Effect of Low-Frequency Magnetic Field Fluctuations on the Confinement of Plasma in the Uragan-3M Torsatron at Rare Collision Frequencies

Abstract

The plasma energy content in the Uragan-3M torsatron was determined by means of diamagnetic measurements under the conditions of its radiofrequency (RF) generation and heating (at near ion-cyclotron frequencies) in the presence of bootstrap current. The balance of power under the fast heating of plasma was considered and used as a basis to estimate the radiofrequency power absorbed by the plasma in the confinement volume. The behavior of energy losses in the discharge period was calculated, and the effect of magnetic field fluctuations on the energy losses within a frequency range of 0.5–70 kHz was discussed.

INTRODUCTION

The production and heating of plasma in the Uragan-3M (U-3M) torsatron [1] are traditionally performed by means of radiofrequency heating at near ion-cyclotron frequencies [2]. One of the regimes provides the generation of low-density and rather hot plasma [3], in which a relatively high longitudinal current is observed [3, 4]. Such a regime is characterized by rare frequencies of collisions between plasma particles (the banana region in the Galeev—Sagdeev curve).

Studies on the behavior of plasma in the regime of rare collisions is of interest not only to understand the confinement of plasma in a certain particular setup, but also may be interesting from the viewpoint of general physics, as the future thermonuclear reactor based on a toroidal trap will operate only in the regime of rare collisions.

The estimation of plasma parameters, in particular, the plasma temperature in this regime in the U-3M torsatron is attended with great difficulties. This is first of all due to the distortion of the distribution function under the conditions of rare collisions. For this reason, it proves useful for the solution of some problems to use diamagnetic measurements, which estimate the total stored energy in the plasma volume. Such measurements are useful, e.g., in the analysis of the power balance.

To use the diamagnetic effect in toroidal setups with external rotational transform, such as the U-3M torsatron, in the presence of a relatively great longitudinal current for plasma diagnosis purposes, it is necessary to know the distribution of the current and rotational transform angle over the cross section of a plasma cord. The estimation of the above-mentioned parameters as such is a complicated problem. This paper presents the conditions for the rather precise estimation of plasma energy content by the method of diamagnetic measurements in stellator systems.

Radiofrequency heating at a relatively high level of power absorbed by plasma is classified among fast heating methods. Fast heating is heating, under which the time of change in the power introduced into plasma is much shorter than the time of change in its parameters, which in turn are much smaller than the skin time for the penetration of the magnetic field of plasma currents into plasma and metallic surrounding elements. As shown in [5–7], the power spent to change the magnetic field in the plasma confinement region should be considered in the power balance under fast heating. It is also notable that the power spent on the heating of plasma under fast heating may attain only 1/3 of the introduced power according to [7].

One reason for excess in the losses of heat and particles from the plasma volume in toroidal magnetic traps over neoclassic theory predictions is the excitation of different instabilities in the plasma. In particular, among such instabilities are different branches of Alfven, geodesic, and drift waves, whose characteristic feature is magnetic field fluctuations. To detect magnetic field fluctuations, a set of magnetic sensors was installed on the U-3M torsatron in one of its small cross sections.

The objective of this paper was to determine the plasma energy content in the regime of heating with rare frequencies of collisions, estimate the radiofrequency power absorbed by the plasma and the losses of energy from the plasma volume, and to establish the relation between the detected magnetic field fluctuations with these losses.

CONDITIONS OF EXPERIMENT AND RESULTS OF STUDIES

Experiments were performed on the U-3M torsatron in the region of radiofrequency heating at a frequency ω = 2πf ≈ 0.8ω_Bi(0), where ω_Bi(0) is the ion-cyclotron frequency on the geometric axis of a torus. In this experiment, the RF generator frequency was  f ≈ 8.6 MHz. The nominal voltage on the generator lamp was 9 kV. The magnetic field on the geometric axis was B ≈ 0.7 T.

Uragan-3M is a three-thread torsatron with large radius of the plasma cord R ≈ 1 m and average last magnetic surface radius a ≈ 0.1 m. The screw-type winding is placed into the vacuum chamber with a volume of nearly 70 m³. The rotational transform angle distribution can be described as follows:

$${{\iota }_{{{\text{st}}}}} = {{\iota }_{0}} + {{\iota }_{2}}{{(r{\text{/}}a)}^{2}}$$

((1))

where ${{\iota }_{0}}$ ≈ 0.22, ${{\iota }_{2}}$ ≈ 0.11, and r is the current average radius of the magnetic surface. The working gas was hydrogen supplied in the flow-through regime. The breakdown in the working gas was implemented at a pressure of nearly 10–5 Torr and an anodic voltage of the generator lamp below the nominal level.

The temporal behavior of the basic parameters of a plasma discharge is illustrated in Fig. 1. The plasma density was measured with a 2-mm interferometer. The plasma energy content was determined by means of diamagnetic measurements. The diamagnetic loop represents two coaxial circular coils, whose diameters differ from each other by 0.01 m. The idea of measurement consists in that the useful signal Φ for both coils is the same, but the signals associated with the passage of currents in the metallic surrounding are proportional to the difference between the surface areas of these coils. The maximum frequency of signals detected by this diamagnetic loop is 20 kHz.

Fig. 1.

Temporal behavior of the basic plasma parameters, such as average plasma density ${{\bar {n}}_{e}}$, plasma energy content P, toroidal magnetic flux change Φ, longitudinal plasma current I, current in RF antenna I_RF, ratio of the plasma current to plasma energy content I/P, sum of the average temperatures of electrons and ions $\overline {{{T}_{e}} + {{T}_{i}}}$ determined from diamagnetic measurements. The moments, at which additional power was turned on and off, and the moment of singularity in the behavior of the plasma energy content are marked with dashed lines.

The equation relating the plasma energy content with the other measured discharge parameters has form

$$P = - \frac{3}{4}RB\Phi \, + \,\frac{3}{2}\pi R\frac{{{{I}^{2}}}}{{{{c}^{2}}}}\, + \,3{{\pi }^{2}}\frac{B}{c}\int\limits_0^a {j\frac{\partial }{{\partial r}}\left( {{{r}^{2}}\int\limits_0^r {{{\iota }_{{{\text{st}}}}}xdx} } \right)dr.} $$

((2))

Here, P = $\frac{3}{2}\int p$dV is the plasma energy content. p = n_ek(T_e + T_i) is the plasma pressure, V is the volume, T_e is the temperature of electrons, T_i ≈ $\sum\nolimits_j^{} {{{\alpha }_{j}}}$T_j, T_j is the temperature of ions of the jth type, α_j is their relative concentration, I = 2π$\int_0^a {jr}$dr is the longitudinal plasma current measured by the Rogowski coil, and j is its density.

Equation (2) represents the well-known relationship derived for non-steep toroidal systems with a/R ≪ 1, where the two first summands on the right side are used in tokamaks (see, e.g., [8, 9]), and the third summand considers the existence of external rotational transform [10]. In the absence of longitudinal current, the toroidal magnetic flux Φ is changed at the expense of diamagnetic currents passing across the magnetic field. In the presence of longitudinal current, the squared current term describes the paramagnetic change in the magnetic flux. In the setups with external rotational transform, the magnetic flux changes its sign depending on the current direction.

As can be seen from Fig. 1, the current at the initial discharge stage is still small, and the diamagnetic component prevails in the change of the toroidal magnetic flux. When the longitudinal current grows, the signal sign is switched. This indicates that the current direction coincides with the magnetic field direction, i.e., the current leads to an increase in rotational transform.

As can be seen from Eq. (2), the change in the toroidal magnetic flux at a given plasma energy content and a rather strong longitudinal current depends on distribution ${{\iota }_{{{\text{st}}}}}$(r) and j(r). This strongly complicates the application of diamagnetic measurements in stellator systems. However, change in the stellator rotational transform angle at low values of parameter

$$\beta = \frac{{8\pi }}{{{{B}^{2}}}}p \ll {{\beta }_{p}} = \frac{1}{2}\iota _{{{\text{st}}}}^{2}(a)\frac{a}{R}$$

in the process of discharge is small. In our experiment, β ≤ 4 × 10^–4, being much lower than β_p ~ 10^–2. For this reason, the calculations of the plasma energy content in the case of diamagnetic measurements should consider only the change in the current distribution with time and use distribution ${{\iota }_{{{\text{st}}}}}$(r) in the form of Eq. (1). The performed calculations have shown that the dependence of P on the longitudinal current distribution is not so crucial. If the current distribution is expressed in form

$$j = {{j}_{0}}{{(r{\text{/}}a)}^{m}}{{[1 - {{r}^{n}}{\text{/}}{{a}^{n}}]}^{g}} + {{j}_{1}},$$

((3))

Eq. (2) in the case of j₁ = 0 can be written as

$$P = - \frac{3}{4}RB\Phi + \frac{3}{2}\pi R\frac{{{{I}^{2}}}}{{{{c}^{2}}}} + 2\pi B{{\iota }_{{{\text{st}}}}}(a)\iota \frac{l}{c}.$$

((4))

In this case, l = 0.27 ± 0.03 for real current distributions (n = 1–3) and the U-3M magnetic system parameters at m = 0 and g = 2 (distribution with a current maximum near the magnetic axis), l = 0.4 at m = 0 and g = 0 (uniform distribution), and l = 0.43 ± 0.05 at m ≠ 0 and g = 2 (current distribution with j(0) = 0).

Hence, it can be seen that the longitudinal magnetic flux in the experiments on the U-3M torsatron with the use of diamagnetic measurements more strongly depends on longitudinal current I than on its distribution along the radius. However, for the plasma energy content to be more precisely determined, it will be useful to know the nature of current and, as a consequence, its distribution.

The currents possible for this experiment in plasma are a bootstrap current [9], current drive [11], and current appearing under fast heating [10].

Bootstrap current appears in toroidal plasma in the regime of rare collisions between plasm particles. The expression for the density of this current in a tokamak has form [9]

$${{j}_{b}} = - 1.46c\sqrt {\frac{R}{r}} \frac{1}{{B\iota }}\frac{{\partial p}}{{\partial r}}.$$

((5))

As can be seen from this expression, the bootstrap current density near the magnetic axis tends to zero, and its highest value takes place in the region of a maximum plasma pressure gradient.

Current drive I_g may appear as a result of interaction between plasma electrons and the RF field providing the heating of plasma in the presence of asymmetry in the propagation of electromagnetic waves along the torus. The expressions for this current can be written in form

$${{I}_{g}} = kW,$$

((6))

where W is the power introduced into the plasma, and k is a certain proportionality coefficient. The antenna used for RF-heating of plasma in the U-3M torsatron is not designed for excitation of electromagnetic waves running in the same direction. The efficiency of the excitation of such current at frequencies close to the ion-cyclotron frequency without the creation of special antennas is low. In addition, the existence of locked particles at rare frequencies of collisions between plasma particles abruptly decreases the current excitation efficiency at near ion-cyclotron frequencies [12].

The spatial distribution of the current drive depends on the RF power absorption region, so the localization of such a current may change in the course of discharge. The direction of the current drive under our conditions is unknown and depends on many factors (magnetic field direction, poloidal and toroidal plasma rotation directions, etc.).

The current appearing under fast heating is a result of change in the poloidal magnetic flux due to the displacement of magnetic surfaces at a fast change in β under the freezing of the magnetic field into the plasma. This current is proportional to β². The current has a maximum value in the phase of growth and descent in β and is zero at the steady-state discharge stage. Estimates show that the maximum value of the current appearing under fast heating does not exceed 50 A for our plasma parameters and magnetic system, so this current is not considered as basic against the background of the detected current of 2 kA.

Let us consider the available experimental data. As a result of measuring the longitudinal current displacement, is has been shown [13] that the displacement of current in the horizontal direction outwards is nearly 4 cm at the initial discharge state and nearly 1 cm at the end discharge stage. This may be explained by the fact that the current at the beginning of discharge is localized near the magnetic axis shifted with respect to the geometric center outwards at 5 cm. By the end of discharge, the major part of current is concentrated near the outer surface shifted outwards at nearly 1 cm.

The longitudinal current value is close to the current calculated by Eq. (5) and depends on the magnetic field direction [4], being in agreement with the dependence described by Eq. (5).

As can be seen from Fig. 1, I/P ≈ const is established after 15 ms, thus being evidence for the dependence of the longitudinal current on the plasma pressure. From Eq. (5) it can be seen that the bootstrap current density is proportional to ∂p/∂r. Expressing the pressure distribution as p = p₀[1 – (r/a)^d], according to Eq. (5), it is possible to obtain

$$\frac{I}{P} = \frac{{0.97}}{{\pi {{a}^{2}}}}\frac{1}{{B({{\iota }_{0}} + 0.45{{\iota }_{2}})}}\sqrt {\frac{a}{R}} \frac{{d + 2}}{{d + 1{\text{/}}2}}.$$

((7))

According to Fig. 1, I/P remains almost unchanged beginning from 15 ms. From this moment, experimental I/P coincides with Eq. (7) for the bootstrap current in tokamaks at d ≈ 2.4–2.8. However, the observed density profile [14] indicates a more peaked pressure profile (i.e., d ≤ 2). In this case, the coefficient for the current density should be decreased by 20–30% to use Eq. (5) for the U-3M torsatron.

The growth of I/P until 15 ms can be explained by the expansion of the region of rare collisions with an increase in plasma temperature.

The performed experimental studies do not give any unambiguous answer to the nature of the observed longitudinal current, so experiments with a stepwise increase in the power introduced into the plasma were performed [3]. It is obvious that the current drive must immediately grow in this case together with time constant L/Ω, where L is the longitudinal current induction, and Ω is its resistance. The bootstrap current must not change at the first time moment, but subsequently grows to the value corresponding to the established pressure. In this experiment, the current remained unchanged at the stepwise power increase moment.

Hence, it is possible to state that only bootstrap current is observed in our discharge.

The discharge parameters are such that the appearance of bootstrap current is quite substantiated.

According to the neoclassic theory [9], the particles are in the region of rare collisions when meeting the condition

$${{\nu }_{j}} \ll \frac{{{{{v}}_{{T{{j}^{i}}}}}}}{R}{{\varepsilon }^{{3/2}}},$$

((8))

where ν_j is the frequency of collisions for particles of the jth type, ${{{v}}_{{Tj}}}$ is their thermal velocity, and ε = $\frac{r}{R}$.

From condition (8) it follows that the conditions of rare collisions for density n_e ≈ 10¹⁸ m^–3 and Z = 1 are T_e ≫ 130 eV and T_i ≫ 100 eV. By analogy with the work [15], where the temporal behavior of basic parameters of a discharge similar to the considered one are discussed, we assume that the cross-sectional average temperatures of ions and electrons at a nominal level of the power introduced into the plasma vary within the range of T_e = 100–300 eV and T_i = 80–200 eV, and the average charge of ions approaches Z = 1 by the end of discharge. Such discharge parameters allow us to state that the regime of rare collisions is implemented in an appreciable discharge cross-section area. Bootstrap current must be observed in such a discharge.

For bootstrap current, j(0) = 0 and l = 0.43 (Eq. (4)) and, moreover, we assume that j₁ = 0 and the drive and fast heating currents are small. Taking into account the aforesaid, Eq. (4) was used to calculate the plasma energy content value given in Fig. 1.

RF power is stepwisely increased in three steps (Figs. 1 and 2); breakdown occurs at 1.8 ms, and the power introduced into the plasma grows at 3 and 6 ms. The plasma energy content and density quickly grow after 3 ms when a relatively small power is introduced. The process of fast increase in the plasma parameters lasts for nearly 1 ms. In this case, the plasma temperature attains 100 eV. Afterwards, the plasma parameters begin to saturate. After 6 ms, the plasma energy content first quickly increases by nearly 30% for 0.5 ms and then quickly decreases (for 2 ms) to the initial plasma energy content values, thereupon, the plasma energy content begins to grow further for the entire period of discharge. The actuation of additional RF power at 6 ms also leads to fast decrease in the density, accelerates growth in the current, and increases the plasma temperature. A continuous slow decrease in the density and an increase in the temperature are further observed until the end of discharge. The plasma energy content is observed to continuously grow from 9 to nearly 30 ms from the beginning of discharge but does not almost change beginning from 38 ms.

Fig. 2.

Temporal behavior of power W absorbed by the plasma, plasma energy content P, and magnetic field fluctuations $\tilde {B}$ (b) within a range of 1.5–70 kHz, and power W_n spent to heat the plasma and change the magnetic field energy at two time scales.

The performed estimations of skin time τ_sk = $\frac{{4\pi \sigma {{a}^{2}}}}{{{{c}^{2}}}}$, where σ is the conductivity, show that the skin time varies at plasma parameters within a range from τ_sk ≈ 10 ms at the initial discharge stage to τ_sk ≈ 40 ms by the end of discharge. From Figs. 1 and 2 it can be seen that the characteristic time of change in the plasma energy content at the beginning of discharge is τ_f ≈ 1 ms at an increase and decrease in the RF pulse front of less than 50 μs, and τ_f ≈ 10 ms in the middle of the discharge, being appreciably lower than τ_sk for the entire duration of discharge. We think that the plasma is frozen into the magnetic field, and the process of heating can be described using the theory developed for fast heating [5, 6].

According to [6], the balance of power under fast plasma heating can be written as

$$\frac{d}{{dt}}\int {\left( {\frac{3}{2}p\, + \,\frac{{{{B}^{2}}}}{{8\pi }}} \right)dV\, + \,\oint {\left( {\frac{5}{2}p{\mathbf{v}}\, + \,{\mathbf{q}}\, + \,\frac{c}{{4\pi }}[{\mathbf{EB}}]} \right)} } d{\mathbf{S}}\, + \,W{\text{*}}{\text{,}}$$

((9))

$$\delta {\mathbf{E}} = \frac{1}{c}[{\mathbf{uB}}],$$

((10))

$$\begin{gathered} \frac{d}{{dt}}\int {\left( {\frac{3}{2}p + \frac{{{{B}^{2}}}}{{8\pi }}} \right)dV = \frac{\partial }{{\partial t}}\int {\left( {\frac{3}{2}p + \frac{{{{B}^{2}}}}{{8\pi }}} \right)dV} } \\ \, + \oint {\left( {\frac{3}{2}p + \frac{{{{B}^{2}}}}{{8\pi }}} \right){\mathbf{u}}d{\mathbf{s}}.} \\ \end{gathered} $$

((11))

Here, v = v₀ + u is the plasma motion velocity, u is the plasma motion velocity under fast heating, and v₀ is the plasma motion velocity under steady-state conditions, E = E₀ + δE is the electrical field, δE is the field appearing under fast heating, E₀ is the electrical field under steady-state conditions, q is the heat flux from the plasma volume, W* is the power spent on the heating of the plasma, the change of the magnetic field, and the losses, which are not associated with elementary processes (convective processes, heat conductivity, and diffusion), and S is the surface area. Equation (10) describes the condition of plasma freezing into the magnetic field.

We assume that the total power absorbed by the plasma in the confinement volume is

$$W = W{\text{*}} + {{W}_{B}},$$

((12))

where W_B is the power spent on the elementary processes (ionization, dissociation, recharging, irradiation, etc.). In addition,

$$\frac{\partial }{{\partial t}}\int {\frac{{B_{{v}}^{2} + B_{r}^{2}}}{{8\pi }}dV = \frac{\partial }{{\partial t}}L\frac{{{{l}^{2}}}}{2},} $$

((13))

$$\frac{c}{{4\pi }}[\delta {\mathbf{EB}}] = {\mathbf{u}}\frac{{{{B}^{2}}}}{{4\pi }},$$

((14))

$$\tau _{E}^{*} = \frac{P}{{\oint {\left( {\frac{5}{2}p{{{\mathbf{v}}}_{0}} + {\mathbf{q}} + \frac{c}{{4\pi }}[{{{\mathbf{E}}}_{0}}{\mathbf{B}}]} \right)d{\mathbf{S}} + {{W}_{0}}} }}.$$

((15))

Here, ${{B}_{{v}}}$ and B_r are the magnetic field components in the quasi-cylindrical of coordinates r, ϑ, and ϕ, where r is the component oriented along the small torus radius, ϑ is the poloidal component along the small torus girdle, ϕ is the component along the large torus girdle, and $\tau _{E}^{*}$ is the time scale characterizing the losses of energy from the plasma volume. Parameter $\tau _{E}^{*}$ coincides with the known value of energy confinement time τ_E at the steady-state discharge stage. However, an appreciable difference associated with the estimation of τ_E may be observed in the dynamic regions.

Taking into account that, according to Eq. (10),

$${{u}_{r}} = - \frac{1}{{2\pi aB}}\frac{{\partial \Phi }}{{\partial t}} - \frac{{c\iota a}}{{2\pi {{R}^{2}}B}}L\frac{{\partial I}}{{\partial t}},$$

((16))

Eq. (9) can rewritten following [6] as

$$\frac{{\partial P}}{{\partial t}} - \frac{3}{4}BR\frac{{\partial \Phi }}{{\partial t}} + \left( {I - \frac{3}{4}\frac{{c\iota {{a}^{2}}B}}{R}} \right)L\frac{{\partial I}}{{\partial t}} + \frac{P}{{\tau _{E}^{*}}} = W.$$

((17))

Here, $\iota$ = ${{\iota }_{{{\text{st}}}}}$(a) + ${{\iota }_{\tau }}$(a) is the total angle of rotary conversion at the plasma boundary and ${{\iota }_{\tau }}$(a) = $\frac{{2IR}}{{c{{a}^{2}}}}$B is the rotational transform angle created by the longitudinal plasma current.

Let us write Eq. (17) in form

$$\frac{{\partial P}}{{\partial t}} + \frac{P}{{\tau _{E}^{*}}} = W + \frac{3}{4}BR\frac{{\partial \Phi }}{{\partial t}} + \left( {\frac{3}{4}\frac{{c\iota {{a}^{2}}B}}{R} - I} \right)L\frac{\partial }{{\partial t}}I.$$

((18))

Then, the right side of Eq. (18) represents the power spent to heat the plasma. At the heating stage, toroidal flux Φ changes in the diamagnetic direction and the power associated with the change of this magnetic flux is subtracted from the introduced power. If Φ changes in the paramagnetic direction, the power is added. A similar situation takes place for change in the magnetic flux of plasma current I. When heating is turned off, everything changes oppositely.

Let us denote the first three summands on the left side of Eq. (17) as W_n, i.e.,

$${{W}_{n}} = \frac{{\partial P}}{{\partial t}} - \frac{3}{4}BR\frac{{\partial \Phi }}{{\partial t}} + \left( {I - \frac{3}{4}\frac{{c\iota {{a}^{2}}B}}{R}} \right)L\frac{{\partial I}}{{\partial t}},$$

((19))

which represent the power spent on the heating of the plasma and the change of the magnetic field energy. In this case, the power absorbed by the plasma will be

$${{W}_{n}} + \frac{P}{{\tau _{E}^{*}}} = W.$$

((20))

The temporal behavior detected for parameter W_n, power absorbed by plasma W, plasma energy content P, and fluctuation in poloidal magnetic field component ${{\tilde {B}}_{{v}}}$ by one of the magnetic probes at radius b = 16.8 cm outside the confinement volume within the frequency range of 1.5–70 kHz, is illustrated in Fig. 2.

Parameter W_n representing the power spent on the heating of the plasma and the change of magnetic field energy (see Eq. (19)) has a complicated temporal behavior. Positive values of parameter W_n correspond to the power spent to increase the plasma and magnetic field energy, while its negative values correspond to the power spent on their decrease under fast heating. A sharp peak of 9.3 kW is observed in W_n at the initial discharge stage and attains 11 kW after actuation of the first stage of additional power at 3 ms. Afterwards, parameter W_n decreases almost to zero. This occurs due to the fact that $\frac{{\partial P}}{{\partial t}}$ → 0, and the longitudinal current becomes positive and begins to grow. After the second stage when additional power is actuated, a short extensive peak up to 7.7 kW coinciding with the peak in the plasma energy content is observed at 6 ms and followed by a strong negative peak up to –6.5 kW correlating with a decrease in the plasma energy content and the average plasma density and an abrupt increase in the longitudinal current (see Fig. 1). Further, parameter W_n varies within the vicinity of zero until the end of heating. This indicates that most power is spent on compensating energy losses from the plasma. After RF heating is turned off, W_n is negative, as the energy accumulating in the plasma and magnetic field decreases.

Time periods with a linear increase in plasma energy content are observed at the initial discharge stage at the RF power actuation moment. Considering this fact, it is possible to estimate parameter $\tau _{E}^{*}$. The current change may be neglected to recast Eq. (20) as

$$2.5\frac{{\partial P}}{{\partial t}} + \frac{P}{{\tau _{E}^{*}}} = W.$$

((21))

From Eq. (21) it can be seen that the power spent on the heating of the plasma and the change of magnetic field W_n = 2.5$\frac{{\partial P}}{{\partial t}}$ under fast heating when the plasma is frozen into the magnetic field and in the absence of longitudinal current. In other words, the power spent to change the magnetic field is 1.5 times higher than the power spent to heat the plasma.

At constant $\tau _{E}^{*}$ and W, the solution of Eq. (21) is known:

$$P = {{P}_{0}}[1 - \exp ( - t{\text{/}}25\tau _{E}^{*})].$$

((22))

In the case of t ≪ 2.5$\tau _{E}^{*}$, Eq. (21) is reduced to form

$$P = {{P}_{0}}\frac{t}{{2.5\tau _{E}^{*}}}.$$

((23))

Equation (23) describes the temporal behavior of the plasma energy content at the initial discharge stage in the linear regions of increase in the plasma energy content.

The available experimental data allow us to estimate the energy losses and the RF power absorbed by the plasma in the confinement volume. At the initial discharge stage, parameter $\tau _{E}^{*}$ can not be less than 1.7 ms to meet condition 2.5$\tau _{E}^{*}$ ≫ τ_E = 0.85 ms. The estimated power spent on the elementary processes (neutral gas ionization, recharging, and neoclassic transfer) is nearly 1.2 kW. For this reason, if the plasma losses can be described by the neoclassic theory, the maximum value of $\tau _{E}^{*}$ ≤ 3 ms. Considering the aforesaid and Eq. (19), the absorbed RF power at the initial discharge stage may be within the range W = 10.5–11.6 kW.

To estimate the power absorbed by plasma W_n after the first stepwise increase, we take the increment after the actuation of the first stage, ΔW_n = 1.9 kW, P = 7.5 J, and minimal $\tau _{E}^{*}$ = 1.1 ms calculated from the linear increase in plasma energy content P after stepwise increase in the power for 3 ms. We obtain the value of 17 kW (ionization + recharging + transfer; 4.6 kW).

Similarly, the power absorbed by the plasma after the second stepwise increase is W ≈ 27–30 kW within the interval of 46–50 ms. This result does not contradict with the data, which were obtained in the earlier bolometric measurements for a similar discharge and gave the power emitted by the plasma at a level of 12—14 kW [16]. The current in the antenna almost does not change after the actuation of the second power stage, so we assume that the power absorbed by the plasma is constant.

The temporal behavior detected for magnetic field fluctuations by 1 of 15 magnetic sensors installed in a cross section distant from the antenna is illustrated in Fig. 2. Probes with corresponding electronic equipment provided the possibility to detect magnetic field fluctuations in the plasma volume at a level of 10^–7 T within the band of frequencies of up to 70 kHz. The frequency range was limited by the signal digitization rate and the length of conducting cables.

Magnetic field fluctuations appear approximately 1.5 ms after breakdown. The appearance of fluctuations coincides with a decrease in the plasma density and plasma energy content growth rate (see Figs. 1 and 2). In other words, certain parameter values (maybe, the value of β) must be attained in the plasma volume for the excitation of fluctuations. Fluctuations disappear immediately after RF heating is turned off. The temporal behavior of $\tilde {B}$ is rather complicated. For example, the amplitude of fluctuations starts to increase from 13 ms after the beginning of discharge. An increase in the amplitude of fluctuations coincides with a decrease in the plasma energy content growth rate. At 15 ms (this moment is marked with a dashed line), the amplitude of fluctuations decreases for 10 ms by several times, and the plasma efficiency growth rate abruptly increases from this moment. By the way, the growth of I/P stops at the same moment (see Fig. 1).

The behavior of $\tau _{E}^{*}$ in the course of RF discharge is illustrated in Fig. 3. This was accomplished using expression

$$\tau _{E}^{*} = \frac{P}{{W - {{W}_{n}}}}.$$

((24))

Fig. 3.

Behavior of $\tau _{E}^{*}$ and magnetic field fluctuation in the course of discharge.

It is worth pointing out that $\tau _{E}^{*}$ ≈ 1.7–3 ms at the beginning of discharge, and $\tau _{E}^{*}$ ≈ 6.7 ms immediately after RF heating is turned off at magnetic field fluctuations below detecting system sensitivity. Parameter $\tau _{E}^{*}$ changes throughout the entire remaining period of discharge from 0.25–0.35 ms to nearly 7.5 ms after emission of an RF pulse (1.5 ms after an abrupt increase in the RF power) and to 0.7–0.8 ms by the end of discharge. A nearly linear growth in $\tau _{E}^{*}$ is observed from 7 to 27 ms except the interval of 13.5–16.5 ms. At 13.5–16.5 ms, the amplitude of fluctuations changes by several times, and the energy confinement time changes by 20% from $\tau _{E}^{*}$ = 0.38–0.43 to 0.43–0.52 ms. The catastrophe associated with an abrupt decrease in the plasma confinement at 3.5 ms correlates with the appearance of high-amplitude magnetic field fluctuations.

As can be seen from Fig. 3, there is no linear relation between the amplitude of magnetic field fluctuations and the behavior of $\tau _{E}^{*}$, but a qualitative correlation between these parameters can clearly be seen.

There can be several reasons for the absence of direct relation between the behaviors of energy losses and amplitude of detected magnetic field fluctuations. First, fluctuations in a different frequency range may be responsible for major losses. Second, magnetic field fluctuations are excited deep in the plasma volume and cannot propagate towards the boundary to be emitted into the ambient space due to some reasons. Third, magnetic field fluctuations are not solely responsible for energy losses from the plasma volume. The losses can also be caused by different mechanisms, but an abrupt increase, which occurs in $\tau _{E}^{*}$ at 15 ms and is interpreted as a transition to the improved confinement regime [16–18], seems to be completely governed by the observed magnetic field fluctuations. It should be noted that the error in estimating the power introduced into the plasma has hardly any effect on the behavior of $\tau _{E}^{*}$ in the course of discharge.

The change of $\tau _{E}^{*}$ effects the plasma density profile (Fig. 4). The density profile was determined from two values measured using the cutoff of the passage of RF waves of different frequency through the plasma volume [14] and Langmuir probes at the boundary [19] such that the density profile between these two points corresponded to the integral measured over the bisecant with a 2-mm interferometer. The possible range of change in the density in Fig. 4 is colored.

Fig. 4.

Plasma density profile at different time moments depending on average magnetic surface radius 〈r〉 normalized to the average radius of the last magnetic surface (1) under RF heating (t ≈ 15 ms) and (2) after RF heating is turned off (t ≈ 50.5 ms).

From Fig. 4 it can be seen that the density profile at the active discharge state is sharp. The maximum gradient lies at the average radius of magnetic surface 〈r〉/a ~ 0.4, while the minimum gradient lies within range 〈r〉/a > 0.8. It seems that the events providing losses from the plasma volume occur only at the plasma cord boundary.

After RF heating is turned off, the density profile becomes flat with a steep gradient at the boundary. Such a profile may be caused by a number of reasons.

First of all, as can be seen from Fig. 2, it may be due to an abrupt decrease occurring in the level of fluctuations in $\tilde {B}$ after RF heating is turned off. Moreover, a decrease in the longitudinal current induces the appearance of the voltage of U = $\frac{\partial }{{\partial t}}$LI ≈ 3 V on the toroidal loop. According to the neoclassic theory, the pinching of plasma must occur at rare collision frequencies in this case [9]. This subject was discussed in [3]. It is obvious that these two reasons must lead to an appreciable improvement in the plasma confinement as observed in the experiment ($\tau _{E}^{*}$ nearly attains 7 ms).

The presented experimental data do not allow us to state that the observed relation between the RF power introduced into the plasma and the losses from the plasma volume is direct. From our viewpoint, the “canonic” profile theory [20, 21] is most suitable for explanation of the available data. Really, the plasma energy content profile seems to be not very different from the canonic profile for the given magnetic configuration at the initial RF discharge stage when the absorbed RF power is small. An increase in the plasma efficiency leads to the transformation of this profile and its appreciable distinction from the canonic one. This results in the excitation of instability and decreases $\tau _{E}^{*}$. An abrupt growth in the power introduced into the plasma at the first moment leads to an increase in the plasma energy content which further sustains a decrease, which may be due to an appreciable distortion in the profile of P and an increase in the losses. The profile of P further approaches the canonic form in the process of discharge, thus leading to the growth of $\tau _{E}^{*}$ in the course of discharge. After RF heating is turned off, the profile becomes canonic with consideration for pinching. In this case, magnetic field fluctuations $\tilde {B}$ immediately disappear, and the confinement abruptly improves.

CONCLUSIONS

(1) The possibility of using diamagnetic measurements for the estimation of plasma energy content in stellator systems with longitudinal current, when β is small, and the nature of current is known, has been demonstrated.

(2) The neoclassic nature of the longitudinal current (bootstrap current) appearing in the studied discharge has been confirmed.

(3) The balance of power under fast plasma heating has been considered and used as a basis to estimate the RF power absorbed by the plasma in the confinement volume.

(4) The behavior of $\tau _{E}^{*}$ characterizing the losses from the plasma in the course of discharge has been calculated. It has been shown that the appearance of magnetic field fluctuations in the process of discharge abruptly decreases $\tau _{E}^{*}$.

(5) It has been shown that the behavior of magnetic field fluctuations within a frequency range of 0.5–70 kHz qualitatively explains the change in heat losses from the confinement volume during the studied RF discharge.

ACKNOWLEDGMENTS

In conclusion, the authors would like to thank M.M. Kozulya for submitting the data on the RF power emitted by the antenna in the course of discharge, R.O. Pavlichenko for kindly furnishing information about the average plasma density, and the team of the U-3M setup for performing the experiment.