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Gravity wave radiation from vortex trains in rotating shallow water

Published online by Cambridge University Press:  26 April 2006

Rupert Ford
Rupert Ford
Affiliation:
Centre for Atmospheric Science, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK Current address: Scripps Institution of Oceanography, 9500 Gilman Drive, La Jolla, CA92093-0225. USA.
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Abstract

Gravity wave radiation by vortical flows in the f-plane shallow-water equations is investigated by direct nonlinear numerical simulation. The flows considered are initially parallel flows, consisting of a single strip in which the potential vorticity differs from the background value. The flows are unstable to the barotropic instability mechanism, and roll up into a train of vortices. During the subsequent evolution of the vortex train, gravity waves are radiated. In the limit of small Froude number, the gravity wave radiation is compared with that predicted by an appropriately modified version of the Lighthill theory of aerodynamic sound generation. It is found that the gravity wave field agrees well with that predicted by the theory, provided typical lengthscales of vortical motions are well within one deformation radius.

It is found that the nutation time for vortices in the train increases rapidly with increasing Froude number in cases where the potential vorticity in the vortices is of the same sign as the background value, whereas the nutation time is almost independent of Froude number in cases where the potential vorticity in the vortices is zero or of opposite sign to the background. Consequently, in the former cases, the unsteadiness of the flow decreases with increasing Froude number, so the effect of the inertial cutoff frequency is increased, leading to an optimal Froude number for gravity wave radiation, above which the intensity of the radiated waves decreases as the Froude number is further increased. It is proposed that the existence of a finite range of interaction within the vortices, for flows with positive vortex potential vorticity, may account for the strong dependence of nutation time on Froude number in those cases. The interaction scale within the vortices becomes infinite in the limit of zero vortex potential vorticity, and so the arguments do not apply in those cases.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 281 , 25 December 1994 , pp. 81 - 118
Copyright
© 1994 Cambridge University Press

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