Editing Apps/Korycansky Papaloizou 1996

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=Korycansky and Papaloizou (1996)=
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! style="height: 125px; width: 125px; background-color:lightgreen;" |[[H_BookTiledMenu#Nonlinear_Dynamical_Evolution_2|<b>Constructing<br />Infinitesimally Thin<br />Nonaxisymmetric<br />Disks</b>]]
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{{ KP96full }} &#8212; hereafter, {{ KP96hereafter }} &#8212; developed a method to find nontrivial, nonaxisymmetric steady-state flows in a <i>two-dimensional</i> setting.  Specifically, they constructed infinitesimally thin steady-state disk structures in the presence of a time-independent, nonaxisymmetric perturbing potential.  While their problem was only two-dimensional and they did not seek a self-consistent solution of the [[PGE/PoissonOrigin|gravitational Poisson equation]], the approach they took to solving the 2D Euler equation in tandem with the continuity equation for a <i>compressible</i> fluid is instructive.  What follows is a summary of their approach.
&nbsp;<br />
&nbsp;<br />

==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.

==Two-Dimensional Planar Flow==

In keeping with their objective to study steady-state flows in infinitesimally thin disks, {{ KP96hereafter }} imposed one additional important constraint on the set of governing equations.  

<font color="darkblue">Assumption #3:</font> {{ KP96hereafter }} set <math>v_z = 0</math> everywhere.

Then, in order to determine the steady-state spatial distribution of the three principal physical variables <math>\rho(x,y)</math>, <math>v_x(x,y)</math>, and <math>v_y(x,y)</math>, they looked for solutions that would simultaneously satisfy the following three PDEs:
* The z-component of the curl of the steady-state Euler equation, that is,
<div align="center">
<math>
[\nabla\times\vec{A}]_z =  0 ;
</math>
</div>
* The steady-state continuity equation, that is, 
<div align="center">
<math>
\vec{v}\cdot\nabla\rho + \rho\nabla\cdot\vec{v} = 0 ; 
</math>
</div>
and,
* The scalar product of <math>\vec{v}</math> with the steady-state Euler equation, that is,
<div align="center">
<math>
\vec{v} \cdot \nabla F_B  = 0 .
</math>
</div>
The algebraic equation of state that they used to supplement this coupled set of governing PDEs is identified in their paper, in the discussion associated with their Eq. (23).  

Drawing from our [[PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product|accompanying discussion]] of how the curl of <math>\vec{A}</math> behaves when <math>v_z = 0</math>, the first of these PDEs takes the form,
<div align="center">
<math>
[\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x  \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] = 0 
</math><br />

<math>
\Rightarrow ~~~~~\vec{v}\cdot\nabla(\zeta_z + 2\Omega) + (\zeta_z + 2\Omega)\nabla\cdot\vec{v} = 0 .
</math>
</div>
This last expression appears as Eq. (2) in {{ KP96hereafter }}.  (For this last expression to be valid it must be understood that, for the inherently 2D problem under investigation by {{ KP96hereafter }}, <math>\nabla</math> is only operating in x and y.) Rewriting this equation, we conclude that,
<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>
But from the steady-state continuity equation we 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, combining the two relations, we conclude that,
<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 {{ KP96hereafter }}'s Eq. (3).  It tells us that, in the steady-state flow whose spatial structure we are seeking, the velocity vector (and also the momentum density vector <math>\rho\vec{v}</math>) must everywhere be tangent to contours of constant <math>(2\Omega + \zeta_z)/\rho</math> &#8212; a scalar quantity that {{ KP96hereafter }} refer to as ''<font color="red">vortensity</font>.''  

<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.  {{ KP96hereafter }} 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 {{ KP96hereafter }}):
<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>

Since, by design, streamlines defined by the momentum-density vector field must trace out lines of constant <math>\Psi</math> and, according to the conclusion drawn above, the momentum density vector must everywhere be tangent to contours of constant ''vortensity'', we can conclude &#8212; as did {{ KP96hereafter }} &#8212; that the ''vortensity'' <math>(\zeta_z + 2\Omega)/\rho</math> must be constant along streamlines.  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>

By the same token, the condition obtained from the scalar product of <math>\vec{v}</math> with the steady-state Euler equation,
<div align="center">
<math>
\vec{v} \cdot \nabla F_B  = 0 ,
</math>
</div>
implies that the Bernoulli function must also be expressible as a function of <math>\Psi</math> alone.  Hence, we can write,
<div align="center">
<math>
\frac{1}{2}v^2 + H + \Phi - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2  = F_B(\Psi) .
</math>
</div>
{{ KP96hereafter }} 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.

==Summary==
In summary, {{ KP96full }} constrain their flow as follows:
# They use the z-component of the curl of the Euler equation;
# They use the compressible version of the continuity equation;
# Instead of taking the divergence of the Euler equation to obtain a Poisson-like equation, they obtain an algebraic constraint on the Bernoulli function (as in our traditional SCF technique) by simply "dotting" <math>\vec{v}</math> into the Euler equation.

=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>

=See Also=
<ul>
  <li>[https://repository.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998, Ph.D. Dissertation, Louisiana State University)], ''The Structure and Stability of Selected, 2-D Self-Gravitating Systems''</li>
</ul>

<table border="1" align="center">
<tr>
  <td align="center" colspan="8" cellpadding="8">
[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Andalib] velocity maps
  </td>
</tr>
<tr>
  <td align="center">Filename:</td>
  <td align="center">vel05</td>
  <td align="center">vel12</td>
  <td align="center">vel14</td>
  <td align="center">vel20</td>
  <td align="center">vel30</td>
  <td align="center">vel36</td>
  <td align="center">vel43</td>
</tr>
<tr>
  <td align="center">Thumbnail:</td>
<td align="center">[[File:Andalib.vel05Small.jpeg|100px|Andalib.vel05]]</td>
<td align="center">[[File:Andalib.vel12Small.jpeg|100px|Andalib.vel12]]</td>
<td align="center">[[File:Andalib.vel14Small.jpeg|100px|Andalib.vel14]]</td>
<td align="center">[[File:Andalib.vel20Small.jpeg|100px|Andalib.vel20]]</td>
<td align="center">[[File:Andalib.vel30Small.jpeg|100px|Andalib.vel30]]</td>
<td align="center">[[File:Andalib.vel36Small.jpeg|100px|Andalib.vel36]]</td>
<td align="center">[[File:Andalib.vel43Small.jpeg|100px|Andalib.vel43]]</td>
</tr>
<tr>
  <td align="center">Model. No.</td>
  <td align="center">??</td>
  <td align="center">??</td>
  <td align="center">D13</td>
  <td align="center">D14</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">C13(?)</td>
</tr>
<tr>
  <td align="center">Fig. No.</td>
  <td align="center">??</td>
  <td align="center">??</td>
  <td align="center">5.16</td>
  <td align="center">5.15</td>
  <td align="center">5.14</td>
  <td align="center">5.13</td>
  <td align="center">5.6(?)</td>
</tr>
</table>


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