A theoretical framework for low-frequency electromagnetic (drift-)kinetic turbulence in a collisionless, multi-species plasma is presented. The result generalises reduced magnetohydrodynamics (RMHD) and kinetic RMHD (Schekochihin et al. 2009) for pre
ssure-anisotropic plasmas, allowing for species drifts---a situation routinely encountered in the solar wind and presumably ubiquitous in hot dilute astrophysical plasmas (e.g. intracluster medium). Two main objectives are achieved. First, in a non-Maxwellian plasma, the relationships between fluctuating fields (e.g., the Alfven ratio) are order-unity modified compared to the more commonly considered Maxwellian case, and so a quantitative theory is developed to support quantitative measurements now possible in the solar wind. The main physical feature of low-frequency plasma turbulence survives the generalisation to non-Maxwellian distributions: Alfvenic and compressive fluctuations are energetically decoupled, with the latter passively advected by the former; the Alfvenic cascade is fluid, satisfying RMHD equations (with the Alfven speed modified by pressure anisotropy and species drifts), whereas the compressive cascade is kinetic and subject to collisionless damping. Secondly, the organising principle of this turbulence is elucidated in the form of a generalised kinetic free-energy invariant. It is shown that non-Maxwellian features in the distribution function reduce the rate of phase mixing and the efficacy of magnetic stresses; these changes influence the partitioning of free energy amongst the various cascade channels. As the firehose or mirror instability thresholds are approached, the dynamics of the plasma are modified so as to reduce the energetic cost of bending magnetic-field lines or of compressing/rarefying them. Finally, it is shown that this theory can be derived as a long-wavelength limit of non-Maxwellian slab gyrokinetics.
We present numerical evidence that in strong Alfvenic turbulence, the critical balance principle---equality of the nonlinear decorrelation and linear propagation times---is scale invariant, in the sense that the probability distribution of the ratio
of these times is independent of scale. This result only holds if the local alignment of the Elsasser fields is taken into account in calculating the nonlinear time. At any given scale, the degree of alignment is found to increase with fluctuation amplitude, supporting the idea that the cause of alignment is mutual dynamical shearing of Elsasser fields. The scale-invariance of critical balance (while all other quantities of interest are strongly intermittent, i.e., have scale-dependent distributions) suggests that it is the most robust of the scaling principles used to describe Alfvenic turbulence. The quality afforded by situ fluctuation measurements in the solar wind allows for direct verification of this fundamental principle.
Magnetic reconnection in strongly magnetized (low-beta), weakly collisional plasmas is investigated using a novel fluid-kinetic model [Zocco & Schekochihin, Phys. Plasmas 18, 102309 (2011)] which retains non-isothermal electron kinetics. It is shown
that electron heating via Landau damping (linear phase mixing) is the dominant dissipation mechanism. In time, electron heating occurs after the peak of the reconnection rate; in space, it is concentrated along the separatrices of the magnetic island. For sufficiently large systems, the peak reconnection rate is $cE_{max}approx 0.2v_AB_{y,0}$, where $v_A$ is the Alfven speed based on the reconnecting field $B_{y,0}$. The island saturation width is the same as in MHD models except for small systems, when it becomes comparable to the kinetic scales.
Experimental data from the Mega Amp Spherical Tokamak (MAST) is used to show that the inverse gradient scale length of the ion temperature R/LTi (normalized to the major radius R) has its strongest local correlation with the rotational shear and the
pitch angle of the magnetic field (or, equivalently, an inverse correlation with q/{epsilon}, the safety factor/the inverse aspect ratio). Furthermore, R/LTi is found to be inversely correlated with the gyro-Bohm-normalized local turbulent heat flux estimated from the density fluctuation level measured using a 2D Beam Emission Spectroscopy (BES) diagnostic. These results can be explained in terms of the conjecture that the turbulent system adjusts to keep R/LTi close to a certain critical value (marginal for the excitation of turbulence) determined by local equilibrium parameters (although not necessarily by linear stability).
Beam Emission Spectroscopy (BES) measurements of ion-scale density fluctuations in the MAST tokamak are used to show that the turbulence correlation time, the drift time associated with ion temperature or density gradients, the particle (ion) streami
ng time along the magnetic field and the magnetic drift time are consistently comparable, suggesting a critically balanced turbulence determined by the local equilibrium. The resulting scalings of the poloidal and radial correlation lengths are derived and tested. The nonlinear time inferred from the density fluctuations is longer than the other times; its ratio to the correlation time scales as $ u_{*i}^{-0.8pm0.1}$, where $ u_{*i}=$ ion collision rate/streaming rate. This is consistent with turbulent decorrelation being controlled by a zonal component, invisible to the BES, with an amplitude exceeding the drift waves by $sim u_{*i}^{-0.8}$.
A 2D linear theory of the instability of Sweet-Parker (SP) current sheets is developed in the framework of Reduced MHD. A local analysis is performed taking into account the dependence of a generic equilibrium profile on the outflow coordinate. The p
lasmoid instability [Loureiro et al, Phys. Plasmas {bf 14}, 100703 (2007)] is recovered, i.e., current sheets are unstable to the formation of a large-wave-number chain of plasmoids ($k_{rm max}Lsheet sim S^{3/8}$, where $k_{rm max}$ is the wave-number of fastest growing mode, $S=Lsheet V_A/eta$ is the Lundquist number, $Lsheet$ is the length of the sheet, $V_A$ is the Alfven speed and $eta$ is the plasma resistivity), which grows super-Alfvenically fast ($gmaxtau_Asim S^{1/4}$, where $gmax$ is the maximum growth rate, and $tau_A=Lsheet/V_A$). For typical background profiles, the growth rate and the wave-number are found to {it increase} in the outflow direction. This is due to the presence of another mode, the Kelvin-Helmholtz (KH) instability, which is triggered at the periphery of the layer, where the outflow velocity exceeds the Alfven speed associated with the upstream magnetic field. The KH instability grows even faster than the plasmoid instability, $gmax tau_A sim k_{rm max} Lsheetsim S^{1/2}$. The effect of viscosity ($ u$) on the plasmoid instability is also addressed. In the limit of large magnetic Prandtl numbers, $Pm= u/eta$, it is found that $gmaxsim S^{1/4}Pm^{-5/8}$ and $k_{rm max} Lsheetsim S^{3/8}Pm^{-3/16}$, leading to the prediction that the critical Lundquist number for plasmoid instability in the $Pmgg1$ regime is $Scritsim 10^4Pm^{1/2}$. These results are verified via direct numerical simulation of the linearized equations, using a new, analytical 2D SP equilibrium solution.
Differential rotation is known to suppress linear instabilities in fusion plasmas. However, even in the absence of growing eigenmodes, subcritical fluctuations that grow transiently can lead to sustained turbulence. Here transient growth of electrost
atic fluctuations driven by the parallel velocity gradient (PVG) and the ion temperature gradient (ITG) in the presence of a perpendicular ExB velocity shear is considered. The maximally simplified case of zero magnetic shear is treated in the framework of a local shearing box. There are no linearly growing eigenmodes, so all excitations are transient. The maximal amplification factor of initial perturbations and the corresponding wavenumbers are calculated as functions of q/epsilon (=safety factor/aspect ratio), temperature gradient and velocity shear. Analytical results are corroborated and supplemented by linear gyrokinetic numerical tests. For sufficiently low values of q/epsilon (<7 in our model), regimes with fully suppressed ion-scale turbulence are possible. For cases when turbulence is not suppressed, an elementary heuristic theory of subcritical PVG turbulence leading to a scaling of the associated ion heat flux with q, epsilon, velocity shear and temperature gradient is proposed; it is argued that the transport is much less stiff than in the ITG regime.
Weak Alfvenic turbulence in a periodic domain is considered as a mixed state of Alfven waves interacting with the two-dimensional (2D) condensate. Unlike in standard treatments, no spectral continuity between the two is assumed and indeed none is fou
nd. If the 2D modes are not directly forced, k^{-2} and k^{-1} spectra are found for the Alfven 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 counterpropagting Alfven 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.
A numerical study of magnetic reconnection in the large-Lundquist-number ($S$), plasmoid-dominated regime is carried out for $S$ up to $10^7$. The theoretical model of Uzdensky {it et al.} [Phys. Rev. Lett. {bf 105}, 235002 (2010)] is confirmed and p
artially amended. The normalized reconnection rate is $ ormEeffsim 0.02$ independently of $S$ for $Sgg10^4$. The plasmoid flux ($Psi$) and half-width ($w_x$) distribution functions scale as $f(Psi)sim Psi^{-2}$ and $f(w_x)sim w_x^{-2}$. The joint distribution of $Psi$ and $w_x$ shows that plasmoids populate a triangular region $w_xgtrsimPsi/B_0$, where $B_0$ is the reconnecting field. It is argued that this feature is due to plasmoid coalescence. Macroscopic monster plasmoids with $w_xsim 10$% of the system size are shown to emerge in just a few Alfven times, independently of $S$, suggesting that large disruptive events are an inevitable feature of large-$S$ reconnection.
We consider the problem of self-regulated heating and cooling in galaxy clusters and the implications for cluster magnetic fields and turbulence. Viscous heating of a weakly collisional magnetised plasma is regulated by the pressure anisotropy with r
espect to the local direction of the magnetic field. The intracluster medium is a high-beta plasma, where pressure anisotropies caused by the turbulent stresses and the consequent local changes in the magnetic field will trigger very fast microscale instabilities. We argue that the net effect of these instabilities will be to pin the pressure anisotropies at a marginal level, controlled by the plasma beta parameter. This gives rise to local heating rates that turn out to be comparable to the radiative cooling rates. Furthermore, we show that a balance between this heating and Bremsstrahlung cooling is thermally stable, unlike the often conjectured balance between cooling and thermal conduction. Given a sufficient (and probably self-regulating) supply of turbulent power, this provides a physical mechanism for mitigating cooling flows and preventing cluster core collapse. For observed density and temperature profiles, the assumed balance of viscous heating and radiative cooling allows us to predict magnetic-field strengths, turbulent velocities and turbulence scales as functions of distance from the centre. Specific predictions and comparisons with observations are given for several different clusters. Our predictions can be further tested by future observations of cluster magnetic fields and turbulent velocities.
