Sharp vorticity gradients in two-dimensional hydrodynamic turbulence

Abstract

The appearance of sharp vorticity gradients in two-dimensional hydrodynamic turbulence and their influence on the turbulent spectra is considered. We have developed the analog of the vortex line representation 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 vorticity discontinuities at high Reynolds numbers. For two-dimensional turbulence in the case of strong anisotropy the vorticity discontinuities can generate spectra with the fall-off at large $k$ proportional to $k^{-3}$ resembling the Kraichnan spectrum for the enstrophy cascade. For turbulence with weak anisotropy the $k$ dependence of the spectrum due to discontinuities coincides with that of the Saffman spectrum: $k^{-4}$. 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 found a spectrum close to the $k^{-3}$-spectrum.