Poloidal Field

A poloidal field relative to a chosen origin is a vector field of the form

where is the position vector from that origin and is a sufficiently smooth scalar function (Backus 1986). For , the equivalent expression is

Here is the gradient, is the curl, and is the Laplacian. A poloidal field is a divergenceless field, also called a solenoidal field, because the divergence of a curl vanishes.

A sufficiently smooth divergenceless field on a spherical shell centered at the chosen origin, with zero flux integral through each concentric sphere, can be decomposed into a poloidal field and a toroidal field (Backus 1986). In general, these components change when the chosen origin changes. This separation is important in geo- and heliophysics, and in particular in dynamo theory and helioseismology.