# One dimensional non-Hausdorff manifolds and foliations of the plane

André Haefliger and Georges Reeb
(translated by Gangotryi Sorcar)

(6 December 1956)

## Foreword from the translator

The original article titled “Variétés (non séparées) à une dimension et
structures feuilletées du plan” was published in 1957 in French in
_L’Enseignement Mathématique_. It establishes a beautiful connection between
foliations of the plane and non-Hausdorff $1$-dimensional manifolds arising
naturally as leaf spaces of the foliations. Since its appearance, this theory
has paved the way for several results concerning dynamical systems and
foliations of the plane and $2$-manifolds. For example, in $1985$ Neumann and
Goel studied unstable dynamical systems of $2$-manifolds. In $2005$, LeRoux
studied homeomorphisms of the plane that are fixed point free and orientation
preserving (also known as Brouwer homeomorphisms or free mappings). In $2012$,
LeRoux, O’Farrell, Roginskaya, and Short developed necessary and sufficient
conditions for certain plane homeomorphisms (Reeb homeomorphisms) to be
embeddable within a flow. More recently, in $2021$, Groisman and Nitecki have
proved a characterization of non-trivial foliations of the plane and used it
to study conjugacy classes of Anosov diffeomorphisms of the plane that are
time-$1$ maps of flows. Sushil Bhunia and the translator are currently working
on classifying conjugacy classes of Brouwer homeomorphisms embeddable in
flows. Haefliger and Reeb’s article inspires many results in the theory of
foliation of $3$-manifolds as well: we refer the interested reader to Danny
Calegari’s book on the topic.

Haefliger and Reeb’s article also has been applied to areas outside
topological dynamics. For example, in $2008$ Baillif and Gabard showed that
connected homogeneous manifolds need not be Hausdorff; in their result, the
non-Hausdorff $1$-dimensional manifolds from this article proved to be useful.
This article has been referenced in $43$ papers to-date and has helped build
many nice results. Only very few of these results are mentioned here due
entirely to the translator’s inability to convey the remainder in a succinct
manner. In the literature, the main theorem of this article has been commonly
referred to as Haefliger-Reeb theory or “a classical result by Haefliger and
Reeb.” However, to the best of our knowledge, no English version of this
article is currently available. We have kept the article entirely unchanged
except to add figures and correct a few typographical errors. We hope that
this translation will be useful for the broader audience.

## Introduction

The concept of a Hausdorff111Usually we say topological manifold instead of
Hausdorff topological manifold. topological manifold is fundamental to a
number of branches of mathematics. Recall its definition:

A Hausdorff $n$-dimensional topological manifold is a topological space
$V^{n}$ such that:

  1. 1.

Any point of $V^{n}$ has an open neighborhood homeomorphic to
$\mathbb{R}^{n}$.

  2. 2.

$V^{n}$ is Hausdorff, i.e. any two points of $V^{n}$ have open neighborhoods
that do not intersect.

The convenience of condition $2$ appears in the study of certain properties of
differential geometry and topology.

For example, a Hausdorff manifold $V^{n}$ with a countable basis is
metrizable. It follows that it is homeomorphic to a subspace of higher
dimensional Euclidean space. Any Hausdorff manifold of one dimension with
countable basis is homeomorphic to $\mathbb{R}$ or $\mathbb{S}^{1}$.

However, it seems useful to also study topological manifolds that are non-
Hausdorff. These spaces arise naturally while answering several
questions222For example, a sheaf defined on a Hausdorff manifold is generally
endowed with a non-Hausdorff manifold structure.. The purpose of this article
is to show how the study of one-dimensional manifolds (usually non-Hausdorff)
allows us to find several properties of foliations of $\mathbb{R}^{2}$.

The first section is devoted to the study of non-Hausdorff manifolds (in
particular those of one dimension). After providing some definitions and
examples (1.1), we establish some properties of $1$-dimensional simply
connected manifolds (1.2) and differentiable structures that can be defined on
them (1.3). These properties are applied in the second section.

The foliations of the plane were studied by Poincaré and many others. The
basic definitions and main results are collected in (2.1). The Theorems 2, 3,
and 4 are due to Kaplan, Kamke, and Wazewsky, become particularly clear in our
opinion if we start from the following fundamental remark: The space of leaves
of a foliation of $\mathbb{R}^{2}$ is a $1$-dimensional manifold (in general
non-Hausdorff) (2.2). These theorems are presented in (2.3).

##  1 Properties of $1$-dimensional manifolds

###  1.1 Definitions and examples

######  Definition 1.

An $n$-dimensional topological manifold $V_{n}$ is a topological space such
that any point $p$ admits an open neighborhood homeomorphism to
$\mathbb{R}^{n}$.

A homeomorphism $h:\mathbb{R}^{n}\rightarrow U\subset V_{n}$ where $U$ is the
image of $h$ is called a chart of $\mathbb{R}^{n}$ in $V_{n}$. The transition
map associated with two charts $h_{i}$ and $h_{j}$ of $\mathbb{R}^{n}$ in
$V_{n}$ with respective images $U_{i}$ and $U_{j}$ is the homeomorphism
$h_{j}^{-1}h_{i}$333If $f$ is a function from a subset $A$ of a set $E$ to a
set $E^{\prime}$ and $f^{\prime}$ is a function from a subset $A^{\prime}$ of
$E^{\prime}$ to a set $E^{\prime\prime}$, we will denote by $f^{\prime}f$ the
function $x\mapsto f^{\prime}[f(x)]$ of the subset of $E$ formed by points $x$
such that $f(x)\in A^{\prime}$.:

$h_{j}^{-1}h_{i}:h_{i}^{-1}(U_{i}\cap U_{j})\rightarrow h_{j}^{-1}(U_{i}\cap
U_{j})$

According to previous definition, there is always a set of charts whose images
cover $V_{n}$. Such a set will be called an atlas of $\mathbb{R}^{n}$ on
$V_{n}$444We are using the terminology and definition of C. Ehresmann..

Hausdorff topological manifolds often have additional structures: orientation,
differentiable structure, complex structure. These notions (as well as, for
example, tangent vectors and tensor products) are defined without appealing to
the Hausdorff axiom, therefore they translate immediately to non-Hausdorff
manifolds.

A general construction process:

######  Definition 2.

We say that a continuous function $p:E\rightarrow E^{\prime}$ “spreads $E$
over $E^{\prime}$” if every point $x\in E$ admits an open neighborhood $U$
such that $p|_{U}$ is a homeomorphism to an open set in $E^{\prime}$.

######  Proposition 1.

Let $V_{n}$ be an $n$-dimensional manifold and let $\rho$ be an equivalence
relation in $V_{n}$ for which every point $x\in V_{n}$ has a neighborhood
$U_{x}$ such that no two points in $U_{x}$ are $\rho$-equivalent. Then the
topological quotient space $V_{n}^{\prime}=V_{n}/\rho$ is an $n$-dimensional
topological manifold, the quotient map $p:V_{n}\rightarrow V_{n}/\rho$ is a
local homeomorphism, and $V_{n}$ is an étale space over $V_{n}/\rho$ (or
equivalently $p$ spreads $V_{n}$ over $V_{n}/\rho$).

Indeed, if $U$ is an open neighborhood of $x$ such that a restriction of
$\rho$ to $U$ is the identity, the restriction of $p$ to $U$ is a
homeomorphism of $U$ to an open set $U^{\prime}$ in
$V_{n}^{\prime}=V_{n}/\rho$; since each point of $V_{n}$ admits a neighborhood
homeomorphic to $\mathbb{R}^{n}$ so will it be for each point of
$V_{n}^{\prime}$.

Note that even if $V_{n}$ is Hausdorff, $V_{n}^{\prime}$ may not be, as the
examples ahead will show.

The procedure of construction of the manifold given in Proposition $1$ is
general in the following sense: Given an $n$-dimensional manifold $V_{n}$,
there is a family of charts $h_{i}:\mathbb{R}^{n}\rightarrow V_{n}$ whose
images cover $V_{n}$ (as $i$ takes all values within some index set $I$); the
manifold $V_{n}$ is therefore isomorphic to a quotient of the space
$I\times\mathbb{R}^{n}$ by the equivalence relation associated to the function
from $I\times\mathbb{R}^{n}$ to $V_{n}$ given by $(i,x)\rightarrow h_{i}(x)$.

######  Definition 3.

A point $x$ in a manifold $V_{n}$ is called a branch point if there is a point
$z\in V_{n}$ ($z\neq x$) which is not “separate” from $x$, i.e. any
neighborhood of $x$ has non-empty intersection with all neighborhoods of $z$.

Note: The relation “$x$ is not separate from $z$” is reflexive and symmetric,
but not transitive in general (see Example $3$ below).

_Examples._

Proposition 1 allows us to construct an array of examples of manifolds. Now,
let us study some examples, limiting ourselves to $1$-dimensional manifolds
that are second countable.

EXAMPLE $1$. Let $R_{1}$ and $R_{2}$ be two copies of the real line
$\mathbb{R}$ and let $\Sigma$ be the disjoint union of $R_{1}$ and $R_{2}$.
Consider an open set $\Omega$ in $\mathbb{R}$ and the equivalence relation
$\rho$ in $\Sigma$ that identifies points of $R_{1}$ and $R_{2}$ that have the
same coordinate $t\in\Omega$ and is reduced to identity for the other points,
and therefore satisfies the conditions of property of Proposition 1. Passing
to the quotient we get a $1$-dimensional manifold (not necessarily Hausdorff).
The branch points are those who have coordinates same as boundary points of
$\Omega$555The coordinate of point $x\in\Sigma/\rho$ is the coordinates of
points of $\Sigma$ that project onto $x$ by the canonical quotient map
$\Sigma\rightarrow\Sigma/\rho$..

It is useful to mention some particular cases.

  1. a)

The simple branching: Here $\Omega=(-\infty,0)$. The branch points have
coordinate $0$.

  2. b)

The lasso: Here $\Omega=(-\infty,-1)\cup(0,\infty)$. The branch points have
coordinates $-1$ and $0$.

  3. c)

The strangled lasso: Here $\Omega=\mathbb{R}\setminus\\{0\\}$. The branch
points have coordinate $0$.

  4. d)

$\Omega$ is the complement of the Cantor’s perfect set. Here the collection of
branch points is uncountable.

EXAMPLE $2$. The loop: Let $\rho$ be the equivalence relation on $\mathbb{R}$
which identifies the points with coordinates $t$ and $-t$ for $\lvert
t\rvert>1$666The original paper says $\lvert t\rvert<1$ which we have
interpreted as a typo since in that case the point $0$ will fail to satisfy
the condition of Proposition 1 and is reduced to identity for the other
points. The quotient space is the loop and the branch points are $1$ and $-1$.

EXAMPLE $3$. The star: Let $\Sigma$ be the topological sum of
$R_{1},R_{2},\ldots,R_{n}$, $n$ copies of $\mathbb{R}$. Let $\rho$ be the
equivalence relation in $\Sigma$ that identifies each point with coordinate
$t>0$ in $R_{i}$ with the point with coordinate $-t$ in $R_{i+1}$ ($1\leq
i\leq n$; assume $R_{n+1}=R_{1}$). The quotient space is a $1$-dimensional
manifold that looks like a star with $n$ branches. The branch points here are
the points with coordinate $0$. Two such points in $R_{i}$ and $R_{j}$ are
separate if and only if $i-j\neq 1$. We could also consider a star with
infinitely many branches.

EXAMPLE $4$: The plume: Example $1a)$ shows how it is possible to “graft” a
copy of $\mathbb{R}$ as a simple branching at a point $t=0$; we can obviously
graft one such branching at a point anywhere in $\mathbb{R}$. If we
simultaneously graft a simple branch at all points in $\mathbb{R}$ with
rational coordinates – it is not productive to describe this process in detail
– one obtains a manifold of $1$-dimension that we call “the plume”. The
straight line $\mathbb{R}$ is the stem on which the branches are grafted. Here
the branch points form a dense set in $\mathbb{R}$.

By grafting a branch at every point of $\mathbb{R}$ we would obtain a manifold
that does not have a countable basis.

EXAMPLE $5$: The compound plume: If in a plume we replaced each branch with a
new plume, we get a new manifold of $1$-dimension that we call the double
plume. In a double plume you can replace each branch with a simple plume, thus
obtaining a triple plume. Repeating this process $n$ times ($n$ being a
natural number) we get the $n$-plume. We can carry out a countable sequence of
these operations to get the $1$-dimensional manifold that is called the
compound plume. The compound plume has the following remarkable property: The
branch point set is countable and everywhere dense in the obtained space.

These examples show the great diversity of connected, second countable,
$1$-dimensional manifolds. A topological classification of these spaces
already seems quite complicated.

Let us give another example which will give a faint idea of the complexity of
non-Hausdorff manifolds of dimension greater than $1$. Let $E_{1}$ and $E_{2}$
be two copies of $\mathbb{R}^{2}$ with the polar coordinate system
$(r,\omega)$. Let $\rho$ be the equivalence relation on the space
$\Sigma=E_{1}\sqcup E_{2}$ that identifies any point of $E_{1}$ with
coordinate $(r,\omega)$, $r<1$ with the point of $E_{2}$ with coordinate
$(r,\frac{\omega}{1-r})$ and is identity for other points. The quotient space
$\Sigma/\rho$ is a $2$-dimensional manifold; the image in $\Sigma/\rho$ of any
point of $E_{1}$ with coordinate $(1,\omega_{0})$ is not Hausdorff from any
point of the image in $\Sigma/\rho$ of the circle $r=1$ of $E_{2}$.

###  1.2 Simply connected $1$-dimensional manifolds

Recall the following definitions:

######  Definition 4.

The pair $(\widetilde{V},p)$ of a topological space $\widetilde{V}$ and a
continuous function $p:\widetilde{V}\rightarrow V$, where $V$ is another
topological space, is called a covering of $V$ if every point of $V$ has an
open neighborhood $U$ such that $p^{-1}(U)$ admits a partition in open subsets
$U_{i}$ such that the restriction of $p$ on each $U_{i}$ is a homeomorphism
onto $U$.

A topological space $V$ will be called simply connected if it is connected and
if, for any connected covering $(\widetilde{V},p)$ of $V$, the projection $p$
is a homeomorphism of $\widetilde{V}$ to $V$.

######  Definition 5.

An $n$-dimensional manifold $V_{n}$ is said to be orientable if there exists
an atlas $A$ of $\mathbb{R}^{n}$ on $V_{n}$ such that any change of charts
associated to two charts in $A$ is an orientation preserving homeomorphism
from an open set of $\mathbb{R}^{n}$ to an open set of $\mathbb{R}^{n}$.

The manifolds constructed in examples $2$ and $3$ (for $n$ odd) are not
orientable.

If a $1$-dimensional manifold $V$ can be “spread” over (for which there exists
a local homeomorphism $V\to\mathbb{R}$) the real line $\mathbb{R}$, then $V$
is necessarily orientable. On the other hand, an orientable $1$-dimensional
manifold does not always spread over $\mathbb{R}$: just consider the case of
the circle. However, it is possible to show the following proposition which
will be useful later.

######  Proposition 2.

Suppose $V$ is a $1$-dimensional simply connected manifold with countable
basis. Then there exists a continuous function $f:V\rightarrow\mathbb{R}$
which is a local homeomorphism.

In the proof, we will use the following lemma:

######  Lemma 1.

If $V$ is a $1$-dimensional simply connected manifold, then for any $x\in V$
$V\setminus\\{x\\}$ has two connected components.

Indeed, let $U$ be a neighborhood of $x$ homeomorphic to an interval;
$U\setminus\\{x\\}$ has two connected components $U_{+}$ and $U_{-}$. Consider
now two copies $V^{\prime}$ and $V^{\prime\prime}$ of $V\setminus\\{x\\}$ and
let $U_{+}^{\prime}$, $U_{-}^{\prime}$ and $U_{+}^{\prime\prime}$,
$U_{-}^{\prime\prime}$ be the copies of $U_{+}$ and $U_{-}$ in $V^{\prime}$
and $V^{\prime\prime}$ respectively. Complete the space $V^{\prime}\sqcup
V^{\prime\prime}$ by two points $x^{\prime}$ and $x^{\prime\prime}$ admitting
respectively the neighborhoods $U^{\prime}$ and $U^{\prime\prime}$ such that
$U^{\prime}\cap V^{\prime}=U_{+}^{\prime}$, $U^{\prime}\cap
V^{\prime\prime}=U_{-}^{\prime\prime}$ and $U^{\prime\prime}\cap
V^{\prime}=U_{-}^{\prime}$, $U^{\prime\prime}\cap
V^{\prime\prime}=U_{+}^{\prime\prime}$ to obtain a space $\overline{V}$ that,
with the canonical projection $p$ on $V$ (in particular
$p(x^{\prime})=p(x^{\prime\prime})=x)$, is a $2$-sheeted cover of $V$777The
proof can be completed by noting that if $V\setminus x$ is connected, then
$\overline{V}$ is connected, hence $V$ is not simply connected..

Conversely, one can show that if $V$ is a $1$-dimensional connected manifold
with the property that the respective complement of every point in $V$ is
disconnected, then $V$ is simply connected.

Let us now proceed to the proof of the proposition. It is possible to find a
countable family of charts $h_{i}$ $(i=1,2,\ldots)$ from $\mathbb{R}$ to $V$
whose images $O_{i}$ cover $V$. Since $V$ is connected we can assume that the
numbering of $O_{i}$ is made such that the union
$\Omega_{n}=\bigcup_{i=1,2,\ldots,n}O_{i}$ is connected for all integers $n$.
Now we will reason by induction. Suppose we define on $\Omega_{n}$ a
continuous function $f_{n}$ that spreads $\Omega_{n}$ over the interval
$(-n,n)$. We will show that $f_{n}$ can be further extended to the function
$f_{n+1}$ that spreads $\Omega_{n+1}$ over the interval $(-n-1,n+1)$.

It follows from the lemma that $\Omega_{n}\cap O_{n+1}$ is connected because
if this were not the case then we could find a point $x$ such that
$V\setminus\\{x\\}$ is connected. So $h_{n+1}^{-1}(\Omega_{n}\cap O_{n+1})=I$
is an open interval of $\mathbb{R}$ on which the function $f_{n}h_{n+1}$ is
continuous, strictly monotonic, and such that $\lvert f_{n}h_{n+1}\rvert<n$.
The function $f_{n}h_{n+1}$ can be extended to a continuous function $\varphi$
on $\mathbb{R}$, strictly monotonic and for which $\lvert\varphi\rvert<n+1$.
The function $\varphi h_{n+1}^{-1}$888The original paper writes $\varphi
h^{-1}$ which is a typo. defined on $O_{n+1}$ and the function $f_{n}$ are
identical on $\Omega_{n}\cap O_{n+1}$, so amalgamating them defines the
desired extension $f_{n+1}$.

###  1.3 Manifolds with differentiable structures

######  Definition 6.

A $C^{r}$-differentiable structure for $r$ a positive integer or $\infty$
(respectively an analytic structure) on an $n$-dimensional manifold $V_{n}$ is
defined by the data of an atlas $A$ of $\mathbb{R}^{n}$ on $V_{n}$ such that,
for any two charts $h_{i},h_{j}\in A$, the transition map $h_{j}^{-1}h_{i}$ is
a homeomorphism that is $r$ times continuously differentiable (resp. analytic)
(i.e., $h_{j}^{-1}h_{i}$ is a $C^{r}$ (resp. analytic) homeomorphism) from an
open set of $\mathbb{R}^{n}$ to an open set of $\mathbb{R}^{n}$.

An $r$-differentiable function on $V_{n}$ is a function
$f:V_{n}\rightarrow\mathbb{R}$ such that for any chart $h_{i}\in A$, the
function $fh_{i}$ is a $C^{r}$ function on $\mathbb{R}^{n}$. An
$r$-differentiable function or $C^{r}$ function on $V_{n}$ is called “rank
$1$” at a point $x\in V_{n}$ if, for one chart $h_{i}$ whose image contains
$x$, the function $fh_{i}$ has at least one nonzero partial derivative at the
point $h_{i}^{-1}(x)$. This definition is obviously independent of the choice
of chart $h_{i}\in A$.

We similarly define the notion of $C^{r}$ functions from a
$C^{r}$-differentiable manifold $V_{n}$ to a $C^{r}$-differentiable manifold
$V_{m}$.

A chart $f:\mathbb{R}^{n}\rightarrow V_{n}$ is called “compatible” with atlas
$A$ if, for all $h\in A$, the transition maps $f^{-1}h$ and $h^{-1}f$ are
$C^{r}$ (or analytic) homeomorphisms from open sets of $\mathbb{R}^{n}$ to
$\mathbb{R}^{n}$. The set of all compatible charts with $A$ form a maximal
atlas $\overline{A}$ generated by $A$. Two sub-atlases of $\overline{A}$
define the same $C^{r}$-differentiable (or analytic) structure on $V_{n}$.

Property $1$: The simple branching (Example $1a)$) is a $1$-dimensional
manifold admitting two non-isomorphic (i.e. distinct) $C^{\infty}$-manifold
structures. (A similar argument applies to the strangled lasso.)

In other words, the simple branching can be given two $C^{\infty}$
differentiable structures such that there is no $C^{\infty}$ homeomorphism
(with a $C^{\infty}$ inverse) from $V$ with the first structure to $V$ with
the second structure.

Using the notations of the Example $1$, $R_{1}$ and $R_{2}$ (identified to
their images in $V$) are two open sets forming a cover of $V$; let the point
of $R_{1}$ (respectively $R_{2}$) of coordinate $t$ be designated $t_{1}$
(respectively $t_{2}$). One can define a $C^{\infty}$-differentiable structure
on $V$ by giving two maps $h_{1}$ and $h_{2}$ on $\mathbb{R}$ to $R_{1}$ and
$R_{2}$ respectively such that $h_{1}^{-1}h_{2}$ and $h_{2}^{-1}h_{1}$ are
$C^{\infty}$-homeomorphisms of the half line $(-\infty,0)$.

First structure: $h_{1}(t)=t_{1}$ and $h_{2}(t)=t_{2}$.

Second structure: $h_{1}(t)=t_{1}$ and $h_{2}(t)=t_{2}^{3}$.

For the first structure, the function which takes the value $t$ at the point
$t_{1}$ or $t_{2}$ is a $C^{\infty}$-function of rank $1$ everywhere on $V$.
On the other hand, for the second structure, any $C^{\infty}$-function $f$ on
$V$ is rank $0$ at the point $t=0$. Indeed, let $f_{1}=fh_{1}$ and
$f_{2}=fh_{2}$: from $f_{1}(t)=f_{2}h_{2}^{-1}h_{1}(t)$ we get
$\left(\frac{d}{dt}f_{1}(t)\right)_{t=0}=0$. Alternatively, one can use
$f_{2}=f_{1}h_{1}^{-1}h_{2}(t)$ to obtain
$\left(\frac{d}{dt}f_{2}(t)\right)_{t=0}=0$ and we thank the referee for
pointing this out. This establishes Property $1$999In a recent article Milnor
built two distinct differentiable structures on $\mathbb{S}^{7}$. Milnor’s
result is global, and on the contrary, here we have a local property relative
to an arbitrary open neighborhood of a pair of points that cannot be
separated..

Note that the two structures defined above are even analytic. It is clear that
we can produce multiple examples. We understand how to easily use the previous
example to construct a $C^{\infty}$-differentiable structure on a compound
plume which highlights the following property:

Property $2$: There exists $1$-dimensional manifolds (for example, the
compound plume) with $C^{\infty}$-differentiable structures such that
continuously differentiable functions on these manifolds are constant
functions.

The pathological properties highlighted above lead to a more strict notion of
“differentiable structure” for which the previous properties are no longer
valid.

######  Definition 7.

A $C^{r}$-differentiable structure on a manifold $V_{n}$ is called _regular_
if for all $C^{r}$-differentiable functions $f$ defined on a neighborhood of
$x\in V_{n}$, there is a $C^{r}$-differentiable function $f^{\prime}$ defined
on $V_{n}$ such that $f$ and $f^{\prime}$ coincide on a neighborhood of $x$.

All $C^{r}$-differentiable structures on Hausdorff manifolds are regular. The
second structure on the simple branching is not regular.

In the proof of Proposition $2$ we will use the lemma below:

######  Lemma 2.

Let $V$ be a manifold with a regular $C^{r}$-differentiable structure. If a
real-valued $C^{r}$ function on $V$ is of rank $1$ at a point $x\in V$, it is
also of rank $1$ at any point $y$ not separated from $x$.

To simplify the notation we will prove this lemma in the case where $V$ is a
$1$-dimensional manifold. Let $h_{1}$ and $h_{2}$ be two charts of
$\mathbb{R}$ in $V$ such that $h_{1}(0)=x$ and $h_{2}(0)=y$. The function
$h=h_{2}^{-1}h_{1}$ (as well as its inverse) is a $C^{r}$-homeomorphism from
an open set $U_{1}$ of $\mathbb{R}$ to an open set $U_{2}$ of $\mathbb{R}$,
the origin belonging to the closure of $U_{1}$ and $U_{2}$; the functions
$f_{1}=fh_{1}$ and $f_{2}=fh_{2}$ are $C^{r}$-differentiable on $\mathbb{R}$
and $f_{1}^{\prime}(0)\neq 0$. Suppose $t_{1},t_{2},\ldots,t_{n},\ldots$ is a
sequence of points in $U_{1}$ such that
$\displaystyle\lim_{n\to\infty}t_{n}=0$; by setting
$\overline{t}_{n}=h(t_{n})$ we also have
$\displaystyle\lim_{n\to\infty}\overline{t}_{n}=0$. As
$f_{1}^{\prime}(t_{n})=f_{2}^{\prime}(\overline{t}_{n})h^{\prime}(t_{n})$ for
all $n$, if
$\displaystyle\lim_{n\to\infty}f_{2}^{\prime}(\overline{t}_{n})=f_{2}^{\prime}(0)$
is $0$, then $\displaystyle\lim_{n\to\infty}h^{\prime}(t_{n})=0$ would be
infinite; but then if $g$ is a $C^{r}$ function on $V$ of rank $1$ at $y$
(such a function always exists by virtue of the assumption of regularity),
with the corresponding notations,
$\displaystyle\lim_{n\to\infty}g_{1}^{\prime}(t_{n})$ would be infinite, which
is impossible. So $f_{2}^{\prime}(0)\neq 0$.

Proposition $2$ can be restated in the following way:

Proposition $2$: _On any $1$-dimensional regular $C^{r}$-differentiable
manifold $V$ that is simply connected and has a countable basis there is a
$C^{r}$-differentiable function with rank $1$ everywhere._

In other words, the manifold $V$ can be spread over $\mathbb{R}$ by a $C^{r}$
function that is rank $1$ everywhere.

We assume the following known lemma:

######  Lemma 3.

Let $f(t)$ be a $C^{r}$ function defined on an open set $I^{\prime}$ in
$\mathbb{R}$ and whose derivative is nonzero at any point in a closed interval
$I\subset I^{\prime}$; the restriction of $f$ to $I$ can be extended to a
$C^{r}$ function on $\mathbb{R}$ whose derivative is nonzero everywhere on
$\mathbb{R}$.

Suppose $A$ is the atlas of $\mathbb{R}$ on $V$ that defines the
$C^{r}$-differentiable structure on $V$. The general setup of the proof is
like that of Proposition $1$ of $1.2$. Let us take the same notations this
time assuming that each $O_{i}$ is the image of the interval $(-1,1)$ by a
homeomorphism $h_{i}$ which extends along a homeomorphism
$\widetilde{h}_{i}\in A$ of $\mathbb{R}$ in $V$. We suppose a $C^{r}$ function
$f_{n}$ of everywhere rank $1$ is defined on $\Omega_{n}$ and that for all
$O_{i}$, $1\leq i\leq n$, the function $f_{n}h_{i}$ which is defined on
$I=(-1,1)$ may be extended to a $C^{r}$ function on $\mathbb{R}$ with rank $1$
everywhere. We will show that $f_{n}$ can be extended to a $C^{r}$ function
$f_{n+1}$ on $\Omega_{n+1}$ with the same properties.

As $\Omega_{n}\cap O_{n+1}$ is connected, $f_{n}h_{n+1}$ is a $C^{r}$ function
over an interval $(t_{0},t_{1})$ contained in $I$ with nonzero derivative. The
sets $h_{n+1}^{-1}(O_{i})$, $1\leq i\leq n$ are open intervals that cover
$(t_{0},t_{1})$. Suppose $O_{k}$ is an open set such that
$h_{n+1}^{-1}(O_{k})$ is an interval of the form $(t_{0},t_{2})$ where
$t_{2}\leq t_{1}$. By the induction hypothesis, the function $f_{n}h_{k}$ can
be extended on $\mathbb{R}$ to a function $\widehat{f}_{k}$ with rank $1$
everywhere. Let $t_{0}^{\prime}$ be the point of the interval $[-1,1]$ defined
by $t_{0}^{\prime}=\displaystyle\lim_{t\to t_{0}}h_{k}^{-1}h_{n+1}(t)$ and let
$x^{\prime}=\widetilde{h}_{k}(t_{0}^{\prime})$; the point
$x=\tilde{h}_{n+1}(t_{0})$ is not Hausdorff from $x^{\prime}$. There exists a
$C^{r}$ function $g$ on $V$ which coincides with
$\widehat{f}_{k}\widetilde{h}_{k}^{-1}$ on a neighborhood of $x^{\prime}$, and
as $g$ is of rank $1$ at $x^{\prime}$, it is also rank $1$ at $x$ (Lemma $2$).
The function which is equal to $gh_{n+1}$ on the interval $(-\infty,t_{0}]$
and to $f_{n}h_{n+1}$ on $(t_{0},t_{1})$ is $C^{r}$ on $(-\infty,t_{1})$
because the two functions $g\widetilde{h}_{n+1}$ and $f_{n}h_{n+1}$ coincide
in $(t_{0},t_{2}^{\prime})$. By repeating the same construction for $t_{1}$,
we get a function defined on $\mathbb{R}$ of rank $1$ on $(t_{0},t_{1})$ and
whose restriction to $(t_{0},t_{1})$ coincides with $f_{n}h_{n+1}$; according
to Lemma $3$, there exists a $C^{r}$ function $\widehat{f}_{n+1}$ that extends
$f_{n}h_{n+1}$ that is everywhere rank $1$. The function $f_{n+1}$ we seek is
equal to $f_{n}$ on $\Omega_{n}$ and to $\widehat{f}_{n+1}h_{n+1}^{-1}$ on
$O_{n}$.

######  Corollary 1.

All differentiable structures on $\mathbb{R}$ are equivalent.

Let $R$ be $\mathbb{R}$ with the usual differentiable structure on
$\mathbb{R}$ and $R^{\prime}$ be $\mathbb{R}$ with a $C^{r}$-differentiable
structure. According to the proposition there exists a $C^{r}$ function of
rank $1$ everywhere from $R^{\prime}$ to $R$ (up to a suitable homothety). As
this function is one-to-one, it is a $C^{r}$ isomorphism of $R^{\prime}$ and
$R$ (provided with the usual $C^{r}$-differentiable structure).

##  2 Foliations of $\mathbb{R}^{2}$

###  2.1 Review of definitions and classic properties

######  Definition 8.

A foliation $(F)$ on a $2$-dimensional manifold $V_{2}$ is defined by an atlas
$A$ from $\mathbb{R}^{2}$ to $V_{2}$ such that if $h_{i}$ and $h_{j}$ are any
two charts in $A$, the transition map $h_{ji}=h_{j}^{-1}h_{i}$ is a
homeomorphism from open set $U_{ji}$ of $\mathbb{R}^{2}$ to an open set of
$\mathbb{R}^{2}$ such that, in a neighborhood of any point of $U_{ji}$, it is
expressed by equations of the form:

$x^{\prime}=g_{ji}(x,y)\hskip 28.45274pty^{\prime}=k_{ji}(y)$ (1)

If the transition maps $h_{ji}$ are $C^{r}$ (resp. analytic), we will say that
the foliation $(F)$ is $C^{r}$ (resp. analytic)101010For a general definition
of foliated manifolds, see [6].

Let’s say that a chart $f$ of $\mathbb{R}^{2}$ in $V_{2}$ is compatible with
$A$ if for any chart $h\in A$, the transition map $f^{-1}h$ is also of the
form $(1)$ (and furthermore is $C^{r}$, resp. analytic, if $(F)$ is $C^{r}$,
resp. analytic). All of the atlases compatible with $A$ form the maximal atlas
generated by $A$ and defines on $V$ the foliation $(F)$. In what follows, we
will assume $A$ is a maximal atlas.

Let $U$ be an open set of $V_{2}$. The set of charts of $A$ whose images lie
in $U$ form an atlas of $\mathbb{R}^{2}$ on $U$ that defines the foliation
$(F_{U})$ induced from $(F)$ on $U$.

If $(F)$ and $(F^{\prime})$ are foliations of $V_{2}$ and $V_{2}^{\prime}$
respectively defined by the maximal atlases $A$ and $A^{\prime}$, an
isomorphism of $(F)$ to $(F^{\prime})$ is a homeomorphism
$\psi:V_{2}\rightarrow V_{2}^{\prime}$ such that $A^{\prime}=\psi A$ (that is
any chart of $A^{\prime}$ is of the form $\psi h$, where $h\in A$).

In the image $O_{i}$ of each chart $h_{i}\in A$, we define an equivalence
relation $\rho_{i}$ whose classes are the images of $h_{i}$ restricted on
straight lines $y=\text{constant}$. From (1) it follows that for any point
$x\in O_{i}\cap O_{j}$, the equivalence relations induced by $\rho_{i}$ and
$\rho_{j}$ coincide in a sufficiently small neighborhood of $x$. Let $\rho$ be
the equivalence relation generated by the relations $\rho_{i}$.

######  Definition 9.

The classes of $\rho$ in $V_{2}$ are called leaves of the foliation $(F)$.

The leaf space, i.e. the quotient space of $V_{2}$ by the equivalence relation
$\rho$ (provided with the quotient topology of that of $V_{2}$) will play an
essential role in the following.

Note that the equivalence relation $\rho$ is open since it is generated by the
$\rho_{i}$, which are open. Recall that for any vector field $E$ defined on a
Hausdorff manifold $V_{2}$ satisfying the following two conditions:

  1. (i)

$E$ is $C^{r}$-differentiable (or analytic),

  2. (ii)

$E(z)\neq 0$ for all $z\in V_{2}$,

there is an associated foliation which is $C^{r}$-differentiable (or
analytic). The leaves are then vector field trajectories of $E$. Reciprocally,
to all $C^{r}$-foliations $(F)$ of a Hausdorff manifold $V_{2}$ 111111The
original paper writes $\mathbb{R}^{2}$ instead of $V_{2}$. We thank the
referee for pointing this out. we can match a vector field $E$ on $V_{2}$
whose trajectories are the leaves of $(F)$.

Example: The curves which are solutions of the differential equation (in polar
coordinates $r$ and $\omega$) $\frac{dr}{d\omega}=r(1-r^{2})$ are the leaves
of an analytic foliation of $\mathbb{R}^{2}\setminus\\{0\\}$. The circle $r=1$
is one leaf around which the other leaves curl asymptotically.

The quotient space of $\mathbb{R}^{2}\setminus\\{0\\}$ by the equivalence
relation $\rho$ associated with the above foliation admits a partition into an
open subspace homeomorphic to two circles and one point whose only
neighborhood is the entire space.

######  Definition 10.

The pair $(O_{i},h_{i})$ formed by a chart $h_{i}\in A$ and its image $O_{i}$
is called a distinguished open set of the foliation $(F)$.

Let us state the main results related to the foliations of $\mathbb{R}^{2}$.
Theorem 1 below is classical; its proof is based on Jordan’s theorem (a
particularly easy application); it basically uses the fact that
$\mathbb{R}^{2}$ is simply connected (or more precisely, that its $\beta_{1}$
(first Betti number) is even).

######  Theorem 1 (Poincaré, Bendixon).

Let $(O_{i},h_{i})$ be a distinguished open set of a foliation of
$\mathbb{R}^{2}$. The image of $h_{i}^{-1}$ of the intersection of $O_{i}$
with any leaf is reduced to the empty set or to a line $y=\mathrm{constant}$.

######  Theorem 2 (Kaplan).

To any foliation of $\mathbb{R}^{2}$ we can associate a real valued function
$\psi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ with the following properties:

  1. (i)

$\psi$ is continuous and does not admit a maximum or a minimum.

  2. (ii)

$\psi$ is constant on the leaves of the foliation.

######  Theorem 3 (Kamke).

Let $(F)$ be a $C^{r}$-foliation of $\mathbb{R}^{2}$. Let $\Omega$ be a
bounded open set of $\mathbb{R}^{2}$. There exists a function
$\psi:\Omega\rightarrow\mathbb{R}$ satisfying the following properties:

  1. (i)

$\psi$ is $C^{r}$ and the gradient of $\psi$ is nonzero at all points of
$\Omega$.

  2. (ii)

$\psi$ is constant on the leaves of the foliation induced by $(F)$ on
$\Omega$.

######  Theorem 4 (Wazewsky).

We can equip $\mathbb{R}^{2}$ with a $C^{\infty}$-foliation such that any
$C^{r}$ function on $\mathbb{R}^{2}$ that is constant on the leaves of the
foliation is reduced to a constant function.

We will not reproduce the proof of Theorem 1 but we will show in 2.2 and 2.3
that Theorems 2, 3, and 4 are consequences of Theorem 1 and the properties of
$1$-dimensional manifolds (as) described in the first part.

Examples of foliations of $\mathbb{R}^{2}$:

  1. 1.

The lines $y=\text{constant}$ are obviously the leaves of a foliation of
$\mathbb{R}^{2}$.

  2. 2.

Let $C$ be a Jordan curve in $\mathbb{R}^{2}$. The previous foliation induced
by $C$ is homeomorphic to an analytic foliation on $\mathbb{R}^{2}$.

  3. 3.

The complement $U$ in $\mathbb{R}^{2}$ of the set of points with coordinates
$x=0$ and $y\geq 0$ is homeomorphic to $\mathbb{R}^{2}$. The connected
components of the level curves of the function $\psi=xy$ are the leaves of an
analytic foliation on $U$.

###  2.2 The leaf space of a foliation of $\mathbb{R}^{2}$

The following proposition is an essential consequence of Theorem 1 of Section
2.1.

######  Proposition 3.

Let $(F)$ be a foliation of $\mathbb{R}^{2}$. The quotient space $V$ of
$\mathbb{R}^{2}$ by the equivalence relation $\rho$ associated with the
foliation (see Definition (9) of Section 2.1) is a $1$-dimensional manifold
that is simply connected and has a countable basis. If $(F)$ is a
$C^{r}$-foliation (or analytic foliation) the leaf space $V$ is provided with
a canonical $C^{r}$ (or analytic) manifold structure.

As $\mathbb{R}^{2}$ is connected and has countable basis, $V$ is also
connected and has countable basis. To show that $V$ is a $1$-dimensional
manifold it will suffice to show that all points $z$ in $V$ admit an open
neighborhood homeomorphic to $\mathbb{R}$. Let $\pi:\mathbb{R}^{2}\rightarrow
V$ be the canonical projection; the leaf $\pi^{-1}(z)$ meets at least one
distinguished open set $O_{i}$. The equivalence relation induced by $\rho$ on
$O_{i}$121212The original paper writes $O^{i}$ but we think it is a typo and
it should be $O_{i}$. is the relation $\rho_{i}$, according to Theorem $1$. So
$\pi(O_{i})$, which is an open neighborhood of $z$ because $\rho$ is an open
equivalence relation, is homeomorphic to $O_{i}/\rho_{i}$ and therefore to
$\mathbb{R}$.

By virtue of Jordan’s theorem, the complement of any leaf (which is a closed
subset of $\mathbb{R}^{2}$) has two connected components; the complement of
any point of $V$ therefore also has two connected components. The property is
equivalent to the fact that $V$ is simply connected (see Lemma 1).

Let $A$ be an atlas that defines a differentiable foliation on
$\mathbb{R}^{2}$, and let $\overline{h}_{i}$ be the restriction of the chart
$h_{i}\in A$ on the line $x=0$. The set of charts $\pi\overline{h}_{i}$ of
$\mathbb{R}$ in $V$ is an atlas that defines a differentiable structure on
$V$.

The reader can construct as an exercise the space of leaves of the foliations
of $\mathbb{R}^{2}$ defined in the above examples.

###  2.3 Applications of results of Section 1 to foliations of
$\mathbb{R}^{2}$

### Proof of Theorem 2:

Theorem 2 of 2.1 (Kaplan) is an immediate consequence of Proposition 2 and
Proposition 3. Let $f$ be a function that spreads $\mathbb{R}^{2}/\rho$ over
$\mathbb{R}$ (i.e. local homeomorphism); the function $\varphi=f\pi$ is a real
valued function on $\mathbb{R}^{2}$ that satisfies the conditions of Theorem
2.

### Proof of Theorem 3:

If the theorem is true for bounded open sets that are simply connected in
$\mathbb{R}^{2}$, it will also be true for all bounded open sets $\Omega$: it
suffices to consider the open set $C$ bounded by a circle which contains
$\Omega$; if $\varphi$ is a function in $C$ which satisfies the condition of
the theorem, it will obviously do so for its restriction to $\Omega$. So
Kamke’s theorem is a consequence of the restated version of Proposition 2 in
Section $1.3$ and of the following proposition:

######  Proposition 4.

The space of leaves $V^{\prime}=\Omega/\rho^{\prime}$ of the foliation
$(F^{\prime})$ on $\Omega$ induced by the foliation $(F)$ is provided with a
regular $C^{r}$-differentiable manifold structure.

Let $\pi$ and $\pi^{\prime}$ respectively be canonical projections of
$\mathbb{R}^{2}$ on $\mathbb{R}^{2}/\rho=V$ and
$\Omega/\rho^{\prime}=V^{\prime}$. The canonical injection
$\Omega\rightarrow\mathbb{R}^{2}$ defines a continuous function
$\psi:V^{\prime}\rightarrow V$ by passing to quotients. Since any
distinguished open set of $\Omega$ is a distinguished open set in
$\mathbb{R}^{2}$, the function $\psi$ spreads $V^{\prime}$ over $V$ (a local
homeomorphism) and it is $C^{r}$ and everywhere rank $1$. Let $x^{\prime}$ be
a point of $V^{\prime}$ and let $x=\psi(x^{\prime})$. As $\Omega$ is
relatively compact, the intersection of the leaf $F=\pi^{-1}(x)$ with the
closure $\overline{\Omega}$ of $\Omega$ is compact. We can then find a
distinguished open set $W$ in $\mathbb{R}^{2}$ sufficiently narrow and
elongated to contain $F\cap\overline{\Omega}$ and such that $W\cap\Omega$ is
saturated by leaves of $\Omega$. Let $U:=\pi(W)$, an open set containing $x$
homeomorphic to an interval. Let $f^{\prime}$ be a $C^{r}$ function defined on
an open set $U^{\prime}$ containing $x^{\prime}\in V^{\prime}$ homeomorphic to
an interval and sufficiently small that $\psi(U^{\prime})\subset U$. The
function $f=f^{\prime}\circ\psi|_{U^{\prime}}^{-1}$ is a $C^{r}$ function
defined on an open set containing $x$. Since $U$ is homeomorphic to an
interval, it is possible to construct a $C^{r}$ function $g$ on $U$ that
coincides with $f$ on an open neighborhood of $x$ and that is zero outside a
compact set $K$ containing $U$. As $\pi^{-1}(K)\cap W\cap\Omega$ is closed in
$\Omega$ and is saturated with leaves of $\Omega$, the function which is equal
to $g\pi$ on $W\cap\Omega$ and zero at other points of $\Omega$ is $C^{r}$,
hence passing to quotients we obtain a $C^{r}$ function $g^{\prime}$ on
$V^{\prime}$ which coincides with $f^{\prime}$ in a neighborhood of
$x^{\prime}$.

### Proof of Theorem 4:

It is generally necessary to assume that $\Omega$ is bounded for Kamke’s
theorem to be true, as the following example shows. Let $O_{1}$ and $O_{2}$ be
two copies of $\mathbb{R}^{2}$ in the disjoint union $O_{1}\sqcup O_{2}$.
Consider the equivalence relation $O_{1}\ni(x_{1},y_{1})\sim(x_{2},y_{2})\in
O_{2}$ for $x_{2}=x_{1}+1/y_{1}$, $y_{2}=y_{1}^{3}$ for all $y_{1}<0$, and
identity for all points such that $y_{1}\geq 0$ or $y_{2}\geq 0$. The quotient
space $E$ is homeomorphic to $\mathbb{R}^{2}$; the two charts $O_{1}$ and
$O_{2}$ form an atlas that defines an analytic foliation on $E$ where the leaf
space is the simple branching with the second differentiable structure defined
in Property $1$ of Section $1.3$.

This example helps to understand the construction of the example by Wazewsky.
In particular, it’s easy to imagine a $C^{\infty}$-foliation with leaf space a
composite plume (Example $5$ of Section $1.1$) equipped with a
$C^{\infty}$-differentiable structure such that any differentiable function on
$V$ is reduced to a constant.

CLASSIFICATION OF THE FOLIATIONS OF $\mathbb{R}^{2}$.

The problem of classification of foliations of $\mathbb{R}^{2}$ was completely
resolved by Kaplan. We will indicate briefly and without proof how our methods
also solve this problem.

It is insufficient to consider only the quotient space $V$ associated with the
foliation to characterize the foliation. For that we need to introduce an
order relation among the branch points. Let $V$ be $1$-dimensional simply
connected oriented manifold and let $A$ be an atlas of $\mathbb{R}$ on $V$
defining an orientation on $V$. Let $x_{1}$ and $x_{2}$ be two points of $V$
which are not separate. We will say that $x_{1}$ and $x_{2}$ are not separated
on the right (resp. left) and write $x_{1}\sim x_{2}$ mod $\lambda^{+}$ (resp.
$\lambda^{-}$) if, given two charts $h_{1}$ and $h_{2}$ of $A$ such that
$h_{1}(0)=x_{1}$ and $h_{2}(0)=x_{2}$, $h_{2}^{-1}h_{1}$ is defined for points
with coordinates greater than $0$ (resp. less than $0$). These two relations
are equivalence relations. Each class contains a finite or countable number of
points.

Let us call a foliation of $\mathbb{R}^{2}$ with consistent orientation on the
leaves an oriented foliation and let us say two oriented foliations are
equivalent if there is a self-homeomorphism of $\mathbb{R}^{2}$ that
transports the oriented leaves of the first foliation to those of the second
oriented foliation. (Notice that any foliation of $\mathbb{R}^{2}$ can be
oriented.) We then show that the leaf space of $V$ of any oriented foliation
of $\mathbb{R}^{2}$ is a $1$-dimensional oriented manifold whose order is
canonically defined in each equivalence class modulo $\lambda^{+}$ or
$\lambda^{-}$. We say that the oriented manifold $V$ is provided with an
ordering.

Two oriented foliations of $\mathbb{R}^{2}$ are equivalent if and only if
there exists a homeomorphism of the leaf space of the first foliation to the
leaf space of the second foliation that preserves orientation.

Finally, to any simply connected $1$-dimensional manifold with countable basis
that is orientable with an ordering, there corresponds a foliation of
$\mathbb{R}^{2}$.

_Acknowledgments:_ We wish to thank André Haefliger for proofreading this
translation. We also thank Indira Chatterji for helpful guidance and
encouragement, and the referee for their detailed feedback and comments.

## References

  * [1] Bendixon, _Sur les courbes défines par des équations différentielles._ Acta Mathematica, 24, (1901), pp. 1–80.
  * [2] Kamke, _Über die partielle Differentialgleichung $fz_{x}+gz_{y}=h$._ Math. Zeitschr., vol. 4 (1936), pp. 56-66 et vol. 42 (1936), pp. 287-300.
  * [3] Kaplan, _Regular curve families filling the plane I, II._ Duke Journ., 7 (1940), pp. 154-185 et 8 (1941), pp. 11-46.
  * [4] J. Milnor, _On manifolds homeomorphic to the $7$ sphere._ J. Annals of Math., 64 (1956), pp. 399–405.
  * [5] Poincaré,  _Sur les courbes défines par des équations différentielles._ Œuvres, vol. I.
  * [6] G. Reeb, _Sur certaines propriétés topologiques des variétés feuilletées._ Act. Scient. et Ind., 1183, Hermann, 1952.
  * [7] T. Wazewsky, _Sur l’équation $Pp+Qq=0$._ Mathematica, 8 (1934), pp. 103-116; 9 (1935), pp. 179-182.


# The sound-ring radiation of expanding vortex clusters

August Geelmuyden  School of Mathematical Sciences, University of Nottingham,
University Park, Nottingham, NG7 2RD, UK  Sebastian Erne  School of
Mathematical Sciences, University of Nottingham, University Park, Nottingham,
NG7 2RD, UK  Vienna Center for Quantum Science and Technology, Atominstitut,
TU Wien, Stadionallee 2, 1020 Vienna, Austria  Sam Patrick  School of
Mathematical Sciences, University of Nottingham, University Park, Nottingham,
NG7 2RD, UK  Department of Physics and Astronomy, University of British
Columbia, Vancouver, British Columbia, V6T 1Z1, Canada  Carlo Barenghi  Joint
Quantum Centre Durham-Newcastle, School of Mathematics, Statistics and
Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK  Silke
Weinfurtner  School of Mathematical Sciences, University of Nottingham,
University Park, Nottingham, NG7 2RD, UK  Centre for the Mathematics and
Theoretical Physics of Quantum Non-Equilibrium Systems, University of
Nottingham, Nottingham, NG7 2RD, UK

###### Abstract

We investigate wave-vortex interaction emerging from an expanding compact
vortex cluster in a two-dimensional Bose-Einstein condensate. We adapt
techniques developed for compact gravitational objects to derive the
characteristic modes of the wave-vortex interaction perturbatively around an
effective vortex flow field. We demonstrate the existence of orbits or sound-
rings, in analogy to gravitational light-rings, and compute the characteristic
spectrum for the out-of-equilibrium vortex cluster. The spectrum obtained from
numerical simulations of a stochastic Gross-Pitaevskii equation exhibiting an
expanding vortex cluster is in excellent agreement with analytical
predictions. Our findings are relevant for 2d-quantum turbulence, the semi-
classical limit around fluid flows, and rotating compact objects exhibiting
discrete circulation.

Introduction.— The interaction between waves and rapidly rotating objects
gives rise to a range of exotic phenomena. A prominent example is that of
compact gravitational objects, such as black holes and neutron stars, which
can exhibit circular orbits of light commonly referred to as light-rings [1].
The existence of light-rings implies that the gravitational field
significantly affects the wave propagation nearby. In particular, the light-
ring is intimately connected to the quasi-normal modes of black holes and
neutron stars [2], which are the damped linear oscillations (ringdown) of a
system as it settles into equilibrium. This black hole relaxation process, or
ringdown, following a black hole merger event has been observed by the LIGO
collaboration [3, 4].

The importance of the light-ring in characterising scattering processes is not
limited to gravitational systems. One can identify analogue or effective
light-rings in hydrodynamic systems. To derive effective light-rings in the
system one studies the wave-propagation of bulk (i.e. sound waves) or
interface waves [5] in rapidly rotating fluids. Recently it was shown that the
free surface of a macroscopic classical vortex flow oscillates at the analogue
of light-ring frequencies [6].

A promising new line of research within the gravitational wave community is to
extend the concept of light-rings from beyond the aftermath of a binary merger
all the way back before the merger begins [7]. This is somewhat surprising
since the light-ring is tied to wave scattering around a stationary
gravitational background, whereas a binary merger refers to a highly non-
linear gravitational field evolution. If the light-ring has predictive power
for the non-linear dynamics in general relativity, might the same then be true
in hydrodynamic systems? We show that this is the case, by adapting the light-
ring methods to predict the spectrum of sound emitted by a rapidly decaying
compact vortex cluster in a two-dimensional Bose-Einstein condensate (BEC).
Here, the effective light-rings correspond to circular trajectories of sound,
or sound-rings.

Figure 1: Simulations of a decaying vortex cluster. Left column: condensate
density at three qualitatively different stages of the decay. initial multi-
winded vortex (A); multi-winded vortex has decayed in many singly winded
vortices forming a disordered cluster (B); cluster has expanded (C). Right
column: enlargements to show the vortex configuration at the three stages
together with the outermost sound-ring radius (black circle) and the minimal
cluster size (white circle).

Being a macroscopic quantum fluid, a BEC is known to exhibit a wide range of
exotic phenomena including wave-propagation of heat (second sound) [8],
perfectly inviscid flow, and quantization of circulation [9]. The latter leads
to vortices in BECs (holes in the condensate) that are discretized and
topologically protected by their phase winding. Whereas prior studies of
effective light-rings have been conducted in draining (spiral) classical
hydrodynamical vortices [6], we show that the notion of light-ring extends to
the purely circular flow around quantized vortices and clusters thereof. As we
will show, the sound-ring of a cluster can extend well outside the cluster
core. Consequently, the cluster exhibits a characteristic spectrum independent
of vortex configuration. This demonstrates the predictive power of sound-rings
for composite rotational systems. Our findings are of direct relevance for
wave-vortex interaction in superfluids, complementing ongoing research in
quantum turbulence in two and three dimensions exhibiting clusters and bundles
of vortices, respectively.

Modelling.— We investigate the non-linear relaxation of a rapidly rotating
disc-shaped BEC through numerical simulations of the two-dimensional (2D)
Stochastic Gross-Pitaevskii Equation (SGPE), [10, 11],

$\begin{split}i\hbar\partial_{t}\Psi=&\
\left(1-i\gamma\right)\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+U(\mathbf{r})-\mu+g|\Psi|^{2}\right]\Psi\\\
&\quad+\eta(\mathbf{r},t)~{}.\end{split}$ (1)

Here, $t$ is time and $\mathbf{r}=(r,\theta)$ are 2D polar coordinates. We
consider a tight transverse confinement, such that dynamics along the the
transverse $z$ direction is effectively frozen. The wavefunction $\Psi$
describes the low-energy coherent modes in the system, coupled to a thermal
bath through the complex noise $\eta$ and damping $\gamma$ [12]. Here $m$ is
the boson mass, $\mu$ is the chemical potential, $g$ is the effective 2D
interaction constant, and $U(\mathbf{r})$ is a rotationally symmetric box-trap
with $U(\mathbf{r})\gg\mu$ for $|\mathbf{r}|\geq r_{B}$, where $r_{B}$ is the
radius of the confining potential [12]. In the following, we consider
dimensionless variables, with length measured in units of the healing length
$\xi=\hbar/\sqrt{m\mu}$, time in units of $\mu/\hbar$, and the wavefunction in
units of $\sqrt{\mu/g}$. In these units, only the external potential $U$ and
noise $\eta$ depend on the microscopic details of the system such that
solutions to Eq. (1) are applicable over a wide range of parameters.

Fig. 1 illustrates a typical evolution for the relaxation of a vortex cluster.
We initialize a cluster of singly winded vortices through the decay of an
energetically and dynamically unstable multiply ($\ell=29$) winded vortex
(stage A) [13]. Here, the stochastic driving $\eta$ seeds the instability,
ensuring spatial disorder of the resulting vortex ensemble by explicitly
breaking rotational symmetry. The initial nonlinear relaxation of the cluster
shows a strong increase in fluctuations transferring energy away from the core
(stage B). Here, the individual vortices form common time-independent sound-
rings extending to a radius $r_{\mathit{sr}}$ far outside the cluster. These
sound-rings determine the low-frequency sound-field outside the cluster,
independent of the nonlinear dynamics within its core. During the subsequent
evolution (stage C) the cluster further expands under the influence of the
damping $\gamma$, which (i) reduces reflection of fluctuations from the
boundaries and (ii) enables us to tune the typical timescale of the expansion.
Note, the presented results are independent of their specific values for
sufficiently small $\eta$ and $\gamma$ [12]. Once vortices have expanded
beyond the largest $r_{\mathit{sr}}$ the common sound-ring disappears,
breaking up into individual sound-rings for each vortex. This removes the
potential barrier enabling the low-frequency sound-field to probe the cluster
core.

Vortex cluster.— We study the fluctuations in the hydrodynamic approximation,
writing the wavefunction as $\Psi=\sqrt{\rho}e^{i\Phi}$, and consider small
fluctuations of the density $\rho=\rho_{0}+\delta\rho$ and phase
$\Phi=\phi_{0}+\delta\phi$, around a coarse-grained background field
$\psi_{0}=\sqrt{\rho_{0}}e^{i\phi_{0}}$. The background $\psi_{0}$ includes
the macroscopic flow of the condensate and, in general, depends on the motion
of individual vortices. However, for a sufficiently high number of vortices we
can define a slowly evolving coarse-grained background, by time-averaging over
the fast rotation of the cluster around its own axis. Therefore, $\psi_{0}$
only evolves due to the slow expansion of the cluster.

Chaotic movement of vortices within the cluster ensures ergodicity, such that
we can determine the time-averaged background through ensemble averages [12].
For a collection of $N$ irrotational vortices with winding $\ell_{j}$,
uniformly distributed within a cluster of size $R$, the coarse-grained
velocity field
$\mathbf{v}\equiv\nabla\phi_{0}=v_{\theta}(r)\,\hat{\mathbf{e}}_{\theta}$ is
given by the Rankine vortex

$v_{\theta}(r)=\begin{cases}\ell r/R^{2},\quad r\leq R\\\ \ell/r,\qquad\
r>R\end{cases}\text{ for }\ell\equiv\sum_{j}^{N}\ell_{j}~{},$ (2)

familiar from classical fluid mechanics [14]. In accordance with Feynman’s
rule [9], the flow within the cluster ($r<R$) resembles solid body rotation.
Far outside the cluster ($r>R$) one recovers, even without the time-averaging,
the irrotational flow field of a single multiply winded vortex. Note, in
particular, that this is independent of the details within the core and only
depends on the net winding $\ell$ of the cluster [12].

In the Thomas-Fermi approximation, $|\nabla\rho_{0}|\ll\rho_{0}$ and
$\nabla\mathbf{v}_{0}^{2}\ll 2n_{0}$, the coarse-grained density background
$\rho_{0}=1-\frac{1}{2}\mathbf{v}^{2}$ is given by

$\rho_{0}=\begin{cases}1+\frac{1}{2}\left(\ell/R\right)^{2}\left(r/R\right)^{2}-\left(\ell/R\right)^{2},\quad
r\leq R,\\\ 1-\frac{1}{2}\left(\ell/r\right)^{2},\qquad\ \ r>R\end{cases}$ (3)

for $\rho_{0}\gtrsim 0$ and $\rho_{0}\approx 0$ otherwise. Hence, the solid
body region is absent from the flow for $R<\ell/\sqrt{2}$ and the cluster
transitions to a single multiply winded vortex (c.f. Fig. 1), requiring
corrections beyond the Thomas-Fermi approximation.

Figure 2: Analytical predictions. Three qualitatively different regions
shaded: waves bound to the core region (red), waves trapped outside (blue),
and waves that communicate between the core and the boundary (yellow). Left
panel: the effective potential $\omega^{\pm}(r)$ (solid black line) for
$m=-15$. The symmetry $-\omega_{D}^{\pm}(m)=\omega_{D}^{\mp}(-m)$ means that
negative frequencies inform us of $m=+15$. Middle panel: radial phase space
trajectories for waves with azimuthal number $m=-15$. Right panel: allowed
frequencies $\omega$ over azimuthal number $m$ for all radii $r$ in the
system. For a single $m=\pm 15$, this corresponds to the left panel projected
onto the frequency axis.

Perturbative expansion.— To model the frequency content of the sound field,
Eq. (1) is linearized about the slowly moving background
$\partial_{t}\phi_{0},\partial_{t}\rho_{0}\ll 1$, i.e. $\partial_{t}R\ll 1$.
The resulting dynamics for the fluctuations is given by

$\displaystyle\left(\partial_{t}+\mathbf{v}\cdot\nabla\right)\delta\rho$
$\displaystyle=-\rho_{0}\nabla^{2}\delta\phi~{},$ (4a)
$\displaystyle\left(\partial_{t}+\mathbf{v}\cdot\nabla\right)\delta\phi$
$\displaystyle=-\left(1-\frac{\nabla^{2}}{4\rho_{0}}\right)\delta\rho~{}.$
(4b)

Here, we neglected $\eta,\gamma\ll 1$, which act as sources and sinks of
fluctuations but do not significantly alter the sound-field frequencies. In
the long wavelength limit, the excitations are phonons described by a single
relativistic Klein-Gordon equation on an effective spacetime determined by the
background flow $\mathbf{v}$. This mathematical equivalence builds the basis
of analogue gravity systems [15].

To model the sound-waves, we apply a WKB approximation, which amounts to
assuming that the solutions to (4) look locally like plane waves

$\begin{bmatrix}\delta\phi\\\
\delta\rho\end{bmatrix}=\begin{bmatrix}\mathcal{A}(r)\\\
\mathcal{B}(r)\end{bmatrix}e^{i\int k_{r}(r)dr+im\theta-i\omega t},$ (5)

with a slowly varying amplitude, i.e. $\partial_{r}k_{r}\ll k_{r}^{2}$,
$\partial_{r}\mathcal{A}\ll k_{r}\mathcal{A}$ and similarly for $\mathcal{B}$.
Here, we have also used the independence of $t$ and $\theta$ in (2) and (3) to
decompose into separate frequency $\omega$ and azimuthal $m$ components.
Inserting this ansatz into (4) and keeping only the dominant terms gives the
local dispersion relation,

$\omega_{D}^{\pm}=\frac{mv_{\theta}}{r}\pm\sqrt{\rho_{0}k^{2}+\tfrac{1}{4}k^{4}},\qquad
k=\sqrt{k_{r}^{2}+\frac{m^{2}}{r^{2}}}$ (6)

where a wave which satisfies the equations of motion must lie either on the
upper branch ($\omega=\omega_{D}^{+}$) or on the lower branch
($\omega=\omega_{D}^{-}$).

Figure 3: Simulation results. Left panel: ensemble-averaged squared amplitudes
for $m=\pm 15$. Central panel: radially averaged spectrum from $r=40$ (see
white vertical dashed line in left panel) to the boundary $r=120$. Right
panel: relative amplitudes of central panel at selected values for $m$. Here
$\Delta m=10$ corresponds to a amplitude of $0.1$ in that $m$-channel. Black
solid horizontal lines are depicting the sound-ring frequencies, and the
shaded region for $m>0$ is the delimiting frequency $\omega^{+}(R)$ from the
initial ($R_{0}=26.2\pm 0.3$) and final ($R_{1}=32.7\pm 0.5$) ensemble-
averaged core size. The black dashed line is the delimiting frequency
$\omega^{+}(r)$ at the average core size $r=R_{0}/2+R_{1}/2$. As expected it
peaks at the frequencies corresponding to core sizes at earlier times.

Using (5) we obtain the radial phase-space trajectory $(r(t),k_{r}(t))$ of a
wave that satisfies the dispersion relation $\omega_{D}^{\pm}$, by following a
trajectory of constant phase [16]. The resulting Hamiltonian dynamics
${\dot{r}=\partial_{k_{r}}\omega_{D}^{\pm}}$ with
${\dot{k}_{r}=-\partial_{r}\omega_{D}^{\pm}}$ inform us that waves stagnate
radially ($\dot{r}=0$) for $k_{r}=0$. We refer to the radius $r_{\mathit{tp}}$
where a mode stagnates as a turning point, and the frequencies

$\omega^{\pm}(r)=\frac{mv_{\theta}}{r}\pm\sqrt{\rho_{0}\frac{m^{2}}{r^{2}}+\frac{m^{4}}{4r^{4}}}$
(7)

of modes that stagnate at $r_{\mathit{tp}}$, as the turnover frequencies
$\omega^{\pm}$. Note that these curves inform us of the location of the
turning points $r_{\mathit{tp}}$ for a given frequency $\omega$ through the
relation ${\omega=\omega^{\pm}(r_{\mathit{tp}})}$. An example of
$\omega^{\pm}(r)$ for ${m=-15}$ is displayed in the left panel of Fig. 2. The
plotted negative frequencies inform us of the co-rotating ($m>0$) case
${m=15}$ by virtue of the symmetry ${-\omega(m)=\omega(-m)}$ in equation (6).

For positive frequency ($\omega>0$) counter-rotating waves, i.e. waves that
rotate in a direction opposite to that of the cluster ($m<0$), the
$\omega^{+}(r)$ frequencies can exhibit a local maximum. This local maximum,
which is an unstable fixed-point in radial phase space (see middle panel in
Fig. 2), is precisely the sound-ring which is situated at,

$r_{\mathit{sr}}=\sqrt{\tfrac{1}{2}(6\ell^{2}-m^{2})+\ell\sqrt{6\ell^{2}-m^{2}}},\quad\omega_{\mathit{sr}}=\omega^{+}(r_{\mathit{sr}}).$
(8)

Here $r_{\mathit{sr}}$ is the radius of the sound-ring and
$\omega_{\mathit{sr}}$ is the frequency of the mode which orbits the system at
this radius, see Fig. 2. Since the effective Hamiltonian is locally flat at
the sound-ring $r_{\mathit{sr}}$, i.e.
$\partial_{r}\omega^{+}|_{r=r_{\mathit{sr}}}=0$, waves which are created in
this region will linger around $r_{\mathit{sr}}$ before eventually dispersing
[17]. Note that the sound-ring only exists when the cluster is sufficiently
compact, i.e. $R<r_{\mathit{sr}}$, and only for azimuthal modes with
${|m|\lesssim\sqrt{6}|\ell|}$. For a singly winded vortex ($|\ell|=1$) the
sound-ring is of the order of the healing length $r_{\mathit{sr}}\sim 1$.

For co-rotating waves $m>0$, for which there is no sound-ring, all frequencies
originating in the core propagate freely to the boundary of the condensate.
These waves have ${\omega>\omega^{+}(R)}$ and occupy the yellow region of Fig.
2. By contrast, waves with $\omega<\omega^{+}(R)$ scatter with the barrier
$\omega^{+}$ in the region $r>R$ and therefore do not probe the core. These
waves occupy the blue region.

The existence of a sound-ring for counter-rotating modes alters this picture.
As in the previous case, we distinguish between two types of waves which
penetrate (yellow)/do not penetrate (blue) the core, with
$\omega_{\mathit{sr}}$ delimiting the two scenarios, see Fig. 2. There are two
important differences from the co-rotating case: (1) Waves with
$\omega_{\mathit{sr}}$ do not travel freely between the core and the boundary.
Instead, they linger at the sound-ring radius $r_{\mathit{sr}}$ before
dispersing both inwards and outwards. (2) There are frequencies below
$\omega_{\mathit{sr}}$ that exist in the core region, which are prevented from
escaping due to the presence of a potential barrier outside the core where the
flow is irrotational. This situation, where waves with
$\omega<\omega_{\mathit{sr}}$ become trapped in a region localised around the
core, i.e. $r\sim R$, is shaded red in Fig. 2. Notice that for sufficiently
compact clusters, the positive branch $\omega_{D}^{+}$ of the dispersion
relation takes negative values. By virtue of the symmetry
$-\omega_{D}^{\pm}(-m)=\omega_{D}^{\mp}(m)$, these waves appear as co-
rotating, positive frequency waves from the lower branch of the dispersion.

In general, a perturbed system will start to settle into a superposition of
resonant modes after the initial perturbation has had time to collide with the
boundaries of the system. The resonant frequencies are computed used a
procedure that is conceptually analogous to computing bound states of the
Schrödinger equation for a potential that contains either one or two wells
within the semi-classical approximation[12]. The frequencies which solve the
resonance conditions are illustrated as points in the right panel of Fig. 2.

Results.— In our simulations, and in experiments, damping is expected to act
more aggressively on higher frequencies. Therefore, the majority of the sound-
energy escaping the cluster must reside in the frequencies delimiting the
yellow and blue regions. Consequently, the peak amplitude for co-rotating
frequencies depends on the core size, and therefore on the time the waves were
emitted from the core. In comparison, the peak at the sound-ring frequencies
remains constant and is thus well resolved. This result is independent of the
validity of the model inside the core.

In Fig. 3 we display the frequency content of the sound field acquired from
our simulations. The left panel illustrates the radial dependence of the sound
in different frequencies for $m=-15$. Superimposing the turnover frequencies
from our model, we see that the data is in excellent agreement with our
prediction for $r>R$, since $\omega^{\pm}$ correctly separates the propagating
and evanescent regions of the system. By contrast, the model fails for $r<R$
since for our choice of $\ell$ (discussed later) waves in the studied
$\omega$-range will be sensitive to vortex dynamics.

Based on this observation, we display the spectrum in the central panel, i.e.
the radial average of the modulus squared over $r\in[r_{a},r_{b}]$ of the left
panel, for each $m$. We have chosen $r_{a}=40$ (vertical white dashed line in
left panel) to be just outside the largest final core in the ensemble. The co-
rotating waves are still affected by their dependence on the core size $R$ at
the time of departure (see Fig. 3). As expected, the counter-rotating power
spectrum in the simulations becomes independent of $r_{a}$ provided
$r_{a}>\mathrm{max}[r_{\mathit{sr}}(m)]$. The dominant features in this signal
are the sound-ring frequencies and the resonant modes living outside the
cluster (corresponding to the blue region in Fig. 2). Resonant frequencies in
the yellow and red regions are not present since these probe the region $r<R$
where our model breaks down.

Conclusion. In this work, we introduced the notion of the sound-ring to
describe the wave dynamics of a compact vortex. By direct simulation of the
full BEC equations of motion, we then verified that the sound-ring of the
average flow field is the dominant signal in the measured spectrum. This
complements recent progress in black hole physics, where the gravitational
waveform produced during binary mergers is determined by the light-ring of the
average one-body spacetime. In our case, the sound-ring plays a dominant role
since it exists in a region of the flow which is highly symmetric and non-
evolving over the course of evolution. The resulting signal is therefore
independent of the non-linear vortex dynamics inside the cluster, provided the
vortices are packed close enough together that the cluster radius is smaller
than the sound-ring, i.e. $R<r_{sr}$.

We expect our findings to be of interest in 2d-quantum turbulence. In a
confined system such as a trapped atomic condensate, the number of possible
vortex configurations is bounded and the entropy reaches a maximum at a finite
value of the energy [18]. Larger energy can be reached only by lower entropy
creating clustering of vortices of the same sign. Therefore, injection of
energy into such a 2d flow will promote the formation of vortex clusters and
sound rings. A coarse-grained velocity field of $\ell$ same-sign vortices will
induce a sound-ring of radius $r_{sr}\sim\ell$, see Eq. (8), in which a
dominant wave signal is localised around $\omega_{sr}$ in the frequency
domain. This is something which could be checked in current experimental and
numerical studies of 2d-quantum turbulence in which vortex clustering can be
induced by evaporative heating, i.e. the annihilation of vortex-antivortex
pairs [19].

Finally, although in our case we do not observe a discrete spectrum of modes
which probe the inside of the cluster, we expect these modes to be excited for
larger vortex clusters with $\ell\gtrsim 100$. In this limit, we believe the
discretization of angular momentum in the underlying micro-structure will be
hidden from the long wavelength modes, which perceive only the average
“classical” Rankine background. These considerations may be of interest for
the corresponding quantum to classical limit around both fluid flows as well
as rotating compact objects exhibiting discrete angular momentum.

Acknowledgements. We thank Steffen Biermann, Thiago Cardoso, Cisco Gooding,
Maurício Richartz, Jörg Schmiedmayer, Vitor B. Silveira, and William G. Unruh
for stimulating discussions. CB and SW acknowledge support provided by the
Science and Technology Facilities Council on Quantum Simulators for
Fundamental Physics (ST/T006900/1) as part of the Quantum Technologies for
Fundamental Physics programme. SW acknowledges support provided by the
Leverhulme Research Leadership Award (RL-2019 - 020), the Royal Society
University Research Fellowship (UF120112) and the Royal Society Enhancement
Grant (RGF/EA/180286), and partial support by the Science and Technology
Facilities Council (Theory Consolidated Grant ST/P000703/1). AG and SW
acknowledge support provided by the Royal Society Enhancement Grant
(RGF/EA/181015). SE and SW acknowledge support from the EPSRC Project Grant
(EP/P00637X/1). SE acknowledges support through the Wiener Wissenschafts- und
TechnologieFonds (WWTF) project No MA16 - 066 (“SEQUEX”), and SE funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No 801110 and the Austrian Federal
Ministry of Education, Science and Research (BMBWF) from an ESQ fellowship. SP
acknowledges support from the Natural Science and Engineering Research Council
of Canada.

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##  Appendix A Supplemental Material

###  A1 Numerical simulations

We model our system with the stochastic Gross-Pitaevskii equation (SGPE),
which in our dimensionless units (see main text) takes the form

$i\partial_{t}\Psi=(1-i\gamma)\left[-\frac{1}{2}\nabla^{2}+U-1+|\Psi|^{2}\right]\Psi+\varepsilon\eta~{}.$
(S1)

Here, the noise $\varepsilon\eta$ is written in terms of the dimensionless
variance $\varepsilon^{2}\in\mathbb{R}$ and a Gaussian white noise term
$\eta$, sampled from a standard complex normal distribution. The SGPE
describes the dynamics of the coherent region $\Psi$ while being in contact
with a thermal bath. The dissipative term $\gamma$ represents the particle
transfer between the coherent region (low-energy modes) and the thermal
reservoir (high-energy modes) and the noise $\eta$ represents the random
nature of incoherent scattering within the system. This leads to the system
approaching a thermal equilibrium state in the long time limit, with the
temperature $T$ determined by the relation $\varepsilon^{2}\sim T\gamma$. Note
that in our adimensionalized variables, the temperature varies with the atomic
species constituting the condensate. For the results presented in the main
text, we choose $\gamma=2.5\cdot 10^{-3}$ and $\varepsilon^{2}=2.5\cdot
10^{-3}$. Provided they are reasonably small, the values for the noise $\eta$
and damping $\gamma$ play little role for the results presented in the main
text. However, for the considered system, their presence plays a delicate role
in the numerical implementation.

The damping $\gamma>0$, which introduces energy loss to the system, makes the
vortex cluster decay faster and adds stability to the evolution. This greatly
reduces the computational cost of a single simulation. Moreover, as a cluster
decays into individual vortices, the energy is released as waves. In the
absence of damping ($\gamma=0$), one would have to wait a very long time for
the cluster to decay to sizes $R\simeq\ell$, and worse, by then the
condensate, and its resonances, would be highly populated by the waves emitted
in the process. Note, however, that choosing $\gamma$ too large results in
overcritical damping of waves and makes the cluster decay too fast for the
sound-ring phenomena to be relevant.

The main role of the noise $\varepsilon\eta$ is to generate realistic vortex
configurations in the early stage of the relaxation by explicitly breaking the
rotational symmetry of the initial state. To avoid imposing a specific vortex
configuration, the initial state of the simulation is a multiply winded
central vortex (defined precisely later). For such a state to decay into a
cluster of individual vortices, there must be some feature in the system that
provides the seeds for breaking the symmetry of the initial state. In the
absence of noise, the only non-rotationally symmetric feature in the
simulation is the cartesian grid, which results in early vortex configurations
inheriting the $\mathbb{Z}_{4}$ symmetries of the discretization. Inclusion of
noise, as e.g. quantum or the considered thermal fluctuations, alleviates this
problem and leads to realistic random vortex configurations.

The numerical discretization $\Psi_{n,i,j}\equiv\Psi(t_{n},x_{i},y_{j})$ of
the wave-function $\Psi$ is done onto a set of equal-sized cartesian voxels

$\\{t_{n}\\}\times\\{x_{i}\\}\times\\{y_{j}\\}\subseteq\mathbb{R}^{N_{t}}\times\mathbb{R}^{1536}\times\mathbb{R}^{1536}~{}.$

To resolve individual vortices, a spatial grid-spacing $\Delta x\equiv
x_{i+1}-x_{i}=1/5$ is chosen. The time-step $\Psi_{n,i,j}$ to $\Psi_{n+1,i,j}$
is performed using a Fourier-spectral Strang-splitting scheme as outlined in
[20]. The contribution from noise $\eta$ is added after each timestep using
$\Psi_{n+1,i,j}\mapsto\Psi_{n+1,i,j}+\varepsilon\eta\Delta t$. Although the
stability of the solver varies with the damping $\gamma$, the presented
simulations are using $\Delta t=5\cdot 10^{-3}$, which is well within the
stable region. To enforce periodicity of $\Psi$ at the boundaries
$x,y\in\\{\pm 1536\Delta x/2\\}$ of the simulation domain, and thus the
adequacy of the Fourier-spectral scheme, a sharp logistic trap of the form
$U(r)\simeq 10/(1+\exp(2(r_{B}-r)))$ with $r_{B}\simeq 614\Delta x$ is used.

Each simulation is initialized in the ground state of the external potential
$U(r)$ by evolving a uniform condensate under the influence of an initial
strong damping (imaginary time evolution). Once completed, the noise and
damping are set to their final small values and we imprint a multiply winded
vortex with $\ell=29$ in the center of the potential. Here we use the
approximate expression

$\Psi_{n,i,j}\mapsto\Psi_{n,i,j}e^{i\ell\theta(x_{i},y_{j})}$ (S2)

with the density adjusted according to (3) with a flat ($\rho_{0}=0$) core
inside $r=\ell/\sqrt{2}$. Note that due to the existence of damping the vortex
core relaxes to its exact form long before the cluster decays into individual
vortices.

###  A2 Post-processing and data analysis

Expectation values are calculated averaging over 51 independent realizations
$\\{\Psi_{a}(t_{n},x_{i},y_{j})\\}$ referred to as an ensemble. Each
$\Psi_{a}$ is cropped around a time window with $512$ time-frames
$t\in[t_{1},t_{2}]$ with $\Delta t=1$, such that the wavefunction has time-
averaged cluster radius $R=(1\pm 0.02)\ell$. The resulting ensemble averaged
cluster size in the window is $R\in[26.2\pm 0.3,32.7\pm 0.5]$. Here, $R$ is
estimated by fitting the wavefunction to the course-grained velocity field in
equation (2). To make sure this fit is adequate, it was tested against vortex-
tracking results and similar fits of the densities $\rho^{(a)}=|\Psi_{a}|^{2}$
to (3). The result of such a vortex tracking procedure is exhibited in Fig.
S1. The vortex trajectories in the right panel are plotted in the frame co-
rotating with the solid body flow in the core. Two features should be noticed
in this figure: (1) The vortex density is slightly larger in the outer regions
of the core. The consequence of this is a modification of the solid body flow
inside the core, wherein the peripheral region rotates faster than the central
region. (2) Even in the co-rotating frame, the vortex trajectories are seen to
be very chaotic.

The density data $\rho^{(a)}$ in the time-window is interpolated onto a polar
mesh and decomposed into the different frequency and azimuthal components
using a temporal Hilbert transform. The result is a complex function

$\rho_{m}^{(a)}(\omega,r)=\int^{t_{2}}_{t_{1}}\int^{2\pi}_{0}\rho^{(k)}(t,r,\theta)e^{-im\theta+i\omega
t}d\theta\,dt.$ (S3)

Finally, we average over a radial window, and over the ensemble, to obtain the
spectrum of the cluster $\langle|\rho_{m}^{(a)}(\omega)|^{2}\rangle_{r,a}$.
This is the quantity plotted in the central panel of Fig. 3. The quantity
plotted in the left panel is the ensemble-averaged (S3) for a given $m$, i.e.
$\langle|\rho_{m}^{(a)}(\omega,r)|^{2}\rangle_{a}$.

Figure S1:  Vortex tracking results. Left panel: density of the first time
frame in in the analysed window of a single realisation. White lines are drawn
along curves of constant phase. White circular dots are the detected phase
singularities, white crosses are the locations of the phase singularities with
inverted winding with respect to the net rotation of the cluster. Middle
panel: phase information of the data shown in the left panel. The dots and
crosses are the same as in the left panel. Right panel: traced trajectories of
phase singularities (vortices) in the co-rotating frame. Black dots are the
initial phase singularities shown in the left and middle panel.

###  A3 The Velocity field of a quantized vortex cluster

For a collection of $N$ vortices at positions
$\\{(x_{k},y_{k})\\}\subset\mathbb{R}^{2}$ with windings
$\\{\ell_{k}\\}\subset\mathbb{Z}\setminus\\{0\\}$, the background velocity
potential may be approximated as the sum of individual (boundary-independent)
windings provided that they are well over a healing length apart,

$\phi_{0}=\sum_{k=1}^{N}\ell_{k}\arctan\left(\frac{y-y_{k}}{x-x_{k}}\right).$
(S4)

The gradient $\mathbf{v}\equiv\nabla\phi_{0}$ can be conveniently written in
terms of complex variables $z=x+iy$ as

$v^{*}=\sum_{k=1}^{N}\frac{i\ell_{k}}{z_{k}-z}$ (S5)

where $*$ denotes the complex conjugate. Now focus on the case where the
center of vorticity is in the origin, i.e. $\sum_{n}\ell_{k}z_{k}=0$, and
impose an ordering
$0\leq|z_{1}|\leq...|z_{M}|\leq|z|\leq|z_{M+1}|...\leq|z_{N}|\equiv R$.
Introduce polar variables $z=re^{i\theta}$ and denote the azimuthal average by
$\langle\cdot\rangle$, then it follows from the residue theorem that

$\langle ve^{-i\theta}\rangle^{*}=\sum_{k=1}^{N}\frac{\ell_{k}}{2\pi
r}\oint_{|z|=r}\frac{dz}{z_{k}-z}=-\frac{i}{r}\sum_{k=1}^{M}\ell_{k}.$ (S6)

so that

$\langle\mathbf{v}\rangle=\frac{\ell_{M}}{r}\mathbf{e}_{\theta}~{}\text{ for
}~{}\ell_{M}\equiv\sum_{k=1}^{M}\ell_{k}~{},$ (S7)

where $\mathbf{e}_{\theta}\equiv ie^{i\theta}$. That is, the azimuthally
averaged velocity field is that of a single, central vortex with a winding
equal to the sum of the windings of the vortices contained inside the circle
of radius $|z|$. In fact, outside the cluster $|z|>|z_{N}|$, this relation
holds to second order in $|z_{N}/z|$ without the need for an azimuthal average

$\mathbf{v}=\frac{\ell}{r}\mathbf{e}_{\theta}+\mathcal{O}\left(\left|\frac{z_{N}}{z}\right|^{2}\right)~{}\text{
for }~{}\ell\equiv\sum_{k=1}^{N}\ell_{k}.$ (S8)

Assuming a uniform distribution of vortices inside $R$, i.e.
$\mathrm{d}\ell\propto\mathrm{d}A$, we may write

$\ell_{M}\simeq\ell\left(\frac{r}{r_{c}}\right)^{2}\text{ where
}r_{c}\equiv\begin{cases}&R\text{ for }r\leq R\\\ &r\text{ for
}r>R\end{cases}$ (S9)

equation (S9) takes the form of a drain-free Rankine vortex
$\mathbf{v}=v_{\theta}(r)\mathbf{e}_{\theta}$ with

$v_{\theta}=\frac{\ell}{r}\left(\frac{r}{r_{c}}\right)^{2}=\begin{cases}&\frac{\ell
r}{R^{2}}\text{ for }r\leq R\\\ &\frac{\ell}{r}\text{ for }r>R\end{cases}$
(S10)

To find an expression for the background solution
$\rho_{0}=1-\frac{1}{2}\mathbf{v}^{2}$ we need to obtain an expression for
$\langle\mathbf{v}^{2}\rangle$ inside the cluster. Note that the need for this
calculation arises because the azimuthally averaged velocity field is no
longer irrotational inside the cluster. The following procedure is equivalent
to the result one would obtain from including the integral of
$\mathbf{v}\times(\nabla\times\mathbf{v})$ from $r$ to infinity in the
Bernoulli equation. The outside is already given from equation (S8), but to
find the value inside the cluster we observe that

$\displaystyle v^{*}v$
$\displaystyle=\sum_{m,n}\frac{\ell_{n}\ell_{m}}{(z_{n}-z)(z_{m}^{*}-z^{*})}\simeq
r^{6}\sum_{m,n}\ell_{n}\ell_{m}\left[\frac{\mathbb{I}_{n\leq
M}}{r^{4}}-\frac{\mathbb{I}_{n>M}}{(z_{n}z^{*})^{2}}\right]\left[\frac{\mathbb{I}_{m\leq
M}}{r^{4}}-\frac{\mathbb{I}_{m>M}}{(z_{m}^{*}z)^{2}}\right]$
$\displaystyle=\left[\frac{\ell_{M}}{r}-\sum_{n>M}\frac{r^{3}\ell_{n}}{(z_{n}z^{*})^{2}}\right]\left[\frac{\ell_{M}}{r}-\sum_{n>M}\frac{r^{3}\ell_{n}}{(z_{n}^{*}z)^{2}}\right]=\frac{\ell_{M}^{2}}{r^{2}}-\frac{\ell_{M}}{r^{2}}\sum_{n>M}\ell_{n}\left[\left(\frac{z}{z_{n}}\right)^{2}+\left(\frac{z^{*}}{z_{n}^{*}}\right)^{2}\right]+\sum_{n>M}\sum_{m>M}\frac{r^{2}\ell_{m}\ell_{n}}{(z_{n}z_{m}^{*})^{2}}~{},$

where $\mathbb{I}_{C}$ is the indicator function. Taking the azimuthal average
removes the entire middle term and all off-diagonal entries in the last term,
leaving only

$\langle v^{*}v\rangle\simeq\frac{\ell_{M}^{2}}{r^{2}}+2\pi
r^{2}\sum_{n>M}\frac{\ell_{n}^{2}}{r_{n}^{4}}$ (S11)

For a uniform vortex distribution, only the diagonal terms $n=m$ in the sum
contributes, leaving us with

$\langle|\mathbf{v}|^{2}\rangle\simeq\frac{\ell^{2}}{r_{c}^{2}}\left[2-\frac{r^{2}}{r_{c}^{2}}\right]=\begin{cases}\frac{\ell^{2}}{R^{2}}\left[2-\frac{r^{2}}{R^{2}}\right]&\text{
for }r\leq R\\\ \frac{\ell^{2}}{r^{2}}&\text{ for }r>R\\\ \end{cases}$ (S12)

This finally allows expressing the background density

$\rho_{0}(r)=1-\frac{1}{2}\frac{\ell^{2}}{r_{c}^{2}}\left[2-\frac{r^{2}}{r_{c}^{2}}\right].$
(S13)

The models for the background density $\rho_{0}(r)$ in equation S13 and
velocity $v_{\theta}(r)$ in S10 is found to be in good agreement with
numerical results (see Fig. S2).

Figure S2:  Measured background. Left panel: measured velocities
$\mathbf{v}=\text{Im}\nabla\ln\Psi_{a}$ at the initial (red dot-dashed) and
final (blue dashed) timeframe. The ensemble averaged azimuthal mean is plotted
with the standard deviation over ensemble shaded. Right panel: azimuthally
averaged density profile $\rho^{(a)}(r)$ at the initial (red dot-dashed) and
final (blue dashed) timeframe. The density is compared to the uniform Thomas-
Fermi density $\sqrt{1-U(r)}$ (yellow dotted). In both panels, the model is
fitted to find the core size $R$ (vertical lines) and plotted as black lines
on top.

###  A4 Computing the resonant modes

Since the system is spatially finite, our model may populate resonant modes.
We compute these by imposing the boundary conditions and accounting for the
mode specific scattering induced by the effective potential $\omega^{\pm}(r)$.
This procedure is conceptually analogous to computing bound states of the
Schrödinger equation for a potential that contains either one or two wells
within the semi-classical approximation.

To leading order, the WKB expansion of the perturbative dynamics (4) gives the
dispersion relation (6). The constraint imposed by the dispersion relation can
be thought of as the requirement that the function

$H\equiv\frac{1}{2}\left(\omega-\frac{mv_{\theta}}{r}\right)^{2}-\frac{1}{2}k^{2}\left(\rho_{0}+\frac{1}{4}k^{2}\right),\
k^{2}\equiv k_{r}^{2}+\frac{m^{2}}{r^{2}}$

is zero, i.e. $H=0$. This is exactly a Hamilton-Jacobi equation with the phase
$\text{Im}(\ln\delta\rho)$ acting as the principal/action function. The WKB
propagation of waves can then be interpreted as the flow in phase space
generated by the Hamiltonian $H$. While the rotational invariance of $H$ allow
us to study the reduced, radial phase $(r,k_{r})$ for each $m$, problems arise
on turning points $\partial_{k_{r}}H=0$. At these points, the WKB assumption
of a phase varying much faster than the amplitude is violated. In this
section, we search for solutions to the wave equation close to a turning
point, and use the asymptotic expansions of these solutions to relate WKB
solutions on opposite sides of the turning point.

To start, we consider a turning point $(r_{0},{k_{r}}_{0})$ and expand the
Hamiltonian around this point

$\tilde{H}\equiv(\partial_{r}H)(r-r_{0})+\frac{1}{2}(\partial_{k_{r}}^{2}H)(k_{r}-{k_{r}}_{0})^{2}~{},$
(S14)

where it is understood that $\partial_{r}H$ and $\partial_{k_{r}}^{2}H$ is
evaluated at the turning point. Next, we relax the WKB assumption by promoting
$k_{r}$ to a differential operator $-i\partial_{r}$. The result is a
Hamiltonian operator

$\displaystyle\mathcal{H}$
$\displaystyle=(\partial_{r}H)(r-r_{0})+\frac{1}{2}(\partial_{k_{r}}^{2}H)({k_{r}}_{0}^{2}+2{k_{r}}_{0}i\partial_{r}-\partial_{r}^{2})$
(S15)

determining the solutions $\phi$ at the turning point through the differential
equation $\mathcal{H}\phi=0$. If $\partial_{k_{r}}^{2}H\neq 0$ at the turning
point, then the equation $\mathcal{H}\phi=0$ takes the form

$\displaystyle-\partial_{r}^{2}\phi+2i{k_{r}}_{0}\partial_{r}\phi+\left[{k_{r}}_{0}^{2}+Q(r-r_{0})\right]\phi=0$
(S16)

for $Q\equiv 2(\partial_{r}H)/(\partial_{k_{r}}^{2}H)$. Factoring out a plane
wave with $\phi\equiv\psi\exp(i{k_{r}}_{0}r)$, and substituting $z\equiv
Q^{1/3}(r-r_{0})$ this can be brought to the form of an Airy equation
$\psi^{\prime\prime}(z)=z\psi(z)$, whose solutions are Airy functions $Ai(z)$
and $Bi(z)$. Therefore, the solution to the wave equation around a turning
point $(r_{0},{k_{r}}_{0})$ is

$\phi=e^{i{k_{r}}_{0}r}\left[C_{1}Ai(z)+C_{2}Bi(z)\right]$ (S17)

for two constants $C_{1}$ and $C_{2}$. For the purpose of matching the WKB
modes on either side of the turning point, the only thing we need from the
Airy functions is their asymptotic expansion:

$\displaystyle Ai(z)$
$\displaystyle\simeq\frac{|z|^{-\frac{1}{4}}}{2\sqrt{\pi}}\begin{cases}e^{-\frac{2}{3}z^{\frac{3}{2}}}&\text{
for }z\rightarrow\infty\\\
2\cos\left(-\frac{2}{3}(-z)^{\frac{3}{2}}+\frac{\pi}{4}\right)&\text{ for
}z\rightarrow-\infty\end{cases}$ (S18a) $\displaystyle Bi(z)$
$\displaystyle\simeq\frac{|z|^{-\frac{1}{4}}}{2\sqrt{\pi}}\begin{cases}2e^{\frac{2}{3}z^{\frac{3}{2}}}&\text{
for }z\rightarrow\infty\\\
2\sin\left(-\frac{2}{3}(-z)^{\frac{3}{2}}+\frac{\pi}{4}\right)&\text{ for
}z\rightarrow-\infty\end{cases}$ (S18b)

Before matching with the WKB solutions close to this point, we first note that
(S14) can be inverted to inform us that if $\tilde{H}=0$ then

$\int k_{r}dr={k_{r}}_{0}r\pm\frac{2}{3}(-z)^{\frac{3}{2}}+\text{const}$

Therefore, we may match WKB modes of the form $\mathcal{A}\exp(i\int k_{r}dr)$
with the asymptotic expansions of (S17) independent of the evolution of the
amplitude. Matching with both $z\sim-\infty$ and $z\sim\infty$ results in the
linear relation

$\displaystyle\left[\begin{matrix}A_{+}\\\
A_{-}\end{matrix}\right]=M\left[\begin{matrix}B_{+}\\\
B_{-}\end{matrix}\right]\text{ where }M\equiv e^{-\frac{\pi
i}{4}}\frac{1}{2}\left[\begin{matrix}i&2\\\ 1&2i\end{matrix}\right]$

relating amplitudes defined by

$B_{+}e^{\frac{2}{3}z^{\frac{3}{2}}}+B_{-}e^{-\frac{2}{3}z^{\frac{3}{2}}}\overset{M}{\mapsto}A_{+}e^{\frac{2}{3}i(-z)^{\frac{3}{2}}}+A_{-}e^{-\frac{2}{3}i(-z)^{\frac{3}{2}}}.$

Now, we introduce naming conventions where $k_{r}^{\pm}$ refers to propagating
modes with positive ($+$) and negative ($-$) radial wavenumber $k_{r}$.
Likewise, $\tilde{k}_{r}^{\pm}$ refers to the evanescent modes that grow ($-$)
and decay ($+$) with radius $r$. To match, we note that $z<0$ whenever $Q>0$
and $r<r_{0}$, or $Q<0$ and $r>r_{0}$. Then, for $Q>0$ we may associate
$A_{\mp}$ above with the propagating WKB modes $k_{r}^{\pm}$, and $B_{\pm}$
with the evanescent WKB modes $\tilde{k}_{r}^{\mp}$ (notice the different
sign). For $Q<0$, the direction of $z$ is opposite to that of $r$ so $M$ maps
inner amplitudes to outer amplitudes, and $M^{-1}$ maps outer amplitudes to
inner amplitudes. That is, $M^{-1}$ maps outer amplitudes $A_{\pm}$
corresponding to $k_{r}^{\pm}$ into inner amplitudes $B_{\pm}$ corresponding
to $\tilde{k}_{r}^{\pm}$.

Staying with this notation, we may depict the transition as a graphical
mnemonic (with radius growing towards the right)

$\displaystyle M\equiv\frac{1}{2}e^{-\frac{\pi
i}{4}}\left[\begin{matrix}i&2\\\ 1&2i\end{matrix}\right]$ $\displaystyle
M^{-1}\equiv\frac{1}{2}e^{-\frac{\pi i}{4}}\left[\begin{matrix}2&2i\\\
i&1\end{matrix}\right]$

for $Q>0$ and $Q<0$ respectively. Here, arrows point to the right if the
phase, real or imaginary part, increases with radius, and to the left if it
decreases. That is, arrows on propagating lines indicate the direction of
increasing phase with $r$, i.e. the direction of propagation of a positive
frequency (and positive branch of the dispersion relation) modes. The arrows
on the evanescent modes indicate the direction in which the mode decays.

We note that if we define the flipping matrix $F$ by $F(a,b)=(b,a)$ then $MF$
flips the right arms of the diagram (swaps column vectors of $M$), and $FM$
swaps the left arms of the diagram (swaps row vectors of $M$). It is therefore
convenient to define the matrix $T\equiv FMF$ such that the diagrams take on
consistent “arrow” conventions:

$\displaystyle T\equiv FMF=\frac{1}{2}e^{-\frac{\pi
i}{4}}\left[\begin{matrix}2i&1\\\ 2&i\end{matrix}\right]$
$\displaystyle\tilde{T}\equiv M^{-1}=\frac{1}{2}e^{-\frac{\pi
i}{4}}\left[\begin{matrix}2&2i\\\ i&1\end{matrix}\right]$

Modes with frequencies larger than the sound-ring frequency (yellow modes) are
then determined by the following diagram

$\tilde{T}$$B$ (S19)

where the crossed circle $\otimes$ indicates an amplitude that is zero. Here,
the $B$ refers to the boundary condition. We shall consider an energy-
conserving boundary condition of the form $A_{-}=e^{2i\eta}A_{+}$. In our
case, the WKB propagator from radius $r_{i}$ to $r_{j}$ is given by

$\displaystyle\mathcal{W}_{ji}\equiv
Q_{ji}\left[\begin{matrix}e^{+i\Phi_{ji}}&0\\\
0&e^{-i\Phi_{ji}}\end{matrix}\right]\text{ for
}\Phi_{ji}\equiv\int_{r_{i}}^{r_{j}}k_{r}dr$

where $Q_{ij}$ is a geometrical amplitude factor satisfying $Q_{ij}Q_{ji}=1$.
The equation corresponding to diagram (S19) then takes the form

$\left[\begin{matrix}0\\\
\tilde{b}_{-}\end{matrix}\right]=\tilde{T}\mathcal{W}_{IB}A_{+}\left[\begin{matrix}1\\\
e^{2i\eta}\end{matrix}\right],$

where $r_{I}$ is the turning point inside the cluster core, and $r_{B}$ is the
outer boundary of the condensate. For this to permit non-trivial solutions, we
must have

$\cos\left(\eta+\frac{\pi}{4}+\int_{r_{I}}^{r_{B}}k_{r}dr\right)=0~{}.$ (S20)

This is the resonance condition for the modes that may propagate freely
between the core and the boundary – the yellow region in Fig. 2.

For frequencies smaller than the sound-ring frequency, the wave needs to
tunnel through the effective potential $\omega_{\pm}(r)$. This happens due to
the introduction of a new pair of turning points $r_{1}$ and $r_{2}$. That is,
the diagram for the corresponding modes is

$\tilde{T}$$T$$\tilde{T}$$B$ (S21)

whose equation is given by

$\left[\begin{matrix}0\\\
\tilde{b}_{-}\end{matrix}\right]=\tilde{T}\mathcal{W}_{I1}T\mathcal{W}_{J2}\tilde{T}\mathcal{W}_{2B}A_{+}\left[\begin{matrix}1\\\
e^{2i\eta}\end{matrix}\right]$

The result is

$4\cot\left(\Phi_{1I}\right)\cot\left(\eta+\frac{\pi}{4}+\Phi_{B2}\right)=\exp\left(-2\int_{r_{1}}^{r_{2}}|k_{r}|dr\right)$
(S22)

For a severe evanescent separation $|k_{r}(r_{2}-r_{1})|\gg 1$ this condition
splits into two separate conditions

$\displaystyle\cos\left(\int_{r_{I}}^{r_{1}}k_{r}dr\right)=0$ (S23a)
$\displaystyle\cos\left(\eta+\frac{\pi}{4}+\int_{r_{2}}^{r_{B}}k_{r}dr\right)=0$
(S23b)

which for a Neumann boundary condition $\eta=0$ are exactly the equations used
to compute the expected resonant frequencies in Fig. 2. Equation (S23a)
corresponds to resonant modes in the red regions, (S23b) to the blue regions,
and (S20) for the yellow region. Note that unlike the yellow condition (S20),
the blue condition only takes the form (S23b) in the case of severe evanescent
separation. Additionally, one may encounter cases where $r_{B}<r_{2}$ for
which there are no blue modes, and the red modes take their asymptotic form
(S23a). If $r_{B}>r_{2}$ and the right side of (S22) is not negligible (weak
evanescent separation), then there is no natural concept of red and blue
modes, but rather a set of combined resonant modes.