Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows.

This paper presents a complete theoretical framework for studying turbulence and transport in rapidly rotating tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio ε = ρi/α of the gyroradius to the equilibrium scale length. Proceeding order by order in this expansion, a set of coupled multiscale equations is developed. They describe 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 interplay between the equilibrium and the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equilibrium is obtained from the generalized Grad-Shafranov equation for a rotating plasma, determining the magnetic flux function from the mean pressure and velocity profiles of the plasma. The slow (resistive-timescale) 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 (in space) cascade of the free energy of the fluctuations (i.e. there is no turbulence spreading). Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical and fluctuating corrections to the equilibrium Maxwellian act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived from the transport equations. Total energy, thermal, kinetic and magnetic, 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 interspecies temperature differences. This equilibration is found to include both turbulent and collisional contributions. Finally, this framework is condensed, in the low-Mach-number limit, to a more concise set of equations suitable for numerical implementation.

Turbulent transport and heating of trace heavy ions in hot magnetized plasmas.

Scaling laws for the transport and heating of trace heavy ions in low-frequency magnetized plasma turbulence are derived and compared with direct numerical simulations. The predicted dependences of turbulent fluxes and heating on ion charge and mass number are found to agree with numerical results for both stationary and differentially rotating plasmas. Heavy ion momentum transport is found to increase with mass, and heavy ions are found to be preferentially heated, implying a mass-dependent ion temperature for very weakly collisional plasmas and for partially ionized heavy ions in strongly rotating plasmas.

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/ϵ (magnetic field pitch/inverse aspect ratio) is a critical control parameter for sheared tokamak turbulence. By reducing q/ϵ, 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 (γ(E), q/ϵ, R/L(T)), where γ(E) is the perpendicular flow shear and R/L(T) is the normalized inverse temperature gradient scale. The extent to which it can be constructed from linear theory is discussed.

Gyrokinetic simulations of solar wind turbulence from ion to electron scales.

A three-dimensional, nonlinear gyrokinetic simulation of plasma turbulence resolving scales from the ion to electron gyroradius with a realistic mass ratio is presented, where all damping is provided by resolved physical mechanisms. The resulting energy spectra are quantitatively consistent with a magnetic power spectrum scaling of k(-2.8) as observed in in situ spacecraft measurements of the "dissipation range" of solar wind turbulence. Despite the strongly nonlinear nature of the turbulence, the linear kinetic Alfvén wave mode quantitatively describes the polarization of the turbulent fluctuations. The collisional ion heating is measured at subion-Larmor radius scales, which provides evidence of the ion entropy cascade in an electromagnetic turbulence simulation.

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.

Effect of quasihelical symmetry on trapped-electron mode transport in the HSX stellarator.

This Letter presents theory-based predictions of anomalous electron thermal transport in the Helically Symmetric eXperiment stellarator, using an axisymmetric trapped-electron mode drift wave model. The model relies on modifications to a tokamak geometry that approximate the quasihelical symmetry in the Helically Symmetric eXperiment (particle trapping and local curvature) and is supported by linear 3D gyrokinetic calculations. Transport simulations predict temperature profiles that agree with experimental profiles outside a normalized minor radius of rho>0.3 and energy confinement times that agree within 10% of measurements. The simulations can reproduce the large measured electron temperatures inside rho<0.3 if an approximation for turbulent transport suppression due to shear in the radial electric field is included.

Dimits shift in realistic gyrokinetic plasma-turbulence simulations.

In simulations of turbulent plasma transport due to long wavelength (k perpendicular rhoi < or = 1) electrostatic drift-type instabilities, we find a persistent nonlinear up-shift of the effective threshold. Next-generation tokamaks will likely benefit from the higher effective threshold for turbulent transport, and transport models should incorporate suitable corrections to linear thresholds. The gyrokinetic simulations reported here are more realistic than previous reports of a Dimits shift because they include nonadiabatic electron dynamics, strong collisional damping of zonal flows, and finite electron and ion collisionality together with realistic shaped magnetic geometry. Reversing previously reported results based on idealized adiabatic electrons, we find that increasing collisionality reduces the heat flux because collisionality reduces the nonadiabatic electron microinstability drive.

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 Alfvén waves above the ion gyroscale (spectral index -5/3) and of kinetic Alfvén 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.

Small-scale turbulence in a closed-field-line geometry.

Plasma turbulence due to small-scale entropy modes is studied with gyrokinetic simulations in a simple closed-field-line geometry, the Z pinch, in low-beta parameter regimes that are stable to ideal interchange modes. We find an enormous variation in the nonlinear dynamics and particle transport as a function of two main parameters, the density gradient and the plasma collisionality. This variation is explained in part by the damping and stability properties of spontaneously formed zonal flows in the system. As in toroidal systems, the zonal flows can lead to a strong nonlinear suppression of transport below a critical gradient that is determined by the stability of the zonal flows.

X-point collapse and saturation in the nonlinear tearing mode reconnection.

We study the nonlinear evolution of the resistive tearing mode in slab geometry in two dimensions. We show that, in the strongly driven regime (large delta'), a collapse of the X point occurs once the island width exceeds a certain critical value approximately 1/delta'. A current sheet is formed and the reconnection is exponential in time with a growth rate proportional eta(1/2), where eta is the resistivity. If the aspect ratio of the current sheet is sufficiently large, the sheet can itself become tearing-mode unstable, giving rise to secondary islands, which then coalesce with the original island. The saturated state depends on the value of delta'. For small delta', the saturation amplitude is proportional delta' and quantitatively agrees with the theoretical prediction. If delta' is large enough for the X-point collapse to have occurred, the saturation amplitude increases noticeably and becomes independent of delta'.

Giant electron tails and passing electron pinch effects in tokamak-core turbulence.

The anomalous particle transport in a tokamak core is believed to be linked to the advection of magnetically trapped electrons alone, owing to the passing electrons maintaining a thermal equilibrium along the field lines. Surprisingly, in nonlinear numerical studies, the radial flux of passing electrons rivals that of the trapped ones. The strong interaction of passing electrons and electric fluctuations is mediated by long tails of the modes along the magnetic field, which are generated by the passing electrons in the first place.

Prediction of significant tokamak turbulence at electron gyroradius scales.

The experimental conditions under which tokamak turbulence at hyperfine (electron gyroradius) scales is predicted to be significant and observable are described. The first quantitative predictions of fluctuation amplitudes, spectral features, and the associated electron energy transport are presented. A novel theoretical model which quantitatively describes the boundaries of the high-amplitude streamer transport regime is presented and shown to explain the gyrokinetic simulation results. This model uniquely includes consideration of two distinct secondary instabilities.

Generation and stability of zonal flows in ion-temperature-gradient mode turbulence.

We address the mechanisms underlying zonal flow generation and stability in turbulent systems driven by the electrostatic ion-temperature-gradient (ITG) mode. In the case of zonal flow stability, we show the poloidal flows typical of numerical simulations become unstable when they exceed a critical level. Near marginal stability of the linear ITG mode, the system can generate zonal flows that are sufficiently weak to remain stable and sufficiently strong to suppress the linear ITG mode. This stable region corresponds to the parameter regime of the nonlinear Dimits up-shift.

Electron temperature gradient turbulence.

The first toroidal, gyrokinetic, electromagnetic simulations of small scale plasma turbulence are presented. The turbulence considered is driven by gradients in the electron temperature. It is found that electron temperature gradient (ETG) turbulence can induce experimentally relevant thermal losses in magnetic confinement fusion devices. For typical tokamak parameters, the transport is essentially electrostatic in character. The simulation results are qualitatively consistent with a model that balances linear and secondary mode growth rates. Significant streamer-dominated transport at long wavelengths occurs because the secondary modes that produce saturation become weak in the ETG limit.

Circulating blood volume in appropriate and small for gestational age full term and preterm polycythaemic infants.

Relative circulating volume (CV) was determined in 31 polycythaemic newborn infants who underwent a partial plasma exchange transfusion (PPET). The infants were divided in four sub-groups: appropriate for gestational age (AGA) term and preterm infants, and small for gestational age (SGA) term and preterm infants. CV was calculated using a regression line between the number of exchange steps and central venous haematocrit values obtained before the procedure and after every exchange step. The median relative CV in all polycythaemic newborn infants was 94.0 (range 69-143) ml/kg. Irrespective of gestational age and birthweight the median relative CV in AGA infants was 86.5 (range 69-107) ml/kg, which differed significantly from the value of 106 (range 85-143) ml/kg found in SGA infants. If a PPET is indicated we recommend to calculate the exchange volume in AGA infants using a relative CV of 86 ml/kg and in SGA infants using a relative CV of 106 ml/kg.