First mathematical proof for a key law of turbulence in fluid mechanics

So here goes: in turbulence, scaling laws are a big thing. Specifically, it's extremely useful to know how the power density in a turbulent flow is distributed between small and large whirls.

For the distribution of energy, we have the well-known Kolmogorov scaling law, that the power density scales with wavenumber to the power of -5/3. This holds approximately for all except the very largest whorls, down to the scale where viscosity begins to dominate, where there are no smaller whorls. This is known as the Kolmogorov scale. Due to a phenomenon called intermittency, we have strong reason to believe it is not exactly -5/3.

These papers are concerned with the power spectrum of scalar mixing. Imagine dropping ink into a turbulent water flow, how does it distribute between small and large whorls?

Batchelor [1] arrived at the result in 1958 that the power spectrum scales with wavenumber to the power of -1. This result has always been thought to be much more robust than the Kolmogorov scaling, since different approaches have given exactly the same answer.

The papers linked here give a proof that this relationship is exact. From a quick skim, they are extremely technical in the mathematical sense, invoking measure theory and Sobolev spaces and Itô calculus.

Certainly these results are very interesting. In some sense the big question is whether their approaches can transfer to solving outstanding questions about other phenomena in turbulence.

[1] https://www.cambridge.org/core/journals/journal-of-fluid-mec...

"To understand fluid flow, scientists must first understand turbulence."

[...]

"Since its introduction in 1959, physicists have debated the validity and scope of Batchelor’s law, which helps explain how chemical concentrations and temperature variations distribute themselves in a fluid. For example, stirring cream into coffee creates a large swirl with small swirls branching off of it and even smaller ones branching off of those. As the cream mixes, the swirls grow smaller and the level of detail changes at each scale. Batchelor’s law predicts the detail of those swirls at different scales."

My Comments: Classical physicists look at matter as being composed of atoms (neutrons, protons, electrons), String Theorists look at matter as being composed of vibrating strings, Quantum Physicists look at matter as being composed of quanta, that is, the possibility of a measurement in a given place and time. I personally have considered looking at matter as the presence or absence of force in a given space at a given scale -- but perhaps there's another possible view here -- looking at matter as solidified fluids of various different scales where as you go to smaller scales, the fluids move faster / are less viscuous... If that view turns out to have any truth to it (not saying that it does; it's all theoretical), then understanding fluid mechanics, and more specifically, turbulence, might turn out to have an additional application in helping to understand those small spaces / interactions in matter...

Related: "An Introduction To Fluid Dynamics by G.K. Batchelor, University Of Cambridge": https://www.cambridge.org/core/books/an-introduction-to-flui...