second-order turning problems in oceanography开题资料.pdf

Deep-Sea Research, 1975, Vol. 22, pp. 837 to 852. Pergamon Press. Printed in Great Britain.
Second-order turning point problems in oceanography
RONALD SMITH*
(Received 6 August 1974; final revision received 17 April 1975; accepted 17 April 1975)
Almtract--Short-wave asymptotic solutions are obtained for three problems: small-scale internal
waves near a local minimum of the Brunt-V~ii~l/i frequency, shallow water waves near a minimum
diameter of a channel, and continental shelf waves near a local minimum of the cut-off frequency.
The common property of these problems is that for waves of a special frequency there is a position P
such that locally the component of group velocity towards P is directly proportional to the distance
from P.
1. INTRODUCTION infinite wave amplitude. In particular, if there is a
IN TIlE oceans there are numerous examples of point x = 0 (say) such that locally the component
wave motions to which the Liouville-Crreen (or of group velocity towards x = 0 is directly
.) approximation is applicable, . in the proportional to x,
direction of propagation the length scale of . c(x) = xcl + O(x2),
variations in the propagating medium greatly then the Liouville-Green solution (lb) would
exceeds a typical wavelength and the wave can suggest that it takes an infinite time for wave
be represented as being locally sinusoidal with an energy to reach x = 0 and that ultimately the
amplitude and wavelength which vary on the wave amplitude has an inverse square-root
same length scale as the propagating medium. singularity at x = 0. The aim of this paper is to
For such waves the group velocity is of central illustrate that such second-order turning points
importance, being the velocity at which the wave have relevance