Shell Models of Magnetohydrodynamic Turbulence

Shell models of hydrodynamic turbulence originated in the seventies. Their main aim was to describe the statistics of homogeneous and isotropic turbulence in spectral space, using a simple set of ordinary differential equations. In the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged based on the same principles as their hydrodynamic counter-part but also incorporating interactions between magnetic and velocity fields. In recent years, significant improvements have been made such as the inclusion of non-local interactions and appropriate definitions for helicities. Though shell models cannot account for the spatial complexity of MHD turbulence, their dynamics are not over simplified and do reflect those of real MHD turbulence including intermittency or chaotic reversals of large-scale modes. Furthermore, these models use realistic values for dimensionless parameters (high kinetic and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing extended inertial range and accurate dissipation rate. Using modern computers it is difficult to attain an inertial range of three decades with direct numerical simulations, whereas eight are possible using shell models. In this review we set up a general mathematical framework allowing the description of any MHD shell model. The variety of the latter, with their advantages and weaknesses, is introduced. Finally we consider a number of applications, dealing with free-decaying MHD turbulence, dynamo action, Alfven waves and the Hall effect.Comment: published in Physics Report

Occurrence Rates and Heating Effects of Tangential and Rotational Discontinuities as Obtained from Three-dimensional Simulation of Magnetohydrodynamic Turbulence

In solar wind, magnetohydrodynamic (MHD) discontinuities are ubiquitous and often found to be at the origin of turbulence intermittency. They may also play a key role in the turbulence dissipation and heating of the solar wind. The tangential (TD) and rotational (RD) discontinuities are the two most important types of discontinuities. Recently, the connection between turbulence intermittency and proton thermodynamics has been being investigated observationally. Here we present numerical results from three-dimensional MHD simulation with pressure anisotropy and define new methods to identify and to distinguish TDs and RDs. Three statistical results obtained about the relative occurrence rates and heating effects are highlighted: (1) RDs tend to take up the majority of the discontinuities along with time; (2) the thermal states embedding TDs tend to be associated with extreme plasma parameters or instabilities, while RDs do not; (3) TDs have a higher average T as well as perpendicular temperature $T_\perp$. The simulation shows that TDs and RDs evolve and contribute to solar wind heating differently. These results will inspire our understanding of the mechanisms that generate discontinuities and cause plasma heating.Comment: 5 Figures, Submitted to Astrophys. J. Lett., in the process of refereein

Reconnection Studies Under Different Types of Turbulence Driving

We study a model of fast magnetic reconnection in the presence of weak turbulence proposed by Lazarian and Vishniac (1999) using three-dimensional direct numerical simulations. The model has been already successfully tested in Kowal et al. (2009) confirming the dependencies of the reconnection speed $V_{rec}$ on the turbulence injection power $P_{inj}$ and the injection scale $l_{inj}$ expressed by a constraint $V_{rec} \sim P_{inj}^{1/2} l_{inj}^{3/4}$ and no observed dependency on Ohmic resistivity. In Kowal et al. (2009), in order to drive turbulence, we injected velocity fluctuations in Fourier space with frequencies concentrated around $k_{inj}=1/l_{inj}$, as described in Alvelius (1999). In this paper we extend our previous studies by comparing fast magnetic reconnection under different mechanisms of turbulence injection by introducing a new way of turbulence driving. The new method injects velocity or magnetic eddies with a specified amplitude and scale in random locations directly in real space. We provide exact relations between the eddy parameters and turbulent power and injection scale. We performed simulations with new forcing in order to study turbulent power and injection scale dependencies. The results show no discrepancy between models with two different methods of turbulence driving exposing the same scalings in both cases. This is in agreement with the Lazarian and Vishniac (1999) predictions. In addition, we performed a series of models with varying viscosity $\nu$. Although Lazarian and Vishniac (1999) do not provide any prediction for this dependence, we report a weak relation between the reconnection speed with viscosity, $V_{rec}\sim\nu^{-1/4}$.Comment: 19 pages, 9 figures. arXiv admin note: text overlap with arXiv:0903.205

Astrophysical turbulence modeling

The role of turbulence in various astrophysical settings is reviewed. Among the differences to laboratory and atmospheric turbulence we highlight the ubiquitous presence of magnetic fields that are generally produced and maintained by dynamo action. The extreme temperature and density contrasts and stratifications are emphasized in connection with turbulence in the interstellar medium and in stars with outer convection zones, respectively. In many cases turbulence plays an essential role in facilitating enhanced transport of mass, momentum, energy, and magnetic fields in terms of the corresponding coarse-grained mean fields. Those transport properties are usually strongly modified by anisotropies and often completely new effects emerge in such a description that have no correspondence in terms of the original (non coarse-grained) fields.Comment: 88 pages, 26 figures, published in Reports on Progress in Physic