Neoclassical Viscosity Research Articles

Screening of resonant magnetic perturbations by flows in tokamaks

The non-linear reduced four-field RMHD model in cylindrical geometry was extended to include plasma rotation, neoclassical poloidal viscosity and two fluid diamagnetic effects. Interaction of the static resonant magnetic perturbations (RMPs) with the rotating plasmas in tokamaks was studied. The self-consistent evolution of equilibrium electric field due to RMP penetration is taken into account in the model. It is demonstrated that in the pedestal region with steep pressure gradients, mean flows perpendicular to the magnetic field, which includes and electron diamagnetic components plays an essential role in RMP screening by plasma. Generally, the screening effect increases for lower resistivity, stronger rotation and smaller RMP amplitude. Strong screening of central islands was observed limiting RMP penetration to the narrow region near the separatrix. However, at certain plasma parameters and due to the non-linear evolution of the radial electric field produced by RMPs, the rotation can be compensated by electron diamagnetic rotation locally. In this case, RMPs can penetrate and form magnetic islands. Typical plasma parameters and RMPs spectra on DIII-D, JET and ITER were used in modelling examples presented in the paper.

Recent progress in RF heating and long-pulse experiments on EAST

New capabilities including radio-frequency (RF) powers and diagnostics have been developed since the last IAEA meeting. Divertor performance has been systematically assessed for both single null and double null configurations with normal and reversed BT directions to address asymmetries correlated with classic drift and the issues of heat load on divertor plates. 1 MW lower hybrid wave power injection has produced fully non-inductive diverted plasma discharges. Such discharges have been readily extended to well over 1 min duration. With the assistance of lower hybrid current drive (LHCD), EAST can be operated at 1 MA with both RTEFIT/Iso-flux or pre-programmed control algorithms. A later operation mode could produce an improved confinement plasma by increasing the ohmic power through regulating the loop voltage. Both direct electron and ion heating by ion cyclotron resonant frequency power have been observed for the first time on EAST. Intrinsic toroidal rotation and momentum study in ohmic and LHCD plasmas reveals the importance of the balance between neoclassical viscosity and neutral friction at the edge and finds clear linkage between edge and core toroidal rotation. The expected capabilities on EAST in the next two years will allow access to high-performance regimes, such as H-modes, and enable specific physics studies.

A Dip Structure in the Intrinsic Toroidal Rotation Near the Edge of the Ohmic Plasmas in EAST

Ion's toroidal velocity, vt, in both the outermost 4 cm of the confined region and the scrap-off layer of Ohmic L-mode plasmas in EAST was measured using Mach probes. At about 1 cm inside the separatrix a local minimum in vt was observed, from which a cocurrent rotation increased both inwards and outwards. The radial width of the vt dip was 1 cm to 2 cm, and both the density and electron temperature profiles exhibited steep gradients at this dip position. It was observed in both divertor and limiter configurations. To find out its origin, the toroidal torques induced by neutral friction, neoclassical viscosity, collisional perpendicular shear viscosity, ion orbit loss and turbulent Reynolds stress were estimated using the measured parameters. Our results indicate that in this particular parameter regime the neutral friction was the dominant damping force. The calculated cocurrent toroidal torque by the neoclassical viscosity dominates over those from the collisional perpendicular shear viscosity, ion orbit loss and turbulent Reynolds stress. These results are potentially important for the understanding of boundary conditions for the intrinsic toroidal momentum in tokamak plasmas.

Neoclassical Toroidal Viscosity Calculations in Tokamaks Using aδfMonte Carlo Simulation and Their Verifications

Neoclassical toroidal viscosities (NTVs) in tokamaks are investigated using a δf Monte Carlo simulation, and are successfully verified with a combined analytic theory over a wide range of collisionality. A Monte Carlo simulation has been required in the study of NTV since the complexities in guiding-center orbits of particles and their collisions cannot be fully investigated by any means of analytic theories alone. Results yielded the details of the complex NTV dependency on particle precessions and collisions, which were predicted roughly in a combined analytic theory. Both numerical and analytic methods can be utilized and extended based on these successful verifications.

Neoclassical Transport Modeling Compatible with a Two-Fluid Transport Equation System

A neoclassical transport model compatible with a system of two-fluid equations is proposed, with an emphasis on the heat flux contribution. The model is based on the moment approach and is capable of accurately reproducing important neoclassical properties through a simple expression of the neoclassical viscosity tensor, with the aid of the NCLASS module. Applying neoclassical transport theory in a fluid context, we confirm the reproducibility of first-order flows, poloidal flows, neoclassical resistivity, bootstrap current and particle flux. c

A Monte‐Carlo‐Based Calculation of Neoclassical Flows and Viscosity for Nonaxisymmetric Toroidal Plasmas

AbstractThis paper reports the Monte‐Carlo calculation of the neoclassical diffusion and viscosity coefficients with the incompressible E × B drift. A recently developed Monte‐Carlo code, based on the Einstein‐Helfand relation, is extended for this purpose. Present status of a Monte‐Carlo‐based moment transport code to evaluate the neoclassical flows for multiple species plasmas is also discussed (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An approximate analytic expression for neoclassical toroidal plasma viscosity in tokamaks

An approximate analytic expression for neoclassical toroidal plasma viscosity in tokamaks that have error fields or magnetohydrodynamic activities is presented. The expression smoothly joins transport fluxes or plasma viscosity in all the known collisionality regimes derived from the solution of the bounce averaged drift kinetic equation and should be useful in modelling results of existing and future tokamak experiments. It also incorporates some of the extensions of the known expressions to include the effects of finite ∇B drift in the non-resonant transport processes. Here, B is the magnitude of the magnetic field. The toroidal momentum balance equation is a nonlinear function of the radial electric field when the neoclassical plasma viscosity is dominant. It can have bifurcated solutions for the radial electric field and may lead to better plasma confinement as a result.

Transport Simulation of Helical Plasmas Using the TASK/TX Code

It is widely accepted that the radial electric field strongly affects plasma confinement through the transport process. We have analysed the time evolution of the radial electric field and the radial transport in plasmas in a helical magnetic configuration by extending the TASK/TX code [M. Honda and A. Fukuyama, J. Comput. Phys. 227, 2808 (2008)]. TASK/TX is a one-dimensional dynamic transport code originally developed for axisymmetric plasmas. A set of flux-surface averaged fluid equations is solved simultaneously, rather than a set of diffusion equations based on the flux-gradient relations. TASK/TX consists of one-dimensional two fluids (electron and ion) equations, Maxwell's equations and diffusion equations of neutrals. To apply TASK/TX to helical plasmas, we have included two additional effects; helical neoclassical viscosity force and diffusion due to magnetic braiding. Plasma transport simulation is carried out mainly using the LHD parameters. We obtained negative radial electric field (ion root) with ion heating and positive radial electric field (electron root) with electron heating. Effects of magnetic braiding are also studied.

Moment approach to the bootstrap current in nonaxisymmetric toroidal plasmas using δf Monte Carlo methods

To evaluate the bootstrap current in nonaxisymmetric toroidal plasmas quantitatively, a δf Monte Carlo method is incorporated into the moment approach. From the drift-kinetic equation with the pitch-angle scattering collision operator, the bootstrap current and neoclassical conductivity coefficients are calculated. The neoclassical viscosity is evaluated from these two monoenergetic transport coefficients. Numerical results obtained by the δf Monte Carlo method for a model heliotron are in reasonable agreement with asymptotic formulae and with the results obtained by the variational principle.

Neoclassical Viscosities in NCSX and QPS with Few Toroidal Periods and Low Aspect Ratios

Previously reported benchmarking examples of the analytical formulae of neoclassical viscosities were presented implicitly assuming applications in a future integrated simulation system of the Large Helical Device (LHD). Therefore, the assumed toroidal period numbers were mainly N = 10. However, in this type of calculation, an implicit (or sometimes explicit) assumption of ι/N « 1 is sometimes included. This assumption is included not only in simplified bounce-averaged drift kinetic equations for ripple diffusions, but also in the equation before the averaging for non-bounce-averaged effects determining neoclassical parallel viscosity and banana-plateau diffusions. For clarifying the applicability of the analytical methods for configurations with extremely low toroidal period numbers (required for low aspect ratios), we show recent benchmarking examples in the National Compact Stellarator Experiment (NCSX) with N = 3 and the Quasi-Poloidal Stellarator (QPS) with N = 2.

Interpretation of edge pedestal rotation measurements in DIII-D

A novel methodology for inferring experimental toroidal angular momentum transfer rates from measured toroidal rotation velocities and other measured quantities has been developed and applied to analyze rotation measurements in the DIII-D [J. Luxon, Nucl. Fusion 42, 6149 (2002)] edge pedestal. The experimentally inferred values have been compared with predictions based on atomic physics processes and on neoclassical toroidal viscosity. The poloidal rotation velocities have been calculated from poloidal momentum balance using neoclassical parallel viscosity and a novel retention of all terms in the poloidal momentum balance, and compared with measured values in the DIII-D edge pedestal.

Nonlinear Simulation of Ion Temperature Gradient Driven Drift Wave Turbulence

The gyro-fluid model proposed by Ottaviani et. al is extended, in which the ion parallel flow and ion neoclassical viscosity are taken into account. Using this model, the ion temperature gradient driven drift wave turbulence is analyzed. The saturation level and ion heat flux are evaluated in case with profile relaxation. The multi-scale interaction between the micro-scale and meso-scale is observed in our simulations. The avalanche process and turbulence spreading occur in the nonlinear phase. In addition, the various kind of parallel scheme are tested to improve the performance of the code. We find a way which gives better parallel efficiency of the spectral code.

Analytical Methods to Calculate Flows and Diffusions Driven by Neoclassical Viscosities in Helical Plasmas

Methods to obtain monoenergetic viscosity coefficients by combining analytical approximations of the linearized drift kinetic equation are studied for a previously formulated full neoclassical transport matrix in general nonsymmetric toroidal plasmas. A unified analytical treatment of two coefficients due to the non-bounce-averaged radial drifts of guiding centers is shown. These coefficients were previously obtained by a direct numerical calculation of the kinetic equation in the three-dimensional (3-D) phase-space (pitch-angle, poloidal and toroidal angles). In a present study, the radial drift term in the equation is divided into three parts, and then the perturbed distribution and the resulting monoenergetic coefficients are expressed by superposed components, which can be calculated by combining analytical methods. An analytical expression for the boundary layer correction to the parallel viscosity in the 1/ν regime also is newly derived to complete the full matrix without a numerical calculation in 3-D phase-space. Analytical results given by adding these components approximately reproduce results of the direct numerical calculation of the kinetic equation.

Viscous relaxation of drift-Alfvén waves in tokamaks and its application for triggering improved confinement regimes

Viscous damping of the drift-Alfvén modes in tokamaks and its application for triggering improved confinement regimes are considered. It is shown that these waves are effectively damped by the neoclassical viscosity with the upper limit for their frequency ω equal to the collision frequency of slow ions νi∕ϵ, where ϵ is the inverse aspect ratio. Quasistationary plasma poloidal and toroidal velocities and current drive produced by damping of these waves are estimated. Evaluations of the viscosity effects on the Alfvén waves which were induced by the dynamic ergodic divertor in the Tokamak Experiment for Technology Oriented Research tokamak [K. H. Finken et al., Phys. Rev. Lett. 94, 015003 (2005)] are made.

Rotation Velocities and Radial Electric Field in the Plasma Edge

A set of equations for the calculation of plasma rotation and the radial electric field in the edge plasma is presented. The equations are based on particle and momentum balance and explicitly incorporate neoclassical viscosity and atomic physics effects, and allow for the inclusion of anomalous effects. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A neoclassical calculation of toroidal rotation profiles and comparison with DIII-D measurements

Momentum and particle balance and neoclassical viscosity were applied to calculate the radial profile of toroidal rotation velocity in several DIII-D [J. Luxon, Nucl. Fusion 42, 614 (2002)] discharges in a variety of energy confinement regimes (low-mode, low-mode with internal transport barrier, high-mode, and high-mode with quiescentd double barrier). Calculated toroidal rotation velocities generally were found to (over) predict measured values to well within a factor of 2.

Generation of zonal flows by ion-temperature-gradient and related modes in the presence of neoclassical viscosity

Generation of zonal flows by primary waves that are more complex than those considered in the standard drift-wave model is studied. The effects of parallel ion velocity and ion perturbed temperature and the part of the nonlinear mode interaction proportional to the ion pressure are taken into account. This generalization of the standard model allows the analysis of generation of zonal flows by a rather wide variety of primary modes, including ion temperature gradients, ion sound, electron drift, and drift-sound modes. All the listed effects, which are present in the slab geometry model, are complemented by effects of neoclassical viscosity inherent to toroidal geometry. We show that the electrostatic potential of secondary small-scale modes is expressed in terms of a nonlinear shift of the mode frequency and interpret this shift in terms of the perpendicular and parallel Doppler, nonlinear Kelvin-Helmholtz (KH), and nonlinear ion-pressure-gradient effects. A basic assumption of our model is that the primary modes form a nondispersive monochromatic wave packet. The analysis of zonal-flow generation is performed following an approach similar to that of convective-cell theory. Neoclassical zonal-flow instabilities are separated into fast and slow ones, and these are divided into two varieties. The first of them is independent of the nonlinear KH effect, while the second one is sensitive to it.

Neoclassical generation of toroidal zonal flow by drift wave turbulence

Zonal-flow instabilities due to drift-wave turbulence in the presence of toroidicity-induced parallel (neoclassical) viscosity and allowing for the toroidal flow are studied. It is shown that, as a result of the neoclassical viscosity a new type of zonal-flow instability is possible, leading to the generation of the considerable toroidal zonal flow. The toroidal instability is complementary to the previously studied instability resulting in the poloidal flow generation and occurs as a second branch of the general dispersion relation describing the evolution of the poloidal and toroidal flow. Nonlinear saturation of the new instability is studied. It is shown that saturated zonal toroidal velocity, generated in this instability, is large compared to the mean cross-field drift velocity as the ratio q∕ϵ, where q is the safety factor and ϵ is the inverse aspect ratio. In addition to the broad turbulent spectrum of drift waves, a monochromatic wave packet is considered. It is revealed that for the case of sufficiently strong neoclassical viscosity such a wave packet is subjected to generation of the toroidal zonal flow due to instability of hydrodynamic type.

Assessment of Flow Drive by Use of Ion Bernstein Wave on Heliotron J and EAST Devices

The possibility of driving poloidal flows by use of ion Bernstein wave is assessed for Heliotron J and EAST devices by means of ray tracing analysis. Sheared poloidal flow is expected to suppress plasma turbulences due to the decorrelation of the waves. In Heliotron J and EAST plasma, the rays of Ion Bernstein Wave travel into the central region with oscillations along the magnetic lines of force and their power is absorbed by ions at the cyclotron resonance layers. The momentum input has been estimated by calculating the momentum change of rays and the poloidal flow has been estimated using neoclassical viscosities. The wave momentum changes its sign as it propagates inward, depositing sheared momentum to the plasma, and therefore causes sheared poloidal flows.

Magnetohydrodynamics equilibria with toroidal and poloidal flow

In the present work, the effects of flow on tokamak equilibria are investigated, focusing in particular on the effects of poloidal flows. It is shown that discountinuous transonic equilibria with a pedestal structure can be obtained for relatively low values of the poloidal velocity. Equilibria with poloidal flow of the order of the poloidal Alfvén speed are shown to develop inverted Shafranov shift. Since the rotation is damped by the neoclassical poloidal viscosity, a quasi-omnigenous solution for equilibria with large rotation is also derived in order to minimize the flow damping. In this solution, the magnetic field is construed to be a function of the poloidal magnetic flux Ψ up to a small correction by an appropriate choice of the flow profiles. All numerical results are obtained with the code FLOW [L. Guazzotto, R. Betti, J. Manickam, and S. Kaye, Phys. Plasmas 11, 604 (2004)].