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Effective plasma radius: Difference between revisions
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Various alternative definitions are possible. | Various alternative definitions are possible. | ||
== | == Flux-based normalized effective radius == | ||
If the [[Flux surface|flux surfaces]] are known (typically, by calculating the [[MHD equilibrium|Magneto-Hydrodynamic equilibrium]]), and assuming the existence of toroidally nested flux surfaces, the simplest procedure is to define the mean radius as a function of some flux quantity (i.e., any quantity that is constant on a flux surface). | If the [[Flux surface|flux surfaces]] are known (typically, by calculating the [[MHD equilibrium|Magneto-Hydrodynamic equilibrium]]), and assuming the existence of toroidally nested flux surfaces, the simplest procedure is to define the mean radius as a function of some flux quantity (i.e., any quantity that is constant on a flux surface). | ||
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where ''ψ<sub>N</sub>'' is the normalized poloidal flux, such that it is 0 on the magnetic axis and 1 at the Last Closed Flux Surface (LCFS). | where ''ψ<sub>N</sub>'' is the normalized poloidal flux, such that it is 0 on the magnetic axis and 1 at the Last Closed Flux Surface (LCFS). | ||
== Effective radius based on field lines == | |||
If the flux surfaces are not known, the effective radius of a surface traced out by a field line can be found by following the field line and calculating the geometric mean of the distance between points on the field line and the magnetic axis. The mean should be weighed with ''1/B'' in order to account for the variation of the field strength along the flux surface. | If the flux surfaces are not known, the effective radius of a surface traced out by a field line can be found by following the field line and calculating the geometric mean of the distance between points on the field line and the magnetic axis. The mean should be weighed with ''1/B'' in order to account for the variation of the field strength along the flux surface. | ||
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It can be used for magnetic configurations with [[Magnetic island|magnetic islands]], although this requires applying some special treatment for points inside the islands. It may be argued that assigning an effective radius to spatial points inside a magnetic island is not very useful, since such points are topologically disconnected from the main plasma volume. Similarly, the definition of an effective radius in ergodic magnetic zones is ambiguous, since the concept of flux surface has no meaning inside an ergodic zone. | It can be used for magnetic configurations with [[Magnetic island|magnetic islands]], although this requires applying some special treatment for points inside the islands. It may be argued that assigning an effective radius to spatial points inside a magnetic island is not very useful, since such points are topologically disconnected from the main plasma volume. Similarly, the definition of an effective radius in ergodic magnetic zones is ambiguous, since the concept of flux surface has no meaning inside an ergodic zone. | ||
<ref>[http://dx.doi.org/10.1016/j.jcp.2008.02.026 B. Seiwald et al, ''Optimization of energy confinement in the 1/ν regime for stellarators'', Journal of Computational Physics '''227''', 12 (2008) 6165-6183]</ref> | <ref>[http://dx.doi.org/10.1016/j.jcp.2008.02.026 B. Seiwald et al, ''Optimization of energy confinement in the 1/ν regime for stellarators'', Journal of Computational Physics '''227''', 12 (2008) 6165-6183]</ref> | ||
== Effective radius based on volume or surface == | |||
To obtain the dimensional effective radius ''r''<sub>eff</sub> (in meters) of a flux surface, several possibilities exist. | |||
* Use the volume ''V(ψ)'' enclosed in a flux surface (''V'' = 2 π<sup>2</sup>''Rr''<sub>eff</sub><sup>2</sup>) | |||
* Use the surface area ''S(ψ)'' of a flux surface (''S'' = 4 π<sup>2</sup>''Rr''<sub>eff</sub>) | |||
Here, ''R'' is the [[Toroidal coordinates|major radius]] of the [[:Wikipedia:Torus|torus]]. | |||
One can avoid making an (arbitrary) choice for ''R'' by defining | |||
* ''r''<sub>eff</sub> = 2''V/S'' or | |||
* ''r''<sub>eff</sub> = ''dV/dS'' | |||
== Hybrid definitions == | |||
* Use the flux-based normalized effective radius ρ<sub>eff</sub> defined above and multiply by the mean radius of the LCFS (e.g., by following field lines) | |||
== References == | == References == | ||
<references /> | <references /> | ||