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Effective plasma radius: Difference between revisions
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where ''ψ<sub>N</sub>'' is the normalized toroidal 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 toroidal 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 ''r''<sub>eff</sub> (in meters) of a flux surface, it is common to make the assumption that the shape of the flux surface does not deviate much from a [[:Wikipedia:Torus|torus]]. In this case, several possibilities exist to define a radius: | |||
* Based on the volume ''V(ψ)'' enclosed in a flux surface (using ''V'' = 2 π<sup>2</sup>''Rr''<sub>eff</sub><sup>2</sup>) | |||
* Based on the surface area ''S(ψ)'' of a flux surface (using ''S'' = 4 π<sup>2</sup>''Rr''<sub>eff</sub>) | |||
Here, ''R'' is the [[Toroidal coordinates|major radius]] of the [[:Wikipedia:Torus|torus]]. | |||
Particularly in helical systems, choosing a value of ''R'' may be inappropriate (since the magnetic axis is not a circle, and the shape of the flux surfaces deviates from that of a torus). | |||
One can avoid making an (arbitrary) choice for ''R'' by defining | |||
* ''r''<sub>eff</sub> = 2''V/S'' | |||
This still implicitly assumes the surfaces are very similar to a torus. | |||
A different approach is offered by recognizing that the flux surfaces are topological toroids of a single parameter. | |||
Then, the surface area and volume corresponding to such surfaces are related via a differential equation (''dV = S dr''). | |||
Assuming only that ''S'' is linear in ''r''<sub>eff</sub> (or ''V'' is cuadratic in ''r''<sub>eff</sub>), it follows that | |||
''dr = (dS/S) dV/dS = dr/r dV/dS'', so: | |||
* ''r''<sub>eff</sub> = ''dV/dS'' | |||
This definition is more general, although its validity is subject to the mentioned assumption. A fully general definition follows from | |||
* <math>r_{\rm eff} = \int_0^V{dV'/S(V')}</math> | |||
but it requires knowledge of the full equilibrium in terms of the function ''S(V)''. | |||
== Effective radius based on poloidal cross sections == | == Effective radius based on poloidal cross sections == | ||