Singularities in Multifractal Turbulence—Dissipation Networks and Their Degeneration

We suggest that large-scale turbulence dissipation is concentrated along caustic networks (that appear due to vortex sheet instability in three-dimensional space), leading to an effective fractal dimension Deff = 5/3 of the network backbone and a turbulence intermittency exponent μ = 1/6. Actually, Deff < 5/3 and μ > 1/6 due to singularities on these caustic networks. It is shown (using the theory of caustic singularities) that the strongest (however, stable on the backbone) singularities lead to Deff = 4/3 (an elastic backbone) and to μ = 1/3. Thus, there is a restriction of the network fractal variability: 4/3 < Deff < 5/3, and consequently: 1/6 < μ < 1/3. Degeneration of these networks into a system of smooth vortex filaments: Deff = 1, leads to μ = 1/2. After degeneration, the strongest singularities of the dissipation field, ε, lose their power-law form, while the smoother field lnε takes it. It is shown (using the method of multifractal asymptotics) that the probability distribution of the dissipation changes its form from exponential-like to log-normal-like with this degeneration, and that the multifractal asymptote of the field lnε is related to the multifractal asymptote of the energy field. Finally, a phenomenon of acceleration of large-scale turbulent diffusion of passive scalar by the singularities is briefly discussed. All results are based on experimental data.