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
6.2.6. Quasi-linear absorption on electrons
Consider the quasi-linear deformation of the electron distribution from 1-D ( v) analysis of the Fokker–Planck equation, with application to lower hybrid current drive (LHCD). Wave absorption may be computed assuming a Maxwellian distribution for the electrons where feexp[ (v / v th e, ) ]2 and
, 1/2
vth e(2 /T me e) . However, it is well known that LH waves damping on “tail”
electrons at v ~ 3v th e, (see Ref. [6.54]) will locally (in velocity space) modify the distribution function through acceleration by the RF electric field.
FIG. 6.8. Local modification of the electron distribution function as LH waves damp on “tail”
electrons.
This modification of the electron distribution results in a “flattening”
in velocity space which in turn can modify the local rate of Landau damping which is proportional to fe / v (see for example Fig. 6.8). Fisch [6.55, 6.56]
recognized that it was possible to employ RF waves in the LHRF regime to create non-thermal particle distributions of the type shown in Fig. 6.8 which could be used to drive net toroidal current in a tokamak, thus making steady state operation feasible. Phased arrays of waveguides can be used to launch travelling LH waves (e.g. 90° phasing) that will impart momentum preferentially to electrons in one
direction. Furthermore, the process is highly efficient as the current carrying electrons are less collisional than the bulk electrons, being created at v3vth e, , so they would retain their momentum for a longer time than thermal electrons (v=vth e, ). It is possible to obtain quantitative estimates of the efficiency of this process by following the 1-D (v) solution method of Fisch [6.55]. Consider the 1-D ( v) electron Fokker–Planck equation given by Eqs (6.100–6.103):
(v ) (v ) ( )
v v v
e RF e e e
e
f eE
D f C f f
t m
(6.100)
2,
2 1
( ) (v ) v v
v 2 v 2 v
eff e
e e th e
Z f
C f f
(6.101)
3
0 ,
(v ) vth e/ v
(6.102)
4
0 3
,
log ( )
2 v
pe e
e th e
n
(6.103)
where DRF(v ) is the RF or quasi-linear diffusion coefficient [6.50] due to the RF waves and is proportional to ERF2 , and C(fe) is the high velocity form of the Braginskii collision operator [6.57]. The third term on the right hand side of Eq. (6.100) corresponds to the Dreicer effect [6.58] whereby tail electrons in the presence of a DC electric field can be accelerated enough to overcome the electron collisional drag and “run away” to very high velocities. In deriving Eqs (6.100–6.103), a Maxwellian distribution was assumed in the perpendicular direction where fe(v , v ) f0exp[ (v / v ) ] (v ) th e, 2 fe in order to integrate the 2-D ( v, v) Fokker–Planck equation over perpendicular velocities [6.55, 6.59]. Physically, Eqs (6.100–6.103) describe a competition between quasi-linear diffusion in parallel velocity space and collisions which tend to restore the distribution to a Maxwellian. A steady state solution of Eq. (6.101) can be found by assuming that DRF(v ) is non-zero within a finite region of parallel velocity space (v1vv2, where v1 and v2 are lower and upper parallel velocity plateau limits), neglecting the DC field term and taking (v ) 0fe and fe/ v 0 at
v :
2
1
v 0
v
( ) exp 12 ( )
e e
RF
f w f wdw
D w
(6.104)
2,
( ) 2 ( ) / ( )v
RF 2 RF th e
D w D w w
Z
(6.105)
where wv / v th e, is the normalized parallel phase speed. If
2,
(v ) / [ (v )v ] 1
RF th e
D , that is, if the quasi-linear diffusion is strong compared to the collisional diffusion, then a plateau forms in the distribution function similar to what is depicted in Fig. 6.8. The normalization constant Fe0 is chosen to conserve number density. Equations (6.104) and (6.105) can be used in the definitions of the driven RF current density (jRF) and RF power dissipation (SRF) given below (Eqs (6.106) and (6.107)) to derive a simple expression for the normalized current drive efficiency [6.55]:
(v ) (v ) v
RF e e
j n e f d
∞
-∞
= ∫
(6.106)
1 2
(v ) v (v ) (v ) v
2 v v
RF e e e
S n m D f d
∞
-∞
∂ ∂
= ∫ ∂ ∂
(6.107)
, 2 2
2 1 1 2
0 2,
/ ( v ) 1 / log ( / )
2
/ ( v )
RF e th e
eff e
RF e e th e
j n e
w w w w
Z
S n m
η= n = + - (6.108)
where w1=v / v1 th e, and w2 =v / v2 th e,. For LH current drive the upper parallel velocity plateau limit v2 is typically determined by either wave accessibility or the extent of the launcher power spectrum and the lower parallel velocity limit v1 is determined by the condition for strong Landau damping, i.e. v1≈3vth e, . Physically we see that efficiency of the current drive process is higher for higher phase velocity LH waves, which is intuitively consistent with the notion that these waves impart more momentum to electrons than slower waves. However, for higher phase velocities, accessibility (Eq. (6.58)) dictates lower values of
2 / 2
pe ce
or higher magnetic field for a given value of the density (which may be determined by the fusion power requirement ~nTE).
The physical picture of current drive described above turns out to be a simplified view of the actual process. Numerical solutions of the Fokker–Planck equation with a model diffusion operator [6.59] and an elegant analysis by Fisch and Boozer [6.60] revealed that 2-D (v, v) velocity space effects are responsible for more than half of the driven current owing to the creation of an asymmetric resistivity from preferential heating of electrons. During LH current drive the asymmetry is driven by pitch angle scattering of electrons from the parallel to the perpendicular direction, while in electron cyclotron current drive (ECCD) the asymmetry arises from direct perpendicular heating of electrons.
The resulting normalized efficiency is given by [6.60]:
, 3
2 2
0 ,
/ ( v ) ( ) 4
5
/ ( ) ( )
RF e th e s
RF e e th e s eff
j n e s wu
Z
S n m v s u
η n
= = ⋅ ∇
+
⋅ ∇
(6.109)
where u2 w2x2, x=v / v^ th e, and s is a velocity displacement vector.
The enhancement in LH current drive efficiency that comes from including 2-D velocity space effects can be seen by evaluating Eqs (6.108) and (6.109) for a narrow spectrum of LH waves where (w2 w1), ( / ) 1 w1 , s e z
and parallel to B
is taken to be along the z–direction. Taking w2w w1 yields
1D (2 / 3)w2
η ≈ and η2D≈(4 / 3)w2, so that 2-D effects result in an approximate factor of two increase in the current drive efficiency. Physically, this enhancement in efficiency occurs because of enhanced pitch angle scattering at high parallel phase speed that results in a large effective perpendicular temperature in the distribution function and a concomitant decrease in collisionality. This effect has been studied extensively using numerical models of varying detail[6.59, 6.61–6.63] and an example of this for a broad spectrum of LH waves is shown in Fig. 6.9 [6.64].
FIG. 6.9. 2-D Fokker–Planck simulation of the electron distribution function during LH current drive in Alcator C-Mod showing perpendicular broadening effect due to pitch angle scattering. Here is the relativistic factor defined as 1-(v / c)2-1/ 2, where c is the speed of light, and v is the total velocity. Reprinted from Ref. [6.64]. Copyright (2011), American Institute of Physics.
6.2.6.2. Electron cyclotron current drive
From the discussion in the preceding section it can be seen that waves that preferentially heat electrons in the perpendicular direction can be used to drive current simply through the creation of an asymmetric plasma resistivity. Indeed this is the case with electron cyclotron current drive (ECCD). If one assumes that the velocity displacement vector (s) in Eq. (6.109) is in the direction perpendicular to B
, then it is straightforward to show that 2D for ECCD is about ắ the value it is for LHCD [6.60].
However, this estimate omits the effects of particle trapping. EC waves accelerate electrons primarily in the direction perpendicular to the magnetic field, which increases their magnetic moment and hence their tendency to be trapped in the magnetic well. Ohkawa [6.65] exploited this effect to show that enhanced trapping of electrons travelling in one toroidal direction would result in a net
toroidal current. Unfortunately, this current is opposite in sign to the current due to the asymmetric resistivity that is generated by the same EC wave [6.66].
These trapping effects [6.67, 6.68], as well as other important physical effects like momentum conserving corrections to the background collision operator [6.61] have been investigated in detail through a Green’s function treatment of the Fokker–Planck equation that takes advantage of the self-adjoint property of the collision operator [6.69]
( )
3
v 3 RF RF
RF RF
j p d p
S d p
χ
∂ ⋅ G
∂
= G ⋅
∫
∫
(6.110)
QL e
RF D f
p
(6.111)
Here ( ) p is the Green’s function that solves the Spitzer–Họrm problem, namely (C fm ) q fv m, where C is the high velocity collision operator and fm is the perturbed distribution function. The advantage of this approach for treating ECCD [6.70] is that a full solution of the Fokker–Planck equation is not necessary in order to compute jRF because the RF wave induced flux can be computed directly by taking fe = fe0, since there is very little distortion of the distribution function in ECCD. An example of this technique applied to estimate the effect of particle trapping on the EC current drive efficiency is shown in Fig. 6.10 left [6.67]. It can be seen that trapping significantly reduces the efficiency of ECCD at /r R0.1 on the outside of the flux surfaces (LFS of tokamak). This is to be contrasted with methods that generate much faster electrons, such as LHCD, which is shown in Fig. 6.10 right.
FIG. 6.10. Left: Non-relativistic ECCD current drive efficiency from an adjoint treatment versus inverse aspect ratio for the inside and outside extremes of a flux surface. Taken from Cohen [6.67]. Right: Non-relativistic LHCD current drive efficiency from an adjoint treatment vs. inverse aspect ratio for inside and outside extremes of flux surfaces. In this figure j is the driven EC current density normalized to its thermal value and Pd is the dissipated ECRF power density normalized to its thermal value following the definitions used in Eqs (6.108) and (6.109). Reprinted from Ref. [6.67]. Copyright (2011), American Institute of Physics.
A clear picture of the physics involved in ECCD can be obtained from Fokker–Planck code calculations. Figure 6.11 [6.66] shows the results of a calculation using the CQL3D Fokker–Planck code for ECCD applied at 0.4 at a poloidal location directly above the discharge centre. The arrows in Fig. 6.11 show the net flux in velocity space due to a wave–plasma interaction in the region shaded white. These fluxes are the resultant of the flux due to the EC wave interaction combined with the collisional relaxation flux, forming convective cells in velocity space. Although the EC wave accelerates the electrons in the perpendicular direction in the interaction region, the net flux is along the resonance curve, with a significant parallel component. This figure shows that even though the interaction region is far from the trapping boundary, the interaction of electrons with the magnetic well can still be important. The total driven current can be found from the first moment of the distribution function associated with the fluxes of Fig. 6.11 and it will be shown later that the Fokker–
Planck model is the best fit to the experiment.
FIG. 6.11. Velocity space diagram for ECCD. The yellow circles are contours of constant v / vth e, , the green lines separate the circulating particles below the lines from the trapped particles above the lines, and the red curves are the resonance condition at each side of a small range about the minor radius where the wave interaction is strongest. The white shading shows where the wave–particle interaction is strongest. The arrows indicate the direction of the flux in this velocity space due to the wave and to collisional relaxation. The length of the arrows is proportional to the logarithm of the local flux. Reprinted from Ref. [6.66]. Copyright (2011), American Institute of Physics.
6.2.6.3. Current drive by the fast ICRF waves
It is also possible to drive current with low frequency waves in the ICRF regime. Although these waves may have a low parallel phase speed /k, they can be regarded as having a high momentum content, as measured by the parallel wavenumber (k) [6.71]. For example, fast magnetosonic waves at
ci can damp via direct electron Landau damping, transit time magnetic pumping or cyclotron damping.
Furthermore, since these waves cause minimal distortion to the distribution function when they damp, the adjoint treatment method of the previous section can also be used to estimate the current drive efficiency [6.61] as shown in Fig. 6.10. The right hand scale in Fig. 6.12 gives the CD efficiency for a plasma with ne10 m20 –3, Te10 keV and R01m. The dashed lines in Fig. 6.12 correspond to the normalized efficiency from an asymptotic analysis of the adjoint solution of the Fokker–Planck equation. For (v / v ) th e, 21 the leading terms from this analysis given by Eqs (31) and (32a) from Ref. [6.61] for Landau damped and cyclotron damped waves respectively are:
, 2 ,
0 2,
4(v / v ) / ( v )
/ ( v ) 5
RF e th e th e
eff
RF e e th e
j n e
Z
S n mn = + + ⋅⋅⋅
(Landau damped waves) (6.112)
, 2 ,
0 2,
3(v / v ) / ( v )
5
( v )
RF e th e th e RF e e th e eff
j n e
Z
S n mn = + + ⋅⋅⋅
(cyclotron damped waves) (6.113)
It is interesting to note that these expressions are to leading order identical to the estimates given Ref. [6.60].
FIG. 6.12. Current drive efficiency obtained versus normalized parallel phase speed for low frequency waves damping via Landau (“L”) and cyclotron (“C”) damping (reproduced from Ref. [6.61]). Shown as dashed lines are asymptotic results for (v / v ) th e, 21 using Eqs (31) and (32a) from Ref. [6.61]. On the left y-axis in this figure jRF [A.m–2] is the driven RF current density, PRF [W.m–3] is the dissipated RF power density, q [C] is the charge, ρt=me th ev , is the thermal momentum value, and nt is the thermal value of the collision frequency given by Eq. (6.103). On the right y-axis in this figure IRF [A] is the integrated driven RF current and WRF [W] is the integrated RF power dissipation. Reprinted from Ref. [6.61]. Copyright (2011), American Institute of Physics.