# Hydrodynamics of Quantum Vortices on a Closed Surface

Yanqi Xiong  Graduate School of China Academy of Engineering Physics, Beijing
100193, China  Xiaoquan Yu<EMAIL_ADDRESS>Graduate School of China Academy
of Engineering Physics, Beijing 100193, China  Department of Physics, Centre
for Quantum Science, and Dodd-Walls Centre for Photonic and Quantum
Technologies, University of Otago, Dunedin, New Zealand

###### Abstract

We develop a neutral vortex fluid theory on closed surfaces with zero genus.
The theory describes collective dynamics of many well-separated quantum
vortices in a superfluid confined on a closed surface. Comparing to the case
on a plane, the covariant vortex fluid equation on a curved surface contains
an additional term proportional to Gaussian curvature multiplying the
circulation quantum. This term describes the coupling between topological
defects and curvature in the macroscopic level. For a sphere, the simplest
nontrivial stationary vortex flow is obtained analytically and this flow is
analogous to the celebrated zonal Rossby-Haurwitz wave in classical fluids on
a nonrotating sphere. For this flow the difference between the coarse-grained
vortex velocity field and the fluid velocity field generated by vortices is
solely driven by curvature and vanishes in the corresponding vortex flow on a
plane when the radius of the sphere goes to infinity.

##  I Introduction

Fluids on curved surfaces exhibit rich phenomena which are absent on a plane.
The interplay between geometry, topology and fluid dynamics has been explored
extensively in diverse platforms, including quantum Hall liquids Son (2013);
Wiegmann (2013a); Cho  _et al._ (2014); Gromov and Abanov (2014); Gromov  _et
al._ (2015); Can  _et al._ (2015), active matter Mickelin  _et al._ (2018);
Shankar  _et al._ (2017); Rank and Voigt (2021), classical fluids Bogomolov
(1977); Reuther and Voigt (2015, 2018); Samavaki and Tuomela (2020), and soft
materials Sigurdsson and Atzberger (2016); Levine  _et al._ (2004); Rower  _et
al._ (2022).

The coupling between geometric potentials induced by curvature and quantum
vortices plays an essential role in determining properties of superfluids on a
curved surface Vitelli and Turner (2004); Turner  _et al._ (2010). For a
superfluid film, a curved surface is realized by the underlying substrate
Turner  _et al._ (2010). Recent experimental advances in Bose-Einstein
condensates (BECs) in International Space Station Aveline  _et al._ (2020) now
allow ultracold atomic bubbles Carollo  _et al._ (2022), providing a promising
possibility to investigate a bubble trapped superfluid experimentally.
Motivated by the experimental progress, research interests on few body vortex
dynamics on curved surfaces have been renewed Bereta  _et al._ (2021);
Caracanhas  _et al._ (2022); Padavić  _et al._ (2020), adding different
perspectives on a more mathematical treatment of point vortex dynamics on
curved surfaces Hally (1980); Kimura (1999); Dritschel and Boatto (2015).
However, the effects of curvature and topology on collective dynamics of
quantum vortices remain unexplored, motivating us to consider vortex fluids on
curved surfaces. Furthermore, static vortex distributions influenced by
curvature remains a challenge Turner  _et al._ (2010), especially when the
vortex number is large. Examining stationary solutions of such vortex fluid
equations would provide a feasible way to tackle this problem.

A vortex fluid is a coarse-grained model for a system consisting of a large
number of point vortices and its dynamical equations describe collective
dynamics of well-separated quantum vortices at large scales Wiegmann and
Abanov (2014a); Yu and Bradley (2017). The theory reveals several emergent
properties. For instance, a binary vortex fluid is compressible Yu and Bradley
(2017) while a chiral vortex fluid is incompressible Wiegmann and Abanov
(2014a); there exists an odd viscous tensor and the circulation quantum plays
the role of the nondissipative odd viscosity coefficient. The theory also
predicts a universal long-time dynamics of the vorticity distribution in a
dissipative superfluid and this prediction has been verified in experiments
Stockdale  _et al._ (2020). However, on a finite region with boundaries,
boundary conditions are difficult to incorporate in general, hence a closed
surface is a better venue for vortex fluids. Vortex fluids are also closely
related to quantum Hall liquids Wiegmann (2013b) and fractons Doshi and Gromov
(2021); Grosvenor  _et al._ (2021).

In this paper we develop a vortex fluid theory on orientable closed surfaces
with zero genus. For a closed surface, the total voticity must vanish and
hence we consider binary vortex fluids containing equal number of vortices and
anti-vortices. On a plane, the momentum flux tensor of the vortex fluid
contains an emergent odd viscous tensor and a quantum pressure like stress
tensor Yu and Bradley (2017), preventing applying the minimal coupling
principle directly to derive the covariant vortex fluid equation on a curved
surface. We overcome this difficulty by introducing an auxiliary tensor which
is mathematically equivalent to the original momentum flux tensor, however, is
more readily amenable for applying the minimal coupling substitution. After
the minimal coupling substitution and rewriting the equation in terms of the
original momentum flux tensor, we obtain the vortex fluid equation on a closed
surface in isothermal coordinates. The emergent curvature term plays the role
of a source term in the vortex fluid equation and hence might be referred to
as curvature anomaly. The generalized relation between the superfluid velocity
field generated by the vortices and the coarse-grained vortex velocity field
induces the equation of motion (EOM) of point vortices on closed surfaces,
verifying the minimal coupling approach. A connection between the odd viscous
tensor and Euler characteristic of the closed surface is obtained. For a
sphere, an exact stationary vortex flow solution determined by Gaussian
curvature is found, whose vorticity exhibits the profile of a vortex-dipole in
spherical coordinates and its velocity distribution has the profile of a
Kaufmann vortex in stereographic coordinates. It should be noted that the
obtained vortex fluid equation holds also for infinitely large curved
surfaces, where the vortex system does not have to be neutral.

##  II Quantum vortices and vortex fluids on a plane

In a superfluid, the circulation of a vortex is quantized in units of
circulation quantum $\kappa\equiv 2\pi\hbar/m$ Pitaevskii and Stringari
(2016), and the vorticity has a singularity at the vortex core
$\mathbf{r}_{i}$:
$\omega(\mathbf{r})=\nabla\times\mathbf{u}=\kappa\sigma_{i}\delta(\mathbf{r}-\mathbf{r}_{i})$
with sign $\sigma_{i}=\pm 1$ for singly charged vortices. Here $m$ is the
atomic mass and $\mathbf{u}$ is the fluid velocity generated by the vortex at
$\mathbf{r}=\mathbf{r}_{i}$. This quantization arises from the single-
valuedness of the macroscopic superfluid wave function. It ensures that the
vorticity of a quantum vortex concentrates around the core region in the
dynamics, which is not the general case for classical fluids Eyink and
Sreenivasan (2006). Hence when the mean separation between quantum vortices is
much larger than the vortex core size $\ell$, the point vortex model governs
the dynamics of quantum vortices Onsager (1949); Eyink and Sreenivasan (2006);
Novikov (1975); Aref  _et al._ (1999), provided vortex annihilation can be
neglected. In this regime, a superfluid at low temperatures is nearly
incompressible.

Let us introduce complex coordinates $z=x^{1}+ix^{2}$,
$\partial_{z}\equiv(\partial_{1}-i\partial_{2})/2,$
$\partial_{\bar{z}}\equiv(\partial_{1}+i\partial_{2})/2$ and complex velocity
$u\equiv u^{1}-iu^{2}$. For a system containing $N_{+}$ singly-charged quantum
vortices and $N_{-}$ anti-vortices, the superfluid velocity $u$ generated by
these vortices and the vortex velocity $v_{i}\equiv d\bar{z}_{i}/dt$ read

$\displaystyle
u=-\frac{1}{2\pi}\sum^{N}_{j=1}\frac{i\kappa\sigma_{j}}{z-z_{j}},\quad
v_{i}=-\frac{1}{2\pi}\sum^{N}_{j,j\neq
i}\frac{i\kappa\sigma_{j}}{z_{i}(t)-z_{j}(t)},$ (1)

where $u=2i\partial_{z}\psi$, the stream function
$\psi(z)=-\kappa/2\pi\sum_{i}\sigma_{i}\log\left|(z-z_{i})/\ell\right|$, and
$N=N_{+}+N_{-}$ is the total number of vortices. The vorticity is
$\omega(\mathbf{r})=\kappa\sum_{i}\sigma_{i}\delta(\mathbf{r}-\mathbf{r}_{i})$.
The above fluid velocity $u$ appears to be a singular solution of
incompressible two-dimensional (2D) Euler or Hemlholtz equation G.R.Kirchhoff
(1876) : $\partial_{t}\omega+\mathbf{u}\cdot\nabla\omega=0$, which describes
2D nonviscous incompressible classical fluids .

In the point vortex regime, the slow motion of vortices is nearly decoupled
from fast degree of freedom–acoustic modes. In this regime, a large number of
well-separated quantum vortices are almost isolated and can be treated as a
fluid Wiegmann and Abanov (2014b); Yu and Bradley (2017). On a plane, the
corresponding hydrodynamical equation is Yu and Bradley (2017)

$\displaystyle\partial_{t}(\rho
v^{\alpha})+\partial_{\beta}T^{\alpha\beta}+\rho\partial^{\alpha}p=0,$ (2)

where $\rho(\mathbf{r})\equiv\sum_{i}\delta(\mathbf{r}-\mathbf{r}_{i})$ is
vortex number density,
$\sigma(\mathbf{r})\equiv\sum_{i}\sigma_{i}\delta(\mathbf{r}-\mathbf{r}_{i})=\kappa^{-1}\omega$
is vortex charge density, $v^{\alpha}$ is vortex velocity field defined as
$\rho
v^{\alpha}\equiv\sum_{i}\delta(\mathbf{r}-\mathbf{r}_{i})v^{\alpha}_{i}$, $p$
is the fluid pressure, $T^{\alpha\beta}=\rho
v^{\alpha}v^{\beta}-\Pi^{\alpha\beta}$ is the momentum flux tensor, and

$\displaystyle\Pi^{\alpha\beta}=-\eta^{2}\sigma\partial^{\beta}\left(\frac{1}{\rho}\partial^{\alpha}\sigma\right)-8\pi\eta^{2}\sigma^{2}\delta^{\alpha\beta}-\sigma\tau^{\alpha\beta}$
(3)

is the emergent Cauchy stress tensor with $\eta=\kappa/8\pi$. This emergent
tensor $\Pi^{\alpha\beta}$ is absent in the Euler equation and describes
emergent macroscopic effects of discrete quantum vortices. In particular,

$\displaystyle\tau^{\alpha\beta}=-\eta\left(\epsilon^{\alpha}_{\gamma}\partial^{\beta}v^{\gamma}+\epsilon^{\beta}_{\gamma}\partial^{\gamma}v^{\alpha}\right)$
(4)

is the nondissipative odd viscous tensor and $\eta$ is identified as the odd
viscosity coefficient. Here $\epsilon^{1}_{2}=1,\epsilon^{2}_{1}=-1$, and
$\epsilon^{1}_{1}=\epsilon^{2}_{2}=0$. The presence of $\tau^{\alpha\beta}$ in
Eq. (3) is due to that in a vortex system the parity symmetry is broken,
namely $\eta\rightarrow-\eta$ under the parity transformation
$(x^{1},x^{2})\rightarrow(-x^{1},x^{2})\,\text{or}\,(x^{1},-x^{2})$. The odd
viscosity effects in 2D fluids are very rich Avron (1998); Fruchart  _et al._
(2023) and have been investigated in quantum Hall systems Avron  _et al._
(1995); Hoyos and Son (2012); Bradlyn  _et al._ (2012); Abanov (2013), chiral
active matter Souslov  _et al._ (2019); Banerjee  _et al._ (2017); Lou  _et
al._ (2022), chiral superfluids Hoyos  _et al._ (2014), 2D vortex matter
Wiegmann and Abanov (2014b); Yu and Bradley (2017); Bogatskiy and Wiegmann
(2019); Bogatskiy (2019); Moroz  _et al._ (2018) and classical fluids Ganeshan
and Abanov (2017); Monteiro and Ganeshan (2021).

The vortex fluid theory describes emergent collective dynamics of well-
separated quantum votices at large scales and is valid whenever the point
vortex model (PVM) Eq. (1) is applicable. Vortex annihilations due to
collisions, dissipative vortex dipole decay and boundary loss would modify the
PVM. For a low temperature BEC containing many vortices, annihilations due to
collisions limit dominantly the PVM approach. However, the collision rate
depends on the vortex distribution Yu and Bradley (2017) and for largely
vorticity-polarized states, including the vortex shear flow Yu and Bradley
(2017), Onsager clustered states Onsager (1949); Reeves  _et al._ (2014); Yu
_et al._ (2016); Gauthier  _et al._ (2019); Billam  _et al._ (2014); Skaugen
and Angheluta (2016), and the enstrophy cascade Reeves  _et al._ (2017),
vortex number losses are negligible.

##  III Vortex fluids on closed surfaces

The wisdom on deriving laws of physics in curved spacetime from those in flat
spacetime is the so-called minimal coupling (MC) principle. For our situation,
it means the following substitution:

$\displaystyle\delta_{\mu\nu}\rightarrow
g_{\mu\nu};\quad\partial_{\mu}\rightarrow\nabla_{\mu},$ (5)

where $g_{\mu\nu}$ the metric on the surface, and $\nabla_{\mu}$ is Levi-
Civita covariant derivative. When acting a vector field $V^{\nu}$,

$\displaystyle\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma^{\nu}_{\mu\lambda}V^{\lambda},$
(6)

where

$\displaystyle\Gamma^{\nu}_{\mu\lambda}=\frac{1}{2}g^{\nu\rho}\left(\frac{\partial
g_{\rho\mu}}{\partial x^{\lambda}}+\frac{\partial g_{\rho\lambda}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\lambda}}{\partial x^{\rho}}\right)$ (7)

is the connection coefficient–Christoffel symbol. The second covariant
derivatives do not commute, namely

$\displaystyle(\nabla_{\alpha}\,\nabla_{\beta}-\nabla_{\beta}\,\nabla_{\alpha})V^{\mu}=R^{\mu}_{\nu\alpha\beta}V^{\nu},$
(8)

where $R^{\mu}_{\nu\alpha\beta}$ is Riemann curvature tensor.

Unless specified, in the following we use isothermal coordinates

$\displaystyle
ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=h(x^{1},x^{2})[(dx^{1})^{2}+(dx^{2})^{2}],$
(9)

namely, $g_{12}=g_{21}=0$ and $g_{11}=g_{22}=h(x^{1},x^{2})$, where
$h(x^{1},x^{2})$ is a positive function and exists locally for 2D surfaces
Chern (1955). In isothermal coordinates, calculations are considerably
simplified. For instance, $g^{\alpha\beta}=\delta^{\alpha\beta}h^{-1}$ and
$v^{\alpha}=g^{\alpha\beta}v_{\beta}=h^{-1}v_{\alpha}$.

We define the vortex number density and vortex charge density on a curved
surface as

$\displaystyle\rho(x^{\mu})=\frac{1}{\sqrt{\det
g_{\mu\nu}}}\sum_{i}\delta(x^{\mu}-x^{\mu}_{i}),$ (10)
$\displaystyle\sigma(x^{\mu})=\frac{1}{\sqrt{\det
g_{\mu\nu}}}\sum_{i}\sigma_{i}\delta(x^{\mu}-x^{\mu}_{i}).$ (11)

The assumption of absence of vortex annihilation ensures the the following
continuity equations:

$\displaystyle\partial_{t}\rho+\nabla_{\mu}J_{n}^{\mu}=0,\quad\partial_{t}\sigma+\nabla_{\mu}J_{c}^{\mu}=0$
(12)

where

$\displaystyle J^{\mu}_{n}$ $\displaystyle=\frac{1}{\sqrt{\det
g_{\mu\nu}}}\sum_{i}\delta(\mathbf{r}-\mathbf{r}_{i})v^{\mu}_{i}\equiv\rho
v^{\mu},$ (13) $\displaystyle J^{\mu}_{c}$ $\displaystyle=\frac{1}{\sqrt{\det
g_{\mu\nu}}}\sum_{i}\delta(\mathbf{r}-\mathbf{r}_{i})\sigma_{i}v^{\mu}_{i}\equiv\rho
w^{\mu},$ (14)

are the currents for charge and number, respectively.

We use Eq. (5) to obtain the relation between $u$ and $v$ on a curved surface
from it on a plane Yu and Bradley (2017) :

$\displaystyle\rho v^{\mu}=\rho
u^{\mu}-\eta\epsilon^{\mu}_{\nu}g^{\nu\alpha}\nabla_{\alpha}\sigma,$ (15)
$\displaystyle\rho w^{\mu}=\sigma
u^{\mu}-\eta\epsilon^{\mu}_{\nu}g^{\nu\alpha}\nabla_{\alpha}\rho.$ (16)

Consequently,
$\omega_{v}-\omega=\eta\nabla_{\mu}(\frac{1}{\rho}\nabla^{\mu}\sigma)$, where
$\omega_{v}=\epsilon^{\gamma}_{\alpha}\nabla_{\gamma}v^{\alpha}$ and
$\omega=\epsilon^{\mu}_{\nu}\nabla_{\mu}u^{\nu}=8\pi\eta\sigma$. The vortex
fluid is compressible and
$\nabla_{\mu}v^{\mu}=-\eta\epsilon^{\mu}_{\nu}\nabla_{\mu}\left(\frac{1}{\rho}\nabla^{\nu}\sigma\right)\neq
0$ foo . Note that here
$\epsilon^{\beta}_{\alpha}\equiv\epsilon^{\beta}_{\,\,\,\,\alpha}=g^{\beta\gamma}\epsilon_{\gamma\alpha}$
and $\epsilon_{\gamma\alpha}=\sqrt{\det
g_{\mu\nu}}\tilde{\epsilon}_{\gamma\alpha}$ is the Levi-Civita tensor and
$\tilde{\epsilon}_{\gamma\alpha}$ is the Levi-Civita symbol. In isothermal co-
ordinates, the tensor $\epsilon^{\beta}_{\alpha}$ used here takes the same
value as what is introduced previously [below Eq. (4)]. For a scalar $f$,
$\nabla_{\alpha}f=\partial_{\alpha}f$, in complex coordinates, Eqs. (15) (16)
become

$\displaystyle\rho v=\rho u-2i\eta\frac{1}{h}\partial_{z}\sigma,$ (17)
$\displaystyle\rho w=\sigma u-2\eta i\frac{1}{h}\partial_{z}\rho.$ (18)

The above relations reveal that the velocity of a vortex at position
$\mathbf{r}$ is the fluid velocity excluding the flow generated by the vortex
itself at $\mathbf{r}$. The superfluid velocity field $u$ is irregular at a
vortex core and subtracting the pole at the vortex core leads to a regular
vortex velocity field $v$.

There is no solid reason why the MC principle must lead to correct results
Reall (2013). Hence justification is needed. To verify Eqs. (17)(18), let us
apply the relation (17), which is for coarse-grained variables, to discrete
point vortices. The fluid velocity generated by these point vortices on a
closed surface is $u=2ih^{-1}\partial_{z}\psi$ with the stream function

$\displaystyle\psi(z)=8\pi\eta\sum_{i}\sigma_{i}G(z,z_{i}),$ (19)

where $G(z,z_{i})$ is the Green’s function satisfying Dritschel and Boatto
(2015)

$\displaystyle\Delta G(z,z_{i})=-\delta_{z,z_{i}}+1/\Omega,$ (20)

$\Delta\equiv\nabla^{\mu}\nabla_{\mu}$, $\Omega$ is the area of the surface
and

$\displaystyle\delta_{z,z_{k}}\equiv h^{-1}\delta(z-z_{k}).$ (21)

The fluid velocity at $z=z_{k}$ is

$\displaystyle u_{z\rightarrow z_{k}}=\frac{16\pi\eta i}{h}\left[\sum_{i\neq
k}\sigma_{i}\partial_{z}G(z,z_{i})|_{z=z_{k}}+\sigma_{k}\lim_{z\rightarrow
z_{k}}\partial_{z}G(z,z_{k})\right],$ (22)

where the last part is the contribution from the vortex at $z=z_{k}$ itself
and contains a pole. To analyze the last term in Eq. (22), it is useful to
isolate the logarithmic singularity of the Green’s function Gustafsson (2022):

$\displaystyle
G(z,z_{k})=\frac{1}{2\pi}\left[-\log|z-z_{k}|+H(z,z_{k})\right],$ (23)

where $H(z,z_{k})=H(z_{k},z)$ is a regular function. Expanding in a power
series in $z$ around $z_{k}$, we obtain

$\displaystyle H(z,z_{k})=h_{0}(z_{k})+\frac{1}{2}h_{1}(z_{k})\,(z-z_{k})+{\rm
H.c.}+{\mathcal{O}}(|z-z_{k}|^{2})$

and

$\displaystyle\partial_{z}H(z,z_{k})=\frac{1}{2}h_{1}(z_{k})+{\mathcal{O}}(|z-z_{k}|)=\frac{1}{2}\partial_{z_{k}}h_{0}(z_{k})+{\mathcal{O}}(|z-z_{k}|).$

Here $h_{0}(z_{k})=H(z_{k},z_{k})$ and
$h_{1}(z_{k})=\partial_{z}H_{z,z_{k}}|_{z=z_{k}}$. Let us now analyze the
singular term in $\partial_{z}\sigma$. Noting that $2/(\pi
h)\partial_{\bar{z}}\partial_{z}\log|z-z_{k}|=\delta_{z,z_{k}}$ and re-
arranging derivatives , we obtain

$\displaystyle\lim_{z\rightarrow
z_{k}}\partial_{z}\sigma=-\sigma_{k}\delta_{z,z_{k}}\partial_{z}\log
h|_{z=z_{k}}-2\sigma_{k}\frac{1}{z-z_{k}}\delta_{z,z_{k}},$ (26)

where we have used $\partial_{\bar{z}}(1/z)=\pi\delta(\mathbf{r})$. Hence the
singular terms $\propto 1/(z-z_{k})$ in Eq. (22) and Eq. (26) cancel and the
remaining finite part in Eq. (17) gives rise precisely, by recognizing
$v(z=z_{k})=d\bar{z}_{k}(t)/dt$ and $\lim_{z\rightarrow
z_{k}}\rho=\delta_{z,z_{k}}$, the EOM of point vortices on closed surfaces
with zero genus Dritschel and Boatto (2015):

$\displaystyle\sigma_{k}h\frac{d\bar{z}_{k}(t)}{dt}=8\pi\eta
i\left[2\sum_{i\neq
k}\sigma_{k}\sigma_{i}\partial_{z}G(z,z_{i})|_{z=z_{k}}+\partial_{z_{k}}R_{\rm
robin}(z_{k})\right],$ (27)

where $R_{\rm robin}(z_{k})\equiv(1/2\pi)[h_{0}(z_{k})+\log\sqrt{h(z_{k})}]$
is the celebrated Robin function Gustafsson (2022).

Note that Eq. (27) holds for infinitely large curved surfaces as well Hally
(1980), and hence so do Eqs. (17)(18). For an infinitely large surface,
$R_{\rm robin}(z_{k})=(1/2\pi)\log\sqrt{h(z_{k})}$. In contrast to the
scenario on a plane, on a curved surface the self-energy of a vortex is
position dependent and a single vortex may move driven by the geometrical
potential (Robin function) Turner  _et al._ (2010). It was not an easy task to
obtain the EOM of point vortices on closed surfaces Dritschel and Boatto
(2015). From the vortex fluid point of view, it is somewhat striking that
relation (17) naturally generalized from it on a plane could lead to Eq. (27).

##  IV Dynamical equations of vortex fluids on closed surfaces

The Euler equation on a curved surface can be obtained from its form on a
plane applying the MC principle Reuther and Voigt (2015); Mickelin  _et al._
(2018):

$\displaystyle\partial_{t}u^{\alpha}+\nabla_{\beta}{\cal T}^{\alpha\beta}=0,$
(28)

where the momentum flux tensor ${\cal
T}^{\alpha\beta}=u^{\alpha}u^{\beta}+pg^{\alpha\beta}$ (hereafter we set the
fluid (mass) density $n=1$). Unlike the case of Euler equation, we can not
apply the MC principle to Eq. (2) directly. The reason is that there are terms
containing second derivatives of vectors in Eq. (2). On a plane, the order of
derivatives of these terms are interchangeable, namely:
$\partial_{\beta}\partial^{\alpha}\partial^{\beta}\sigma=\partial^{\alpha}\partial_{\beta}\partial^{\beta}\sigma$
and
$\partial_{\beta}\partial^{\gamma}v^{\alpha}=\partial^{\gamma}\partial_{\beta}v^{\alpha}$.
However on a curved surface,
$\nabla_{\beta}\nabla^{\alpha}\nabla^{\beta}\sigma\neq\nabla^{\alpha}\nabla_{\beta}\nabla^{\beta}\sigma$,
and
$\nabla_{\beta}\nabla^{\gamma}v^{\alpha}\neq\nabla^{\gamma}\nabla_{\beta}v^{\alpha}$.
At this stage, there is no preferred order for which the MC substitution
should be applied.

Our strategy is to search for another tensor $Q^{\alpha\beta}$ such that

$\displaystyle 1)\text{ it does not contain derivatives of vectors};$
$\displaystyle
2)\,\partial_{\beta}T^{\alpha\beta}=\partial_{\beta}Q^{\alpha\beta}.$

To do so, it is convenient to use complex coordinates, in which, Eq. (2)
becomes

$\displaystyle\partial_{t}(\rho
v)+\partial_{z}T_{z\bar{z}}+\partial_{\bar{z}}T+\rho\partial_{z}(2p)=0,$ (29)

where

$\displaystyle T$ $\displaystyle=\rho
vv+4\eta^{2}\sigma\partial_{z}\left(\frac{1}{\rho}\partial_{z}\sigma\right)-4i\eta\sigma\partial_{z}v,$
(30) $\displaystyle T_{z\bar{z}}$ $\displaystyle=\rho
v\bar{v}+16\pi\eta^{2}\sigma^{2}+4\eta^{2}\sigma\partial_{\bar{z}}\left(\frac{1}{\rho}\partial\sigma\right)$
(31) $\displaystyle=\rho
v\bar{v}+4i\eta\sigma\partial_{\bar{z}}v-4\eta^{2}\sigma\partial_{\bar{z}}\left(\frac{1}{\rho}\partial\sigma\right).$

In the last step we have used $\partial_{\bar{z}}u=-4\pi i\eta\sigma$ and
$u=v+2i\eta\partial_{z}\sigma/\rho$ Yu and Bradley (2017).

Let us now define

$\displaystyle Q_{z\bar{z}}$ $\displaystyle\equiv\rho v\bar{v}-4i\eta
v\partial_{\bar{z}}\sigma+4\eta^{2}\frac{1}{\rho}\partial_{\bar{z}}\sigma\partial\sigma,$
(32) $\displaystyle Q$ $\displaystyle\equiv\rho vv+4i\eta
v\partial_{z}\sigma-4\eta^{2}\frac{1}{\rho}\partial_{z}\sigma\partial_{z}\sigma.$
(33)

Clearly condition 1) is satisfied. Since
$T_{z\bar{z}}-Q_{z\bar{z}}=4i\eta\partial_{\bar{z}}(\sigma
v)-4\eta^{2}\partial_{\bar{z}}[(\sigma/\rho)\partial_{z}\sigma]$ and
$T-Q=-4i\eta\partial_{z}(\sigma
v)+4\eta^{2}\partial_{z}[(\sigma/\rho)\partial_{z}\sigma]$, it is easy to
verify that
$\partial_{z}Q_{z\bar{z}}+\partial_{\bar{z}}Q=\partial_{z}T_{z\bar{z}}+\partial_{\bar{z}}T$
which is the complex form of condition 2). Hence $Q^{\alpha\beta}$ defined in
Eqs. (32) (33) is the tensor we search for.

It is now ready to apply the MC principle to obtain the vortex fluid equation
on a closed surface :

$\displaystyle\partial_{t}(\rho
v^{\alpha})+\nabla_{\beta}Q^{\alpha\beta}+\rho\nabla^{\alpha}p=0$ (34)

where

$\displaystyle Q^{\alpha\beta}=\rho v^{\alpha}v^{\beta}+2\eta
v^{\alpha}\epsilon^{\beta}_{\mu}\nabla^{\mu}\sigma+\eta^{2}\frac{1}{\rho}\epsilon^{\alpha}_{\mu}\epsilon^{\beta}_{\nu}\nabla^{\mu}\sigma\nabla^{\nu}\sigma$
(35)

and the pressure $p$ is determined by
$\nabla_{\mu}(u^{\nu}\nabla_{\nu}u^{\mu})=-\nabla_{\mu}\nabla^{\mu}p$.

It is crucial that the momentum flux tensor includes the odd viscous tensor
$\tau^{\alpha\beta}(\partial_{\mu}\rightarrow\nabla_{\mu})$. For this purpose,
we need to write the dynamical equation in terms of $T^{\alpha\beta}$:

$\displaystyle\partial_{t}\left(\rho
v^{\alpha}\right)+\nabla_{\beta}T^{\alpha\beta}+\rho\nabla^{\alpha}p=\eta
K\left(\eta\frac{\sigma}{\rho}\nabla^{\alpha}\sigma-2\sigma\epsilon^{\alpha}_{\beta}v^{\beta}\right),$
(36)

where

$\displaystyle T^{\alpha\beta}=\rho
v^{\alpha}v^{\beta}+\eta^{2}\sigma\nabla^{\beta}(\frac{1}{\rho}\nabla^{\alpha}\sigma)+8\pi\eta^{2}\sigma^{2}g^{\alpha\beta}+\sigma\tau^{\alpha\beta},$
(37)

$K=R_{1212}/\det g_{\mu\nu}=R_{1212}/h^{2}$ is Gaussian curvature. Here we
have used $\epsilon^{\mu}_{\nu}\nabla_{\mu}u^{\nu}=8\pi\eta\sigma$ and Eq.
(15).

Comparing to Eq. (2), the conspicuous feature of Eq. (36) is that the
combination of Gaussian curvature and the circulation quantum/odd viscosity
plays the role of the coefficient of a source term. The presence of this
additional term might be referred to as curvature anomaly. The momentum flux
tensor $T^{\alpha\beta}$ is not symmetric for binary vortex fluids and it can
not be symmetrized in the usual way due to that its anti-symmetric part

$\displaystyle T^{12}-T^{21}=\eta^{2}\sigma h^{-1}\nabla_{\mu}v^{\mu}$ (38)

is not a total divergence. The hydrodynamics equation (36) is invariant under
the following scaling transformation $x\rightarrow\lambda x$,
$t\rightarrow\lambda^{2}t$, $\rho\rightarrow\lambda^{-2}\rho$ ,
$\sigma\rightarrow\lambda^{-2}\sigma$,
$v^{\alpha}\rightarrow\lambda^{-1}v^{\alpha}$ $K\rightarrow\lambda^{-2}K$,
$p\rightarrow\lambda^{-2}p$. The vortex core size $\ell$ plays the role of the
ultraviolet cut-off of the hydrodynamics theory.

Since the odd viscous tensor $\tau^{\alpha\beta}$ is of fundamental impotence
and appears in a large class of fluids Fruchart  _et al._ (2023) , it is
worthwhile exploring its properties on a curved surface. From the definition
of $\tau^{\alpha\beta}$, one obtains

$\displaystyle\Delta v_{\alpha}\nabla_{\beta}\tau^{\alpha\beta}=-\eta
K\epsilon^{\alpha}_{\beta}v^{\beta}\Delta v_{\alpha}.$ (39)

For a closed orientable surface, due to Gauss-Bonnet theorem, we have

$\displaystyle\int ds\frac{\Delta
v_{\alpha}\nabla_{\beta}\tau^{\alpha\beta}}{\epsilon^{\alpha}_{\beta}v^{\beta}\Delta
v_{\alpha}}=-\eta\int dsK=-2\pi\eta\chi({\cal M}),$ (40)

where $\chi({\cal M})=2(2-g)$ is Euler characteristic, and $g$ is the genus of
the surface. It should be noted that Eq. (40) holds for any value of $g$.
Connecting Eq. (40) to physical observable deserves future investigations. The
hydrodynamic equation (36) can be verified by substituting Eqs. (15) (16) into
Eq. (28).

##  V Vortex flow on a sphere

We consider vortex fluids on a sphere embedded in $\mathbb{R}^{3}$. We
introduce the Cartesian coordinates

$\displaystyle\xi=R\sin\theta\cos\phi,\quad\eta=R\sin\theta\sin\phi,\quad\zeta=R\cos\theta,$
(41)

where $R$ is the radius, $\theta$ is the polar angle and $\phi$ is the
azimuthal angle. On a sphere, stereographic coordinates $z=x^{1}+ix^{2}$ are
isothermal coordinates and are related to the spherical coordinates by
$z=R\tan(\theta/2)e^{i\phi}$ (projection from the south pole). In terms of
$z$, the Riemannian metric reads

$\displaystyle h=\frac{4R^{4}}{(R^{2}+|z|^{2})^{2}}$ (42)

and in spherical coordinates

$\displaystyle ds^{2}=R^{2}d\theta^{2}+R^{2}\sin^{2}\theta d\phi^{2}.$ (43)

###  V.1 Conserved quantities

It is known that for point vortices on a sphere, the quantities

$\displaystyle L_{\xi}$ $\displaystyle=\kappa
R^{2}\sum_{j}\sigma_{j}\sin\theta_{j}\cos\phi_{j},$ (44) $\displaystyle
L_{\eta}$ $\displaystyle=\kappa
R^{2}\sum_{j}\sigma_{j}\sin\theta_{j}\sin\phi_{j},$ (45) $\displaystyle
L_{\zeta}$ $\displaystyle=\kappa R^{2}\sum_{j}\sigma_{j}\cos\theta_{j},$ (46)

are conserved Kimura (1999). In terms of collective variables,

$\displaystyle L_{\xi}$ $\displaystyle=\kappa R^{2}\int
ds\,\sigma\sin\theta\cos\phi,$ (47) $\displaystyle L_{\eta}$
$\displaystyle=\kappa R^{2}\int ds\,\sigma\sin\theta\sin\phi,$ (48)
$\displaystyle L_{\zeta}$ $\displaystyle=\kappa R^{2}\int
ds\,\sigma\cos\theta.$ (49)

These conserved quantities are directly related to the corresponding fluid
angular momentum $\int ds\,\mathbf{r}\times\mathbf{u}$ which is associated
with the $\textrm{SO}(3)$ symmetry. In stereographic coordinates, they become

$\displaystyle L_{\xi}$ $\displaystyle=\kappa R\int dx^{1}dx^{2}h^{3/2}\sigma
x^{1},\quad L_{\eta}=\kappa R\int dx^{1}dx^{2}h^{3/2}\sigma x^{2},$ (50)
$\displaystyle L_{\zeta}$ $\displaystyle=-\frac{\kappa}{2}\int
dx^{1}dx^{2}h^{3/2}\sigma|z|^{2}+\frac{\kappa R^{2}}{2}\int
dx^{1}dx^{2}h^{3/2}\sigma.$ (51)

Then it is easy to notice that, as $R\rightarrow\infty$,

$\displaystyle L_{\xi}/R\propto P_{x^{2}}$
$\displaystyle=-\kappa\sum_{i}\sigma_{i}x^{1}_{i}=-\kappa\int
dx^{1}dx^{2}\sigma x^{1},$ (52) $\displaystyle L_{\eta}/R\propto P_{x^{1}}$
$\displaystyle=\kappa\sum_{i}\sigma_{i}x^{2}_{i}=\kappa\int dx^{1}dx^{2}\sigma
x^{2},$ (53)

where $P_{x^{2}}$ and $P_{x^{1}}$ are components of canonical momentum of
vortices on a plane. Also, as $R\rightarrow\infty$,

$\displaystyle L_{\zeta}\propto
L=\kappa\sum_{i}\sigma_{i}|\mathbf{r}_{i}|^{2}=\kappa\int
dx^{1}dx^{2}\sigma|z|^{2}$ (54)

which is the canonical angular momentum of the point-vortex system on a plane.
Hence there is a one-to-one correspondence between conserved quantities on a
plane and on a sphere.

It is worthwhile to mention that the enstrophy

$\displaystyle H\equiv\int ds\,\omega^{2}$ (55)

is conserved in any closed surface with zero genus, as

$\displaystyle\frac{dH}{dt}=-2\int ds\omega u^{\mu}\nabla_{\mu}\omega=-\int
ds\,\nabla_{\mu}(u^{\mu}\omega^{2})=0.$ (56)

However the symmetry associated with this conservation law is not obvious
Marjieh  _et al._ (2022).

###  V.2 Stationary vortex flows

For constant vortex density $\rho=\rho_{0}$ on a surface with constant
Gaussian curvature $K=K_{0}$, the vortex fluid becomes incompressible
$\nabla_{\mu}v^{\mu}=0$ and Eq.(36) becomes

$\displaystyle\partial_{t}\omega_{v}+\frac{1}{\rho_{0}}\epsilon^{\gamma}_{\alpha}\nabla_{\gamma}\nabla_{\beta}T^{\alpha\beta}=\frac{2\eta
K_{0}}{\rho_{0}}v^{\beta}\nabla_{\beta}\sigma.$ (57)

For a sphere, $K_{0}=1/R^{2}$, and we find a stationary solution of Eq. (57)

$\displaystyle\sigma$
$\displaystyle=\rho_{0}\frac{K^{-1}_{0}-{|{z}|}^{2}}{K^{-1}_{0}+{|{z}|}^{2}},$
(58) $\displaystyle\quad v^{1}$
$\displaystyle=-\left(4\pi\eta\rho_{0}-K_{0}\eta\right)x^{2},\quad
v^{2}=\left(4\pi\eta\rho_{0}-K_{0}\eta\right)x^{1}.$ (59)

Note that $\sigma(z=0)=\rho_{0}=-\sigma(z=\infty)$. For this flow
$\tau^{\alpha\beta}=0$, $L_{\xi}=L_{\eta}=0$ and $L_{\zeta}=4/3\pi
R^{4}\kappa\rho_{0}$. The modulus of the vortex velocity field is

$\displaystyle|v|=\sqrt{v_{1}v^{1}+v_{2}v^{2}}=\frac{2R^{2}|4\pi\eta\rho_{0}-K_{0}\eta||z|}{R^{2}+|z|^{2}},$
(60)

having the profile of a Kaufmann vortex. For $|z|\ll R$, $|v|\propto|z|$,
while $|v|\propto 1/|z|$ for $|z|\gg R$. The maximum value of $|v|$ is reached
at $|z|=R$. The anomalous correction to the fluid velocity is

$\displaystyle v^{1}-u^{1}=K_{0}\eta x^{2},\quad v^{2}-u^{2}=-K_{0}\eta x^{1}$
(61)

and its modulus is
$(v^{1}-u^{1})(v_{1}-u_{1})+(v^{2}-u^{2})(v_{2}-u_{2})=hK^{2}_{0}\eta^{2}|z|^{2}=4K_{0}\eta|z|^{2}/(R^{2}+|z|^{2})^{2}$.
The vorticity of the vortex velocity field also has an anomalous correction
that is proportional to $K_{0}$

$\displaystyle\omega_{v}-\omega=-2K_{0}\eta\frac{K^{-1}_{0}-|z|^{2}}{K^{-1}_{0}+|z|^{2}}.$
(62)

When $R\rightarrow\infty$, $K_{0}\rightarrow 0$, $\sigma\rightarrow\rho_{0}$
for $z\neq\infty$ ($\rho_{0}$ keeps a constant as $R\rightarrow\infty$), and
this corresponds to rigid body rotation of a chiral vortex flow on a plane.
The oppositely charged vortices accumulate at $z=\infty$. It is important to
note that the anomalous corrections, i.e., the differences between $v$ and $u$
(or $\omega_{v}$ and $\omega$), are proportional to curvature and vanish as
$K_{0}\rightarrow 0$.

Figure 1: Schematic of the stationary vortex flow on a sphere. The arrows
represent the vortex velocity field $\mathbf{v}$ and the background color
shows the renormalized vorticity of the vortex fluid
$\omega_{v}(\theta)/|\omega_{v}(0)|$.

It is helpful to express this stationary flow using spherical coordinates, for
which $\mathbf{v}=v^{\theta}\partial_{\theta}+v^{\phi}\partial_{\phi}$ and

$\displaystyle\sigma=\rho_{0}\cos\theta,\quad
v^{\phi}=R(4\pi\eta\rho_{0}-K_{0}\eta),\quad v^{\theta}=0.$ (63)

The modulus of the vortex velocity field is

$\displaystyle|\mathbf{v}|=\sqrt{v^{\phi}v_{\phi}+v^{\theta}v_{\theta}}=R|(4\pi\eta\rho_{0}-K_{0}\eta)\sin\theta\,|,$
(64)

which vanishes at the poles and reaches the maximum at the equator (see Fig.
1). Since $u^{\phi}=4\pi\eta R\rho_{0}$ and $u^{\theta}=0$, we have
$|v-u|^{2}=(v^{\phi}-u^{\phi})(v_{\phi}-u_{\phi})=K_{0}\eta^{2}\sin^{2}\theta$.
The vorticity of the vortex fluid reads

$\displaystyle\omega_{v}$
$\displaystyle=2\left(4\pi\rho_{0}-K_{0}\right)\eta\cos\theta.$ (65)

and the correction is $\omega_{v}-\omega=-2K_{0}\eta\cos\theta$. Due to
compactness of the sphere,
$\omega_{v}(\theta=0)=-\omega_{v}(\theta=\pi)=2\left(4\pi\rho_{0}-K_{0}\right)$,
the vorticity of this vortex flow has the profile of a vortex-dipole. It is
worthwhile mentioning that the vortex flows we found here are analogous to
zonal Rossby–Haurwitz flows in Euler fluids on a sphere Tritton (2012);
Nualart (2023), which play an important role in analyzing dynamics of
atmosphere of Earth Rossby (1939); Platzman (1968); Haurwitz (1940).

##  VI Conclusion

We generalize the vortex fluid theory on a plane to closed surfaces with zero
genus. The dynamical equation is derived using the minimal coupling principle
from it on a flat surface. An additional curvature term emerges and describes
the interaction between topological defects and curvature in the
hydrodynamical level. Since the vortex fluid equation contains second
derivatives of vectors, there is an ambiguity for applying the minimal
coupling principle directly. Our method does get over this difficulty and
provides a feasible recipe to investigate other complex fluids on curved
surfaces. For a sphere, a nontrivial stationary vortex flow is found
analytically. It poses a challenge to find analytic solutions for surfaces
with non-constant Gaussian curvature. For surfaces with Gaussian curvature
weakly depending on positions [$K(x^{\mu})=K_{0}+\delta K(x^{\mu})$ with
$\delta K(x^{\mu})\ll K_{0}$], it might be possible to treat $\delta
K(x^{\mu})$ as perturbations and investigate the effects of non-constant
Gaussian curvature. These are interesting topics which are worthwhile
exploring in the feature.

The theory developed in this work leads to a broad understanding of the
interaction between topological defects and curvature, and provides a
theoretical framework for investigating rich phenomena involving a large
number of quantum vortices Tononi  _et al._ (2022); Kanai and Guo (2021); Li
and Efimkin (2023); Saito and Hayashi (2023) in bubble trapped Bose-Einstein
condensates Tononi  _et al._ (2020); Tononi and Salasnich (2019).

## Acknowledgment

We acknowledge J. Frauendiener, B. Feng, A. M. Mateo, L. A. Williamson, P. B.
Blakie and A. S. Bradley for useful discussions. X.Y. acknowledges support
from the National Natural Science Foundation of China (Grant No. 12175215),
the National Key Research and Development Program of China (Grant No. 2022YFA
1405300) and NSAF (Grant No. U2330401).

## References

  * Son (2013) Dam Thanh Son, “Newton-cartan geometry and the quantum hall effect,” (2013), arXiv:1306.0638 [cond-mat.mes-hall] .
  * Wiegmann (2013a) P. B. Wiegmann, “Hydrodynamics of euler incompressible fluid and the fractional quantum hall effect,” Phys. Rev. B 88, 241305 (2013a).
  * Cho  _et al._ (2014) Gil Young Cho, Yizhi You, and Eduardo Fradkin, “Geometry of fractional quantum hall fluids,” Phys. Rev. B 90, 115139 (2014).
  * Gromov and Abanov (2014) Andrey Gromov and Alexander G. Abanov, “Density-curvature response and gravitational anomaly,” Phys. Rev. Lett. 113, 266802 (2014).
  * Gromov  _et al._ (2015) Andrey Gromov, Gil Young Cho, Yizhi You, Alexander G. Abanov, and Eduardo Fradkin, “Framing anomaly in the effective theory of the fractional quantum hall effect,” Phys. Rev. Lett. 114, 016805 (2015).
  * Can  _et al._ (2015) Tankut Can, Michael Laskin, and Paul B. Wiegmann, “Geometry of quantum hall states: Gravitational anomaly and transport coefficients,” Annals of Physics 362, 752–794 (2015).
  * Mickelin  _et al._ (2018) Oscar Mickelin, Jonasz Słomka, Keaton J. Burns, Daniel Lecoanet, Geoffrey M. Vasil, Luiz M. Faria, and Jörn Dunkel, “Anomalous chained turbulence in actively driven flows on spheres,” Phys. Rev. Lett. 120, 164503 (2018).
  * Shankar  _et al._ (2017) Suraj Shankar, Mark J. Bowick, and M. Cristina Marchetti, “Topological sound and flocking on curved surfaces,” Phys. Rev. X 7, 031039 (2017).
  * Rank and Voigt (2021) M. Rank and A. Voigt, “Active flows on curved surfaces,” Physics of Fluids 33 (2021), 10.1063/5.0056099, 072110.
  * Bogomolov (1977) VA Bogomolov, “Dynamics of vorticity at a sphere,” Fluid Dynamics 12, 863–870 (1977).
  * Reuther and Voigt (2015) S. Reuther and A. Voigt, “The interplay of curvature and vortices in flow on curved surfaces,” Multiscale Modeling & Simulation 13, 632–643 (2015).
  * Reuther and Voigt (2018) Sebastian Reuther and Axel Voigt, “Solving the incompressible surface Navier-Stokes equation by surface finite elements,” Physics of Fluids 30 (2018), 10.1063/1.5005142, 012107.
  * Samavaki and Tuomela (2020) Maryam Samavaki and Jukka Tuomela, “Navier–stokes equations on riemannian manifolds,” Journal of Geometry and Physics 148, 103543 (2020).
  * Sigurdsson and Atzberger (2016) Jon Karl Sigurdsson and Paul J. Atzberger, “Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes,” Soft Matter 12, 6685–6707 (2016).
  * Levine  _et al._ (2004) Alex J. Levine, T. B. Liverpool, and F. C. MacKintosh, “Mobility of extended bodies in viscous films and membranes,” Phys. Rev. E 69, 021503 (2004).
  * Rower  _et al._ (2022) David A. Rower, Misha Padidar, and Paul J. Atzberger, “Surface fluctuating hydrodynamics methods for the drift-diffusion dynamics of particles and microstructures within curved fluid interfaces,” Journal of Computational Physics 455, 110994 (2022).
  * Vitelli and Turner (2004) Vincenzo Vitelli and Ari M. Turner, “Anomalous coupling between topological defects and curvature,” Phys. Rev. Lett. 93, 215301 (2004).
  * Turner  _et al._ (2010) Ari M. Turner, Vincenzo Vitelli, and David R. Nelson, “Vortices on curved surfaces,” Rev. Mod. Phys. 82, 1301–1348 (2010).
  * Aveline  _et al._ (2020) David C Aveline, Jason R Williams, Ethan R Elliott, Chelsea Dutenhoffer, James R Kellogg, James M Kohel, Norman E Lay, Kamal Oudrhiri, Robert F Shotwell, Nan Yu, _et al._ , “Observation of bose–einstein condensates in an earth-orbiting research lab,” Nature 582, 193–197 (2020).
  * Carollo  _et al._ (2022) Ryan A Carollo, David C Aveline, Brendan Rhyno, Smitha Vishveshwara, Courtney Lannert, Joseph D Murphree, Ethan R Elliott, Jason R Williams, Robert J Thompson, and Nathan Lundblad, “Observation of ultracold atomic bubbles in orbital microgravity,” Nature 606, 281–286 (2022).
  * Bereta  _et al._ (2021) Sálvio J. Bereta, Mônica A. Caracanhas, and Alexander L. Fetter, “Superfluid vortex dynamics on a spherical film,” Phys. Rev. A 103, 053306 (2021).
  * Caracanhas  _et al._ (2022) Mônica A. Caracanhas, Pietro Massignan, and Alexander L. Fetter, “Superfluid vortex dynamics on an ellipsoid and other surfaces of revolution,” Phys. Rev. A 105, 023307 (2022).
  * Padavić  _et al._ (2020) Karmela Padavić, Kuei Sun, Courtney Lannert, and Smitha Vishveshwara, “Vortex-antivortex physics in shell-shaped bose-einstein condensates,” Phys. Rev. A 102, 043305 (2020).
  * Hally (1980) David Hally, “Stability of streets of vortices on surfaces of revolution with a reflection symmetry,” Journal of Mathematical Physics 21, 211–217 (1980).
  * Kimura (1999) Y. Kimura, “Vortex motion on surfaces with constant curvature,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 455 (1999).
  * Dritschel and Boatto (2015) David Gerard Dritschel and Stefanella Boatto, “The motion of point vortices on closed surfaces,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, 20140890 (2015).
  * Wiegmann and Abanov (2014a) Paul Wiegmann and Alexander G. Abanov, “Anomalous hydrodynamics of two-dimensional vortex fluids,” Phys. Rev. Lett. 113, 034501 (2014a).
  * Yu and Bradley (2017) Xiaoquan Yu and Ashton S. Bradley, “Emergent Non-Eulerian Hydrodynamics of Quantum Vortices in Two Dimensions,” Phys. Rev. Lett. 119, 185301 (2017).
  * Stockdale  _et al._ (2020) Oliver R. Stockdale, Matthew T. Reeves, Xiaoquan Yu, Guillaume Gauthier, Kwan Goddard-Lee, Warwick P. Bowen, Tyler W. Neely, and Matthew J. Davis, “Universal dynamics in the expansion of vortex clusters in a dissipative two-dimensional superfluid,” Phys. Rev. Res. 2, 033138 (2020).
  * Wiegmann (2013b) P. B. Wiegmann, “Hydrodynamics of euler incompressible fluid and the fractional quantum hall effect,” Phys. Rev. B 88, 241305 (2013b).
  * Doshi and Gromov (2021) Darshil Doshi and Andrey Gromov, “Vortices as fractons,” Communications Physics 4, 44 (2021).
  * Grosvenor  _et al._ (2021) Kevin T. Grosvenor, Carlos Hoyos, Francisco Peña Benitez, and Piotr Surówka, “Hydrodynamics of ideal fracton fluids,” Phys. Rev. Res. 3, 043186 (2021).
  * Pitaevskii and Stringari (2016) Lev Pitaevskii and Sandro Stringari,  _Bose-Einstein Condensation and Superfluidity_ , Vol. 164 (Oxford University Press, 2016).
  * Eyink and Sreenivasan (2006) Gregory L. Eyink and Katepalli R. Sreenivasan, “Onsager and the theory of hydrodynamic turbulence,” Rev. Mod. Phys. 78, 87–135 (2006).
  * Onsager (1949) L. Onsager, “Statistical hydrodynamics,” Il Nuovo Cimento (1943-1954) 6, 279–287 (1949).
  * Novikov (1975) EA Novikov, “Dynamics and statistics of a system of vortices,” Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 68, 1868–1882 (1975).
  * Aref  _et al._ (1999) H Aref, PL Boyland, MA Stremler, and DL Vainchtein, “Turbulent statistical dynamics of a system of point vortices,” in  _Fundamental Problematic Issues in Turbulence_ (Springer, 1999) pp. 151–161.
  * G.R.Kirchhoff (1876) G.R.Kirchhoff, in  _Vorlesungen über mathematische Physik_ (Teubner, Leipzig, 1876).
  * Wiegmann and Abanov (2014b) Paul Wiegmann and Alexander G. Abanov, “Anomalous hydrodynamics of two-dimensional vortex fluids,” Phys. Rev. Lett. 113, 034501 (2014b).
  * Avron (1998) JE Avron, “Odd viscosity,” Journal of statistical physics 92, 543–557 (1998).
  * Fruchart  _et al._ (2023) Michel Fruchart, Colin Scheibner, and Vincenzo Vitelli, “Odd viscosity and odd elasticity,” Annual Review of Condensed Matter Physics 14, 471–510 (2023).
  * Avron  _et al._ (1995) J. E. Avron, R. Seiler, and P. G. Zograf, “Viscosity of quantum hall fluids,” Phys. Rev. Lett. 75, 697–700 (1995).
  * Hoyos and Son (2012) Carlos Hoyos and Dam Thanh Son, “Hall viscosity and electromagnetic response,” Phys. Rev. Lett. 108, 066805 (2012).
  * Bradlyn  _et al._ (2012) Barry Bradlyn, Moshe Goldstein, and N. Read, “Kubo formulas for viscosity: Hall viscosity, ward identities, and the relation with conductivity,” Phys. Rev. B 86, 245309 (2012).
  * Abanov (2013) Alexander G Abanov, “On the effective hydrodynamics of the fractional quantum hall effect,” Journal of Physics A: Mathematical and Theoretical 46, 292001 (2013).
  * Souslov  _et al._ (2019) Anton Souslov, Kinjal Dasbiswas, Michel Fruchart, Suriyanarayanan Vaikuntanathan, and Vincenzo Vitelli, “Topological waves in fluids with odd viscosity,” Phys. Rev. Lett. 122, 128001 (2019).
  * Banerjee  _et al._ (2017) Debarghya Banerjee, Anton Souslov, Alexander G Abanov, and Vincenzo Vitelli, “Odd viscosity in chiral active fluids,” Nature communications 8, 1573 (2017).
  * Lou  _et al._ (2022) Xin Lou, Qing Yang, Yu Ding, Peng Liu, Ke Chen, Xin Zhou, Fangfu Ye, Rudolf Podgornik, and Mingcheng Yang, “Odd viscosity-induced hall-like transport of an active chiral fluid,” Proceedings of the National Academy of Sciences 119, e2201279119 (2022).
  * Hoyos  _et al._ (2014) Carlos Hoyos, Sergej Moroz, and Dam Thanh Son, “Effective theory of chiral two-dimensional superfluids,” Phys. Rev. B 89, 174507 (2014).
  * Bogatskiy and Wiegmann (2019) A. Bogatskiy and P. Wiegmann, “Edge wave and boundary layer of vortex matter,” Phys. Rev. Lett. 122, 214505 (2019).
  * Bogatskiy (2019) A Bogatskiy, “Vortex flows on closed surfaces,” Journal of Physics A: Mathematical and Theoretical 52, 475501 (2019).
  * Moroz  _et al._ (2018) Sergej Moroz, Carlos Hoyos, Claudio Benzoni, and Dam Thanh Son, “Effective field theory of a vortex lattice in a bosonic superfluid,” SciPost Phys. 5, 039 (2018).
  * Ganeshan and Abanov (2017) Sriram Ganeshan and Alexander G. Abanov, “Odd viscosity in two-dimensional incompressible fluids,” Phys. Rev. Fluids 2, 094101 (2017).
  * Monteiro and Ganeshan (2021) Gustavo M. Monteiro and Sriram Ganeshan, “Nonlinear shallow water dynamics with odd viscosity,” Phys. Rev. Fluids 6, L092401 (2021).
  * Reeves  _et al._ (2017) Matthew T. Reeves, Thomas P. Billam, Xiaoquan Yu, and Ashton S. Bradley, “Enstrophy cascade in decaying two-dimensional quantum turbulence,” Phys. Rev. Lett. 119, 184502 (2017).
  * Reeves  _et al._ (2014) Matthew T. Reeves, Thomas P. Billam, Brian P. Anderson, and Ashton S. Bradley, “Signatures of coherent vortex structures in a disordered two-dimensional quantum fluid,” Phys. Rev. A 89, 053631 (2014).
  * Yu  _et al._ (2016) Xiaoquan Yu, Thomas P. Billam, Jun Nian, Matthew T. Reeves, and Ashton S. Bradley, “Theory of the vortex-clustering transition in a confined two-dimensional quantum fluid,” Phys. Rev. A 94, 023602 (2016).
  * Gauthier  _et al._ (2019) Guillaume Gauthier, Matthew T. Reeves, Xiaoquan Yu, Ashton S. Bradley, Mark A. Baker, Thomas A. Bell, Halina Rubinsztein-Dunlop, Matthew J. Davis, and Tyler W. Neely, “Giant vortex clusters in a two-dimensional quantum fluid,” Science 364, 1264–1267 (2019).
  * Billam  _et al._ (2014) T. P. Billam, M. T. Reeves, B. P. Anderson, and A. S. Bradley, “Onsager-kraichnan condensation in decaying two-dimensional quantum turbulence,” Phys. Rev. Lett. 112, 145301 (2014).
  * Skaugen and Angheluta (2016) Audun Skaugen and Luiza Angheluta, “Vortex clustering and universal scaling laws in two-dimensional quantum turbulence,” Phys. Rev. E 93, 032106 (2016).
  * Chern (1955) Shing-Shen Chern, “An elementary proof of the existence of isothermal parameters on a surface,” Proceedings of the American Mathematical Society 6, 771–782 (1955).
  * (62) _Note that its integral over a closed surface vanishes, namely $\int ds\,\nabla_{\mu}v^{\mu}=0$ _.
  * Reall (2013) Harvey Reall, “Part 3 general relativity,” University of Cambridge 65 (2013).
  * Gustafsson (2022) Björn Gustafsson, “Vortex pairs and dipoles on closed surfaces,” Journal of Nonlinear Science 32, 62 (2022).
  * Marjieh  _et al._ (2022) Raja Marjieh, Natalia Pinzani-Fokeeva, and Amos Yarom, “Enstrophy from symmetry,” SciPost Phys. 12, 085 (2022).
  * Tritton (2012) David J Tritton,  _Physical fluid dynamics_ (Springer Science & Business Media, 2012).
  * Nualart (2023) Marc Nualart, “On zonal steady solutions to the 2d euler equations on the rotating unit sphere,” (2023), arXiv:2201.05522 [math.AP] .
  * Rossby (1939) Carl-Gustaf Rossby, “Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action,” J. mar. Res. 2, 38–55 (1939).
  * Platzman (1968) George W Platzman, “The rossby wave,” Quarterly Journal of the Royal Meteorological Society 94, 225–248 (1968).
  * Haurwitz (1940) Bernhard Haurwitz, “The motion of atmospheric disturbances on the spherical earth,” Journal of Marine Research 3, (3). (1940).
  * Tononi  _et al._ (2022) Andrea Tononi, Axel Pelster, and Luca Salasnich, “Topological superfluid transition in bubble-trapped condensates,” Phys. Rev. Research 4, 013122 (2022).
  * Kanai and Guo (2021) Toshiaki Kanai and Wei Guo, “True mechanism of spontaneous order from turbulence in two-dimensional superfluid manifolds,” Phys. Rev. Lett. 127, 095301 (2021).
  * Li and Efimkin (2023) Guangyao Li and Dmitry K. Efimkin, “Equatorial waves in rotating bubble-trapped superfluids,” Phys. Rev. A 107, 023319 (2023).
  * Saito and Hayashi (2023) Hiroki Saito and Masazumi Hayashi, “Rossby–haurwitz wave in a rotating bubble-shaped bose–einstein condensate,” Journal of the Physical Society of Japan 92, 044003 (2023).
  * Tononi  _et al._ (2020) A. Tononi, F. Cinti, and L. Salasnich, “Quantum bubbles in microgravity,” Phys. Rev. Lett. 125, 010402 (2020).
  * Tononi and Salasnich (2019) A. Tononi and L. Salasnich, “Bose-einstein condensation on the surface of a sphere,” Phys. Rev. Lett. 123, 160403 (2019).