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The quasi-geostrophic theory of the thermal shallow water equations

Published online by Cambridge University Press:  16 April 2013

Emma S. Warneford  and
Paul J. Dellar
Emma S. Warneford
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: dellar@maths.ox.ac.uk
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Abstract

The thermal shallow water equations provide a depth-averaged description of motions in a fluid layer that permits horizontal variations in material properties. They typically arise through an equivalent barotropic approximation of a two-layer system, with a spatially varying density contrast due to an evolving temperature field in the active layer. We formalize a previous derivation of the quasi-geostrophic (QG) theory of these equations, by performing a direct asymptotic expansion for small Rossby number. We then present a second derivation as the small Rossby number limit of a balanced model that projects out high-frequency dynamics due to inertia-gravity waves. This latter derivation has wider validity, not being restricted to mid-latitude $\beta $-planes. We also derive their local energy conservation equation from the QG limit of a thermal shallow water pseudo-energy conservation equation. This derivation involves the ageostrophic correction to the leading-order geostrophic velocity that is eliminated in the usual derivation of a closed evolution equation for the QG potential vorticity. Finally, we derive the non-canonical Hamiltonian structure of the thermal QG equations from a decomposition in Rossby number of a pseudo-energy and Poisson bracket for the thermal shallow water equations.

JFM classification

Mathematical Foundations: Hamiltonian theory Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 723 , 25 May 2013 , pp. 374 - 403
Copyright
©2013 Cambridge University Press 

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