Editing ThreeDimensionalConfigurations/Challenges (section)

===Compressible===

Now in our [[#AndalibBernoulli|above discussion of Andalib's work]], the steady-state form of the Euler equation was formulated as,

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

<tr>
  <td align="right">
<math>\nabla \biggl[H + \Phi_\mathrm{grav} + F_B(\Psi) + \frac{1}{2}u^2 - \frac{1}{2}\Omega_f^2 (x^2 + y^2) \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0 \, .</math>
  </td>
</tr>
</table>
It is easy to appreciate that,
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  <td align="right">
<math>~\nabla F_B(\Psi) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\boldsymbol\zeta +  2{\vec{\Omega}}_f )\times \bold{u} \, .</math>
  </td>
</tr>
</table>
As we have shown, in the case of the incompressible (Riemann S-type ellipsoid) models, 
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  <td align="right">
<math>~(\boldsymbol\zeta +  2{\vec{\Omega}}_f )\times \bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\nabla \biggl\{
\frac{\lambda^2(a^2 + b^2)}{2} \biggl[ \frac{x^2}{a^2} + \frac{y^2}{b^2}   \biggr] 
+
\biggl[ \Omega_f \lambda\biggl( \frac{b}{a} x^2 + \frac{a}{b}y^2 \biggr)  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>

If we attempt to directly relate these two expressions, we must acknowledge that,
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<tr>
  <td align="right">
<math>~F_B(\Psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\lambda^2(a^2 + b^2)}{2} \biggl[ \frac{x^2}{a^2} + \frac{y^2}{b^2}   \biggr] 
+
\biggl[ \Omega_f \lambda\biggl( \frac{b}{a} x^2 + \frac{a}{b}y^2 \biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x^2}{a^2} \biggl[\frac{1}{2}\lambda^2(a^2 + b^2) + \Omega_f \lambda a b \biggr]
+
\frac{y^2}{b^2} \biggl[ \frac{1}{2}\lambda^2(a^2 + b^2) + \Omega_f \lambda a b \biggr] \, .
</math>
  </td>
</tr>
</table>

As we have discussed above, Andalib (1998) found that some interesting model sequences could be constructed if he adopted the functional form,
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<tr>
  <td align="right">
<math>~F_B(\Psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_0 \Psi + \frac{1}{2} C_1 \Psi^2 \, .</math>
  </td>
</tr>
</table>
Evidently, the incompressible (uniform-density) Riemann S-type ellisoids can be retrieved from our derived compressible-model formalism if we set, <math>~C_1 = 0</math>, and,

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  <td align="right">
<math>~\Psi(x,y)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x^2}{a^2} + \frac{y^2}{b^2} \, ,</math>
  </td>
</tr>
</table>
with,
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<tr>
  <td align="right">
<math>~C_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{1}{2}\lambda^2(a^2 + b^2) + \Omega_f \lambda a b \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\zeta_z}{2}\biggl(  \frac{a^2 b^2}{a^2 + b^2}\biggr)\biggl[ 2\Omega_f - \zeta_z \biggr] \, .</math>
  </td>
</tr>
</table>

<font color="red">Afterthought:</font>&nbsp; Because we want <math>~\Psi(x,y)</math> to go to zero at the surface, it likely will be better to set,
<div align="center">
<math>~\Psi \equiv 1 - \biggl[ \frac{x^2}{a^2} + \frac{y^2}{b^2} \biggr] \, ,</math>
</div>
then adjust the sign of <math>~C_0</math> and add a constant (zeroth-order term) to the definition of <math>~F_B(\Psi)</math>.