Editing ThreeDimensionalConfigurations/RiemannStype (section)

====Adopted Velocity Flow-Field====

As {{ Ou2006 }} has pointed out [text that is taken directly from that publication appears here in an orange-colored font], <font color="orange">the velocity field of a Riemann S-type ellipsoid as viewed from a frame rotating with angular velocity <math>~{\vec{\Omega}}_f = \boldsymbol{\hat{k}} \Omega_f</math> takes the following form:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\vec{v}}_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda \biggl[ \boldsymbol{\hat{\imath}} \biggl(\frac{a}{b}\biggr)y - \boldsymbol{\hat{\jmath}} \biggl(\frac{b}{a}\biggr)x \biggr]  \, ,</math>
  </td>
</tr>
</table>

{{ LL96 }}, p. 700, &sect;2, Eq. (1)<br />
{{ Ou2006 }}, p. 550, &sect;2, Eq. (3)
</div>


<font color="orange">where <math>~\lambda</math> is a constant that determines the magnitude of the internal motion of the fluid, and the origin of the x-y coordinate system is at the center of the ellipsoid.  This velocity field <math>~{\vec{v}}_\mathrm{rot}</math> is designed so that velocity vectors everywhere are always aligned with elliptical stream lines by demanding that they be tangent to the</font> equi-''effective''-potential <font color="orange"> contours, which are concentric ellipses.</font>

<table border="1" align="center" cellpadding="10" width="65%"><tr><td align="left">
Plugging Ou's expression for  <math>~{\vec{v}}_\mathrm{rot}</math> into the expression on the left-hand side of the steady-state Euler equation, we see that for Riemann S-type ellipsoids,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~({\vec{v}}_\mathrm{rot} \cdot \nabla){\vec{v}}_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \lambda\biggl(\frac{a}{b}\biggr)y \frac{\partial}{\partial x} - \lambda\biggl(\frac{b}{a}\biggr)x \frac{\partial}{\partial y} \biggr] 
\biggl[\boldsymbol{\hat{\imath}}\lambda\biggl(\frac{a}{b}\biggr)y - \boldsymbol{\hat{\jmath}} \lambda\biggl( \frac{b}{a}\biggr)x \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \boldsymbol{\hat{\jmath}} \biggl[ \lambda\biggl(\frac{a}{b}\biggr)y\biggr] \frac{\partial}{\partial x}  
\biggl[ \lambda\biggl( \frac{b}{a}\biggr)x \biggr]
- \boldsymbol{\hat{\imath}}
\biggl[ \lambda\biggl(\frac{b}{a}\biggr)x\biggr] \frac{\partial}{\partial y}  
\biggl[\lambda\biggl(\frac{a}{b}\biggr)y \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\lambda^2\biggl[ \boldsymbol{\hat{\imath}} x + \boldsymbol{\hat{\jmath}} y \biggr] 
=
-\nabla\biggl[\frac{1}{2}\lambda^2(x^2 + y^2)  \biggr] \, .
</math>
  </td>
</tr>
</table>


----

Alternatively, from a [[PGE/Euler#in_terms_of_the_vorticity:|separate discussion of vector identities]] we realize that,
<div align="center">
<math>
(\vec{v}\cdot\nabla)\vec{v} = \frac{1}{2}\nabla(\vec{v}\cdot \vec{v}) + \vec{\zeta}\times \vec{v} ,
</math>
</div>
where, <math>\vec\zeta \equiv \nabla\times\vec{v}</math> is the fluid vorticity.  Plugging in Ou's expression for <math>~{\vec{v}}_\mathrm{rot}</math>, we find that &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec{\zeta} = \nabla\times {\vec{v}}_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\boldsymbol{\hat{k}} \lambda \biggl[ \frac{b}{a} + \frac{a}{b} \biggr] \, ;</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
{{ Ou2006 }}, p. 551, &sect;2, Eq. (17)
</td></tr>
<tr>
  <td align="right">
<math>~\vec{\zeta} \times {\vec{v}}_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\lambda^2\biggl[\boldsymbol{\hat{\jmath}} \biggl(1 + \frac{a^2}{b^2}\biggr)y + \boldsymbol{\hat{\imath}} \biggl(1 + \frac{b^2}{a^2}\biggr)x \biggr] \, ; </math> &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{2}\nabla( {\vec{v}}_\mathrm{rot} \cdot  {\vec{v}}_\mathrm{rot} )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda^2\biggl[ \boldsymbol{\hat{\imath}} \biggl(\frac{b}{a}\biggr)^2x + \boldsymbol{\hat{\jmath}} \biggl(\frac{a}{b}\biggr)^2y  \biggr] \, .</math>
  </td>
</tr>
</table>

Hence, we again appreciate that, for Riemann S-type ellipsoids,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~( {\vec{v}}_\mathrm{rot} \cdot \nabla){\vec{v}}_\mathrm{rot} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\lambda^2\biggl[ \boldsymbol{\hat{\imath}} x + \boldsymbol{\hat{\jmath}} y \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\nabla\biggl[\frac{1}{2}\lambda^2(x^2 + y^2)  \biggr] \, .</math>
  </td>
</tr>
</table>

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


The steady-state Euler-equation specification therefore becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-\nabla\biggl[\frac{1}{2} \lambda^2(x^2 + y^2)  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \nabla\biggl[ H + \Phi_\mathrm{grav} - \frac{1}{2} \Omega_f^2(x^2 + y^2)\biggr] - \nabla\biggl[\Omega_f \lambda \biggl(\frac{b}{a}x^2 + \frac{a}{b}y^2 \biggr)  \biggr] \, .
</math>
  </td>
</tr>
</table>

{{ Ou2006 }}, p. 550, &sect;2, Eq. (5)
</div>

<font color="orange">Hence, within the configuration the following Bernoulli's function must be uniform in space:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
H + \Phi_\mathrm{grav} - \frac{1}{2} \Omega_f^2(x^2 + y^2) 
- \frac{1}{2} \lambda^2(x^2 + y^2)  
+ \Omega_f \lambda \biggl(\frac{b}{a}x^2 + \frac{a}{b}y^2 \biggr)  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
C_B \, ,
</math>
  </td>
</tr>
</table>

{{ Ou2006 }}, p. 550, &sect;2, Eq. (6)
</div>

<font color="orange">where <math>~C_B</math> is a constant.</font>  It is customary to define an effective potential which is the sum of the gravitational potential and the system's centrifugal potential (as viewed from the rotating frame), namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_\mathrm{eff} \equiv \Phi_\mathrm{grav} + \Psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Phi_\mathrm{grav} - \frac{1}{2} \Omega_f^2(x^2 + y^2) 
- \frac{1}{2} \lambda^2(x^2 + y^2)  
+ \Omega_f \lambda \biggl(\frac{b}{a}x^2 + \frac{a}{b}y^2 \biggr) \, ,
</math>
  </td>
</tr>
</table>

{{ Ou2006 }}, p. 550, &sect;2, Eq. (7)
</div>
in which case the statement of detailed force balance in Riemann S-type ellipsoids can be rewritten in the following deceptively simpler form:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~H + \Phi_\mathrm{eff}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_B \, .</math>
  </td>
</tr>
</table>

{{ Ou2006 }}, p. 550, &sect;2, Eq. (8)
</div>