Editing Appendix/Ramblings/ForCohlHoward (section)

=Lebovitz and Lifschitz=
==Invitation==

<table border="1" align="center" width="65%" cellpadding="8">
<tr>
<td align="left">
On 8 May 2023, Stephen Sorokanich sent the following email to Joel:

"<font color="darkgreen">Dr. Cohl and I are wondering if you would like to join our discussion of &hellip;"</font>
<div align="center">{{ LL96bfigure }}</div>

"<font color="darkgreen">This paper is very interesting to us because the authors consider Eulerian perturbations to the steady-state S-type ellipsoid velocity and pressure fields. The resulting linear equations in the Eulerian frame seem ideal for us to compare with our current finite element simulations of the nonlinear system. This is the only paper I've found by Lebovitz and Lifschitz where the stability analysis is carried out in an Eulerian frame. 

There's some nice analysis in the paper too. The linearized equations are solved using the "geometrical-optics approximation," which reduces the 3-dimensional linearized Euler equations to a system of ordinary differential equations. These solutions are used to draw conclusions for the stability of the S-type ellipsoids. Maybe there are other approximations to this system?</font>"

NOTE from Joel:  The referenced paper is obviously related to the companion publication &hellip;
<div align="center">{{ LL96figure }}</div>We have already prepared a [[ThreeDimensionalConfigurations/Stability/RiemannEllipsoids#Euler_Equation|MediaWiki chapter]] whose focus is on this "{{ LL96hereafter }}" publication.

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

==WKB (geometrical-optics) Method==

As Stephen has pointed out, in {{ LL96b }}, the linearized equations are solved using the "geometrical-optics approximation."  I am not familiar with this so-called "geometrical-optics" method, but in the last couple of paragraphs of &sect;1 of their paper, {{ LL96bhereafter }} refer to it also as the "WKB" approximation/method.  I ''have'' been exposed to this alternate terminology; the WKB approximation has been used by astronomers in connection with efforts to understand the onset and development of "spiral-arm" structures in disk galaxies.  Here is an excerpt from [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>]. 

<table border="1" align="center" width="80%" cellpadding="8">
<tr><td align="left">
<div align="center">'''The Tight-Winding Approximation'''<br />&sect;6.2.2 (p. 352) of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] </div>
<font color="darkgreen">"&hellip; In the early 1960s a number of workers, notably A. J. Kalnajs, C. C. Lin, and A. Toomre, realized that for tightly wound density waves (''i.e.,'' waves whose radial wavelength is much less than the radius</font> [of the galaxy's disk]) <font color="darkgreen">the long-range coupling</font> [due to gravity] <font color="darkgreen">is negligible, the response is determined locally, and the relevant solutions are analytic.  As we shall see, this '''tight-winding''' or '''WKB approximation'''<sup>&dagger;</sup> is an indispensable tool for understanding the origin and evolution of density waves in galaxies."

----

<sup>&dagger;</sup>Named  after the closely related [https://en.wikipedia.org/wiki/WKB_approximation Wentzel-Kramers-Brillouin approximation] of quantum mechanics.</font>
</td></tr></table>

==Governing Linearized Equations==
In our [[ThreeDimensionalConfigurations/Stability/RiemannEllipsoids#EulerRotating|accompanying discussion]] we present a derivation of the &hellip;

<table border="1" align="center" cellpadding="8" width="80%">
<tr><td align="center" bgcolor="lightgreen">{{ Lebovitz89afigure }}</td></tr>
<tr><td align="left">
<div align="center">
<font color="#770000">'''Mixed Lagrangian/Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>
</div>
<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)
  </td>
</tr>
</table>
</td></tr>
</table>

When rewritten with an alternate set of variable notations &#8212; specifically, <math>(\mathbf{u}, p, \Phi_\mathrm{L96}) \rightarrow (\mathbf{V}, P, \Psi)</math> &#8212; and recognizing that <math>D/Dt = (\partial/\partial t + \mathbf{V}\cdot \nabla)</math>, this key relation becomes,
<table border="1" align="center" cellpadding="8" width="80%">
<tr><td align="center" bgcolor="yellow">{{ LL96bfigure }}</td></tr>
<tr><td align="left">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>\biggl[\frac{\partial}{\partial t} + \mathbf{V}\cdot \nabla\biggr]\mathbf{V} + 2\boldsymbol\omega \boldsymbol\times \mathbf{V}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ \rho^{-1} \nabla P + \mathbf\nabla \{ \Psi + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \}
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LL96b }}, &sect;2, p. 929, Eq. (2.1)
  </td>
</tr>
</table>
</td></tr>
</table>

Next, following {{ LL96bhereafter }}, we let each variable be represented by the sum of an unperturbed, steady-state component (subscript zero) and a small perturbation,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\mathbf{V}</math></td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="left"><math>\mathbf{V}_0 + a_0(\pi G \rho_0)^{1 / 2}\mathbf{u} \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\rho</math></td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="left"><math>\rho_0 </math> &nbsp; &nbsp; &nbsp; (''i.e.,'' remains unchanged)</td>
</tr>

<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="left"><math>P_0 + (\pi G \rho_0^2 a_0^2) p \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\Psi</math></td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="left"><math>\Psi_0 + (\pi G a_0^2 \rho_0)\psi \, ,</math></td>
</tr>
</table>

where <math>a_0</math> is a characteristic length scale.  As a result, we have,
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{\partial}{\partial t} + [\mathbf{V}_0 + a_0(\pi G \rho_0)^{1 / 2}\mathbf{u}]\cdot 
\nabla\biggr][\mathbf{V}_0 + a_0(\pi G \rho_0)^{1 / 2}\mathbf{u}] 
+ 2\boldsymbol\omega \boldsymbol\times [\mathbf{V}_0 + a_0(\pi G \rho_0)^{1 / 2}\mathbf{u}]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ \rho_0^{-1} \nabla [P_0 + (\pi G \rho_0^2 a_0^2)p] + \mathbf\nabla \{ [\Psi_0 + (\pi G \rho_0^2 a_0)\psi ] + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \}
\, .
</math>
  </td>
</tr>
</table>

Then, subtracting off the unperturbed steady-state component, namely,
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>(\mathbf{V}_0 \cdot \nabla) \mathbf{V}_0 + 2\boldsymbol\omega \boldsymbol\times \mathbf{V}_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ \rho_0^{-1} \nabla P_0 + \mathbf\nabla \{ \Psi_0 + \tfrac{1}{2} |\boldsymbol\omega \boldsymbol\times \mathbf{x}|^2 \}
\, ,
</math>
  </td>
</tr>
</table>
we obtain,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>
a_0(\pi G\rho_0)^{1 / 2}
\frac{\partial \mathbf{u}}{\partial t}  
+
\biggl[ \mathbf{V}_0 \cdot \nabla\biggr]a_0(\pi G\rho_0)^{1 / 2}\mathbf{u} 
+
\biggl[ a_0(\pi G\rho_0)^{1 / 2}\mathbf{u}\cdot \nabla\biggr]\mathbf{V}_0 
+
(\pi G \rho_0 a_0^2)
\cancelto{\mathrm{small}}{\biggl[ \mathbf{u}\cdot \nabla\biggr]\mathbf{u} }
+ 2\boldsymbol\omega \boldsymbol\times \mathbf{u} (a_0^2 \pi G\rho_0)^{1 / 2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ (\pi G \rho_0^2 a_0^2)\rho_0^{-1} \nabla p + (\pi G \rho_0 a_0^2)\mathbf\nabla \psi
\, .
</math>
  </td>
</tr>
</table>
Finally, dividing thru by <math>(\pi G \rho_0 a_0)</math>, 
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>
(\pi G\rho_0)^{-1 / 2}
\frac{\partial \mathbf{u}}{\partial t}  
+
\biggl[ \mathbf{V}_0 \cdot \nabla\biggr](\pi G\rho_0)^{-1 / 2}\mathbf{u} 
+
\biggl[ (\pi G\rho_0)^{-1 / 2}\mathbf{u}\cdot \nabla\biggr]\mathbf{V}_0 
+ 2\boldsymbol\omega \boldsymbol\times \mathbf{u} (\pi G\rho_0)^{-1 / 2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ a_0 \nabla p + a_0\mathbf\nabla \psi
\, ,
</math>
  </td>
</tr>
</table>
and adopting the characteristic timescale, <math>t_0 \equiv (\pi G \rho_0)^{-1 / 2}</math>, we have the dimensionless Euler equation,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>
t_0 \underbrace{\biggl[
\frac{\partial }{\partial t}  
+
\mathbf{V}_0 \cdot \nabla \biggr]}_{D/Dt}\mathbf{u} 
+
t_0 \overbrace{[ \mathbf{u}\cdot \nabla ]\mathbf{V}_0}^{L\mathbf{u}} 
+ 2t_0 ~\overbrace{\boldsymbol\omega \boldsymbol\times \mathbf{u}}^{\Omega\mathbf{u} }</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
-~ a_0 \nabla p + a_0\mathbf\nabla \psi
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LL96b }}, &sect;2, p. 929, Eqs. (2.4) &amp; (2.7)
  </td>
</tr>
</table>