The free oscillations of fluid on a hemisphere bounded by meridians of longitude

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Abstract
A precise calculation is presented of the normal modes of oscillation of an ocean of uniform depth which is bounded by two meridians of longitude separated by an angle of 180°. The calculation takes full account of the horizontal divergence of the motion, and so is applicable to both barotropic and baroclinic modes of oscillation.At small values of the parameter (defined fully in § 1) the calculation yields both the familiar gravity waves and also the nondivergent planetary waves computed in an earlier paper (Longuet-Higgins 1966). At large, positive values of e, corresponding to baroclinic waves, new types of oscillation appear in which the flux of energy is concentrated near the equator, the circuit being completed by Kelvin waves along the meridianal boundaries. The calculated frequencies are compared with asymptotic expressions derived from a recent beta-plane analysis by D. W. Moore. Solutions are also found corresponding to negative values of e. These must be included in a complete calculation of the response of the ocean to external forces. At small values of e these solutions resemble the planetary waves. At large (negative) values of e they represent almost-inertial motions concentrated near the poles, having a phase-velocity towards the east and an amplitude modulated so as to vanish at the boundaries. The calculations are relevant to the real ocean in so far as they show the kinds of oscillation that might be expected in any ocean basin including any section of the equator (or including a pole). They also indicate the degree of accuracy to be expected in computing the frequencies of the normal modes by beta-plane methods.

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