Skip to main content Accessibility help
×
Home

Trapped edge waves in stratified rotating fluids: numerical and asymptotic results

  • ALEXANDER T. I. ADAMOU (a1), R. V. CRASTER (a1) and STEFAN G. LLEWELLYN SMITH (a2)

Abstract

The existence of trapped edge waves in a rotating stratified fluid with non-constant topography is studied using asymptotic and numerical techniques. A refinement of the classical WKBJ method is employed that is uniform at both the shoreline and caustic, where the classical approximation is singular, and is also uniform over long distances from the shore. This approach requires the use of comparison equations and it is shown that the two used previously in the literature are asymptotically equivalent in terms of the wave amplitude, but have small differences in the predicted wave frequencies. These asymptotic results, and results using shallow-water theory, are then compared to results from a careful numerical study of the nonlinear differential eigenvalue problem, allowing their range of practical applicability to be assessed. This numerical approach is also used to investigate whether trapping occurs in non-trivial and realistic geometries in the internal gravity wave band, which has been an open question for some time.

Copyright

References

Hide All
Abramowitz, M. & Stegun, I. A. (Ed.) 1974 Handbook of Mathematical Functions. Dover.
Adamou, A. T. I., Gridin, D. & Craster, R. V. 2005 Acoustic quasi-modes in slowly-varying cylindrical tubes. Q. J. Mech. Appl. Maths 58, 419438.
Ball, F. K. 1967 Edge waves in an ocean of finite depth. Deep-Sea Res. 14, 7988.
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.
Dale, A. C. & Sherwin, T. J. 1996 The extension of baroclinic coastal-trapped wave theory to superinertial frequencies. J. Phys. Oceanogr. 26, 23052315.
Davies, E. B. & Parnovski, L. 1998 Trapped modes in acoustic waveguides. Q. J. Mech. Appl. Maths 51, 477492.
Duclos, P. & Exner, P. 1995 Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Maths Phys. 7, 73102.
Evans, D. V. 1988 Mechanisms for the generation of edge waves over a sloping beach. J. Fluid Mech. 186, 379391.
Evans, D. V. 1989 Edge waves over a sloping beach. Q. J. Mech. Appl. Maths 42, 131142.
Evans, D. V., Levitin, M. & Vassiliev, D. 1994 Existence theorems for trapped modes. J. Fluid Mech. 261, 2131.
Greenspan, H. P. 1970 A note on edge waves in a stratified fluid. Stud. Appl. Maths 49, 381388.
Gridin, D., Adamou, A. T. I. & Craster, R. V. 2004 Electronic eigenstates in quantum rings: Asymptotics and numerics. Phys. Rev. B 69, 155317.
Gridin, D., Adamou, A. T. I. & Craster, R. V. 2005 a Trapped modes in curved elastic plates. Proc. R. Soc. Lond. A 461, 11811197.
Gridin, D., Craster, R. V. & Adamou, A. T. I. 2005 b Trapped modes in bent elastic rods. Wave Motion 42, 352366.
Kaplunov, J. D., Rogerson, G. A. & Tovstik, P. E. 2005 Localized vibration in elastic structures with slowly varying thickness. Q. J. Mech. Appl. Maths 58, 645664.
Kravtsov, Y. A. 1964 A modification of the geometrical optics method. Radiofizika 7, 664673, in Russian.
Langer, R. E. 1931 On the asymptotic solutions of ordinary differential equations with an application to the Bessel functions of large order. Trans. Am. Maths Soc. 33, 2364.
Linton, C. & Ratcliffe, K. 2004 Bound states in coupled guides. I. Two dimensions. J. Maths Phys. 45, 13591379.
Llewellyn Smith, S. G. 2004 Stratified rotating edge waves. J. Fluid Mech. 498, 161170.
Ludwig, D. 1966 Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Maths 19, 215250.
McIver, M. 1999 Uniqueness below a cut-off frequency for the two-dimensional linear water-wave problem. Proc. R. Soc. Lond. A 455, 14351441.
McKee, W. D. 1973 Internal-inertia waves in a fluid of variable depth. Proc. Camb. Phil. Soc. 73, 205213.
Miles, J. 1989 Edge waves on a gently sloping beach. J. Fluid Mech. 199, 125131.
Munk, W. H., Snodgrass, F. & Wimbush, M. 1970 Tides off shore: Transition from California coastal to deep-sea waters. Geophys. Fluid Dyn. 1, 161235.
Muzylev, S. V., Bulgakov, S. N. & Duran-Matute, M. 2005 Edge capillary-gravity waves on a sloping beach. Phys. Fluids 17, 048103.
Muzylev, S. V. & Odulo, A. B. 1980 Waves in a rotating stratified fluid on a sloping beach. Dokl. Akad. Nauk. SSSR 250, 331335.
Ou, H. W. 1980 On the propagation of free topographic Rossby waves near continental margins. I. Analytic model for a wedge. J. Phys. Oceanogr. 10, 10511060.
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.
Pringle, J. M. & Brink, K. H. 1999 High-frequency internal waves on a sloping shelf. J. Geophys. Res. 104, 52835299.
Saint-Guily, B. 1968 Ondes de frontière dans un bassin tournant dont le fond est incliné. C. R. Acad. Sci. Paris 266, 12911293.
Shen, M. C. & Keller, J. B. 1975 Uniform ray theory of surface, internal and acoustic wave propagation in a rotating ocean or atmosphere. SIAM J. Appl. Maths 28, 857875.
Shen, M. C., Meyer, R. E. & Keller, J. B. 1968 Spectra of water waves in channels and around islands. Phys. Fluids 11, 22892304.
Smith, R. 1977 Propagation in slowly-varying wave-guides. SIAM J. Appl. Maths 33, 3950.
Stokes, G. G. 1846 Report on recent researches in hydrodynamics. Brit. Ass. Rep. 1, 120.
Sun, S. M. & Shen, M. C. 1994 Linear water waves over a gently sloping beach. Q. J. Appl. Maths 52, 243259.
Ursell, F. 1952 Edge waves on a sloping beach. Proc. R. Soc. Lond. A 214, 7997.
Whitham, G. B. 1979 Lectures on Wave Propagation, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 61. New Delhi: Narosa.
Zhevandrov, P. 1991 Edge waves on a gently sloping beach: uniform asymptotics. J. Fluid Mech. 233, 483493.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Related content

Powered by UNSILO

Trapped edge waves in stratified rotating fluids: numerical and asymptotic results

  • ALEXANDER T. I. ADAMOU (a1), R. V. CRASTER (a1) and STEFAN G. LLEWELLYN SMITH (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.