Hydrodynamic Entropy and Emergence of Order in Two-dimensional Euler Turbulence Mahendra K. Verma1and Soumyadeep Chatterjee1y

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Hydrodynamic Entropy and Emergence of Order in

Two-dimensional Euler Turbulence

Mahendra K. Verma1, ∗and Soumyadeep Chatterjee1, †

1Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India

(Dated: November 29, 2022)

Using numerical simulations, we show that the asymptotic states of two-

dimensional (2D) Euler turbulence exhibit large-scale ﬂow structures due to nonzero

energy transfers among small wavenumber modes. These asymptotic states, which

depend on the initial conditions, are out of equilibrium, and they are diﬀerent from

the predictions of Onsager and Kraichnan. We propose “hydrodynamic entropy” to

quantify order in 2D Euler turbulence; we show that this entropy decreases with

time, even though the system is isolated with no dissipation and no contact with a

heat bath.

I. INTRODUCTION

Euler turbulence remains unsolved till date. In this paper, we address the energy ﬂux and

entropy of two-dimensional (2D) Euler turbulence. The equations of incompressible Euler

ﬂow are

∂tu+u· ∇u=−∇p;∇ · u= 0,(1)

where u, p are the velocity and pressure ﬁelds respectively [1,2]. The above system is

isolated, as it lacks external force and viscous dissipation. Consequently, the thermodynamic

entropy of an Euler ﬂow remains constant [1]. However, when we solve Euler equations with

an ordered initial condition, structurally, three-dimensional (3D) Euler turbulence becomes

more random during its evolution [3], whereas 2D Euler turbulence tends to become more

orderly [4,5]. Hence, the entropy of Euler turbulence needs a reexamination.

∗mkv@iitk.ac.in

†inspire.soumya@gmail.com

arXiv:2210.06445v2 [cond-mat.stat-mech] 28 Nov 2022

Onsager [6] modelled 2D Euler ﬂow using a collection of point vortices interacting via log-

arithmic potential. Onsager showed that for large energy, 2D Euler turbulence exhibits

“negative temperature” and a large cluster of same-circulation vortices. Recently, Gau-

thier et al. [7] observed such giant vortices in an experiment involving 2D quantum ﬂuid,

thus providing an experimental veriﬁcation of Onsager’s theory. Billam et al. [8] developed

a ﬁrst-principles realization of Onsager’s vortex model in a 2D superﬂuid. Miller [9] and

Robert [10] extended Onsager’s theory to continuum version of 2D Euler turbulence and

computed entropy for the ﬂow. Using tools of equilibrium statistical mechanics, Bouchet

and Simonnet [11], and Bouchet and Venaille [12] derived multiple stationary states, namely

a dipole and unidirectional ﬂow (shear layer), for 2D Euler turbulence. Pakter and Levin

[13] provide a contrary viewpoint and showed that 2D Euler turbulence is out of equilibrium,

and that a system of interacting vortices becomes trapped in a nonequilibrium stationary

state; these results deviate from the predictions of Onsager [6]. Refer to review articles by

Eyink and Sreenivasan [14], and Bouchet and Venaille [12] for an extensive discussion.

Lee [15] and Kraichnan [16] provide an alternative framework for Euler turbulence. They

showed that the evolution of Fourier modes of Euler equation follows Liouville’s theorem,

and that the equilibrium solutions of Euler turbulence are

E(k) = k2

β−γk2for 3D; (2)

E(k) = k

β+γk2for 2D,(3)

where βand γare constants. Here, the Fourier modes form a microcanonical ensemble. The

derivation of Eqs. (2,3) involves two competing conservation laws: kinetic energy (Rdru2/2)

and kinetic helicity (Rdr(u·ω)) in 3D, and kinetic energy and enstrophy (Rdrω2/2) in 2D,

where ωis the vorticity ﬁeld. For some combinations of energy and enstrophy, 2D Euler

turbulence yields β < 0 or “negative temperature” [4,17]. Kraichnan and Montgomery [17]

provide a detailed review of 2D Euler turbulence.

For a δ-correlated random velocity ﬁeld as an initial condition, both 2D and 3D Euler

turbulence follow the energy spectra of Eqs. (3,2) with γ≈0 [18,19]. In addition, for

an initial condition with large-scale structures, 3D Euler turbulence asymptotes to E(k) of

Eq. (2) [3]. However, E(k) of 2D Euler turbulence diﬀers from Eq. (3) for coherent velocity

ﬁeld as an initial condition. For example, for enstrophy-dominated 2D Euler turbulence,

Fox and Orszag [4] reported deviations from Eq. (3) at small wavenumbers. For parameters

where β+γk2≈0, Seyler et al. [5] observed large vortex structures, similar to those in

a discrete vortex system [20]. Dritschel et al. [21] studied the unsteady nature of 2D ﬂow

structures on a sphere. Robert and Sommeria [22], and Bouchet and Venaille [12] analyzed

such structures in the framework of equilibrium statistical mechanics.

The works of Fox and Orszag [4], Seyler et al. [5], Pakter and Levin [13], Bouchet and

Simonnet [11], Dritschel et al. [21], and Modin and Viviani [23] indicate that 2D Euler

turbulence is out of equilibrium, contrary to the assumptions of Onsager [6] and Kraichnan

[16]. Bouchet and Venaille [12] argued that even though nonequilibrium steady states of

2D Euler turbulence often break detailed balance, under weak force and zero viscosity,

they may be described by microcanonical measures and entropy functional. Bouchet and

coworkers [11,12,24], and Modin and Viviani [23] explained the structures of 2D Euler

turbulence in this framework. In this paper, we advance this theme by carefully examining

the energy transfers and energy ﬂux of 2D Euler turbulence. We show that the small

wavenumber modes exhibit nonzero energy transfers, hence break the detailed balance, which

is a stringent criterion for equilibrium. Thus, we demonstrate the nonequilibrium nature

of 2D Euler turbulence. We also quantify the order of the structures using hydrodynamic

entropy.

The outline of the paper is as follows. In Sec. II, we describe nonequilibrium nature of 2D

Euler turbulence. We propose hydrodynamic entropy in Sec. III to quantify this nature. We

conclude the paper in Sec. IV.

II. NONEQUILIBRIUM NATURE OF 2D EULER TURBULENCE

Prior to a detailed discussion on 2D Euler turbulence, we summarize the energy spectrum

and ﬂux of 3D Euler turbulence. Cichowlas et al. [3] simulated 3D Euler turbulence with

Taylor-Green vortex as an initial condition. For such simulations, in the early phase, the

energy ﬂows from large scales to small scales, and the energy ﬂux is positive. After several

eddy turnover times, the system approaches equilibrium with vanishing energy ﬂux. Refer

to Appendix A for details.

For 2D Euler turbulence, we performed pseudo-spectral simulations on a (2π)2box with a

M2grid. Here, M= 512. We dealiase the code by setting all the modes outside the sphere

of radius M/3 to zero. To conserve the total energy, we time evolve Eq. (1) using position-

extended Forest-Ruth-like (PEFRL) scheme [25,26] with time step = 10−4. We carried out

three runs with the following initial conditions:

1. Run A: The initial velocity proﬁle is taken as (sin 11xcos 11y+ηx,−cos 11xsin 11y+

ηy), where (ηx, ηy) is random noise. We take |ηx|  1 and |ηy|  1.

2. Run B: The initial nonzero velocity Fourier modes are u(1,0) = (0,1), u(0,1) = (1,0),

and u(1,1) = (−i, i).

3. Run C: The initial nonzero velocity Fourier modes are u(10,0) = (0,1), u(0,10) =

(1,0), and u(10,10) = (−i, i).

FIG. 1. For Runs A, B, and C of 2D Euler turbulence: (a,c,e) the initial states, (b,d,f) the ﬁnal

states respectively. Here we plot the velocity ﬁeld over the density plots of the vorticity ﬁeld.

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HydrodynamicEntropyandEmergenceofOrderinTwo-dimensionalEulerTurbulenceMahendraK.Verma1,andSoumyadeepChatterjee1,y1DepartmentofPhysics,IndianInstituteofTechnologyKanpur,Kanpur208016,India(Dated:November29,2022)Usingnumericalsimulations,weshowthattheasymptoticstatesoftwo-dimensional(2D)Eulerturbulenc...

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Hydrodynamic Entropy and Emergence of Order in Two-dimensional Euler Turbulence Mahendra K. Verma1and Soumyadeep Chatterjee1y

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