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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: [email protected]
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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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References

REFERENCES

Barth, J. A. 1989 Stability of a coastal upwelling front. Part 1. Model development and a stability theorem. J. Geophys. Res. 94, 10844–10856.CrossRefGoogle Scholar
Boss, E., Paldor, N. & Thompson, L. 1996 Stability of a potential vorticity front: from quasi-geostrophy to shallow water. J. Fluid Mech. 315, 65–84.CrossRefGoogle Scholar
Bouchut, F., Le Sommer, J. & Zeitlin, V. 2004 1D rotating shallow water: nonlinear semi-geostrophic adjustment, slow manifold and nonlinear wave phenomena. Part 2. High-resolution numerical simulations. J. Fluid Mech. 514, 35–63.CrossRefGoogle Scholar
Bouchut, F. & Morales, T. 2008 An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. M2AN 42, 683–698.CrossRefGoogle Scholar
Bouchut, F. & Zeitlin, V. 2010 A robust well-balanced scheme for multi-layer shallow water equations. Discrete Continuous Dyn. Syst. B 13, 739–758.CrossRefGoogle Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.CrossRefGoogle Scholar
Castro, M. J., LeFloch, P. G., Munos-Ruiz, M. L. & Parés, C. 2008 Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. J. Comput. Phys. 227, 8107–8129.CrossRefGoogle Scholar
Griffiths, R. W. & Linden, P. F. 1982 Part I. Density-driven boundary currents. Geophys. Astrophys. Fluid Dyn. 19, 159–187.CrossRefGoogle Scholar
Gula, J., Plougonven, R. & Zeitlin, V. 2009 a Ageostrophic instabilities of fronts in a channel in the stratified rotating fluid. J. Fluid Mech 627, 485–507.CrossRefGoogle Scholar
Gula, J. & Zeitlin, V. 2010 Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part 1. Passive lower layer. J. Fluid Mech. 659, 69–93.CrossRefGoogle Scholar
Gula, J., Zeitlin, V. & Plougonven, R. 2009 b Instabilities of two-layer shallow-water flows with vertical shear in the rotating annulus. J. Fluid Mech 638, 27–47.CrossRefGoogle Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. 111 (470), 877–946.CrossRefGoogle Scholar
Iga, K. 1997 Instability of a front with a layer of uniform potential vorticity. J. Meteorol. Soc. Japan 75, 1–11.CrossRefGoogle Scholar
Killworth, P. D., Paldor, N. & Stern, M. E. 1984 Wave propagation and growth on a surface front in a two-layer geostrophic current. J. Mar. Res. 42, 761–785.CrossRefGoogle Scholar
Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81–106.CrossRefGoogle Scholar
Kubokawa, A. 1988 Instability and nonlinear evolution of a desity-driven coastal current with a surface front in a two-layer ocean. Geophys. Astrophys. Fluid Dyn. 40, 195–223.CrossRefGoogle Scholar
Leblond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.Google Scholar
LeSommer, J., Medvedev, S., Plougonven, R. & Zeitlin, V. 2003 Singularity formation during relaxation of jets and fronts towards the state of geostrophic equilibrium. Commun. Nonlinear Sci. Numer. Simul. 8, 415–442.CrossRefGoogle Scholar
Lyapidevsky, V. Y. & Teshukov, V. M. 2000 Mathematical models of long wave propagation in inhomogeneous fluid (in Russian). Siberian Branch of the Russian Academy of Science, Novosibirsk. ISBN: 5-7692-0340-4.Google Scholar
Paldor, N. & Ghil, M. 1991 Shortwave instabilities of coastal currents. Geophys. Astrophys. Fluid Dyn. 58, 225–241.CrossRefGoogle Scholar
Paldor, N. & Killworth, P. D. 1987 Instabilities of a two-layer coupled front. Deep-Sea Res. 34, 1525–1539.CrossRefGoogle Scholar
Poulin, F. J. & Flierl, G. R. 2003 The nonlinear evolution of barotropically unstable jets. J. Phys. Oceanogr. 33, 2173–2192.2.0.CO;2>CrossRefGoogle Scholar
Ripa, P. 1991 General stability conditions for a multi-layer model. J. Fluid Mech. 222, 119–137.CrossRefGoogle Scholar
Sakai, S. 1989 Rossby–Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176.CrossRefGoogle Scholar
Scherer, E. & Zeitlin, V. 2008 Instability of coupled geostrophic density fronts and its nonlinear evolution. J. Fluid Mech. 613, 309–327.CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral Methods in Matlab. SIAM.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar