1. magnetic confinement - ENEA - Fusione

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1. MAGNETIC CONFINEMENT
1.3 Plasma Theory
T e (3/4)T i , the zonal flow spontaneous
generation takes place in the wave cut-off region, i.e., with greatly reduced
effectiveness. In the case of AITG, similar conclusions may be drawn on the basis of
the mode dispersion relation [1.61]. Compression effects for AITG result in a ≈(1-
ω ∗pi /ω) scaling in the zonal flow growth rate. Thus, spontaneous excitation of zonal
flows by AITG is possible only for ω>ω ∗pi , which is typical of moderately unstable
AITG. For strong instability, detailed analyses still need to be carried out. On the
basis of the analytic dispersion relation, however, it is possible to conclude that -
sufficiently above threshold and for sufficiently low frequency - spontaneous
excitation of zonal flows is inhibited. If confirmed, this fact will have a strong impact
on the anomalous transport associated with AITG.
1.3.4 Drift and drift-Alfvén wave structures near a minimum-q
surface
Theoretical investigations of drift and drift-Alfvén mode structures near a minimumq
surface and eigenmode analyses that assume small but finite magnetic shear can be
discussed within a unified mathematical formulation. For the sake of clarity, the
problem is studied for the case where toroidal mode coupling can be consistently
neglected. The toroidal problem can be analysed in exactly the same fashion, but
with greater technical complexity.
It is part of common wisdom to assume that, within the usual ballooning formalism
[1.62], the translational invariance of radial mode structures breaks down for different
poloidal Fourier modes when magnetic shear vanishes. This is evidently true.
However, as shown in [1.63, 1.64], only the separation of spatial scales between
equilibrium quantities and wavelengths is really needed for the analysis of high-n
mode structures. This separation of scales is still valid for high-n modes (n being the
toroidal mode number) both near and at a minimum-q surface, where magnetic shear
vanishes by definition. Only this fairly general assumption is made in the following
treatment.
The strength of the formalism employing the separation of scales is based on the fact
that the fast radial scale and the spatial coordinate along the magnetic field can be
considered Fourier conjugate variables. This is obvious from the following identities:
[1.61] F. Zonca, et al.,
Phys. Plasmas 6, 1917,
(1999)
[1.62] J.W. Connor, R.J.
Hastie, and J.B. Taylor,
Phys. Rev. Lett. 40, 396
(1978)
[1.63] F. Zonca, Continuum
damping of toroidal
Alfvén eigenmodes in
finite-beta tokamak
equilibria, Ph.D. thesis.
Princeton University,
Plasma Physics Lab.,
Princeton N.J. (1993)
[1.64] F. Zonca and L.
Chen, Phys. Fluids B5,
3668 (1993)
'
∂ '
qR0k||; mn , = nq0( r −r0)
⇒i s ; q0
≠0
,
∂κr
''
nq
S
qR0k mn qR0k mn r 0
0 r r0 2 2
∂
'
||; , = ||; , ( ) + ( − ) ⇒ ΩAm
, - ; q
n
22 0 = 0
2 2 ∂κ
r
(2)
Here, the notation r 0 denotes the radial coordinate of the considered flux surface,
where q=q 0 , q’=q 0 ’ , etc. Meanwhile, κ r =(r 0 /m)κ r , s≡r 0 q 0 ’ /q 0 , Ω A,m =nq 0 -m,
S≡√r 0 q 0 ” /q 0 , and δφ m (κr), the Fourier Transform of the fluctuating field δΨ m (r), is
given by
∞
1 ⎛ m ⎞ m
δφm( κr) = ∫ exp i ⎜ κr( r−
r0) ⎟ δψm( r) d ( r−
r0)
2π
−∞ ⎝r0
⎠ r0
(3)
Here, s and S are, respectively, the usual and generalised magnetic shear definitions.
Note that for q 0
” < 0 the definition of S would change accordingly and that (2)
assumes Ω A,m =nq 0 -m=0, for q 0 ’ ≠0, as is always possible. From (2) and (3), it is
readily shown that