Feasibility Studies of Alpha-Channeling in Mirror Machines Page: 6 of 12
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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
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
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
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:
k = OD
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
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 . 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, () . 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 function
4I(r, z, kR, k) = kRR(r, z) + ky(r, z),
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'
kR - brkz -bzk~r k,
k t 0,
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-
kR -OD/OR, R = OD/OIkR,
ky =-OD/8, = OD/Okq.
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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.