User:Tohline/2DStructure/AxisymmetricInstabilities

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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
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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.

Schwarzschild Criterion

Thermal convection arises when the condition,

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

<math>~> </math>

<math>~0 </math>

     [stable] ,

[ T78 ], p. 167, Eq. (43)

is violated, where, <math>~s</math> is the local specific entropy of the fluid. In the simplest case of spherically symmetric configurations, this means that the specific entropy must increase outward.

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
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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