Elliptical Vortex

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<br/><br/><br/><br/>
.title[Wavelike Oscillations of an Elliptical Shallow-Water Vortex]
<br/><br/><br/>
.author[Jonathan M. Lilly]
.institution[The Planetary Science Institute, Tucson, Arizona]
<br/><br/>
.date[February 23, 2026]<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.-->

<!-- class: center
##Eddies Induce Long-Distance Transport

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

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

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##A Portrait of Two Anticyclonic Eddies

These are important players in the ocean circulation. <br/> 
We would like a simple model for their dynamics.

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##A Simple Model for a Large Anticyclone

A freely evolving elliptical paraboloid (a.k.a., an elliptical bowl) on an $f$-plane under shallow water dynamics.

Four modes of variability:  (i) orbital flow or swirl within  a fixed ellipse,  (ii) precession, (iii) change in size, and (iv) deformation.

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##History

A problem with ancient roots: Dirichelet (1860),  Kirchhoff (1876), Bassett (1888), Lamb (1932), Goldsbrough (1930), **Ball (1963)**

Exact solutions to all the modes variability have been found.

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)]

However, unlike the famous Kida (1981) vortex, these shallow-water vortex solutions are  rarely used today.

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##The Pulson Spirograph

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##The Importance of Exact Solutions

3. As **building blocks** for more complete and complex theories
2. As idealized **test cases** and initial conditions
1. As **foundations** for our own intuitive understanding

Exact solutions of the nonlinear equations of motion are rare!

This kind of theoretical work (with pen and paper) can teach us things that no AI, or even simulation, could.

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##Why Are These Not More Widely Used?

- Extremely abstract Hamiltonian formulation (Holm)  
- Formidable algebra, often unintuitive variables (everyone else)
- Disagreement between Holm's and Young's solutions

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

The idea here is to simplify, reconcile, and extend these solutions by appealing to fundamentals of **ellipse geometry**.

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##Approach

Following Holm (1991), we're going to use a Lagrangian formulation.   We need to derive an expression for the Lagrangian

where $K'$ and $P'$ are the kinetic and potential energies in the inertial (nonrotating) frame of reference.

This will be done by explicitly expressing gemetric properties of the eddy in terms of ellipse parameters.

After this, the solution becomes almost trivial.

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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 (Lilly, 2018).  This turns out to be a key.

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

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##First Ingredient: Ellipse-Flow Equivalance

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

**If you know the flow, you know exactly how the ellipse parameters will evolve, and vice-versa.**

We can readily move back and forth between Eulerian parameters and the Lagrangian (fluid-following) ellipse parameters.

(a) divergence, (b) vorticity, (c) normal strain, and (d) shear strain

<!-- Lagrangian parameters on the left, Eulerian on the right:

<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>

$\rho = \sqrt{ab}$, $\eta=a/b$, $\theta=$ orientation, $\phi$ = particle position   
$\delta = $ divergence, $\zeta =$ vorticity, $\tilde\nu= $ normal strain, $\tilde\sigma$ = shear strain-->

This result is termed *ellipse-flow equivalance* (Lilly, 2018).

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##Second Ingredient: Integral Properties

Building on ellipse–flow equivalence, one can find explicit expressions for all important integral properites of the eddy in terms of the ellipse parameters and their rates of change:

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

where  $\phi(t)$ is the parcel phase angle, $\theta(t)$ is the ellipse orientation angle, and the *moment of inertia* and *aspect ratio angle* are

<div>\[ I(t)\equiv \frac{1}{3}\frac{a^2+b^2}{2},\quad\quad \chi(t) \equiv \frac{\pi}{4}-\arctan\left(\frac{b}{a}\right) \]</div>

The kinetic energy $K'$ contains of four non-interacting terms: **circulation**, **precession**, **pulsation**, and **deformation**.

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##Third Ingredient: Potential Energy

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

Inertial potential energy is not the same as in the rotating frame,

<div>\[P'(t)  \ne P(t) = \iint_\mathcal{D} \frac{1}{2} gh^2  \, \mathrm{d}A.\]</div>

Instead the correct expression in the inertial frame is

<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 arising from the absorbtion of the centrifugal force in the rotating frame.

For the elliptical vortex, with $h_*$ being the central thickness,
<div>\[P'(t) = \frac{1}{3}gh_*  +  \frac{1}{8}f^2I .\]</div>

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##Origin of the Reversed Centrifugal Force

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.

This point does not appear to be well understood in the literature!

This reversed centrifugal force is a component of true gravity due to the Earth's change in curvature under rotation. </br>
.cite[Durran (1993), van der Toorn and Zimmerman (2008), Early (2012) ]

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##The Lagrangian Function

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

<div>\[L(t)  = \frac{1}{2}I\left\{\left[\dot \phi  +\cos2\chi\left(\dot \theta +{\textstyle\frac{1}{2}} f\right)\right]^2+ \sin^22\chi\left(\dot \theta+{\textstyle\frac{1}{2}}f\right)^2\right.\]</div>
<div>\[\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.+\frac{1}{4}\frac{ \dot I^2}{I^2}+\dot\chi^2- 2\frac{C_o}{ I^2\cos2\chi}-\frac{1}{4}f^2\right\}.\]</div>

in agreement with Holm (1991).

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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##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>\[\frac{\mathrm{d}}{\mathrm{d} t}\left\{I\left[\left(\dot \theta +{\textstyle\frac{1}{2}} f\right)\cos2\chi+  \dot \phi\right]\right\}=0\]</div>

<div>\[\frac{\mathrm{d}}{\mathrm{d} t}\left\{I\left[\left(\dot \theta +{\textstyle\frac{1}{2}} f\right)+  \dot \phi\cos2\chi\right]\right\}=0\]</div>

which can shown to be conservation of circulation and angular momentum, respectively.

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

These two conservation laws combine to give

<div>\[\dot\phi = \frac{\Pi_o-\cos2\chi M_o}{I\sin^22\chi},\quad\quad\quad\dot\theta + {\textstyle\frac{1}{2}}f = \frac{ M_o-\cos2\chi   \Pi_o}{I\sin^22\chi}  \]</div>

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

Here $\Pi_o$ and $M_o$ are the conserved values of the absolute circulation and absolute angular momentum, respectively.

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>\[\dot\phi = \frac{\Pi_o-\cos2\chi_o M_o}{I_o\sin^22\chi_o},\quad\quad\quad\dot\theta + {\textstyle\frac{1}{2}}f = \frac{ M_o-\cos2\chi_o   \Pi_o}{I_o\sin^22\chi_o}  \]</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.  Note that $\chi_o$  is a function of $\Pi_o$ and $M_o$.

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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>\[I(t)= I_o+J_o \cos ft, \quad\quad\quad I_o\equiv\frac{4 E_o}{f^2}\]</div>

where $J_o$ 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:

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>\[ \rho(t) = \rho_o\sqrt{1 +\epsilon_o\cos ft }\]</div>

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

where $\rho=\sqrt{ab}$, $\epsilon_o\equiv J_o/I_o$, and $\zeta_o\equiv \Pi_o/(\pi \rho_o^2)$ is the vorticity.

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

This is new, very simple formulation for the Lagrangian solution.

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##The Pulson Spirograph

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##The Pulson Solution

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## The Pulsrodon

If the vortex is not circular but the aspect ratio angle is constant, $\chi=\chi_o$, we have the *pulsrodon* solution.

Note that this can only happen for particular choices of the aspect ration angle.  That is, $\chi_o$ is a known function of the circulation $\Pi_o$ and angular momementum $M_o$.

This also has an exact analytic solution, including simple expressions for particle paths, only slightly more complicated than that for the pulson case—a new result.

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

To get the full solution, with a deformational mode, you still need to integrate an ODE, which we won't show here.

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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 later this year.

Key open  question: how and to what extent do the pulsating modes of variability exchange energy with the internal wave field?

Let's make sure we keep doing theory!

Thanks!