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

Strategy

"Let $\mathbf{𝐮}(\mathbf{𝐱}),p(\mathbf{𝐱}),\rho (\mathbf{𝐱})$ 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 $\mathit{\omega }=\omega {\mathbf{𝐞}}_3$ 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 $\mathit{\xi }$."

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

"This basic equation [is of the form],"     ${\mathit{\xi }}_{tt}+A{\mathit{\xi }}_t+B\mathit{\xi }+{\rho }^{-1}\mathrm{\nabla }\mathrm{\Delta }p=0$ … Eq. (10).

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

"We introduce for the solution space $\mathrm{\Sigma }$ a basis $\{{\xi }_i\}$ the first $N$ vectors $\{\xi {\}}_{i=1}^N$ of which represent a basis for ${\mathrm{\Sigma }}_n$, the space of solenoidal vector polynomials of degree not exceeding $n$, as in L89a, L89b. It is easily found (see L89a) that $N=N(n)=(n+1)(n+2)(2n+9)/6$. Since ${\mathrm{\Sigma }}_n$ is invariant under the operators $A$ and $B$, we seek solutions of Eq. (10) in this space:" $\mathit{\xi }(\mathbf{𝐱},t)=\sum _{i=1}^N{c}_i(t){\xi }_i$ … 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

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

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

$\frac{D\mathbf{𝐮}}{Dt}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}$ $=$ $-{\rho }^{-1}\mathrm{\nabla }p+\mathrm{\nabla }\{{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2\}.$ 📚 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, $D/Dt$, that appears on the left-hand side of this Lagrangian representation of the Euler equation can be replaced by its Eulerian counterpart, $\partial /\partial t$, via the operator relation,

$\frac{D}{Dt}$	$\leftrightarrow$	$\frac{\partial }{\partial t}+(\mathbf{𝐮}\cdot \mathrm{\nabla }).$
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, $\partial \mathbf{𝐮}/\partial t\to 0$, that is,

$\frac{D\mathbf{𝐮}}{Dt}$

$\to$

${\xcancel{\frac{\partial \mathbf{𝐮}}{\partial t}}}^0+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮},$

in which case the following relation holds:

Steady-State Flow as viewed from a Rotating Reference Frame
$(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}$	$=$	$-{\rho }^{-1}\mathrm{\nabla }p+\mathrm{\nabla }\{{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2\}.$

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 $t=0$, 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,

${\mathbf{𝐮}}_0(\mathbf{𝐱})$

$=$

$[\hat{\mathit{ı}}{u}_x(\mathbf{𝐱})+\hat{\mathit{ȷ}}{u}_y(\mathbf{𝐱})+\hat{\mathbf{k}}{u}_z(\mathbf{𝐱}){]}_0,$

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. For example,

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

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) LBO67, p. 293, Eq. (1)

Following L89a and applying the operator, $\mathrm{\Delta }$, to each side of the Euler equation, we can write,

$\mathrm{\Delta }\left\{\frac{D\mathbf{𝐮}}{Dt}\right\}+\mathrm{\Delta }\left\{2\mathit{\omega }\mathit{\times }\mathbf{𝐮}\right\}$

$=$

$-\mathrm{\Delta }\left\{{\rho }^{-1}\mathrm{\nabla }p\right\}+\mathrm{\Delta }\left\{\mathrm{\nabla }\right[{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}.$

LHS

With the assurance provided by L89a that $\mathrm{\Delta }$ commutes with the Lagrangian time-derivative, $D/Dt$ — see also the paragraph immediately preceding Eq. (4) in LBO67 — and that

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

we can immediately appreciate that,

LHS

$=$

$\frac{D}{Dt}\left[\mathrm{\Delta }\mathbf{𝐮}\right]+2\mathit{\omega }\mathit{\times }\left[\mathrm{\Delta }\mathbf{𝐮}\right]$ = $\frac{D}{Dt}\left[\frac{D\mathit{\xi }}{Dt}\right]+2\mathit{\omega }\mathit{\times }\left[\frac{D\mathit{\xi }}{Dt}\right].$

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

(Lagrangian) Perturbed Euler Equation
$\frac{D^2\mathit{\xi }}{D{t}^2}+2\mathit{\omega }\mathit{\times }\left[\frac{D\mathit{\xi }}{Dt}\right]$	$=$	$-\mathrm{\Delta }\left\{{\rho }^{-1}\mathrm{\nabla }p\right\}+\mathrm{\Delta }\left\{\mathrm{\nabla }\right[{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}.$
L89a, §2, p. 223, Eq. (5)

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

LHS	$=$	$\frac{D}{Dt}[{\mathit{\xi }}_t+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]+2\mathit{\omega }\mathit{\times }[{\mathit{\xi }}_t+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]$
	$=$	$[\frac{\partial }{\partial t}+(\mathbf{𝐮}\cdot \mathrm{\nabla })][{\mathit{\xi }}_t+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]+2\mathit{\omega }\mathit{\times }[{\mathit{\xi }}_t+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]$
	$=$	${\mathit{\xi }}_{tt}+\frac{\partial }{\partial t}[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]+(\mathbf{𝐮}\cdot \mathrm{\nabla })[{\mathit{\xi }}_t+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]+2\mathit{\omega }\mathit{\times }[{\mathit{\xi }}_t+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]$

where we have adopted L89a's shorthand notation,

${\mathit{\xi }}_t\equiv \frac{\partial \mathit{\xi }}{\partial t},$

and,

${\mathit{\xi }}_{tt}\equiv \frac{{\partial }^2\mathit{\xi }}{\partial t^2}.$

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 $(\mathbf{𝐮}\cdot \mathrm{\nabla })$ commutes with the Eulerian time-derivative, that is,

$\frac{\partial }{\partial t}[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]\to (\mathbf{𝐮}\cdot \mathrm{\nabla }){\mathit{\xi }}_t,$

which means we may write,

LHS	$=$	${\mathit{\xi }}_{tt}+2\{(\mathbf{𝐮}\cdot \mathrm{\nabla }){\mathit{\xi }}_t+\mathit{\omega }\mathit{\times }{\mathit{\xi }}_t\}+\{(\mathbf{𝐮}\cdot \mathrm{\nabla }{)}^2\mathit{\xi }+2\mathit{\omega }\mathit{\times }[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]\}.$
L89a, §2, p. 224, immediately preceding Eq. (10)

RHS

Next, L89a introduces the Eulerian-change operator, $\delta$ (which commutes with $\mathrm{\nabla }$),

$\delta F$	$=$	$F(\mathbf{𝐱},t)-F_0(\mathbf{𝐱},t).$
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), $\mathrm{\Delta }F$ $=$ $\delta F+\mathit{\xi }\cdot \mathrm{\nabla }F.$ 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 $=-\mathrm{\Delta }\left\{{\rho }^{-1}\mathrm{\nabla }p\right\}$	$=$	${\rho }^{-2}\mathrm{\Delta }\rho \left[\mathrm{\nabla }p\right]-{\rho }^{-1}\mathrm{\nabla }\left[\mathrm{\Delta }p\right]$
	$=$	${\rho }^{-1}\left[\frac{\mathrm{\Delta }\rho }{\rho }\right]\mathrm{\nabla }p-{\rho }^{-1}\mathrm{\nabla }[\delta p+(\mathit{\xi }\cdot \mathrm{\nabla })p]$
	$=$	$-{\rho }^{-1}[\mathrm{\nabla }\cdot \mathit{\xi }]\mathrm{\nabla }p-{\rho }^{-1}[\mathrm{\nabla }\delta p+(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p]$
L89a, Appendix B, p. 239, Eq. (B.2)
	$=$	$-{\rho }^{-1}\{\mathrm{\nabla }\delta p+(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p+(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p\}$
	$=$	$-{\rho }^{-1}\left\{\mathrm{\nabla }\right[\underset{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{o}}{\underbrace{\mathrm{\Delta }p}}-\mathit{\xi }\cdot \mathrm{\nabla }p]+(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p+(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p\}$
L89a, Appendix B, p. 240, Eq. (B.3)
	$=$	$-{\rho }^{-1}\mathrm{\nabla }(\mathrm{\Delta }p)+{\rho }^{-1}\{\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)-(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p-(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p\}.$
Comments: In order to move from the 2nd to the 3rd line of this derivation, it seems that L89a employs the relation:   $[\mathrm{\Delta }\rho /\rho ]=-\mathrm{\nabla }\cdot \mathit{\xi }.$ This relation strongly resembles the continuity equation which, in Lagrangian form, is $[D\rho /Dt]=-\rho \mathrm{\nabla }\cdot \mathbf{𝐮}.$ In order to move from the 2nd to the 3rd line of this derivation, L89a seems to be acknowledging that, $\mathrm{\nabla }$ commutes with $(\mathit{\xi }\cdot \mathrm{\nabla }).$ A typographical error appears in Eq. (B.3) of L89a; $\mathrm{\Delta }\rho$ appears in the publication whereas, as noted here in the fifth line of this derivation, the term should be $\mathrm{\Delta }p$.

2nd term on RHS $=+\mathrm{\Delta }\left\{\mathrm{\nabla }\right[{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}$	$=$	$\mathrm{\nabla }\{\mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\mathrm{\Delta }[\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}$
	$=$	$\mathrm{\nabla }\{[\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\mathit{\xi }\cdot \mathrm{\nabla }{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}]+{\xcancel{\delta [\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]}}^0+\mathit{\xi }\cdot \mathrm{\nabla }[\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}$
	$=$	$\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\mathrm{\nabla }\{(\mathit{\xi }\cdot \mathrm{\nabla }){\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\mathit{\xi }\cdot \mathrm{\nabla }[\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}$
	$=$	$\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\mathit{\xi }\cdot \mathrm{\nabla }\left\{\mathrm{\nabla }\right[{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}.$
Comment: A term in the 2nd row of this derivation goes to zero because there is no Eulerian variation in either of the vectors, $\mathit{\omega }$ or $\mathbf{𝐱}$.

Adding these two "RHS" terms together gives,

RHS

$=$

$-{\rho }^{-1}\mathrm{\nabla }(\mathrm{\Delta }p)+{\rho }^{-1}\{\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)-(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p-(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p\}+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\mathit{\xi }\cdot \mathrm{\nabla }\left\{\mathrm{\nabla }\right[{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}.$

That is to say,

RHS	$=$	$-{\rho }^{-1}\mathrm{\nabla }(\mathrm{\Delta }p)+L\mathit{\xi },$
L89a, §2, p. 224, Eq. (8)

where the operator, $L$, is defined such that,

$L\mathit{\xi }$	$\equiv$	${\rho }^{-1}\{\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)-(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p-(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p\}+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\mathit{\xi }\cdot \mathrm{\nabla }\left\{\mathrm{\nabla }\right[{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2]\}.$
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
$\mathrm{\nabla }\{{\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+\frac{1}{2}|\mathit{\omega }\mathit{\times }\mathbf{𝐱}{|}^2\}$	$=$	${\rho }^{-1}\mathrm{\nabla }p+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}.$

Hence, the operator,

$L\mathit{\xi }$	$=$	${\rho }^{-1}\{\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)-(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p-(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p\}+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+(\mathit{\xi }\cdot \mathrm{\nabla })\{{\rho }^{-1}\mathrm{\nabla }p+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}\}$
	$=$	${\rho }^{-1}\{\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)-(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p\}-{\rho }^{-1}\{(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p\}+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+(\mathit{\xi }\cdot \mathrm{\nabla })\left\{{\rho }^{-1}\mathrm{\nabla }p\right\}+(\mathit{\xi }\cdot \mathrm{\nabla })\{(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}\}$
	$=$	${\rho }^{-1}\{\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)-(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p\}-{\rho }^{-1}(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p+{\rho }^{-1}(\mathit{\xi }\cdot \mathrm{\nabla })\mathrm{\nabla }p-{\rho }^{-2}\mathrm{\nabla }p(\mathit{\xi }\cdot \mathrm{\nabla })\rho +\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+(\mathit{\xi }\cdot \mathrm{\nabla })\{(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}\}$
	$=$	${\rho }^{-1}\{\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)-(\mathrm{\nabla }\cdot \mathit{\xi })\mathrm{\nabla }p\}-{\xcancel{{\rho }^{-2}\mathrm{\nabla }p(\mathit{\xi }\cdot \mathrm{\nabla })\rho }}^0+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+(\mathit{\xi }\cdot \mathrm{\nabla })\{(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}\},$
📚 Lebovitz (1989b), §2, p. 227, Eq. (4)

where, following the lead of 📚 Lebovitz (1989b), the term containing $\mathrm{\nabla }\rho$ 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,

$\mathrm{\nabla }\cdot \mathit{\xi }=0.$
 
LL96, §3.1, p. 703, Eq. (13)

Hence,

$L\mathit{\xi }$	$=$	${\rho }^{-1}\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+(\mathit{\xi }\cdot \mathrm{\nabla })[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}].$
LL96, §3.1, p. 703, Eq. (17)

Summary

Finally, setting LHS = RHS, we have,

${\mathit{\xi }}_{tt}+2\{(\mathbf{𝐮}\cdot \mathrm{\nabla }){\mathit{\xi }}_t+\mathit{\omega }\mathit{\times }{\mathit{\xi }}_t\}+\{(\mathbf{𝐮}\cdot \mathrm{\nabla }{)}^2\mathit{\xi }+2\mathit{\omega }\mathit{\times }[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]\}$	$=$	$-{\rho }^{-1}\mathrm{\nabla }(\mathrm{\Delta }p)+L\mathit{\xi }$
	$=$	$-{\rho }^{-1}\mathrm{\nabla }(\mathrm{\Delta }p)+{\rho }^{-1}\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+(\mathit{\xi }\cdot \mathrm{\nabla })[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}].$

Following L89a, this may be rewritten as,

$0$	$=$	${\mathit{\xi }}_{tt}+\underset{M{\mathit{\xi }}_t}{\underbrace{2\{(\mathbf{𝐮}\cdot \mathrm{\nabla }){\mathit{\xi }}_t+\mathit{\omega }\mathit{\times }{\mathit{\xi }}_t\}}}+\underset{\mathrm{\Lambda }\mathit{\xi }}{\underbrace{\{(\mathbf{𝐮}\cdot \mathrm{\nabla }{)}^2\mathit{\xi }+2\mathit{\omega }\mathit{\times }[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]\}-\stackrel{L\mathit{\xi }}{\overbrace{\{{\rho }^{-1}\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+(\mathit{\xi }\cdot \mathrm{\nabla })[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}]\}}}}}+{\rho }^{-1}\mathrm{\nabla }(\mathrm{\Delta }p)$
	$=$	${\mathit{\xi }}_{tt}+M{\mathit{\xi }}_t+\mathrm{\Lambda }\mathit{\xi }+{\rho }^{-1}\mathrm{\nabla }(\mathrm{\Delta }p),$
L89a, §2, p. 224, Eq. (10)

where,

$M\mathit{\xi }$	$\equiv$	$2\{(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }+\mathit{\omega }\mathit{\times }\mathit{\xi }\},$	and,	$\mathrm{\Lambda }\mathit{\xi }$	$\equiv$	$\{(\mathbf{𝐮}\cdot \mathrm{\nabla }{)}^2\mathit{\xi }+2\mathit{\omega }\mathit{\times }[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]\}-L\mathit{\xi }.$
L89a, §2, p. 224, Eq. (11) L89b, §2, p. 227, Eq. (2)

---

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

$0$	$=$	${\mathit{\xi }}_{tt}+A{\mathit{\xi }}_t+B\mathit{\xi }+{\rho }^{-1}\mathrm{\nabla }(\mathrm{\Delta }p).$
LL96, §3.1, p. 701, Eq. (10)

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

$A\mathit{\xi }$

$=$

$2\{(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }+\mathit{\omega }\mathit{\times }\mathit{\xi }\},$

and,

$B\mathit{\xi }$

$=$

$\{(\mathbf{𝐮}\cdot \mathrm{\nabla }{)}^2\mathit{\xi }+2\mathit{\omega }\mathit{\times }[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\xi }]\}-L\mathit{\xi },$

where,

$L\mathit{\xi }$	$=$	${\rho }^{-1}\mathrm{\nabla }(\mathit{\xi }\cdot \mathrm{\nabla }p)+\mathrm{\nabla }\delta {\mathrm{\Phi }}_{\mathrm{L}\mathrm{8}\mathrm{9}}+(\mathit{\xi }\cdot \mathrm{\nabla })[(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathbf{𝐮}+2\mathit{\omega }\mathit{\times }\mathbf{𝐮}].$
LL96, §3.1, p. 703, Eq. (17)

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

$D$	$=$	$\mathbf{𝐮}\cdot \mathrm{\nabla },$	and,	$\mathrm{\Omega }\mathit{\xi }$	$=$	$\mathit{\omega }\times \mathit{\xi },$
LL96, §3.1, p. 703, Eq. (15)

the matrix operator, $B$, can be rewritten as,

$B$	$=$	$D^2+2\mathrm{\Omega }D-L.$
LL96, §3.1, p. 703, Eq. (16)

See Also

• A library of ellipsoidal harmonics may be available from George Dassios
• 📚 N. R. Lebovitz (1961, ApJ, Vol. 134, pp. 500 - 536) titled, The Virial Tensor and Its Application to Self-Gravitating Fluids
• 📚 D. Lynden-Bell & P. Ostriker (1967, MNRAS, Vol. 136, pp. 293 - 310) titled, On the Stability of Differentially Rotating Bodies
• 📚 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
• 📚 N. R. Lebovitz, & A. Lifschitz (1996, ApJ, Vol. 458, pp. 699 - 713) titled, New Global Instabilities of the Riemann Ellipsoids
• 📚 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.