, due to the drift c, is equilibrated by the average Coriolis force, proportional to β. The results are valid for both homogeneous and stratified cases and the lowest-order solution need not be an axisymmetric vortex. The classical β-plane approximation predicts correctly the leading order of c/β, but makes large errors in the O(R−1) term of the vortex structure.

A method is developed to construct the correct O(R−1) term, starting from any steady solution of the f-plane equations, as the O(R0) term. The expansion is exemplified starting with a homogeneous fluid, solid body rotating at an anticyclonic rate −νf0, with 0 < ν < 1. To O(R−1) particle orbits and isobaths belong to different families of nonconcentric circles. A water column moves faster and becomes taller the farther away it is from the equator. In order to keep its potential vorticity, the water column experiences changes of relative vorticity equal to −(2 − ν)/(3 − 3ν) times the variations of the ambient vorticity (Coriolis parameter). The physics of this solution is compared with that of a circular and rigid disk, studied in Part I.

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

Email: ripa@cicese.mx

Abstract

Zonally propagating solutions of the primitive equations for an isolated volume of fluid are considered. In a moving stereographic projection (from the antipode of the center of mass) geometric distortion enters at O(R−2), with R the radius of the earth, whereas planet curvature effects are O(R−1). The imbalance between the centrifugal force and the poleward gravitational force, due to the drift c, is equilibrated by the average Coriolis force, proportional to β. The results are valid for both homogeneous and stratified cases and the lowest-order solution need not be an axisymmetric vortex. The classical β-plane approximation predicts correctly the leading order of c/β, but makes large errors in the O(R−1) term of the vortex structure.

A method is developed to construct the correct O(R−1) term, starting from any steady solution of the f-plane equations, as the O(R0) term. The expansion is exemplified starting with a homogeneous fluid, solid body rotating at an anticyclonic rate −νf0, with 0 < ν < 1. To O(R−1) particle orbits and isobaths belong to different families of nonconcentric circles. A water column moves faster and becomes taller the farther away it is from the equator. In order to keep its potential vorticity, the water column experiences changes of relative vorticity equal to −(2 − ν)/(3 − 3ν) times the variations of the ambient vorticity (Coriolis parameter). The physics of this solution is compared with that of a circular and rigid disk, studied in Part I.

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

Email: ripa@cicese.mx

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