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We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh–Bénard convection between free-slip boundaries. We focus on the ability of the convection to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers ( $\\mathit{Pr}$ ) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number ( $\\mathit{Ra}$ ) sufficiently, and we explore the resulting convection for $\\mathit{Ra}$ up to $10^{10}$ . When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as $\\mathit{Ra}\\rightarrow \\infty$ . The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with $\\mathit{Ra}$ . When the large-scale shear is present with $\\mathit{Pr}\\lesssim 2$ , the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with $\\mathit{Ra}$ for $\\mathit{Pr}=1$ . When the shear is present with $\\mathit{Pr}\\gtrsim 3$ , the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of $\\mathit{Ra}$ , but the growth rates are slower than any previously reported for Rayleigh–Bénard convection without large-scale shear. We find that the Nusselt numbers grow proportionally to $\\mathit{Ra}^{0.077}$ when $\\mathit{Pr}=3$ and to $\\mathit{Ra}^{0.19}$ when $\\mathit{Pr}=10$ . Analogies with tokamak plasmas are described.

It is proposed that critical balance – a scale-by-scale balance between the linear propagation and nonlinear interaction time scales – can be used as a universal scaling conjecture for determining the spectra of strong turbulence in anisotropic wave systems. Magnetohydrodynamic (MHD), rotating and stratified turbulence are considered under this assumption and, in particular, a novel and experimentally testable energy cascade scenario and a set of scalings of the spectra are proposed for low-Rossby-number rotating turbulence. It is argued that in neutral fluids the critically balanced anisotropic cascade provides a natural path from strong anisotropy at large scales to isotropic Kolmogorov turbulence at very small scales. It is also argued that the k−2⊥ spectra seen in recent numerical simulations of low-Rossby-number rotating turbulence may be analogous to the k−3/2⊥ spectra of the numerical MHD turbulence in the sense that they could be explained by assuming that fluctuations are polarised (aligned) approximately as inertial waves (Alfvén waves for MHD).

We argue that the hypothesis of preservation of shape of dimensionless second- and third-order correlations during decay of incompressible homogeneous magnetohydrodynamic (MHD) turbulence requires, in general, at least two independent similarity length scales. These are associated with the two Elsässer energies. The existence of similarity solutions for the decay of turbulence with varying cross-helicity implies that these length scales cannot remain in proportion, opening the possibility for a wide variety of decay behaviour, in contrast to the simpler classic hydrodynamics case. Although the evolution equations for the second-order correlations lack explicit dependence on either the mean magnetic field or the magnetic helicity, there is inherent implicit dependence on these (and other) quantities through the third-order correlations. The self-similar inertial range, a subclass of the general similarity case, inherits this complexity so that a single universal energy spectral law cannot be anticipated, even though the same pair of third-order laws holds for arbitrary cross-helicity and magnetic helicity. The straightforward notion of universality associated with Kolmogorov theory in hydrodynamics therefore requires careful generalization and reformulation in MHD.

Laboratory observations show how a macroscopic magnetohydrodynamic plasma instability drives a fine-scale secondary instability that is associated with magnetic reconnection. Magnetic reconnection gathers speed Magnetic reconnection is an energy-releasing phenomenon in plasma physics involving the breakage and reconnection of lines of magnetic field. Although localized diffusion of field lines is part of the process, observed reconnection rates are typically much faster than can be accounted for using classical electrical resistivity. It is thought that the field diffusion underlying fast reconnection results instead from some combination of non-magnetohydrodynamic processes that become important on the 'microscopic' scales of the ion Larmor radius or ion skin depth. Auna Moser and Paul Bellan report laboratory experimental observations demonstrating one possible mechanism by which a cascade of instabilities from a distinct, macroscopic-scale magnetohydrodynamic instability couples to a distinct, microscopic-scale instability associated with fast magnetic reconnection. The essential components of this mechanism have been observed separately in nature. Magnetic reconnection, the process whereby magnetic field lines break and then reconnect to form a different topology, underlies critical dynamics of magnetically confined plasmas in both nature 1 , 2 , 3 , 4 and the laboratory 5 , 6 , 7 , 8 , 9 . Magnetic reconnection involves localized diffusion of the magnetic field across plasma, yet observed reconnection rates are typically much higher than can be accounted for using classical electrical resistivity 10 . It is generally proposed 10 that the field diffusion underlying fast reconnection results instead from some combination of non-magnetohydrodynamic processes that become important on the ‘microscopic’ scale of the ion Larmor radius or the ion skin depth. A recent laboratory experiment 11 demonstrated a transition from slow to fast magnetic reconnection when a current channel narrowed to a microscopic scale, but did not address how a macroscopic magnetohydrodynamic system accesses the microscale. Recent theoretical models 12 and numerical simulations 13 , 14 suggest that a macroscopic, two-dimensional magnetohydrodynamic current sheet might do this through a sequence of repetitive tearing and thinning into two-dimensional magnetized plasma structures having successively finer scales. Here we report observations demonstrating a cascade of instabilities from a distinct, macroscopic-scale magnetohydrodynamic instability to a distinct, microscopic-scale (ion skin depth) instability associated with fast magnetic reconnection. These observations resolve the full three-dimensional dynamics and give insight into the frequently impulsive nature of reconnection in space and laboratory plasmas.

A weak turbulence theory is derived for magnetohydrodynamics (MHD) under rapid rotation and in the presence of a uniform large-scale magnetic field which is associated with a constant Alfvén velocity $\\def \\xmlpi #1{}\\def \\mathsfbi #1{\\boldsymbol {\\mathsf {#1}}}\\let \\le =\\leqslant \\let \\leq =\\leqslant \\let \\ge =\\geqslant \\let \\geq =\\geqslant \\def \\Pr {\\mathit {Pr}}\\def \\Fr {\\mathit {Fr}}\\def \\Rey {\\mathit {Re}}{\\boldsymbol {b}}_{{0}}$ . The angular velocity ${\\boldsymbol{\\Omega}}_{{0}}$ is assumed to be uniform and parallel to ${\\boldsymbol {b}}_{{0}}$ . Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length $d \\equiv b_0/\\varOmega _0$ . In the large-scale limit ( $kd \\to 0$ , with $k$ being the wavenumber) the left- and right-handed waves tend to the inertial and magnetostrophic waves, respectively, whereas in the small-scale limit ( $kd \\to + \\infty $ ) pure Alfvén waves are recovered. By using a complex helicity decomposition, the asymptotic weak turbulence equations are derived which describe the long-time behaviour of weakly dispersive interacting waves via three-wave interaction processes. It is shown that the nonlinear dynamics is mainly anisotropic, with a stronger transfer perpendicular than parallel to the rotation axis. The general theory may converge to pure weak inertial/magnetostrophic or Alfvén wave turbulence when the large- or small-scale limits are taken, respectively. Inertial wave turbulence is asymptotically dominated by the kinetic energy/helicity, whereas the magnetostrophic wave turbulence is dominated by the magnetic energy/helicity. For both regimes, families of exact solutions are found for the spectra, which do not correspond necessarily to a maximal helicity state. It is shown that the hybrid helicity exhibits a cascade whose direction may vary according to the scale $k_f$ at which the helicity flux is injected, with an inverse cascade if $k_fd < 1$ and a direct cascade otherwise. The theory is relevant to the magnetostrophic dynamo, whose main applications are the Earth and the giant planets, such as Jupiter and Saturn, for which a small ( ${\\sim }10^{-6}$ ) Rossby number is expected.

In this paper we review some theoretical aspects of the dynamics of the mesoscale filaments extending along the magnetic field lines in the edge plasma, which are often called ‘blobs’. We start with a brief historical survey of experimental data and the main ideas on edge and SOL plasma transport, which finally evolved into the modern paradigm of convective very-intermittent cross-field edge plasma transport. We show that both extensive analytic treatments and numerical simulations demonstrate that plasma blobs with enhanced pressure can be convected coherently towards the wall. The mechanism of convection is related to an effective gravity force (e.g. owing to magnetic curvature effects), which causes plasma polarization and a corresponding E× B convection. The impacts of different effects (e.g. X-point magnetic geometry, plasma collisionality, plasma beta, etc.) on blob dynamics are considered. Theory and simulation predict, both for current tokamaks and for ITER, blob propagation speeds and cross-field sizes to be of the order of a few hundred meters per second and a centimeter, respectively, which are in reasonable agreement with available experimental data. Moreover, the concept of blobs as a fundamental entity of convective transport in the scrape-off layer provides explanations for observed outwards convective transport, intermittency and non-Gaussian statistics in edge plasmas, and enhanced wall recycling in both toroidal and linear machines.

Two-dimensional gyrokinetics is a simple paradigm for the study of kinetic magnetised plasma turbulence. In this paper, we present a comprehensive theoretical framework for this turbulence. We study both the inverse and direct cascades (the ‘dual cascade’), driven by a homogeneous and isotropic random forcing. The key characteristic length of gyrokinetics, the Larmor radius, divides scales into two physically distinct ranges. For scales larger than the Larmor radius, we derive the familiar Charney–Hasegawa–Mima equation from the gyrokinetic system, and explain its relationship to gyrokinetics. At scales smaller than the Larmor radius, a dual cascade occurs in phase space (two dimensions in position space plus one dimension in velocity space) via a nonlinear phase-mixing process. We show that at these sub-Larmor scales, the turbulence is self-similar and exhibits power-law spectra in position and velocity space. We propose a Hankel-transform formalism to characterise velocity-space spectra. We derive the exact relations for third-order structure functions, analogous to Kolmogorov's four-fifths and Yaglom's four-thirds laws and valid at both long and short wavelengths. We show how the general gyrokinetic invariants are related to the particular invariants that control the dual cascade in the long- and short-wavelength limits. We describe the full range of cascades from the fluid to the fully kinetic range.

Purely helical absolute equilibria of incompressible neutral fluids and plasmas (electron, single-fluid and two-fluid magnetohydrodynamics) are systematically studied with the help of helical (wave) representation and truncation, for genericities and specificities about helicity. A unique chirality selection and amplification mechanism and relevant insights, such as the one-chiral-sector-dominated states, among others, about (magneto)fluid turbulence follow.

We analyse the anisotropy of homogeneous turbulence in an electrically conducting fluid submitted to a uniform magnetic field, for low magnetic Reynolds number, in the quasi-static approximation. We interpret contradictory earlier predictions between linearized theory and simulations: in the linear limit, the kinetic energy of transverse velocity components, normal to the magnetic field, decays faster than the kinetic energy of the axial component, along the magnetic field (Moffatt, J. Fluid Mech., vol. 28, 1967, p. 571); whereas many numerical studies predict a final state characterized by dominant energy of transverse velocity components. We investigate the corresponding nonlinear phenomenon using direct numerical simulation (DNS) of freely decaying turbulence, and a two-point statistical spectral closure based on the eddy-damped quasi-normal Markovian (EDQNM) model. The transition from the three-dimensional turbulent flow to a ‘two-and-a-half-dimensional’ flow (Montgomery & Turner, Phys. Fluids, vol. 25, 1982, p. 345) is a result of the combined effects of short-time linear Joule dissipation and longer time nonlinear creation of polarization anisotropy. It is this combination of linear and nonlinear effects which explains the disagreement between predictions from linearized theory and results from numerical simulations. The transition is characterized by the elongation of turbulent structures along the applied magnetic field, and by the strong anisotropy of directional two-point correlation spectra, in agreement with experimental evidence. Inertial equatorial transfers in both DNS and the model are presented to describe in detail the most important equilibrium dynamics. Spectral scalings are maintained in high-Reynolds-number turbulence attainable only with the EDQNM model, which also provides simplified modelling of the asymptotic state of quasi-static magnetohydrodynamic (MHD) turbulence.

We investigate the steepening of the magnetic fluctuation power law spectra observed in the inner Solar wind for frequencies higher than 0.5 Hz. This high frequency part of the spectrum may be attributed to dispersive nonlinear processes. In that context, the long-time behavior of weakly interacting waves is examined in the framework of three-dimensional incompressible Hall magnetohydrodynamic (MHD) turbulence. The Hall term added to the standard MHD equations makes the Alfvén waves dispersive and circularly polarized. We introduce the generalized Elsässer variables and, using a complex helicity decomposition, we derive for three-wave interaction processes the general wave kinetic equations; they describe the nonlinear dynamics of Alfvén, whistler and ion cyclotron wave turbulence in the presence of a strong uniform magnetic field $B_0 \\^{e}_{\\Vert}$. Hall MHD turbulence is characterized by anisotropies of different strength: (i) for wavenumbers $\\textit{kd}_{\\rm i}\\,{\\gg}\\,1$ ($d_{\\rm i}$ is the ion inertial length) nonlinear transfers are essentially in the direction perpendicular ($\\perp$) to ${\\bf B}_0$; (ii) for $\\textit{kd}_{\\rm i}\\,{\\ll}\\,1$ nonlinear transfers are exclusively in the perpendicular direction; (iii) for $\\textit{kd}_{\\rm i} \\sim 1$, a moderate anisotropy is predicted. We show that electron and standard MHD turbulence can be seen as two frequency limits of the present theory but the standard MHD limit is singular; additionally, we analyze in detail the ion MHD turbulence limit. Exact power law solutions of the master wave kinetic equations are given in the small- and large-scale limits for which we have, respectively, the total energy spectra $E(k_{\\perp},k_{\\Vert}) \\sim k_{\\perp}^{-5/2} |k_{\\Vert}|^{-1/2}$ and $E(k_{\\perp},k_{\\Vert}) \\sim k_{\\perp}^{-2}$. An anisotropic phenomenology is developed to describe continuously the different scaling laws of the energy spectrum; one predicts $E(k_{\\perp},k_{\\Vert}) \\sim k_{\\perp}^{-2} |k_{\\Vert}|^{-1/2} (1+k_{\\perp}^2d_{\\rm i}^2)^{-1/4}$. Non-local interactions between Alfvén, whistler and ion cyclotron waves are investigated; a non-trivial dynamics exists only when a discrepancy from the equipartition between the large-scale kinetic and magnetic energies happens.