A new integral property of inertial waves in rotating fluid spheres

Abstract

The problem of fluid motion in the form of inertial waves in an incompressible inviscid fluid contained in a rotating sphere is governed by the Poincaré equation, a second-order hyperbolic partial differential equation. Its explicit general analytical solution in terms of a double Poincaré polynomial was found by Zhang et al . ( Zhang et al . 2001 J. Fluid Mech . 437 , 103–119), describing the pressure p mnK and the velocity u mnK of spherical inertial waves, where the triple indices ( mnK ) are indicative of the azimuthal, vertical and radial structures, respectively. On the basis of the general explicit solution, we reveal a new intriguing integral property of the spherical inertial waves for all possible values of m , l and n , where M and K are related to the degree of the double Poincaré polynomial and denotes the complex conjugate of u mlM . A mathematical proof of the vanishing of the integral involving the construction of two auxiliary recurrence relations is presented. Furthermore, a comparison with the corresponding integral for rotating cylinders is made, showing a fundamental difference between the two systems.