Toroidal and Poloidal Field Representation for Convective Flow within a Sphere

In spherical regions, an arbitrary solenoidal velocity field can be developed into a series of toroidal and poloidal fields. These fundamental vector fields, expressed in terms of spherical Bessel functions and spherical harmonics, have certain orthogonality properties which prove useful in treating convective flow problems within spheres. The utility of this velocity field 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.