aainstitutetext: Max Planck Institute for Empirical Aesthetics, Frankfurt am
Main 60322, Germanybbinstitutetext: Institute for Theoretical Physics, KU
Leuven Celestijnenlaan 200D, Leuven B-3001, Belgiumccinstitutetext: Department
of Physics, Technion, Haifa 32000, Israel

# Enstrophy from symmetry

Raja Marjieh b Natalia Pinzani-Fokeeva c and Amos Yarom
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study symmetry principles associated with the approximately conserved
enstrophy current, responsible for the inverse energy cascade in non
relativistic $2+1$ dimensional turbulence. We do so by identifying the
accidental symmetry associated with enstrophy current conservation in a
recently realized effective action principle for hydrodynamics. Our analysis
deals with both relativistic and non relativistic effective actions and their
associated symmetries.

##  1 Introduction

Non relativistic incompressible fluid flow in two spatial dimensions supports
an approximately conserved enstrophy charge whose existence plays a crucial
role in generating the inverse energy cascade of turbulent flow Kraichnan .
Planar, non relativistic and compressible flow also supports an approximately
conserved enstrophy charge as long as the equation of state is barotropic (the
pressure is a function of the mass, or particle number density), see, e.g.,
AncoDar . Likewise, an approximately conserved enstrophy charge was shown to
be present in relativistic and conformal invariant fluid flow in $2+1$
dimensions Carrasco:2012nf .

Charges which are conserved under the equations of motion are tied, via
Noether’s theorem, to symmetries of the underlying action. Thus, it stands to
reason that there exists an approximate symmetry responsible for the
approximate conservation of enstrophy. The goal of this work is to identify
the symmetry associated with enstrophy conservation in relativistic and non
relativistic fluid flows using a recently discovered action principle for
fluids Haehl:2015foa ; Crossley:2015evo ; Haehl:2015uoc . (See also
Haehl:2016pec ; Haehl:2016uah ; Glorioso:2016gsa ; Jensen:2017kzi ;
Gao:2017bqf ; Glorioso:2017fpd ; Glorioso:2017lcn ; Jensen:2018hhx ;
Haehl:2018uqv ; Jensen:2018hse ; Gao:2018bxz ; Haehl:2018lcu ;
Glorioso:2019koh .)

In local theories charge conservation follows from current conservation. It is
interesting to contrast the approximate conservation of the enstrophy current
with the behavior of the entropy current. Recall that the entropy current is
given by $J_{s}^{\mu}=su^{\mu}+\mathcal{O}(\partial)$ where $s$ is the entropy
density, $u^{\mu}$ is the velocity field ($u^{\mu}=\gamma(1,\,v^{i})$ in a
relativistic setting and the same with $\gamma=1$ in a non relativistic one),
and $\mathcal{O}(\partial)$ denotes corrections which include derivatives of
hydrodynamic variables. The entropy current is conserved in the absence of
dissipative terms and its divergence is positive semi-definite otherwise.
Thus, at leading order in a derivative expansion one may view the entropy
current as being approximately conserved in the sense that
$\partial_{\mu}J_{s}^{\mu}=\mathcal{O}(\partial^{2})$ under the equations of
motion. That is to say, the leading order contribution to the entropy current
is of zeroth order in a derivative expansion, but, under the equations of
motion, its divergence is second order. The approximate conservation of the
enstrophy current is of a similar type. As we will see shortly, the enstrophy
current is second order in derivatives but, under the equations of motion, its
divergence is fourth order. The analogy between the enstrophy current and the
entropy current may run deeper than a comparison of their approximate
conservation at leading order. In planar, non relativistic, incompressible
flow the enstrophy charge is conserved in the absence of dissipation but its
time derivative is negative semi-definite once dissipation comes into play.

Be that as it may, it is possible to identify the symmetry principle
responsible for the conservation of the entropy current in the absence of
dissipative terms using the hydrodynamic effective action Haehl:2014zda ;
Haehl:2015pja ; deBoer:2015ija ; Crossley:2015tka ; Sasa:2015zga . In fact,
one can identify the mechanism responsible for its full non conservation for
generic, dissipative, fluids Glorioso:2016gsa ; Jensen:2018hhx ; Haehl:2018uqv
. This raises the hope that a similar construction may be generated in order
to better understand the enstrophy current. In this work we take a first step
in this direction and find the symmetry associated with enstrophy conservation
at leading order in the derivative expansion. Along the way we provide a
rudimentary construction of the effective action for Galilean fluids, offering
a slightly different perspective on it than the recent comprehensive work of
Jain:2020vgc . In addition, we identify an enstrophy current in generic
relativistic fluids, generalizing the result of Carrasco:2012nf . We will not
discuss negativity of enstrophy production in a relativistic setting but will
comment on its possible realization where relevant.

Our work is organized as follows. In Section 2 we discuss the structure of the
relativistic and Galilean enstrophy currents. Our result for the relativistic
enstrophy current generalizes that of Carrasco:2012nf , relevant for an
uncharged conformal fluid, while our expression for the non relativistic
enstrophy current has been cast in a manifestly covariant form by using the
Newton-Cartan formalism. In Section 3 we discuss how a conserved enstrophy
current arises as a Noether current of an effective action for fluid dynamics
in 2+1 dimensions for both relativistic and Galilean fluid flows. We end with
an outlook and discussion in Section 4. A review of the traditional approach
to the construction of the enstrophy charge and a summary of Newton-Cartan
geometry and its relation to Galilean invariant hydrodynamics have been
relegated to the appendices.

##  2 The enstrophy current

In $2+1$ dimensional incompressible non relativistic fluid flow the enstrophy
charge is given by

$W=\frac{1}{2}\int\sqrt{g}\,\omega_{ij}\omega^{ij}d^{2}x\,,$ (1)

where $\omega_{ij}$ is the vorticity two form

$\omega_{ij}=\partial_{i}v_{j}-\partial_{j}v_{i}\,,$ (2)

with $v_{i}$ the velocity of a fluid element. The argument that $W$ is time
independent in the absence of dissipation and negative semidefinite otherwise
can be found in any textbook on hydrodynamics, e.g., inbook . We present the
canonical derivation of this result in appendix A for completeness.

The total enstrophy $W$ in (1) may be interpreted as a volume integral over an
enstrophy density which may be thought of as the zero component of an
enstrophy current,

$j_{(1)}^{\mu}=\omega_{ij}\omega^{ij}u_{G}^{\mu}$ (3)

with

$u_{G}^{\mu}=(1,\,\vec{v})\,.$ (4)

The reason for the parenthetical $(1)$ in (3) will become clear presently. The
subscript $G$ in (4) stands for Galilean, to be distinguished from its
Lorentzian counterpart which we will introduce shortly. The enstrophy current
$j_{(1)}^{\mu}$ satisfies $\nabla_{\mu}j_{(1)}^{\mu}=0$ at leading order in a
derivative expansion and $\nabla_{\mu}j_{(1)}^{\mu}\leq 0$ otherwise.

To be somewhat more precise, there exists not one, but a family of enstrophy
currents usually written in the form

$j_{(n)}^{\mu}=(\omega_{ij}\omega^{ij})^{n}u_{G}^{\mu}\,,$ (5)

with $n$ a positive integer. As was the case for $j_{(1)}^{\mu}$,
$\nabla_{\mu}j_{(n)}^{\mu}=0$ at leading order in the derivative expansion,
and, as long as $n>0$, $\nabla_{\mu}j_{(n)}^{\mu}\leq 0$ in the presence of
dissipation. While non-standard, it is straightforward to argue that the
currents (5) can be replaced with

$j_{h}^{\mu}=h(\omega_{ij}\omega^{ij})u_{G}^{\mu}$ (6)

where $h$ is a monotonically increasing function of its argument; one finds
that $\nabla_{\mu}j_{h}^{\mu}\leq 0$ with a strict equality once the viscosity
is set to zero.

Slightly less familiar is the family of conserved enstrophy currents
associated with non relativistic, inviscid, compressible, and barotropic
fluids,

$j_{G}^{\mu}=\frac{g(s/\rho)}{s^{2n-1}}\left(\omega_{ij}\omega^{ij}\right)^{n}u^{\mu}_{G}\,.$
(7)

Here, $\rho$ is the particle number density, $s$ the entropy density, and $g$
an arbitrary function. The barotropic condition states that $P=P(\rho)$ where
$P$ is the pressure. Note that in incompressible flow the particle number
density and the entropy density are constant so that (3) takes the same form
as (7) in its regime of validity. The expression (7) can be replaced by

$j_{G\,h}^{\mu}=h(s/\rho,\,\omega_{ij}\omega^{ij}/s^{2})su^{\mu}_{G}.$ (8)

Following standard conventions we will, throughout this work, consider the
version of the enstrophy current given in (7). Where relevant we will comment
on the alternate form $j_{G\,h}^{\mu}$ mentioned above.

An uncharged, inviscid, relativistic, conformal fluid in $2+1$ dimensions also
possesses a conserved enstrophy current given by

$J^{\mu}_{\hbox{\tiny conformal}}=\frac{\Omega^{2}}{\epsilon^{2/3}}u^{\mu}$
(9)

with $\epsilon$ the energy density, $u^{\mu}$ a relativistic velocity field
and $\Omega^{2}=\Omega_{\mu\nu}\Omega^{\mu\nu}$ where
$\Omega_{\mu\nu}=\nabla_{\mu}\left(\epsilon^{1/3}u_{\nu}\right)-\nabla_{\nu}\left(\epsilon^{1/3}u_{\mu}\right)$.
See Carrasco:2012nf . In what follows we will generalize this result. In
particular, we will argue that, in the presence of a $U(1)$ global symmetry,
one can write a conserved enstrophy current for generic fluids of the form

$J^{\mu}=\frac{g(s/\rho)}{s^{2n-1}}(\Omega^{2})^{n}u^{\mu}$ (10)

with

$\Omega_{\mu\nu}=\partial_{\mu}\left(Tf\left(\frac{\mu}{T}\right)u_{\nu}\right)-\partial_{\nu}\left(Tf\left(\frac{\mu}{T}\right)u_{\mu}\right)\,.$
(11)

Here, $T$ is the temperature, $\mu$ the chemical potential, $\rho$ a $U(1)$
charge density, and $f$ an arbitrary function of its argument. The current
$J^{\mu}$ is conserved as long as the pressure, $P$, satisfies
$P(T,\,\mu)=p(Tf(\mu/T))$. In the presence of an external electric field, $f$
becomes linear in $\mu$, and $\Omega_{\mu\nu}$ in (11) receives a contribution
linear in the field strength, (see (26)). As was the case for the Galilean
enstrophy current one may replace the family of conserved currents (10) with

$J^{\mu}_{h}=h(s/\rho,\Omega^{2}/s^{2})su^{\mu}\,.$ (12)

###  2.1 The relativistic enstrophy current

Recall that the dynamical fields of hydrodynamic theory can be chosen to be
the temperature $T$, a velocity field $u^{\mu}$ satisfying
$u_{\mu}u^{\mu}=-1$, and a chemical potential $\mu$ if a conserved charge is
present. The energy momentum tensor and other conserved currents of the theory
can be expressed in terms of the dynamical fields and their derivatives. This
description is usually made manifest in terms of a derivative expansion. For
instance,

$\displaystyle\begin{split}T^{\mu\nu}&=\epsilon(T,\mu)u^{\mu}u^{\nu}+P(T,\mu)\left(g^{\mu\nu}+u^{\mu}u^{\nu}\right)+\mathcal{O}(\partial)\,,\\\
J_{c}^{\mu}&=\rho(T,\mu)u^{\mu}+\mathcal{O}(\partial)\,.\end{split}$ (13)

Here, $\epsilon(T,\mu)$, $P(T,\mu)$ and $\rho(T,\mu)$ are functions of the
temperature and chemical potential which, in equilibrium, reduce to the energy
density, pressure and charge density respectively. The entropy density
$s(T,\mu)$ and charge density $\rho(T,\mu)$ satisfy

$s=\left(\frac{\partial P}{\partial
T}\right)_{\mu}\,,\qquad\rho=\left(\frac{\partial
P}{\partial\mu}\right)_{T}\,,$ (14)

and

$\epsilon+P=sT+\rho\mu\,.$ (15)

In the inviscid limit, energy-momentum and charge current conservation,
$\nabla_{\mu}T^{\mu\nu}=F^{\nu\mu}J_{c\,\mu}$ and $\nabla_{\mu}J_{c}^{\mu}=0$,
lead to the equations of motion $E=0$, $E^{\prime}=0$, and $E_{\mu}=0$ with

$\displaystyle\begin{split}{E}&=-\nabla_{\mu}(su^{\mu})\,,\\\
{E}_{\mu}&=\frac{(P+\epsilon)}{T}(P^{\alpha}_{\mu}\partial_{\alpha}T+Ta_{\mu})-\rho
V_{\mu}\,,\\\ E^{\prime}&=-\nabla_{\mu}\left(\rho
u^{\mu}\right)\,.\end{split}$ (16)

Here $a_{\mu}=u^{\alpha}\nabla_{\alpha}u_{\mu}$ is the acceleration,
$P_{\alpha\beta}=g_{\alpha\beta}+u_{\alpha}u_{\beta}$ is a projection matrix
and

$V_{\mu}=F_{\mu\nu}u^{\nu}-TP^{\nu}_{\mu}\partial_{\nu}\left(\frac{\mu}{T}\right)\,,$
(17)

with $F_{\mu\nu}$ an external field strength.

Suppose we find a closed two-form $\Omega_{\mu\nu}dx^{\mu}dx^{\nu}$ which is
orthogonal to the velocity field, $\Omega_{\mu\nu}u^{\nu}=0$, (at least under
the equations of motion). Such a two-form satisfies

$\mathcal{L}_{u}\Omega_{\mu\nu}=0$ (18)

under the equations of motion, with $\mathcal{L}_{u}$ the Lie derivative in
the $u^{\mu}$ direction. Using (18) and

$\nabla_{\mu}u_{\alpha}=\frac{1}{2}\sigma_{\mu\alpha}+\frac{1}{2}\omega_{\mu\alpha}+\frac{1}{d}\Theta
P_{\mu\alpha}-u_{\mu}a_{\alpha}\,,$ (19)

where $d$ is the number of spatial dimensions and

$\displaystyle\begin{split}\frac{1}{2}\sigma_{\mu\nu}&=\frac{1}{2}P_{\mu}{}^{\alpha}P_{\nu}{}^{\beta}\left(\nabla_{\alpha}u_{\beta}+\nabla_{\beta}u_{\alpha}\right)-\frac{1}{d}P_{\mu\nu}\nabla_{\alpha}u^{\alpha}\,,\\\
\omega_{\mu\nu}&=P_{\mu}{}^{\alpha}P_{\nu}{}^{\beta}\left(\nabla_{\alpha}u_{\beta}-\nabla_{\beta}u_{\alpha}\right)\,,\\\
\Theta&=\nabla_{\alpha}u^{\alpha}\,,\end{split}$ (20)

we find that under the equations of motion

$\nabla_{\alpha}\left(u^{\alpha}\Omega_{\mu\nu}\right)=\frac{1}{2}\left(\sigma_{\nu}{}^{\alpha}\Omega_{\alpha\mu}+\sigma_{\alpha\mu}\Omega_{\nu}{}^{\alpha}\right)-\left(u_{\nu}a^{\alpha}\Omega_{\alpha\mu}+u_{\mu}a^{\alpha}\Omega_{\nu\alpha}\right)+\Theta\,\Omega_{\mu\nu}\left(1-\frac{2}{d}\right)\,.$
(21)

In two spatial dimensions the first and last term on the right-hand-side of
(21) vanish. The former follows by denoting

$\displaystyle\begin{split}\sigma_{\nu}{}^{\alpha}\Omega_{\alpha\mu}+\Omega_{\nu}{}^{\alpha}\sigma_{\alpha\mu}&=\epsilon_{\nu\mu\rho}u^{\rho}\sigma\,,\\\
\Omega_{\mu\nu}=\epsilon_{\mu\nu\rho}u^{\rho}\omega\,,\end{split}$ (22)

where $\epsilon_{\mu\nu\rho}$ is the Levi-Civita tensor, and noting that

$\sigma\propto\epsilon^{\mu\nu\rho}u_{\rho}\sigma_{\nu}{}^{\alpha}\epsilon_{\alpha\mu\lambda}u^{\lambda}\omega=0\,.$
(23)

Thus, in two spatial dimensions, and after imposing the equations of motion,

$\nabla_{\alpha}\left(u^{\alpha}\Omega_{\mu\nu}\right)=-\left(u_{\nu}a^{\alpha}\Omega_{\alpha\mu}+u_{\mu}a^{\alpha}\Omega_{\nu\alpha}\right)\,.$
(24)

It is now straightforward to argue that $J^{\mu}$ given in (10) is conserved
for any value of $n$ under the equations of motion. Note that if $n<0$, then
$J^{\mu}$ is ill defined in equilibrium. Also, $J^{\mu}=su^{\mu}$ coincides
with the (inviscid) entropy current for $n=0$ and $g=1$. Likewise,
$J^{\mu}=\rho u^{\mu}$ coincides with the charge current for $n=0$ and
$g=s/\rho$. The first term on the right-hand side of (21) bears a striking
similarity to the vortex stretching term of non relativistic incompressible
flow (see equation (96) in appendix A). Therefore, it is sensible to identify
$J^{\mu}$ of (10) with $n=1$ and $g=1$ with the enstrophy current and
$J^{\mu}$ with larger $n$ with its higher moments.

Using (24) one can also show that the current $J_{h}^{\mu}$ defined in (12) is
also conserved under the equations of motion. It is also possible to
generalize (10) to fluids with multiple $U(1)$ charges whereby
$J^{\mu}=g\left(\frac{s}{\rho_{1}}\ldots\frac{s}{\rho_{n}}\right)\frac{(\Omega^{2})^{n}}{s^{2n-1}}u^{\mu}$,
with $\rho_{i}$ the various charge densities, is conserved. One might be
tempted to construct an additional conserved current by contracting (24) with
$\epsilon^{\mu\nu\rho}u_{\rho}$ to generate

$J^{\mu}_{H}=H(s/\rho,\varpi/s)su^{\mu}$ (25a) with
$\varpi=u_{\rho}\epsilon^{\mu\nu\rho}\Omega_{\mu\nu}.$ (25b)

It is straightforward to show that $\Omega^{2}=\frac{1}{2}\varpi^{2}+{\cal
O}(E)$ implying that $J_{H}^{\mu}$ and $J_{h}^{\mu}$ are equivalent under the
equations of motion.

The enstrophy current (10) was derived on the premise that a closed two-form
$\Omega=\Omega_{\mu\nu}dx^{\mu}dx^{\nu}$, satisfying
$\Omega_{\mu\nu}u^{\nu}=0$ is available. To find it, let us start with the
most general exact two-form which is first order in derivatives

$\Omega_{\mu\nu}=\partial_{\mu}\left(Tf(T,\,\nu)u_{\nu}\right)-\partial_{\nu}\left(Tf(T,\,\nu)u_{\mu}\right)+\theta
F_{\mu\nu}\,,$ (26)

where $\theta$ is a constant, $\nu=\mu/T$ and $f$ is an arbitrary function of
its variables. A somewhat lengthy computation yields

${\Omega}_{\mu\nu}u^{\nu}=-\frac{fT}{P+\epsilon}{E}_{\mu}+\left(\frac{f\rho
T}{P+\epsilon}-\frac{\partial
f}{\partial\nu}\right)TP^{\alpha}_{\mu}\partial_{\alpha}\nu-T\frac{\partial
f}{\partial T}P^{\alpha}_{\mu}\partial_{\alpha}T-\left(\frac{f\rho
T}{P+\epsilon}-\theta\right)F_{\mu\nu}u^{\nu}\,.$ (27)

In order for the penultimate term on the right-hand side of (27) to vanish we
need that

$\frac{\partial f}{\partial T}=0\,.$ (28)

Solving for both (28) and the requirement that the second term on the right-
hand-side of (27) vanish, we find that the equation of state must take the
form

$P(T,\,\mu)=p(Tf(\mu/T))\,.$ (29)

Requiring that (27) vanishes under the equations of motion implies, in
addition, that

$f(\nu)=\theta\nu+\theta_{0}$ (30)

with $\theta_{0}$ an integration constant.

Let us summarize our findings. In the presence of an external electromagnetic
field, a charged fluid must have an equation of state of the form (29) with
(30) in order to possess a conserved enstrophy current. In the absence of an
electromagnetic field, we must satisfy the somewhat less restrictive
condition, (29), in order for $J^{\mu}$ of (10) to be conserved. Note that a
charged conformal fluid and any neutral fluid will automatically have an
equation of state of the form (29) and therefore possess a conserved enstrophy
current (10).

###  2.2 The Galilean enstrophy current

The conserved enstrophy current for Galilean fluids can be constructed by
borrowing the techniques used to construct the relativistic enstrophy current.
A key feature of the construction of the relativistic enstrophy current was
the existence of a closed two-form $\Omega_{\mu\nu}dx^{\mu}dx^{\nu}$ which was
orthogonal to the velocity field under the equations of motion. With this two-
form at hand, and the decomposition (19), we were lead to (21) from which
enstrophy conservation in $2+1$ dimensions followed.

To construct such a Galilean invariant two-form, and consequently a conserved
enstrophy current, we use the Newton-Cartan formalism which allows one to
couple a Galilean invariant theory to a background spacetime in a covariant
way. Galilean boost invariance is ensured by requiring a certain “Milne
invariance” of the background geometry. We summarize this and other salient
features of the Newton-Cartan geometry in appendix B.1. Briefly, Newton-Cartan
geometry is characterized by a spatial metric $h^{\mu\nu}$, two timelike
vectors $n_{\mu}$ and $\bar{n}^{\mu}$ such that $h^{\mu\nu}n_{\mu}=0$ and
$n_{\mu}\bar{n}^{\mu}=1$, and a gauge field $A_{\mu}$. From these data one
constructs an inverse metric $\bar{h}_{\mu\nu}$ and a projection
$P^{\mu}{}_{\nu}$ via (106b). Fluid dynamics in a background Newton-Cartan
geometry can be described by introducing a timelike Galilean velocity field
$u^{\mu}_{G}$ which satisfies $u^{\mu}_{G}n_{\mu}=1$. We briefly review
hydrodynamics in a Newton-Cartan geometry in appendix B.2. The interested
reader is referred to Jensen:2014ama  for a detailed exposition.

In a generic Newton-Cartan background geometry the equivalent of the
decomposition (19) is

$\displaystyle\begin{split}\tilde{\nabla}_{\mu}u_{G}^{\nu}&=\frac{1}{2}\sigma_{\mu}{}^{\nu}+\frac{1}{2}\omega_{\mu}{}^{\nu}+\frac{2}{d}\tilde{P}_{\mu}^{\nu}\Theta+n_{\mu}a^{\nu}\,,\end{split}$
(31)

with the combinations

$\displaystyle\begin{split}\sigma_{\mu}{}^{\nu}&=\tilde{h}_{\mu\alpha}\tilde{P}^{\nu}_{\beta}\left(\tilde{\nabla}^{\alpha}u_{G}^{\beta}+\tilde{\nabla}^{\beta}u_{G}^{\alpha}\right)-\frac{2}{d}\tilde{P}_{\mu}^{\nu}\Theta\,,\\\
\omega_{\mu}{}^{\nu}&=\tilde{h}_{\mu\alpha}\tilde{P}^{\nu}_{\beta}\left(\tilde{\nabla}^{\alpha}u_{G}^{\beta}-\tilde{\nabla}^{\beta}u_{G}^{\alpha}\right)\,,\\\
a^{\nu}&=u_{G}^{\alpha}\tilde{\nabla}_{\alpha}u_{G}^{\nu}\,,\\\
\Theta&=\tilde{\nabla}_{\mu}u_{G}^{\mu}\,.\end{split}$ (32)

Here, $\tilde{h}_{\mu\nu}$ and $\tilde{P}^{\mu}{}_{\nu}$ are given in (118),
$\tilde{\nabla}_{\mu}$ is the Milne invariant covariant derivative constructed
in (119), $\tilde{\nabla}^{\alpha}=h^{\alpha\beta}\tilde{\nabla}_{\beta}$ and
in obtaining (32) we made repeated use of
$\tilde{\nabla}_{\mu}h^{\alpha\beta}=0$ together with
$n_{\mu}\tilde{\nabla}_{\alpha}u_{G}^{\mu}=0$. The latter follows from the
requirement that $\tilde{\nabla}_{\mu}n_{\nu}=0$. It is important to keep in
mind that in the Newton-Cartan formalism the Christoffel connection has
torsion, see (119). Following Jensen:2014ama , we have chosen it to be
timelike.

Using the Cartan formula ${\cal L}_{u}\Omega_{\mu\nu}=0$, we find that, under
the equations of motion,

$\displaystyle\begin{split}\Omega^{\alpha\beta}\tilde{\nabla}_{\mu}(u_{G}^{\mu}\Omega_{\alpha\beta})&=-\Omega^{\alpha\beta}\Omega_{\alpha\mu}\sigma^{\mu}_{\beta}+\Theta\Omega^{2}\left(1-\frac{2}{d}\right)-2\Omega^{\alpha\beta}\tilde{T}^{\nu}_{\alpha\mu}u_{G}^{\mu}\Omega_{\nu\beta}\,,\\\
\end{split}$ (33)

where $\tilde{T}^{\mu}_{\alpha\beta}$ is the torsion tensor, and we have
defined

$\Omega^{\mu\nu}=h^{\mu\alpha}h^{\nu\beta}\Omega_{\alpha\beta}\,,\qquad\Omega^{2}=\Omega^{\mu\nu}\Omega_{\mu\nu}\,.$
(34)

Since torsion is timelike,
$\tilde{T}^{\mu}_{\alpha\beta}=-u_{G}^{\mu}F^{(n)}_{\alpha\beta}$, c.f.,
(119), the last term on the right-hand-side of (33) vanishes under the
equations of motion,

$-2\Omega^{\alpha\beta}\tilde{T}^{\nu}_{\alpha\mu}u_{G}^{\mu}\Omega_{\nu\beta}=2\Omega^{\alpha\beta}F^{(n)}_{\alpha\mu}u_{G}^{\mu}\left(u^{\nu}_{G}\Omega_{\nu\beta}\right)=0\,.$
(35)

Thus, (33) reduces to

$\displaystyle\begin{split}\Omega^{\alpha\beta}\tilde{\nabla}_{\mu}(u_{G}^{\mu}\Omega_{\alpha\beta})&=-\Omega^{\alpha\beta}\Omega_{\alpha\mu}\sigma^{\mu}_{\beta}+\Theta\Omega^{2}\left(1-\frac{2}{d}\right)\,.\end{split}$
(36)

Equation (36) is the Galilean equivalent of (21): the last term on its right
clearly vanishes in $d=2$ spatial dimensions. The first term on the right-
hand-side of (36) is a vortex stretching term which, as we will now show, also
vanishes in $d=2$ spatial dimensions. Let us work in a coordinate system
where, locally, $u_{G}^{\mu}=(1,0)$. In this coordinate system we have
$\Omega_{\alpha\mu}=\delta^{i}_{\alpha}\delta^{j}_{\mu}\epsilon_{ij}\,\omega$
with $\epsilon_{ij}$ the Levi-Civita tensor and $\omega$ a real number. It is
now straightforward to compute

$\Omega^{\mu\alpha}\Omega_{\mu\beta}=\frac{1}{2}\Omega^{2}\tilde{P}^{\alpha}_{\beta}$
(37)

from which

$\Omega^{\alpha\beta}\Omega_{\alpha\mu}\sigma^{\mu}_{\beta}=0$ (38)

follows.

Using (36), we find that in 2+1 dimensions and under the equations of motion

$\displaystyle\begin{split}(\tilde{\nabla}_{\mu}-\tilde{\cal
G}_{\mu})J_{G}^{\mu}&=0\,,\end{split}$ (39)

where $\tilde{\cal G}_{\mu}$ was defined in (124) and $J_{G}^{\mu}$ is given
by

$J_{G}^{\mu}=g\left(\frac{\rho}{s}\right)\frac{(\Omega^{2})^{n}}{s^{2n-1}}u_{G}^{\mu}\,.$
(40)

Equation (40) is a covariant version of (7). In obtaining (39) we repeatedly
used the fact that $\tilde{\cal G}_{\mu}u_{G}^{\mu}=0$ and
$\tilde{\nabla}_{\alpha}h^{\mu\nu}=0$.111 If the torsion tensor is not
timelike then $\tilde{\cal G}_{\mu}u^{\mu}_{G}\neq 0$. Nevertheless, it is
possible to show that the enstrophy current (40) is conserved, in the sense of
(39), as long as a closed two-form $\Omega_{\mu\nu}dx^{\mu}dx^{\nu}$
orthogonal to the velocity field exists. As in the relativistic case,
conservation of

$J_{G\,h}^{\mu}=h(s/\rho,\Omega^{2}/s^{2})su^{\mu}_{G}$ (41)

also follows from (36). Moreover, contraction of (36) with
$\epsilon^{\mu\nu\rho}n_{\rho}$ leads to a conserved
$J^{\mu}_{G\,H}=H(s/\rho,\varpi/s)su^{\mu}_{G}$ where
$\varpi=n_{\rho}\epsilon^{\mu\nu\rho}\Omega_{\mu\nu}$. Since
$\Omega^{2}=\frac{1}{2}\varpi^{2}+{\cal O}(E)$, $J^{\mu}_{G\,h}$ and
$J^{\mu}_{G\,H}$ are equivalent up to terms proportional to the equations of
motion.

It remains to find a closed and velocity orthogonal $\Omega_{\mu\nu}$. The
most general $U(1)$ and Milne invariant closed two-form $\Omega_{\mu\nu}$ that
can be constructed using the Newton-Cartan data is given by

$\Omega_{\mu\nu}=\tilde{F}_{\mu\nu}+\partial_{\mu}(qn_{\nu})-\partial_{\nu}(qn_{\mu})\,,$
(42)

(up to a multiplicative constant which we set to 1 without loss of generality)
where $q$ is a generic function of the entropy density, $s$, and particle
number density, $\rho$, and $\tilde{F}_{\mu\nu}$ is the Milne invariant field
strength defined in (114). Contracting one of the indices of (42) with the
velocity field and using the equations of motion (125) we find

$\displaystyle\begin{split}\Omega_{\mu\nu}u_{G}^{\nu}&=\left(\frac{1}{\rho}\frac{\partial
P}{\partial\rho}+\frac{\partial
q}{\partial\rho}\right)\tilde{P}^{\alpha}_{\mu}\partial_{\alpha}\rho+\left(\frac{1}{\rho}\frac{\partial
P}{\partial s}+\frac{\partial q}{\partial
s}\right)\tilde{P}^{\alpha}_{\mu}\partial_{\alpha}s+\left(q+\frac{(P+\epsilon)}{\rho}\right)F^{(n)}_{\mu\nu}u_{G}^{\nu}\,.\end{split}$
(43)

In the absence of torsion,
$F_{\mu\nu}^{(n)}=\partial_{\mu}n_{\nu}-\partial_{\nu}n_{\mu}=0$, we find that
the right-hand-side of (43) vanishes for an equation of state of the form

$P=P(\rho)$ (44)

and

$q=-\int\frac{1}{\frac{\partial P}{\partial\mu}}\frac{\partial
P}{\partial\rho}d\rho=-\mu+c_{0}(T)\,.$ (45)

In the presence of torsion we need to require, in addition, that

$\rho=\rho(\mu+c(T))$ (46)

(where $c_{0}(T)=-Tc^{\prime}(T)$).

So far we have considered generic flows. In the context of fluid flow at low
velocities it is also interesting to consider subsonic flow where the fluid
becomes incompressible, see, e.g., muller1998low . In this limit the particle
number density becomes constant, so that the equations of motion reduce to the
incompressible Navier-Stokes equations. Put differently, incompressible flow
can be thought of as a particular class of solutions to the equations of
motion where the particle number density, and consequently the entropy
density, are constant, and the pressure becomes an independent function of the
coordinates. In torsionless, incompressible flow, equation (43) is
automatically satisfied for an arbitrary choice of $q$. In the presence of
torsion we must require $q=-(P+\epsilon)/\rho$.

To relate the covariant expressions (40), (42) and (44) to (5) and (7) we take
the flat, torsionless, spacetime limit of (40) defined in (126). The enstrophy
current $J^{\mu}_{G}$ in (40) reduces to (7). For incompressible flow, the
particle number density, and therefore the entropy density become constant in
which case (40) reduces to (5) up to an overall constant.

##  3 Enstrophy from symmetry

As stated in the introduction, it stands to reason that the enstrophy current
of hydrodynamics is a result of an emergent symmetry of the theory. In what
follows, we will use a recently developed formalism which allows one to
construct an effective action for hydrodynamics Haehl:2015foa ;
Crossley:2015evo ; Haehl:2015uoc  in order to identify the symmetry associated
with enstrophy conservation. We will start with the relativistic enstrophy
current for which the effective action has been studied in detail and then
move on to the non relativistic theory where some extra ingredients are
necessary in order to construct the effective action and derive the symmetry
associated with (approximate) enstrophy conservation.

###  3.1 Relativistic enstrophy from symmetry

An effective action for an ideal charged fluid can be written in terms of a
set of dynamical fields $X^{\mu}(\sigma)$ and $C(\sigma)$,

$S_{eff}(X^{\mu}\,,C;\,\beta^{i},\,\Lambda_{\beta},\,g_{\mu\nu},\,A_{\mu})=\int\sqrt{-|g_{ij}|}P(T,\,\mu)d^{d+1}\sigma\,.$
(47)

The function $X^{\mu}(\sigma)$ may be thought of as a dynamical Eulerian
coordinate specifying the location of the fluid in a target space and
$C(\sigma)$ an equivalent function specifying its phase under a global $U(1)$
symmetry. The parameters $\beta^{i}$ and $\Lambda_{\beta}$ specify the
configuration of the fluid in the far past, and $g_{\mu\nu}$ and $A_{\mu}$
specify the metric and $U(1)$ flavor field of the target space where the fluid
resides. The fields $g_{ij}$, $T$ and $\mu$ are defined via

$g_{ij}(X)=\partial_{i}X^{\mu}\partial_{j}X^{\nu}g_{\mu\nu}(X)\,,\qquad\beta^{i}g_{ij}\beta^{j}=-T^{-2}\,,\qquad\frac{\mu}{T}=\beta^{i}\left(\partial_{i}X^{\mu}A_{\mu}(X)+\partial_{i}C\right)+\Lambda_{\beta}$
(48)

and $P$ is a real function. By computing the stress tensor one finds that $P$
can be identified with the pressure, $T$ with the temperature and $\mu$ with
the chemical potential. Other actions for ideal or inviscid fluids can be
found in 1911AnP…341..493H ; PhysRev.94.1468 ; Carter ; Dubovsky:2005xd ;
Andersson:2006nr ; Dubovsky:2011sj . We have chosen (47) since by doubling the
fields (and adding appropriate ghosts) the action can be extended to include
non dissipative fluids. We refer the reader to Glorioso:2018wxw  for an
extensive discussion.

We claim that the following transformation of the dynamical fields

$\displaystyle\begin{split}\delta
X^{\mu}&=\frac{1}{Ts^{2}}\Omega^{2}u^{\mu}-\frac{2}{sp^{\prime
2}}E^{\mu}\Theta-\frac{4}{sp^{\prime}}P^{\mu\beta}{\Omega}_{\beta\alpha}a^{\alpha}+\frac{4}{p^{\prime}}P^{\mu\beta}\nabla_{\alpha}\left(\frac{1}{s}{\Omega}^{\alpha}{}_{\beta}\right)\\\
\delta C&=\frac{\mu}{Ts^{2}}{\Omega}^{2}-A_{\alpha}\delta
X^{\alpha}\,,\end{split}$ (49)

is a symmetry of the action in $2+1$ dimensions. Here
$u^{\mu}=\partial_{i}X^{\mu}\beta^{i}T$, and the remaining objects are related
to $u^{\mu}$, $T$, $\mu$ and $P$ as in section 2. For instance, $p^{\prime}$
is the derivative of $p$ with respect to its argument (see (29)). Further, the
symmetry (49) leads to a conserved current

$J^{\prime\,\mu}=\frac{\Omega^{2}}{s}u^{\mu}+\frac{4}{sp^{\prime}}\Omega^{\mu\nu}E_{\nu}$
(50)

which we may identify with the enstrophy current (10) with $g=1$ and $n=1$
once the equations of motion are satisfied. We will generalize (49) and the
associated (50) to obtain the class of currents (10) shortly.

To see that (49) is indeed a symmetry and leads to (50) let us consider a
generic variation $\delta X^{\mu}$ and $\delta C$ of the action. The equations
of motion for $\delta X^{\mu}$ are energy-momentum conservation in the target
space and the equation of motion for $\delta C$ is current conservation. Thus,

$\delta_{X}S_{eff}=-\int
d^{d+1}\sigma\sqrt{-|g_{ij}|}\bigg{(}\left(\nabla_{\mu}T^{\mu}{}_{\nu}-F_{\nu}{}^{\mu}J_{c\,\mu}+A_{\nu}\nabla_{\mu}J_{c}^{\mu}\right)\delta
X^{\nu}+\nabla_{\mu}J^{\mu}_{c}\delta
C+\left(\begin{subarray}{c}\hbox{total}\\\
\hbox{derivative}\end{subarray}\right)\bigg{)}\,.$ (51)

If the transformations $\delta X^{\mu}$ and $\delta C$ are a symmetry of the
action, then $\delta_{X}S_{eff}=0$ independently of the equations of motion.
Therefore, if $\delta X^{\mu}$ and $\delta C$ are symmetries,

$\left(\nabla_{\mu}T^{\mu}{}_{\nu}-F_{\nu}{}^{\mu}J_{c\,\mu}+A_{\nu}\nabla_{\mu}J_{c}^{\mu}\right)\delta
X^{\nu}+\nabla_{\mu}J_{c}^{\mu}\delta C=\nabla_{\mu}S^{\mu}$ (52)

with $S^{\mu}$ a local current. The symmetries which will generate the
enstrophy current should lead to $S^{\mu}=J^{\mu}$ up to possible extra terms
proportional to the equations of motion. Using the expression for $J^{\mu}$ in
(10) with $g=1$ and $n=1$ we find

$\nabla_{\mu}J^{\mu}=\frac{1}{s^{2}}E{\Omega}^{2}-\frac{2}{sp^{\prime
2}}(E^{\alpha}E_{\alpha})\Theta+\frac{4}{sp^{\prime}}{\Omega}_{\alpha\beta}a^{\alpha}E^{\beta}+\frac{4}{p^{\prime}}\nabla_{\alpha}\left(\frac{1}{s}{\Omega}^{\alpha\beta}\right)E_{\beta}-4\nabla_{\alpha}\left(\frac{\Omega^{\alpha\beta}E_{\beta}}{p^{\prime}s}\right)\,.$
(53)

Inserting (53) into (52) and solving for $\delta X^{\mu}$ and $\delta C$ will
give us transformations which can not be written in terms of positive powers
of the equations of motion or their derivatives. To remedy this, we use
$S^{\mu}=J^{\prime\mu}$ which leads to

$\displaystyle\begin{split}&(E_{\alpha}-TEu_{\alpha}-T\mu
E^{\prime}u_{\alpha})\delta X^{\alpha}-E^{\prime}(A_{\alpha}\delta
X^{\alpha}+\delta C)=\\\ &+\frac{1}{s^{2}}E{\Omega}^{2}-\frac{2}{sp^{\prime
2}}(E^{\alpha}E_{\alpha})\Theta+\frac{4}{sp^{\prime}}{\Omega}_{\alpha\beta}a^{\alpha}E^{\beta}+\frac{4}{p^{\prime}}\nabla_{\alpha}\left(\frac{1}{s}{\Omega}^{\alpha\beta}\right)E_{\beta}\,.\end{split}$
(54)

(Note that covariance of (LABEL:E:OffshellX) is ensured due to $\delta
C\rightarrow-\delta X^{\alpha}\partial_{\alpha}\Lambda$ under gauge
transformations.) One can check that the $\delta X^{\mu}$ and $\delta C$ given
in (49) satisfy (LABEL:E:OffshellX).

Symmetries associated with conserved currents constructed from higher powers
of $\Omega^{2}$ as in (10), can be obtained in a similar fashion. Using
$S^{\mu}=J^{\mu}+\mathcal{O}(E)$ in (52) we find that

$\displaystyle\begin{split}\delta
X^{\mu}=&\frac{(2n-1)g}{Ts^{2n}}({\Omega}^{2})^{n}u^{\mu}-\frac{g^{\prime}}{Ts^{2n-1}\rho}(\Omega^{2})^{n}u^{\mu}-\frac{2ng}{s^{2n-1}p^{\prime
2}}(\Omega^{2})^{n-1}E^{\mu}\Theta\\\
&-\frac{4ng}{s^{2n-1}p^{\prime}}(\Omega^{2})^{n-1}P^{\mu}{}_{\beta}{\Omega}^{\beta\alpha}a_{\alpha}+\frac{4n}{p^{\prime}}P^{\mu}{}_{\beta}\nabla_{\alpha}\left(\frac{g}{s^{2n-1}}(\Omega^{2})^{n-1}{\Omega}^{\alpha\beta}\right)\,,\\\
\delta C=&\frac{(2n-1)\mu g}{Ts^{2n}}({\Omega}^{2})^{n}-\frac{\mu
g^{\prime}}{Ts^{2n-1}\rho}(\Omega^{2})^{n}-\frac{g^{\prime}}{s^{2(n-1)}\rho^{2}}(\Omega^{2})^{n}-A_{\alpha}\delta
X^{\alpha}\,,\end{split}$ (55)

leads to the conserved current

${J}^{\prime\,\mu}=g\left(\frac{s}{\rho}\right)\frac{(\Omega^{2})^{n}}{s^{2n-1}}u^{\mu}+g\left(\frac{s}{\rho}\right)\frac{4n}{s^{2n-1}p^{\prime}}(\Omega^{2})^{n-1}\Omega^{\mu\nu}E_{\nu}\,.$
(56)

For $n=1$ and $g=1$ we recover (49) and (50) as expected. For completeness we
note that

$\displaystyle\begin{split}\delta
X^{\mu}&=\frac{1}{T}\left(\frac{2\dot{h}}{s^{2}}\Omega^{2}-\frac{s}{\rho}h^{\prime}-h\right)u^{\mu}-\frac{4\dot{h}}{sp^{\prime}}P^{\mu\alpha}\Omega_{\alpha\beta}a^{\beta}-\frac{2\dot{h}}{sp^{\prime
2}}\Theta
E^{\mu}+\frac{4}{p^{\prime}}P^{\mu}{}_{\beta}\nabla_{\alpha}\left(\frac{\dot{h}}{s}\Omega^{\alpha\beta}\right)\,,\\\
\delta
C&=\frac{\mu}{T}\left(\frac{2\dot{h}}{s^{2}}\Omega^{2}-\frac{s}{\rho}h^{\prime}-h\right)-\frac{s^{2}}{\rho^{2}}h^{\prime}-A_{\alpha}\delta
X^{\alpha}\end{split}$ (57)

generate $J_{h}^{\mu}$ as defined in (12). In (57) we have defined
$h^{\prime}$ and $\dot{h}$ to be the derivatives with respect to the first and
second argument of $h$ respectively.

Unfortunately, neither (49) nor (55) nor (57) seem to provide a physically
meaningful insight into the symmetry responsible for enstrophy conservation.
The simplest expression we were able to extract from (55) is

$\displaystyle\begin{split}\delta
X^{\alpha}=\frac{1}{\sqrt{2}}\frac{4}{p^{\prime}}\epsilon^{\alpha\beta\rho}u_{\beta}a_{\rho}\,,\qquad\delta
C=-A_{\alpha}\delta X^{\alpha}\,,\end{split}$ (58)

(up to the equations of motion), obtained by setting $n=1/2$ and $g=1$. The
transformation (58) is associated with
$J^{\prime\mu}_{H}=\frac{1}{\sqrt{2}}\left(\varpi
u^{\mu}+\frac{2}{p^{\prime}}\epsilon^{\mu\alpha\beta}E_{\alpha}u_{\beta}\right)$
obtained from (25) with $H=\frac{1}{\sqrt{2}}\varpi/s$. It is unclear whether
the divergence of the latter current is sign definite.

So far we have worked with a Lagrange description of the fluid. It is possible
to relate the symmetry (49) to a symmetry of the Eulerian degrees of freedom,
$u^{\mu}$, $T$ and $\mu$. This symmetry can be easily found by considering the
pushforwards of the initial state data through the dynamical degrees of
freedom $X^{\mu}$ and $C$,

$\beta^{\mu}=\partial_{i}X^{\mu}\beta^{i}(\sigma(X))\,,\qquad\bar{\Lambda}_{\beta}=\Lambda_{\beta}(\sigma(X))+\beta^{\mu}\partial_{\mu}C(\sigma(X))$
(59)

which, under a change $\delta X^{\mu}$ and $\delta C$, transform as

$\displaystyle\begin{split}\delta\beta^{\mu}=&-\mathcal{L}_{\delta
X}\beta^{\mu}\,,\qquad\delta\bar{\Lambda}_{\beta}=-\mathcal{L}_{\delta
X}\bar{\Lambda}_{\beta}+\beta^{\mu}\partial_{\mu}\delta C\,,\end{split}$ (60)

where $\mathcal{L}_{\delta X}$ is the Lie derivative in the $\delta X^{\mu}$
direction with $\delta X^{\mu}$ and $\delta C$ defined in (49). Using (60) and
the definitions of the Eulerian variables in the target space

$T=\frac{1}{\sqrt{-\beta^{\mu}\beta^{\nu}g_{\mu\nu}}}\,,\qquad
u^{\mu}=T\beta^{\mu}\,,\qquad\mu=u^{\mu}A_{\mu}+\bar{\Lambda}_{\beta}\,,$ (61)

the symmetry (49), or more generally (55), acts on the conventional degrees of
freedom as

$\displaystyle\begin{split}\delta u^{\mu}=&-TP^{\mu}_{\nu}{\cal L}_{\delta
X}\beta^{\nu}\,,\qquad\delta T=-T^{2}u_{\nu}{\cal L}_{\delta
X}\beta^{\nu}\,,\\\ \delta\mu=&-\mu Tu_{\nu}{\cal L}_{\delta
X}\beta^{\nu}-TA_{\nu}{\cal L}_{\delta X}\beta^{\nu}-T{\cal L}_{\delta
X}\bar{\Lambda}_{\beta}+u^{\mu}\partial_{\mu}\delta C\,,\end{split}$ (62)

while it is inert on the target space sources

$\delta g_{\mu\nu}=0\,,\qquad\delta A_{\mu}=0\,.$ (63)

###  3.2 Galilean enstrophy from symmetry

The procedure described in the previous section for obtaining the symmetry
which generates the relativistic enstrophy current can be readily generalized
to Galilean invariant systems. In what follows we first describe the
ingredients required to construct an effective action for Galilean invariant
fluids and then proceed to identify the symmetry associated with approximate
conservation of enstrophy.

####  3.2.1 A Galilean effective action for hydrodynamics

A Schwinger-Keldysh effective action for Galilean fluids can be constructed
from a higher dimensional relativistic one by equipping the latter with a null
Killing vector PhysRevD.31.1841 . This procedure was carried out in detail in
Jain:2020vgc . Here, we will use an alternate construction similar to the one
used to formulate the Schwinger-Keldysh effective action for relativistic
fluids, or any infrared action for that matter. Namely, we identify the
symmetries and dynamical fields associated with the fluid and then construct
the most general action compatible with those symmetries. Since a full
construction of the Schwinger-Keldysh effective action for Galilean fluids is
available in Jain:2020vgc  and since the various conceptual hurdles for
constructing effective actions for fluids were described in detail in
Haehl:2015foa ; Crossley:2015evo ; Haehl:2015uoc ; Haehl:2016pec ;
Haehl:2016uah ; Glorioso:2016gsa ; Jensen:2017kzi ; Gao:2017bqf ;
Glorioso:2017fpd ; Glorioso:2017lcn ; Jensen:2018hhx ; Haehl:2018uqv ;
Jensen:2018hse ; Gao:2018bxz ; Haehl:2018lcu ; Glorioso:2019koh , we will be
somewhat sparse in our exposition.

In a Newton-Cartan background geometry, the effective action should be
invariant under coordinate reparameterizations, $x^{\mu}\rightarrow
x^{\mu}+\xi^{\mu}$, the $U(1)$ gauge symmetry with parameter $\Lambda$, and
Milne boosts with parameter $\psi_{\nu}$. When acting on the the Newton-Cartan
data, these transformations take the form

$\displaystyle\begin{split}\delta_{\chi}n_{\mu}&={\cal L}_{\xi}n_{\mu}\,,\\\
\delta_{\chi}h^{\mu\nu}&={\cal L}_{\xi}h^{\mu\nu}\,,\\\
\delta_{\chi}\bar{n}^{\mu}&={\cal
L}_{\xi}\bar{n}^{\mu}+h^{\mu\nu}\psi_{\nu}\,,\\\ \delta_{\chi}A_{\mu}&={\cal
L}_{\xi}A_{\mu}+\partial_{\mu}\Lambda+P_{\mu}^{\nu}\psi_{\nu}-\frac{1}{2}n_{\mu}\psi^{2}\,,\\\
\delta_{\chi}\bar{h}_{\mu\nu}&={\cal
L}_{\xi}\bar{h}_{\mu\nu}-\left(n_{\mu}P_{\nu}^{\lambda}+n_{\nu}P^{\lambda}_{\mu}\right)\psi_{\lambda}+n_{\mu}n_{\nu}\psi^{2}\,,\end{split}$
(64)

where $\psi^{2}=\psi_{\nu}\psi_{\rho}h^{\nu\rho}$ and $\delta_{\chi}$ denotes
a target space coordinate reparameterization, a $U(1)$ gauge transformation
and a Milne transformation. The inverse metric $\bar{h}_{\mu\nu}$ is defined
in (106b).

The dynamical fields of the Galilean invariant effective action for fluid
dynamics are given by the coordinates $X^{\mu}(\sigma)$ and a phase
$C(\sigma)$. As is the case for relativistic fluid dynamics, the $X^{\mu}$
fields parameterize worldlines of fluid elements. They provide a mapping
between a parameter space specified by the coordinate $\sigma^{i}$ which we
refer to as a worldvolume and the space where the fluid elements live in,
which we refer to as the target space. Similarly, $C(\sigma)$ is the field
that captures the local phase of each fluid element. The astute reader will
note that in addition to $X^{\mu}(\sigma)$ and $C(\sigma)$, one may have
included a field $\phi_{\mu}(\sigma)$ which maps Milne transformations from
the target space to the worldvolume. Worldvolume quantities which are not
Milne invariant could then be rendered as such by modifying them with
appropriate factors of $\phi_{\mu}(\sigma)$. As we shall see shortly all
worldvolume quantities are explicitly Milne invariant so that
$\phi_{\mu}(\sigma)$ will not appear in the effective action.

The dynamical variables are bifundamental fields and as such transform under
the target space symmetries as well as under the corresponding symmetries
induced on the worldvolume: worldvolume reparameterizations labeled by
$\hat{\xi}^{i}$, worldvolume $U(1)$ gauge transformations with parameter
$\hat{\Lambda}$, and worldvolume Milne boosts parameterized by
$\hat{\psi}_{i}$,

$\displaystyle\begin{split}\delta_{(\chi,\hat{\chi})}X^{\mu}(\sigma)&=-\xi^{\mu}(X(\sigma))+\hat{\xi}^{i}(\sigma)\partial_{i}X^{\mu}(\sigma)\,,\\\
\delta_{(\chi,\hat{\chi})}C(\sigma)&=-\Lambda(X(\sigma))+\hat{\Lambda}(\sigma)+{\cal
L}_{\hat{\xi}}C(\sigma)\,,\end{split}$ (65)

with $\delta_{\hat{\chi}}$ denoting worldvolume transformations. Had we added
$\phi_{\mu}(\sigma)$, we would have found

$\delta_{(\chi,\hat{\chi})}\phi_{\mu}(\sigma)=-P_{\mu}^{\nu}(X(\sigma))\psi_{\nu}(X(\sigma))+P^{\nu}_{\mu}(X(\sigma))(\partial_{i}X^{\nu})^{-1}\hat{\psi}_{i}(\sigma)+\hat{\xi}^{i}(\sigma)\partial_{i}\phi_{\mu}(\sigma)-{\cal
L}_{\xi}\phi_{\mu}(\sigma)$ (66)

As should be clear from (65), $\delta X^{\mu}$ and $\delta C$ are both
invariant under worldvolume Milne transformations.

In addition to the dynamical fields, the effective action will depend on the
initial state data which specifies the equilibrium state of the system in the
infinite past. This consists of a timelike Killing vector,
$\beta^{i}(\sigma)$, specifying the initial velocity and temperature, a gauge
Killing parameter, $\Lambda_{\beta}(\sigma)$, associated with the initial
chemical potential, and a Milne boost one-form $\psi^{i}_{\beta}(\sigma)$.
Since the system is in equilibrium in the infinite past the mapping between
the target space and worldvolume is trivial. Thus,

$\delta_{\beta}n_{\mu}(t=-\infty)=0\,,\quad\delta_{\beta}h^{\mu\nu}(t=-\infty)=0\,,\quad\delta_{\beta}\bar{n}^{\mu}(t=-\infty)=0\,,\quad\delta_{\beta}A_{\mu}(t=-\infty)=0\,,$
(67)

where $\delta_{\beta}$ collectively denotes a worldvolume transformation given
in (72) with parameters $\\{\beta^{i},\Lambda_{\beta},\psi_{i}^{\beta}\\}$.
Worldvolume coordinate reparameterizations and $U(1)$ gauge transformations
acting on the initial data take the form:222Note that it is always possible to
choose a gauge where the parameters specifying the initial data are fixed. A
common choice is the static gauge where $\beta^{i}=b(1,\vec{0})$, with $b$ a
constant, $\Lambda_{\beta}=0$ and $\psi_{i}^{\beta}=0$. As a result
worldvolume transformations of the initial data will be restricted to a subset
preserving the static gauge. We refrain from choosing a gauge in order to
retain an explicitly covariant formulation of the action.

$\displaystyle\begin{split}\delta_{\hat{\chi}}\beta^{i}&={\cal
L}_{\hat{\xi}}\beta^{i}\,,\\\ \delta_{\hat{\chi}}\Lambda_{\beta}&={\cal
L}_{\hat{\xi}}\Lambda_{\beta}-\beta^{i}\partial_{i}\hat{\Lambda}\,,\\\
\delta_{\hat{\chi}}{\psi}^{\beta}_{i}&={\cal
L}_{\hat{\xi}}{\psi}^{\beta}_{i}-{\cal
L}_{\beta}{{\hat{\psi}}_{i}}+{\hat{\psi}}_{i}\,.\end{split}$ (68)

Notice that $\beta^{i}$ and $\Lambda_{\beta}$ are invariant under Milne boosts
while $\psi_{i}^{\beta}$ transforms non trivially under it.

The local effective action for Galilean fluids, $S_{eff}$, is constructed from
worldvolume and target space invariant combinations of the dynamical fields
and initial state data. In practice, it is convenient to define the target
space invariant quantities

$\displaystyle\begin{split}n_{i}(\sigma)&=\partial_{i}X^{\mu}n_{\mu}(X)\,,\\\
h^{ij}(\sigma)&=(\partial_{i}X^{\mu})^{-1}(\partial_{j}X^{\nu})^{-1}h^{\mu\nu}(X)\,,\\\
\tilde{A}_{i}(\sigma)&=\partial_{i}X^{\mu}\tilde{A}_{\mu}(X)+\partial_{i}C\,,\\\
\tilde{h}_{ij}(\sigma)&=\partial_{i}X^{\mu}\partial_{j}X^{\nu}\tilde{h}_{\mu\nu}(X)\,,\end{split}$
(69)

where the tilde’d quantities

$\displaystyle\begin{split}\tilde{A}_{\mu}=&A_{\mu}+u_{G\,\mu}-\frac{1}{2}n_{\mu}u_{G}^{2}\,,\\\
\tilde{h}_{\mu\nu}=&\bar{h}_{\mu\nu}-u_{G\,\mu}n_{\nu}-u_{G\,\nu}n_{\mu}+n_{\mu}n_{\nu}u_{G}^{2}\,,\end{split}$
(70)

with

$u^{\mu}_{G}(X)=\frac{1}{\beta^{i}n_{i}}\beta^{j}\partial_{j}X^{\mu}\,,\qquad
u_{G\,\mu}=\bar{h}_{\mu\nu}u^{\nu}_{G}\,,\qquad
u^{2}_{G}=u_{G\,\mu}u^{\mu}_{G}\,,$ (71)

are Milne invariant. Note that had we not used the target space Milne
invariant variables $\tilde{A}_{\mu}$ and $\tilde{h}_{\mu\nu}$ in (69), we
would have been forced to use $\phi_{\mu}$ to ensure target space Milne
invariance of $\tilde{A}_{i}$ and $\tilde{h}_{ij}$. It is the absence of
$\phi_{\mu}$ on the right-hand-side of (69) that ensures that it does not
appear in the effective action. It is straightforward to show that the target
space invariant combinations (69) transform under worldvolume
reparameterizations and $U(1)$ gauge transformations induced by the
transformations of the dynamical fields (65) as

$\displaystyle\begin{split}\delta_{\hat{\chi}}n_{i}&={\cal
L}_{\hat{\xi}}n_{i}\,,\\\ \delta_{\hat{\chi}}h^{ij}&={\cal
L}_{\hat{\xi}}h^{ij}\,,\\\ \delta_{\hat{\chi}}\tilde{A}_{i}&={\cal
L}_{\hat{\xi}}\tilde{A}_{i}+\partial_{i}\hat{\Lambda}\,,\\\
\delta_{\hat{\chi}}\tilde{h}_{ij}&={\cal
L}_{\hat{\xi}}\tilde{h}_{ij}\,,\end{split}$ (72)

and are invariant under worldvolume Milne boosts.

The symmetries on the worldvolume can be maintained by requiring the action to
be a scalar that depends only on $U(1)$ gauge invariant and Milne invariant
quantities. At leading order in derivatives, the unique scalar invariants are

$\displaystyle\begin{split}T=\frac{1}{\beta^{i}n_{i}}\,,\qquad\mu=T\beta^{i}\tilde{A}_{i}+T\Lambda_{\beta}\end{split}$
(73)

corresponding respectively to the temperature and the chemical potential.
Keeping all the symmetries intact, we find that the most general effective
action for Galilean fluids at leading order in derivatives is

$S_{eff}=\int d^{d+1}\sigma\sqrt{\gamma}\,P(T,\mu)\,,$ (74)

where the measure is given by the (Milne invariant) determinant of
$\gamma_{ij}=\partial_{i}X^{\mu}\partial_{j}X^{\nu}\bar{h}_{\mu\nu}+n_{i}n_{j}$
and $P$ is a generic function of the temperature $T$ and chemical potential
$\mu$.

To get a feel for this formulation of Galilean hydrodynamics let us derive the
equations of motion for (ideal) Galilean fluids by varying the effective
action with respect to the dynamical variables. A generic variation of the
effective action (74) is given by

$\delta S_{eff}=\int
d^{d+1}\sigma\sqrt{\gamma}\left(P\frac{1}{\sqrt{\gamma}}\delta\sqrt{\gamma}+s\,\delta
T+\rho\,\delta\mu\right)$ (75)

where we have defined

$s=\left(\frac{\partial P}{\partial
T}\right)_{\mu}\qquad\hbox{and}\qquad\rho=\left(\frac{\partial
P}{\partial\mu}\right)_{T}\,.$ (76)

In order to write the variations specified in (75) in terms of variations of
the dynamical variables, we first use (73) to write

$\displaystyle\begin{split}\delta T&=-Tu^{i}_{G}\delta n_{i}\,,\\\
\delta\mu&=u^{i}_{G}\delta\tilde{A}_{i}-\mu u^{i}_{G}\delta n_{i}\,,\\\
\frac{1}{\sqrt{\gamma}}\delta\sqrt{\gamma}&=\frac{1}{\sqrt{\tilde{\gamma}}}\delta\sqrt{\tilde{\gamma}}=u_{G}^{i}\delta
n_{i}+\frac{1}{2}\tilde{\gamma}^{ij}\delta{\tilde{h}}_{ij}\,,\end{split}$ (77)

where we have defined $u^{i}_{G}=T\beta^{i}$ and

$\tilde{\gamma}_{ij}=n_{i}n_{j}+\tilde{h}_{ij}\,,\qquad\tilde{\gamma}^{ij}=u_{G}^{i}u_{G}^{j}+h^{ij}\,.$
(78)

To derive the last expression in (77) we have used

$\delta\tilde{h}_{ij}=\tilde{P}^{i}_{k}\tilde{P}^{j}_{l}\delta\bar{h}_{kl}-(u_{G\,i}-n_{i}u_{G}^{2})\tilde{P}^{k}_{j}\delta
n_{k}-(u_{G\,j}-n_{j}u_{G}^{2})\tilde{P}^{k}_{i}\delta n_{k}\,,\qquad\delta
u_{G}^{i}=-u_{G}^{i}u_{G}^{k}\delta n_{k}\,,$ (79)

with $u^{2}_{G}=u_{G\,i}u^{i}_{G}$ and
$\tilde{P}^{i}_{j}=h^{ik}\tilde{h}_{kj}=\delta^{i}_{j}-u^{i}_{G}n_{j}$. In
writing the generic variations in (77) we have not included variations with
respect to the initial state data $\beta^{i}$ and $\Lambda_{\beta}$ since they
do not depend on the dynamical variables.

Next, consider

$\displaystyle\begin{split}\delta n_{i}&=\partial_{i}X^{\mu}{\cal L}_{\delta
X}n_{\mu}\,,\\\ \delta\tilde{A}_{i}&=\partial_{i}X^{\mu}{\cal L}_{\delta
X}\tilde{A}_{\mu}+\partial_{i}\delta C\,,\\\
\delta\tilde{h}_{ij}&=\partial_{i}X^{\mu}\partial_{j}X^{\nu}\tilde{h}_{\mu\nu}\end{split}$
(80)

where ${\cal L}_{\delta X}$ is the Lie derivative along $\delta X^{\mu}$.
Inserting (80) into (77) and then into (75) we find

$\displaystyle\begin{split}\delta S_{eff}=-\int
d^{d+1}\sigma\sqrt{\gamma}\,\bigg{(}\left(E_{\mu}+(TE+\mu
E^{\prime})\,n_{\rho}\right)\delta X^{\rho}-E^{\prime}\left(\delta
C+\tilde{A}_{\rho}\delta X^{\rho}\right)\bigg{)}\end{split}$ (81)

up to total derivatives. In writing (81) we have repeatedly used the relation

$\frac{1}{\sqrt{\gamma}}\partial_{\mu}(\sqrt{\gamma}\,V^{\mu})=(\nabla_{\mu}-{\cal
G}_{\mu})V^{\mu}=(\tilde{\nabla}_{\mu}-\tilde{\cal G}_{\mu})V^{\mu}$ (82)

with $\tilde{\cal G}_{\mu}$ defined in (124). Satisfyingly, the expressions
for $E$, $E^{\prime}$ and $E_{\mu}$ coincide with those in (125). We reproduce
them here for convenience,

$\displaystyle\begin{split}{E}_{\mu}&=\tilde{P}^{\alpha}_{\mu}\partial_{\alpha}P-\rho\tilde{F}_{\mu\alpha}u_{G}^{\alpha}+(P+\epsilon)F^{(n)}_{\mu\alpha}u^{\alpha}_{G}\,,\\\
{E}&=-(\tilde{\nabla}_{\mu}-\tilde{\cal G}_{\mu})(s\,u_{G}^{\mu})\,,\\\
{E}^{\prime}&=-(\tilde{\nabla}_{\mu}-\tilde{\cal
G}_{\mu})(\rho\,u_{G}^{\mu})\,.\end{split}$

####  3.2.2 Extracting the Galilean enstrophy from symmetry

The transformations of $\delta X^{\mu}$ and $\delta C$ which generate the
symmetry associated with enstrophy conservation must satisfy

$(E_{\mu}+(TE+\mu E^{\prime})n_{\mu})\delta X^{\mu}-E^{\prime}(\delta
C+\tilde{A}_{\rho}\delta X^{\rho})=(\tilde{\nabla}_{\mu}-\tilde{\cal
G}_{\mu})S^{\mu}\,,$ (83)

with

$S^{\mu}=J_{G}^{\mu}+{\cal O}(E)\,.$ (84)

In 2+1 dimensions the expression in (33) reduces to

$\Omega^{\alpha\beta}\tilde{\nabla}_{\mu}(u^{\mu}_{G}\Omega_{\alpha\beta})=-2\Omega^{\mu\nu}\tilde{\nabla}_{\mu}\left(\frac{1}{\rho}E_{\nu}\right)+2\Omega^{\mu\nu}F_{\mu\rho}^{(n)}u^{\rho}\left(\frac{1}{\rho}E_{\nu}\right)\,,$
(85)

and conservation of the enstrophy current defined in (40) reads

$\displaystyle\begin{split}(\tilde{\nabla}_{\mu}-\tilde{\cal
G}_{\mu})J_{G}^{{\mu}}=&\frac{(2n-1)g}{s^{2n}}(\Omega^{2})^{n}E+\frac{g^{\prime}}{\rho\,s^{2n-1}}\left(\frac{s}{\rho}E^{\prime}-E\right)(\Omega^{2})^{n}\\\
&-\frac{4ng}{s^{2n-1}}(\Omega^{2})^{n-1}\Omega^{\alpha\beta}\tilde{\nabla}_{\alpha}\left(\frac{1}{\rho}E_{\beta}\right)+\frac{4ng}{s^{2n-1}\rho}(\Omega^{2})^{n-1}\Omega^{\alpha\beta}F^{(n)}_{\alpha\mu}u_{G}^{\mu}E_{\beta}\,.\end{split}$
(86)

Defining

$J^{\prime\mu}_{G}=J^{\mu}_{G}+\frac{4ng}{s^{2n-1}\rho}(\Omega^{2})^{n-1}\Omega^{\mu\beta}E_{\beta}\,,$
(87)

we find

$\displaystyle\begin{split}(\tilde{\nabla}_{\mu}-\tilde{\cal
G}_{\mu})J_{G}^{\prime\mu}&=\frac{(2n-1)g}{s^{2n}}(\Omega^{2})^{n}E+\frac{g^{\prime}}{\rho\,s^{2n-1}}\left(\frac{s}{\rho}E^{\prime}-E\right)(\Omega^{2})^{n}\\\
&+\frac{1}{\rho}E_{\beta}\tilde{\nabla}_{\alpha}\left(\frac{4ng}{s^{2n-1}}(\Omega^{2})^{n-1}\Omega^{\alpha\beta}\right)+\frac{4ng}{s^{2n-1}\rho}(\Omega^{2})^{n-1}\Omega^{\alpha\beta}F^{(n)}_{\alpha\mu}u_{G}^{\mu}E_{\beta}\,.\end{split}$
(88)

It is now straightforward to show that

$\displaystyle\begin{split}\delta
X^{\mu}&=\frac{1}{T}\frac{(2n-1)g}{s^{2n}}(\Omega^{2})^{n}u^{\mu}_{G}-\frac{g^{\prime}}{T\rho\,s^{2n-1}}(\Omega^{2})^{n}u^{\mu}_{G}\\\
&+\frac{1}{\rho}\tilde{P}^{\mu}_{\beta}\tilde{\nabla}_{\alpha}\left(\frac{4ng}{s^{2n-1}}(\Omega^{2})^{n-1}\Omega^{\alpha\beta}\right)+\frac{4ng}{s^{2n-1}\rho}(\Omega^{2})^{n-1}\Omega^{\alpha\mu}F_{\alpha\nu}^{(n)}u^{\nu}_{G}\,,\\\
\delta
C&=\frac{\mu}{T}\frac{(2n-1)g}{s^{2n}}(\Omega^{2})^{n}-\frac{\mu}{T}\frac{g^{\prime}}{\rho\,s^{2n-1}}(\Omega^{2})^{n}-\frac{g^{\prime}}{\rho^{2}\,s^{2n-2}}(\Omega^{2})^{n}-\tilde{A}_{\rho}\delta
X^{\rho}\end{split}$ (89)

satisfy the condition (83) with $S^{\mu}$ given by $J^{\prime\mu}_{G}$,
defined in (87). For completeness we note that

$\displaystyle\begin{split}\delta
X^{\mu}&=\frac{1}{T}\left(\frac{2\dot{h}}{s^{2}}\Omega^{2}-\frac{s}{\rho}h^{\prime}-h\right)u_{G}^{\mu}+\frac{4}{\rho}\tilde{P}^{\mu}{}_{\beta}\tilde{\nabla}_{\alpha}\left(\frac{\dot{h}}{s}\Omega^{\alpha\beta}\right)+\frac{4\dot{h}}{s\rho}\tilde{P}^{\mu}{}_{\beta}\Omega^{\alpha\beta}F^{(n)}_{\alpha\nu}u^{\nu}_{G}\\\
\delta
C&=\frac{\mu}{T}\left(\frac{2\dot{h}}{s^{2}}\Omega^{2}-\frac{s}{\rho}h^{\prime}-h\right)-\frac{s^{2}}{\rho^{2}}h^{\prime}-\tilde{A}_{\alpha}\delta
X^{\alpha}\end{split}$ (90)

lead to the conservation of $J_{G\,h}^{\mu}$ defined in (41).

##  4 Discussion and outlook

In this work we used the recently discovered effective action for
hydrodynamics to determine the approximate symmetry responsible for the
approximately conserved enstrophy current in $2+1$ dimensional relativistic
and Galilean flow. In the process of our analysis, we have identified a
mechanism which allows for the construction of the enstrophy current and used
it to generalize previously known results regarding its form.

The mechanism we identified for constructing the enstrophy current relies on
the existence of a closed two-form $\Omega_{\mu\nu}dx^{\mu}dx^{\nu}$
orthogonal to the velocity field, $\Omega_{\mu\nu}u^{\nu}=0$, at least under
the equations of motion. Once such a two-form is available the existence of
the enstrophy current is guaranteed. We believe that this mechanism can be
used to construct an enstrophy current for fluid flows which are not
relativistic or Galilean. Fluid dynamics in the absence of boost invariance
has been studied recently in deBoer:2017ing ; deBoer:2020xlc  and may be
relevant to a variety of physical systems, see, e.g., PhysRevE.58.4828 .

Our current analysis neglected dissipation, which, in the Galilean case, leads
to a non trivial but sign definite change in enstrophy charge over time. This
fact, together with conservation of energy, is a key ingredient in the
argument leading to the inverse energy cascade in turbulent flow (see Appendix
A). It is not known whether a relativistic enstrophy current whose divergence
is sign (semi-)definite exists. In order to study this problem one would start
with $J^{\mu}$ in (10) (setting, say, $n=1$) and consider
$\mathcal{O}(\partial^{3})$ corrections to it such that its divergence is sign
(semi-)definite up to $\mathcal{O}(\partial^{4})$. The existence of a
relativistic enstrophy current with a sign (semi-)definite divergence may have
implications for relativistic turbulence in $2+1$ dimensions.

In the context of holography, the existence of an enstrophy current for
conformal $2+1$ dimensional fluid flow implies its dual manifestation in
asymptotically AdS4 black brane geometries. More precisely, as is the case for
entropy, one may expect that asymptotically AdS4 black branes possess a
geometric quantity that captures enstrophy conservation in the boundary
theory. There are several approaches to this problem in the literature
Adams:2013vsa ; Eling:2013sna ; Green:2013zba  which may serve as an excellent
starting point for fully addressing this issue.

## Acknowledgments

We would like to thank A. Frishman and H. Liu for useful discussions. NPF is
supported by the Bijzonder Onderzoeksfonds 2020 at KU Leuven. AY is supported
in part by an Israeli Science Foundation excellence center grant 2289/18 and a
Binational Science Foundation grant 2016324.

##  Appendix A The non relativistic enstrophy charge

As discussed in the main text it is straightforward to argue that the
enstrophy is conserved in inviscid $2+1$ dimensional incompressible flow.
Consider the Navier-Stokes equation for incompressible fluids in the absence
of random forces

$\displaystyle\begin{split}\partial_{t}\vec{v}+\vec{v}\cdot\vec{\nabla}\vec{v}+\vec{\nabla}P&=\frac{1}{R}\nabla^{2}\vec{v}\,,\\\
\vec{\nabla}\cdot\vec{v}&=0\,,\end{split}$ (91)

where $R$ is the Reynolds number, $P$ is the pressure, and $\vec{v}$ is the
velocity field. We start by making two observations. By dotting the Navier
Stokes equation into $\vec{v}$ we find that

$\frac{1}{2}\partial_{t}v^{2}+\frac{1}{2}\vec{\nabla}\cdot\left(\vec{v}v^{2}\right)+\vec{\nabla}\left(\vec{v}P\right)=\frac{1}{R}\left(-\frac{1}{2}\omega_{ij}\omega^{ij}+\nabla_{j}\left(v_{i}\nabla^{j}v^{i}-v_{i}\nabla^{i}v^{j}\right)\right)$
(92)

where

$v^{2}=\vec{v}\cdot\vec{v}\,,\qquad\omega_{ij}=\partial_{i}v_{j}-\partial_{j}v_{i}\,,\qquad$
(93)

and we have used the incompressibility condition. Integrating (92) we find

${\partial_{t}}E=-\frac{1}{R}W$ (94)

where

$E=\frac{1}{2}\int\sqrt{g}\,v^{2}d^{d}x,\qquad\hbox{and}\qquad
W=\frac{1}{2}\int\sqrt{g}\,\omega_{ij}\omega^{ij}d^{d}x$ (95)

are referred to as the total energy and the total enstrophy respectively. In
obtaining (94) we have assumed that the fluid is on a manifold without a
boundary. We will not consider manifolds with boundaries in the remainder of
this work. When $R^{-1}=0$ then, unsurprisingly, energy is conserved.

To understand the role of enstrophy in establishing the dynamics of the
theory, let us consider the equation of motion for $\omega_{ij}$. By taking a
derivative of (91) we obtain

$\partial_{t}\omega_{ij}+\nabla_{k}\left(v^{k}\omega_{ij}\right)+\frac{1}{2}\left(\omega_{ik}\sigma^{k}_{j}+\sigma_{ik}\omega^{k}_{j}\right)=\\\
\frac{1}{R}\nabla^{2}\omega_{ij}$ (96)

where

$\sigma_{ij}=\nabla_{i}v_{j}+\nabla_{j}v_{i}\,.$ (97)

The third term from the left is referred to as a ‘vortex stretching’ term and
it vanishes in $2$ spatial dimensions. Indeed, let

$\omega_{ik}\sigma^{k}_{j}+\sigma_{ik}\omega^{k}_{j}=\epsilon_{ij}s$ (98)

and also

$\omega_{ij}=\epsilon_{ij}\omega\,.$ (99)

Then,

$s\propto\epsilon^{ij}\omega_{ik}\sigma^{k}_{j}=\omega\epsilon^{ij}\epsilon_{ik}\sigma^{k}_{j}=\sigma^{j}_{j}=0$
(100)

where the last equality follows from the incompressibility condition. The
enstrophy production equation reads

$\partial_{t}W=\int\sqrt{g}\,\omega_{ji}\omega^{i}{}_{k}\sigma^{kj}d^{d}x-\frac{1}{R}P$
(101)

where $P$ is the Palinstrophy,

$P=\int\sqrt{g}\,\nabla_{k}\omega_{ij}\nabla^{k}\omega^{ij}d^{d}x\,.$ (102)

In the presence of the vortex stretching term the rate of change of $W$ is not
sign definite. In this case experimental results and indirect theoretical
arguments lead to

$\lim_{R^{-1}\rightarrow 0}\frac{W}{R}=e_{0}$ (103)

where $e_{0}$ is a constant. With some work, (see, e.g., frisch1995turbulence
) one can show that (103) leads to the Kolmogorov energy cascade in turbulent
flow. Once the vortex stretching term is absent, it is easy to show that
$\partial_{t}W\leq 0$. Since the enstrophy is a positive quantity, it can not
diverge if it were initially finite and (103) is no longer valid. Instead one
finds, via (94), that energy will be conserved at large Reynolds number
leading, eventually, to an inverse energy cascade (and also a direct enstrophy
cascade) in two dimensional turbulent flow.

We also note in passing that higher moments of the enstrophy are also
monotonically decreasing and conserved when $R^{-1}=0$, viz.

$\partial_{t}\int\sqrt{g}\left(\omega_{ij}\omega^{ij}\right)^{n}d^{2}x=-\frac{n}{R}\int\sqrt{g}\left(\omega_{ij}\omega^{ij}\right)^{n-1}\nabla_{k}\omega_{ij}\nabla^{k}\omega^{ij}d^{2}x\,.$
(104)

whenever $n>0$. Alternately,

$\partial_{t}\int\sqrt{g}\,h\left(\omega_{ij}\omega^{ij}\right)d^{2}x=-\frac{1}{R}\int\sqrt{g}\,h^{\prime}\left(\omega_{ij}\omega^{ij}\right)\nabla_{k}\omega_{ij}\nabla^{k}\omega^{ij}d^{2}x\,.$
(105)

is negative as long as $h$ is a monotonically increasing function.

##  Appendix B Newton-Cartan geometry and hydrodynamics

Galilean invariant dynamics in a curved background, and Galilean invariant
hydrodynamics in particular, is properly described by Newton-Cartan geometry.
In what follows we will briefly summarize key elements of the Newton-Cartan
formalism developed in Jensen:2014aia  and then use it to recast Galilean
hydrodynamics in a manifestly covariant form. See Jensen:2014ama .

###  B.1 Newton-Cartan geometry

In $d+1$ spacetime dimensions, the independent Newton-Cartan background data
can be taken to be the set $(n_{\mu},h^{\mu\nu},A_{\mu},\,\bar{n}^{\mu})$,
where $n_{\mu}$ is a nowhere vanishing one-form which defines the local time
direction, $h^{\mu\nu}$ is a rank $d$ positive semi-definite symmetric tensor
which satisfies $h^{\mu\nu}n_{\mu}=0$ and can be seen as defining the
(inverse) spatial metric, $A_{\mu}$ is a $U(1)$ gauge field associated with
the conservation of particle number, and $\bar{n}^{\mu}$ is related to
$n_{\mu}$ via

$\bar{n}^{\mu}n_{\mu}=1.$ (106a) Based on the Newton-Cartan data, one can
define a positive-definite spacetime metric $\gamma^{\mu\nu}$ (and its inverse
$\gamma_{\mu\nu}$), a rank $d$ (spatial) metric $\bar{h}_{\mu\nu}$ and a
projector $P^{\mu}{}_{\nu}$ via
$\gamma^{\mu\nu}=\bar{n}^{\mu}\bar{n}^{\nu}+h^{\mu\nu}\,,\qquad\bar{h}_{\mu\nu}=\gamma_{\mu\nu}-n_{\mu}n_{\nu}\,,\qquad
P^{\mu}{}_{\nu}=h^{\mu\rho}\bar{h}_{\nu\rho}=\delta^{\mu}_{\nu}-\bar{n}^{\mu}n_{\nu}\,.$
(106b)

Note that $\bar{h}_{\mu\nu}\bar{n}^{\nu}=0$ and
$P^{\mu}{}_{\nu}\bar{n}^{\nu}=P^{\mu}{}_{\nu}n_{\mu}=0$.

In Newton-Cartan theory different choices of $\bar{n}^{\mu}$ are equivalent.
This is a result of the requirement that the underlying theory is Galilean
invariant. In practice, we require that the action is invariant under a
transformation $\bar{n}^{\mu}\to\bar{n}^{\prime\mu}$ obtained via

$\bar{n}^{\prime\mu}=\bar{n}^{\mu}+h^{\mu\nu}\psi_{\nu}$ (107)

with $\psi_{\nu}$ a transverse one-form, $\psi_{\nu}\bar{n}^{\nu}=0$. The
transformation (107) is referred to as a Milne boost. The action of Milne
boosts on the metric and gauge field is given by

$\displaystyle\begin{split}\bar{h}_{\mu\nu}^{\prime}&=\bar{h}_{\mu\nu}-({n}_{\mu}P^{\rho}{}_{\nu}+n_{\nu}P^{\rho}{}_{\mu})\psi_{\rho}+n_{\mu}n_{\nu}h^{\alpha\beta}\psi_{\alpha}\psi_{\beta}\,,\\\
A_{\mu}^{\prime}&=A_{\mu}+P^{\nu}{}_{\mu}\psi_{\nu}-\frac{1}{2}n_{\mu}h^{\nu\rho}\psi_{\nu}\psi_{\rho}\,,\end{split}$
(108)

with $h^{\mu\nu}$ and $n_{\mu}$ invariant. Invariance under Milne
transformations is a key requirement used to construct Galilean invariant
theories in a curved background.

The covariant measure appearing in spacetime integrals is
$d^{d+1}x\,\sqrt{\gamma}$, where $\gamma={\rm det}(\gamma_{\mu\nu})$, which
can be shown to be Milne invariant. The derivative that reduces to a boundary
term under the integral is given by the combination

$({\nabla}_{\mu}-{\cal
G}_{\mu})V^{\mu}=\frac{1}{\sqrt{\gamma}}\partial_{\mu}(\sqrt{\gamma}\,V^{\mu})\,,$
(109)

where we have defined

${\cal
G}_{\mu}={T}^{\alpha}_{\mu\alpha}=-F^{(n)}_{\mu\alpha}\bar{n}^{\alpha}\,$
(110)

with

$F_{\mu\nu}^{(n)}=\partial_{\mu}n_{\nu}-\partial_{\nu}n_{\mu}\,.$ (111)

A covariant derivative ${\nabla}_{\mu}$ can be constructed by requiring
compatibility with the Newton-Cartan data,

${\nabla}_{\mu}n_{\nu}=0\,,\qquad{\nabla}_{\mu}h^{\alpha\beta}=0\,,$ (112)

and restricting the torsion to be timelike,
$\bar{h}_{\lambda\rho}{T}^{\lambda}_{\mu\nu}=0$,

$\displaystyle\begin{split}{\Gamma}^{\lambda}_{\mu\nu}&=\bar{n}^{\lambda}\partial_{\nu}n_{\mu}+\frac{1}{2}h^{\lambda\rho}\left(\partial_{\mu}\bar{h}_{\nu\rho}+\partial_{\nu}\bar{h}_{\mu\rho}-\partial_{\rho}\bar{h}_{\mu\nu}\right)+\frac{1}{2}h^{\lambda\rho}\left(n_{\mu}{F}_{\nu\rho}+n_{\nu}{F}_{\mu\rho}\right)\,,\\\
{T}_{\mu\nu}^{\lambda}&={\Gamma}^{\lambda}_{\mu\nu}-{\Gamma}^{\lambda}_{\nu\mu}=-\bar{n}^{\lambda}F_{\mu\nu}^{(n)}\,,\end{split}$
(113)

where we have defined the field strength

${F}_{\mu\nu}=\partial_{\mu}{A}_{\nu}-\partial_{\nu}{A}_{\mu}$ (114)

and we used the conventions

${\nabla}_{\mu}V^{\alpha}_{\beta}=\partial_{\mu}V^{\alpha}_{\beta}+{\Gamma}^{\alpha}_{\rho\mu}V^{\rho}_{\beta}-{\Gamma}_{\beta\mu}^{\rho}V_{\rho}^{\alpha}\,.$
(115)

Our construction closely follows that of Jensen:2014aia . A more general
analysis can be found in Hartong:2015zia . The connection (113) is not Milne
invariant. Unfortunately, using only the Newton-Cartan data it is not possible
to construct a connection that is both Milne invariant and gauge invariant,
see Jensen:2014aia . As we will see shortly, when discussing the hydrodynamic
theory, one can use the velocity field as additional data in order to
construct Milne and $U(1)$ invariant connections, see Jensen:2014ama .

###  B.2 Fluids on a Newton-Cartan background

Let us now consider a fluid in a curved Newton-Cartan background geometry.
Following Jensen:2014ama , we equip our theory with a Milne-invariant timelike
velocity vector field $u^{\mu}_{G}$ normalized such that
$u_{G}^{\mu}n_{\mu}=1$. We also define the lower index counterpart of
$u^{\mu}_{G}$ and its norm as

$u_{G\,\mu}=\bar{h}_{\mu\nu}u_{G}^{\nu}\,,\qquad
u_{G}^{2}=u_{G\,\mu}u_{G}^{\mu}$ (116)

which transform under Milne boosts as

$\displaystyle\begin{split}u_{G\,\mu}^{\prime}&=u_{G\,\mu}-P_{\mu}^{\nu}\psi_{\nu}+n_{\mu}h^{\nu\rho}(\psi_{\nu}\psi_{\rho}-u_{G\,\nu}\psi_{\rho})\,,\\\
(u^{\prime}_{G})^{2}&=u_{G}^{2}+h^{\mu\nu}\psi_{\mu}\psi_{\nu}-2h^{\mu\nu}u_{G\,\mu}\psi_{\nu}\,.\end{split}$
(117)

With these quantities at hand, it is straightforward to construct the Milne
invariant combinations

$\displaystyle\begin{split}\tilde{h}_{\mu\nu}&=\bar{h}_{\mu\nu}-(u_{G\,\mu}n_{\nu}+u_{G\,\nu}n_{\mu})+u_{G}^{2}n_{\mu}n_{\nu}\,,\\\
\tilde{A}_{\mu}&=A_{\mu}+u_{G\,\mu}-\frac{1}{2}n_{\mu}u_{G}^{2}\,,\\\
\tilde{P}^{\mu}{}_{\nu}&=h^{\mu\rho}\tilde{h}_{\rho\nu}=\delta^{\mu}{}_{\nu}-u_{G}^{\mu}n_{\nu}\,,\end{split}$
(118)

which satisfy $\tilde{h}_{\mu\nu}u_{G}^{\nu}=0$ and
$\tilde{P}^{\mu}{}_{\nu}u_{G}^{\nu}=\tilde{P}^{\mu}{}_{\nu}n_{\mu}=0$.

Using the velocity field $u_{G}^{\mu}$ we can define a Milne and $U(1)$ gauge
invariant connection compatible with the Newton-Cartan data

$\displaystyle\begin{split}\tilde{\Gamma}^{\lambda}_{\mu\nu}&=u_{G}^{\lambda}\partial_{\nu}n_{\mu}+\frac{1}{2}h^{\lambda\rho}\left(\partial_{\mu}\tilde{h}_{\nu\rho}+\partial_{\nu}\tilde{h}_{\mu\rho}-\partial_{\rho}\tilde{h}_{\mu\nu}\right)+\frac{1}{2}h^{\lambda\rho}\left(n_{\mu}\tilde{F}_{\nu\rho}+n_{\nu}\tilde{F}_{\mu\rho}\right)\,,\\\
\tilde{T}_{\mu\nu}^{\lambda}&=\tilde{\Gamma}^{\lambda}_{\mu\nu}-\tilde{\Gamma}^{\lambda}_{\nu\mu}=-u_{G}^{\lambda}F_{\mu\nu}^{(n)}\,,\end{split}$
(119)

where we have defined the field strength

$\tilde{F}_{\mu\nu}=\partial_{\mu}\tilde{A}_{\nu}-\partial_{\nu}\tilde{A}_{\mu}\,.$
(120)

The constitutive relations for Galilean fluids at leading order in a
derivative expansion for the Milne-invariant stress-energy tensor ${\cal
T}^{\mu\nu}$, energy current ${\cal E}^{\mu}$ and particle number current
$J^{\mu}_{c}$ are

$\displaystyle\begin{split}{\cal
T}^{\mu\nu}&=P\,h^{\mu\nu}+\rho\,u^{\mu}_{G}u_{G}^{\nu}+{\cal
O}(\partial)\,,\\\ {\cal E}^{\mu}&=\epsilon\,u^{\mu}_{G}+{\cal
O}(\partial)\,,\\\ J^{\mu}_{c}&=\rho\,u^{\mu}_{G}+{\cal
O}(\partial)\,,\end{split}$ (121)

where $P$ is the pressure function, $\rho$ is the particle number density, and
$\epsilon$ is the energy density. All these quantities are generic functions
of the (Milne-invariant) temperature, $T$, and chemical potential, $\mu$, and
satisfy the thermodynamic relations

${\epsilon}=Ts+\mu\rho-P\,,\qquad d\epsilon=Tds+\mu d\rho$ (122)

where $s$ is the entropy density.

The equations of motion for Galilean fluids in a curved Newton-Cartan
background are captured by the conservation of the stress-energy tensor and
currents

$\displaystyle\begin{split}(\tilde{\nabla}_{\nu}-\tilde{\cal G}_{\nu}){\cal
T}^{\mu\nu}=&-h^{\mu\rho}F_{\rho\nu}^{(n)}{\cal E}^{\nu}\,,\\\
(\tilde{\nabla}_{\mu}-2\tilde{\cal G}_{\mu}){\cal
E}^{\mu}=&-\frac{1}{2}\left(\tilde{h}_{\rho\mu}{\cal
T}^{\mu\nu}\tilde{\nabla}_{\nu}u^{\rho}_{G}+\tilde{h}_{\rho\nu}{\cal
T}^{\mu\nu}\tilde{\nabla}_{\mu}u^{\rho}_{G}\right)\,,\\\
(\tilde{\nabla}_{\mu}-\tilde{\cal G}_{\mu})J^{\mu}_{c}=&0\,.\end{split}$ (123)

Here,

$\tilde{\cal
G}_{\mu}=\tilde{T}^{\alpha}{}_{\mu\alpha}=-F^{(n)}_{\mu\alpha}u^{\alpha}_{G}\,,$
(124)

and $\tilde{\nabla}_{\mu}$ is the covariant derivative defined with the Milne
invariant connection (119). See Jensen:2014ama . The leading order equations
of motion for Galilean fluids can be obtained by inserting the expressions
(121) in (123). After some massaging, one can rewrite these equations in the
form $E_{\mu}=0$, $E=0$, and $E^{\prime}=0$, where

$\displaystyle\begin{split}{E}_{\mu}&=\tilde{P}^{\alpha}_{\mu}\partial_{\alpha}P-\rho\tilde{F}_{\mu\alpha}u^{\alpha}_{G}+(P+\epsilon)F^{(n)}_{\mu\alpha}u^{\alpha}_{G}\,,\\\
\tilde{E}&=-(\tilde{\nabla}_{\mu}-\tilde{\cal G}_{\mu})(s\,u_{G}^{\mu})\,,\\\
\tilde{E}^{\prime}&=-(\tilde{\nabla}_{\mu}-\tilde{\cal
G}_{\mu})(\rho\,u_{G}^{\mu})\,.\end{split}$ (125)

By taking the flat spacetime limit

$n_{\mu}=(1,0)\,,\qquad
h^{\mu\nu}=\delta^{ij}\delta_{i}^{\mu}\delta_{j}^{\nu}\,,\qquad
u^{\mu}_{G}=(1,v^{i})\,,\qquad A_{\mu}=0\,,$ (126)

where $v^{i}$ is the usual fluid velocity in Cartesian coordinates and
$i=1,\dots d$ label the spatial coordinates, equations (125) reduce to the
conventional Euler equation, continuity equation and entropy conservation.

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