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

Journal of Fluid Mechanics

Volumn 659, Issue , 2010, Pages 69-93

  Gula, J ^a   Zeitlin, V ^a  

^a LABORATOIRE DE MÉTÉOROLOGIE DYNAMIQUE   (France)

Author keywords

gravity currents; shallow water flows; waves in rotating fluids

Indexed keywords

BUOYANCY; EQUATIONS OF MOTION; LINEAR STABILITY ANALYSIS; NONLINEAR EQUATIONS; VORTEX FLOW;

BAROCLINIC TWO-LAYER MODEL; BOUNDARY AND INITIAL CONDITIONS; COHERENT VORTEX STRUCTURES; GRAVITY CURRENTS; SHALLOW WATER EQUATIONS; SHALLOW WATER FLOW; WAVE-MEAN-FLOW INTERACTION; WAVES IN ROTATING FLUIDS;

OCEAN CURRENTS;

BOUNDARY CONDITION; BUOYANCY; COASTAL CURRENT; FINITE VOLUME METHOD; FLOW STABILITY; FLOW STRUCTURE; NUMERICAL MODEL; OCEAN WAVE; ROTATING FLOW; ROTATING FLUID; SHALLOW-WATER EQUATION; VORTEX FLOW; VORTICITY; WAVE-CURRENT INTERACTION;

EID: 77957154314     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/S0022112010002405     Document Type: Article

References (41)

1. 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.
2. Barth, J. A. 1989a Stability of a coastal upwelling front, 1. Model development and a stability theorem. J. Geophys. Res. 94, 10844-10856.
3. Barth, J. A. 1989b Stability of a coastal upwelling front, 2. Model results and comparison with observations. J. Geophys. Res. 94, 10857-10883.
4. Bennett, J. R. 1973 A theory of large-amplitude Kelvin waves. J. Phys. Oceanogr. 3, 57-60.
5. 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.
6. Bouchut, F. 2004 Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources. Frontiers in Mathematics Series, Birkḧauser.
7. Bouchut, F. 2007 Efficient numerical finite-volume schemes for shallow-water models. In Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances (ed. V. Zeitlin), vol.2, pp. 189-256. Edited Series on Advances in Nonlinear Science and Complexity, Elsevier.
8. 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.
9. Bouchut, F., Le Sommer, J. & Zeitlin, V. 2005 Breaking of equatorial waves. Chaos 15, 0135031-01350313519
10. 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.
11. Dahl, O. H. 2005 Development of perturbations on a buoyant coastal current. J. Fluid Mech. 527, 337-351.
12. Fedorov, A. V. & Melville, W. K. 1995 Propagation and breaking of nonlinear Kelvin waves. J. Phys. Oceanogr. 25, 2519-2531.
13. Fedorov, A. V. & Melville, W. K. 2000 Kelvin fronts on the equatorial thermocline. J. Phys. Oceanogr. 30, 1692-1705.
14. Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117, 343-377.
15. Griffiths, R. W. & Linden, P. F. 1982 Laboratory experiments in fronts. Part I: Density-driven boundary currents. Geophys. Astrophys. Fluid Dyn. 19, 159-187.
16. 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).
17. 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.
18. Iga, K. 1993 Reconsideration of Orlanski's instability theory of frontal waves. J. Fluid Mech. 255, 213-236.
19. Iga, K. 1999 Critical layer instability as a resonance between a non-singular mode and continuous modes. Fluid Dyn. Res. 25, 63-86.
20. Jones, W. L. 1967 Propagation of internal gravity waves in fluids with shear flow and rotation. J. Fluid Mech. 30, 439-448.
21. Killworth, P. D. 1983 Long-wave instability of an isolated front. Geophys. Astrophys. Fluid Dyn. 25, 235-258.
22. 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.
23. Killworth, P. D. & Stern, M. E. 1982 Instabilities on density-driven boundary currents and fronts. Geophys. Astrophys. Fluid Dyn. 22, 1-28.
24. Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81-106.
25. Kubokawa, A. 1985 Instability of a geostrophic front and its energetics. Geophys. Astrophys. Fluid Dyn. 33, 223-257.
26. 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.
27. 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.
28. 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.
29. Meacham, S. P. & Stephens, J. C. 2001 Instabilities of gravity currents along a slope. J. Phys. Oceanogr. 31, 30-53.
30. Paldor, N. 1983 Stability and stable modes of coastal fronts. Geophys. Astrophys. Fluid Dyn. 27, 217-228.
31. Paldor, N. 1988 Amplitude-wavelength relations of nonlinear frontal waves on coastal currents. J. Phys. Oceanogr. 18, 753-760.
32. Paldor, N. & Ghil, M. 1991 Shortwave instabilities of coastal currents. Geophys. Astrophys. Fluid Dyn. 58, 225-241.
33. Poulin, F. J. & Flierl, G. R. 2003 The nonlinear evolution of barotropically unstable jets. J. Phys. Oceanogr. 33, 2173-2192.
34. Pratt, L. J., Helfrich, K. & Leen, D. 2008 On the stability of ocean overflows. J. Fluid Mech. 602, 241-266.
35. 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.
36. 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.
37. Satomura, T. 1981 An investigation of shear instability in a shallow water. J. Met. Soc. Japan 59, 148-167.
38. Scherer, E. & Zeitlin, V. 2008 Instability of coupled geostrophic density fronts and its nonlinear evolution. J. Fluid Mech. 613, 309-327.
39. Ste rn, M. E. 1980 Geostrophic fronts, bores, breaking and blocking waves. J. Fluid Mech. 99, 687-703.
40. Tre fe th e n, L. N. 2000 Spectral Methods in Matlab. SIAM.
41. Vanneste J. 1998 A nonlinear critical layer generated by the interaction of free Rossby waves. J. Fluid Mech. 371, 319-344.