Editing ThreeDimensionalConfigurations/Stability/RiemannEllipsoids (section)

===Specifically Perturb Riemann S-Type Ellipsoids===

Now let's assume that the initial equilibrium configuration is a steady-state, Riemann S-Type ellipsoid.  Then, [[#Steady-State_Unperturbed_Flows|from above]], we know that,
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<font color="#770000">'''Steady-State Flow<br />as viewed from a Rotating Reference Frame'''</font> 
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<math> 
\mathbf\nabla \{ \Phi_\mathrm{L89} + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \}
</math>
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<math>=</math>
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<math>\rho^{-1} \nabla p + (\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u} \, .</math>
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Hence, the operator,

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<math>L\boldsymbol\xi</math>
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<math>=</math>
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<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\{ 
\rho^{-1} \nabla p + (\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u} 
\biggr\} 
</math>
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&nbsp;
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<math>=</math>
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<math> 
\rho^{-1} \biggl\{  
\nabla ( \boldsymbol\xi \cdot \nabla p )
- (\nabla \cdot \boldsymbol\xi ) \nabla p 
\biggr\}
-
\rho^{-1} \biggl\{  
(\boldsymbol\xi \cdot \nabla)\nabla p  
\biggr\}
+
\nabla \delta \Phi_\mathrm{L89}
+ (\boldsymbol\xi \cdot \mathbf\nabla ) \biggl\{ 
\rho^{-1} \nabla p  
\biggr\} 
+ ( \boldsymbol\xi \cdot \mathbf\nabla ) \biggl\{ 
(\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u} 
\biggr\} 
</math>
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&nbsp;
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<math>=</math>
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<math> 
\rho^{-1} \biggl\{  
\nabla ( \boldsymbol\xi \cdot \nabla p )
- (\nabla \cdot \boldsymbol\xi ) \nabla p 
\biggr\}
- \rho^{-1}  (\boldsymbol\xi \cdot \nabla)\nabla p  
+ \rho^{-1} (\boldsymbol\xi \cdot \mathbf\nabla ) \nabla p  
-\rho^{-2} \nabla p(\boldsymbol\xi \cdot \mathbf\nabla ) \rho
+
\nabla \delta \Phi_\mathrm{L89}
+ ( \boldsymbol\xi \cdot \mathbf\nabla ) \biggl\{ 
(\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u} 
\biggr\} 
</math>
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&nbsp;
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<math>=</math>
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<math> 
\rho^{-1} \biggl\{  
\nabla ( \boldsymbol\xi \cdot \nabla p )
- (\nabla \cdot \boldsymbol\xi ) \nabla p 
\biggr\}
- \cancelto{0}{\rho^{-2} \nabla p(\boldsymbol\xi \cdot \mathbf\nabla ) \rho}
+
\nabla \delta \Phi_\mathrm{L89}
+ ( \boldsymbol\xi \cdot \mathbf\nabla ) \biggl\{ 
(\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u} 
\biggr\} \, ,
</math>
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{{ Lebovitz89b }}, &sect;2, p. 227, Eq. (4)
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where, following the lead of {{ Lebovitz89b }}, the term containing <math>\nabla\rho</math> has been set to zero because, throughout a Riemann ellipsoid, <font color="green">"&hellip; the unperturbed density is spatially uniform &hellip;"</font>

In addition, following the lead of {{ LL96 }}, <font color="green">"&hellip; we consider here the incompressible case and therefore adjoin to [the perturbed Euler equation] the expression of mass conservation &hellip;"</font> namely,
<div align="center"><math>\nabla \cdot \boldsymbol\xi = 0 \, .</math><br />&nbsp;<br />
{{ LL96hereafter }}, &sect;3.1, p. 703, Eq. (13)</div>
Hence,

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<math>L\boldsymbol\xi</math>
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<math>=</math>
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<math> 
\rho^{-1} \nabla ( \boldsymbol\xi \cdot \nabla p )
+
\nabla \delta \Phi_\mathrm{L89}
+ ( \boldsymbol\xi \cdot \mathbf\nabla ) [ 
(\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u} 
] \, .
</math>
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{{ LL96hereafter }}, &sect;3.1, p. 703, Eq. (17)
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