The stability of an incompressible electrically conducting fluid rotating about an axis when current flows parallel to the axis

We perform a local stability analysis of rotational flows in the presence of a constant vertical magnetic field and an azimuthal magnetic field with a general radial dependence. Employing the short-wavelength approximation we develop a unified framework for the investigation of the standard, the helical, and the azimuthal version of the magnetorotational instability, as well as of current-driven kink-type instabilities. Considering the viscous and resistive setup, our main focus is on the case of small magnetic Prandtl numbers which applies, e.g., to liquid metal experiments but also to the colder parts of accretion disks. We show that the inductionless versions of MRI that were previously thought to be restricted to comparably steep rotation profiles extend well to the Keplerian case if only the azimuthal field slightly deviates from its current-free (in the fluid) profile. We find an explicit criterion separating the pure azimuthal inductionless magnetorotational instability from the regime where this instability is mixed with the Tayler instability. We further demonstrate that for particular parameter configurations the azimuthal MRI originates as a result of a dissipation-induced instability of the Chandrasekhar's equipartition solution of ideal magnetohydrodynamics.

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“…Another particular case ω Az = 0 and m = 0 yields the following extension of the stability condition by Michael (1954):…”

Section: Connection To Known Stability and Instability Criteriamentioning

confidence: 99%

Local instabilities in magnetized rotational flows: a short-wavelength approach

Kirillov

Stefani

Fukumoto³

2014

J. Fluid Mech.

189

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“…Without density stratification the ideal Taylor-Couette flow with imposed azimuthal magnetic field is stable against axisymmetric disturbances, if (Michael 1954). This condition is the combination of the Rayleigh stability condition for differential rotation without magnetic field and the magnetic field stability condition without rotation.…”

Section: Sri With Azimuthal Magnetic Fieldmentioning

confidence: 99%

Stratorotational instability in MHD Taylor-Couette flows

Rüdiger¹,

Shalybkov

2008

A&A

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Aims. The stability of the dissipative Taylor-Couette flow with a stable axial density stratification and a prescribed azimuthal magnetic field is considered. Methods. Global nonaxisymmetric solutions of the linearized MHD equations with toroidal magnetic field, density stratification, and differential rotation are found for both insulating and conducting cylinders. Results. Hydrodynamic calculations for various gap widths show that flat rotation laws such as the Kepler rotation are always unstable against SRI. Quasigalactic rotation laws, however, are stable for wide gaps. The influence of a current-free toroidal magnetic field on SRI strongly depends on the magnetic Prandtl number Pm: SRI is supported by Pm > 1 and it is suppressed by Pm < ∼ 1. For rotation laws that are too flat, when the hydrodynamic SRI ceases, a smooth transition exists to the instability that the toroidal magnetic field produces in combination with the differential rotation. For the first time this nonaxisymmetric azimuthal magnetorotational instability (AMRI) has been computed in the presence of an axial density gradient. If the magnetic field between the cylinders is not current-free, then the Tayler instability occurs, too. The transition from the nonmagnetic centrifugal instability to the magnetic Tayler instability in the presence of differential rotation and density stratification proves to be complex. Most spectacular is the "ballooning" of the stability domain by the density stratification: already a small rotation stabilizes magnetic fields against the Tayler instability. An azimuthal component of the electromotive force u × B for the instability only exists for density-stratified flows. The related α-effect for magnetic-influenced SRI with Kepler rotation appears to be positive for negative dρ/dz < 0.

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“…He showed that this problem has a remarkable analogy with that of the stability of a density stratified fluid at rest under gravity. Michael [2] extended this problem to the case of a perfectly conducting liquid with an electric current distribution parallel to the axis of cylinders and found that Rayleigh's analogy holds in a slightly modified form. Using this analogy, Howard and Gupta [3] investigated the stability of nondissipative swirling flow of an incompressible fluid between two concentric cylinders with respect to axisymmetric disturbances.…”

Section: Introductionmentioning

confidence: 99%

“…(21) of [4] are replaced by a02 + VA2, \p and N"2 respectively, where N2 is the square of the adiabatic Brunt-Vaisala frequency. The expression for \p is Michael's discriminant [2], while Nlr2 may be recognized as the square of the adiabatic Brunt-Vaisala frequency modified by the presence of the circular magnetic field.…”

Section: Introductionmentioning

confidence: 99%

On the stability of swirling flow in magnetogasdynamics

Dandapat¹,

Gupta²

1975

Quart. Appl. Math.

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Introduction.The stability of the steady circular nondissipative flow of an incompressible fluid betwen two concentric cylinders was first studied by Rayleigh [1] who assumed the disturbances to be axisymmetric. He showed that this problem has a remarkable analogy with that of the stability of a density stratified fluid at rest under gravity. Michael [2] extended this problem to the case of a perfectly conducting liquid with an electric current distribution parallel to the axis of cylinders and found that Rayleigh's analogy holds in a slightly modified form. Using this analogy, Howard and Gupta [3] investigated the stability of nondissipative swirling flow of an incompressible fluid between two concentric cylinders with respect to axisymmetric disturbances. They found that stability is ensured if a Richardson number based on the swirl velocity and the shear in the axial flow exceeds ^ everywhere. Recently Howard [4] using a modification of the analysis due to Chimonas [5] on compressible stratified shear flow, derived a Richardson-number theorem for the linear stability to axisymmetric perturbations of compressible nondissipative swirling flow.The present note is an extension of Howard 's [4] problem to the case of a perfectly conducting fluid permeated by an axial distribution of electric current. It is important to note that in a compressible swirling flow, the swirl velocity distribution V(r) not only plays a role similar to that in incompressible flows, but also gives rise, through

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The stability of an incompressible electrically conducting fluid rotating about an axis when current flows parallel to the axis

Cited by 66 publications

References 4 publications

Local instabilities in magnetized rotational flows: a short-wavelength approach

Local instabilities in magnetized rotational flows: a short-wavelength approach

Stratorotational instability in MHD Taylor-Couette flows

On the stability of swirling flow in magnetogasdynamics

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