Finite Beta Research Articles

Practical Bayesian estimation of a finite beta mixture through gibbs sampling and its applications

This paper deals with a Bayesian analysis of a finite Beta mixture model. We present approximation method to evaluate the posterior distribution and Bayes estimators by Gibbs sampling, relying on the missing data structure of the mixture model. Experimental results concern contextual and non-contextual evaluations. The non-contextual evaluation is based on synthetic histograms, while the contextual one model the class-conditional densities of pattern-recognition data sets. The Beta mixture is also applied to estimate the parameters of SAR images histograms.

Theoretical considerations of doublet-like configuration in LHD

In order to investigate the separatrix structure inside the plasma, the magnetohydrodynamic (MHD) equilibrium and the high-energy particle orbit are studied for the Large Helical Device plasmas with two magnetic axes located within the two lobes of a figure-eight magnetic separatrix. Magnetic axes are split for strongly elongated cross section by the external quadrupole field. MHD equilibria without assuming existence of nested flux surfaces are obtained from the HINT code. For the vertical elongation, the figure-eight structure evolves due to the finite beta effect and the structure of magnetic field lines outside the separatrix is ergodized. For the horizontal elongation, the Shafranov shift is very large Δ/a > 0.5. To see MHD equilibrium beta limit, high beta equilibrium (∼5%) is studied. The magnetic axis shifts extremely to the outside of the torus and a new figure-eight separatrix appears. The particle orbit on the separatrix is studied by solving the guiding-centre drift equation on the rectangular grid. If the reflecting point of the trapped particle is close to the separatrix, the banana width is very large implying large, nonlocal neoclassical transport.

Observation of high-frequency waves during strong tearing mode activity in FTU plasmas without fast ions

MHD spectroscopy in FTU has revealed high frequency (HF) oscillations between 30 and 70 kHz that accompany the development of large m0 = −2, n0 = −1 islands in ohmic plasmas. This frequency range is one order of magnitude above the island rotation frequency, one order of magnitude below the first toroidal gap in the Alfvén continuum and of the same order of the low frequency gap introduced by finite beta effects. The HF spectrum is organized in pairs of modes with opposite toroidal numbers, n = 1 and n = −1 for the stronger ones, which form standing waves in the island rest frame. The poloidal structure of HF modes is |m| = 2 for the main lines. HF lines appear when the level of poloidal field perturbation produced by the island at the plasma edge exceeds 0.2%. The absence of energetic ions in FTU ohmic plasmas, the strict correlation between HF and tearing modes and the existence of a threshold indicate that HF modes tap their energy from the tearing mode by a non-linear coupling mechanism.

On the possibility of the sheath-driven, finite-beta modes localized near the divertor plate

It is shown that, in a finite beta plasma, there may exist sheath-driven modes whose amplitude decreases exponentially with the distance from the divertor plate. The modes are sensitive to the radial tilt of the divertor plate. The short-wavelength branch of the instability, with the cross-field wavelength of the order of a few ion gyro-radii, is present in the case of a ‘positive’ tilt of the divertor plate, whereas the long-wavelength branch, with of the order of the 10 or so gyro-radii is unstable for the opposite sign of the tilt. The parallel e-folding length becomes less than the distance from the plate to the X-point (thereby making the mode insensitive to the processes near the X-point and the upper scrape-off layer) at plasma betas exceeding (2–3) × 10−4. A detailed analysis of the dispersion relations is provided. The features of the modes that can be used for their experimental identification are discussed. It is pointed out that the analogue of these modes may also exist in linear plasma devices with shaped end electrodes.

Formation and healing of n = 1 magnetic islands in LHD equilibrium

Magnetic islands with the toroidal mode number n = 1, e.g. m/n = 1/1 and 2/1 islands, in a Large Helical Device (LHD) equilibrium are studied using the three-dimensional MHD equilibrium code, HINT. In order to accomplish this purpose, the HINT code has been improved. The equilibrium analysis, in particular an analysis of the LHD equilibrium with an m/n = 1/1 island, is required for the local island divertor experiment, in order to understand the magnetic structures of field lines, i.e. flux surfaces, islands and ergodic field lines. We find that the m/n = 2/1 island can be healed for a finite equilibrium beta, while the m/n = 1/1 island is not healed and is surrounded with ergodic field lines for finite-β. From the latter result, we can conjecture that the island divertor concept is effective even for finite equilibrium beta-values, but the performance of the island divertor is deteriorated for finite-β because of the existence of the ergodic zone between the closed surfaces (i.e. the core region) and the m/n = 1/1 island. We also find that the width of the m/n = 1/1 island depends on the equilibrium beta value and that the island located at the inside of the torus has the advantage of retaining its width.

Magnetic structure at the edge of a compact stellarator (NCSX)

The magnetic field between the plasma surface and wall of the National Compact Stellarator (NCSX), is mapped via field line tracing in finite beta (4%) configurations with part (20%) of the rotational transform generated by the bootstrap current. The edge plasma magnetics are more stellarator-like, with a complex 3D structure, and less like the ordered 2D symmetric structure of a tokamak. The field lines make a transition from ergodically covering a surface to ergodically covering a volume, as the distance from the last closed magnetic surface is increased. The first few toroidal transits from the starting points are highly structured, with significant flux expansion at the bean shaped cross-section. The later toroidal transits become moderately stochastic, but with Kolmogorov lengths that can be made smaller than the connection if the separation between wall and plasma is sufficiently large.

Approximate process-parameter-dependent study of the logarithmic domain lossy integrator harmonic distortion levels

This paper estimates approximately and sheds light in an insightful way upon the impact of the basic bipolar process parameters upon harmonic distortion for the log-domain lossy integrator case. A step-by-step, symbolic, transistor-level distortion calculation is elaborated. The determination of the harmonic distortion levels present at the integrator's output is based upon the exploitation on the medium complexity Charge-Control-Model (CCM) for the Bipolar Junction Transistor (BJT). Results correlating the input signal strength with the impact of basic BJT non-idealities (e.g. finite beta values, parasitic base and emitter resistances) upon the output linearity levels are provided.

Bifurcation in the MHD behaviour of a self-organizing system: the reversed field pinch (RFP)

Within the framework of MHD modelling the RFP is shown to develop turbulent or laminar regimes switching from the former to the latter in a continuous way depending on the strength of dissipative forces (the higher they are the more laminar is the corresponding regime). In either of these cases interesting features can be observed such as the occurrence of quasi-periodic relaxation events involving reconnection processes, or the formation of stationary helical symmetric configurations. The first case corresponds to the conventional turbulent dynamo in the RFP where perturbations with multiple helical harmonic content are present. The second case corresponds to a global single helical deformation of the current channel. This simpler configuration is associated with a laminar electrostatic dynamo field and may also be found as a solution of a helical Ohmic equilibrium problem where a finite beta is necessary. The continuity of the transition between the two regimes suggests that the simple helical symmetric solution can provide a fruitful intuitive description of the RFP dynamo in general. Many of the MHD predictions are in good agreement with experimental findings and suggest possible improvements for the confinement properties of the RFP configuration.

Hall dynamics and resistive tearing instability

The stability of the neutral current sheet is analyzed in the frame of Hall dynamics. The problem is treated analytically and results are verified by numerical processing. A new version of tearing instability is found and described. It scales more favorably with the Lundquist number and has smaller wavelengths. Combined Hall magnetohydrodynamic analysis shows that Hall effects dominate the diffusion layer and growth rate of the instability in a relatively large range of sheet widths. The effect of finite beta on instability is described.

Beam emission spectroscopy measurement for density fluctuations in compact helical system

A system of beam emission spectroscopy (BES) which has been proposed as a method for the measurement of long wavelength plasma density fluctuations has been applied to compact helical system (CHS). Spatial channels of BES consist of 16 optical fibers with object lenses focused on the plasma minor radius in CHS. Spatial resolution which corresponds to the width of a sightline and the spatial pitch between sightlines are 1.0 and 1.1 cm on the imaging surface, respectively. The hydrogen neutral beam for heating with the acceleration voltage of about 28 keV is used for the probe beam. The avalanche photodiode detectors combined with a 100 kHz low-pass filter are used as photon detectors. The temporal evolution of the inverse density scale length Ln−1 and the density fluctuation were measured simultaneously at the edge region where the beam attenuation was negligible. As an initial result, the rapid increase of the local plasma density gradient near the plasma edge accompanied by a coherent-like density fluctuation of several kilohertz was detected by using this system when the edge particle transport barrier was formed. Measuring the density fluctuations at the inner regions considering the beam profile, its attenuation, and normalized radii of the sightlines with finite beta shift will be a subject of study in the future.

Numerical study of zonal flow dynamics and electron transport in electron temperature gradient driven turbulence

The electron temperature gradient (ETG) driven turbulence in tokamak core plasmas is numerically investigated based on a three-dimensional gyrofluid model with adiabatic ion response. Attentions are focused on the zonal flow dynamics in ETG fluctuations and the resultant electron heat transport. A high electron energy confinement mode is found in the weak magnetic shear region, which is relevant to the self-organization behavior of turbulence through the enhanced zonal flow dynamics rather than the weak shear stabilization of ETG fluctuations. It is demonstrated that the weak shear favors the enhancement of zonal flows in ETG turbulence. The enhanced zonal flows may be plausibly limited by the excitation of a secondary Kelvin–Helmholtz mode, which is observed from the spatial spectral analyses and time-frequency wavelet analyses of turbulent fluctuations. Electromagnetic ETG simulations show that while the Ohkawa’s scaling of anomalous electron transport with beta, χe∝1/βe, is reproduced in the moderate shear plasma, the finite beta effect may reverse the Ohkawa’s scaling in weak shear ETG turbulence due to the reduction of zonal flow generation by the magnetic Reynolds stress. Further, it is shown that the toroidal coupling enhances the zonal flow while it destabilizes the ETG mode. Hence, the electron transport seems insensitive to the toroidicity in the weak shear plasma due to the complex destabilizing and stabilizing competitions.

Effects of charge non-neutrality and finite beta on the electron temperature gradient modes in Tokamaks

Local kinetic analysis of the electron temperature gradient (ETG) mode in Tokamaks indicates that the effects of charge non-neutrality are significant in the parameter regime of Tokamaks. The maximum growth rate occurs at (k/kDe)2 [Formula: see text] 0.5 when the electron temperature and density are varied over a wide range. The growth rate becomes dependent on the β factor even though the ETG mode is predominantly electrostatic. Finite β stabilization of the ETG mode requires a large ballooning parameter so as to cause an effective drift reversal. Mixing length estimate yields an electron thermal diffusivity χe [Formula: see text] qνTe (c/ωpe)2 [Formula: see text] / Ln where c/ωpe is the electron skin depth. PACS Nos.: 52.65.Tt, 52.55.Fa, 52.35.Py

Full radius linear and nonlinear gyrokinetic simulations for tokamaks and stellarators: zonal flows, applied E × B flows, trapped electrons and finite beta

The aim of this paper is to report on recent advances made in global gyrokinetic simulations of ion temperature gradient (ITG) modes and other microinstabilities. The nonlinear development and saturation of ITG modes and the role of E × B zonal flows are studied with a global nonlinear δf formulation that retains parallel nonlinearity and thus allows for a check of the energy conservation property as a means of verifying the quality of the numerical simulation. Due to an optimized loading technique, the conservation property is satisfied with an unprecedented quality well into the nonlinear stage. The zonal component of the perturbation evolves to a quasi-steady state with regions of ITG suppression, strongly reduced radial energy flux and steepened effective temperature profiles alternating with regions of higher ITG mode amplitudes, larger radial energy flux and flattened effective temperature profiles. A semi-Lagrangian approach free of statistical noise is proposed as an alternative to the nonlinear δf formulation. An ASDEX-Upgrade experiment with an internal transport barrier is analysed with a global gyrokinetic code that includes trapped electron dynamics. The weakly destabilizing effect of trapped electron dynamics on ITG modes in an axisymmetric bumpy configuration modelling W7-X is shown in global linear simulations that retain the full electron dynamics. Finite β effects on microinstabilities are investigated with a linear global spectral electromagnetic gyrokinetic formulation. The radial global structure of electromagnetic modes shows a resonant behaviour with rational q values.

Recent advances in the LHD experiment

In the first four years of the LHD experiment, several encouraging results have emerged, the most significant of which is that MHD stability and good transport are compatible in the inward shifted axis configuration. The observed energy confinement at this optimal configuration is consistent with ISS95 scaling with an enhancement factor of 1.5. The confinement enhancement over the smaller heliotron devices is attributed to the high edge temperature. We find that the plasma with an average beta of 3% is stable in this configuration, even though the theoretical stability conditions of Mercier modes and pressure driven low-n modes are violated. In the low density discharges heated by NBI and ECR, internal transport barrier (ITB) and an associated high central temperature (>10 keV) are seen. The radial electric field measured in these discharges is positive (electron root) and expected to play a key role in the formation of the ITB. The positive electric field is also found to suppress the ion thermal diffusivity as predicted by neoclassical transport theory. The width of the externally imposed island is found to decrease when the plasma is collisionless with finite beta and increase when the plasma is collisional. The ICRF heating in LHD is successful and a high energy tail (up to 500 keV) has been detected for minority ion heating, demonstrating good confinement of the high energy particles. The magnetic field line structure unique to the heliotron edge configuration is confirmed by measuring the plasma density and temperature profiles on the divertor plate. A long pulse (2 min) discharge with an ICRF power of 0.4 MW has been demonstrated and the energy confinement characteristics are almost the same as those in short pulse discharges.

Modified two‐stream instability in the foot of high Mach number quasi‐perpendicular shocks

The modified two‐stream instability in the foot of supercritical quasi‐perpendicular shock wave is investigated. A linear analysis shows that the instability can sufficiently grow during the shock reformation cycle for the case of a realistic ion to electron mass ratio. The wave‐particle interactions of the Landau type of both the electrons and ions are essential in a finite beta plasma and lead to a reduction of the growth rate with increasing beta. The magnetic polarization in terms of wavevector is also analyzed. Additional one‐dimensional full‐particle electromagnetic simulations in a periodic system reveal some important nonlinear wave properties. The wave spectra indicate a lower cascade due to wave‐wave interactions. The parallel phase space distribution of electrons correlates well with the wave profile of the magnetic field component perpendicular to both the ambient magnetic field and the wavevector. The nonlinearly generated waves lead to electron heating parallel to the magnetic field.

Finite beta effects on the toroidal field ripple in three-dimensional tokamak equilibria

In order to investigate finite beta effects on the toroidal field ripple in a tokamak, three-dimensional free-boundary magnetohydrodynamics (MHD) equilibria are studied for a tokamak configuration that has an aspect ratio of 3.5 and an almost circular cross-section. Equilibria that are consistent with the non-axisymmetric external magnetic field are obtained by the VMEC code with a prescribed safety factor profile. To study tokamak configurations with a free-boundary constraint, we modify the boundary condition to specify the plasma position and size in the VMEC code. From the equilibria obtained, the change of the toroidal field ripple due to the finite pressure is calculated and the three-dimensional MHD effect on the ripple is estimated. In a finite pressure equilibrium, the toroidal ripple is changed by the equilibrium plasma current. The influence of the purely three-dimensional MHD effect on the toroidal field ripple is not large but the ripple is also changed by the variation of the axisymmetric component in finite pressure equilibria. In a medium-beta equilibrium, high energy particle confinement changes significantly due to the variation of the ripple well depth along the field line, because of the Shafranov shift, the change of the magnetic field line pitch due to the Pfirsch–Schlüter current, and the above mentioned finite pressure effect on the field ripple.

Radiation-condensation instability in a self-gravitating dusty astrophysical plasma.

Properties of radiation-condensation instability in a partially-ionized self-gravitating dusty astrophysical plasmas are studied. For this purpose, new dispersion relations for coupled dusty plasma and condensation modes in both unmagnetized and magnetized plasmas are derived. The dispersion relations are numerically analyzed to investigate the interplay between self-gravitation and impurity losses, as well as to study the effects of the external magnetic field and finite plasma beta on instabilities we found.

On the stabilization of ion temperature gradient modes by finite beta in noncircular toroidal devices

The stability of electromagnetic ion temperature gradient driven (ITG) modes is investigated using an advanced fluid model allowing for arbitrary elongation (κ) and finite beta effects and Shafranov shift. The ITG eigenvalue equation is solved numerically and the stability properties are discussed and compared in both the circular and noncircular geometry. It is found that elongation and finite beta effects reduce the growth rates and that the ITG mode is stabilized for large beta values as elongation is increased. Elongation increases the beta threshold of the ITG mode with larger growth rates as compared with circular geometry.

Analysis of the critical electron temperature gradient in Tore Supra

The Tore Supra [Equipe Tore Supra presented by R. Aymar in Plasma Physics and Controlled Nuclear Fusion Research, Proc. 12th Int. Conf., Nice, 1988 (IAEA, Vienna, 1989), Vol. 1, p. 9] database of fast wave electron heating discharges is analyzed with respect to the role of the critical electron temperature gradient. The experimental evidence for the critical gradient is presented from power balance thermal flux qe(∇Te) relation extrapolated to zero in the measured gradient ∇Te. The fluctuation spectra are also known to extrapolate to low levels versus ∇Te. The inferred critical gradient (∇Te)c is defined by the offset linear extrapolation of the thermal flux to zero values. Histograms of the anomalous fluxes before and after being normalized to the theoretical models are constructed. For a wide range of heating powers, the electron power balance radial heat flux is shown to be well described by Te3/2 (1/LTe−1/Lc), where Lc=(Te/∇Te)c is the gradient scale length from linear eigenmodes in a sheared magnetic field. The relative deviations of the normalized heat fluxes as a function of the excess of the temperature gradient beyond the critical gradient give one method of assessing the agreement between the models and the data. A second method is the predictive studies in which the transport codes are used to investigate the impact of the critical gradient at low and high radio frequency power levels with the model thermal diffusivities. The dominant dependence of the inferred critical temperature gradient is its proportionality to the magnetic shear as expected from the robust electron Landau damping on thermal electrons in electron temperature gradient turbulence. The evidence for the electrostatic density independent thermal diffusivity scaling versus the finite beta electromagnetic density dependent thermal diffusivity is shown in some detail. The results strongly favor the electromagnetic turbulent diffusivity formula.

Magnetic topology of a candidate NCSX plasma boundary configuration

A candidate magnetic topology of the plasma boundary of the proposed compact stellarator national compact stellarator experiment (NCSX) is investigated using field-line tracing with diffusion. The required magnetic fields are obtained from a free-boundary equilibrium using the magnetic fields from external coils and bootstrap plasma currents inside the last closed magnetic surface (LCMS). These results are used to calculate the magnetic fields of the finite beta equilibria inside and outside the LCMS in a form suitable for field-line tracing. Poincaré plots of field lines that diffuse outwards from starting points just inside the LCMS indicate an ergodic divertor region. Intersections of field lines with a simple limiting surface show contained patches suitable for divertor control. Undesirable regions of sharply inclined angle of intersection with the limiting surface are localized, indicating the suitability of the configuration for optimized divertor design techniques. We also discuss physics implications of field-line lengths in the divertor region.