Self-consistent modeling of radio-frequency plasma generation in stellarators

A self-consistent model of radio-frequency (RF) plasma generation in stellarators in the ion cyclotron frequency range is described. The model includes equations for the particle and energy balance and boundary conditions for Maxwell’s equations. The equation of charged particle balance takes into account the influx of particles due to ionization and their loss via diffusion and convection. The equation of electron energy balance takes into account the RF heating power source, as well as energy losses due to the excitation and electron-impact ionization of gas atoms, energy exchange via Coulomb collisions, and plasma heat conduction. The deposited RF power is calculated by solving the boundary problem for Maxwell’s equations. When describing the dissipation of the energy of the RF field, collisional absorption and Landau damping are taken into account. At each time step, Maxwell’s equations are solved for the current profiles of the plasma density and plasma temperature. The calculations are performed for a cylindrical plasma. The plasma is assumed to be axisymmetric and homogeneous along the plasma column. The system of balance equations is solved using the Crank-Nicholson scheme. Maxwell’s equations are solved in a one-dimensional approximation by using the Fourier transformation along the azimuthal and longitudinal coordinates. Results of simulations of RF plasma generation in the Uragan-2M stellarator by using a frame antenna operating at frequencies lower than the ion cyclotron frequency are presented. The calculations show that the slow wave generated by the antenna is efficiently absorbed at the periphery of the plasma column, due to which only a small fraction of the input power reaches the confinement region. As a result, the temperature on the axis of the plasma column remains low, whereas at the periphery it is substantially higher. This leads to strong absorption of the RF field at the periphery via the Landau mechanism.     

Electrostatic plasma turbulence and spitzer longitudinal conductivity in tokamaks

An analysis has revealed that there may be three radically different steady states of a tokamak plasma: (i) discharges in which the electron and ion transport can be described by neoclassical theory; (ii) discharges with the Spitzer longitudinal conductivity, neoclassical ion transport, and “anomalous” electron transport; and (iii) discharges in which the electron transport and ion transport are both “anomalous.” The dimensionless parameters responsible for the occurrence of the three types of discharges are determined. In accordance with the criteria derived for the achievement of different steady states, discharges of the second type are most typical of modern tokamaks and discharges of the third type can occur only if the turbulence is sufficiently strong. Discharges of the first type cannot occur in the range of the working parameters of present-day tokamaks and future tokamak reactors, but they can be ignited in a large class of magnetic confinement systems. The physical reason for the onset of different types of discharges is associated with the fact that turbulent fluctuations play very different roles in the dynamics of the ion and electron components of a finite-size magnetized plasma. The question of the self-consistency between the profiles is considered. A law is derived according to which the thermal diffusivity increases toward the plasma edge.     

Effect of helical magnetic fields on plasma stability in tokamaks

Physical mechanisms for destabilization of MHD perturbations by external quasistatic magnetic fields and rotating helical magnetic fields in a tokamak plasma are identified using a numerical model of tearing modes in a viscous high-temperature plasma. The critical conditions for the onset of MHD perturbations and their dynamic model are compared with the experimental results from the JET tokamak. The model is used to predict how the stray magnetic fields will influence plasma stability in a tokamak reactor (ITER). __________ Translated from Fizika Plazmy, Vol. 26, No. 8, 2000, pp. 675–682. Original Russian Text Copyright ￠ 2000 by Savrukhin.     

Metal droplet erosion and shielding plasma layer under plasma flows typical of transient processes in tokamaks

It is shown that the shielding plasma layer and metal droplet erosion in tokamaks are closely interrelated, because shielding plasma forms from the evaporated metal droplets, while droplet erosion is caused by the shielding plasma flow over the melted metal surface. Analysis of experimental data and theoretical models of these processes is presented.     

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.     

Studies of fluctuations in the high-temperature plasma of modern stellarators by the microwave scattering technique

Microwave scattering diagnostics are described that allow direct measurements of the turbulent processes in the high-temperature plasma of magnetic confinement systems. The first physical results are presented from fluctuation measurements carried out in 2000–2001 in three stellarators: L-2M (Institute of General Physics, Moscow), LHD (National Institute of Fusion Science, Toki), and TJ-II (CIEMAT, Madrid). Plasma density fluctuations in the axial (heating) regions of the L-2M and LHD stellarators were measured from microwave scattering at the fundamental harmonic of the heating gyrotron radiation. In the TJ-II stellarator, a separate 2-mm microwave source was used to produce a probing beam; the measurements were performed at the middle of the plasma radius. Characteristic features of fluctuations, common for all three devices, are revealed by the methods of statistical and spectral analysis. These features are the wide frequency Fourier and wavelet spectra, autocorrelation functions with slowly decreasing tails, and non-Gaussian probability distributions of the magnitudes and the increments in the magnitude of fluctuations. Observations showed the high level of coherence between turbulent fluctuations in the central region and at the edge of the L-2M plasma. The drift-dissipative instability and the instability driven by trapped electrons are examined as possible sources of turbulence in a high-temperature plasma.     

Potential surface waves in anisotropic plasma

The dependences of the frequency and damping rate of a potential surface wave on the wavenumber and the degree of anisotropy of a bi-Maxwellian electron distribution characterized by different temperatures along and across the plasma surface are established. It is demonstrated that the influence of electron thermal motion along the plasma surface on the surface wave properties is similar to the influence of thermal motion on the properties of a bulk Langmuir wave. On the contrary, thermal motion across the surface qualitatively affects the dispersion relation and substantially increases the damping rate.     

Predictive modeling of competitive biosorption equilibrium data

This paper compares regression and neural network modeling approaches to predict competitive biosorption equilibrium data. The regression approach is based on the fitting of modified Langmuir-type isotherm models to experimental data. Neural networks, on the other hand, are non-parametric statistical estimators capable of identifying patterns in data and correlations between input and output. Our results show that the neural network approach outperforms traditional regression-based modeling in correlating and predicting the simultaneous uptake of copper and cadmium by a microbial biosorbent. The neural network is capable of accurately predicting unseen data when provided with limited amounts of data for training. Because neural networks are purely data-driven models, they are more suitable for obtaining accurate predictions than for probing the physical nature of the biosorption process.     

Bifurcations of axisymmetric plasma equilibrium in a tokamak

Bifurcation of solutions to the Grad–Shafranov-type equation for helically symmetric plasma near the threshold for tearing instability are analyzed. Quadratic and cubic nonlinearities were added to the linear dependence of the current density on the helical flux. Depending on the character of nonlinearity, two types of bifurcation can be observed, the “small” and the “large” ones. The small bifurcation is typical of cubic nonlinearity and reveals itself in the growth of the magnetic island from zero as the profile parameter increases above the instability threshold. The large bifurcation is typical of quadratic nonlinearity and causes jumplike formation of a large-scale magnetic island upon exceeding the instability threshold. As the profile parameter decreases below the instability threshold, the large-scale island continues to persist for some time (the hysteresis effect) and then suddenly disappears.     

Theoretically modeling microarray with the chemical equilibrium and thermodynamics

To date, the idea that microarray may shed the light on cellular processes by identifying groups of genes that appear to be co-expressed seems to remain a dream. This is partly because that there are some blank (meaning the knowledge is unavailable) or even erroneous areas in the fundamental theory in this field. This paper attempts to present the digest of microarray hybridization system with chemical thermodynamics, theoretically clarifying some misunderstandings and looking for answers to some critical questions around this technology, such as the mechanisms and conditions of quantitative measuring by hybridization reaction, the reasons of inconsistency of the data and the analysis results and the solutions, how to analyze the data, etc. A theoretical model for the next generation of microarray is proposed. We believe that this model is universal, laying the foundation for microarray technology from array design through the data analysis.     

Equilibrium of rotating and nonrotating plasmas in tokamaks

The equilibrium of a tokamak plasma with a toroidal flow is discussed. It is shown that the centrifugal force of this rotation always reduces the equilibrium beta limit. An opposite view is analyzed, and the arguments supporting this view are considered. It is shown that, although the equilibrium conditions may be locally improved through a special choice of the profile of the plasma rotation velocity, toroidal rotation, on the whole, has a negative effect on the plasma equilibrium. However, under typical tokamak conditions, a decrease in the equilibrium β limit due to plasma rotation is insignificant and, consequently, the effect of the rotation of a hot plasma on its equilibrium can be neglected.     

Evolution of discontinuous solutions to transport equations in stellarators

It was shown earlier that, in the range of rare collisions, transport equations for stellarators allow steady discontinuous solutions for the ambipolar electric field and for the plasma density and temperature gradients. Moreover, such solutions are non-single-valued; that is, their explicit form depends on the initial values of the ambipolar electric field. The time-independent transport equations are derived under the conventional quasineutrality condition; i.e., it is assumed that the electron and ion densities, N _e and N _i , are related by the relationship N _e = ZN _i (where Z is the ion charge number). In other words, the plasma charge density is assumed to be much less than the product e _i N _i . Under typical conditions, the corresponding inequality is satisfied by a large margin. However, if the electric field E has discontinuities, then it can be seen from the equation ▿·E = 4πρ that, at the discontinuity points, the charge density becomes infinite and the relationship N _e = ZN _i fails to hold, so it is necessary to replace it with N _e = ZN _i + ρ/e _e . In the transport equations, this latter replacement produces additional terms, proportional to the second radial derivative of the field E. With these additional terms, the steady solutions are modified substantially. First, the ambipolar field and the derivatives of the density and temperatures all become continuous functions of the coordinates, a result that seems to be quite obvious. The second, not-so-obvious result is that the steady solutions become single-valued, i.e., independent of the initial values of the ambipolar electric field. It turns out that, in this case, two regimes are possible, depending on the values of the plasma parameters. In the first regime, the solution is unique and is independent of the initial conditions. In the second regime, two steady solutions can exist, depending on the initial conditions. One of the solution is similar to that obtained in the first regime, and the other differs from the first one both in the ambipolar field profile and in the dependence of the density and temperatures on the minor plasma radius. It cannot be excluded that different plasma confinement modes revealed in experiments are associated with the existence of such solutions.     

Single-mode magnetic structures in a plasma with anisotropic pressure

The equations of vortex electron anisotropic hydrodynamics are used to show that, in a plasma with anisotropic pressure, the Weibel instability of short-wavelength perturbations gives rise to a large-amplitude quasi-harmonic magnetic field varying periodically as a function of time. The computed field parameters agree well with the proposed analytic estimates.     

Chemical equilibrium modeling of copper precipitation in a hyper-concentrated solid-liquid system

Rapid progress has been made recently in the understanding of heavy metal sorption and speciation on sediment and soils. One aspect that was overlooked in the previous studies was the process of pollutant transformation and transportation in hyper-concentrated solid-liquid systems. In this paper, batch experiments on copper sorption in association with loess at high sediment concentrations were conducted. However, some reaction mechanisms were difficult to determine experimentally due to the limitations of speciation extraction methods. In an additional study, the MINTEQA2 chemical equilibrium model was used to calculate the speciation and precipitation of copper sorption by loess to give quantitative predictions and detailed information about the reaction process. The experiments and the modeling simulation were made under the same sorption conditions, with sediment concentrations ranging from 50 to 200 kg/m³ and adsorbates of CuSO₄ and Cu(NO₃)₂, in order to compare their results. The modeling results clearly supported the experimental results, fully explained the mechanisms of the effects of chemical form and sediment concentration on the copper sorption, and strengthened the dominant role of carbonates among the main components of loess in the process of copper sorption.     

Analytical modeling of the hysteresis phenomenon in guinea pig ventricular myocytes

In the present study, we have demonstrated hysteresis phenomena in the excitability of single, enzymatically dissociated guinea pig ventricular myocytes. Membrane potentials were recorded with patch pipettes in the whole-cell current clamp configuration. Repetitive stimulation with depolarizing current pulses of constant cycle length and duration but varying strength led to predictable excitation (1:l) and non-excitation (1:0) patterns depending on current strength. In addition, transition between patterns depended on the direction of current intensity change and stable hysteresis loops were obtained in stimulus:response pattern vs. current intensity plots in 14 cells. Increase of pulse duration and decrease of stimulation rate contributed to a reduction in hysteresis loop areas. Changes in amplitude and shape of the subthreshold responses during the transitions from one stable pattern to the other, suggested that activity led to an increase in membrane resistance, particularly in the voltage domain between resting potential, and threshold. Therefore, we modelled the dynamic behaviour of the single cells as a function of diastolic membrane resistance, using previously published analytical solutions. Numerical iteration of the analytical model equations closely reproduced the experimental hysteresis loops in both qualitative and quantitative ways. In particular, the effect of stimulation frequency on the model was similar to the experimental findings. The overall study suggests that the excitability pattern of guinea pig ventricular myocytes accounts for hysteresis and bistabilities when current intensity is allowed to fluctuate around threshold levels.     

Current generation by helicons and lower hybrid waves in modern tokamaks and reactors ITER and DEMO. Scenarios, modeling and antennae

The innovative concept and 3D full-wave code modeling the off-axis current drive by radio-frequency (RF) waves in large-scale tokamaks, ITER and DEMO, for steady-state operation with high efficiency is proposed. The scheme uses the helicon radiation (fast magnetosonic waves at high (20–40) ion cyclotron frequency harmonics) at frequencies of 500–700 MHz propagating in the outer regions of the plasmas with a rotational transform. It is expected that the current generated by helicons, in conjunction with the bootstrap current, ensure the maintenance of a given value of the total current in the stability margin q(0) ≥ 2 and q(a) ≥ 4, and will help to have regimes with a negative magnetic shear and internal transport barrier to ensure stability at high normalized plasma pressure β _N > 3 (the so-called advanced scenarios) of interest for the commercial reactor. Modeling with full-wave three-dimensional codes PSTELION and STELEC showed flexible control of the current profile in the reactor plasmas of ITER and DEMO, using multiple frequencies, the positions of the antennae and toroidal wave slow down. Also presented are the results of simulations of current generation by helicons in the DIII-D, T-15MD, and JT-60AS tokamaks. Commercially available continuous-wave klystrons of the MW/tube range are promising for commercial stationary fusion reactors. The compact antennae of the waveguide type are proposed, and an example of a possible RF system for today’s tokamaks is given. The advantages of the scheme (partially tested at lower frequencies in tokamaks) are a significant decline in the role of parametric instabilities in the plasma periphery, the use of electrically strong resonator-waveguide type antennae, and substantially greater antenna-plasma coupling.     

A nonlinear single-mode structure in a plasma with anisotropic temperature

An exact solution is derived to the equations of vortex electron anisotropic hydrodynamics for a plasma that is unstable against the Weibel instability driven by the electron temperature anisotropy. This solution describes saturation of the Weibel instability in the single-mode regime with an arbitrary wavelength and corresponds to a standing helical wave of magnetic perturbations in which the amplitude of the generated magnetic field varies periodically over time. The longitudinal and transverse (with respect to the rotating anisotropy axis) plasma temperatures are subject to the same periodic variations. In this case, the maximum magnetic field energy can be on the order of the plasma thermal energy.