Editing Apps/Korycansky Papaloizou 1996 (section)

==Governing Steady-State Equations==

{{ KP96hereafter }} begin with the standard set of [[PGE|principal governing equations]], but choose to work from the set that is expressed in terms of a [[PGE/RotatingFrame#Rotating_Reference_Frame|rotating frame of reference]]. (Throughout the presentation on this page, it is to be understood that all variables are viewed from a rotating frame even though the subscript notation "rot" does not appear in the equations.)  Their Eq. (1), for example, comes from the

<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Continuity Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\frac{\partial\rho}{\partial t}  + \nabla \cdot (\rho {\vec{v}}) = 0</math> ,
</div>
and the,
<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\frac{\partial\vec{v}}{\partial t} + ({\vec{v}}\cdot \nabla) {\vec{v}} = - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] - 2{\vec{\Omega}}_f \times {\vec{v}} </math> .
</div>
And their Eq. (7) can be derived straightforwardly from the

<div align="center">
Euler Equation<br />
written <font color="#770000">'''in terms of the Vorticity'''</font> and<br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr] + ({\vec{\zeta}}+2{\vec\Omega}_f) \times {\vec{v}}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}v^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .
</div>


<font color="darkblue">Assumption #1:</font> {{ KP96hereafter }} align the angular velocity vector of the rotating frame of reference with the z-axis of a Cartesian coordinate system.  Specifically, they set 
<div align="center">
<math>{\vec{\Omega}}_f = \hat{k}\Omega</math>.  
</div>

<font color="darkblue">Assumption #2:</font> Because {{ KP96hereafter }} are seeking steady-state solutions, they set all Eulerian time-derivatives to zero.

Hence, the steady-state versions of the Euler and continuity equations shown above give rise to Eq. (1) of KP96, namely,
<div align="center">
<math>
(\vec{v}\cdot \nabla)\vec{v}  + 2\Omega\hat{k}\times\vec{v} + \frac{1}{\rho}\nabla P + \nabla \biggl[\Phi -\frac{1}{2}\omega^2 \varpi^2  \biggr] = 0 ,
</math>

<math>
\nabla\cdot(\rho \vec{v}) = \vec{v}\cdot\nabla\rho + \rho\nabla\cdot\vec{v} = 0 .
</math>
</div>
And, if written in terms of the vorticity, our steady-state Euler equation becomes essentially Eq. (7) of KP96, namely,
<div align="center">

<math>
0 = ({\vec{\zeta}}+2\Omega{\hat{k}}) \times {\vec{v}} + \frac{1}{\rho} \nabla P + \nabla \biggl[\Phi + \frac{1}{2}v^2 - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2 \biggr] 
</math><br />

<math>
= - {\vec{v}}\times({\vec{\zeta}}+2\Omega{\hat{k}})  +  \nabla \biggl[\frac{1}{2}v^2 + H + \Phi - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2 \biggr] ,
</math>
</div>

where, in this last expression, we have replaced the gradient of the pressure by the gradient of the [[SR#Barotropic_Structure|enthalpy]] via the relation, <math>\nabla H = \nabla P/\rho </math>.
Note that the {{ KP96hereafter }} 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>\hat{z}</math> is used instead of <math>\hat{k}</math> to denote a unit vector in the z-coordinate direction;
* the vorticity vector is written as <math>\hat{z}\omega</math> instead of <math>\vec\zeta</math>;
* <math>W</math> is used instead of {{Math/VAR_Enthalpy01}} to denote the enthalpy; and
* <math>\Phi_g</math> represents the combined, time-independent gravitational and centrifugal potential, that is, <math>\Phi_g = (\Phi - |\Omega\hat{k} \times \vec{x}|^2/2)</math>.

Up to this point, only the two assumptions itemized above have been imposed on the key governing equations.  Hence, although {{ KP96hereafter }} apply these equations to the study of a two-dimensional flow problem, our derived forms for the equations can serve to describe a fully 3D problem. 

Staying with this generalized approach, let's examine a few more aspects of these governing relations before focusing in on the more restrictive, 2D problem that has been tackled in {{ KP96hereafter }}.  First, let's rewrite the steady-state Euler equation in the form,
<div align="center">
<math>
\nabla F_B  + \vec{A} = 0 ,
</math>
</div>
where, the scalar "Bernoulli" function,
<div align="center">
<math>
F_B  \equiv  \frac{1}{2}v^2 + H + \Phi - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2 ;
</math>
</div>
and,
<div align="center">
<math>
\vec{A} \equiv ({\vec{\zeta}}+2{\vec\Omega}_f) \times {\vec{v}} ,
</math>
</div>
is the vector involving a nonlinear cross-product of the velocity that has been introduced in our accompanying [[PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product|general discussion of the Euler equation]] as viewed from a rotating frame of reference.

===Scalar Product of Velocity and Euler===
If we dot the vector <math>\vec{v}</math> into the steady-state Euler equation, we obtain the expression,
<div align="center">
<math>
\vec{v} \cdot \nabla F_B  = 0 .
</math>
</div>
The vector <math>\vec{A}</math> disappears as a result of the dot product with <math>\vec{v}</math> because <math>\vec{A}</math> is necessarily everywhere perpendicular to <math>\vec{v}</math>.

===Curl of Euler===

If, on the other hand, we take the curl of the steady-state Euler equation, we obtain the expression,
<div align="center">
<math>
\nabla\times\vec{A} = 0 .
</math>
</div>
In this case the gradient of the Bernoulli function disappears because the curl of any gradient is zero.  This vector equation provides three independent physical constraints on our system, as all three Cartesian components of the curl of <math>\vec{A}</math> must independently be zero.  Expressions for the three components of <math>\nabla\times\vec{A}</math> can be found in our accompanying [[PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product|general discussion of the Euler equation]] as viewed from a rotating frame of reference.