MHD Turbulence: Nonlocal, Anisotropic, Nonuniversal?

Summary. Kolmogorov’s theory and philosophy of turbulence are based on a num-ber of assumptions that have become standard notions with which one approaches turbulence in many, including non-hydrodynamic, systems. However, it turns out that in MHD turbulence, locality of interactions in scale space, isotropy of small scales or even universality cannot be taken for granted and, in fact, can be shown to fail. This note focuses on these unconventional aspects of MHD turbulence and on the related phenomenon of small-scale dynamo using a combination of simple physical reasoning and numerical evidence. MHD turbulence is the turbulence of a highly conducting fluid in which chaotic velocity and magnetic-field fluctuations exist and interact in a broad range of spatial and temporal scales. It is the prevailing form of turbulence that is presumed to occur and in many cases is, indeed, observed in astrophys-ical plasmas: examples are solar wind, interstellar medium, accretion discs, galaxy clusters etc. (see, e.g., a recent review [22] and references therein). It is important to remember that the MHD approximation is strictly only appropriate for low-frequency fluctuations at collisional scales, although for strongly magnetised plasmas, it can under certain assumptions be extended to collisionless scales as well [27]. In this short note, we shall not venture beyond the incompressible MHD equations, which read ∂tu+ u · ∇u = −∇p+B · ∇B+ ν∇2u+ f, (1) ∂tB+ u · ∇B = B · ∇u+ η∇2B, (2) where u is velocity, B magnetic field, p the total pressure determined by the incompressibility condition ∇ · u = 0, ν viscosity, η magnetic diffusivity, and f a body force (we use units in which p is scaled by ρ and B by 4piρ, where ρ = const is the density of the medium). The body force models some system-specific energy input at the outer scale `0. As usual, we assume that the properties of turbulence at scales much smaller than `0 are not sensitiv