Quantitative theory and kinetic simulation of the fluctuation dynamo in a collisionless, weakly magnetized plasma: the effective-collision-rate regime sequence, saturation, and effective viscosities at intracluster-medium conditions

In a hot, weakly magnetized, effectively collisionless plasma such as the intracluster medium (ICM), the fluctuation dynamo — the amplification of a seed magnetic field by turbulence — is regulated not by Coulomb collisions but by pitch-angle scattering off microinstabilities (firehose and mirror), giving a field-dependent effective collision rate. St-Onge and Kunz (2018) simulated this regime kinetically (with a hybrid-kinetic code, Pegasus, in which ions are treated as kinetic particles and electrons as a massless charge-neutralizing fluid) and identified a chain of open problems: the quantitative capture of the predicted sequence of effective-collision-rate regimes before saturation ($\nu_{\rm eff}\sim\Omega_i\rightarrow\nu_{\rm eff}\sim k_\parallel v_{\mathrm{thi}}\rightarrow\nu_{\rm eff}\sim S\beta_i$, where $\Omega_i$ is the ion gyrorotation rate, $k_\parallel$ the field-parallel wavenumber, $v_{\mathrm{thi}}$ the ion thermal speed, $S$ the flow shear rate, and $\beta_i$ the ratio of ion thermal to magnetic pressure), needed to test self-accelerating (explosive) dynamo theories; the saturated spectral peak location, the persistence of the gap between the parallel-viscous scale and the magnetic-energy peak, and the evolution of folds into ribbons at higher resolution; the mapping from kinetic scattering physics to effective anisotropic viscosities ($\mathrm{Re}_\parallel$, $\mathrm{Re}_\perp$) in astrophysical parameter regimes; and the sensitivity of growth rates, spectra, and saturation to driving-scale parameters. These four issues remain unresolved.

Background

The fluctuation dynamo is the stochastic amplification of a weak seed magnetic field by turbulent plasma motion; it is the standard mechanism invoked to explain microgauss magnetic fields in galaxy clusters. In the intracluster medium (ICM), the dilute, hot plasma is weakly collisional (the Coulomb mean free path is comparable to the turbulent injection scale) and weakly magnetized ($\beta_i$, the ratio of ion thermal pressure to magnetic pressure, is $\sim 10^2$), so magnetohydrodynamic (MHD) closures fail: field-line stretching drives pressure anisotropy $\Delta p = p_\perp - p_\parallel$ with respect to the magnetic-field direction at the rate set by the flow shear $S$, and the plasma responds by firehose and mirror microinstabilities (kinetic instabilities triggered when the anisotropy violates threshold conditions) whose fluctuating fields provide the pitch-angle scattering that breaks the anisotropy. The net effect is an effective collision rate $\nu_{\rm eff}$ set by the microinstability scattering rather than by Coulomb collisions, entering a Braginskii-MHD-like description (Braginskii viscosity is the field-parallel viscous stress of a magnetized weakly collisional plasma) with effective parallel Reynolds number $\mathrm{Re}_\parallel$. Schekochihin and Cowley (2006) and Mogavero and Schekochihin (2014) argued that because $\nu_{\rm eff}$ decreases as the growing magnetic field tightens the microinstability thresholds, the effective $\mathrm{Re}_\parallel$ grows with the field, so the dynamo growth rate may itself accelerate — an explosive dynamo that could magnetize clusters faster than a constant-rate dynamo. Melville, Schekochihin and Kunz (2016) confirmed parts of this scattering-rate prescription in kinetic simulations, and Rincon, Califano, Schekochihin and Valentini (2016) observed growth rates rising as the initial $\beta_i$ fell in low-resolution hybrid-kinetic simulations. St-Onge and Kunz (2018) then performed the definitive hybrid-kinetic simulation study of this regime (ions kinetic, electrons a massless isothermal fluid; Ohmic and hyper-resistivity representing unresolved electron-scale dissipation), verifying the $\nu_{\rm eff}\sim S\beta_i$ regime near saturation but lacking the scale separation and resolution to resolve the earlier transitions $\nu_{\rm eff}\sim\Omega_i$ and $\nu_{\rm eff}\sim k_\parallel v_{\mathrm{thi}}$ (scattering dominated by ion-Larmor-scale fluctuations versus by parallel phase-mixing), and observing folded magnetic sheets (field reversals compressed into thin structures) tending toward ribbon-like structures whose saturated spectrum peaked well above the resistive scale — with the open question of whether that peak (inferred observationally to sit near kpc scales in clusters) is set by kinetic-scale physics. The four unresolved issues listed above jointly ask for a quantitative, microphysically grounded account of the collisionless fluctuation dynamo from kinematic growth to saturation at ICM-relevant parameters; the effective anisotropic viscosities ($\mathrm{Re}_\parallel$, $\mathrm{Re}_\perp$) that such an account fixes are the key microphysical inputs for cluster simulations of viscous heating, X-ray cold fronts, and rising buoyant bubbles.

Problem Statement

Establish a quantitative, microphysically grounded account of the fluctuation dynamo — the stochastic amplification of a weak seed magnetic field by turbulence — in a collisionless, weakly magnetized plasma at intracluster-medium-relevant conditions, comprising four jointly linked targets from St-Onge and Kunz (2018): (1) resolve and quantitatively characterize the sequence of effective-collision-rate regimes before saturation, $\nu_{\rm eff}\sim\Omega_i\rightarrow\nu_{\rm eff}\sim k_\parallel v_{\mathrm{thi}}\rightarrow\nu_{\rm eff}\sim S\beta_i$ (where $\nu_{\rm eff}$ is the effective ion pitch-angle scattering rate set by microinstability turbulence rather than Coulomb collisions, $\Omega_i$ the ion gyrorotation rate, $k_\parallel$ the wavenumber along the magnetic field, $v_{\mathrm{thi}}$ the ion thermal speed, $S$ the characteristic turbulent shear rate, and $\beta_i$ the ratio of ion thermal pressure to magnetic pressure), with enough scale separation and resolution to decide whether dynamo growth accelerates (is explosive, as predicted by the self-accelerating effective-viscosity theory) or stays approximately constant; (2) determine the saturated spectral peak location of the magnetic energy, whether the gap between the parallel-viscous scale (the scale at which field-parallel viscous stresses damp the turbulent cascade) and the magnetic-energy peak persists or closes, and to what degree folded magnetic sheets (field reversals compressed into thin structures elongated in one direction) evolve into ribbons (elongated in two directions), using higher resolution; (3) establish the quantitative mapping from kinetic scattering physics (microinstability-regulated pitch-angle scattering) to the effective anisotropic viscosities, i.e. accurate values of the effective Reynolds numbers $\mathrm{Re}_\parallel$ and $\mathrm{Re}_\perp$ (dimensionless ratios of advective to viscous transport along and across the local magnetic field) as functions of astrophysical plasma parameters; and (4) quantify the sensitivity of measured growth rates, spectra, and saturation levels to driving-scale parameters such as $k_F\rho_{i0}$ (the ratio of the turbulent driving scale, $1/k_F$, to the initial ion gyroradius $\rho_{i0}$) and the sonic Mach number $M$ (the ratio of flow speed to sound speed). The four components are jointly required and serve the single overarching objective of a quantitative account of the collisionless dynamo from kinematic growth to saturation; resolving only some of the four targets does not resolve the task.

References

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