Dynamics and transport of instabilities in magnetized quasi-Keplerian Taylor-Couette flows

The stability and transition to turbulence in canonical shear flows have since long been an outstanding scientific problem. One of the most exciting examples of shear flow is Keplerian motion of gas and dust in accretion disks. Although the Keplerian velocity profile is linearly stable, the presence of magnetic fields gives rise to the magnetorotational instability (MRI). MRI is considered one of the most powerful sources of turbulence in hydrodynamically stable quasi-Keplerian flows, however obtaining observational evidence of its operation is challenging. Although the linear stability of Keplerian flows with applied external magnetic fields has been studied for decades, the influence of the instability on the outward angular momentum transport, an inherent prerequisite for accretion to occur, is still far from understood. The aim of this thesis was to provide a better understanding of angular momentum transport and nonlinear properties of the MRI. Motivated by recent laboratory experiments, the MRI driven by an azimuthal magnetic field in an electrically conducting fluid sheared between two concentric rotating cylinders (Taylor--Couette flow) was explored. The instability was studied numerically with both linear stability analysis and fully resolved direct numerical simulations of the Navier--Stokes and induction equations. It was found that at low magnetic Prandtl numbers, as those in liquid metals, the laminar Couette flow becomes unstable to a wave rotating in the azimuthal direction and standing in the axial direction via a supercritical Hopf bifurcation. Subsequently, the flow features a catastrophic transition to spatio-temporal chaos which is mediated by a subcritical Hopf bifurcation. The results are in quantitative agreement with the PROMISE experiment and dramatically extend its realizable parameter range. Subsequently, the enhancement of angular momentum transport by turbulent stresses in the highly turbulent flow regimes was determined. One regime is dominated by magnetically triggered inertial waves, with transport mostly due to velocity fluctuations, and another by magnetocoriolis waves, where magnetic field fluctuations prevail. The magnetic Reynolds number defines the type of turbulence, with a crossover around the critical value of $100$. The results give a comprehensive picture of transport enhancement by MRI spanning from low (as in liquid metals) to high (as in plasma) magnetic Prandtl numbers. In the latter case the existence of a finite-amplitude dynamo was demonstrated. This suggests that accretion disks can operate self-sustaining MHD-turbulence and thereby transport angular momentum efficiently without the need of considering imposed magnetic fields.