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Flux surface (view source)

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:<math>\vec B \cdot \vec n = 0</math>
:<math>\vec B \cdot \vec n = 0</math>


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


:<math>\vec B \cdot \vec \nabla f = 0</math>
:<math>\vec B \cdot \vec \nabla f = 0</math>


In three dimensions, the only closed flux surface corresponding to a ''non-vanishing'' vector field is a topological toroid.
In three dimensions, the only closed flux surface corresponding to a ''non-vanishing'' vector field is a topological toroid.
<ref>The Poincaré-Hopf Theorem.</ref>
<ref>[[:Wikipedia:Hairy_ball_theorem|The Poincaré-Hopf Theorem]].</ref>
This fact lies at the basis of the design of magnetic confinement devices.
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).
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
 
:<math>F = \int_S{\vec B \cdot \vec n dS}</math>
 
 
[[File:Flux_definition.png|250px|thumb|right|Diagram showing the surfaces defining the poloidal (red) and toroidal (blue) flux]]
When ''B'' is a magnetic field with toroidal nested flux surfaces, two magnetic fluxes can be defined from two corresponding surfaces.
<ref>R.D. Hazeltine, J.D. Meiss, ''Plasma Confinement'', Courier Dover Publications (2003) {{ISBN|0486432424}}</ref>
The poloidal flux is defined by
 
:<math>\psi = \int_{S_p}{\vec B \cdot \vec n dS}</math>
 
where ''S<sub>p</sub>'' is a ring-shaped ribbon stretched between the magnetic axis and the flux surface ''f''.
(Complementarily, ''S<sub>p</sub>'' can be taken to be a surface spanning the central hole of the torus.<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetically confined plasmas'', Rev. Mod. Phys. '''76''' (2005) 1071 - 1141]</ref>)
Likewise, the toroidal flux is defined by
 
:<math>\phi = \int_{S_t}{\vec B \cdot \vec n dS}</math>
 
where ''S<sub>t</sub>'' is a poloidal section of the flux surface.
It is natural to use &psi; or &phi; 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 ==
== References ==
<references />
<references />
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