The understanding of the nonlinear energy cascade and the spatial small-scale structure of incompressible MHD turbulence is largely based on phenomenology and direct numerical simulation. This chapter brie.y summarizes the current level of knowledge and highlights recent developments in this field of research. Particular stress is put on the di.erence between isotropic turbulence and configurations permeated by a mean magnetic field. Properly taking into account spatial anisotropy of the turbulence induced by the magnetic field is an important challenge to MHD turbulence theory at present. The physical picture of the turbulent energy cascade is still under discussion. While there is ample evidence of Kolmogorov scaling in numerical simulations of globally isotropic MHD turbulence, recent high-resolution simulations where a strong mean magnetic field is imposed on the turbulent flow show Iroshnikov-Kraichnan-like scaling for field-perpendicular fluctuations (cf. Sect. 3.4). This rules out the Goldreich Sridhar model which is the first anisotropic phenomenology of MHD turbulence and predicts field perpendicular Kolmogorov spectra. These findings have been corroborated by results of EDQNM closure theory calculations which give a simple relation between the residual and the total energy spectrum veri.ed by numerical simulation. A recent enhancement of the GS-model by Boldyrev might however explain the observed behaviour. The intermittent small-scale structure, which is probed by higher-order two-point statistics, is visible in the structure-function scaling exponents. In isotropic MHD turbulence, their characteristic behaviour can be matched well by a Log-Poisson model which takes into account that the energy cascade is Kolmogorov-like and that the most singular dissipative structures are quasi-two-dimensional current and vorticity sheets. The model can be generalized to reproduce the structure-function scalings parallel and perpendicular to an applied mean magnetic .eld. Numerical simulations with increasing field strength show, furthermore, that the system becomes gradually twodimensional in the field-perpendicular direction while the dissipative structures turn out to be more homogeneous and less intermittent along the mean field. This behaviour is the consequence of the alignment of dissipative current and vorticity sheets with the mean magnetic field. © Spring-Verlag Berlin Heidelberg 2009.
