# Nuclear three-body short-range correlations in coordinate space

Ronen Weiss  Theoretical Division, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA  Stefano Gandolfi  Theoretical Division, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA

###### Abstract

We study the effects of three-nucleon short-range correlations on nuclear
coordinate-space densities. For this purpose, novel three-body densities are
calculated for ground state nuclei using the auxiliary-field diffusion Monte
Carlo method. The results are analyzed in terms of the Generalized Contact
Formalism, extended to include three-body correlations, revealing the
universal behavior of nucleon triplets at short distances. We identify the
quantum numbers of such correlated triplets and extract scaling factors of
triplet abundances that can be compared to upcoming inclusive electron
scattering data.

††preprint: LA-UR-22-32748

Short-range correlations (SRCs) are an integral part of strongly interacting
many-body quantum systems, including nuclei and atomic systems. Strong SRCs
between nucleons pose one of the main challenges in the description of nuclei.
Accounting for the impact of SRC physics is crucial for the description of
two-body densities and momentum distributions [1, 2, 3, 4, 5, 6], electron and
neutrino scattering [7, 8, 9, 10, 11], spectroscopic factors [12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24], neutrinoless double beta decay matrix
elements [25, 26, 27], neutron star properties [28, 29, 30, 31] and more.

The properties of nuclear SRC pairs, i.e. two nucleons found close together
inside a nucleus, have been studied thoroughly in the last decades [32, 33,
34]. The universal features of SRC pairs and the dominance of neutron-proton
$(np)$ pairs have been established based on both experimental studies, using
mainly large momentum-transfer quasi-elastic electron- and proton-scattering
reactions [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54], and ab-initio many-body calculations [2, 3, 55, 4, 56, 57, 58,
59]. A connection to the internal structure of nucleons in nuclei was also
revealed [60, 33, 61, 62, 63, 64, 65, 41].

Unlike two-body SRCs, the features and importance of three-nucleon SRCs are
mostly unknown. There are currently significant efforts to study such
correlated triplets experimentally [66, 67, 68], but there has been no clear
identification of SRC triplets. Similarly, nuclear many-body ab-initio
calculations that allow direct access to triplet properties have not been
performed so far. Theoretical studies of three-nucleon SRCs are important for
guiding the experimental efforts and data analysis and for revealing the
properties and impact of such triplets.

Following the development of the contact theory for zero-range interactions
[69, 70, 71, 72, 73], the Generalized Contact Formalism (GCF) was introduced
to study nuclear SRCs [74, 5, 75]. Based on the asymptotic factorization of
nuclear wave functions, the GCF allows us to identify the impact of SRC pairs
on many different quantities [5, 6, 76, 9, 54, 77, 52, 78, 53, 44, 27, 75, 74,
79], resulting in an overall comprehensive and consistent picture of two-
nucleon SRCs.

In this work we extend the GCF to include the description of three-nucleon
SRCs. We first discuss the expected factorized form of the many-body wave
function when three nucleons are found close to each other and the possible
quantum numbers of such triplets. We then define three-body nuclear contacts,
describing the abundance of SRC triplets, and derive an asymptotic expression
for the three-body nuclear density. Finally, we present quantum Monte Carlo
calculations of such densities, verify the predictions of the GCF and extract
scaling factors of triplet abundances that are connected to inclusive
electron-scattering cross sections.

When two nucleons are close to each other in a nucleus with $A$ nucleons, the
many-body wave function $\Psi$ factorizes to a two-body part, describing the
correlated pair, and a function describing the rest of the nucleons in the
system [5]

$\Psi\xrightarrow[r_{ij}\rightarrow
0]{}\sum_{\alpha}\varphi^{\alpha}(\bm{r}_{ij})A^{\alpha}(\bm{R}_{ij},\\{\bm{r}_{k}\\}_{k\neq
i,j}).$ (1)

This is the GCF factorization. Here, $\bm{r}_{k}$ denotes the single-nucleon
coordinate, $\bm{r}_{ij}\equiv\bm{r}_{j}-\bm{r}_{i}$ and
$\bm{R}_{ij}\equiv(\bm{r}_{i}+\bm{r}_{j})/2$ are the relative and center-of-
mass (CM) coordinates, and $\alpha$ denotes the quantum numbers of the pair.
$\varphi^{\alpha}(\bm{r}_{ij})$ describes the dynamics of the correlated pair
and is defined as the solution of the zero-energy two-body Schrödinger
equation. As such it is universal, i.e. nucleus-independent, but depends on
the nucleon-nucleon interaction model. This factorization was verified using
ab-initio calculations [5, 6, 76, 80], and is supported by renormalization-
group arguments [81, 82, 83] and the Coupled-Cluster expansion [84].

The above two-body factorization is valid when none of the remaining $A-2$
nucleons is close to the correlated pair. If one nucleon is close enough to
such a pair, i.e. when three nucleons are close to each other, we expect the
many-body nuclear wave function to factorize in the following way

${\Psi}\xrightarrow[x_{ij},x_{ijl}\rightarrow
0]{}\sum_{\beta}{\varphi}_{ijl}^{\beta}(\bm{{x}}_{ij},\bm{{x}}_{ijl}){B}_{ijl}^{\beta}(\bm{R}_{ijl},\\{\bm{r}_{m}\\}_{m\neq
i,j,l}).$ (2)

We used here the Jacobi coordinates $\bm{x}_{ij}\equiv\bm{r}_{j}-\bm{r}_{i}$
and $\bm{x}_{ijl}\equiv\bm{r}_{l}-(\bm{r}_{i}+\bm{r}_{j})/2$ and the triplet
CM coordinate $\bm{R}_{ijl}\equiv(\bm{r}_{i}+\bm{r}_{j}+\bm{r}_{l})/3$.
${\varphi}_{ijl}^{\beta}$ describes the dynamics of the SRC triplet and is
defined as a zero-energy solution of the three-body Schrödinger equation with
quantum numbers given by $\beta$ (with the same nuclear-interaction model used
to define $\Psi$). We note that there could be multiple independent zero-
energy solutions with the same quantum numbers due to different possible
boundary conditions. We assume here that a single three-body wave function per
channel (i.e. per quantum numbers $\beta$) is sufficient, similar to the case
of two-nucleon SRCs [85, 86], and we provide below numerical evidence for this
claim based on ab-initio calculations. The connection of three-nucleon SRCs to
the zero-energy three-body eigenstates is discussed also in Ref. [84].

For realistic nuclear interactions, each channel $\beta$ is defined by the
quantum numbers
$\beta=(\pi_{\beta},j_{\beta},m_{\beta},t_{\beta},t_{z,\beta})$, where
$\pi_{\beta}$ is the parity, $j_{\beta}$ and $m_{\beta}$ are the total angular
momentum and its projection, and $t_{\beta}$ and $t_{z,\beta}$ are the total
isospin of the triplet and its projection. At short distances, we expect to
see a dominant contribution of channels that include a zero angular-momentum
component $\ell=0$. This is possible only for channels with positive parity
$\pi=+$. Since the total spin of three nucleons is either $s=1/2$ or $s=3/2$,
the dominant channels should have $j=1/2$ or $j=3/2$. Isospin value of $t=3/2$
is suppressed due to Pauli blocking, as three nucleons cannot be in the same
location with $t=3/2$. Therefore, $t=1/2$ is expected to be dominant at short
distances. Similarly, the symmetric $s=3/2$ component is also suppressed due
to Pauli blocking. Thus, $s=1/2$ and therefore $j=1/2$ is expected to be
dominant at short distances. To conclude, among the infinite number of three-
body channels in Eq. (2), we expect the dominant channels to be with the
quantum number $\pi=+,j=1/2,t=1/2$ (and $m=\pm 1/2$ and $t_{z}=\pm 1/2$).
These channels correspond to the quantum numbers of 3He and 3H ground states.
It means also that proton-proton-proton $(ppp)$ and neutron-neutron-neutron
$(nnn)$ triplets are expected to be suppressed at short distances compared to
proton-proton-neutron $(ppn)$ and proton-neutron-neutron $(pnn)$ triplets.
Notice that while the $np$ dominance of two-body SRCs is caused by the tensor
force [87, 88], here it is the Pauli principle that leads to the $t=1/2$
dominance for three-body SRCs. See more details regarding the structure of
${\varphi}_{ijl}^{\beta}$ and the dominant channels in the supplementary
materials.

Based on Eq. (2), we can now define the three-nucleon contact matrix

$C_{3}^{\beta\gamma}(JM)=\frac{A(A-1)(A-2)}{6}\langle
B_{123}^{\beta}|B_{123}^{\gamma}\rangle.$ (3)

$J$ and $M$ above are the total angular momentum and projection of $\Psi$. The
combinatorical factor is suitable assuming $\Psi$ is fully anti-symmetric.
Three-body contacts were similarly defined for the zero-range limit [89, 90]
and for Helium atoms [91]. Based on orthogonality properties of the
$B_{ijl}^{\beta}$ functions, we can conclude that if $m_{\beta}\neq
m_{\gamma}$, then $C_{3}^{\beta\gamma}(JM)=0$. Similarly, if $t_{z,\beta}\neq
t_{z,\gamma}$ or $\pi_{\beta}\neq\pi_{\gamma}$, then
$C_{3}^{\beta\gamma}(JM)=0$ as well, assuming $\Psi$ has well defined isospin
$T$ and parity. This is generally not the case for $j$ or $t$. Nevertheless,
for $J=0$ nuclei, we do get $C_{3}^{\beta\gamma}(00)=0$ if $j_{\beta}\neq
j_{\gamma}$, and for $T=0$, we get $C_{3}^{\beta\gamma}(JM)=0$ if
$t_{\beta}\neq t_{\gamma}$. We can also show that the sum over $m_{\beta}$ of
diagonal contacts $\sum_{m_{\beta}}C_{3}^{\beta\beta}(JM)$ is independent of
$M$. Derivations appear in the supplementary materials.

It is useful to define also the $M$-averaged three-body nuclear contacts

$C_{3}^{\beta\gamma}=\frac{1}{2J+1}\sum_{M}C^{\beta\gamma}(JM).$ (4)

The averaged contacts are diagonal in both $j$ and $m$, i.e.
$C_{3}^{\beta\gamma}=0$ if $m_{\beta}\neq m_{\gamma}$ or $j_{\beta}\neq
j_{\gamma}$, and they are also independent of $m_{\beta}$ and $m_{\gamma}$.
See derivations in the supplementary materials. The averaged contacts inherit
the properties of $C_{3}^{\beta\gamma}(JM)$ regarding parity and the isospin
projection $t_{z}$. Notice that the averaged contacts are still generally not
diagonal in $t_{\beta}$.

To study the implications of SRC triplets, we consider in this work three-body
densities in coordinate space. Specifically, we consider the three-body
density describing the probability of finding three nucleons inside a nucleus
in a triangle with sides of length $r_{12}$, $r_{13}$ and $r_{23}$

$\displaystyle\rho_{3}(r_{12},r_{13},r_{23})=\frac{A(A-1)(A-2)}{6}$
$\displaystyle\times\langle\Psi|\delta(|\bm{r}_{1}-\bm{r}_{2}|-r_{12})\delta(|\bm{r}_{1}-\bm{r}_{3}|-r_{13})\delta(|\bm{r}_{2}-\bm{r}_{3}|-r_{23})|\Psi\rangle.$
(5)

In the limit of $r_{12},r_{13},r_{23}\rightarrow 0$, we can use Eqs. (2)-(4)
to obtain

$\displaystyle\rho_{3}(r_{12},r_{13},r_{23})\xrightarrow[r_{12},r_{13},r_{23}\rightarrow
0]{}\sum_{\pi_{\beta},j_{\beta},t_{\beta}}\left(\sum_{m_{\beta},t_{z,\beta}}C_{3}^{\beta}\right)\langle\varphi_{123}^{\beta}|$
$\displaystyle\times\delta(|\bm{r}_{1}-\bm{r}_{2}|-r_{12})\delta(|\bm{r}_{1}-\bm{r}_{3}|-r_{13})\delta(|\bm{r}_{2}-\bm{r}_{3}|-r_{23})|\varphi_{123}^{\beta}\rangle.$
(6)

$\rho_{3}(r_{12},r_{13},r_{23})$ is independent of $M$ and therefore we could
use here the M-averaged contacts, which are diagonal in
$\pi_{\beta},j_{\beta},m_{\beta},t_{z,\beta}$. In addition, the three-body
part in Eq. (Nuclear three-body short-range correlations in coordinate space)
is diagonal in $t_{\beta}$ and, therefore, we are left only with the diagonal
contacts $C_{3}^{\beta}\equiv C_{3}^{\beta\beta}$. Finally, the three-body
part is also independent of $m_{\beta}$ and $t_{z,\beta}$. Eq. (Nuclear three-
body short-range correlations in coordinate space) reveals the interpretation
of the diagonal three-body contacts $C_{3}^{\beta}$ \- they measure the
probability of finding three-nucleons close together with quantum numbers
$\beta$.

$\pi_{\beta},j_{\beta}$ and $t_{\beta}$ are the quantum numbers governing the
$r$-dependence of $\rho_{3}$ at short distances. As discussed before, a single
set of values with $\pi_{\beta}=+,j_{\beta}=1/2,t_{\beta}=1/2$ should be
dominant. Therefore, we expect to find a universal behavior of $\rho_{3}$ at
short-distances for all nuclei, i.e. the same $r$-dependence with only a
global scaling factor that depends on the nucleus. We can also consider
isospin-projected densities $\rho_{3}^{t}(r_{12},r_{13},r_{23})$, by inserting
the appropriate three-body projection operator in Eq. (Nuclear three-body
short-range correlations in coordinate space). For these quantities, we expect
to see a dominance of $t=1/2$ over $t=3/2$ as discussed above.

In order to verify these GCF predictions regarding three-nucleons SRCs, we now
turn to ab-initio calculations of $\rho_{3}(r_{12},r_{13},r_{23})$ for the
ground-state nuclei 3He, 4He, 6Li and 16O. We used the auxiliary-field
diffusion Monte Carlo (AFDMC) method [92, 93] combined with the N2LO local
chiral interaction with the E1 parametrization of the three-body force [94,
95, 96, 93]. We focus here on the $R_{0}=1.0$ fm cutoff but show also some
results for $R_{0}=1.2$ fm.

We first start with investigating $\rho_{3}^{t}$ in order to compare the
$t=1/2$ and $t=3/2$ densities. The three-body density for 6Li is shown in Fig.
1 for the equilateral triangle, i.e. $\rho_{3}(r,r,r)$ as a function of $r$.
We can clearly see that at short distances the $t=1/2$ component is dominant.
As $r$ increases the contribution of $t=3/2$ triplets grows. We note that the
total number of $t=1/2$ and $t=3/2$ triplets in 6Li is $16$ and $4$
respectively. The contribution of $t=1/2$ triplets in 6Li for $r<1$ fm is
larger than $95\%$ of all triplets, i.e. significantly larger than the total
combinatorial part of such triplets which is $80\%$. $t=1/2$ dominance is seen
also for other geometries, e.g. isosceles triangles, and also for 16O (3He and
4He include only $t=1/2$ triplets). These results agree with our expectation
due to the Pauli exclusion rule.

Figure 1:  6Li three-body density for equilateral triangles as a function of
the triangle side using the AFDMC method and the N2LO(1.0) interaction.
Projections to $t=1/2$ and $t=3/2$ triplets are shown together with the total
density.

We can now focus on the $t=1/2$ component and compare the behavior of
different nuclei. We present in Fig. 2 the density $\rho_{3}^{t=1/2}$ for all
available nuclei for both equilateral and isosceles triangle geometries using
the N2LO(1.0) interaction. For the isosceles triangle, we fix the base to be
of length $a=0.85$ fm. The 4He, 6Li and 16O calculations are rescaled so that
their shape at short distances can be compared to 3He. Only for plotting
purposes, the densities of the isosceles triangle are multiplied by minus one
to separate them from the equilateral-triangle results. We can see that, for
each of the geometries, the $r$-dependence of $\rho_{3}^{t=1/2}$ is the same
at short distances ($r\lesssim 1.1$ fm) for all nuclei, as all densities
coincide with the 3He density. This shows the universal behavior of SRC
triplets as predicted by the GCF. Indeed, a single $t=1/2$ channel is dominant
here (otherwise the densities would not coincide) due to the dominance of
$\ell=0$ at short distances. Also, as assumed in Eq. (2), a single three-body
wave function for the dominant quantum numbers is sufficient as all nuclei
behave like the bound 3He. It should be emphasized that the same scaling
factor is applied to both the equilateral and isosceles cases for each
nucleus, in agreement with Eq. (Nuclear three-body short-range correlations in
coordinate space). The same behavior is seen for other configurations
involving three particles close together, e.g. other isosceles triangles as
long as the base and legs lengths are small. This result is an important
validation of the asymptotic three-body factorization of the many-body wave
function, Eq. (2). We also include in Fig. 2 the 3He equilateral-triangle
density using the N2LO(1.2) interaction. We can see that the short-distance
behavior in this case is different. This shows that the three-body wave
functions of the GCF ${\varphi}_{ijl}^{\beta}$ indeed depend on the model of
the interaction.

Figure 2:  AFDMC $\rho_{3}^{t=1/2}$ densities for 3He, 4He, 6Li and 16O for
both equilateral triangle and isosceles triangle using the N2LO(1.0)
interaction (circles). For the latter, the base is fixed at length of $0.85$
fm and the densities are multiplied by minus one (see text for details). The
4He, 6Li and 16O densities are each multiplied by a scaling factor (the same
factor for both geometries). 3He equilateral-triangle density is shown also
for the N2LO(1.2) interaction (dashed line).

Despite the dominance of the $t=1/2$ component, we can also look into the
properties of the $t=3/2$ density. We note that among the nuclei considered in
this work, only 6Li and 16O contain a $t=3/2$ three-body component. The
equilateral-triangle and isosceles-triangle $\rho_{3}^{t=3/2}$ densities are
shown in Fig. 3 with rescaled 16O densities, similar to Fig. 2. We can see
that also for this case a universal short-distance behavior exists. This
indicates that the asymptotic factorization holds for $t=3/2$ triplets and
that there is a single dominant $t=3/2$ channel. The latter is expected to
have the quantum numbers $\pi=+$ and $j=1/2$. Calculations for additional
nuclei with $t=3/2$ component are needed to validate these conclusions.

Figure 3:  The same as Fig. 2, but for the $t=3/2$ component. 16O density is
rescaled.

The scaling factors used in Figs. 2 and 3 are equal to a sum of three-nucleon
contact values (relative to 3He). For $t=1/2$, it is the sum of contacts

$C_{3}^{t=1/2}\equiv\sum_{m,t_{z}}C_{3}^{\beta_{1/2}},$ (7)

where $\beta_{1/2}$ denotes the quantum numbers of the leading $t=1/2$ three-
body channel. Notice that both $ppn$ and $pnn$ triplets (i.e. $t_{z}=\pm 1/2$)
are included here. As mentioned above, such contact values are proportional to
the probability of finding correlated triplets in a given nucleus. Similar
probabilities can be accessible using large momentum transfer quasi-elastic
inclusive electron-scattering experiments. For the case of two-body SRCs, pair
abundances were extracted from inclusive measurements at appropriate
kinematics, defining $a_{2}$ as the per-nucleon cross section ratio with
respect to the deuteron [35, 36, 37, 38, 39, 40, 41, 42, 43]. There are
ongoing experimental efforts to extract such three-nucleon abundances [66]. In
this case, a cross section ratio can be defined as [68]

$a_{3}(A,Z)=\frac{3}{A}\frac{\sigma_{eA}}{(\sigma_{e^{3}\textrm{He}}+\sigma_{e^{3}\textrm{H}})/2},$
(8)

where $\sigma_{eA}$ is the inclusive electron-scattering cross section off
nucleus A (with $A$ nucleons, $Z$ protons) at kinematics dominated by three-
body SRCs. Interpreting $a_{3}$ as the ratio of three-nucleon SRC abundances,
it is connected to the nuclear contacts for a symmetric nucleus $A$ by

$a_{3}(A,Z)=\frac{3}{A}\frac{C_{3}^{t=1/2}(A)}{C_{3}^{t=1/2}(^{3}\textrm{He})}.$
(9)

We consider here only the leading contribution of $t=1/2$ triplets. It should
be noted that, similar to the case of two-body SRCs [78], different effects
can influence this interpretation of $a_{3}$, such as the CM motion of
triplets, excitation energy of the $A-3$ system and contribution of $t=3/2$
triplets. In addition, the dependence of $a_{3}$ on kinematic variables (such
as x-bjorken and $Q^{2}$) should cancel in the ratio if we consider a
symmetric nucleus in the numerator and the sum of 3He and 3H in the
denominator (ignoring the above effects), due to similar contribution of $ppn$
and $nnp$ triplets in the numerator and denominator. For the case of non-
symmetric nuclei or if using only the 3He cross section in the denominator, a
more careful analysis of the reaction is needed.

Contact ratios extracted from the AFDMC calculations are presented in Table 1.
Their values were fitted to the equilateral three-body density based on Eq.
(Nuclear three-body short-range correlations in coordinate space), where only
the leading channel (for each value of $t$) is considered. Since we are
looking on contact ratios, there is no need to calculate the functions
$\varphi_{ijl}^{\beta}$. Uncertainties were estimated by varying the lower and
upper limits of the fitting range between $0.1-0.4$ fm and $1-1.2$ fm,
respectively. The $t=1/2$ contact ratio provides a prediction for the value of
$a_{3}$ for 4He, 6Li and 16O. We can see that the per-nucleon $t=1/2$ ratios
for 4He and 16O are similar, consistent with a 4He-cluster structure of 16O.
It is also interesting to note that the per-nucleon $t=1/2$ ratio for 6Li is
smaller than that of 4He. Results for additional nuclei are needed in order
study the $A$-dependence of $a_{3}$.

In a recent work [68, 97, 98], Sargsian et. al. suggested a connection between
two-body and three-body abundances
$a_{3}(A)=1.12a_{2}(A)^{2}/a_{2}(^{3}\mathrm{He})$, leading to a value of
$a_{3}(^{4}\mathrm{He})\approx 3.15$ based on the experimental values of
$a_{2}$ from Ref. [63]. This is smaller than the value we obtained here (table
1). We emphasize that in the GCF approach three-nucleon abundances are
generally independent of two-nucleon abundances. Inclusive experimental
measurements might be able to clear this issue.

contact ratio | $A=4$ | $A=6$ | $A=16$
---|---|---|---
$t=1/2$ | $3.8\pm 0.3$ | $3.1\pm 0.3$ | $4.2\pm 0.5$
$t=3/2$ | - | - | $5.4\pm 0.8$
Table 1:  Per nucleon three-body contact ratio for $t=1/2$ (with respect to
3He) and for $t=3/2$ (with respect to 6Li), i.e.
$\frac{3}{A}\frac{C_{3}^{t=1/2}(A)}{C_{3}^{t=1/2}(^{3}\textrm{He})}$, and
$\frac{6}{A}\frac{C_{3}^{t=3/2}(A)}{C_{3}^{t=3/2}(^{6}\textrm{Li})}$,
respectively.

To summarize, we have studied here the properties of three-nucleon SRCs in
coordinate space. The GCF was extended to include correlated triplets,
described as an isolated and universal subsystem within the many-body nucleus.
The leading $t=1/2$ channel was identified and three-nucleon contacts were
defined. Using novel AFDMC ab-initio calculations of three-body densities, the
$t=1/2$ dominance and universality of such triplets at short-distances was
established numerically. Specifically, we found that three nucleons at short
distances behave like the bound 3He wave function. We have also extracted the
values of the leading $t=1/2$ and $t=3/2$ contact ratios, describing the
scaling of SRC triplet abundances. The connection to inclusive electron-
scattering cross sections was discussed.

This work opens the path for additional studies of SRC triplets, including
their impact on two-body densities, momentum distributions, spectral
functions, electron and neutrino scattering off nuclei and neutrinoless double
beta decay matrix elements. The sensitivity of SRC triplet properties to the
nuclear interaction model and specifically to the three-nucleon force should
also be further studied. This is also an important step towards a systematic
short-range expansion of the nuclear wave function.

We would like to thank J. Carlson, J. Martin, S. Novario and I. Tews, for
helpful discussions. The work of R.W. was supported by the Laboratory Directed
Research and Development program of Los Alamos National Laboratory under
project number 20210763PRD1. The work of S. G. was supported by U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, under
Contract No. DE-AC52-06NA25396, by the DOE NUCLEI SciDAC Program, and by the
DOE Early Career Research Program. Computer time was provided by the Los
Alamos National Laboratory Institutional Computing Program, which is supported
by the U.S. Department of Energy National Nuclear Security Administration
under Contract No. 89233218CNA000001.

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††institutetext: Department of Applied Mathematics and Theoretical Physics
University Cambridge, CB3 0WA, UK

# A Gauge Theory for Shallow Water

David Tong<EMAIL_ADDRESS>
###### Abstract

The shallow water equations describe the horizontal flow of a thin layer of
fluid with varying height. We show that the equations can be rewritten as a
$d=2+1$ dimensional gauge theory with a Chern-Simons term. The theory contains
two Abelian gauge fields, corresponding to the conserved height and conserved
vorticity of the fluid. In a certain linearised approximation, the shallow
water equations reduce to relativistic Maxwell-Chern-Simons theory. This
describes Poincaré waves. The chiral edge modes of the theory are identified
as coastal Kelvin waves.

##  1 Introduction

The shallow water equations describe the dynamics of a thin layer of fluid
whose height is much smaller than its horizontal extent. The basic equations,
and a number of extensions, are ubiquitous in modelling of the atmosphere,
oceans, rivers, and lakes.

####

The variables of the shallow water equations are the height $h(x,y,t)$ of the
fluid, and the horizontal velocity ${\bf u}(x,y,t)$. Writing the components of
the 2d velocity vector as $u^{i}$, with $i=1,2$, the equations are

$\displaystyle\frac{Dh}{Dt}=-h\nabla\cdot{\bf u}\ \ \ {\rm and}\ \ \
\frac{Du^{i}}{Dt}=f\epsilon^{ij}u^{j}-g\frac{\partial{h}}{\partial{x^{i}}}$
(1.1)

Here the material time derivative, $Dh/Dt=\partial h/\partial t+{\bf
u}\cdot\nabla h$, tracks the variation of height with the flow. Similarly,
$Du^{i}/Dt=\partial u^{i}/\partial t+({\bf u}\cdot\nabla)u^{i}$.

####

The shallow water equations contain two parameters: the gravitational
acceleration $g$, and the Coriolis parameter $f$. On Earth, at latitude
$\theta$, the Coriolis parameter is given by $f=2\sin\theta\,({\rm
day})^{-1}$. A derivation of the shallow water equations, together with many
detailed applications can be found in the textbook vallis . A whirlwind tour
of some highlights can be found in Section 4.3 of the lecture notes mylectures
.

####

In recent years, a number of surprising parallels have been found between the
shallow water equations, at least in their linearised form, and topological
phases of matter. With a healthy dose of hindsight, the story goes back to the
work of William Thomson in 1879 who showed that the linearised shallow water
equations admit chiral edge modes, now known as coastal Kelvin waves kelvin .
These waves are exponentially localised at the coast and propagate clockwise
around land masses in the Northern hemisphere and anti-clockwise in the
Southern hemisphere. Similar chiral modes in the shallow water equations are
also found localised at the equator, now propagating from west to east matsuno
; long . At a calculational level, these chiral waves are strikingly
reminiscent of the chiral fermion zero modes of the Dirac equation that were
discovered a century later jr .

####

In the quantum world, chiral edge modes are the smoking gun for a topological
phase of matter laughlin ; halperin ; jeff ; zhang . This is also true for the
classical shallow water equations, as first shown in the beautiful paper
equator . We will now give a cartoon version of the argument.

####

The linearised shallow water equations admit a band of solutions known as
Poincaré waves, whose frequency $\omega$ is related to their wavevector ${\bf
k}$ through the dispersion relation

$\displaystyle\omega^{2}=c^{2}{\bf k}^{2}+f^{2}$ (1.2)

Here the speed is given by $c=\sqrt{gH}$, with $H$ the average height of the
fluid. (The derivation of this dispersion relation will be reviewed in Section
3.) For non-vanishing Coriolis force, $f\neq 0$, the frequency has a gap. But
the gap closes at the equator where $f=0$. Given such a gapped band of
solutions, labelled by 2d momentum ${\bf k}$, one can follow lattice models
and compute the Chern number by integrating a suitable Berry curvature over
momentum space, á la TKNN tknn and Haldane haldane . This Chern number is non-
vanishing and depends on the sign of $f$. This means that the Chern class
jumps as we cross the equator. This change in the underlying topology can be
viewed as responsible for the existence of chiral equatorial waves.

####

The cartoon sketched above is sloppy for a number of reasons, not least the
fact that momentum space in this problem is non-compact. This is in sharp
contrast to the original uses of a Chern number which was for lattice models
where momentum space is a compact Brillouin zone tknn ; haldane . This
compactness is necessary for the Chern number to be an integer, and hence
topological. To circumvent this issue, the original paper equator used a
slightly more subtle topological invariant, and subsequent papers have
explored the possibility of regulating the momentum integral in some way
hallvisc1 ; hallvisc2 , although not without inducing further diversions anom
. Meanwhile, a topological explanation for the original coastal Kelvin waves
has also been presented, involving spatially varying $f({\bf x})$ and mean
height $H({\bf x})$ coast . These complications notwithstanding, the main
lesson is clear: chiral modes observed in shallow water owe their existence to
topology. Further studies of topological properties of the shallow water
system can be found in lamb ; tauber1 ; jay ; tauber2 .

####

The purpose of this work is to provide an effective field theory description
of shallow water dynamics in the hope that it may make topological properties
more manifest. We do this by writing the shallow water equations as a gauge
theory. This gauge theory involves Chern-Simons terms, which are the
archetypal example of a topological field theory. In particular, Chern-Simons
theories describe quantum Hall states zhk ; wenniu .

####

In Section 2, we show that the non-linear shallow water equations have a
natural rendering as a theory of two interacting Abelian gauge fields. The two
magnetic fields are associated to the height and vorticity in the shallow
water equations and the conservation of these quantities in the gauge theory
language follows from the Bianchi identity. The two gauge fields are coupled
through a Chern-Simons term. We will see that the vorticity gauge field must
be formulated in a so-called Clebsch parameterisation and therein hides many
of the subtleties of the system.

####

In Section 3, we turn to the linearised equations. We will show that the
effective field theory describing Poincaré waves (1.2) is given by
relativistic Maxwell-Chern-Simons theory. The coastal Kelvin waves arise as
chiral edge modes of this Chern-Simons theory.

##  2 A Gauge Theory for Shallow Water

In this section we formulate the shallow water equations as a gauge theory in
$d=2+1$ dimensions. Our strategy will be to identify conserved currents in the
shallow water equations and replace them with gauge fields. Rather than simply
write down the Lagrangian, we will instead proceed in smaller steps, and
occasional missteps, to build some intuition for the gauge theoretic
formulation. The more impatient reader can find the final answer in (2.9) and
(2.11).

####

To understand the physics, we focus on conserved quantities. The shallow water
equations (1.1) enjoy a number of conservation laws, but for our immediate
purposes we need just two. Both are conserved quantities in the sense that
they admit a current obeying the continuity equation,

$\displaystyle\partial_{\mu}J^{\mu}=\frac{\partial{J^{0}}}{\partial{t}}+\nabla\cdot{\bm{J}}=0$

Throughout, we use $\mu=0,1,2$ to denote space and time indices, and $i=1,2$
to denote only spatial indices. Note that, unlike in relativistic systems,
$x^{0}=t$ with no additional factor of speed. This means that $x^{0}$ and
$x^{i}$ have different dimensions.

####

The first conserved quantity is the height $h$ of the fluid. Indeed, the first
equation in (1.1) is a continuity equation with

$\displaystyle J^{0}=h\ \ \ {\rm and}\ \ \ {\bm{J}}=h{\bf u}$ (2.3)

This is a manifestation of the conservation of mass of the fluid. The second
conserved quantity is associated to the vorticity, defined in two dimensions
as

$\displaystyle\zeta=\frac{\partial{u^{2}}}{\partial{x}}-\frac{\partial{u^{1}}}{\partial{y}}$

The presence of the Coriolis force means that there is a slight modification
to the conserved current, which is given by

$\displaystyle\tilde{J}^{0}=\zeta+f\ \ \ {\rm and}\ \ \
\tilde{\bm{J}}=(\zeta+f){\bf u}$ (2.4)

Importantly, the two conserved currents (2.3) and (2.4) are not fully
independent: they obey the relation
${\bm{J}}/J^{0}=\tilde{\bm{J}}/\tilde{J}^{0}={\bf u}$. Part of the challenge
will be to reproduce this relation when the currents are replaced by gauge
fields.

####

We now explain what it means to replace a conserved current with a gauge
field. We denote our gauge field as $A_{\mu}$, with $\mu=0,1,2$. In $d=2+1$
dimensions, the electric field is a vector $E_{i}$, while the magnetic field
$B$ is a (pseudo)-scalar,

$\displaystyle
E_{i}=\frac{\partial{A_{i}}}{\partial{t}}-\frac{\partial{A_{0}}}{\partial{x^{i}}}\
\ \ \ {\rm and}\ \ \ \
B=\frac{\partial{A_{2}}}{\partial{x^{1}}}-\frac{\partial{A_{1}}}{\partial{x^{2}}}$

Both are invariant under gauge transformations $A_{\mu}\rightarrow
A_{\mu}+\partial_{\mu}\theta$.

####

Abelian gauge fields in $d=2+1$ dimensions are special because they come with
an associated conserved, global charge. This follows simply from the Bianchi
identity which reads

$\displaystyle\epsilon^{\mu\nu\rho}\partial_{\mu}\partial_{\nu}A_{\rho}=0\ \ \
\Rightarrow\ \ \
\frac{\partial{B}}{\partial{t}}-\frac{\partial{E_{2}}}{\partial{x^{2}}}+\frac{\partial{E_{1}}}{\partial{x^{2}}}=0$
(2.5)

This has the form of the continuity equation, with the magnetic field as the
charge and the components of the electric field as the current,

$\displaystyle J^{0}=B\ \ \ {\rm and}\ \ \ {\bm{J}}=(-E_{2},E_{1})\ \ \ {\rm
or}\ \ \ J^{i}=-\epsilon^{ij}E_{j}$

To formulate a gauge theory description of shallow water, we will think of
conserved currents in terms of gauge fields. Ultimately we will introduce
gauge fields associated to both the height (2.3) and the vorticity (2.4). But
we will build up slowly, and we start with just a single gauge field $A_{\mu}$
that we identify as the height current, meaning that we take

$\displaystyle B=h\ \ \ {\rm and}\ \ \ E_{i}=\epsilon_{ij}hu^{j}$ (2.6)

This single gauge field includes all degrees of freedom of the shallow water
system, with the velocity given by $u^{i}=-\epsilon^{ij}E_{j}/B$. The first of
the shallow water equations (1.1) is automatically obeyed by virtue of the
Bianchi identity (2.5). It remains to write the second shallow water equation
in the language of the gauge theory. For this, we will adopt an action
principle. We will make a first attempt at writing down an action for the
shallow water system and see that it fails. But the way in which it fails will
be instructive.

####

To motivate the action, we first note that the shallow water system has a
conserved energy, given by

$\displaystyle{\cal E}=\frac{1}{2}h{\bf
u}^{2}+\frac{1}{2}gh^{2}=\frac{{\bm{E}}^{2}}{2B}+\frac{1}{2}gB^{2}$

This takes the form of kinetic energy plus potential energy, so a natural
guess for the action is

$\displaystyle S_{\rm fail}=\int dt\,d^{2}x\
\left(\frac{{\bm{E}}^{2}}{2B}-\frac{1}{2}gB^{2}\right)$ (2.7)

The fact that there is an inverse power of $B$ in the kinetic term should
cause no more concern that an inverse power of the metric in the Einstein-
Hilbert action: the dynamics should be thought of as an expansion around a
non-vanishing fluid height, meaning that $B\neq 0$.

####

To derive the equations of motion, we vary the action with respect to
$A_{\mu}$. The equation of motion for $A_{0}$ is Gauss’ law and reads

$\displaystyle\frac{\partial}{\partial x^{i}}\left(\frac{E^{i}}{B}\right)=0\ \
\ \Rightarrow\ \ \ \zeta=0$ (2.8)

This is the first way in which the action (2.7) fails: Gauss’ law only allows
irrotational flows with vanishing vorticity. This restriction notwithstanding,
we fare slightly better with the equations of motion for $A_{i}$. These are

$\displaystyle\frac{\partial{}}{\partial{t}}\left(\frac{E_{i}}{B}\right)+\frac{1}{2}\epsilon_{ij}\frac{\partial{}}{\partial{x^{j}}}\left(\frac{{\bm{E}}^{2}}{B^{2}}\right)+g\epsilon_{ij}\frac{\partial{B}}{\partial{x^{j}}}=0$

When translated into the the variables $h$ and ${\bf u}$, the equation becomes

$\displaystyle\dot{u}^{i}+\frac{1}{2}\frac{\partial{{\bf
u}^{2}}}{\partial{x^{i}}}=-g\frac{\partial{h}}{\partial{x^{i}}}\ \ \
\Rightarrow\ \ \
\frac{Du^{i}}{Dt}+\epsilon^{ij}u^{j}\zeta=-g\frac{\partial{h}}{\partial{x^{i}}}$

With the Gauss’ law constraint $\zeta=0$, this gives us something close to the
shallow water equation (1.1). We see that, in addition to the fact the
vorticity is forced to vanish, the action does not accommodate the Coriolis
force. Of course, this should be no surprise as we made no attempt to include
it.

####

With this failure behind us, we can now see how to resolve both issues. Again,
our first attempt will not be a complete success. But our second will be.
Clearly we need to include some additional degrees of freedom, charged under
$A_{\mu}$, so that Gauss’ law (2.8) receives an extra contribution, breathing
life into the otherwise frozen vorticity. As we’ve seen, the vorticity itself
is also a conserved quantity which motivates us to introduce a second,
dynamical gauge field $\tilde{A}_{\mu}$. We couple this to the first through a
Chern-Simons term. At the same time, we also introduce the Coriolis parameter
$f$ through a background charge. The resulting action is

$\displaystyle S=\int dt\,d^{2}x\
\left(\frac{{\bm{E}}^{2}}{2B}-\frac{1}{2}gB^{2}+fA_{0}-\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}\tilde{A}_{\rho}\right)$
(2.9)

The Gauss law for this revised action is

$\displaystyle\frac{\partial{}}{\partial{x^{i}}}\left(\frac{E_{i}}{B}\right)+f-\tilde{B}=0\
\ \ \Rightarrow\ \ \ \tilde{B}=\zeta+f$ (2.10)

This looks more promising: the new magnetic field
$\tilde{B}=\partial_{1}\tilde{A}_{2}-\partial_{2}\tilde{A}_{1}$ is identified
with the conserved charge (2.4) associated to vorticity. But we’re not yet out
of the water. If we treat $\tilde{A}_{\mu}$ as a dynamical gauge field then we
must also implement the equation of motion for $\tilde{A}_{0}$ and this would
seem to fix $B=0$ or, equivalently, $h=0$. That’s not helpful.

####

There is a rather charming resolution to this, again specific to gauge fields
in $d=2+1$ dimensions. Any 3d vector field can be written in a so-called
Clebsch parameterisation. For our gauge field $\tilde{A}_{\mu}$, this means
that we introduce three functions $\chi$(x,t), $\beta({\bf x},t)$ and
$\alpha({\bf x},t)$ and write,

$\displaystyle\tilde{A}_{\mu}=\partial_{\mu}\chi+\beta\partial_{\mu}\alpha$
(2.11)

This parameterisation is non-linear and non-unique. It does not often arise in
quantum field theory but there is a long history of using this
parameterisation in fluid mechanics, typically for the velocity field in three
spatial dimensions. Here we will see that it will serve our purposes nicely.
We vary the action (2.9) not with respect to $\tilde{A}_{\mu}$, but instead
with respect to the component fields $\chi$, $\beta$ and $\alpha$. The field
$\chi$ is pure gauge and does not contribute to the action. Meanwhile, the
equations of motion for $\beta$ and $\alpha$ are

$\displaystyle B\dot{\alpha}+\epsilon^{ij}E_{i}\partial_{j}\alpha=0\ \ \ {\rm
and}\ \ \ B\dot{\beta}+\epsilon^{ij}E_{i}\partial_{j}\beta=0$ (2.12)

From this we can determine the electric field $\tilde{E}_{i}$. It is defined
by

$\displaystyle\tilde{E}_{i}=\partial_{t}\tilde{A}_{i}-\partial_{i}\tilde{A}_{0}=\dot{\beta}\partial_{i}\alpha-(\partial_{i}\beta)\dot{\alpha}$

From the equations of motion (2.12), we have

$\displaystyle\tilde{E}_{i}$ $\displaystyle=$
$\displaystyle-\frac{E_{k}}{B}\epsilon^{kj}\left(\partial_{j}\beta\partial_{i}\alpha-\partial_{i}\beta\partial_{j}\alpha\right)=\frac{E_{i}}{B}\tilde{B}=\epsilon_{ij}u^{j}(\zeta+f)$

This is what we wanted. The electric field is not independent of our other
conserved quantities, but is fixed by the magnetic field $\tilde{B}$ and the
original gauge field $A_{\mu}$. This is the property of the vorticity current
(2.4) that we previously highlighted.

####

The fact that working with the Clebsch parameterisation in an action gives
equations that are less restrictive than working directly with a vector field
was previously noted in clebsch . The observation that it allows us to
introduce currents that are not fully independent is related to remarks made
in jackiw about the role the Clebsch parameterisation plays in the symplectic
structure of the Euler equation.

####

It remains only to compute the equation of motion for the original spatial
gauge field, $A_{i}$. This is

$\displaystyle\frac{\partial{}}{\partial{t}}\left(\frac{E_{i}}{B}\right)+\frac{1}{2}\epsilon_{ij}\frac{\partial{}}{\partial{x^{j}}}\left(\frac{{\bm{E}}^{2}}{B^{2}}\right)+g\epsilon_{ij}\frac{\partial{B}}{\partial{x^{j}}}-\epsilon_{ij}\tilde{E}_{j}=0$

This time, when translated into the the variables $h$ and ${\bf u}$, this
equation coincides with the second shallow water equation (1.1). This confirms
that the gauge theory action (2.9), with the Clebsch parameterisation (2.11),
does indeed describe the shallow water system. An alternative derivation of
this action is given in Appendix A.

##  3 The Linearised Shallow Water Equations

Much interesting physics, including aspects of topology, can be found in the
linearised version of the shallow water equations. We expand about a fluid of
constant, average height $H$, and write

$\displaystyle h({\bf x},t)=H+\eta({\bf x},t)$

Then, keeping only terms linear in $\eta$ and ${\bf u}$, the shallow water
equations (1.1) become

$\displaystyle\frac{\partial{\eta}}{\partial{t}}+H\nabla\cdot{\bf u}=0\ \ \
{\rm and}\ \ \
\frac{\partial{u^{i}}}{\partial{t}}=f\epsilon^{ij}u^{j}-g\frac{\partial{\eta}}{\partial{x^{i}}}$
(3.13)

In this section we first review some properties of the solutions to (3.13) and
then turn to their description in the language of gauge theory.

###  3.1 Poincaré Waves

It is straightforward to find solutions to the linearised equations (3.13) by
looking at Fourier modes. We write

$\displaystyle\eta={\eta}_{0}e^{i(\omega t-{\bf k}\cdot{\bf x})}\ \ \ {\rm
and}\ \ \ {\bf u}={{\bf u}}_{0}e^{i(\omega t-{\bf k}\cdot{\bf x})}$

The linearised shallow water equations (3.13) can then be written in a way
that looks suggestively like the Dirac equation

$\displaystyle{\cal H}\Psi=\omega\Psi\ \ \ {\rm where}\ \ \ \ {\cal
H}=\left(\begin{array}[]{ccc}0&ck_{x}&ck_{y}\\\ ck_{x}&0&-if\\\
ck_{y}&if&0\end{array}\right)\ \ \ {\rm and}\ \ \
\Psi=\left(\begin{array}[]{c}\sqrt{g/H}\eta_{0}\\\ u_{0}\\\
v_{0}\end{array}\right)$ (3.20)

Here we’ve written the two-component velocity as ${{\bf u}}_{0}=(u_{0},v_{0})$
and defined $c=\sqrt{gH}$ which we recognise as the speed of surface waves in
the absence of the Coriolis force. We will see that $c$ plays a similar role
in the present context.

####

The eigenvalue problem (3.20) is easily solved. There are two, gapped bands
with frequencies

$\displaystyle\omega=\pm\sqrt{c^{2}{\bf k}^{2}+f^{2}}$ (3.21)

These are known as Poincaré waves. For small wavelengths, $c|{\bf k}|\gg f$,
these waves travel with speed $c$. The long-wavelength modes are affected by
the presence of the Coriolis force.

####

In addition to the Poincaré waves, there is a flat band of solutions to
(3.20), with

$\displaystyle\omega=0$

This reflects the fact that there are static solutions to the full non-linear
equations (1.1) beyond the obvious $h={\rm constant}$. These solutions have a
non-trivial profile in one direction, say $h=h(y)$, with the gravitational
force balanced by a corresponding velocity profile $fu=-\partial h/\partial y$
generating a Coriolis force. This is known as geostrophic balance. These
static solutions manifest themselves in the linear problem as a flat band.

####

There is one other feature of the linearised shallow water equations (3.13)
that we will need. This follows from conservation laws. The linearised version
of the conservation of height (2.3) and vorticity (2.4) becomes

$\displaystyle\frac{\partial{\eta}}{\partial{t}}+H\nabla\cdot{\bf u}=0\ \ \
{\rm and}\ \ \ \frac{\partial{\zeta}}{\partial{t}}+f\nabla\cdot{\bf u}=0$

Eliminating $\nabla\cdot{\bf u}$ from both of these expressions gives us the
surprisingly powerful formula

$\displaystyle\dot{Q}=0\ \ \ {\rm with}\ \ \ Q=H\zeta-f\eta$ (3.22)

This is much stronger than most conservation laws: it is telling us that there
is a function $Q({\bf x})$ that does not change in time. The function $Q$ is
called potential vorticity111The existence of a function that is invariant in
time is special to the linearised equations. In the full non-linear equations,
potential vorticity is defined as $Q=(\zeta+f)/h$. It is not conserved in the
sense that it obeys a continuity equation, but it is does have vanishing
material derivative: $DQ/Dt=0$..

####

We can ask: what is the potential vorticity evaluated on the solutions to the
eigenvalue problem (3.20)? It is simple to check that the flat band, with
$\omega=0$, has

$\displaystyle Q\sim\sqrt{\frac{H}{g}}\left(c^{2}{\bf k}^{2}+f^{2}\right)$

Meanwhile, the Poincaré waves with dispersion relation (3.21) have

$\displaystyle Q=0$

This fact will be important in the next section when we write down an
effective field theory for the Poincaré waves.

###  3.2 A Linearised Gauge Theory

We now return to our gauge theory (2.9). We will see how the above phenomena
arise in this language. We linearise the $A_{\mu}$ gauge fields as

$\displaystyle A_{\mu}=\hat{A}_{\mu}+\delta A_{\mu}\ \ \ {\rm with}\ \ \
\hat{A}_{0}=0\ \ \ {\rm and}\ \ \
\partial_{1}\hat{A}_{2}-\partial_{2}\hat{A}_{1}=H$

We’re going to abuse notation slightly and refer to the fluctuation $\delta
A_{\mu}$ simply as $A_{\mu}$. (In other words, we make the substitution,
$A_{\mu}\rightarrow\hat{A}_{\mu}+A_{\mu}$.) This will make the subsequent
equations somewhat clearer. The translation (2.6) to the fluid variables then
becomes

$\displaystyle B=\eta\ \ \ {\rm and}\ \ \ E_{i}=H\epsilon_{ij}u^{j}$ (3.23)

which ensures that the first equation in (3.13) is obeyed courtesy of the
Bianchi identity. Meanwhile, we also linearise the Clebsch gauge field
$\tilde{A}_{\mu}$, with components

$\displaystyle\beta=\hat{\beta}+p\ \ \ {\rm and}\ \ \ \alpha=\hat{\alpha}+q\ \
\ {\rm with}\ \ \
\partial_{1}\hat{\beta}\,\partial_{2}\hat{\alpha}-\partial_{2}\hat{\beta}\,\partial_{1}\hat{\alpha}=f$
(3.24)

The fluctuations are written as $\delta\beta=p$ and $\delta\alpha=q$ to avoid
an overload of $\partial\delta$’s in the expressions below. Expanding the
action (2.9), we find that the terms linear in fluctuations vanish, as they
must, while the terms quadratic in fluctuations read

$\displaystyle S=\int dt\,d^{2}x\
\left(\frac{1}{2H}{\bm{E}}^{2}-\frac{1}{2}gB^{2}-Hp\dot{q}+\epsilon^{ij}E_{i}\left(q\,\partial_{j}\hat{\beta}-p\,\partial_{j}\hat{\alpha}\right)\right)$
(3.25)

The action is both simple and rather opaque. We will attempt to shed some
light. We first look at the equations of motion. Gauss’ law is

$\displaystyle\partial_{i}E_{i}=H\epsilon^{ij}\left(\partial_{i}\hat{\beta}\,\partial_{j}q-\partial_{i}\hat{\alpha}\,\partial_{j}p\right)$
(3.26)

while the equation of motion for $A_{i}$ is

$\displaystyle\dot{E}_{i}=-gH\epsilon_{ij}\partial_{j}B-H\epsilon^{ij}\left(\partial_{j}\hat{\beta}\,\dot{q}-\partial_{j}\hat{\alpha}\,\dot{p}\right)$
(3.27)

Finally, the equations of motion for $p$ and $q$ read

$\displaystyle H\dot{q}=-\epsilon^{ij}E_{i}\partial_{j}\hat{\alpha}\ \ \ {\rm
and}\ \ \ H\dot{p}=-\epsilon^{ij}E_{i}\partial_{j}\hat{\beta}$ (3.28)

We can substitute (3.28) into (3.27) to find

$\displaystyle\dot{E}_{i}=\epsilon_{ij}\left(fE_{j}-gH\partial_{j}B\right)$

This coincides with the second linearised shallow water equation in (3.13)
using the dictionary (3.23).

####

We can now ask how this gauge theoretic formulation captures the wave
phenomena described in Section 3.1. First, the geostrophic flat band
$\omega=0$ arises from static solutions with

$\displaystyle A_{0}=-\frac{c^{2}}{f}B$

Here our interest lies more with the Poincaré waves. We would like to extract
from (3.25) an effective theory that describes the dynamics of the Poincaré
waves. We can do this using the observation that they have vanishing potential
vorticity (3.22). In the gauge theory language, this becomes the statement

$\displaystyle Q=H\zeta-f\eta=0\ \ \ \Rightarrow\ \ \ \partial_{i}E_{i}=fB$
(3.29)

But Gauss’ law (3.26) already gives us an expression for the divergence of the
electric field: when restricted to Poincaré waves, the right-hand side of
(3.26) must also equal $fB$. We can solve this straightforwardly. We work in
$A_{0}=0$ gauge, and identify

$\displaystyle H{q}=-\epsilon^{ij}{A}_{i}\partial_{j}\hat{\alpha}\ \ \ {\rm
and}\ \ \ Hp=-\epsilon^{ij}A_{i}\partial_{j}\hat{\beta}$

This solves the equations of motion (3.28) and ensures that Gauss’ law (3.26)
coincides with the requirement (3.29) of vanishing potential vorticity.
Restricting to such solutions, we can eliminate $q$ and $p$ from the action
(3.25). Disregarding total derivatives, this gives us an effective action
governing Poincaré waves,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int dt\,d^{2}x\
\left(\frac{1}{2H}{\dot{A}_{i}}^{2}-\frac{1}{2}gB^{2}+\frac{f}{2H}\epsilon^{ij}{A}_{i}\dot{A}_{j}\right)$

where we have used the condition
$\epsilon_{ij}\partial_{i}\hat{\beta}\partial_{j}\hat{\alpha}=f$. This
effective action must be accompanied by Gauss’ law, now in the form (3.29).
This is easily achieved by reinstating $A_{0}$ in the action so that its
equation of motion reproduces the potential vorticity constraint. The action
then becomes

$\displaystyle S=\int dt\,d^{2}x\
\frac{1}{2H}\left({\bm{E}}^{2}-c^{2}B^{2}-f\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}\right)$
(3.30)

This is a very familiar: it is Maxwell-Chern-Simons theory. Here, it governs
the dynamics of Poincaré waves. It is simple to check that the excitations of
this theory have the dispersion relation (3.21), with the profiles of the
electric and magnetic fields coinciding with the velocity and height of
Poincaré waves, using the dictionary (3.23).

###  3.3 Coastal Kelvin Waves in Maxwell-Chern-Simons

Viewed as a quantum field theory, the action (3.30) flows at low energies to a
topological field theory. Indeed, it is perhaps the simplest topological field
theory and actions of this type are well known to describe quantum Hall fluids
zhk ; wenniu . Here, of course, we treat (3.30) as a purely classical theory.
Nonetheless, a remnant of its topological nature survives in chiral edge
modes.

####

For concreteness, we consider the theory with a spatial boundary at $x=0$.
Varying the action then gives, in addition to the bulk term, a boundary term

$\displaystyle\delta S=\delta S_{\rm bulk}+\frac{1}{H}\int dt\,dy\
\left[-(E_{1}-2fA_{2})\delta A_{0}-(c^{2}B+2fA_{0})\delta A_{2}\right]$ (3.31)

The bulk term $\delta S_{\rm bulk}$ vanishes when the equations of motion are
obeyed. But to have a well defined variational principle we must impose
boundary conditions to ensure that the boundary term also vanishes. We will
describe our boundary condition of choice shortly.

####

In the presence of a boundary, Chern-Simons theory comes with an additional
concern. The Chern-Simons action is invariant under gauge transformations only
up to a total derivative. This means that the theory risks a failure of gauge
invariance in the presence of a boundary. Under a gauge transformation
$A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\theta$, the action (3.30)
transforms as

$\displaystyle S\rightarrow S+\frac{f}{2H}\int dt\,dy\ \theta\,E_{2}$ (3.32)

There are various ways to deal with this lack of gauge invariance, depending
on the boundary condition that we impose to ensure that (3.31) vanishes. In
the context of the quantum Hall effect, one typically restricts to gauge
transformations $\theta$ that vanish on the boundary. But restricting the
allowed gauge transformations resurrects modes that were previously gauge
redundancies seiberg . These are the chiral modes on the boundary. The chiral
modes that arise in this way are closely related to the anomaly inflow
arguments that underlie the bulk/boundary correspondence for topological
insulators.

####

The chiral modes that arise as coastal Kelvin waves are somewhat different
because we impose different boundary conditions. This time, the boundary term
in (3.31) vanishes by setting $A_{0}=A_{2}=0$ on the boundary. This ensures
that $E_{2}=0$ on the boundary so the gauge variation of the action (3.32)
also vanishes. In electromagnetism, this is the boundary condition appropriate
for a conductor. In fluid mechanics it is the relevant boundary condition
because it means that ${\bf u}\cdot\hat{\bf x}=0$, so that no fluid flows into
the boundary. The resulting chiral edge modes are no longer due to anomaly
inflow, but instead follow from the conservation of charge.

####

The bulk equations of motion of Maxwell-Chern-Simons theory are

$\displaystyle\partial_{i}E_{i}=fB\ \ \ {\rm and}\ \ \
\dot{E}_{i}=-c^{2}\epsilon_{ij}\partial_{j}B+f\epsilon_{ij}E_{j}$

We can look for solutions with $A_{0}=A_{2}=0$ everywhere. We make the ansatz

$\displaystyle A_{1}=A(x)\,e^{i(\omega t-ky)}$

The bulk equations tell us that

$\displaystyle\dot{E}_{1}=-c^{2}\partial_{2}B\ \ \ $
$\displaystyle\Rightarrow$ $\displaystyle\ \ \ \omega^{2}=c^{2}k^{2}\ \ \
\Rightarrow\ \ \ \omega=\pm ck$

and

$\displaystyle\partial_{1}E_{1}=fB\ \ \ $ $\displaystyle\Rightarrow$
$\displaystyle\ \ \ \omega A^{\prime}=kfA\ \ \ \Rightarrow\ \ \
A^{\prime}=\pm\frac{f}{c}A$

For $f>0$, corresponding to the Northern hemisphere, a fluid restricted to lie
in the region $x>0$ has the normalisable solution only if

$\displaystyle\omega=-ck\ \ \ \Rightarrow\ \ \ A(x)\sim e^{-fx/c}$

This is the chiral, coastal Kelvin wave, first found in kelvin . It travels in
the direction of decreasing $y$ which, in this context, means southward. Said
differently, the wave moves around land masses in the clockwise direction in
the Northern hemisphere. The novelty here is to see this wave emerging from
the effective Maxwell-Chern-Simons description.

##  Appendix A An Alternative Derivation of the Gauge Theory

There is a long history of constructing variational principles for fluids in
the Eulerian framework. The results do not usually look like a gauge theory.
In this appendix we provide an alternative derivation of the gauge theory
action (2.9) that makes contact with the more traditional approach.

####

For fluids in $d$ spatial dimensions, one usually starts by introducing a map
from physical spacetime to the positions of fluid parcels.

$\displaystyle\alpha:\mathds{R}^{d}\times\mathds{R}\rightarrow\mathds{R}^{d}$

We write this map as $\alpha^{a}({\bf x},t)$ with $a=1,\ldots,d$. The map must
be a diffeomorphism, with ${\rm det}(\partial\alpha^{a}/\partial x^{i})\neq
0$. A discussion of this approach to fluids can be found in the textbook
mandel . More recently, it has been revived in the context of relativistic
fluids dub ; son .

####

Given a map $\alpha^{a}({\bf x},t)$, the velocity field ${\bf u}$ is defined
by the equation

$\displaystyle\frac{D\alpha^{a}}{Dt}\equiv\frac{\partial{\alpha^{a}}}{\partial{t}}+{\bf
u}\cdot\nabla\alpha^{a}=0$ (A.33)

This captures the idea that the physical label $\alpha^{a}({\bf x},t)$ of the
point in the fluid is unchanged as the fluid moves. The velocity ${\bf u}$
defined in this way is invariant under diffeomorphisms of $\alpha$. Note,
however, that this definition comes with a rather confusing minus sign,
related to active vs passive transformations. This is seen, for example, in
the simplest laminar flow $\mbox{\boldmath$\alpha$}=(x-ut,y)$ which, according
to (A.33), is associated to a velocity field ${\bf u}=(+u,0)$.

####

For application to the shallow water equations, we have $d=2$ dimensions. We
also have the height variable $h(x,y,t)$, which obeys the continuity equation

$\displaystyle\frac{Dh}{Dt}+h\nabla\cdot{\bf u}=0$ (A.34)

reflecting the incompressibility of the underlying fluid.

####

An action for the shallow water equations in these variables was given in
seliger

$\displaystyle S=\int dt\,d^{2}x\left(\frac{1}{2}h{\bf
u}^{2}-\frac{1}{2}gh^{2}-h{\bf u}\cdot{\bf a({\bf
x})}\right)+\phi\left(\frac{Dh}{Dt}+h\nabla\cdot{\bf
u}\right)-h\beta_{a}\frac{D\alpha^{a}}{Dt}$

This action should be varied with respect to $h$, ${\bf u}$, $\phi$,
$\alpha^{a}$ and $\beta_{a}$. Both $\beta_{a}$ and $\phi$ act as Lagrange
multipliers, imposing the conditions (A.33) and (A.34) respectively. We have
introduced a background gauge field ${\bf a}({\bf x})$ whose role will be to
implement the Coriolis force222In quantum Hall physics, there is a convention
that lower-case gauge fields $a_{\mu}$ are dynamical while upper-case gauge
fields $A_{\mu}$ are background. Annoyingly, we have chosen the opposite
convention here. As we will see, this has the advantage that the field
strength $f_{12}=\partial_{1}a_{2}-\partial_{2}a_{1}$ coincides with the
Coriolis parameter, canonically denoted in geophysics as $f$.. This background
field depends only on ${\bf x}$ and not on time. We will determine the form of
${\bf a}({\bf x})$ shortly.

####

The equation of motion for ${\bf u}$ reveals that the velocity has a Clebsch-
like parameterisation,

$\displaystyle{\bf u}=\nabla\phi+\beta_{a}\nabla\alpha^{a}+{\bf a}({\bf x})$
(A.35)

This differs from our previous Clebsch parameterisation for the gauge field
(2.11) in that we have a pair of potentials $(\beta_{a},\alpha_{a})$, with
$a=1,2$, with $\alpha^{a}$ restricted to be a diffeomorphism. In addition, the
velocity gets a contribution from the fixed background ${\bf a}({\bf x})$.

####

The equation of motion for $\alpha^{a}$ reveals a conserved current, involving
the Lagrange multiplier $\beta_{a}$,

$\displaystyle\frac{\partial{(h\beta_{a})}}{\partial{t}}+\nabla\cdot({h\beta_{a}{\bf
u}})=0\ \ \ \Rightarrow\ \ \ \frac{D\beta_{a}}{Dt}=0$ (A.36)

where the second equation follows from the conservation of the height (A.34).
Finally, the equation of motion for $h$ gives

$\displaystyle\frac{D\phi}{Dt}=\frac{1}{2}{\bf u}^{2}-gh-{\bf u}\cdot{\bf a}$
(A.37)

where an additional term has been set to zero courtesy of (A.33). Although it
is not immediately obvious, equations (A.33), (A.35), (A.36) and (A.37) are
equivalent to the second of the shallow water equations (1.1). To see this, we
first use (A.35) to compute the material derivative of the velocity

$\displaystyle\frac{Du_{i}}{Dt}=(\partial_{i}\dot{\phi}+u^{j}\partial_{j}\partial_{i}\phi)+(\dot{\beta}_{a}+u^{j}\partial_{j}\beta_{a})\partial_{i}\alpha^{a}+\beta_{a}(\partial_{i}\dot{\alpha}^{a}+u^{j}\partial_{j}\partial_{i}\alpha^{a})+u^{j}\partial_{j}a_{i}$

Invoking (A.33), (A.36) and (A.37), a short calculation then reveals

$\displaystyle\frac{Du_{i}}{Dt}=-f_{ij}u^{j}-g\partial_{i}h\ \ \ \ {\rm with}\
f_{ij}=\partial_{i}a_{j}-\partial_{j}a_{i}$

We see that the background field strength $f_{ij}$ acts like a Coriolis force.
To reproduce the constant Coriolis force of (1.1), we simply need to find a
background ${\bf a}({\bf x})$ that satisfies $f_{ij}=-f\epsilon_{ij}$. For
example, ${\bf a}=f(y,0)$ or ${\bf a}=\frac{1}{2}f(y,-x)$ both do the job.
Note that any choice necessarily breaks translational symmetry and/or
rotational symmetry, even though the final shallow water equations do not.
This is unavoidable in this formalism. The situation is identical to that of a
charged particle in a constant background magnetic field.

####

We now show how this formalism is related to the gauge theory introduced in
Section 2. The idea is to use a 3d duality transformation, first introduced in
poly and beloved of many a high-energy and condensed matter theorist. To
proceed, it’s useful to first integrate by parts, so that the Lagrangian has
an overall factor of the height $h$. (This reflects the fact that the shallow
water equations arise after integrating over the $z$-direction of the thin
fluid.)

$\displaystyle S=\int dt\,d^{2}x\ h\left(\frac{1}{2}{\bf
u}^{2}-\frac{1}{2}gh-{\bf u}\cdot{\bf a}({\bf
x})-\frac{D\phi}{Dt}-\beta_{a}\frac{D\alpha^{a}}{Dt}\right)$ (A.38)

Note that only derivatives of the Lagrange multiplier $\phi$ appear in this
action. This allows us to exchange $\phi$ for a vector field $C_{\mu}$, by
writing the action as

$\displaystyle S=\int dt\,d^{2}x\ h\left(\frac{1}{2}{\bf
u}^{2}-\frac{1}{2}gh-{\bf u}\cdot{\bf a}({\bf
x})-C_{0}-u^{i}C_{i}-\beta_{a}\frac{D\alpha^{a}}{Dt}\right)+\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}C_{\rho}$

Here we have further introduced a new Lagrange multiplier, $A_{\mu}$. This can
be viewed as a gauge field because the action is invariant under
transformation $A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\theta$. The equation
of motion for $A_{\mu}$ gives the constraint
$\epsilon^{\mu\nu\rho}\partial_{\nu}C_{\rho}=0$ which can be solved locally by
$C_{\mu}=\partial_{\mu}\phi$. This takes us back to our original action
(A.38). Alternatively, we may instead choose to impose the equation of motion
for $C_{\mu}$, which also acts as a Lagrange multiplier. This gives a
constraint on the magnetic field $B=\partial_{1}A_{2}-\partial_{2}A_{1}$ and
electric fields $E_{i}=\dot{A}_{i}-\partial_{i}A_{0}$,

$\displaystyle B=h\ \ \ {\rm and}\ \ \ E_{i}=\epsilon_{ij}hu^{j}$

These coincide with the relations introduced in (2.6). We may then replace $h$
and ${\bf u}$ in the action with our new gauge fields. This gives

$\displaystyle S=\int dt\,d^{2}x\
\left(\frac{{\bm{E}}^{2}}{2B}-\frac{1}{2}gB^{2}+\epsilon^{ij}E_{i}a_{j}({\bf
x})-\epsilon^{\mu\nu\rho}\partial_{\mu}A_{\nu}\,\beta_{a}\partial_{\rho}\alpha^{a}\right)$

In its essential details, this coincides with the action (2.9) presented in
Section 2. The Coriolis term becomes the background charge $fA_{0}$ after an
integration by parts. The form above makes it clear that this term too has its
origin as a Chern-Simons term, now involving a background field. After
integration by parts, the action respects both translation and rotation
symmetry provided that $f$ is constant. Meanwhile, the second gauge field
arises automatically in the Clebsch parameterisation in the above
presentation,

$\displaystyle\tilde{A}_{\mu}=\partial_{\mu}\chi+\beta_{a}\partial_{\mu}\alpha^{a}$

Here $\chi$ is pure gauge and does not contribute to the action. This differs
from (2.11) only in the $a=1,2$ index ascribed to $\alpha$ and $\beta$.
Diffeomorphism invariance ensures that this does not contribute any further
degrees of freedom.

####

There are other equations in fluid mechanics that have a similar structure to
the shallow water equations (1.1) and hence lend themselves to a gauge
theoretic formalism. For example, a compressible fluid obeys the equations

$\displaystyle\frac{\partial{\rho}}{\partial{t}}+\nabla\cdot({\rho{\bf u}})=0\
\ \ {\rm and}\ \ \ \rho\frac{D{\bf u}}{Dt}=-\nabla P$

with an equation of state relating the pressure $P$ to the density $\rho$ of
the form $P=C\rho^{\gamma}$, where $C$ is a constant and $\gamma$ is the ratio
of specific heats. When the fluid lives in $d=2$ spatial dimensions, this too
may be written as gauge theory, with $B=\rho$ and $E_{i}=\epsilon_{ij}\rho
u^{j}$ and with the potential term $B^{2}$ in (2.9) replaced by
$B^{\gamma-1}$. Again, the slightly disconcerting fact that the power $\gamma$
is not an integer is no cause for concern when it is appreciated that this
theory should be expanded around the solution $B=\rho={\rm constant}\neq 0$.

### Acknowledgements

I’m especially grateful to Matt Davison for introducing me to the idea of
topology in fluid dynamics and for guiding me through some basic aspects of
atmospheric and oceanic flows. Thanks also to Jay Armas, Sean Hartnoll and Max
Metlitski for helpful discussions. This work was supported by STFC grant
ST/L000385/1, the EPSRC grant EP/V047655/1 “Chiral Gauge Theories: From Strong
Coupling to the Standard Model”, and a Simons Investigator Award.

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