# When will there be a theory of multifractal turbulence? – Uriel Frisch (Observatoire de la Côte d’ Azur)

Newton Institute

About twenty years after Kolmogorov (1941) developed a theory of fully developped incompressed 3D turbulence, he thought that experimental techniques had made enough progress to test the theory, for example the power-law with exponent -5/3 predicted for the energy spectrum. The theory seemed close to working fine, with however moderately small-scale deviations from the predicted self-similarity. These took the form of intermittent bursts of activity, also seen by Batchelor and Townsend in their 1949 experiments.

Kolmogorov deemed that the 1941 theory was in need of revisiting. He and his collaborators Obukhov and Yaglom developed a number of models intended to match the data more adequately. Mandelbrot suggested that the proper explanation of turbulence required that the energy dissipation would be concentrated on a fractal with some non-integer dimension. Then, in the early 1980, Anselmet et al. performed state-of-the-art measurements of small-scale intermittency for turbulence.

Georgio Parisi and this author looked at the data of Anselmet et al. and found that they could not be explained with a single fractal dissipation, set of a prescribed dimension. They tried a multifractal description, which seemed to fit the data. It took a few years to realize that the multifractal model is the turbulence counterpart of the probabilistic theory of large deviations in finances, due to Cramer 1938. Large deviations are able to capture tiny deviations from the law of large numbers. They played a key role in the foundations of statistical mechanics (Cramer’s rate function is basically the entropy). In the first part of the lecture we shall give some highlights of the Parisi’s and Frisch’s orignal 1983 multifractal approach. The theory of multifractal turbulence was probably one of the many fields of activity of Giorgio Parisi, which convinced the Nobel Committee to grant him the 2021 Physics Nobel Prize.

Nevertheless, “multifractal turbulence” is so far, only a result of fitting experimental (or, later, numerical) data. It would be unreasonable to demand a full mathematical theory of such turbulence. We do not even know if the solution to the Euler/Navier-Stokes equations in 3D, with nice initial data, do remain so for a finite or infinite time. Hence, it will take some time before we can derive multifractality from the basic hydrodynamicalequations. In the mean time, it would be nice to derive multifractality from the Burgers’ equation. The latter is not just a poor-man’s look-alike of the Euler/Navier-Stokes equation, but is also important in condensed matter physics, cosmology and plays an important role in Parisi’s key contributions. In 1992, Sinai provedrigorously that if the initial velocity is a Brownian motion function, then the Lagrange map is a Devil’s staircase with fractal dimension 1/2 (She et al (1992) first obtained this result by numerical simulations). This looks like a monofractal solution, at least for the Lagrangian map. However, as stressed by Khanin (2021), we do not know how the various velocity structure functions scale.

Working with the Burgers’ equation, Frisch, Pandit and Roy (2020) constructed genuine multifractal solutions. However, the multifractal behaviour is seen not at the small scales but at the largest ones. Afew words on how these solutions are constructed. Instead of working with fractional Brownian motion of a single Hearst (scaling) exponent h, one decomposes the initial velocity into octave-band Fourier datawith different Hearst exponents. As will be explained in the second part of my lecture, I shall present briefy the theory and the simulations.