ANZIAM  J.  47 (2005), 21-38
Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 1. Fields with divergence

G. D. McBain
  School of Aerospace
  Mechanical and Mechatronic Engineering
  The University of Sydney

It is shown how to decompose a three-dimensional field periodic in two Cartesian coordinates into five parts, three of which are identically divergence-free and the other two orthogonal to all divergence-free fields. The three divergence-free parts coincide with the mean, poloidal and toroidal fields of Schmitt and Wahl; the present work, therefore, extends their decomposition from divergence-free fields to fields of arbitrary divergence. For the representation of known and unknown fields, each of the five subspaces is characterised by both a projection and a scalar representation. Use of Fourier components and wave coordinates reduces poloidal fields to the sum of two-dimensional poloidal fields, and toroidal fields to the sum of unidirectional toroidal fields.
Download the article in PDF format (size 392 Kb)

Australian Mathematical Publishing Association Inc. ©  Australian MS