Difference between revisions of "User:Tohline/Appendix/Ramblings/Hybrid Scheme Implications"

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This form of the Euler equation also appears early in [Ref05], where we set up a discussion of the paper by [http://adsabs.harvard.edu/abs/1996ApJS..105..181K Korycansky &amp; Papaloizou] (1996, ApJS, 105, 181; hereafter KP96).
This form of the Euler equation also appears early in [Ref05], where we set up a discussion of the paper by [http://adsabs.harvard.edu/abs/1996ApJS..105..181K Korycansky &amp; Papaloizou] (1996, ApJS, 105, 181; hereafter KP96).  But, for now, let's back up a couple of steps and retain the ''total'' time derivative on the left-hand-side.  That is, let's select as the foundation expression the,
<div align="center">
<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>
 
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi 
- 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot}
- {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x})  \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 2, &sect;12, p. 25, Eq. (62)
</td>
</tr>
</table>
</div>
which also served as the foundation of most of our [Ref03] discussions.


==Exercising the Hybrid Scheme==
==Exercising the Hybrid Scheme==

Revision as of 22:31, 26 August 2020

Implications of Hybrid Scheme

Whitworth's (1981) Isothermal Free-Energy Surface
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Background

Key H_Book Chapters

[Ref01]   Inertial-Frame Euler Equation

[Ref02]   Traditional Description of Rotating Reference Frame

[Ref03]   Hybrid Advection Scheme

[Ref04]   Riemann S-type Ellipsoids

[Ref05]   Korycansky and Papaloizou (1996)

Principal Governing Equations

Quoting from [Ref01] … Among the principal governing equations we have included the inertial-frame,

Lagrangian Representation
of the Euler Equation,

LSU Key.png

<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

[EFE], Chap. 2, §11, p. 20, Eq. (38)
[BLRY07], p. 13, Eq. (1.55)

Shifting into a rotating frame characterized by the angular velocity vector,

<math>~\vec{\Omega}_f \equiv \hat\mathbf{k} \Omega_f \, ,</math>

and applying the operations that are specified in the first few subsections of [Ref02], we recognize the following relationships …

<math>~\vec{v}_\mathrm{inertial}</math>

<math>~=</math>

<math>~\vec{v}_\mathrm{rot} + {\vec\Omega}_f \times \vec{x} \, ,</math>

<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{inertial}</math>

<math>~=</math>

<math>~ \biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} + {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) </math>

 

<math>~=</math>

<math>~ \biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2 </math>

 

<math>~=</math>

<math>~ \biggl[ \frac{\partial \vec{v}}{\partial t} \biggr]_\mathrm{rot} + ({\vec{v}}_\mathrm{rot} \cdot \nabla){\vec{v}}_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2 \, .</math>

Making this substitution on the left-hand-side of the above-specified "Lagrangian Representation of the Euler Equation," we obtain what we have referred to also in [Ref02] as the,

Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_\mathrm{rot} + ({\vec{v}}_\mathrm{rot}\cdot \nabla) {\vec{v}}_\mathrm{rot}= - \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}}_\mathrm{rot} \, .</math>

This form of the Euler equation also appears early in [Ref05], where we set up a discussion of the paper by Korycansky & Papaloizou (1996, ApJS, 105, 181; hereafter KP96). But, for now, let's back up a couple of steps and retain the total time derivative on the left-hand-side. That is, let's select as the foundation expression the,

Lagrangian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} </math>

<math>~=</math>

<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) \, ,</math>

[EFE], Chap. 2, §12, p. 25, Eq. (62)

which also served as the foundation of most of our [Ref03] discussions.

Exercising the Hybrid Scheme

Focusing on the advection term that appears on the left-hand-side of this last expression, let's replace the second reference to the rotating-frame velocity with its equivalent expression in terms of the inertial-frame velocity. That is, let's set …

<math>~({\vec{v}}_\mathrm{rot}\cdot \nabla) {\vec{v}}_\mathrm{rot}</math>

<math>~=</math>

<math>~ ({\vec{v}}_\mathrm{rot}\cdot \nabla) [\vec{v}_\mathrm{inertial} - {\vec\Omega}_f \times \vec{x} ]\, . </math>


Whitworth's (1981) Isothermal Free-Energy Surface

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