Non-locality and scaling of extreme events in fluid turbulence*

Intense velocity gradient fluctuations spontaneously develop in turbulent flows at very high Reynolds numbers. The intense fluctuations of vorticity are amplified via the nonlinear coupling with the rate of strain tensor, known as vortex stretching. The relation between strain and vorticity, however, is highly nonlocal. An important observation is that the averaged value of strain, conditioned on vorticity, behaves as a power law.

We investigate this nonlocal coupling with highly resolved direct numerical simulations of isotropic turbulence in periodic domains of up to 12,2883 grid points and Taylor-scale Reynolds number R_λ in the range 140–1300. Specifically, we decompose the strain-rate tensor into local and nonlocal contributions obtained through Biot-Savart integration of vorticity in a sphere of radius R. We find that vorticity is predominantly amplified by the nonlocal strain due to scales beyond a characteristic scale size, which varies as a simple power law of vorticity magnitude. The underlying dynamics preferentially align vorticity with the most extensive eigenvector of nonlocal strain. The remaining local strain aligns vorticity with the intermediate eigenvector and does not contribute significantly to amplification; instead it surprisingly attenuates intense vorticity, leading to breakdown of the observed power law and ultimately also the scale invariance of vorticity amplification.

Based on elementary fluid mechanical considerations, and on the empirical relation between strain and vorticity, we propose a phenomenological description of the intense events in turbulence. Our approach provides an excellent description of the numerical data, which captures very well the observed variations as a function of R_λ.