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 shape-distorting or convective-type motions. Here are a few well-known examples.

Rayleigh-Taylor Instability

A Rayleigh-Taylor, 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.

Poincaré-Wavre Theorem

As [ T78 ] points out — see his pp. 78 - 81 — Poincaré and Wavre were the first to, effectively, prove the following theorem:

For rotating, self-gravitating configurations "any of the following statements implies the three others:   (i) the angular velocity is a constant over cylinders centered about the axis of rotation, (ii) the effective gravity can be derived from a potential, (iii) the effective gravity is normal to the isopycnic surfaces, (iv) the isobaric- and isopycnic-surfaces coincide."

Among other things, this implies that for rotating barotropic configurations not only is the equation of state given by a function of the form, <math>~P = P(\rho)</math>, but it must also be true that,

<math>~\frac{\partial \dot\varphi}{\partial z}</math>	<math>~=</math>	<math>~0 \, .</math>
[ T78 ], §4.3, Eq. (30)

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) see also [ KW94 ], §43.2, Eq. (43.22)
<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) see also [ KW94 ], §43.2, Eq. (43.23)

where, <math>~s</math>, is the local specific entropy, and <math>~j \equiv \dot\varphi \varpi^2</math>, is the local specific angular momentum of the fluid. 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>~j</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) see also [ KW94 ], §6.1, Eq. (6.13) … or … pp. 93 - 98 of [ Shu92 ]

is violated. 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{dj^2}{d\varpi} </math>	<math>~> </math>	<math>~0 </math>	[stable] ,
[ T78 ], §7.3, Eq. (44) see also [ KW94 ], §43.2, Eq. (43.18) … or … pp. 98 - 101 of [ Shu92 ]

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 (1917) criterion for an inviscid, incompressible fluid."

Modeling Implications and Advice

Here are some recommendations to keep in mind as you attempt to construct equilibrium models of self-gravitating astrophysical fluids.

Rayleigh-Taylor instability:  In order to avoid constructing configurations that are subject to the Rayleigh-Taylor instability, be sure that lower density material is never placed "beneath" higher density material. For example …

• When building a bipolytropic configuration, a value for the (imposed) discontinuous jump in the mean-molecular weight will need to be specified at the interface between the envelope and the core. If you choose a ratio, <math>~{\bar\mu}_e/{\bar\mu}_c</math>, that is greater than unity, the resulting equilibrium model will exhibit a discontinuous density jump that makes the density higher at the base of the envelope than it is at the surface of the core. The core/envelope interface of this configuration will be unstable to the Rayleigh-Taylor instability, but you won't know that until and unless you examine the hydrodynamic stability of the configuration.

Schwarzschild criterion:  In order to avoid constructing configurations that violate the Schwarzschild criterion, be sure that the specific entropy of the fluid is uniform (marginally stable) or increases outward throughout the equilibrium structure, where the word "outward" is only meaningful when referenced against the direction that the effective gravity points. For example …

• Suppose that you build a spherically symmetric polytropic configuration whose structural index is, <math>~n</math>, but then you want to test the stability of the configuration assuming that compressions/expansions of individual fluid elements occur along adiabats for which the adiabatic index is, <math>~\gamma \ne (n+1)/n</math>. Although we generally think of polytropes as being homentropic configurations, if <math>~\gamma \ne (n+1)/n</math>, then different fluid elements will, in practice, evolve along adiabats that are characterized by different values of the specific entropy; that is, throughout the equilibrium model, <math>~\nabla s</math> will not be zero. Whether the specific entropy increase (stable) or decreases (unstable) outward will depend on whether you select a value for the evolutionary <math>~\gamma</math> that is greater than (stable) or less than (unstable) <math>~(n+1)/n</math>.

See Also

• Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154) — On the Dynamics of Revolving Fluids

 

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