ThreeDimensionalConfigurations/Stability/RiemannEllipsoids

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 $b/a$ of semiaxes in the equatorial plane, and the ordinate is the ratio $c/a$ 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

• 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 $f\equiv \zeta /{\mathrm{\Omega }}_f=0$ 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, ${\mathrm{\Omega }}_f$.
• 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 $1/f={\mathrm{\Omega }}_f/\zeta =0$ 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, $\zeta /{\mathrm{\Omega }}_f$, in the adjoint configuration $(f_{+})$ is identical to the value of the frequency ratio in the direct configuration $(f_{-})$; specifically, $f_{+}=f_{-}=-(a^2+b^2)/(ab)$. 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, $\zeta /{\mathrm{\Omega }}_f$, in the adjoint configuration $(f_{+})$ is identical to the value of the frequency ratio in the direct configuration $(f_{-})$; specifically, $f_{+}=f_{-}=+(a^2+b^2)/(ab)$. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [EFE].

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):

• (b/a, c/a) = (0.41, 0.385); the chapter name is "Riemann Meets COLLADA & Oculus Rift S"
• (b/a, c/a) = (0.90, 0.333)
• (b/a, c/a) = (0.74, 0.692)
• (b/a, c/a) = (0.28, 0.256)

 
 
 

Self-Adjoint Sequences

What are the expressions that define the upper $(x=-1)$ and lower $(x=+1)$ 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, $x$, that is associated with each point $(b/a,c/a)$ within the horned shaped region is given by the expression,

$1+2Cx+x^2$	$=$	$0,$
📚 Lebovitz & Lifschitz (1996), §2, Eq. (5)

where,

$C$	$=$	$\left[\frac{ab{B}_{12}}{c^2{A}_3-a^2{b}^2{A}_{12}}\right],$
📚 Lebovitz & Lifschitz (1996), §2, Eq. (6)

$A_{12}$	$\equiv$	$-\frac{A_1-A_2}{(a^2-b^2)},$
[ EFE, §21, Eq. (107) ]
$B_{12}$	$\equiv$	$A_2-a^2{A}_{12}.$
[ 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 $x=-1$. That is, it is associated with coordinate pairs $(b/a,c/a)$ for which,

$1-2C+1$	$=$	$0$
$\Rightarrow C$	$=$	$+1$
$\Rightarrow \left[\frac{ab{B}_{12}}{c^2{A}_3-a^2{b}^2{A}_{12}}\right]$	$=$	$+1$
$\Rightarrow ab{B}_{12}$	$=$	$c^2{A}_3-a^2{b}^2{A}_{12}$
$\Rightarrow c^2{A}_3$	$=$	$ab[A_2-a^2{A}_{12}]+a^2{b}^2{A}_{12}$
	$=$	$ab{A}_2+b{a}^2{A}_{12}(b-a)$
	$=$	$ab{A}_2+b{a}^2(a-b)\left[\frac{A_1-A_2}{a^2-b^2}\right]$
$\Rightarrow \left[\frac{c^2}{ab}\right]A_3$	$=$	$A_2+a\left[\frac{A_1-A_2}{a+b}\right]$
$\Rightarrow \left[\frac{c^2(a+b)}{ab}\right]A_3$	$=$	$a{A}_1+b{A}_2.$

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

$c^2(a+b)A_3$	$=$	$a^{2}b{A}_1+a{b}^2[2-(A_1+A_3)]$
	$=$	$a^{2}b{A}_1+2a{b}^2-a{b}^2{A}_1-a{b}^2{A}_3$
$\Rightarrow c^2(a+b)A_3+a{b}^2{A}_3$	$=$	$2a{b}^2+a^{2}b{A}_1-a{b}^2{A}_1$
$\Rightarrow \frac{a}{b}[c^2(a+b)+a{b}^2]A_3$	$=$	$2a^{2}b+a^2(a-b)A_1$
$\Rightarrow [c^2(a+b)+a{b}^2]\left[\frac{(b/a)\sin \theta -(c/a)E(\theta ,k)}{(1-k^2){\sin }^3\theta }\right]$	$=$	$a^{2}b+bc(a-b)\left\{\right[\frac{F(\theta ,k)-E(\theta ,k)}{k^2{\sin }^3\theta }\left]\right\},$

where, $F(\theta ,k)$ and $E(\theta ,k)$ are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

$\theta ={\cos }^{-1}\left(\frac{c}{a}\right)$	and	$k=[\frac{1-(b/a{)}^2}{1-(c/a{)}^2}{]}^{1/2}.$
[ EFE, Chapter 3, §17, Eq. (32) ]

STRATEGY for finding the locus of points that define the upper boundary of the horned-shape region …    Set $a=1$, and pick a value for $0<b<1$; then, using an iterative technique, vary $c$ until the following expression is satisfied: $[c^2(a+b)+a{b}^2]\left[\frac{(b/a)\sin \theta -(c/a)E(\theta ,k)}{(1-k^2){\sin }^3\theta }\right]$ $=$ $a^{2}b+bc(a-b)\left[\frac{F(\theta ,k)-E(\theta ,k)}{k^2{\sin }^3\theta }\right].$ Choose another value of $0<b<1$, then iterate again to find the value of $c$ that corresponds to this new, chosen value of $b$. Repeat!

Lower Boundary

Similarly, the lower boundary is obtained by setting $x=+1$, that is, it is associated with coordinate pairs $(b/a,c/a)$ for which,

$C$	$=$	$-1$
$\Rightarrow \left[\frac{ab{B}_{12}}{c^2{A}_3-a^2{b}^2{A}_{12}}\right]$	$=$	$-1$
$\Rightarrow -ab{B}_{12}$	$=$	$c^2{A}_3-a^2{b}^2{A}_{12}$
$\Rightarrow c^2{A}_3$	$=$	$-ab[A_2-a^2{A}_{12}]+a^2{b}^2{A}_{12}$
	$=$	$-ab{A}_2+b{a}^2{A}_{12}(b+a)$
	$=$	$-ab{A}_2-b{a}^2(a+b)\left[\frac{A_1-A_2}{a^2-b^2}\right]$
$\Rightarrow \left[\frac{c^2}{ab}\right]A_3$	$=$	$-A_2-a\left[\frac{A_1-A_2}{a-b}\right]$
$\Rightarrow \left[\frac{c^2(a-b)}{ab}\right]A_3$	$=$	$A_2(b-a)-a{A}_1+a{A}_2$
	$=$	$b{A}_2-a{A}_1.$

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

$c^2(a-b)A_3$	$=$	$2a{b}^2-a{b}^2{A}_1-a{b}^2{A}_3-a^{2}b{A}_1$
$\Rightarrow c^2(a-b)A_3+a{b}^2{A}_3$	$=$	$2a{b}^2-ab(b+a)A_1$
$\Rightarrow \frac{a}{b}[c^2(a-b)+a{b}^2]A_3$	$=$	$a^2[2b-(b+a)A_1]$
$\Rightarrow [c^2(a-b)+a{b}^2]\left[\frac{(b/a)\sin \theta -(c/a)E(\theta ,k)}{(1-k^2){\sin }^3\theta }\right]$	$=$	$a^{2}b-bc(b+a)\left[\frac{F(\theta ,k)-E(\theta ,k)}{k^2{\sin }^3\theta }\right].$

STRATEGY for finding the locus of points that define the lower boundary of the horned-shape region …    Set $a=1$, and pick a value for $0<b<1$; then, using an iterative technique, vary $c$ until the following expression is satisfied: $[c^2(a-b)+a{b}^2]\left[\frac{(b/a)\sin \theta -(c/a)E(\theta ,k)}{(1-k^2){\sin }^3\theta }\right]$ $=$ $a^{2}b-bc(b+a)\left[\frac{F(\theta ,k)-E(\theta ,k)}{k^2{\sin }^3\theta }\right].$ Choose another value of $0<b<1$, then iterate again to find the value of $c$ that corresponds to this new, chosen value of $b$. Repeat!

Stability Equations

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

$[\frac{d\overrightarrow{v}}{dt}{]}_{rot}=-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }\mathrm{\Phi }\underset{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}}{\underbrace{-2{\overrightarrow{\mathrm{\Omega }}}_f\times {\overrightarrow{v}}_{rot}}}\underset{\mathrm{C}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{l}}{\underbrace{-{\overrightarrow{\mathrm{\Omega }}}_f\times ({\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x})}}.$

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

${\overrightarrow{a}}_{\mathrm{C}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{l}}\equiv -{\overrightarrow{\mathrm{\Omega }}}_f\times ({\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x})=\frac{1}{2}\mathrm{\nabla }[|{\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x}{|}^2],$

the Euler equation becomes,

$[\frac{d\overrightarrow{v}}{dt}{]}_{rot}+2{\overrightarrow{\mathrm{\Omega }}}_f\times {\overrightarrow{v}}_{rot}$

$=$

$-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }\mathrm{\Phi }+\frac{1}{2}\mathrm{\nabla }[|{\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x}{|}^2].$

Except for the adopted sign convention for the gravitational potential, $\mathrm{\Phi }\leftrightarrow -{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}$, 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

$\frac{D\mathbf{𝐮}}{dt}+2\mathit{\omega }\mathbf{\times }\mathbf{𝐮}$ $=$ $-{\rho }^{-1}\mathrm{\nabla }\mathbf{𝐩}+\mathrm{\nabla }\{{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathbf{\times }\mathbf{𝐱}{|}^2\}.$ L89a, §2, p. 223, Eq. (2)

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

Lagrangian Displacement and Linearization

Suppose that, at time $t=0$, the function set $[{\mathbf{𝐮}}_0(\mathbf{𝐱}),{\rho }_0(\mathbf{𝐱}),p_0(\mathbf{𝐱})]$ 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 ${\mathbf{𝐮}}_0(\mathbf{𝐱})$, after an interval of time, $t$, each "Lagrangian" fluid element will move from its initial location, $\mathbf{𝐱}$, to a new position, $\mathbf{𝐱}+\mathit{\xi }$. In general each Lagrangian fluid element will discover that, at its new coordinate location, the "environment" is different. Specifically,

$p_0(\mathbf{𝐱})$	$\to$	$p(\mathbf{𝐱}+\mathit{\xi },t),$
${\rho }_0(\mathbf{𝐱})$	$\to$	$\rho (\mathbf{𝐱}+\mathit{\xi },t),$
${\mathbf{𝐮}}_0(\mathbf{𝐱})$	$\to$	$\mathbf{𝐮}(\mathbf{𝐱}+\mathit{\xi },t).$

With this in mind, L89a introduces a Lagrangian-change operator, $\mathrm{\Delta }$, in order to mathematically indicate that this evolutionary step is being executed for any physical variable, $F$; specifically,

$\mathrm{\Delta }F$	$=$	$F(\mathbf{𝐱}+\mathit{\xi },t)-F_0(\mathbf{𝐱}).$
L89a, §2, p. 223, Eq. (3)

After assuring us that $\mathrm{\Delta }$ commutes with the Lagrangian time-derivative, $D/Dt$, and that

$\mathrm{\Delta }\mathbf{𝐮}$	$=$	$\frac{D\xi }{Dt},$
L89a, §2, p. 223, Eq. (4)

L89a applies the operator, $\mathrm{\Delta }$, to each side of the Euler equation.

See Also

• A library of ellipsoidal harmonics may be available from George Dassios
• 📚 Ou (2006)
• 📚 S. Ou, J. E. Tohline, & P. M. Motl (2007, ApJ, Vol. 665, pp. 1074 - 1083), Further Evidence for an Elliptical Instability in Rotating Fluid Bars and Ellipsoidal Stars.
• 📚 N. R. Lebovitz (1974, ApJ, Vol. 190, pp. 121 - 130), titled, The Fission Theory of Binary Stars. II. Stability to Third-Harmonics Disturbances
• 📚 N. R. Lebovitz (1989a, Geophysical & Astrophysical Fluid Dynamics, Vol. 46:4, pp. 221 - 243) titled, The Stability Equations for Rotating, Inviscid Fluids:   Galerkin Methods and Orthogonal Bases
• 📚 N. R. Lebovitz (1989b, Geophysical & Astrophysical Fluid Dynamics, Vol. 47:1, pp. 225 - 236) titled, Lagrangian Perturbations of Riemann Ellipsoids