Astronomy Colloquium

Magnetohydrodynamic (MHD) turbulence is found e.g. in the solar wind, in the solar convection zone or in the interstellar medium. In that context, direct numerical simulations of three-
dimensional MHD at a Taylor Reynolds number of 1700 on a grid of 15363 points are reported 1,2,3. The flow is incompressible and decaying in time, and the initial condition is a superposition of large scale ABC Beltrami flows for wavenumbers k ≤ 4 and random noise at small scales with a k−3 spectrum, with negligible correlation between the velocity and the magnetic field (ρC ∼ 10−4) and equal kinetic and magnetic energies; finally, no uniform magnetic field is imposed and the magnetic Prandtl number is equal to unity. Parallel current and vorticity sheets form at the same spatial locations, and further destabilize and fold or roll-up after an initial exponential phase. A self-similar evolution of the current and vorticity maxima is found, in which they grow as a cubic power of time; the flow then reaches a finite dissipation rate independent of Reynolds number. A Kelvin-Helmoltz instability of current and vorticity sheets is seen only at the highest Reynolds number, reminiscent of recent observations in the magnetosphere. At peak of dissipation, the total energy spectrum is a combination of two components, each moderately resolved. Isotropy obtains in the large scales, with a spectrum compatible with the Iroshnikov-Kraichnan theory stemming from the weakening of nonlinear interactions due to Alfv´en waves and leading to a ∼ k−3/2 law; scaling of structure functions confirms the non-Kolmogorovian nature of the flow in this range. At small scales, weak turbulence emerges with a k−2 spectrum, the perpendicular direction referring to the local quasi-uniform magnetic field. Finally, local directional alignment of the velocity and magnetic field fluctuations occurs rapidly, both observed in direct numerical simulations and in solar wind data. This relaxation process leads to a local weakening of the nonlinear terms in the small scale vorticity and current structures where alignment takes place. Whether such results are universal is not clear, and several parameters may play a role, such as ρC or the amount of magnetic helicity in the flow. Thus, high-resolution parametric studies are needed in order to understand in detail the interactions of turbulent eddies and Alfven waves and the dynamics of reconnection events. In order to reach higher Reynolds numbers, several possibilities will be evoked if time permits.