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Flux surface: Difference between revisions
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A given surface ''S'' with normal ''n'' is a flux surface of a vector field ''B'' when | A given smooth surface ''S'' with normal ''n'' is a flux surface of a smooth vector field ''B'' when | ||
:<math>\vec B \cdot \vec n = 0</math> | :<math>\vec B \cdot \vec n = 0</math> | ||
everywhere. | 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. | ||
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 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 == | == References == | ||
<references /> | <references /> | ||