Editing Apps/RiemannEllipsoidsCompressible (section)

===Solution Strategy===

<font color="darkblue"><b>Constraint #1:</b></font>
For their two-dimensional disk problem, KP96 focused on the constraint provided by the z-component of the curl of the Euler equation, which can be rewritten as (see above derivation, or Eq. 2 of KP96),
<div align="center">
<math>
\nabla\cdot\vec{v} =-\vec{v} \cdot \biggl[ \frac{\nabla(2\omega + \zeta_z)}{(2\omega + \zeta_z)} \biggr] 
= -\vec{v} \cdot \nabla[\ln(2\omega + \zeta_z)].
</math>
</div>
<font color="darkblue"><b>Constraint #2:</b></font>
But from the continuity equation they also know that,
<div align="center">
<math>
\nabla\cdot\vec{v} = -\vec{v}\cdot\biggl[\frac{\nabla\rho}{\rho} \biggr] = -\vec{v} \cdot \nabla[\ln\rho] .
</math>
</div>
Hence,
<div align="center">
<math>
\vec{v} \cdot \nabla[\ln(2\omega + \zeta_z)]  = \vec{v} \cdot \nabla[\ln\rho] ,
</math>
</div>
that is,
<div align="center">
<math>
\vec{v} \cdot \nabla\ln\biggl[ \frac{(2\omega + \zeta_z)}{\rho} \biggr]  = 0  .
</math>
</div>
This is essentially KP96's Eq. (3).

<font color="darkblue"><b>Introduce stream function:</b></font>
The constraint implied by the continuity equation also suggests that it might be useful to define a stream function in terms of the momentum density &#8212; instead of in terms of just the velocity, which is the natural treatment in the context of incompressible fluid flows.  KP96 do this. They define the stream function, <math>\Psi</math>, such that (see their Eq. 4),
<div align="center">
<math>
\rho\vec{v} = \nabla\times(\hat{k}\Psi)  .
</math>
</div>
in which case,
<div align="center">
<math>
v_x = \frac{1}{\rho} \frac{\partial \Psi}{\partial y} ~~~~~\mathrm{and}~~~~~  
v_y = - \frac{1}{\rho} \frac{\partial \Psi}{\partial x} .
</math>
</div>
This implies as well that the z-component of the fluid vorticity can be expressed in terms
of the stream function as follows (see Eq. 5 of KP96):
<div align="center">
<math>
\zeta_z = - \nabla\cdot \biggl( \frac{\nabla\Psi}{\rho} \biggr) = - \frac{\partial}{\partial x} \biggl[ \frac{1}{\rho} \frac{\partial\Psi}{\partial x} \biggr] - \frac{\partial}{\partial y} \biggl[ \frac{1}{\rho} \frac{\partial\Psi}{\partial y} \biggr].
</math>
</div>

According to KP96, this expression, taken in combination with the conclusion drawn above from the second constraint &#8212; that is, Eq. (3) taken in combination with Eq. (4) from KP96 &#8212; "tell us that the 'vortensity' <math>(\zeta_z + 2\omega)/\rho</math> is constant along streamlines which are lines of constant <math>\Psi</math>."  The vortensity is therefore a function of <math>\Psi</math> alone, so we can write,
<div align="center">
<math>
\frac{\zeta_z + 2\omega}{\rho} = g(\Psi) .
</math>
</div>


<font color="darkblue"><b>Constraint #3:</b></font>

Taking the scalar product of <math>\vec{v}</math> and the following form of the steady-state Euler equation, 
<div align="center">
<math>
2\omega\hat{k}\times\vec{v} - \vec{v}\times(\nabla\times\vec{v}) + \nabla \biggl[\frac{1}{2}v^2 + H + \Phi -\frac{1}{2}\omega^2 R^2  \biggr] = 0 ,
</math>
</div>
we obtain the constraint,
<div align="center">
<math>
\vec{v}\cdot\nabla \biggl[\frac{1}{2}v^2 + H + \Phi -\frac{1}{2}\omega^2 R^2  \biggr] = 0 .
</math>
</div>
When tied with our earlier discussion, this means that the Bernoulli function also must be constant along streamlines.  Hence, we can write,
<div align="center">
<math>
\frac{1}{2}v^2 + H + \Phi -\frac{1}{2}\omega^2 R^2  = F(\Psi) .
</math>
</div>
KP96 then go on to demonstrate that the relationship between the functions <math>g(\Psi)</math> and <math>F(\Psi)</math> is,
<div align="center">
<math>
\frac{dF}{d\Psi} = -g(\Psi) ,
</math>
</div>
which allows the determination of <math>F</math> up to a constant of integration.