Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 1. Fields with divergence

Abstract

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.

Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 2. The Stokes equations

“…The incompressibility constraint (3.2) implies the existence of the Schmitt-Wahl mean-poloidal-toroidal representation [8,17]…”

“…In Part 1 [8] an orthogonal decomposition of doubly periodic (dp.) vector fields into five parts, three of which are divergence-free, was derived.…”