Planetary waves on a rotating sphere

Abstract

Simple solutions are found for barotropic planetary oscillations of a fluid in a $\beta$-plane, both in the unbounded plane and in enclosed basins of various shapes. These are compared with analogous motions of fluid on the surface of a sphere. The motions in a $\beta$-plane are considered in part I. It is shown that waves can exist which may be oriented in any horizontal direction; they drift westwards with a velocity $\beta$/$\kappa^2$, where $\kappa$ = 2$\pi$/wavelength is the absolute wave number. The group velocity of these waves makes an angle 2$\alpha$ with the eastward direction, where $\alpha$ is the angle made by the vector wave number. The reflexion of such waves from a fixed boundary is studied; both the wavelength and orientation of the reflected wave differs from that of the incident wave in general. Westward-drifting motions are typical of motions on an unbounded $\beta$-plane. On the other hand, motions in an enclosed basin can be described as a carrier wave modulated by a real amplitude function f(x, y). The equation for f is equivalent to the equation for a vibrating membrane clamped at the boundaries; normal mode solutions can be obtained explicitly for a variety of shapes of basin, including the rectangle, circle and equilateral triangle. Motions on a sphere are considered in part II. On an unbounded sphere the general solutions are spherical harmonics S$_n$($\theta$', $\phi$') where ($\theta$', $\phi$') denote spherical co-ordinates with respect to some pole P', not necessarily on the axis of rotation. The motions are propagated by a westward drift of the pole P' round a circle of latitude, with angular velocity 2$\Omega$/n(n+1). Solutions in enclosed basins have been found not only when the boundaries of the basin are circles of latitude, but also when the boundaries are meridians of longitude. The validity of the $\beta$-plane approximation is investigated, first by determining the asymptotic forms of the surface harmonics when the wave number n is high; secondly by comparing the periods of the lower modes in an enclosed basin on a sphere with the corresponding periods for the $\beta$-plane. At high wave numbers the solutions in terms of spherical harmonics do generally reduce to motions satisfying the $\beta$-plane equations: but exceptions occur in the neighbourhood of certain caustic lines, where the variation of $\beta$ must be taken into account. Thus it is possible for a wave motion to be trapped in the neighbourhood of a great-circle, the amplitude falling off rapidly to either side. The plane of this great-circle rotates slowly round the axis of rotation of the sphere. On the other hand, in an enclosed basin centred on the equator some of the lower modes of oscillation agree very well with those derived from the $\beta$-plane approximation. Even when the radius of the basin is as great as one quadrant, the periods of the four lowest symmetric modes agree within 10%.