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Effective plasma radius: Difference between revisions
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The definition of the effective plasma radius is non-trivial for stellarators, yet needed for comparing the measurements of diagnostics. | 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. | Various alternative definitions are possible, some of which are discussed below. | ||
The numerical value of the effective radius has hardly more significance than that of providing a flux surface label, | |||
except when plasmas from different machines are compared. | |||
Therefore, this issue is relevant for the elaboration of multi-machine [[Scaling law]]s. | |||
== Normalized effective radius based on flux == | == Normalized effective radius based on flux == | ||
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* Based on the surface area ''S(ψ)'' of a flux surface (using ''S'' = 4 π<sup>2</sup>''Rr''<sub>eff</sub>) | * 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]]. | 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 | 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''). | |||
* ''r''<sub>eff</sub> = ''dV/dS'' | 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 == | |||
A poloidal cross section is a cut of the flux surface with the plane ''φ = cst''. | |||
The result of such a cut is a closed curve, of which its circumference and area are easily determined; an effective plasma radius can then be deduced, assuming the curve deviates only slightly from a circle. | |||
The mean plasma radius can be determined by averaging the result over the angle ''φ''. | |||
While the procedure is adequate for toroidally symmetric plasmas, it is not clear that this is also the case for non-axisymmetric systems, since the flux surface intersects the plane ''φ = cst'' obliquely, possibly leading to an over-estimate of the actual plasma size. | |||
The intersection angle can be deduced from the inner product | |||
:<math>\vec \nabla \psi \cdot \vec \nabla \phi</math> | |||
which is zero for axisymmetic systems (since ''ψ'' does not depend on ''φ''), but non-zero for stellarators. | |||
== Effective radius based on field lines == | == Effective radius based on field lines == | ||
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== See also == | == See also == | ||
* [[: | * [[:Media:Surf_vol.pdf|Comments on the use of the minor radius for stellarators]], where it is argued that it is preferable to use the Volume or Surface directly, instead of the effective radius, in view of the ambiguities in the definition of the latter - at least when making comparisons between different machines, e.g., in the framework of [[Scaling law]]s. | ||
== References == | == References == | ||
<references /> | <references /> | ||