User:Tohline/Appendix/Ramblings/Hybrid Scheme Implications
Implications of Hybrid Scheme
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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,
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<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 …
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<math>~\vec{v}_\mathrm{inertial}</math> |
<math>~=</math> |
<math>~\vec{v}_\mathrm{rot} + {\vec\Omega}_f \times \vec{x} \, ,</math> |
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<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> |
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<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> |
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<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
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<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> |
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[EFE], Chap. 2, §12, p. 25, Eq. (62) |
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which also serves 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 …
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<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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© 2014 - 2021 by Joel E. Tohline |