Azimuthal Magnetorotational Instability at low and high magnetic Prandtl numbers
01 Mar 2017-Magnetohydrodynamics (Institute of Physics, University of Latvia)-Vol. 53, Iss: 1, pp 25-34
TL;DR: In this article, the influence of the instability on the outward angular momentum transport, necessary for the accretion of the disk, has been studied for a Keplerian rotation with Taylor-Couette flow and imposed azimuthal magnetic field using both linear and nonlinear approaches.
Abstract: Magnetorotational instability is considered to be one of the most powerful sources of turbulence in hydrodynamically stable quasi-Keplerian flows, such as those governing the accretion disk flows. Although the linear stability of these flows with an applied external magnetic field has been studied for decades, the influence of the instability on the outward angular momentum transport, necessary for the accretion of the disk, is still not well known. In this work, we model a Keplerian rotation with Taylor-Couette flow and imposed azimuthal magnetic field using both linear and nonlinear approaches. We present scalings of instability with Hartmann and Reynolds numbers via linear analysis and direct numerical simulations for two magnetic Prandtl numbers of 1.4 ·10−6 and 1. Inside the instability domains, modes with different axial wavenumbers dominate, resulting in sub-domains of instabilities which appear different for each Pm. Direct numerical simulations show the emergence of 1- and 2-frequency spatio-temporally oscillating structures for Pm = 1 close the onset of instability, as well as the significant enhancement of the angular momentum transport for Pm = 1, if compared to Pm = 1.4 ·10−6. Figs 7, Refs 11.
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TL;DR: In this article, it was shown that the potential flow subject to axial fields becomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number R m ¯ = R e ⋅ R m (with R e the Reynolds number of rotation, R m its magnetic Reynolds number).
40 citations
Cites background from "Azimuthal Magnetorotational Instabi..."
...g. 37. Note that we always only looked for the maximal values belonging to a given Re. We can thus assume that at least for Re < ˘10 3 the eective viscosity does not exceed the given value. Refs. [90, 91] have suggested though that the e >ective viscosities can become considerably enhanced once Rm ˘102, when the turbulence is eectively triggered twice over, once by having Re su ciently large, and...
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TL;DR: The existence of a finite-amplitude dynamo is demonstrated, whereby a suitable initial condition yields mutually sustaining turbulence and magnetic fields, even though neither could exist without the other.
Abstract: We numerically compute the flow of an electrically conducting fluid in a Taylor-Couette geometry where the rotation rates of the inner and outer cylinders satisfy Ω_o /Ω_i = (r_o /r_ i )^{ −3/2} . In this quasi-Keplerian regime a non-magnetic system would be Rayleigh-stable for all Reynolds numbers Re, and the resulting purely azimuthal flow incapable of kinematic dynamo action for all magnetic Reynolds numbers Rm. For Re = 10^4 and Rm = 10^5 we demonstrate the existence of a finite-amplitude dynamo, whereby a suitable initial condition yields mutually sustaining turbulence and magnetic fields, even though neither could exist without the other. This dynamo solution results in significantly increased outward angular momentum transport, with the bulk of the transport being by Maxwell rather than Reynolds stresses.
22 citations
TL;DR: The magnetorotational instability (MRI) is thought to be a powerful source of turbulence in Keplerian accumulation disks as discussed by the authors, and the authors of this paper investigate the effect of the magnetic field on the angular velocity profiles of magnetic resonance imaging (MRI).
Abstract: The magnetorotational instability (MRI) is thought to be a powerful source of turbulence in Keplerian
accretion disks. Motivated by recent laboratory experiments, we study the MRI driven by an azimuthal
magnetic field in an electrically conducting fluid sheared between two concentric rotating cylinders.
By adjusting the rotation rates of the cylinders, we approximate angular velocity profiles ω ∝ r
q
. We
perform direct numerical simulations of a steep profile close to the Rayleigh line q & −2 and a quasiKeplerian
profile q ≈ −3/2 and cover wide ranges of Reynolds (Re ≤ 4 · 104
) and magnetic Prandtl
numbers (0 ≤ Pm ≤ 1). In the quasi-Keplerian case, the onset of instability depends on the magnetic
Reynolds number, with Rmc ≈ 50, and angular momentum transport scales as √
PmRe2
in the turbulent
regime. The ratio of Maxwell to Reynolds stresses is set by Rm. At the onset of instability both
stresses have similar magnitude, whereas the Reynolds stress vanishes or becomes even negative as
Rm increases. For the profile close to the Rayleigh line, the instability shares these properties as long
as Pm & 0.1, but exhibits a markedly different character if Pm → 0, where the onset of instability is
governed by the Reynolds number, with Rec ≈ 1250, transport is via Reynolds stresses and scales as
Re2
. At intermediate Pm = 0.01 we observe a continuous transition from one regime to the other, with
a crossover at Rm = O(100). Our results give a comprehensive picture of angular momentum transport
of the MRI with an imposed azimuthal field.
10 citations
TL;DR: The magnetorotational instability (MRI) is thought to be a powerful source of turbulence in Keplerian accretion disks as mentioned in this paper, and the authors of this paper have studied the MRI driven by an azimuthal magnetic field in a fluid sheared between two concentric rotating cylinders.
Abstract: The magnetorotational instability (MRI) is thought to be a powerful source of turbulence in Keplerian accretion disks. Motivated by recent laboratory experiments, we study the MRI driven by an azimuthal magnetic field in an electrically conducting fluid sheared between two concentric rotating cylinders. By adjusting the rotation rates of the cylinders, we approximate angular velocity profiles $\omega \propto r^{q}$. We perform direct numerical simulations of a steep profile close to the Rayleigh line $q \gtrsim -2 $ and a quasi-Keplerian profile $q \approx -3/2$ and cover wide ranges of Reynolds ($Re\le 4\cdot10^4$) and magnetic Prandtl numbers ($0\le Pm \le 1$). In the quasi-Keplerian case, the onset of instability depends on the magnetic Reynolds number, with $Rm_c \approx 50$, and angular momentum transport scales as $\sqrt{Pm} Re^2$ in the turbulent regime. The ratio of Maxwell to Reynolds stresses is set by $Rm$. At the onset of instability both stresses have similar magnitude, whereas the Reynolds stress vanishes or becomes even negative as $Rm$ increases. For the profile close to the Rayleigh line, the instability shares these properties as long as $Pm\gtrsim0.1$, but exhibits a markedly different character if $Pm\rightarrow 0$, where the onset of instability is governed by the Reynolds number, with $Re_c \approx 1250$, transport is via Reynolds stresses and scales as $Re^2$. At intermediate $Pm=0.01$ we observe a continuous transition from one regime to the other, with a crossover at $Rm=\mathcal{O}(100)$. Our results give a comprehensive picture of angular momentum transport of the MRI with an imposed azimuthal field.
9 citations
TL;DR: In this paper, the authors studied the stability of Taylor-Couette flows under the influence of large-scale magnetic fields and showed that the potential flow subject to axial fields becomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number.
Abstract: Decades ago S. Lundquist, S. Chandrasekhar, P.H. Roberts and R. J.~Tayler first posed questions about the stability of Taylor-Couette flows of conducting material under the influence of large-scale magnetic fields. These and many new questions can now be answered numerically where the nonlinear simulations even provide the instability-induced values of several transport coefficients. The cylindrical containers are axially unbounded and penetrated by magnetic background fields with axial and/or azimuthal components. The influence of the magnetic Prandtl number $Pm$ on the onset of the instabilities is shown to be substantial. The potential flow subject to axial fields becomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number $\overline{Rm}=\sqrt{Re\cdot Rm}$ (with $Re$ the Reynolds number of rotation, $Rm$ its magnetic Reynolds number). Rotation profiles as flat as the quasi-Keplerian rotation law scale similarly but only for $Pm\gg 1$ while for $Pm\ll 1$ the instability instead sets in for supercritical $Rm$ at an optimal value of the magnetic field. Among the considered instabilities of azimuthal fields, those of the Chandrasekhar-type, where the background field and the background flow have identical radial profiles, are particularly interesting. They are unstable against nonaxisymmetric perturbations if at least one of the diffusivities is non-zero. For $Pm\ll 1$ the onset of the instability scales with $Re$ while it scales with $\overline{Rm}$ for $Pm\gg 1$. - Even superrotation can be destabilized by azimuthal and current-free magnetic fields; this recently discovered nonaxisymmetric instability is of a double-diffusive character, thus excluding $Pm= 1$. It scales with $Re$ for $Pm\to 0$ and with $Rm$ for $Pm\to \infty$.
7 citations