On two-dimensional magnetohydrodynamic turbulence

It is shown that two-dimensional MHD turbulence is in certain respects closer to three-dimensional than to two-dimensional hydrodynamic turbulence. A second-order closure indicates that: at zero viscosity and magnetic diffusivity, a singularity appears at a finite time; there is an energy cascade to small scales and an inverse cascade of squared magnetic potential, in agreement with a conjecture of Fyfe & Montgomery (1976); small-scale magnetic energy acts like a negative eddy viscosity on large-scale magnetic fields;a (iv) upon injection of magnetic energy, a stationary state is obtained which has zero magnetic energy for a positive magnetic diffusivity λ (anti-dynamo theorem); however, this stationary state is preceded by a very long non-zero magnetic energy plateau which probably extends to infinite times as λ → 0. It is suggested that direct numerical simulation of the two-dimensional MHD equations with high resolution (a 512 2 or 1024 2 grid) could lead to a better understanding of the small-scale structure of fully developed turbulence, especially questions of intermittency and geometry.