Thermal barrier ion trapping rate in a tandem mirror

Abstract

An analytic expression for the Coulomb collision induced ion trapping rate in the thermal barrier cell of a tandem mirror has been obtained. The overall system is assumed to be maintained in steady state by particle injection in the central cell and charge-exchange pumping in the barrier cell. For small ratios of bounce-to-collision time scales the problem reduces to a series of boundary value problems in the various regions of phase space. For conditions of interest, pitch-angle trapping is dominant and a Lorentz collision operator describes reasonably well the kinetic problem, which is solved using a square-well approximation. The analytic results are found to agree with numerical results within expected limits, on the order of the inverse of the barrier/mirror ratio (∼ 10 to 20%).

The magnetic mirror approach to fusion

“…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.…”

A variational calculation of the trapping rate in thermal barriers

“…In Ref. [9], the comparison was made on the basis of the same total-ion density, nb, since this is what directly determines the collision frequency, and therefore the trapping rate, in the thermal barrier. Comparison on the basis of a given passing-ion density is a more sensitive test, however, since one has to calculate both the total density and the trapping rate.…”

“…There are two steps in the calculation; making the comparison In the limit of large gb, Jtrap is only weakly dependent on gb and approaches vn^ (which are simply input data in Ref. [9]). For example, Futeh and LoDestro used their code for the calculation of the low-mirror-ratio case (Rb = 2).…”

“…With the input n p = 0.55 X 10" cm" 3 , they obtained n b = 14.68 X 10" cm' 3 , the trapping current Jt= 3.53 X 10" cm" 3 "s" 1 , and gb = 26.7. Can-era and Callen [9] used the code output value nb = 14.68 X 10 11 cm" 3 as their input and obtained their output Jt = 3.09 X 10" cm" 3 -s 1 . This represents an error of 12% in Jt.…”

“…Numerical calculations of the trapping rate in thermal barriers have been done by Futch and LoDestro [8] using Fokker-Planck codes. Carrera and Callen [9] have given an analytical calculation of the trapping rate using a pitch-angle scattering collision operator; diffusion in energy was neglected. In this paper we present a variational calculation which allows a better treatment of the boundary condition and includes the effect of energy diffusion.…”