Editing Apps/Korycansky Papaloizou 1996 (section)

=Suggested Strategy by Joel=
For the time being, I want to continue to discuss only planar flows.  Aside from the Poisson equation and the equation of state, the three key scalar equations are:
# Continuity equation;
# Z-component of the curl of the Euler equation; and
# Scalar product of <math>\vec{v}</math> and Euler.
Let's take one of Shangli's compressible Riemann ellipsoids as our starting guess.  We need to "tilt" each of the velocity vectors slightly so that they everywhere satisfy the compressible equation of state.  By combining the first two of the three key scalar equations, the KP96 study tells us that we can do this '''not''' by orienting them tangent to isodensity contours but, rather, by orienting them tangent to iso-vortensity surfaces.  That is, we need,
<div align="center">
<math>
\vec{v} \cdot \nabla\Lambda  = 0  .
</math>
</div>
where,
<div align="center">
<math>
\Lambda \equiv \ln\biggl[ \frac{(2\Omega + \zeta_z)}{\rho} \biggr]  .
</math>
</div>
Let's rely on this expression '''only''' to tell us how to orient each velocity vector, that is '''only''' to give us the ratio 
<div align="center">
<math>
f \equiv \frac{v_x}{v_y}
</math>
</div>
everywhere.  And let's assume that <math>\Omega</math>, <math>\rho(\vec{x})</math> and <math>\zeta_z(\vec{x})</math> are known from Shangli's model.  Then we can straightforwardly conclude that, at all spatial locations within the configuration,
<div align="center">
<math>
f(\vec{x}) = -\frac{\nabla_y(\Lambda)}{\nabla_x(\Lambda)} .
</math>
</div>
Everywhere along the ''surface'', <math>\nabla\Lambda</math> may not be well-determined; in which case I suspect that we should align the velocity vectors tangent to the surface.

Next, let's use ''either'' the continuity equation ''or'' the z-component of the curl of Euler to determine the updated '''magnitude''', <math>v_0(\vec{x})</math>, of each velocity vector.  Using the continuity equation, for example, we need,
<div align="center">
<math>
\nabla\cdot\vec{v} = -\vec{v}\cdot\nabla(\ln\rho) ,
</math>
</div>
where, in terms of the function <math>f(\vec{x})</math>,
<div align="center">
<math>
\vec{v} = v_0 \biggl[ \hat{i} f ( 1 + f^2 )^{-1/2} + \hat{j} ( 1 + f^2 )^{-1/2} \biggr] .
</math>
</div>
Let me emphasize that, during this step of our SCF iteration cycle, we should assume that the functions <math>\rho(\vec{x})</math> and <math>f(\vec{x})</math> are known throughout the configuration.  The continuity equation can therefore be written as,
<div align="center">
<math>
\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} = - \biggl[ v_x\frac{\partial}{\partial x} 
+ v_y \frac{\partial}{\partial y} \biggr] (\ln\rho) 
</math><br />

<math>
\Rightarrow ~~~~~
f \frac{\partial (\ln v_0)}{\partial x} + \frac{\partial (\ln v_0)}{\partial y} = 
- ( 1 + f^2 )^{1/2} \biggl[ \frac{\partial [f ( 1 + f^2 )^{-1/2}]}{\partial x}
+  \frac{\partial [( 1 + f^2 )^{-1/2}]}{\partial y} \biggr]
- \biggl[ f \frac{\partial}{\partial x} + \frac{\partial}{\partial y} \biggr] (\ln\rho) 
</math>
</div>