On generalised toroidal-poloidal solutions of vector field equations

Source abstract

Abstract The orthogonal coordinate systems ξ i ( i = 1, 2, 3) are determined, in which the gneralised toroidal and poloidal fields, defined respectively by T{ T } = ∇ × { T ∇ξ 1 } and S{ S } = ∇ × T{ S }, have the following three properties: GP1 Decoupling of the vector Helmholtz equation : There exist linear differential operators L 1 and L 2 such that Hu = 0, where H is the vector Helmholtz operator [see equation (1)] and u = T{ T } + S { S }, if and only if L 1 T = 0 and L 2 S = 0. GP2 Orthogonality GP3 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 ξ 1 , ξ 2 and ξ 3 , and type II fields with ξ 1 -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.