Currents generated by lower hybrid waves

Electron currents can be driven in a linear plasma by the absorption of lower‐hybrid waves excited primarily in one direction. Current‐drive has been demonstrated both for collisional and resonant‐electron absorption. The magnitude of the excited current is compared with the predictions from an electron kinetic equation with a Lorentz collision operator in the regime k∥vte/ω≪1.


I. INTRODUCTION
Damping of traveling radio-frequency waves has been proposed as a mechanism for driving steady-state toroidal currents in a tokamak plasma. Lower-hybrid waves are a good possibillty 1 because the waves can be damped strongly by the plasma. Also, their relatively low frequency makes possible the use of high-power sources which are presently available.
As far back as 1952, Thonemann, Cowhig, and Dav-enport2 used Alfv6n waves to drive currents. Several others have demonstrated curre·nt generafion through use of electromagnetic waves 3 " 10 and ion cyclotron 11 waves. Theoretical aspects have been discussed by others, 12 " 18 at least one of whom 1 ' felt the concept not as viable as using ion beams for current drive. Starting about two years ago, Fisch, 1 • 11 • 19 Bers, 18 and Kar-ney19 made convincing arguments that driving toroidal currents via rf waves was feasible. Work by K1fma 20 supports the conclusions of these recent arguments. Decker and Hirschfielct2 1 and, more recently, Wonrf 2 have performed experiments demonstrating that nearly electrostatic lower-hybrid waves can couple by collisionless absorption to electron currents in beam-supported plasmas. McWilliams et al., 23 showed preliminary results indicating that the electrostatic lower-hybrid wave could also drive currents via collisional absorption in isotropic plasmas. There is now evi-dence2Ma that lower-hybrid waves can drive currents in tori. The theory for current drive has now progressed to specific design considerations for reactor grade plasmas. 27 This paper discusses currents driven by lower-hybrid waves not supported by beams. Experimental results show the current generated over a range of plasma and wave parameters, which include regimes where the waves are damped collisionally or damped by colllsionless means. The theory presented here includes a prediction from kinetic theory of the current generated in the plasma when the lower hybrid waves are colllslonally damped.
The outline of the paper is as follows: The theory is discussed in Sec. Il, including a brief outline of lower hybrid wave propagation, and calculations of the current generated under conditions where the wave is damped by collisional or collisionless mechanisms. Section Ill describes the experimental apparatus, plasma parameters, and experimental techniques employed. Section IV presents experimental results of electron currents driven by collisional wave damping and currente generated via colllsionless damping. Section V contains a discussion and conclusions.

II. THEORY
A. Wave propagation Let us model the waves and plasma in a slab geometry With the confining magnetic field in the z direction and the density increasing in the minus x direction away from the antenna. The plasma is homogeneous in the z and y directions. The frequencies are ordered we 1 « w « we• • w,. , where Weo w.,. are the ion and electron gyro-frequencies, w is the wave frequency, and w,. i s the electron plasma frequency. Hence, the ions are essentially unmagnetized and the electrons are strongly magnetized. The region at the edge of the plasma where w > w,. ls assumed to be very thin. Proper matching across this thin layer can be done, and the reader is referred to Brambilla's work2 8 or that done by Chan and Chiu 29 or Wilson and Wong. 3° For a slowly varying plasma density, the dispersion relation for the slow waves has been solved by Briggs and Parker 3~ and Bellan and Porkolab. 32 The local dispersion relation is found to be 2 2 ( !!!! k~) ( ) w = w1 h 1 + m., lJ , 1 where w~h = w!/(1 + w!.,!w;, ), wp 1 is the ion plasma frequency, and ku and k,, are the wavenumbers parallel and perpendicular to the confining magnetic field, respectively. If we further restrict w so that w » wp 1 everywhere in the plasma and w!. « w~, then and it is easlly seen that the wave propagates in the well-known resonance cones. Additionally, there are some restrictions on the wave for accessibility to a given plasma density. These restrictions have been discussed by Golant 33 and StixM and studied experimentally by Motley et al. 35 Kinetic theory 38 yields a complex dispersion relation with a temporal damping rate for the waves given by The first contribution to (3) is v, which represents the collisional damping, e.g., electron-ion and electron-neutral collisions, while the second term is the Landau damping contribution. For a wave launched at a given location, this damping can also be viewed as a damping of each ku component of the wave in space as it propagates away from the antenna. Probes for this experiment were able to measure the square of the electric field, and, hence, it is of interest to calculate the expected E2(ku) as a function of distance from the antenna.
Control of the wave spectrum launched by the antenna is important for driving currents. In Particular, an asymmetric spectrum must be obtained in order to produce a unidirectional current. Figure 1 shows the calculated spectrum for an antenna of principal wavelength 8 cm excited at 70 MHz. A ring antenna similar to that used by Bellan and Porkolab 37 was used in the experiment. The spectrum calculated here was found by as:suming a 90° phase difference between rings (equal amplitude to all rings) and taking the Fourier spectrum of the resulting vacuum electric field at the antenna. A more complete description of the particular antennas used, including the effects of different phasings to the rings, may be found later in the paper. Figure l(a) shows the calculated spectrum at the antenna, while Fig. l(b) shows the effect of Landau damping on the wave as it travels away from the antenna. A temperature of T. = 11.5 eV was used for the calculations because in a later section we observe that an 11.5 eV temperature predicts curves which flt well with the experimental data obtained when a Langmuir probe indicated T., ::10.6 eV. As expected, the shorter wavelengths damp rather quickly, while the longer wavelengths excited by the antenna are less affected.

B. Simple current generation model
Oscillating electromagnetic fields can cause steadystate forces (i.e., time averages are nonzero) to act on plasmas. Examples of this are the ponderomotive force or other mechanisms which can make j • E nonzero, such as Landau damping or collisions. We wish to determine the effect on electron motion when collisions are the mechanism for energy loss in a wave.
A simple model 23 (also obtained by Schmidt 38 ) is developed for order-of-magnitude estimates and to help in visualizing the underlying physical processes involved. The next section will present a kinetic theory applicable to our experiment so that a more precise estimate of the currents generated in a plasma can be obtained.
The plasma is considered to be confined by a uniform magnetic field, B 0 = Boi . The ions are cold and the plasma parameters are such that we.» w J4 » w > w , 1 • The motion of an electron is then governed by the Lorentz force and a collisional term for an rf traveling wave. The experimental conditions were such that o= v/w -kuvt,lw « 1.
We desire to know the long term effects of the wave on the particles and hence divide the velocity and position of the electron into a slow and fast part v=(v)+ov, z={z)+oz, where ( ) denotes a time average over the wave period. Carrying terms to 0 (E 2 ) and expanding about (z) gives  i.e., Ov =A sin(k11(z)wt)+ B cos(ku{z)wt) • (8) Substituting this into Eq. (7) we get the following restraints on A and B: Since O.z is probably of order E, Eq. (9) can be solved to order E, giving A and B. 6v can now be integrated to yield: The time evolution of (v) can now be written (12) where C is determined by the initial conditions. For the condition O « 1, there results a current density of (j) =-ew~k11E 2 / 8rrmw 3 • (13) Hence, for electrons interacting with an rf traveling wave in the presence of a mechanism which can change the electron response from being purely reactive, a net electron flow in the direction of propagation of the wave will occur on time scales t » 1/ w.

C. Full collisional current calculation
The previous section shows that one may expect a damped traveling wave to generate a cu. rrent in a plasma. Let us now develop a model with more detail and apply the results to lower-hybrid waves specifically. Consider a cylindrical plasma of length L in a wiiform magnetic field Bo = Bai with cold ions and electron-ion collisions the dominant collisional process. This model approximates one of the conditions in which the experiments were performed. The electron motion may be approximately described by an electron kinetic equa-tion· with the Lorentz collision operator 39 : v aµ aµ ' (14) where/, is the electron distribution [unction and µ. = cos8, with B the angle of v with respect to the main magnetic field. As in the. previous section (and also for part of the experiment) the problem may be restricted to the regime k uv,/w-vlw= 6 « 1, with w the wave frequency and k11 the wavenumber parallel to B 0 • It will prove convenient to divide the distribution function into slow and fast components !. = F +.f, such that (/): f. 2 11 ..,fdt = O, 0 and also to write Expanding Eq, (14) to varying order in 6 allows f to be found . If we write f = f 0 + J1 +/ 2 + · • ·, where the ordering is r-o", then we find J2 was obtained assuming J~ v,(a/ 0 /az)dz :::::0, since it is reasonable to assume that there is no net potential drop along the plasma column.
When the ( ) operator is applied to Eq. (14) along with the assumption used to get Eq. (18), we find that describes the steady-state solution for the distribution function . This equation amounts to neglecting two terms from the time average of Eq, (14). We find lhat this approximation is good only for very modest powers in our experiment (i.e., powers less than a watt). At higher powers these terms come into play and a ct to inhibit the current flow. (For a further discussion see the Appendix.) The right-hand side of Eq. (19) is almost Legendre's differential equation and hence suggests looking for solutions to F by expanding F in a Legendre series. After performing the algabra one arrives at F=Fo + e2!' (~1 2 µ(v(4µ 2 -l) a;Jo + {10 -16µ. 2 )*2)• (20) where we have taken F 0 to be a Maxwellian. Figure 2 shows a plot of / 1 = F -F 0 versus the perpendicular a nd parallel electron velocities normalized to v,. As expected, F -F 0 is synuuetric for v1 --v1, since for an applied rf field oscillating in the i direction there ls no preferred perpendicular reaction. On the other hand,  F-F 0 is anti-symmetric for vu--v... This is also to be expected, since examination of the right-hand side of Eq. (14) shows that this operator acts to make functions which are even in µ lead to no net contribution to a current source. The figure also shows that the largest perturbations to the distribution function occur around v 1 • One would expect that very fast particles will not collide and, hence, would not contribute to a current. The particular collision model chosen here breaks down for the very slow particles (Iv I -v,. 1 ), but since there are not many electrons meeting that criterion, we suspect they do not substantially change the above picture. Electrons in the region Vt , 1 < Iv I «Vt collide sufficiently quickly with the ions never to contribute momentum to the current (and the ions do not contribute since for a given momentum their contribution to a current is down by the electron to ion mass ratio).
We may now calculate the time-averaged current density induced by the collisionally damped waves, and get For a lower-hybrid wave launched from an antenna and damped before reaching the end of the column, the resulting total current may be found to be suUs from integrals over the velocity distribution and from geometric factors.

. Landau damping currents
In the last section it was shown that a collisionally damped traveling wave can act as a source for generating an electron current in a. plasma. Of more interest to thermonuclear research is the question of driving currents when the waves are damped by means of resonant particle interaction with the wave rather than via collisional processes. Fisch, 1 · 11 • 19 Bers, 18 and Kar-ney19 have proposed that a current of electrons may be generated in a tokamak via resonant interaction with the wave and that the current would then be dissipated by collisions with the bulk electrons which would then allow transfer of the momentum to the ions. The question of producing an electron current via resonant interaction can be addressed on the H-1 device.u This is a linear machine; when currents were generated via resonant interaction it was found that end-loss was the major current sink rather than collisional dissipation.
For the slow wave which is nearly electrostatic, the wave momentum density has a component along the magnetic field of k 11 "' a ( ) where Kh is the Hermitian part of the dielectric tensor.
For the lower-hybrid wave propagating under our experimental conditions this expression reduces to If the waves are now damped by electron Landau damping, an electron current density of j"' (-elmeHk11~/87Tw) (25} will result. ln a toroidal device it is anticipated that the electrons will travel around the torus repeatedly sampling the waves until enough momentum has been gained by the resonant particles to flatten the distribution function over the width of resonant interaction in velocity space. 1 For a linear machine, the current will flow out the end of the device . This current flow entails a readjustment of the plasma sheaths at each end of the column, which can go on until the current approaches j =ne(kT.,/m 1 ) 112 •

A. Plasma source and parameters
The experiments were performed on the linear H-1 device 0 (see Fig. 3) at Princeton University, utilizing a helium, neon, or argon plasma approximately 10 cm in diameter and 200 cm long. The steady-state confining magnetic field, B 0 =Boi, was typically 5-12 kG.  Pump glow plasma, during which the electron density varied from 2x10 12 to 2x10 10 cm -s, and the electron temperature fell from around 12 eV to around 1 eV, depending on the filling gas type and pressure. An example of the density and temperature dependence on delay time is shown in Fig. 4. Over most of the afterglow the ion temperature was thought to be held just above the background gas temperature by charge-exchange cooling.
These plasma parameters allowed three distinct regimes for the propagation of the lower hybrid waves, which we shall call hot, warm, and cold. In the hot regime the waves were damped via Landau damping, with the electron-ion or electron-neutral collisions at least ten times less significant than the Landau damping. The warm plasma was characterized by a temperature high enough so that electron-ion collisions were insignificant, yet low enough so that Landau effects were weak. Hence, in the warm plasma, the waves made a transit down the plasma column without depositing much energy or momentum in the particles. In the cold plasma, the waves were substantially damped by electron-ion collisions. As an example, the plasma described in Fig. 4 was warm for delay times less than about 50 µsec and cold for delay times gr eater than 100 µsec. As a general rule hot plasmas were obtained only at short delay times In helium discharges.

B. Diagnostics and wave source
Electron temperature was inferred from Langmuir probe traces, the blackbody radiation near the e lectron cyclotron frequency,'2 and at late delay times (.?100 µs ec) in the afterglow, by use of an electrostatic energy analyzer (see, for example, Ref. 43). Currents generated by the waves were detected by means of four loops, two on each side of the anterma, oriented to detect a B,/a t and by an electrode placed at the end of the plasma column. Electron density was inferred from the phase shift of an 8.6 mm microwave interferometer and from the angle of propagation of the lower-hybrid waves with respect to B 0 • The lower-hybrid waves were excited by application of 0.2-5.0 µsec bursts (chosen to be many wave periods long, but shorter than the time for most plasma parameters to change) of 60-120 MHz fields to eight pha~ed loops 37 mounted (typically 2 .5 cm apart) on the outside of a 9.4 cm diam fused quartz tube. Control of the phase to each of the loops allowed the wave to be launched with controlled directionality. For example, the wave could be launched either parallel or antiparallel to Bo, or in both directions at once, as can be seen in Fig. 5. Initially, an antenna was used which was not shielded from the plasma. This was found to produce a pattern of divergent current flow in the plasma column when the rf was applied, since the loops then acted as electron sinks. The quartz shielding of the loops solved this problem. The metal end electrode was also found to act as an electron sink if it intercepted the rf resonance cone.
The forward and reflected powers to the antenna were measured by inserting directional couplers in the transmission lines immediately before the loops. All references to power in this article refer to the difference in the power radiated by the antenna with and without plasma in the column. Without plasma, approximately 95% of the forward power was reflected back down the line. With plasma, only about 5%-20% of the power was reflected (for the plasmas studied here). In general, the antenna coupled to both the slow and fast modes in the plasma, but since the accessibility conditions for the two waves are different, it was possible to produce plasmas which minimized the fast wave coupling while still having the right parameters for good s low wave coupling. The lower-hybrid waves were detected using a single or double tip triaxial probe which could be moved in the axial and radial directions. The tips of the double probe were separated radially by less than 1 mm and the sum or difference of the signals to the two tips could be taken, enhancing the fast or slow signal, respectively, due to the relatively long wavelength of the fast wave and the relatively short wavelength of the lower-hybrid wave.

A . Wave identification
For the parameter range studied here the lower-hybrid group and phase velocities are The dispersion relation shows that the wave energy will propagate at an angle 8=wlw,.. with respect to B 0 • Hence, a wave generated by a slow wave structure two wavelengths long in the z direction should propagate into the plasma at the angle 8 and should be approximately two wavelengths long in the axial d. irection and two wavelengths wide ln the radlal direction with 2;\.l.   Also, as seen in Fig. 6{b) an antenna two wavelengths long in z produces waves which are about two wavelengths wide radially.
The backward nature of the lower hybrid wave ls demonstrated in Fig. 7. As seen in Fig. 6(a) the wave fields propagate inward radially. Figure 7 shows that the perpendicular phase velocity is radially outward at a speed of v• =4.1X10 7 cm/sec. For a plasma density of 2 .sx10 10 cm''\ /=70 MHz, and ><rr==10 cm, the dispersion relation yields v. =3.5X 10 7 cm/sec.
The experiment also confirmed v, • vr ""0 for these waves. Using an 8 cm antenna we measured v, rr, as described in the next paragraph, and v ~.L from which k can be inferred. The angle of propagation of the energy was measured as in Fig. 6{a). We found (for f =70 MHz, n. =3X10 10 cm~) that Vr had an angle of 4.5 x 10-2 with respect to B 0 and v. had an angle of ?T/2 -5.03X 10-2 with respect to B 0 , and, hence, vr • v. "'0. measurements gave results indicating a group velocity of roughly twice what we expected. It was then found that sufficient power at the second harmonic (i. e. , 2/) was generated by the amplifier to cause the leading edge of the wave front to travel at Iv, I =2/'A.. With a filter in the transmission line blocking the second harmonic, the wave packet propagated down the column at roughly the expected velocity, as indicated in Fig. 8, which shows the measured group velocity as a function of wave frequency.

. Collisional currents
Section n C showed a calculation predicting an electron current, /,_,, when the lower-hybrid wave ls collisionally damped on the plasma column. For waves incompletely damped on the column, account must be taken for the fraction of the wave lost out the end of the column. The current generated under circumstances of incomplete damping may be approximated by where s is the distance from the antenna to the end of the plasma column and l 0 is the momentum damping distance. Figure 9 shows the current generated as a function of delay time in a neon afterglow plasma. The plasma parameters are those described in Fig. 4. At small delay times where T, was relatively large (several eV, but not large enough for Landau damping}. the current was reduced. The current was also reduced at large delay times where the density was small. Incomplete clamping of the waves was observed at sinall and large delay times and this is consistent with theoreti~l calculations of the damping. Hence, as expected, smaller currents resulted if the waves were only partially absorbed during transit down the plasma column. As an example, consider the current generated at a delay time of 807 µ.sec as indicated in Fig. 9. The plasma parameters are given in Fig. 4 and give I== 0.471,h, as a result of incomplete damping. The current in Fig. 9 for complete damping is about 30 mA. The predicted value of 807 µ.sec is thus 14.5 mA. The observed value Control of the directionality of the current will prove to be important for useful applications, such as driving the toroidal current in a tokamak. Waves driven parallel (antiparallel) to B 0 should cause electron currents driven parallel (antiparallel) to B 0 • This dependence was observed and ls displayed in Fig. 9. Apparently, control of the current directiQn in future uses will thus depend on the ablllty to specify the phasing on the waye launching antenna.
The current as a function of input power is plotted in Fig. 10. At low powers the current is proportional to the input power (as was the spatially averaged signal level, f ~2 dA, of the lower-hybrid wave taken over a plasma cross section) and is roughly half that predicted by the theory. We note that the current measuring probes were calibrated using a cylindrically symmetric current source, while the observed resonance cones · were not always symmetric . Additionally, the power What fraction of the radiated power appears in the lower-hybrid wave ls difficult to determine. Hence, the use of the radiated power in Eq. (22) leads to an overestimate by the theory. Figure 10 also shows evidence of a tendency for the current to saturate at powers exceeding a few watts. It appears that the plasma has difficulty sustaining currents much in excess of i s::::ne(k T.I m 1 ) 112 , which would violate plasma charge neutrality .
Another factor limiting the current at higher powers may be the coupling of the antenna· to the lower-hybrid wave. This will be discussed later in the paper. We also noted that at powers over a few watts the current substantially terminated within 500 nsec of startup. Figure 11 shows the spatial distribution of the electron current, determined with the aid of the moveable B9 loops. The Be loops measured a signal proportional to J.
where R, is the radial position of the probe. By varying R,, we obtain a plot which allows us to find a>It f ar, which is proportional to the current at the specified radius. The current was found to be localized to the radial range over which the lower-hybrid wave existed and was found to be essentially uniform in the z direction.
The theory predicts that the current ls dependent on An . This parameter was varied by changing the phasing of the antenna loops. One can see the results in predicted by theory. In the region where the current appears to saturate, the dependence on wavelength decreased.
End effects can sometimes be important in linear devices. With waves launched in just one direction in the column, asymmetric heating of the electrons toward one end may cause a net current to be generated by changing the end sheath potentials. A calculation (see the Appendix} suggests that such a.n effect, for the parameters of Fig. 10, would not generate more than l mA/ W, which is much smaller than the observed currents.

C. Landau regime identification
Lower-hybrid waves can experience substantial Landau damping in a transit down the plasma column when the wave frequency, wavelength, and plasma temperature are such that the ima. ginary part of k11, ku, 1 = 1T 112 k 11a3 exp(-a 2 ), (26) with <l';::::; w/(kuv 1 }, is large. This occurs roughly when a~ 3 . Due to the exponential dependence on cil in Eq. (26), this onset of lAndau damping ls a rather sudden function of a. Figure 13 shows the imaginary part of the wavenumber in the parallel direction plotted versus wave frequency for several different parallel wavelengths in an 11.5 eV plasma. For a fixed wavelength, one sees the strong dependence of the damping on frequency. The dots on the graph represent the points at which a;::::;3. ately at the turnoff of the plasma source of 13 eV). Hence, a rapidly increased with delay time in the afterglow and Landau damping was i. mportant only for the first few microseconds of the afterglow. For these plasma parameters the electron-ion collision frequency ls less than 1.2x10 5 sec -t and the electron-neutral collision frequency" is less than 7Xla5 sec-1 • Therefore, electron-neutral and electron-ion collisions will not significantly damp the lower-hybrid wave ln its transit down the plasma column.
At low powers the wave was significantly damped in its transit, as shown in Fig. 14 temperature plasmas. The theory uses Landau damping as the mechanism and is calculated using the appropriate power spectrum for the antenna, as shown in Sec. llA. One sees that the observed damping is compatible with Landau damping (10 < T. < 13 eV) .
A good check on the Landau damping hypothesls is to measure the damping length as a function of wave frequency. The results of such a test are shown ln Fig.  15. Collisional damping effects are calculated and shown for comparison. Once again the calculated damping rates are shown with the experimental data and one sees that the observed damping is consistent with a plasma temperature around 11.5 eV.

D. Currents in Landau regime
In the previous section we showed that at very early delay times in a helium discharge afterglow, the plasma and wave parameters were such that electron Landau damping Was the strongest damping mechanism. Under these circumstances the wave momentum in a lowerhybrid wave should be delivered to the electrons, resulting in a net current if a unidirectional wave were excited. The plasma parameters were thus arranged as described in the last section: n, =3.7X10 10 cm-3, T, =10.6 eV, B 0 = 6 kG, and neutral density no=5.3 X10-t Torr.
As we found in the case of the currents driven by means of collisional damping, the direction of the electron flow induced by the waves was found to be parallel (antiparallel) to B 0 when the waves were launched parallel (antlparallel) to B 0 •  Equation (27) predicts a 1// 2 dependence for the current generated by the wave when Landau damped. Figure 17 shows that the current generated experi· mentally is a monotonically decreasingfunction offreqquency. There are two reasons to expect this; (1) the current should decrease as shown in Eq. (27), and (2) the wave damping is incomplete at the higher frequencies. The figure also shows the current predicted by Eq. (27) (multiplied by the 0.25 factor discussed herein), which assumes complete damping of the wave momentum, and also a prediction taking into account the actual amount of wave momentum damped as measured and shown in Fig. 15. The actual current lies between these two lines for the higher frequencies. It seems likely that higher k u components of the wave still attenuate at the higher frequencies and thus contribute to a small current even though the principal wavenumber component radiated by the antenna does not decay during a transit down the plasma column at these frequencies.

V. DISCUSSION
For damped lower-hybrid waves in a plasma where the ions are cold, the momentum is expected to be transferred to the electrons from the waves. A theory has been developed for the case where the waves are c olllsionally damped and the resultant electron current is predicted. This current is proportional to the input power (at low power levels). Linear machines can drive currents up to about ; .. , the sound current, at which point it is suspected that ambipolar effects limit the total current. The induced electron flow from the wave damping is in the direction of the flow of the waves (i.e. , parallel or antiparallel to the confining magnetic field).
Theory predicts that waves which can damp via resonant interaction with the electrons will induce a net electron current carried by the resonant electrons. We have observed that lower-hybrid waves subjected to electron Landau damping induce such a current. Once again, the current is proportional to the input power (at low power levels); the electrons are given a momentum component in the direction of the wave flow. Evidently, the ability to specify the antenna phasing; i.e., the ability to produce a unidirectional wave, is crucial for generating substantial amounts of net current.
Fisch and others have predicted that these currents may be used as a method for driving steady-state toroidal currents in tokamaks. The exact magnitude which can be achieved is open to speculation at the moment. It is not clear how well quasi-linear diffusion will be able to flatten the resonant region of the distribution function in the face of collisional diffusion. Also, the theoretical models usually assume that the lower-hybrid wave is present everywhere in the plasma, in contradiction to the expected resonance cone behavior. As a result, induced currents may drive instabillties in the plasma. On the positive side, since wave trapping broadens the resonant region, the maximum current which can be driven by the lower hybrid is probab-R. McWilliams and R. W. Motley ly not limited to the level predicted by Fisch. In quasilinear theory, once enough power has been put into the particles to flatten the r~sonant region and maintain the plateau against collisional losses, no additional current may be driven even if the power is increased. Trapping would allow a larger current to be driven once the trapping width starfs to exceed the inherent resonant width of the antenna. However, due to the uncertainty of the radiated antenna spectrum and poor definition of the resonance cone structure in a tokamak, it may be difficult to determine when trapping will be dominant.
Current drive by lower-hybrid wave excitation has already been demonstrated in toroidal devices. Recent work by LaHaye, et al. 2 5 on the General Atomic Octopole, by Wong, Horton, and Ono 24 on the Princeton ACT-1 device, and by Yamamoto et al. 26 on the JFT-2 tokamak have shown that unidirectional electron currents can be driven with an order of magnitude improvement in the amps/watt figure over the linear results. This is to be expected since the interacting electrons are not lost out the ends (as in the linear H-1 device), but circle around the torus and can sample the wave many times before losing the momentum gained in one transit.

ACKNOWLEDGMENTS
It is a pleasure to acknowledge discussions with W. M. Hooke and E. J. Valeo, the laboratory assistance of G. Anania, C. Gwinn, and L. Olson, and the technical help of J. Frangipani. This work was supported by the Department of Energy Contract No. DE-AC02-76-CH03073.

APPENDIX A: COUPLING VERSUS POWER
As mentioned earlier, a direct measure of the absolute power resident in the lower-hybrid wave is difficult to achieve. So nearly all references to power in this paper are to the power emitted by the antenna with a plasma in the column as compared with no plasma in the column. As has been shown, there is a tendency for currents driven by the wave to saturate as the power is increased. This is probably due to limitations in how much current can be driven out of a linear device without disturbing charge neutrality.
We have noticed one other effect which will cause the current to increase at less than a linear rate in power. Chan and Chiu 211 have predicted that ponderomotive forces near the antenna will cause a decrease in the coupling efficiency of the antenna. An rf field exerts a force on the plasma causing it to be pushed away from the anteruia until the plasma pressure can balance the radiation force. This reduction of density near the antenna will result in poorer coupling to the plasma. Hence, the net energy density of the lower-hybrid wave is lower than that predicted by linear theory. Figure  18 contains a plot of the detected power in the lowerhybrid wave versus input power for a cold (0.7 eV) argon plasma. One sees that at power levels above a few watts the coupled power is dropping below the linear prediction and that at powers above around 20 W, the Wilson suggests' 6 that for our experiment, when 1} I v2n..., is a few tenths or greater than the coupling efficiency into the lower-hybrid wave should start to decrease. This would arise for power levels of 20-40 W for the plasma parameters of Fig. 18. Decyk, Morales, and Dawson 47 have done simulations of lower-hybrid heating which show such effects for a capacitor plate antenna. While ponderomotive processes probably would not lead to a saturation of the current, they could cause the current to be smaller than that predicted by linear coupling theories.

APPENDIX B: SOME OTHER CURRENT SOURCES IN A COLLISIONAL PLASMA
Another mechanism may possibly drive currents in a linear device where the lower-hybrid wave is collisionally damped. This current could result from the differences occurring at the two end sheaths of the column when only one end of the column is heated, as would be the case when the wave is launched in only one direction down the plasma.
For convention, a positive current is taken to be a current traveling from sheath 1 to sheath 2. For the following discussion subscripts e, i, 1, 2 refer to electrons, ions, sheath 1, and sheath 2, respectively. In equilibrium, with no lower-hybrid wave present, plasma flows out both ends of the column with where v 1 =(2kT.lm 1 ) 112 and, of course, ii =h· If a lower-hybrid wave is launched toward (and collisionally R. McWilliams and R. W. Motley damps before getting to} sheath 2, then the plasma potential at each end will respond, in addition to Ti increasing, to the changed plasma. For small changes in temperature toward sheath 2 we may calculate the new currents out the ends to find for a column ini~ially at T and potential V . This current is in the same direction as the current calculated in the section describing the collisional current calculation and the relative magnitudes should now be compared for the experimental conditions on H-1.
For the neon plasma giving the results plotted in Fig.  10 and, hence, az 9.25Xl0" 1 • AT may be estimated from ~NAT=Pll.t, using At= 300 nsec, since that was the turn-on time for the observed currents. P is the total input power and N is the total number of electrons involved in the heating. Using these estimates, the total current driven by thermal effects under these conditions is approxi- Phys. Fluids, Vol. 24, No. 11, November 1981 mately /=4.1 x lO"'A/W, which is about an order of magnitude less than that which was observed. Thus, sheath effects may contribute to the current but are considerably less than those described in the collisional current section.
At this point it is also appropriate to consider the terms dropped in order to obtain Eq. (19). The two neglected terms are v ·VF and -eE/m • aF/av, where F is the slowly varying distribution function and E represents the slowly varying, i.e. , de electric field. Comparing these terms with the collisional term on the right-hand side of Eq. (14}, we get v· VF(v;11 ...£._(l µ.2) aF)" where l is the mean-free-path and L is the plasma column length, and (eE/m)(aF/av) eE For the collisional plasmas studied here the first term ls typically 0.25. The second term may be dropped for plasma conditions where the particle acceleration is substantially retarded by collisional drag forces. Evidently, as sufficient charge leaves the column this second term will become stronger and lead to conditions inhibiting further electron flow out of the column (as was observed and was discussed in the experimental obs~rvations).