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Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part 2. Active lower layer

Published online by Cambridge University Press:  22 October 2010

J. GULA ,
V. ZEITLIN  and
F. BOUCHUT
J. GULA*
Affiliation:
Laboratoire de Météorologie Dynamique, ENS and University P. and M. Curie, 24 rue Lhomond, 75231 Paris, France
V. ZEITLIN
Affiliation:
Laboratoire de Météorologie Dynamique, ENS and University P. and M. Curie, 24 rue Lhomond, 75231 Paris, France
F. BOUCHUT
Affiliation:
Laboratoire d'Analyse et de Mathématiques Appliquées, University Paris-Est and CNRS, 5 boulevard Descartes, 77454 Marne-la-Vallée, France
*
†Email address for correspondence: gula@lmd.ens.fr
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Abstract

This paper is the second part of the work on linear and nonlinear stability of buoyancy-driven coastal currents. Part 1, concerning a passive lower layer, was presented in the companion paper Gula & Zeitlin (J. Fluid Mech., vol. 659, 2010, p. 69). In this part, we use a fully baroclinic two-layer model, with active lower layer. We revisit the linear stability problem for coastal currents and study the nonlinear evolution of the instabilities with the help of high-resolution direct numerical simulations. We show how nonlinear saturation of the ageostrophic instabilities leads to reorganization of the mean flow and emergence of coherent vortices. We follow the same lines as in Part 1 and, first, perform a complete linear stability analysis of the baroclinic coastal currents for various depths and density ratios. We then study the nonlinear evolution of the unstable modes with the help of the recent efficient two-layer generalization of the one-layer well-balanced finite-volume scheme for rotating shallow water equations, which allows the treatment of outcropping and loss of hyperbolicity associated with shear, Kelvin–Helmholtz type, instabilities. The previous single-layer results are recovered in the limit of large depth ratios. For depth ratios of order one, new baroclinic long-wave instabilities come into play due to the resonances among Rossby and frontal- or coastal-trapped waves. These instabilities saturate by forming coherent baroclinic vortices, and lead to a complete reorganization of the initial current. As in Part 1, Kelvin fronts play an important role in this process. For even smaller depth ratios, short-wave shear instabilities with large growth rates rapidly develop. We show that at the nonlinear stage they produce short-wave meanders with enhanced dissipation. However, they do not change, globally, the structure of the mean flow which undergoes secondary large-scale instabilities leading to coherent vortex formation and cutoff.

JFM classification

Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Shallow water flows Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 665 , 25 December 2010 , pp. 209 - 237
Copyright
Copyright © Cambridge University Press 2010

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