Intermittency, stochastic Universal Multifractals and the deterministic Scaling Gyroscope Cascade model

Intermittency is a fundamental feature of turbulence and more generally of geophysics, where its ubiquity is increasingly recognized. It corresponds to the concentration of the activity of a field, e.g. the vorticity of a flow, into very small fractions of the physical space. This induces strongly non-Gaussian fluctuations over a wide range of space-time scales. Multifractality corresponds to the fact that this concentration for increasing level of activity, in fact increasing singular behaviour, is supported by fractal sets of decreasing dimensions (and increasing codimensions). This is a general outcome of the (stochastic) multiplicative cascade models and of the universal multifractals, which statistics are defined with the help of two physically meaningful parameters:

• the ‘mean codimension’ C₁ ≥ 0 measures the mean concentration of the activity (C₁ = 0 for a non-intermittent field);
• the ‘multifractality index’ α ∈ (0, 2) measures how fast increases the concentration of the activity with the activity level (α=0 correspond to the monofractal case with a unique singularity / codimension C₁, α= 2 corresponds to another exceptional case, the so-called ‘Log-normal’ model)

Multifractal analysis of various turbulence data, especially from lab experiments and atmospheric in-situ/remotely sensed data, have rather constantly yielded estimates of α ≈ 1.5 and C₁ ≈ 0.25 , although error bars are difficult to assess. However, the relation between stochastic cascades and the deterministic Navier-Stokes equations have often been brought into question. We therefore analysed in more details the relation between stochastic multiplicative cascades, namely their universality case, and the deterministic Scaling Gyroscope Cascade (SGC, [1]), whose philosophy is rather different: it is based on a parsimonious discretisation of the Fourier transform of the Bernoulli’s form of the Navier-Stokes equations:

(∂/∂t -vΔ)u(x,t)=u(x,t)∧w(x,t)-grad(α), w(x,t)=curl(u(x,t)).

The discretization of the Bernoulli’s form is performed along a dyadic tree structure in a 2D cut: each eddy of velocity uⁱ_mhas two interacting sub-eddies of velocities u²ⁱ⁻¹_m+1and u²ⁱ_m+1, where m indexes the cascade level of wave-number k_m = 2^m, i ∈ [1, 2^m] being the eddy location. This discretization preserves many symmetries, including the most important one: the non trivial ‘detailed energy conservation’, i.e. that nonlinearly transferred within the triad of a parent eddy and its two children.

We have performed numerous SGC simulations with a constant forcing at a low wave number, a number of cascade levels as high as N = 15 and a duration of 150 largest eddy turnover times. All these simulations display an extreme space-time intermittency. Their multifractal analysis confirms in a very robust manner the estimated α ≈ 1.5 , which is a very important result: it brings into question more than ever the relevance of the often used of the log-normal model, at least for hydrodynamic turbulence. We will present at the conference a similarly robust estimate of C₁ after having clarified a recently noted, unexpected sensibility to simulation details.

Keywords: intermittency; the SGC model; multifractal 

Reference:

[1]Chigirinskaya Y, Schertzer D. Cascade of scaling gyroscopes: Lie structure, universal multifractals and self-organized criticality in turbulence[M]//Stochastic Models in Geosystems. Springer, New York, NY, 1997: 57-81.