User:Tohline/2DStructure/AxisymmetricInstabilities

Axisymmetric Instabilities to Avoid

Here we draw heavily from the extensive discussion of instabilities that appears in [T78]

When constructing rotating equilibrium configurations that obey a barotropic equation of state, keep in mind that certain parameter ranges should be avoided because they will lead to structures that are unstable toward the dynamical development of local, convective-type motions. Here are a few well-known examples.

Rayleigh-Taylor Instability

This bouyancy-driven instability arises when the condition,

<math>~(- \vec{g} ) \cdot \nabla\rho</math>

<math>~< </math>

<math>~0 </math>

[stable] ,

is violated, where, <math>~\vec{g}</math> is the local gravitational acceleration vector. In the simplest case of spherically symmetric configurations, this means that the mass density must decrease outward.

Høiland Criterion

As is stated on p. 166 of [ T78 ], in rotating barotropic configurations, axisymmetric stability requires the simultaneous satisfaction of the following pair of conditions:

<math>~\biggl(\frac{1}{\varpi^3} \biggr) \frac{\partial j^2}{\partial \varpi} + \frac{1}{c_P} \biggl( \frac{\gamma - 1}{\Gamma_3 - 1}\biggr) (- \vec{g} ) \cdot \nabla s</math>	<math>~></math>	<math>~0 </math>	[stable] ;
[ T78 ], §7.3, Eq. (41)
<math>~-g_z \biggl[ \frac{\partial j^2}{\partial \varpi} \biggl(\frac{\partial s}{\partial z} \biggr) - \frac{\partial j^2}{\partial z} \biggl(\frac{\partial s}{\partial \varpi} \biggr)\biggr]</math>	<math>~></math>	<math>~0 </math>	[stable] .
[ T78 ], §7.3, Eq. (42)

According to [ T78 ] — see p. 168 — this pair of mathematically expressed conditions has the following meaning:

"A baroclinic star in permanent rotation is dynamically stable with respect to axisymmetric motions if and only if the two following conditions are satisfied:   (i) the entropy per unit mass, <math>~s</math>, never decreases outward, and (ii) on each surface <math>~s</math> = constant, the angular momentum per unit mass, <math>~\Omega \varpi^2</math>, increases as we move from the poles to the equator."

Schwarzschild Criterion

In the case of nonrotating equilibrium configurations, the Høiland Criterion reduces to the Schwarzschild criterion. That is, thermal convection arises when the condition,

<math>~(- \vec{g} ) \cdot \nabla s</math>	<math>~> </math>	<math>~0 </math>	[stable] ,
[ T78 ], §7.3, Eq. (43)

is violated, where, <math>~s</math> is the local specific entropy of the fluid. This means that, in order for a spherical system to be stable against dynamical convective motions, the specific entropy must increase outward.

Solberg/Rayleigh Criterion

In the case of an homentropic equilibrium configuration, the Høiland Criterion reduces to the Solberg criterion. That is, an axisymmetric exchange of fluid "rings" will occur on a dynamical time scale if the condition,

<math>~\frac{d}{d\varpi} \biggl( \Omega^2 \varpi^4 \biggr)</math>	<math>~> </math>	<math>~0 </math>	[stable] ,
[ T78 ], §7.3, Eq. (44)

is violated. This means that, for stability, the specific angular momentum must necessarily increase outward. As [ T78 ] points out, this "Solberg criterion generalizes to homentropic bodies the well-known Rayleigh criterion for an inviscid, incompressible fluid."

See Also

• T. Fukushima (2016, AJ, 152, article id. 35, 31 pp.) — Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects
• W.-T. Kim & S. Moon (2016, ApJ, 829, article id. 45, 22 pp.) — Equilibrium Sequences and Gravitational Instability of Rotating Isothermal Rings
• D. Petroff & S. Horatschek (2008, MNRAS, 389,156 - 172) — Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity
• P. H. Chavanis (2006, International Journal of Modern Physics B, 20, 3113 - 3198) — Phase Transitions in Self-Gravitating Systems
• M. Lombardi & G. Bertin (2001, Astronomy & Astrophysics, 375, 1091 - 1099) — Boyle's Law and Gravitational Instability
• J. W. Woodward, J. E. Tohline, & I. Hachisu (1994, ApJ, 420, 247 - 267) — The Stability of Thick, Self-Gravitating Disks in Protostellar Systems
• I. Bonnell & P. Bastien (1991, ApJ, 374, 610 - 622) — The Collapse of Cylindrical Isothermal and Polytropic Clouds with Rotation
• J. E. Tohline & I. Hachisu (1990, ApJ, 361, 394 - 407) — The Breakup of Self-Gravitating Rings, Tori, and Thick Accretion Disks
• F. Schmitz (1988, Astronomy & Astrophysics, 200, 127 - 134) — Equilibrium Structures of Differentially Rotating Self-Gravitating Gases
• P. Veugelen (1985, Astrophysics & Space Science, 109, 45 - 55) — Equilibrium Models of Differentially Rotating Polytropic Cylinders
• M. A. Abramowicz, A. Curir, A. Schwarzenberg-Czerny, & R. E. Wilson (1984, MNRAS, 208, 279 - 291) — Self-Gravity and the Global Structure of Accretion Discs
• P. Bastien (1983, Astronomy & Astrophysics, 119, 109 - 116) — Gravitational Collapse and Fragmentation of Isothermal, Non-Rotating, Cylindrical Clouds
• Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875) — Another Equilibrium Sequence of Self-Gravitating and Rotating Incompressible Fluid
• J. E. Tohline (1980, ApJ, 236, 160 - 171) — Ring Formation in Rotating Protostellar Clouds
• T. Fukushima, Y. Eriguchi, D. Sugimoto, & G. S. Bisnovatyi-Kogan (1980, Progress of Theoretical Physics, 63, 1957 - 1970) — Concave Hamburger Equilibrium of Rotating Bodies
• J. Katz & D. Lynden-Bell (1978, MNRAS, 184, 709 - 712) — The Gravothermal Instability in Two Dimensions
• P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584- 597) — Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid (includes torus with non-uniform rotation)

1. Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an equilibrium incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"

• C. J. Hansen, M. L. Aizenman, & R. L. Ross (1976, ApJ, 207, 736 - 744) — The Equilibrium and Stability of Uniformly Rotating, Isothermal Gas Cylinders
• C.-Y. Wong (1974, ApJ, 190, 675 - 694) — Toroidal Figures of Equilibrium
• C.-Y. Wong (1973, Annals of Physics, 77, 279 - 353) — Toroidal and Spherical Bubble Nuclei
• J. Ostriker (1964, ApJ, 140, 1056) — The Equilibrium of Polytropic and Isothermal Cylinders
• J. Ostriker (1964, ApJ, 140, 1067) — The Equilibrium of Self-Gravitating Rings
• J. Ostriker (1964, ApJ, 140, 1529) — On the Oscillations and the Stability of a Homogeneous Compressible Cylinder
• J. Ostriker (1965, ApJ Supplements, 11, 167) — Cylindrical Emden and Associated Functions
• Gunnar Randers (1942, ApJ, 95, 88) — The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.
• Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154) — On the Dynamics of Revolving Fluids
• F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106) — The Potential of an Anchor Ring. Part II.
  1. In this paper, Dyson derives the gravitational potential inside the ring mass distribution
• F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95) — The Potential of an Anchor Ring. Part I.
  1. In this paper, Dyson derives the gravitational potential exterior to the ring mass distribution
• S. Kowalewsky (1885, Astronomische Nachrichten, 111, 37) — Zusätze und Bemerkungen zu Laplace's Untersuchung über die Gestalt der Saturnsringe
• Poincaré (1885a, C. R. Acad. Sci., 100, 346), (1885b, Bull. Astr., 2, 109), (1885c, Bull. Astr. 2, 405). — references copied from paper by Wong (1974)

 

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