User:Tohline/Appendix/Ramblings/Hybrid Scheme Implications

Implications of Hybrid Scheme

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,

<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>

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