# Thermal disequilibration of ions and electrons by collisionless plasma turbulence

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Edited by Ramesh Narayan, Harvard University, Cambridge, MA, and approved November 30, 2018 (received for review July 19, 2018)

## Significance

Large-scale astrophysical processes inject energy into turbulent motions and electromagnetic fields, which carry this energy to small scales and eventually thermalize it. How this energy is partitioned between ions and electrons is important both in plasma physics and in astrophysics. Here we determine this energy partition via gyrokinetic turbulence simulations and provide a simple prescription for the ion-to-electron heating ratio. We find that turbulence promotes disequilibration of the species: When magnetic energy density is greater than the thermal energy density, electrons are preferentially heated, whereas when it is smaller, ions are. This is a relatively rare example of nature promoting an ever more out-of-equilibrium state in an environment where particle collisions are not frequent enough to equalize the temperatures of the species.

## Abstract

Does overall thermal equilibrium exist between ions and electrons in a weakly collisional, magnetized, turbulent plasma? And, if not, how is thermal energy partitioned between ions and electrons? This is a fundamental question in plasma physics, the answer to which is also crucial for predicting the properties of far-distant astronomical objects such as accretion disks around black holes. In the context of disks, this question was posed nearly two decades ago and has since generated a sizeable literature. Here we provide the answer for the case in which energy is injected into the plasma via Alfvénic turbulence: Collisionless turbulent heating typically acts to disequilibrate the ion and electron temperatures. Numerical simulations using a hybrid fluid-gyrokinetic model indicate that the ion–electron heating-rate ratio is an increasing function of the thermal-to-magnetic energy ratio,

In many astrophysical plasma systems, such as accretion disks, the intracluster medium, and the solar wind, collisions between ions and electrons are extremely infrequent compared to dynamical processes and even compared to collisions within each species. In the effective absence of interspecies collisions, it is an open question whether there is any mechanism for the system to self-organize into a state of equilibrium between the two species and, if not, what sets the ion-to-electron temperature ratio. This is clearly an interesting plasma–physics question on a fundamental level, but it is also astrophysically important for interpreting observations of plasmas from the heliosphere to the Galaxy and beyond. Historically, the posing of this question 20 y ago in the context of radiatively inefficient accretion flows and in particular of our own Galactic Center, Sagittarius A* (Sgr A*) [in which preferential ion heating was invoked to explain low observed luminosity (1⇓–3)], has prompted a flurry of research and porting of analytical and numerical machinery developed in the context of fusion plasmas and of fundamental turbulence theories to astrophysical problems (see, e.g., refs. 4⇓⇓⇓⇓⇓⇓⇓–12, but also ref. 13 and references therein for an alternative strand of investigations). In more recent years, heating prescriptions resulting from these investigations have increasingly been in demand for global models aiming to reproduce observations quantitatively (e.g., refs. 14 and 15 and references therein).

In a nonlinear plasma system, turbulence is generally excited by large-scale free-energy sources (e.g., the Keplerian shear flow in a differentially rotating accretion disk), then transferred to ever smaller scales in the position–velocity phase space via a “turbulent cascade,” and finally converted into thermal energy of plasma particles via microscale dissipation processes. This turbulent heating is not necessarily distributed evenly between ions and electrons. It may, in principle, lead to either thermal disequilibration or equilibration between ions and electrons, depending on how the ion-to-electron heating ratio changes with the ratio of their temperatures,

This task requires a number of assumptions, many of which are quite simplistic, but are made here to distill what we consider to be the most basic features of the problem at hand. We assume that the large-scale free-energy injection launches a cascade of perturbations that are anisotropic with respect to the direction of the ambient mean magnetic field and whose characteristic frequencies are Alfvénic—we know both from theory (6, 16) and detailed measurements in the solar wind (17) that this is what inertial-range turbulence in a magnetized plasma would look like. This means that the particles’ cyclotron motion can be averaged out at all spatial scales, all the way to the ion Larmor radius and below. This “gyrokinetic” (GK) approximation (4, 18) leaves out any heating mechanisms associated with cyclotron resonances (because frequencies are low) and with shocks (19) (because sonic perturbations are ordered out). The amplitude of the fluctuations is assumed to be asymptotically small relative to the mean field, and thus stochastic heating (20) and any other mechanisms relying on finite-amplitude fluctuations (21⇓⇓⇓–25) are also absent. Furthermore, we assume that ions and electrons individually are near Maxwellian equilibria, but at different temperatures. This excludes any heating mechanisms associated with pressure anisotropies (26⇓–28) or significant nonthermal tails in the particle distribution functions (29, 30). We note that reconnection is allowed within the GK model, and so the results obtained here include any heating, ion or electron, that might occur in reconnecting sheets spontaneously formed within the turbulent dynamics. [Note, however, that the width of the inertial range that we can afford is necessarily modest. It therefore remains an open question whether reconnecting structures that emerge in collisionless plasma turbulence in extremely wide inertial ranges (31, 32) are capable of altering any of the features of ion–electron energy partition reported here.] Although the GK approximation may be viewed as fairly crude [e.g., it may not always be appropriate to neglect high-frequency fluctuations at ion Larmor scales (33)], it does a relatively good job of quantitatively reproducing solar wind observations (5); see ref. 34 for a detailed discussion of the applicability of the GK model to solar wind. In any event, such a simplification is crucial for carrying out multiple kinetic turbulence simulations at reasonable computational cost.

It can be shown that in GK turbulence, Alfvénic and compressive (slow-wave–like) perturbations decouple energetically in the inertial range (6). In the solar wind, the compressive perturbations are energetically subdominant in the inertial range (17), although it is not known how generic a situation this is. [For example, turbulence in accretion flows is mostly driven by the magnetorotational instability (MRI) (35). The partition of compressive and Alfvénic fluctuations in MRI-driven turbulence is an open question.] At low

## Numerical Approach

An Alfvénic turbulent cascade starts in the magnetohydrodynamic (MHD) inertial range, where ions and electrons move in concert. Therefore, it is not possible to determine the energy partition between species within the MHD approximation. This approximation breaks down and the two species decouple at the ion Larmor scale, *Center*).

Thus, the energy partition is decided around the ion Larmor scale, where the electron kinetic effects are not important (at least in the asymptotic limit of small electron-to-ion mass ratio). We may therefore determine this partition within a hybrid model in which ions are treated gyrokinetically and electrons as an isothermal fluid (6). The isothermal electron fluid equations are derived from the electron GK equation via an asymptotic expansion in the electron-to-ion mass ratio

Our hybrid GK code (12) [based on AstroGK (8), an Eulerian

In the hybrid code, the phase space of the ion distribution function is spanned by

To model the large-scale energy injection, we use an oscillating Langevin antenna (36), which excites Alfvén waves (AWs) by driving an external parallel current. We set the driven modes to have the oscillation frequency

The ions have a fully conservative linearized collision operator, including pitch-angle scattering and energy diffusion (37, 38). The collision frequency is chosen to be

## Energy Partition

The main result of our simulations is given in Fig. 1, which shows the dependence of the ratio of the time-averaged ion and electron heating rates *Left* shows that

### Low Beta.

In the limit *Left*).

The scale where the ion heating occurs is apparent in Fig. 2, *Bottom*. For low to moderate

### High Beta.

In the opposite limit of high

The physics behind this result are more complicated. In a high-*Right* [this resembles the subviscous cascade in high-magnetic Prandtl-number MHD and, similarly to it (40), might be exhibiting a *Left* and *Center*) are very similar to what has been observed both in numerical simulations (5, 9, 10, 33, 42, 43) and in solar wind observations (17) at

Thus, there is a finite wave-number interval of strong damping around *D*). Exactly what fraction it will be is what our numerical study tells us. We do not have a quantitative theory that would explain why

### Relation to Standard Model Based on Linear Damping.

It is instructive to compare *i*) continuity of the magnetic-energy spectrum across the ion–Larmor-scale transition, (*ii*) linear Landau damping as the rate of free-energy dissipation leading to ion heating, and (*iii*) critical balance between linear propagation and nonlinear decorrelation rates. As evident in Fig. 1, *Left*, *Inset*, the model gives a broadly correct qualitative trend, but produces some noticeable quantitative discrepancies: notably, much lower ion heating at low

This is perhaps not surprising, for a number of reasons. First, the Landau damping rate is not, in general, a quantitatively good predictor of the rate at which linear phase mixing would dissipate free energy in a driven system (47). Indeed, we have found that an approximation such as *Bottom*. Second, at high

### Temperature Disequilibration.

Apart from the *Right*). Some dependence on

Overall, we see that whether ions and electrons are already disequilibrated or not makes relatively little difference to the heating rates—there is no intrinsic tendency in the collisionless system to push the two species toward equilibrium with each other (except at

### Fitting Formula.

For a researcher who is interested in using these results in global models (as in, e.g., refs. 14 and 15), here is a remarkably simple fitting formula, which, without aspiring to ultrahigh precision, works quite well over the parameter range that we have investigated (Fig. 1, *Right*):

## Phase-Space Cascades

One of the more fascinating developments prompted by the interest in energy partition in plasma turbulence has been the realization that, in a kinetic system, we are dealing with a free-energy cascade through the entire phase space, with energy travelling from large to small scales in both position and velocity space (6, 33, 44, 45, 48⇓⇓⇓⇓⇓–54). This is inevitable because the plasma collision operator is a diffusion operator in phase space and so the only way for a kinetic system to have a finite rate of dissipation at very low collisionality is to generate small phase-space scales—just like a hydrodynamic system with low viscosity achieves finite viscous dissipation by generating large flow-velocity gradients. The study of velocity-space cascades in kinetic systems is still in its infancy—but advances in instrumentation and computing mean that the amount of available information on such cascades in both real (space) physical plasmas (52) and their numerical counterparts (33, 46, 54) is rapidly increasing. Let us then investigate the nature of the phase-space cascade in our ion-heating simulations.

In low-frequency (GK) turbulence, there are two routes for the velocity-space cascade: Linear phase mixing, also known as Landau damping (55), produces small scales in the distribution of the velocities parallel to the magnetic field (

We use the Hermite–Laguerre spectral decomposition of the gyroaveraged perturbed distribution function

### Low Beta.

At low *C*; this is because ions’ thermal motion is slow compared to the phase speed of the Alfvénic perturbations), so most of the ion entropy is cascaded simultaneously to large *A*) before being thermalized by collisions, giving rise to (perpendicular) ion heating. The Fourier–Laguerre spectrum contains little energy at high ℓ when

### High Beta.

In contrast, at high *D*) by linear phase mixing, as is indeed confirmed by the characteristic *E*; at low

## Discussion

To discuss an example of astrophysical consequences of our findings, let us return briefly to the curious case of low-luminosity accretion flows—most famously, the supermassive black hole Sgr A* at our Galaxy’s center. Two classes of theory have been advanced to explain the observed low-luminosity, each corresponding to a distinct physical scenario: The first scenario has

On a broader and perhaps more fundamental level, we have shown that turbulence is capable of pushing weakly collisional plasma systems away from interspecies thermal equilibrium—depending on whether

## Acknowledgments

We thank S. Balbus, B. Chandran, S. Cowley, W. Dorland, C. Gammie, G. Howes, M. Kunz, N. Loureiro, A. Mallet, R. Meyrand, F. Parra, and E. Quataert for fruitful discussions and suggestions. This work was supported by the Science and Technology Facilities Council Grant ST/N000919/1. A.A.S. was also supported in part by the Engineering and Physical Sciences Research Council (EPSRC) Grant EP/M022331/1. For the simulations reported here, the authors acknowledge the use of ARCHER through the Plasma High-End Computing Consortium EPSRC Grant EP/L000237/1 under Projects e281-gs2, the EUROfusion High Performance Computing (HPC) (Marconi–Fusion) under Project MULTEI, the Cirrus UK National Tier-2 HPC Service at the Edinburgh Parallel Computing Centre funded by the University of Edinburgh and EPSRC (EP/P020267/1), and the University of Oxford’s Advanced Research Computing facility.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: yohei.kawazura{at}physics.ox.ac.uk.

Author contributions: M.B. and A.A.S. designed research; Y.K., M.B., and A.A.S. performed research; Y.K. and M.B. contributed new reagents/analytic tools; Y.K., M.B., and A.A.S. analyzed data; and Y.K., M.B., and A.A.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).

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