Editing Apps/RiemannEllipsoidsCompressible (section)

===Governing Steady-State Equations===

As in our above [http://www.vistrails.org/index.php/User:Tohline/PGE/RotatingFrame#Example_of_Riemann_S-Type_Ellipsoids preamble to discussion of Riemann S-Type Ellipsoids], KP96 set <math>{\vec{\omega}} = \hat{k}\omega</math>.  Hence, their steady-state Euler equation and steady-state continuity equation become (see their Eq. 1 or their Eq. 7),
<div align="center">
<math>
(\vec{v}\cdot \nabla)\vec{v}  + 2\omega\hat{k}\times\vec{v} + \nabla \biggl[H + \Phi -\frac{1}{2}\omega^2 R^2  \biggr] = 0 ,
</math>

<math>
\nabla\cdot(\rho \vec{v}) = \vec{v}\cdot\nabla\rho + \rho\nabla\cdot\vec{v} = 0 .
</math>
</div>
Note that the KP96 notation is slightly different from ours:
* <math>\Sigma</math> is used in place of <math>\rho</math> to denote a two-dimensional <i>surface</i> density;
* <math>\Omega</math> is used instead of <math>\omega</math> to denote the angular frequency of the rotating reference frame;
* <math>W</math> is used instead of <math>H</math> to denote the fluid enthalpy; and
* <math>\Phi_g</math> represents the combined, time-independent gravitational and centrifugal potential, that is, <math>\Phi_g = (\Phi - \omega^2 R^2/2)</math>.

Using the <font color="darkgreen">vector identity</font>, 
<div align="center">
<math>
(\vec{v}\cdot \nabla)\vec{v} = \frac{1}{2}\nabla(v^2) - \vec{v}\times(\nabla\times\vec{v}) ,
</math>
</div>
the above steady-state Euler equation can also be written as,
<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>
Up to this point, no assumptions have been made regarding the behavior of the vector flow-field; we have only chosen to align the <math>\vec{\omega}</math> with the coordinate unit vector, <math>\hat{k}</math>.  In particular, these derived forms for the steady-state Euler and continuity equations can serve to describe a fully 3D problem.

Before proceeding further we should emphasize that, in the context of the Euler equation written in this form (i.e., the form preferred by KP96), the vector <math>\vec{A}</math> defined in the preamble, above, should be written,
<div align="center">
<math>
\vec{A} = 2\omega\hat{k}\times\vec{v} +(\nabla\times\vec{v})\times\vec{v} + \nabla \biggl[\frac{1}{2}v^2 \biggr] .
</math>
</div>