RADIOFREQUENCY WAVES, HEATING AND CURRENT DRIVE IN MAGNETICALLY CONFINED PLASMAS
6.2. THEORY OF RF WAVE PROPAGATION IN A MAGNETIZED PLASMA The theory of wave propagation in magnetized plasmas has been
6.2.5. Quasi-linear absorption of ICRF waves on ions
As seen above, typically ICRF wave power is absorbed by a small fraction of minority ions, which then collisionally equilibrates with the bulk plasma (majority ion species) and this will automatically lead to a highly distorted non-Maxwellian distribution of the minority species. This means that while the
above absorption models may initially be quite accurate, ultimately we must consider a more accurate model based on quasi-linear theory [6.3, 6.43]. Similar processes may occur also for cyclotron harmonic absorption and since the power absorption is proportional to bi( 1)l , for l2 the more energetic particles will preferentially absorb the RF power and hence become even more energetic, thus developing a perpendicular ion tail, balanced by collisions with the bulk. Again, quasi-linear theory is necessary for an accurate description of this process.
To obtain the true distribution function f of a species of charged particles, one must solve a Fokker–Planck equation, including quasi-linear (QL) diffusion and collisional drag and diffusion of the form [6.3, 6.43]:
QL v v 1 v v v v
2
f f f f
t t
(6.90)
where the 2nd and 3rd terms on the RHS correspond to collisional terms and
QL v QL v
f D f
t
(6.91)
is the wave induced quasi-linear term. Although the waves may be coherent, the transiting particles lose phase memory as they pass around the torus hundreds of times even if they experience only rare collisions. In general this is a difficult problem which has been solved analytically only in a few instances. For example, the case of a minority species distribution in the steady state in the presence of ICRF heating and collisional drag has been determined by Stix [6.3, 6.43]. In the steady state the result is the characterization of a high energy minority tail by an effective temperature
3/2
R ( )
1 1 1 1
(1 ) (1 ) 1 /
j e j e
eff e j j j
T T T
T T T R E E
(6.92) where j stands for minority ion species and
2 ,
,
R v
v
j j th e
j e th j
n Z
n (6.93)
,
1/2 2 4
v
8 ln
th e e
m P n nZ e
(6.94)
and P is the average power per unit volume deposited. Here the majority ion species is characterized by density ni, temperature Tj, charge eZj and thermal speed
, 1/2
vth j (2 /T mj j) ; electrons are characterized by density ne and thermal speed vth,e; the minority species being accelerated by cyclotron resonance are designated by n, m, v, Z and E m v / 22 . We note that for 0 the minority ion species
are characterized by a temperature close to that of the majority ion temperature, whereas for EEj( ) , Teff Te(1) and ion acceleration is entirely balanced by electron drag. As will be shown below, the experimental verification of this theory has been one of the early triumphs of ICRF experiments on tokamaks. The results are summarized in Fig. 6.7 [6.3, 6.43].
FIG. 6.7. Fokker–Planck energy distribution calculated for RF excitation at the minority cyclotron resonance for different normalized RF power densities [6.43].
Let us now ignore collisions and examine in more detail the quasi-linear diffusion term in the presence of RF waves. In particular, the power absorbed by charged particles can be calculated as follows:
2 0
3
QL
v v
abs m2 f
P d
t
(6.95)
The quasi-linear evolution of the distribution function in a magnetized plasma has been given by Kennel and Engelmann nearly three decades ago and the expression in 3-D is given in Ref. [6.50]. For simplicity, we shall skip details not applicable here and we readily deduce the following relevant expression for ion cyclotron resonance absorption [6.47]:
0(v) f
t
2 2 2 2 2 0
2 1
1 v v v 1
v v v v
8
l ci l ci
k l f
Z e E J
m k k
(6.96)
where J and δ are Bessel and delta functions respectively, and the summation is over ion cyclotron harmonics. Using cylindrical coordinates and azimuthal symmetry, we may readily proceed with the integration in Eq. (6.96):
2 2 2 2 2 2
0 0 1
0
(v ) v v v
4 v
abs res l
ci
Z e k
P f E f J d
m k
(6.97) where vres ( l ci) /k is the resonant velocity determined by the δ function vanishing argument. For a Maxwellian, Eq. (6.97) can be readily integrated for small arguments of the Bessel function and we obtain [6.47]
2 2 2
1/2 ( 1) ( 1) 2 2
, ,
( )
16 v ( 1)!2 exp v
pi l ci
abs i l
th i th i
E l l
P b
k l k
(6.98) which is exactly the same result as Eq. (6.83). We can now use Eq. (6.97) to integrate over a Maxwellian distribution to all orders of the finite ion Larmor radius. In particular, taking the derivative and integrating by parts, Eq. (6.97) can be integrated exactly and we obtain [6.47]
2 2
1 1
1/2 ,
( ) ( ) ( )
16 v
pi i
abs l i l i l i
th i
E l b
P I b I b I b
k l
×
× 2
2 2,
exp v
i ci
b
th i
e l
k
(6.99)
where Il is the modified Bessel function of order l and argument bi. The small Larmor radius limit of Eq. (6.99) results from the first term of the square bracket (expand Il1( )bi ) and it agrees with Eq. (6.98). It is easy to generalize Eq. (6.99) to include absorption on an energetic minority ion species. Using the bulk plasma to calculate the polarization and dividing Pabs with S, we can easily obtain the transmission coefficient 2, due to an energetic minority component or at a cyclotron harmonic. We note that in obtaining Eq. (6.99) we kept only the contribution from E. However, it was shown that for a tail or minority energy of the order of 100 keV or more, contributions from E can be just as important, especially for higher harmonics [6.51]. An evolution of the actual distribution function at harmonics is the topic of present day research using full wave codes and iterating them with a Fokker–Planck code [6.52].
Furthermore, for energetic ions of the order of 100 keV, finite-width banana orbits may be important and this is covered by Monte Carlo codes at the present time [6.53]. These examples show the power of using quasi-linear theory for calculating power absorption to all orders of the Larmor radius and for arbitrary distribution functions. In particular, larger values of bi lead to stronger
absorption and the result is that cyclotron harmonic resonances in the plasma lead to energetic ion tail production due to quasi-linear diffusion.
We can also use Eq. (6.95) to calculate power absorption due to electron Landau damping and electron transit time magnetic pumping. It has been shown [6.47] that for a Maxwellian plasma the result for the quasi-linear power absorption is identical to Eq. (6.75), and the damping rates are identical to those obtained previously. In general, due to collisional drag, using the fast magnetosonic wave it is difficult to distort the distribution function from a Maxwellian for / v k 2, which is the usual region of interest for reasonable single pass damping (i.e. 10% per pass). Thus, for most cases of practical interest for fast wave electron absorption the results obtained earlier (Eqs (6.77–6.81)) usually suffice.