On generalised toroidal-poloidal solutions of vector field equations

Abstract

AbstractThe orthogonal coordinate systems ξi(i = 1, 2, 3) are determined, in which the gneralised toroidal and poloidal fields, defined respectively by T{T} = ∇ × {T∇ξ1} and S{S} = ∇ × T{S}, have the following three properties:GP1 Decoupling of the vector Helmholtz equation: There exist linear differential operators L1 and L2 such that Hu = 0, where H is the vector Helmholtz operator [see equation (1)] and u = T{T} + S {S}, if and only if L1T = 0 and L2S = 0.GP2 OrthogonalityGP3 Closure: ∇ × S{S} is a T field.Two choices of T and S fields are considered: type I fields with potentials T and S, which may depend on ξ1, ξ2 and ξ3, and type II fields with ξ1-independent potentials. It is shown that properties GP1–GP3 only hold for type I fields in spherical and cylindrical coordinate systems, and for type II fields in azimuthal and cylindrical coordinate systems with axisymmetric and two-dimensional potentials, respectively. Analogues of GP1 for the vector wave and diffusion equations, and the Navier equation of linear elasticity, are also only true in the same four cases. Generalisations of type I and II T and S fields to arbitrary coordinate systems are indicated.