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Wave propagation in rotating shallow water in the presence of small-scale topography

Published online by Cambridge University Press:  28 July 2021

E.J. Goldsmith*
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
J.G. Esler
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
*
Email address for correspondence: [email protected]

Abstract

The question of how finite-amplitude, small-scale topography affects small-amplitude motions in the ocean is addressed in the framework of the rotating shallow water equations. The extent to which the dispersion relations of Poincaré, Kelvin and Rossby waves are modified in the presence of topography is illuminated, using a range of numerical and analytical techniques based on the method of homogenisation. Both random and regular periodic arrays of topography are considered, with the special case of regular cylinders studied in detail, because this case allows for highly accurate analytical results. The results show that, for waves in a $\beta$-channel bounded by sidewalls, and for steep topographies outside of the quasi-geostrophic regime, topography acts to slow Poincaré waves slightly, Rossby waves are slowed significantly and Kelvin waves are accelerated for long waves and slowed for short waves, with the two regimes being separated by a narrow band of resonant wavelengths. The resonant band, which is due to the excitation of trapped topographic Rossby waves on each seamount, may affect any of the three wave types under the right conditions, and for physically reasonable results requires regularisation by Ekman friction. At larger topographic amplitudes, for cylindrical topography, a simple and accurate formula is given for the correction to the Rossby wave dispersion relation, which extends previous results for the quasi-geostrophic regime.

Type
JFM Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Allaire, G. 2012 A brief introduction to homogenization and miscellaneous applications. ESAIM: Proc. 37, 149.10.1051/proc/201237001CrossRefGoogle Scholar
Balagurov, B. & Kashin, V.A. 2001 The conductivity of a 2d system with a doubly periodic arrangement of circular inclusions. Tech. Phys. 46, 101106.10.1134/1.1340893CrossRefGoogle Scholar
Benilov, E.S. 2000 Waves on the beta-plane over sparse topography. J. Fluid Mech. 423, 263273.10.1017/S0022112000001890CrossRefGoogle Scholar
Biello, J.A. & Majda, A.J. 2005 A new multiscale model for the Madden–Julian oscillation. J. Atmos. Sci. 62 (6), 16941721.10.1175/JAS3455.1CrossRefGoogle Scholar
Bühler, O. & Holmes-Cerfon, M. 2011 Decay of an internal tide due to random topography in the ocean. J. Fluid Mech. 678, 271293.10.1017/jfm.2011.115CrossRefGoogle Scholar
Gill, A.E. 1982 Atmosphere–Ocean Dynamics. Academic Press.Google Scholar
Godin, Y.A. 2013 Effective complex permittivity tensor of a periodic array of cylinders. J. Math. Phys. 54 (5), 053505.10.1063/1.4803490CrossRefGoogle Scholar
Hara, T. & Mei, C.C. 1987 Bragg scattering of surface waves by periodic bars: theory and experiment. J. Fluid Mech. 178, 221241.10.1017/S0022112087001198CrossRefGoogle Scholar
Holmes, M.H. 2012 Introduction to Perturbation Methods. Springer.Google Scholar
Hu, X. & Chan, C.T. 2005 Refraction of water waves by periodic cylinder arrays. Phys. Rev. Lett. 95, 154501.10.1103/PhysRevLett.95.154501CrossRefGoogle ScholarPubMed
Jansons, K.M. & Johnson, E.R. 1988 Topographic Rossby waves above a random array of sea mountains. J. Fluid Mech. 191, 373388.10.1017/S0022112088001612CrossRefGoogle Scholar
Keller, J.B. 1963 Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders. J. Appl. Phys. 34 (4), 991993.10.1063/1.1729580CrossRefGoogle Scholar
Li, Y. & Mei, C.C. 2014 Scattering of internal tides by irregular bathymetry of large extent. J. Fluid Mech. 747, 481505.10.1017/jfm.2014.159CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1967 On the trapping of wave energy round islands. J. Fluid Mech. 29 (4), 781821.10.1017/S0022112067001181CrossRefGoogle Scholar
McPhedran, R. & McKenzie, D. 1980 Electrostatic and optical resonances of arrays of cylinders. Appl. Phys. 23, 223235.10.1007/BF00914904CrossRefGoogle Scholar
McPhedran, R.C., Poladian, L. & Milton, G.W. 1988 Asymptotic studies of closely spaced, highly conducting cylinders. Proc. R. Soc. Lond. A 415 (1848), 185196.Google Scholar
Mei, C.C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315335.10.1017/S0022112085000714CrossRefGoogle Scholar
Mei, C.C. & Vernescu, B. 2010 Homogenization Methods for Multiscale Mechanics. World Scientific.10.1142/7427CrossRefGoogle Scholar
Naciri, M. & Mei, C.C. 1988 Bragg scattering of water waves by a doubly periodic seabed. J. Fluid Mech. 192, 5174.10.1017/S0022112088001788CrossRefGoogle Scholar
Nandakumaran, A. 2007 An overview of homogenization. J. Ind. Inst. Sci. 87, 475484.Google Scholar
Paldor, N., Rubin, S. & Mariano, A.J. 2007 A consistent theory for linear waves of the shallow-water equations on a rotating plane in midlatitudes. J. Phys. Oceanogr. 37 (1), 115128.10.1175/JPO2986.1CrossRefGoogle Scholar
Paldor, N. & Sigalov, A. 2008 Trapped waves on the mid-latitude $\beta$-plane. Tellus A 60 (4), 742748.10.1111/j.1600-0870.2008.00332.xCrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.10.1007/978-1-4612-4650-3CrossRefGoogle Scholar
Rayleigh, Lord 1892 On the influence of obstacles arranged in rectangular order upon the properties of a medium. Philos. Mag. 34 (211), 481502.10.1080/14786449208620364CrossRefGoogle Scholar
Rhines, P. & Bretherton, F. 1973 Topographic Rossby waves in a rough-bottomed ocean. J. Fluid Mech. 61 (3), 583607.10.1017/S002211207300087XCrossRefGoogle Scholar
Rosales, R. & Papanicolaou, G. 1983 Gravity waves in a channel with a rough bottom. Stud. Appl. Maths 68, 89102.10.1002/sapm198368289CrossRefGoogle Scholar
Trefethen, L.N. 2000 Spectral Methods in MATLAB. SIAM.10.1137/1.9780898719598CrossRefGoogle Scholar
Vallis, G.K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.10.1017/CBO9780511790447CrossRefGoogle Scholar
Vanneste, J. 2000 a Enhanced dissipation for quasi-geostrophic motion over small-scale topography. J. Fluid Mech. 407, 105122.10.1017/S0022112099007430CrossRefGoogle Scholar
Vanneste, J. 2000 b Rossby wave frequency change induced by small-scale topography. J. Phys. Oceanogr. 30 (7), 18201826.10.1175/1520-0485(2000)030<1820:RWFCIB>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar