On the schwarzschild criterion in accretion disk theory

Elstner, D.; Rüdiger, G.; Tschäpe, R.

doi:10.1080/03091928908218531

Cited by 4 publications

(3 citation statements)

References 6 publications

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“…Obviously dS/dz > 0 is a sufficient condition for stability of Kepler disks after the Solberg-Høiland criterion also for the combined action of vertical density stratification and differential Kepler rotation. The same result have been derived in earlier papers by Livio & Shaviv (1977), Abramowicz et al (1984) and Elstner et al (1989).…”

Section: Introduction: the Solberg-høiland Criterionsupporting

confidence: 89%

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

Rüdiger

Arlt

Shalybkov

2002

A&A

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Abstract. 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-Høiland 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 Sect. 2. This confirms the results of the Solberg-Høiland 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.

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Section: Introduction: the Solberg-høiland Criterionsupporting

confidence: 89%

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

Rüdiger

Arlt

Shalybkov

2002

A&A

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“…Equation (2) provides both the Rayleigh condition for stability, ∂(R 4 Ω 2 )/∂R > 0, for isentropic axial stratification and the Schwarzschild criterion for stability, ∂S/∂z > 0, for resting fluids. If, however, a fluid with ∂P/∂z < 0 rotates with a stable rotation law Ω = Ω (R), then also the second stability Solberg-Høiland condition (3) is fulfilled with the Schwarzschild criterion ∂S ∂z > 0 (4) [17,8]. A combination of stable axial density stratifications and centrifugally-stable rotation laws should thus also be stable.…”

Section: Introductionmentioning

confidence: 99%

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

Rüdiger

Seelig

Schultz

et al. 2017

Geophysical & Astrophysical Fluid Dynamics

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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 max only slightly exceeds the shear value of the quasi-uniform flow with U φ ≃ 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.

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“…In general, however, the coefficients of the dispersion relation (10) are complex. To find the stability criterion the method by Elstner, Rüdiger & Tschäpe (1989) is applied. For ideal fluids Blokland et al (2005) used a very similar approach.…”

Section: Dispersion Relation For Very Small Gapsmentioning

confidence: 99%

Helical magnetorotational instability of Taylor‐Couette flows in the Rayleigh limit and for quasi‐Kepler rotation

Rüdiger¹,

Schultz²

2008

Astron Nachr

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The magnetorotational instability (MRI) of differential rotation under the simultaneous presence of axial and azimuthal components of the (current-free) magnetic field is considered. For rotation with uniform specific angular momentum the MHD equations for axisymmetric perturbations are solved in a local short-wave approximation. All the solutions are overstable for Bz · B φ = 0 with eigenfrequencies approaching the viscous frequency. For more flat rotation laws the results of the local approximation do not comply with the results of a global calculation of the MHD instability of TaylorCouette flows between rotating cylinders. -With B φ and Bz of the same order the traveling-mode solutions are also prefered for flat rotation laws such as the quasi-Kepler rotation. For magnetic Prandtl number Pm → 0 they scale with the Reynolds number of rotation rather than with the magnetic Reynolds number (as for standard MRI) so that they can easily be realized in MHD laboratory experiments. -Regarding the nonaxisymmetric modes one finds a remarkable influence of the ratio B φ /Bz only for the extrema. For B φ ≫ Bz and for not too small Pm the nonaxisymmetric modes dominate the traveling axisymmetric modes. For standard MRI with Bz ≫ B φ , however, the critical Reynolds numbers of the nonaxisymmetric modes exceed the values for the axisymmetric modes by many orders so that they are never prefered.

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On the schwarzschild criterion in accretion disk theory

Cited by 4 publications

References 6 publications

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

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

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

Helical magnetorotational instability of Taylor‐Couette flows in the Rayleigh limit and for quasi‐Kepler rotation

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