# 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 on *S*.
In other words, the magnetic field does not *cross* the surface *S* anywhere, i.e., the magnetic flux traversing *S* is zero.
It is then possible to define a scalar *flux function* (*f*) such that its value is constant on the surface *S*, and

- $ \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.
^{[1]}
This fact lies at the basis of the design of magnetic confinement devices.

Assuming the flux surfaces have this toroidal topology, the function *f* defines a set of *nested* surfaces, so it makes sense to use this function to label the flux surfaces, i.e., *f* may be used as a "radial" coordinate. Each toroidal surface *f* encloses a volume *V(f)*.
The surface corresponding to an infinitesimal volume *V* is essentially a line that corresponds to
the *toroidal axis* (called *magnetic axis* when *B* is a magnetic field).

The flux *F* through an arbitrary surface *S* is given by

- $ F = \int_S{\vec B \cdot \vec n dS} $

When *B* is a magnetic field with toroidal nested flux surfaces, two magnetic fluxes can be defined from two corresponding surfaces.
^{[2]}
The poloidal flux is defined by

- $ \psi = \int_{S_p}{\vec B \cdot \vec n dS} $

where *S _{p}* is a ring-shaped ribbon stretched between the magnetic axis and the flux surface

*f*. (Complementarily,

*S*can be taken to be a surface spanning the central hole of the torus.

_{p}^{[3]}) Likewise, the toroidal flux is defined by

- $ \phi = \int_{S_t}{\vec B \cdot \vec n dS} $

where *S _{t}* is a poloidal section of the flux surface.
It is natural to use ψ or φ to label the flux surfaces instead of the unphysical label

*f*.

## See also

- MHD equilibrium
- Toroidal coordinates
- Flux coordinates
- Rotational transform
- Magnetic shear
- Effective plasma radius
- Separatrix
- Flux tube

## References

- ↑ The Poincaré-Hopf Theorem.
- ↑ R.D. Hazeltine, J.D. Meiss,
*Plasma Confinement*, Courier Dover Publications (2003) ISBN 0486432424 - ↑ A.H. Boozer,
*Physics of magnetically confined plasmas*, Rev. Mod. Phys.**76**(2005) 1071 - 1141