Modes In Toroidal Geometry Research Articles

Heuristic theory of nonlocally broken gyro-Bohm scaling

Global gyrokinetic simulations of ion temperature gradient turbulent transport with piecewise-flat profiles are given to illustrate the breaking of gyro-Bohm scaling by a nonlocal mechanism. The nonlocal drainage of the turbulence from unstable regions spreading into stable (or less unstable) regions breaks the gyro-Bohm scaling toward Bohm in unstable regions and toward super-gyro-Bohm in stable (or less unstable) regions. A heuristic model for this nonlocal process is formulated in terms of a nonlocal growth rate resulting from a locally weighted radial average of the local linear ballooning mode growth rate. A nonlocality length L measured in ion gyroradii provides the exponential scale for the local weighting. The nonlocal growth rate can be incorporated into a local gyro-Bohm-scaled transport model in place of the local growth rate. The resulting nonlocal transport model will provide some transport in stable regions. A heuristic theory of this nonlocal transport mechanism based on the partial formation of global modes in toroidal geometry is detailed. The theory argues that the nonlocality length L increases with relative gyroradius and decreases with the linear growth rate.

Theoretical modelling of the feedback stabilization of external MHD modes in toroidal geometry

A theoretical framework for understanding thefeedback mechanism for stabilization of external MHD modes has beenformulated. Efficient computational tools - the GATOstability code coupled with a substantially modified VACUUM code - have been developed to effectively design viable feedbacksystems against these modes. The analysis assumed a thin resistiveshell and a feedback coil structure accurately modelled in θ andϕ, albeit with only a single harmonic variation in ϕ. Timeconstants and induced currents in the enclosing resistive shell arecalculated. An optimized configuration based on an idealized modelhas been computed for the DIII-D device. Up to 90% of theeffectiveness of an ideal wall can be achieved.

The Linear Evolution of Tearing Mode in Toroidal Geometry

A set of linearly-reduced MHD equations in toroidal geometry has been solved numerically in flux coordinate with toroidal coupling. In the case of q > 1 on the magnetic axis, where q is the safety factor, the result shows that an unstable 2/1 tearing mode destabilizes both 1/1 and 3/1 modes. The 1/1 and 3/1 modes contribute local perturbations on the resonant surface of q = 2. And the 2/1 mode also contributes a local perturbation on the resonant surface of q = 3.

Sheared-flow modes in toroidal geometry

Using a Fourier–Bessel representation for the fluctuating (turbulent) electrostatic potential, an equation governing the sheared-flow modes in toroidal geometry is derived from the gyrokinetic Poisson equation, where both the adiabatic and nonadiabatic responses of the electrons are taken into account. It is shown that the principal geometrical effect on sheared-flow modes of the electrostatic potential is due to the flux-surface average of 1/B, where B is the magnetic field strength.

Toroidal mode structure in weak and reversed magnetic shear plasmas and its role in the internal transport barrier

It is becoming clear that anomalous transport in an L-mode tokamak plasma is dominated by radially extended non-local modes in toroidal geometry. This indicates that various improved modes such as the internal transport barrier (ITG) formed in a recent reversed magnetic shear experiment can be achieved by suppressing such a global structure of the toroidal mode. We investigate the role of plasma shear rotation and negative/weak and also reversed magnetic shear on these non-local modes by employing our toroidal simulation code together with a non-local theory. From simulations, in addition to the overall reduction of the wave excitation in the entire negative magnetic shear region, we find that the weak or zero magnetic shear which appears near the minimum q (safety factor) surface breaks up toroidal coupling. This leads to a discontinuity (or gap) in wave excitation around the -surface, leading to the emergence of the transport barrier. The plasma shear rotation which is observed in the experiment enhances the discontinuity so that the performance of the discontinuity is increased. We also investigate the structure of the self-generated radial electric field induced by such ITG modes and the related plasma shear rotation (referred to as the zonal flow) which influence the fluctuation level in the steady state.

Finite element approach to global gyrokinetic Particle-In-Cell simulations using magnetic coordinates

We present a fully-global linear gyrokinetic simulation code (GYGLES) aimed at describing the unstable spectrum of the ion-temperature-gradient modes in toroidal geometry. We formulate the Particle-In-Cell method with finite elements defined in magnetic coordinates, which provides excellent numerical convergence properties. The poloidal mode structure corresponding to k || = 0 is extracted without approximation from the equations, which reduces drastically the numerical resolution needed. The code can simulate routinely modes with both very long and very short toroidal wavelengths, can treat realistic (MHD) equilibria of any size, and runs on a massively parallel computer.

Stability analysis of the ideal m=n=1 kink mode in toroidal geometry by direct expansion of the hydromagnetic equations

The stability condition and growth rate of the ideal, internal m=n=1 kink mode in a toroidal plasma are calculated by means of a direct inverse-aspect-ratio expansion of the magnetohydrodynamic (MHD) equations, rather than by using the energy principle. The analysis is based on the use of computer-aided algebra. The general equation for incompressible MHD oscillations in a low-β circular tokamak is derived to lowest order in the inverse aspect ratio. This derivation generalizes the Pfirsch-Schlüter factor for the kinetic energy in toroidal geometry.

Stability analysis of electromagnetic interchange modes in toroidal geometry

An investigation is presented of how the stability of collisionless electromagnetic interchange modes depends on εn (the ratio of the magnetic drift frequency to the diamagnetic drift frequency), the ion temperature gradient and the electron temperature gradient. A linear two fluid model in toroidal geometry is used. The eigenvalue problem is solved analytically and then the complex frequency is solved numerically from the dispersion relation. Comparison is made with Mercier's criterion, in the magnetohydrodynamic limit. The most important observed effects are:1. When εn increases Mercier's criterion becomes increasingly incorrect. The toroidal system becomes more stable than Mercier's criterion predicts. εn is large in regions where we have flat density profiles, Ln » LB (the characteristic scale length of density and magnetic field inhomogeneities).2. Finite ηi (Ln/LTi) may cause instability below the critical pressure gradient in the Mercier criterion.

Improved plasma performance in tokamaks with negative magnetic shear.

A tokamak plasma configuration is reported that simultaneously improves on the maximum stable plasma pressure, the bootstrap current contribution, and kinetic stability to temperature and density gradient driven modes in toroidal geometry. It is characterized by negative magnetic shear in the plasma interior and a peaked pressure profile. Stability to the ideal low-n external kink modes requires a conducting shell at 1.3 times the plasma minor radius. This novel plasma configuration is promising for improved plasma performance in advanced tokamak experiments.

Two-fluid equations for low-n singular modes in the low-frequency regime

Braginskii’s two-fluid equations are employed to obtain the eigenmode equations governing low-n singular modes in toroidal geometry. Collisional effects are neglected. The equilibrium electric field is taken into account. The frequency of the modes is assumed to be lower than that of the ion acoustic wave along the magnetic field lines. In this low-frequency regime it is shown that the pressure of the ion fluid tends to be constant along the magnetic field lines, as also does that of the electron fluid, and the parallel electric field tends to vanish. The finite-gyroradius (FGR) effect is recovered by taking into consideration the diamagnetic drift velocity and the gyroviscosity of the ion fluid. The results show that the toroidal effect gives rise not only to the so-called apparent mass effect, but also to an enhancement of the FGR and equilibrium electric field effects. The toroidicity-induced FGR and equilibrium electric field effects are shown to be even more important than those exhibited in the cylindrical equilibrium model.

Ion-temperature-gradient modes in noncircular tokamak geometry

The work of Xu and Rosenbluth [Phys. Fluids B 3, 627 (1991)] on ion-temperature-gradient modes in toroidal geometry is extended to noncircular cross sections to examine the scaling with the elongation κ. The growth rate, wave numbers, and stability thresholds are calculated for a flat density profile characteristic of H-mode operation, and a mixing-length estimate of the ion thermal diffusivity χi is calculated. It is found that in the plasma core, growth rates decrease as the central elongation κ increases, due to finite-Larmor-radius effects. The calculated χi is approximately of the form χi=DTi3/2[(LTi)c−LTi] where D and (LTi)c depend on τ=(Te/Ti) and geometry, and LTi=‖Ti/∇Ti‖. Preliminary evidence indicates fair agreement with experimental data near the plasma edge, where LTi≪ (LTi)c, but the fit is poorer in the core where LTi∼(LTi)c and the theoretical χi is very sensitive to the accuracy in calculating (LTi)c.

Toroidal kinetic ηi-mode study in high-temperature plasmas

A new kinetic integral equation for the study of the ion-temperature-gradient-driven mode in toroidal geometry is developed that includes the ion toroidal (curvature and magnetic gradient) drift motion ωD, the mode coupling from finite k∥ due to the toroidal feature of the sheared magnetic configuration. The integral equation allows the stability study for arbitrary k∥vi/(ω − ωD) and k⊥ ρi. A systematic parameter study is carried out for the low β circular flux surface equilibrium. Possible correlations between the unstable mode characteristics and some experimental results such as the fluctuation spectrum and the anomalous ion thermal transport measurements are discussed.

Collapse of Alfven modes into ballooning-type modes due to toroidal nonaxisymmetry

The effect of nonaxisymmetry on Alfven modes in toroidal geometry is investigated numerically. A model equation is used which simplifies the analysis on the resonant surfaces. Alfven modes, characterized by their poloidal and toroidal node numbers (m,n), are found to exist in moderate nonaxisymmetry as well. For fixed (m,n), however, the mode collapses from a global feature on the resonant surface into an infinitely thin Alfven ballooning mode along a field line, if the nonaxisymmetry exceeds a critical threshold or, with given nonaxisymmetry, if the poloidal variation does so. Alfven balloonings are polarized within the magnetic surfaces and are stable.

Neoclassical viscosity effects on resistive magnetohydrodynamic modes in toroidal geometry

The flux-surface-averaged linearized resistive magnetohydrodynamic (MHD) boundary-layer equations including the compressibility, diamagnetic drift, and neoclassical viscosity terms are derived in toroidal geometry. These equations describe the resistive layer dynamics of resistive MHD modes over the collisionality regime between the banana plateau and the Pfirsch–Schlüter. From the resulting equations, the effects of neoclassical viscosity on the stability of the tearing and resistive ballooning modes are investigated numerically. Also, a study is given for the problem of how the neoclassical resistive MHD mode is generated as the collisionality is reduced. It is shown that the neoclassical viscosity terms give a significant destabilizing effect for the tearing and resistive ballooning modes. This destabilization comes mainly from the reduction of the stabilizing effect of the parallel ion sound compression by the ion neoclassical viscosity. In the banana-plateau collisionality limit, where the compressibility is negligible, the dispersion relations of the tearing and resistive ballooning modes reduce to the same form, with the threshold value of the driving force given by Δc=0. On the other hand, with the finite neoclassical effect it is found that the neoclassical resistive MHD instability is generated in agreement with previous results. Furthermore, it is shown that this later instability can be generated in a wide range of the collisionality including near the Pfirsch–Schlüter regime as well as the banana-plateau regime, suggesting that this mode is a probable cause of anomalous transport.

Micro-tearing stability in Tokamaks

The resonant layer equations governing tearing and micro-tearing modes in toroidal geometry are formulated and solved for a plasma in the banana regime and for wavelengths in the intermediate collisionality range nu e< omega *e< nu e/ epsilon . It is shown that stability of such tearing modes is determined by a competition between destabilizing trapped electron dissipation and stabilizing effects arising from ion magnetization and collisional broadening of the passing-electron Landau resonance. Analytic stability criteria are derived, but for realistic parameters these modes are predicted to be stable.

Magnetohydrodynamic stability and nonlinear evolution of the m=1 mode in toroidal geometry for safety factor profiles with an inflection point

Magnetohydrodynamic (MHD) stability and nonlinear evolution of the m=1 mode are studied in toroidal geometry for safety factor profiles having an inflection point at the q=1 surface. For ideally stable cases, linear growth rates of the resistive m=1 mode are decreased by any combination of the following variations: decreasing the aspect ratio, decreasing shear at the q=1 surface, increasing S, decreasing q0, and/or decreasing beta. Transitions from resistive kink to tearing mode to complete stability are accompanied by increasing localization as the mode is stabilized. The ideal stability of the m=1 mode for low-shear inflection-point profiles is broken at very low, but finite, beta by the appearance of a radially localized ideal interchange mode. As beta is increased, the m=1 mode becomes a global ideal internal kink. These ideal modes are stabilized by decreasing the aspect ratio and, to a lesser extent, by increasing shear at the q=1 surface. Nonlinear evolution of the m=1 mode is found to follow the Kadomtsev complete reconnection process or to be stable, depending on the linear stability of the mode. However, all nonlinear calculations were carried out at Lundquist number S=105 and hence contain substantial resistive kink contributions. Consequently, nonlinear evolution in the m=1 tearing mode regime is not examined. Even for cases in which ideal effects dominate the linear mode, Kadomtsev reconnection is observed. The nonlinear stability of linearly stable modes was also uniformly observed, even for several cases having large initial perturbations. Hence complete linear stabilization resulting from a combination of toroidal effects and low shear at the q=1 surface can suppress sawteeth, as described by the Kadomtsev reconnection process, in tokamaks having safety factor profiles with q0&amp;lt;1 at the magnetic axis; but, for the S values considered, nonlinear calculations of unstable modes result in complete Kadomtsev reconnection.

Toroidal ion-pressure-gradient-driven drift instabilities and transport revisited

A unified theory of ion-pressure-gradient-driven drift wave instabilities and transport is presented, which ties the long-wavelength trapped-ion mode to the moderate-wavelength hydrodynamic mode in toroidal geometry. An analytic dispersion relation that retains ion drift resonances, and keeps the leading-order contribution from finite Larmor radius effects and parallel compressibility, is derived. Results indicate that the slab and toroidal branches of these instabilities are of comparable importance, and are both strong candidates to explain the observed anomalous ion loss in toroidal fusion devices. However, it is concluded that in the limit of flat-density profiles characteristic of H-mode discharges, the stabilizing influence of perpendicular compressibility is insufficient to corroborate an improvement, if any, in ion confinement quality. Mixing-length expressions for the fluctuation amplitudes and both electron and ion transport coefficients are derived. Results also indicate that the heretofore experimentally observed favorable current scaling of the energy confinement time may saturate in low ion-collisionality discharges. Finally, it is shown that a population of energetic trapped particles, such as those that may be produced during radio frequency or perpendicular neutral beam heating, can significantly exacerbate the instability. Several suggestions for experiments are made to help in differentiating among various anomalous transport scenarios.

Nonlinear evolution of the internal kink mode in toroidal geometry for shaped tokamak plasmas

The nonlinear evolution of the internal kink mode is studied in toroidal geometry for noncircular cross-section tokamak plasmas. The study is focused on very low-shear and hollow q profiles, with q(ρ)≥1 for which the internal kink is unstable in the latter case, even at β=0. The nonlinear evolution is dominated by ideal magnetohydrodynamics (MHD), and the instability saturates, giving a quasihelical shift to the magnetic axis. The nonlinear saturation is caused by increased field line bending. Time scales of 103 τHp and axis shifts of 20% are reached when changes in q on the order of 3×10−3 from the marginal profile are produced.

Toroidal internal kink stability in tokamaks with ultra flat q profiles

Linear stability properties of the ideal internal kink mode in toroidal geometry are calculated for equilibria in which the q(r) profile is very flat and q ≈ 1 in the core region of the plasma. Marginal stability criteria and growth rates are calculated analytically in the large aspect ratio limit and compared with numerical results from the FAR code. The theory is developed for m = n = 1 modes and for higher m (= n) modes. The temperature perturbation due to adiabatic expansion in the linear phase of the 1/1 mode is calculated and compared with experimental data, showing that the predictions from ideal MHD theory cannot account for the observed temperature increase on JET. Comparison with collisionless theory suggests that the observed temperature increase can be accounted for by compression of the trapped particles.

Stability of ideal and resistive internal kink modes in toroidal geometry

The stability of the ideal and resistive m=1 internal modes is investigated for tokamak equilibria having a variety of different q(r) profiles, including nonmonotonic q(r) with multiple q=1 surfaces. Detailed comparisons between analytic theory and numerical results from a linear toroidal magnetohydrodynamic code are presented. Particular attention is paid to the study of equilibria that are near marginal stability.