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Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow-water model. Part 1. Passive lower layer

Published online by Cambridge University Press:  30 June 2010

J. GULA  and
V. ZEITLIN
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
*
†Email address for correspondence: gula@lmd.ens.fr
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Abstract

Buoyancy-driven coastal currents, which are bounded by a coast and a surface density front, are ubiquitous and play essential role in the mesoscale variability of the ocean. Their highly unstable nature is well known from observations, laboratory and numerical experiments. In this paper, we revisit the linear stability problem for such currents in the simplest reduced-gravity model and study nonlinear evolution of the instability by direct numerical simulations. By using the collocation method, we benchmark the classical linear stability results on zero-potential-vorticity (PV) fronts, and generalize them to non-zero-PV fronts. In both cases, we find that the instabilities are due to the resonance of frontal and coastal waves trapped in the current, and identify the most unstable long-wave modes. We then study the nonlinear evolution of the unstable modes with the help of a new high-resolution well-balanced finite-volume numerical scheme for shallow-water equations. The simulations are initialized with the unstable modes obtained from the linear stability analysis. We found that the principal instability saturates in two stages. At the first stage, the Kelvin component of the unstable mode breaks, forming a Kelvin front and leading to the reorganization of the mean flow through dissipative and wave–mean flow interaction effects. At the second stage, a new, secondary unstable mode of the Rossby type develops on the background of the reorganized mean flow, and then breaks, forming coherent vortex structures. We investigate the sensitivity of this scenario to the along-current boundary and initial conditions. A study of the same problem in the framework of the fully baroclinic two-layer model will be presented in the companion paper.

JFM classification

Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Waves in rotating fluids

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Papers
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Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 69 - 93
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. & Perthame, B. 2004 A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065.CrossRefGoogle Scholar
Barth, J. A. 1989 a Stability of a coastal upwelling front, 1. Model development and a stability theorem. J. Geophys. Res. 94, 10844–10856.CrossRefGoogle Scholar
Barth, J. A. 1989 b Stability of a coastal upwelling front, 2. Model results and comparison with observations. J. Geophys. Res. 94, 10857–10883.CrossRefGoogle Scholar
Bennett, J. R. 1973 A theory of large-amplitude Kelvin waves. J. Phys. Oceanogr. 3, 57–60.2.0.CO;2>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. 2004 Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources. Frontiers in Mathematics Series, Birkhäuser.CrossRefGoogle Scholar
Bouchut, F. 2007 Efficient numerical finite-volume schemes for shallow-water models. In Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances (ed. Zeitlin, V.), vol. 2, pp. 189–256. Edited Series on Advances in Nonlinear Science and Complexity, Elsevier.CrossRefGoogle Scholar
Bouchut, F., Le Sommer, J. & Zeitlin, V. 2004 One-dimensional 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., Le Sommer, J. & Zeitlin, V. 2005 Breaking of equatorial waves. Chaos 15, 013503-1–013503-19.CrossRefGoogle ScholarPubMed
Bouchut, F. & Zeitlin, V. 2010 A robust well-balanced scheme for multi-layer shallow water equations. Discrete Contin. Dyn. Syst. Ser. B 13, 739–758.Google Scholar
Dahl, O. H. 2005 Development of perturbations on a buoyant coastal current. J. Fluid Mech. 527, 337–351.CrossRefGoogle Scholar
Fedorov, A. V. & Melville, W. K. 1995 Propagation and breaking of nonlinear Kelvin waves. J. Phys. Oceanogr. 25, 2519–2531.2.0.CO;2>CrossRefGoogle Scholar
Fedorov, A. V. & Melville, W. K. 2000 Kelvin fronts on the equatorial thermocline. J. Phys. Oceanogr. 30, 1692–1705.2.0.CO;2>CrossRefGoogle Scholar
Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117, 343–377.CrossRefGoogle Scholar
Griffiths, R. W. & Linden, P. F. 1982 Laboratory experiments in fronts. Part I: Density-driven boundary currents. Geophys. Astrophys. Fluid Dyn. 19, 159–187.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 2. Active lower layer. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Hayashi, Y.-Y. & Young, W. R. 1987 Stable and unstable shear modes of rotating parallel flows in shallow water. J. Fluid Mech. 184, 477–504.CrossRefGoogle Scholar
Iga, K. 1993 Reconsideration of Orlanski's instability theory of frontal waves. J. Fluid Mech. 255, 213–236.CrossRefGoogle Scholar
Iga, K. 1999 Critical layer instability as a resonance between a non-singular mode and continuous modes. Fluid Dyn. Res. 25, 63–86.CrossRefGoogle Scholar
Jones, W. L. 1967 Propagation of internal gravity waves in fluids with shear flow and rotation. J. Fluid Mech. 30, 439–448.CrossRefGoogle Scholar
Killworth, P. D. 1983 Long-wave instability of an isolated front. Geophys. Astrophys. Fluid Dyn. 25, 235–258.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
Killworth, P. D. & Stern, M. E. 1982 Instabilities on density-driven boundary currents and fronts. Geophys. Astrophys. Fluid Dyn. 22, 1–28.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. 1985 Instability of a geostrophic front and its energetics. Geophys. Astrophys. Fluid Dyn. 33, 223–257.CrossRefGoogle Scholar
Kubokawa, A. 1986 Instability caused by the coalescence of two modes of a one-layer coastal current with a surface front. J. Oceanogr. Soc. Japan 42, 373–380.CrossRefGoogle Scholar
Kubokawa, A. & Hanawa, K. 1984 A theory of semigeostrophic gravity waves and its application to the intrusion of a density current along a coast. Part 1. J. Oceanogr. Soc. Japan 40, 247–259.CrossRefGoogle Scholar
LeSommer, J., Reznik, G. & Zeitlin, V. 2004 Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane. J. Fluid Mech. 515, 135–170.Google Scholar
Meacham, S. P. & Stephens, J. C. 2001 Instabilities of gravity currents along a slope. J. Phys. Oceanogr. 31, 30–53.2.0.CO;2>CrossRefGoogle Scholar
Paldor, N. 1983 Stability and stable modes of coastal fronts. Geophys. Astrophys. Fluid Dyn. 27, 217–228.CrossRefGoogle Scholar
Paldor, N. 1988 Amplitude–wavelength relations of nonlinear frontal waves on coastal currents. J. Phys. Oceanogr. 18, 753–760.2.0.CO;2>CrossRefGoogle Scholar
Paldor, N. & Ghil, M. 1991 Shortwave instabilities of coastal currents. Geophys. Astrophys. Fluid Dyn. 58, 225–241.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
Pratt, L. J., Helfrich, K. & Leen, D. 2008 On the stability of ocean overflows. J. Fluid Mech. 602, 241–266.CrossRefGoogle Scholar
Ripa, P. 1983 General stability conditions for zonal flows in a one-layer model on the β-plane or the sphere. J. Fluid Mech. 126, 463–489.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
Satomura, T. 1981 An investigation of shear instability in a shallow water. J. Met. Soc. Japan 59, 148–167.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
Stern, M. E. 1980 Geostrophic fronts, bores, breaking and blocking waves. J. Fluid Mech. 99, 687–703.CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral Methods in Matlab. SIAM.CrossRefGoogle Scholar
Vanneste, J. 1998 A nonlinear critical layer generated by the interaction of free Rossby waves. J. Fluid Mech. 371, 319–344.CrossRefGoogle Scholar