ThreeDimensionalConfigurations/Stability/RiemannEllipsoids

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Lebovitz & Lifschitz (1996)

Lebovitz & Lifschitz
(1996)

Here we review the work of 📚 N. R. Lebovitz, & A. Lifschitz (1996, ApJ, Vol. 458, pp. 699 - 713) titled, "New Global Instabilities of the Riemann Ellipsoids," and discuss various extensions that have been made to this work. Note that a good summary of the research efforts that preceded (and inspired) the work of 📚 Lebovitz & Lifschitz (1996) can be found in the introductory section of 📚 S. Ou, J. E. Tohline, & P. M. Motl (2007, ApJ, Vol. 665, pp. 1074 - 1083).

We were prompted to tackle this review in response to an email received in December 2021 from Howard Cohl.  
 

Background

In Figure 1, the abscissa is the ratio of semiaxes in the equatorial plane, and the ordinate is the ratio of the vertical semiaxis to the larger of the equatorial semi axes. This diagram shows what 📚 Lebovitz & Lifschitz (1996) — hereafter, LL96 — refer to as "the horn-shaped region of existence of S-type ellipsoids and the Jacobi family;" it underpins all four panels of the LL96 Figure 2.

Figure 1: The Horn-Shaped Region of S-type Ellipsoids

EFE Diagram02

  • Jacobi sequence — the smooth curve that runs through the set of small, dark-blue, diamond-shaped markers; the data identifying the location of these markers have been drawn from §39, Table IV of [EFE]. The small red circular markers lie along this same sequence; their locations are taken from our own determinations, as detailed in Table 2 of our accompanying discussion of Jacobi ellipsoids. All of the models along this sequence have and are therefore solid-body rotators, that is, there is no internal motion when the configuration is viewed from a frame that is rotating with frequency, .
  • Dedekind sequence — a smooth curve that lies precisely on top of the Jacobi sequence. Each configuration along this sequence is adjoint to a model on the Jacobi sequence that shares its (b/a, c/a) axis-ratio pair. All ellipsoidal figures along this sequence have and are therefore stationary as viewed from the inertial frame; the angular momentum of each configuration is stored in its internal motion (vorticity).
  • The X = -1 self-adjoint sequence — At every point along this sequence, the value of the key frequency ratio, , in the adjoint configuration is identical to the value of the frequency ratio in the direct configuration ; specifically, . The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [EFE].
  • The X = +1 self-adjoint sequence — At every point along this sequence, the value of the key frequency ratio, , in the adjoint configuration is identical to the value of the frequency ratio in the direct configuration ; specifically, . The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [EFE].

EFE Diagram identifying example models from Ou (2006)

Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram. The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by 📚 S. Ou (2006, ApJ, Vol. 639, pp. 549 - 558) and that we have also chosen to use as examples.

Four example models of equilibrium Riemann S-Type ellipsoids (click each parameter-pair to go to a related chapter discussion):

 
 
 

Self-Adjoint Sequences

What are the expressions that define the upper and lower boundaries of the horned shaped region of equilibrium S-Type Riemann Ellipsoids? Well, as we have discussed in an associated chapter, the value of the parameter, , that is associated with each point within the horned shaped region is given by the expression,

📚 Lebovitz & Lifschitz (1996), §2, Eq. (5)

where,

📚 Lebovitz & Lifschitz (1996), §2, Eq. (6)

[ EFE, §21, Eq. (107) ]

[ EFE, §21, Eq. (105) ]
See also the note immediately following §21, Eq. (127)

Upper Boundary

The upper boundary of the horn-shaped region is obtained by setting . That is, it is associated with coordinate pairs for which,

 

 

Now, from the expressions for A1, A2, and A3, we can furthermore write,

 

where, and are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

      and      

[ EFE, Chapter 3, §17, Eq. (32) ]

STRATEGY for finding the locus of points that define the upper boundary of the horned-shape region …    Set , and pick a value for ; then, using an iterative technique, vary until the following expression is satisfied:

Choose another value of , then iterate again to find the value of that corresponds to this new, chosen value of . Repeat!

Lower Boundary

Similarly, the lower boundary is obtained by setting , that is, it is associated with coordinate pairs for which,

 

 

 

Now, from the expressions for A1, A2, and A3, we can furthermore write,

STRATEGY for finding the locus of points that define the lower boundary of the horned-shape region …    Set , and pick a value for ; then, using an iterative technique, vary until the following expression is satisfied:

Choose another value of , then iterate again to find the value of that corresponds to this new, chosen value of . Repeat!

Stability Equations

Strategy

"Let represent the velocity field, pressure, and density, respectively, of an inviscid fluid mass in a steady state relative to a reference frame rotating with angular velocity about an axis fixed in space (the z-, or x3-, axis) … The stability of this steady state is determined, in linear approximation, by the solutions, with arbitrary initial data, of the … equation [governing the time-dependent behavior of] the Lagrangian displacement ."

— Drawn from the first paragraph of §2 (p. 226) in 📚 Lebovitz (1989b).

"This basic equation [is of the form],"     … Eq. (10).

— Drawn from the first paragraph of §3.1 (p. 701) in 📚 Lebovitz & Lifschitz (1996).

"We introduce for the solution space a basis the first vectors of which represent a basis for , the space of solenoidal vector polynomials of degree not exceeding , as in L89a, L89b. It is easily found (see L89a) that . Since is invariant under the operators and , we seek solutions of Eq. (10) in this space:"

… Eq. (18)

— Drawn from the first paragraph of §3.2 (p. 703) in 📚 Lebovitz & Lifschitz (1996).

Here we will closely follow the derivation found in 📚 N. R. Lebovitz (1989a, Geophysical & Astrophysical Fluid Dynamics, Vol. 46:4, pp. 221 - 243), hereafter L89a.

Euler Equation

From our initial overarching presentation of the principal governing equation, we draw an expression for the,

Lagrangian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

Moving the term that accounts for the Coriolis acceleration to the left-hand side of this expression, and realizing that the centrifugal acceleration may be rewritten in the form,

Centrifugal Acceleration

the Euler equation becomes,

Except for the adopted sign convention for the gravitational potential, , this precisely matches Equation (2) of L89a, namely,

N. R. Lebovitz (1989a)
The Stability Equations for Rotating, Inviscid Fluids:   Galerkin Methods and Orthogonal Bases
Geophysical & Astrophysical Fluid Dynamics, Vol. 46:4, pp. 221 - 243

📚 Lebovitz (1989a), §2, p. 223, Eq. (2)
📚 Lebovitz & Lifschitz (1996b), §2, p. 929, Eq. (2.1)

In what follows, we will adopt the L89a variable notation.

Steady-State Unperturbed Flows

As we have discussed in a much broader context, the so-called Lagrangian (or "material") time derivative, , that appears on the left-hand side of this Lagrangian representation of the Euler equation can be replaced by its Eulerian counterpart, , via the operator relation,

LBO67, §1, p. 294, Eq. (4)

Furthermore, if our unperturbed fluid configuration is in steady-state, this will be reflected in the Euler equation by setting, , that is,

in which case the following relation holds:

Steady-State Flow
as viewed from a Rotating Reference Frame

This relationship between structural variables in the context of steady-state unperturbed flows will be used below.

Lagrangian Displacement and Linearization

Suppose that, at time , the function set properly describes the properties of a — as yet unspecified — geometrically extended, fluid configuration. According to the Euler equation and, in particular, as dictated by the flow-field,

after an interval of time, , each "Lagrangian" fluid element will move from its initial location, , to a new position, . In general each Lagrangian fluid element will discover that, at its new coordinate location, the "environment" is different. For example,

With this in mind, L89a introduces a Lagrangian-change operator, , in order to mathematically indicate that this evolutionary step is being executed for any physical variable, . Specifically,

L89a, §2, p. 223, Eq. (3)
LBO67, p. 293, Eq. (1)

Following L89a and applying the operator, , to each side of the Euler equation, we can write,

LHS

With the assurance provided by L89a that commutes with the Lagrangian time-derivative, — see also the paragraph immediately preceding Eq. (4) in LBO67 — and that

L89a, §2, p. 223, Eq. (4)

we can immediately appreciate that,

LHS

=

Hence, we obtain the (still, exact nonlinear),

(Lagrangian) Perturbed Euler Equation

L89a, §2, p. 223, Eq. (5)

For later reference, notice that the LHS may further be rewritten as,

LHS

 

 

where we have adopted L89a's shorthand notation,

    and,    

Finally, if … the unperturbed solution … is steady — as is the case in the context of our study of the stability of Riemann S-Type ellipsoids (see more, below) — then commutes with the Eulerian time-derivative, that is,

which means we may write,

LHS

L89a, §2, p. 224, immediately preceding Eq. (10)

RHS

Next, L89a introduces the Eulerian-change operator, (which commutes with ),

L89a, §2, p. 224, Eq. (6)
LBO67, p. 293, Eq. (2)

Without immediate proof, L89a states that the relationship between the Lagranian-change operator and the Eulerian-change operator is, to lowest order (linear),

L89a, §2, p. 224, Eq. (7)
LBO67, p. 294, Eq. (3)

Introducing this mapping into the right-hand side of the perturbed Euler equation gives:

1st term on RHS

 

 

L89a, Appendix B, p. 239, Eq. (B.2)

 

 

L89a, Appendix B, p. 240, Eq. (B.3)

 

Comments:

  1. In order to move from the 2nd to the 3rd line of this derivation, it seems that L89a employs the relation:   This relation strongly resembles the continuity equation which, in Lagrangian form, is
  2. In order to move from the 2nd to the 3rd line of this derivation, L89a seems to be acknowledging that, commutes with
  3. A typographical error appears in Eq. (B.3) of L89a; appears in the publication whereas, as noted here in the fifth line of this derivation, the term should be .

2nd term on RHS

 

 

 

Comment:

  1. A term in the 2nd row of this derivation goes to zero because there is no Eulerian variation in either of the vectors, or .

Adding these two "RHS" terms together gives,

RHS

That is to say,

RHS

L89a, §2, p. 224, Eq. (8)

where the operator, , is defined such that,

L89a, Appendix B, p. 240, Eq. (B.4)

Are our four comments correct?

Specifically Perturb Riemann S-Type Ellipsoids

Now let's assume that the initial equilibrium configuration is a steady-state, Riemann S-Type ellipsoid. Then, from above, we know that,

Steady-State Flow
as viewed from a Rotating Reference Frame

Hence, the operator,

 

 

 

📚 Lebovitz (1989b), §2, p. 227, Eq. (4)

where, following the lead of 📚 Lebovitz (1989b), the term containing has been set to zero because, throughout a Riemann ellipsoid, "… the unperturbed density is spatially uniform …"

In addition, following the lead of 📚 Lebovitz & Lifschitz (1996), "… we consider here the incompressible case and therefore adjoin to [the perturbed Euler equation] the expression of mass conservation …" namely,


 
LL96, §3.1, p. 703, Eq. (13)

Hence,

LL96, §3.1, p. 703, Eq. (17)

Summary

Finally, setting LHS = RHS, we have,

 

Following L89a, this may be rewritten as,

 

L89a, §2, p. 224, Eq. (10)

where,

      and,      

L89a, §2, p. 224, Eq. (11)
L89b, §2, p. 227, Eq. (2)




Note that in LL96, "the basic equation" appears in the form,

LL96, §3.1, p. 701, Eq. (10)

This means that the matrix operators, & , found in 📚 Lebovitz (1989b) and re-derived herein, have simply been renamed in LL96. That is to say, in LL96,

      and,      

where,

LL96, §3.1, p. 703, Eq. (17)

They also point out that, after adopting the shorthand notation,

      and,      

LL96, §3.1, p. 703, Eq. (15)

the matrix operator, , can be rewritten as,

LL96, §3.1, p. 703, Eq. (16)

See Also


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