G. Ruediger, R. Hollerbach
Liu et al. [Phys. Rev. E 74, 056302 (2006)] have presented a WKB analysis of the helical magnetorotational instability (HMRI), and claim that it does not exist for Keplerian rotation profiles. We show that if radial boundary conditions are included, the HMRI can exist even for rotation profiles as flat as Keplerian, provided only that at least one of the boundaries is sufficiently conducting.
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We investigate magnetic Taylor--Couette flow in the presence of an imposed axial magnetic field. First we calculate nonlinear steady axisymmetric solutions and determine how their strength depends on the applied magnetic field. Then we perturb these solutions to find the critical Reynolds numbers for the appearance of wavy modes, and the related wavespeeds, at increasing magnetic field strength. We find that values of imposed magnetic field which alter only slightly the transition from circular--Couette flow to Taylor--vortex flow, can shift the transition from Taylor--vortex flow to wavy mode…
We study the stability of cylindrical Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields, and show that adding an azimuthal field profoundly alters the previous results for purely axial fields. For small magnetic Prandtl numbers Pm, the critical Reynolds number Re_c for the onset of the magneto-rotational instability becomes independent of Pm, whereas for purely axial fields it scales as Pm^{-1}. For typical liquid metals, Re_c is then reduced by several orders of magnitude, enough that this new design should succeed in realizing this instability in the laborat…
Recent experiments have shown that it is possible to study a fundamental astrophysical process such as dynamo action in controlled laboratory conditions using simple MHD flows. In this paper we explore the possibility that Taylor-Couette flow, already proposed as a model of the magneto-rotational instability of accretion discs, can sustain generation of magnetic field. Firstly, by solving the kinematic dynamo problem, we identify the region of parameter space where the magnetic field's growth rate is higher. Secondly, by solving simultaneously the coupled nonlinear equations which govern veloc…
Taylor--Couette flow in the presence of a magnetic field is a problem belonging to classical hydromagnetics and deserves to be more widely studied than it has been to date. In the nonlinear regime the literature is scarce. We develop a formulation suitable for solution of the full three dimensional nonlinear hydromagnetic equations in cylindrical geometry, which is motived by the formulation for the magnetic field. It is suitable for study at finite magnetic Prandtl numbers and in the small Prandtl number limit, relevant to laboratory liquid metals. The method is used to determine the onset of…
Azimuthal magnetorotational instability is a mechanism that generates nonaxisymmetric field pattern. Nonlinear simulations in an infinite Taylor-Couette system with current-free external field show, that not only the linearly unstable mode m=1 appears, but also an inverse cascade transporting energy into the axisymmetric field is possible. By varying the Reynolds number of the flow and the Hartmann number for the magnetic field, we find that the ratio between axisymmetric (m=0) and dominating nonaxisymmetric mode (m=1) can be nearly free chosen. On the surface of the outer cylinder this mode d…
The linear marginal instability of an MHD Taylor-Couette flow of infinite vertical extension is considered. For hydrodynamically unstable flows the minimum Reynolds number exists even without a magnetic field, but there are also solutions with smaller Reynolds numbers for certain (weak) magnetic fields. The magnetic field can, therefore, destabilize the rotating flow. This instability, however, can only exist for not too small magnetic Prandtl numbers. Also the situation for nonaxisymmetric disturbances is considered. Without magnetic fields such modes possess higher Reynolds numbers than the …
We describe an instability of viscoelastic Couette-Taylor flow that is directly analogous to the magnetorotational instability (MRI) in astrophysical magnetohydrodynamics, with polymer molecules playing the role of magnetic field lines. By determining the conditions required for the onset of instability and the properties of the preferred modes, we distinguish it from the centrifugal and elastic instabilities studied previously. Experimental demonstration and investigation should be much easier for the viscoelastic instability than for the MRI in a liquid metal. The analogy holds with the case…
The linear marginal instability of an axisymmetric MHD Taylor-Couette flow of infinite vertical extension is considered. The dependence of the flow stability on magnetic Prandtl number, Pm, and gap-width between rotating cylinders is investigated. There is an unstability for flows with resting outer cylinder even without a magnetic field. Nevertheless, there are solutions with smaller critical Reynolds numbers for certain (weak) magnetic fields. These solutions, however, exist for not too small Pm or large enough gap. For hydrodynamically stable flow, we found that a magnetic field amplitude a…
The linear stability theory of Taylor-Couette flows (unbounded in_z_) is described including magnetic fields, Hall effect or a density stratification in order to prepare laboratory experiments to probe the stability of differential rotation in astrophysics. If an axial field is present then the magnetorotational instability (MRI) is investigated also for small magnetic Prandtl numbers. For rotating outer cylinder beyond the Rayleigh line characteristic minima are found for magnetic Reynolds number of the order of 10 and for Lundquist numbers of order 1. The inclusion of extra toroidal current-…
We study experimentally the flow of a liquid metal confined between differentially rotating cylinders, in the presence of externally imposed axial and azimuthal magnetic fields. For increasingly large azimuthal fields a wave-like disturbance arises, traveling along the axis of the cylinders. The wavelengths and speeds of these structures, as well as the field strengths and rotation rates at which they arise, are broadly consistent with theoretical predictions of such a traveling wave magnetorotational instability.
The magnetorotational instability (MRI) in cylindrical Taylor-Couette flow with external helical magnetic field is simulated for infinite and finite aspect ratios. We solve the MHD equations in their small Prandtl number limit and confirm with time-dependent nonlinear simulations that the additional toroidal component of the magnetic field reduces the critical Reynolds number from $O(10^6)$ (axial field only) to $O(10^3)$ for liquid metals with their small magnetic Prandtl number. Computing the saturated state we obtain velocity amplitudes which help designing proper experimental setups. Exper…
We study the magnetorotational instability in cylindrical Taylor-Couette flow, with the (vertically unbounded) cylinders taken to be perfect conductors, and with externally imposed spiral magnetic fields. The azimuthal component of this field is generated by an axial current inside the inner cylinder, and may be slightly stronger than the axial field. We obtain an instability beyond the Rayleigh line, for Reynolds numbers of order 1000 and Hartmann numbers of order 10, and independent of the (small) magnetic Prandtl number. For experiments with R_out = 2 R_in = 10 cm and Omega_out = 0.27 Omega…
The turbulent transport of angular momentum plays an important role in many astrophysical objects, but its modelization is still far from satisfactory. We discuss here what can be learned from laboratory experiments. We analyze the results obtained by Wendt (1933) and Taylor (1936) on the classical Couette-Taylor flow, in the case where angular momentum increases with distance from the rotation axis, which is the most interesting for astrophysical applications. We show that when the gap between the coaxial cylinders is wide enough, the criterion for the onset of the finite amplitude instabilit…
We study the saturation near threshold of the axisymmetric magnetorotational instability (MRI) of a viscous, resistive, incompressible fluid in a thin-gap Taylor-Couette configuration. A vertical magnetic field, Keplerian shear and no-slip, conducting radial boundary conditions are adopted. The weakly non-linear theory leads to a real Ginzburg-Landau equation for the disturbance amplitude, like in our previous idealized analysis. For small magnetic Prandtl number (P \ll 1), the saturation amplitude scales as P^{2/3} while the magnitude of angular momentum transport scales as P^{4/3}. The diffe…
In the paper we comment on (Rüdiger & Shalybkov, Phys. Rev. E. 69, 016303 (2004) (RS)), the instability of the Taylor--Couette flow interacting with a homogeneous background field subject to Hall effect is studied. We correct a falsely generalizing interpretation of results presented there which could be taken to disprove the existence of the Hall--drift induced magnetic instability described in Rheinhardt and Geppert, Phys. Rev. Lett. 88, 101103. It is shown that in contrast to what is suggested by RS, no additional shear flow is necessary to enable such an instability with a non--potential m…
The linear stability of MHD Taylor-Couette flow of infinite vertical extension is considered for various magnetic Prandtl numbers Pm. The calculations are performed for a wide gap container with \hatη=0.5 with an axial uniform magnetic field excluding counterrotating cylinders. For both hydrodynamically stable and unstable flows the magnetorotational instability produces characteristic minima of the Reynolds number for certain (low) magnetic field amplitudes and Pm> 0.01. For Pm <~ 1 there is a characteristic magnetic field amplitude beyond which the instability sets in in form of nonaxisymmet…
Aims: To investigate the stability and angular momentum transport by the strato-rotational instability in the nonlinear regime. Methods: The hydrodynamic compressible equations are solved in a cartesian box in which the outer cylinder is embedded. Gravity along the rotation axis leads to density stratification. No-slip boundary conditions are used in the radial direction, while free-slip conditions are used on the two ends of the cylinders. Results: The strato-rotational instability is confirmed and the Reynolds stress is shown to transport angular momentum away from the axis. However, the gro…
The stability problem of MHD Taylor-Couette flows with toroidal magnetic fields is considered in dependence on the magnetic Prandtl number. Only the most uniform (but not current-free) field with B\_in = B\_out has been considered. For high enough Hartmann numbers the toroidal field is always unstable. Rigid rotation, however, stabilizes the magnetic (kink-)instability. The axial current which drives the instability is reduced by the electromotive force induced by the instability itself. Numerical simulations are presented to probe this effect as a possibility to measure the turbulent conducti…
Toroidal magnetic fields subject to the Tayler instability can transport angular momentum. We show that the Maxwell and Reynolds stress of the nonaxisymmetric field pattern depend linearly on the shear in the cylindrical gap geometry. Resulting angular momentum transport also scales linear with shear. It is directed outwards for astrophysical relevant flows and directed inwards for superrotating flows with dOmega/dR>0. We define an eddy viscosity based on the linear relation between shear and angular momentum transport and show that its maximum for given Prandtl and Hartmann number depends lin…
We investigate numerically a traveling wave pattern observed in experimental magnetized Taylor-Couette flow at low magnetic Reynolds number. By accurately modeling viscous and magnetic boundaries in all directions, we reproduce the experimentally measured wave patterns and their amplitudes. Contrary to previous claims, the waves are shown to be transiently amplified disturbances launched by viscous boundary layers rather than globally unstable magnetorotational modes.
The stability of density-stratified viscous Taylor-Couette flows is considered using the Boussinesq approximation but without any use of the short-wave approximation. The flows which are unstable after the Rayleigh criterion (\hat μ<\hat η^2, with \hat μ=Ω_{out}/Ω_{in} and \hat η= R_{in}/R_{out}) now develop overstable axisymmetric Taylor vortices. For the considered wide-gap container we find the nonaxisymmetric modes as the most unstable ones. The nonaxisymmetric modes are unstable also beyond the Rayleigh line. For such modes the instability condition seems simply to be \hatμ<1 as stressed …
The linear marginal instability of an axisymmetric MHD Taylor-Couette flow of infinite vertical extension is considered. For flows with a resting outer cylinder there is a well-known characteristic Reynolds number even without magnetic field but for sufficiently weak magnetic fields there are solutions with smaller Reynolds numbers so that a characteristic minimum exists. The minimum only exists, however, for not too small magnetic Prandtl numbers. For small magnetic Prandtl numbers one only finds the typical magnetic suppression of the instability. We are here particularly interested in the c…
We report experimental observation of an instability in a Couette-Taylor flow of a polymer fluid in a thin gap between two coaxially rotating cylinders in a regime where their angular velocity decreases with the radius while the specific angular momentum increases with the radius. In the considered regime, neither the inertial Rayleigh instability nor the purely elastic instability are possible. We propose that the observed "elasto-rotational" instability is an analog of the magnetorotational instability which plays a fundamental role in astrophysical Keplerian accretion disks.
We consider the linear stability of dissipative MHD Taylor-Couette flow with imposed toroidal magnetic fields. The inner and outer cylinders can be either insulating or conducting; the inner one rotates, the outer one is stationary. The magnetic Prandtl number can be as small as 10-5, approaching realistic liquid-metal values. The magnetic field destabilizes the flow, except for radial profiles of B$_ϕ$(R) close to the current-free solution. The profile with B$_{in}$=B$_{out}$ (the most uniform field) is considered in detail. For weak fields the TC-flow is stabilized, until for moderately stro…
The stability of dissipative Taylor-Couette flows with an axial stable density stratification and a prescribed azimuthal magnetic field is considered. Global nonaxisymmetric solutions of the linearized MHD equations with toroidal magnetic field, axial density stratification and differential rotation are found for both insulating and conducting cylinder walls. Flat rotation laws such as the quasi-Kepler law are unstable against the nonaxisymmetric stratorotational instability (SRI). The influence of a current-free toroidal magnetic field depends on the magnetic Prandtl number Pm: SRI is support…
The magnetorotational instability (MRI) of differential rotation under the simultaneous presence of axial and azimuthal components of the (current-free) magnetic field is considered. For rotation with uniform specific angular momentum the MHD equations for axisymmetric perturbations are solved in a local short-wave approximation. All the solutions are overstable for B_z \cdot B_ϕ\neq 0 with eigenfrequencies approaching the viscous frequency. For more flat rotation laws the results of the local approximation do not comply with the results of a global calculation of the MHD instability of Taylor…
The linear stability of MHD Taylor-Couette flow of infinite vertical extension is considered for liquid sodium with its small magnetic Prandtl number Pm of order of 10^-5 and an uniform axial magnetic field. The sign of the constant a in the basic rotation law Ω=a+b/R^2 strongly influences the presented results. It is negative for resting outer cylinder. The main point here is that the subcritical excitation which occurs for large Pm disappears for small Pm. This is the reason that the existence of the magnetorotational instability remained unknown over decades. For rotating outer cylinder the…
We study magnetic effects induced by rigidly rotating plates enclosing a cylindrical MHD Taylor-Couette flow at the finite aspect ratio $H/D=10$. The fluid confined between the cylinders is assumed to be liquid metal characterized by small magnetic Prandtl number, the cylinders are perfectly conducting, an axial magnetic field is imposed $\Ha \approx 10$, the rotation rates correspond to $\Rey$ of order $10^2-10^3$. We show that the end-plates introduce, besides the well known Ekman circulation, similar magnetic effects which arise for infinite, rotating plates, horizontally unbounded by any w…
Hollerbach and Rudiger have reported a new type of magnetorotational instability (MRI) in magnetized Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields. The salient advantage of this "helical'' MRI (HMRI) is that marginal instability occurs at arbitrarily low magnetic Reynolds and Lundquist numbers, suggesting that HMRI might be easier to realize than standard MRI (axial field only). We confirm their results, calculate HMRI growth rates, and show that in the resistive limit, HMRI is a weakly destabilized inertial oscillation propagating in a unique direction al…
The helical magnetorotational instability of the magnetized Taylor-Couette flow is studied numerically in a finite cylinder. A distant upstream insulating boundary is shown to stabilize the convective instability entirely while reducing the growth rate of the absolute instability. The reduction is less severe with larger height. After modeling the boundary conditions properly, the wave patterns observed in the experiment turn out to be a noise-sustained convective instability. After the source of the noise resulted from unstable Ekman and Stewartson layers is switched off, a slowly-decaying in…
We consider the magnetorotational instability (MRI) of a hydrodynamically stable Taylor-Couette flow with a helical external magnetic field in the inductionless approximation defined by a zero magnetic Prandtl number ($\Pm=0)$. This leads to a considerable simplification of the problem eventually containing only hydrodynamic variables. First, we point out that the energy of any perturbation growing in the presence of magnetic field has to grow faster without the field. This is a paradox because the base flow is stable without the magnetic while it is unstable in the presence of a helical magne…
We investigate the instability and nonlinear saturation of temperature-stratified Taylor-Couette flows in a finite height cylindrical gap and calculate angular-momentum transport in the nonlinear regime. The model is based on an incompressible fluid in Boussinesq approximation with a positive axial temperature gradient applied. While both ingredients itself, the differential rotation as well as the stratification due to the temperature gradient, are stable, together the system becomes subject of the stratorotational instability and nonaxisymmetric flow pattern evolve. This flow configuration t…
In this paper we present axisymmetric nonlinear simulations of magnetized Ekman and Stewartson layers in a magnetized Taylor-Couette flow with a centrifugally stable angular-momemtum profile and with a magnetic Reynolds number below the threshold of magnetorotational instability. The magnetic field is found to inhibit the Ekman suction. The width of the Ekman layer is reduced with increased magnetic field normal to the end plate. A uniformly-rotating region forms near the outer cylinder. A strong magnetic field leads to a steady Stewartson layer emanating from the junction between differential…
We analyze numerically the magnetorotational instability of a Taylor-Couette flow in a helical magnetic field (HMRI) using the inductionless approximation defined by a zero magnetic Prandtl number (Pm=0). The Chebyshev collocation method is used to calculate the eigenvalue spectrum for small amplitude perturbations. First, we carry out a detailed conventional linear stability analysis with respect to perturbations in the form of Fourier modes that corresponds to the convective instability which is not in general self-sustained. The helical magnetic field is found to extend the instability to a…
A recent paper [R. Hollerbach and G. Rudiger, Phys. Rev. Lett. 95, 124501 (2005)] has shown that the threshold for the onset of the magnetorotational instability (MRI) in a Taylor-Couette flow is dramatically reduced if both axial and azimuthal magnetic fields are imposed. In agreement with this prediction, we present results of a Taylor-Couette experiment with the liquid metal alloy GaInSn, showing evidence for the existence of the MRI at Reynolds numbers of order 1000 and Hartmann numbers of order 10.
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