Editing 2DStructure/AxisymmetricInstabilities (section)

==H&oslash;iland Criterion==

<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<font color="indianred">
"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>
</td></tr>
<tr><td align="right">
&#8212; Drawn from p. 475 of [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], ARAA, 5, 465 
</td></tr></table>

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:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<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>
  </td>
  <td align="center">
<math>~></math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
  <td align="center">&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. (41)
<br />see also<br />[ <b>[[Appendix/References#KW94|<font color="red">KW94</font>]] </b>], &sect;43.2, Eq. (43.22)
  </td>
</tr>

<tr>
  <td align="right">
<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>
  </td>
  <td align="center">
<math>~></math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
  <td align="center">&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. (42)
<br />see also<br />[ <b>[[Appendix/References#KW94|<font color="red">KW94</font>]] </b>], &sect;43.2, Eq. (43.23)
  </td>
</tr>
</table>
</div>

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:
<table border="0" align="center" width="75%" cellpadding="5">
<tr><td align="left"><font color="darkgreen">
"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."
</font></td></tr>
</table>

===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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(- \vec{g} ) \cdot \nabla s</math>
  </td>
  <td align="center" width="20px">
<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>]
  </td>
</tr>
</table>
</div>
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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dj^2}{d\varpi} </math>
  </td>
  <td align="center" width="20px">
<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>]
  </td>
</tr>
</table>
</div>
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>