MHD Turbulence: A Biased Review

This review puts the developments of the last few years in the context of the canonical time line (Kolmogorov to Iroshnikov-Kraichnan to Goldreich-Sridhar to Boldyrev). It is argued that Beresnyak's objection that Boldyrev's alignment theory violates the RMHD rescaling symmetry can be reconciled with alignment if the latter is understood as an intermittency effect. Boldyrev's scalings, recovered in this interpretation, are thus an example of a physical theory of intermittency in a turbulent system. Emergence of aligned structures brings in reconnection physics, so the theory of MHD turbulence intertwines with the physics of tearing and current-sheet disruption. Recent work on this by Loureiro, Mallet et al. is reviewed and it is argued that we finally have a reasonably complete picture of MHD cascade all the way to the dissipation scale. This picture appears to reconcile Beresnyak's Kolmogorov scaling of the dissipation cutoff with Boldyrev's aligned cascade. These ideas also enable some progress in understanding saturated MHD dynamo, argued to be controlled by reconnection and to contain, at small scales, a tearing-mediated cascade similar to its strong-mean-field counterpart. On the margins of this core narrative, standard weak-MHD-turbulence theory is argued to require adjustment - and a scheme for it is proposed - to take account of the part that a spontaneously emergent 2D condensate plays in mediating the Alfven-wave cascade. This completes the picture of the MHD cascade at large scales. A number of outstanding issues are surveyed, concerning imbalanced MHD turbulence (for which a new theory is proposed), residual energy, subviscous and decaying regimes of MHD turbulence (where reconnection again features prominently). Finally, it is argued that the natural direction of research is now away from MHD and into kinetic territory.Comment: 188 pages, 49 figures; (re)submitted to JPP; this version is substantially modified from v1, especially secs 7.3, 8.2, 11, 12.4, 13.4 and appendices B.3, C.5, C.

Thermal disequilibration of ions and electrons by collisionless plasma turbulence

Does overall thermal equilibrium exist between ions and electrons in a weakly collisional, magnetised, 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 discs around black holes. In the context of discs, 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 Alfv\'enic 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, $\beta_\mathrm{i}$: it ranges from $\sim0.05$ at $\beta_\mathrm{i}=0.1$ to at least $30$ for $\beta_\mathrm{i} \gtrsim 10$. This energy partition is approximately insensitive to the ion-to-electron temperature ratio $T_\mathrm{i}/T_\mathrm{e}$. Thus, in the absence of other equilibrating mechanisms, a collisionless plasma system heated via Alfv\'enic turbulence will tend towards a nonequilibrium state in which one of the species is significantly hotter than the other, viz., hotter ions at high $\beta_\mathrm{i}$, hotter electrons at low $\beta_\mathrm{i}$. Spectra of electromagnetic fields and the ion distribution function in 5D phase space exhibit an interesting new magnetically dominated regime at high $\beta_i$ and a tendency for the ion heating to be mediated by nonlinear phase mixing ("entropy cascade") when $\beta_\mathrm{i}\lesssim1$ and by linear phase mixing (Landau damping) when $\beta_\mathrm{i}\gg1

Fluidization of collisionless plasma turbulence

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

Firehose and Mirror Instabilities in a Collisionless Shearing Plasma

Hybrid-kinetic numerical simulations of firehose and mirror instabilities in a collisionless plasma are performed in which pressure anisotropy is driven as the magnetic field is changed by a persistent linear shear $S$. For a decreasing field, it is found that mostly oblique firehose fluctuations grow at ion Larmor scales and saturate with energies $\sim$$S^{1/2}$; the pressure anisotropy is pinned at the stability threshold by particle scattering off microscale fluctuations. In contrast, nonlinear mirror fluctuations are large compared to the ion Larmor scale and grow secularly in time; marginality is maintained by an increasing population of resonant particles trapped in magnetic mirrors. After one shear time, saturated order-unity magnetic mirrors are formed and particles scatter off their sharp edges. Both instabilities drive sub-ion-Larmor--scale fluctuations, which appear to be kinetic-Alfv\'{e}n-wave turbulence. Our results impact theories of momentum and heat transport in astrophysical and space plasmas, in which the stretching of a magnetic field by shear is a generic process.Comment: 5 pages, 8 figures, accepted for publication in Physical Review Letter

Magneto-immutable turbulence in weakly collisional plasmas

We propose that pressure anisotropy causes weakly collisional turbulent plasmas to self-organize so as to resist changes in magnetic-field strength. We term this effect "magneto-immutability" by analogy with incompressibility (resistance to changes in pressure). The effect is important when the pressure anisotropy becomes comparable to the magnetic pressure, suggesting that in collisionless, weakly magnetized (high-$\beta$) plasmas its dynamical relevance is similar to that of incompressibility. Simulations of magnetized turbulence using the weakly collisional Braginskii model show that magneto-immutable turbulence is surprisingly similar, in most statistical measures, to critically balanced MHD turbulence. However, in order to minimize magnetic-field variation, the flow direction becomes more constrained than in MHD, and the turbulence is more strongly dominated by magnetic energy (a nonzero "residual energy"). These effects represent key differences between pressure-anisotropic and fluid turbulence, and should be observable in the $\beta\gtrsim1$ turbulent solar wind.Comment: Accepted for publication in J. Plasma Phy

Weak Alfvén-wave turbulence revisited

Weak Alfvénic turbulence in a periodic domain is considered as a mixed state of Alfvén waves interacting with the two-dimensional (2D) condensate. Unlike in standard treatments, no spectral continuity between the two is assumed, and, indeed, none is found. If the 2D modes are not directly forced, k−2 and k−1 spectra are found for the Alfvén waves and the 2D modes, respectively, with the latter less energetic than the former. The wave number at which their energies become comparable marks the transition to strong turbulence. For imbalanced energy injection, the spectra are similar, and the Elsasser ratio scales as the ratio of the energy fluxes in the counterpropagating Alfvén waves. If the 2D modes are forced, a 2D inverse cascade dominates the dynamics at the largest scales, but at small enough scales, the same weak and then strong regimes as described above are achieved

Reconnection-controlled decay of magnetohydrodynamic turbulence and the role of invariants

We present a new theoretical picture of magnetically dominated, decaying turbulence in the absence of a mean magnetic field. We demonstrate that such turbulence is governed by the reconnection of magnetic structures, and not by ideal dynamics, as has previously been assumed. We obtain predictions for the magnetic-energy decay laws by proposing that turbulence decays on reconnection timescales, while respecting the conservation of certain integral invariants representing topological constraints satisfied by the reconnecting magnetic field. As is well known, the magnetic helicity is such an invariant for initially helical field configurations, but does not constrain non-helical decay, where the volume-averaged magnetic-helicity density vanishes. For such a decay, we propose a new integral invariant, analogous to the Loitsyansky and Saffman invariants of hydrodynamic turbulence, that expresses the conservation of the random (scaling as $\mathrm{volume}^{1/2}$) magnetic helicity contained in any sufficiently large volume. Our treatment leads to novel predictions for the magnetic-energy decay laws: in particular, while we expect the canonical $t^{-2/3}$ power law for helical turbulence when reconnection is fast (i.e., plasmoid-dominated or stochastic), we find a shallower $t^{-4/7}$ decay in the slow `Sweet-Parker' reconnection regime, in better agreement with existing numerical simulations. For non-helical fields, for which there currently exists no definitive theory, we predict power laws of $t^{-10/9}$ and $t^{-20/17}$ in the fast- and slow-reconnection regimes, respectively. We formulate a general principle of decay of turbulent systems subject to conservation of Saffman-like invariants, and propose how it may be applied to MHD turbulence with a strong mean magnetic field and to isotropic MHD turbulence with initial equipartition between the magnetic and kinetic energies.Comment: 30 pages, 15 figures, accepted by Phys. Rev.

Diffusion of passive scalar in a finite-scale random flow

We consider a solvable model of the decay of scalar variance in a single-scale random velocity field. We show that if there is a separation between the flow scale k_flow^{-1} and the box size k_box^{-1}, the decay rate lambda ~ (k_box/k_flow)^2 is determined by the turbulent diffusion of the box-scale mode. Exponential decay at the rate lambda is preceded by a transient powerlike decay (the total scalar variance ~ t^{-5/2} if the Corrsin invariant is zero, t^{-3/2} otherwise) that lasts a time t~1/\lambda. Spectra are sharply peaked at k=k_box. The box-scale peak acts as a slowly decaying source to a secondary peak at the flow scale. The variance spectrum at scales intermediate between the two peaks (k_box0). The mixing of the flow-scale modes by the random flow produces, for the case of large Peclet number, a k^{-1+delta} spectrum at k>>k_flow, where delta ~ lambda is a small correction. Our solution thus elucidates the spectral make up of the ``strange mode,'' combining small-scale structure and a decay law set by the largest scales.Comment: revtex4, 8 pages, 4 figures; final published versio