Editing ThreeDimensionalConfigurations/Stability/RiemannEllipsoids (section)

====LHS====
With the assurance provided by {{ Lebovitz89ahereafter }} that <math>\Delta</math> commutes with the Lagrangian time-derivative, <math>D/Dt</math> &#8212; see also the paragraph immediately preceding Eq. (4) in {{ LBO67hereafter }} &#8212; and that
<table border="0" align="center" cellpadding="3">
<tr>
  <td align="right"><math>\Delta \mathbf{u}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{D\boldsymbol\xi}{Dt} \, ,</math></td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, §2, p. 223, Eq. (4)
  </td>
</tr>
</table>

we can immediately appreciate that,
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<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="right">
<math>\frac{D}{Dt} \biggl[\Delta\mathbf{u} \biggr] 
+ 2\boldsymbol\omega \boldsymbol\times \biggl[ \Delta \mathbf{u}\biggr] </math>
=
<math>\frac{D}{Dt} \biggl[\frac{D\boldsymbol{\xi}}{Dt} \biggr] 
+ 2\boldsymbol\omega \boldsymbol\times \biggl[\frac{D\boldsymbol{\xi}}{Dt} \biggr] \, .</math>
  </td>
</tr>
</table>
Hence, we obtain the (still, exact nonlinear),
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  <td align="center" colspan="3">
<font color="#770000">'''(Lagrangian) Perturbed Euler Equation'''</font>
  </td>
</tr>
<tr>
  <td align="right">
<math>\frac{D^2\boldsymbol{\xi}}{Dt^2}  
+ 2\boldsymbol\omega \boldsymbol\times \biggl[\frac{D\boldsymbol{\xi}}{Dt} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \Delta\biggl\{ \rho^{-1} \nabla p \biggr\} 
+ \Delta\biggl\{\mathbf\nabla \biggl[ \Phi_\mathrm{L89} + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr]\biggr\}
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, §2, p. 223, Eq. (5)
  </td>
</tr>
</table>

For later reference, notice that the LHS may further be rewritten as,
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<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{D}{Dt} \biggl[\boldsymbol{\xi}_t + (\mathbf{u}\cdot \nabla)\boldsymbol\xi \biggr] 
+ 2\boldsymbol\omega \boldsymbol\times \biggl[\boldsymbol\xi_t 
+ (\mathbf{u}\cdot \nabla)\boldsymbol\xi  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\partial }{\partial t} + (\mathbf{u}\cdot \nabla)\biggr]
\biggl[\boldsymbol{\xi}_t + (\mathbf{u}\cdot \nabla)\boldsymbol\xi \biggr] 
+ 2\boldsymbol\omega \boldsymbol\times \biggl[\boldsymbol\xi_t 
+ (\mathbf{u}\cdot \nabla)\boldsymbol\xi  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\xi}_{tt} + \frac{\partial }{\partial t} \biggl[(\mathbf{u}\cdot \nabla)\boldsymbol\xi \biggr] 
+
(\mathbf{u}\cdot \nabla)
\biggl[\boldsymbol{\xi}_t + (\mathbf{u}\cdot \nabla)\boldsymbol\xi \biggr] 
+ 2\boldsymbol\omega \boldsymbol\times \biggl[\boldsymbol\xi_t 
+ (\mathbf{u}\cdot \nabla)\boldsymbol\xi  \biggr] 
</math>
  </td>
</tr>
</table>

where we have adopted {{ Lebovitz89ahereafter }}'s shorthand notation,
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<tr>
  <td align="right">
<math>
\boldsymbol\xi_{t} \equiv \frac{\partial \boldsymbol\xi}{\partial t}  \, ,
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and, &nbsp; &nbsp;
  </td>
  <td align="right">
<math>
\boldsymbol\xi_{tt} \equiv \frac{\partial^2 \boldsymbol\xi}{\partial t^2}  \, .
</math>
  </td>
</tr>
</table>
Finally, if <font color="green">&hellip; the unperturbed solution &hellip; is steady</font> &#8212; as is the case in the context of our study of the stability of Riemann S-Type ellipsoids (see more, below) &#8212; then <math>(\mathbf{u}\cdot \nabla)</math> commutes with the Eulerian time-derivative, that is,
<div align="center"><math>\frac{\partial}{\partial t}\biggl[ (\mathbf{u}\cdot \nabla) \boldsymbol\xi \biggr] ~\rightarrow ~  (\mathbf{u}\cdot \nabla) \boldsymbol\xi_t \, ,</math></div>
which means we may write,
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<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\xi}_{tt} + 2\{ (\mathbf{u}\cdot \nabla)\boldsymbol\xi_t 
+ \boldsymbol\omega \boldsymbol\times \boldsymbol\xi_t \}
+ \{ (\mathbf{u}\cdot \nabla)^2\boldsymbol\xi 
+ 2\boldsymbol\omega \boldsymbol\times [(\mathbf{u}\cdot \nabla)\boldsymbol\xi ] \} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, §2, p. 224, immediately preceding Eq. (10)
  </td>
</tr>
</table>