Kinetic Turbulence in Collisionless High-$\beta$ Plasmas

We present results from three-dimensional hybrid-kinetic simulations of Alfvénic turbulence in a high-$\beta$, collisionless plasma. The key feature of such turbulence is the interplay between local wave-wave interactions between the fluctuations in the cascade and the nonlocal wave-particle interactions associated with kinetic microinstabilities driven by anisotropy in the thermal pressure (namely, firehose, mirror, and ion cyclotron). We present theoretical estimates for, and calculate directly from the simulations, the effective collisionality and plasma viscosity in pressure-anisotropic high-$\beta$ turbulence, demonstrating that, for strong Alfvénic turbulence, the effective parallel-viscous scale is comparable to the driving scale of the cascade. Below this scale, the kinetic-energy spectrum indicates an Alfvénic cascade with a slope steeper than $-5/3$ due to the anisotropic viscous stress. The magnetic-energy spectrum is shallower than $-5/3$ near the ion-Larmor scale due to fluctuations produced by the firehose instability. Most of the cascade energy ($\approx 80\%–90\%$) is dissipated as ion heating through a combination of Landau damping and anisotropic viscous heating. Our results have implications for models of particle heating in low-luminosity accretion onto supermassive black holes, the effective viscosity of the intracluster medium, and the interpretation of near-Earth solar-wind observations.

Popular Summary

Many space and astrophysical plasmas—such as the solar-wind, black-hole accretion flows, and the intracluster medium (ICM) of galaxy clusters—are weakly collisional or even collisionless. Such plasmas can be driven away from local thermodynamic equilibrium and, if the ratio of thermal pressure to magnetic pressure (known as the plasma “beta”) is high enough, can become unstable to various microinstabilities. These instabilities grow rapidly at small scales, but regulate the large-scale transport properties of these plasmas, such as the plasma viscosity. The enormous scale separation—up to 12 orders of magnitude in astrophysical plasmas—makes studying this interplay between macroscales and microscales extremely challenging. Our work is the first self-consistent study of this interplay in collisionless high-beta turbulence.

We construct an analytical model for the effective collisionality provided by kinetic microinstabilities and for its associated viscosity, and use state-of-the-art numerical simulations to verify it. Our results are consistent with recent observations of turbulence in the high-beta ICM that imply a suppression of the plasma viscosity there by at least an order of magnitude. We investigate the turbulent spectrum, determine the turbulent dissipation mechanisms responsible for the energization of ions, and provide testable predictions for various astrophysical systems.

While much progress has been made to understand certain aspects of turbulence and transport in these kinds of plasmas, our study marks the first rigorous effort to understand the interplay between microscales and macroscales in a new and complicated (though presumably commonplace) regime of plasma turbulence. Our results can be used to construct more realistic subgrid models for fluid numerical simulations of the ICM, black-hole accretion flows, and the solar wind as well as to interpret current and future observations of these systems.

Figure 1

Qualitative picture of how the cascade proceeds in low-$\beta$ (a) and high-$\beta$ (b) kinetic turbulence. For $\beta \lesssim 1$, the energy flux, injected at some large scale $L$, remains constant in the inertial range, and is eventually dissipated by ions and electrons at the corresponding kinetic scales. In contrast, high-$\beta$ plasma allows for the nonlocal energy transfer by kinetic microinstabilities. The effective viscosity of such a plasma can convert bulk kinetic energy into thermal energy. The goal of this paper is to examine the effects of this physics on the turbulent cascade and on the distribution of energy between species.

Figure 2

Time evolution of the magnetic-field strength along the guide field ($\delta B_z\approx \delta B_{\parallel }$, upper row) and perpendicular to the guide field ($\sqrt{\delta B_x^2+\delta B_y^2}\approx \delta B_{\perp }$, lower row). We show three snapshots: $t\approx 0.4t_{\text{cross}}$, when we see the first mirror fluctuations (see also Fig. 16 in Appendix pp1 and further discussion there); $t\approx 0.6t_{\text{cross}}$, when we detect AIC fluctuations (see also Fig. 17); and in the quasisteady state ($t\approx 6.7t_{\text{cross}}$), when we can instead identify firehose fluctuations (see also Fig. 6).

Figure 3

(a) Time evolution of box-averaged perpendicular and parallel temperatures of the plasma ($\delta T_{\perp ,\parallel }\equiv {\overline{T}}_{\perp ,\parallel }-T_{i0}$) and box-averaged temperature anisotropy $\overline{\mathrm{\Delta }T}={\delta T}_{\perp }-{\delta T}_{\parallel }$ in ${\beta }_{i0}=16$ simulation. Different stages of the simulation are marked with vertical lines. See text for more details about individual stages: Sec. 3b1 and Appendix pp1 for mirror and AIC stage, Sec. 3b2 for Landau-damping stage, and Sec. 3b3 for quasisteady-state firehose stage. (b) Evolution of box-averaged magnetic moment of particles ($\delta \mu \equiv \overline{\mu }-{\mu }_{\mathrm{th}i0}$). During the mirror, AIC, and Landau-damping stages, the average magnetic moment decreases due to scattering off mirror and AIC fluctuations. In the quasisteady state, it slowly increases, because of ion heating and coupling of perpendicular and parallel temperatures due to scattering of ions off firehose fluctuations.

Figure 4

Time evolution of the magnetic-energy spectra in the ${\beta }_{i0}=16$ simulation. In the initial phase of the simulation (red line, Sec. 3b1), mirror instability is triggered, and the spectrum has a bump at kinetic scales. In the Landau-damping stage (green and orange lines, Sec. 3b2), the spectrum is steeper than $k_{\perp }^{-5/3}$ due to dissipation of turbulence. In the quasisteady state (blue line, Sec. 3b3), firehose fluctuations are produced, and the spectrum becomes slightly shallower than $k_{\perp }^{-5/3}$.

Figure 5

A probability density function of pressure anisotropy and parallel plasma $\beta$ for simulations with ${\beta }_{i0}=4$ and ${\beta }_{i0}=16$. Histograms are normalized so that the integral over ${\beta }_{\parallel }$ and $p_{\perp }/p_{\parallel }-1$ equals 1. Black dots indicate the positions of these simulations at the start of each run. Dashed and dot-dashed lines represent the threshold of various kinetic microinstabilities (see text for more detail). Despite being initially driven toward positive values, the quasisteady-state pressure anisotropy is negative, and is close to the $-1.4/{\beta }_{\parallel }$ threshold of kinetic firehose instability (dot-dashed line).

Figure 6

Snapshots of the magnetic-field-strength fluctuations (a),(b), fluctuations of the flow velocity (c),(d), and of pressure anisotropy normalized to the magnetic pressure (e),(f), all taken in the quasisteady state of ${\beta }_{i0}=16$ simulation. There is much more small-scale structure in the magnetic field relative to the flow velocity, which indicates that the effective magnetic Prandtl number is large. Mirror and AIC fluctuations are not apparent, unlike in the earlier stages of the same simulation (Figs. 16 and 17). Instead, there are small-scale oblique modes in the regions with large negative pressure anisotropy (e.g., near $z\approx 25{\rho }_{i0},y\approx 75{\rho }_{i0}$ and $z\approx 130{\rho }_{i0},y\approx 10{\rho }_{i0}$). We associate these fluctuations with the firehose instability.

Figure 7

Energy-transfer functions [Eqs. (20)–(22)] due to the Reynolds (a), Maxwell (b), and viscous stresses (c) in the quasisteady state of our ${\beta }_{i0}=16$ simulation. There is a considerable amount of nonlocal energy transfer due to the firehose instability, which is not present during the early stages of the simulation (see Appendix pp2 for an analogous plot featuring the earlier stages).

Figure 8

(a),(b) Spectra of parallel (orange) and perpendicular (green) pressures, pressure anisotropy (purple), and the sum of electron pressure and magnetic pressure (gray). We define the effective viscous scale ${\ell }_{\nu ,\text{eff}}$ as the scale at which the pressure anisotropy fluctuation amplitude $\sim k_{\perp }^{1/2}\mathrm{\Delta }{p}_k$ peaks. This scale is close to the driving scale, in line with the analytical prediction of Sec. 2. (c),(d) Spectral indices as a function of wave number for the spectra from upper panels.

Figure 9

(a),(b) Quasisteady-state spectra of magnetic (blue) and kinetic (red) energies in the ${\beta }_{i0}=16$ (a) and ${\beta }_{i0}=4$ (b) simulations. These spectra have similar slopes near the driving scale, but deviate from one another at scales below the effective viscous scale (defined in Sec. 3d and in the caption of Fig. 8, and indicated by the dotted lines). The kinetic-energy spectra are steeper than $-5/3$ because of the anisotropic viscous stress; the slope of the kinetic-energy spectrum becomes close to $-2$. The magnetic-energy spectrum becomes shallower toward $k_{\perp }{\rho }_{i0}\sim 1$ because firehose fluctuations are injected at these scales in the quasisteady state. (c),(d) Spectral indices as functions of wave number for the magnetic- and kinetic-energy spectra. (e),(f) Parallel wave number ($k_{\parallel }$) of the fluctuations as a function of their perpendicular wave number ($k_{\perp }$) obtained from magnetic-field structure-function analysis. The value of spectral anisotropy $k_{\parallel }/k_{\perp }$ agrees with critical-balance predictions.

Figure 10

Comparison between the spectra of pressure anisotropy (blue lines) and $p\widehat{\mathit{b}}\widehat{\mathit{b}}:\mathbf{\nabla }\mathit{u}$ (red lines), the latter of which is proportional to the rate of growth of the magnetic field [see Eq. (6)]. Above the effective viscous scale ${\ell }_{\nu ,\text{eff}}$ (vertical dotted line), the pressure anisotropy is proportional to $p\widehat{\mathit{b}}\widehat{\mathit{b}}:\mathbf{\nabla }\mathit{u}$. The proportionality coefficient between the two (the “Braginskii estimate” for the effective collision frequency) is $⟨{\nu }_{\text{eff}}⟩\sim 0.01{\mathrm{\Omega }}_{i0}$. At scales smaller than ${\ell }_{\nu ,\text{eff}}$, this effective collisionality is not large enough relative to the dynamical frequencies, so the pressure anisotropy deviates significantly from the Braginskii estimate. Note the bump in $\widehat{\mathit{b}}\widehat{\mathit{b}}:\mathbf{\nabla }\mathit{u}$ near $k_{\perp }{\rho }_{i0}\sim 1$, which corresponds to firehose fluctuations in quasisteady state (the additional peak at $k_{\perp }{\rho }_{i0}\approx 3$ is due to particle noise).

Figure 11

Trajectory of a tracked particle from the ${\beta }_{i0}=16$ simulation with single-mode driving (see Sec. 3a3). This particle moves through a region with firehose-unstable pressure anisotropy (c),(d). The same region shows small-scale oblique magnetic-field fluctuations (a),(b), which we interpret as firehose modes. As the particle passes through the region, its magnetic moment starts to break (e), and it experiences pitch-angle scattering at almost constant total energy (f). The gray shaded regions in (e) and (f) indicate the period of time over which the trajectory in (a)–(d) is plotted.

Figure 12

Histograms of collision times of tracked particles in the ${\beta }_{i0}=16$ (a) and ${\beta }_{i0}=4$ (c) simulations averaged over the duration of the simulations. Solid lines denote the probabilities obtained with the collisionality diagnostic described in Sec. 3e, which uses all particles in the simulation (Sec. 3e2). (b),(d) Comparison of effective collisionalities obtained by computing $\mathrm{\Delta }{p}_{\mathit{k}}$ and $(p\widehat{\mathit{b}}\widehat{\mathit{b}}:\mathbf{\nabla }\mathit{u}{)}_{\mathit{k}}$ at large scales ($k_{\perp }{\ell }_{\nu ,\text{eff}}<1$; see Fig. 10) and those obtained from $⟨{\tau }_{\mathrm{coll}}⟩$ based on the distributions in (a) and (c). A good agreement between these independent methods indicates that the collisionality is similar to the Braginskii estimate [Eq. (26)]. Vertical dashed and dot-dashed lines in (b) show the values of time for the snapshots in Figs. 16 and 17.

Figure 13

The dependence of anisotropic viscous heating $\mathbf{\Pi }:\mathbf{\nabla }\mathit{u}$ in the ${\beta }_{i0}=16$ simulation on time (a) and wave number (b). Heating is separated into local (red line) and nonlocal (blue line) components, as defined in Sec. 3f, with total heating indicated by the green dashed line. The purple dot-dashed line denotes the nonlocal transfer from kinetic to magnetic energy, which we attribute to the growth of small-scale firehose fluctuations.

Figure 14

(a) Wave number dependence of particle energization during three time intervals in the ${\beta }_{i0}=16$ simulation. The first two stages have a considerable amount of heating at $k_{\perp }{\rho }_{\star }\sim 1$, where Landau damping is expected to be important. Subviscous heating is suppressed in the quasisteady state, and most of the heating occurs close to the effective viscous scale. (b) Phase-space dependence of parallel energization of particles during the same time intervals. The peaks at $w_{\parallel }\sim \pm v_{A0}$ (vertical dotted lines), indicative of Landau damping, are present during the early stages, but disappear in the quasisteady state.

Figure 15

(a) The ratio of ion to electron energization $Q_i/Q_e$ as a function of plasma $\beta$ from several hybrid-kinetic simulations (${\beta }_{i0}=1/9$ simulation from Ref. [80], ${\beta }_{i0}=1$ and 0.3 simulations from Ref. [79], and ${\beta }_{i0}=16$ and 4 simulations herein). Ion heating is measured directly from the evolution of the thermal energy of the particles. Electron heating is inferred from hyperresistive dissipation near the grid scale. For comparison, the dashed line shows the results from the hybrid-gyrokinetic simulations of Ref. [78]. (b) Time dependence of ion heating ($Q_i$) and hyperresistive dissipation (${ϵ}_{\eta }$) in the high-$\beta$ hybrid-kinetic simulations. All lines are normalized to the total energy dissipation $\dot{ϵ}$ averaged over the final $3.5t_{\text{cross}}$ of each simulation.

Figure 16

(a),(b) Snapshots of parallel magnetic-field fluctuations $\delta B_{\parallel }$ (relative to ${\mathit{B}}_0$) during the early stage of the ${\beta }_{i0}=16$ simulation ($t\approx 0.4t_{\text{cross}}$). Panel (a) shows a slice perpendicular to the guide field; panel (b) shows a slice along the guide field. There is a clear “cross” pattern in both slices, indicating oblique mirror fluctuations. (c) The distribution function of points in the box as a function of their $\beta$ and pressure anisotropy. Dashed lines show the thresholds for mirror [18, 19, 20] and fluid-firehose [14, 15, 16, 17] instabilities; dot-dashed line represents a threshold for the AIC instability [21, 74] and kinetic firehose instabilities [45, 84]. In this snapshot, the majority of the box is above the mirror threshold. The black dot indicates the initial position of the simulation box. (d) Sixth-order structure function of the fluid velocity- and magnetic-field fluctuations. High-order structure functions are chosen to highlight intense small-scale fluctuations, which for this snapshot are mirror modes: oblique modes producing kinetic-range peaks in both parallel and perpendicular structure functions of parallel magnetic-field fluctuations of parallel magnetic field.

Figure 17

The same as Fig. 16, but for a slightly later time $t\approx 0.6t_{\text{cross}}$, at which quasiparallel fluctuations are manifest. The perpendicular component of magnetic field is shown instead of the parallel one. Those fluctuations are produced by the AIC instability. The structure functions show two predominant modes of fluctuations: oblique fluctuations in $\delta B_{\parallel }$ caused by mirror instability and quasiparallel fluctuations in $\delta B_{\perp }$, which are AIC waves.

Figure 18

Nonlocal energy transfer functions averaged over the quasisteady state due to the Reynolds stress (left), Maxwell stress (center), and anisotropic viscous stress (right) in ${\beta }_{i0}=16$ simulation. Most of the transfer in the early stages of the simulation is local, but there is considerable nonlocal transfer mediated by the Maxwell stress in the quasisteady state. Panels (j)–(l) show the remaining transfer terms in the momentum equation as well as terms in the induction equation and the equation for thermal energy. Note that terms associated with compressions and gradients of isotropic pressures are smaller than “advectionlike” terms.

Figure 19

Parallel (a),(c) and perpendicular (b),(d) ion distribution function at the end of the ${\beta }_{i0}=16$ (a),(b) and ${\beta }_{i0}=4$ (c),(d) simulations. Parallel distribution functions (a),(c) have a flat core, which we associate with the Landau damping and the firehose instability [45]. Dashed lines show best-fit Maxwellians to $f(w_{\parallel })$ at $w_{\parallel }\approx 1.6v_{\mathrm{th}i0}$ (in particular, from 1.59 to $1.61v_{\mathrm{th}i0}$ to exclude the flattened part of the core) and to $f(w_{\perp })$ at $w_{\perp }<v_{\mathrm{th}i0}$. (e)–(h) The same distribution functions, plotted on logarithmic scale. The core and the tail of the distributions may be approximated with Gaussians (straight lines in these coordinates). It is apparent that the wings of the distribution functions are much more isotropic than the cores ($T_{\parallel }^{\mathrm{tail}}\sim T_{\perp }^{\mathrm{tail}}$, while $T_{\parallel }^{\mathrm{core}}>T_{\perp }^{\mathrm{core}}$).