G. W. Hammett

Comparisons and physics basis of tokamak transport models and turbulence simulations

The predictions of gyrokinetic and gyrofluid simulations of ion-temperature-gradient (ITG) instability and turbulence in tokamak plasmas as well as some tokamak plasma thermal transport models, which have been widely used for predicting the performance of the proposed International Thermonuclear Experimental Reactor (ITER) tokamak [Plasma Physics and Controlled Nuclear Fusion Research, 1996 (International Atomic Energy Agency, Vienna, 1997), Vol. 1, p. 3], are compared. These comparisons provide information on effects of differences in the physics content of the various models and on the fusion-relevant figures of merit of plasma performance predicted by the models. Many of the comparisons are undertaken for a simplified plasma model and geometry which is an idealization of the plasma conditions and geometry in a Doublet III-D [Plasma Physics and Controlled Nuclear Fusion Research, 1986 (International Atomic Energy Agency, Vienna, 1987), Vol. 1, p. 159] high confinement (H-mode) experiment. Most of the mo...

ASTROPHYSICAL GYROKINETICS: KINETIC AND FLUID TURBULENT CASCADES IN MAGNETIZED WEAKLY COLLISIONAL PLASMAS

This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulent motions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the inertial range above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvenic fluctuations and a passive cascade of density and magnetic-field-strength fluctuations. The former are governed by the reduced magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvenic component (in the collisional limit, these compressive fluctuations become the slow and entropy modes of the conventional MHD). In the dissipation range below ion gyroscale, there are again two cascades: the kinetic-Alfven-wave (KAW) cascade governed by two fluid-like electron reduced magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations both in space and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was Landau-damped at the ion gyroscale to collisional scales in the phase space and leads to ion heating. The KAW energy is similarly damped at the electron gyroscale and converted into electron heat. Kolmogorov-style scaling relations are derived for all of these cascades. The relationship between the theoretical models proposed in this paper and astrophysical applications and observations is discussed in detail.This paper presents a theoretical framework for understand ing plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and densit y fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulent motions at weakly collisional an d collisionless scales. This paper focuses on turbulence in a strong mean magnetic field (the guide field). T he key assumptions behind the theory developed here are that the turbulent fluctuations are anisotropic wit h respect to the mean field and that their frequency is low compared to the ion cyclotron frequency. The turbulen ce is assumed to be stirred (forced) at some system-specific outer scale L. The energy injected at this scale has to be dissipated into h eat, which ultimately cannot be accomplished without collisions. A kinetic cascadedevelops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic c as ade in various scale ranges depends on the physics of plasma fluctuations that can exist there. There ar four special scales that separate physically distinct regimes: the electron gyroscale ρe, the ion gyroscaleρi , the mean free path λmfpi and the electron heat diffusion scale (mi/me)λmfpi (me andmi are electron and ion masses). In each of the scale ranges sepa rated by these scales, a number of physically meaningful and rigorously ju stifiable simplifications of the fully kinetic plasma description are possible. These are derived systematicall y vi a hierarchy of asymptotic expansions. The result is that, in each scale range, the fully kinetic proble m is reduced to a more physically transparent and computationally tractable system of equations, which are d erived in a rigorous way. In the “inertial range” above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvénic fluctuations and a passive cascade of density and magnetic-field-strength fluc tuations. The former are governed by two fluid-like Reduced Magnetohydrodynamic (RMHD) equations at both the c ollisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines as ociated with the Alfvénic component (in the collisional limit, these passive fluctuations become the sl ow and entropy modes of the conventional MHD). In the “dissipation range” between the ion and electron gyroscales, there are again two cascades: the kineticAlfvén-wave (KAW) cascade governed by two fluid-like Electr on Reduced Magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations b oth in space and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was dampe d by collisionless wave-particle interaction at the ion gyroscle to collisional scales and leads to ion heating. The KAW energy is similarly damped at the electron gyroscale and is converted into electron heat. The relation ship between the theoretical models proposed in this paper and astrophysical applications and observations is d i cussed in detail. Subject headings: magnetic fields—methods: analytical—MHD—plasmas—turbul ence

A Model of Turbulence in Magnetized Plasmas: Implications for the Dissipation Range in the Solar Wind

This paper studies the turbulent cascade of magnetic energy in weakly col- lisional magnetized plasmas. A cascade model is presented, based on the assumptions of local nonlinear energy transfer in wavenumber space, critical balance between linear propagation and nonlinear interaction times, and the applicability of linear dissipation rates for the nonlinearly turbulent plasma. The model follows the nonlinear cascade of energy from the driving scale in the MHD regime, through the transition at the ion Lar- mor radius into the kinetic Alfven wave regime, in which the turbulence is dissipated by kinetic processes. The turbulent fluctuations remain at frequencies below the ion cy- clotron frequency due to the strong anisotropy of the turbulent fluctuations, kk ≪ k⊥ (implied by critical balance). In this limit, the turbulence is optimally described by gy- rokinetics; it is shown that the gyrokinetic approximation is well satisfied for typical slow solar wind parameters. Wave phase velocity measurements are consistent with a kinetic Alfven wave cascade and not the onset of ion cyclotron damping. The conditions under which the gyrokinetic cascade reaches the ion cyclotron frequency are established. Cas- cade model solutions imply that collisionless damping provides a natural explanation for the observed range of spectral indices in the dissipation range of the solar wind. The dis- sipation range spectrum is predicted to be an exponential fall off; the power-law behav- ior apparent in observations may be an artifact of limited instrumental sensitivity. The cascade model is motivated by a programme of gyrokinetic simulations of turbulence and particle heating in the solar wind.

Astrophysical Gyrokinetics: Basic Equations and Linear Theory

Magnetohydrodynamic (MHD) turbulence is encountered in a wide variety of astrophysical plasmas, including accretion disks, the solar wind, and the interstellar and intracluster medium. On small scales, this turbulence is often expected to consist of highly anisotropic fluctuations with frequencies small compared to the ion cyclotron frequency. For a number of applications, the small scales are also collisionless, so a kinetic treatment of the turbulence is necessary. We show that this anisotropic turbulence is well described by a low-frequency expansion of the kinetic theory called gyrokinetics. This paper is the first in a series to examine turbulent astrophysical plasmas in the gyrokinetic limit. We derive and explain the nonlinear gyrokinetic equations and explore the linear properties of gyrokinetics as a prelude to nonlinear simulations. The linear dispersion relation for gyrokinetics is obtained, and its solutions are compared to those of hot-plasma kinetic theory. These results are used to validate the performance of the gyrokinetic simulation code GS2 in the parameter regimes relevant for astrophysical plasmas. New results on global energy conservation in gyrokinetics are also derived. We briefly outline several of the problems to be addressed by future nonlinear simulations, including particle heating by turbulence in hot accretion flows and in the solar wind, the magnetic and electric field power spectra in the solar wind, and the origin of small-scale density fluctuations in the interstellar medium.

Gyrofluid turbulence models with kinetic effects

Nonlinear gyrofluid equations are derived by taking moments of the nonlinear, electrostatic gyrokinetic equation. The principal model presented includes evolution equations for the guiding center n, u∥, T∥, and T⊥ along with an equation expressing the quasineutrality constraint. Additional evolution equations for higher moments are derived that may be used if greater accuracy is desired. The moment hierarchy is closed with a Landau damping model [G. W. Hammett and F. W. Perkins, Phys. Rev. Lett. 64, 3019 (1990)], which is equivalent to a multipole approximation to the plasma dispersion function, extended to include finite Larmor radius effects (FLR). In particular, new dissipative, nonlinear terms are found that model the perpendicular phase mixing of the distribution function along contours of constant electrostatic potential. These ‘‘FLR phase‐mixing’’ terms introduce a hyperviscositylike damping ∝k⊥2‖Φkk×k’‖, which should provide a physics‐based damping mechanism at high k⊥ρ which is potentially as important as the usual polarization drift nonlinearity. The moments are taken in guiding center space to pick up the correct nonlinear FLR terms and the gyroaveraging of the shear. The equations are solved with a nonlinear, three‐dimensional initial value code. Linear results are presented, showing excellent agreement with linear gyrokinetic theory.

Field-aligned coordinates for nonlinear simulations of tokamak turbulence

Turbulence in tokamaks is characterized by long parallel wavelengths and short perpendicular wavelengths. A coordinate system for nonlinear fluid, gyrokinetic ‘‘Vlasov,’’ or particle simulations is presented that exploits the elongated nature of the turbulence by resolving the minimum necessary simulation volume: a long thin twisting flux tube. It is very similar to the ballooning representation, although periodicity constraints can be incorporated in a manner that allows E×B nonlinearities to be evaluated efficiently with fast Fourier transforms (FFT’s). If the parallel correlation length is very long, however, enforcing periodicity can introduce artificial correlations, so periodicity should not necessarily be enforced in the poloidal angle at θ=±π. This method is applied to high resolution three‐dimensional simulations of toroidal ion temperature gradient (ITG) driven turbulence, which predict fluctuation spectra and ion heat transport similar to experimental measurements.

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.

Quantitative predictions of tokamak energy confinement from first-principles simulations with kinetic effects

A first‐principles model of anomalous thermal transport based on numerical simulations is presented, with stringent comparisons to experimental data from the Tokamak Fusion Test Reactor (TFTR) [Fusion Technol. 21, 1324 (1992)]. This model is based on nonlinear gyrofluid simulations, which predict the fluctuation and thermal transport characteristics of toroidal ion‐temperature‐gradient‐driven (ITG) turbulence, and on comprehensive linear gyrokinetic ballooning calculations, which provide very accurate growth rates, critical temperature gradients, and a quasilinear estimate of χe/χi. The model is derived solely from the simulation results. More than 70 TFTR low confinement (L‐mode) discharges have been simulated with quantitative success. Typically, the ion and electron temperature profiles are predicted within the error bars, and the global energy confinement time within ±10%. The measured temperatures at r/a≂0.8 are used as a boundary condition to predict the temperature profiles in the main confinement ...

Landau fluid models of collisionless magnetohydrodynamics

A closed set of fluid moment equations including models of kinetic Landau damping is developed which describes the evolution of collisionless plasmas in the magnetohydrodynamic parameter regime. The model is fully electromagnetic and describes the dynamics of both compressional and shear Alfven waves, as well as ion acoustic waves. The model allows for separate parallel and perpendicular pressures p∥ and p⊥, and, unlike previous models such as the Chew–Goldberger–Low theory, correctly predicts the instability threshold for the mirror instability. Both a simple 3+1 moment model and a more accurate 4+2 moment model are developed, and both could be useful for numerical simulations of astrophysical and fusion plasmas.

Shearing rate of time-dependent E×B flow

Theory of E×B shear suppression of turbulence in toroidal geometry [Phys. Plasmas 2, 1648 (1995)] is extended to include fast time variations of the E×B flows often observed in nonlinear simulations of tokamak turbulence. It is shown that the quickly time varying components of the E×B flows, while they typically contribute significantly to the instantaneous E×B shearing rate, are less effective than the slowly time varying components in suppressing turbulence. This is because the shear flow pattern changes before eddies get distorted enough. The effective E×B shearing rate capturing this important physics is analytically derived and estimated from zonal flow statistics of gyrofluid simulation. This provides new insights into understanding recent gyrofluid and gyrokinetic simulations that yield a reduced, but not completely quenched, level of turbulence in the presence of turbulence-driven zonal flows.