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A concise review is given of astrophysically motivated experimental and theoretical research on Taylor-Couette flow. The flows of interest rotate differentially with the inner cylinder faster than the outer, but are linearly stable against Rayleigh's inviscid centrifugal instability. At shear Reynolds numbers as large as [Formula: see text], hydrodynamic flows of this type (quasi-Keplerian) appear to be nonlinearly stable: no turbulence is seen that cannot be attributed to interaction with the axial boundaries, rather than the radial shear itself. Direct numerical simulations agree, although they cannot yet reach such high Reynolds numbers. This result indicates that accretion-disc turbulence is not purely hydrodynamic in origin, at least insofar as it is driven by radial shear. Theory, however, predicts linear magnetohydrodynamic (MHD) instabilities in astrophysical discs: in particular, the standard magnetorotational instability (SMRI). MHD Taylor-Couette experiments aimed at SMRI are challenged by the low magnetic Prandtl numbers of liquid metals. High fluid Reynolds numbers and careful control of the axial boundaries are required. The quest for laboratory SMRI has been rewarded with the discovery of some interesting inductionless cousins of SMRI, and with the recently reported success in demonstrating SMRI itself using conducting axial boundaries. Some outstanding questions and near-future prospects are discussed, especially in connection with astrophysics. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (Part 2)'.

Taylor-Couette flow (TCF) is often used as a simplified model for complex rotating flows in the interior of stars and accretion discs. The flow dynamics in these objects is influenced by magnetic fields. For example, quasi-Keplerian flows in Taylor-Couette geometry become unstable to a travelling or standing wave in an external magnetic field if the fluid is conducting; there is an instability even when the flow is hydrodynamically stable. This magnetorotational instability leads to the development of chaotic states and, eventually, turbulence, when the cylinder rotation is sufficiently fast. The transition to turbulence in this flow can be complex, with the coexistence of parameter regions with spatio-temporal chaos and regions with quasi-periodic behaviour, involving one or two additional modulating frequencies. Although the unstable modes of a periodic flow can be identified with Floquet analysis, here we adopt a more flexible equation-free data-driven approach. We analyse the data from the transition to chaos in the magnetized TCF and identify the flow structures related to the modulating frequencies with dynamic mode decomposition; this method is based on approximating nonlinear dynamics with a linear infinite-dimensional Koopman operator. With the use of these structures, one can construct a nonlinear reduced model for the transition. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)'.

We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. In the presence of the magnetic field, the hydrodynamically stable flow can demonstrate non-axisymmetric azimuthal magnetorotational instability (AMRI) both in the diffusionless case and in the double-diffusive case with viscous and ohmic dissipation. Performing stability analysis of amplitude transport equations of short-wavelength approximation, we find that the threshold of the diffusionless AMRI via the Hamilton-Hopf bifurcation is a singular limit of the thresholds of the viscous and resistive AMRI corresponding to the dissipative Hopf bifurcation and manifests itself as the Whitney umbrella singular point. A smooth transition between the two types of instabilities is possible only if the magnetic Prandtl number is equal to unity, Pm =1. At a fixed Pm ≠1, the threshold of the double-diffusive AMRI is displaced by finite distance in the parameter space with respect to the diffusionless case even in the zero dissipation limit. The complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities, the double-diffusive system reduces to a marginally stable system which is either Hamiltonian or parity-time-symmetric., Competing Interests: I declare I have no competing interests.

This paper demonstrates an equivalence between rotating magnetized shear flows and a stressed elastic beam. This results from finding the same form of dynamical equations after an asymptotic reduction of the axis-symmetric magnetorotational instability (MRI) under the assumption of almost-critical driving. The analysis considers the MRI dynamics in a non-dissipative near-equilibrium regime. Both the magnetic and elastic systems reduce to a simple one-dimensional wave equation with a non-local nonlinear feedback. Under transformation, the equation comprises a large number of mean-field interacting Duffing oscillators. This system was the first proven example of a strange attractor in a partial differential equation. Finding the same reduced equation in two natural applications suggests the model might result from other applications and could fall into a universal class based on symmetry.