# Publications by Alexander Schekochihin

## Stochastic transport of high-energy particles through a turbulent plasma

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The interplay between charged particles and turbulent magnetic fields is crucial to understanding how cosmic rays propagate through space. A key parameter which controls this interplay is the ratio of the particle gyroradius to the correlation length of the magnetic turbulence. For the vast majority of cosmic rays detected at the Earth, this parameter is small, and the particles are well confined by the Galactic magnetic field. But for cosmic rays more energetic than about 30 EeV, this parameter is large. These highest energy particles are not confined to the Milky Way and are presumed to be extragalactic in origin. Identifying their sources requires understanding how they are deflected by the intergalactic magnetic field, which appears to be weak, turbulent with an unknown correlation length, and possibly spatially intermittent. This is particularly relevant given the recent detection by the Pierre Auger Observatory of a significant dipole anisotropy in the arrival directions of cosmic rays of energy above 8 EeV. Here we report measurements of energetic-particle propagation through a random magnetic field in a laser-produced plasma. We characterize the diffusive transport of these particles and recover experimentally pitch-angle scattering measurements and extrapolate to find their mean free path and the associated diffusion coefficient, which show scaling-relations consistent with theoretical studies. This experiment validates these theoretical tools for analyzing the propagation of ultra-high energy cosmic rays through the intergalactic medium.

## Fluidization of collisionless plasma turbulence

Proceedings of the National Academy of Sciences National Academy of Sciences (0)

In a collisionless, magnetized plasma, particles may stream freely along magnetic-field lines, leading to phase "mixing" of their distribution function and consequently to smoothing out of any "compressive" fluctuations (of density, pressure, etc.,). This rapid mixing underlies Landau damping of these fluctuations in a quiescent plasma-one of the most fundamental physical phenomena that make plasma different from a conventional fluid. Nevertheless, broad power-law spectra of compressive fluctuations are observed in turbulent astrophysical plasmas (most vividly, in the solar wind) under conditions conducive to strong Landau damping. Elsewhere in nature, such spectra are normally associated with fluid turbulence, where energy cannot be dissipated in the inertial scale range and is therefore cascaded from large scales to small. By direct numerical simulations and theoretical arguments, it is shown here that turbulence of compressive fluctuations in collisionless plasmas strongly resembles one in a collisional fluid and does have broad power-law spectra. This "fluidization" of collisionless plasmas occurs because phase mixing is strongly suppressed on average by "stochastic echoes", arising due to nonlinear advection of the particle distribution by turbulent motions. Besides resolving the long-standing puzzle of observed compressive fluctuations in the solar wind, our results suggest a conceptual shift for understanding kinetic plasma turbulence generally: rather than being a system where Landau damping plays the role of dissipation, a collisionless plasma is effectively dissipationless except at very small scales. The universality of "fluid" turbulence physics is thus reaffirmed even for a kinetic, collisionless system.

## A solvable model of Vlasov-kinetic plasma turbulence in Fourier-Hermite phase space

Journal of Plasma Physics Cambridge University Press (0)

A class of simple kinetic systems is considered, described by the 1D Vlasov--Landau equation with Poisson or Boltzmann electrostatic response and an energy source. Assuming a stochastic electric field, a solvable model is constructed for the phase-space turbulence of the particle distribution. The model is a kinetic analog of the Kraichnan--Batchelor model of chaotic advection. The solution of the model is found in Fourier--Hermite space and shows that the free-energy flux from low to high Hermite moments is suppressed, with phase mixing cancelled on average by anti-phase-mixing (stochastic plasma echo). This implies that Landau damping is an ineffective dissipation channel at wave numbers below a certain cut off (analog of Kolmogorov scale), which increases with the amplitude of the stochastic electric field and scales as inverse square of the collision rate. The full Fourier--Hermite spectrum is derived. Its asymptotics are $m^{-3/2}$ at low wave numbers and high Hermite moments ($m$) and $m^{-1/2}k^{-2}$ at low Hermite moments and high wave numbers ($k$). The energy distribution and flows in phase space are a simple and, therefore, useful example of competition between phase mixing and nonlinear dynamics in kinetic turbulence, reminiscent of more realistic but more complicated multi-dimensional systems that have not so far been amenable to complete analytical solution.

## Experimental Signatures of Critically Balanced Turbulence in MAST

ArXiv (0)

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 $\sim \nu_{*i}^{-0.8}$.

## Microstability physics as illuminated in the spherical tokamak

Plasma Physics and Controlled Fusion **47** (2005)

Spherical tokamaks (STs) have attractive features for fusion, and there is considerable interest in understanding their transport properties which depend on the underlying microinstabilities. STs are capable of operation with low magnetic fields and exhibit large inhomogeneity in the toroidal magnetic field. These factors strongly affect particle dynamics and the potency of magnetic perturbations, which correspondingly impact on the microstability properties of STs. This paper reviews previous microstability studies in ST plasma configurations and presents gyrokinetic microstability calculations for a range of ST equilibria, using the gyrokinetic code GS2. Microstability properties of L-mode and H-mode equilibria, from the MAST experiment at Culham, are compared. In MAST the shearing rates of equilibrium E × B flows usually exceed the growth rates of microinstabilities with k⊥ρi < 1 (including ion temperature gradient, ITG, driven drift waves) and are generally smaller than the growth rates of shorter wavelength modes with k ⊥ρi > 1 (electron temperature gradient, ETG, driven drift waves). Electromagnetic effects are significant at mid-radius in these MAST equilibria, where the local β ≥ 0.1. At k ⊥ρi < 1, strongly electromagnetic modes dominate over ITG instabilities, and these modes are found to have tearing parity in the H-mode plasma and twisting parity in the L-mode plasma. Numerical experiments have been carried out to assess the properties of the tearing parity modes and to probe the underlying physical drive mechanism. At shorter wavelengths the electromagnetic effects can significantly stabilize the ETG instabilities. Nonlinear electron scale microturbulence calculations for two surfaces of a MAST H-mode plasma suggest that significant electron heat transport can be carried via this mechanism. In an extremely high β ST equilibrium, which has been proposed as the basis of a conceptual ST power plant, electrostatic instabilities are fully stabilized, but tearing parity modes are predicted to be unstable over-wide range of length scales. © 2005 IOP Publishing Ltd.

## Nonlinear tearing mode reconnection

32nd EPS Conference on Plasma Physics 2005, EPS 2005, Held with the 8th International Workshop on Fast Ignition of Fusion Targets - Europhysics Conference Abstracts **1** (2005) 193-196

## Plasma instabilities and magnetic field growth in clusters of galaxies

ASTROPHYSICAL JOURNAL **629** (2005) 139-142

## The onset of a small-scale turbulent dynamo at low magnetic Prandtl numbers

ASTROPHYSICAL JOURNAL **625** (2005) L115-L118

## Self-Similar Turbulent Dynamo

Physical Review Letters **92** (2004)

The amplification of magnetic fields in a highly conducting fluid is studied numerically. During growth, the magnetic field is spatially intermittent: it does not uniformly fill the volume, but is concentrated in long thin folded structures. Contrary to a commonly held view, intermittency of the folded field does not increase indefinitely throughout the growth stage if diffusion is present. Instead, as we show, the probability-density function (PDF) of the field-strength becomes self-similar. The normalized moments increase with magnetic Prandtl number in a powerlike fashion. We argue that the self-similarity is to be expected with a finite flow scale and system size. In the nonlinear saturated state, intermittency is reduced and the PDF is exponential. Parallels are noted with self-similar behavior recently observed for passive-scalar mixing and for map dynamos. © 2004 The American Physical Society.

## Critical Magnetic Prandtl Number for Small-Scale Dynamo

Physical Review Letters **92** (2004) 545021-545024

The existence of small-scale dynamo in homogeneous incompressible turbulence with magnetic Prandtl number is discussed. A series of numerical simulations showing that the critical magnetic Reynolds number Rmc for the nonhelical small-scale dynamo depends on the Reynolds number Re is discussed. The dynamo is shut down if the magnetic Prandtl number Pr m = Rm/Re is less than some critical value Prm,c≲1. It is found that the existence of the dynamo depends on the exact inertial-range scaling of the velocity field and/or only manifests itself at very large Rm.

## Saturated State of the Nonlinear Small-Scale Dynamo

Physical Review Letters **92** (2004)

We consider the problem of incompressible, forced, nonhelical, homogeneous, and isotropic MHD turbulence with no mean magnetic field and large magnetic Prandtl number. This type of MHD turbulence is the end state of the turbulent dynamo, which generates folded fields with small-scale direction reversals. We propose a model in which saturation is achieved as a result of the velocity statistics becoming anisotropic with respect to the local direction of the magnetic folds. The model combines the effects of weakened stretching and quasi-two-dimensional mixing and produces magnetic-energy spectra in remarkable agreement with numerical results at least in the case of a one-scale flow. We conjecture that the statistics seen in numerical simulations could be explained as a superposition of these folded fields and Alfvén-like waves that propagate along the folds. © 2004 The American Physical Society.

## From small-scale dynamo to isotropic MHD turbulence

Astrophysics and Space Science **292** (2004) 141-146

We consider the problem of incompressible, forced, nonhelical, homogeneous, isotropic MHD turbulence with no mean magnetic field. This problem is essentially different from the case with externally imposed uniform mean field. There is no scale-by-scale equipartition between magnetic and kinetic energies as would be the case for the Alfvén-wave turbulence. The isotropic MHD turbulence is the end state of the turbulent dynamo which generates folded fields with small-scale direction reversals. We propose that the statistics seen in numerical simulations of isotropic MHD turbulence could be explained as a superposition of these folded fields and Alfvén-like waves that propagate along the folds.

## Structure of small-scale magnetic fields in the kinematic dynamo theory.

Phys Rev E Stat Nonlin Soft Matter Phys **65** (2002) 016305-

A weak fluctuating magnetic field embedded into a a turbulent conducting medium grows exponentially while its characteristic scale decays. In the interstellar medium and protogalactic plasmas, the magnetic Prandtl number is very large, so a broad spectrum of growing magnetic fluctuations is excited at small (subviscous) scales. The condition for the onset of nonlinear back reaction depends on the structure of the field lines. We study the statistical correlations that are set up in the field pattern and show that the magnetic-field lines possess a folding structure, where most of the scale decrease is due to the field variation across itself (rapid transverse direction reversals), while the scale of the field variation along itself stays approximately constant. Specifically, we find that, though both the magnetic energy and the mean-square curvature of the field lines grow exponentially, the field strength and the field-line curvature are anticorrelated, i.e., the curved field is relatively weak, while the growing field is relatively flat. The detailed analysis of the statistics of the curvature shows that it possesses a stationary limiting distribution with the bulk located at the values of curvature comparable to the characteristic wave number of the velocity field and a power tail extending to large values of curvature where it is eventually cut off by the resistive regularization. The regions of large curvature, therefore, occupy only a small fraction of the total volume of the system. Our theoretical results are corroborated by direct numerical simulations. The implication of the folding effect is that the advent of the Lorentz back reaction occurs when the magnetic energy approaches that of the smallest turbulent eddies. Our results also directly apply to the problem of statistical geometry of the material lines in a random flow.

## A model of nonlinear evolution and saturation of the turbulent MHD dynamo

New Journal of Physics **4** (2002)

The growth and saturation of magnetic field in conducting turbulent media with large magnetic Prandtl numbers are investigated. This regime is very common in low-density hot astrophysical plasmas. During the early (kinematic) stage, weak magnetic fluctuations grow exponentially and concentrate at the resistive scale, which lies far below the hydrodynamic viscous scale. The evolution becomes nonlinear when the magnetic energy is comparable to the kinetic energy of the viscous-scale eddies. A physical picture of the ensuing nonlinear evolution of the MHD dynamo is proposed. Phenomenological considerations are supplemented with a simple Fokker-Planck model of the nonlinear evolution of the magnetic-energy spectrum. It is found that, while the shift of the bulk of the magnetic energy from the subviscous scales to the velocity scales may be possible, it occurs very slowly - at the resistive, rather than dynamical, timescale (for galaxies, this means that the generation of large-scale magnetic fields cannot be explained by this mechanism). The role of Alfvénic motions and the implications for the fully developed isotropic MHD turbulence are discussed.

## The small-scale structure of magnetohydrodynamic turbulence with large magnetic Prandtl numbers

ASTROPHYSICAL JOURNAL **576** (2002) 806-813

## Spectra and growth rates of fluctuating magnetic fields in the kinematic dynamo theory with large magnetic Prandtl numbers

ASTROPHYSICAL JOURNAL **567** (2002) 828-852

## Spectra and Growth Rates of Fluctuating Magnetic Fields in the Kinematic Dynamo Theory with Large Magnetic Prandtl Numbers

Astrophys. J. **567** (2002) 828-828

The existence of a weak galactic magnetic field has been repeatedly confirmed by observational data. The origin of this field has not as yet been explained in a fully satisfactory way and represents one of the main challenges of the astrophysical dynamo theory. In both the galactic dynamo theory and the primordial-origin theory, a major influence is exerted by the small-scale magnetic fluctuations. This article is devoted to constructing a systematic second-order statistical theory of such small-scale fields. The statistics of these fields are studied in the kinematic approximation and for the case of large Prandtl numbers, which is relevant for the galactic and protogalactic plasma. The advecting velocity field is assumed to be Gaussian and short-time correlated. Theoretical understanding of this kinematic dynamo model is a necessary prerequisite for any prospective nonlinear dynamo theory. The theory is developed for an arbitrary degree of compressibility and formally in d dimensions, which generalizes the previously known results, elicits the structure of the solutions, and uncovers a number of new effects. The magnetic energy spectra are studied as they grow and spread over scales during the initial stage of the field amplification. Exact Green's-function solutions are obtained. The spectral theory is supplemented by the study of magnetic-field correlation functions in the configuration space, where the dynamo problem can be mapped onto a particular one-dimensional quantum-mechanical problem. The latter approach is most suitable for the description of the kinematic dynamo in the long-time limit, i.e. when the magnetic excitation has spread over all scales present in the system. A simple way of calculating the growth rates of the magnetic fields in this long-time limit is proposed.

## Finite-correlation-time effects in the kinematic dynamo problem

Physics of Plasmas **8** (2001) 4937-4953

Most of the theoretical results on the kinematic amplification of small-scale magnetic fluctuations by turbulence have been confined to the model of white-noise-like (δ-correlated in time) advecting turbulent velocity field. In this work, the statistics of the passive magnetic field in the diffusion-free regime are considered for the case when the advecting flow is finite-time correlated. A new method is developed that allows one to systematically construct the correlation-time expansion for statistical characteristics of the field such as its probability density function or the complete set of its moments. The expansion is valid provided the velocity correlation time is smaller than the characteristic growth time of the magnetic fluctuations. This expansion is carried out up to first order in the general case of a d-dimensional arbitrarily compressible advecting flow. The growth rates for all moments of the magnetic-field strength are derived. The effect of the first-order corrections due to the finite correlation time is to reduce these growth rates. It is shown that introducing a finite correlation time leads to the loss of the small-scale statistical universality, which was present in the limit of the δ-correlated velocity field. Namely, the shape of the velocity time-correlation profile and the large-scale spatial structure of the flow become important. The latter is a new effect, that implies, in particular, that the approximation of a locally-linear shear flow does not fully capture the effect of nonvanishing correlation time. Physical applications of this theory include the small-scale kinematic dynamo in the interstellar medium and protogalactic plasmas. © 2001 American Institute of Physics.