# 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 [42], 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, andy 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

ALPHA-CHANNELING

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(17)

and + b

- 2

b -n n2

where

a 1 - _'e-_ nIn(a,)

S X(Lj - nQ,)'

2 6-A, [2In(X)

b Z I - J e - n I(A + 2A2 (In

nQ, Ad ;2 ne A(Ir -I') 2 e

d + PC.

w w-nQ,, JQ(18)

(19)

(20)

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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.