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Instabilities of coupled density fronts and their nonlinear evolution in the two-layer rotating shallow-water model: influence of the lower layer and of the topography

Published online by Cambridge University Press:  25 January 2013

Bruno Ribstein  and
Vladimir Zeitlin
Bruno Ribstein*
Affiliation:
LMD, Université Pierre et Marie Curie, and ENS, 24 rue Lhomond, 75005 Paris, France
Vladimir Zeitlin
Affiliation:
LMD, Université Pierre et Marie Curie, and ENS, 24 rue Lhomond, 75005 Paris, France Institut Universitaire de France, 75005 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

We undertake a detailed analysis of linear stability of geostrophically balanced double density fronts in the framework of the two-layer rotating shallow-water model on the $f$-plane with topography, the latter being represented by an escarpment beneath the fronts. We use the pseudospectral collocation method to identify and quantify different kinds of instabilities resulting from phase locking and resonances of frontal, Rossby, Poincaré and topographic waves. A swap in the leading long-wave instability from the classical barotropic form, resulting from the resonance of two frontal waves, to a baroclinic form, resulting from the resonance of Rossby and frontal waves, takes place with decreasing depth of the lower layer. Nonlinear development and saturation of these instabilities, and of an instability of topographic origin, resulting from the resonance of frontal and topographic waves, are studied and compared with the help of a new-generation well-balanced finite-volume code for multilayer rotating shallow-water equations. The results of the saturation for different instabilities are shown to produce very different secondary coherent structures. The influence of the topography on these processes is highlighted.

JFM classification

Geophysical and Geological Flows: Gravity currents Instability: Instability Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 716 , 10 February 2013 , pp. 528 - 565
Copyright
©2013 Cambridge University Press

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References

Balmforth, N. J. 1999 Shear instability in shallow water. J. Fluid Mech. 387, 97–127.Google Scholar
Bouchut, F. 2007 Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances (ed. Zeitlin, V.). Elsevier.Google Scholar
Bouchut, F., Scherer, E. & Zeitlin, V. 2008 Nonlinear adjustment of a front over escarpment. Phys. Fluids 20, 016602.CrossRefGoogle Scholar
Bouchut, F. & Zeitlin, V. 2010 A robust well-balanced scheme for multi-layer shallow water equations. Disc. Cont. Dyn. Syst. B 13, 739–758.Google Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.Google Scholar
Castro, M. J., Le Floch, 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.Google Scholar
Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117, 343–377.Google Scholar
Griffiths, R. W. & Linden, P. F. 1982 Laboratory experiments on fronts. Part 1. Density driven boundary currents. Geophys. Astrophys. Fluid Dyn. 19, 159–187.Google 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 I. Passive lower layer. J. Fluid Mech. 659, 69–93.CrossRefGoogle Scholar
Gula, J., Zeitlin, V. & Bouchut, F. 2010 Instabilities of buoyancy driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part II. Active lower layer. J. Fluid Mech. 665, 209–237.Google Scholar
Iga, K. 1999 Critical layer instability as a resonance between a non-singular mode and continous modes. Fluid Dyn. Res. 25, 63–86.Google Scholar
Killworth, P. D., Paldor, N. & Stern, M. E. 1982 Instabilities on density-driven boundary currents and fronts. Geophys. Astrophys. Fluid Dyn. 23, 1–28.Google 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.Google Scholar
Knessl, C. & Keller, J. B. 1995 Stability of linear shear flows in shallow water. J. Fluid Mech. 303, 203–214.Google Scholar
Kubokawa, A. 1988 Instability and nonlinear evolution of a density-driven coastal current with a surface front in a two-layer ocean. Geophys. Astrophys. Fluid Dyn. 40, 195–223.Google Scholar
Lambaerts, J., Lapeyre, G. & Zeitlin, V. 2012 Moist vs dry baroclinic instability in a simplified two-layer atmospheric model with condensation and latent heat release. J. Atmos. Sci. 69.Google Scholar
Le Sommer, J., Medvedev, S. B., Plougonven, R. & Zeitlin, V. 2003 Singularity formation during relaxation of jets and fronts toward the state of geostrophic equilibrium. Commun. Nonlinear Sci. Numer. Simul. 8, 415–442.Google Scholar
Longuet-Higgins, M. S. 1967 On the trapping of waves along a discontinuity of depth in rotating ocean. J. Fluid Mech. 31, 417–434.Google Scholar
Longuet-Higgins, M. S. 1968 Double Kelvin waves with continuous depth profiles. J. Fluid Mech. 34, 49–80.CrossRefGoogle Scholar
Paldor, N. 1983 Linear stability and stable modes of geostrophic fronts. Geophys. Astrophys. Fluid Dyn. 27, 217–228.Google Scholar
Paldor, N. & Ghil, M. 1990 Finite-wavelength instabilities of a coupled density front. J. Phys. Oceanogr. 20, 114–123.Google Scholar
Paldor, N. & Killworth, P. D. 1987 Instabilities of a two-layer coupled front. Deep-Sea Res. 34, 1525–1539.Google Scholar
Reznik, G. M., Zeitlin, V. & BenJelloul, M. 2001 Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow water model. J. Fluid Mech. 445, 93–120.Google Scholar
Ripa, P. 1987 On the stability of elliptical vortex solutions of the shallow-water equations. J. Fluid Mech. 183, 343–363.Google Scholar
Ripa, P. 1990 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.Google Scholar
Scherer, E. & Zeitlin, V. 2008 Instability of coupled geostrophic density fronts and its nonlinear evolution. J. Fluid Mech. 613, 309–327.Google Scholar
Stern, M. E., Whitehead, J. A. & Hua, B. L. 1982 The intrusion of a density current along the coast of a rotating fluid. J. Fluid Mech. 123, 237–265.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in Matlab. SIAM.CrossRefGoogle Scholar
Vanneste, J. 1996 Rossby wave interaction in a shear flow with critical levels. J. Fluid Mech. 323, 317–338.Google Scholar
Vanneste, J. 1998 A nonlinear critical layer generated by the interaction of free Rossby waves. J. Fluid Mech. 371, 319–344.Google Scholar