Effective plasma radius: Difference between revisions

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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.

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''eff (or ''V'' is cuadratic in ''r''eff), it follows that:

Assuming only that ''S'' is linear in ''r''eff (or ''V'' is cuadratic in ''r''eff), it follows that

''dr = (dS/S) dV/dS = dr/r dV/dS'', so:

* ''r''eff = ''dV/dS''

This definition is more general.

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 ==

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== See also ==

* [[:~~File~~: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.

* [[: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 ==

@@ Line 27: / Line 27: @@
 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.
+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:
+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.
+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 ''&phi; = 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 ''&phi;''.
+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 ''&phi; = 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 ''&psi;'' does not depend on ''&phi;''), but non-zero for stellarators.
 == Effective radius based on field lines ==
@@ Line 46: / Line 60: @@
 == See also ==
-* [[:File: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.
+* [[: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 />