Effective plasma radius

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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, 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 laws.

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 toroidal magnetic flux. The normalized effective radius is defined by

$ \rho_{\rm eff} = \sqrt{\psi_N} $

where ψN 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 reff (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 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 π2Rreff2)
  • Based on the surface area S(ψ) of a flux surface (using 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, and the shape of the flux surfaces deviates from that of a torus). One can avoid making an (arbitrary) choice for R by defining

  • reff = 2V/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 reff (or V is cuadratic in reff), it follows that dr = (dS/S) dV/dS = dr/r dV/dS, so:

  • reff = dV/dS

This definition is more general, although its validity is subject to the mentioned assumption. A fully general definition follows from

  • $ r_{\rm eff} = \int_0^V{dV'/S(V')} $

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

$ \vec \nabla \psi \cdot \vec \nabla \phi $

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 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.

See also

References

  1. B. Seiwald et al, Optimization of energy confinement in the 1/ν regime for stellarators, Journal of Computational Physics 227, 12 (2008) 6165-6183