Effective plasma radius: Difference between revisions
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Various alternative definitions are possible. | Various alternative definitions are possible. | ||
== | == Normalized effective radius based on flux == | ||
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 | == Effective radius based on flux == | ||
To obtain the dimensional effective radius ''r''<sub>eff</sub> (in meters) of a flux surface, several possibilities exist. | To obtain the dimensional effective radius ''r''<sub>eff</sub> (in meters) of a flux surface, several possibilities exist. | ||
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* ''r''<sub>eff</sub> = 2''V/S'' or | * ''r''<sub>eff</sub> = 2''V/S'' or | ||
* ''r''<sub>eff</sub> = ''dV/dS'' | * ''r''<sub>eff</sub> = ''dV/dS'' | ||
== 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. | |||
This procedure, while general in principle, still assumes that the field lines lie on flux surfaces. | |||
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> | |||
== Hybrid definitions == | == Hybrid definitions == | ||
* Use the flux-based normalized effective radius ρ<sub>eff</sub> defined above and multiply by the mean radius of the LCFS | * Use the flux-based normalized effective radius ρ<sub>eff</sub> defined above and multiply by the mean field-line based radius of the LCFS. | ||
== References == | == References == | ||
<references /> | <references /> | ||
Revision as of 09:52, 4 May 2010
The definition of the effective plasma radius is non-trivial for stellarators, yet needed for comparing the measurements of diagnostics. Various alternative definitions are possible.
Normalized effective radius based on flux
If the flux surfaces are known (typically, by calculating the 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).
E.g., at TJ-II, magnetic equilibria can be obtained from the VMEC code (see TJ-II:Magnetic coordinates). It returns ψ, the poloidal magnetic flux. The normalized effective radius is defined by
where ψN 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 flux
To obtain the dimensional effective radius reff (in meters) of a flux surface, several possibilities exist.
- Use the volume V(ψ) enclosed in a flux surface (V = 2 π2Rreff2)
- Use the surface area S(ψ) of a flux surface (S = 4 π2Rreff)
Here, R is the major radius of the torus. Particularly in helical systems, choosing a value of R may be inappropriate (since the magnetic axis is not a circle).
One can avoid making an (arbitrary) choice for R by defining
- reff = 2V/S or
- reff = dV/dS
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.
This procedure, while general in principle, still assumes that the field lines lie on flux surfaces. It can be used for magnetic configurations with 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. [1]
Hybrid definitions
- Use the flux-based normalized effective radius ρeff defined above and multiply by the mean field-line based radius of the LCFS.