Zero-Turbulence Manifold in a Toroidal Plasma

Sheared toroidal flows can cause bifurcations to zero-turbulent-transport states in tokamak plasmas. The maximum temperature gradients that can be reached are limited by subcritical turbulence driven by the parallel velocity gradient. Here it is shown that q/\epsilon (magnetic field pitch/inverse aspect ratio) is a critical control parameter for sheared tokamak turbulence. By reducing q/\epsilon, far higher temperature gradients can be achieved without triggering turbulence, in some instances comparable to those found experimentally in transport barriers. The zero-turbulence manifold is mapped out, in the zero-magnetic-shear limit, over the parameter space (\gamma_E, q/\epsilon, R/L_T), where \gamma_E is the perpendicular flow shear and R/L_T is the normalised inverse temperature gradient scale. The extent to which it can be constructed from linear theory is discussed.Comment: 5 Pages, 4 Figures, Submitted to PR

Transition to subcritical turbulence in a tokamak plasma

Tokamak turbulence, driven by the ion-temperature gradient and occurring in the presence of flow shear, is investigated by means of local, ion-scale, electrostatic gyrokinetic simulations (with both kinetic ions and electrons) of the conditions in the outer core of the Mega-Ampere Spherical Tokamak (MAST). A parameter scan in the local values of the ion-temperature gradient and flow shear is performed. It is demonstrated that the experimentally observed state is near the stability threshold and that this stability threshold is nonlinear: sheared turbulence is subcritical, i.e. the system is formally stable to small perturbations, but, given a large enough initial perturbation, it transitions to a turbulent state. A scenario for such a transition is proposed and supported by numerical results: close to threshold, the nonlinear saturated state and the associated anomalous heat transport are dominated by long-lived coherent structures, which drift across the domain, have finite amplitudes, but are not volume filling; as the system is taken away from the threshold into the more unstable regime, the number of these structures increases until they overlap and a more conventional chaotic state emerges. Whereas this appears to represent a new scenario for transition to turbulence in tokamak plasmas, it is reminiscent of the behaviour of other subcritically turbulent systems, e.g. pipe flows and Keplerian magnetorotational accretion flows.Comment: 16 pages, 5 figures, accepted to Journal of Plasma Physic

Transport Bifurcation in a Rotating Tokamak Plasma

The effect of flow shear on turbulent transport in tokamaks is studied numerically in the experimentally relevant limit of zero magnetic shear. It is found that the plasma is linearly stable for all non-zero flow shear values, but that subcritical turbulence can be sustained nonlinearly at a wide range of temperature gradients. Flow shear increases the nonlinear temperature gradient threshold for turbulence but also increases the sensitivity of the heat flux to changes in the temperature gradient, except over a small range near the threshold where the sensitivity is decreased. A bifurcation in the equilibrium gradients is found: for a given input of heat, it is possible, by varying the applied torque, to trigger a transition to significantly higher temperature and flow gradients.Comment: 4 pages, 4 figures, submitted to PR

Multiscale Gyrokinetics for Rotating Tokamak Plasmas: Fluctuations, Transport and Energy Flows

This paper presents a complete theoretical framework for plasma turbulence and transport in tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio of the gyroradius to the equilibrium scale length. Proceeding order-by-order in this expansion, a framework for plasma turbulence is developed. It comprises an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equillibrium is obtained from the Grad-Shafranov equation for a rotating plasma and the slow (resistive) evolution of the magnetic field is given by an evolution equation for the safety factor q. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the high-flow gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local cascade of free energy. Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical corrections and the fluctuations act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived. Total energy is conserved and there is no net turbulent heating. Entropy is produced by the action of fluxes flattening gradients, Ohmic heating, and the equilibration of mean temperatures. Finally, this framework is condensed, in the low-Mach-number limit, to a concise set of equations suitable for numerical implementation.Comment: 113 pages, 3 figure

Nonlinear phase mixing and phase-space cascade of entropy in gyrokinetic plasma turbulence

Electrostatic turbulence in weakly collisional, magnetized plasma can be interpreted as a cascade of entropy in phase space, which is proposed as a universal mechanism for dissipation of energy in magnetized plasma turbulence. When the nonlinear decorrelation time at the scale of the thermal Larmor radius is shorter than the collision time, a broad spectrum of fluctuations at sub-Larmor scales is numerically found in velocity and position space, with theoretically predicted scalings. The results are important because they identify what is probably a universal Kolmogorov-like regime for kinetic turbulence; and because any physical process that produces fluctuations of the gyrophase-independent part of the distribution function may, via the entropy cascade, result in turbulent heating at a rate that increases with the fluctuation amplitude, but is independent of the collision frequency.Comment: Revtex, 4 pages, 3 figures; replaced to match published versio

Kinetic Simulations of Magnetized Turbulence in Astrophysical Plasmas

This letter presents the first ab initio, fully electromagnetic, kinetic simulations of magnetized turbulence in a homogeneous, weakly collisional plasma at the scale of the ion Larmor radius (ion gyroscale). Magnetic and electric-field energy spectra show a break at the ion gyroscale; the spectral slopes are consistent with scaling predictions for critically balanced turbulence of Alfven waves above the ion gyroscale (spectral index -5/3) and of kinetic Alfven waves below the ion gyroscale (spectral indices of -7/3 for magnetic and -1/3 for electric fluctuations). This behavior is also qualitatively consistent with in situ measurements of turbulence in the solar wind. Our findings support the hypothesis that the frequencies of turbulent fluctuations in the solar wind remain well below the ion cyclotron frequency both above and below the ion gyroscale.Comment: 4 pages, 3 figures, submitted to Physical Review Letter

Dissipation-Scale Turbulence in the Solar Wind

We present a cascade model for turbulence in weakly collisional plasmas that follows the nonlinear cascade of energy from the large scales of driving in the MHD regime to the small scales of the kinetic Alfven wave regime where the turbulence is dissipated by kinetic processes. Steady-state solutions of the model for the slow solar wind yield three conclusions: (1) beyond the observed break in the magnetic energy spectrum, one expects an exponential cut-off; (2) the widely held interpretation that this dissipation range obeys power-law behavior is an artifact of instrumental sensitivity limitations; and, (3) over the range of parameters relevant to the solar wind, the observed variation of dissipation range spectral indices from -2 to -4 is naturally explained by the varying effectiveness of Landau damping, from an undamped prediction of -7/3 to a strongly damped index around -4.Comment: 6 pages, 2 figures, accepted for publication in AIP Conference Proceedings on "Turbulence and Nonlinear Processes in Astrophysical Plasmas

Ion-scale turbulence in MAST: anomalous transport, subcritical transitions, and comparison to BES measurements

We investigate the effect of varying the ion temperature gradient (ITG) and toroidal equilibrium scale sheared flow on ion-scale turbulence in the outer core of MAST by means of local gyrokinetic simulations. We show that nonlinear simulations reproduce the experimental ion heat flux and that the experimentally measured values of the ITG and the flow shear lie close to the turbulence threshold. We demonstrate that the system is subcritical in the presence of flow shear, i.e., the system is formally stable to small perturbations, but transitions to a turbulent state given a large enough initial perturbation. We propose that the transition to subcritical turbulence occurs via an intermediate state dominated by low number of coherent long-lived structures, close to threshold, which increase in number as the system is taken away from the threshold into the more strongly turbulent regime, until they fill the domain and a more conventional turbulence emerges. We show that the properties of turbulence are effectively functions of the distance to threshold, as quantified by the ion heat flux. We make quantitative comparisons of correlation lengths, times, and amplitudes between our simulations and experimental measurements using the MAST BES diagnostic. We find reasonable agreement of the correlation properties, most notably of the correlation time, for which significant discrepancies were found in previous numerical studies of MAST turbulence.Comment: 67 pages, 37 figures. Submitted to PPC

Gyrokinetic simulation of entropy cascade in two-dimensional electrostatic turbulence

Two-dimensional electrostatic turbulence in magnetized weakly-collisional plasmas exhibits a cascade of entropy in phase space [Phys. Rev. Lett. 103, 015003 (2009)]. At scales smaller than the gyroradius, this cascade is characterized by the dimensionless ratio D of the collision time to the eddy turnover time measured at the scale of the thermal Larmor radius. When D >> 1, a broad spectrum of fluctuations at sub-Larmor scales is found in both position and velocity space. The distribution function develops structure as a function of v_{perp}, the velocity coordinate perpendicular to the local magnetic field. The cascade shows a local-scale nonlinear interaction in both position and velocity spaces, and Kolmogorov's scaling theory can be extended into phase space.Comment: 8 pages, 10 figures, Conference paper presented at 2009 Asia-Pacific Plasma Theory Conference. Ver.2 includes corrected typos & updated reference