  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.
