A Non-Unitary Conformal Field Theory Approach to Two-Dimensional Turbulence Jun Nian1Xiaoquan Yu23Jinwu Ye45

2025-04-24 0 0 333.3KB 38 页 10玖币

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A Non-Unitary Conformal Field Theory Approach to

Two-Dimensional Turbulence

Jun Nian,1Xiaoquan Yu,2,3Jinwu Ye4,5

1International Centre for Theoretical Physics Asia-Paciﬁc,

University of Chinese Academy of Sciences, 100190 Beijing, China

2Graduate School of China Academy of Engineering Physics, Beijing 100193, China

3Department of Physics, Centre for Quantum Science,

and Dodd-Walls Centre for Photonic and Quantum Technologies,

University of Otago, Dunedin, New Zealand

4Institute for Theoretical Sciences, Westlake University, Hangzhou, 310024, Zhejiang, China

5Department of Physics and Astronomy, Mississippi State University, MS, 39762, USA

Fluid turbulence is a far-from-equilibrium phenomenon and remains one of

the most challenging problems in physics. Two-dimensional, fully developed

turbulence may possess the largest possible symmetry, the conformal symme-

try. We focus on the steady-state solution of two-dimensional bounded turbu-

lent ﬂow and propose a c=0boundary logarithmic conformal ﬁeld theory

for the inverse energy cascade and another bulk conformal ﬁeld theory in the

classical limit c→ −∞ for the direct enstrophy cascade. We show that these

theories give rise to the Kraichnan-Batchelor scaling k−3and the Kolmogorov-

Kraichnan scaling k−5/3for the enstrophy and the energy cascades, respec-

tively, with the expected cascade directions, ﬂuxes, and fractal dimensions. We

also made some new predictions for future numerical simulations and experi-

ments to test.

arXiv:2210.06762v1 [hep-th] 13 Oct 2022

Introduction

Fluid turbulence happens at scales ranging from a water tab and a geographical storm to the

size of a galaxy. Although the underlying hydrodynamic equation, i.e., the Navier-Stokes equa-

tion, has been known for over a century, many fundamental questions on turbulence remain

unresolved. In the fully developed homogeneous turbulence, there is a scale range called the

inertial range, in which the statistically stationary turbulence forms a self-similar steady ﬂow.

Kolmogorov ﬁrst found that assuming a constant energy ﬂux per mass, the three-dimensional

(3d) turbulence in the inertial range has a kinetic energy spectrum obeying the scaling law

E(k)∼k−5/3(1, 2). Unlike the 3d case (3), the two-dimensional (2d) turbulence conserves both

energy and enstrophy in the inviscid limit. Due to these two conservation laws, Kraichnan pro-

posed two cascades in the inertial range for 2d turbulence (4): one with constant energy ﬂux

and kinetic energy spectrum k−5/3extending from the intermediate injection scale toward the

largest scale available; the other with constant enstrophy ﬂux and kinetic energy spectrum k−3

extending from the injection scale down to the viscous scale, known as the inverse energy cas-

cade and the direct enstrophy cascade, respectively. These scaling laws have been extensively

tested in numerical simulations (5–10), suggesting a possible underlying conformal symmetry.

Conformal ﬁeld theory (CFT) is a powerful tool for understanding (1+1)d quantum and 2d

classical critical phenomena. Since the turbulence is a driven-dissipative system, the conven-

tional unitary CFT is unsuitable for describing its two scaling laws in the inertial range. So,

ﬁnding a non-unitary CFT description for the turbulence in the inertial range is desirable. This

program was initiated by Polyakov (11,12). However, despite many numerical attempts (5–10),

none of the CFT approaches (13–18) reproduced the precise scalings (k−5/3and k−3) and the

directions of two cascades simultaneously. There is also an interesting approach using the

AdS/CFT correspondence (19, 20), but it is based on the assumption that the boundary turbu-

lence is described by a unitary CFT. The fact that turbulence is a driven-dissipative system,

therefore non-unitary, makes this approach questionable.

In this work, we resolve this long-standing problem of ﬁnding the correct non-unitary CFT

description for the 2d turbulence in the inertial range, which produces the precise scalings k−5/3

and k−3, the directions of two cascades and also the fractal dimensions in a uniﬁed picture. We

also establish some intrinsic relations between the three length scales (L, `, a) in the inertial

range in Fig. 1, which can be tested in future numerical simulations and experiments.

For the direct enstrophy cascade, we propose a new CFT description, the semiclassical W2

conformal ﬁeld theory. It is a 2d non-unitary CFT with Virasoro algebra and can be obtained

from the minimal models (p0,p) by taking the limit pﬁnite and p0→ ∞. We ﬁnd that it can pro-

vide the precise Kraichnan-Batchelor scaling E(k)∼k−3+O(1/c)for the direct enstrophy cascade

as the central charge c→ −∞. In addition, this CFT also correctly predicts a constant enstrophy

ﬂux, a vanishing energy ﬂux, and almost no fractals in the vorticity cluster boundaries. For the

inverse energy cascade, we ﬁnd that the corresponding CFT is a c=0 boundary logarithmic

CFT, which is a direct sum of the 2d CFTs, i.e., ((Q=1)-Potts model) ⊕(O(N=0) model). These

two coexisting CFTs describe the energy cascade. Using this coexisting boundary logarithmic

CFT description, we can derive the precise Kolmogorov-Kraichnan scaling k−5/3, a constant en-

ergy ﬂux, a vanishing enstrophy ﬂux, and also the two coexisting fractal dimensions 7

4and 4

3of

the vorticity cluster boundaries, fully consistent with the previous numerical results (6). Finally,

we show that the inﬁnite conserved quantities in our non-unitary CFTs on both cascades are in

the one-to-one correspondence with those in the classical Korteweg-De Vries (KdV) equation.

Conformal Field Theory Approach to 2D Turbulence

Let us ﬁrst review some aspects of the 2d turbulence. The Navier-Stokes equation (NSE) is

∂u

∂t+u· ∇u=−1

ρ∇p+ν∇2u+1

ρf,(1)

where uis the ﬂuid velocity, ρis the mass density, pis the pressure, νis the viscosity, and fis the

stirring force. The turbulence solution emerges at large Reynolds numbers 1 Re ≡ρL|u|/ν.

Let us denote the inertial range of turbulence by [ko,kc], where kc∼a−1with a small length

scale cutoﬀa, and ko∼L−1with Ldenoting the system size. On the right-hand side of the NSE,

the term ν∇2uis more relevant when close to the UV cutoﬀkc, while the external stirring force

fis important near the injection scale ki≡`−1, where the energy is fed to the system. In terms

of the vorticity ω=αβ∂βuα, the (2+1)d incompressible (∇ · u=0) NSE becomes

˙ω+αβ ∂αψ ∂β∂2ψ=ν ∂2ω+F,(2)

where ψdenotes the stream function, ω=∂2ψ,uα=αβ∂βψand F=αβ∂βfα/ρ for a constant

density ρ. We focus on the NSE of vorticity for most parts of the paper and only occasionally

use the NSE of velocity.

As discussed by T.D. Lee in (21), 2d turbulence cannot support the usual Kolmogorov direct

energy cascade of 3d turbulence. Kraichnan proposed a dual cascade picture by considering

two diﬀerent constants of the 2d turbulence, the energy transfer rate J(E)and the enstrophy

transfer rate J(H), where the enstrophy is H≡Rd2xω2. By imposing one of the two constants,

Kraichnan found that for kokki, the energy transfers at a constant rate J(E)from small

scales (large k) to large scales (small k), which is called the inverse energy cascade, while for

kikkcthe enstrophy transfers at a constant rate J(H)from large scales (small k) to small

scales (large k), which is called the direct enstrophy cascade. The energy spectrum has the

scaling E(k)∼k−5/3for the inverse energy cascade and E(k)∼k−3for the direct enstrophy

cascade (4) (see Fig. 1).

Figure 1: The scalings of 2d turbulence: the blue line has E(k)∼k−5/3, while the yellow line

has E(k)∼k−3. The arrows denote the cascade directions.

The simple scalings in the inertial range strongly suggest the existence of an eﬀective con-

formal symmetry. Polyakov’s pioneer work (11, 12) opened up the possibility of using 2d con-

formal ﬁeld theories to study 2d turbulence. The main idea is to search for a CFT that captures

far-from-equilibrium ﬂuctuations of the vorticity ﬁeld in the inertial range, in which the incom-

pressible inviscid (ν=0) NSE is satisﬁed as an operator equation, ˙ω=−αβ ∂αψ ∂β∂2ψ=0,

by identifying the ﬂuctuating stream function ψwith a primary ﬁeld in the CFT. This is based

on the assumption that the system possesses the biggest symmetry in the inertial range, i.e., the

conformal symmetry, a possible emergent symmetry through chaos. Let ψbe a primary opera-

tor with the algebraic fusion rule ψ×ψ=φ+···, where φis a primary operator with either the

highest or the lowest conformal weight depending on the boundary conditions. Consequently,

−αβ ∂αψ ∂β∂2ψhas a CFT interpretation ∼ |a|2(hφ−2hψ)L−2¯

−1−¯

L−2L2

−1φ, where L−nand ¯

L−n

are Virasoro generators (22). If this CFT expression vanishes when a→0, the incompressible

NSE as an operator equation ˙ω=0 will automatically be satisﬁed. This can happen when (i)

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ANon-UnitaryConformalFieldTheoryApproachtoTwo-DimensionalTurbulenceJunNian,1XiaoquanYu,2;3JinwuYe4;51InternationalCentreforTheoreticalPhysicsAsia-Pacic,UniversityofChineseAcademyofSciences,100190Beijing,China2GraduateSchoolofChinaAcademyofEngineeringPhysics,Beijing100193,China3DepartmentofPhysics,C...

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A Non-Unitary Conformal Field Theory Approach to Two-Dimensional Turbulence Jun Nian1Xiaoquan Yu23Jinwu Ye45

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