Editing 2DStructure/AxisymmetricInstabilities (section)

=Axisymmetric Instabilities to Avoid=
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When constructing rotating equilibrium configurations that obey a [[SR#Time-Independent_Problems|barotropic equation of state]], keep in mind that certain physical variable profiles 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.
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==Rayleigh-Taylor Instability==
Referencing both p. 101 of [ <b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]] </b>], and volume I, p. 410 of [ <b>[[Appendix/References#P00|<font color="red">P00</font>]] </b>], a [https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability Rayleigh-Taylor] instability is a bouyancy-driven instability that arises when a heavy fluid rests on top of a light fluid in an effective gravitational field, <math>~\vec{g}</math>.   In the simplest case of spherically symmetric, self-gravitating configurations, the condition for stability against a Rayleigh-Taylor instability may be written as,
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<math>~(- \vec{g} ) \cdot \nabla\rho</math>
  </td>
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<math>~< </math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
  <td align="center">&nbsp; &nbsp;&nbsp;&nbsp;[stable] ,</td>
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that is to say, the mass density ''must decrease outward.''  In an expanded discussion, [ <b>[[Appendix/References#P00|<font color="red">P00</font>]] </b>] &#8212; see pp. 410 - 413 &#8212; derives a dispersion relation that describes the development of the Rayleigh-Taylor instability in a plane-parallel fluid layer that initially contains a discontinuous jump/drop in the density. 

In addition, [ <b>[[Appendix/References#P00|<font color="red">P00</font>]] </b>] &#8212; see pp. 413 - 416 &#8212; and [ <b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]] </b>] &#8212; see pp. 101 - 105 &#8212; both discuss circumstances that can give rise to the so-called ''Kelvin-Helmholtz'' instability.  It is another common two-fluid instability, but one that depends on the existence of transverse velocity flows and that is independent of the gravitational field.

==Poincar&eacute;-Wavre Theorem==
As [ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>] points out &#8212; see his pp. 78 - 81 &#8212; Poincar&eacute; and Wavre were the first to, effectively, prove the following theorem:
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<tr><td align="left">For rotating, self-gravitating configurations<font color="darkgreen"> "any of the following statements implies the three others: &nbsp; (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."
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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,
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<math>~\frac{\partial \dot\varphi}{\partial z}</math>
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<math>~=</math>
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<math>~0 \, .</math>
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  <td align="center" colspan="3">
[ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>], &sect;4.3, Eq. (30)
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NOTE:  We should investigate how this theorem comes into play in the context of our [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|accompanying discussion of Type 1 Riemann ellipsoids]].  These are equilibrium triaxial, uniform-density configurations in which the system's internal vorticity vector does not align with the tumble-axis of the ellipsoid and, therefore apparently <math>~\dot\varphi</math> is not independent of <math>~z</math>.

==H&oslash;iland Criterion==

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"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion &#8212; that <math>~\varpi^4 \Omega^2</math> increase outwards &#8212; is necessary and sufficient for stability to axisymmetric disturbances.  In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain.  Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland H&oslash;iland (1941)] (see also [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'' (1965)], and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa.  The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>
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&#8212; Drawn from p. 475 of [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], ARAA, 5, 465 
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As is stated on p. 166 of [ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>], in rotating barotropic configurations, axisymmetric stability requires the simultaneous satisfaction of the following pair of conditions:
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<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>
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<math>~></math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
  <td align="center">&nbsp; &nbsp;&nbsp;&nbsp;[stable] ;</td>
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<tr>
  <td align="center" colspan="4">
[ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>], &sect;7.3, Eq. (41)
<br />see also<br />[ <b>[[Appendix/References#KW94|<font color="red">KW94</font>]] </b>], &sect;43.2, Eq. (43.22)
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<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>
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<math>~></math>
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  <td align="left">
<math>~0 </math>
  </td>
  <td align="center">&nbsp; &nbsp;&nbsp;&nbsp;[stable] .</td>
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[ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>], &sect;7.3, Eq. (42)
<br />see also<br />[ <b>[[Appendix/References#KW94|<font color="red">KW94</font>]] </b>], &sect;43.2, Eq. (43.23)
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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 [ <b>[[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>~j</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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<math>~(- \vec{g} ) \cdot \nabla s</math>
  </td>
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<math>~> </math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
  <td align="left">&nbsp; &nbsp;&nbsp;&nbsp;[stable] ,</td>
</tr>
<tr>
  <td align="center" colspan="4">
[ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>], &sect;7.3, Eq. (43)
<br />see also<br />[ <b>[[Appendix/References#KW94|<font color="red">KW94</font>]] </b>], &sect;6.1, Eq. (6.13) &hellip; or &hellip; pp. 93 - 98 of [ <b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]] </b>]
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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 H&oslash;iland Criterion reduces to the Solberg criterion.  That is, an axisymmetric exchange of fluid "rings" will occur on a dynamical time scale if the condition,
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<math>~\frac{dj^2}{d\varpi} </math>
  </td>
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<math>~> </math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
  <td align="left">&nbsp; &nbsp;&nbsp;&nbsp;[stable] ,</td>
</tr>
<tr>
  <td align="center" colspan="4">
[ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>], &sect;7.3, Eq. (44)
<br />see also<br />[ <b>[[Appendix/References#KW94|<font color="red">KW94</font>]] </b>], &sect;43.2, Eq. (43.18) &hellip; or &hellip; pp. 98 - 101 of [ <b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]] </b>]
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is violated.  This means that, for stability, the specific angular momentum ''must necessarily increase outward.''  As [ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>] points out, this <font color="darkgreen">"Solberg criterion generalizes to homentropic bodies the well-known [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Rayleigh (1917)] criterion for an inviscid, incompressible fluid."</font>