Breaking phenomena in incompressible fluids as a route to the Kolmogorov and Kraichnan spectra

In this talk we demonstrate that breaking phenomena in incompressible fluids can be considered as a route to the Kolmogorov spectrum [Kolmogorov] and the Kraichnan spectrum [Kraichnan] for three- and two-dimensional developed hydrodynamic turbulence respectively.
For two-dimensional turbulence we study the appearance of sharp vorticity gradients and their influence on the turbulent spectra [3,4]. We have developed the analog of the vortex line representation [5] as a transformation to the curvilinear system of coordinates moving together with the di-vorticity lines. Compressibility of this mapping can be considered as the main reason for the formation of the sharp vorticity gradients at high Reynolds numbers. In the case of strong anisotropy the sharp vorticity gradients can generate spectra which fall off as $k^{-3}$ at large k resembling the Kraichnan spectrum for the enstrophy cascade. For weak anisotropy the spectrum due to the sharp gradients coincides with the Saffman spectrum [5]: $E(k)\sim k^{-4}$. We have compared the analytical predictions with direct numerical solutions of the two-dimensional Euler equation for decaying turbulence. We observe that the di-vorticity is reaching very high values and is distributed locally in space along piecewise straight lines. Thus, indicating strong anisotropy and accordingly we find a spectrum close to the $k^{-3}$-spectrum [3,4]. In the numerical experiments [7] for the 8192 x 8192 grid points we observe the spectra with strong angular dependence which can be interpreted as a set of jets with their both weak and strong overlapping. The structure functions of second and third orders show a good correspondence to the Kraichnan direct cascade picture with the constant enstrophy flux. Powers $\zeta_n$ for higher structure functions grow weaker the linear dependence relative to $n$ demonstrating the intermittency property.
Recent numerical experiments in the framework of the Euler equations for two colliding Lamb vortex dipoles Orlandi and Co. [8] testify to favor of the collapse appearance when the vorticity becomes infinite in a finite time according to the law $(t_0-t)^{-1}$, the collapse region vanishes like $(t_0-t)^{1/2}$, and the velocity component parallel to the vorticity blows up proportionally to $(t_0-t)^{-1/2}$. During the collapse the region of the maximal vorticity represents the pancake-like structure. In this paper it is shown that all these self-similarities can be obtained from the analysis of the singularity while breaking of vortex lines. In the collapse instant the vorticity $\Omega$ gets the singularity of the Kolmogorov type: $\Omega\sim x^{-2/3}$ where $x$ coincides with the direction of the breaking [9].
Acknowledgements. This work was supported by the grant of the Government of the Russian Federation (contract No. 11.G34.31.0035 dated November 25, 2010 between the Russian Ministry of Education and Sciences, NSU, and leading scientist), by the RFBR (grant No. 12-01-00943), by the program "Fundamental problems of nonlinear dynamics" of the RAS Presidium, and by the grant No. Nsh 6170.2012.2 for state support of leading scientific schools of the RF.
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