New possibilities for the mathematical modeling of turbulent transport processes in plasma

A new mathematical model is proposed for the probability distributions of the characteristics of the processes observed in turbulent plasmas. The model is based on formal theoretical considerations related to probabilistic limit theorems for a nonhomogeneous random walk and has the form of a finite mixture of Gaussian distributions. The reliability of the model is confirmed by the results of a statistical analysis of the experimental data on density fluctuations in high-temperature plasmas of the L-2M, LHD, and TJ-II stellarators and the local fluctuating flux in the TAU-1 linear device and in the edge plasma of the L-2M stellarator with the use of the estimation-maximization algorithm. It is shown that low-frequency structural turbulence in a magnetized plasma is related to non-Brownian transport, which is determined by the characteristic temporal and spatial scales of the ensembles of stochastic plasma structures. Mechanisms that could be responsible for the random nature of time samples of the local turbulent flux in TAU-1 are indicated. A new physical concept of the intermittence of plasma turbulent pulsations is developed on the basis of the statistical separation of mixtures in terms of the model proposed. The intermittence of plasma pulsations is shown to be associated with the generation of plasma structures (solitons and vortices) and their nonlinear interaction, as well as with their damping and drift.     

Modeling of transport processes in stellarators

A relatively simple model of transport processes in stellarators is constructed that is based on neoclassical theory and accounts for anomalous heat losses. The model is used to perform calculations for the L-2M, ATF, CHS, and LHD stellarators over a wide range of plasma densities and absorbed powers. The plasma energy lifetimes calculated for these devices coincide to within factors on the order of unity with those obtained from the ISS95 and LHD empirical scalings.     

Effect of electron-cyclotron resonance heating conditions on the local parameters of short-wavelength plasma turbulence in the L-2M stellarator

Changes in the energy spectra of short-wavelength (k _s ≈ 35 cm^?1) plasma density fluctuations in the local region of the plasma column (r/a = 0.5–0.6) of the L-2M stellarator were studied by the method of collective scattering of 150-GHz radiation. The plasma was heated at the second harmonic of the electron gyrofrequency, the microwave heating power being in the range of 90–170 kW. A sector limiter was introduced in the peripheral plasma (r/a ≥ 0.8), and the Shafranov shift of the magnetic axis was varied by varying the vertical magnetic field. The results of measurements were averaged over 9–16 discharges. It is found that an increase in the heating power and/or the introduction of the sector limiter in the plasma lead to an increase in the energy of density fluctuations, which correlates with a decrease in the plasma energy lifetime. In the spectra of fluctuations, a broad spectral band in the range of 3–50 kHz was observed in which the spectral density was one order of magnitude higher than in the rest of the spectrum. Analysis of the Fourier spectra showed that the introduction of the sector limiter in the plasma resulted in an increase in both the spectral density of fluctuations in the range of 3–50 kHz and the fraction of quasi-coherent structures in turbulent density fluctuations.     

Stability and variations of plasma parameters in the L-2M stellarator during excitation of the induction current in the regime of ECR plasma heating

Results are presented from experimental studies of variations in the plasma parameters during the excitation of a multiaxis magnetic configuration by the induction current (up to 17 kA) in the basic magnetic configuration of the L-2M stellarator in the regime of ECR heating at a microwave power of ～200 kW (～1 MW m^?3) and an average plasma density of (1–2) × 10¹⁹ m^?3. The current direction was chosen to reduce the net rotational transform (the so-called “negative“ current). The current was high enough for the rotational transform to change its sign inside the plasma column. Computer simulations of the L-2M magnetic structure showed that the surface with a zero rotational transform is topologically unstable and gives rise to magnetic islands, i.e., to a multiaxis magnetic configuration. Magnetic measurements showed that, at negative currents above 10 kA, intense bursts of MHD oscillations with a clearly defined toroidal mode number n = 0 were observed in the frequency range of several kilohertz. Unfortunately, the experimental data are insufficient to draw the final conclusion on the transverse structure of these oscillations. The radial temperature profiles along the stellarator major radius in the equatorial plane were studied. It is found that the electron temperature decreases by a factor of 1.3 in the plasma core (r/a ≤ 0.6) and that the temperature jump is retained near the boundary. A change in turbulent fluctuations of the plasma density during the excitation of a negative current was studied using wave scattering diagnostics. It is found that the probability density function of the increments of fluctuations in the plasma core differs from a Gaussian distribution. The measured distribution is heavy-tailed and broadens in the presence of the current. It is found that the spectrum of turbulent fluctuations and their Doppler shift near the plasma boundary are nonuniform in the radial direction. This may be attributed to the shear of the poloidal velocity. The experimental results indicate that the formation of regions with a zero rotational transform in the plasma core somewhat intensifies plasma transport.     

Reflection and backscattering of microwaves under doubling of the plasma density and displacement of the gyroresonance region during electron cyclotron resonance heating of plasma in the l-2M stellarator

Reflection and backscattering of high-power (400 kW) gyrotron radiation creating and heating plasma at the second harmonic of the electronic cyclotron frequency in the L-2M stellarator have been investigated experimentally. The effect of the displacement of the gyroresonance region from the axis of the plasma column under doubling of the plasma density on the processes of reflection and backscattering of microwave radiation has been examined. A near doubling of short-wavelength (k _⊥ ≈ 30 cm^–1) turbulent density fluctuations squared is observed. The change in the energy confinement time under variations of plasma parameters and characteristics of short-wavelength turbulence is discussed. A discrepancy between the measured values of the reflection coefficient from the electron cyclotron resonance heating region and predictions of the one-dimensional model is revealed.     

Second harmonic of gyrotron radiation: New potentialities of plasma diagnostics

It is shown that the relative intensity of the second harmonic of gyrotron radiation on the axis of a microwave beam after quasi-optical filtering in a four-mirror quasi-optical transmission line is about ?50 dB of the total radiation intensity. The second-harmonic radiation is used in collective-scattering diagnostics of turbulent density fluctuations in the plasma column of the L-2M stellarator. At an electron temperature of 0.8–1.0 keV and average plasma density of 2.0–2.5×10¹³ cm^?3 (a plasma energy of about 0.6 kJ), which was achieved after the boronization of the vacuum chamber, spatiotemporal structures in plasma density fluctuations were observed in the central region of the plasma column. The correlation time between the structures was found to be on the order of 1 ms. It is shown that the spectrum of the signal from the second-harmonic scattering extends to higher frequencies compared to that from the fundamental-harmonic scattering.     

Toroidal inhomogeneity of plasma density fluctuations during ECR plasma heating in the L-2M stellarator

Correlation between short-wavelength (k_⊥ ≈ 20–30 cm^–1) and long-wavelength (k_⊥ ≈ 1–2 cm^–1) plasma density fluctuations in two poloidal cross sections of the stellarator chamber separated by 1/14 or 5/14 of the torus perimeter was studied using collective scattering of radiation of two 75-GHz gyrotrons and radiation of a 37-GHz Doppler reflectometer at an ECR heating power density of 1.6–3.2 MW/m³. It is found that excitation of turbulent fluctuations is bursty in character and that fluctuations excited in different L-2M cross sections are uncorrelated. It is shown that the energy of turbulent fluctuations is modulated by a low frequency of 5–20 kHz. An idea is put forward that anomalous transport is toroidally inhomogeneous.     

Displacement of the electron cyclotron resonance heating region and time evolution of the characteristics of short-wavelength turbulence in the 3D magnetic configuration of the L-2M stellarator

Reflection of the heating extraordinary microwave incident obliquely onto the surface of the electron cyclotron resonance (ECR) at the second harmonic of the electron gyrofrequency in the 3D magnetic configuration of the L-2M stellarator was studied experimentally. The plasma was heated using two gyrotrons with a total power of 600–700 kW, the specific heating power being 2.4–2.8 MW/m³. The displacement of the ECR region in the course of heating was monitored by measuring the phase of the reflected extraordinary wave. It is found that the growth of the plasma density is accompanied by the displacement of the ECR heating region from the center of the plasma column toward its periphery. The coefficient of reflection of the heating microwave beam from the ECR region was measured. The spectra of short-wavelength (k _s ≈ 30 cm^?1) plasma density fluctuations were explored by analyzing backscattered microwave radiation. A tenfold increase in the energy of short-wavelength density fluctuations and the growth of the spectral density of fluctuations in the frequency range of 0.3–1.5 MHz were observed.     

Could reversed-field pinches and quasi-helical stellarators benefit from transport suppression in tokamaks?

The trapped particle theory of turbulent transport successfully explains key features of tokamak transport: the canonical L-mode, supershort plasma profiles, and the transport suppression by negative magnetic shear and poloidal rotation. Here, this theory is applied to reversed-field pinch (RFP) profiles, which can be justified if the magnetic fluctuations are suppressed, and to stellarators. A canonical density profile for RFPs is suggested, and it is found that no analogue of the transport suppression by negative shear in tokamaks is possible in RFPs. In quasi-helical stellarators, on the other hand, it appears possible to create an analogue of the tokamak reversed shear mode in the entire plasma volume.     

Dependence of the energy lifetime on the type of anomalous losses in stellarators

A relatively simple model of transport process in stellarators that was proposed earlier by the author on the basis of neoclassical theory makes it possible to determine the density and temperature profiles of the plasma components, the ambipolar electric field profile, and the particle and energy lifetimes from the given device parameters and given particle and energy sources with allowance for anomalous losses. The results of numerical simulations carried out with this model for the L-2M, ATF, CHS, and LHD stellarators over broad ranges of plasma densities and absorbed powers showed that the plasma energy lifetimes in these devices coincide to within factors on the order of two with those found from empirical scalings. A specific model of anomalous losses was chosen for calculations. Results are presented from simulations with a more general form of the anomalous thermal conductivity. Namely, the thermal conductivity is chosen to be K _j ^(a) ≈ N ^α T _j ^β B ₀ ^?γ , where N(r) is the plasma density and T _j (r) is the temperature of the jth plasma component (j = e, i). The parameters α, β, and γ are set equal to α = 1, β = 2, and γ = 1; α = 0.5, β = 2.5, and γ = 1; α = 1.5, β = 2, and γ = 2; α = 1, β = 2.5, and γ = 2; and α = 1.5, β = 2.5, and γ = 2. The simulations have been done for the L-2M and LHD stellarators. It is found that, in all the five models, the calculated energy lifetimes τ _c are essentially independent of the functional form of the anomalous thermal conductivity and coincide to within a factor on the order of two with those following from the LHD scaling.     

Structure of fluctuations in the floating potential and ion saturation current in the edge plasma of the L-2M stellarator

Results are presented from the measurements of the ion saturation current, the floating potential, and their fluctuations in the edge plasma of the L-2M stellarator. Distinguishing features in the distribution of the ion saturation current and the floating potential near the separatrix are revealed and examined. Based on the cross correlation measurements with probes positioned at different toroidal angles, it is concluded that fluctuations in the ion saturation current are related to fast vertical displacements of the plasma column, whereas fluctuations in the floating potential have the form of waves propagating in the radial direction.     

On the phase shift between electric potential and plasma density fluctuations in the edge turbulence

In some cases, the phase shift between fluctuations of the electric potential and plasma density helps to identify the instability that governs the turbulent state. In this paper, the basic experimental and theoretical results that denote the possibility (or impossibility) of such identification are briefly discussed. The experimental data based on measurements of the phase shift between the floating potential and ion saturation current fluctuations in the L-2M stellarator—a system with externally imposed magnetic surfaces—are presented (Shchepetov, Kholnov, Fedyanin, et al., Plasma Phys. Controlled Fusion 50, 045001 (2008)). It is shown that the observed phase shift Ω varies in a wide range from π to 0, gradually decreasing with deepening inside the plasma. A number of arguments are presented suggesting that Ω ≈ π can indicate that the process is nonlocal, i.e., oscillations at a given spatial point are driven and mainly determined by the processes localized outside of the observation point. We note that, within the framework of the magnetohydrodynamic theory, plasma was definitely unstable with respect to resistive interchange modes in all cases under study. It is demonstrated experimentally that the widespread notion that the phase shift Ω ≈ π/2 is characteristic of only resistive interchange modes is hardly universal. The experimental results are analyzed on the basis of analytical estimates.     

Energy Loss and Microturbulence under Multipulse ECR Plasma Heating at the L-2M Stellarator

Plasma Physics Reports - In experiments on multipulse on-axis electron cyclotron resonance heating (ECRH) of plasma by a series of microwave pulses at the L-2M stellarator, several phases of plasma...     

Measurements of Microwave Power Absorbed during ECR Plasma Heating at the L-2M Stellarator

Plasma Physics Reports - The results are presented from measurements of the power absorbed in plasma of the L-2M stellarator in experiments on the electron cyclotron resonance (ECR) plasma heating...     

The use of Doppler reflectometry in the L-2M stellarator

Results are presented from measurements of the plasma rotation velocity and plasma density fluctuations in the L-2M stellarator by the method of Doppler reflectometry. Specific problems that arise when applying this diagnostics to the stellarator are revealed. The poloidal plasma velocity at the periphery of the plasma column is determined. The results of measurements are well reproducible.     

Kinetic Simulations of EC Plasma Heating and Current Drive in the L-2M Stellarator

The ECHLAB code, intended for a self-consistent numerical analysis of the evolution of the electron distribution function and the spatial structure of the electromagnetic field during EC plasma heating in a stellarator, is described. The results from calculations of plasma heating and current drive under conditions corresponding to experiments on EC plasma heating by an X2-mode in the L-2M stellarator are presented. It is shown that, at the existing level of microwave power, the energy deposition region displaces only slightly during heating. The energy is mainly absorbed by relatively fast passing electrons. The influence of locally trapped electrons on the efficiency of current drive is insignificant.     

Stochastic electron acceleration in plasma waves driven by a high-power subpicosecond laser pulse

The mechanism of stochastic electron acceleration and heating by a picosecond laser pulse in underdense plasma is studied using particle-in-cell simulations and theoretical models. The formation of wide electron energy spectra in the simultaneously acting laser and plasma fields is analyzed. It is shown that electron scattering by turbulent plasma fluctuations excited through stimulated forward Raman scattering plays a governing role in the formation of high-energy tails in the electron distribution function.     

Spectra of low-frequency modulation of gyrotron radiation during electron-cyclotron resonance heating of plasma in the L-2M stellarator

Results from experimental studies of the modulation of the gyrotron power during electron cyclotron resonance heating of plasma L-2M stellarator are presented. It is shown that the modulation spectrum consists of separate spectral bands, among which a 20-kHz peak with a spectral density exceeding by one order of magnitude the spectral density of the other peaks is observed. This can be explained by the gyrotron operation being affected by the wave reflected from long-wavelength plasma fluctuations.     

Effect of vacuum chamber boronization on the plasma parameters in the L-2M stellarator

After boronization of the vacuum chamber of the L-2M stellarator, radiative losses from ohmically and ECR heated plasmas were reduced by a factor of 3–4. Under these conditions, radiative losses in the ECRH regime comprise only 10–15% of the input microwave power. Some effects have been detected that were not observed previously: a substantial increase in the gradient of the electron temperature near the separatrix, a preferentially outward-directed radial turbulent particle flux (both throughout the discharge phase and from shot to shot), and a longer (by a factor of 2–3) duration of the plasma cooling phase.     

Determination of the plasma effective charge from the soft X-ray spectrum and plasma conductivity at the L-2M stellarator after boronization of the vacuum chamber

Boronization of the vacuum chamber wall results in a considerable change in the composition of the plasma generated in working pulses of the L-2M stellarator and, accordingly, in the plasma effective charge. The paper presents results of measurements of the plasma effective charge carried out by two methods in the ohmic heating mode: from the data on the plasma conductivity and from the soft X-ray spectrum of plasma emission. Comparison of the values of the plasma effective charge obtained by these two methods makes it possible to determine the conditions in which the two values are in good agreement. Under these conditions, the plasma effective charge can be correctly estimated from spectral measurements.