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 [ | 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 & Papaloizou] (1996, ApJS, 105, 181; hereafter KP96). | ||
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Revision as of 22:00, 25 August 2020
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).
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