# Feasibility Studies of Alpha-Channeling in Mirror Machines Page: 5 of 12

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3

companied by the particle ejection into the loss cone can

be observed in this system. In order to affect a wide range

of a particle pitch angles, it is further proposed to use

several diffusion paths with different slopes as shown in

Fig. 1. More general configurations of diffusion paths,

including intersecting networks of paths, were consid-

ered [22], however, the highest a-channeling efficiency

was demonstrated for the system of paths depicted on

Fig. 1. In the following section, a rough numerical op-

timization of the a-channeling efficiency with respect to

the rf region parameters is outlined.

III. OPTIMIZATION WITH RESPECT TO RF

REGION PARAMETERS

The conceptual picture of a-channeling discussed in

the previous section can be affected by such non-ideal ef-

fects as an inhomogeneous background magnetic field, a

finite wave spectrum, a finite diffusion path width (essen-

tial to affect a large portion of the a particle population),

nonlinear effects, and particle phase correlations. The in-

fluence of these effects on the feasibility of a-channeling

in practical mirror machines can be studied by perform-

ing a rough numerical optimization of the a-channeling

efficiency with respect to the rf region parameters. As a

first step, the numerical optimization discussed in Ref. 14

was performed without accounting for wave dispersion in

mirror machine plasmas. The next step, of course, is to

search for plasma waves with parameters close to opti-

mal, as discussed in Sec. IV.

The most straightforward approach to the simulation

of the a particle dynamics interacting with a wave in a

mirror machine is a solution of the Newtonian equations

of the a particle motion:

m = rv x B+ qE, (5)

C

where m is the a particle mass, q is the charge, c is

the speed of light, and B, E are the fixed magnetic

and electric fields correspondingly. Being the most accu-

rate model of the particle motion considered in Ref. 14,

Eq. (5) requires the largest computational time. An ap-

proach requiring less computational resources, based on

an approximate model, is to calculate the resonant in-

teraction of a particles with an electrostatic wave (see

Eqs. (2 5) of Ref. 14).

Simulations of a large number of particles can be per-

formed by simulating a particle random-walk in reduced

phase space, a state in which is described by the particlemagnetic moment p, the parallel momentum ply, the ra-

dial position r and the time t when the particle passes the

midplane. The change of the particle state s= ( , p l, r, t)

on a single axial bounce is given in this model by expres-

sion:n+1 n + f(W.) + d(W.) 7,

(6)

where f and d define, respectively, the averaged and

stochastic components of the kick that the particle re-

ceives from the wave in a single axial bounce time, w is

a 4-dimensional vector such that (w2) 0, (wwj) ='2,

and %2 is the particle state on the nth midplane cross-

ing. To further simplify the model, one can neglect the

random terms in the time excursion and assume that the

particle radial excursion Ar is related to the change of

the particle magnetic moment Ap by [23]:A nrnQ'

where f is the wave azimuthal wave number. The re-

maining components of f and d can be either found by

employing the bounce-averaged quasilinear diffusion the-

ory [24], or calculated numerically for each rf region using

Eq. (5) or Eqs. (2 5) of Ref. 14.

Numerical simulations of a particle motion in a mir-

ror machine in the presence of several electrostatic waves

were presented in Ref. 14, where it was shown that up

to 60% of the total a particle energy can, in principle,

be extracted from a practical open-ended fusion device.

Specifically, simulations show that 60% of the initial a

particle energy can be redirected to the wave in 300 ms.

The remaining 40% of the energy is accounted in two

ways: (a) some of this energy leaves the device with the

unconfined a particles and (b) the deeply trapped a par-

ticles simply stay in the device and give up their energy

to the other plasma species. A rough optimization of a-

channeling efficiency with respect to rf region parameters

(ignoring restrictions imposed by the plasma dispersion

relation) was performed and the optimal wave parame-

ters were identified. The fuel ion ejection by the rf regions

was shown to be weak, while the ion injection mechanism

employing the energy of the same waves which were used

for a-channeling [12] was demonstrated [14]. It should

be emphasized that these simulations were done for ideal

waves, just to answer the question of whether any waves,

not necessarily realistic waves, could produce a signifi-

cant a-channeling effect. With the answer to this ques-

tion positive, it is then important to find suitable waves.

We outline such a search in the next section.(7)

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Zhmoginov, A. I. & Fisch, N. J. Feasibility Studies of Alpha-Channeling in Mirror Machines, report, March 19, 2010; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc1013916/m1/5/: accessed March 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.