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Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane

Published online by Cambridge University Press:  24 March 2010

ANDREW L. STEWART  and
PAUL J. DELLAR
ANDREW L. STEWART
Affiliation:
OCIAM, Mathematical Institute, 24–29 St Giles', Oxford, OX1 3LB, UK
PAUL J. DELLAR*
Affiliation:
OCIAM, Mathematical Institute, 24–29 St Giles', Oxford, OX1 3LB, UK
*
†Email address for correspondence: [email protected]
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Abstract

We derive equations to describe the flow of multiple superposed layers of inviscid, incompressible fluids with constant densities over prescribed topography in a rotating frame. Motivated by geophysical applications, these equations incorporate the complete Coriolis force. We do not make the widely used ‘traditional approximation’ that omits the contribution to the Coriolis force from the locally horizontal part of the rotation vector. Our derivation is performed by averaging the governing Euler equations over each layer, and from two different forms of Hamilton's variational principle that differ in their treatment of the coupling between layers. The coupling may be included implicitly through the map from Lagrangian particle labels to particle coordinates, or explicitly by adding terms representing the work done on each layer by the pressure exerted by the layers above. The latter approach requires additional terms in the Lagrangian, but extends more easily to many layers. We show that our equations obey the expected conservation laws for energy, momentum and potential vorticity. The conserved momentum and potential vorticity are modified by non-traditional effects. The vertical component of the rotation vector that appears in the potential vorticity for each layer under the traditional approximation is replaced by the component perpendicular to the layer's midsurface. The momentum includes an additional contribution that reflects changes in angular momentum caused by changes in a fluid element's distance from the rotation axis as it is displaced vertically. Again, this effect is absent in the traditional approximation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 651 , 25 May 2010 , pp. 387 - 413
Copyright
Copyright © Cambridge University Press 2010

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