Interpretation of Two-Dimensional Turbulence Energy Spectrum in Terms of Quasi-Singularity in Some Vortex Cores

We propose a new geometrical interpretation of two-dimensional turbulence energy spectrum, supposing the presence of at least one cusplike axisymmetric coherent structure in the flow. Such a coherent structure is a scaling distribution of vorticity which presents, instead of a characteristic radius, a range of radii corresponding to all scales where the associated energy spectrum has a power law behaviour. We compute a relation between the spectral slope and the exponent of the singularity in the Euler limit (Reynolds number tending to infinity). We predict a sensitivity of the two-dimensional flow dynamics to the regularity of the initial vorticity field: if it is initially regular, i.e. smooth, the coherent structures formed will be regular presenting a flat core, and if it is initially nonregular they will have a cusplike shape with only the vortex core regularized by dissipation, which corresponds to quasi-singularities.