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>

Now, recalling that <math>~\mathbf{u'} = (\mathbf{v} - \mathbf{\hat{e}}_\varphi \varpi \Omega_f)</math>, let's make the substitutions …

<math>~u'_\varpi \rightarrow v_\varpi \, ,</math>

<math>~u'_\varphi \rightarrow (v_\varphi - \varpi\Omega_f) \, ,</math>      and,

<math>~u'_z \rightarrow v_z \, .</math>

This mapping gives,

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

Steady-State Velocity Field for Jacobi Ellipsoids

In steady-state, the (Lagrangian time-derivative) operator on the left-hand-side of all three component equations maps to the following operator:

<math>~\mathbf{u'} \cdot \nabla</math>	<math>~=</math>	<math>~\sum_{i=1}^3 u'_i \frac{\partial}{\partial x_i} \, ,</math>	(in Cartesian coordinates);
<math>~\mathbf{u'} \cdot \nabla</math>	<math>~=</math>	<math>~ u'_\varpi \frac{\partial}{\partial \varpi} + \frac{u'_\varphi}{\varpi} \frac{\partial}{\partial \varphi} + u'_z \frac{\partial}{\partial z} \, ,</math>	(in cylindrical coordinates);

We know, as well, that,

<math>~u'_\varpi = u'_x \cos\varphi + u'_y \sin\varphi \, ,</math>

and,

<math>~u'_\varphi = u'_y \cos\varphi - u'_x \sin\varphi \, .</math>

Hence, the cylindrical-coordinate-based operator may be rewritten as,

<math>~\mathbf{u'} \cdot \nabla</math>

<math>~=</math>

<math>~ ( u'_x \cos\varphi + u'_y \sin\varphi ) \frac{\partial}{\partial \varpi} + ( u'_y \cos\varphi - u'_x \sin\varphi )\frac{1}{\varpi} \frac{\partial}{\partial \varphi} + u'_z \frac{\partial}{\partial z} \, .</math>

Drawing from [ Ref04 ] … As Ou(2006) has pointed out, the velocity field of a Riemann S-type ellipsoid as viewed from a frame rotating with angular velocity <math>~{\vec{\Omega}}_f = \boldsymbol{\hat{k}} \Omega_f</math> takes the following form:

<math>~{\mathbf{u'}}</math>

<math>~=</math>

<math>~\lambda \biggl[ \boldsymbol{\hat{\imath}} \biggl(\frac{a}{b}\biggr)y - \boldsymbol{\hat{\jmath}} \biggl(\frac{b}{a}\biggr)x \biggr] \, ,</math>

Ou(2006), p. 550, §2, Eq. (3)

where <math>~\lambda</math> is a constant that determines the magnitude of the internal motion of the fluid, and the origin of the x-y coordinate system is at the center of the ellipsoid. This velocity field, <math>~\mathbf{u'}</math>, is designed so that velocity vectors everywhere are always aligned with elliptical stream lines by demanding that they be tangent to the equi-effective-potential contours, which are concentric ellipses. Hence, for Riemann S-type ellipsoids, we have,

<math>~u'_x = \lambda\biggl(\frac{a}{b}\biggr)y = \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \, ;</math>

<math>~u'_y = -\lambda\biggl(\frac{b}{a}\biggr)x = -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \, ;</math>

<math>~u'_z = 0 \, .</math>

So, for the velocity flow that underpins Riemann S-type ellipsoids, the cylindrical-coordinate-based operator is

<math>~\mathbf{u'} \cdot \nabla</math>	<math>~=</math>	<math>~ \biggl[ \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \cos\varphi -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \sin\varphi \biggr] \frac{\partial}{\partial \varpi} + \biggl[ -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \cos\varphi - \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \sin\varphi \biggr] \frac{1}{\varpi} \frac{\partial}{\partial \varphi} </math>
	<math>~=</math>	<math>~ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi} \, .</math>

And, given that,

<math>~\mathbf{\hat{e}}\Omega_f \varpi</math>

<math>~=</math>

<math>~ \Omega_f \varpi \biggl[ \boldsymbol{\hat{\jmath}} \cos\varphi - \boldsymbol{\hat{\imath}} \sin\varphi \biggr] </math>

the inertial-frame velocity components are,

<math>~ v_x = \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi - \Omega_f \varpi \sin\varphi = \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin\varphi \, ;</math>

<math>~ v_y = -\lambda\biggl(\frac{b}{a}\biggr) \varpi\cos\varphi + \Omega_f\varpi \cos\varphi = \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos\varphi \, ;</math>

<math>~v_z = 0 \, .</math>

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation