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On the dispersion relation for trapped internal waves

Published online by Cambridge University Press:  26 April 2006

B. C. Barber
B. C. Barber
Affiliation:
Space Sector, Defence Research Agency, Farnborough, Hants, GU14 6TD, UK
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Abstract

An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp = 1/CpO + k/ωmax + ∈(k) where cp is the wave phase speed for a particular mode, CpO is the phase speed at k = 0, ωmax is the maximum possible wave angular frequency and ωmax ≤ Nmax where Nmax is the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) = o(k) for k large, and is bounded for finite k. In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp ≈ (∫∞0N2(y)ydy)-½ + k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functions N2(x) which is taken to be a class of realvalued functions of a real variable x where O ≤ x ∞ such that N2(x) = O(e-βx) as x → ∞ and 1/β is an arbitrary length scale. It is demonstrated that N2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax = Nmax. More generally, it is demonstrated that when k → ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that if N2(x) is O(e-βx) as x → ∞ and has a maximum value N2max then a sufficient condition for 1/cp ∼ k/Nmax to hold for large k for the lowest mode is that N2(t)/t is convex for O ≤ t ≤ 1 where t = e-βx.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 31 - 49
Copyright
© 1993 Cambridge University Press

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References

Cheney, E. W. 1966 Introduction to Approximation Theory. McGraw-Hill.
Eckman, V. W. 1906 On dead water. In The Norwegian North Polar Expedition 1893–1896, Scientific Results (ed. F. Nansen), vol. V, chap. XV. Longmans, Green and Co.
Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1953 Higher Transcendental Functions, vol. 1. McGraw-Hill.
Garrett, C. & Munk, W. 1972 Space-time scales of internal waves. Geophys. Fluid Dyn. 2, 225–264.Google Scholar
Hardy, G. H., Littlewood, J. E. & PÓlya, G. 1959 Inequalities. Cambridge University Press.
Keller, J. B. & Munk, W. H. 1970 Internal wave wakes of a body moving in a stratified fluid. Phys. Fluids 13, 1425–1431.Google Scholar
Krauss, W. 1966 Methoden und Ergebnisse der Theoretischen Ozeanographie: Band II, Interne Wellen. Gebrüder Borntraeger.
Ince, E. L. 1926 Ordinary Differential Equations. Longmans, Green and Co.
Lee-Bapty, I. P. 1991 An equation to model internal-wave wakes in the region of the Mach cone. Tech. Rep. RAE TR 91055. Defence Research Agency, Farnborough, Hants, UK.
Lee-Bapty, I. P. 1992 Structure of internal-wave wakes: linear theory. Tech. Rep. DRA TR 92005. Defence Research Agency, Farnborough, Hants, UK.
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.
Miloh, T. & Tulin, M. P. 1988 Dead water phenomena: a non-linear theory of wave disturbances due to a ship in a shallow thermocline. Tech. Rep. 88-29. Ocean Engng Lab., University of California, Santa Barbara, USA.
Perry, J. R. 1992 Experimental results in SAR imaging of ship wakes. Technical Report DRA TR 92044, Defence Research Agency, Farnborough, Hants, UK.
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes. Cambridge University Press.
Slater, L. J. 1960 Confluent Hypergeometric Functions. Cambridge University Press.
Stapleton, N. R. & Perry, J. R. 1992 Synthetic aperture radar imagery of ship generated internal waves during the UK-US Loch Linnhe series of experiments. 12th Annual Intl Geoscience and Remote Sensing Symp. (IGARSS '92), Houston, Texas, USA, 1992, pp. 1338–1340.
Tricomi, F. G. 1954 Funzioni Ipergeometriche Confluenti. Edizioni Cremonese.
Watson, G., Chapman, R. D. & Apel, J. R. 1992 Measurements of the internal wave wake of a ship in a highly stratified sea loch. J. Geophys. Res. 97, C61 9689–9703.Google Scholar
Watson, G. N. 1944 Theory of Bessel Functions. Cambridge University Press.
Whittaker, E. T. 1918 A formula for the solution of algebraic or transcendental equations. Proc. Edinburgh Math. Soc. 36, 103–106.Google Scholar
Yih, C.-S. 1965 Dynamics of Nonhomogenous Fluids. Macmillan.