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.

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 poloidal magnetic flux. The normalized effective radius is defined by

$\rho _{\rm {eff}}={\sqrt {\psi _{N}}}$

where ψ_N is the normalized poloidal 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 r_eff (in meters) of a flux surface, several possibilities exist.

• From the volume V(ψ) enclosed in a flux surface (using V = 2 π²Rr_eff²)
• From the surface area S(ψ) of a flux surface (using S = 4 π²Rr_eff)

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 is not close to a toroid).

One can avoid making an (arbitrary) choice for R by defining

• r_eff = 2V/S (still implicitly assumes the surfaces are near toroids) or
• r_eff = dV/dS (more general)

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.