Flux surface: Difference between revisions
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:<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>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''. 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 with zero volume is the ''toroidal axis'' (called ''magnetic axis'' when ''B'' is a magnetic field). | |||
== References == | == References == | ||
<references /> | <references /> | ||
Revision as of 14:59, 19 August 2009
A given smooth surface S with normal n is a flux surface of a smooth vector field B when
everywhere. Defining a scalar flux function (f) such that its value is constant on the surface S, this can be rewritten
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.
If a single vector field B has several such toroidal flux surfaces, they must necessarily be nested. 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 with zero volume is the toroidal axis (called magnetic axis when B is a magnetic field).
References
- ↑ The Poincaré-Hopf Theorem.