We provide a tutorial on the paradigms and tools of magnetohydrodynamic (MHD) turbulence. The principal paradigm is that of a turbulent cascade from large scales to small, resulting in power law behavior for the frequency power spectrum for magnetic fluctuations EBðf Þ. We will describe five useful statistical tools for MHD turbulence in the time domain: the temporal autocorrelation function, t...

In this paper we develop adaptive numerical schemes for certain nonlinear variational problems. The discretization of the variational problems is done by a suitable frame decomposition of the solution, i.e., a complete, stable, and redundant expansion. The discretization yields an equivalent nonlinear problem on ℓ2(N), the space of frame coefficients. The discrete problem is then adaptively sol...

It is shown that two-dimensional MHD turbulence is in certain respects closer to threedimensional than to two-dimensional hydrodynamic turbulence. A second-order closure indicates that: (i) a t zero viscosity and magnetic diffusivity, a singularity appears a t a finite time; (ii) there is an energy cascade to small scales and an inverse cascade of squared magnetic potential, in agreement with a...

Recent developments in the derivation of reduced models for weakly compressible magnetohydrodynamic (MHD) turbulence are discussed. A four-field system of equations has been derived from the compressible magnetohydrodynamic (MHD) equations to describe turbulence in the interstellar medium and the solar wind. These equations apply to a plasma permeated by a spatially varying mean magnetic field ...

Global motion of magnetohydrodynamic fluid in a domain bounded by a free surface and under the external electrodynamic field is proved. The motion is such that velocity and magnetic field are small in H3-space.

We study the afterglow phases of a GRB through relativistic magnetohydrodynamic simulations. The evolution of a relativistic shell propagating into a homogeneous external medium is followed. We focus on the effect of the magnetization of the ejecta on the initial phases of the ejecta-external medium interaction. In particular we are studying the condition for the existence of a reverse shock in...