User:Tohline/Appendix/Ramblings/Hybrid Scheme Implications
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Implications of Hybrid Scheme
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Background
Key H_Book Chapters
[Ref01] InertialFrame Euler Equation
[Ref02] Traditional Description of Rotating Reference Frame
[Ref03] Hybrid Advection Scheme
[Ref04] Riemann Stype Ellipsoids
[Ref05] Korycansky and Papaloizou (1996)
Principal Governing Equations
Quoting from [Ref01] … Among the principal governing equations we have included the inertialframe,
Lagrangian Representation
of the Euler Equation,

[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,
and applying the operations that are specified in the first few subsections of [Ref02], we recognize the following relationships …












Making this substitution on the lefthandside of the abovespecified "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
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 lefthandside. That is, let's select as the foundation expression the,
Lagrangian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame



[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 , instead of , to represent the fluid velocity vector as viewed from the rotating frame of reference. Our foundation expression becomes,



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



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 components.
NOTE: For the time being, we will write the velocity vector in terms of generic components, namely,
But, eventually, we want to explicitly insert the rotatingframe velocity that underpins the equilibrium properties of Riemann Stype 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 cylindricalcoordinate components. 
The timederivative on the lefthandside of our foundation expression becomes,









We also recognize that, when expressed in cylindrical coordinates,












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
















(mult. thru by ϖ) 







Now, recalling that , let's make the substitutions …

and, 

This mapping gives,




















SteadyState Velocity Field for Jacobi Ellipsoids
In steadystate, the (Lagrangian timederivative) operator on the lefthandside of all three component equations maps to the following operator:



(in Cartesian coordinates); 



(in cylindrical coordinates); 
We know, as well, that,

and, 

Hence, the cylindricalcoordinatebased operator may be rewritten as,



Drawing from [ Ref04 ] … As Ou(2006) has pointed out, the velocity field of a Riemann Stype ellipsoid as viewed from a frame rotating with angular velocity takes the following form:



Ou(2006), p. 550, §2, Eq. (3)
where is a constant that determines the magnitude of the internal motion of the fluid, and the origin of the xy coordinate system is at the center of the ellipsoid. This velocity field, , is designed so that velocity vectors everywhere are always aligned with elliptical stream lines by demanding that they be tangent to the equieffectivepotential contours, which are concentric ellipses. Hence, for Riemann Stype ellipsoids, we have,



So, for the velocity flow that underpins Riemann Stype ellipsoids, the cylindricalcoordinatebased operator is






And, given that,



the inertialframe velocity components are,



That is,












Note, as well, that,






Finally, then, we find that the lefthandside of the momentumcomponent expressions are,







































Try Again
An example equation of motion is,
Here we will focus only on the lefthandside of this equation, examining various ways the vector acceleration may be mathematically expressed. We will consider, in particular, building a model in a curvilinear (cylindrical), rather than a Cartesian, coordinate base; and viewing the model's evolution in a rotating, rather than inertial, frame of reference.
Inertial Frame
As viewed from a cylindricalcoordinatebased inertial reference frame, we are interested in specifying the location,
[BT87], p. 646, Appendix §1.B.2, Eq. (1B18)
of a Lagrangian fluid element at time — hereafter denoted by the subscript, — as well as at later times. Although the position vector, , does not explicitly display a dependence on the azimuthal coordinate angle, , it is important to realize that the orientation in space of the unit vector, , does depend on the value of this coordinate angle.
At any point in time, the instantaneous velocity of this Lagrangian fluid element will correspond precisely with the (total) timederivative of its instantaneous position vector, that is,










[BT87], p. 647, Appendix §1.B.2, Eq. (1B23) 
In carrying out this time differentiation, the last term on the righthandside accounts for the aforementioned dependence of on . Similarly, the following component breakdown of the Lagrangian fluid element's acceleration takes into account the dependence of on :













[BT87], p. 647, Appendix §1.B.2, Eq. (1B24) 
Let's rewrite the velocity vector as,



and (the second line of) this acceleration expression as,

Now, if is a vector quantity that characterizes some property of a fluid element — such as momentum density, velocity, or vorticity — the difference between the Lagrangian and Eulerian timederivatives of that vector quantity is given by the expression,



where the various elements of this righthandside mathematical operator can be obtained by replacing with in the socalled convective operator.
Convective Operator in Cylindrical Coordinates
We will adopt the following, more compact notation:
where the operator,

In particular, if we are examining the behavior of the fluid velocity , we find that,





where the operator,



Notice that the pair of "curvature terms" that appear in this expression are identical to the pair of curvature terms that appear in the acceleration expression, above. We conclude, therefore, that for each of the three separate (cylindricalcoordinatebased) components of the vector acceleration, the relationship between the Lagrangian (total) and Eulerian (partial) time derivative is, respectively,
: 



: 



: 



Rotating Frame
Drawing from an accompanying discussion of rotating reference frames, let's build our model in a cylindrical coordinate system that is spinning about its axis with a timeindependent angular velocity, . Furthermore, let's use — instead of — to represent the velocity as viewed in the rotating frame. We know that,





and,

[BT87], p. 664, Appendix §1.D.3, Eq. (1D43) 
Note that, in the particular case being considered here,

We may therefore also write,





where the operator,



In numerical simulations that are carried out on a cylindrical grid and in a rotating reference frame, it is customary to group the "curvature terms" with the fictitious acceleration to obtain,












treating the ensemble as an additional "source" of acceleration.
Example from the Literature
(see an accompanying related discussion) We begin with the version of the Euler equation that has just been derived, namely,
and that, after multiplying the standard Lagrangian representation of the continuity equation through by , we have,
Radial Component: Multiplying the component of this Euler equation through by , gives,
Azimuthal Component: Multiplying the component of this Euler equation through by , gives,
Then, multiplying through by , gives,

Hybrid Scheme
In our hybrid scheme, we will continue to use — that is, an advection operator that incorporates the rotatingframe velocity, — but we will switch all other velocity references to the inertialframe velocity, , and its components. This will be done via the abovedeclared mapping,



that is, , , and . The Euler equation becomes,



where we recognize that,



Now, given that,



and,



the Euler equation becomes,






Also, if we multiply the standard Lagrangian representation of the continuity equation through by , we have,












Vertical Component: Multiplying the component of our modified Euler equation through by , then incorporating this version of the continuity equation, gives,






Radial Component: Multiplying the component of our modified Euler equation through by , then incorporating the continuity equation, gives,






Azimuthal Component: Multiplying the component of our modified Euler equation through by , then incorporating the continuity equation, gives,






Then, multiplying through by , we have,












Compare
Here we consider which formalism is best suited for modeling a fully threedimensional, nonaxisymmetric configuration that is spinning about (usually) its shortest axis with a uniform and timeindependent frequency and which, when viewed from a frame that is rotating with that frequency, exhibits a nontrivial but nevertheless steadystate internal flow. Examples are Riemann Stype ellipsoids, and binary stars in circular orbits.
It is most desirable to choose a formalism that recognizes the steadystate nature of the flow. In the vast majority of cases being considered here, this rules out using any scheme that is designed around an inertialframe coordinate base. (As a counterexample, Dedekind ellipsoids can be constructed in the inertial frame because for all models along the Dedekind equilibrium sequence.) It is quite reasonable, however, to adopt a rotating, cylindricalcoordinate base as has been described above and as is summarized immediately below.
Traditional Rotating, CylindricalCoordinate Summary

In this scheme, all of the velocities and associated momentum densities in all three components of the Euler equation are expressed in terms of the rotatingframe velocity vector, , or its cylindricalcoordinatebased components, . When the configuration's distorted (nonaxisymmetric) shape is largely supported by rapid rotation, this scheme provides an advantage over other — for example, inertialframebased — schemes because the fraction of the fluid's total momentum that is being advected across the grid is often quite small. There is a penalty to be paid, however. Additional "source" terms appear on the righthandside of the radial and azimuthalcomponent expressions; they are nonlinear in the velocity and introduce crosstalk between the component expressions.
Hybrid Scheme Summary

For Riemann Stype ellipsoids,












Hence,












And,















To within an additive constant — see, for example, our associated discussion of Maclaurin spheroids — the gravitational potential and the enthalpy are, respectively,






Hence,












Vertical Component: Because , it must be true that . This, in turn means,



Azimuthal Component: In steadystate, the partial timederivative must be zero, so we require,





















This expression can be used either (a) to give in terms of a set of known quantities and the unknown parameter, ; or (b) to give in terms of a set of known quantities and the unknown parameter, .
Recognizing from here that, and






this last expression can be rewritten as,



[ EFE, Chapter 7, §48, Eq. (34) ] 
Radial Component: In steadystate, this partial timederivative also must be zero, so we require,





















Also,






Hence,



















































[ EFE, Chapter 7, §48, Eq. (33) ] 
© 2014  2020 by Joel E. Tohline 