Magnetized Taylor-Couette Flow Research Articles

Nonlinear evolution of magnetorotational instability in a magnetized Taylor-Couette flow: Scaling properties and relation to upcoming DRESDYN-MRI experiment

Nonlinear evolution of magnetorotational instability in a magnetized Taylor-Couette flow: Scaling properties and relation to upcoming DRESDYN-MRI experiment

Transition to chaos and modal structure of magnetized Taylor-Couette flow.

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)'.

STUDY OF THE MAGNETIC TAYLOR-COUETTE FLOW WITH THE AXIAL FLOW IN ROTATING MACHINERY UNDER QUADRUPOLE MAGNETIC FIELD

In this study, the effect of applying a magnetic field resulting from quadrupole magnets on the hydrodynamics and heat transfer in magnetic Taylor-Couette flow with axial flow was investigated numerically for the first time. The studied geometry consisted of two vertical concentric cylinders. The inner cylinder rotated at a specific angular velocity to create the Taylor-Couette flow, while the outer cylinder was stationary. The outer cylinder was also considered to be under constant heat flux boundary conditions. The magnetic nanofluid was a water-based nanofluid consisting of iron oxide nanoparticles with diameters of 8 nm and the magnetic field was produced by quadrupole magnets. The modeling of the problem was implemented by using the two-phase mixture model and control volume technique. The obtained results indicated that in the absence of a magnetic field and using our considered range of Taylor numbers, the flow consisted of many vortices, which is called the Taylor vortex flow. By applying the magnetic field, the ferrofluid flow was pushed toward the outer cylinder wall, which also considerably increased the stability of the flow. Moreover, by imposing the magnetic field, the local heat transfer coefficient increased and the heat transfer coefficient decreased as the Taylor number of the flow increased.

From helical to standard magnetorotational instability: Predictions for upcoming liquid sodium experiments

This work is a first preparatory theoretical study for upcoming magnetorotational instability (MRI) experiments with liquid sodium planned within the DRESDYN project. Therefore, we focus on the characteristics of MRI for typical values of the main parameters of the basic magnetized Taylor-Couette (TC) flow achievable in these experiments. In contrast to previous attempts at finding MRI in the laboratory, our results clearly indicate that standard MRI and its helically modified version can be detected for the first time in the DRESDYN-TC device, including the astrophysically important Keplerian rotation, despite the very small magnetic Prandtl number of liquid sodium.

Analysis of azimuthal magnetorotational instability of rotating magnetohydrodynamic flows and Tayler instability via an extended Hain-Lüst equation.

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-Lüst differential equation for the radial Lagrangian displacement that incorporates the effects of the axial and azimuthal magnetic fields, differential rotation, viscosity, and electrical resistivity. We apply the Wentzel-Kramers-Brillouin method to the extended Hain-Lüst equation and derive a comprehensive dispersion relation for the local stability analysis of the flow to three-dimensional disturbances. We confirm that in the limit of low magnetic Prandtl numbers, in which the ratio of the viscosity to the magnetic diffusivity is vanishing, the rotating flows with radial distributions of the angular velocity beyond the Liu limit, become unstable subject to a wide variety of the azimuthal magnetic fields, and so is the Keplerian flow. In the analysis of the dispersion relation we find evidence of a new long-wavelength instability which is caught also by the numerical solution of the boundary value problem for a magnetized Taylor-Couette flow.

Suppression of magnetorotational instability in viscous resistive magnetized Taylor–Couette flow

Magnetorotational instability (MRI) is an instability that is responsible for accretion, the phenomenon observed in astrophysical disks, for example around black holes. MRI can be modeled using the equations of MHD. A typical linear analysis in the past made use of the small Prandtl number approximation, which results in dropping one term from these equations. Previously, one of the authors (Herron) showed that the small magnetic Prandtl number approximation suppresses MRI in axisymmetric viscous resistive magnetized Taylor–Couette flow when one has no-slip velocity boundary conditions, and insulating magnetic boundary conditions. We follow up here with a proof that MRI is still suppressed with perfectly conducting magnetic boundary conditions on the cylinders.

Quasi-two-dimensional nonlinear evolution of helical magnetorotational instability in a magnetized Taylor–Couette flow

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 simulations, in this paper we investigate the nonlinear development and saturation properties of the helical magnetorotational instability (HMRI)—a relative of the standard MRI—in a magnetized Taylor–Couette flow at very low magnetic Prandtl number (correspondingly at low magnetic Reynolds number) relevant to liquid metals. For simplicity, the ratio of azimuthal field to axial field is kept fixed. From the linear theory of HMRI, it is known that the Elsasser number, or interaction parameter determines its growth rate and plays a special role in the dynamics. We show that this parameter is also important in the nonlinear problem. By increasing its value, a sudden transition from weakly nonlinear, where the system is slightly above the linear stability threshold, to strongly nonlinear, or turbulent regime occurs. We calculate the azimuthal and axial energy spectra corresponding to these two regimes and show that they differ qualitatively. Remarkably, the nonlinear state remains in all cases nearly axisymmetric suggesting that this HMRI-driven turbulence is quasi two-dimensional in nature. Although the contribution of non-axisymmetric modes increases moderately with the Elsasser number, their total energy remains much smaller than that of the axisymmetric ones.

Absolute versus convective helical magnetorotational instabilities in Taylor–Couette flows

We study magnetic Taylor–Couette flow in a system having nondimensional radii ri = 1 and ro = 2, and periodic in the axial direction with wavelengths . The rotation ratio of the inner and outer cylinders is adjusted to be slightly in the Rayleigh-stable regime, where magnetic fields are required to destabilize the flow, in this case triggering the axisymmetric helical magnetorotational instability (HMRI). Two choices of imposed magnetic field are considered, both having the same azimuthal component , but differing axial components. The first choice has Bz = 0.1, and yields the familiar HMRI, consisting of unidirectionally traveling waves. The second choice has , and yields HMRI waves that travel in opposite directions depending on the sign of Bz. The first configuration corresponds to a convective instability, the second to an absolute instability. The two variants behave very similarly regarding both linear onset as well as nonlinear equilibration.

Destabilization of rotating flows with positive shear by azimuthal magnetic fields.

According to Rayleigh's criterion, rotating flows are linearly stable when their specific angular momentum increases radially outward. The celebrated magnetorotational instability opens a way to destabilize those flows, as long as the angular velocity is decreasing outward. Using a local approximation we demonstrate that even flows with very steep positive shear can be destabilized by azimuthal magnetic fields which are current free within the fluid. We illustrate the transition of this instability to a rotationally enhanced kink-type instability in the case of a homogeneous current in the fluid, and discuss the prospects for observing it in a magnetized Taylor-Couette flow.

NOISE-SUSTAINED CONVECTIVE INSTABILITY IN A MAGNETIZED TAYLOR-COUETTE FLOW

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 inertial oscillation is observed in the simulation. We reach the conclusion that the experiments completed to date have not yet reached the regime of absolute instability.

The Magneto-Rotational Instability Near Threshold: Spatio-Temporal Amplitude Equation and Saturation

We show, by means of a perturbative weakly nonlinear analysis, that the axisymmetric magneto-rotational instability (MRI) in a magnetic Taylor-Couette (mTC) flow in a thin-gap gives rise, for very small magnetic Prandtl numbers (P_m), to a real Ginzburg-Landau equation for the disturbance amplitude. The saturation amplitude A_s is found to scale in this regime as P^\delta, with 1/2<\delta<2/3 (depending on the boundary conditions adopted). The asymptotic results are shown to comply with numerical calculations performed by using a spectral code. They suggest that the transport due to the nonlinearly developed MRI may be vanishingly small for P_m << 1.

Magnetized Ekman layer and Stewartson layer in a magnetized Taylor-Couette flow

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-momentum 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 differentially rotating rings at the endcaps. The Stewartson layer becomes thinner with larger Reynolds number and penetrates deeper into the bulk flow with stronger magnetic field and larger Reynolds number. However, at Reynolds number larger than a critical value approximately 600 , axisymmetric, and perhaps also nonaxisymmetric, instabilities occur and result in a less prominent Stewartson layer that extends less far from the boundary.

Comment on “Helical magnetorotational instability in magnetized Taylor-Couette flow”

Liu [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.

Theory of current-driven instability experiments in magnetic Taylor-Couette flows

We consider the linear stability of dissipative magnetic 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(phi)(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 Taylor-Couette flow is stabilized, until for moderately strong fields the m=1 azimuthal mode dramatically destabilizes the flow again so that a maximum value for the critical Reynolds number exists. For sufficiently strong fields (as measured by the Hartmann number) the toroidal field is always unstable, even for the nonrotating case with Re=0 . The electric currents needed to generate the required toroidal fields in laboratory experiments are a few kA if liquid sodium is used, somewhat more if gallium is used. Weaker currents are needed for wider gaps, so a wide-gap apparatus could succeed even with gallium. The critical Reynolds numbers are only somewhat larger than the nonmagnetic values; hence such experiments would work with only modest rotation rates.

Compensation of instabilities in magnetic Taylor-Couette flow

The axisymmetric linear stability of the Taylor-Couette flow with an azimuthal magnetic field is considered. It is shown that a flow with the combination of a linearly unstable rotation and a linearly unstable azimuthal magnetic field can be linearly stable. The flow stabilization takes place for both ideal and dissipative flows. For dissipative flow the stabilization exists only for a combination of counter-rotating cylinders and a counterdirecting azimuthal magnetic field on cylinders. The effect can be important for the problem of a plasma confinement by the magnetic field.

Traveling waves in a magnetized Taylor-Couette flow

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.

Helical magnetorotational instability in magnetized Taylor-Couette flow

Hollerbach and Rüdiger 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), and that it might be relevant to cooler astrophysical disks, especially those around protostars, which may be quite resistive. We confirm previous results for marginal stability and calculate HMRI growth rates. We show that in the resistive limit, HMRI is a weakly destabilized inertial oscillation propagating in a unique direction along the axis. But we report other features of HMRI that make it less attractive for experiments and for resistive astrophysical disks. Large axial currents are required. More fundamentally, instability of highly resistive flow is peculiar to infinitely long or periodic cylinders: finite cylinders with insulating endcaps are shown to be stable in this limit, at least if viscosity is neglected. Also, Keplerian rotation profiles are stable in the resistive limit regardless of axial boundary conditions. Nevertheless, the addition of a toroidal field lowers thresholds for instability even in finite cylinders.

Nonlinear simulations of magnetic Taylor-Couette flow with currentfree helical magnetic fields

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 timedependent 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. Experiments with liquid gallium require axial field ∼ 50 Gauss and axial current ∼ 4 kA for the toroidal field. It is sufficient that the vertical velocity uz of the flow can be measured with a precision of 0.1mm/s. We also show that the endplates enclosing the cylinders do not destroy the traveling wave instability which can be observed as presented in earlier studies. For TC containers without and with endplates the angular momentum transport of the MRI instability is shown as to be outwards.

Comment on “Linear instability of magnetic Taylor-Couette flow with Hall effect”

In the paper we comment on [Rudiger and Shalybkov (RS), Phys. Rev. E 69, 016303 (2004)], the instability of the Taylor-Couette flow interacting with a homogeneous background field subject to the 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 (2002). 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 nonpotential magnetic background field, whereas for a curl-free one it is. In the latter case, the instabilities found in RS in situations where neither a hydrodynamic nor a magnetorotational instability exists are demonstrated to be most likely magnetic instead of magnetohydrodynamic. Further, some minor inaccuracies are clarified.

Linear instability of magnetic Taylor-Couette flow with Hall effect.

The influence of the Hall effect on the linear marginal stability of a molecular hydrodynamic Taylor-Couette flow in the presence of an axial uniform magnetic field is considered. The Hall effect leads to the situation that the Taylor-Couette flow becomes unstable for any ratio of the angular velocities of the inner and outer cylinders. The instability, however, does not exist for both signs of the axial magnetic field B0. For positive shear dOmega/dR the Hall instability exists for negative Hartmann number and for negative shear dOmega/dR the Hall instability exists for positive Hartmann number. For negative shear, of course, the Hall instability combines with the magnetorotational instability, resulting in a rather complex bifurcation diagram. In this case the critical magnetic Reynolds numbers with Hall effect are much lower than without Hall effect. In order to verify the presented shear-Hall instability at the laboratory with experiments using liquid metals, one would need rather large magnetic fields ( approximately 10(7) G).