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Variations on a beta-plane: derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere

Published online by Cambridge University Press:  14 March 2011

PAUL J. DELLAR
PAUL J. DELLAR*
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK
*
†Email address for correspondence: dellar@maths.ox.ac.uk
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Abstract

Starting from Hamilton's principle on a rotating sphere, we derive a series of successively more accurate β-plane approximations. These are Cartesian approximations to motion in spherical geometry that capture the change with latitude of the angle between the rotation vector and the local vertical. Being derived using Hamilton's principle, the different β-plane approximations each conserve energy, angular momentum and potential vorticity. They differ in their treatments of the locally horizontal component of the rotation vector, the component that is usually neglected under the traditional approximation. In particular, we derive an extended set of β-plane equations in which the locally vertical and locally horizontal components of the rotation vector both vary linearly with latitude. This was previously thought to violate conservation of angular momentum and potential vorticity. We show that the difficulty in maintaining these conservation laws arises from the need to express the rotation vector as the curl of a vector potential while approximating the true spherical metric by a flat Cartesian metric. Finally, we derive depth-averaged equations on our extended β-plane with topography, and show that they coincide with the extended non-traditional shallow-water equations previously derived in Cartesian geometry.

JFM classification

Geophysical and Geological Flows: Ocean circulation Geophysical and Geological Flows: Rotating flows Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 174 - 195
Copyright
Copyright © Cambridge University Press 2011

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