arXiv subjects

G. Rüdiger

Publications and source records attributed to G. Rüdiger.

At least 37 records · Page 2Linked to original sources

arXiv2009

Stratorotational instability in Taylor-Couette flow heated from above

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 transports angular momentum outwards and will therefor be relevant for astrophysical applications. The belonging viscosity $α$ coefficient is of the order of unity if the results are adapted to the size of an accretion disc. The strength of the stratification, the fluids Prandtl number and the boundary conditions applied in the simulations are well-suited too for a laboratory experiment using water and a small temperature gradient below five Kelvin. With such a rather easy realizable set-up the SRI and its angular momentum transport could be measured in an experiment.

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arXiv2009

Stability of latitudinal differential rotation in stars

The question is addressed whether stellar differentially rotating radiative zones (like the solar tachocline) excite nonaxisymmetric r-modes which can be observed. To this end the hydrodynamical stability of latitudinal differential rotation is studied. The amount of rotational shear required for the instability is estimated in dependence of the character of radial stratification and the flow patterns excited by the instability are found. The eigenvalue equations for the nonaxisymmetric disturbances are formulated in 3D and then solved numerically. Radial displacements and entropy disturbances are included. The equations contain the 2D approximation of strictly horizontal displacements as a special limit. The critical magnitude of the latitudinal differential rotation for onset of the instability is considerably reduced in the 3D theory compared to the 2D approximation. The instability requires a subadiabatic stratification. It does not exist in the bulk of convection zone with almost adiabatic stratification but may switch on near its base in the region of penetrative convection. Growth rates and symmetry types of the modes are computed in dependence on the rotation law parameters. The S1 mode with its transequatorial toroidal vortices is predicted as the dominating instability mode. The vortices show longitudinal drift rates retrograde to the basic rotation which are close to that of the observed weak r-mode signatures at the solar surface.

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arXiv2008

Toroidal field instability and eddy viscosity in Taylor-Couette flows

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 linear on the magnetic Reynolds number Rm. For Rm=1000 the eddy viscosity is of the size of 30 in units of the microscopic value.

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arXiv2008

Diamagnetic pumping near the base of a stellar convection zone

The property of inhomogeneous turbulence in conducting fluids to expel large-scale magnetic fields in the direction of decreasing turbulence intensity is shown as important for the magnetic field dynamics near the base of a stellar convection zone. The downward diamagnetic pumping confines a fossil internal magnetic field in the radiative core so that the field geometry is appropriate for formation of the solar tachocline. For the stars of solar age, the diamagnetic confinement is efficient only if the ratio of turbulent magnetic diffusivity of the convection zone to the (microscopic or turbulent) diffusivity of the radiative interiour is larger than 10^5. Confinement in younger stars require still larger diffusivity ratio. The observation of persistent magnetic structures on young solar-type stars can thus provide evidences for the nonexistence of tachoclines in stellar interiors and on the level of turbulence in radiative cores.

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arXiv2008

Helicity generation and alpha-effect by Vandakurov-Tayler instability with z-dependent differential rotation

We investigate in isothermal MHD simulations the instability of toroidal magnetic fields resulting by the action of z-dependent differential rotation on a given axial field B^0 in a cylindrical enclosure where in particular the helicity of the resulting nonaxisymmetric flow is of interest. The idea is probed that helicity H is related to the external field and the differential rotation as H ~ B^0_i B^0_j Omega_i,j. The observed instability leads to a nonaxisymmetric solution with dominating mode m=1. With the onset of instability both kinematic and current helicity are produced which fulfill the suggested relation. Obviously, differential rotation dOmega/dz only needs an external axial field B^0_z to produce remarkable amounts of the helicities. Any regular time-dependency of the helicity could not be found. The resulting axial alpha-effect is mainly due to the current helicity, the characteristic time scale between both the values is of order of the rotation time. If the axial field is switched off then the helicity and the alpha-effect disappear.

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arXiv2007

Stability of toroidal magnetic fields in the solar tachocline and beneath

Stability of toroidal magnetic field in a stellar radiation zone is considered for the cases of uniform and differential rotation. In the rigidly rotating radiative core shortly below the tachocline, the critical magnetic field for instability is about 600 G. The unstable disturbances for slightly supercritical fields have short radial scales ~1 Mm. Radial mixing produced by the instability is estimated to conclude that the internal field of the sun can exceed the critical value of 600 G only marginally. Otherwise, the mixing is too strong and not compatible with the observed lithium abundance. Analysis of joint instability of differential rotation and toroidal field leads to the conclusion that axisymmetric models of the laminar solar tachocline are stable to nonaxisymmetric disturbances. The question of whether sun-like stars can posses tachoclines is addressed with positive answer for stars with rotation periods shorter than about two months.

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arXiv2005

Differential rotation and meridional flow in the solar convection zone and beneath

The influence of the basic rotation on anisotropic and inhomogeneous turbulence is discussed in the context of differential rotation theory. An improved representation for the original turbulence leads to a $\Lam$-effect which complies with the results of 3D numerical simulations. The resulting rotation law and meridional flow agree well with both the surface observations ($\partial\Om/\partial r<0$ and meridional flow towards the poles) and with the findings of helioseismology. The computed equatorward flow at the bottom of convection zone has an amplitude of about 10 m/s and may be significant for the solar dynamo. The depth of the meridional flow penetration into the radiative zone is proportional to $ν^{0.5}_\mathrm{core}$, where $ν_\mathrm{core}$ is the viscosity beneath the convection zone. The penetration is very small if the tachocline is laminar.

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arXiv2004

Seed fields for galactic dynamos by the magnetorotational instabilities

A linear but global numerical model for the magnetorotational instability (MRI) in disk geometry is considered to estimate the instability parameters for galaxies. Similarity rules suggested by a local analysis are applied to reveal a universal behavior approached by the results of global calculations for large magnetic Prandtl numbers. The findings are used to estimate the MRI characteristics for galaxies with their very large magnetic Prandtl numbers which cannot be attained in any numerical simulations. The resulting minimum field for the instability, Bmin 10-25 G, is small compared to any seed fields currently discussed. The growth times of MRI are estimated to be on the order of the rotation period of the inner rigidly-rotating core, i.e. ~100 Myr. Global MRI excites preferentially the magnetic field modes of quadrupolar symmetry.

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arXiv2004

Turbulent magnetic Prandtl number and magnetic diffusivity quenching from simulations

Forced turbulence simulations are used to determine the turbulent kinematic viscosity, nu_t, from the decay rate of a large scale velocity field. Likewise, the turbulent magnetic diffusivity, eta_t, is determined from the decay of a large scale magnetic field. In the kinematic regime, when the field is weak, the turbulent magnetic Prandtl number, nu_t/eta_t, is about unity. When the field is nonhelical, eta_t is quenched when magnetic and kinetic energies become comparable. For helical fields the quenching is stronger and can be described by a dynamical quenching formula.

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arXiv2003

The stability of protostellar disks with Hall effect and buoyancy

The stability properties of inviscid protostellar disks are examined taking into account the Hall effect and buoyancy. Depending on the parameters, different types of instabilities can exist in different regions of disks. In a very low ionized region, the instability associated with baroclinic effects of buoyancy is likely most efficient. The Hall-driven shear instability can lead to destabilization of regions with a higher ionization. The magnetorotational instability modified by buoyancy can only be a destabilizing factor in regions with strong magnetic field or a relatively high conductivity ($σB^2 / ρ> a_e Ω$, with $a_e$ the magnetization parameter of electrons).

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arXiv2002

Do spherical alpha^2-dynamos oscillate?

The question is answered whether kinematic alpha^2-shell-dynamos are able to produce a cyclic activity or not. The alpha-effect is allowed to be latitudinally inhomogeneous and/or anisotropic, but it is assumed as radially uniform in the turbulent shell. For a symmetric alpha-tensor we only find oscillatory solutions if i) the alpha_zz vanishes or is of the opposite sign as alpha_{phi phi}, ii) the alpha-effect is strongly concentrated to the equatorial region and iii) the alpha-effect is concentrated to a rather thin outer shell. In the other cases almost always the nonaxisymmetric field mode S1 possesses the lowest dynamo number which slowly drifts along the azimuthal direction. Also uniform but anisotropic alpha-effect (alpha_zz = 0) leads to the nonaxisymmetric solutions as it is confirmed by the Karlsruhe dynamo experiment. However, one of the antisymmetric parts of the alpha-tensor basically plays the role of a differential rotation in the induction equation. Using for the radial profile of this effect the results of anumerical simulation for the alpha-tensor of the solar convection zone, one indeed finds the possibility of oscillating alpha^2-dynamos even without the existence of real nonuniform plasma rotation.

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arXiv2002

The linear MHD Taylor-Couette instability for liquid sodium

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 limiting case a=0 plays an exceptional role. The hydromagnetic instability exists with certain Reynolds numbers at certain Hartmann numbers of the magnetic field. These Reynolds numbers exactly scale with Pm^-1/2 resulting in moderate values of order 10^4 for Pm=10^-5. However, already for the smallest positive value of a the Reynolds numbers start to scale as 1/Pm leading to much higher values of order 10^6 for Pm=10^-5. Also nonaxisymmetric modes have been considered. With vacuum boundary conditions their excitation is always more difficult than the excitation of axisymmetric modes; we never observed a crossover of the lines of marginal stability. For conducting walls, however, such crossovers exist for both resting and rotating outer cylinders, and this might be essential for future dynamo experiments.

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arXiv2002

The influence of Hall drift on the magnetic fields of neutron stars

We consider the evolution of magnetic fields under the influence of Hall drift and Ohmic decay. The governing equation is solved numerically, in a spherical shell with r_i/r_o = 0.75. Starting with simple free decay modes as initial conditions, we then consider the subsequent evolution. The Hall effect induces so-called helicoidal oscillations, in which energy is redistributed among the different modes. We find that the amplitude of these oscillations can be quite substantial, with some of the higher harmonics becoming comparable to the original field. Nevertheless, this transfer of energy to the higher harmonics is not sufficient to significantly accelerate the decay of the original field, at least not at the R_B = O(100) parameter values accessible to us, where this Hall parameter R_B measures the ratio of the Ohmic timescale to the Hall timescale. We do find clear evidence though of increasingly fine structures developing for increasingly large R_B, suggesting that perhaps this Hall-induced cascade to ever shorter lengthscales is eventually sufficiently vigorous to enhance the decay of the original field. Finally, the implications for the evolution of neutron star magnetic fields are discussed.

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arXiv2002

Differential rotation and meridional flow in the solar supergranulation layer: Measuring the eddy viscosity

We measure the eddy viscosity in the outermost layers of the solar convection zone by comparing the rotation law computed with the Reynolds stress resulting from f-plane simulations of the angular momentum transport in rotating convection with the observed differential rotation pattern. The simulations lead to a negative vertical and a positive horizontal angular momentum transport. The consequence is a subrotation of the outermost layers, as it is indeed indicated both by helioseismology and the observed rotation rates of sunspots. In order to reproduce the observed gradient of the rotation rate a value of about 1.5 x 10^{13} cm/s for the eddy viscosity is necessary. Comparison with the magnetic eddy diffusivity derived from the sunspot decay yields a surprisingly large magnetic Prandtl number of 150 for the supergranulation layer. The negative gradient of the rotation rate also drives a surface meridional flow towards the poles, in agreement with the results from Doppler measurements. The successful reproduction of the abnormally positive horizontal cross correlation (on the northern hemisphere) observed for bipolar groups then provides an independent test for the resulting eddy viscosity.

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arXiv2002

The stability of MHD Taylor-Couette flow

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 axisymmetric modes. The situation is changed in the MHD regime. For modest magnetic fields the excitation of modes with m=1 is easier than the excitation of modes with m=0 if critical magnetic amplitudes are exceeded.

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arXiv2002

Angular momentum transport in the solar supergranulation layer

The eddy viscosity in the solar supergranulation layer is derived from the observed rotational shear by computing theoretical rotation laws for the outermost parts of the solar convection zone using the results from numerical simulations of rotating convection as input. By varying the eddy viscosity, the results can be tuned to match the observations. The value of 1.5 X 10^13 cm^2/s found for the eddy viscosity is considerably larger than the eddy magnetic diffusivity derived from the sunspot decay. The results are checked by comparison of the horizontal cross correlations of the velocity fluctuations with the observed Ward profile.

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arXiv2002

Nonaxisymmetric patterns in the linear theory of MHD Taylor-Couette instability

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 nonaxisymmetric spirals with the azimuthal number m=1. Obviously, the magnetic field is able to excite nonaxisymmetric configurations despite of the tendency of differential rotation to favor axisymmetric magnetic fields which is known from the dynamo theory. If Pm is too big or too small, however, the axisymmetric mode with m=0 appears to be the most unstable one possessing the lowest Reynolds numbers -- as it is also true for hydrodynamic Taylor-Couette flow or for very weak fields. That the most unstable mode for modest Pm proves to be nonaxisymmetric must be considered as a strong indication for the possibility of dynamo processes in connection with the magnetorotational instability.

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arXiv2002

Hydrodynamic stability in accretion disks under the combined influence of shear and density stratification

The hydrodynamic stability of accretion disks is considered. The particular question is whether the combined action of a (stable) vertical density stratification and a (stable) radial differential rotation gives rise to a new instability for nonaxisymmetric modes of disturbances. The existence of such an instability is not suggested by the well-known Solberg-Hoiland criterion. It is also not suggested by a local analysis for disturbances in general stratifications of entropy and angular momentum which is presented in our Section 2 confirming the results of the Solberg-Hoiland criterion also for nonaxisymmetric modes within the frame of ideal hydrodynamics but only in the frame of a short-wave approximation for small m. As a necessary condition for stability we find that only conservative external forces are allowed to influence the stable disk. As magnetic forces are never conservative, linear disk instabilities should only exist in the magnetohydrodynamical regime which indeed contains the magnetorotational instability as a much-promising candidate. To overcome some of the used approximations in a numerical approach,the equations of the compressible adiabatic hydrodynamics are integrated imposing initial nonaxisymmetric velocity perturbations with m=1 to m=200. Only solutions with decaying kinetic energy are found. The system always settles in a vertical equilibrium stratification according to pressure balance with the gravitational potential of the central object. keywords: accretion disks -- hydrodynamic instabilities -- turbulence

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