Whistlers, helicons, and lower hybrid waves: The physics of radio frequency wave propagation and absorption for current drive via Landau damping

This tutorial is concerned with plasma waves in the lower hybrid range of frequencies (LHRF) for current drive applications in tokamaks. Magnetic confinement of particles and heat in tokamaks is a result of the poloidal field associated with the toroidal plasma current, so tokamak reactors require plasma current on the order of 10 MA.¹ Most of the current in a steady-state tokamak reactor is thought to be self-driven by the subtle “bootstrap” effect,¹ but reactor studies consistently show that a significant fraction of the total current in reactors must be driven by other means, and for a steady-state device those means must be other than inductive (transformer action). These reactor studies^2,3 also show that a substantial part of this non-inductive current has to be driven at a radial location midway between the magnetic axis and the edge of the plasma, a location that will be herein termed “mid-radius.”

Wave current drive is a subject that has a long and successful history,⁴ but many interesting challenges remain. The basic principle is to transfer energy from waves to electrons in a toroidally asymmetric way in velocity space. Figure 1, from the seminal 1979 paper by Karney and Fisch,⁵ shows a plateau of energetic electrons on one side of the velocity distribution function but not the other, constituting the asymmetric feature that is carrying the current well away from thermal velocities. The damping of those waves in a reactor, which is necessarily in a nearly collisionless regime, therefore must be by a mechanism that dominates in the low collisionality limit.

There are several reasons why this topic is of interest. First of all, rf current drive is an outstanding success in plasma physics, in which theory suggested an effect that was soon established experimentally and then scaled up to a useful level within just a few years. Below, we will review some of the successes of the most-firmly established form of current drive with waves, which is lower hybrid current drive. Although this topic goes back to the 1970s, recently there has been renewed interest in the subject primarily because of the newly constructed superconducting tokamaks in Asia. Second, the linkages between this tokamak-relevant area and areas of space science and to the physics of rf plasma sources are intrinsically interesting, and some of these connections will be briefly discussed in this paper. The last topic is a variation on “conventional” lower hybrid current drive called “helicon current drive” that is about to be experimentally tested in the DIII-D tokamak. Although current drive by this method was suggested many years ago and studied in tokamaks without much success,⁶ conditions in those experiments were such that a strong effect would have in fact contradicted theoretical expectations. The upcoming experiment will test helicon current drive for the first time under conditions that are predicted to yield a strong and easily measured effect. The helicon current drive technique has important advantages for many steady-state reactor designs based on the Advanced Tokamak.

Figure 2 (adapted from Ref. 7), from the JT-60U device in Japan, shows a 3.5 MA discharge in which about 5 MW of rf power in the form of lower hybrid waves was able to sustain all of the current, as indicated by the slightly negative surface voltage at the low density of $1.2 × 10 19 m − 3$⁠. Similar results were obtained at about the same time on the JET device, where 3 MA of current was fully sustained by lower hybrid waves, though a true steady-state with fully equilibrated electric field was not reached in those experiments.⁸ For very long pulses, one must utilize a superconducting machine to fully take advantage of the non-inductive current drive, and a small high-field (8 T) superconducting device in Japan, TRIAM-1M, ran discharges at very low density for over three hours⁹ (over $10 4$ s) duration. In another superconducting tokamak, the Tore Supra device at Cadarache, discharges of over six minutes duration at an electron density of 1.5 × 10¹⁹m⁻³ were achieved and a total energy of over a GJ was deposited and removed, in a truly steady-state situation.¹⁰ 

In the remainder of this paper, wave damping mechanisms that are available for electron current drive are enumerated, and we show how the properties of those damping mechanisms set the requirements on the choice of wave, the wave frequency, and the wavelength. That leads to a discussion of what waves are useful for this purpose, which motivates an aside about jargon. It turns out that the same wave goes by different names in different fields of physics. We describe why there is particular interest in what is referred to as the lower hybrid range of frequencies and describe wave propagation in that range. An important point is that the properties of the damping mechanism that is being used and a characteristic of the wave propagation known as “wave accessibility” impose limits on the wavelength that is launched. Those limits in turn imply that coupling to these waves with a practical antenna located in the vacuum region away from the plasma surface is difficult. Finally, an upcoming experiment on the DIII-D tokamak on the helicon wave is described.

The processes that are available for collisionless absorption in a magnetized plasma can be divided into parallel and perpendicular, referring to the component of the electron velocity with which the waves are interacting with respect to the direction of the static magnetic field. Landau damping occurs if a wave with an electric field component along the static magnetic field propagates with a wavenumber along the field $k | | = k ⋅ B 0 / | B 0 |$ and so has a phase velocity along that direction $v ph | | = ω / k | |$⁠, and an electron is drifting with a velocity component along the magnetic field of $v | |$⁠. From elementary plasma wave physics,^11,12 the interaction averages out to zero except for electrons that are moving with very nearly the same parallel velocity as the wave, because those are the only electrons that experience a nearly dc electric field in their own frame of reference. The interaction depends on the relative sign of the wave phase velocity and the electron's drift velocity, since if the particle and wave are moving in opposite directions, there is no net interaction. That provides an opportunity for interacting with electrons that are moving in one direction by launching a spectrum of waves that has parallel phase velocities predominantly in only one direction. The resonance condition illustrated in Fig. 3(a) is then just $v ph = v | |$⁠, which can be written in the familiar form $ω − k | | v | | = 0$⁠, which is the Landau damping resonance condition.²⁹ The other possibility is cyclotron damping, which we call a perpendicular process, because the interaction is between the wave electric field component perpendicular to the static magnetic field and electrons with a non-zero component of velocity perpendicular to the B-field line around which the electrons are gyrating. If the wave has a perpendicular component of electric field and that component is elliptically or circularly polarized and is rotating around the magnetic field lines, and the electric field vector's rotation frequency $ω / ( 2 π )$ matches the natural gyration frequency of the electrons $Ω e / 2 π$⁠, the interaction between the electron and the wave will be maximum $( ω = Ω e )$⁠. If the wave has non-zero $k | |$⁠, as shown in Fig. 3(b), the wave frequency experienced by the electron is Doppler-shifted to $ω − k | | v | |$ and the generalized resonance condition becomes $ω − k | | v | | = Ω e$⁠. Notice that there is a handedness to the resonance condition, in that the sense of the rotation of the field vector and the electron's gyration must agree for resonant interaction. For example, since electrons gyrate in a right-handed sense around the magnetic field lines, if the wave polarization is purely left-handed at the Doppler-shifted gyro-resonance, there is no cyclotron interaction. In more generality,¹¹ it turns out that the cyclotron interaction occurs at integer multiples (harmonics) of the gyrofrequency if the wave has non-zero wavenumber $k ⊥$ perpendicular to the field lines so that the most general resonance condition can be written as $ω − k | | v | | = ℓ Ω e$⁠, where $ℓ$ is an integer.

As is evident from the above, different components of the wave electric fields are involved in the two classes of damping. In the case of cyclotron damping, the perpendicular part is the relevant one. We are interested in electron current drive, which is done by asymmetrically interacting with electrons with different signs of parallel velocity, which at first glance would appear to be possible only with the parallel interaction. Despite this intuitive idea, it turns out that that parallel asymmetry can be achieved using the perpendicular interaction (cyclotron damping), because of the presence of the Doppler shift term in the resonance condition. That is the story of electron cyclotron current drive (ECCD),¹³ which is not the primary topic in this paper. At frequencies well below the lowest-order electron cyclotron resonance (⁠ $ℓ = 1$⁠), the available collisionless damping mechanism is (⁠ $ℓ = 0$⁠) Landau damping, where the involved electric field component is along the static magnetic field lines. (The other possible $ℓ = 0$ interaction, TTMP,²⁹ which is not usually of practical significance in the tokamak context, involves the electric field component perpendicular to the field lines but produces electron acceleration along the field lines similar to the Landau interaction.)

So far, the discussion has been of purely linear Landau damping, but to perform the calculation correctly, one must take another step, which is to take into account the interplay between the collisions that do take place, albeit infrequently, and the forces resulting from the wave-particle interaction. This is the topic of the theory of current drive,⁴ an application of “quasilinear theory” in which absorption of wave energy drives diffusion in velocity space and the effect of collisions balances the wave-induced flux in steady-state.^4,11 This theory is obtained by solving the Fokker-Planck equation that describes these effects. Many aspects of this model have been validated by detailed comparison between theory and experiment in different regimes. Figure 4 documents this agreement in three examples. First, a 1985 example from the Princeton Large Torus¹⁴ that shows the comparison between the current drive theory for lower hybrid waves and experimental results from a database of 250 discharges that have non-steady current, both ramp-up and ramp-down as well as steady-state conditions. Excellent agreement is found with only two adjustable parameters in that model. Figure 4(b) shows a similar theory/experiment comparison from a case in which the current is driven using a completely different wave, this being a wave in the ion cyclotron range of frequencies in the DIII-D device, in a variety of different discharge conditions at different electron temperatures.¹⁵ The cross-hatched region on the plot indicates the range of the theoretical prediction, and again good agreement between theory and experiment is found across the whole range. Finally, Fig. 4(c) illustrates the case of electron cyclotron current drive on DIII-D,¹⁶ and here points on the 45° line would indicate perfect agreement between theory and experiment. Again, good agreement between the predictions of the current drive model and the experimental results is found for a variety of discharge conditions. These disparate cases show that independent of the type of wave used, the quasilinear theory of current drive captures in detail the wave damping physics and the effect of collisions on the driven current.

As is reviewed in Refs. 4 and 13, the effects of collisions on the current drive process are considerably more involved than might be immediately apparent. Since the collision frequency decreases as the speed of the test particle relative to that of the field particles on which it is slowing down to the third power, investing toroidal momentum into electrons that are moving along the field lines at a few times the thermal velocity can be quite efficient, because of their strongly reduced collision frequency. Since the collision frequency can be reduced by increasing either the parallel velocity of the test particle or its perpendicular energy, this leads to the possibility of driving currents by increasing the perpendicular energy of electrons that have one sign of parallel velocity, even though the electrons are not pushed by the waves in the parallel direction! This process is known as the “Fisch-Boozer” mechanism. Unfortunately, in a toroidal geometry, with its concomitant toroidal magnetic field inhomogeneity and the resulting magnetic trapping, another current drive mechanism that comes into play, the “Ohkawa” mechanism, has the opposite sign as the Fisch-Boozer mechanism and thus reduces the net current drive efficiency significantly for deposition far from the magnetic axis. All of these effects are treated self-consistently in the standard quasilinear current drive model.

Most of these cases shown in Fig. 4 involved current drive in the region of the magnetic axis, and we wish to move towards the more challenging task of mid-radius current drive. Hence, we need to apply our understanding of wave propagation and current drive efficiency to find the optimum set of wave parameters, because we will generally find conflict among the multiple requirements on the waves. In comparing the applications of different waves for current drive, two often-conflicting factors are efficiency and accessibility. As just described, the efficiency of current drive by the parallel mechanism scales as the square of the parallel velocity (as long as it is fast but non-relativistic electrons with which the waves are interacting), because the faster the particles are, the less collisional they are. If the waves are interacting with electrons with a high parallel velocity and a more-or-less average perpendicular velocity, those are the electrons furthest away from the boundary in velocity space between passing and mirror-trapped electrons, so magnetic trapping through the Ohkawa effect only slightly reduces the current drive efficiency. However, in some parameter regimes, waves with very high (but subluminal) parallel phase velocity often cannot propagate from the edge of the plasma, where the wave is excited, to the region in the plasma where the damping and current drive is desired. In such a case, one would say that the high phase velocity waves have poor accessibility to the desired location. There is often this kind of conflict between the most efficient waves for current drive and an accessible path for the waves to propagate from the edge of the plasma to the damping location, and this tension leads to the existence of optima.

An example of such a comparison between current drive using different damping mechanisms, in this case in the reactor design Fusion Nuclear Science Facility–Advanced Tokamak (FNSF-AT), is shown in Fig. 5. Here, a comparison is shown between ECCD launched in the conventional way from the low field side, ECCD launched nearly vertically, and a wave called the “helicon” or the “fast wave in the lower hybrid range of frequencies,” launched from above the midplane on the low field side. In this case, one sees that the helicon drives current at a minor radius farther away from the magnetic axis than either of the optimized ECCD cases, with an efficiency that is at least 50% higher than the best ECCD case. This high efficiency in the desired radial location is the reason that fast waves in the lower hybrid range of frequencies were chosen in the mid-1990s in the ARIES-AT reactor study² for the task of mid-radius current drive.

Unfortunately, waves that experience Landau damping in the plasma are necessarily radially evanescent in the vacuum region outside the confined plasma. This constitutes a coupling challenge, because in the case of linear wave excitation, at least, the antenna is near the first wall, in very nearly a vacuum region. Since the maximum parallel phase velocity that can be resonant with any electrons is less than the speed of light in vacuum, for non-zero Landau damping we must launch waves with $v ph | | < c$⁠, or the parallel index of refraction $n | | = k | | c / ω > 1$⁠. It is a simple exercise to show that waves with $| n | | | > 1$ have wave amplitudes that decrease exponentially in the radial (perpendicular) direction in the vacuum region adjacent to the antenna, and that the rapidity of that decay increases with $| n | | |$ at fixed frequency, or with frequency at fixed $| n | | |$⁠. Figure 6 illustrates a full-wave calculation where the wave amplitude is chosen to be unity at the antenna surface; the results exhibit much lower amplitude at the plasma for the higher value of $n | |$⁠. The more rapid decay means that if we wish to couple a certain level of power without exceeding some maximum electric field in the antenna (above which electrical breakdown of the antenna or excessive rf/materials interaction might result), there is a practical maximum value of $| n | | |$⁠.

We need to excite the wave with an antenna in the vacuum region, and the wave must propagate from the antenna region to the desired location of damping/current drive inside the plasma without losing too much power along the way, so that all of the wave energy is deposited in a radial zone that is reasonably well-defined. Now, we consider the waves that are available with $| n | | | > 1$ for tokamak parameters. For a concrete example, we examine the cold-plasma dispersion relation (while the damping is crucially dependent on non-zero temperature, the propagation of the waves from the antenna through the plasma is well-described by the cold-plasma model up to the point where the damping starts to be important) at a fixed magnetic field of 1.5 T, an electron density of $5 × 10 19 m − 3$⁠, the ion species being deuterium, and a parallel wavenumber of $0 . 4 cm − 1$⁠. We ask the question: at a particular $k ⊥$⁠, how many different frequencies satisfy the cold-plasma dispersion relation? The answer is five, i.e., there are up to five propagating solutions,¹⁷ as shown by the intersections of the vertical dotted line in Fig. 7(a) with the branches of the dispersion relation. There is one solution at low frequencies, below the ion cyclotron fundamental frequency, one at intermediate frequencies, and three at frequencies in the neighborhood of the electron cyclotron frequency. However, in many practical situations, $k | |$ is fixed but we also fix the wave frequency, because we have a rf source that operates at that particular frequency. So then, we ask the different question: how many distinct real (propagating) $k ⊥$ values satisfy the cold-plasma dispersion relation at a fixed frequency and $k | |$ (and plasma parameters)? The answer is zero, one, or two, depending on what frequency is chosen. Let us focus on the range of frequencies between the ion and electron cyclotron frequency, for reasons that will soon become apparent. Figure 7(b) shows that the horizontal dotted line at 0.65 GHz has two intersections with the dispersion curve. Where there are two solutions, they may be unambiguously distinguished by their radial phase velocities $ω / k ⊥$⁠. The one with larger $k ⊥$ (lower radial phase velocity) is called the “slow wave” and the other is called the “fast wave.” Even in a range where there is only one propagating wave, we refer to it as the fast wave because it smoothly connects to the fast wave in the range where both waves propagate. The advantage of this sort of dispersion diagram in which $ω$ is plotted against $k ⊥$ at a fixed $k | |$ is that the radial group velocity, which is the velocity at which wave energy propagates across the magnetic field, is just the slope $d ω / dk ⊥$ of the curve at a given point. One sees that at the radial cutoff, where $k ⊥$ goes to zero (at about 50 MHz for these plasma parameters and this $k | |$⁠), the radial group velocity of the fast wave also goes to zero; that the slow root has a low phase velocity (the curve is close to horizontal); and that the sign of the group velocity of the slow wave is opposite to the sign of its radial phase velocity, i.e., it is a (radially) “backward” wave. Furthermore, one sees that for this value of $k | |$ and these plasma parameters, the two solutions are part of the same curve and that at a particular value of $k ⊥$ the two solutions join, at a point where the radial group velocity vanishes. Since the two waves are indistinguishable there, this is called a mode conversion point.

This frequency range logarithmically between the ion and electron cyclotron frequencies is referred to as the “lower hybrid range of frequencies” because the wave frequency is near a wave resonance called the lower hybrid resonance (LHR).^11,17 The propagation of cold-plasma waves in a uniform plasma or in a plasma with inhomogeneous density in one direction perpendicular to a straight, uniform static magnetic field is summarized in  Appendix A, where the condition for the LHR is derived. At the LHR, represented by a horizontal dashed line on the dispersion diagram in Fig. 7(b), the slow root asymptotically approaches zero group velocity, and the wave's perpendicular phase velocity $ω / k ⊥$ also goes to zero as $k ⊥$ diverges. This means that a wave launched from lower density can never reach the lower hybrid resonance density, only approach it asymptotically. As shown in  Appendix A, if $ω pe 2 ≫ Ω e 2$⁠, i.e., at high density and low magnetic field, the LHR frequency is approximately equal to the geometric mean of the electron and ion cyclotron frequencies (what is referred to as the geometric-mean-gyrofrequency), so that $ω LHR ≈ ω gmg ≡ Ω i Ω e$⁠, while in the opposite limit of $ω pe 2 ≪ Ω e 2$ (low density, high field), the LHR frequency approaches the ion plasma frequency $ω pi$⁠.

How does the lower hybrid resonance affect the propagation of the slow and fast branches? Instead of plotting the solution of the cold-plasma dispersion relation at a fixed density, Fig. 8 exhibits the solution as a function of density at fixed magnetic field, ion species, frequency, and $k | |$ (hence $n | |$⁠). What is plotted on the ordinate is proportional³⁰ to $k ⊥ 2$⁠, so where the value is negative, the wave is radially evanescent and where it is positive, the wave is propagating. The point at which $k ⊥$ vanishes is the radial cut-off. Hence, one sees that for one range of densities, only the slow wave is propagating, while in the adjacent higher range of densities, both waves are propagating. At the lower hybrid resonance density, the slow wave $k ⊥$ diverges, while the fast wave is unaffected by that density and continues to propagate to arbitrarily high densities. The different response of the two waves at the lower hybrid resonance results from the difference in wave polarizations. As long as the two branches have different values of $k ⊥$ and hence are distinct, their polarizations differ, and only the slow wave polarization excites the lower hybrid resonance.³³ 

One can visualize some other aspects from yet another variation of the dispersion plot, where instead of holding $k | |$ fixed for the whole plot, we fix $n | |$ (at a fixed set of plasma parameters), the rationale being that $n | |$ is the parameter that enters into considerations of Landau damping. We select a value of $n | |$ so that the parallel phase velocity is about one-third of the speed of light, i.e., $n | | = 3$⁠, with the resulting plot being exhibited in Fig. 9. The curve corresponding to the fast branch is asymptotic to a vertical line down to low frequencies. This is what would be termed the compressional Alfvén wave at low frequency, and the perpendicular phase velocity (and group velocity) is the Alfvén velocity $v A = B / μ 0 n e m i = c ( Ω i / ω pi )$⁠. As the frequency increases, well above the ion cyclotron frequency the curve starts to deviate from the vertical line, in this case above about 100 MHz. In this range, the nomenclature for the wave depends on the application of plasma physics with which one is concerned; in the different fields of rf plasma sources, magnetosphere/space physics, and magnetic fusion, the terminology differs. In the fusion context, the fast branch or the mainly electromagnetic wave is sometimes referred to simply as the “fast wave in the lower hybrid range of frequencies,” while in the rf plasma source literature, where the fact that the geometry is bounded is crucial, the same wave tends to be called the “helicon,” and in the magnetosphere context, it is almost always referred to as a “whistler” wave, for reasons described below. The other, radially backward branch, is traditionally called “the lower hybrid” wave or sometimes the “slow” wave in the tokamak context (if the other branch is being mentioned in the same context), while in the study of rf plasma sources the same wave is a surface wave in most practical cases, and it is called a “Trivelpiece-Gould” mode. In the magnetosphere literature, in the older papers both branches are sometimes considered to be part of the whistler, while in the more recent literature, the slow branch is referred to as the lower hybrid wave or the “quasi-electrostatic” wave. The damping mechanisms in the different fields, although their relative importance varies substantially from field to field, are the same mechanisms—collisional damping and Landau damping, in some cases on a non-thermal population.

The situation regarding wave excitation is quite different in the three different applications of plasma physics. In tokamaks, waves are launched at a particular frequency, which is determined by the source frequency. We are interested in steady-state situations, or at least ones in which the time scale on which the plasma parameters vary is very much longer than the inverse of the wave frequency, so the bandwidth is extremely narrow. Because current drive is the goal, ideally one launches waves with a controlled sign and magnitude of $k | |$⁠. This is done by launching the waves with a toroidally extended, phased structure. In a tokamak with conventional aspect ratio, the toroidal direction is almost the same as the direction of the static magnetic field, because the tilt angle of the magnetic field lines is fairly shallow (especially at the high-field, small-major-radius side, referred to as the “inboard” side). The fact that the toroidal angle is ignorable (axisymmetry) implies that the toroidal wave number is conserved as the wave propagates, and since the toroidal and parallel directions are nearly the same, $n | |$ is also approximately conserved. However, in space plasmas, waves of interest are excited by impulsive events in the upper atmosphere: lightning. A stroke of lightning is nearly a delta function in time and well-localized in space relative to the spatial scales of interest, so it excites a wide band of frequencies and wavelengths. The frequencies that one measures at a remote location relative to the location of the stroke of lightning are determined by the propagation of the waves through the magnetosphere, acting like a delay line or band-pass filter. It turns out that the fast waves in the lower hybrid range [at typical magnetospheric plasma parameters, the lower hybrid range coincides with the very low frequency (VLF) (3–30 kHz) range, corresponding to audio frequencies] disperse—higher frequencies have higher group velocities parallel to the magnetic field lines and hence are received before the lower frequencies arrive at the detector. So, the received audio frequency tone (after detection and conversion to sound) after an amplifier whistles downwards and that is why these waves are called whistlers in this context.^11,17 In rf plasma sources, the waves are again excited at a specific frequency, often 13.56 MHz, by antennas that are not very k-specific. The wavelengths that propagate are determined by the boundary conditions in the finite geometry: the waves that will fit at a particular density and field, i.e., eigenmodes. The word “helicon” came from solid-state plasmas—in 1960, a French solid-state physicist (P. Aigrain) dubbed these waves “helicons”¹⁸ referring to the polarization of the electric field of the wave, which rotates around the static magnetic field line. The tip of the electric field vector traces out a helix rotating in a right-handed sense. Strictly speaking, the polarization fits the “helicon” description (that is, the wave electric field is right-hand circularly polarized), however, only for wave propagation exactly along the static magnetic field lines (⁠ $k ⊥$ = 0),¹¹ so it is appropriate for a typical rf plasma source geometry. The perpendicular wavenumber $k ⊥$ vanishes in a magnetic confinement context only at the location where the wave is cut off, so the wave is circularly polarized only where the wave is being excited by the antenna fields. As the wave propagates to higher density, the wave quickly becomes nearly linearly polarized, with the wave electric field primarily in the perpendicular direction, so the term “helicon” seems less apt.

Although the plasma parameters—the actual values of the densities and magnetic fields—are very different in the three different fields, the dimensionless frequencies of interest are in the same neighborhood. That is, for the wave frequencies of interest, often $ω / Ω i$ is very large, $ω / Ω e$ is very small, and $ω pe / Ω e$ is on the order of unity or larger, and the modes with which we are concerned are in the lower hybrid range of frequencies (Table I.)

Wave accessibility to high densities in the lower hybrid range of frequencies is strongly dependent on the wavelength along the magnetic field, i.e., at a fixed frequency, on $n | |$⁠. Consider the comparison in Fig. 8 of (a) $n | | = 3$ with (b) $n | | = 1.4$⁠, at 0.5 GHz, $B 0 = 1.5 T$⁠, deuterium (⁠ $ω / Ω i = 44$⁠, $ω / Ω e = 0.012$⁠). We are motivated to reduce the magnitude of $n | |$ for two reasons: first, to increase the efficiency of the current drive by interacting with more energetic and hence less collisional electrons and also to reduce the rapidity of the radial evanescence in the vacuum region adjacent to the antenna. But at the lower value of $n | |$⁠, the two branches approach each other as density increases and then merge, so in that case the two branches are coupled to each other and at a particular density are not distinct. A full-wave solution in the whole domain shows¹⁹ that energy launched towards higher density on one branch will reflect back down the density gradient in the other branch. This phenomenon is symmetric with respect to which mode is initially launched inwards at the particular value of $n | |$⁠. For the lower value of $n | |$⁠, higher values of density than the one at which the two branches merge are inaccessible from the low density side.

The minimum value of $n | |$ for which a given density is accessible depends strongly on the magnetic field, with higher magnetic fields allowing accessibility to lower $n | |$⁠. The accessibility criteria for waves in the lower hybrid range of frequencies are derived in detail in  Appendix B for a cold plasma in a straight, uniform static magnetic field. We may use that strong dependence of accessibility on magnetic field strength to improve the current drive efficiency with slow waves. In a given tokamak design, the highest toroidal field strengths are available at the inboard edge of the plasma, so if the technical problems associated with placing a wave-launching structure on the high-field side can be solved, the best wave accessibility is available from that launch position. As shown in Ref. 20, for one version of the FDF reactor design with given density and temperature profiles, waves launched from the (heretofore conventional) low field side that can deposit their energy on electrons before hitting the accessibility limit can penetrate only to a normalized minor radius of about 0.9, while launching from the high field side allows penetration and damping at a normalized minor radius of 0.6. The wider window between the accessibility and strong absorption limits arises from the improved accessibility as field increases—the damping limit does not depend on field strength. A number of other possible advantages of high-field-side launch were discussed in Ref. 21.

The remainder of this paper is primarily concerned with the fast wave (whistler, helicon) and the slow wave in the LHRF. By comparison to the slow wave at the same value of $n | |$⁠, the fast wave has a lower value of $E | |$ for the same power flux, so the Landau damping of the fast wave is weaker, permitting penetration to higher temperature at a given $n | |$ before the onset of strong absorption. As the wave frequency is raised at a fixed $n | |$⁠, the Landau damping increases (essentially due to the larger $E | |$ in the wave polarization, which in turn stems from the greater importance of the non-zero electron inertia as $ω / Ω e$ increases), so the radial location of the damping in a given temperature profile can be adjusted by changing the wave frequency. This motivates study of the changes in wave propagation for fast waves as the wave frequency rises from the ion cyclotron range into the LHRF. This is done by calculating the angle $ψ$ between the ray direction, which is the direction of the group velocity vector $v gr = ∂ ω / ∂ k$⁠, and the static magnetic field as a function of $n | |$ with frequency as a parameter. We use parameters typical of plasmas in the DIII-D tokamak as a concrete example. For $n e = 3 × 10 19 m − 3$ (in deuterium) and $B 0 = 1.5 T$⁠, Fig. 10 shows that for 20 MHz fast waves $( ω / Ω D = 1.31 )$ the angle $ψ$ monotonically decreases from $π / 2$ (group velocity directed across the field lines) at $n | | = 0$ to wholly parallel to the field lines (⁠ $ψ$ = 0) at the cut-off $n | |$ of about 43. As the frequency is increased at fixed density and field, the curve flattens out in the middle, becomes non-monotonic, and develops a minimum at low $n | |$⁠. At a certain frequency, that minimum touches the x-axis, i.e., another value of n_|| appears, other than the cutoff, at which the ray direction becomes parallel to the magnetic field. This frequency, 0.527 GHz in this case, is just the LHR frequency, and at slightly higher frequencies the slow branch propagates at a very small angle to the magnetic field, almost independent of n_|| away from the immediate neighborhood of the $n | |$ at which the angle is zero (which is the mode-conversion point).

The ray angle $ψ$ can then be used to compute ray paths in a plane-stratified geometry, in an “unwrapped” model of the DIII-D tokamak where the ordinate in Fig. 11 is the direction of the stratification of the density and the magnetic field strength (the major radius direction assuming that waves are launched from the outboard midplane) and the abscissa corresponds to the toroidal direction (ignoring the toroidal curvature).³¹ At 90 MHz $( ω / Ω D ∼ 8 )$⁠, the fast wave ray path makes an angle of about 20° to the magnetic field, close to the well-known maximum angle for a whistler wave of $tan − 1 ( 8 − 0.5 ) = 19 . 5 °$⁠.^11,17 At 500 MHz (⁠ $ω / Ω D ∼ 44$⁠, in the LHRF), at similar values of $n | |$⁠, the ray angle is much smaller, around 1°. At the same frequency and $n | |$⁠, the slow wave propagates at a still smaller angle to the magnetic field and penetrates only very slowly. The slow wave can reach the LHR density only asymptotically, which for these parameters occurs at $B 0 = 1.22 T$⁠, $n e = 5 × 10 19 m − 3$⁠.

We survey the wave accessibility in the full three-dimensional geometry in an axisymmetric tokamak equilibrium using ray-tracing, because the complicated effect of the evolution of the poloidal wavenumber³² in toroidal geometry has important consequences for wave propagation in a finite aspect ratio torus,²² which cannot be easily addressed analytically and have no equivalent in simpler geometries. These calculations are carried out with the GENRAY ray-tracing code,²³ and we use them to find the optimum frequency for an experiment using the fast wave branch in the DIII-D tokamak,²⁴ in which the goal is to maximize current drive at mid-radius in a particular class of high-beta discharges. The equilibrium and kinetic profiles are taken from a discharge that has been obtained in DIII-D (the density and magnetic field profiles also having been used in the plane-stratified example earlier in this paper). The predicted driven current and the radial location of the peak in the driven current are shown in Fig. 12 as functions of the fast wave frequency at a fixed $n | |$ (launched) of 3, from the outboard midplane. At frequencies near 100 MHz, despite the rather high ion cyclotron harmonic number (around 9 or 10), the high ion temperature and the high $k ⊥$ of the wave at the low toroidal field and high density (that is, large values of $k ⊥ ρ i$⁠, the finite-ion-gyroradius parameter that determines the strength of the ion cyclotron interaction) yield strong ion damping (on both thermal and fast ions) at the 9th deuterium cyclotron harmonic layer that the wave first traverses on its way towards smaller major radius. Complete ion absorption prevents wave penetration to the region where electron Landau damping would become significant. As the frequency is raised, and the ion cyclotron harmonic number increases, ion damping decreases, along with stronger electron damping at higher frequencies, so that the predicted current drive rises to a maximum at 400–500 MHz. Around the optimum, the predicted current drive efficiency is about 0.06 A/W, with the peak occurring at a normalized minor radius of ∼0.6. The current drive “engineering efficiency” parameter $n ¯ e I C D R 0 / P r f$ that is often used for comparison would be about $0.07 × 10 20$ Am⁻²/W. At still higher frequencies, accessibility becomes poor, so the fast waves (also true of slow waves at the same launched $n | |$ and poloidal launch location) cannot propagate to the desired mid-radius region. Furthermore, the more rapid radial decay of the wave field amplitude in the vacuum region (not addressed by the ray-tracing calculations) at higher frequencies would make excitation of the waves at high power levels significantly more difficult.

Next, we display the ray paths for slow and fast waves for some specific cases. Launching a slow wave at 0.65 GHz from the outside [Fig. 13(a)] yields an extremely slowly penetrating ray, which encounters mode conversion points and evolves in a nearly unpredictable way. However, for a 5 GHz slow wave launched from the inside (the high-field side), much-improved accessibility results in a straightforward ray path with penetration and absorption around normalized minor radius of ∼0.7 within a half of a toroidal turn [Fig. 13(b)]. A 90 MHz fast wave has no problem penetrating radially but is completely absorbed by ions at the 9th harmonic before any significant electron damping has occurred [Fig. 14(a)], while a 500 MHz fast wave or “helicon” from the outside is absorbed at the desired location within about a half-turn toroidally [Fig. 14(b)].

The current drive model in GENRAY, which is based on the classical current drive theory discussed above, has been compared to direct solution of the bounce-averaged Fokker-Planck equation using the CQL3D code²⁵ for some of these fast wave cases, with satisfactory agreement in the predicted current drive efficiency and the radial location of that driven current having been obtained at power levels (in the quasilinear code) up to several MWs.²⁴ 

In summary, electron current drive can be efficiently produced by Landau damping of a toroidally asymmetric wave spectrum, and the damping and the current drive processes are reasonably well understood. To drive current at mid-radius in a reactor-scale plasma with good efficiency, investigation of variations on already-proven current drive methods is needed. For electron current drive in tokamaks via Landau damping, the optimal frequency lies in the lower hybrid range of frequencies, where two distinct wave branches exist, known as the slow or lower hybrid wave and the fast wave, also known as the whistler or helicon wave. Wave accessibility from the edge of the plasma to the desired damping location has been examined; accessibility for the helicon branch to the mid-radius region is excellent from the low-field side, while accessibility for the slow wave may require launch from the high-field side.²¹ Experimentally, the fast wave in the lower hybrid range of frequencies (helicon, whistler) has been an underexplored possibility and is at present the subject of a study on the DIII-D tokamak. The application of this wave for mid-radius current drive extrapolates very well to use in a class of steady-state fusion reactors based on the Advanced Tokamak, thus providing a strong motivation for this study.

This paper is dedicated to the memory of Professor T. H. Stix (1924-2001) and Dr. Victor L. Vdovin (1937-2015). The author's understanding of the topic of this paper was significantly improved by many discussions with these two distinguished colleagues. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Award No. DE-FC02-04ER54698. DIII-D data shown in this paper can be obtained in digital format by following the links at https://fusion.gat.com/global/D3D_DMP.

Here, we outline the theory of cold plasma waves in a uniform and then in a one-dimensional slab geometry. This topic is fully covered in standard textbooks^1,11,17 but is summarized here for convenience.

We are interested in linear (small amplitude) wave propagation in a steady-state situation in which the parameters of the medium vary in only one direction, which we will call the x-direction in a Cartesian coordinate system. This direction corresponds to the minor radial direction in a toroidal magnetic confinement device such as a tokamak, stellarator, or RFP. We imagine that the static magnetic field lines are straight and parallel, so we ignore magnetic shear and the direction of the static magnetic field defines the z-direction. With reference to the case of a tokamak of not too low an aspect ratio at reasonably high values of the safety factor “q,” we think of the y-direction as being the poloidal direction and the z-direction as the toroidal direction.

In the cold-plasma approximation, we ignore the motion of the charged particles in the absence of the wave fields that results from the particles' thermal energy, i.e., the particles' drift along the magnetic field lines and their gyration around those field lines. We also omit the effect of collisions, though a simple collision model can easily be added later. Hence, there is no energy dissipation in the cold plasma model. It is by no means obvious that this set of assumptions can lead to a model relevant to experimental situations, just as in the realm of magnetohydrodynamic (MHD) theory, it is not obvious that the ideal MHD (no dissipation) theory has relevance to experimental realizable situations. Discussion of the justification and range of applicability of the cold plasma theory is deferred to textbooks.

The relevant wave equation is derived from Faraday's law and Ampere's law (with the displacement current term), which in MKS units are $∇ × E 1 = − ∂ B 1 ∂ t$ and $1 μ 0 ∇ × B 1 = + ε 0 ∂ E 1 ∂ t + j 1$⁠, where the relationship between the first-order current and the first-order (wave) electric field E₁ (we assume no zero order electric field) is through a conductivity tensor via $j 1 = σ ⋅ E 1$⁠. Note that in a magnetic field, the current due to an electric field is not generally in the same direction as the electric field, so that the conductivity is a tensor rather than a scalar. The key step in the derivation is the solution of the equation of motion for each species (ions, electrons) to derive that conductivity tensor. The linearized force balance (so the second-order term $v ⋅ ∇ v$ disappears) for the each pressureless (cold) species s of charge q_s and mass m_s is identical to the single-particle equation: $m s ∂ v s ∂ t = q s ( E 1 + v s × B 0 )$⁠. If we for the moment assume that the plasma is uniform in all directions, we can take the wave electric and magnetic fields to vary as $exp [ i ( k x x + k z z − ω t ) ]$⁠, where k_x and k_z do not depend on position (the spatial symmetry means they are conserved) and where ω does not vary in time and is real (no damping or wave growth in time). Note that in a uniform plasma, we are entitled to choose the coordinates such that k_y = 0 and the wave fields vary only in the x-direction (perpendicular to B₀) and in the z-direction (parallel to B₀). A key feature of the cold plasma model is that, since the equation of motion for the small-amplitude response has no spatial derivatives, the response does not depend on the wavelength of the fields, only on the wave frequency and the local plasma parameters (this is not true for the hot plasma model). This is why when we derive equations for the wave fields in the case where the parameters vary in the x-direction, we will be able to use the same dielectric tensor at each value of x that we are obtaining for the uniform case. Then, the equation of motion for the response of each species to the first-order wave fields at a particular k_x and k_z at a particular frequency replaces the time derivative with the multiplicative factor −iω. The solution for the component along the zero-order magnetic field (⁠ $− i ω m s v z = q s E z$⁠) is trivial, while to solve for the relationship between the perpendicular electric field components E_x and E_y and the velocity components v_x and v_y requires more algebra. Having determined the velocities, the currents are just the sum of the products of the velocities, densities, and the charges of each species. Thus, j_z is the sum of $i q s 2 n s m s ω E z$ over the charged particle species, so that the components of the conductivity tensor σ_zx and σ_zy vanish and $σ z z = i ε 0 ω ∑ s ω p s 2$⁠, in which the plasma frequency for each species $ω p s 2 ≡ n s q s 2 ε 0 m s$ has been introduced. The perpendicular currents j_x and j_y each have terms in both E_x and E_y; it is left as an exercise for the reader to show that $σ x x = σ y y = i ε 0 ω ∑ s ω p s 2 ω 2 − Ω s 2$⁠, in which the angular gyrofrequency for each species $Ω s ≡ q s B 0 m s$⁠, and that $σ x y = − σ y x = − ε 0 ∑ s Ω s ω p s 2 ω 2 − Ω s 2$⁠. In the expressions for the non-vanishing off-diagonal conductivity components, the gyrofrequency carries the sign of the charge, i.e., for a static magnetic field pointing in the positive z-direction, the electron gyrofrequency $Ω e$ is negative. Since there is no response perpendicular to the magnetic field lines to an electric field component along those field lines, $σ x z = σ y z = 0$ and vice-versa, so $σ z x = σ z y = 0$⁠.

Returning to the ω  $≪$ Ω_e case where the lower hybrid resonance is the relevant branch, it is easy to see that in the Ion Cyclotron Range of Frequencies $ω ∼ n Ω i$ with n a small integer, the lower hybrid resonance occurs at very low densities, in the far scrape-off layer in typical magnetic fusion devices, so only the fast wave propagates in the confined plasma in that frequency range. Raising the frequency into the range $Ω i ≪ ω ≪ Ω e$ pushes the lower hybrid resonance density into the confined plasma so that both slow and fast waves can propagate in a large volume of the plasma. This range is referred to as the “lower hybrid range of frequencies” for this reason.

We have arrived at the picture of cold-plasma wave propagation in the LHRF in which either slow or fast waves are excited from the plasma edge with rf electric fields in the z- or y-directions, respectively, then the waves propagate across the static magnetic field lines towards increasing density, gradually at shorter and shorter perpendicular wavelength as the density increases. If the wave frequency is less than the geometric mean gyrofrequency $Ω g m g$⁠, the slow wave can encounter a density layer at which the perpendicular wavelength goes to zero, which is the lower hybrid resonance. The fast wave is unaffected by the lower hybrid resonance, so at higher densities only the fast wave can propagate. If $ω > Ω g m g$⁠, no lower hybrid resonant density exists, so that both slow and fast branches continue to propagate to arbitrarily high density. What is missing from this description, however, is that for a particular value of n_z, conserved in this geometry due to the symmetry in the z-direction, at some point the coupling terms may grow so large that the polarizations of the two waves become indistinguishable from one another. At such a mode-coupling point, further analysis shows that the group velocity in the x-direction vanishes and the wave cannot penetrate to higher density. Instead, the wave is reflected in the opposite mode and proceeds back down the density gradient towards the boundary. At the value of n_z being considered, higher density is “inaccessible.” We can evaluate the critical value of n_z by a somewhat elaborate analysis, first mentioned in Ref. 27, briefly explained in the 1962 edition of Ref. 11, made more precise in Ref. 28, and detailed in the following. The underlying assumption is that the static magnetic field strength is constant, while the density rises from zero to a maximum value as x increases.

We have gone into detail with this derivation of the accessibility criterion because some of those details are not readily available in the literature (though given in a compressed form in Ref. 11 and in more detail in Ref. 1). However, it must be observed that these analytic expressions do not directly apply to more realistic situations, where the magnetic field strength varies along the ray trajectory, as in, for example, a finite-aspect-ratio toroidal geometry. Also, the assumption that the wavenumber is precisely zero in the direction mutually perpendicular to the direction of the density gradient and to the straight, uniform magnetic field (what would correspond to the poloidal direction in a torus) is not a very good one in a more realistic geometry with magnetic shear, poloidal curvature, etc.