Monofractality in the Solar Wind at Electron Scales: Insights from Kinetic Alfvén Waves Turbulence.

The breakdown of scale invariance in turbulent flows, known as multifractal scaling, is considered a cornerstone of turbulence. In solar wind turbulence, a monofractal behavior can be observed at electron scales, in contrast to larger scales where multifractality always prevails. Why scale invariance appears at electron scales is a challenging theoretical puzzle with important implications for understanding solar wind heating and acceleration. We investigate this long-standing problem using direct numerical simulations of three-dimensional electron reduced magnetohydrodynamics. Both weak and strong kinetic Alfvén waves turbulence regimes are studied in the balanced case. After recovering the expected theoretical predictions for the magnetic spectra, a higher-order multiscale statistical analysis is performed. This study reveals a striking difference between the two regimes, with the emergence of monofractality only in weak turbulence, whereas strong turbulence is multifractal. This result, combined with recent studies, shows the relevance of collisionless weak KAW turbulence to describe the solar wind at electron scales.

Direct Evidence of a Dual Cascade in Gravitational Wave Turbulence.

We present the first direct numerical simulation of gravitational wave turbulence. General relativity equations are solved numerically in a periodic box with a diagonal metric tensor depending on two space coordinates only, g_{ij}≡g_{ii}(x,y,t)δ_{ij}, and with an additional small-scale dissipative term. We limit ourselves to weak gravitational waves and to a freely decaying turbulence. We find that an initial metric excitation at intermediate wave number leads to a dual cascade of energy and wave action. When the direct energy cascade reaches the dissipative scales, a transition is observed in the temporal evolution of energy from a plateau to a power-law decay, while the inverse cascade front continues to propagate toward low wave numbers. The wave number and frequency-wave-number spectra are found to be compatible with the theory of weak wave turbulence and the characteristic timescale of the dual cascade is that expected for four-wave resonant interactions. The simulation reveals that an initially weak gravitational wave turbulence tends to become strong as the inverse cascade of wave action progresses with a selective amplification of the fluctuations g_{11} and g_{22}.

Proof of the zeroth law of turbulence in one-dimensional compressible magnetohydrodynamics and shock heating.

The zeroth law is one of the oldest conjectures in turbulence that is still unproven. Here, we consider weak solutions of one-dimensional compressible magnetohydrodynamics and demonstrate that the lack of smoothness of the fields introduces a dissipative term, named inertial dissipation, into the expression of energy conservation that is neither viscous nor resistive in nature. We propose exact solutions assuming that the kinematic viscosity and the magnetic diffusivity are equal, and we demonstrate that the associated inertial dissipation is positive and equal on average to the mean viscous dissipation rate in the limit of small viscosity, proving the conjecture of the zeroth law of turbulence and the existence of an anomalous dissipation. As an illustration, we evaluate the shock heating produced by discontinuities detected by Voyager in the solar wind around 5 AU. We deduce a heating rate of ∼10^{-18}Jm^{-3}s^{-1}, which is significantly higher than the value obtained from the turbulent fluctuations. This suggests that collisionless shocks can be a dominant source of heating in the outer solar wind.

Turbulence of Weak Gravitational Waves in the Early Universe.

We study the statistical properties of an ensemble of weak gravitational waves interacting nonlinearly in a flat space-time. We show that the resonant three-wave interactions are absent and develop a theory for four-wave interactions in the reduced case of a 2.5+1 diagonal metric tensor. In this limit, where only plus-polarized gravitational waves are present, we derive the interaction Hamiltonian and consider the asymptotic regime of weak gravitational wave turbulence. Both direct and inverse cascades are found for the energy and the wave action, respectively, and the corresponding wave spectra are derived. The inverse cascade is characterized by a finite-time propagation of the metric excitations-a process similar to an explosive nonequilibrium Bose-Einstein condensation, which provides an efficient mechanism to ironing out small-scale inhomogeneities. The direct cascade leads to an accumulation of the radiation energy in the system. These processes might be important for understanding the early Universe where a background of weak nonlinear gravitational waves is expected.

Chiral exact relations for helicities in Hall magnetohydrodynamic turbulence.

Besides total energy, three-dimensional incompressible Hall magnetohydrodynamics (MHD) possesses two inviscid invariants, which are the magnetic helicity and the generalized helicity. Exact relations are derived for homogeneous (nonisotropic) stationary Hall MHD turbulence (and also for its inertialess electron MHD limit) with nonzero helicities and in the asymptotic limit of large Reynolds numbers. The universal laws are written only in terms of mixed second-order structure functions, i.e., the scalar product of two different increments. It provides, therefore, a direct measurement of the dissipation rates for the corresponding invariant flux. This study shows that the generalized helicity cascade is strongly linked to the left polarized fluctuations, while the magnetic helicity cascade is linked to the right polarized fluctuations.

Direct Evidence of the Transition from Weak to Strong Magnetohydrodynamic Turbulence.

One of the most important predictions in magnetohydrodynamics is that in the presence of a uniform magnetic field b_{0}e[over ^]_{∥} a transition from weak to strong wave turbulence should occur when going from large to small perpendicular scales. This transition is believed to be a universal property of several anisotropic turbulent systems. We present, for the first time, direct evidence of such a transition using a decaying three-dimensional direct numerical simulation of incompressible balanced magnetohydrodynamic turbulence with a grid resolution of 3072^{2}×256. From large to small scales, the change of regime is characterized by (i) a change of slope in the energy spectrum going from approximately -2 to -3/2, (ii) an increase of the ratio between the wave and nonlinear times, with a critical ratio of χ_{c}∼1/3, (iii) a modification of the isocontours of energy revealing a transition from a purely perpendicular cascade to a cascade compatible with the critical-balance-type phenomenology, and (iv) an absence followed by a dramatic increase of the communication between Alfvén modes. The changes happen at approximately the same transition scale and can be seen as manifest signatures of the transition from weak to strong wave turbulence. Furthermore, we observe a significant nonlocal three-wave coupling between strongly and weakly nonlinear modes resulting in an inverse transfer of energy from small to large scales.

Theory for helical turbulence under fast rotation.

Recent numerical simulations have shown the strong impact of helicity on homogeneous rotating hydrodynamic turbulence. The main effect can be summarized through the law n+ñ=-4, where n and ñ are the power law indices of the one-dimensional energy and helicity spectra, respectively. We investigate this rotating turbulence problem in the small Rossby number limit by using the asymptotic weak turbulence theory derived previously. We show that the empirical law is an exact solution of the helicity equation where the power law indices correspond to perpendicular (to the rotation axis) wave number spectra. It is proposed that when the cascade towards small scales tends to be dominated by the helicity flux the solution tends to ñ=-2, whereas it is ñ=-3/2 when the energy flux dominates. The latter is compatible with the solution previously observed numerically and derived theoretically in the weak turbulence regime when only the energy equation is used, whereas the former solution is constrained by a locality condition.

Exact relation with two-point correlation functions and phenomenological approach for compressible magnetohydrodynamic turbulence.

Compressible isothermal magnetohydrodynamic turbulence is analyzed under the assumption of statistical homogeneity and in the asymptotic limit of large kinetic and magnetic Reynolds numbers. Following Kolmogorov we derive an exact relation for some two-point correlation functions which generalizes the expression recently found for hydrodynamics. We show that the magnetic field brings new source and flux terms into the dynamics which may act on the inertial range similarly as a source or a sink for the mean energy transfer rate. The introduction of a uniform magnetic field simplifies significantly the exact relation for which a simple phenomenology may be given. A prediction for axisymmetric energy spectra is eventually proposed.

Anomalous k⊥(-8/3) spectrum in electron magnetohydrodynamic turbulence.

Electron magnetohydrodynamic turbulence is investigated under the presence of a relatively strong external magnetic field b0e∥ and through three-dimensional direct numerical simulations. Our study reveals the emergence of a k⊥(-8/3) scaling for the magnetic energy spectrum at scales k∥(D)≤k⊥≤k⊥(D), where k∥(D) and k⊥(D) are, respectively, the typical largest dissipative scales along and transverse to the b0 direction. Unlike standard magnetohydrodynamic, this turbulence regime is characterized by filaments of electric currents parallel to b0. The anomalous scaling is in agreement with a heuristic model in which the transfer in the parallel direction is negligible. Implications for solar wind turbulence are discussed.

Spontaneous chiral symmetry breaking of Hall magnetohydrodynamic turbulence.

Hall magnetohydrodynamics (MHD) is investigated through three-dimensional direct numerical simulations. We show that the Hall effect induces a spontaneous chiral symmetry breaking of the turbulent dynamics. The normalized magnetic polarization is introduced to separate the right- (R) and left-handed (L) fluctuations. A classical k(-7/3) spectrum is found at small scales for R magnetic fluctuations which corresponds to the electron MHD prediction. A spectrum compatible with k(-11/3) is obtained at large-scales for the L magnetic fluctuations; we call this regime the ion MHD. These results are explained heuristically by rewriting the Hall MHD equations in a succinct vortex dynamical form. Applications to solar wind turbulence are discussed.

Exact relation for correlation functions in compressible isothermal turbulence.

Compressible isothermal turbulence is analyzed under the assumption of homogeneity and in the asymptotic limit of a high Reynolds number. An exact relation is derived for some two-point correlation functions which reveals a fundamental difference with the incompressible case. The main difference resides in the presence of a new type of term which acts on the inertial range similarly as a source or a sink for the mean energy transfer rate. When isotropy is assumed, compressible turbulence may be described by the relation -2/3ε(eff)r = F(r)(r), where F(r) is the radial component of the two-point correlation functions and ε(eff) is an effective mean total energy injection rate. By dimensional arguments, we predict that a spectrum in k(-5/3) may still be preserved at small scales if the density-weighted fluid velocity ρ(1/3)u is used.

Two-dimensional state in driven magnetohydrodynamic turbulence.

The dynamics of the two-dimensional (2D) state in driven three-dimensional (3D) incompressible magnetohydrodynamic turbulence is investigated through high-resolution direct numerical simulations and in the presence of an external magnetic field at various intensities. For such a flow the 2D state (or slow mode) and the 3D modes correspond, respectively, to spectral fluctuations in the plane k(∥)=0 and in the area k(∥)>0. It is shown that if initially the 2D state is set to zero it becomes nonnegligible in few turnover times, particularly when the external magnetic field is strong. The maintenance of a large-scale driving leads to a break for the energy spectra of 3D modes; when the driving is stopped, the previous break is removed and a decay phase emerges with Alfvénic fluctuations. For a strong external magnetic field the energy at large perpendicular scales lies mainly in the 2D state, and in all situations a pinning effect is observed at small scales.

Exact vectorial law for homogeneous rotating turbulence.

Three-dimensional hydrodynamic turbulence is investigated under the assumptions of homogeneity and weak axisymmetry. Following the kinematics developed by E. Lindborg [J. Fluid Mech. 302, 179 (1995)] we rewrite the von Kármán-Howarth equation in terms of measurable correlations and derive the exact relation associated with the flux conservation. This relation is then analyzed in the particular case of turbulence subject to solid-body rotation. We make the ansatz that the development of anisotropy implies an algebraic relation between the axial and the radial components of the separation vector r and we derive an exact vectorial law which is parametrized by the intensity of anisotropy. A simple dimensional analysis allows us to fix this parameter and find a unique expression.

von Kármán-Howarth equations for Hall magnetohydrodynamic flows.

The von Kármán-Howarth equations are derived for three-dimensional Hall magnetohydrodynamics in the case of a homogeneous and isotropic turbulence. From these equations, we derive exact scaling laws for the third-order correlation tensors. We show how these relations are compatible with previous heuristic and numerical results. These multiscale laws provide a relevant tool to investigate the nonlinear nature of the high-frequency magnetic field fluctuations in the solar wind or, more generally, in any plasma where the Hall effect is important.

Energy decay laws in strongly anisotropic magnetohydrodynamic turbulence.

We investigate the influence of a uniform magnetic field B(0)=B(0)e( parallel) on energy decay laws in incompressible magnetohydrodynamic (MHD) turbulence. The nonlinear transfer reduction along B(0) is included in a model that distinguishes parallel and perpendicular directions, following a phenomenology of Kraichnan. We predict a slowing down of the energy decay due to anisotropy in the limit of strong B(0), with distinct power laws for energy decay of shear- and pseudo-Alfvén waves. Numerical results from the kinetic equations of Alfvén wave turbulence recover these predictions, and MHD numerical results clearly tend to follow them in the lowest perpendicular planes.

Development of anisotropy in incompressible magnetohydrodynamic turbulence.

We present a set of three-dimensional direct numerical simulations of incompressible decaying magnetohydrodynamic turbulence in which we investigate the influence of an external uniform magnetic field B0 . A parametric study in terms of B0 intensity is made where, in particular, we distinguish the shear-from the pseudo-Alfvén waves dynamics. The initial kinetic and magnetic energies are equal with a negligible cross correlation. Both the temporal and spectral effects of B0 are discussed. A subcritical balance is found between the Alfvén and nonlinear times with both a global and a spectral definition. The nonlinear dynamics of strongly magnetized flows is characterized by a different k_{ perpendicular} spectrum (where B0 defines the parallel direction) if it is plotted at a fixed k_{ parallel} (two-dimensional spectrum) or if it is integrated (averaged) over all k_{ parallel} (one-dimensional spectrum). In the former case a much wider inertial range is found with a steep power law, closer to the wave turbulence prediction than the Kolmogorov one such as in the latter case. It is believed that the averaging effect may be a source of difficulty to detect the transition towards wave turbulence in natural plasmas. Another important result of this paper is the formation of filaments reported within current and vorticity sheets in strongly magnetized flows, which modifies our classical picture of dissipative sheets in conductive flows.

Weak inertial-wave turbulence theory.

A weak wave turbulence theory is established for incompressible fluids under rapid rotation using a helicity decomposition, and the kinetic equations for energy E and helicity H are derived for three-wave coupling. As expected, nonlinear interactions of inertial waves lead to two-dimensional behavior of the turbulence with a transfer of energy and helicity mainly in the direction perpendicular to the rotation axis. For such a turbulence, we find, analytically, the anisotropic spectra E approximately k(-5/2)(perpendicular)k(-1/2)(parallel), H approximately k(-3/2)(perpendicular)k(-1/2)(parallel), and we prove that the energy cascade is to small scales. At lowest order, the wave theory does not describe the dynamics of two-dimensional (2D) modes which decouples from 3D waves.