Magnetized Spherical Couette Flow Research Articles

Dynamic transitions of the magnetized spherical Couette flow between its base state and the return flow instability

The transition between the stable base state of the magnetized spherical Couette (MSC) flow and the return flow instability is experimentally investigated. The experiments are conducted using an MSC setup consisting of insulating spheres with the ratio of the inner to the outer radii r i/r o = 0.5, Reynolds number Re = 1000 and Hartmann number Ha ∈ [25, 29]. The transition is characterized by changes in the power spectra of the azimuthal modes in the flow as Ha is dynamically changed. The transition occurs in the interval Ha ∈ [26.5, 27.5]. The evolution of the power spectra of the azimuthal modes exhibits hysteretic effect depending on whether Ha is increased or decreased within the experimental interval. The power spectra in the azimuthal modes m ∈ {3, 4} increases and remains dominant as Ha is increased, while the power spectra in m ∈ {2, 4} are dominant while the flow is time dependent due to return flow instability as Ha is decreased.

Modulated rotating waves and triadic resonances in spherical fluid systems: The case of magnetized spherical Couette flow

The existence of triadic resonances in the magnetized spherical Couette system 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 to 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 analyzed and interpreted using spherical harmonics, and the results are compared with previous studies in spherical geometry.

Experimental investigation of the return flow instability in magnetized spherical Couette flows

We conduct magnetized spherical Couette (MSC) flow experiments in the return flow instability regime with GaInSn as the working fluid, the ratio of the inner to the outer sphere radii ri/ro = 0.5, the Reynolds number Re = 1000, and the Hartmann number Ha ∈ [27.5, 40]. Rotating waves with different azimuthal wavenumbers m ∈ {2, 3, 4} manifest in certain ranges of Ha in the experiments, depending on whether the values of Ha were fixed or varied from different initial values. These observations demonstrate the multistability of rotating waves, which we attribute to the dynamical system representing the state of the MSC flow tending to move along the same solution branch of the bifurcation diagram when Ha is varied. In experiments with both fixed and varying Ha, the rotation frequencies of the rotating waves are consistent with the results of nonlinear stability analysis. A brief numerical investigation shows that differences in the azimuthal wavenumbers of the rotating waves that develop in the flow also depend on the azimuthal modes that are initially excited.

Saturation of nonaxisymmetric instabilities of magnetized spherical Couette flow.

We numerically investigate the saturation of the hydromagnetic instabilities of a magnetized spherical Couette flow. Previous simulations demonstrated a region where the axisymmetric flow, calculated from a 2D simulation, was linearly unstable to nonaxisymmetric perturbations. Full, nonlinear, 3d simulations showed that the saturated state would consist only of harmonics of one azimuthal wave number, though there were bifurcations and transitions as nondimensional parameters (Re, Ha) were varied. Here, the energy transfer between different azimuthal modes is formulated as a network. This demonstrates a mechanism for the saturation of one mode and for the suppression of other unstable modes. A given mode grows by extracting energy from the axisymmetric flow, and then saturates as the energy transfer to its second harmonic equals this inflow. At the same time, this mode suppresses other unstable modes by facilitating an energy transfer to linearly stable modes.

Modes and instabilities in magnetized spherical Couette flow

AbstractSeveral teams have reported peculiar frequency spectra for flows in a spherical shell. To address their origin, we perform numerical simulations of the spherical Couette flow in a dipolar magnetic field, in the configuration of the$DTS$experiment. The frequency spectra computed from time-series of the induced magnetic field display similar bumpy spectra, where each bump corresponds to a given azimuthal mode number$m$. The bumps appear at moderate Reynolds number (${\simeq }2600$) if the time-series are long enough (${\gt }300$rotations of the inner sphere). We present a new method that permits retrieval of the dominant frequencies for individual mode numbers$m$, and extraction of the modal structure of the full nonlinear flow. The maps of the energy of the fluctuations and the spatio-temporal evolution of the velocity field suggest that fluctuations originate in the outer boundary layer. The threshold of instability is found at${\mathit{Re}}_{c} = 1860$. The fluctuations result from two coupled instabilities: high-latitude Bödewadt-type boundary layer instability, and secondary non-axisymmetric instability of a centripetal jet forming at the equator of the outer sphere. We explore the variation of the magnetic and kinetic energies with the input parameters, and show that a modified Elsasser number controls their evolution. We can thus compare with experimental determinations of these energies and find a good agreement. Because of the dipolar nature of the imposed magnetic field, the energy of magnetic fluctuations is much larger near the inner sphere, but their origin lies in velocity fluctuations that are initiated in the outer boundary layer.

Magnetic induction maps in a magnetized spherical Couette flow experiment

The DTS experiment is a spherical Couette flow experiment with an imposed dipolar magnetic field. Liquid sodium is used as a working fluid. In a series of measurement campaigns, we have obtained data on the mean axisymmetric velocity, the mean induced magnetic field and electric potentials. All these quantities are coupled through the induction equation. In particular, a strong omega-eff ect is produced by di fferential rotation within the fluid shell, inducing a significant azimuthal magnetic field. Taking advantage of the simple spherical geometry of the experiment, I expand the azimuthal and meridional fields into Legendre polynomials and derive the expressions that relate all measurements to the radial functions of the velocity field for each harmonic degree. For small magnetic Reynolds numbers Rm the relations are linear, and the azimuthal and meridional equations decouple. Selecting a set of measurements for a given rotation frequency of the inner sphere (Rm = 9.4), I invert simultaneously the velocity and the magnetic data and thus reconstruct both the azimuthal and the meridional fields within the fluid shell. The results demonstrate the good internal consistency of the measurements, and indicate that turbulent non-axisymmetric fluctuations do not contribute significantly to the axisymmetric magnetic induction.

Instabilities in magnetized spherical Couette flow

We report three-dimensional numerical simulations of the flow of an electrically conducting fluid in a spherical shell when a magnetic field is applied. Different spherical Couette configurations are investigated by varying the rotation ratio between the inner and the outer sphere, the geometry of the imposed field, and the magnetic boundary conditions on the inner sphere. Either a Stewartson layer or a Shercliff layer, accompanied by a radial jet, can be generated depending on the rotation speeds and the magnetic-field strength, and various nonaxisymmetric destabilizations of the flow are observed. We show that instabilities arising from the presence of boundaries present striking similarities with the magnetorotational instability (MRI). To this end, we compare our numerical results to experimental observations of the Maryland experiment [D. R. Sisan et al., Phys. Rev. Lett. 93, 114502 (2004)], which claimed to observe MRI in a similar setup.