Editing ThreeDimensionalConfigurations/Stability/RiemannEllipsoids

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=Lebovitz &amp; Lifschitz (1996)=
{| class="InstabilitiesOfRiemannEllipsoids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:white;" |[[H_BookTiledMenu#Ellipsoidal_&_Ellipsoidal-Like_2|<b>Lebovitz &amp; Lifschitz<br />(1996)</b>]]
|}

Here we review the work of {{ LL96full }} titled, "New Global Instabilities of the Riemann Ellipsoids," and discuss various extensions that have been made to this work. Note that a good summary of the research efforts that preceded (and inspired) the work of {{ LL96 }} can be found in the introductory section of {{ OTM2007full }}. 

We were prompted to tackle this review in response to an email received in December 2021 from [[Appendix/Ramblings/ForCohlHoward|Howard Cohl]].
&nbsp;<br />
&nbsp;<br />

==Background==
In Figure 1, the abscissa is the ratio <math>b/a</math> of semiaxes in the equatorial plane, and the ordinate is the ratio <math>c/a</math> of the vertical semiaxis to the larger of the equatorial semi axes.  This diagram shows what {{ LL96 }} &#8212; hereafter, {{ LL96hereafter }} &#8212; refer to as "<font color="darkgreen">the horn-shaped region of existence of S-type ellipsoids and the Jacobi family</font>;" it underpins all four panels of the {{ LL96hereafter }} Figure 2.

<table border="1" align="center" cellpadding="8"><tr><td align="center">
'''Figure 1:''' The ''Horn-Shaped'' Region of S-type Ellipsoids</td></tr>
<tr><td align="center">
[[File:EFEdiagram02.png|350px|EFE Diagram02]]
</td></tr></table>

<ul>
<li>''Jacobi'' sequence &#8212; the smooth curve that runs through the set of small, dark-blue, diamond-shaped markers; the data identifying the location of these markers have been drawn from &sect;39, Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].  The small red circular markers lie along this same sequence; their locations are taken from our own determinations, as detailed in [[ThreeDimensionalConfigurations/JacobiEllipsoids#Table2|Table 2]] of our accompanying discussion of Jacobi ellipsoids.  All of the models along this sequence have <math>~f \equiv \zeta/\Omega_f = 0</math> and are therefore solid-body rotators, that is, there is no internal motion when the configuration is viewed from a frame that is rotating with frequency, <math>~\Omega_f</math>.</li>
<li>''Dedekind'' sequence &#8212; a smooth curve that lies precisely on top of the ''Jacobi'' sequence. Each configuration along this sequence is ''adjoint'' to a model on the ''Jacobi'' sequence that shares its (b/a, c/a) axis-ratio pair.  All ellipsoidal figures along this sequence have <math>~1/f = \Omega_f/\zeta = 0</math> and are therefore stationary as viewed from the ''inertial'' frame; the angular momentum of each configuration is stored in its internal motion (vorticity).</li>
<li>The X = -1 ''self-adjoint'' sequence &#8212; At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration  <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = -(a^2+b^2)/(ab)</math>.  The data identifying the location of the small, solid-black markers along this sequence have been drawn from &sect;48, Table VI of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].</li>
<li>The X = +1 ''self-adjoint'' sequence &#8212; At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration  <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = +(a^2+b^2)/(ab)</math>.  The data identifying the location of the small, solid-black markers along this sequence have been drawn from &sect;48, Table VI of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].</li>
</ul>

<table border="0" align="left" cellpadding="25"><tr><td align="center">
[[File:EFEdiagram4.png|200px|EFE Diagram identifying example models from Ou (2006)]]
</td></tr></table>
Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram.  The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by {{ Ou2006full }} and that [[ThreeDimensionalConfigurations/RiemannStype#Models_Examined_by_Ou_(2006)|we have also chosen to use as examples]].

Four example models of equilibrium Riemann S-Type ellipsoids (click each parameter-pair to go to a related chapter discussion):
<ul>
<li><b>[[ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS|(b/a, c/a) = (0.41, 0.385)]]</b>; the chapter name is "Riemann Meets COLLADA &amp; Oculus Rift S"</li>
<li><b>[[Appendix/Ramblings/RiemannB90C333|(b/a, c/a) = (0.90, 0.333)]]</b></li>
<li><b>[[Appendix/Ramblings/RiemannB74C692|(b/a, c/a) = (0.74, 0.692)]]</b></li>
<li><b>[[Appendix/Ramblings/RiemannB28C256|(b/a, c/a) = (0.28, 0.256)]]</b></li>
</ul>
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
==Self-Adjoint Sequences==

What are the expressions that define the upper <math>(x = -1)</math> and lower <math>(x = +1)</math> boundaries of the ''horned shaped'' region of equilibrium S-Type Riemann Ellipsoids? Well, as we have [[ThreeDimensionalConfigurations/RiemannStype#Based_on_Virial_Equilibrium|discussed in an associated chapter]], the value of the parameter, <math>x</math>, that is associated with each point <math>(b/a, c/a)</math> within the ''horned shaped'' region is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
1 +2Cx + x^2
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">{{ LL96 }}, &sect;2, Eq. (5)</td></tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>C</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a b B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]  \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">{{ LL96 }}, &sect;2, Eq. (6)</td></tr>
</table>

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~A_{12}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (107)</font> ]</td></tr>

<tr>
  <td align="right">
<math>~B_{12}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~A_2 - a^2A_{12} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (105)</font> ]<br />See also the ''note'' immediately following &sect;21, Eq. (127)</td></tr>
</table>

===Upper Boundary===
The upper boundary of the ''horn-shaped'' region is obtained by setting <math>x = -1</math>.  That is, it is associated with coordinate pairs <math>(b/a, c/a)</math> for which,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
1 - 2C + 1
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ C
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>+1</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl[ \frac{a b B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>+1</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ a b B_{12} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c^2 A_3 - a^2 b^2 A_{12}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~c^2 A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a b [ A_2 - a^2A_{12}]  + a^2 b^2 A_{12}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a b A_2  + b a^2 A_{12} (b  - a)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a b A_2  + b a^2 (a - b )\biggl[\frac{A_1-A_2}{a^2 - b^2} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{c^2}{ab}\biggr] A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>A_2  + a \biggl[\frac{A_1-A_2}{a+b} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{c^2(a+b)}{ab}\biggr] A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>aA_1 + bA_2 \, .</math>
  </td>
</tr>
</table>
Now, from the [[ThreeDimensionalConfigurations/RiemannStype#Riemann_S-Type_Ellipsoids|expressions for A<sub>1</sub>, A<sub>2</sub>, and A<sub>3</sub>]], we can furthermore write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
c^2(a+b) A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b A_1 + ab^2 [2 -(A_1 + A_3)] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b A_1 + 2ab^2 - ab^2 A_1 - ab^2 A_3 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ c^2(a+b) A_3 + ab^2 A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 2ab^2 + a^2 b A_1 - ab^2 A_1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{a}{b}\biggl[c^2(a+b)+ ab^2 \biggr]A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 2a^2b + a^2(a  - b)A_1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl[c^2(a+b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> a^2b + bc(a  - b)
\biggl\{
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr]
\biggr\} \, ,</math>
  </td>
</tr>
</table>
where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;17, Eq. (32)</font> ]</td></tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">STRATEGY</font> for finding the locus of points that define the upper boundary of the horned-shape region &hellip; &nbsp; &nbsp;Set <math>a = 1</math>, and pick a value for <math>0 < b < 1</math>; then, using an iterative technique, vary <math>c</math> until the following expression is satisfied:

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

<tr>
  <td align="right">
<math>
\biggl[c^2(a+b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> a^2b + bc(a  - b)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr]
\, .</math>
  </td>
</tr>
</table>
Choose another value of <math>0 < b < 1</math>, then iterate again to find the value of <math>c</math> that corresponds to this new, chosen value of <math>b</math>.  Repeat!
</td></tr></table>

===Lower Boundary===

Similarly, the lower boundary is obtained by setting <math>x = +1</math>, that is, it is associated with coordinate pairs <math>(b/a, c/a)</math> for which,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
C
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-1</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl[ \frac{a b B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-1</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~- a b B_{12} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c^2 A_3 - a^2 b^2 A_{12}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~c^2 A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- a b [ A_2 - a^2A_{12}]  + a^2 b^2 A_{12}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-a b A_2  + b a^2 A_{12} (b  + a)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-a b A_2  - b a^2 (a + b )\biggl[\frac{A_1-A_2}{a^2 - b^2} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{c^2}{ab}\biggr] A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-A_2  - a \biggl[\frac{A_1-A_2}{a-b} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~\biggl[ \frac{c^2(a-b)}{ab}\biggr] A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>A_2(b-a) - aA_1 + aA_2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>bA_2 - aA_1 \, .</math>
  </td>
</tr>
</table>

Now, from the [[ThreeDimensionalConfigurations/RiemannStype#Riemann_S-Type_Ellipsoids|expressions for A<sub>1</sub>, A<sub>2</sub>, and A<sub>3</sub>]], we can furthermore write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
c^2(a-b) A_3  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2ab^2  - ab^2 A_1 - ab^2 A_3 - a^2 b A_1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ c^2(a-b) A_3  + ab^2 A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2ab^2 - ab(b + a)A_1  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{a}{b}\biggl[ c^2(a-b)+ ab^2 \biggr]A_3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2\biggl[ 2b - (b + a)A_1 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl[ c^2(a-b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b - bc(b + a) 
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">STRATEGY</font> for finding the locus of points that define the lower boundary of the horned-shape region &hellip; &nbsp; &nbsp;Set <math>a = 1</math>, and pick a value for <math>0 < b < 1</math>; then, using an iterative technique, vary <math>c</math> until the following expression is satisfied:

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

<tr>
  <td align="right">
<math>
\biggl[ c^2(a-b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b - bc(b + a) 
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, .
</math>
  </td>
</tr>
</table>
Choose another value of <math>0 < b < 1</math>, then iterate again to find the value of <math>c</math> that corresponds to this new, chosen value of <math>b</math>.  Repeat!
</td></tr></table>

==Stability Equations==

===Strategy===

<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"Let <math>\mathbf{u}(\mathbf{x}), p(\mathbf{x}), \rho(\mathbf{x})</math> represent the velocity field, pressure, and density, respectively, of an inviscid fluid mass in a steady state relative to a reference frame rotating with angular velocity <math>\boldsymbol{\omega} = \omega \mathbf{e}_3</math> about an axis fixed in space (the z-, or x<sub>3</sub>-, axis) &hellip; The stability of this steady state is determined, in linear approximation, by the solutions, with arbitrary initial data, of the &hellip; equation [governing the time-dependent behavior of] the Lagrangian displacement <math>\boldsymbol\xi</math>."
</font>
</td></tr>
<tr><td align="right">
— Drawn from the first paragraph of &sect;2 (p. 226) in {{ Lebovitz89b }}. 
</td></tr></table>

<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"This basic equation
</font>[is of the form],"  &nbsp; &nbsp; <math>\boldsymbol\xi_{tt} + A \boldsymbol\xi_t + B\boldsymbol\xi + \rho^{-1}\nabla \Delta p = 0 </math> &hellip; Eq. (10).
</td></tr>
<tr><td align="right">
— Drawn from the first paragraph of &sect;3.1 (p. 701) in {{ LL96 }}. 
</td></tr>
</table>

<table border="0" cellpadding="3" align="center" width="80%">
<tr><td align="left">
<font color="darkgreen">
"We introduce for the solution space <math>\Sigma</math> a basis <math>\{\xi_i\}</math> the first <math>N</math> vectors <math>\{\xi\}_{i=1}^N</math> of which represent a basis for <math>\Sigma_n</math>, the space of solenoidal vector polynomials of degree not exceeding <math>n</math>, as in {{ Lebovitz89ahereafter }}, {{ Lebovitz89bhereafter }}. It is easily found (see {{ Lebovitz89ahereafter }}) that <math>N = N(n) = (n+1)(n+2)(2n+9)/6</math>.  Since <math>\Sigma_n</math> is invariant under the operators <math>A</math> and <math>B</math>, we seek solutions of Eq. (10) in this space:"</font><br />
<div align="center"><math>\boldsymbol\xi(\mathbf{x}, t) = \sum_{i=1}^{N} c_i(t) \xi_i</math> &hellip; Eq. (18)</div>
</td></tr>
<tr><td align="right">
— Drawn from the first paragraph of &sect;3.2 (p. 703) in {{ LL96 }}. 
</td></tr>
</table>

Here we will closely follow the derivation found in {{ Lebovitz89afull }}, hereafter {{ Lebovitz89ahereafter }}.

===Euler Equation===
From our initial overarching presentation of the principal governing equation, we draw an expression for the,

<div align="center">
<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi 
~\underbrace{- ~2{\vec{\Omega}}_f \times {\vec{v}}_{rot}}_\mathrm{Coriolis} 
~\underbrace{- ~{\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})}_\mathrm{Centrifugal} 
\, .</math>
</div>

Moving the term that accounts for the Coriolis acceleration to the left-hand side of this expression, and realizing that the centrifugal acceleration may be rewritten in the form,
<div align="center">
<font color="#770000">'''Centrifugal Acceleration'''</font> 

<math>
{\vec{a}}_\mathrm{Centrifugal} \equiv - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})
= \frac{1}{2} \nabla\biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2  \biggr] \, , 
</math>
</div>
the Euler equation becomes,
<table border="0" cellpadding="8" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> - \frac{1}{\rho} \nabla P - \nabla \Phi 
+ ~\frac{1}{2} \nabla\biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2  \biggr] 
\, .</math>
  </td>
</tr>
</table>
Except for the adopted sign convention for the gravitational potential, <math>\Phi \leftrightarrow -\Phi_\mathrm{L89}</math>, this precisely matches Equation (2) of {{ Lebovitz89ahereafter }}, namely,
<div align="center" id="EulerRotating">
<table border="1" align="center" cellpadding="8" width="80%">
<tr><td align="center" bgcolor="lightgreen">{{ Lebovitz89afigure }}</td></tr>
<tr><td align="left">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>\frac{D\mathbf{u}}{Dt} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ \rho^{-1} \nabla p + \mathbf\nabla \{ \Phi_\mathrm{L89} + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \}
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89a }}, §2, p. 223, Eq. (2)<br />
{{ LL96b }}, &sect;2, p. 929, Eq. (2.1)
  </td>
</tr>
</table>
</td></tr>
</table>
</div>

In what follows, we will adopt the {{ Lebovitz89ahereafter }} variable notation.

===Steady-State Unperturbed Flows===
[[PGE/Euler#in_terms_of_velocity:_2|As we have discussed in a much broader context]], the so-called Lagrangian (or "material") time derivative, <math>D/Dt</math>, that appears on the left-hand side of this Lagrangian representation of the Euler equation can be replaced by its Eulerian counterpart, <math>\partial/\partial t</math>, via the operator relation,

<table border="0" cellpadding="8" align="center">
<tr>
  <td align="right">
<math>\frac{D}{Dt} 
</math>
  </td>
  <td align="center"><math>\leftrightarrow</math></td>
  <td align="left">
<math> 
\frac{\partial}{\partial t} + (\mathbf{u} \cdot \nabla)
\, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LBO67hereafter }}, §1, p. 294, Eq. (4)
  </td>
</tr>
</table>
Furthermore, if our unperturbed fluid configuration is in steady-state, this will be reflected in the Euler equation by setting, <math>\partial \mathbf{u}/\partial t \rightarrow 0</math>, that is,

<table border="0" cellpadding="8" align="center">
<tr>
  <td align="right">
<math>\frac{D\mathbf{u}}{Dt} 
</math>
  </td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="left">
<math> 
\cancelto{0}{\frac{\partial \mathbf{u}}{\partial t}} + (\mathbf{u} \cdot \nabla)\mathbf{u}
\, ,</math>
  </td>
</tr>
</table>
in which case the following relation holds:
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="center" colspan="3">
<font color="#770000">'''Steady-State Flow<br />as viewed from a Rotating Reference Frame'''</font> 
  </td>
</tr>
<tr>
  <td align="right">
<math>(\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ \rho^{-1} \nabla p + \mathbf\nabla \{ \Phi_\mathrm{L89} + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \}
\, .
</math>
  </td>
</tr>
</table>
This relationship between structural variables in the context of steady-state unperturbed flows [[#Specifically_Perturb_Riemann_S-Type_Ellipsoids|will be used below]].

===Lagrangian Displacement and Linearization===

Suppose that, at time <math>t = 0</math>, the function set <math>[\mathbf{u}_0(\mathbf{x}), \rho_0(\mathbf{x}), p_0(\mathbf{x})]</math> properly describes the properties of a &#8212; as yet unspecified &#8212; geometrically extended, fluid configuration. <!-- Now suppose that the entire fluid configuration is "perturbed." -->  According to the Euler equation and, in particular, as dictated by the flow-field,
<table border="0" align="center" cellpadding="3">
<tr>
  <td align="right"><math>\mathbf{u}_0(\mathbf{x})</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[\boldsymbol{\hat\imath} u_x(\mathbf{x}) + \boldsymbol{\hat\jmath} u_y(\mathbf{x}) + \mathbf{\hat{k}} u_z(\mathbf{x}) \biggr]_0
\ ,
</math>
  </td>
</tr>
</table>

after an interval of time, <math>t</math>, each "Lagrangian" fluid element will move from its initial location, <math>\mathbf{x}</math>, to a new position, <math>\mathbf{x} + \boldsymbol\xi</math>.  In general each Lagrangian fluid element will discover that, at its new coordinate location, the "environment" is different.  For example,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>p_0(\mathbf{x})</math></td>
  <td align="center"><math>~~\rightarrow~~</math></td>
  <td align="left"><math>p(\mathbf{x} + \boldsymbol\xi,t) \, ,</math></td>
</tr>
<tr>
  <td align="right"><math>\rho_0(\mathbf{x})</math></td>
  <td align="center"><math>~~\rightarrow~~</math></td>
  <td align="left"><math>\rho(\mathbf{x} + \boldsymbol\xi,t) \, ,</math></td>
</tr>
<tr>
  <td align="right"><math>[u_i(\mathbf{x})]_0</math></td>
  <td align="center"><math>~~\rightarrow~~</math></td>
  <td align="left"><math>u_i(\mathbf{x} + \boldsymbol\xi,t) \, .</math></td>
</tr>
</table>
With this in mind, {{ Lebovitz89ahereafter }} <font color="red">introduces a ''Lagrangian-change operator''</font>, <math>\Delta</math>, in order to mathematically indicate that this evolutionary step is being executed for any physical variable, <math>F</math>.  Specifically, 
<table border="0" align="center" cellpadding="3">
<tr>
  <td align="right"><math>\Delta F</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>F(\mathbf{x} + \boldsymbol\xi,t) - F_0(\mathbf{x}) \, .</math></td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, §2, p. 223, Eq. (3)<br />
{{ LBO67hereafter }}, p. 293, Eq. (1)
  </td>
</tr>
</table>

Following {{ Lebovitz89ahereafter }} and applying the operator, <math>\Delta</math>, to each side of the Euler equation, we can write,
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>\Delta \biggl\{ \frac{D\mathbf{u}}{Dt} \biggr\} 
+ \Delta\biggl\{ 2\boldsymbol\omega \boldsymbol\times \mathbf{u} \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>
</table>

====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,
<table border="0" cellpadding="3" align="center">
<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),
<table border="0" cellpadding="3" align="center">
<tr>
  <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,
<table border="0" cellpadding="3" align="center">

<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,
<table border="0" cellpadding="3" align="center">
<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,
<table border="0" cellpadding="3" align="center">

<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>

====RHS====

Next, {{ Lebovitz89ahereafter }} <font color="red">introduces the ''Eulerian-change operator''</font>, <math>\delta</math> (which commutes with <math>\nabla</math>),
<table border="0" align="center" cellpadding="3">
<tr>
  <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>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, §2, p. 224, Eq. (6)<br />
{{ LBO67hereafter }}, p. 293, Eq. (2)
  </td>
</tr>
</table>

<table border="1" width="80%" cellpadding="8" align="center">
<tr><td align="left">
Without immediate proof, {{ Lebovitz89ahereafter }} states that the relationship between the ''Lagranian-change operator'' and the ''Eulerian-change operator'' is, to lowest order (linear),
<table border="0" align="center" cellpadding="3">
<tr>
  <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>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, §2, p. 224, Eq. (7)<br />
{{ LBO67hereafter }}, p. 294, Eq. (3)
  </td>
</tr>
</table>

</td></tr>
</table>

Introducing this mapping into the right-hand side of the perturbed Euler equation gives:
<table border="0" cellpadding="3" align="center" width="80%">

<tr>
  <td align="right">
1<sup>st</sup> term on RHS <math>= - \Delta\biggl\{ \rho^{-1} \nabla p \biggr\}</math>
  </td>
  <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>

<tr>
  <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>
  </td>
</tr>

<tr>
  <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>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, Appendix B, p. 239, Eq. (B.2)
  </td>
</tr>

<tr>
  <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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <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>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, Appendix B, p. 240, Eq. (B.3)
  </td>
</tr>

<tr>
  <td align="right">
&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>
  </td>
</tr>
<tr>
  <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>
  </font></td>
</tr>
</table>

<table border="0" cellpadding="3" align="center" width="80%">

<tr>
  <td align="right">
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>
  </td>
  <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>
  </td>
</tr>

<tr>
  <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>
  </td>
</tr>

<tr>
  <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>
  </td>
</tr>

<tr>
  <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>
  </td>
</tr>
<tr>
  <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>
  </font></td>
</tr>
</table>
Adding these two "RHS" terms together gives,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
RHS
  </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\}
+
\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>
  </td>
</tr>
</table>
That is to say,

<table border="0" cellpadding="3" align="center">

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

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>L\boldsymbol\xi</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <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>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, Appendix B, p. 240, Eq. (B.4)
  </td>
</tr>
</table>

<font color="red">Are our four comments correct?</font>

===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,
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="center" colspan="3">
<font color="#770000">'''Steady-State Flow<br />as viewed from a Rotating Reference Frame'''</font> 
  </td>
</tr>
<tr>
  <td align="right">
<math> 
\mathbf\nabla \{ \Phi_\mathrm{L89} + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho^{-1} \nabla p + (\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u} \, .</math>
  </td>
</tr>
</table>
Hence, the operator,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>L\boldsymbol\xi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <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\{ 
\rho^{-1} \nabla p + (\mathbf{u} \cdot \nabla)\mathbf{u} + 2\boldsymbol\omega \boldsymbol\times \mathbf{u} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ Lebovitz89b }}, &sect;2, p. 227, Eq. (4)
  </td>
</tr>
</table>
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,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>L\boldsymbol\xi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ LL96hereafter }}, &sect;3.1, p. 703, Eq. (17)
  </td>
</tr>
</table>

===Summary===

Finally, setting LHS = RHS, we have,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<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>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \rho^{-1} 
\nabla ( \Delta p )
+ L\boldsymbol\xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \rho^{-1} 
\nabla ( \Delta p )
+ 
\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>
  </td>
</tr>
</table>

Following {{ Lebovitz89ahereafter }}, this may be rewritten as,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math> 0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\xi}_{tt} +
\underbrace{ 
2\{ (\mathbf{u}\cdot \nabla)\boldsymbol\xi_t 
+ \boldsymbol\omega \boldsymbol\times \boldsymbol\xi_t \} 
}_{M {\boldsymbol\xi}_t}
+ 
\underbrace{
\{ (\mathbf{u}\cdot \nabla)^2\boldsymbol\xi 
+ 2\boldsymbol\omega \boldsymbol\times [(\mathbf{u}\cdot \nabla)\boldsymbol\xi ] \} 
-
\overbrace{
\{\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} 
]\}
}^{L\boldsymbol\xi}
}_{\Lambda \boldsymbol\xi}
+ \rho^{-1} 
\nabla ( \Delta p )
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\boldsymbol{\xi}_{tt} +
M {\boldsymbol\xi}_t + \Lambda \boldsymbol\xi + \rho^{-1} \nabla ( \Delta p ) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ Lebovitz89ahereafter }}, &sect;2, p. 224, Eq. (10)
  </td>
</tr>
</table>
where,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>M\boldsymbol\xi</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
2\{ (\mathbf{u}\cdot \nabla)\boldsymbol\xi 
+ \boldsymbol\omega \boldsymbol\times \boldsymbol\xi \} \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\Lambda\boldsymbol\xi</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\{ (\mathbf{u}\cdot \nabla)^2\boldsymbol\xi 
+ 2\boldsymbol\omega \boldsymbol\times [(\mathbf{u}\cdot \nabla)\boldsymbol\xi ] \} 
- L \boldsymbol\xi \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="7">
{{ Lebovitz89ahereafter }}, &sect;2, p. 224, Eq. (11)<br />
{{ Lebovitz89bhereafter }}, &sect;2, p. 227, Eq. (2)
  </td>
</tr>
</table>


----


Note that in {{ LL96hereafter }}, <font color="green">"the basic equation"</font> appears in the form,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>0</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\boldsymbol{\xi}_{tt} +
A {\boldsymbol\xi}_t + B \boldsymbol\xi + \rho^{-1} \nabla ( \Delta p ) \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ LL96hereafter }}, &sect;3.1, p. 701, Eq. (10)
  </td>
</tr>
</table>
This means that the matrix operators, <math>M</math> &amp; <math>\Lambda</math>, found in {{ Lebovitz89b }} and re-derived herein, have simply been renamed in {{ LL96hereafter }}.  That is to say, in {{ LL96hereafter }},

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>A\boldsymbol\xi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\{ (\mathbf{u}\cdot \nabla)\boldsymbol\xi 
+ \boldsymbol\omega \boldsymbol\times \boldsymbol\xi \} \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>B\boldsymbol\xi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\{ (\mathbf{u}\cdot \nabla)^2\boldsymbol\xi 
+ 2\boldsymbol\omega \boldsymbol\times [(\mathbf{u}\cdot \nabla)\boldsymbol\xi ] \} 
- L \boldsymbol\xi \, ,
</math>
  </td>
</tr>
</table>
where,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>L \boldsymbol\xi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ LL96hereafter }}, &sect;3.1, p. 703, Eq. (17)
  </td>
</tr>
</table>
They also point out that, after adopting the shorthand notation,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>D</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\mathbf{u}\cdot \nabla \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\Omega \boldsymbol\xi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\boldsymbol\omega \times \boldsymbol\xi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ LL96hereafter }}, &sect;3.1, p. 703, Eq. (15)
  </td>
</tr>
</table>
the matrix operator, <math>B</math>, can be rewritten as,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>B</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
D^2 + 2\Omega D - L \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ LL96hereafter }}, &sect;3.1, p. 703, Eq. (16)
  </td>
</tr>
</table>

=See Also=
<ul>
<li>A library of ellipsoidal harmonics may be available from [https://en.wikipedia.org/wiki/George_Dassios George Dassios]</li>
<li>{{ Lebovitz61full }} titled, ''The Virial Tensor and Its Application to Self-Gravitating Fluids''</li>
<li>{{ LBO67full }} titled, ''On the Stability of Differentially Rotating Bodies''</li>
<li>{{ Lebovitz74full }} titled, ''The Fission Theory of Binary Stars. II. Stability to Third-Harmonics Disturbances''</li>
<li>{{ Lebovitz89afull }} titled, ''The Stability Equations for Rotating, Inviscid Fluids: &nbsp; Galerkin Methods and Orthogonal Bases''</li>
<li>{{ Lebovitz89bfull }} titled, ''Lagrangian Perturbations of Riemann Ellipsoids''</li>
<li>{{ LL96full }} titled, ''New Global Instabilities of the Riemann Ellipsoids''</li>
<li>{{ Ou2006 }}</li>
<li>{{ OTM2007full }}, ''Further Evidence for an Elliptical Instability in Rotating Fluid Bars and Ellipsoidal Stars.''</li>
</ul>


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