Comparative Experiments on Lower Hybrid Wave Heating of Ions in High-Density Hydrogen and Deuterium Plasmas at the FT-2 Tokamak

Abstract

In the framework of the isotope effect studies at the FT-2 tokamak, the efficiencies were compared of the lower hybrid wave heating of the ion components of the hydrogen and deuterium plasmas with high densities (close to the Greenwald limit, 〈n_e〉 ≤ 10²⁰ m^–3). It was experimentally ascertained that, in accordance with the theoretical concepts, the efficient axial heating of the deuterium plasma ions occurs, as opposed to the peripheral heating of the hydrogen plasma ions. Such an isotope effect occurs due to the different localization of the plasma-RF wave interaction regions. The distinctive feature of these experiments is the fact that, in deuterium plasma, during the preliminary ohmic heating, the dependence of the energy lifetime on density τ_Е(n) is linear (LOC dependence), and, with increasing density, the transition to the improved ohmic confinement mode occurs. In hydrogen plasma, on the contrary, the transition to the saturation mode is observed. In this study, the considerable decrease in τ_Е was revealed, observed during the additional lower hybrid wave heating in both hydrogen and deuterium plasmas.

DISTINCTIVE FEATURES OF MEASURING AND MODELING THE DISCHARGE PARAMETERS AT HIGH DENSITIES

A. The complex approach was used to reconstruct the density profiles, which included both the local Thomson scattering (TS) measurements along the vertical chord shifted 1.5 cm from the center and the integrated phase measurements performed with the help of the microwave interferometer with seven vertical sounding chords.

During the interferometry measurements, the plasma was probed using the ordinary wave with a frequency of f_i = 135–138 GHz. The dependence of the measures phase shift on the plasma density is as follows:

$$\Delta \Phi (t) = \frac{{2\pi L}}{\lambda }\left( {1 - {{{\left( {1 - {{{{n}_{e}}(t)} \mathord{\left/ {\vphantom {{{{n}_{e}}(t)} {{{n}_{c}}}}} \right. \kern-0em} {{{n}_{c}}}}} \right)}}^{{1/2}}}} \right),$$

(A.1)

where L is the path length in the plasma, λ is the wavelength of the probe radiation, and ${{n}_{c}} = {{{{m}_{e}}\pi f_{i}^{2}} \mathord{\left/ {\vphantom {{{{m}_{e}}\pi f_{i}^{2}} {{{e}^{2}}}}} \right. \kern-0em} {{{e}^{2}}}}$ is the critical density [6]. The distinctive feature of the interferometry measurements in the high-density plasma is the strong refraction of the probe beam. The ray trajectories became curved, so the dependence of the phase shift on the density turns out to be nonlinear that, along with an increase in the path travelled, additionally contributes to the measured phase difference [6]. In the expression relating the electron density to the experimentally measured phase,

$$n(t) = {{n}_{c}}\left( {\frac{{\Phi (t)\lambda }}{{\pi L}} - {{{\left( {\frac{{\Phi (t)\lambda }}{{2\pi L}}} \right)}}^{2}}} \right),$$

(A.2)

the quadratic term becomes important. The procedure for reconstructing the density profile using data of the chord measurements consists in the sequential accumulation (from the base to the apex) of the cylindrical layers of equal density. In the case of the straight rays, in the linear approximation, the density, radius, and horizontal shift of each layer are chosen using the condition that the phases at the layer edges coincide with those on the approximating profile. In the case of the curved rays, in expression (2), the quadratic term must be taken into account. In addition, the analytical calculations of the ray trajectory between the probing and receiving horn antennas are required. Since, under conditions of the strong refraction, the maximum signal power may correspond not to the central ray, the next step in processing the measurement results is the ray tracing calculations in the geometrical optics approximation. In these calculations, the density profile obtained in the first stage is used, and the real antenna patterns are taken into account. Then, the integrated phase is calculated for a given set of rays. The density profile is reconstructed as a result of the iterative procedure, during which the experimentally measured phases are compared with the calculated ones. In calculations, the rays forming the beam are considered separately, and the diffraction effects are not taken into account. This procedure considerably increases the reliability of the radial density profile reconstruction.

The interferometry measurements were used to calibrate the relative densities obtained using the local Thomson scattering diagnostics. The calibration was performed by means of correlating the Thomson scattering data with the integrated density measured by the interferometer along the vertical chord coinciding with the axis of the laser sounding. Moreover, to increase the reliability, the absolute calibration of the density profile measured using the TS diagnostics was usually performed in the initial discharge stage at the relatively low electron densities, when the refraction effects are small.

B. The ion temperature profiles T_i (r) were measured using the five-channel nuclear particle analyzer (NPA) [14], which can scan the plasma from shot to shot in the vertical plane. The ion temperature was determined by means of analyzing the recorded energy distributions. Under conditions of the FT-2 tokamak, the applicability of this method was confirmed by the calculations using the ASCOT code [7]. As a result, it was shown that the direct determination of the ion temperature is possible in the plasma density range of 〈n_e〉 ≤ (4–5) × 10¹⁹ m^–3. Using the calculations, it is also possible to determine the region of the RF power direct absorption from data on the source localization of the high-energy charge exchange atoms with energies Е_СХ ≥ 2 keV. In the denser plasma with 〈n_e〉 ≥ (5–6) × 10¹⁹ m^–3, the flow of charge exchange atoms from the axial region of the plasma column decreases when it passes through the peripheral plasma layers. This effect is important, since it complicates the direct measurements of the T_i(r) profile. Therefore, to correct the ion temperatures determined from data on the NPA measurements, the model calculations were additionally performed using the DOUBLE-MC Monte Carlo code, which involve the entire experimental data array [8]. To simulate the charge exchange atom flow along the observation line, the following integral expression is considered:

$$\begin{gathered} \Gamma _{0}^{{}}(E) \\ = \int\limits_0^L {{{n}_{i}}(x)\sum\limits_j {\left[ {n_{j}^{0}(x){{{\left\langle {\sigma _{j}^{0}v} \right\rangle }}_{{{{v}_{j}}}}}} \right]} {\kern 1pt} {\kern 1pt} {{f}_{i}}(E,x){\kern 1pt} \mu (E,x){\kern 1pt} dx} , \\ \end{gathered} $$

(A.3)

where E is the energy of particles (the plasma atoms and corresponding ions escaping from the plasma); n_i(x) is the ion density of the basic plasma; $n_{j}^{0}(x)$ is the density of donors of the j type contributing to the neutralization process; ${{\langle \sigma _{j}^{0}v\rangle }_{{vj}}}$ is the corresponding reaction rate averaged over the distribution of the relative velocity of ions and donors; f_i(E, x) is the ion energy distribution function; μ(E, x) is the factor characterizing the atom flow loss due to the processes resulting in their repeated ionization; and x is the coordinate along the observation line. Summation is performed over the basic processes resulting in the neutralization of plasma ions. For the FT-2 tokamak, the most significant are the charge exchange processes between the ions and plasma neutrals, as well as the radiation recombination of ions and electrons. The integration is performed along the chord, the length L of which is determined by the intersections of the observation line with the boundaries of the poloidal section of the plasma volume. The f_i(E) ion energy distribution function is set to be the Maxwellian function, and the density and temperature profiles of ions and electrons are assumed to be the double parabolic functions approximating the experimental data. The expression for the atom flow loss factor has the following form:

$$\mu (E,x) = \exp \left\{ { - \int\limits_x^L {\sum\limits_k {\left[ {n_{k}^{i}(l)\frac{{{{{\left\langle {\sigma _{k}^{i}v} \right\rangle }}_{{{{v}_{k}}}}}}}{v}} \right]} \,} dl} \right\},$$

(A.4)

where $n_{k}^{i}(l)$ is the density of the corresponding plasma component that contributes to the process k resulting in the ionization; ${{\langle \sigma _{k}^{i}v\rangle }_{{vk}}}$ is the reaction rate averaged over the distribution of the relative velocity of the emitted atoms and the plasma component k; $v$ is the velocity of particles (the plasma atoms and corresponding ions escaping from the plasma); and l is the coordinate along the observation line. Summation is performed over the basic processes contributing to the ionization of atoms escaping from the plasma center. In the case of the FT-2 tokamak, the most significant processes are the charge exchange processes between the ions and plasma neutrals, as well as the ionizations by the electron and ion impact.

The upper boundary of the energy range, used in determining the ion temperature in the ОН regime, was limited to 1.6 keV. This is due to a decrease in the signals of the NPA detection channels to the noise level, which occurs with increasing energy, which, in turn, can be explained by the exponential decay of the ion energy distribution function at high energies. Since, at low energies, the loss factor (3) is rather high, the energy flows come to the NPA mainly from the peripheral plasma regions, where the ion temperature is lower than in the axial region. As a result, for the chosen energy range E_CX ≤ 1.6 keV, the measured ion temperatures T_{ion, eff} turn out to be considerably lower than the true temperatures T_ion, which was demonstrated by the model calculations using the DOUBLE-MC code. The calculations were used to correct the temperature estimate by means of introducing the correction coefficients K = T_{ion, eff}  /T_ion. The coefficients were determined by means of comparing the set and calculated profiles T_ion(r). The obtained dependence of the correction coefficients K(n, x) on the density were used to make the scaling corrections to the experimental te-mperatures T_{ion, eff} (r, t) measured in the axial region 0 ≤ r ≤ 3.5 cm. As an example, for the D- and H-plasmas, the coefficients K(n, x) calculated for the central chord of the plasma section, х = 0, are shown in Fig. 11.

Fig. 11.

Correction coefficients K(n, x) for the central chord of the plasma cross section calculated under assumption that the edge temperature of the working gas is T_e w = 3–5 eV. Curves 1 and 2 correspond to the H- and D‑plasmas, respectively.

The modeling using the DOUBLE_MC code was performed in two stages. In the first stage, the neutralization target was calculated, i.e., the density of all donors involved in the neutralization. Using the Monte Carlo method, the penetration of neutral particles through the peripheral plasma was calculated. Next, the target was calculated, which formed as a result of the recombination processes. To do this, we used the equations of the ionization balance in the approximation of the corona equilibrium. In the second stage, the energy distribution of the escaping atoms was calculated in accordance with expression (2). We also performed calculations of the auxiliary parameters, such as, for example, the luminosity function characterizing the probability of production of atom with a certain energy in a given point of the plasma and its further escape from this point. Thus, the use of the DOUBLE-MC code made it possible to establish the correspondence between the energy distributions of atoms escaping from the plasma and plasma ions. By means of varying the ion temperature used in the simulations, we can establish the correspondence between the calculated and experimental energy distributions of the escaping atoms. At densities of 〈n_e〉 ≤ (4–5) × 10¹⁹ m^–3, the simulations performed in accordance with the above scenario made it possible to directly determine the ion temperatures over the entire plasma section, and at high densities (in the HDR regime), it was possible only in the radial range of 3.5 cm ≤ r ≤ 8 cm, where a decrease in the charge exchange atom flow from the axial regions is not important.

In the axial regions of the plasma column, the profiles were corrected using the correction coefficients K(n, x).

Thus, in the HDR regime, under conditions of the plasma density increasing with time, it was necessary to correct the ion temperature measured in the axial regions. The correction was performed using the coefficients K(n, x) in both the OH regime and the LHH regime, when, as a result of interaction between the RF wave and the plasma, the high-energy particles (E_CX ≥ 2 keV) appeared in the energy distribution of ions.

We also note that the technique for determining the ion temperature profiles form data on the charge exchange atom flows can be sensitive to the effects of the magnetic field rippling, which consist in the loss of the so-called locally trapped ions. Indeed, at the FT-2 facility, the magnetic field rippling is rather high (10% in the limiter region). However, as the simulations using the ASCOT code [7] show, at the FT-2 tokamak, the loss of the locally trapped ions is negligible. This is due to the strong poloidal rotation of the plasma resulting in the considerable changes in the ion drift orbits. As a result, the effect of the magnetic field rippling on the ion temperature measurements can be neglected.