COUPLING TO FAST WAVES NEAR THE LOWER HYBRID FREQUENCY

Calculations show that the fast wave near the lower hybrid frequency (we;<< w <<Wee) may be launched efficiently into large tokamak plasmas. Coupling efficiency is calculated for waves, which may be useful for current drive experiments in the Prince ton Large Torus, the Tokamak Fusion Test Reactor, and reactor grade plasmas.


INTRODUCTION
Significant advances have been made with lower hybrid (LH) waves for generating steady-state currents in tokamaks. 1 Work has proceeded swiftly from small linear devices 2 • 3 generating < I A to large tokamaks4-7 generating over 100 kA. The tokamak experiments appear to work well only for densities where the wave frequency is well above the LH frequency, w 2 ;.:::: 4w[H = 4w;;l(l + w;e l wJe)· Additionally, the highenergy densities cause some uncertainty about the future role of parametric decay 8 • 9 and turbulence. 10 The fast wave near the LH frequency is unlikely, according to cold plasma theory, to produce significant ion heating. H ence, most earlier work concentrated on the slow wave. The waveguide arrays necessary for launching the fast wave are also more complicated than t hose needed for the slow wave. With strong attention given now to LH current drive in addition to ion heating, it is worthwhile to examine the feasibility of fast wave current drive (FWCD). Providing accessibility conditions are satisfied, the fast wave propagates through the LH resonance layer and therefore has favorab le prospects for generating good current distribution with FWCD.
In t his paper, we address the question of coupling efficiency from an external launching structure into FUSION 13 We calculate transmission coefficients for fast waves with parameters useful for FWCD.

WAVE PHYSICS
The geometry for launching the fast wave from a grill antenna array is shown in Fig. I. The waveguide grill (see Brambilla 14 ) is oriented with the waveguideinduced vacuum electric field in the ±y direction when the confining magnetic field is in the z direction and plasma density increases in the x direction. An approximate dispersion relation for the fast wave is (when (1) where N = cklw The wave is evanescent in the plasma until N} > 0.
The fast wave is evanescent until densities are reached satisfying (approximately)

COUPLING MODEL
The evanescent region in the low-density plasma edge can lead to an imperfect coupling of energy from the waveguide into the fast wave in the plasma. Only if a substantial fract ion of energy can be transmitted to the fast wave is FWCD feasible. The question of coupling from a waveguide to a plasma was addressed in a general formalism by Bers and Theilhaber, 1 s and the specific case of Ny = 0 was discussed by Golant 11 and Theilhaber and Bers. 13 When Ny* 0, the boundary conditions are changed and there is some coupling with the density gradient. As shown later, the launching of waves with a finite Ny has a substantial advantage over the previous cases because of much improved coupling. We calculate the transmission coefficient of a single mode fast wa ve from the antenna into the plasma, while the technique of Bram bill a 14 should be used when a superposition of modes is studied.
A linear density rise is used , as shown in Fig. 1, with a vacuum gap between the waveguide, positioned at x = -x,., and the plasma. The polarization of the launched wave is in they direction with the waveguide 284 grill built and phased to control ·Ny and N,. The energy in the reflected wave from the surface is assumed not to interfere with the antenna operation and not to reenter the plasma after further possible reflections. This assumption could be removed when a Brambilla-style superposition of modes is calculated .
. Ey= AU[ia,exp(-i~).J2au] , (8) where A is the amplitude to be determined by the boundary conditions. Making tangential components of E and H continuous across the vacuum/plasma interface gives The where r(z) is the gamma functio n.

APPLICATION TO LARGE TOKAMAKS
For tokamaks the transmission coefficient is found in a manner similar to the above calculations. Complications arise since the density profile is no longer a linear function of x. Hence, the theory was modified and computer codes were developed. The density profile was taken to be parabolic from r = 0 to r = a with an edge plasma with exponentially decreasing density going as exp[-(ra) ld] from r =a tor= a+ sand vacuum from r =a + s to r = a+ s + Xu where the mouth of the waveguide grill is located. Figure 2 shows the transmission coefficient for a Princeton Large Torus (PL T)-type plasma using f = 800 MHz, a= 42 cm, d = s = I cm, Xu= 10-2 cm, and n(a) = 5 x 10 12 cm-3 . The transmission reaches a maximum for N , = l. 7 and Ny=l. The maximum current drive efficiency for LH waves on the PL T has occurred for the coupler with N , = 2. It is thought that reducing N, to :S2 might increase the LH current d rive. Hence, the fast wave has maximum coupling efficiency in a range where good FWCD may be expected on the PLT.
For the Tokamak Fusion Test Reactor (TFTR), calculations were done for two modes of operation.
Both modes use d = s = l cm, Xu = 10-2 cm, and f = 800 MHz. Figure 3 shows coupling to a lowdensity, low-field plasma with n(a) = 1.6 x 10 12 cm -3 , (n) =2x10 13  this case may be as high as -800Jo . Hence, the fast wave couples better with a nominal operating parameters plasma in TFTR than the low-density, low-field preliminary parameters plasmas.
For a reactor plasma running at B = 30 kG, f = 800 MHz, and n ( a ) = 3 x 10 13 cm-3 , with a toroidal belt limiter the wave launching structure might be placed in an edge plasma with d = 1.4 cm, s = 2 cm, and Xu= io-2 cm. Figure 5 shows that the transmission coefficient reaches a maximum with N i = 2.15 and Ny = -l.5. A good coupling of 850Jo is achieved for this combination.

CONCLUSIONS
Waveguide arrays can be used to launch fast waves suitable for current drive in large tokamaks. By optimizing the wave spectrum, coupling coefficients as high as 850Jo can be achieved, with values in excess of 600Jo easily realizable in general for large tokamak plasmas. Accessibility criteria do not hinder the wave penetration into the plasma. For the experimental arrangements considered here, the evanescent layer is sufficiently thin that surface reflection should not be a substantial problem. For the optimal wave spectrum that couples the highest fraction of wave power into the plasma, the fast wave (which propagates directly through the LH layer if accessibility is satisfied) should allow good current profiles to be established or maintained in large tokamak plasmas.