Elliptical Vortices

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<br/><br/><br/><br/>
.title[Analytic solutions for elliptical vortices]
<br/><br/><br/>
.author[Jonathan M. Lilly]
.institution[The Planetary Science Institute, Tucson, Arizona]
<br/><br/>
.date[November 21, 2022]<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 transport and mixing.

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

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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 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>\[\frac{\partial}{\partial t}\mathbf{u} =   -\mathbf{u} \mathbf{\cdot} \nabla \mathbf{u} -f  \hat{\mathbf{k}}\times  \mathbf{u}- g\nabla h\]</div>

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

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

<div>\[\mathbf{u}(\mathbf{x},t) = \mathbf{U}(t) \mathbf{x}\]</div>

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

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.

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

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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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##The Importance of Analytic Solutions

Exact solutions of the nonlinear equations of motion are rare!

While simplified, these have a number of important applications:

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

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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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##How I Got Interested

A kinematic model for particle trajectories in the vicinity of an evolving eddy using ideas from signal processing. *No Dynamics.* <br/>.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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##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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##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.

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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##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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##Decomposition of a Linear Flow

The flow tensor $\mathbf{U}(t)$ can be decomposed as .cite[(Lilly 2018, 2023)]
<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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##Ellipse Parameters

.left-column[<img style="width:100%"  src="../figures/kinematics-schematic-single.jpg">]
.right-column[Semi-axes $a$ and $b$, orientation angle $\theta$, and particle phase $\phi$.

Alternatively, replace axes $a$ and $b$ with geometric mean radius $\rho\equiv\sqrt{ab}$ and aspect ratio $\eta \equiv a/b$.]

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##Ellipse–Flow Equivalence

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

It can be shown that 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 the flow that must be generating 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.

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

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##Proof

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.

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##A Generalization of Stokes' Theorem

Recall the divergence and Stokes' theorems

<div>\[\iint_\mathcal{D}  \nabla \cdot  \mathbf{u}\, \mathrm{d} A=\oint_\mathcal{C} \mathbf{u} \cdot \mathrm{d}\mathbf{n},\quad\quad\quad
 \iint_\mathcal{D}  \hat{\mathbf{k}}\cdot\nabla \times \mathbf{u}\, \mathrm{d} A=\oint_\mathcal{C} \mathbf{u} \cdot  \mathrm{d} \mathbf{x}\]</div>

relating the area integral over domain $\mathcal{D}$ to the line integral over its bounding surface $\mathcal{C}$.  Here $\mathrm{d}\mathbf{n}=-\hat{\mathbf{k}}\times\mathrm{d}\mathbf{x}$ is a differential normal.

One may show that the *tensor-valued* identity .cite[(Lilly 2018, 2023)],

<div>\[\iint_\mathcal{D}  \nabla \mathbf{u}\, \mathrm{d}A=\oint_\mathcal{C} \mathbf{u} \otimes\mathrm{d} \mathbf{n}\]</div>

the *gradient tensor theorem*, compactly includes the divergence theorem, Kelvin-Stokes' theorem, and two strain theorems.

Here $\mathbf{u} \otimes\mathrm{d} \mathbf{n}$ is the outer product, vs. the inner product $\mathbf{u} \cdot \mathrm{d} \mathbf{n}$.<br/>
(We can think of this as $\mathbf{x}\otimes \mathbf{y}\sim \mathbf{x}  \mathbf{y}^T$ as opposed to $\mathbf{x}\cdot \mathbf{y}\sim \mathbf{y}^T \mathbf{x}$.)

Although implied by the generalized Stokes theorem of differential geometry, this form, surprisingly, appears to be a new result.

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##A Generalization of Stokes' Theorem

It can be shown  that ellipse–flow equivalence is a manisfestion of a generalized version of Stokes' thereom. .cite[(Lilly 2018, 2023)]

That is, when the gradient tensor theorem is applied to an elliptical region in a linear flow, the result is ellipse–flow equivalence.

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##Change of Variables

At this point it is convenient to change ellipse variables to the moment of inertia $I$ and aspect ratio angle $\chi$,

<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>

where $\chi$ is defined such that $\chi=0$ for the circular case $a=b$, while $\chi=\pm\pi/4$ corresponds to $b$ and $a$ vanishing.

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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 finds .cite[(Lilly 2018, 2023) ]

<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>

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

Note that this is purely kinematic.  There has as of yet been no consideration of forces!

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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!

This is wrong:

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

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##The Shallow-Water Equations

Beginning with the nonrotating shallow water equations, we transform to the rotating frame in the usual way
<div>\[\frac{\partial}{\partial t}\mathbf{u}' =   -\mathbf{u}' \mathbf{\cdot} \nabla \mathbf{u}'  - g'\nabla h\]</div>

<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>

giving the rotating shallow water equations with a centrifugal term
<div>\[\frac{\partial}{\partial t}\mathbf{u}=   -\mathbf{u} \mathbf{\cdot} \nabla \mathbf{u} -f  \hat{\mathbf{k}}\times  \mathbf{u} -  g'\nabla h + \boxed{\frac{1}{4} f^2 \mathbf{x}}.\]</div>

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

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

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##The Shallow-Water Equations

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

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

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

and the omitted centrifugal term reappears with opposite sign!

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

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

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

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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.

In fact, this negative 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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##Potential Energy

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

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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. We have

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

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

<div>\[ h_*(t) = \frac{2V_o}{3\pi} \frac{1}{I_o +J_o\cos\left(f(t-t_o)\right) }\]</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>\[\phi(t)=-\frac{ft}{2} + \frac{1}{2}\frac{\zeta_o}{f}\frac{1}{\sqrt{1-\epsilon_o^2}}\arctan\left(\frac{ \epsilon_o+\tan \frac{1}{2}f(t-t_o)}{\sqrt{1-\epsilon_o^2}}\right)+\phi_o\]</div>

where $\epsilon_o\equiv J_o/I_o$, and $\zeta_o\equiv \Pi_o/(\pi \rho_o^2)$ is the vorticity.

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

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##The Pulson Spirograph

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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, similar to that for the pulson case—a new result.

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## Recoverability

Consider the kinematic model of a signal generated by a particle orbiting a time-varying ellipse.

From Bedrosian (1963), ellipse parameters for signals above the dotted line are exactly recoverable from a single trajectory.  Recoverability of the exact vortex solutions is a next step.

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name: conclusions
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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!

For all submitted papers as well as my software toolbox, kindly visit my [{website}](http://www.jmlilly.net).

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