Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-22T05:37:14.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Double Kelvin waves with continuous depth profiles

Published online by Cambridge University Press:  28 March 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Oregon State University, Corvallis, Oregon
Get access Rights & Permissions [Opens in a new window]

Abstract

The possibility of long waves in a rotating ocean being trapped along a straight discontinuity in depth was demonstrated in a recent paper (Longuet-Higgins 1968). The analysis is now extended to the situation where the depth varies continuously, in a zone separating two regions of different depths. The trapping of waves in the transition zone is investigated, taking full account of the horizontal divergence of the motion.

If the profile of the depth is assumed to be monotonic, then it is shown that the trapped waves always travel along the transition zone with the shallower water to their right in the northern hemisphere and to their left in the southern hemisphere. The wave period must always exceed a pendulum-day. The period is also bounded below by a quantity depending inversely on the maximum bottom gradient.

By allowing the width W of the transition zone to vary, asymptotic forms for the trapped modes are obtained, both as W → 0 and as W → ∞. In the limit as W → 0 the depth becomes discontinuous, and it is shown that the lowest mode then becomes a double Kelvin wave (Longuet-Higgins 1968) propagated along the discontinuity. The periods of the higher modes, on the other hand, all tend to infinity; these modes become steady currents.

Numerical calculations of the trapped modes are presented for two different laws of depth in the transition zone. It is found that as W → 0 the lowest mode is insensitive to the form of the depth profile. Higher modes depend on the details of the profile. Hence the lowest mode is the most likely to be observed in the real ocean.

The dispersion relation is also investigated. It is shown that the group-velocity of all modes must change sign at some point in the range of wave-numbers, if the divergence is taken into account. When the divergence was neglected the lowest mode appeared to be exceptional, in that the group-velocity was always in the same direction. This anomaly is now removed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 34 , Issue 1 , 16 October 1968 , pp. 49 - 80
Copyright
© 1968 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antosiewicz, H. A. 1964 Bessel functions of fractional order. Chapter 10, pp. 435–478 in Handbook of Mathematical Functions, U.S. Nat. Bur. Standards Appl. Math. Ser. 55, Washington, D. C. 1046 pp.
Buchwald, V. T. & Adams, J. K. 1968 The propagation of continental shelf waves. Proc. Roy. Soc. A, in press.Google Scholar
Eckart, C. 1951 Surface waves on water of variable depth. U.S. Office of Naval Research. Wave Rept. no. 100. (Notes of lectures given at the Scripps Institution of Oceanography.)Google Scholar
Fox, L. 1950 Brit. Ass. Math. Tables, vol. 6: Bessel Functions. Pt. I. Functions of orders zero and unity. Cambridge University Press.
LONGUET-HIGGINS, M. S. 1965 Some dynamical aspects of ocean currents Quart. J. Roy. Meteor. Soc. 91, 425–457.Google Scholar
LONGUET-HIGGINS, M. S. 1967 On the trapping of wave energy round islands J. Fluid Mech. 29, 781–821.Google Scholar
LONGUET-HIGGINS, M. S. 1968 On the trapping of waves along a discontinuity of depth in a rotating ocean J. Fluid Mech. 31, 417.Google Scholar
Miller, J. C. P. 1946 The Airy integral. Brit. Ass. Adv. Sci. Mathematical Tables Part vol. B. Cambridge University Press.
Olver, F. W. J. 1955 The asymptotic solution of linear differential equations of the second order for large values of a parameter. Phil. Trans A 247, 307–327.Google Scholar
Olver, F. W. J. 1964 Bessel functions of integer order. Chapter 9, pp. 355–434 in Handbook of Mathematical Functions, U.S. Nat. Bur. Standards Appl. Math. Ser. 55. Washington, D.C. 1046 pp.
Phillips, N. A. 1965 Elementary Rossby waves Tellus, 17, 295–301.Google Scholar
Rhines, P. B. 1967 The influence of bottom topography on long-period waves in the ocean. Ph.D. dissertation, Cambridge University.
Shinbrot, M. 1963 Note on a nonlinear eigenvalue problem Proc. Am. Math. Soc. 14, 552–558.Google Scholar