# Publications by Alexander Schekochihin

## Fast collisionless reconnection and electron heating in strongly magnetized plasmas

Physical Review Letters **111** (2013)

Magnetic reconnection in strongly magnetized (low-beta), weakly collisional plasmas is investigated by using a novel fluid-kinetic model [Zocco and Schekochihin, Phys. Plasmas 18, 102309 (2011)] which retains nonisothermal electron kinetics. It is shown that electron heating via Landau damping (linear phase mixing) is the dominant dissipation mechanism. In time, electron heating occurs after the peak of the reconnection rate; in space, it is concentrated along the separatrices of the magnetic island. For sufficiently large systems, the peak reconnection rate is ≈0.2vABy,0, where v A is the Alfvén speed based on the reconnecting field B y,0. The island saturation width is the same as in magnetohydrodynamics models except for small systems, when it becomes comparable to the kinetic scales. © 2013 American Physical Society.

## Linear structures in the core of the Coma cluster of galaxies

Science **341** (2013) 1365-1368

The hot x-ray-emitting plasma in galaxy clusters is predicted to have turbulent motion, which can contribute around 10% of the cluster's central energy density. We report deep Chandra X-ray Observatory observations of the Coma cluster core, showing the presence of quasi-linear high-density arms spanning 150 kiloparsecs, consisting of low-entropy material that was probably stripped from merging subclusters. Two appear to be connected with a subgroup of galaxies at a 650-kiloparsec radius that is merging into the cluster, implying coherence over several hundred million years. Such a long lifetime implies that strong isotropic turbulence and conduction are suppressed in the core, despite the unrelaxed state of the cluster. Magnetic fields are presumably responsible. The structures seen in Coma present insight into the past billion years of subcluster merger activity.

## Measurement and physical interpretation of the mean motion of turbulent density patterns detected by the beam emission spectroscopy system on the mega amp spherical tokamak

Plasma Physics and Controlled Fusion **54** (2012)

The mean motion of turbulent patterns detected by a two-dimensional beam emission spectroscopy (BES) diagnostic on the mega amp spherical tokamak (MAST) is determined using a cross-correlation time delay method. Statistical reliability of the method is studied by means of synthetic data analysis. The experimental measurements on MAST indicate that the apparent mean poloidal motion of the turbulent density patterns in the lab frame arises because the longest correlation direction of the patterns (parallel to the local background magnetic fields) is not parallel to the direction of the fastest mean plasma flows (usually toroidal when strong neutral-beam injection is present). This effect is particularly pronounced in a spherical tokamak because of the relatively large mean rotation and large magnetic pitch angle. The experimental measurements are consistent with the mean motion of plasma being toroidal. The sum of all other contributions (mean poloidal plasma flow, phase velocity of the density patterns in the plasma frame, non-linear effects, etc) to the apparent mean poloidal velocity of the density patterns is found to be negligible. These results hold in all investigated L-mode, H-mode and internal transport barrier discharges. The one exception is a high-poloidal-beta (the ratio of the plasma pressure to the poloidal magnetic field energy density) discharge, where a large magnetic island exists. In this case BES detects very little motion. This effect is currently theoretically unexplained. © 2012 IOP Publishing Ltd.

## Weak Alfvén-wave turbulence revisited.

Phys Rev E Stat Nonlin Soft Matter Phys **85** (2012) 036406-

Weak Alfvénic turbulence in a periodic domain is considered as a mixed state of Alfvén waves interacting with the two-dimensional (2D) condensate. Unlike in standard treatments, no spectral continuity between the two is assumed, and, indeed, none is found. If the 2D modes are not directly forced, k(-2) and k(-1) spectra are found for the Alfvén waves and the 2D modes, respectively, with the latter less energetic than the former. The wave number at which their energies become comparable marks the transition to strong turbulence. For imbalanced energy injection, the spectra are similar, and the Elsasser ratio scales as the ratio of the energy fluxes in the counterpropagating Alfvén waves. If the 2D modes are forced, a 2D inverse cascade dominates the dynamics at the largest scales, but at small enough scales, the same weak and then strong regimes as described above are achieved.

## EIDOSCOPE: Particle acceleration at plasma boundaries

Experimental Astronomy **33** (2012) 491-527

We describe the mission concept of how ESA can make a major contribution to the Japanese Canadian multi-spacecraft mission SCOPE by adding one cost-effective spacecraft EIDO (Electron and Ion Dynamics Observatory), which has a comprehensive and optimized plasma payload to address the physics of particle acceleration. The combined mission EIDOSCOPE will distinguish amongst and quantify the governing processes of particle acceleration at several important plasma boundaries and their associated boundary layers: collisionless shocks, plasma jet fronts, thin current sheets and turbulent boundary layers. Particle acceleration and associated cross-scale coupling is one of the key outstanding topics to be addressed in the Plasma Universe. The very important science questions that only the combined EIDOSCOPE mission will be able to tackle are: 1) Quantitatively, what are the processes and efficiencies with which both electrons and ions are selectively injected and subsequently accelerated by collisionless shocks? 2) How does small-scale electron and ion acceleration at jet fronts due to kinetic processes couple simultaneously to large scale acceleration due to fluid (MHD) mechanisms? 3) How does multi-scale coupling govern acceleration mechanisms at electron, ion and fluid scales in thin current sheets? 4) How do particle acceleration processes inside turbulent boundary layers depend on turbulence properties at ion/electron scales? EIDO particle instruments are capable of resolving full 3D particle distribution functions in both thermal and suprathermal regimes and at high enough temporal resolution to resolve the relevant scales even in very dynamic plasma processes. The EIDO spin axis is designed to be sun-pointing, allowing EIDO to carry out the most sensitive electric field measurements ever accomplished in the outer magnetosphere. Combined with a nearby SCOPE Far Daughter satellite, EIDO will form a second pair (in addition to SCOPE Mother-Near Daughter) of closely separated satellites that provides the unique capability to measure the 3D electric field with high accuracy and sensitivity. All EIDO instrumentation are state-of-the-art technology with heritage from many recent missions. The EIDOSCOPE orbit will be close to equatorial with apogee 25-30 RE and perigee 8-10 RE. In the course of one year the orbit will cross all the major plasma boundaries in the outer magnetosphere; bow shock, magnetopause and magnetotail current sheets, jet fronts and turbulent boundary layers. EIDO offers excellent cost/benefits for ESA, as for only a fraction of an M-class mission cost ESA can become an integral part of a major multi-agency L-class level mission that addresses outstanding science questions for the benefit of the European science community. © 2011 Springer Science+Business Media B.V.

## Subcritical fluctuations and suppression of turbulence in differentially rotating gyrokinetic plasmas

PLASMA PHYSICS AND CONTROLLED FUSION **54** (2012) ARTN 055011

## X-ray surface brightness and gas density fluctuations in the Coma cluster

Monthly Notices of the Royal Astronomical Society **421** (2012) 1123-1135

X-ray surface brightness fluctuations in the core (650 × 650 kpc) region of the Coma cluster observed with XMM-Newton and Chandra are analysed using a 2D power spectrum approach. The resulting 2D spectra are converted to 3D power spectra of gas density fluctuations. Our independent analyses of the XMM-Newton and Chandra observations are in excellent agreement and provide the most sensitive measurements of surface brightness and density fluctuations for a hot cluster. We find that the characteristic amplitude of the volume filling density fluctuations relative to the smooth underlying density distribution varies from 7-10 per cent on scales of ~500kpc down to ~5 per cent on scales of ~30kpc. On smaller spatial scales, projection effects smear the density fluctuations by a large factor, precluding strong limits on the fluctuations in 3D. On the largest scales probed (hundreds of kpc), the dominant contributions to the observed fluctuations most likely arise from perturbations of the gravitational potential by the two most massive galaxies in Coma, NGC4874 and NGC4889, and the low-entropy gas brought to the cluster by an infalling group. Other plausible sources of X-ray surface brightness fluctuations are discussed, including turbulence, metal abundance variations and unresolved sources. Despite a variety of possible origins for density fluctuations, the gas in the Coma cluster core is remarkably homogeneous on scales from ~500 to ~30kpc. © 2012 The Authors Monthly Notices of the Royal Astronomical Society © 2012 RAS.

## Critical balance in magnetohydrodynamic, rotating and stratified turbulence: Towards a universal scaling conjecture

Journal of Fluid Mechanics **677** (2011) 134-153

It is proposed that critical balance a scale-by-scale balance between the linear propagation and nonlinear interaction time scales can be used as a universal scaling conjecture for determining the spectra of strong turbulence in anisotropic wave systems. Magnetohydrodynamic (MHD), rotating and stratified turbulence are considered under this assumption and, in particular, a novel and experimentally testable energy cascade scenario and a set of scalings of the spectra are proposed for low-Rossby-number rotating turbulence. It is argued that in neutral fluids the critically balanced anisotropic cascade provides a natural path from strong anisotropy at large scales to isotropic Kolmogorov turbulence at very small scales. It is also argued that the k-2 spectra seen in recent numerical simulations of low-Rossby-number rotating turbulence may be analogous to the k-3/2 spectra of the numerical MHD turbulence in the sense that they could be explained by assuming that fluctuations are polarised (aligned) approximately as inertial waves (Alfvén waves for MHD). © 2011 Cambridge University Press.

## Anisotropy of Alfvénic turbulence in the solar wind and numerical simulations

Monthly Notices of the Royal Astronomical Society **415** (2011) 3219-3226

We investigate the anisotropy of Alfvénic turbulence in the inertial range of slow solar wind and in both driven and decaying reduced magnetohydrodynamic simulations. A direct comparison is made by measuring the anisotropic second-order structure functions in both data sets. In the solar wind, the perpendicular spectral index of the magnetic field is close to -5/3. In the forced simulation, it is close to -5/3 for the velocity and -3/2 for the magnetic field. In the decaying simulation, it is -5/3 for both fields. The spectral index becomes steeper at small angles to the local magnetic field direction in all cases. We also show that when using the global rather than local mean field, the anisotropic scaling of the simulations cannot always be properly measured. © 2011 The Authors Monthly Notices of the Royal Astronomical Society © 2011 RAS.

## Large-scale magnetic field generation by randomly forced shearing waves

Physical Review Letters **107** (2011)

A rigorous theory for the generation of a large-scale magnetic field by random nonhelically forced motions of a conducting fluid combined with a linear shear is presented in the analytically tractable limit of low magnetic Reynolds number (Rm) and weak shear. The dynamo is kinematic and due to fluctuations in the net (volume-averaged) electromotive force. This is a minimal proof-of-concept quasilinear calculation aiming to put the shear dynamo, a new effect recently found in numerical experiments, on a firm theoretical footing. Numerically observed scalings of the wave number and growth rate of the fastest-growing mode, previously not understood, are derived analytically. The simplicity of the model suggests that shear dynamo action may be a generic property of sheared magnetohydrodynamic turbulence. © 2011 American Physical Society.

## Critically balanced ion temperature gradient turbulence in fusion plasmas

Physical Review Letters **107** (2011)

Scaling laws for ion temperature gradient driven turbulence in magnetized toroidal plasmas are derived and compared with direct numerical simulations. Predicted dependences of turbulence fluctuation amplitudes, spatial scales, and resulting heat fluxes on temperature gradient and magnetic field line pitch are found to agree with numerical results in both the driving and inertial ranges. Evidence is provided to support the critical balance conjecture that parallel streaming and nonlinear perpendicular decorrelation times are comparable at all spatial scales, leading to a scaling relationship between parallel and perpendicular spatial scales. This indicates that even strongly magnetized plasma turbulence is intrinsically three dimensional. © 2011 American Physical Society.

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

Physical Review Letters **107** (2011)

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 insitu 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. © 2011 American Physical Society.

## Turbulent transport in tokamak plasmas with rotational shear

Physical Review Letters **106** (2011)

Nonlinear gyrokinetic simulations are 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×B shear value for plasma confinement. Local maxima in the momentum fluxes are observed, implying the possibility of bifurcations in the E×B shear. The critical temperature gradient for the onset of turbulence increases with flow shear at low flow shears; at higher flow shears, the dependence of heat flux on temperature gradient becomes less stiff. The turbulent Prandtl number is found to be largely independent of temperature and flow gradients, with a value close to unity. © 2011 American Physical Society.

## Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas

Physics of Plasmas **18** (2011)

A minimal model for magnetic reconnection and, generally, low-frequency dynamics in low-beta plasmas is proposed. The model combines analytical and computational simplicity with physical realizability: it is a rigorous limit of gyrokinetics for plasma beta of order the electron-ion mass ratio. The model contains collisions and can be used both in the collisional and collisionless reconnection regimes. It includes gyrokinetic ions (not assumed cold) and allows for the topological rearrangement of the magnetic field lines by either resistivity or electron inertia, whichever predominates. The two-fluid dynamics are coupled to electron kinetics-electrons are not assumed isothermal and are described by a reduced drift-kinetic equation. The model, therefore allows for irreversibility and conversion of magnetic energy into electron heat via parallel phase mixing in velocity space. An analysis of the exchanges between various forms of free energy and its conversion into electron heat is provided. It is shown how all relevant linear waves and regimes of the tearing instability (collisionless, semicollisional, and fully resistive) are recovered in various limits of our model. An efficient way to simulate our equations numerically is proposed, via the Hermite representation of the velocity space. It is shown that small scales in velocity space will form, giving rise to a shallow Hermite-space spectrum, whence it is inferred that, for steady-state or sufficiently slow dynamics, the electron heating rate will remain finite in the limit of vanishing collisionality. © 2011 American Institute of Physics.

## Anisotropy of imbalanced Alfvénic turbulence in fast solar wind

Physical Review Letters **106** (2011)

We present the first measurement of the scale-dependent power anisotropy of Elsasser variables in imbalanced fast solar wind turbulence. The dominant Elsasser mode is isotropic at lower spacecraft frequencies but becomes increasingly anisotropic at higher frequencies. The subdominant mode is anisotropic throughout. There are two distinct subranges exhibiting different scalings within what is normally considered the inertial range. The low Alfvén ratio and the different scaling of the Elsasser modes suggests an interpretation of the observed discrepancy between the velocity and magnetic field scalings, the total energy is dominated by the latter. These results do not appear to be fully explained by any of the current theories of incompressible imbalanced MHD turbulence. © 2011 American Physical Society.

## Momentum injection in tokamak plasmas and transitions to reduced transport

Physical Review Letters **106** (2011)

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. © 2011 American Physical Society.

## Transport bifurcation induced by sheared toroidal flow in tokamak plasmas

PHYSICS OF PLASMAS **18** (2011) ARTN 102304

## A non-linear theory of the parallel firehose and gyrothermal instabilities in a weakly collisional plasma

Monthly Notices of the Royal Astronomical Society **413** (2011) 7-38

Weakly collisional magnetized cosmic plasmas have a dynamical tendency to develop pressure anisotropies with respect to the local direction of the magnetic field. These anisotropies trigger plasma instabilities at scales just above the ion Larmor radius ρ i and much below the mean free path λ mfp. They have growth rates of a fraction of the ion cyclotron frequency, which is much faster than either the global dynamics or even local turbulence. Despite their microscopic nature, these instabilities dramatically modify the transport properties and, therefore, the macroscopic dynamics of the plasma. The non-linear evolution of these instabilities is expected to drive pressure anisotropies towards marginal stability values, controlled by the plasma beta β i. Here this non-linear evolution is worked out in an ab initio kinetic calculation for the simplest analytically tractable example - the parallel (k ⊥ = 0) firehose instability in a highbeta plasma. An asymptotic theory is constructed, based on a particular physical ordering and leading to a closed non-linear equation for the firehose turbulence. In the non-linear regime, both the analytical theory and the numerical solution predict secular (∝t) growth of magnetic fluctuations. The fluctuations develop a k {double pipe}-3 spectrum, extending from scales somewhat larger than ρ i to the maximum scale that grows secularly with time (∝t 1/2); the relative pressure anisotropy (p ⊥ - p {double pipe})/p {double pipe} tends to the marginal value -2/β i. The marginal state is achieved via changes in the magnetic field, not particle scattering. When a parallel ion heat flux is present, the parallel 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 magnitude of the ion heat flux. The non-linear evolution of the GTI also features secular growth of magnetic fluctuations, but the fluctuation spectrum is eventually dominated by modes around a maximal scale ~ρ il T/λ 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 our interest in the dynamics of galaxy cluster plasmas (which are used as the main astrophysical example), but its relevance to solar wind and accretion flow plasmas is also briefly discussed. © 2011 The Authors. Monthly Notices of the Royal Astronomical Society © 2011 RAS.

## A thermally stable heating mechanism for the intracluster medium: turbulence, magnetic fields and plasma instabilities

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **410** (2011) 2446-2457

## Two-dimensional gyrokinetic turbulence

Journal of Fluid Mechanics **664** (2010) 407-435

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 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. © 2010 Cambridge University Press.