# Feasibility Studies of Alpha-Channeling in Mirror Machines Page: 6 of 12

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4

IV. SEARCH FOR WAVES

The numerical optimizations of the a-channeling ef-

ficiency discussed in Sec. III were performed under the

assumption that ki < kL and Lo Q. The simulations

confirmed that in order to maximize the extracted en-

ergy, the wave performing a-channeling should interact

with the deeply-trapped a particles and kL must satisfy

kLp 1. Furthermore, the wave amplitude should be suf-

ficiently large to expel a particles out of the device over a

time much smaller than the typical a-electron collisional

relaxation time.

If the wave damping is strong, significant energy is

required to excite the waves, possibly even larger than

the energy extracted from a particles using this wave.

Therefore, the fusion reaction rate affects the maximum

allowed wave damping rate. Ideally, the wave damping

rate should be approximately equal to the wave growth

rate due to the a particle energy extraction. In the fol-

lowing, we assume that the wave growth rate is small, so

that the excitation of a localized weakly-damped mode

trapped in the device core is necessary. This confines our

search significantly.

Weakly-damped modes satisfying these conditions can

be found assuming the validity of the geometrical optics

approximation and restricting the search to the waves

propagating primarily along (quasi-longitudinal waves)

or transverse (quasi-transverse waves) to the magnetic

field lines. We also assume that the device is axisym-

metric and fix the azimuthal wave number f. A method

of identifying such modes is described in Ref. 13, where

several potential mode candidates existing in practical

mirror designs are identified. Here, using a more sys-

tematic approach than that used in Ref. 13, we rederive

the ray-tracing equations for the quasi-longitudinal and

the quasi-transverse waves, making more clear the mode

search algorithm.

The ray-tracing equations for quasi-longitudinal and

quasi-transverse waves can be derived by considering first

the ray-tracing equations in (r, z, kT, kz) coordinates:OD

r= ,

Ok

OD ,OD

kr= ,r

k = OD

kz2 Oz'where D(k, kz, r, z; ) is a local dispersion relation

tained from the original D(k, ky, kz, x, y, z; ) by sub

tuting x - r cos 0, y - r sin 0, kx = k cos 0 -ra si

and ky = k sin 0 + fr-1 cos 0. The motions of the w(8)

packet along and across the magnetic field lines can be

decoupled by introducing new coordinates (R, y) such

that R and y are constant along the magnetic field lines

and along the curves orthogonal to them correspond-

ingly. Requiring that R(r, z = 0) = r, (r = 0, z) = z,

OR/Or -abz, OR/Oz = abr, Oy/Or = )b,, and

Oy/Oz = , where b = (b, bz) is a unit vector field

directed along the magnetic field lines, one can calcu-

late functions a and ). In particular, considering a

divergence-free magnetic field given by B = -rB(z)'/2

and Bz(r, z) = B(z), where B(z) is a function defining

magnetic field profile along the device axis, one obtains:

B(z) r2' r4 10

a = B(0) (1 882 ) + O L4 (10)

B"(z) 3r2B'2(z) 4

=1 -r2 + 82(z) + (11)

8(z) /B() L

where we assumed that the characteristic device radius is

much smaller than the device length L. The important

property of this coordinate transformation is that the

basis vectors eR and e. are orthogonal when the original

basis (e5, ez) is orthonormal [40]. The metric tensor gzj in

the new coordinates is diagonal and gRR -- " e - a-2,

n - e 2. Note that even though the lines of

constant R coincide with the lines of constant magnetic

flux, the new coordinates (R, y) are different from the flux

coordinates (p, () [25]. In particular, the magnetic flux

p is a non-linear function of R and the linear coordinate

along the magnetic field line ( is equal to y only in the

lowest order in r/L.

Making a canonical transformation in D from (r, z) to

(R, y) using a generating function4I(r, z, kR, k) = kRR(r, z) + ky(r, z),

(12)

one obtains that kT = kR OR/Or + kn Oy/Or and k,

kR OR/Oz + kn Oy/Oz. Substituting the expressions for

R(r, z) and n(r, z) here, one finally obtains for kR and k'

[41]:kR - brkz -bzk~r k,

a a

berkr+bzkz kI

k t 0,(13)

(14)where k bk and kn-k2-kl.

(9) The new coordinates (R, y) and the new momenta

(kR, k) form a canonical set of coordinates and there-

ob- fore:'sti-

[10,

avekR -OD/OR, R = OD/OIkR,

ky =-OD/8, = OD/Okq.(15)

(16)

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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/m1/6/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.