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It is shown that the magnetic-field coils of a stellarator can, at least in principle, be substantially simplified by the use of permanent magnets. Such magnets cannot create toroidal magnetic flux but they can be used to shape the plasma and thus to create poloidal flux and rotational transform, thereby easing the requirements on the magnetic-field coils. As an example, a quasiaxisymmetric stellarator configuration is constructed with only 8 circular coils (all identical) and permanent magnets.

This white paper, submitted for the Plasma 2020 Decadal Survey, concerns the physics of weakly collisional, high-beta plasmas -- plasmas in which the thermal pressure dominates over the magnetic pressure and in which the inter-particle collision time is comparable to the characteristic timescales of bulk motions. This state of matter, although widespread in the Universe, remains poorly understood: we lack a predictive theory for how it responds to perturbations, how it transports momentum and energy, and how it generates and amplifies magnetic fields. Such topics are foundational to the scientific study of plasmas, and are of intrinsic interest to those who regard plasma physics as a fundamental physics discipline. But these topics are also of extrinsic interest: addressing them directly informs upon our understanding of a wide variety of space and astrophysical systems, including accretion flows around supermassive black holes, the intracluster medium (ICM) between galaxies in clusters, and regions of the near-Earth solar wind. Specific recommendations to advance this field of study are discussed., Comment: 6 pages, submitted for the Plasma 2020 Decadal Survey

Martin David Kruskal was one of the most versatile theoretical physicists of his generation and is distinguished for his enduring work in several different areas, most notably plasma physics, a memorable detour into relativity, and his pioneering work in nonlinear waves. In the latter, together with Norman Zabusky, he invented the concept of the soliton and, with others, developed its application to classes of partial differential equations of physical significance., Comment: 24 pages, 3 figures and 2 tables

Tokamak stability to, potentially explosive, `ballooning' displacements of elliptical magnetic flux tubes is examined in large aspect ratio equilibrium. Above a critical pressure gradient the energy stored in the plasma may be lowered by finite (but not infinitesimal) displacements of such tubes (metastability). Above a higher pressure gradient, the linear stability boundary, such tubes are linearly and nonlinearly unstable. The flux tube displacement can be of the order of the pressure gradient scale length. Plasma transport from displaced flux tubes may result in rapid loss of confinement., Comment: 4 pages, 6 figures

A theoretical framework for low-frequency electromagnetic (drift-)kinetic turbulence in a collisionless, multi-species plasma is presented. The result generalises reduced magnetohydrodynamics (RMHD) and kinetic RMHD (Schekochihin et al. 2009) for pressure-anisotropic plasmas, allowing for species drifts---a situation routinely encountered in the solar wind and presumably ubiquitous in hot dilute astrophysical plasmas (e.g. intracluster medium). Two main objectives are achieved. First, in a non-Maxwellian plasma, the relationships between fluctuating fields (e.g., the Alfven ratio) are order-unity modified compared to the more commonly considered Maxwellian case, and so a quantitative theory is developed to support quantitative measurements now possible in the solar wind. The main physical feature of low-frequency plasma turbulence survives the generalisation to non-Maxwellian distributions: Alfvenic and compressive fluctuations are energetically decoupled, with the latter passively advected by the former; the Alfvenic cascade is fluid, satisfying RMHD equations (with the Alfven speed modified by pressure anisotropy and species drifts), whereas the compressive cascade is kinetic and subject to collisionless damping. Secondly, the organising principle of this turbulence is elucidated in the form of a generalised kinetic free-energy invariant. It is shown that non-Maxwellian features in the distribution function reduce the rate of phase mixing and the efficacy of magnetic stresses; these changes influence the partitioning of free energy amongst the various cascade channels. As the firehose or mirror instability thresholds are approached, the dynamics of the plasma are modified so as to reduce the energetic cost of bending magnetic-field lines or of compressing/rarefying them. Finally, it is shown that this theory can be derived as a long-wavelength limit of non-Maxwellian slab gyrokinetics., Comment: 61 pages, accepted to Journal of Plasma Physics; Abstract abridged

The eruption of multiple flux tubes in a magnetised plasma atmosphere is proposed as a mechanism for explosive release of energy in plasmas. Linearly stable isolated flux tubes are shown to be metastable in a box model magnetised atmosphere in which ends of the field lines are embedded in conducting walls. The energy released by destabilising such field lines can be a significant fraction of the gravitational energy stored in the system. This energy can be released in a fast dynamical time., Comment: 40 pages, 18 figures

Slow dynamical changes in magnetic-field strength and invariance of the particles' magnetic moments generate ubiquitous pressure anisotropies in weakly collisional, magnetized astrophysical plasmas. This renders them unstable to fast, small-scale mirror and firehose instabilities, which are capable of exerting feedback on the macroscale dynamics of the system. By way of a new asymptotic theory of the early nonlinear evolution of the mirror instability in a plasma subject to slow shearing or compression, we show that the instability does not saturate quasilinearly at a steady, low-amplitude level. Instead, the trapping of particles in small-scale mirrors leads to nonlinear secular growth of magnetic perturbations, $\delta B/B \propto t^{2/3}$. Our theory explains recent collisionless simulation results, provides a prediction of the mirror evolution in weakly collisional plasmas and establishes a foundation for a theory of nonlinear mirror dynamics with trapping, valid up to $\delta B/B =O(1)$., Comment: 5 pages, submitted

Calculations of tearing mode stability in tokamaks split conveniently into an external region, where marginally stable ideal MHD is applicable, and a resonant layer around the rational surface where sophisticated kinetic physics is needed. These two regions are coupled by the stability parameter. Pressure and current perturbations localized around the rational surface alter the stability of tearing modes. Equations governing the changes in the external solution and - are derived for arbitrary perturbations in axisymmetric toroidal geometry. The relationship of - with and without pressure flattening is obtained analytically for four pressure flattening functions. Resistive MHD codes do not contain the appropriate layer physics and therefore cannot predict stability directly. They can, however, be used to calculate -. Existing methods (Ham et al. 2012 Plasma Phys. Control. Fusion 54 025009) for extracting - from resistive codes are unsatisfactory when there is a finite pressure gradient at the rational surface and favourable average curvature because of the Glasser stabilizing effect. To overcome this difficulty we introduce a specific pressure flattening function that allows the earlier approach to be used. The technique is first tested numerically in cylindrical geometry with an artificial favourable curvature. Its application to toroidal geometry is then demonstrated using the toroidal tokamak tearing mode stability code T7 (Fitzpatrick et al. 1993 Nucl. Fusion 33 1533) which uses an approximate analytic equilibrium. The prospects for applying this approach to resistive MHD codes such as MARS-F (Liu et al. 2000 Phys. Plasmas 7 3681) which utilize a fully toroidal equilibrium are discussed., Comment: 26 pages, 5 figures. This is an author-created, un-copyedited version of an article submitted for publication in Plasma Physics & Controlled Fusion. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it

We have extended our study of the competition between the drive and stabilization of plasma microinstabilities by sheared flow to include electromagnetic effects at low plasma $\beta$ (the ratio of plasma to magnetic pressure). The extended system of characteristic equations is formulated, for a dissipative fluid model developed from the gyrokinetic equation, using a twisting mode representation in sheared slab geometry and focusing on the ion temperature gradient mode. Perpendicular flow shear convects perturbations along the field at the speed we denote as $Mc_s$ (where $c_s$ is the sound speed). $M > 1/ \sqrt{\beta}$ is required to make the system characteristics unidirectional and inhibit eigenmode formation, leaving only transitory perturbations in the system. This typically represents a much larger flow shear than in the electrostatic case, which only needs $M>1$. Numerical investigation of the region $M < 1/\sqrt{\beta}$ shows the driving terms can conflict, as in the electrostatic case, giving low growth rates over a range of parameters. Also, at modest drive strengths and low $\beta$ values typical of experiments, including electromagnetic effects does not significantly alter the growth rates. For stronger flow shear and higher $\beta$, geometry characteristic of the spherical tokamak mitigates the effect of an instability of the shear Alfv\'{e}n wave, driven by the parallel flow shear., Comment: 23 pages, 12 figures

In this paper, we extend the multiscale approach developed in [Abel et. al., Rep. Prog. Phys., submitted] by exploiting the scale separation between ions and the electrons. The gyrokinetic equation is expanded in powers of the electron to ion mass ratio, which provides a rigorous method for deriving the reduced electron model. We prove that ion-scale electromagnetic turbulence cannot change the magnetic topology, and argue that to lowest order the magnetic field lies on fluctuating flux surfaces. These flux surfaces are used to construct magnetic coordinates, and in these coordinates a closed system of equations for the electron response to ion-scale turbulence is derived. All fast electron timescales have been eliminated from these equations. We also use these magnetic surfaces to construct transport equations for electrons and for electron heat in terms of the reduced electron model., Comment: 62 pages, 1 figure

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 figures

Beam Emission Spectroscopy (BES) measurements of ion-scale density fluctuations in the MAST tokamak are used to show that the turbulence correlation time, the drift time associated with ion temperature or density gradients, the particle (ion) streaming time along the magnetic field and the magnetic drift time are consistently comparable, suggesting a "critically balanced" turbulence determined by the local equilibrium. The resulting scalings of the poloidal and radial correlation lengths are derived and tested. The nonlinear time inferred from the density fluctuations is longer than the other times; its ratio to the correlation time scales as $\nu_{*i}^{-0.8\pm0.1}$, where $\nu_{*i}=$ ion collision rate/streaming rate. This is consistent with turbulent decorrelation being controlled by a zonal component, invisible to the BES, with an amplitude exceeding the drift waves' by $\sim \nu_{*i}^{-0.8}$., Comment: 6 pages, 4 figures, submitted to PRL

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 PRL

Differential rotation is known to suppress linear instabilities in fusion plasmas. However, even in the absence of growing eigenmodes, subcritical fluctuations that grow transiently can lead to sustained turbulence. Here transient growth of electrostatic fluctuations driven by the parallel velocity gradient (PVG) and the ion temperature gradient (ITG) in the presence of a perpendicular ExB velocity shear is considered. The maximally simplified case of zero magnetic shear is treated in the framework of a local shearing box. There are no linearly growing eigenmodes, so all excitations are transient. The maximal amplification factor of initial perturbations and the corresponding wavenumbers are calculated as functions of q/\epsilon (=safety factor/aspect ratio), temperature gradient and velocity shear. Analytical results are corroborated and supplemented by linear gyrokinetic numerical tests. For sufficiently low values of q/\epsilon (<7 in our model), regimes with fully suppressed ion-scale turbulence are possible. For cases when turbulence is not suppressed, an elementary heuristic theory of subcritical PVG turbulence leading to a scaling of the associated ion heat flux with q, \epsilon, velocity shear and temperature gradient is proposed; it is argued that the transport is much less stiff than in the ITG regime., Comment: 36 pages in IOP latex style; 12 figures; submitted to PPCF

Recent gyrokinetic stability calculations have revealed that the spherical tokamak is susceptible to tearing parity instabilities with length scales of a few ion Larmor radii perpendicular to the magnetic field lines. Here we investigate this 'micro-tearing' mode in greater detail to uncover its key characteristics, and compare it with existing theoretical models of the phenomenon. This has been accomplished using a full numerical solution of the linear gyrokinetic-Maxwell equations. Importantly, the instability is found to be driven by the free energy in the electron temperature gradient as described in the literature. However, our calculations suggest it is not substantially affected by either of the destabilising mechanisms proposed in previous theoretical models. Instead the instability is destabilised by interactions with magnetic drifts, and the electrostatic potential. Further calculations reveal that the mode is not significantly destabilised by the flux surface shaping or the large trapped particle fraction present in the spherical tokamak. Its prevalence in spherical tokamak plasmas is primarily due to the higher value of plasma {\beta}, and the enhanced magnetic drifts due to the smaller radius of curvature., Comment: 12 pages, 12 figures

First-principles numerical simulations are used to describe a transport bifurcation in a differentially rotating tokamak plasma. Such a bifurcation is more probable in a region of zero magnetic shear than one of finite magnetic shear because in the former case the component of the sheared toroidal flow that is perpendicular to the magnetic field has the strongest suppressing effect on the turbulence. In the zero-magnetic-shear regime, there are no growing linear eigenmodes at any finite value of flow shear. However, subcritical turbulence can be sustained, owing to the transient growth of modes driven by the ion temperature gradient (ITG) and the parallel velocity gradient (PVG). Nonetheless, in a parameter space containing a wide range of temperature gradients and velocity shears, there is a sizeable window where all turbulence is suppressed. Combined with the relatively low transport of momentum by collisional (neoclassical) mechanisms, this produces the conditions for a bifurcation from low to high temperature and velocity gradients. The path of this bifurcation is mapped out using interpolation from a large number of simulations. Numerical simulations are also used to construct a parametric model which accurately describes the combined effect of the temperature gradient and the flow gradient over a wide range of their values. Using this parametric model, it is shown that in this reduced-transport state, heat is transported almost neoclassically, while momentum transport is dominated by subcritical PVG turbulence. It is further shown that for any given input of torque, there is an optimum input of heat which maximises the temperature gradient. The parametric model describes both the behaviour of the subcritical turbulence and the complicated effect of the flow shear on the transport stiffness. It may prove useful for transport modelling of tokamaks with sheared flows., Comment: 17 pages, 13 figures, PoP in press

The effect of momentum injection on the temperature gradient in tokamak plasmas is studied. A plausible scenario for transitions to reduced transport regimes is proposed. The transition happens when there is sufficient momentum input so that the velocity shear can suppress or reduce the turbulence. However, it is possible to drive too much velocity shear and rekindle the turbulent transport. The optimal level of momentum injection is determined. The reduction in transport is maximized in the regions of low or zero magnetic shear., Comment: 4 pages, 5 figures

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 PRL

Nonlinear gyrokinetic simulations have been conducted to investigate turbulent transport in tokamak plasmas with rotational shear. At sufficiently large flow shears, linear instabilities are suppressed, but transiently growing modes drive subcritical turbulence whose amplitude increases with flow shear. This leads to a local minimum in the heat flux, indicating an optimal E x B shear value for plasma confinement. Local maxima in the momentum fluxes are also observed, allowing for the possibility of bifurcations in the E x B shear. The sensitive dependence of heat flux on temperature gradient is relaxed for large flow shear values, with the critical temperature gradient increasing at lower flow shear values. The turbulent Prandtl number is found to be largely independent of temperature and flow gradients, with a value close to unity., Comment: 4 pages, 5 figures, submitted to PRL

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 references

We consider the problem of self-regulated heating and cooling in galaxy clusters and the implications for cluster magnetic fields and turbulence. Viscous heating of a weakly collisional magnetised plasma is regulated by the pressure anisotropy with respect to the local direction of the magnetic field. The intracluster medium is a high-beta plasma, where pressure anisotropies caused by the turbulent stresses and the consequent local changes in the magnetic field will trigger very fast microscale instabilities. We argue that the net effect of these instabilities will be to pin the pressure anisotropies at a marginal level, controlled by the plasma beta parameter. This gives rise to local heating rates that turn out to be comparable to the radiative cooling rates. Furthermore, we show that a balance between this heating and Bremsstrahlung cooling is thermally stable, unlike the often conjectured balance between cooling and thermal conduction. Given a sufficient (and probably self-regulating) supply of turbulent power, this provides a physical mechanism for mitigating cooling flows and preventing cluster core collapse. For observed density and temperature profiles, the assumed balance of viscous heating and radiative cooling allows us to predict magnetic-field strengths, turbulent velocities and turbulence scales as functions of distance from the centre. Specific predictions and comparisons with observations are given for several different clusters. Our predictions can be further tested by future observations of cluster magnetic fields and turbulent velocities., Comment: 12 pages, 4 figures, 1 table; substantial revision with essential new content

Weakly collisional plasmas dynamically develop pressure anisotropies with respect to the magnetic field. These anisotropies trigger plasma instabilities at scales just above the ion Larmor radius \rho_i and much below the mean free path \lambda_{mfp}. They have growth rates of a fraction of the ion cyclotron frequency - much faster than either the global dynamics or local turbulence. The instabilities dramatically modify the transport properties and, therefore, the macroscopic dynamics of the plasma. Their nonlinear evolution drives pressure anisotropies towards marginal stability, controlled by the plasma beta \beta_i. Here this nonlinear evolution is worked out for the simplest analytically tractable example - the parallel firehose instability. In the nonlinear regime, both analytical theory and the numerical solution predict secular growth of magnetic fluctuations. They develop a k^{-3} spectrum, extending from scales somewhat larger than \rho_i to the maximum scale that grows secularly with time (~t^{1/2}); the relative pressure anisotropy (\pperp-\ppar)/\ppar tends to the marginal value -2/\beta_i. The marginal state is achieved via changes in the magnetic field, not particle scattering. When a parallel ion heat flux is present, the firehose mutates into the new gyrothermal instability (GTI), which continues to exist up to firehose-stable values of pressure anisotropy, which can be positive and are limited by the heat flux. The nonlinear evolution of the GTI also features secular growth of magnetic fluctuations, but the spectrum is eventually dominated by modes around the scale ~\rho_i l_T/\lambda_{mfp}, where l_T is the scale of the parallel temperature variation. Implications for momentum and heat transport are speculated about. This study is motivated by the dynamics of galaxy cluster plasmas., Comment: 34 pages, replaced with the version published in MNRAS

Both global dynamics and turbulence in magnetized weakly collisional cosmic plasmas are described by general magnetofluid equations that contain pressure anisotropies and heat fluxes that must be calculated from microscopic plasma kinetic theory. It is shown that even without a detailed calculation of the pressure anisotropy or the heat fluxes, one finds the macroscale dynamics to be generically unstable to microscale Alfvenically polarized fluctuations. Two instabilities are considered in detail: the parallel firehose instability (including the finite-Larmor-radius effects that determine the fastest growing mode) and the gyrothermal instability (GTI). The latter is a new result - it is shown that a parallel ion heat flux destabilizes Alfvenically polarized fluctuations even in the absence of the negative pressure anisotropy required for the firehose. The main conclusion is that both pressure anisotropies and heat fluxes trigger plasma microinstabilities and, therefore, their values will likely be set by the nonlinear evolution of these instabilities. Ideas for understanding this nonlinear evolution are discussed. It is argued that cosmic plasmas will generically be "three-scale systems," comprising global dynamics, mesoscale turbulence and microscale plasma fluctuations. The astrophysical example of cool cores of galaxy clusters is considered and it is noted that observations point to turbulence in clusters being in a marginal state with respect to plasma microinstabilities and so it is the plasma microphysics that is likely to set the heating and conduction properties of the intracluster medium. In particular, a lower bound on the scale of temperature fluctuations implied by the GTI is derived., Comment: 10 pages, MNRAS tex style, 1 figure

Two-dimensional numerical simulations of the effect of background turbulence on 2D resistive magnetic reconnection are presented. For sufficiently small values of the resistivity ($\eta$) and moderate values of the turbulent power ($\epsilon$), the reconnection rate is found to have a much weaker dependence on $\eta$ than the Sweet-Parker scaling of $\eta^{1/2}$ and is even consistent with an $\eta-$independent value. For a given value of $\eta$, the dependence of the reconnection rate on the turbulent power exhibits a critical threshold in $\epsilon$ above which the reconnection rate is significantly enhanced., Comment: Accepted to MNRAS

Two-dimensional gyrokinetics is a simple paradigm for the study of kinetic magnetised plasma turbulence. In this paper, we present a comprehensive theoretical framework for this turbulence. We study both the inverse and direct cascades (the `dual cascade'), driven by a homogeneous and isotropic random forcing. The key characteristic length of gyrokinetics, the Larmor radius, divides scales into two physically distinct ranges. For scales larger than the Larmor radius, we derive the familiar Charney--Hasegawa--Mima (CHM) equation from the gyrokinetic system, and explain its relationship to gyrokinetics. At scales smaller than the Larmor radius, a dual cascade occurs in phase space (two dimensions in position space plus one dimension in velocity space) via a nonlinear phase-mixing process. We show that at these sub-Larmor scales, the turbulence is self-similar and exhibits power law spectra in position and velocity space. We propose a Hankel-transform formalism to characterise velocity-space spectra. We derive the exact relations for third-order structure functions, analogous to Kolmogorov's four-fifths and Yaglom's four-thirds laws and valid at both long and short wavelengths. We show how the general gyrokinetic invariants are related to the particular invariants that control the dual cascade in the long- and short-wavelength limits. We describe the full range of cascades from the fluid to the fully kinetic range., Comment: Being considered for publication in the Journal of Fluid Mechanics. Version 3 contains more cross referencing including forward referencing, to make the paper easier to read. Final pre-production version

A detailed numerical study of magnetic reconnection in resistive MHD for very large, previously inaccessible, Lundquist numbers ($10^4\le S\le 10^8$) is reported. Large-aspect-ratio Sweet-Parker current sheets are shown to be unstable to super-Alfv\'enically fast formation of plasmoid (magnetic-island) chains. The plasmoid number scales as $S^{3/8}$ and the instability growth rate in the linear stage as $S^{1/4}$, in agreement with the theory by Loureiro et al. [Phys. Plasmas {\bf 14}, 100703 (2007)]. In the nonlinear regime, plasmoids continue to grow faster than they are ejected and completely disrupt the reconnection layer. These results suggest that high-Lundquist-number reconnection is inherently time-dependent and hence call for a substantial revision of the standard Sweet-Parker quasi-stationary picture for $S>10^4$., Comment: submitted to Phys. Rev. Lett

Howes et al. Reply to Comment on "Kinetic Simulations of Magnetized Turbulence in Astrophysical Plasmas" arXiv:0711.4355, Comment: 1 page Reply to Matthaeus et al. Comment (PRL, 101, 149501, 2008) on our PRL 100, 065004 (2008) [ arXiv:0711.4355 ]

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 version

A new analytically and numerically manageable model collision operator is developed specifically for turbulence simulations. The like-particle collision operator includes both pitch-angle scattering and energy diffusion and satisfies the physical constraints required for collision operators: it conserves particles, momentum and energy, obeys Boltzmann's H-theorem (collisions cannot decrease entropy), vanishes on a Maxwellian, and efficiently dissipates small-scale structure in the velocity space. The process of transforming this collision operator into the gyroaveraged form for use in gyrokinetic simulations is detailed. The gyroaveraged model operator is shown to have more suitable behavior at small scales in phase space than previously suggested models. A model operator for electron-ion collisions is also presented., Comment: revtex, 12 pages

Numerical simulations of forced turbulence in elongated shearing boxes are carried out to demonstrate that a nonhelical turbulence in conjunction with a linear shear can give rise to a mean-field dynamo. Exponential growth of magnetic field at scales larger than the outer (forcing) scale of the turbulence is found. Over a range of values of the shearing rate S spanning approximately two orders of magnitude, the growth rate of the magnetic field is proportional to the imposed shear, gamma ~ S, while the characteristic spatial scale of the field is l_b ~ S^(-1/2). The effect is quite general: earlier results for the nonrotating case by Yousef et al. 2008 (PRL 100, 184501) are extended to shearing boxes with Keplerian rotation; it is also shown that the shear dynamo mechanism operates both below and above the threshold for the fluctuation dynamo. The apparently generic nature of the shear dynamo effect makes it an attractive object of study for the purpose of understanding the generation of magnetic fields in astrophysical systems., Comment: 13 pages, 10 figures, version published in Astronomische Nachrichten

This paper describes a conceptual framework for understanding kinetic plasma turbulence as a generalized form of energy cascade in phase space. It is emphasized that conversion of turbulent energy into thermodynamic heat is only achievable in the presence of some (however small) degree of collisionality. The smallness of the collision rate is compensated by the emergence of small-scale structure in the velocity space. For gyrokinetic turbulence, a nonlinear perpendicular phase mixing mechanism is identified and described as a turbulent cascade of entropy fluctuations simultaneously occurring at spatial scales smaller than the ion gyroscale and in velocity space. Scaling relations for the resulting fluctuation spectra are derived. An estimate for the collisional cutoff is provided. The importance of adequately modeling and resolving collisions in gyrokinetic simulations is biefly discussed, as well as the relevance of these results to understanding the dissipation-range turbulence in the solar wind and the electrostatic microturbulence in fusion plasmas., Comment: iop revtex style, 14 pages, 1 figure; submitted to PPCF; invited talk for EPS Conference on Plasma Physics, Crete, June 2008; Replaced to match published version

A new approach to high pressure magnetically-confined plasmas is necessary to design efficient fusion devices. This paper presents an equilibrium combining two solutions of the Grad-Shafranov equation, which describes the magnetohydrodynamic equilibrium in toroidal geometry. The outer equilibrium is paramagnetic and confines the inner equilibrium, whose strong diamagnetism permits to balance large pressure gradients. The existence of both equilibria in the same volume yields a dual equilibrium structure. Their combination also improves free-boundary mode stability.

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 Letters

The feasibility of a mean-field dynamo in nonhelical turbulence with superimposed linear shear is studied numerically in elongated shearing boxes. Exponential growth of magnetic field at scales much larger than the outer scale of the turbulence is found. The charateristic scale of the field is l_B ~ S^{-1/2} and growth rate is gamma ~ S, where S is the shearing rate. This newly discovered shear dynamo effect potentially represents a very generic mechanism for generating large-scale magnetic fields in a broad class of astrophysical systems with spatially coherent mean flows., Comment: 4 pages, 5 figures; replaced with revised version that matches the published PRL

In turbulent high-beta astrophysical plasmas (exemplified by the galaxy cluster plasmas), pressure-anisotropy-driven firehose and mirror fluctuations grow nonlinearly to large amplitudes, dB/B ~ 1, on a timescale comparable to the turnover time of the turbulent motions. The principle of their nonlinear evolution is to generate secularly growing small-scale magnetic fluctuations that on average cancel the temporal change in the large-scale magnetic field responsible for the pressure anisotropies. The presence of small-scale magnetic fluctuations may dramatically affect the transport properties and, thereby, the large-scale dynamics of the high-beta astrophysical plasmas., Comment: revtex, 4 pages, 1 figure; replaced to match published version

This paper is a detailed report on a programme of simulations used to settle a long-standing issue in the dynamo theory and demonstrate that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm>>1 and small magnetic Prandtl number Pm<<1. The dependence of the critical Rm_c vs. the hydrodynamic Reynolds number Re is obtained for 11. The stability curve Rm_c(Re) (and, it is argued, the nature of the dynamo) is substantially different from the case of the simulations and liquid-metal experiments with a mean flow. It is not as yet possible to determine numerically whether the growth rate is ~Rm^{1/2} in the limit Re>>Rm>>1, as should be the case if the dynamo is driven by the inertial-range motions. The magnetic-energy spectrum in the low-Pm regime is qualitatively different from the Pm>1 case and appears to develop a negative spectral slope, although current resolutions are insufficient to determine its asymptotic form. At 1

We present a theoretical framework for plasma turbulence in astrophysical plasmas (solar wind, interstellar medium, galaxy clusters, accretion disks). The key assumptions are that the turbulence is anisotropic with respect to the mean magnetic field and frequencies are low compared to the ion cyclotron frequency. The energy injected at the outer scale scale has to be converted into heat, which ultimately cannot be done without collisions. A KINETIC CASCADE develops that brings the energy to collisional scales both in space and velocity. Its nature depends on the physics of plasma fluctuations. In each of the physically distinct scale ranges, the kinetic problem is systematically reduced to a more tractable set of equations. In the "inertial range" above the ion gyroscale, the kinetic cascade splits into a cascade of Alfvenic fluctuations, which are governed by the RMHD equations at both the collisional and collisionless scales, and a passive cascade of compressive fluctuations, which obey a linear kinetic equation along the moving field lines associated with the Alfvenic component. In the "dissipation range" between the ion and electron gyroscales, there are again two cascades: the kinetic-Alfven-wave (KAW) cascade governed by two fluid-like Electron RMHD equations and a passive phase-space cascade of ion entropy fluctuations. The latter cascade brings the energy of the inertial-range fluctuations that was damped by collisionless wave-particle interaction 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 these cascades. Astrophysical and space-physical applications are discussed in detail., Comment: aastex using emulateapj, 65 pages, 10 figures; replaced to match published version

Current sheets formed in magnetic reconnection events are found to be unstable to high-wavenumber perturbations. The instability is very fast: its maximum growth rate scales as S^{1/4} v_A/L, where L is the length of the sheet, v_A the Alfven speed and S the Lundquist number. As a result, a chain of plasmoids (secondary islands) is formed, whose number scales as S^{3/8}., Comment: revtex, 4 pages, 4 figures

Direct numerical simulations of incompressible nonhelical randomly forced MHD turbulence are used to demonstrate for the first time that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm>>1 and small magnetic Prandtl number Pm<<1. The dependence of the critical Rm_c for dynamo on the hydrodynamic Reynolds number Re is obtained for 16000 compared to Rm_c~60 for Pm>=1. Is is not as yet possible to determine numerically whether the growth rate of the magnetic energy is ~Rm^{1/2} in the limit Rm->infinity, as it should if the dynamo is driven by the inertial-range motions at the resistive scale., Comment: revtex, 4 pages, 4 figures; final published version: minor edits, slightly improved numerical results

Theoretical approaches to low-frequency magnetized turbulence in collisionless and weakly collisional astrophysical plasmas are reviewed. The proper starting point for an analytical description of these plasmas is kinetic theory, not fluid equations. The anisotropy of the turbulence is used to systematically derive a series of reduced analytical models. Above the ion gyroscale, it is shown rigourously that the Alfven waves decouple from the electron-density and magnetic-field-strength fluctuations and satisfy the Reduced MHD equations. The density and field-strength fluctuations (slow waves and the entropy mode in the fluid limit), determined kinetically, are passively mixed by the Alfven waves. The resulting hybrid fluid-kinetic description of the low-frequency turbulence is valid independently of collisionality. Below the ion gyroscale, the turbulent cascade is partially converted into a cascade of kinetic Alfven waves, damped at the electron gyroscale. This cascade is described by a pair of fluid-like equations, which are a reduced version of the Electron MHD. The development of these theoretical models is motivated by observations of the turbulence in the solar wind and interstellar medium. In the latter case, the turbulence is spatially inhomogeneous and the anisotropic Alfvenic turbulence in the presence of a strong mean field may coexist with isotropic MHD turbulence that has no mean field., Comment: IOP latex, 16 pages, 1 figure; invited topical review at International Congress on Plasma Physics, Kiev 22-26 May 2006; accepted for publication in Plasma Phys. Control. Fusion

Observations of galaxy clusters show that the intracluster medium (ICM) is likely to be turbulent and is certainly magnetized. The properties of this magnetized turbulence are determined both by fundamental nonlinear magnetohydrodynamic interactions and by the plasma physics of the ICM, which has very low collisionality. Cluster plasma threaded by weak magnetic fields is subject to firehose and mirror instabilities. These saturate and produce fluctuations at the ion gyroscale, which can scatter particles, increasing the effective collision rate and, therefore, the effective Reynolds number of the ICM. A simple way to model this effect is proposed. The model yields a self-accelerating fluctuation dynamo whereby the field grows explosively fast, reaching the observed, dynamically important, field strength in a fraction of the cluster lifetime independent of the exact strength of the seed field. It is suggested that the saturated state of the cluster turbulence is a combination of the conventional isotropic magnetohydrodynamic turbulence, characterized by folded, direction-reversing magnetic fields and an Alfv\'en-wave cascade at collisionless scales. An argument is proposed to constrain the reversal scale of the folded field. The picture that emerges appears to be in qualitative agreement with observations of magnetic fields in clusters., Comment: revtex, 9 pages, 5 figures; invited talk for the 47th APS DPP Meeting, Denver, CO, Oct 2005; minor corrections to match the published version

We propose a model of magnetic-field growth in galaxy clusters whereby the field is amplified by a factor of about 10^8 over a cosmologically short time of ~10^8 yr. Our model is based on the idea that the viscosity of the intracluster medium during the field-amplification epoch is determined not by particle collisions but by plasma microinstabilities: these give rise to small-scale fluctuations, which scatter particles, increasing their effective collision rate and, therefore, the effective Reynolds number. This gives rise to a bootstrap effect as the growth of the field triggers the instabilities which increase the Reynolds number which, in turn, accelerates the growth of the field. The growth is explosive and the result is that the observed field strength is reached over a fraction of the cluster lifetime independent of the exact strength of the seed field (which only needs to be above ~10^{-15} G to trigger the explosive growth)., Comment: latex (AN style), 5 pages, 2 figures

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 ~1/Delta'. A current sheet is formed and the reconnection is exponential in time with a growth rate ~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 ~Delta' and quantitatively agrees with the theoretical prediction. If Delta' is large enough for the X-point collapse to have occured, the saturation amplitude increases noticeably and becomes independent of Delta'., Comment: revtex4, 4 pages, 18 figures

We discuss the current understanding of the most basic properties of astrophysical MHD turbulence and trace the origins of the modern views and theoretical uncertainties to the ideas set forth in 1950s and 1960s by Iroshnikov, Kraichnan, Batchelor, Schlueter and Biermann. Universal aspects of the theory are emphasised. Two main astrophysical applications are touched upon: turbulence in the solar wind and in clusters of galaxies. These are, in a certain (very approximate) sense, two ``pure'' cases of small-scale turbulence, where theoretical models of the two main regimes of MHD turbulence -- with and without a strong mean magnetic field -- can be put to the test. They are also good examples of a complication that is more or less generic in astrophysical plasmas: the MHD description is, in fact, insufficient for astrophysical turbulence and plasma physics must make an entrance., Comment: latex, 30 pages, 7 figures. Minor corrections. Version with better-resolved Figs. 1, 4, 6 is available from http://www.damtp.cam.ac.uk/user/as629/ or by e-mail upon request. Invited chapter in "Magnetohydrodynamics: Historical Evolution and Trends", S. Molokov et al., Eds, Springer (2007), pp. 85-115

We show that under very general conditions, cluster plasmas threaded by weak magnetic fields are subject to very fast growing plasma instabilities driven by the anisotropy of the plasma pressure (viscous stress) with respect to the local direction of the magnetic field. Such an anisotropy will naturally arise in any weakly magnetized plasma that has low collisionality and is subject to stirring. The magnetic field must be sufficiently weak for the instabilities to occur, viz., beta>Re^{1/2}. The instabilities are captured by the extended MHD model with Braginskii viscosity. However, their growth rates are proportional to the wavenumber down to the ion gyroscale, so MHD equations with Braginskii viscosity are not well posed and a fully kinetic treatment is necessary. The instabilities can lead to magnetic fields in clusters being amplified from seed strength of ~10^{-18} G to dynamically important strengths of ~10 microG on cosmologically trivial time scales (~10^8 yr). The fields produced during the amplification stage are at scales much smaller than observed. Predicting the saturated field scale and structure will require a kinetic theory of magnetized cluster turbulence., Comment: aastex with emulateapj5; 4 pages; minor corrections to match final published version

We study numerically the dependence of the critical magnetic Reynolds number Rmc for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror--symmetric. We are interested in the regime of low magnetic Prandtl number Pm=Rm/Re<1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rmc->const as Re->infinity (a small-scale dynamo exists at low Pm) or Rmc/Re=Pmc->const as Re->infinity (no small-scale dynamo exists at low Pm). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, Rmc grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rmc~500 for Re~3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity and Smagorinsky large-eddy viscosity, we find that Rmc is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. 2004, PRL 92, 054502 and Haugen et al. 2004, PRE, 70, 016308 and the first detailed scan of the numerically accessible part of the stability curve Rmc(Re)., Comment: 4 pages, emulateapj aastex, 2 figures; final version as published in ApJL (but with colour figures)

Cluster plasmas are magnetised already at very low magnetic field strength. Low collisionality implies that conservation of the first adiabatic invariant results in an anisotropic viscous stress (Braginskii viscosity) or, equivalently, anisotropic plasma pressure. This triggers firehose and mirror instabilities, which have growth rates proportional to the wavenumber down to scales of the order of ion Larmor radius. This means that MHD equations with Braginskii viscosity are not well posed and fully kinetic description is necessary. In this paper, we review the basic picture of small-scale dynamo in the cluster plasma and attempt to reconcile it with the existence of plasma instabilities at collisionless scales., Comment: latex, 6 pages, 1 figure; Proceedings of "Magnetized Plasma in Galaxy Evolution", Sep. 27th-Oct. 1st 2004, Jagiellonian University, Krakow, Poland, Eds. K. T. Chyzy, R.-J. Dettmar, K. Otmianowska-Mazur and M. Soida

This is a brief account of our numerical study of the tearing mode reconnection. We demonstrate two main points. First, we show that, given sufficiently small resistivity, the Rutherford regime always exists; larger values of Delta' require smaller values of resistivity. Rutherford's negligible-inertia assumption is validated and the asymptotically linear dependence of the time derivative of the island width on the resistivity and Delta' is confirmed. Second, we find that, at large Delta', the Rutherford regime is followed by a nonlinear stage of fast growth linked to X-point collapse and formation of a current sheet. This causes the reconnection to become Sweet-Parke (SP) like. The signature resistivity^{1/2} scaling of the effective island growth rate is, indeed, found in this nonlinear stage. The SP stage culminates in the saturation of the mode, which can, thus, be achieved much faster than via Rutherford regime., Comment: latex, 4 pages, 8 figures; Paper 4-208, Proceedings of the 31st EPS Conference on Plasma Physics, London 28 June - 2 July 2004