<figure><img src="image_1.png" /></figure>

HAL
open science

# Coexistence of Two Equilibrium Configurations in 2D turbulence

Wesley Agoua, Xi-Yuan Yin, Tong Wu, Wouter J T Bos

► To cite this version:

Wesley Agoua, Xi-Yuan Yin, Tong Wu, Wouter J T Bos. Coexistence of Two Equilibrium Configurations in 2D turbulence. Physical Review Fluids, 2025, 10 (3), pp.034604. 10.1103/PhysRevFluids.10.034604. hal-04933280

HAL Id: hal-04933280

https://hal.science/hal-04933280v1

Submitted on 6 Feb 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Coexistence of Two Equilibrium Configurations in 2D turbulence

Wesley Agoua,<sup>1,2</sup> Xi-Yuan Yin,<sup>1,3</sup> Tong Wu,<sup>1,4</sup> and Wouter J. T. Bos<sup>1</sup>

(Dated: 6 February 2025)

In the past, two families of statistical mechanics approaches have been applied to the two-dimensional Euler equations. The first one is formulated in Fourier space and considers the Galerkin truncated dynamics. The other one is formulated in physical space and considers either point-vortices or coarse-grained vorticity. We show that in a Galerkin truncated system both methods describe a part of the flow. A condensate can be identified assuming that it can be characterized by an unspecified functional relation between the vorticity and the stream-function. It is shown, a posteriori, that this function is a hyperbolic sine relation, as predicted by point-vortex statistical mechanics. The energy spectrum associated with the condensate is well described by an exponential function and the tails of the probability density function of the vorticity are following a power law. Analytical arguments to explain these observations are proposed. After removing the condensate, the remaining field can be described by Fourier-statistical mechanics.

## I. INTRODUCTION

An impressive feature of two-dimensional turbulence is its tendency to self-organize, forming large scale structures¹. These structures are observed in freely decaying two-dimensional turbulent flows², but they also survive in the presence of forcing³ and in thin layer turbulence (e.g.⁴⁻⁶). In the presence of forcing, the formation of condensates at the largest size of the domain can be associated with the inverse energy cascade⁷. For freely decaying turbulence, this explanation is somewhat more tenuous.

A heuristic explanation of the tendency to self-organize in freely evolving two-dimensional flow was given by Onsager in 1949⁸, who suggested applying statistical mechanics to a point-vortex system, introduced by Helmholtz⁹, and predicted that point vortices, under certain constraints, should have the tendency to clump together, forming large clusters of like-signed vortices.

An alternative statistical mechanics approach in turbulence to predict the most probable state

consists in considering, instead of vortices in physical space, the dynamics of modes, after a convenient decomposition⁷. The most common decomposition in turbulence research is the Fourier-decomposition, and once a flow-field is decomposed onto a large, but finite number of modes, the statistically most probable state can be determined by maximizing a well-chosen Boltzmann entropy, accounting for physical constraints, such as energy conservation.

These two approaches (vorticity based versus Fourier-mode based) rely on two different finite-dimensional approximations and only partially describe the full system, albeit in a complementary way. Indeed, as we shall discuss, the structure-based statistical mechanics corresponds to a system of finitely many point vortices where no truncation is applied on the number of modes. In contrast, in Fourier-statistical mechanics a truncation is imposed and ergodicity is assumed. Ergodicity means, in this case, that all positions in phase space are visited with an equal probability. In the presence of long-living structures, it seems that ergodicity is vi-

<footnote><sup>1)</sup> Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, LMFA, 69340 Ecully France</footnote>

<footnote><sup>2)</sup> LEGI - CNRS - Université Grenoble Alpes, 38058 Grenoble, France</footnote>

<footnote><sup>3)</sup> Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany</footnote>

<footnote><sup>4)</sup> Theoretical Physics I, University of Bayreuth, 95447 Bayreuth, Germany</footnote>
olated. In fact, our numerical results, also supported by previous numerical experiments$^{10,11}$, suggest that in fact a superposition of the two equilibria predicted from both statistical approaches is present. This coexistence was investigated previously in the more general case of long-range interacting systems with varying interaction range$^{12}$. In that study the violent relaxation towards a space-filling structure was studied in a more general system of which the Euler equations are a special case. In the present article we reproduce certain of their results and further investigate the 2D Euler case. In particular, we determine the shape of the spectral energy density of the condensate and the probability density function (PDF) of its vorticity. We explain, by analytical reasoning why these quantities have this shape.

In the next section, we will briefly recall some major results of both statistical mechanics approaches. Subsequently, in Sec. III, we will show numerical experiments where both statistical mechanics approaches fail, but with an appropriate decomposition of the flow, each method allows us to explain parts of the observations. In Sec. IV, we argue, as a consequence of the functional relation between the stream function and the vorticity, that the condensate energy spectrum in the freely decaying case can be expected to have exponential decay with respect to wavenumber and that its associated vorticity is described by a PDF with powerlaw tails.

In Sec. V, before concluding, we propose to describe the phenomenology of the system on the level of the Fourier spectra.

## II. TWO RESULTS OF STATISTICAL MECHANICS APPLIED TO TWO-DIMENSIONAL TURBULENCE

The equations we study here are the 2D Euler equations describing an incompressible inviscid fluid flow,

$$ \partial_t u + (u \cdot \nabla)u = -\nabla p, \quad (1a) $$

$$ \nabla \cdot u = 0. \quad (1b) $$

with $u$ the velocity and $p$ the pressure. We restrict this study to flows in a periodic box. Let $\omega = \nabla \times u$ be the vorticity field and define $\psi$ to be the stream function given by $-\Delta\psi = \omega$ where $\Delta$ denotes the Laplacian operator. We have that $u = \nabla \times \psi e_3$, where $e_3$ is the unit vector normal to the plane. Then, taking the curl of (1) we can write the vorticity equation:

$$ \partial_t \omega + \{\omega, \psi\} = 0, \quad (2) $$

where the Poisson bracket is given by

$$ \{\omega, \psi\} = \frac{\partial \omega}{\partial x} \frac{\partial \psi}{\partial y} - \frac{\partial \omega}{\partial y} \frac{\partial \psi}{\partial x} = ((\nabla \times \psi e_3) \cdot \nabla)\omega. \quad (3) $$

We will briefly review two analytical predictions that were obtained by applying statistical mechanics to the above equations. We will first discuss the statistical mechanics of a truncated set of Fourier modes. After that, we recall an important result associated with point-vortex statistical mechanics.

In the following, we will distinguish the wavenumber spectrum of the kinetic energy $E(k)$ from the kinetic energy $E$ by the wavenumber argument $k$. Similarly, we distinguish the enstrophy spectrum $W(k)$ from the enstrophy $W$.

### A. Fourier Statistical Mechanics

For three-dimensional Euler-turbulence, statistical mechanics on a truncated set of Fourier-modes yields, if the only constraint of the system is energy conservation, an equipartition of energy between all the modes. This means
that, statistically, every Fourier mode contains the same amount of energy. This result was first predicted by Lee¹³ who observed that the Fourier-transformed truncated Euler equations satisfy a Liouville-theorem.

Let us now consider the two-dimensional case of the Euler equations, governed by the weak form,

$$ (\partial_t \omega + \{\omega, \psi\}, \phi)_{L^2} = 0 \quad \text{for all } \phi \in V_K, \quad (4) $$

where $V_K$ is the space of test functions $\phi$ consisting of all Fourier modes with wavevectors below a truncation frequency $K$. By construction, the finite-dimensional system conserves total circulation, enstrophy and energy. However, Casimir conservation (integrals of functions of the vorticity other than total circulation and enstrophy) is lost, indicating divergence from the full system as the true solution starts to populate wavenumbers higher than $K$.

In his seminal paper on two-dimensional turbulence, Kraichnan⁷ applied statistical mechanics to the system (4), and predicted that the equilibrium energy spectrum should behave as

$$ E(k) = \frac{2\pi k}{\alpha + \beta k^2}. \quad (5) $$

The parameters $\alpha$ and $\beta$ can be determined by considering the conservation of energy and enstrophy of the system (see for instance¹⁴ for analytical expressions relating $\alpha$ and $\beta$ to $E, W, k_0, k_{max}$). For further details on this prediction, we refer to the review² or a more recent manuscript¹⁵.

The prediction (5) was verified in early simulations of the two-dimensional Euler equations. Indeed, the development of pseudospectral methods in the 1970s allowed to solve non-dissipative simulations of truncated Euler dynamics. It was shown by¹⁶,¹⁷ that the theory of Kraichnan roughly predicts the correct energy spectrum (5) of the equilibrium state. It was furthermore observed that "substantial deviations from equilibrium are present at low wavenumbers. These are probably explained by the long dynamical time scale of large-scale turbulent eddies, so that continuing the calculation

to much longer times may give relaxation"¹⁶. However, more recent numerical investigations showed that these deviations persist even at very long times with no hint of relaxation towards the supposed equilibrium¹²,¹⁸,¹⁹. In particular, when the ratio of energy to enstrophy is large, noticeable discrepancy is observed for small wavenumbers. To understand the origin of this discrepancy we need to review certain key results of an alternative statistical mechanics approach.

## B. Point-vortex statistical mechanics

The structure-based statistical mechanics proposed by Onsager⁸ was further explored by Joyce and Montgomery²⁰,²¹, and more recently by the works of Miller²²,²³, Robert and Sommeria²⁴. A concise review can be found in the article of Eyink and Sreenivasan²⁵, and more comprehensive reviews in² and²⁶. The point-vortex and coarse-grained vorticity statistical mechanics are approaches based on the observation that enstrophy, and small vorticity patches in general, are advected conservatively in the two-dimensional Euler equations.

In the point-vortex approach used by Joyce and Montgomery and studied numerically by Lundgren and Pointin²⁷, the flow is phenomenologically modeled by an ensemble of a large number of $N$ point-vortices of circulation $\Gamma_i$ (with $\sum_{i=1}^N \Gamma_i = 0$ in our case) at positions $x_i(t)$ at time $t$. The vorticity distribution is then given by

$$ \omega_N(x, t) = \sum_{i=1}^{N} \Gamma_i \delta_{x_i(t)}(x). \quad (6) $$

The point-vortices are advected under the influence of the velocity field induced by all the other point-vortices, which conserves by construction the total circulation. Furthermore the so-defined system is Hamiltonian, conserving in addition energy. We focus on the case of zero circulation. The limit of infinitely many point vortices may be compared with the geometric formulation of Arnold²⁸ where the configuration
space for the Euler equations is the space of all volume preserving diffeomorphisms.

Joyce and Montgomery postulated that the most likely point-vortex configuration arising from the Hamiltonian system are microstates, which maximize the Gibbs-Boltzmann entropy $S(\omega) = \int_{\Omega} -\omega \log \omega \, dx$. Therefore, given an energy level $E$, the most likely vorticity microstate can be obtained by applying a Lagrangian multiplier method for the constrained optimization problem (maximizing entropy, subject to given energy level). This procedure yields a non-trivial relation between the vorticity and the streamfunction $\psi$,

$$ \omega = \beta \sinh(\lambda\psi), \quad (7) $$

for some $\beta$ and $\lambda$ positive ($\beta\lambda < 0$ is not possible in a periodic box). We note that due to the functional dependence between $\omega$ and $\psi$, the advection term in Eq. (1) vanishes identically, making these solutions steady states of the Euler equations. This equilibrium corresponds to a state where like-signed vortices have clustered together, forming two blobs, or clusters of vortices$^{21,27}$.

### C. Are Fourier and point-vortex approaches compatible?

It is interesting that the result (7), which strictly applies to the Hamiltonian system of point-vortices, is also observed to a very good approximation as a transient for the two-dimensional Navier-Stokes equation, i.e. a dissipative system$^{29}$. Even though different results are observed for certain initial conditions$^{30}$ and the data can in certain cases be equally well fitted by functions different from a sinh$^{31}$, the result (7) is widely observed, even in three-dimensional systems in the absence of vortex-stretching$^{32}$.

One explanation of this somewhat surprising result is that it corresponds to a physical space structure, consisting of two counter-rotating vortices which retain the energy at large scales, limiting the amount of dissipation. Indeed, if the third order structure functions

satisfy $\langle |u(x+r) - u(x)|^3 \rangle^{1/3} \sim C|r|^{\alpha}$ for any $\alpha > 1/3$, then by the first half of Onsager's conjecture shown by Constantin, E and Titi$^{33}$ (and also previous works by Eyink$^{34}$), total viscous dissipation goes to 0 in the inviscid limit. This is the case for solution of the Euler equations in 2D with smooth initial condition as the regularity is known to be preserved by the theorem of Beale, Kato and Majda$^{35}$, so that for small enough $\nu$, the dynamics of large-scale structures is practically non-dissipative.

Therefore, assuming that we may compare the $k_{max} \to \infty$ limit of the Galerkin-truncated system with the $\nu \to 0$ limit of the non-truncated Navier-Stokes equations, it is then plausible to postulate that the large-scale structure is stable in the truncated system, as long as this truncation is applied at large-enough wavenumber. If this assumption is valid, and a large-scale stable structure is present in the flow, ergodicity is most certainly lost since the large-structure will impede the system to equally sample all possible positions in phase-space.

Note that the limit $k_{max} \to \infty$ is discussed in [14]. It is argued there that, if the vorticity distribution is averaged over small distances, the relaxation of systems with infinite $k_{max}$, and with large finite $k_{max}$, should be equivalent. It is also suggested that states in which the vorticity field is dominated by a few large eddies, cannot be reached from an initial state with energy at large scales, but sufficiently smaller than the domain size, for $k_{max}$ tending to infinity. Indeed enstrophy equipartition implies that for large $k_{max}$ entrophy is dominated by contributions around $k_{max}$.

This section contains a number of unresolved issues. First, would a large-scale structure, formed in a truncated system, be regular enough? what is the corresponding meaning of regularity for truncated solutions? And how to qualify the notions "large-enough" $k_{max}$, or small enough $\nu$? In order to advance on the understanding of these notions, we will carry out numerical experiments. A previous numerical study of the system was carried out by Venaille et al.$^{12}$ and we refer to their paper for results on
a more general model of long-range interaction and additional explanations.

### III. NUMERICAL EXPERIMENTS OF TRUNCATED 2D EULER DYNAMICS

We carry out pseudo-spectral numerical simulations of the Galerkin truncated Euler equations. We anticipate that the results are not fully described by either of the two statistical methods, rather, for the same flow, one part can be described by point-vortex results and the rest by a Fourier-equilibrium, satisfying Eq. (5). We then introduce a physical-space procedure to disentangle the two co-existing equilibrium flows and give analytic evidence for the shape of the energy spectrum of the condensate and the PDF of its vorticity.

#### A. Numerical simulations

We perform numerical experiments for the Galerkin truncated Euler equations (4). Numerical simulations are performed with the parallel pseudospectral code GHOST³⁶ on a 2π-periodic square box, so that the largest wavelength corresponds to the wavenumber $k_{min} = 1$. The initial condition for the velocity field is given in Fourier space by a Gaussian shaped spectrum distribution

$$E(k) = C e^{-(k-k_0)^2/2\sigma^2}, \quad (8)$$

for some real numbers C and σ to be chosen to determine the shape of the initial spectrum.

The system we consider has two important control parameters. The first one is the spatial resolution. We will present results for $k_{max}/k_{min} = 169$, corresponding to a resolution of 512² grid-points, since the 2/3-method is used to dealiase.

The other parameter defining the final state is the ratio of enstrophy to energy, which can be expressed as a characteristic wavenumber

$$k_u = \sqrt{\frac{W}{E}}. \quad (9)$$

The minimum value of this wavenumber is $k_u = 1$, corresponding to the case where the energy spectrum is concentrated on the first wavenumber. The largest value of $k_u$ is analogously obtained when all energy is concentrated on $k_{max}$, in which case $k_u/k_{max} = 1$. In Eq. (8), we set $k_0 = 4$, $\sigma = 2$ and C is chosen so that $\int_1^{k_{max}} E(k) dk \equiv E = 1$. All Fourier-phases are chosen randomly so that the initial vorticity-field is in general structureless.

In Fig. 1 we show steady-state spectra obtained for simulations for initial conditions with typical wavenumber $k_u$ in the range $k_u \in [10, 100]$. We observe that the long-time shape is in agreement with the theoretical predictions at large wavenumbers for all cases. For small wavenumbers, discrepancies are observed. Whereas for the case $k_u = 100$, this discrepancy might be attributed to the discrete lattice effect associated with the limited number of Fourier-modes in the lowest wavenumber shells, for the cases with the smaller values of $k_u$ this discrepancy becomes more important. In particular for the case $k_u = 10$, it is difficult to attribute the difference between prediction and observation to non-converged statistics.

These observations confirm previous results¹²,¹⁶,¹⁸,¹⁹: the large wavenumbers are well predicted by Kraichnan's theory, but the deviation is clearly noticeable in the small wavenumbers, in particular in the case where $k_u/k_{max}$ is small. Indeed, it seems that with enough separation between the box size and the truncation scale, a robust and persistent condensate can form and that this structure leads to violation of ergodicity in Fourier-modes. In the next section we will focus on the case where $k_u = 10$ (Figure 1(a)), where the deviation from Eq. (5) is most significant.

#### B. Identification of the condensate

Visualization of the vorticity field in 2D provides a good hint to the underlying reason for the deviation of numerical simulations from the statistical predictions. In Fig. 2(a) we observe that a well-defined dipole is present in the vor-
<figure><img src="image_3.png" /><figcaption>FIG. 1. Equilibrium energy spectrum for different values of $k_u$ with Gaussian initial fields.</figcaption></figure>

ticity field. This dipole is persistent and observed at all time-instant s during the statistically steady-state of this simulation. We therefore hypothesize that the flow can be decomposed into a condensate and an incoherent part. This leads to the questions of how to define this decomposition and whether the results of statistical mechanics discussed in sections IIA and IIB may be relevant to the decomposition.

We first decompose the flow and extract the condensate. Since the dipole is persistent, a first attempt could consist in time-averaging the flow to identify the condensate. Since the position and orientation of the dipole is not necessarily stationary, this would necessitate a Lagrangian averaging, introducing the need to determine the position of the dipole a-priori. This is not the way we proceed. Alternatively it is possible to isolate large scale features of the solution through a low-pass filter, but it is not immediately clear how this filter should be chosen and whether this choice artificially determines the statistics of the remaining small scales. A more adapted way would be coherent-vortex

extraction$^{3738}$, which still relies on a filtering threshold and the shape and properties of the wavelets.

In the present work, we use a more physical way, using the assumption that the condensate is close to a steady-solution of the Euler equations. If we assume that the condensates arise from a maximum entropy configuration as discussed in Sec. IIB, we can further assume a functional relation between the vorticity and stream function, without specifying the shape of this function. We therefore attempt the following decomposition: Given a proposed function $f$ for the functional relationship $\omega = f(\psi)$, we define the condensate-vorticity as $\bar{\omega} = f(\psi)$. and an oscillation part $\omega' = \omega - \bar{\omega}$:

$$ \bar{\omega} = f(-\Delta^{-1}\omega), \quad (10) $$

$$ \omega' = \omega - f(-\Delta^{-1}\omega). \quad (11) $$

We stress that we do not prescribe any particular function $f$. However our assumption is that this function is constant in time and that the temporal fluctuations of $\omega'$ are of zero average. We obtain then our estimate of $f$ by assessing
from the data

$$ \langle f(\psi) \rangle = \langle \Delta\psi \rangle. \qquad (12) $$

We apply this method to the data obtained from our numerical experiment shown in figure 2(a). In Fig. 2(b) we show a scatter-plot of the data, i.e., every black dot corresponds to a data point $(\psi(x, y, t), \omega(x, y, t))$. In this representation, the information on the spatial dependence in $(x, y)$ is lost. Instead, only the distribution of the $\omega$-values along level sets of $\psi$ is represented. Averaging $\omega$ conditional on the value of $\psi$ for a sufficiently long time yields the red solid curve, which is our estimate of $f$. We note that for this particular realization, the range of the stream function is wider in the negative values. This could be due to the initial condition, or, more plausibly, due to insufficiently long simulation time. We also note that, although the Joyce-Montgomery statistical theory predicts a sinh-relation between the vorticity and the stream function, to the author's knowledge, the range of the positive and negative branches remains unknown. In light of the estimates in Sec. IV, it is not excluded that the positive and negative vortex blobs forming the condensate have different radii, resulting in an uneven range for the stream function.

The fitted function $f$ now in turn provides a self-consistently determined, and physically motivated low-pass filter by defining the condensate $\bar{\omega}(x, y, t)$, at every point in space and time through the relation (10). The incoherent part is the remaining part of the flow, $\omega'(x, y, t) = \omega(x, y, t) - \bar{\omega}(x, y, t)$. The resulting condensate and incoherent fields are shown in Fig. 3(a,b). We show that the condensate is a well-defined smooth dipole-structure. Furthermore, the noise-part seems to be completely structureless. In Fig. 3(b) we show the energy spectra associated with the two contributions. The incoherent part, containing less energy, is well described by a Kraichnan equilibrium Eq. (5), with a close to $k^{-1}$ scaling over most of the wavenumber range, associated with enstrophy equipartition. The noise is also homogeneous; the distribution of the fluctuating vorticity $\omega'$, as shown in Fig. 4(b), is close to

Gaussian, indicative of a lack of coherent structures. On the other hand, the spectrum of the condensate $E_0(k)$ in Fig. 3(b) falls off rapidly close to exponentially. Indeed, the energy spectrum associated with the condensate is well approximated by

$$ E_0(k) \sim \exp(-ck), \qquad (13) $$

with $c = O(1)$ (see Sec. IV for an explanation of the origin of this type of spectrum associated with the sinh-Poisson condensate). This part of the flow is associated with large-scale coherent structures whose non-Gaussian distribution (see Fig. 4(a)) follows closely the approximations for the sinh-Poisson condensate derived in Sec. IV and the corresponding $\bar{\omega}^{-3/2}$ scaling analytically derived in Sec. IV B. We note that in Fig. 3(b), the condensate spectrum deviates somewhat from the sinh-Poisson spectrum. Indeed, the condensate isolated from the realization does not exactly satisfy the sinh-Poisson relation, which can lead to small differences between the theoretical spectrum and the analytical estimate. It remains, however, that the spectrum decays exponentially, indicative of the smoothness of the large-scale dipole structure.

It is interesting to note the similarities and differences of the system compared to forced two-dimensional turbulence where a condensate forms. The incoherent part is in these flows also characterized by a $k^{-1}$ spectrum$^3$ (also observed in the non-forced case$^{39}$). The spectrum associated with the condensate is generally observed to exhibit a scaling proportional to $k^{-3}$ (see e.g. Refs. [3, 40, and 41]), in contrast to the exponential spectrum of the condensate in the present case. The difference in these latter cases is that there is a continuous flux of energy towards the condensate, which is absent in the present system.

The decomposition carried out shows that we can describe the condensate part of the flow by a sinh relation between stream-function and (filtered) vorticity. It is therefore in the point-vortex system, in decaying 2D Navier-Stokes turbulence$^{29}$, and, on average, in truncated Euler at high resolution for large values of $E/W$ that we observe this phenomenology. Simulta-
<figure><img src="image_4.png" /><figcaption>FIG. 2. Results of direct numerical simulations of the Galerkin truncated 2D Euler equations. (a) A visualization of the vorticity field for the case $k_{max} = 170$, $k_u = 28.4$. (b) Scatter plot between vorticity $\omega(x, y, t)$ and stream function $\psi(x, y, t)$ for a given time-instant. The solid line is a time-average over a long-enough (approx. 26 turnover time) time-interval during the statistically steady equilibrium state, yielding a functional relation $\omega = \bar{f}(\psi)$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 3. (a) The condensate vorticity (left) decomposed from the remaining incoherent (right). (b) The time-averaged energy spectrum associated with both contributions. The predictions from Kraichnan's equilibrium theory for the incoherent part and from the point-vortex system for the sinh-Poisson condensate are also shown in full lines (see Sec IV for a closed form approximation).</figcaption></figure>

neously, the remaining, noise-like vorticity field
can be described by the Kraichnan Fourier equi-
librium.

IV. ESTIMATES FOR THE ENERGY
SPECTRUM AND PROBABILITY DENSITY
FUNCTION OF THE CONDENSATE

In experiments and simulations of two-
dimensional or thin-layer turbulence the en-
ergy spectrum of a flow in the presence of a
condensate was previously assessed. In these

studies³⁻⁵,⁴²,⁴³ the condensate was argued to be
associated with an energy distribution propor-
tional to $k^{-3}$, in contrast to the observations
in the present investigation. We will illustrate
now that in the present case, in the unforced
Euler equations, when the condensate is well
described by a hyperbolic sine relation, the en-
ergy spectrum should be of exponential shape.

The sinh relation requires solutions of the
nonlinear Boltzmann-Poisson equation $-\Delta\psi =
\sinh(\Lambda\psi)$. However, for $\Lambda > 0$, the typical max-
imum principle and energy estimates cannot be
applied to guarantee uniqueness of the solu-
<figure><img src="image_6.png" /><figcaption>FIG. 4. The distribution of the vorticity field compared against theoretical predictions. (a) Distribution of the vorticity condensate $\bar{\omega}$ and distribution of a sinh-Poisson condensate obtained from Eq. 18. The probability densities follow the predicted $\bar{\omega}^{-3/2}$ scaling predicted in Sec. IV B. (b) Distribution of the incoherent $\omega'$ showing a Gaussian distribution.</figcaption></figure>

tion. Motivated by numerical observations of vortex pairs, if we assume that $\psi$ is axisymmetric around its local extrema, we write a model ODE satisfying the sinh-Poisson relation in a small circular domain,

$$ \partial_r^2 \psi + \frac{1}{r} \partial_r \psi + \exp(\psi) = 0. \quad (14) $$

There exist families of solutions of this ODE with general form

$$ \psi(r) = \log \left( \frac{2c_1^2 c_2 r^{c_1-2}}{(c_2 + r^{c_1})^2} \right), \quad (15) $$

where $c_1, c_2$ are arbitrary positive constants.

These solutions are generally not regular, implying a singularity at the vortex core. We argue that these singular cases should be ruled out as limiting states of the unforced Euler equations, as the extrema of the vorticity field are preserved in time. If the condensate is thought of as the infinite-time coarse-grained limit of the vorticity field, then the range of the vorticity values will shrink, ruling out condensate states with singular point vortices. In equation (15), taking $c_1 = 2$ reveals a possible existence of a regular stream function of the form

$$ \psi(r) = -2 \log (1 + (ar)^2) + \log(8a^2) \quad (16) $$

to leading order as $r \to 0$, for $a = c_2^{-1/2}$, with

vorticity of the form

$$ \omega(r) = 8a^2 (1 + (ar)^2)^{-2}. \quad (17) $$

The parameter $a$ essentially fixes the radius of the vortex blobs.

A. Estimate of the energy spectrum

We use Eq. (17) as an ansatz to propose a sinh-Poisson approximation for the stream function:

$$ \psi_B(x, y) = \log \left( \frac{1 + 2a^2 (2 - \cos(x) - \cos(y))}{1 + 2a^2 (2 + \cos(x) + \cos(y))} \right), \quad (18) $$

which essentially consists of two vortex blobs of opposite signs at $(0, 0)$ and $(\pi, \pi)$ respectively (see figure 5(a)). The idea being that $\cos(x) + \cos(y) \approx 2 - (x^2 + y^2)/2$ near $(0, 0)$ so that $2 - (\cos(x) + \cos(y))$ approximates $r^2/2$ near $(0, 0)$ while being strictly positive elsewhere. Similarly, $2 + (\cos(x) + \cos(y))$ approximates $r^2/2$ near $(\pi, \pi)$ and is strictly positive elsewhere. Expression (18) consist therefore of compositions of the logarithm in Eq. (16), with trigonometric functions bounded away from zero. Hence $\psi_B$ is a periodic real-analytic function on the torus for any $a$, with $\pm(1 + (ar)^2)^{-2}$ behaviour near $(0, 0)$ and $(\pi, \pi)$
<figure><img src="image_7.png" /><figcaption>FIG. 5. sinh-Poisson approximation (18). (a) Visualization of the vorticity field for $a^2 = 5$. (b) Scatter plots of $\psi_B$ vs $\omega_B$ for $a^2 = 5, 10$ and $15$ reveal a close agreement with the hyperbolic sine relation. (c) The energy spectrum of the sinh-Poisson approximation shows a clear exponential decay ($\approx \exp(-ck)$) with rate of decay $c = 0.88, 0.63$ and $0.51$ respectively.</figcaption></figure>

and smooth transition between these two vortex cores. It can be shown that for periodic real analytic functions, the coefficients of the Fourier series have at least exponential decay$^{44,45}$ as $k \to \infty$, with the rate-of-decay depending on the distance of the closest complex pole to the real axis. Therefore, in the absence of singular point-vortices, our ansatz (18) shows that there exists condensate states consisting of two counter-rotating vortices approximating the sinh-Poisson vorticity-stream relation and whose energy spectrum decays (at least) exponentially. In fact, we expect this to hold for more general functions $f$, since the regularity of the condensate relies only on the boundedness of the vorticity (maximum condensate vorticity should be less than the maximum initial vorticity) and the smoothness of $f$.

We check numerically that the $\psi_B-\omega_B$ relation approximates well a hyperbolic sine in figure 5(b) and the energy spectrum has exponential decay in frequency space as shown in figure 5(c). We stress that these results do not constitute a mathematical proof, but rather a plausible explanation that, if the condensate is described by a sinh relation, its spectrum should be close to exponential. We also note the discrepancy of these results with existing findings on equilibrium statistics of the forced two-dimensional Navier-Stokes or similar dissipative systems$^3$ where a $k^{-3}$ spectrum is observed, possibly due to the presence of forcing.

## B. Analytical estimate of the probability density function

Eq. (17) also allows us to determine the shape of the PDF of the vorticity. Indeed, considering a single axisymmetric vorticity distribution of form Eq. (17), in a circular domain of surface $S_{tot}$. The probability to find a value of the vorticity between $\omega$ and $\omega + \delta\omega$ is proportional to

$$ \delta\omega p(\omega) = \delta S(\omega)/S_{tot}. \quad (19) $$

where $\delta S(\omega)$ is the part of the surface where the vorticity takes a value between $\omega$ and $\omega + \delta\omega$. We have therefore that

$$ p(\omega) \sim \frac{dS(\omega)}{d\omega}. \quad (20) $$

Eq. (17) can be inverted to express $r$ as a function of $\omega$,

$$ r = a^{-1} \left( \sqrt{\frac{8a^2}{\omega}} - 1 \right)^{1/2} . \quad (21) $$

Since in our axisymmetric approximation $S(r) = \pi r^2$ this allows to compute $dS(\omega)/d\omega$

$$ \frac{dS(\omega)}{d\omega} = \frac{\partial S}{\partial r} \frac{\partial r}{\partial \omega} = - \frac{\sqrt{2\pi}}{a} \omega^{-3/2}. \quad (22) $$
We find therefore that

$$p(\omega) \sim \omega^{-3/2}, \qquad (23)$$

with a normalization factor which depends on the domain size. Furthermore, $p(\omega)$ is bounded by the values $|\omega| \le 8a^2$. This expression is also shown in Fig. 4(a).

## V. PHENOMENOLOGY OF THE COEXISTING EQUILIBRIUM FLOWS

We describe here the observed phenomenology from the point of view of energy spectra and make a hypothesis on the dynamics of the Galerkin truncated Euler equations in the limit of infinite truncation wavenumber. The observations from the numerical experiments can be summarized as follows:

1. For a given conserved total energy $E$ and total enstrophy $W$, the long-time solution approaches a steady energy distribution. For large enough $k_{max}$, the solution admits a splitting into a condensate part with total energy and enstrophy $E_0, W_0$ and an incoherent part with total energy and enstrophy $E_1, W_1$.

2. The energy spectrum of the condensate is a truncation of some spectrum $g(k)$ with finite second moment: $\int_1^\infty k^2 g(k) dk < \infty$. The exponential spectrum observed for the condensate in our numerical simulation satisfies this requirement.

3. The incoherent part satisfies an enstrophy equidistribution.

Assuming these observations hold in the limit of $k_{max} \to \infty$, we see that the enstrophy spectrum for the incoherent part is

$$W_1(k) = \frac{2W_1}{k_{\max}^2} k \qquad (24)$$

with an associated energy spectrum

$$E_1(k) = \frac{2W_1}{k_{\max}^2} k^{-1}. \qquad (25)$$

The incoherent energy is then given by

$$E_1 = 2W_1 \frac{\ln(k_{\max}) - 1}{k_{\max}^2} \le 2W \frac{\ln(k_{\max}) - 1}{k_{\max}^2} \qquad (26)$$

which tends to 0 as $k_{max}$ tends to infinity. Therefore, the conservation (or boundedness) of total enstrophy implies that in the limit of $k_{max} \to \infty$, all the energy is contained in the condensate.

This is consistent with the picture$^{46,47}$, where solutions of the full Euler equations in 2D approach a stable asymptotic state through the effect of mixing. The asymptotic steady state should have the same energy as the initial state but part of the enstrophy will be lost to mixing so that the vorticity only converges in an averaged coarse-grained sense.

## VI. CONCLUSION

In this paper, we investigated the observation that in certain Galerkin truncated pseudospectral simulations of the Euler equations, the solution disagrees with the classical predictions from Fourier statistical mechanics. The cause of this discrepancy is the tendency of 2D turbulence to form large scale structures. The formation of these structures is well observed in the viscous system and can be predicted using the statistical mechanics applied to the inviscid point-vortex system. It seems that for the truncated equations, with large enough scale separation between the flow's energetic scales and the truncation scale, the system approximates well the full non-truncated system, allowing a condensate to form.

From our simulations, it seems that the mean flow created by the condensate does not change statistical properties of the superposed Kraichnan equilibrium state, which, with sufficient scale separation, is not able to perturb the condensate. Thereby, both parts of the flow can co-exist.

We note that our procedure to extract the condensate is still fundamentally a filtering operation where the filter is provided by the shape
of $f$. Indeed, linearizing the operator $f \circ -\Delta^{-1}$:
$$ \omega \mapsto f(-\Delta^{-1}\omega) = \bar{\omega} $$
we see that modes with high wavenumbers $k$ are damped by a factor of $\max|f'|k^{-2}$. On the other hand, lower wavenumbers can be amplified, however, assuming good fit of the vorticity-stream data at lower wavenumbers by the function $f$, large-scale structures are mostly unaffected by this filtering. The filter used in the present system differs from scale based filters in that no shape or threshold is prescribed, it is uniquely based on the assumptions that there exists a functional relation $f$ and that a large-scale condensate has already formed. In fact, we do not expect this procedure to work without either of these two assumptions.

We believe that the main cause for difference between 2D and 3D Galerkin truncated Euler equations is the vanishing energy transfer at large wavenumbers for the 2D case as opposed to the 3D case. This means that the amount of energy re-injected to the large scales by the Galerkin system becomes limited as $k_{max} \to \infty$ and so the incoherent energy can be controlled. From the point of view of energy spectra, the conservation of both energy and enstrophy in 2D controls a characteristic wavenumber $\sqrt{\overline{W}/E}$, preventing its escape to infinity in frequency space. It would therefore be interesting to investigate this problem on more complicated systems such as band-forced 2D Navier-Stokes equations, where structures are observed at scales different from the domain-size (e.g. refs$^{48-50}$) or the turbulence without vortex stretching models in 3D$^{51}$. Lastly, one interesting direction to examine is the link to mixing. Indeed, one may see the persistence of the vortex structures as some sort of “immunity to mixing”: as mixing breaks down large scales structures by transferring them to small scales, a stable persistent structure at large scales must experience less mixing by the flow. In this regard, the study of long-time behaviours in 2D flows is related to the study of scalar mixing and the influence thereon of the correlation with the vorticity or stream functions$^{52}$.

**Acknowledgments.** For the purpose of Open Access, a CC-BY public copyright licence has

been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

The authors acknowledge a helpful referee who pointed us to ref. 12, where some of our results were already obtained and discussed.

All DNS simulations were carried out using the facilities of the PMCS2I (École Centrale de Lyon). The authors acknowledge partial funding from the Agence Nationale de la Recherche (ANR), project CM2E, grant ANR-20-CE46-0010-01 and financial support from the French Federation for Magnetic Fusion Studies (FR-FCM) and the Eurofusion consortium, funded by the Euratom Research and Training Programme under Grant Agreement No.633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

<footnote n="1">¹Guido Boffetta and Robert E Ecke. Two-dimensional turbulence. *Annual review of fluid mechanics*, 44(1):427–451, 2012.</footnote>

<footnote n="2">²Robert H Kraichnan and David Montgomery. Two-dimensional turbulence. *Reports on Progress in Physics*, 43(5):547, 1980.</footnote>

<footnote n="3">³M Chertkov, C Connaughton, I Kolokolov, and V Lebedev. Dynamics of energy condensation in two-dimensional turbulence. *Physical review letters*, 99(8):084501, 2007.</footnote>

<footnote n="4">⁴H Xia, M Shats, and Gregory Falkovich. Spectrally condensed turbulence in thin layers. *Physics of Fluids*, 21(12), 2009.</footnote>

<footnote n="5">⁵Adrian van Kan and Alexandros Alexakis. Condensates in thin-layer turbulence. *Journal of Fluid Mechanics*, 864:490–518, 2019.</footnote>

<footnote n="6">⁶Stefano Musacchio and Guido Boffetta. Condensate in quasi-two-dimensional turbulence. *Physical Review Fluids*, 4(2):022602, 2019.</footnote>

<footnote n="7">⁷Robert H Kraichnan. Inertial ranges in two-dimensional turbulence. *Physics of fluids*, 10(7):1417, 1967.</footnote>

<footnote n="8">⁸Lars Onsager. Statistical hydrodynamics. *Il Nuovo Cimento (1943-1954)*, 6(Suppl 2):279–287, 1949.</footnote>

<footnote n="9">⁹Hermann von Helmholtz. On integrals of the hydrodynamical equations, which express vortex-motion. *The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*, 33(226):485–512, 1867.</footnote>

<footnote n="10">¹⁰S Fox and PA Davidson. The competition between quadratic and integral invariants in inviscid truncated two-dimensional and quasigeostrophic shallow-water turbulence. *Physics of Fluids*, 21(12), 2009.</footnote>

<footnote n="11">¹¹Klas Modin and Milo Viviani. Canonical scale separa-</footnote>
tion in two-dimensional incompressible hydrodynamics. *Journal of Fluid Mechanics*, 943:A36, 2022.

¹²A Venaille, Thierry Dauxois, and Stefano Ruffo. Violent relaxation in two-dimensional flows with varying interaction range. *Physical Review E*, 92(1):011001, 2015.

¹³T.D. Lee. On some statistical properties of hydrodynamical and magnetohydrodynamical fields. *Q. Appl. Math*, 10:69, 1952.

¹⁴Robert H Kraichnan. Remarks on turbulence theory. *Advances in Mathematics*, 16(3):305–331, 1975.

¹⁵Adrian van Kan, Alexandros Alexakis, and Marc Brachet. Geometric microcanonical theory of two-dimensional truncated euler flows. *Philosophical Transactions of the Royal Society A*, 380(2226):20210049, 2022.

¹⁶Douglas G. Fox and Steven A. Orszag. Inviscid dynamics of two-dimensional turbulence. *The Physics of Fluids*, 16(2):169–171, February 1973.

¹⁷C. E. Seyler, Jr., Yehuda Salu, David Montgomery, and Georg Knorr. Two-dimensional turbulence in inviscid fluids or guiding center plasmas. *The Physics of Fluids*, 18(7):803–813, July 1975.

¹⁸Mahendra K. Verma and Soumyadeep Chatterjee. Hydrodynamic entropy and emergence of order in two-dimensional Euler turbulence. *Physical Review Fluids*, 7(11):114608, November 2022.

¹⁹S. Fox and P. A. Davidson. Integral invariants of two-dimensional and quasigeostrophic shallow-water turbulence. *Physics of Fluids*, 20(7):075111, July 2008.

²⁰Glenn Joyce and David Montgomery. Negative temperature states for the two-dimensional guiding-centre plasma. *Journal of Plasma Physics*, 10(1):107–121, August 1973.

²¹David Montgomery and Glenn Joyce. Statistical mechanics of “negative temperature” states. *The Physics of Fluids*, 17(6):1139–1145, June 1974.

²²Jonathan Miller. Statistical mechanics of euler equations in two dimensions. *Physical review letters*, 65(17):2137, 1990.

²³Jonathan Miller, Peter B Weichman, and MC Cross. Statistical mechanics, euler's equation, and jupiter's red spot. *Physical Review A*, 45(4):2328, 1992.

²⁴Raoul Robert and Joel Sommeria. Statistical equilibrium states for two-dimensional flows. *Journal of Fluid Mechanics*, 229:291–310, 1991.

²⁵Gregory L. Eyink and Katepalli R. Sreenivasan. Onsager and the theory of hydrodynamic turbulence. *Reviews of Modern Physics*, 78(1):87–135, January 2006.

²⁶Freddy Bouchet and Antoine Venaille. Statistical mechanics of two-dimensional and geophysical flows. *Physics Reports*, 515(5):227–295, June 2012.

²⁷TS Lundgren and YB Pointin. Statistical mechanics of two-dimensional vortices. *Journal of statistical physics*, 17(5):323–355, 1977.

²⁸Vladimir I Arnold. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. *Annales de l'institut Fourier*, 16(1):319–361, 1966.

²⁹David Montgomery, William H Matthaeus, W Troy Stribling, Daniel Martinez, and Sean Oughton. Relaxation in two dimensions and the “sinh-poisson” equation. *Physics of Fluids A: Fluid Dynamics*, 1992.

³⁰Enrico Segre and Shigeo Kida. Late states of incompressible 2d decaying vorticity fields. *Fluid dynamics research*, 23(2):89, 1998.

³¹Rubén A Pasmanter. On long-lived vortices in 2-d viscous flows, most probable states of inviscid 2-d flows and a soliton equation. *Physics of Fluids*, 6(3):1236–1241, 1994.

³²Tong Wu, Tomos David, and Wouter JT Bos. Point-vortex statistical mechanics applied to turbulence without vortex stretching. *Journal of Statistical Mechanics: Theory and Experiment*, 2023(11):113203, 2023.

³³Peter Constantin, Weinan E, and Edriss S Titi. Onsager's conjecture on the energy conservation for solutions of euler's equation. 1994.

³⁴Gregory L Eyink. Energy dissipation without viscosity in ideal hydrodynamics i. fourier analysis and local energy transfer. *Physica D: Nonlinear Phenomena*, 78(3-4):222–240, 1994.

³⁵J Thomas Beale, Tosio Kato, and Andrew Majda. Remarks on the breakdown of smooth solutions for the 3-d euler equations. *Communications in Mathematical Physics*, 94(1):61–66, 1984.

³⁶Pablo D. Mininni, Duane Rosenberg, Raghu Reddy, and Annick Pouquet. A hybrid MPI–OpenMP scheme for scalable parallel pseudospectral computations for fluid turbulence. *Parallel Computing*, 37(6):316–326, June 2011.

³⁷Marie Farge, Kai Schneider, and Nicholas Kevlahan. Non-gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis. *Physics of Fluids*, 11(8):2187–2201, 1999.

³⁸Kai Schneider and Oleg V Vasilyev. Wavelet methods in computational fluid dynamics. *Annual review of fluid mechanics*, 42(1):473–503, 2010.

³⁹Carsten Beta, Kai Schneider, and Marie Farge. Wavelet filtering to study mixing in 2d isotropic turbulence. *Communications in Nonlinear Science and Numerical Simulation*, 8(3-4):537–545, 2003.

⁴⁰Vadim Borue. Inverse energy cascade in stationary two-dimensional homogeneous turbulence. *Physical review letters*, 72(10):1475, 1994.

⁴¹Chuong V Tran and John C Bowman. Robustness of the inverse cascade in two-dimensional turbulence. *Physical Review E*, 69(3):036303, 2004.

⁴²Chi-kwan Chan, Dhrubaditya Mitra, and Axel Brandenburg. Dynamics of saturated energy condensation in two-dimensional turbulence. *Physical Review E—Statistical, Nonlinear, and Soft Matter Physics*, 85(3):036315, 2012.

⁴³Lei Fang and Nicholas T Ouellette. Spectral condensa-
tion in laboratory two-dimensional turbulence. *Physical Review Fluids*, 6(10):104605, 2021.

<sup>44</sup>L Grafakos. *Classical Fourier Analysis*. Springer, 2008.

<sup>45</sup>Vladimir I Arnold. *Geometrical methods in the theory of ordinary differential equations*, volume 250. Springer Science & Business Media, 2012.

<sup>46</sup>Michele Dolce and Theodore D Drivas. On maximally mixed equilibria of two-dimensional perfect fluids. *Archive for Rational Mechanics and Analysis*, 246(2):735-770, 2022.

<sup>47</sup>Alexander Shnirelman. On the long time behavior of fluid flows. *Procedia IUTAM*, 7:151-160, 2013.

<sup>48</sup>Javier Jiménez and Alan Guegan. Spontaneous generation of vortex crystals from forced two-dimensional homogeneous turbulence. *Physics of Fluids*, 19(8), 2007.

<sup>49</sup>Martin James, Wouter JT Bos, and Michael Wilczek. Turbulence and turbulent pattern formation in a minimal model for active fluids. *Physical Review Fluids*, 3(6):061101, 2018.

<sup>50</sup>Adrian van Kan, Benjamin Favier, Keith Julien, and Edgar Knobloch. Spontaneous suppression of inverse energy cascade in instability-driven 2-d turbulence. *Journal of Fluid Mechanics*, 952:R4, 2022.

<sup>51</sup>Tong Wu and Wouter J. T. Bos. Statistical mechanics of the euler equations without vortex stretching. *Journal of Fluid Mechanics*, 929:A11, 2021.

<sup>52</sup>Xi-Yuan Yin, Wesley Agoua, Tong Wu, and Wouter JT Bos. Influence of the vorticity-scalar correlation on mixing. *Physical Review Fluids*, 9(10):104502, 2024.