Effective plasma radius

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 ψ_N is the normalized toroidal flux, such that it is 0 on the magnetic axis and 1 at the Last Closed Flux Surface (LCFS).

Very true! Makes a change to see smooene spell it out like that. :)

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.

Hybrid definitions

• Use the flux-based normalized effective radius ρ_eff defined above and multiply by the mean field-line based radius of the LCFS.

See also

• 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 laws.