Singularities in multifractal turbulence dissipation networks and their degeneration

Abstract

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 D eff = 5/3 of the network backbone and a turbulence intermittency exponent µ = 1/6. Actually, D ef f < 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 D eff = 4/3 (an elastic backbone) and to µ = 1/3. Thus, there is a restriction of the network fractal variability: 4/3 < D eff < 5/3, and consequently: 1/6 < µ < 1/3.Degeneration of these networks into a system of smooth vortex filaments: D ef f = 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.

Intermittency of Internal Wave Shear and Turbulence Dissipation

“…Extremely large I ε and I χ measured for oceanic turbulence are in the range 3-7, Baker and Gibson (1987). These values are consistent with the third universal similarity hypothesis for turbulence of Kolmogorov (1962) and a length scale range over 3-7 decades from viscous or diffusive to buoyancy or Coriolis force domination, where the measured universal intermittency constant µ = 0.44, Gibson (1991a), is a result of singularities in multifractal turbulence dissipation networks and their degeneration, Bershadskii and Gibson (1994). The extreme intermittency of small scale internal wave shears in the ocean is a fossil turbulence remnant of the extreme intermittency of ocean turbulence, and the waves themselves may be fossil turbulence.…”

“…Exponential distributions have been predicted rather than lognormal, and claims of multifractal behavior of turbulence with no universal intermittency coefficient µ. Bershadskii and Gibson (1994) show that µ = 0.5 is a limiting value to be expected at very high Reynolds numbers Re, and that the lognormality of ε at such high Re follows from multifractal asymptotics. Turbulence dissipation at low Re is concentrated on caustic networks that appear due to vortex sheet instability in three dimensional space, leading to an effective fractal dimension D of 5/3 of the network backbone without caustic singularities and a turbulence intermittency exponent µ = 1/6.…”

“…According to the intermittency hypothesis of Kolmogorov (1962), random variables of turbulence in the ocean such as ε are highly intermittent because the equations of turbulent momentum and vorticity are highly nonlinear and the range of scales of the turbulence cascade is enormous. Universal intermittency parameters of turbulence have been investigated in terms of singularities on dissipation networks, or caustics, and their degeneration, by Bershadskii and Gibson (1994), using multifractal asymptotics. The hallmark of intermittent random variables is a self-similar nonlinear cascade over a wide range of scales.…”

Turbulence in the Ocean, Atmosphere, Galaxy, and Universe

“…49, no. 5, May 1996, 299-315 0.4-0.5 characteristic of natural flows is related to the evolution of different classes of geometrical singularities of the multifractal dissipation field networks and their degeneration as the Reynolds number of the flows becomes asymptotically large, where µ → 1/2 as shown by Bershadskii and Gibson (1994). Estimates of L O = r exp (σ 2 lnεr /µ) are extremely sensitive to the value of µ; for example, if σ 2 lnεr is 5 for r = 1 meter typical of the ocean seasonal thermocline scrambled horizontally by a storm, then the macroscale of turbulence is 100 km for µ = 0.43, which is realistic.…”

Multifractal Critical Chaos Generated by Two Interacting Particles in a Disordered Chain: Anderson Model

“…2 On the other hand, the universal Feigenbaum scenario of the chaos appearance 3 was widely discussed in literature as an alternative to the lognormal description of the turbulence intermittency. Therefore, it is interesting to find a signature of the universal critical chaos in the Anderson model as well.…”