On the trapping of waves along a discontinuity of depth in a rotating ocean

Abstract

It is shown that, according to the linearized theory of long waves in a rotating, unbounded sea, if there is a discontinuity in depth along a straight line separating two regions each of uniform depth, then wave motions may exist which are propagated along the discontinuity and whose amplitude falls off exponentially to either side. Thus the discontinuity acts as a kind of wave-guide. The period of the waves is always greater than the inertial period. The wave period also exceeds the period of Kelvin waves in the deeper medium. As the ratio of the depth tends to infinity, the wave period tends to the inertial period or to the Kelvin wave period, whichever is the greater. On the other hand as the wavelength decreases (within the limits of shallow-water theory) so the waves tend to the non-divergent planetary waves found recently by Rhines. In an infinite ocean of uniform depth free waves with period greater than a pendulum-day cannot normally be propagated without attenuation (if the Coriolis parameter is constant). But non-uniformities of depth provide a means whereby such energy may be channelled over great distances with little attenuation. It is suggested that a gradually diminishing discontinuity will act as a chromatograph, each position along the discontinuity being marked by waves of a particular period.