# Why is there no UV catastrophe (divergence) in turbulence?

I have just read that as the Reynolds number is increased, the separation of macroscopic and microscopic scales increases and that this also means that there is no UV catastrophy (or equivalently UV devergence?) in turbulence.

I do not understand what this means, for example does the increased separation of the macro and microscales just mean that the spectrum is broadening? And what exactly would an UV catastrophe mean in the context of turbulence and how can I see (technically and mathematically) that it does in fact not exist? I have only a rough intuition what it could mean when considering the turbulence problem from a QFT approach, namely that it from this point of view the infinit Reynolds number limit should be renormalizable (?)...

Any comments that would help me to understand what the sentence in the first paragraph exactly means would be appreciated.

It would help to know the context in which you read the phrase you have a problem understanding.

Regarding the scaling vs Reynolds problem, this part is pretty straighforward. The classical statistical theory of turbulence says that the energy cascades from the low wave number modes (typical scale L) to the high wave numbers modes (typical scale µ) where it is dissipated by viscosity.

In a statistical steady state the energy produced at L must be dissipated at µ. Now it is established that L/µ = Re^(3/4). So one clearly sees that as Reynolds increases, µ decreases and is 0 in the infinite Re limit. So this relation can indeed be described (imho misleadingly) by "separation of macro and micro increases".

From the dynamical point of view at infinite Re, the flow becomes inviscid and the Navier Stokes equations are substituted by the Euler equation.

The "UV catastrophy" is more mysterious. Formally one could say that as µ goes to 0, the energy density of the dissipative domain goes to infinity (UV divergence ?). However to that 2 remarks :

1) Navier Stokes is continuous but when µ arrives at molecular scales, the flow is no more continuous and Navier Stokes breaks down.

2) It is true that the spectral energy density increases at high wave numbers. But the nature shows us that there is actually no divergence - you will never see a very small vortex spinning infinitely fast.

The fluid equivalent to the Ultraviolet Catastrophe is the transition to turbulence. Just as Maxwell's Equations break down at high frequencies and are replaced by a stochastic model (Quantum Mechanics), the Navier-Stokes Equations break down at high Reynolds Number and are replaced by a stochastic model (Kolmogorov Cascade). Reference: Reid, J. (2019) The Fluid Catastrophe, Cambridge Scholars Publishing, Newcastle upon Tyne.