Elliptical Vortices

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<br/><br/><br/><br/>
.title[Exact solutions for elliptical vortices]
<br/><br/><br/>
.author[Jonathan M. Lilly]
.institution[Planetary Science Institute, Tucson, Arizona]
<br/><br/>
.date[September 24, 2025]<br/>
<br/>
.note[Created with [{Liminal}](https://github.com/jonathanlilly/liminal) using [{Remark.js}](http://remarkjs.com/) + [{Markdown}](https://github.com/adam-p/markdown-here/wiki/Markdown-Cheatsheet) +  [{KaTeX}](https://katex.org)]

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##Multiscale Variability in the Oceans

Sea surface height slope magnitude, from a model by H. Simmons.

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##Eddies Induce Long-Distance Transport

The ultimate goal is to quantitfy the impact of coherent eddies on fluid trapping, transport, and mixing.

An isolated anticyclonic eddy on a beta plane, from a quasigeostrophic model by Jeffrey Early.

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##Lagrangian Observations

A primary research interest has been creating tools to analyze  Lagrangian or freely-drifting observations of the ocean currents.

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##Lagrangian Vortex Extraction

See Lilly and Pérez-Brunius (2021) and references therein.

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##Focus on Large Anticyclones

Sea surface temperature near the Gulf Stream.<br/>

MODIS Aqua satellite image from the [Earth Scan Lab](https://www.esl.lsu.edu/imagery/gallery/modis-sensor-demo/),
Coastal Studies Institute, Louisiana State University.

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##A Simple Model for a Large Anticyclone

A starting place for understanding eddy dynamics is to examine freely evolving elliptical paraboloid (a.k.a., an elliptical bowl) on an $f$-plane under shallow water dynamics.

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##Problem Setup

The equations we wish to solve are
<div>\[\frac{\partial}{\partial t}\mathbf{u} =   -\mathbf{u} \mathbf{\cdot} \mathbf{\nabla} \mathbf{u} -f  \hat{\mathbf{k}}\times  \mathbf{u}- g\mathbf{\nabla} h\]</div>

<div>\[\frac{\partial}{\partial t} h  = - \mathbf{u}\mathbf{\cdot} \mathbf{\nabla}   h -  h\mathbf{\nabla} \mathbf{\cdot}  \mathbf{u}\]</div>

subject to the following assumptions for the structure of $\mathbf{u}$ and $h$

<div>\[\mathbf{u}(\mathbf{x},t) = \mathbf{U}(t) \mathbf{x}, \quad\quad\quad h(\mathbf{x},t) =h_*(t)  -  \mathbf{x} \mathbf{H}(t) \mathbf{x}.\]</div>

Here $\mathbf{x}$ is the horizontal position vector relative to the ellipse center, $h_\*(t)$ is central eddy thickness, and $\mathbf{U}(t)$ and $\mathbf{H}(t)$ are $2\times 2$ tensors called the *flow tensor* and *curvature tensor*.

Note that velocity  $\mathbf{u}$  is linear in $\mathbf{x}$ while thickness $h$ is quadratic.

These describe the evolution of fluid mass of an elliptic paraboloid on a rotating plane subject to shallow water dynamics under the assumption that the flow is a linear function of position.

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##History of the Problem

This problem has roots roots extending back more nearly 150 years, to the two-dimensional Kirchhoff (1876) vortex, see also Bassett (1888), Lamb (1932), and Kida (1981).

For the shallow water vortex, essential foundational work was done by Goldsbrough (1930) and especially Ball (1963).

Beginning in the 1980's, the dynamics of freely evolving shallow water anticyclone was then examined by Cushman-Roisin et al. (1985), Cushman-Roisin (1987), Ripa (1987), Rogers (1989), and Kirwan and Liu (1991).

This work cultimated in complete solutions by Young (1986) and Holm (1991).  However, these solutions are rarely used today.

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##Shallow-Water Anticyclones on an $f$-Plane

It is now known that a freely-evolving shallow-water vortex has four modes of variability:

(i) orbital motion or swirl—a flow along the periphery of a fixed ellipse, (ii) change in size, (iii) deformation, and (iv) precession.

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##A Taxonomy of Solutions

1. Precession only: the rotational solution or *rodon*  <br/>.cite[Cushman-Roisin et al. (1985), Cushman-Roisin  (1987), Ripa (1991), Kirwan and Liu (1991)]
2. Change in size only, at $f$, for a circular vortex: the *pulson* <br/>.cite[Cushman-Roisin et al. (1985), Cushman-Roisin  (1987), Kirwan and Liu (1991)]
3. Rotation + pulsation for an elliptical vortex: the *pulsrodon* <br/>.cite[Rogers (1989)]
4. Rotation, pulsation, and deformation: the complete solution <br/>.cite[Young (1986), Holm (1991)]

“Pulson” and “rodon” names were coined by Kirwan and Liu (1991).

All solutions apart from #4 are analytic, i.e., you can just write then down in closed form in terms of elementary functions.

The deformation mode in #4 is solved with a simple numeric integration of an ODE.

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##Why Are These Not More Widely Used?

Unfortunately, the solutions are not in a ready-to-use form.

Particularly for the pulsrodon solution of Rogers (1989) and the complete solution of Young (1986), the algebra is quite formidable, with solutions  presented in terms of unintuitive (to me) variables.

The complete solution of Holm (1991), on the other hand, requires a solid knowlege of Hamiltonian mechanics to understand and is extremely dense and (to me) intuitively opaque.

The Young and Holm solutions should be equivalent.  However, they are very difficult to compare because they use two  different variable sets.

What is clear is that they do not agree.  Concerningly, while Young finds the moment of inertial oscillates at $f$, Holm finds $\frac{1}{2}f$, in contradiction to Ball's (1963) theorem on inertial pulsation.

More work is needed to make this important and hard-won knowledge readily accessible!

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##The Importance of Exact Solutions

Exact solutions of the nonlinear equations of motion are rare!

While simplified, these have a number of important applications:

4. As a means to refine our understanding of fundamental principles
1. As easy-to-conceptualize foundations for further investigations
2. As idealized test cases and initial conditions
3. As building blocks for more complete and complex theories

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##How I Got Interested

A kinematic model for particle trajectories in the vicinity of an evolving eddy using ideas from signal processing. .cite[Lilly and Olhede, (2010), “Bivariate Instantaneous Frequency and Bandwidth.”]

The parameterization of a particle path is identical that used in Holm (1991)—including the same variable names.

A connection between Lagrangian paths and vortex dynamics?-->

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##Steps to the Solution

1. Converting between Eulerian and Lagrangian coordiantes
1. Potential energy in an inertial frame
1. Moments of an elliptical fluid body 
1. Formation of the Lagrangian 
1. Writing down the solution from the Lagrangian

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##An Approach Via Ellipse Kinematics

Our approach to first derive intermediate results regarding the fundamental kinematic properties of an ellispe in a linear flow.

That is, we consider geometric properites of an evolving ellipse independent of underlying dynamics.

In particular, we need to derive an expression for the Lagrangian

in terms of the ellipse parameters. Here $K'$ and $P'$ are kinetic and potential energies of the elliptical vortex in the inertial frame.

Using these results, the solution will become almost trivial.

The goal of this portion is to enable use to case the entire problem in terms of the variables describing the ellipse.

The next few slides will create some key kinematic tools.

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##Ellipse Parameters

Semi-axes $a$ and $b$, orientation angle $\theta$, and particle phase $\phi$

Alternatively, replace  $a$ and $b$ with geometric mean radius $\rho\equiv\sqrt{ab}$ and aspect ratio $\eta \equiv a/b$ or aspect ratio angle $\chi(t) \equiv \arctan\left(\frac{b}{a}\right)$.

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##A Linear Flow

A linear flow $\mathbf{u}=\mathbf{U}\mathbf{x}$ has spatial derivatives that are constant.

<div>\[\delta\equiv \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \quad\quad\zeta\equiv\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\quad\quad
\nu \equiv  \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\quad\quad\sigma \equiv\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}
\]</div>

<img style="width:100%"  src="../figures/kinematics-ijkl-vectors-edited.png">
     Divergence            Vorticity          Normal Strain      Shear Strain

We can then decompose the flow into a sum of these four patterns.

Linear flows are fundamental because any almost any flow is locally linear to first order in a Taylor series!

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##An Ellipse in a Linear Flow

Let an ellipse of particles be advected by a linear flow $\mathbf{u}=\mathbf{U}\mathbf{x}$.

It is well known .cite[(see e.g., Kida (1981))] that a linear flow will transform an elliptical region into a different elliptical region.

This shows that divergence expands the ellipse, vorticity rotates, normal strain deforms, and shear strain both rotates and deforms.

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##Ellipse–Flow Equivalence

In fact, there is a one-to-one correspondence between flow derivatives and the rates of change of the ellipse parameters:

<div>\[\frac{\dot \rho}{\rho} = \frac{1}{2}\delta\quad\quad  \frac{\dot \eta}{\eta}=\tilde\nu, \quad\quad \dot \theta  = \frac{1}{2}\zeta + \frac{1}{2}\frac{\eta^2+1}{\eta^2-1}\tilde\sigma\quad\quad \dot\phi = -\frac{1}{2} \frac{2\eta}{\eta^2-1} \tilde \sigma \]</div>

Here $\tilde\nu$ and $\tilde\sigma$ are the strain terms in the frame of the ellipse.

*Given the flow parameters, you know exactly how the ellipse parameters will evolve.  Conversely, if you watch an ellipse evolve, you know exactly the flow that must be advecting it.*

This means for linear flows, we can readily move back and forth between Eulerian parameters and the Lagrangian (fluid-following) parameters of an ellipse.

This result is termed *ellipse–flow equivalence*.

.cite[Lilly (2018), “Kinematics of a fluid ellipse in a linear flow.”]

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##Moments of an Elliptical Paraboloid

One may show that the volume and moment of inertia of an elliptical paraboloid are, respectively, 
<div>\[
V \equiv\iint_\mathcal{A} h\, \mathrm{d} A=2\pi a b \hbar,\quad\quad I(t) \equiv \frac{1}{V} \iint_\mathcal{A} \|\bm{x}\|^2 \,h\,\mathrm{d} A =\frac{1}{3}\frac{a^2+b^2}{2} 
\]</div>

where $\hbar$ is the depth at the eddy center.  The absolution circulation (with a nonstandard $1/3$) and absolute angular momentum

<div>\[\Pi \equiv \frac{1}{3}\frac{1}{2\pi}\oint_\mathcal{C} \left(\bm{u}+{\textstyle\frac{1}{2}}f\bm{k}\times\bm{x}\right)\cdot\mathrm{d}\bm{x}\]</div>

<div>\[ M\equiv \frac{1}{V}\iint_\mathcal{A} \,\left(\bm{k}\cdot\bm{x}\times \bm{u}+{\textstyle\frac{1}{2}} f\|\bm{x}\|^2\right) h \, \mathrm{d} A 
\]</div>

take on the symmetric forms, with $\xi \equiv \frac{2ab}{a^b+b^2}$ being the circularity,

<div>\[\Pi = \left[\dot \phi +\xi\left(\dot \theta +{\textstyle\frac{1}{2}} f\right)\right] I,\quad\quad\quad M =\left[\xi\dot \phi+\left(\dot\theta + {\textstyle\frac{1}{2}} f \right) \right] I\]</div>

in terms of the rates of change of the ellipse parameters.

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##Angular Momentum vs. Circulation

These are different quantities, which become identitical for a circle.

“Fluid-body” motion, involving a change in $\phi$, is compared with solid-body motion, involving a change in $\theta$.

$\Pi$ and $M$ involve different combinations of $\dot \phi$ and $\dot \theta$.

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##Kinetic Energy

We need to know the kinetic energy in the inertial (nonrotating) frame, denoted $K'$.

Building on ellipse–flow equivalence, one can express the kinetic energy in terms of the ellipse parameters as

<div>\[ K'(t) \equiv  \iint_\mathcal{D} \frac{1}{2}\left\|\mathbf{u}+{\textstyle\frac{1}{2}}f\hat{\mathbf{k}}\times \mathbf{x}\right\|^2 h \,\mathrm{d} A= \\ \frac{1}{2}I\Bigg\{\,\overset{\mathrm{circulation}}{\overbrace{\left[\dot \phi  + \left(\dot \theta +{\textstyle\frac{1}{2}} f\right)\sin 2\chi\right]^2}}+\overset{\mathrm{precession}}{\overbrace{\left(\dot \theta+{\textstyle\frac{1}{2}}f\right)^2\cos^2 2\chi}}+\overset{\mathrm{pulsation}}{\overbrace{\frac{1}{4}\frac{ \dot I^2}{I^2}}}+\overset{\mathrm{deformation}}{\overbrace{\dot\chi^2}}\,\Bigg\}\]</div>

such that the kinetic energy contains of four non-interacting terms: circulation, precession, pulsation, and deformation.

Recall that $\chi(t) \equiv \arctan\left(\frac{b}{a}\right)$.  For a circle $a=b$,  we have $\chi=\pi/4$ such that $\sin 2\chi=1$ and $\cos 2\chi =0$.

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##Potential Energy

Determining the correct potential energy in the inertial frame for the shallow-water  vortex is surprisingly subtle—and informative!

Potential energy $P$ in the rotating frame is simply the integral of $\frac{1}{2}gh^2$ over the vortex body:

<div>\[P(t)  =  \iint_\mathcal{D} \frac{1}{2} gh^2  \, \mathrm{d}A. \]</div>

However, this is not the same as the potential energy $P'$ in the inertial frame: $P'\ne P$.

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##The Rotating Shallow-Water Equations

Beginning with the nonrotating shallow water equations
<div>\[\frac{\partial}{\partial t}\mathbf{u}' =   -\mathbf{u}' \mathbf{\cdot} \mathbf{\nabla} \mathbf{u}'  - g'\mathbf{\nabla} h\]</div>

where the primes denote quantities in the *inertial* frame,  we transform to the rotating frame in the usual way,

<div>\[\mathbf{u}=\mathbf{u}' +{\textstyle\frac{1}{2}} f \hat{\mathbf{k}} \times \mathbf{x}, \quad\quad
\left. \frac{\mathrm{d}}{\mathrm{d} t}\right|_F\mathbf{u}=\left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{F'}\mathbf{u}' +f\hat{\mathbf{k}} \times\mathbf{x}- \frac{1}{4} f^2 \mathbf{x}.\]</div>

This gives the rotating shallow water equations

<div>\[\frac{\partial}{\partial t}\mathbf{u}=   -\mathbf{u} \mathbf{\cdot} \mathbf{\nabla}\mathbf{u} -f  \hat{\mathbf{k}}\times  \mathbf{u} -  g' \mathbf{\nabla} h + \boxed{\frac{1}{4} f^2 \mathbf{x}}\]</div>

which include the  centrifugal term, a fictitious force which appears to push radially outward.

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##The Rotating Shallow-Water Equations

On physical grounds, it is argued that this term should be absorbed into gravity, leading to the usual form
<div>\[\frac{\partial}{\partial t}\mathbf{u}=   -\mathbf{u} \mathbf{\cdot}\mathbf{\nabla} \mathbf{u} -f  \hat{\mathbf{k}}\times  \mathbf{u} -  g\mathbf{\nabla}h \]</div>

in which no centrifugal term appears.  Here $g$ is now the *effective* gravity.

This removal of the centrifugal term amounts to *redefining horizontal surfaces as curved* and then neglecting that curvature.

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##The Nonrotating Shallow-Water Equations

But if in rotating frame we have no centifugal term
<div>\[\frac{\partial}{\partial t}\mathbf{u}=   -\mathbf{u} \mathbf{\cdot} \mathbf{\nabla} \mathbf{u} -f  \hat{\mathbf{k}}\times  \mathbf{u} -  g\mathbf{\nabla} h \]</div>

then transforming this system back to the inertial frame, we have
<div>\[\frac{\partial}{\partial t}\mathbf{u}' =   -\mathbf{u}' \mathbf{\cdot} \mathbf{\nabla} \mathbf{u}'  - g\mathbf{\nabla} h-\boxed{\frac{1}{4} f^2 \mathbf{x}}\]</div>

and the omitted centrifugal term reappears with opposite sign!

This amounts to a force that pushes fluid parcels radially *inward*.

The reversed centrifugal force is a *true force* due to a component of gravity along “flat” surfaces that have been deformed by rotation. .cite[Durran (1993), van der Toorn & Zimmerman (2008), Early (2012) ]

Locations far from the rotation axis thus become locations of high potential energy.

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##Equipotential Surfaces

The absorbtion of the centrifugal force in the rotating frame amounts to a redefinition of the horizontal, causing the centrifugal term to re-appear in the inertial frame with opposite sign.

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##Potential Energy in the Inertial Frame

For the rotating shallow water equations, the potential energy of a fluid volume in the inertial or nonrotating frame is therefore

<div>\[P'(t)  \equiv \iint_\mathcal{D} \frac{1}{2} gh^2  \, \mathrm{d}A +\boxed{ \iint_\mathcal{D} \frac{1}{8} f^2 \|\mathbf{x}\|^2 h\, \mathrm{d} A} \]</div>

where the boxed term is a reversed centrifugal energy term arising from the omission of the centrifugal force in the rotating frame.

This impact of the steps leading to the rotating shallow water equations on the *inertial* frame potential energy does  not appear to be well understood in the literature.

This is found to become, for the elliptical vortex
<div>\[P'(t) = \frac{1}{3}gh_*  +  \frac{1}{8}f^2I .\]</div>

We can now form the Lagrangian function.

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##The Lagrangian Function

The Lagrangian for the shallow-water vortex $L(t)  = K'-P'$ is found to be, with $C_o\equiv \frac{2}{9\pi}  gV_o$,

<div>\[L(t)  = \frac{1}{2}I\left\{\left[\dot \phi  +\sin2\chi\left(\dot \theta +{\textstyle\frac{1}{2}} f\right)\right]^2+ \cos^22\chi\left(\dot \theta+{\textstyle\frac{1}{2}}f\right)^2\right.\]</div>
<div>\[\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+\frac{1}{4}\frac{ \dot I^2}{I^2}+\dot\chi^2- 2\frac{C_o}{ I^2\sin2\chi}-\frac{1}{4}f^2\right\}.\]</div>

in agreement with Holm (1991), using a very different approach.

We can now readily find the equations of motion from the Euler-Lagrange equations

<div>\[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L'}{\partial \dot{q_i}} =\frac{\partial L'}{\partial q_i}\]</div>

for $i=1,2,3,4$ with the $q_i$ being the ellipse parameters $\phi$, $\theta$, $I$, $\chi$.

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##Lagrangian and Eulerian Evolution

Solutions for the ellipse parameters $I$, $\chi$, $\theta$, and $\phi$ are found from the Lagrangian.

The Eulerian evolution is then found from ellipse–flow equivalence, which in terms of the above parameters becomes

<div>\[\delta = \frac{\dot I}{I} - 2\dot \chi \cot  2\chi,\quad\quad\quad\zeta = 2 \left(\frac{1}{\sin 2\chi} \dot \phi + \dot \theta\right)\]</div>

<div>\[\tilde \nu = \frac{2\dot \chi}{\sin 2\chi},\quad\quad\quad \tilde\sigma = -2\dot \phi \cot 2\chi \]</div>

where $\tilde \nu $ and $\tilde \sigma$ are the normal and shear strain in the reference frame of the eddy.

Thus, once you have the ellipse parameters, the Eulerian fields follow immediately.

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##The $\theta$ and $\phi$ Dynamics

Since the Lagrangian contains neither $\phi$ nor  $\theta$—only their rates of change—the Euler-Lagrange equations for these two coordinates

<div>\[ \frac{\mathrm{d}}{\mathrm{d} t}\frac{\partial L}{\partial \dot{\phi}}=\frac{\partial L}{\partial \phi},  \quad\quad\quad   \frac{\mathrm{d}}{\mathrm{d} t}\frac{\partial L}{\partial \dot{\theta}} =\frac{\partial L}{\partial \theta}\] </div>

give at once

<div>\[\frac{\mathrm{d}}{\mathrm{d} t}\left\{I\left[\left(\dot \theta +{\textstyle\frac{1}{2}} f\right)\sin2\chi+  \dot \phi\right]\right\}=0\]</div>

<div>\[\frac{\mathrm{d}}{\mathrm{d} t}\left\{I\left[\left(\dot \theta +{\textstyle\frac{1}{2}} f\right)+  \dot \phi\sin2\chi\right]\right\}=0\]</div>

which are recognized as conservation of circulation $\Pi$ and angular momentum $M$, respectively, with

<div>\[\Pi\equiv I\left[\left(\dot \theta +{\textstyle\frac{1}{2}} f\right)\sin2\chi+  \dot \phi\right],\quad\quad\quad M \equiv I\left[\dot \theta +{\textstyle\frac{1}{2}} f+  \dot \phi\sin2\chi\right].\]</div>

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##The $\theta$ and $\phi$ Dynamics

These two conservation laws combine to give

<div>\[\dot\phi = \frac{\Pi-\sin2\chi M}{I\cos^22\chi},\quad\quad\quad\dot\theta + {\textstyle\frac{1}{2}}f = \frac{ M-\sin2\chi   \Pi}{I\cos^22\chi}  \]</div>

for the rates of change of particle phase $\phi$ and orientation angle $\theta$.

If the moment of inertia $I$ and aspect ratio angle $\chi$ are both constant, $\phi(t)$ and $\theta(t)$ both progress at uniform rates.

If $I$ is not constant, conservation of circulation and angular momentum  require the rotation rates to become faster as $I$ becomes smaller—like a spinning ice skater.

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##The Rodon Solution

For both $I$ and $\chi$ being constant, we have

<div>\[\dot\phi = \frac{\Pi-\sin2\chi_o M}{I_o\cos^22\chi_o},\quad\quad\quad\dot\theta + {\textstyle\frac{1}{2}}f = \frac{ M-\sin2\chi_o   \Pi}{I_o\cos^22\chi_o}  \]</div>

which immediately integrate to yield

<div>\[\phi(t) = \omega_\phi t+\phi_o,\quad\quad\theta(t) = \omega_\theta t+\theta_o  -{\textstyle\frac{1}{2}}f \]</div>

with particle paths $z(t)=x(t)+\mathrm{i}y(t)$ given by

<div>\[z(t)=\mathrm{e}^{\mathrm{i}\left( \omega_\theta t+\theta_o-\frac{1}{2}f\right)}
  \left\{a_o \cos \left(\omega_\phi t+\phi_o\right)+\mathrm{i}b_o \sin \left(\omega_\phi t+\phi_o\right)\right\}\]
</div>

This is the *rodon* solution of a uniform rotating ellipse of constant size and shape.

Note that $\chi_o$  is a special value of the eccentricity aspect ratio for which the ellipse shape is stationary; this is a function of $\Pi$ and $M$.

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##The Moment of Inertia $I$ Dynamics

The Euler-Lagrange equation for the moment of inertia,
<div>\[ \frac{\mathrm{d}}{\mathrm{d} t}\frac{\partial L}{\partial \dot{I}}=\frac{\partial L}{\partial I}\]</div>

leads to, with $E_o\equiv K'+P'$
<div>\[\ddot I  +f^2 I= 4  E_o \]</div>

such that the moment of inertia undergoes simple hamonic motion at the inertial frequency $f$, or *inertial pulsation*.  The solution is

<div>\[I(t)= I_o\left[1+\epsilon_o \cos \left(f(t-t_o)\right)\right], \quad\quad\quad I_o\equiv\frac{4 E_o}{f^2}\]</div>

where $\epsilon_o$ is a free parameter setting the oscillation magnitude.

This remarkable result traces itself back to Ball (1963).  It agrees with Cushman-Roisin et al. (1985) and Young (1987).  Holm (1991) incorrectly gets a $f/2$ rather than $f$ due to the omission of a term.

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## The Pulson Solution

If the vortex is circular, we have the *pulson* solution:

<div>\[I(t)= I_o\left[1+\epsilon_o \cos \left(f(t-t_o)\right)\right]\]</div>

<div>\[ h_*(t) = \frac{2V_o}{3\pi} \frac{1}{I_o\left[1 +\epsilon_o\cos\left(f(t-t_o)\right) \right]}\]</div>

From conservation of volume $V\_o=\frac{1}{2}\pi a b h\_\*=\frac{1}{2}\pi \rho^2 h\_\*$ and conservation of absolute circulation, one readily finds

<div>\[ \rho(t) = \rho_o\sqrt{1 +\epsilon_o\cos\left(f(t-t_o)\right) }\]</div>

<div>\[\phi(t)=-\frac{ft}{2} + \frac{1}{2}\frac{\zeta_o}{f}\frac{1}{\sqrt{1-\epsilon_o^2}}\arctan\left(\frac{\tan \frac{1}{2}f(t-t_o)+ \epsilon_o}{\sqrt{1-\epsilon_o^2}}\right)+\phi_o\]</div>

where $\zeta_o\equiv \Pi/(\pi \rho_o^2)$ is the vorticity.

Particle paths are very simply given by $z(t)=\sqrt{\rho(t)}\mathrm{e}^{\mathrm{i}\phi(t)}$.

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##The Pulson Spirograph

<img style="width:75%"  src="../figures/pulson-spirograph.png">

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##The Pulson's Time Evolution

<img style="width:70%"  src="../figures/pulson-lineplots.png">

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## The Pulsrodon

More generally, if the vortex is not circular but the aspect ratio angle is constant, $\chi=\chi_o$, we have the *pulsrodon* solution, also for particular choices of $\chi_o$.

<div>\[I(t)= I_o\left[1+\epsilon_o \cos \left(f(t-t_o)\right)\right]\]</div>

<div>\[ 
\phi(t)  =\frac{1}{f} \frac{\Pi -\xi_o  M }{\left(1-\xi_o ^2\right)I_o } \frac{2}{\sqrt{1-\epsilon_o ^2}}\arctan\left(\frac{\tan {\textstyle\frac{1}{2}}ft+\epsilon_o }{\sqrt{1-\epsilon_o ^2}}\right) +\phi_o \]</div>

<div>\[ \theta(t) = \frac{1}{f} \frac{M -\xi_o    \Pi }{\left(1-\xi_o ^2\right)I_o } \frac{2}{\sqrt{1-\epsilon_o ^2}}\arctan\left(\frac{\tan {\textstyle\frac{1}{2}}ft+\epsilon_o }{\sqrt{1-\epsilon_o ^2}}\right)+\theta_o - {\textstyle\frac{1}{2}}ft\]</div>

This is simply a consequence of the moment of inertial equation together with conservation of circulation and angular momentum.

This simple analytic solution is a new result.

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## The Deformation Dynamics

The deformation dynamics are somewhat more complicated and required the numerical solution of an ODE following .cite[Holm (1991)].  Once $\chi(t)$ is found, this can be converted into $\xi(t)$ and inserted into the previous equations to find the other other parameters.

This involves defining the forcing function

<div>\[  P(\chi)\equiv 2\cos 2\chi \left[ \frac{\left(\Pi_\star- M_\star\sin2\chi\right)\left(M_\star-\Pi_\star\sin2\chi\right)  }{\cos^42\chi}+\frac{1}{\sin^22\chi} \right]\]</div>

and introducing $X(\tau(t))\equiv \chi(t)$ as a function of a nondimensional time variable $\tau = t /(I/C)$.  Then $X$ satisfies an oscillator equation

<div>\[  \ddot X=  P\left(X\right)\]</div>

with respect to the nondimensional time.

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## A Historical Note

The elliptical vortex is closely related to, and derives from an archaic approach to modeling planetary bodies as self-gravitating homogeneous fluid ellipsoids.  This line of work dates back to an 1860 paper by Dirichlet, published posthumously with assistance from Dedekind, see Lamb (1945).

In 1987 S. Chandrasekhar published a book, *Elliptical Figures of Equilibrium*, revisiting this older work.  He writes:

“But the author may be permitted to state his reason for devoting a substantial part of nine years to the problems treated in this book.

“The subject had attracted the attention of a long succession of distinguished mathematicians and astronomers—albeit of an earlier and more relaxed age. But the subject, nevertheless, had been left in an incomplete state with many gaps and omissions and some plain errors and misconceptions.  It seemed a pity that it should be allowed to remain in that destitute state.  Whether the effort expended in its rehabilitation was worth the time, the author cannot presume to judge.”

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## A Historical Note

Curiously, it was at virtually the same time as Chandrasekhar's survey on seemingly obsolete topic that Cushman-Roisin and collaborators (1985, 1987)  “initiated a new direction of research”, in the words of Kirwan and Liu (1991), by applying what is essentially a two-dimensional version of Dirichlet's model to the study of ocean eddies, which was then extended by Young (1986) and Holm (1991).

This coincidence could be taken to indicate the value  of learning from the past.

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## Conclusions

1. Classical analytic solutions for shallow-water vortices have been rederived in a (hopefully) more accessible manner.
1. The solution approach relies of first establishing the fundamental kinematic properties of an ellipse in a linear flow.
1. Future work will use these solutions to determine what modes of variability are recoverable from individual trajectories.

A set of several papers on this work is expected to be completed early next year.

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#  Thank you!

.center[.footnote[P.S.  Like the way this presentation looks? Check out [{Liminal}](https://github.com/jonathanlilly/liminal).]]

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##Decomposition of a Linear Flow

The flow tensor $\mathbf{U}(t)$ can be decomposed as .cite[(Lilly, 2018)]
<div>\[\mathbf{U}(t)=\frac{1}{2}\left(\delta \mathbf{I}+\zeta\mathbf{J}+\nu\mathbf{K}+\sigma\mathbf{L}\right)
\]</div>

where we have introduced the $\mathbf{I}\mathbf{J}\mathbf{K}\mathbf{L}$ tensor basis. These $2\times 2$ tensors are represented as the matrices
<div>\[\left[\mathbf{I}\right]_S = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix},\quad
\left[\mathbf{J}\right]_S=\begin{bmatrix}  0& -1 \\ 1 & 0\end{bmatrix},\quad
\left[\mathbf{K}\right]_S=\begin{bmatrix} 1 & 0 \\0& -1\end{bmatrix},\quad
\left[\mathbf{L}\right]_S=\begin{bmatrix} 0 & 1 \\1 & 0\end{bmatrix}\]</div>

in a Cartesian coordinate system $S$.

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##Proof of Ellipse–Flow Equivalence

Write the ellipse boundary in terms of the *ellipse vector*

<div>\[\breve{\mathbf{x}}(t)\equiv  \mathbf{R}\left(\theta\right) \begin{bmatrix}a \cos\phi\\b\sin\phi\end{bmatrix}\]</div>

where $\mathbf{R}\left(\theta\right)$ is the rotation matrix

<div>\[\mathbf{R}\left(\theta\right)\equiv \begin{bmatrix}  \cos\theta &-\sin \theta\\ \sin \theta & \cos\theta\end{bmatrix}. \]</div>

Then rexpress $\breve{\mathbf{x}}(t)$ in terms of $\rho$ and $\eta$ as

<div>\[\breve{\mathbf{x}}(t)= \frac{\rho}{\sqrt{\eta}}\,\mathbf{R}\left(\theta\right)\begin{bmatrix}\eta\cos\phi\\  \sin\phi \end{bmatrix}\]</div>

and take the time derivative $\frac{\mathrm{d}}{\mathrm{d} t}\breve{\mathbf{x}}=\frac{\partial\breve{\mathbf{x}}}{\partial \rho}\frac{\mathrm{d}\rho}{\mathrm{d} t}+\frac{\partial\breve{\mathbf{x}}}{\partial \eta}\frac{\mathrm{d}\eta}{\mathrm{d} t}+\frac{\partial\breve{\mathbf{x}}}{\partial \theta}\frac{\mathrm{d}\theta}{\mathrm{d} t}+\frac{\partial\breve{\mathbf{x}}}{\partial \phi}\frac{\mathrm{d}\phi}{\mathrm{d} t}$.

Identifying $\mathbf{U}\mathbf{x}=\frac{\mathrm{d}}{\mathrm{d} t}\breve{\mathbf{x}}$, ellipse–flow equivalence follows.