Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis

Journal of Fluid Mechanics

Volumn 624, Issue , 2009, Pages 191-224

  Alam, Mohammad Reza ^a   Liu, Yuming ^a   Yue, Dick K P ^a  

^a Massachusetts Institute of Technology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FLUID DYNAMICS; OCEANOGRAPHY; TOPOGRAPHY; WATER WAVES;

INCIDENT WAVE FREQUENCIES; NONLINEAR INTERACTIONS; NONLINEAR RESONANCE; PERTURBATION ANALYSIS; REGULAR PERTURBATIONS; RESONANCE CONDITION; RESONANT INTERACTION; UNIFORMLY VALID SOLUTION;

RESONANCE;

BOTTOM TOPOGRAPHY; INTERNAL WAVE; NONLINEARITY; OCEAN WAVE; RESONANCE; RIPPLE; THEORETICAL STUDY; WAVE DIRECTION; WAVE PROPAGATION;

EID: 65649134763     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/S0022112008005478     Document Type: Article

References (43)

1. ALAM, M.-R. 2008 Interaction of waves in a two-layer density stratified fluid. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA.
2. ALAM, M.-R., LIU, Y. & YUE, D. K. P. 2009 Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation. J. Fluid Mech. 624, 225-253.
3. ALAM, M.-R. & MEI, C. C. 2007 Attenuation of long interfacial waves over a randomly rough seabed. J. Fluid Mech. 587, 73-96.
4. ANTENUCCI, J. P. & IMBERGER, J. 2001 On internal waves near the high-frequency limit in an enclosed basin. J. Geophys. Res. 106 (C 10), 22465-22474.
5. ARDHUIN, F. & HERBERS, T. H. C. 2002 Bragg scattering of random surface gravity waves by irregular seabed topography. J. Fluid Mech. 451, 1-33.
6. Ardhuin, F. & Magne, R. 2007 Scattering of surface gravity waves by bottom topography with a current. J. Fluid Mech. 576, 235-264.
7. BAINES, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.
8. BAINES, P. G. 1997 Topographic Effects in Stratified Flows. In Cambridge Monographs on Mechanics (ed. G. K. Batchelor & L. B. Freud). Cambridge University Press.
9. BALL, F. 1964 Energy transfer between external and internal gravity waves. J. Fluid Mech. 19, 465-478.
10. BELZONS, M., GUAZZELLE, E. & PARODI, O. 1988 Gravity-waves on a rough bottom - experimental-evidence of one-dimensional localization. J. Fluid Mech. 186, 539-558.
11. BOEGMAN, L., IMBERGER, J., IVEY, G. N. & ANTENUCCI, J. P. 2003 High-frequency internal waves in large stratified lakes. Limnol. Oceanogr. 48 (2), 895-919.
12. CHEN, Y. & LIU, P. L.-F. 1996 On interfacial waves over random topography. Wave Mot. 24 (2), 169-184.
13. CUMMINS, P. F., VAGLE, S., ARMI, L. & FARMER, D. M. 2003 Stratified flow over topography: Upstream influence and generation of nonlinear internal waves. Proc. R. Soc. Lond. A 459 (2034), 1467-1487.
14. DAVIES, A. G. 1982 The reflection of wave energy by undulation on the seabed. Dyn. Atmos. Oceans 6, 207-232.
15. DIAS, F. & VANDEN-BROECK, J.-M. 2003 On internal fronts. J. Fluid Mech. 479, 145-154.
16. FARMER, D & ARMI, L 1999 The generation and trapping of solitary waves over topography. Science 283 (5399), 188-190.
17. GARGETTT, A. E. & HUGHES, B. A. 1972 On the interaction of surface and internal waves. J. Fluid Mech. 52 (01), 179-191.
18. GARRETT, C. & MUNK, W. 1975 Space-time scales of internal waves: A progress report. J. Geophys. Res. 80 (C3), 291-298.
19. GARRETT, C. & MUNK, W. 1979 Internal waves in the ocean. Annu. Rev. Fluid Mech. 11 (1), 339-369.
20. GASTER, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222-224.
21. GUAZZELLI, E., REY, V. & BELZONS, M. 1992 Higher-order bragg reflection of gravity surface waves by periodic beds. J. Fluid Mech. 245, 301-317.
22. HASSELMANN, K. 1966 Feynman diagrams and interaction rules of wave-wave scattering processes. 4 (1), 1-32.
23. HEATHERSHAW, A. D. & DAVIES, A. G. 1985 Resonant wave reflection by transverse bedforms and its relation to beaches and offshore bars. Mar. Geol. 62, 321-338.
24. HILL, D. F. 2004 Weakly nonlinear cubic interaction between surface waves and interfacial waves: An analytic solution. Phys. Fluids 16 (3), 839-842.
25. HILL, D. F. & FODA, M. A. 1996 Subharmonic resonance of short internal standing waves by progressive surface waves. J. Fluid Mech. 321, 217-233.
26. HILL, D. F. & FODA, M. A. 1998 Subharmonic resonance of oblique interfacial waves by a progressive surface wave. Proc. R. Soc. Lond. A 454 (1972), 1129-1144.
27. HOLT, J. T. & THORPE, S. A. 1997 The propagation of high frequency internal waves in the Celtic Sea. Deep-Sea Res. Pt I 44 (12), 2087-2116.
28. JAMALI, M. 1998 Surface wave interaction with with oblique internal waves. PhD thesis, University of British Columbia, Vancouver, Canada.
29. JAMALI, M., LAWRENCE, G. A. & SEYMOUR, B. 2003 A note on the resonant interaction between a surface wave and two interfacial waves. J. Fluid Mech. 491, 1-9.
30. KIRBY, J. T. 1986 A general wave equation for waves over rippled beds. J. Fluid Mech. 162, 171-186.
31. KUDRYAVTSEV, V. N. 1994 The coupling of wind and internal waves: modulation and friction mechanisms. J. Fluid Mech. 278, 33-62.
32. LAMB, H. 1932 Hydrodynamics. Dover.
33. LIU, Y. & YUE, D. K. P. 1998 On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297-326.
34. MCKEE, W. D. 1996 Bragg resonances in two-layer fluid. J. Austral. Math. Soc. B 37 (3), 334-345.
35. MEI, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315-335.
36. MEI, C. C., STIASSNIE, MICHAEL & YUE, DICK K.-P. 2005 Theory and Applications of Ocean Surface Waves. Part 1,2. Advanced Series on Ocean Engineering, vol. 23. World Scientific.
37. MELVILLE, W. K. & HELFRICH, KARL R. 1987 Transcritical two-layer flow over topography. J. Fluid Mech. 178, 31-52.
38. MIROPOL'SKY, Y. Z. 2001 Dynamics of Internal Gravity Waves in the Ocean: Atmospheric and Oceanographic Sciences Library (trans. and ed. O. D. Shishkina), vol. 24. Kluwer Academic Publishers.
39. OLBERS, D. J. & HERTERICH, K. 1979 The spectral energy transfer from surface waves to internal waves. J. Fluid Mech. 92 (2, 349-379.
40. THORPE, S. A. 1975 The excitation, dissipation, and interaction of internal waves in the deep ocean. J. Geophys. Res 80 (3), 328-337.
41. THORPE, S. A., KEEN, J. M., JIANG, R. & LEMMIN, U. 1996 High-frequency internal waves in lake Geneva. Phil. Trans. 354 (1705), 237-257.
42. VELETSOS, A. S. & SHIVAKUMAR, P. 1993 Sloshing response of layered liquids in rigid tanks. Tech Rep. BNL-52378, Brookhaven National Lab., Upton, NY, USA.
43. WEN, F. 1995 Resonant generation of internal waves on the soft sea bed by a surface water wave. Phys. Fluids 7 (8), 1915-1922.