Editing ThreeDimensionalConfigurations/Stability/RiemannEllipsoids (section)

====RHS====

Next, {{ Lebovitz89ahereafter }} <font color="red">introduces the ''Eulerian-change operator''</font>, <math>\delta</math> (which commutes with <math>\nabla</math>),
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  <td align="right"><math>\delta F</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>F(\mathbf{x},t) - F_0(\mathbf{x}, t) \, .</math></td>
</tr>
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  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, §2, p. 224, Eq. (6)<br />
{{ LBO67hereafter }}, p. 293, Eq. (2)
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Without immediate proof, {{ Lebovitz89ahereafter }} states that the relationship between the ''Lagranian-change operator'' and the ''Eulerian-change operator'' is, to lowest order (linear),
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  <td align="right"><math>\Delta F</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\delta F + \boldsymbol\xi \cdot \nabla F \, .</math></td>
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  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, §2, p. 224, Eq. (7)<br />
{{ LBO67hereafter }}, p. 294, Eq. (3)
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Introducing this mapping into the right-hand side of the perturbed Euler equation gives:
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1<sup>st</sup> term on RHS <math>= - \Delta\biggl\{ \rho^{-1} \nabla p \biggr\}</math>
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  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\rho^{-2}\Delta\rho \biggl[\nabla p \biggr] - \rho^{-1} \nabla \biggl[\Delta p \biggr] 
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\rho^{-1}\biggl[\frac{\Delta\rho}{\rho} \biggr] \nabla p - \rho^{-1} \nabla \biggl[\delta p + (\boldsymbol\xi \cdot \nabla) p \biggr] 
</math>
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \rho^{-1}\biggl[\nabla \cdot \boldsymbol\xi \biggr] \nabla p 
- \rho^{-1} \biggl[\nabla \delta p + (\boldsymbol\xi \cdot \nabla)\nabla p \biggr] 
</math>
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  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, Appendix B, p. 239, Eq. (B.2)
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \rho^{-1} \biggl\{ \nabla \delta p + (\nabla \cdot \boldsymbol\xi ) \nabla p 
+ (\boldsymbol\xi \cdot \nabla)\nabla p  
\biggr\}
</math>
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  <td align="right">
&nbsp;
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  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \rho^{-1} \biggl\{ \nabla \biggl[ \underbrace{\Delta p}_{\mathrm{fixed}~\mathrm{typo}} - \boldsymbol\xi \cdot \nabla p \biggr] + (\nabla \cdot \boldsymbol\xi ) \nabla p 
+ (\boldsymbol\xi \cdot \nabla)\nabla p  
\biggr\}
</math>
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  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, Appendix B, p. 240, Eq. (B.3)
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&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \rho^{-1} 
\nabla ( \Delta p )
+ \rho^{-1} \biggl\{  
\nabla ( \boldsymbol\xi \cdot \nabla p )
- (\nabla \cdot \boldsymbol\xi ) \nabla p 
- (\boldsymbol\xi \cdot \nabla)\nabla p  
\biggr\} \, .
</math>
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  <td align="left" colspan="3"><font color="green">
Comments: 
<ol type="1"><li>
In order to move from the 2<sup>nd</sup> to the 3<sup>rd</sup> line of this derivation, it seems that {{ Lebovitz89ahereafter }} employs the relation: &nbsp; <math>[\Delta\rho/\rho] = - \nabla\cdot \boldsymbol\xi \, .</math>  This relation strongly resembles the continuity equation which, in Lagrangian form, is <math>[D\rho/Dt] = -\rho \nabla\cdot \mathbf{u} \, .</math></li>
<li>In order to move from the 2<sup>nd</sup> to the 3<sup>rd</sup> line of this derivation, {{ Lebovitz89ahereafter }} seems to be acknowledging that, <math>\nabla</math> commutes with <math>(\boldsymbol\xi \cdot \nabla) \, .</math> </li>
<li>A typographical error appears in Eq. (B.3) of {{ Lebovitz89ahereafter }}; <math>\Delta\rho</math> appears in the publication whereas, as noted here in the fifth line of this derivation, the term should be <math>\Delta p</math>.</li>
</ol>
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2<sup>nd</sup> term on RHS <math> =
+ \Delta\biggl\{\mathbf\nabla \biggl[ \Phi_\mathrm{L89} + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr]\biggr\}
</math>
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  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\mathbf\nabla  \biggl\{\Delta \Phi_\mathrm{L89} + \Delta\biggl[\tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr]\biggr\}
</math>
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\mathbf\nabla  \biggl\{[\delta \Phi_\mathrm{L89} + \boldsymbol\xi\cdot \nabla\Phi_\mathrm{L89}] 
+ \cancelto{0}{\delta\biggl[\tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr] }
+ \boldsymbol\xi \cdot \nabla \biggl[\tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr]

\biggr\}
</math>
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\nabla \delta \Phi_\mathrm{L89}
+ \mathbf\nabla  \biggl\{ (\boldsymbol\xi\cdot \nabla) \Phi_\mathrm{L89} 
+ \boldsymbol\xi \cdot \nabla \biggl[\tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr]
\biggr\}
</math>
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\nabla \delta \Phi_\mathrm{L89}
+ \boldsymbol\xi \cdot \mathbf\nabla  \biggl\{ \nabla \biggl[ \Phi_\mathrm{L89} 
+ \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr]
\biggr\} \, .
</math>
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  <td align="left" colspan="3"><font color="green">
Comment: 
<ol type="1" start="4"><li>
A term in the 2<sup>nd</sup> row of this derivation goes to zero because there is no ''Eulerian'' variation in either of the vectors, <math>\boldsymbol\omega</math> or <math>\mathbf{x}</math>.
</li>
</ol>
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Adding these two "RHS" terms together gives,

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  <td align="right">
RHS
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  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \rho^{-1} 
\nabla ( \Delta p )
+ \rho^{-1} \biggl\{  
\nabla ( \boldsymbol\xi \cdot \nabla p )
- (\nabla \cdot \boldsymbol\xi ) \nabla p 
- (\boldsymbol\xi \cdot \nabla)\nabla p  
\biggr\}
+
\nabla \delta \Phi_\mathrm{L89}
+ \boldsymbol\xi \cdot \mathbf\nabla  \biggl\{ \nabla \biggl[ \Phi_\mathrm{L89} 
+ \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr]
\biggr\} \, .
</math>
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That is to say,

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  <td align="right">
RHS
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  <td align="center">
<math>=</math>
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  <td align="left">
<math> 
- \rho^{-1} 
\nabla ( \Delta p )
+ L\boldsymbol\xi \, ,
</math>
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  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, &sect;2, p. 224, Eq. (8)
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where the operator, <math>L</math>, is defined such that,

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<math>L\boldsymbol\xi</math>
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  <td align="center">
<math>\equiv</math>
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  <td align="left">
<math> 
\rho^{-1} \biggl\{  
\nabla ( \boldsymbol\xi \cdot \nabla p )
- (\nabla \cdot \boldsymbol\xi ) \nabla p 
- (\boldsymbol\xi \cdot \nabla)\nabla p  
\biggr\}
+
\nabla \delta \Phi_\mathrm{L89}
+ \boldsymbol\xi \cdot \mathbf\nabla  \biggl\{ \nabla \biggl[ \Phi_\mathrm{L89} 
+ \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \biggr]
\biggr\} \, .
</math>
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  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, Appendix B, p. 240, Eq. (B.4)
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<font color="red">Are our four comments correct?</font>