MHD COUETTE FLOWS: Experiments and Models

Magneto‐Couette Instabilities — Astrophysics, Theory and Experiments
View Description Hide Description

Rotation of conducting fluid in magnetic field
View Description Hide Description

Characterization of the magnetorotational instability from a turbulent background state
View Description Hide DescriptionExperiments in spherical Couette flow (flow between concentric rotating spheres) with an imposed magnetic field have yielded induced magnetic fields consistent with the magnetorotational instability. This might be expected due to the decreasing rotation rate profile in the base state. The observation is at odds though with existing theory, in that the base state has a significant turbulent component. We characterize the observed induced magnetic fields, as well as the velocity disturbance underlying the instability. The saturated state shows a variety of patterns and dynamics depending on applied magnetic field strength and rotation rate. The observed phase diagram is in qualitative agreement with linear stability theory. We also compare the observed stability diagram with that of MHD instabilities calculated by Hollerbach and Skinner.

Magnetorotational Instability in a Short Couette Flow of Liquid Gallium
View Description Hide DescriptionA concise review is given of an experimental project to study magnetorotational instability (MRI) in a short Couette geometry using liquid gallium. Motivated by the astrophysical importance and lack of direct observation of MRI in nature and in the laboratory, a theoretical stability analysis was performed to predict the required experimental parameters. Despite the long‐wavelength nature of MRI, local analysis agrees excellently with global eigenmode calculations when periodic boundary conditions are used in the axial direction. To explore the effects of rigidly rotating vertical boundaries (endcaps), a prototype water experiment was conducted using dimensions and rotation rates favored by the above analysis. Significant deviations from the expected Couette flow profiles were found. The cause of the discrepancy was investigated by nonlinear hydrodynamic simulations using realistic boundary conditions. It was found that Ekman circulation driven by the endcaps transports angular momentum and qualitatively modifies the azimuthal flow. Based on this new understanding, a new design was made to incorporate two independently driven rings at each endcap. Simulations were used to optimize the design by minimizing Ekman circulation while remaining within engineering capabilities. The new apparatus, which has been constructed and assembled, is currently being tested with water and will be ready for the MRI experiment with gallium soon. This development process illustrates the value of interplay between experiment, simulation, and analytic insight.

Laboratory astrophysics as exemplified by the Riga dynamo experiment
View Description Hide DescriptionIt has been proposed to investigate the magnetorotational instability at a large scale liquid sodium facility. This sort of laboratory astrophysics is encouraged by the recent successful dynamo experiments. We report on our experiences with the Riga dynamo experiment where magnetic field self‐excitation is achieved in a cylindrical vessel filled with approximately 2 m^{3} of liquid sodium which can reach flow velocities up to 20 m/s. The main experimental results on the kinematic and the saturation regime are compared with numerical modelling. Some focus is also laid on the spectra of the magnetic field and the pressure.

Convection in the rotating spherical shell under the central Force Field: 3D flow and bifurcation analysis
View Description Hide DescriptionConvection in a spherical shell under the influence of a radial force field is an important problem in the geophysical and astrophysical framework. To turn out the earth gravity field which breaks the simulated central force field, such a experiment has to be performed in a microgravity environment. The radial force field is produced by applying a voltage difference between the inner and outer spheres: the dielectrophoretic force. This last field differs from the earth’s gravity one. This paper aims at simulating the bifurcated dynamics in this framework. In particular, motions with a 3D structure are presented. The most of simulation results are corroborated by the bifurcation analysis.

Magnetic Field Induction in a Toroidal Screw Flow of Liquid Gallium
View Description Hide DescriptionThe magnetic field induced by the nonstationary screw flow of gallium in a toroidal channel is investigated experimentally. Some typical configurations of the imposed magnetic field are considered. The induced field is measured by sensors placed outside of the channel and by the moving probes in the shell. The induction effects observed are attributed to the mean screw flow. An upper bound to the action of the small‐scale flow perturbations is estimated.

Linear theory of MHD Taylor‐Couette flow
View Description Hide DescriptionThe linear theory of MHD Taylor‐Couette flow (subject to an axial magnetic field and unbounded in z) is presented in order to prepare laboratory experiments to probe the MRI. Only stationary flow patterns are considered but also with nonaxisymmetry and for small magnetic Prandtl numbers. If the outer cylinder is at rest for a small interval of magnetic field amplitudes subcritical excitation is found but only for Pm > 1. For rotating outer cylinder beyond the Rayleigh line the situation is different. Characteristic minima are found for the Reynolds number of the inner cylinder for Hartmann numbers of order 10 for Pm = 1 and 1000 for Pm = 10 ^{−5}. The minimal magnetic Reynolds number in all these cases is of order 10 (see Fig. 7). For liquid sodium (Pm = 10 ^{−5}) the characteristic values for a container with R _{in} = R _{out}/2 = 10 cm are 20 Hz for the inner rotation frequency and 1700 Gauss for the magnetic field amplitude.

Hydromagnetic instabilities in Taylor‐Couette flow at finite and infinite aspect ratios
View Description Hide DescriptionWe present calculations of instabilities in hydromagnetic Taylor‐Couette flow at finite and infinite aspect ratios. In the former case we are concerned with the existence of anomalous modes; in the latter case we describe the magneto‐rotational instability as well as kinematic and fully self‐consistent dynamos.

MRI in Taylor‐Dean flows
View Description Hide DescriptionThe magnetorotational instability (MRI) can destabilize hydrodynamically stable flows which are characterized by an angular momentum that is increasing with the radius and by an angular velocity that is decreasing with radius. Its astrophysical importance comes from the fact that the Kepler flow with Ω(r) ∼ r ^{−3/2} exactly such a behaviour. In order to investigate MRI in a laboratory experiment, the Taylor‐Couette flow with Ω(r) = A + Br ^{2} with A > 0 has been proposed as a substitute for the Kepler flow. In this paper we consider the Taylor‐Dean flow as another example of a flow profile which can exhibit the necessary radial dependence. Taylor‐Dean flows are a combination of the traditional Taylor‐Couette flow with an additional flow that is produced by an azimuthal force. Special focus is laid on the case that the Taylor‐Couette part of the flow is a rigid body rotation and the magnitude of the Dean flow is adjusted in such a way that in the outer part of the flow the conditions for MRI are fulfilled. Based on the dispersion relation derived by Ji, Kageyama and Goodman, in combination with some preliminary global instability analysis, we give some first estimates for the physical parameters of a Taylor‐Dean MRI experiment with liquid sodium.

End‐effects in rapidly rotating cylindrical Taylor‐Couette flow
View Description Hide DescriptionWe present numerical simulations of the flow in a rapidly rotating cylindrical annulus. We show that at the rotation rates relevant to the magneto‐rotational instability, the flow is strongly constrained by the Taylor‐Proudman theorem. As a result, it is controlled almost entirely by the end‐plates. We then consider two possible options for minimizing these end‐effects, namely (i) simply taking a very long cylinder, and (ii) splitting the end‐plates into a series of differentially rotating rings. Regarding option (i), we show that the cylinder would have to be hundreds of times as long as it is wide before end‐effects become unimportant in the interior. Since this is clearly not feasible, we turn to option (ii), and show that in order to obtain a smooth angular velocity profile, the end‐plates would have to be split into around ten rings. If the end‐plates are split into fewer rings, perhaps 3–5, the angular velocity profile will not be smooth, but will instead consist of a series of Stewartson layers at the boundaries from one ring to the next. We suggest therefore that the instabilities one obtains in this system will be the familiar Kelvin‐Helmholtz instabilities of these Stewartson layers, rather than the magneto‐rotational instability. At best, one might hope to obtain the MRI superimposed on these Kelvin‐Helmholtz modes. Any subsequent interpretation of results is thus likely to be quite complicated.

Shearing and embedding box simulations of the magnetorotational instability
View Description Hide DescriptionTwo different computational approaches to the magnetorotational instability (MRI) are pursued: the shearing box approach which is suited for local simulations and the embedding box approach whereby a Taylor Couette flow is embedded in a box so that numerical problems with the coordinate singularity are avoided. New shearing box simulations are presented and differences between regular and hyperviscosity are discussed. Preliminary simulations of spherical nonlinear Taylor Couette flow in an embedding box are presented and the effects of an axial field on the background flow are studied.

Magnetic field generation in Kolmogorov turbulence
View Description Hide DescriptionWe analyze the initial, kinematic stage of magnetic field evolution in an isotropic and homogeneous turbulent conducting fluid with a “rough” velocity field, v(l) ∼ l ^{α}, α < 1. This regime is relevant to the problem of magnetic field generation in turbulent fluids with small magnetic Prandtl number, i.e. with ohmic resistivity much larger than viscosity. Our interest in motivated by the following question: suppose that smooth fluctuations of velocity field are able to amplify a weak magnetic field, would the magnetic field be still amplified if the fluid motion becomes strongly turbulent, i.e. non‐smooth? Quite paradoxically, turbulence can be dangerous for magnetic field generation. We propose that the smaller the magnetic Prandtl number, the larger the magnetic Reynolds number that is needed to excite magnetic fluctuations. This implies that numerical or experimental investigations of magnetohydrodynamical turbulence with small Prandtl numbers need to achieve extremely high resolution in order to describe magnetic phenomena adequately.

Taylor‐Couette flow with an imposed magnetic field — linear and nonlinear results
View Description Hide DescriptionUsing numerical simulations we investigate the (in)‐stability and saturation behaviour of moderately compressible, cylindrical Taylor‐Couette flow in the presence of a uniform axial magnetic field. For Rayleigh‐stable configurations, we find magnetically induced Taylor vortices as predicted by linear theory, with both axisymmetric and non‐axisymmetric solutions, depending on the Hartmann number.
The flow shows clear indications of the magneto‐rotational instability which is well‐known from numerical simulations in accretion disc geometry. In the saturated state, the structure of the flow and the magnetic field can be very different from the linear phase of the instability.

Differential rotation in precession driven flow
View Description Hide DescriptionMagnetorotational instability (MRI) occurs for suitable dependencies of the angular velocity of a fluid with distance from a rotation axis. In the case of precession driven flow, this dependence is not well known. Precession driven flow does become unstable in experiments and numerical simulations. There is one well understood mechanism based on triad resonances by which these flows can become unstable. However, in the geophysical application and other contexts not directly accessible to simulation, other mechanisms such as MRI may become predominant.

Taylor‐Couette flow stability: effect of vertical density stratification and azimuthal magnetic fields
View Description Hide DescriptionAlso vertical density stratification and/or azimuthal magnetic fields essentially change the stability properties of the Taylor‐Couette flow. Our results confirm the conclusion of Molemaker, McWilliams & Yavneh that vertically stratified Taylor‐Couette flows are unstable against nonaxisymmetric disturbances if the angular velocity decreases with radius but not the angular momentum. Moreover, azimuthal magnetic fields can destabilize MHD Taylor‐Couette flows with arbitrary rotation law for appropriate magnetic field geometry and amplitude. Such experiments in the laboratory might be much‐promising cases due to the independence of the stability properties on the (very small) magnetic Prandtl number.

Vortices and waves in planar and disk flows
View Description Hide DescriptionWe present and analyse a linear, non‐resonant phenomenon of wave generation by vortices in smooth shear flows. The phenomenon is closely related to non‐normality of linear dynamics of perturbations in such flows and can be well interpreted by the use of the non‐modal approach, i.e. by following in time the linear dynamics of spatial Fourier harmonics of the vortex mode perturbations. We will consider both a planar two‐dimensional flow with linear velocity profile and a keplerian disk flow. We will also present numerical simulations in both configurations. These results may be important for the dynamics of accretion disks.

Magnetic shear‐flows in stars
View Description Hide DescriptionMany main‐sequence stars exhibit extensive radiative zones. Some of these may rotate differentially and have large‐scale meridional circulations, while the solar radiative core rotates rigidly. We are concerned with three topics: The generation of magnetic fields by a dynamo effect of these large‐scale motions, the stability of differential rotation if magnetic fields are present initially, and the formation of the solar tachocline being a thin transition layer from rigid solar‐core rotation to differential rotation of the outer convective shell. We conclude that dynamo‐generation of magnetic fields is unlikely in stellar radiative envelopes. This finding supports the view that the fields of magnetic Ap stars are fossil. If they do exist from the beginning of the stellar life, they will make a differential rotation unstable if the angular velocity decreases with axis distance. This is the magneto‐rotational instability. It is found that the time‐scale of turning a differential rotation into a rigid one is about 10–100 million years. In the solar radiative core, the angular velocity gradient is positive and the magneto‐rotational instability is not found. Nevertheless, magnetic fields will suppress differential rotation by the Lorentz force and reduce the transition between the differentially rotating outer convection zone and the core to a very thin layer. The field strength of the poloidal core magnetic field has to be of the order of 10 G in order to produce the solar tachocline thickness.

Instability of periodic MHD shear flows
View Description Hide DescriptionThe stability of periodic MHD shear flows generated by an external transversal periodic force in magnetized plasma is studied. It is shown that the temporal behaviour of magnetosonic wave spatial Fourier harmonics in such flows is governed by Mathieu equation. Consequently the harmonics with the half frequency of the shear flows grow exponentially in time. Therefore the periodic shear motions are unstable to the perturbations of compressible magnetosonic waves. The motions represent the kinetic part of the transversal oscillation in magnetized plasma. Therefore due to the instability of periodic shear motions, the transversal oscillations may quickly be damped, so transferring their energy to compressible magnetosonic perturbations.