Three temperatures distribution function model

by Yves Peysson (CEA/IRFM)

April 18, 2019
The three temperatures model has been introduced for simulating an anistropic tail in the electron momentum distribution function driven by the quasi-electrostatic wave at the Lower Hybrid frequency in tokamak. It is a very crude model which was used to characterize the fast electron bremsstrahlung during these non-inductive regimes, which extends up to high photon energies corresponding to hard X rays.

With the emergence of refined models based on the use of Fokker-Planck solvers, the three temperatures model is almost obsolete, and its application is restricted to benchmark non-thermal bremsstrahlung codes designed for analysing corresponding diagnostics.

Let define the 3-D distribution function f(ψ,p,ξ), where ψ is the poloidal magnetic flux coordinate (radial position), p the momentum value and ξ = p∥∕p the pitch-angle relative to the local magnetic field direction. Here, f is considered to be homogeneous on a magnetic flux surface, and the trapped electron population is neglected. According to the three temperatures model,

(1)

where fM(ψ,p,ξ) is the thermal bulk, and f3T(ψ,p,ξ) is the non-thermal part. By definition, both fM and f3T are normalized to unity, so that f automatically satisfies this condition. The relative fraction of fast electrons which is roughly given by the ratio λ3T∕λM is usually very small, of the order of 1 × 10-3.

1 The Maxwellian distribution function
2 The non-maxwellian tail
3 Calculation the distribution

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