Editing ThreeDimensionalConfigurations/RiemannStype (section)

==Initial Thoughts==

Let's examine the elliptical trajectory of a Lagrangian particle that is moving in the equatorial plane of a Riemann S-Type ellipsoid. As viewed in a frame that is spinning about the Z-axis at angular frequency, <math>~\Omega</math>, the trajectory is defined by,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
r^2
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{x}{a} \biggr)^2 + \biggl(\frac{y}{b}\biggr)^2 \, ,
</math>
  </td>
</tr>

</table>
where <math>~0 < r \le 1</math>.  (The surface of the relevant ellipsoid is associated with the value, <math>~r=1</math>.)

Let's choose a pair of axis ratios &#8212; for example, <math>~b/a = 0.28</math> and <math>~c/a = 0.231</math> &#8212; then, from Table 1 of our [[#Models_Examined_by_Ou_.282006.29|above discussion]], draw the associated value of either <math>~\lambda</math> or <math>~\zeta</math> that corresponds to the Jacobi-like equilibrium configuration &#8212; in this example, <math>~\lambda = -0.04714</math> and <math>~\zeta = +0.18156</math>.  Then, for any point <math>~(x,y)</math> inside of the ellipsoid, the fluid's velocity components (as viewed from the rotating frame of reference) are,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
v_x  = \frac{dx}{dt}  = \lambda \biggl( \frac{ay}{b} \biggr) = -0.16836 ~y
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~
v_y  = \frac{dy}{dt}  = - \lambda \biggl( \frac{bx}{a} \biggr) = + 0.01320~x \, .
</math>
  </td>
</tr>
</table>

Alternatively, we have,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
u_x = \frac{dx}{dt} = Q_1  y = - \biggl[ 1 + \frac{b^2}{a^2} \biggr]^{-1}\zeta ~y = -0.16836 ~y
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~
u_y  = \frac{dy}{dt}= Q_2 x  = + \biggl[ 1 + \frac{a^2}{b^2} \biggr]^{-1}\zeta ~x = + 0.01320~x \, .
</math>
  </td>
</tr>
</table>

Now, each Lagrangian fluid element's motion is oscillatory in both the <math>~x</math> and <math>~y</math> coordinate directions.  So let's see how this plays out.  Suppose,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
x = x_\mathrm{max} \cos(\varphi t)
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~
y = y_\mathrm{max} \sin(\varphi t) \, .
</math>
  </td>
</tr>
</table>

Then,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{dx}{dt} = -  x_\mathrm{max}\varphi \sin(\varphi t) = -  \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr) \varphi y = - \varphi \biggl(\frac{ay}{b}\biggr)
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~
\frac{dy}{dt} = y_\mathrm{max} \varphi \cos(\varphi t) = + \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \varphi x = + \varphi \biggl(\frac{bx}{a}\biggr) \, .
</math>
  </td>
</tr>
</table>

Hence our functional representation of the time-dependent behavior of both <math>~x</math> and <math>~y</math> works perfectly if, for each orbit inside of or on the surface of the configuration, we set <math>~\varphi = - \lambda</math> and if the ratio <math>~y_\mathrm{max}/x_\mathrm{max} = (b/a)</math>.  Hooray!