We investigate the transfer of energy from large scales to small scales in fully developed forced three-dimensional MHD-turbulence by analyzing the results of direct numerical simulations in the absence of an externally imposed uniform magnetic field. Our results show that the transfer of kinetic energy from the large scales to kinetic energy at smaller scales, and the transfer of magnetic energy from the large scales to magnetic energy at smaller scales, are local, as is also found in the case of neutral fluids, and in a way that is compatible with Kolmogorov (1941) theory of turbulence. However, the transfer of energy from the velocity field to the magnetic field is a highly non-local process in Fourier space. Energy from the velocity field at large scales can be transfered directly into small scale magnetic fields without the participation of intermediate scales. Some implications of our results to MHD turbulence modeling are also discussed.
We investigate the influence of a uniform magnetic field ${\bf B_0}=B_0 e//$ 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 {\it à la} 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…
The ideal invariants present in the formalism of magnetohydrodynamics (MHD), i.e. global quantities that are conserved in the absence of sources and dissipative effects, play an important role in various theoretical and numerical studies of MHD turbulence. The fluxes of these ideal invariants represent separate channels that transfer the information across different scales in a turbulent system. Once a statistically stationary state of turbulence is reached, the amount of any ideal invariant quantity introduced in the system by a forcing mechanism equals the amount of the same quantity removed…
It is shown, using results of recent direct numerical simulations, that spectral properties of distributed chaos in MHD turbulence with zero mean magnetic field are similar to those of hydrodynamic turbulence. An exception is MHD spontaneous breaking of space translational symmetry, when the stretched exponential spectrum $\exp(-k/k_β)^β$ has $β=4/7$.
The purpose of this work is to investigate the spectral properties of the developed isotropic (non-Alfven) MHD turbulence stationary excited by an external force, which injects the cross helicity into the flow simultaneously with the energy. It is shown that the cross helicity blocks the spectral energy transfer in MHD turbulence and results in energy accumulation in the system. This accumulation proceeds until the vortex intensification compensates the decreasing efficiency of nonlinear interactions. The formula for estimating the average turbulence energy is obtained for the set ratio betwee…
We report on a comparison of high-resolution numerical simulations of Lagrangian particles advected by incompressible turbulent hydro- and magnetohydrodynamic (MHD) flows. Numerical simulations were performed with up to $1024^3$ collocation points and 10 million particles in the Navier-Stokes case and $512^3$ collocation points and 1 million particles in the MHD case. In the hydrodynamics case our findings compare with recent experiments from Mordant et al. [1] and Xu et al. [2]. They differ from the simulations of Biferale et al. [3] due to differences of the ranges choosen for evaluating the…
We review the current status of research in MHD turbulence theory and numerical experiments and their applications to astrophysics and solar science. We introduce general tools for studying turbulence, basic turbulence models, MHD equations and their wave modes. Subsequently, we cover the theories and numerics of Alfvenic turbulence, imbalanced turbulence, small-scale dynamos and models and numerics for supersonic MHD turbulence.
The scale locality of energy fluxes for magnetohydrodynamics (MHD) is investigated numerically for stationary states of turbulence. Two types of forces are used to drive turbulence, a kinetic force that acts only on the velocity field and a kinetic-inductive forcing mechanism, which acts on the velocity and magnetic fields alike. The analysis is performed in spectral space, which is decomposed into a series of shells following a power law for the boundaries. The triadic transfers occurring among these shells are computed and the fluxes and locality functions are recovered by partial summation …
The free decay of MHD turbulence at large Reynolds numbers is studied numerically using a shell model. We study the statistical properties based on representative sample of realisations (128 realisations for each type of initial conditions) over the period of $10^5$ large-scale turnover times. The performed simulations show that the force-free non-helical MHD turbulence can demonstrate two different scenarios of evolution in spite of similar initial conditions. Within the first scenario, the cross-helicity accumulation is so fast that the energy cascade vanishes before significant magnetic ene…
In this paper we describe the mode-to-mode energy transfers and energy fluxes of MHD turbulence. These energy transfers are very useful for understanding the turbulence dynamics, as well as for applications such as dynamo. We illustrate how the energy fluxes provide valuable insights into the mechanism of growth of the large-scale magnetic energy in dynamo.
The transport of heavy, neutral or charged, point-like particles by incompressible, resistive magnetohydrodynamic (MHD) turbulence is investigated by means of high-resolution numerical simulations. The spatial distribution of such impurities is observed to display strong deviations from homogeneity, both at dissipative and inertial range scales. Neutral particles tend to cluster in the vicinity of coherent vortex sheets due to their viscous drag with the flow, leading to the simultaneous presence of very concentrated and almost empty regions. The signature of clustering is different for charge…
Magnetohydrodynamic (MHD) turbulence is pervasive in astrophysical systems. Recent high-resolution numerical simulations suggest that the energy spectrum of strong incompressible MHD turbulence is $E(k_{\perp})\propto k_{\perp}^{-3/2}$. So far, there has been no phenomenological theory that simultaneously explains this spectrum and satisfies the exact analytic relations for MHD turbulence due to Politano & Pouquet. Indeed, the Politano-Pouquet relations are often invoked to suggest that the spectrum of MHD turbulence instead has the Kolmogorov scaling -5/3. Using geometrical arguments and nume…
We revisit the issue of spectral slope of MHD turbulence in the inertial range and argue that numerics favor Goldreich-Sridhar -5/3 slope rather than -3/2 slope. We also did precision measurements of anisotropy of MHD turbulence and determined the anisotropy constant C_A=0.34 of Alfvenic turbulence. Together with previously measured Kolmogorov constant C_K=4.2, or 3.3 for purely Alfvenic case, it constitutes a full description of MHD cascade in terms of spectral quantities, which is of high practical value for astrophysics.
Role of the dissipative invariants in isotropic homogeneous MHD turbulence has been studied using the distributed chaos approach and results of direct numerical simulations. Effects of weak imposed mean magnetic field and magnetic fields in undisturbed solar photosphere are also briefly discussed in this context.
An extended inertial range dominated by the cross-helicity effects has been studied for forced (statistically steady) and for freely decaying magnetohydrodynamic MHD turbulence (with and without imposed/mean magnetic field) using the spatio-temporal distributed chaos approach. Good agreement with results of direct numerical simulations, laboratory measurements in MHD wind (plasma) tunnel, measurements in the Earth's magnetosheath and in the solar wind has been established. A spontaneous breaking of local reflection (mirror) symmetry has been briefly discussed for the MHD turbulence with zero a…
Using results of laboratory experiments, direct numerical simulations, geomagnetic and solar observations, it is shown that high moments of helicity distribution can dominate power spectra of the magnetic field generated by the magnetohydrodynamic (MHD) dynamo in swirling turbulence even for the cases with zero global helicity. The notion of helical distributed chaos has been used for this purpose.
Magnetized turbulence is ubiquitous in many astrophysical and terrestrial systems but no complete, uncontested theory even in the simplest form, magnetohydrodynamics (MHD), exists. Many theories and phenomenologies focus on the joint (kinetic and magnetic) energy fluxes and spectra. We highlight the importance of treating kinetic and magnetic energies separately to shed light on MHD turbulence dynamics. We conduct an implicit large eddy simulation of subsonic, super-Alfvénic MHD turbulence and analyze the scale-wise energy transfer over time. Our key finding is that the kinetic energy spectrum…
In hydrodynamic turbulence, the kinetic energy injected at large scales cascades to the inertial range, leading to a constant kinetic energy flux. In contrast, in magnetohydrodynamic (MHD) turbulence, a fraction of kinetic energy is transferred to the magnetic energy. Consequently, for the same kinetic energy injection rate, the kinetic energy flux in MHD turbulence is reduced compared to its hydrodynamic counterpart. This leads to relative weakening of the nonlinear term ($\la | ({\bf u}\cdot \nabla) {\bf u} | \ra$, where ${\bf u}$ is the velocity field) and turbulent drag, but strengthening …
A phenomenology of isotropic magnetohydrodynamic turbulence subject to both rotation and applied magnetic field is presented. It is assumed that the triple correlations decay-time is the shortest between the eddy turn-over time and the ones associated to the rotating frequency and Alfvén wave period. For $Pm=1$ it leads to four kinds of piecewise spectra, depending on the four parameters, injection rate of energy, magnetic diffusivity, rotation rate and applied field. With a shell model of MHD turbulence (including rotation and applied magnetic field), spectra for $Pm \le 1$ are presented, tog…
Recent numerical and observational studies revealed that spectra of magnetic and velocity fluctuations in MHD turbulence have different scaling indexes. This intriguing feature has been recently explained in the case of weak MHD turbulence, that is, turbulence consisting of weakly interacting Alfven waves. However, astrophysical turbulence is strong in majority of cases. In the present work, we propose a unifying picture that allows one to address weak and strong MHD turbulence on the same footing. We argue that magnetic and kinetic energies are different in both weak and strong MHD turbulence…
We present a model for compressible sub-Alfvenic isothermal magnetohydrodynamic (MHD) turbulence in low-beta plasmas and numerically test it. We separate MHD fluctuations into 3 distinct families - Alfven, slow, and fast modes. We find that, production of slow and fast modes by Alfvenic turbulence is suppressed. As a result, Alfven modes in compressible regime exhibit scalings and anisotropy similar to those in incompressible regime. Slow modes passively mimic Alfven modes. However, fast modes show isotropy and a scaling similar to acoustic turbulence.
Leveling of MHD turbulence imbalance in shear flows
Abastumani Astrophysical Observatory（阿巴斯图马尼天体物理观测站）；Department of Physics, Faculty of Exact and Natural Sciences, Tbilisi State University（第比利斯国立大学精确与自然科学学院物理系）；Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗斯多夫亥姆霍兹中心）；M. Nodia Institute of Geophysics（M.诺迪亚地球物理研究所）
Turbulence is the most common state of astrophysical flows. In typical astrophysical fluids, turbulence is accompanied by strong magnetic fields, which has a large impact on the dynamics of the turbulent cascade. Recently, there has been a significant breakthrough on the theory of magnetohydrodynamic (MHD) turbulence. For the first time we have a scaling model that is supported by both observations and numerical simulations. We review recent progress in studies of both incompressible and compressible turbulence. We compare Iroshnikov-Kraichnan and Goldreich-Sridhar models, and discuss scalings…
Magnetized turbulence is ubiquitous in many astrophysical and terrestrial plasmas but no universal theory exists. Even the detailed energy dynamics in magnetohydrodynamic (MHD) turbulence are still not well understood. We present a suite of subsonic, super-Alfvénic, high plasma-beta MHD turbulence simulations that only vary in their dynamical range, i.e., in their separation between the large-scale forcing and dissipation scales, and their dissipation mechanism (implicit large eddy simulation, ILES, versus and direct numerical simulation, DNS). Using an energy transfer analysis framework we ca…
In this short review, we present the main known features of MHD Turbulence at Low Magnetic Reynolds number, for which the flow isn't intense nor electrically conductive enough to disturb an externally applied magnetic field. The emphasis is deliberately placed on the very specific physical mechanisms of these flows, rather than their numerical modelling. We also focus on homogeneous magnetic fields which have received most attention. Since the basic properties of these flows have been thoroughly reviewed a number of times, this review is deliberately biased towards flows in bounded domains, in…
Magnetohydrodynamics (MHD) provides the simplest description of magnetic plasma turbulence in a variety of astrophysical and laboratory systems. MHD turbulence with nonzero cross helicity is often called imbalanced, as it implies that the energies of Alfvén fluctuations propagating parallel and anti-parallel the background field are not equal. Recent analytical and numerical studies have revealed that at every scale, MHD turbulence consists of regions of positive and negative cross helicity, indicating that such turbulence is inherently locally imbalanced. In this paper, results from high reso…
The question is addressed to what extent incompressible magnetohydrodynamics (MHD) can describe random magnetic and velocity fluctuations measured in the solar wind. It is demonstrated that distributions of spectral indices for the velocity, magnetic field, and total energy obtained from high resolution numerical simulations are qualitatively and quantitatively similar to solar wind observations at 1 AU. Both simulations and observations show that in the inertial range the magnetic field spectrum E_b is steeper than the velocity spectrum E_v with E_b >~ E_v and that the residual energy E_R = E…
Wavelet analysis and compression tools are reviewed and different applications to study MHD and plasma turbulence are presented. We introduce the continuous and the orthogonal wavelet transform and detail several statistical diagnostics based on the wavelet coefficients. We then show how to extract coherent structures out of fully developed turbulent flows using wavelet-based denoising. Finally some multiscale numerical simulation schemes using wavelets are described. Several examples for analyzing, compressing and computing one, two and three dimensional turbulent MHD or plasma flows are pres…
We examine the changes in kinetic energy dissipation of a turbulent channel flow caused by a spanwise magnetic field. The numerical study is based on our simulation data from [Krasnov, Zikanov, Schumacher, Boeck, Phys. Fluids 20, 095105 (2008)] obtained by direct and large eddy simulations. We find that the Joule dissipation can exceed the viscous dissipation in the weakly dissipative bulk region, but remains comparatively small in the turbulence-generating near-wall region.
It is shown that the second-order moment of helicity distribution (the Levich-Tsinober invariant of inviscid hydrodynamics) can be preserved as an adiabatic invariant in magnetohydrodynamics. The distributed chaos approach and inertial range phenomenology have been developed for the magnetohydrodynamic processes dominated by this adiabatic invariant, and some consequences of the theoretical consideration have been compared with results of direct numerical simulations and measurements in the solar wind. Helical scaling in interstellar MHD turbulence has been also briefly discussed in this conte…
A nested polyhedra model has been developed for magnetohydrodynamic (MHD) turbulence. Driving only the velocity field at large scales with random, divergence free forcing results in a clear, stationary $k^{-5/3}$ spectrum for both kinetic and magnetic energies. Since the model naturaly effaces disparate scale interactions, does not have a guide field and avoids injecting any sign of helicity by random forcing, the resulting three dimensional k-spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. …
Turbulence is the most common state of astrophysical flows. In typical astrophysical fluids, turbulence is accompanied by strong magnetic fields, which has a large impact on the dynamics of the turbulent cascade. Recently, there has been a significant breakthrough on the theory of magnetohydrodynamic (MHD) turbulence. For the first time we have a scaling model that is supported by both observations and numerical simulations. We review recent progress in studies of both incompressible and compressible turbulence. We compare Iroshnikov-Kraichnan and Goldreich-Sridhar models, and discuss scalings…
An open issue in turbulence theory is related to the determination of the exact evolution equation for the probability density associated to the relevant (stochastic) fluid fields. Such an equation in the usual approaches to turbulence reproduces, at most in an approximate sense, the correct fluid equations. In this paper we present a statistical model which applies to an incompressible, resistive and quasi-neutral magnetofluid. The approach is based on the formulation of an inverse kinetic theory (IKT) for the full set of MHD equations appropriate for an incompressible, viscous, quasi-neutral…
Measurements of intensity and polarization of diffuse Galactic synchrotron emission as well as starlight polarization reveal power law spectra of fluctuations. We show that these fluctuations can arise from magnetohydrodynamic (MHD) turbulence in the Galactic disk and halo. To do so we take into account the converging geometry of lines of sight for the observations when the observer is within the turbulent volume. Assuming that the intensity of turbulence changes along the line of sight, we get a reasonable fit to the observed synchrotron data. As for the spectra of polarized starlight we get …
The aim of the present work is to derive rigorous estimates for turbulent MHD flow quantities such as the size and anisotropy of the dissipative scales, as well as the transition between 2D and 3D state. To this end, we calculate an upper bound for the attractor dimension of the motion equations, which indicates the number of modes present in the fully developed flow. This method has already been used successfully to derive such estimates for 2D and 3D hydrodynamic turbulence as a function of the $\mathcal L_\infty$ norm of the dissipation, as in \cite{doering95}. We tackle here the problem of…
This paper introduces an experimental apparatus, which drives turbulence electrically in a liquid metal pervaded by a high magnetic field. Unlike past magnetohydrodynamic (MHD) setups involving a shallow confinement, the experiment presented here drives turbulence whose dimensionality can be set anywhere between three-dimensional and quasi two-dimensional. In particular, we show that the dimensionality and componentality of the turbulence thus generated are in fact completely fixed by the single parameter lz(li)/h, which quantifies the competition between the solenoidal component of the Lorent…
In the period of 1948-1955, Chandrasekhar wrote four papers on magnetohydrodynamic (MHD) turbulence, which are the first set of papers in the area. The field moved on following these pioneering efforts. In this paper, I will briefly describe important works of MHD turbulence, starting from those by Chandrasekhar.
We investigate three-dimensional magnetohydrodynamics turbulence in the presence of both a large-scale velocity and non-uniform magnetic field. By assuming a turbulence driven by an external forcing with both helical and non-helical spectra, we investigate the combined effect of these two shears on turbulence intensity and turbulent transport represented by turbulent diffusivities (turbulent viscosity, $α$ and $β$ effect) in analogy with Reynolds. We show that turbulence viscosity and diffusivity are quenched for strong flow shear and magnetic field in the limits of weak and strong magnetic fi…
In this paper, we study the development of anisotropy in strong MHD turbulence in the presence of a large scale magnetic field B 0 by analyzing the results of direct numerical simulations. Our results show that the developed anisotropy among the different components of the velocity and magnetic field is a direct outcome of the inverse cascade of energy of the perpendicular velocity components u? and a forward cascade of the energy of the parallel component u k . The inverse cascade develops for a strong B0, where the flow exhibits a strong vortical structure by the suppression of fluctuations …
The spectral slope of strong MHD turbulence has recently been a matter of controversy. While Goldreich-Sridhar model (1995) predicts Kolmogorov's -5/3 slope of turbulence, shallower slopes were often reported by numerical studies. We argue that earlier numerics was affected by driving due to a diffuse locality of energy transfer in MHD case. Our highest-resolution simulation (3072^2x1024) has been able to reach the asymptotic -5/3 regime of the energy slope. Additionally, we found that so-called dynamic alignment, proposed in the model with -3/2 slope, saturates and therefore can not affect as…
Steady Low $R_m$ MHD turbulence is investigated here through estimates of upper bounds for attractor dimension. A flow between two parallel walls with an imposed perpendicular magnetic field is considered. The flow is defined by its maximum velocity and the intensity of the magnetic field. Given the corresponding Reynolds and Hartmann numbers, we derive an upper bound for the dimension of the attractor and find out which modes must be chosen to achieve this bound. Thier, are compared to heuristics estimates for the size of the smallest scales and anisotropy of the turbulence. We identify bound…
Performing a stable, long duration simulation of driven MHD turbulence with a high thermal Mach number and a strong initial magnetic field is a challenge to high-order Godunov ideal MHD schemes because of the difficulty in guaranteeing positivity of the density and pressure. We have implemented a robust combination of reconstruction schemes, Riemann solvers, limiters, and Constrained Transport EMF averaging schemes that can meet this challenge, and using this strategy, we have developed a new Adaptive Mesh Refinement (AMR) MHD module of the ORION2 code. We investigate the effects of AMR on sev…
Magnetised plasma turbulence pervades the universe and is likely to play an important role in a variety of astrophysical settings. Magnetohydrodynamics (MHD) provides the simplest theoretical framework in which phenomenological models for the turbulent dynamics can be built. Numerical simulations of MHD turbulence are widely used to guide and test the theoretical predictions; however, simulating MHD turbulence and accurately measuring its scaling properties is far from straightforward. Computational power limits the calculations to moderate Reynolds numbers and often simplifying assumptions ar…
We present the first study of the multiscaling of time-dependent velocity and magnetic-field structure functions in homogeneous, isotropic magnetohydrodynamic (MHD) turbulence in three dimensions. We generalize the formalism that has been developed for analogous studies of time-dependent structure functions in fluid turbulence to MHD. By carrying out detailed numerical studies of such time-dependent structure functions in a shell model for three-dimensional MHD turbulence, we obtain both equal-time and dynamic scaling exponents.
The search of ways to generalize the theory of strong MHD turbulence for the case of non-zero cross-helicity (or energy imbalance) has attracted considerable interest recently. In our earlier publications we performed three-dimensional numerical simulations and showed that some of existing models are inconsistent with numerics. In this paper we focused our attention on low-imbalance limit and performed new high-resolution simulations. The results strongly suggest that in the limit of small imbalances we smoothly transition to a standard Goldreich-Sridhar (1995) balanced model. We also claim th…
One of the most important predictions in magnetohydrodynamics (MHD) is that in the presence of a uniform magnetic field $\textbf{b}_{0}$ 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 thanks to a three-dimensional direct numerical simulation of incompressible balanced MHD turbulence with a grid resolution of $3072^2 \times 256$. From large to small-scales, the chang…
