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

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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.''
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==
==H&oslash;iland Criterion==
Thermal convection arises when the condition,
 
According to [ <b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]] </b>] &#8212; see p. 168 &#8212; this pair of mathematically expressed conditions has the following meaning:
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"A baroclinic star in permanent rotation is dynamically stable with respect to axisymmetric motions if and only if the two following conditions are satisfied:  &nbsp; (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."
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===Schwarzschild Criterion===
In the case of nonrotating equilibrium configurations, the H&oslash;iland Criterion reduces to the Schwarzschild criterion.  That is, thermal convection arises when the condition,
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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.''
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.''


==Poincar&eacute;-Wavre Theorem==
==Poincar&eacute;-Wavre Theorem==


==Lord Rayleigh Instability==
==Lord Rayleigh Instability==


=See Also=
=See Also=

Revision as of 19:14, 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
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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.

Høiland Criterion

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 ], p. 167, 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.

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

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