Compressible magnetohydrodynamic turbulence: mode coupling, scaling relations, anisotropy, viscosity-damped regime and astrophysical implications

We present numerical simulations and explore scalings and anisotropy of compressible magnetohydrodynamic (MHD) turbulence. Our study covers both gas-pressure-dominated (high β) and magnetic-pressure-dominated (low β) plasmas at different Mach numbers. In addition, we present results for super-Alfvénic turbulence and discuss in what way it is similar to sub-Alfvénic turbulence. We describe a technique of separating different magnetohydrodynamic modes (slow, fast and Alfvén) and apply it to our simulations. We show that, for both high- and low-β cases, Alfvén and slow modes reveal a Kolmogorov   k ^−5/3  spectrum and scale-dependent Goldreich–Sridhar anisotropy, while fast modes exhibit a   k ^−3/2  spectrum and isotropy. We discuss the statistics of density fluctuations arising from MHD turbulence in different regimes. Our findings entail numerous astrophysical implications ranging from cosmic ray propagation to gamma ray bursts and star formation. In particular, we show that the rapid decay of turbulence reported by earlier researchers is not related to compressibility and mode coupling in MHD turbulence. In addition, we show that magnetic field enhancements and density enhancements are marginally correlated. Addressing the density structure of partially ionized interstellar gas on astronomical-unit scales, we show that the viscosity-damped regime of MHD turbulence that we reported earlier for incompressible flows persists for compressible turbulence and therefore may provide an explanation for these mysterious structures.     

MHD compressive turbulence in the solar wind and the nearly incompressible approach

At MHD scales density fluctuation in the solar wind generally have a relative amplitude less than 0.1. The nearly incompressible MHD theory would seem then appropriate to describe a major part of the compressive turbulence at these scales. As a test of the theory, we focus on the scaling properties of density fluctuations with turbulent Mach numbers and on the level of correlation between density and temperature fluctuations. Our findings do not appear in favour of an extended applicability of the nearly incompressible theory to MHD compressive turbulence in the solar wind.     

The impact of correlated noise on SuperWASP detection rates for transiting extrasolar planets

The evolution of the Alfvén turbulence due to three-wave interactions is discussed using kinetic theory for a collisionless, thermal plasma. There are three low-frequency modes, analogous to the three modes of compressible magnetohydrodynamics (MHD). When only Alfvén waves are considered, the known anisotropy of turbulence in incompressible MHD theory is reproduced. Inclusion of a fast mode wave leads to the separation of turbulence into two regimes: small wave numbers where three-wave processes involving a fast mode are dominant, and large wave numbers where the three Alfvén wave process is dominant. Possible application of the anisotropic Alfvén turbulence to the interstellar medium and dissipation of magnetic energy in magnetars are discussed.     

From Small-Scale Dynamo to Isotropic MHD Turbulence

We consider the problem of incompressible, forced, nonhelical, homogeneous, isotropic MHD turbulence with no mean magnetic field. This problem is essentially different from the case with externally imposed uniform mean field. There is no scale-by-scale equipartition between magnetic and kinetic energies as would be the case for the Alfvén-wave turbulence. The isotropic MHD turbulence is the end state of the turbulent dynamo which generates folded fields with small-scale direction reversals. We propose that the statistics seen in numerical simulations of isotropic MHD turbulence could be explained as a superposition of these folded fields and Alfvén-like waves that propagate along the folds.     

Cosmic ray scattering in compressible turbulence

We study the scattering of low-energy cosmic rays (CRs) in a turbulent, compressive magnetohydrodynamic (MHD) fluid. We show that compressible MHD modes – fast or slow waves with wavelengths smaller than CR mean free paths induce cyclotron instability in CRs. The instability feeds the new small-scale Alfvénic wave component with wavevectors mostly along magnetic field, which is not a part of the MHD turbulence cascade. This new component gives feedback on the instability through decreasing the CR mean free path. We show that the ambient turbulence fully suppresses the instability at large scales, while wave steepening constrains the amplitude of the waves at small scales. We provide the energy spectrum of the plane-parallel Alfvénic component and calculate mean free paths of CRs as a function of their energy. We find that for the typical parameters of turbulence in the interstellar medium and in the intercluster medium the new Alfvénic component provides the scattering of the low-energy CRs that exceeds the direct resonance scattering by MHD modes. This solves the problem of insufficient scattering of low-energy CRs in the turbulent interstellar or intracluster medium that was reported in the literature.     

Coronal heating by selective decay of MHD turbulence

We suggest that the solar corona is stationarily heated by Joule dissipation of magnetohydrodynamic (MHD) turbulence. This turbulence is continuously being fed by subphotospheric convective motions at very low wavenumbers where the plasma can be considered ideal. The cascade process due to the nonlinear interaction of the turbulent modes, effectively transfers energy to even shorter wavenumbers until Joule effect becomes dominant and converts the cascading magnetic energy into heat.Assuming that a stationary spectrum is established, we show that MHD turbulence naturally provides a way of enhancing the Joule dissipation of macroscopic magnetic stresses and a heating rate can be derived.     

Hypersonic Swizzle Sticks: Protostellar Turbulence, Outflows and Fossil Outflow Cavities

The expected lifetimes for molecular clouds has become a topic of considerable debate as numerical simulations have shown that MHD turbulence, the nominal means of support for clouds against self-gravity, will decay on short timescales. Thus it appears that either molecular clouds are transient features or they are resupplied with turbulent energy through some means. Jets and molecular outflows are recognized as a ubiquitous phenomena associated with star formation. Stars however form not isolation but in clusters of different density and composion. The ubiquity and high density of outflows from young stars in clusters make them an intriguing candidate for the source of turbulence energy in molecular clouds. In this contribution we present new studies, both observational and theoretical, which address the issue of jet/outflow interactions and their abilityto drive turbulent flows in molecular clouds. Our studies focus on scales associated with young star forming clusters. In particular we first show that direct collisions between active outflows are not effective at stirring the ambient medium. We then show that fossil cavities from “extinct” outflows may provide the missing link in terms of transferring momentum and energy to the cloud.     

Resonant behaviour of MHD waves on magnetic flux tubes

The resonances that appear in the linear compressible MHD formulation of waves are studied for equilibrium states with flow. The conservation laws and the jump conditions across the resonance point are determined for 1D cylindrical plasmas. For equilibrium states with straight magnetic field lines and flow along the field lines the conserved quantity is the Eulerian perturbation of total pressure. Curvature of the magnetic field lines and/or velocity field lines leads to more complicated conservation laws. Rewritten in terms of the displacement components in the magnetic surfaces parallel and perpendicular to the magnetic field lines, the conservation laws simply state that the waves are dominated by the parallel motions for the modified slow resonance and by the perpendicular motions for the modified Alfvén resonance.The conservation laws and the jump conditions are then used for studying surface waves in cylindrical plasmas. These waves are characterized by resonances and have complex eigenfrequencies when the classic true discontinuity is replaced by a nonuniform layer. A thin non-uniform layer is considered here in an attempt to obtain analytical results. An important result related to earlier work by Hollweg et al. (1990) for incompressible planar plasmas is found for equilibrium states with straight magnetic field lines and straight velocity field lines. For these equilibrium states the incompressible and compressible surface waves have the same frequencies at least in the long wavelength limit and there is an exact correspondence with the planar case. As a consequence, the conclusions formulated by Hollweg et al. still hold for the straight cylindrical case. The effects of curvature are subsequently considered.     

A statistical description of astrophysical turbulence

The properties of the ISM indicate that it is turbulent. However, the ISM turbulence is radically different from that in incompressible fluids. That is why it is so important to study it through observations. The relevant study still poses a challenging problem. In the present paper recent results based on a statistical approach to the problem are surveyed. Although it was pointed out long ago (see Kaplanet al., 1970) that random 3D motions of the ISM gas result in fluctuations of the observed electromagnetic emission, it is only recently that the problem of recovering statistical properties of the ISM turbulence from the line integrated data was given an adequate mathematical treatment. Here by the example of studying turbulence in HI, it is shown that the inverse problem can be solved uniquely using a realistic model of the ISM. The application of theoretical conclusions to existing data explains some facts which used to be considered inconsistent with turbulent behaviour and reveals unexpected features of the ISM turbulence.     

Compressible turbulence in Hall Magnetohydrodynamics

By direct numerical simulations we investigate the nonlinear dynamics of a compressible Hall Magnetohydrodynamic (MHD) plasma. At small scales, where the Hall effect dominates, we found an increase of the compressibility of the system and the breakdown of the strong link between velocity and magnetic fields, typical of usual MHD. Moreover, we find that small-scale fluctuations are characterized by an anti-correlation between density and magnetic field intensity. These features characterize the excitation of a quasi-perpendicular magnetosonic turbulence that can be interpreted as the small-scale signature of the break-down of the MHD nonlinear energy cascade due to Hall effect. Fluctuations with the same properties, based on measurements by Cluster spacecraft in space plasma turbulence during different magnetopause crossings, have been recently observed.     

A Review of the Theory of Incompressible MHD Turbulence

This brief review provides an introduction to key ideas in the theory of incompressible magnetohydrodynamic (MHD) turbulence.     

Particle acceleration and the decay of soft X-ray non-thermal line broadening in solar flares

The relationship between the production of -ray emitting particles and non-thermal soft X-ray line broadening is investigated. A model of particle acceleration via the stochastic interaction with MHD turbulence is assumed and the time development of the wave energy density derived under the condition of energy conservation between waves and particles. The inferred numbers and energy distribution of accelerated protons for four -ray flares are used to define the wave energy density and its temporal development. The presence of Alfvén wave turbulence is considered as the source of the non-thermal motions in the ambient plasma. These motions are observed as excess widths in the soft X-ray line emission from these events. The decay of the waves via the particle acceleration process is compared with the observed decays of this non-thermal line broadening. Our results show that both the -ray emission and excess soft X-ray line widths in these flares can be explained by the single physical phenomenon of Alfvén wave turbulence.     

Observation of Turbulence in Solar Surface Convection: I. Line Parameter Correlations

By using slit observations of solar photospheric lines shifted by 0.4 arcsec, a 2D field on the Sun was scanned to obtain a 16-minute time series of 2D line-parameter variations. The aim was to investigate in detail the occurrence of turbulence that can be measured by line-width variations extracted from the line profiles. The continuum-intensity variation served as a proxy for granular (bright) and intergranular (dark) areas. The results show that turbulence is not limited to the intergranular space but is also produced by horizontal motions that may become supersonic, leading to turbulence. These motions lead to brightenings, as predicted by theoretical models. Thus, enhanced line-width variations are found to occur in both bright and dark areas. A Sobel filter served to detect the areas where strong gradients in the line parameters occur. By applying this filter to the different line-parameter variations over the 2D field observed, we can determine whether there exists a similarity of these strong-gradient patterns with other parameters that characterize granular motions such as intensity variations or velocity fluctuations.     

Magnetic field reconnection in a compressible plasma

An extension of Sonnerup's model for the magnetic field-line reconnection for a compressible plasma is given. The plasma is considered to be only slightly compressible so that the leading wave in Sonnerup's model can still be taken to be a thin discontinuity. The flow is assumed to occur under adiabatic conditions, and de Hoffmann-Teller jump conditions are used to connect the state variables across the shocks. The compressibility effects are found to increase the reconnection rate. The signaling problem is finally considered to study the evolution of MHD waves in a compressible, dissipative plasma so as to investigate the conditions under which the assumption of MHD waves in a compressible plasma to be thin discontinuities is valid.     

Dynamo theory and liquid metal MHD experiments

High values of magnetic Reynolds number R_m are characteristic not only to astrophysics, but also to other interesting objects, including liquid metal (LM) flows. Since sixties LM experiments have been performed illustrating important predictions of the dynamo theory, for example, about the existence and features of the α effect. Consideration of so called “laminar” dynamos provides a theoretical base for direct experimental realization and examination of the dynamo process. First step results, gathered at subcritical conditions, confirm the statement that self-excitation in LM experiments can be achieved practically today. In such devices as LM (sodium) cooled fast breeders R_m can reach values up to 50 and specific MHD phenomena have been observed in operating fast reactors. Cautions against crisis like processes have been expressed. It is important for the dynamo theory to understand what kind of perturbed motion is able to coexist with the generated magnetic field. Fundamentally new ideas here are issuing from the theory of 2D MHD turbulence. LM MHD served for the first direct proves, confirming, that the predicted surprising features of 2D turbulence can be observed in reality. It is worth incorporating these already not new ideas in the dynamo theory. In such a way a field for new solutions could be established.     

Linear MHD Sausage Waves in Compressible Magnetically Twisted Flux Tubes

Oscillations of magnetic flux tubes are of great importance as they contain information about the geometry and fine structure of the flux tubes. Here we derive and analytically solve in terms of Kummer’s functions the linear governing equations of wave propagation for sausage surface and body modes (m=0) of a magnetically twisted compressible flux tube embedded in a compressible uniformly magnetized plasma environment in cylindrical geometry. A general dispersion relation is obtained for such flux tubes. Numerical solutions for the phase velocity are obtained for a wide range of wavenumbers and for varying magnetic twist. The effect of magnetic twist on the period of oscillations of sausage surface modes for different values of the wavenumber and vertical magnetic field strength is calculated for representative photospheric and coronal conditions. These results generalize and extend previous studies of MHD waves obtained for incompressible or for compressible but nontwisted flux tubes. It is found that magnetic twist may change the period of sausage surface waves of the order of a few percent when compared to counterparts in straight nontwisted flux tubes. This information will be most relevant when high-resolution observations are used for diagnostic exploration of MHD wave guides in analogy to solar-interior studies by means of global eigenoscillations in helioseismology.     

The magnetic nature of solar flares

The main challenge for the theory of solar eruptions has been to understand two basic aspects of large flares. These are the cause of the flare itself and the nature of the morphological features which form during its evolution. Such features include separating ribbons of H emission joined by a rising arcade of soft x-ray loops, with hard x-ray emission at their summits and at their feet. Two major advances in our understanding of the theory of solar flares have recently occurred. The first is the realisation that a magnetohydrodynamic (MHD) catastrophe is probably responsible for the basic eruption and the second is that the eruption is likely to drive a reconnection process in the field lines stretched out by the eruption. The reconnection is responsible for the ribbons and the set of rising soft x-ray loops, and such a process is well supported by numerical experiments and detailed observations from the Japanese satellite Yohkoh. Magnetic energy conversion by reconnection in two dimensions is relatively well understood, but in three dimensions we are only starting to understand the complexity of the magnetic topology and the MHD dynamics which are involved. How the dynamics lead to particle acceleration is even less well understood. Particle acceleration in flares may in principle occur in a variety of ways, such as stochastic acceleration by MHD turbulence, acceleration by direct electric fields at the reconnection site, or diffusive shock acceleration at the different kinds of MHD shock waves that are produced during the flare. However, which of these processes is most important for producing the energetic particles that strike the solar surface remains a mystery. Received 2 January 2001 / Published online 17 July 2001     

Solar Flare Statistics with a One-Dimensional Mhd Model

The dynamics of dissipative events in coronal loops is modeled with the compressible MHD equations in one space dimension in slab geometry, with a forcing that mimics the footpoint motions. Using a set of numerical simulations, the statistics of strong velocity and magnetic field gradients that develop, leading to a bursty dissipation, are analyzed. Agreement with existing observations of X-ray solar flares obtains concerning the power-law distribution of the luminosity histogram, including when the dynamics are simplified to that of the Burgers equation; in that latter case, this allows for recasting the analysis in terms of avalanche-type models.