The existence of triadic resonances in the magnetized spherical Couette system (MSC) is related to the development of modulated rotating waves, which are quasiperiodic flows understood in terms of bifurcation theory in systems with symmetry. In contrast to previous studies in spherical geometry the resonant modes are not inertial waves but related with the radial jet instability which is strongly equatorially antisymmetric. We propose a general framework in which triadic resonances are generated through successive Hopf bifurcations from the base state. The study relies on an accurate frequency analysis of different modes of the flow, for solutions belonging to two different bifurcation scenarios. The azimuthal and latitudinal nonlinear coupling among the resonant modes is analysed and interpreted using spherical harmonics and the results are compared with previous studies in spherical geometry.
Taylor-Couette flow is often used as a simplified model for complex rotating flows in the interior of stars and accretion disks. 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 tr…
We present a study devoted to a detailed description of modulated rotating waves (MRW) in the magnetized spherical Couette system. The set-up consists of a liquid metal confined between two differentially rotating spheres and subjected to an axially applied magnetic field. When the magnetic field strength is varied, several branches of MRW are obtained by means of three dimensional direct numerical simulations (DNS). The MRW originate from parent branches of rotating waves (RW) and are classified according to Rand's (Arch. Ration. Mech. Anal 79:1-37, 182) and Coughling & Marcus (J. Fluid Mech.…
Rotating waves (RW) bifurcating from the axisymmetric basic magnetized spherical Couette (MSC) flow are computed by means of Newton-Krylov continuation techniques for periodic orbits. In addition, their stability is analysed in the framework of Floquet theory. The inner sphere rotates whilst the outer is kept at rest and the fluid is subjected to an axial magnetic field. For a moderate Reynolds number ${\rm Re}=10^3$ (measuring inner rotation) the effect of increasing the magnetic field strength (measured by the Hartmann number ${\rm Ha}$) is addressed in the range ${\rm Ha}\in(0,80)$ correspo…
A conducting Taylor-Couette flow with quasi-Keplerian rotation law containing a toroidal magnetic field serves as a mean-field dynamo model of the Tayler-Spruit-type. The flows are unstable against nonaxisymmetric perturbations which form electromotive forces defining $α$ effect and eddy diffusivity. If both degenerated modes with $m=\pm 1$ are excited with the same power then the global $α$ effect vanishes and a dynamo cannot work. It is shown, however, that the Tayler instability produces finite $α$ effects if only an isolated mode is considered but this intrinsic helicity of the single-mode…
The Derviche Tourneur Sodium experiment, a spherical Couette magnetohydrodynamics experiment with liquid sodium as the medium and a dipole magnetic field imposed from the inner sphere, recently underwent upgrades to its diagnostics to better characterize the flow and induced magnetic fields with global rotation. In tandem with the upgrades, a set of direct numerical simulations were run with the xshells code to give a more complete view of the fluid and magnetic dynamics at various rotation rates of the inner and outer spheres. These simulations reveal several dynamic regimes, determined by th…
Magnetorotational instability (MRI) is one of the fundamental processes in astrophysics, driving angular momentum transport and mass accretion in a wide variety of cosmic objects. Despite much theoretical/numerical and experimental efforts over the last decades, its saturation mechanism and amplitude, which sets the angular momentum transport rate, remains not well understood, especially in the limit of high resistivity, or small magnetic Prandtl numbers typical to interiors (dead zones) of protoplanetary disks, liquid cores of planets and liquid metals in laboratory. Using direct numerical si…
The long therm behavior of chaotic flows is investigated by means of time dependent frequency analysis. The system under test consists of an electrically conducting fluid, confined between two differentially rotating spheres. The spherical setup is exposed to an axial magnetic field. The classical Fourier Transform method provides a first estimation of the time dependence of the frequencies associated to the flow, as well as its volume-averaged properties. It is however unable to detect strange attractors close to regular solutions in the Feigenbaum as well as Newhouse-Ruelle-Takens bifurcatio…
Nonlinear evolution of magnetorotational instability in a magnetized Taylor-Couette flow: scaling properties and relation to upcoming DRESDYN-MRI experiment
We present an optimization-based method to efficiently calculate accurate nonlinear models of Taylor vortex flow. We use the resolvent formulation of McKeon & Sharma (2010) to model these Taylor vortex solutions by treating the nonlinearity not as an inherent part of the governing equations but rather as a triadic constraint which must be satisfied by the model solution. We exploit the low rank linear dynamics of the system to calculate an efficient basis for our solution, the coefficients of which are then calculated through an optimization problem where the cost function to be minimized is t…
The stability of conducting Taylor-Couette flows under the presence of toroidal magnetic background fields is considered. For strong enough magnetic amplitudes such magnetohydrodynamic flows are unstable against nonaxisymmetric perturbations which may also transport angular momentum. In accordance with the often used diffusion approximation one expects the angular momentum transport vanishing for rigid rotation. In the sense of a nondiffusive $Λ$ effect, however, even for {\em rigidly} rotating $z$-pinches an axisymmetric angular momentum flux appears which is directed outward (inward) for lar…
We reveal and investigate a new type of linear axisymmetric helical magnetorotational instability which is capable of destabilizing viscous and resistive rotational flows with radially increasing angular velocity, or positive shear. This instability is double-diffusive by nature and is different from the more familiar helical magnetorotational instability, operating at positive shear above the Liu limit, in that it works instead for a wide range of the positive shear when ${\rm (i)}$ a combination of axial/poloidal and azimuthal/toroidal magnetic fields is applied and ${\rm (ii)}$ the magnetic…
We consider a differentially rotating flow of an incompressible electrically conducting and viscous fluid subject to an external axial magnetic field and to an azimuthal magnetic field that is allowed to be generated by a combination of an axial electric current external to the fluid and electrical currents in the fluid itself. In this setting we derive an extended version of the celebrated Hain-Lust differential equation for the radial Lagrangian displacement that incorporates the effects of the axial and azimuthal magnetic fields, differential rotation, viscosity, and electrical resistivity.…
Non-axisymmetric modes of magnetorotational and possible hydrodynamical instabilities in the upcoming DRESDYN-MRI experiments -- linear and nonlinear dynamics
