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 serves as the foundation of most of our [Ref03] discussions.

Exercising the Hybrid Scheme

Focus on Tracking Angular Momentum

Let's begin by using <math>~\mathbf{u'}</math>, instead of <math>~{\vec{v}}_\mathrm{rot}</math>, to represent the fluid velocity vector as viewed from the rotating frame of reference. Our foundation expression becomes,

<math>~\frac{d \bold{u'}}{dt} </math>

<math>~=</math>

<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec\Omega}_f \times \bold{u}' - {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) \, ,</math>

where we appreciate that we can move from the Lagrangian to an Eulerian representation by employing the operator substitution,

<math>~\frac{d}{dt}</math>

<math>~\rightarrow</math>

<math>~\frac{\partial}{\partial t} + \mathbf{u'} \cdot \nabla </math>

Next, using [Ref03] as a guide, let's focus on tracking angular momentum. We need to break the vector momentum equation, as well as the velocity vectors, into their <math>~(\bold{\hat{e}}_\varpi, \bold{\hat{e}}_\varphi, \bold{\hat{k}})</math> components.

NOTE: For the time being, we will write the velocity vector in terms of generic components, namely, <math>~\bold{u}' = \bold{\hat{e}}_\varpi u'_\varpi + \bold{\hat{e}}_\varphi u'_\varphi + \bold{\hat{k}}u'_z \, .</math> But, eventually, we want to explicitly insert the rotating-frame velocity that underpins the equilibrium properties of Riemann S-type ellipsoids. In Chap. 7, §47, Eq. 1 (p. 130) of [EFE], this is given in Cartesian coordinates, so we will need to convert his expressions to the equivalent cylindrical-coordinate components.

The time-derivative on the left-hand-side of our foundation expression becomes,

<math> \frac{d\mathbf{u'}}{dt} </math>	<math>~=~</math>	<math> \frac{d}{dt} [ \mathbf{\hat{e}}_\varpi u'_\varpi + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}} u'_z ] </math>
	<math>~=~</math>	<math> \mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt} + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt} + \mathbf{\hat{k}} \frac{d u'_z}{dt} + ( u'_\varpi) \frac{d}{dt}\mathbf{\hat{e}}_\varpi + ( u'_\varphi) \frac{d}{dt}\mathbf{\hat{e}}_\varphi </math>
	<math>~=~</math>	<math> \mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt} + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt} + \mathbf{\hat{k}} \frac{d u'_z}{dt} + \mathbf{\hat{e}}_\varphi(u'_\varpi) \frac{u'_\varphi}{\varpi} - \mathbf{\hat{e}}_\varpi(u'_\varphi) \frac{u'_\varphi}{\varpi} \, . </math>

We also recognize that, when expressed in cylindrical coordinates,

<math> ~{\vec{\Omega}}_f \times \vec{x} </math>	<math>~=~</math>	<math> {\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi \varpi + \mathbf{\hat{k}}z) = \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, , </math>
<math> {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) </math>	<math>~=~</math>	<math> \hat{\mathbf{k}} \Omega_f \times ( \mathbf{\hat{e}}_\varphi \Omega_f \varpi ) = - \mathbf{\hat{e}}_\varpi \Omega_f^2 \varpi \, , </math>
<math> {\vec{\Omega}}_f \times {\mathbf{u'}} </math>	<math>~=~</math>	<math> {\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi u'_\varpi + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}}u'_z) = \mathbf{\hat{e}}_\varphi \Omega_f u'_\varpi - \mathbf{\hat{e}}_\varpi \Omega_f u'_\varphi \, , </math>
<math> {\vec{v}}_\mathrm{inertial} </math>	<math>~=~</math>	<math> \mathbf{u'} + \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, . </math>

The set of scalar momentum-component equations is obtained by "dotting" each unit vector into the vector equation.

<math>\mathbf{\hat{e}}_\varpi:</math>	<math>~\frac{d u'_\varpi}{dt} - \frac{(u'_\varphi)^2}{\varpi} </math>	<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + 2 \biggl[ \Omega_f u'_\varphi \biggr] + \Omega_f^2 \varpi </math>
<math>~\Rightarrow ~~~ \frac{d u'_\varpi}{dt} </math>		<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + \frac{1}{\varpi} \biggl[ (u'_\varphi)^2 + 2 \Omega_f u'_\varphi \varpi + \Omega_f^2 \varpi^2 \biggr]</math>
		<math>~=</math>	<math>~ - \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + \frac{1}{\varpi} (u'_\varphi + \Omega_f \varpi)^2 \, ; </math>
<math>\mathbf{\hat{e}}_\varphi:</math>	<math>~\frac{d u'_\varphi}{dt} + \frac{u'_\varpi u'_\varphi}{\varpi} </math>	<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \nabla \Phi - 2\biggl[ \Omega_f u'_\varpi \biggr] </math>
(mult. thru by ϖ)   <math>~\Rightarrow ~~~\frac{d (\varpi u'_\varphi )}{dt} </math>		<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi - 2 \Omega_f \varpi u'_\varpi \, ; </math>
<math>\mathbf{\hat{k}}:</math>	<math>~\frac{d u'_z}{dt} </math>	<math>~=</math>	<math>~- \mathbf{\hat{k}} \cdot \frac{\nabla P }{\rho} - \mathbf{\hat{k}} \cdot \nabla \Phi \, . </math>

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