Originally developed for the representation of elastic and viscous vibrations (Love 1944, p. 280; Lamb 1932, pp. 632-7) and magnetic vector potentials (Elsasser 1946), the poloidal-toroidal decomposition has also been used occasionally for fluid velocity fields (Lamb 1932, pp. 594-602; Sherman 1968; Joseph 1971; Busse 1975; Munson & Menguturk 1975; Chandrasekhar 1981, pp. 225-6; Forte & Peltier 1987; Bajer & Moffatt 1990; Palaniappan et al. 1992; Padmavathi, Raja Sekhar & Amaranth 1998). The technique is analogous to the introduction of a stream-function in two-dimensional or axisymmetric flow (Moffatt 1978, pp. 20-1); if the flow is axisymmetric, the scalar defining the poloidal part is directly related to Stokes's stream-function (see §B.5). The essential properties of the decomposition are summarized here; further details can be found in the works of Chadwick and Trowbridge (1967), Moffatt (1978, pp. 17-22), Chandrasekhar (1981, pp. 622-6), Backus (1986) and Backus, Parker and Constable (1996, ch. 5). Basically, a toroidal field is a solenoidal field with no radial component and a poloidal field is a solenoidal field with a toroidal curl.