arXiv subjects

G. Rüdiger

Publications and source records attributed to G. Rüdiger.

At least 19 recordsLinked to original sources

arXiv2023

The gap-size influence on the excitation of magnetorotational instability in cylindrical Couette flows

The excitation conditions of the magnetorotational instability are studied for axially unbounded Taylor-Couette flows of various gap widths between the cylinders. The cylinders are considered as made from both perfect-conducting or insulating material and the conducting fluid with a finite but small magnetic Prandtl number rotates with a quasi-Keplerian velocity profile. The solutions are optimized with respect to the wave number and the Reynolds number of the rotation of the inner cylinder. For the axisymmetric modes we find the critical Lundquist number of the applied axial magnetic field the lower the wider the gap between the cylinders. A similar result is obtained for the induced cell structure: the wider the gap the more spherical the cells are. The marginal rotation rate of the inner cylinder -- for fixed size of the outer cylinder -- always possesses a minimum for not too wide and not too narrow gap widths. For perfect-conducting walls the minimum lies at $r_{\rm in}\simeq 0.4$ while it is at $r_{\rm in}\simeq 0.5$ for insulating walls where $r_{\rm in}$ is the normalized radius of the inner cylinder. The lowest magnetic field amplitudes to excite the instability are required for Taylor-Couette flows between perfect-conducting cylinders with gaps corresponding to $r_{\rm in}\simeq 0.2$. For even wider and also for very thin gaps the needed magnetic fields and rotation frequencies are shown to become rather huge. Also the nonaxisymmetric modes with $|m|=1$ have been considered. Their excitation generally requires stronger magnetic fields and higher magnetic Reynolds numbers in comparison to those for the axisymmetric modes which is true for wide-gap containers with $r_{\rm in} \lesssim 0.3$.

physics.flu-dyn↗

arXiv2022

On the toroidal-velocity anti-dynamo theorem under the presence of nonuniform electric conductivity

Laminar electrically conducting Couette flows with the hydrodynamically stable quasi-Keplerian rotation profile and non-uniform conductivity are probed for dynamo instability. In spherical geometry the equations for the poloidal and the toroidal field components completely decouple, resulting in free decay, regardless of the spatial distribution of the electric conductivity. In cylindrical geometry the poloidal and toroidal components do not decouple, but here also we do not find dynamo excitations for the cases that the electric conductivity only depends on the radius or -- much more complex -- that it only depends on the azimuthal or the axial coordinate. The transformation of the plane-flow dynamo model of Busse \& Wicht (1992) to cylindrical or spherical geometry therefore fails. It is also shown that even the inclusion of axial flows of both directions does {\em not} support the dynamo mechanism. The Elsasser toroidal-velocity antidynamo theorem, according to which dynamos without any radial velocity component cannot work, is thus not softened by non-uniform conductivity distributions.

physics.flu-dyn↗

arXiv2021

Destabilization of super-rotating Taylor-Couette flows by current-free helical magnetic fields

In an earlier paper we showed that the combination of azimuthal magnetic fields and super-rotation in Taylor-Couette flows of conducting fluids can be unstable against non-axisymmetric perturbations if the magnetic Prandtl number of the fluid is $Pm\neq 1$. Here we demonstrate that the addition of a weak axial field component may allow axisymmetric perturbation patterns for $Pm$ of order unity depending on the boundary conditions. The axisymmetric modes only occur for magnetic Mach numbers (of the azimuthal field) of order unity, while higher values are necessary for non-axisymmetric modes. The typical growth time of the instability and the characteristic time scale of the axial migration of the axisymmetric mode are long compared with the rotation period, but short compared with the magnetic diffusion time. The modes travel in the positive or negative $z$-direction along the rotation axis depending on the sign of $B_ϕB_z$. We also demonstrate that the azimuthal components of flow and field perturbations travel in phase if $|B_ϕ|\gg |B_z|$, independent of the form of the rotation law. Within a short-wave approximation for thin gaps it is also shown (in an Appendix) that for {\em ideal} fluids the considered helical magnetorotational instability (HMRI) only exists for rotation laws with negative shear.

physics.flu-dyn↗

arXiv2020

Electrodynamics of turbulent fluids with fluctuating electric conductivity

The influence of fluctuating conductivity on the coefficients known from the mean-field electrodynamics is considered. If the conductivity fluctuations are assumed as uncorrelated with the turbulent velocity field then only the effective magnetic diffusivity of the fluid is reduced and the decay time of a large-scale magnetic field is increased. If the fluctuations of conductivity and flow are correlated in a certain direction then an additional diamagnetic pumping effect results transporting magnetic field in opposite direction to the resistivity flux vector $\langle η'\vec{u}'\rangle$. Even for homogeneous turbulence fields in the presence of rotation an alpha effect appears. With the characteristic values of the outer core of the Earth or the solar convection zone, however, the dynamo number of the alpha effect never reaches supercritical values to operate as an $α^2$-dynamo.

physics.plasm-ph↗

arXiv2020

Large-scale dynamo action of magnetized Taylor-Couette flows

A conducting Taylor-Couette flow with quasi-Keplerian rotation law containing a toroidal magnetic field serves as a mean-field dynamo model of the Tayler-Spruit-type. The flows are unstable against nonaxisymmetric perturbations which form electromotive forces defining $α$ effect and eddy diffusivity. If both degenerated modes with $m=\pm 1$ are excited with the same power then the global $α$ effect vanishes and a dynamo cannot work. It is shown, however, that the Tayler instability produces finite $α$ effects if only an isolated mode is considered but this intrinsic helicity of the single-mode is too low for an $α^2$ dynamo. Moreover, an $α\Om$ dynamo model with quasi-Keplerian rotation requires a minimum magnetic Reynolds number of rotation of ${\rm Rm}\simeq 2.000$ to work. Whether it really works depends on assumptions about the turbulence energy. For a steeper-than-quadratic dependence of the turbulence intensity on the magnetic field, however, dynamos are only excited if the resulting magnetic eddy diffusivity approximates its microscopic value, $η_{\rm T}\simeq η$. By basically lower or larger eddy diffusivities the dynamo instability is suppressed.

astro-ph.SR↗

arXiv2019

Two types of axisymmetric helical magnetorotational instability in rotating flows with positive shear

We reveal and investigate a new type of linear axisymmetric helical magnetorotational instability which is capable of destabilizing viscous and resistive rotational flows with radially increasing angular velocity, or positive shear. This instability is double-diffusive by nature and is different from the more familiar helical magnetorotational instability, operating at positive shear above the Liu limit, in that it works instead for a wide range of the positive shear when ${\rm (i)}$ a combination of axial/poloidal and azimuthal/toroidal magnetic fields is applied and ${\rm (ii)}$ the magnetic Prandtl number is not too close to unity. We study this instability first with radially local WKB analysis and then confirm its existence using a global stability analysis of the magnetized flow between two rotating cylinders with conducting or insulating boundaries. From an experimental point of view, we also demonstrate the presence of the new instability in a magnetized viscous and resistive Taylor-Couette flow with positive shear for such values of the flow parameters, which can be realized in upcoming experiments at the DRESDYN facility. Finally, this instability might have implications for the dynamics of the equatorial parts of the solar tachocline and dynamo action there, since the above two necessary conditions for the instability to take place are satisfied in this region. Our global stability calculations for the tachocline-like configuration, representing a thin rotating cylindrical layer with the appropriate boundary conditions -- conducting inner and insulating outer cylinders -- and the values of the flow parameters, indicate that it can indeed arise in this case with a characteristic growth time comparable to the solar cycle period.

physics.flu-dyn↗

arXiv2019

Antisolar differential rotation of slowly rotating cool stars

Rotating stellar convection transports angular momentum towards the equator, generating the characteristic equatorial acceleration of the solar rotation while the radial flux of angular momentum is always inwards. New numerical box simulations for the meridional cross-correlation $\langle u_θu_ϕ\rangle $, however, reveal the angular momentum transport towards the poles for slow rotation and towards the equator for fast rotation. The explanation is that for slow rotation a negative radial gradient of the angular velocity always appears, which in combination with a so-far neglected rotation-induced off-diagonal eddy viscosity term $ν_\bot$ provides "antisolar rotation" laws with a decelerated equator. Similarly, the simulations provided positive values for the rotation-induced correlation $\langle u_r u_θ\rangle $, which is relevant for the resulting latitudinal temperature profiles (cool or warm poles) for slow rotation and negative values for fast rotation. Observations of the differential rotation of slowly rotating stars will therefore lead to a better understanding of the actual stress-strain relation, the heat transport, and the underlying rotating convection.

astro-ph.SR↗

arXiv2019

Non-diffusive angular momentum transport in rotating $\bf z$-pinches

The stability of conducting Taylor-Couette flows under the presence of toroidal magnetic background fields is considered. For strong enough magnetic amplitudes such magnetohydrodynamic flows are unstable against nonaxisymmetric perturbations which may also transport angular momentum. In accordance with the often used diffusion approximation one expects the angular momentum transport vanishing for rigid rotation. In the sense of a nondiffusive $Λ$ effect, however, even for {\em rigidly} rotating $z$-pinches an axisymmetric angular momentum flux appears which is directed outward (inward) for large (small) magnetic Mach numbers. The internal rotation in a magnetized rotating tank can thus never be uniform. Those particular rotation laws are used to estimate the value of the instability-induced eddy viscosity for which the nondiffusive $Λ$ effect and the diffusive shear-induced transport compensate each other. The results provide the Shakura-Sunyaev viscosity ansatz leading to numerical values linearly growing with the Reynolds number of rotation.

physics.plasm-ph↗

arXiv2018

The turbulent pressure of magnetoconvection for slow and rapid rotation

Motivated by recent simulations of sunspot formation, we extend the theory of the pressure difference between magnetized and non-magnetized gas by Dicke to include rotating turbulence. While the (vertical) background field provides a positive-definite magnetic pressure difference between the magnetized and the unmagnetized gas, Reynolds stress and Maxwell stress of turbulence strongly modify this result. With the quasilinear approximation we demonstrate that the influence of the turbulence differs between the high-conductivity and the low-conductivity limits. Sufficiently small magnetic Reynolds numbers lead to magnetic pressure suppression where indeed the pressure excess can even assume negative values. Box simulations of magnetoconvection subject to a vertical magnetic field carried out with the Nirvana code confirm this overall picture. They also demonstrate how a global rotation {\em reduces} the negative magnetic pressure effect. For rapid rotation the total magnetic pressure difference caused by large-scale magnetic fields { and} turbulence even fully disappears for small field strengths. Magnetic fields of moderate strength thus neither reduce nor enhance the turbulence pressure of rapidly rotating convection. Consequences of this phenomenon for the star formation efficiency are shortly discussed.

astro-ph.SR↗

arXiv2018

Cycle period, differential rotation and meridional flow for early M dwarf stars

Recent observations suggest the existence of two characteristic cycle times for early-type M stars dependent on the rotation period. They are of order one year for the fast rotators ($P_{\rm rot}<1$ day) and of order 4 years for the slower rotators. Additionally, the equator-to-pole differences of the rotation rates with $δΩ$ up to 0.03 rad d$^{-1}$ are known from Kepler data for the fast-rotating stars. These values are well-reproduced by the theory of large-scale flows in rotating convection zones on the basis of the $Λ$ effect. The resulting amplitudes $u^{\rm m}$ of the bottom value of the meridional circulation allows the calculation of the travel time from pole to equator at the base of the convection zone of early-type M stars. These travel times strongly increase with rotation period and they always exceed the observed cycle periods. Therefore, the operation of an advection-dominated dynamo in early M dwarfs, where the travel time must always be shorter than the cycle period, is not confirmed by our model nor the data.

astro-ph.SR↗

arXiv2017

The stratorotational instability of Taylor-Couette flows of moderate Reynolds numbers

In view of new experimental data the instability against adiabatic nonaxisymmetric perturbations of a Taylor-Couette flow with an axial density stratification is considered in dependence of the Reynolds number Re of rotation and the Brunt-Väisälä number Rn of the stratification. The flows at and beyond the Rayleigh limit become unstable between a lower and an upper Reynolds number (for fixed Rn). The rotation can thus be too slow or too fast for the stratorotational instability. The upper Reynolds number above which the instability decays, has its maximum value for the potential flow (driven by cylinders rotating according to the Rayleigh limit) and decreases strongly for flatter rotation profiles finally leaving only isolated islands of instability in the (Rn/Re) map. The maximal possible rotation ratio $μ_{\rm max}$ only slightly exceeds the shear value of the quasi-uniform flow with $U_ϕ\simeq$const. Along and between the lines of neutral stability the wave numbers of the instability patterns for all rotation laws beyond the Rayleigh limit are mainly determined by the Froude number Fr which is defined by the ratio between Re and Rn. The cells are highly prolate for Fr>1 so that measurements for too high Reynolds numbers become difficult for axially bounded containers. The instability patterns migrate azimuthally slightly faster than the outer cylinder rotates.

physics.flu-dyn↗

arXiv2016

Nonaxisymmetric MHD instabilities of Chandrasekhar states in Taylor-Couette geometry

We consider axially periodic Taylor-Couette geometry with insulating boundary conditions. The imposed basic states are so-called Chandrasekhar states, where the azimuthal flow $U_ϕ$ and magnetic field $B_ϕ$ have the same radial profiles. Mainly three particular profiles are considered: the Rayleigh limit, quasi-Keplerian, and solid-body rotation. In each case we begin by computing linear instability curves and their dependence on the magnetic Prandtl number Pm. For the azimuthal wavenumber m=1 modes, the instability curves always scale with the Reynolds number and the Hartmann number. For sufficiently small Pm these modes therefore only become unstable for magnetic Mach numbers less than unity, and are thus not relevant for most astrophysical applications. However, modes with m>10 can behave very differently. For sufficiently flat profiles, they scale with the magnetic Reynolds number and the Lundquist number, thereby allowing instability also for the large magnetic Mach numbers of astrophysical objects. We further compute fully nonlinear, three-dimensional equilibration of these instabilities, and investigate how the energy is distributed among the azimuthal (m) and axial (k) wavenumbers. In comparison spectra become steeper for large m, reflecting the smoothing action of shear. On the other hand kinetic and magnetic energy spectra exhibit similar behavior: if several azimuthal modes are already linearly unstable they are relatively flat, but for the rigidly rotating case where m=1 is the only unstable mode they are so steep that neither Kolmogorov nor Iroshnikov-Kraichnan spectra fit the results. The total magnetic energy exceeds the kinetic energy only for large magnetic Reynolds numbers Rm>100.

astro-ph.SR↗

arXiv2016

Mixing of a passive scalar by the instability of a differentially rotating axial pinch

The mixing of a passive scalar like lithium, beryllium or temperature fluctuations due to the magnetic Tayler instability of a rotating axial pinch is considered. Our study is carried out within a Taylor-Couette setup for two rotation laws: quasi-Kepler and solid-body rotation. The minimum magnetic Prandtl number used is 0.05 while the molecular Schmidt number Sc of the fluid varies between 0.1 and 2. An effective diffusivity coefficient for the mixing is numerically measured by the decay process of a global concentration peak located between the cylinder walls. We find that only models with Sc>0.1 do provide finite eddy diffusivity values. We also find that for quasi-Kepler rotation at a magnetic Mach number Mm~2 the flow transits from the slow-rotation regime to the fast-rotation regime. For fixed Reynolds number the relation between the normalized eddy diffusivity and the Schmidt number of the fluid is always linear so that also a linear relation between the instability-induced diffusivity and the molecular viscosity results just in the sense proposed by Schatzman (1977). The numerical value of the coefficient in this relation will reach a maximum at Mm~2 and will decrease for Mm>>1 implying that only toroidal magnetic fields of order kG can exist in the solar tachocline.

astro-ph.SR↗

arXiv2015

Subcritical excitation of the current-driven Tayler instability by super-rotation

It is known that in a hydrodynamic Taylor-Couette system uniform rotation or a rotation law with positive shear ('super-rotation') are linearly stable. It is also known that a conducting fluid under the presence of a sufficiently strong axial electric-current becomes unstable against nonaxisymmetric disturbances. It is thus suggestive that a cylindric pinch formed by a homogeneous axial electric-current is stabilized by rotation laws with $dΩ/dR \geq 0$. However, for magnetic Prandtl numbers Pm$\neq 1$ and for slow rotation also rigid rotation and super-rotation support the instability by lowering their critical Hartmann numbers. For super-rotation in narrow gaps and for modest rotation rates this double-diffusive instability even exists for toroidal magnetic fields with rather arbitrary radial profiles, the current-free profile $B_ϕ\propto 1/R$ included. For rigid rotation and for super-rotation the sign of the azimuthal drift of the nonaxisymmetric hydromagnetic instability pattern strongly depends on the magnetic Prandtl number. The pattern counterrotates with the flow for Pm$\ll 1$ and it corotates for Pm$\gg 1$ while for rotation laws with negative shear the instability pattern migrates in the direction of the basic rotation for all Pm. An axial electric-current of minimal 3.6 kAmp flowing inside or outside the inner cylinder suffices to realize the double-diffusive instability for super-rotation in experiments using liquid sodium as the conducting fluid between the rotating cylinders. The limit is 11 kAmp if a gallium alloy is used.

astro-ph.SR↗

arXiv2015

Axisymmetry vs. nonaxisymmetry of a Taylor-Couette flow with azimuthal magnetic fields

The instability of a supercritical Taylor-Couette flow of a conducting fluid with resting outer cylinder under the influence of a uniform axial electric current is investigated for magnetic Prandtl number Pm=1. In the linear theory the critical Reynolds number for axisymmetric perturbations is not influenced by the current-induced axisymmetric magnetic field but all axisymmetric magnetic perturbations decay. The nonaxisymmetric perturbations with m=1 are excited even without rotation for large enough Hartmann numbers ("Tayler instability"). For slow rotation their growth rates scale with the Alfvén frequency of the magnetic field but for fast rotation they scale with the rotation rate of the inner cylinder. In the nonlinear regime the ratio of the energy of the magnetic m=1 modes and the toroidal background field is very low for the non-rotating Tayler instability but it strongly grows if differential rotation is present. For super-Alfvénic rotation the energies in the m=1 modes of flow and field do not depend on the molecular viscosity, they are almost in equipartition and contain only 1.5% of the centrifugal energy of the inner cylinder. The geometry of the excited magnetic field pattern is strictly nonaxisymmetric for slow rotation but it is of the mixed-mode type for fast rotation -- contrary to the situation which has been observed at the surface of Ap stars.

physics.flu-dyn↗

arXiv2011

The angular momentum transport by standard MRI in quasi-Kepler cylindric Taylor-Couette flows

The instability of a quasi-Kepler flow in dissipative Taylor-Couette systems under the presence of an homogeneous axial magnetic field is considered with focus to the excitation of nonaxisymmetric modes and the resulting angular momentum transport. The excitation of nonaxisymmetric modes requires higher rotation rates than the excitation of the axisymmetric mode and this the more the higher the azimuthal mode number m. We find that the weak-field branch in the instability map of the nonaxisymmetric modes has always a positive slope (in opposition to the axisymmetric modes) so that for given magnetic field the modes with m>0 always have an upper limit of the supercritical Reynolds number. In order to excite a nonaxisymmetric mode at 1 AU in a Kepler disk a minimum field strength of about 1 Gauss is necessary. For weaker magnetic field the nonaxisymmetric modes decay. The angular momentum transport of the nonaxisymmetric modes is always positive and depends linearly on the Lundquist number of the background field. The molecular viscosity and the basic rotation rate do not influence the related α-parameter. We did not find any indication that the MRI decays for small magnetic Prandtl number as found by use of shearing-box codes. At 1 AU in a Kepler disk and a field strength of about 1 Gauss the α proves to be (only) of order 0.005.

astro-ph.SR↗

arXiv2011

The differential rotation of G dwarfs

A series of stellar models of spectral type G is computed to study the rotation laws resulting from mean-field equations. The rotation laws of the slowly rotating Sun, the fast rotating MOST stars epsilon Eri and kappa1 Cet and the rapid rotators R58 and LQ Lup can easily be reproduced. We also find that differences in the depth of the convection zone cause large differences in the surface rotation law and that the extreme surface shear of HD 171488 can only be explained with a artificially shallow convection layer. We also check the thermal wind equilibrium in fast-rotating G dwarfs and find that the polar subrotation (dOmega/dz<0) is due to the barocline effect and that the equatorial superrotation (dOmega/dr>0) is due to the Lambda effect as part of the Reynolds stresses. In the bulk of the convection zones where the meridional flow is slow and smooth the thermal wind equilibrium actually holds between the centrifugal and the pressure forces. It does not hold, however, in the bounding shear layers including the equatorial region where the Reynolds stresses dominate.

astro-ph.SR↗

arXiv2010

Cross helicity and turbulent magnetic diffusivity in the solar convection zone

In a density-stratified turbulent medium the cross helicity is considered as a result of the interaction of the velocity fluctuations and a large-scale magnetic field. By means of a quasilinear theory and by numerical simulations we find the cross helicity and the mean vertical magnetic field anti-correlated. In the high-conductivity limit the ratio of the helicity and the mean magnetic field equals the ratio of the magnetic eddy diffusivity and the (known) density scale height. The result can be used to predict that the cross helicity at the solar surface exceeds the value of 1 Gauss km/s. Its sign is anti-correlated with that of the radial mean magnetic field. Alternatively, we can use our result to determine the value of the turbulent magnetic diffusivity from observations of the cross helicity.

astro-ph.SR↗