ANZIAM J.
47 (2005), 2138

Plane poloidaltoroidal decomposition of doubly periodic vector fields. Part 1. Fields with divergence


Abstract

It is shown how to decompose a threedimensional
field periodic in two Cartesian coordinates into
five parts, three of which are identically
divergencefree and the other two orthogonal to
all divergencefree fields. The three
divergencefree parts coincide with the mean,
poloidal and toroidal fields of Schmitt and Wahl;
the present work, therefore, extends their
decomposition from divergencefree 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 twodimensional
poloidal fields, and toroidal fields to the sum
of unidirectional toroidal fields.

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