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Fast and slow resonant triads in the two-layer rotating shallow water equations

Published online by Cambridge University Press:  02 July 2018

Alex Owen Open the ORCID record for Alex Owen [Opens in a new window] ,
Roger Grimshaw Open the ORCID record for Roger Grimshaw [Opens in a new window]  and
Beth Wingate
Alex Owen*
Affiliation:
College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter EX4 4QF, UK
Roger Grimshaw
Affiliation:
College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter EX4 4QF, UK Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
Beth Wingate
Affiliation:
College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter EX4 4QF, UK
*
†Email address for correspondence: [email protected]
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Abstract

In this paper, we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect that cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode, with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multiscale asymptotic expansion based on a small-amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored, with an emphasis on the parameter regime with small Rossby and Froude numbers.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Shallow water flows Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 850 , 10 September 2018 , pp. 18 - 45
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Copyright
© 2018 Cambridge University Press 

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