# Publications by Alexander Schekochihin

## 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.

## Transport bifurcation in a rotating tokamak plasma

Physical Review Letters **105** (2010)

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 nonzero 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. © 2010 The American Physical Society.

## 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.

## Fast magnetic reconnection in the plasmoid-dominated regime

Physical Review Letters **105** (2010)

A conceptual model of resistive magnetic reconnection via a stochastic plasmoid chain is proposed. The global reconnection rate is shown to be independent of the Lundquist number. The distribution of fluxes in the plasmoids is shown to be an inverse-square law. It is argued that there is a finite probability of emergence of abnormally large plasmoids, which can disrupt the chain (and may be responsible for observable large abrupt events in solar flares and sawtooth crashes). A criterion for the transition from the resistive magnetohydrodynamic to the collisionless regime is provided. © 2010 The American Physical Society.

## Anisotropy of solar wind turbulence between ion and electron scales

Physical Review Letters **104** (2010)

The anisotropy of turbulence in the fast solar wind, between the ion and electron gyroscales, is directly observed using a multispacecraft analysis technique. Second order structure functions are calculated at different angles to the local magnetic field, for magnetic fluctuations both perpendicular and parallel to the mean field. In both components, the structure function value at large angles to the field S is greater than at small angles in the perpendicular component S /S =5±1 and in the parallel component S /S >3, implying spatially anisotropic fluctuations, k >k. The spectral index of the perpendicular component is -2.6 at large angles and -3 at small angles, in broad agreement with critically balanced whistler and kinetic Alfvén wave predictions. For the parallel component, however, it is shallower than -1.9, which is considerably less steep than predicted for a kinetic Alfvén wave cascade. © 2010 The American Physical Society.

## Magnetofluid dynamics of magnetized cosmic plasma: Firehose and gyrothermal instabilities

Monthly Notices of the Royal Astronomical Society **405** (2010) 291-300

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 Alfvénically polarized fluctuations. Two instabilities that can be treated this way are considered in detail: the parallel firehose instability (including the finite Larmor radius effects that determine the growth rate and scale of 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 Alfvénically polarized fluctuations even in the absence of the negative pressure anisotropy required for the firehose. The main physical conclusion is that both pressure anisotropies and heat fluxes associated with the macroscale dynamics trigger plasma microinstabilities and, therefore, their values will likely be set by the non-linear evolution of these instabilities. Ideas for understanding this non-linear 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 quantitatively and it is noted that observations point to turbulence in clusters (velocity, magnetic and temperature fluctuations) 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. © 2010 The Authors. Journal compilation © 2010 RAS.

## Power and spectral index anisotropy of the entire inertial range of turbulence in the fast solar wind

Monthly Notices of the Royal Astronomical Society: Letters **407** (2010)

We measure the power and spectral index anisotropy of magnetic field fluctuations in fast solar wind turbulence from scales larger than the outer scale down to the ion gyroscale, thus covering the entire inertial range. We show that the power and spectral indices above the outer scale of turbulence are approximately isotropic. The turbulent cascade causes the power anisotropy at smaller scales manifested by anisotropic scalings of the spectrum: close to k-5/3 across and k-2 along the local magnetic field, consistent with a critically balanced Alfvénic turbulence. By using data at different radial distances from the Sun and calculating the radial dependence of the ratio of the outer scale to the ion gyroscale, we show that the width of the inertial range does not change with heliocentric distance. At the smallest scales of the inertial range, close to the ion gyroscale, we find an enhancement of power parallel to the magnetic field direction coincident with a decrease in the perpendicular power. This is most likely related to energy injection by ion kinetic modes such as the firehose instability and also marks the beginning of the kinetic range, sometimes called the dissipation range, of solar wind turbulence. © 2010 The Authors. Journal compilation © 2010 RAS.

## Gyrokinetic simulation of entropy cascade in two-dimensional electrostatic turbulence

J. Plasma Fusion Res. SERIES **9** (2010) 509-509

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.

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

Physical Review Letters **103** (2009)

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. © 2009 The American Physical Society.

## Formation of plasmoid chains in magnetic reconnection

Physical Review Letters **103** (2009)

A detailed numerical study of magnetic reconnection in resistive MHD for very large, previously inaccessible, Lundquist numbers (104≤Sa≤108) is reported. Large-aspect-ratio Sweet-Parker current sheets are shown to be unstable to super-AlfvÃ©nically fast formation of plasmoid (magnetic-island) chains. The plasmoid number scales as S3/8 and the instability growth rate in the linear stage as S1/4, in agreement with the theory by Loureiro etÂ al.. 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 quasistationary picture for S>104. © 2009 The American Physical Society.

## Probing magnetic turbulence by synchrotron polarimetry: Statistics and structure of magnetic fields from Stokes correlators

Monthly Notices of the Royal Astronomical Society **398** (2009) 1970-1988

We describe a novel technique for probing the statistical properties of cosmic magnetic fields based on radio polarimetry data. Second-order magnetic field statistics like the power spectrum cannot always distinguish between magnetic fields with essentially different spatial structure. Synchrotron polarimetry naturally allows certain fourth-order magnetic field statistics to be inferred from observational data, which lifts this degeneracy and can thereby help us gain a better picture of the structure of the cosmic fields and test theoretical scenarios describing magnetic turbulence. In this work we show that a fourth-order correlator of specific physical interest, the tension force spectrum, can be recovered from the polarized synchrotron emission data. We develop an estimator for this quantity based on polarized emission observations in the Faraday rotation free frequency regime. We consider two cases: a statistically isotropic field distribution, and a statistically isotropic field superimposed on a weak mean field. In both cases the tension force power spectrum is measurable; in the latter case, the magnetic power spectrum may also be obtainable. The method is exact in the idealized case of a homogeneous relativistic electron distribution that has a power-law energy spectrum with a spectral index of p = 3, and assumes statistical isotropy of the turbulent field. We carry out numerical tests of our method using synthetic polarized emission data generated from numerically simulated magnetic fields. We show that the method is valid, that it is not prohibitively sensitive to the value of the electron spectral index p, and that the observed tension force spectrum allows one to distinguish between e.g. a randomly tangled magnetic field (a default assumption in many studies) and a field organized in folded flux sheets or filaments. © 2009 RAS.

## Gyrokinetic simulations of spherical tokamaks

Plasma Physics and Controlled Fusion **51** (2009)

This paper reviews transport and confinement in spherical tokamaks (STs) and our current physics understanding of this that is partly based on gyrokinetic simulations. Equilibrium flow shear plays an important role, and we show how this is consistently included in the gyrokinetic framework for flows that greatly exceed the diamagnetic velocity. The key geometry factors that influence the effectiveness of turbulence suppression by flow shear are discussed, and we show that toroidal equilibrium flow shear can sometimes entirely suppress ion scale turbulence in today's STs. Advanced nonlinear simulations of electron temperature gradient (ETG) driven turbulence, including kinetic ion physics, collisions and equilibrium flow shear, support the model that ETG turbulence can explain electron heat transport in many ST discharges. © 2009 IOP Publishing Ltd.

## Turbulent magnetic reconnection in two dimensions

Monthly Notices of the Royal Astronomical Society: Letters **399** (2009)

Two-dimensional numerical simulations of the effect of background turbulence on two-dimensional resistive magnetic reconnection are presented. For sufficiently small values of the resistivity (η) and moderate values of the turbulent power (∈), the reconnection rate is found to have a much weaker dependence on η than the Sweet-Parker scaling of η1/2 and is even consistent with an η independent value. For a given value of η, the dependence of the reconnection rate on the turbulent power exhibits a critical threshold in ∈ above which the reconnection rate is significantly enhanced. © 2009 The Authors. Journal compilation © 2009 RAS.

## Magnetic turbulence in clusters of galaxies

Revista Mexicana de Astronomia y Astrofisica: Serie de Conferencias **36** (2009) 209-215

Galaxy clusters are large laboratories for magnetic plasma turbulence which permit us to confront our theoretical concepts of magnetogenesis with detailed observations. Magnetic turbulence in clusters can be studied via the radio-synchrotron emission from the intra-cluster medium in the form of cluster radio relics and halos. The power spectrum of turbulent magnetic fields can be examined via Faraday rotation analysis of extended radio sources. In case of the Hydra A cool core, the observed magnetic spectrum can be understood in terms of a turbulence-mediated feedback loop between gas cooling and the jet activity of the central galaxy. Finally, methods to measure higher-order statistics of the magnetic field using Stokes-parameter correlations are discussed, which permit us to determine the power spectrum of the magnetic tension force. This fourth-order statistical quantity offers a way to discriminate between different magnetic turbulence scenarios and different field structures using radio polarimetric observations. © 2009: Instituto de Astronomía.

## Astrophysical gyrokinetics: Kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas

Astrophysical Journal, Supplement Series **182** (2009) 310-377

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 Alfvénic 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 Alfvénic 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-Alfvén-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. © 2009. The American Astronomical Society. All rights reserved.

## Linearized model fokker-planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests

Physics of Plasmas **16** (2009)

A set of key properties for an ideal dissipation scheme in gyrokinetic simulations is proposed, and implementation of a model collision operator satisfying these properties is described. This operator is based on the exact linearized test-particle collision operator, with approximations to the field-particle terms that preserve conservation laws and an H -theorem. It includes energy diffusion, pitch-angle scattering, and finite Larmor radius effects corresponding to classical (real-space) diffusion. The numerical implementation in the continuum gyrokinetic code GS2 [Kotschenreuther, Comput. Phys. Comm. 88, 128 (1995)] is fully implicit and guarantees exact satisfaction of conservation properties. Numerical results are presented showing that the correct physics is captured over the entire range of collisionalities, from the collisionless to the strongly collisional regimes, without recourse to artificial dissipation. © 2009 American Institute of Physics.

## Linearized model Fokker-Planck collision operators for gyrokinetic simulations. I. Theory

Physics of Plasmas **15** (2008)

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. Model operators for electron-ion and ion-electron collisions are also presented. © 2008 American Institute of Physics.

## Gyrokinetic turbulence: A nonlinear route to dissipation through phase space

Plasma Physics and Controlled Fusion **50** (2008)

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 for by the emergence of a 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 modelling and resolving collisions in gyrokinetic simulations is briefly 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. © 2008 IOP Publishing Ltd.

## MHD turbulence: Nonlocal, anisotropic, nonuniversal?

Solid Mechanics and its Applications **4** (2008) 347-354

Kolmogorov's theory and philosophy of turbulence are based on a number of assumptions that have become standard notions with which one approaches turbulence in many, including non-hydrodynamic, systems. However, it turns out that in MHD turbulence, locality of interactions in scale space, isotropy of small scales or even universality cannot be taken for granted and, in fact, can be shown to fail. This note focuses on these unconventional aspects of MHD turbulence and on the related phenomenon of small-scale dynamo using a combination of simple physical reasoning and numerical evidence. © 2008 Springer.