Isotopic effect in experiments on lower hybrid current drive in the FT-2 tokamak

To analyze factors influencing the limiting value of the plasma density at which lower hybrid (LH) current drive terminates, the isotopic factor (the difference in the LH resonance densities in hydrogen and deuterium plasmas) was used for the first time in experiments carried out at the FT-2 tokamak. It is experimentally found that the efficiency of LH current drive in deuterium plasma is appreciably higher than that in hydrogen plasma. The significant role of the parametric decay of the LH pumping wave, which hampers the use of the LH range of RF waves for current drive at high plasma densities, is confirmed. It is demonstrated that the parameters characterizing LH current drive agree well with the earlier results obtained at large tokamaks.


INTRODUCTION
The development of noninductive methods for sustaining a quasi-steady current in tokamaks is one of the most important problems in designing fusion reactors. Unfortunately, the most efficient method for sustaining the plasma current by means of lower hybrid (LH) waves can be implemented only at relatively low plasma densities not exceeding a certain density limit . The existence of the density limit for LH current drive (LHCD) was attributed to various mechanisms: linear absorption by ions, which increases as the density tends to the resonance LH value; collision losses; scattering by drift turbulence; parametric instabilities; etc. The density limit effect has been studied and discussed for several past decades; however, it has not received comprehensive physical explanation [1].
For a long time, the interaction of LH waves with plasma has been studied at the FT-2 experimental tokamak with the major radius of the toroidal chamber m, the radius of the poloidal limiter m, the magnetic field at the center of the chamber T, the range of the ohmic plasma current kA, the discharge duration ms, the RF pulse duration ms, the pumping wave frequency MHz, and the launched RF power kW [2]. Here, we present results of comparative FT-2 experiments with hydrogen and deuterium plasmas intended to reveal the mechanism governing the value. For this purpose, we employed the influence of the isotopic composition of the working gas on the plasma parameters determining the LHCD efficiency, which is expressed as , where is the mean plasma density along the central chord and is the normalized LH-driven current, with being the current generated by the RF wave.
The paper is organized as follows. The experimental dependences of the LHCD efficiency on the basic parameters and the data on the density limit for hydrogen and deuterium plasmas are presented in Section 2. In Section 3, the obtained dependences of and on the plasma parameters, as well as the maximum values of these quantities, are compared with results of experiments carried out at large tokamaks and the role of the isotopic factor is analyzed. The main results of the work are summarized in the Conclusions.

DEPENDENCE OF THE LHCD EFFICIENCY
ON THE MAIN PLASMA PARAMETERS In order to reveal the mechanism governing the density limit in LHCD experiments ( T, MHz) with deuterium and hydrogen plas- . Above this density, the interaction of the LH wave with electrons is replaced with direct absorption by ions [3]. Estimates show that, under the given experimental conditions, the value in deuterium plasma ( m -3 ) is substantially higher than that in hydrogen plasma ( m -3 , see Appendix A). Therefore, in hydrogen and deuterium plasmas, one of the mechanisms responsible for LHCD termination-absorption of the LH wave by plasma ions-comes into play at substantially different values of the density.
LHCD termination may be caused by various parametric instabilities. In the literature, two types of instability are discussed. One of them leads to the decay of the pumping wave into a number of satellite waves the frequencies of which are shifted from the pumping frequency by a value multiple of the ion cyclotron frequency . The other type of instability leads to the widening of the frequency spectrum of the pumping wave and the excitation of the so-called ionic quasi-mode. In both cases, as a result of parametric instability, satellite waves with longitudinal slow-down factors greater than that of the pumping wave are excited. These waves can be absorbed at the periphery of the discharge already at relatively low plasma densities [2,4,5].  [6] up to a certain density value , above which it rapidly decreases and vanishes at .

Hydrogen Plasma
At relatively low values of the discharge current ( kA), LHCD in hydrogen plasma is suppressed at densities substantially lower than (Fig. 1). In this case, we observe the cooling of the electron component during LHCD (according to the experimental data, the electron temperature T e on the axis of the discharge decreases from ~400 to ~300 eV), which is explained by an increase in the density and a reduction in the ohmic heating power (due the reduction in U pl ) during the LH pulse. The reduction in T e and the increase in the plasma density should lead to a reduction in the parametric instability threshold [7] and, accordingly, the earlier suppression of LHCD. The development of parametric instability is experimentally confirmed by the rise of the first and second satellites in the frequency spectrum of  the pumping wave at [2,5]. The satellite frequency ( ) is down-shifted from the pumping wave frequency. The shift is approximately equal to a multiple of the ion cyclotron frequency ( MHz for T). At a high plasma current ( kA), when the electron temperature on the axis is substantially higher, eV, the frequency spectrum of the pumping wave at n e < n LH has no satellites caused by parametric instabilities [2]. In this case, the experimentally observed density limit increases and approaches the resonance value ( Fig. 1), at which the transverse refractive coefficient in the cold plane approximation tends to infinity (i.e., the condition for the LH resonance is satisfied) and, according to theoretical estimates, the interaction of the LH wave with electrons is replaced with direct absorption by ions. Under the given experimental conditions, we have m -3 . With allowance for thermal motion of charged particles, the condition for absorption by ions begins at a slightly lower density corresponding to the point of linear conversion, m -3 , where a "cold" LH wave is converted into a "warm" plasma mode [3] (see Appendix B). In our estimates, we used the value , corresponding to the calculated slow-down spectrum of the pumping wave near the antenna, which was calculated for the effective plasma charge and ion and electron temperatures of eV and eV (see Fig. 3 in [2]). an energy of eV on the plasma density for kA. The sharp increase in this flux after the plasma density reaches a value close to the above estimate, m -3 , may be an indication of achieving the resonance conditions for the absorption of the LH wave by ions. The appearance of an intense flux of high-energy atoms indicates the emergence of the region of linear conversion, near which the slowed down LH waves ( , where v ph is the phase velocity) are absorbed by ions [3,6].

Deuterium Plasma
In deuterium plasma, for the resonance density and the density corresponding to the point of linear conversion (calculated for , , eV, and eV), we have m -3 , i.e., they are substantially higher than in hydrogen plasma [4]. Therefore, it might be expected that, in experiments with a hot deuterium plasma ( kA, eV), LHCD would vanish at greater values of the density, close to 10 20 m -3 . However, the experimental results contradict these expectations. As is seen from Fig. 2, the density limit m -3 is substantially lower than the densities and in deuterium. Nevertheless, the sharp increase in the charge-exchange atomic flux (1575 eV) is observed at densities higher than those in hydrogen plasma. The measured dependence (Fig. 2) is characterized by an appreciable scatter in the experimental points and has no kink. Apparently, under these conditions, it is possible to find the density value at which the sharp increase in the charge-exchange atomic flux begins, if we assume that, at this density, the characteristic density scale of the flux variation, , is minimal. In addition to , Fig. 3 also presents the dependences of on the densities of hydrogen and deuterium plasmas obtained from the experimental data approximated by the least-squares method. The arrows mark the minimum values of . The corresponding values of the plasma density determine the transition to the mode in which the interaction of the LH wave with plasma ions (protons/deuterons) becomes the main mechanism of RF power absorption. However, the density m -3 of fast neutrals (FNs) determined in this way for deuterium exceeds the value m -3 , at which LHCD terminates. Apparently, LHCD vanishes due to another mechanism. In particular, it could be caused by a substantial increase in as the LH wave penetrates into the plasma column [3].  However, the observed difference between and in deuterium is small and is probably caused by experimental errors. Therefore, in this case, LHCD may vanish due to the interaction of the LH wave with plasma ions.
The reason for such interaction may be the parametric decay of the pumping wave at the periphery of the discharge, which is observed experimentally with the help of an RF probe [2,4]. The frequency spectra of probe signals measured in deuterium plasma during an RF discharge at densities of and m -3 are presented in Fig. 4. The spectra contain several satellites of the pumping wave with the frequencies ( ), where MHz is the ion cyclotron frequency in deuterium (corresponding to T). We see that the amplitudes of the satellites and, accordingly, the role of parametric processes increase with increasing plasma density. In addition, with an increase in the density, a certain widening of the spectral components and is observed, which is apparently caused by the excitation of instability related to induced scattering of the LH pumping wave by ions (the so-called instability with the ionic quasi-mode) [8].
Thus, as the plasma density increases and the periphery of the discharge cools down, the parametric processes are activated. As a result, the absorption of the LH wave by electrons is replaced with direct absorption of the RF power by plasma ions, which is Another possible explanation of the relatively low density m -3 might be the presence in the spectrum of the pumping wave of strongly slowed down waves with , as it follows from model calculations performed in [2]. The experimental confirmation of the excitation of slowed down LH waves with in plasma requires additional studies with the use of diagnostics capable of measuring the spatial and time spectra of LH waves inside the tokamak plasma. One of such diagnostics is that based on enhanced scattering at the upper hybrid resonance [9]. The wavenumbers in this diagnostics are resolved by the time-of-flight [10] and correlation [11] methods. As applied to the study of the propagation of LH waves in a tokamak, the time-of-flight method was used at the FT-1 facility [12] and the first results obtained for the FT-2 tokamak at T were reported in [13,14].
The influence of the hydrogen admixture in experiments with deuterium plasma also cannot be excluded. With the help of monitor spectral observations, the presence of hydrogen in deuterium plasma was detected even after a long series of experiments. The content of hydrogen incoming, apparently, from the chamber wall was determined from the ratio of the spectral line intensities, H β /D β . In some cases, in the end of the discharge, the near-wall region contained up to 15-20% of neutral hydrogen. The influence of the hydrogen admixture on the value of or in deuterium plasma requires additional studies.

VALUES AND FUNCTIONAL DEPENDENCES OF AND
The data obtained in the described experiments were compared with the results obtained at large tokamaks. Such comparison has shown that a number of data characterizing the LHCD effect are in good agreement with the available results. Both the functional dependences of and on the plasma parameters and their values proved to be close to the results obtained at the TORE SUPRA [15] and FTU [16] facilities. For instance, according to Fig. 3, in deuterium plasma at kA and m -3 , we have A m -2 W -1 . This value at eV and agrees well with the experimental dependence presented in [15] and summarizing the result obtained at four facilities. The experimental dependences of the LH-driven current on the parameters , , and obtained in the present work are similar to those in [16]. For example, for the initial chord-averaged density m -3 , the dependence of on (where is the power absorbed in plasma) is close to linear (Fig. 5). On the other hand, in the interval kW, depends on as (ΔP RF -15 kW) α (where 15 kW is the minimum power required for the generation of ), which is very close to the weak power dependence (where ), observed in [16]. In our case, the approximation of the experimental data by a power-law dependence yields close values: for hydrogen and for deuterium. Most probably, the difference in the values of α for hydrogen and deuterium is caused by experimental errors.
As a whole, our experimental study of the influence of the plasma isotopic composition on the LHCD efficiency has demonstrated that the efficiency in deuterium plasma is higher than in hydrogen plasma. As is seen from Fig. 3, within the density range from m -3 to m -3 in deuterium, we have А m -2 W -1 , whereas, for hydrogen plasma, A m -2 W -1 .

CONCLUSIONS
The results of experiments carried out at the FT-2 tokamak allow us to conclude that the most probable reason for LHCD termination in both hydrogen and deuterium plasmas at a relatively low plasma current of kA and, accordingly, a low electron temperature is an additional reduction in T e at the periphery of the plasma column during the LH pulse. This results in a lower threshold for the parametric decay ( ) of the pumping wave. The parametric decay satellites, the frequencies of which are down-shifted, ( ), are slowed down more than the pumping wave. Therefore, the plasma density , at which linear conversion of slowed down satellite waves occurs, is lower than that for the pumping wave; hence, the LH wave after parametric decay can be absorbed by ions even at the plasma periphery, without penetrating into the plasma column.
At a higher plasma current ( kA) and higher electron temperature ( eV), the density at which the LHCD in hydrogen plasma terminates is close to the resonance value ( m -3 ). After the plasma density reaches this value, the interaction of the LH wave with electrons is replaced with direct absorption by ions.
The resonance value of the density for deuterium is substantially higher, m -3 ; however, the obtained value m -3 is smaller by more than one-half. Nevertheless, in contrast to hydrogen plasma, the density at which the sharp increase in the high-energy charge-exchange atomic flux, m -3 , takes place proved to be higher than , i.e., there is an appreciable gap between the values of and . Apparently, the main reason for LHCD termination in this case is the parametric decay of the pumping wave. The experimental results confirm that parametric processes intensify with increasing density during the RF pulse. Nevertheless, we cannot exclude the influence of a substantial slowing down (up to ) of the pumping wave as it propagates into the plasma column. The experimental confirmation of such influence requires special studies.
As a result of the experimental study of the influence of the plasma isotopic composition on the LHCD efficiency, it is established that the efficiency in deuterium plasma is higher than in hydrogen plasma. As is seen from Fig. 3  Here, and are the electron and ion cyclotron frequencies, respectively; is the electron plasma frequency; , where c i is the relative number density of the ith ion species; and Z i and M i are the charge and mass of ions (in units of the proton charge and mass) [3].
The necessary condition for the presence of the LH resonance in plasma is . The plasma density corresponding to the LH resonance is determined from the condition ε = 0, i.e., LC