1. magnetic confinement - ENEA - Fusione

1. MAGNETIC CONFINEMENT
47
1.3 Plasma Theory
1
0.8
0.6
0.4
0.2
(m-1,n)
non-local
continuum damping
(m,n)
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5
x
Fig. 1.44 - Radial structure Fig 1.44 of the shear
Alfvén continuous spectrum for the (m,
n) and (m-1, n) modes in the case
1≥Ω A,m +Ω A,m-1 >>-r 0 /R 0 . The value of
q 2 0 R 2 0 k 2
|| is shown vs. S≡√n ” 0 (r-r 0 ). The
frequencies of the (m, n) and (m-1, n)
modes are also shown as they are
expected from (8).
1
0.8
0.6
non-local
continuum damping
2
for ω d >ω B >>ω. Here, λ H =(k ⊥ ν ⊥ )/ω cH , ΘF 0H =(2ω∂/∂ν 2 +k×b•∇
/ω cH )F 0H , F 0H is the fast hydrogen tail distribution function, and
assuming deeply trapped banana orbits [1.70], for which the mode
drive is expected to be the strongest. For Ω A,m +Ω A,m-1 >>r 0 /R 0 ,
toroidal coupling between the (m,n) and (m-1,n) modes can be
neglected (see fig. 1.44). The two modes then satisfy the following
approximate dispersion relations, derived from a variational
principle [1.67]:
Sπ
⎛
2n
Λ
⎞
ΩAm
Ω
m
, + = −
/ / ⎜
1
52 12 2
n ⎝S
Ω ⎟
2
Am , ⎠
Sπ
⎛
2n
Λ
⎞
ΩAm
Ω
m
, − = −1
−1 −
/ / ⎜
1
52 12 2
n ⎝S
Ω ⎟
2
Am , −1
⎠
where Λ m-1 ≅>ω; thus, the fast ions are characterised by
negative compressibility, which causes the mode frequency to be
shifted upward, contrary to the general case for which fast ion
compression shifts the mode frequency downward [1.65-1.67]. For
this reason, only the (m,n) mode can exist just above the local
maximum of the Alfvén continuum at r 0 , but not the (m-1,n) mode
just below the local minimum of the continuum. The condition for
the existence of the (m,n)mode is Λ m /Ω A,m >S 2 /2n, which, as a
consequence of (8) is independent of n [1.68] and of the mode
frequency. This condition is a lower bound on the fast hydrogen tail
particle density, and for the present parameters
[S=1.54,–n H /(R 0 ∂ r n H )=0.13] it gives n H /n e >3.1%, consistent with
the experimental values (n H /n e =4%). Changing the fast-ion nonresonant
response, the (m-1,n) mode instead of the (m,n) can be
excited [1.70].
(9)
0.4
0.2
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5
x
Fig. 1.45 - Same as fig. 1.44 but for
Fig 1.45
–r 0 /R 0 ≤Ω A,m +Ω A,m-1 ≤r 0 /R 0 .
[1.70] F. Zonca, et al.,
Energetic particle mode
stability in tokamaks with
hollow q-profiles, to be
published on Phys. of
Plasmas
2
Since Ω A,m +ΩA ,m-1 →0 + (which may occur when, as in the
experiment, q 0 drops), as in figure 1.45, toroidal coupling effects
become important. Two branches are still present as in (9), one of
which strongly continuum damped (odd mode [1.67]), and the
other (even mode [1.67]) satisfying the modified dispersion
relation:
which can be straightforwardly derived from within the theoretical
approach of [1.67] in a simple but still relevant limiting case, and where
ε 0 ≡2(r 0 /R 0 +∆’),Ω A,m ≅-1/2. Note that on the left-hand side of (10), the fourth root of
the quantity in parentheses – and not the square root as in the usual TAE case – is
due to the local minimum in the q-profile [1.67]. The existence condition for the
nearly undamped mode of (10) is exactly the same as that discussed for (9). Clearly,
in both cases the exponentially small continuum damping, due to the non-local
interaction with the (m,n)mode continuum, and other kinetic damping mechanisms
must be evaluated and compared with the resonant drive associated with fast ions
[1.67] before the existence of these modes is demonstrated on a rigorous basis. The
complete expressions of non-local continuum damping and fast ion resonant and non-
⎡
π
ε0
2 4 2 14
⎛
2 Λ
Ω −⎜
⎞⎤
/
S
⎛
n
⎞
Ω −14
/ ⎟ = m −1
⎣⎢ ⎝ ⎠⎦⎥ 32 / 12 / ⎜ 2
2 ⎝ Ω ⎟
n S Am , ⎠
(10)