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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. | 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'' | * ''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 | This definition is more general, although its validity is subject to the mentioned assumption. A fully general definition follows from | ||
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which is zero for axisymmetic systems (since ''ψ'' does not depend on ''φ''), but non-zero for stellarators. | which is zero for axisymmetic systems (since ''ψ'' does not depend on ''φ''), but non-zero for stellarators. | ||
== 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 == | ||