# Two-dimensional turbulence above topography: condensation transition and
selection of minimum enstrophy solutions

Basile Gallet1<EMAIL_ADDRESS>1Université Paris-Saclay, CNRS, CEA,
Service de Physique de l’Etat Condensé, 91191 Gif-sur-Yvette, France.

###### Abstract

We consider two-dimensional flows above topography, revisiting the selective
decay (or minimum-enstrophy) hypothesis of Bretherton and Haidvogel. We derive
a ‘condensed branch’ of solutions to the variational problem where a domain-
scale condensate coexists with a flow at the (smaller) scale of the
topography. The condensate arises through a supercritical bifurcation as the
conserved energy of the initial condition exceeds a threshold value, a
prediction that we quantitatively validate using Direct Numerical Simulations
(DNS). We then consider the forced-dissipative case, showing how weak forcing
and dissipation select a single dissipative state out of the continuum of
solutions to the energy-conserving system predicted by selective decay. As the
forcing strength increases, the condensate arises through a supercritical
bifurcation for topographic-scale forcing and through a subcritical
bifurcation for domain-scale forcing, both predictions being quantitatively
validated by DNS. This method provides a way of determining the equilibrated
state of forced-dissipative flows based on variational approaches to the
associated energy-conserving system, such as the statistical mechanics of 2D
flows or selective decay.

###### keywords:

Variational approaches, Quasi-geostrophic turbulence, Ocean processes

##  1 Introduction

Rapidly rotating flows above topography are both a standard model for various
large-scale oceanic flows and a fascinating playground to study how disorder
disrupts the large-scale self-organization of two-dimensional flows. A first
oceanic situation of interest is the Antarctic Circumpolar Current (ACC),
where turbulent eddies are channeled by and somewhat locked to the topography,
with important consequences for both the turbulent transport and the speed of
the ACC itself (Straub, 1993; Nadeau & Straub, 2009; Abernathey & Cessi, 2014;
Nadeau & Ferrari, 2015; Constantinou & Young, 2017; Constantinou, 2018;
Constantinou & Hogg, 2019; Barthel et al., 2022; Yung et al., 2022). At a more
local scale, the Lofoten vortex has attracted significant attention (Köhl,
2007; Søiland & Rossby, 2013; Trodahl et al., 2020; LaCasce et al., 2024)
because its direction of rotation is inconsistent with the natural tendency
for flows to homogenize potential vorticity (Rhines & Young, 1982).

Early on, the study of rotating flows above topography has been connected to
the search for general organizing principles governing the evolution of quasi-
two-dimensional flows. Bretherton & Haidvogel (1976) (BH in the following)
applied ‘selective decay’ to such flows, an organizing principle borrowed from
plasma physics (Kruskal & Kulsrud, 1958; Taylor, 1974; Montgomery et al.,
1978). According to the selective decay principle, the forward-cascading
invariant should be minimized for a given (conserved) value of the inverse-
cascading invariant. Based on the inverse energy cascade and forward enstrophy
cascade of 2D flows, BH thus conjectured that the system evolves towards a
state of minimum enstrophy, while conserving its initial energy. They report
reasonable, although imperfect, agreement with their numerical simulations. A
second, more recent variational approach is the statistical mechanics of two-
dimensional flows (Miller, 1990; Robert, 1991; Robert & Sommeria, 1991;
Bouchet & Venaille, 2012). The approach has been applied to several two
dimensional fluid systems, predicting various branches of solutions with
different flow structures (Naso et al., 2011), together with ‘phase
transitions’ between these various states (Chavanis & Sommeria, 1996; Venaille
& Bouchet, 2009). In its most general form, the theory is based on the
conservation of all the invariants of the system: energy, all the moments of
the vorticity field, together with linear momentum, angular momentum or
circulation, depending on the geometry of interest. However, for most
practical computations the high-order (potential-)vorticity moments are
discarded (e.g. Salmon et al. (1976); Venaille (2012)) and the theory ends up
being equivalent to selective decay (Carnevale & Frederiksen, 1987), namely
enstrophy minimization subject to the conservation of some first and second-
order invariants (see Brands et al. (1999) and Nycander & Lacasce (2004) for
examples where all the invariants are simultaneously considered). While they
provide invaluable insight, an important limitation of such variational
approaches is that they apply to energy-conserving systems only, as opposed to
forced-dissipative ones. Bridging this gap is a central motivation for the
present study.

Recently, Siegelman & Young (2023) (SY in the following) revisited two-
dimensional turbulence above topography, with a focus on the difference
between the low-energy flows and the high-energy flows. In qualitative
agreement with selective decay, enstrophy robustly decreases with time in
their numerical simulations, while energy is conserved to a very good
approximation. However, SY also notice that the flows attained in the long-
time limit depart from the predictions of selective decay. For low initial
energy, the numerical solutions exhibit intense cyclones above topographic
mounts and anticyclones above topographic depressions that are not predicted
by selective decay. Instead, such pinned vortices result from the tendency for
anticyclones (cyclones) to migrate down-slope (up-slope), resulting in a
vortex segregation mechanism that opposes PV homogenization (Carnevale et al.,
1991; Köhl, 2007; Trodahl et al., 2020; Solodoch et al., 2021). Above a
threshold value of the initial energy, these vortices are no longer pinned to
the topography, and instead they roam within the numerical domain.

In the present study, we consider a spatial structure for the topography that
differs from both BH and SY. While these authors include topography at all
scales, we restrict attention to topography at smaller scale than the extent
of the fluid domain (strictly, although no scale separation is needed). Our
goal is threefold. First, we wish to continue the program initiated by SY and
further characterize the transition from the low energy states to the high
energy states: how does the flow transition from a topographically-locked
state to the typical large-scale condensates that characterize weakly damped
two-dimensional flows (Kraichnan, 1975; Gallet & Young, 2013; Laurie et al.,
2014)? Secondly, we want to assess the predictive skill of the aforementioned
variational approaches: do they predict a sharp bifurcation or a smooth
crossover between the two types of flows? Is the transition continuous or
discontinuous? Can the theory quantitatively predict the strength of the
large-scale condensate? Thirdly, we want to address the limitation of the
variational approaches mentioned above, namely that they were designed for
energy-conserving systems only, whereas natural and laboratory flows are
typically forced and dissipative. While the conserved invariants of the system
are the natural control parameters of the variational theories, such
quantities are no longer conserved in the presence of forcing and dissipation.
How to make quantitative predictions in this context? How to leverage the
variational approaches to predict the equilibrated state of the forced-
dissipative system?

We introduce the theoretical setup in section 2, before focusing on the
energy-conserving system in section 3. We predict a transition to condensation
based on selective decay. In section 4, we report numerical simulations that
quantitatively validate the predicted amplitude of the large-scale condensate.
We turn to the forced-dissipative system in section 5. We show how weak
forcing and damping select a single state out of the continuum of solutions to
the energy-conserving system predicted by selective decay. We compare the
predicted amplitude of the condensate to Direct Numerical Simulations (DNS) of
the forced dissipative system, obtaining very good agreement for both
topographic-scale forcing and domain-scale forcing. We conclude in section 6.

##  2 Rapidly rotating flows above topography

We consider a shallow-water flow above weak topography in a rapidly rotating
frame, described within the $f$-plane quasi-geostrophic approximation. The
horizontal velocity field ${\bf u}(x,y,t)$ stems from a streamfunction, ${\bf
u}=-\bnabla\times[\psi(x,y,t){\bf e}_{z}]$. The evolution of the system is
governed by the material conservation of potential vorticity (PV) $q(x,y,t)$:

$\displaystyle\partial_{t}q+J(\psi,q)={\cal F}(x,y)+{\cal D}\,,$ (1)
$\displaystyle q=\Delta\psi+\eta(x,y)\,,$ (2)

where $J(f,g)=(\partial_{x}f)(\partial_{y}g)-(\partial_{y}f)(\partial_{x}g)$
denotes the Jacobian operator, $\Delta$ denotes the Laplacian operator and
$\eta(x,y)$ denotes the ‘topographic’ PV. For a layer of fluid of local depth
$H+h(x,y)$, where $h(x,y)\ll H$ denotes the mean-zero fluctuations in
bathymetry, the topographic PV is $\eta(x,y)=-f_{0}\,h(x,y)/H$, with $f_{0}$
the Coriolis parameter. We consider equations (1-2) inside a doubly periodic
domain, $(x,y)\in[0,2\pi L]^{2}$, with $2\pi L$ greater than the greatest
wavelength of the topography (strictly, although no scale separation is
required). We include forcing and dissipation on the right-hand side of
equation (1). The forcing term ${\cal F}(x,y)$ corresponds to the wind-stress
curl in a typical ocean setting (Salmon, 1998). The dissipative term consists
of linear ‘Ekman’ bottom drag (Pedlosky, 1979) and hyperviscosity, ${\cal
D}=-\kappa\Delta\psi-\nu\Delta^{3}\psi$. In the following we consider both the
‘energy-conserving case’, characterized by ${\cal F}=0$, $\kappa=0$ and small
hyperviscosity $\nu$, and the ‘forced dissipative case’, characterized by
${\cal F}\neq 0$ and $\kappa\neq 0$.

##  3 Energy-conserving case: transition to condensation

When the right-hand side of (1) is set to zero, ${\cal F}={\cal D}=0$, the
system possesses two quadratic invariants. Denoting space average as
$\left<\cdot\right>$, the first invariant is the kinetic energy
$E=\left<|\bnabla\psi|^{2}\right>$, where we omit the prefactor $1/2$ to
alleviate the algebra. The second invariant is the enstrophy
$Q=\left<q^{2}\right>$, omitting again the prefactor $1/2$. In a similar
fashion to two-dimensional turbulence in the absence of topography, one
expects the transient dynamics of the system to be characterized by an inverse
cascade of energy together with a forward cascade of enstrophy. If a small
hyperviscosity $\nu\neq 0$ is included, we expect $Q$ to be robustly
dissipated once enstrophy has reached the small viscous dissipative scale. The
Kolmogorov-Kraichnan dual cascade phenomenology further suggests that such
‘anomalous’ dissipation of enstrophy is maintained as $\nu\to 0$, whereas the
energy $E$ is conserved in that limit. That enstrophy is strongly dissipated
by weak small-scale damping while energy is approximately conserved is clearly
illustrated by the recent DNS of SY. This intuition led BH to conjecture that
the system eventually reaches the minimum-enstrophy state associated with the
conserved energy of the initial condition. This state is obtained by
extremalizing the functional ${\cal L}\\{\psi\\}=Q+\mu E$, where $\mu$ is a
Lagrange multiplier ensuring energy conservation. This variational problem
leads to:

$\displaystyle q=\Delta\psi+\eta=\mu\psi\,.$ (3)

Note that the same relation can be deduced from the statistical mechanics of
2D flows, if one retains only the invariants $E$ and $Q$. An argument in favor
of selective decay as opposed to the statistical mechanics of conservative
systems is that $Q$ consistently decreases during the transient phase observed
in DNS.

An appealing aspect of enstrophy minimization is that fields satisfying (3)
are steady solutions to equation (1) with ${\cal F}={\cal D}=0$. Additionally,
the solution to (3) corresponding to the absolute enstrophy minimum is stable
according to Arnol’d’s stability criterion (Arnol’d, 1978). The following
subsections aim at computing this absolute minimizer.

###  3.1 The Bretherton-Haidvogel branch

The standard approach to solving equation (3) is the one proposed by BH. They
assume that $\mu>-k_{0}^{2}$, where $k_{0}=1/L$ denotes the minimum (gravest)
wavenumber of the square domain. Introducing the Fourier decompositions of
$\eta$ and $\psi$ as:

$\displaystyle\eta(x,y)=\sum_{{\bf k}L\in\mathbb{Z}^{2}}\hat{\eta}_{\bf
k}e^{i{\bf k}\cdot\bf{x}}\,,\qquad\psi(x,y,t)=\sum_{{\bf
k}L\in\mathbb{Z}^{2}}\hat{\psi}_{\bf k}(t)e^{i{\bf k}\cdot\bf{x}}\,,$ (4)

equation (3) has a unique solution obtained in spectral space:

$\displaystyle\hat{\psi}_{\bf k}=\frac{\hat{\eta}_{\bf k}}{\mu+k^{2}}\,.$ (5)

From (5) the energy $E$ and the enstrophy $Q$ can be expressed in terms of
$\mu$. This leads to a parametric curve for $Q$ versus $E$, and one thus
predicts (among other things) the final enstrophy in terms of the initial
energy of the system.

###  3.2 Monoscale topography

Although not a necessary assumption, restricting attention to monoscale
topography provides an illustrative example. We thus consider that the
topography is peaked around a single wavenumber $k_{\eta}$, such that
$\Delta\eta=-k_{\eta}^{2}\eta$. For such topography (5) leads to:

$\displaystyle\psi=\frac{\eta}{\mu+k_{\eta}^{2}}\,,\qquad
E=\frac{k_{\eta}^{2}}{(\mu+k_{\eta}^{2})^{2}}\left<\eta^{2}\right>\,,\qquad
Q=\frac{\mu^{2}}{(\mu+k_{\eta}^{2})^{2}}\left<\eta^{2}\right>\,.$ (6)

It proves convenient to introduce the energy scale
$E_{\eta}=\left<\eta^{2}\right>/k_{\eta}^{2}$ associated with the topography
(following SY, $E_{\eta}$ also corresponds to the energy of a flow with
homogenized PV, see discussion in section 6). Denoting the dimensionless
energy as $\tilde{E}=E/E_{\eta}$, one can eliminate $\mu$ to express $Q$ in
terms of $E$:

$\displaystyle\frac{Q}{\left<\eta^{2}\right>}$ $\displaystyle=$
$\displaystyle\left(\sqrt{\tilde{E}}-1\right)^{2}\,.$ (7)

###  3.3 The condensed branch

Figure 1: The amplitude $\tilde{A}$ of the condensate as a function of the
(conserved) energy of the initial condition $\tilde{E}$. The theory predicts
the existence of a condensed branch for $\tilde{E}>1$ in the scale separation
limit (slightly larger threshold for finite $\tilde{k}_{\eta}$). The DNS are
performed using $\tilde{k}_{\eta}=12$. The data points agree well with the
predicted amplitude, for the two initial conditions introduced in section 4.
Some departure from the theory remains in the immediate vicinity of the
threshold for condensation, as a consequence of the extra vortices pinned to
the topography.

When the system has a size greater than the largest scale of the topography
(when $k_{0}=1/L<k_{\eta}$ for monoscale topography) we argue that the BH
branch above is not always the absolute minimizer of the enstrophy. Indeed,
consider equation (3) with $\mu=-k_{0}^{2}$:

$\displaystyle\Delta\psi+k_{0}^{2}\psi=-\eta\,.$ (8)

The operator on the left-hand side is not invertible anymore and the solution
is (restricting attention to monoscale topography, the generalization to an
arbitrary spectrum for the topography being straightforward):

$\displaystyle\psi(x,y)$ $\displaystyle=$
$\displaystyle\psi_{\eta}(x,y)+A\,\psi_{0}(x,y)\,,$ (9)
$\displaystyle\psi_{\eta}(x,y)$ $\displaystyle=$
$\displaystyle\frac{\eta(x,y)}{k_{\eta}^{2}-k_{0}^{2}}\,.$ (10)

In equation (9), $\psi_{0}(x,y)$ is a harmonic function at the gravest scale
of the domain (that is, $\Delta\psi_{0}=-k_{0}^{2}\psi_{0}$) normalized such
that $\left<\psi_{0}^{2}\right>=1$, and $A\geq 0$ denotes the amplitude of
this large-scale ‘condensate’. $A$ and $\psi_{0}(x,y)$ are related to the two
gravest Fourier coefficients of $\psi$ through
$A\,\psi_{0}(x,y)=\hat{\psi}_{(1/L,0)}e^{ix/L}+\hat{\psi}_{(0,1/L)}e^{iy/L}+\text{c.c.}$,
that is:

$\displaystyle
A=\sqrt{2|\hat{\psi}_{(1/L,0)}|^{2}+2|\hat{\psi}_{(0,1/L)}|^{2}}\,.$ (11)

The amplitude $A$ of the large-scale condensate can be expressed in terms of
the conserved energy $E$ of the initial condition:

$\displaystyle
E=A^{2}\left<|\bnabla\psi_{0}|^{2}\right>+\left<|\bnabla\psi_{\eta}|^{2}\right>\,,$
(12)

that is,

$\displaystyle
A=\sqrt{\frac{E-\left<|\bnabla\psi_{\eta}|^{2}\right>}{\left<|\bnabla\psi_{0}|^{2}\right>}}=\frac{1}{k_{0}}\sqrt{E-\left<|\bnabla\psi_{\eta}|^{2}\right>}\,.$
(13)

For monoscale topography the expression of
$\left<|\bnabla\psi_{\eta}|^{2}\right>$ reads:

$\displaystyle\frac{\left<|\bnabla\psi_{\eta}|^{2}\right>}{E_{\eta}}=\frac{\tilde{k}_{\eta}^{4}}{(\tilde{k}_{\eta}^{2}-1)^{2}}\,,$
(14)

where $\tilde{k}_{\eta}=k_{\eta}/k_{0}$, and after substitution in (13) one
obtains the following expression for the dimensionless amplitude of the
condensate $\tilde{A}=k_{0}A/\sqrt{E_{\eta}}$:

$\displaystyle\tilde{A}=\sqrt{\tilde{E}-\tilde{E}_{c}}\,,\qquad\text{with
}\quad\tilde{E}_{c}=\frac{\tilde{k}_{\eta}^{4}}{(\tilde{k}_{\eta}^{2}-1)^{2}}\,.$
(15)

This prediction takes a particularly simple form in the limit of scale
separation between the domain size and the wavelength of the topography,
$\tilde{k}_{\eta}\gg 1$, which leads to $\tilde{E}_{c}\simeq 1$:

$\displaystyle\tilde{A}=\sqrt{\tilde{E}-1}\,.$ (16)

As illustrated by the plot in figure 1, equation (16) displays a transition to
large-scale condensation above a critical value $E_{\eta}$ of the energy of
the initial condition. An interpretation of this result is that the topography
plays the role of a reservoir that can host a small-scale flow of energy up to
$E_{\eta}$. Any energy beyond $E_{\eta}$ spills out of this reservoir and is
subject to inverse energy transfers. This excess energy eventually ends up
accumulating in a large-scale condensate, hence the nonzero value of $A$.

The enstrophy associated with the condensed branch is:

$\displaystyle\frac{Q}{\left<\eta^{2}\right>}=\frac{1}{\tilde{k}_{\eta}^{2}}\left(\tilde{E}-1-\frac{1}{\tilde{k}_{\eta}^{2}-1}\right)\,,$
(17)

valid only when $A$ exists, that is for $\tilde{E}>\tilde{E}_{c}$. In the
scale separation limit this expression reduces to:

$\displaystyle\frac{Q}{\left<\eta^{2}\right>}=\frac{1}{\tilde{k}_{\eta}^{2}}\left(\tilde{E}-1\right)\,,$
(18)

valid for $\tilde{E}>1$.

Figure 2: Enstrophy $Q$ in the (quasi-)stationary state as a function of the
(conserved) energy of the initial condition $\tilde{E}$, using the same
symbols as in figure 1. Also shown are the results of the minimization: BH
branch in blue and condensed branch in red, for $\tilde{k}_{\eta}=12$. The
absolute minimum (over the branches) is shown as a black dashed line. The DNS
data is close to the BH branch for $\tilde{E}\leq 1$ and close to the
condensed branch for $\tilde{E}\geq 1$. The inset is a zoom on the data for
$\tilde{E}\geq 1$. The departure from the absolute minimum near the
condensation threshold is due to the extra vortices pinned to the topography.

In figure 2 we show the enstrophy $Q$ as a function of the energy $E$ for both
the BH branch and the condensed branch. One can check that the enstrophies
$Q(E)$ evaluated on the two branches (7) and (18) are equal and tangent to one
another at the critical value $\tilde{E}=\tilde{E}_{c}$. For
$\tilde{E}<\tilde{E}_{c}$ the absolute enstrophy minimizer is the BH branch,
while for $\tilde{E}>\tilde{E}_{c}$ the absolute minimizer corresponds to the
condensed branch. Absolute minimization of enstrophy thus predicts a
transition to condensation as $E$ exceeds $E_{c}$, the amplitude of the
condensate being given by (15). These predictions are particularly compact in
the limit of scale separation, where the dimensionless threshold for
condensation reduces to $\tilde{E}_{c}=1$ and the amplitude of the condensate
is given by (16).

##  4 Numerical simulations

Figure 3: For the topography shown on the left-hand panel, a numerical run
initialized with energy $\tilde{E}=0.743$ displays no condensation (center
column), whereas a run initialized with $\tilde{E}=1.44$ settles in a
condensed state in the long-time limit (right-hand column).

To test the predictions above, we perform DNS of equations (1-2) by time-
stepping $\psi$ using a pseudo-spectral solver with a fourth-order Runge-Kutta
time-stepping scheme. The resolution is $1024^{2}$ before de-aliasing
following the $2/3$ rule. We first consider the enstrophy-decaying energy-
conserving case, ${\cal F}=0$ and $\kappa=0$ with a small value of the
hyperviscosity $\nu k_{0}^{3}/\sqrt{E_{\eta}}=2\times 10^{-10}$. The
topography is approximately monoscale and centered around the dimensionless
wavenumber $\tilde{k}_{\eta}=12$. Specifically, we pick random values for the
Fourier coefficients of all the modes such that $12-1/4\leq kL\leq 12+1/4$.
All the numerical runs are performed using the same realization of the
topography, displayed in the left-hand panel of figure 3.

We consider two types of initial conditions:

  * •

The ‘$\psi$ initial condition’: we set some initial complex Fourier amplitude
at wavenumber $(k_{x}L,k_{y}L)=(0,2)$ for $\psi$ to match the target initial
value of $\tilde{E}$.

  * •

The ‘$\phi$ initial condition’: we set some initial complex Fourier amplitude
at wavenumber $(k_{x}L,k_{y}L)=(0,2)$ for a field $\phi(x,y)$, before
initializing $\psi$ as $\psi(x,y,t)=\psi_{\eta}(x,y,t)+\phi(x,y,t)$.

Both types of initial conditions correspond to initial perturbations at rather
large scale. Starting from the $\psi$ initial condition, the flow
spontaneously develops smaller structures through interaction with the
topography. By contrast, the $\phi$ initial condition readily includes the
small-scale flow structure predicted by enstrophy minimization. This initial
condition only allows to achieve an initial energy $\tilde{E}\geq 1$. By
contrast, one can achieve any positive initial value for the energy
$\tilde{E}$ using the $\psi$ initial condition.

As discussed by SY, in such simulations the enstrophy rapidly decays from its
initial value. By contrast, the energy decreases extremely slowly and is
conserved to a very good approximation. We wish to characterize the
(quasi-)stationary state obtained once the transient has subsided. In figure 3
we show snapshots obtained at the end time of two simulations differing in the
value of the initial energy. The streamfunction $\psi$ agrees qualitatively
with the theory: the simulation with $\tilde{E}<1$ displays no large-scale
condensation, whereas the simulation with $\tilde{E}>1$ displays a strong
large-scale condensate. By contrast, in line with previous studies the PV $q$
exhibits strong isolated vortices that are not predicted by enstrophy
minimization (Zhao et al., 2019; Siegelman & Young, 2023; LaCasce et al.,
2024). When condensation arises, however, these vortices appear to be rather
sparsely distributed above the faint background PV structure of the condensed
branch, given by (3) with $\mu=-k_{0}^{2}$. There is thus hope for large-scale
condensation to be quantitatively predicted by enstrophy minimization despite
the emergence of the isolated vortices.

To characterize the condensation transition quantitatively, we extract from
the DNS the amplitude $A$ as defined in equation (11). In figure 1 we show the
resulting dimensionless condensate amplitude $\tilde{A}$ as a function of the
dimensionless initial energy $\tilde{E}$. The numerical data are in very good
agreement with the theoretical prediction from enstrophy minimization. The
$\phi$ initial condition leads to particularly good agreement because this
initial condition is close to the enstrophy minimizer, thus preventing the
emergence of strong isolated PV vortices. The more generic $\psi$ initial
condition also leads to very good quantitative agreement. There is some
departure from the theoretical prediction in the immediate vicinity of the
bifurcation threshold only, as a consequence of the isolated vortices pinned
to the topography.

In figure 2, we plot the equilibrated enstrophy $Q$ extracted from the DNS.
Above the transition $Q$ is significantly lower than the enstrophy of the BH
branch, because the latter is not the absolute minimizer. The equilibrated $Q$
lies extremely close to the prediction of the condensed branch for DNS
initialized using the $\phi$ initial condition. It is slightly greater than
(but close to) this absolute minimum for the $\psi$ initial condition, as a
result of the extra isolated vortices. Once again, this slight departure from
the absolute minimum of $Q$ has only a modest impact on the value of the
condensate amplitude $A$, which closely follows the theoretical prediction.

##  5 Forced-dissipative case

The natural objection to the variational approaches to computing the
statistically steady states of fluid flows is that these approaches are
usually based on conservative dynamics, while real flows are forced and
dissipative. For instance, the statistical mechanics of 2D flows assumes that
energy and enstrophy are conserved and keep their initial values, while DNS
point to a transient phase with a robust decrease in enstrophy. In that
respect, selective decay (enstrophy minimization) is one step closer to
realistic flows, allowing for such a net decrease in enstrophy as the system
evolves towards the minimizer, in line with the forward enstrophy cascade of
2D flows. However, selective decay also assumes that energy is conserved and
set by the initial condition, whereas natural and laboratory flows typically
receive energy from some external forcing and dissipate it through a
combination of damping mechanisms. To make contact between such forced-
dissipative systems and the variational framework discussed above, we now
consider equation (1) with nonzero forcing ${\cal F}$ injecting energy into
the system, and nonzero friction coefficient $\kappa$ extracting energy from
the system (we also retain the hyperviscous coefficient). The goal is at a
fundamental level: more than describing realistic flows of geophysical
interest, we wish to establish a direct quantitative connection between the
variational approach described above and forced-dissipative systems. We thus
illustrate how weak forcing and dissipation select a unique state out of the
continuum of solutions to the energy conserving problem predicted by selective
decay.

###  5.1 Selection of minimum-enstrophy states

Figure 4: Condensate amplitude as a function of the forcing expressed in terms
of the Grashof number. The DNS data points are in good agreement with the
theoretical prediction (30) associated with the selection of minimum-enstrophy
solutions (dashed line).

Consider equation (1) with weak forcing and damping:

$\displaystyle\partial_{t}q+J(\psi,q)=\epsilon[{\cal F}^{(1)}(x,y,t)+{\cal
D}^{(1)}]\,,$ (19) $\displaystyle q=\Delta\psi+\eta$ (20)

where $\epsilon\ll 1$ is a small parameter, ${\cal
D}^{(1)}=-\kappa^{(1)}\Delta\psi+\nu^{(1)}\Delta^{3}\psi$, and the topography
$\eta$ is ${\cal O}(1)$. Expand the fields in powers of $\epsilon$:

$\displaystyle q(x,y,t)$ $\displaystyle=$ $\displaystyle
q^{(0)}(x,y,T)+\epsilon q^{(1)}(x,y,T)+\dots\,,$ (21)
$\displaystyle\psi(x,y,t)$ $\displaystyle=$
$\displaystyle\psi^{(0)}(x,y,T)+\epsilon\psi^{(1)}(x,y,T)+\dots\,,$ (22)

where we have introduced the slow time variable $T=\epsilon t$, that is,
$\partial_{t}\to\partial_{t}+\epsilon\partial_{T}$.

Collecting terms of order ${\cal O}(\epsilon^{0})$ leads to the set of
equations discussed in the preceding section:

$\displaystyle\partial_{t}q^{(0)}+J(\psi^{(0)},q^{(0)})=0\,,\qquad
q^{(0)}=\Delta\psi^{(0)}+\eta\,.$ (23)

We consider a solution to these equations that lies on the condensed branch,
allowing the amplitude of the condensate to vary slowly in time:

$\displaystyle\psi^{(0)}(x,y,T)$ $\displaystyle=$
$\displaystyle\psi_{\eta}(x,y)+A(T)\,\psi_{0}(x,y)\,.$ (24)

Following the traditional approach, the slow evolution equation for the
amplitude $A(T)$ can be obtained from a solvability condition at next order.
An alternate and more enlightening way of obtaining this amplitude equation
consists in writing the energy evolution equation to lowest order. Upon
multiplying equation (19) with $\psi$ before averaging over $x$ and $y$ and
neglecting terms of order $\epsilon^{2}$, one obtains:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}T}\left(\frac{\tilde{A}^{2}}{2}\right)$
$\displaystyle\simeq$ $\displaystyle-\frac{\left<{\cal
F}^{(1)}\eta\right>}{\left<\eta^{2}\right>}-\kappa^{(1)}-\nu^{(1)}k_{\eta}^{4}-\frac{\left<{\cal
F}^{(1)}\psi_{0}\right>}{k_{0}\sqrt{E_{\eta}}}\tilde{A}-\kappa^{(1)}\tilde{A}^{2}\,,$
(25)

where, for brevity, we restrict attention to the scale separation limit
$\tilde{k}_{\eta}\gg 1$. After dividing by $\kappa^{(1)}$, equation (25) can
be recast in terms of the original (non-expanded) variables as:

$\displaystyle\frac{1}{2\kappa}\frac{\mathrm{d}}{\mathrm{d}t}\tilde{A}^{2}$
$\displaystyle=$ $\displaystyle Gr_{\eta}-1-\frac{\nu
k_{\eta}^{4}}{\kappa}-\tilde{A}^{2}+Gr_{0}\tilde{A}\,.$ (26)

For body-forced flows the dimensionless strength of the forcing is usually
quantified by a Grashof number (Doering & Gibbon, 1995). In equation (26) we
have thus introduced a topographic-scale Grashof number $Gr_{\eta}$ and a
domain-scale Grashof number $Gr_{0}$, defined respectively as:

$\displaystyle Gr_{\eta}=-\frac{\left<{\cal
F}\eta\right>}{\kappa\left<\eta^{2}\right>}\,\qquad\text{and}\qquad
Gr_{0}=-\frac{\left<{\cal F}\psi_{0}\right>}{k_{0}\kappa\sqrt{E_{\eta}}}\,.$
(27)

The forcing thus enters the evolution equation (26) for the amplitude of the
condensate in two different ways: through its correlation with the smaller-
scale topography and through its correlation with the domain-scale condensate.
In the following we consider these two situations independently.

###  5.2 Topographic-scale forcing: continuous transition to condensation

Consider first the situation where there is no forcing at large-scale,
$\left<{\cal F}\psi_{0}\right>=0$ and therefore $Gr_{0}=0$. Equation (26)
reduces to:

$\displaystyle\frac{1}{2\kappa}\frac{\mathrm{d}}{\mathrm{d}t}\tilde{A}^{2}$
$\displaystyle=$ $\displaystyle Gr_{\eta}-Gr_{\eta}^{(c)}-\tilde{A}^{2}\,,$
(28)

where we have introduced a critical Grashof number $Gr_{\eta}^{(c)}=1+{\nu
k_{\eta}^{4}}/{\kappa}$.

For $Gr_{\eta}<Gr_{\eta}^{(c)}$, $\tilde{A}^{2}$ decreases and becomes
negative according to this equation. The ansatz (24) is then invalid and
instead the system ends up on the non-condensed BH branch. By contrast, a
steady solution $\tilde{A}\geq 0$ exists for $Gr_{\eta}\geq Gr_{\eta}^{(c)}$.
We thus predict the following value for the equilibrated amplitude of the
condensate:

$\displaystyle\tilde{A}$ $\displaystyle=$ $\displaystyle 0\,\qquad\text{for
}Gr_{\eta}<Gr_{\eta}^{(c)}\,,$ (29) $\displaystyle\tilde{A}$ $\displaystyle=$
$\displaystyle\sqrt{Gr_{\eta}-Gr_{\eta}^{(c)}}\,\qquad\text{for }Gr_{\eta}\geq
Gr_{\eta}^{(c)}\,.$ (30)

This prediction corresponds to a supercritical transition to condensation as
the dimensionless forcing strength $Gr_{\eta}$ increases. The bifurcation
threshold reduces to the compact expression $Gr_{\eta}^{(c)}=1$ in the limit
where hyperviscosity damps energy at a negligible rate as compared to
friction, $\nu k_{\eta}^{4}\ll\kappa$.

As discussed above, the forcing enters the amplitude equation (28) through its
correlation with the topography. We first illustrate this situation
numerically using a forcing that is perfectly correlated with the topography:

$\displaystyle{\cal F}(x,y)=-Gr_{\eta}\,\kappa\,\eta(x,y)\,.$ (31)

We consider the same realization of the topography as in the preceding
sections, together with damping coefficients
$\kappa/(k_{0}\sqrt{E_{\eta}})=0.0021$ and $\nu
k_{0}^{3}/\sqrt{E_{\eta}}=2.8\times 10^{-8}$. The corresponding critical
Grashof number is $Gr_{\eta}^{(c)}\simeq 1.28$. We perform a suite of
numerical simulations for various values of $Gr_{\eta}$, extracting the
amplitude of the condensate once the transient has subsided. We show
$\tilde{A}$ as a function of $Gr$ in figure 4, together with the theoretical
prediction (30). The agreement is very good, both for the threshold value
$Gr_{\eta}^{(c)}$ and for the amplitude of the condensate above threshold.
Together with the simple forcing described above, we have considered a second
forcing structure where ${\cal F}(x,y)$ is proportional to
$\eta(x,y)+\xi(x,y)$, where $\xi(x,y)$ is a random field with the same
statistics as $\eta(x,y)$ but corresponding to another realization (the
realizations of $\eta$ and $\xi$ are the same for the entire suite of
simulations). Even though the resulting forcing ${\cal F}(x,y)$ is less
correlated to the topography than in the previous suite of simulations, the
condensate amplitude is again well predicted by equation (30), as shown in
figure 4.

###  5.3 Large-scale forcing: discontinuous transition to condensation

Consider now the situation where the forcing entering the amplitude equation
(26) is at large scale only, $Gr_{\eta}=0$ and $Gr_{0}\neq 0$. For simplicity,
we also consider that the hyperviscous contribution to this amplitude equation
is negligible, $\nu k_{\eta}^{4}\ll\kappa$. Equation (26) reduces to:

$\displaystyle\frac{1}{2\kappa}\frac{\mathrm{d}}{\mathrm{d}t}\tilde{A}^{2}$
$\displaystyle=$ $\displaystyle Gr_{0}\tilde{A}-1-\tilde{A}^{2}\,,$ (32)

with steady solutions:

$\tilde{A}_{\pm}=\frac{1}{2}\left(Gr_{0}\pm\sqrt{Gr_{0}^{2}-4}\right)\,.$ (33)

These solutions exist only for $Gr_{0}\geq 2$, while for $Gr_{0}<2$ the system
lies on the uncondensed BH branch, with $\tilde{A}=0$. One can furthermore
check that $A_{-}$ is unstable according to equation (32), while $A_{+}$ is a
stable branch of solutions. We thus predict a subcritical bifurcation to
large-scale condensation as $Gr_{0}$ exceeds the threshold value
$Gr_{0}^{(c)}=2$. It is interesting to notice that the same supercritical
bifurcation of the energy-conserving system induces either a supercritical or
a subcritical bifurcation in the weakly forced-dissipative system, depending
on the structure of the forcing.

Figure 5: Condensate amplitude as a function of the large-scale forcing
amplitude expressed in terms of $Gr_{0}$. The DNS data points are in good
agreement with the theoretical prediction $A_{+}$ in (33) associated with the
selection of minimum-enstrophy solutions (dashed line).

We investigated this situation numerically with the same resolution and the
same realization of the topography as before. The friction coefficient is
still $\kappa/(k_{0}\sqrt{E_{\eta}})=0.0021$, while we employ the much smaller
value of the hyperviscosity $\nu k_{0}^{3}/\sqrt{E_{\eta}}=2\times 10^{-10}$,
which ensures that $\nu k_{\eta}^{4}\ll\kappa$. The forcing is:

${\cal F}(y)=-\sqrt{2E_{\eta}}k_{0}\kappa\,Gr_{0}\,\cos(y)\,,$ (34)

associated with a structure $\psi_{0}=\sqrt{2}\cos y$ for the large-scale
flow. To investigate possible bistability, the initial condition is either
very low amplitude fluctuations or the end state of a previous simulation.

The amplitude of the condensate is shown as a function of $Gr_{0}$ in figure
5. Once again, the agreement with the theory is excellent at finite distance
from the instability threshold, with slight discrepancies near threshold. In
particular, the amplitude of the condensate is very well captured by the
upper-branch in (33) for $Gr_{0}\geq 2$. We observe only a very narrow region
of bistability, possibly because the uncondensed branch has other directions
of instability, besides direct condensation following (32). Additionally, the
region of stability of the condensed branch extends somewhat below $Gr_{0}=2$,
down to $Gr_{0}\simeq 1.8$. This $10\%$ correction in threshold value may be a
consequence of the extra vortices pinned to the topography. As for the
previous cases considered above, we conclude that the amplitude of the
condensate is accurately predicted by the theory, except in the immediate
vicinity of the instability threshold where the pinned vortices impact the
dynamics of the system.

###  5.4 Structure of the condensate

We have mainly discussed the amplitude of the condensate, setting aside its
degenerate structure $\psi_{0}(x,y)$. The latter depends on the relative
magnitudes of the complex Fourier coefficients $\hat{\psi}_{(1/L,0)}$ and
$\hat{\psi}_{(0,1/L)}$, as well as on their phases. In simulations of the
energy-conserving system, the precise structure of the condensate (see e.g.
figure 3) seems to be determined predominantly by the precise realization of
the topography and, to a lesser extent, by the initial condition. When driving
the system with the domain-scale forcing (34) together with moderately small
friction, the condensate always adopts the large-scale structure of the
forcing, $\psi_{0}(x,y)=\sqrt{2}\cos(y)$, and the theoretical predictions are
accurately satisfied. However, if $\epsilon$ on the right-hand side of
equation (19) is made extremely small (very weak forcing and dissipation),
there seems to be a competition between the forcing, which favors the
structure $\psi_{0}(x,y)=\sqrt{2}\cos(y)$, and the precise realization of the
topography, which favors an alternate structure, such as the one visible in
figure 3. The prediction (33) for the condensate amplitude does not hold if
the system adopts this alternate condensate structure. An in-depth
characterization of such phase dynamics in the very-weak-drag regime would
deserve a dedicated study that goes beyond the scope of the present article.
One way to lift the degeneracy of the condensate structure consists in
including weak domain-scale topography in a perturbative fashion, as described
in appendix A. Then a single condensate structure $\psi_{0}(x,y)$ proportional
to the domain-scale topography arises, the condensate amplitude being still
determined by forcing and dissipation (or by the initial energy for the
energy-conserving system).

##  6 Conclusion

We have characterized the transition to large-scale condensation for 2D flows
over smaller-scale topography, revisiting the selective decay or minimum-
enstrophy hypothesis of BH. We have computed a condensed branch of solutions
to the minimization problem. This branch of solutions is the absolute
enstrophy minimizer above some threshold value of the (conserved) energy. As
this control parameter increases, selective decay predicts a continuous
transition to large-scale condensation. This bifurcation adds to the various
‘phase transitions’ of quasi-2D fluid systems predicted based on the
statistical mechanics of 2D flows, such as the transition in flow structure as
the aspect ratio of a 2D rectangular fluid domain is varied (Chavanis &
Sommeria, 1996; Bouchet & Venaille, 2012). The present bifurcation is also
very similar to the emergence of a Rossby-wave in the enstrophy-minimizing
state of a channel flow on the $\beta$ plane (Young, 1987), the connection
between the two problems being apparent if $\beta$ is interpreted as a uniform
topographic slope.

For simplicity, we have illustrated the present theory using monoscale
topography, but the approach is easily extended to more complex topography. As
compared to the recent study of flows above topography by SY, the main
qualitative difference is probably that SY include large topographic
variations at the domain scale. SY report a transition from isolated pinned
vortices to freely roaming vortices as the initial energy exceeds a threshold
value $E_{\sharp}=\left<|\bnabla\Delta^{-1}\\{\eta\\}|^{2}\right>$, omitting
the prefactor $1/2$ for consistency with our definition for $E$ (SY identify
$E_{\sharp}$ as the energy of a flow with homogeneous PV $q=0$, which for
monoscale topography reduces to the energy scale $E_{\eta}$ that we introduced
on dimensional grounds). By contrast, in the absence of topography at the
domain scale we observe a transition to condensation as the initial energy
exceeds a threshold value $E_{c}=\left<|\bnabla\psi_{\eta}|^{2}\right>$.
Interestingly, while $E_{c}\neq E_{\sharp}$ for arbitrary topography, $E_{c}$
and $E_{\sharp}$ reduce to the same expression in the limit of separation
between the topographic scale and the domain scale. This indicates that SY’s
study and the present study may correspond to two manifestations of the same
transition. An appealing aspect of the present setup is that selective decay
provides quantitative predictions for the amplitude of the condensate above
threshold.

Indeed, another goal of the present study is to assess the predictive skill of
such a variational approach through comparison with DNS. DNS of the energy-
conserving system show that selective decay quantitatively predicts the
amplitude of the condensate. For initial conditions tailored to be close to
the absolute minimizer, the amplitude of the condensate is in excellent
agreement with the theory. For more generic initial conditions isolated
vortices arise, polluting the potential vorticity field $q$. The vortices
affect the very near-threshold behavior of the condensate amplitude only, the
agreement remaining very good at larger distance from threshold. That the
vortices predominantly affect the near-threshold behavior probably originates
from the fact that they become more scarce as condensation proceeds. Indeed,
the isolated vortices are located at the local extrema of the total
streamfunction, as discussed in SY. In the absence of large-scale
condensation, the streamfunction resembles the topography, with many such
extrema. By contrast, at finite distance above threshold the condensate
dominates the total streamfunction, and the latter has fewer local extrema to
host isolated vortices (see figure 3).

The above-mentioned dependence on initial conditions is perhaps a complication
of the energy-conserving system. After all, natural and laboratory flows are
typically forced and dissipative, with energy supplied by an external source
and removed by some combination of damping mechanisms. The energy of the
equilibrated state is then an emergent property of the system, as opposed to
the initial energy of the system being a control parameter of the variational
approaches to conservative flows. As such, the variational approaches to
energy-conserving systems cannot predict the strength of forced-dissipative
flows. With the goal of bridging the gap between these two situations, we have
introduced a perturbative expansion that illustrates how weak forcing and
dissipation select the equilibrated state of the flow out of the continuum of
states predicted by selective decay (or by the statistical mechanics of 2D
flows). We have restricted attention to steady forcing for simplicity, but the
approach probably holds for slowly varying forcing.

This selection mechanism is very similar to – and was inspired by – the
selection of solitary wave solutions to the complex Ginzburg-Landau equation
by weak dissipation (Fauve & Thual, 1990). The theoretical predictions for the
resulting amplitude of the condensate agree quantitatively with DNS of the
forced-dissipative system. Interestingly, the same continuous transition to
condensation for the energy-conserving system turns into a discontinuous
transition when the system is forced at large scale, while it remains
continuous when the system is forced at the smaller topographic scale.

This method provides an avenue for predicting the equilibrated state of
forced-dissipative flows based on variational approaches initially designed
for conservative systems (selective decay or statistical mechanics). Its
robustness and predictive skill remain to be further assessed through
application to various systems of interest, be they numerical simulations,
laboratory experiments or observational data.

Acknowledgements. I would like to thank W.R. Young and A. Venaille for
insightful discussions, and S. Fauve for raising my interest in the selection
of conservative solutions in weakly forced dissipative systems.

Funding. This research is supported by the European Research Council under
grant agreement FLAVE 757239.

Declaration of interests. The author reports no conflict of interest.

Author ORCID. https://orcid.org/0000-0002-4366-3889.

##  Appendix A Including weak domain-scale topography

Weak domain-scale topography can be included through the following Poincaré-
Lindstedt expansion:

$\displaystyle\eta(x,y)$ $\displaystyle=$
$\displaystyle\eta^{(0)}(x,y)+\epsilon\eta^{(1)}(x,y)\,,$ (35)
$\displaystyle\psi(x,y,t)$ $\displaystyle=$
$\displaystyle\psi^{(0)}(x,y)+\epsilon\psi^{(1)}(x,y)+\dots\,,$ (36)
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle-
k_{0}^{2}+\epsilon\mu^{(1)}\,,$ (37)

where $\eta^{(0)}(x,y)$ denotes the monoscale topography adopted throughout
the main text, $\eta^{(1)}(x,y)$ denotes domain-scale topography on the
gravest modes ${\bf k}=(1/L,0)$ and ${\bf k}=(0,1/L)$, and we have expanded
the Lagrange multiplier $\mu$ around its value $-k_{0}^{2}$ on the condensed
branch. To order ${\cal O}(1)$, equation (3) yields
$\Delta\psi^{(0)}+k_{0}^{2}\psi^{(0)}=-\eta^{(0)}$, with solution
$\psi^{(0)}=A\psi_{0}(x,y)+\psi_{\eta}(x,y)$. To order ${\cal O}(\epsilon)$
equation (3) yields
$\Delta\psi^{(1)}+k_{0}^{2}\psi^{(1)}=\mu^{(1)}\psi^{(0)}-\eta^{(1)}$. The
solvability condition is obtained by demanding that the right-hand side of
this equation have no projection onto the gravest Fourier modes, which simply
yields $\mu^{(1)}A\psi_{0}=\eta^{(1)}$. Using $\left<\psi_{0}^{2}\right>=1$ to
eliminate $\mu^{(1)}$, we obtain the following expression for the structure of
the condensate:

$\displaystyle\psi_{0}(x,y)$ $\displaystyle=$
$\displaystyle\pm\frac{\eta^{(1)}(x,y)}{\sqrt{\left<(\eta^{(1)})^{2}\right>}}\,.$
(38)

Weak domain-scale topography thus lifts the degeneracy of the condensate
structure. The amplitude $A$ of the condensate is still determined following
the procedures described in the main text (initial energy for the energy-
conserving system ; forcing and damping for the forced-dissipative system).

## References

  * Abernathey & Cessi (2014) Abernathey, Ryan & Cessi, Paola 2014 Topographic enhancement of eddy efficiency in baroclinic equilibration. Journal of Physical Oceanography 44 (8), 2107–2126.
  * Arnol’d (1978) Arnol’d, Vladimir Igorevich 1978 Mathematical methods of classical mechanics. Springer Science & Business Media.
  * Barthel et al. (2022) Barthel, Alice, Hogg, Andrew M, Waterman, Stephanie & Keating, Shane 2022 Baroclinic control of southern ocean eddy upwelling near topography. Geophysical Research Letters 49 (7), e2021GL097491.
  * Bouchet & Venaille (2012) Bouchet, Freddy & Venaille, Antoine 2012 Statistical mechanics of two-dimensional and geophysical flows. Physics reports 515 (5), 227–295.
  * Brands et al. (1999) Brands, H, Chavanis, PH, Pasmanter, R & Sommeria, J 1999 Maximum entropy versus minimum enstrophy vortices. Physics of Fluids 11 (11), 3465–3477.
  * Bretherton & Haidvogel (1976) Bretherton, Francis P & Haidvogel, Dale B 1976 Two-dimensional turbulence above topography. Journal of Fluid Mechanics 78 (1), 129–154.
  * Carnevale et al. (1991) Carnevale, GF, Kloosterziel, RC & Van Heijst, GJF 1991 Propagation of barotropic vortices over topography in a rotating tank. Journal of Fluid Mechanics 233, 119–139.
  * Carnevale & Frederiksen (1987) Carnevale, George F & Frederiksen, Jorgen S 1987 Nonlinear stability and statistical mechanics of flow over topography. Journal of Fluid Mechanics 175, 157–181.
  * Chavanis & Sommeria (1996) Chavanis, Pierre-Henri & Sommeria, Joel 1996 Classification of self-organized vortices in two-dimensional turbulence: the case of a bounded domain. Journal of Fluid Mechanics 314, 267–297.
  * Constantinou (2018) Constantinou, Navid C 2018 A barotropic model of eddy saturation. Journal of Physical Oceanography 48 (2), 397–411.
  * Constantinou & Hogg (2019) Constantinou, Navid C & Hogg, Andrew McC 2019 Eddy saturation of the southern ocean: A baroclinic versus barotropic perspective. Geophysical Research Letters 46 (21), 12202–12212.
  * Constantinou & Young (2017) Constantinou, Navid C & Young, William R 2017 Beta-plane turbulence above monoscale topography. Journal of Fluid Mechanics 827, 415–447.
  * Doering & Gibbon (1995) Doering, Charles R & Gibbon, John D 1995 Applied analysis of the Navier-Stokes equations. Cambridge university press.
  * Fauve & Thual (1990) Fauve, Stephan & Thual, Olivier 1990 Solitary waves generated by subcritical instabilities in dissipative systems. Physical review letters 64 (3), 282.
  * Gallet & Young (2013) Gallet, Basile & Young, William R 2013 A two-dimensional vortex condensate at high reynolds number. Journal of Fluid Mechanics 715, 359–388.
  * Köhl (2007) Köhl, Armin 2007 Generation and stability of a quasi-permanent vortex in the lofoten basin. Journal of Physical Oceanography 37 (11), 2637–2651.
  * Kraichnan (1975) Kraichnan, Robert H 1975 Statistical dynamics of two-dimensional flow. Journal of Fluid Mechanics 67 (1), 155–175.
  * Kruskal & Kulsrud (1958) Kruskal, Martin David & Kulsrud, RM 1958 Equilibrium of a magnetically confined plasma in a toroid. The Physics of Fluids 1 (4), 265–274.
  * LaCasce et al. (2024) LaCasce, JH, Palóczy, A & Trodahl, M 2024 Vortices over bathymetry. Journal of Fluid Mechanics 979, A32.
  * Laurie et al. (2014) Laurie, Jason, Boffetta, Guido, Falkovich, Gregory, Kolokolov, Igor & Lebedev, Vladimir 2014 Universal profile of the vortex condensate in two-dimensional turbulence. Physical review letters 113 (25), 254503.
  * Miller (1990) Miller, Jonathan 1990 Statistical mechanics of euler equations in two dimensions. Physical review letters 65 (17), 2137\.
  * Montgomery et al. (1978) Montgomery, David, Turner, Leaf & Vahala, George 1978 Three-dimensional magnetohydrodynamic turbulence in cylindrical geometry. The Physics of Fluids 21 (5), 757–764.
  * Nadeau & Ferrari (2015) Nadeau, Louis-Philippe & Ferrari, Raffaele 2015 The role of closed gyres in setting the zonal transport of the antarctic circumpolar current. Journal of Physical Oceanography 45 (6), 1491–1509.
  * Nadeau & Straub (2009) Nadeau, Louis-Philippe & Straub, David N 2009 Basin and channel contributions to a model antarctic circumpolar current. Journal of physical oceanography 39 (4), 986–1002.
  * Naso et al. (2011) Naso, Aurore, Chavanis, Pierre-Henri & Dubrulle, Bérengère 2011 Statistical mechanics of fofonoff flows in an oceanic basin. The European Physical Journal B 80, 493–517.
  * Nycander & Lacasce (2004) Nycander, J & Lacasce, JH 2004 Stable and unstable vortices attached to seamounts. Journal of Fluid Mechanics 507, 71–94.
  * Pedlosky (1979) Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer.
  * Rhines & Young (1982) Rhines, Peter B & Young, William R 1982 Homogenization of potential vorticity in planetary gyres. Journal of Fluid Mechanics 122, 347–367.
  * Robert (1991) Robert, Raoul 1991 A maximum-entropy principle for two-dimensional perfect fluid dynamics. Journal of Statistical Physics 65, 531–553.
  * Robert & Sommeria (1991) Robert, Raoul & Sommeria, Joel 1991 Statistical equilibrium states for two-dimensional flows. Journal of Fluid Mechanics 229, 291–310.
  * Salmon (1998) Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.
  * Salmon et al. (1976) Salmon, Rick, Holloway, Greg & Hendershott, Myrl C 1976 The equilibrium statistical mechanics of simple quasi-geostrophic models. Journal of Fluid Mechanics 75 (4), 691–703.
  * Siegelman & Young (2023) Siegelman, Lia & Young, William R 2023 Two-dimensional turbulence above topography: Vortices and potential vorticity homogenization. Proceedings of the National Academy of Sciences 120 (44), e2308018120.
  * Søiland & Rossby (2013) Søiland, H & Rossby, T 2013 On the structure of the lofoten basin eddy. Journal of Geophysical Research: Oceans 118 (9), 4201–4212.
  * Solodoch et al. (2021) Solodoch, Aviv, Stewart, Andrew L & McWilliams, James C 2021 Formation of anticyclones above topographic depressions. Journal of Physical Oceanography 51 (1), 207–228.
  * Straub (1993) Straub, David N 1993 On the transport and angular momentum balance of channel models of the antarctic circumpolar current. Journal of physical oceanography 23 (4), 776–782.
  * Taylor (1974) Taylor, J Brian 1974 Relaxation of toroidal plasma and generation of reverse magnetic fields. Physical Review Letters 33 (19), 1139.
  * Trodahl et al. (2020) Trodahl, Marta, Isachsen, Pål Erik, Lilly, Jonathan M, Nilsson, Johan & Kristensen, Nils Melsom 2020 The regeneration of the lofoten vortex through vertical alignment. Journal of Physical Oceanography 50 (9), 2689–2711.
  * Venaille (2012) Venaille, Antoine 2012 Bottom-trapped currents as statistical equilibrium states above topographic anomalies. Journal of Fluid Mechanics 699, 500–510.
  * Venaille & Bouchet (2009) Venaille, Antoine & Bouchet, Freddy 2009 Statistical ensemble inequivalence and bicritical points for two-dimensional flows and geophysical flows. Physical review letters 102 (10), 104501\.
  * Young (1987) Young, WR 1987 Selective decay of enstrophy and the excitation of barotropic waves in a channel. Journal of Atmospheric Sciences 44 (19), 2804–2812.
  * Yung et al. (2022) Yung, Claire K, Morrison, Adele K & Hogg, Andrew McC 2022 Topographic hotspots of southern ocean eddy upwelling. Frontiers in Marine Science 9, 855785.
  * Zhao et al. (2019) Zhao, Bowen, Chieusse-Gérard, Emma & Flierl, Glenn 2019 Influence of bottom topography on vortex stability. Journal of Physical Oceanography 49 (12), 3199–3219.