Quasi-linear theory of perpendicular ion heating by critically balanced turbulence

TL;DR

This work develops a quasi-linear framework to calculate perpendicular ion heating in collisionless, anisotropic RMHD turbulence with arbitrary imbalance. By modeling the wavevector-frequency spectrum as a function of cross-helicity $\sigma_{\rm c}$ and integrating it into RMHD-appropriate diffusion coefficients, it derives a general heating rate $Q_\perp \propto \xi_{\rho,\textrm{th}}^{3} F(\xi_{\rho,\textrm{th}};\sigma_{\rm c})$ that smoothly connects stochastic-heating-like behavior in balanced turbulence to cyclotron-resonant heating in the imbalanced regime. The suppression factor $F$ captures how the spectrum narrows with increasing imbalance, reducing heating at small $\xi_{\rho,\textrm{th}}$ and reproducing exponential-like reductions in the appropriate limits; analytic results in the limiting cases reinforce the link between the two heating channels. The findings provide quantitative, testable predictions for simulations and spacecraft data, clarifying how turbulence imbalance shapes ion heating and the partitioning of turbulent energy.

Abstract

In collisionless astrophysical plasmas, turbulence mediates the partitioning of free energy among cascade channels and its dissipation into ion and electron heat. The resulting ion heating is often anisotropic, with ions observed to be preferentially heated perpendicular to the local magnetic field; understanding the mechanisms responsible for this heating is a key step in understanding the evolution of such plasmas. In this paper, we use the framework of quasi-linear theory to compute analytically the heating rates of ions interacting with turbulent, large-scale Alfvénic fluctuations. We show how the imbalance of the turbulence (the difference in energies between Alfvénic fluctuations travelling parallel and antiparallel to the magnetic field) modifies the spatiotemporal spectrum of these fluctuations, allowing the heating mechanism to transition between two commonly-studied mechanisms: stochastic heating in balanced turbulence to resonant-cyclotron heating in imbalanced turbulence. The resultant heating rate is found to have a general form regardless of the level of imbalance, exhibiting a suppression related to the conservation of the ions' magnetic moment at small turbulent amplitudes and recovering previous empirical results in a formal calculation. The results of this work help to consolidate our qualitative understanding of ion heating within astrophysical plasmas, as well as yielding specific quantitative predictions to analyse simulations and observations.

Figures (22)

• Figure 1: The 2D spectrum model \ref{['eq:RMHDSpecNorm']} with the Goldreich1995-fv critical balance scaling $s_{\rm CB} = 2/3$, $C=1$, and $\xi_{\rho,\textrm{th}}=0.5$ (with critical balance line $k_\| d_{\rm i}=0.5(k_\perp\rho_{\rm i})^{2/3}$). The grey boundaries represent the spectrum cutoff at $|k_\| d_{\rm i}| > 1$ or $k_\perp\rho_{\rm i} > 1$ assumed in \ref{['eq:RMHDSpecNorm']}. Side panels show slices through the spectra (normalised to the maximum value along the slice), showing the individual scalings of $k_\perp$ and $k_\|$ above and below the CB line.
• Figure 2: Slices at constant $k_\perp L_\perp = 10$ through the model wavevector-frequency spectrum of RMHD turbulence (\ref{['eq:RMHDFunctionalForm']}, top), showing it qualitatively reproduces features of the RMHD simulations presented in Appendix \ref{['app:RMHDModel']} (bottom). The turbulence has imbalance $\sigma_{\rm c} = 0,\ 0.59$, and $0.96$ (left, middle and right columns respectively), and the model sets $\xi_{\rho,\textrm{th}} = 1$. The solid black lines correspond to zero frequency and the Alfvén dispersion relation $\omega_{\rm A} = \pm k_zv_{\rm A}$; the red dashed line is the critical balance scaling $k_\|^{\rm CB}$, such that everything with $k_\| < k_\|^{\rm CB}$ is within the CB cone. The frequencies are normalised to the outer scale Alfvén frequency of the simulations $\omega_{\rm A0}=v_{\rm A}/L_z$.
• Figure 3: Slices at constant $k_z L_z = 10$ through the model wavevector-frequency spectrum of RMHD turbulence (\ref{['eq:RMHDFunctionalForm']}, top), showing it qualitatively reproduces features of the RMHD simulations presented in Appendix \ref{['app:RMHDModel']} (bottom). The turbulence has imbalance $\sigma_{\rm c} = 0,\ 0.59$, and $0.96$ (left, middle and right columns respectively), and the model sets $\xi_{\rho,\textrm{th}} = 1$. Solid black lines represent the centre of the Alfvén dispersion relation $\omega_{\rm A}=\pm k_zv_{\rm A}$ at the given value of $k_z$, and the red dashed line shows where $k_\perp = (k_\|^{\rm CB})^{3/2}$. Frequencies are normalised to the outer scale Alfvén frequency $\omega_{\rm A0}=v_{\rm A}/L_z$.
• Figure 4: The perpendicular heating rate $Q_\perp$ in the fully imbalanced limit, where the general quasi-linear theory of velocity-space diffusion reduces to diffusion along contours of constant energy in the wave frame Kennel1966-rx. $Q_\perp$ is calculated numerically from \ref{['eq:QprpImbalancedCase']} using the imbalanced limit of the generalised diffusion coefficient $\mathcal{D}$\ref{['eq:DppImbalancedTurbGS95Spec']} for different values of $\beta_{\rm i}$ (with $C=1$ in $\tilde{\mathcal{E}}_{\rm 2D}$, \ref{['eq:RMHDSpecNorm']}). Dotted lines show only to the contribution of fluctuations below the CB cone to $Q_\perp$.
• Figure 5: The perpendicular heating rate $Q_\perp$ in the balanced limit for different values of $\beta_{\rm i}$, calculated numerically using \ref{['eq:QprpBalancedCase']} with $C=5$ in $\tilde{\mathcal{E}}_{\rm 2D}$\ref{['eq:RMHDSpecNorm']}. To better highlight the suppression, only the contribution from modes below the CB cone are considered. The black dotted line shows the $\beta_{\rm i}$-independent analytic expression for $Q_\perp$\ref{['eq:QprpBalancedLowBeta']} (obtained in the limit $|\tilde{v}_\|| \ll 1$ and $\tilde{k}_\|\ll \tilde{k}_\|^{\rm CB}$) with $\hat{c}_1\approx 5$ and $\hat{c}_2 \approx 0.3$. These show qualitatively similar behaviour to the empirical stochastic heating formula \ref{['eq:SHRate']}Chandran2010-ow, shown by the blue dashed line with the same values for $c_1$ and $c_2$.