A variational calculation of the trapping rate in thermal barriers

Abstract

A variational calculation of the trapping rate and trapped-ion density in thermal barriers is presented. The effects of diffusion in energy as well as pitch-angle scattering are retained. The variational formulation uses the actual trapped/passing boundary in velocity space. The boundary condition is that the trapped-ion distribution function match the passing-ion distribution function, which is taken to be a Maxwellian, on the boundary. The results compare well with the two-dimensional Fokker-Planck code calculations by Futch and LoDestro. The CPU time for a variational calculation is less than 0.1 s using the CRAY-I computer, while a typical Fokker-Planck code calculation takes 10–20 min.

The magnetic mirror approach to fusion

“…We give here the Li-Emmert results (Ref. [217], second part); these provide good analytical fits to the more detailed F-P results.…”

“…Futch and LoDestro [215] have performed F-P code calculations and then fitted their results approximately with analytical expressions. Carrera and Callen [216] and Li and Emmert [217] have approached the problem by making analytical approximations, finding expressions that rather accurately reproduce the Futch-LoDestro F-P results. More recently, Devoto et al [218] have performed extensive new F-P code calculations and have provided analytical fits to their results.…”

“…, taken from TableIof Ref [217],. where the analytical result of Li and Emmert agrees closely with the F-P result, is the following:Input data: n p = 1.1 x 10 11 cm' 3 ; T p = 15 keV; R b = 2.0; tf> b = 40 keV; p = 1.0 s" 1 .…”

Low-energy beam pumping in TARA reactor design

“…Assuming equilibrium with a constant pumping rate v^, we may write the trapping current Jt as Jt = "L n t L (1) where n ^ is the trapped-particle density in the lowenergy beam pumping region. Using the variational principle [5,6], we may write another expression for the trapping current…”

“…Here, gb is the ratio of total density to passing particle density at the bottom of the barrier. We have analysed this problem by using a variational method which has been applied previously to the case of a well-pumped thermal barrier [5,6]. We apply this method to a two-step square well model which approximates the local pumping magnetic geometry.…”