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
6.2.7. Wave propagation in toroidal geometry, geometric optics
The calculation of wave propagation and absorption of waves in the toroidal geometry of tokamak confinement systems has been an area of active research
in fusion plasmas for the past few decades. Considerable effort is required to formulate the wave propagation and absorption in toroidal geometry and computer implementation is almost always needed to arrive at a solution. Owing to the short wavelengths () relative to the device size (a), in the lower hybrid range of frequencies (LHRF) and electron cyclotron range of frequencies (ECRF) ray tracing has been the preferred method. Here “^” refers to the direction perpendicular to the applied magnetic field (B
). In contrast, full-wave treatments have been used in the ion cyclotron range of frequencies (ICRF) where a. These efforts allowed detailed comparisons between simulation and experiment, including simulation of non-thermal particle distributions that are ubiquitous to ICRF heating and LH current drive experiments. The remainder of this section discusses the use of ray tracing techniques to simulate LH wave propagation and quasi-linear electron Landau damping with an application to LH current drive given as an example. Full-wave solutions of wave propagation and absorption in the ICRF regime are then reviewed and examples of this technique applied to minority heating and mode conversion are given. Finally, we discuss recent developments in the use of full-wave techniques to treat LH wave propagation in order to include potentially important effects not included in the geometric optics limit such as focusing and diffraction of waves at caustic surfaces and proper reconstruction of wave fronts at plasma cut-offs.
6.2.7.1. Geometric optics and the ray equations
An important property of plane waves is that their direction of propagation and their amplitude remain constant everywhere in space and time. In general, electromagnetic (EM) waves do not fit this description. Consider, however, a small region of space called a wave surface [6.72] on which the EM wave possesses the properties of a plane wave. On this wave surface the propagation direction and amplitude of the EM wave remain constant over the distance of a wavelength. Then a vector normal to this surface (the wave normal) would define the direction of propagation of the EM wave. The concept of a ray can then be introduced as a curve whose tangents at each point correspond to a wave normal, i.e. the direction of wave propagation. On each wave surface the electric and magnetic fields can be represented in terms of a phase function S x t( , )
known as the eikonal
( , ) 0exp[ ( , )]
E x t =E iS x t (6.114)
( , ) 0exp[ ( , )]
B x t =B iS x t (6.115)
where E0
and B0
are in general weakly varying functions of space and time with the eikonal function carrying the most important (zero-order) information
about the propagation direction. Some understanding of the eikonal function can be gained by assuming that over a small region of space and interval of time
( , )
S x t can be expanded in a Taylor series as:
( , ) 0 S S
S x t S x t
x t
(6.116)
By analogy with plane waves we can write S S 0 k x , where
k S S
x
(6.117)
S
t
(6.118)
The fact that k
can be expressed as the gradient of the eikonal is important since this also implies that k
is irrotational (i.e. that k 0
). For wave propagation in a plasma which is not varying in time and for which there is a spatial inhomogeneity in only one direction, it can be shown that the local dispersion relation provides a unique mapping between the wave position and its wavenumber. Thus for a wave propagating in a plasma which is varying in only the x-direction, the propagation is uniquely determined from the local dispersion relation D x k x0( , ( ), ) 0x , where ky and kz are constants. If however the plasma is varying in two or more spatial dimensions as is the case in a tokamak, then one must use the fact that k
is irrotational in addition to the local dispersion in order to determine ( )k x
uniquely, i.e.
0 k
(6.119)
0( , , ) 0
D x k (6.120)
It was shown by Weinberg [6.73] that the solution in phase space for a wave that satisfies Eqs (6.119) and (6.120) is given by the ray equations
0 0
/ /
D k
dx
dt D
(6.121)
0 0
/ / dk D dx
dt D
(6.122) Equation (6.121) is the group velocity equation for a wave packet with wave vector k
and frequency , while Eq. (6.122) can be viewed as the multi- dimensional analog of Snell’s Law. The fact that more than a single component of k
will vary in a plasma with more than one dimension of inhomogeneity can be seen to follow immediately from Eq. (6.122).
6.2.7.2. Modifications to wave accessibility and absorption in toroidal geometry We saw earlier that LH wave accessibility and absorption depends critically on the parallel wavenumber (k), which is typically imposed by the waveguide launcher at the plasma edge. Consider a toroidal geometry defined by the coordinates (r, q, f), where r is the minor radial position, q is the poloidal angle and f is the toroidal angle. The choice of wave momenta that keeps the ray equations in canonically conjugate form are k( , , )k m nr
, where kr is the radial wavenumber, m is the poloidal mode number and n is the toroidal mode number [6.74, 6.75]. The toroidal component of the magnetic field is assumed to vary as
0/ [1 ( / 0)cos ]
B B r R and the density, temperature and poloidal magnetic field component are assumed only to be functions of r. The ray equations Eqs (6.121) and (6.122) have the straightforward consequence that k now varies due to the non-conservation of the poloidal mode number since D0/ 0 and
/ r r r (m / ) (n / ) /
k k B B e k B e r B e R B B
(6.123) This variation in parallel wavenumber makes a priori predictions of LH wave accessibility and absorption difficult in toroidal geometry and this effect was well documented numerically in early toroidal ray tracing calculations[6.75–6.77]. An example of the variation of the parallel refractive index (N=k c /w) in toroidal geometry is shown in Fig. 6.13. These simulations were performed using the GENRAY ray tracing code [6.78] for parameters typical of the LH current drive experiment in Alcator C-Mod (f0 4.6 GHz, ne(0) 7 10 m 19 –3, (0) 5.3 TB = and Te(0) 2.33 keV ) [6.79]. A group or “bundle” of ten rays was launched from the low field side of the tokamak near the plasma cut-off in a narrow range of parallel refractive index 1.58- ≤N≤ -1.56. Linear electron Landau damping is computed along the ray paths and for these parameters the absorption is quite low on the first path of the rays into the plasma, since w/ ( v ) 6k th e, > near the core for these parameters. Consequently the rays continue to propagate through a focal point or caustic near the plasma centre and then propagate back to the plasma periphery where they reflect from a cut-off. As the ray bundle approaches the cut-off, two interesting phenomena can be seen. First the trajectories start to spread spatially (Fig. 6.13 left). Second, the N of the rays increases rapidly after reflection from the cut-off and the rays are strongly absorbed as they reach the hotter core plasma where the condition for strong electron Landau damping is satisfied, i.e. w/ ( v ) 3k th e, ≤ (see Eqs (6.62, 6.63)). This dramatic modification of the parallel wavenumber at reflection layers near the top and bottom of the tokamak cross-section was also observed in early ray tracing studies in circular plasmas [6.75].
FIG. 6.13. Left: LH ray tracing calculation using the GENRAY code for Alcator C-Mod parameters – poloidal projection of rays. Right: Evolution of parallel refractive index versus poloidal distance along ray paths for rays shown in the left figure.
6.2.7.3. Numerical simulations of lower hybrid current drive using coupled Fokker–Planck and ray tracing calculations
As pointed out earlier, the distortion of the electron distribution as LH waves damp on tail electrons makes it necessary to solve the full 3-D (r, v, v) Fokker–Planck equation
(v ) ( ,v ,v ) 1
v v v
e e e e
RF e F
e
f f eE f f
D C f r
t m r r r
(6.124) The fourth term on the right hand side of Eq. (6.124) is a diffusion operator with fast electron diffusivity (F). Equation (6.124) has been solved with and without the radial diffusion operator by carrying out a non-linear iteration where the quasi-linear RF diffusion coefficient is evaluated using a toroidal ray tracing code [6.63, 6.80] and the diffusion coefficient is then used in a Fokker–Planck code to solve for the perturbed electron distribution. The new distribution function is then used in the ray tracing to recompute the RF diffusion coefficient, which is then used to resolve the Fokker–Planck equation for a new distribution function. This type of iterative computation was first carried out with 1-D (v) Fokker–Planck codes and toroidal ray tracing calculations [6.63, 6.81], but neglecting diffusion. Later, simulations were performed [6.82, 6.83]
using a 1-D (v) Fokker–Planck code to evaluate the RF wave-induced flux and then convolving that with an adjoint solution of the Fokker–Planck equation (see Eqs (6.110) and (6.111)). More recently, 2-D (v, v) and 3-D (r, v, v) solutions have been obtained using the CQL3D/GENRAY codes [6.78, 6.80] and the DKE/LUKE model [6.84]. Indeed the distribution function shown in Fig. 6.9 is the result of iteration between the GENRAY ray tracing code and the CQL3D bounce averaged Fokker–Planck code. The non-thermal electron distribution shown in Fig. 6.9 has been used in a synthetic diagnostic code to simulate the
hard x ray spectra measured by a horizontally viewing hard x ray camera during LH current drive experiments on Alcator C-Mod [6.64].