Flux surface

A given smooth surface S with normal n is a flux surface of a smooth vector field B when

${\vec {B}}\cdot {\vec {n}}=0$

everywhere. Defining a scalar flux function (f) such that its value is constant on the surface S, this can be rewritten

${\vec {B}}\cdot {\vec {\nabla }}f=0$

In three dimensions, the only closed flux surface corresponding to a non-vanishing vector field is a topological toroid. This fact lies at the basis of the design of magnetic confinement devices.

If a single vector field B has several such toroidal flux surfaces, they must necessarily be nested (since they cannot intersect) or be disjoint. Ignoring the latter possibility, it then makes sense to use the function f to label the flux surfaces, so f may be used as an effective "radial" coordinate. The toroidal surface enclosing a volume of zero size is the toroidal axis (called magnetic axis when B is a magnetic field).