Difference between revisions of "User:Tohline/2DStructure/AxisymmetricInstabilities"

From VistrailsWiki
Jump to navigation Jump to search
Line 166: Line 166:
* [http://adsabs.harvard.edu/abs/1965ApJS...11..167O J. Ostriker (1965, ApJ Supplements, 11, 167)] — ''Cylindrical Emden and Associated Functions''
* [http://adsabs.harvard.edu/abs/1965ApJS...11..167O J. Ostriker (1965, ApJ Supplements, 11, 167)] — ''Cylindrical Emden and Associated Functions''
* [http://adsabs.harvard.edu/abs/1942ApJ....95...88R Gunnar Randers (1942, ApJ, 95, 88)] — ''The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.''
* [http://adsabs.harvard.edu/abs/1942ApJ....95...88R Gunnar Randers (1942, ApJ, 95, 88)] — ''The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.''
* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] — ''On the Dynamics of Revolving Fluids''
* [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106)] &#8212; ''The Potential of an Anchor Ring. Part II.'' <ol type="a"><li>In this paper, Dyson derives the gravitational potential ''inside'' the ring mass distribution</li></ol>
* [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106)] &#8212; ''The Potential of an Anchor Ring. Part II.'' <ol type="a"><li>In this paper, Dyson derives the gravitational potential ''inside'' the ring mass distribution</li></ol>
* [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95)] &#8212; ''The Potential of an Anchor Ring. Part I.''<ol type="a"><li>In this paper, Dyson derives the gravitational potential ''exterior to'' the ring mass distribution</li></ol>
* [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95)] &#8212; ''The Potential of an Anchor Ring. Part I.''<ol type="a"><li>In this paper, Dyson derives the gravitational potential ''exterior to'' the ring mass distribution</li></ol>

Revision as of 20:57, 11 August 2017

Axisymmetric Instabilities to Avoid

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


Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

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."

Poincaré-Wavre Theorem

Lord Rayleigh Instability

See Also

  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) …"


 

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation