Numerical simulations of turbulence in 3D plane incompressible Couette flow

Author: George Mamatsashvili
Co-authors: G. Chagelishvili
Keywords: Couette flow, plasma instability, turbulence, transient growth, accretion disc
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We did numerical simulations of subcritical turbulence in 3D Couette flow and investigated the dynamics of non-axisymmetric perturbations and turbulence. The flow corresponds to vertically stratified Keplerian plasma discs threaded by a weak non-zero net vertical magnetic field in the local shearing box approximation. Perturbations are decomposed into shearing waves, or spatial harmonics whose temporal evolution is then followed via numerical integration of the ideal MHD equations of the shearing box. There are two basic modes in the system -- inertia-gravity waves and magnetic mode, which displays the magnetorotational instability (MRI). As distinct from previous related studies, we introduce ``eigen-variables'' characterizing each (counter-propagating) component of the inertia-gravity and magnetic modes, which are governed by a set of four first order \emph{coupled} ordinary differential equations. This allowed us to identify a new process of linear coupling of the two above non-axisymmetric modes due to the flow shear. We also carried out a comparative analysis of the dynamics of non-axisymmetric and axisymmetric magnetic mode perturbations. It is demonstrated that the growth of ``optimal'' and close-to-optimal non-axisymmetric harmonics of this mode, having transient nature, can prevail over the exponential growth of axisymmetric ones (i.e., over the axisymmetric MRI) during dynamical time. A possible implication of this result for axisymmetric channel solutions emerging in numerical simulations is discussed. In particular, the formation of the (axisymmetric) channel may be affected/impeded by non-axisymmetric modes already at the early linear stage leading to its untimely disruption -- the outcome strongly depends on the amplitude and spectrum of initial perturbation. So, this competition may result in an uncertainty in the magnetic mode's non-linear dynamics. Although we consider incompressible perturbations, in the final part, speculate on the dynamics in the compressible case. It is shown that a maximum growth of non-axisymmetric magnetic mode occurs at vertical wavelengths close to the flow scale-height for which compressibility effects are important. This indirectly suggests that compressibility plays a role in the dynamics of the non-axisymmetric MRI and, ultimately, in the resulting turbulent state.