ANZIAM J.
47 (2005), 21-38
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Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 1. Fields with divergence
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Abstract
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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.
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Australian Mathematical Publishing Association Inc.
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Australian MS
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