Specific anisotropy of nonlinear processes and self-sustenance of MRI-turbulence in accretion discs

We investigate the sustenance and dynamical balances of MRI-turbulence in accretion disks with a zero net magnetic flux. Zero net flux MRI has attracted a great interest in the last decade, because of its importance in MRI-dynamo in disks. It is unique, in the sense that there is no characteristic length-scale for MRI to grow purely exponentially and hence the instability is instead of a subcritical type, being energetically powered by linear nonmodal, or transient growth of perturbations. This transient growth of MRI is not, however, able to ensure a long-term sustenance of the turbulence and necessitates nonlinear feedback replenishing such transiently growing modes. To examine the existence of such a nonlinear feedback and ultimately understand the whole self-sustenance process of MRI-turbulence, we first performed simulations and then a detailed analysis of the turbulence dynamics in Fourier space. We showed that the disk flow shear gives rise to anisotropy of nonlinear processes in Fourier space. As a result, the key nonlinear process for the sustenance appears to be a topologically new type of angular (i.e., over wavevector orientations) redistribution of modes in Fourier space – the nonlinear transverse cascade – in contrast to the well-known direct/inverse cascades in the absence of shear in classical theories of isotropic turbulence. Moreover, the transverse cascade is a generic nonlinear process in different kinds of shear flows. The sustenance of zero net flux MRI-turbulence relies on the interplay between the two basic processes -- linear transient growth of MRI and the nonlinear transverse cascade. They mostly operate at length scales comparable to the box size (disk scale height), which we call the vital area of the turbulence in Fourier space. Base on this self-sustenance scheme we give a physical interpretation of the dependence (sensitivity) of the zero net flux MRI-turbulence with respect to magnetic Prandtl number in terms of competition between transverse and direct cascades.