Toroidal and Poloidal Field Representation for Convective Flow Within a Sphere.
A method for determining the onset of convection in a fluid confined within a sphere and heated from below. The velocity field is analyzed in terms of toroidal and poloidal fields, expressed in spherical Bessel functions and spherical harmonics, which have certain orthogonality properties that are useful in treating convective flow problems within spheres. The utility of this representation is demonstrated by considering the stability of a nonuniformly heated fluid in a spherical cavity. A variational principle is presented, equivalent to the eigenvalue problem for the critical Rayleigh number (the stability criterion). This principle forms the basis for an approximate method of determining upper bounds to the critical Rayleigh number. It is found that a class of three-dimensional disturbances is more unstable than either the simplest poloidal (axisymmetric) disturbance mode or the simplest toroidal (two-dimensional) disturbance mode. The numerical results are compared with previously published analyses. 39 pp. Refs.