Feasibility Studies of Alpha-Channeling in Mirror Machines Page: 7 of 12
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The dispersion relation D(kn, kI,, r, z) can be transformed
to the new variables by replacing kIl with )k,, kn with
akR, and by substituting the known expressions for
r(R, y) and z(R, y). Note that the system (15), (16) is
identical to the more complicated system of equations (3-
6) of Ref. 13. Here, however, the curvatures of the mag-
netic field lines and the lines orthogonal to them enter
the wave packet motion equations transparently through
a and ), and the derivation of the approximate equations
can be carried out by replacing these dependencies with
the lowest-order terms from Eqs. (10) and (11).
In the general case, the dynamics governed by the
two-dimensional Hamiltonian D(kR, kq, R, y) can be very
complex. However, there are two special cases, in which
it can be simplified significantly. In these cases, the
dynamics in kR and R variables is much slower (quasi-
longitudinal wave) or much faster (quasi-transverse wave)
compared to the dynamics in k and n.
Consider first a quasi-longitudinal wave. In this case,
one can find the longitudinal dynamics approximately
by fixing values of kR and R and solving the result-
ing dispersion relation D(k, y; kR, R) 0. The slow
transverse dynamics in (kR, R) variables can then be cal-
culated either by requiring that the adiabatic invariant
In(kR, R) f k dy associated with the longitudinal ray
motion is conserved or by averaging Eqs. (15) over the
fast longitudinal bounces. If the magnetic field profile
B(z) is fixed, one can find all ray trajectories localized
axially while passing R Ro by considering k(y; kR);
namely by solving D(k, y; kR, Ro) 0 for different val-
ues of kR. If, however, B(z) can also be varied to search
for the localized wave trajectories, the problem becomes
much more complex. This problem can be simplified
by noticing that in the lowest order a and ) depend
on B but not on its higher derivatives. Assuming also
that the plasma temperature and the line density of the
plasma are constant along the device , one can see
that D(k, n; kR, R) depends on y through B(n) only.
Therefore, all the information about the axially localized
ray trajectories can be extracted from k(B; kR) solving
D(k, B; kR, Ro) 0. While details of this method and
its application to the search for weakly-damped axially-
and radially-localized modes can be found in Ref. 13, the
essential elements of the search algorithm can be seen
directly from the present reformulation.
Consider now a quasi-transverse wave. In this case, by
analogy with the quasi-longitudinal wave, one can find
the transverse dynamics first by fixing values of k, and
y and solving the dispersion relation D(kR, R; k, n) 0.
The slow longitudinal dynamics in (k, y) variables can
then be calculated from the conservation of the transverse
adiabatic invariant IR(ky, y) f kR dR or by averaging
Eqs. (16) over the fast oscillations in (kR, R) coordinates
[the averaged equations are equivalent to Eqs. (8) and (9)
of Ref. 13]. If the magnetic field profile B(z) is fixed, the
axially-localized modes can be identified by studying a
"phase portrait" in (k, y) coordinates. This portrait can
be calculated by solving IR(k, y) I for different values
of I. In the case when the magnetic field profile B is also
allowed to vary, one can use the same approach employed
above. Specifically, after substituting the lowest-order
expressions for a and ) in D and assuming again that
the plasma temperature and the line density are constant
along the device axis, the dispersion relation will depend
on y through B(n) only. Therefore, the search for axially
localized ray trajectories can be performed by studying
kI (B; I) solving IR(k, B) I for various values of I. The
search for weakly-damped modes can be simplified even
further recalling that kI < kn for the waves of interest.
In this case, assuming that the wave is localized near the
midplane, one can expand D in powers of k and n and
consider the lowest-order terms only. This approximation
and its application to the search of the quasi-transverse,
axially-localized modes are similarly discussed in greater
detail in Ref. 13.
V. MODES SUITABLE FOR
The quasi-longitudinal and the quasi-transverse waves
suitable for a-channeling were identified in two practical
open-ended fusion devices [13, 16, 17] using the methods
outlined in Sec. IV. Both modes satisfied the fast wave
dispersion relation reading
and + b
b -n n2
a 1 - _'e-_ nIn(a,)
S X(Lj - nQ,)'
2 6-A, [2In(X)
b Z I - J e - n I(A + 2A2 (In
d ;2 ne A(Ir -I') 2 e
d + PC.
w w-nQ,, JQ
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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/m1/7/: accessed April 18, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.