We study the statistical properties of the vorticity field in two-dimensional turbulence. The field is described in terms of the infinite Lundgren–Monin–Novikov (LMN) chain of equations for multi-point probability density functions (pdf’s) of vorticity. We perform a Lie group analysis of the first equation in this chain using the direct method based on the canonical Lie-Bäcklund transformations devised for integro-differential equations. We analytically show that the conformal group is broken for the first LMN equation i.e. for the 1-point pdf at least for the inviscid case but the equation is still conformally invariant on the associated characteristic with zero-vorticity. Then, we demonstrate that this characteristic is conformally transformed. We find this outcome coincides with the numerical results about the conformal invariance of the statistics of zero-vorticity isolines, see e.g. Falkovich (2007 Russian Math. Surv. 63 497–510). The conformal symmetry can be understood as a ‘local scaling’ and its traces in two-dimensional turbulence were already discussed in the literature, i.e. it was conjectured more than twenty years ago in Polyakov (1993 Nucl. Phys. B 396 367–85) and clearly validated experimentally in Bernard et al (2006 Nat. Phys. 2 124–8).
It was clearly validated experimentally in Bernard et al (2006 Nat. Phys. 2 124–8) that the zero-vorticity isolines in 2D turbulence belong to the class of conformal invariant (Schram–Löwner evolution) curves with . The diffusion coefficient classifies the conformally invariant random curves. With this motivation, we performed a Lie group analysis in Grebenev et al (2017 Phys. A: Math. Theor. 50 5502–44) of the first of the inviscid Lundgren–Monin–Novikov (LMN) equations for 2D vorticity fields. This equation describes the evolution of the 1-point probability density function (PDF) . We proved that the conformal group (CG) is not admitted by the 1-point PDF equation itself, however it is permitted under the condition . The main focus of the present work is to prove explicitly the CG invariance of the zero-vorticity Lagrangian path, which is the characteristic of the inviscid LMN hierarchy truncated to the first equation. We also show the CG invariance of the separation and coincidence properties of the PDFs. Besides the derivation of the CG invariance of the zero-vorticity Lagrangian path, we demonstrate that the infinitesimal operator admitted by the characteristic equations forms a Lie algebra which is the Witt algebra, whose central extension represents exactly the Virasoro algebra. The numerical value of the central charge c occurring here could not be calculated. Other mathematical tools need to be involved to link this analytical study with the previous analyses by Bernard et al, who report the value c = 0, which corresponds to for the .
We consider a Leith model of turbulence (Leith C 1967 Phys. Fluids 10 1409) in which the energy spectrum obeys a nonlinear diffusion equation. We analytically prove the existence of a self-similar solution with a power-law asymptotic on the low-wavenumber end and a sharp boundary on the high-wavenumber end, which propagates to infinite wavenumbers in a finite-time t*. We prove that this solution has a power-law asymptotic with an anomalous exponent x*, which is less than the Kolmogorov value, x* > 5/3. This is a result that was previously discovered by numerical simulations in Connaughton and Nazarenko (2004 Phys. Rev. Lett. 92 044501). We also prove the convergence to this self-similar solution of the spectrum evolving from an arbitrary finitely supported initial data as t → t*.
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