Origin of power-law tails in the local viscous growth-rate distribution

Determine the physical mechanism responsible for the power-law tails observed in the probability distribution of the local viscous contribution to the decorrelator growth rate, defined as \(\lambda_\eta(\mathbf{x}) = [\nu\,\delta \mathbf{u}(\mathbf{x})\cdot \nabla^2 \delta \mathbf{u}(\mathbf{x})] / (|\delta \mathbf{u}(\mathbf{x})|^2/2)\), during the exponential-growth phase of the decorrelator in three-dimensional, forced, homogeneous and isotropic Navier–Stokes turbulence.

Background

The authors introduce spatially local growth-rate fields for the decorrelator by defining $$\lambda_S(\mathbf{x}) \equiv [\delta \mathbf{u}\cdot \mathbf{S} \cdot \delta \mathbf{u}]/(|\delta \mathbf{u}|^2/2)$$ and $$\lambda_\eta(\mathbf{x}) \equiv [\nu\,\delta \mathbf{u}\cdot \nabla^2 \delta \mathbf{u}]/(|\delta \mathbf{u}|^2/2)$$ during the exponential-growth phase. They report that the empirical distributions of these local exponents exhibit exponential tails for $\lambda_S(\mathbf{x})$ and power-law tails for $\lambda_\eta(\mathbf{x})$.

These heavy tails suggest a connection to intermittency, but the detailed mechanism that yields the observed power-law behavior in $\lambda_\eta(\mathbf{x})$ is not established in the paper. The authors explicitly defer the determination of this origin to future work, motivating a focused paper of the statistics of $\beta_S$ and $\beta_\eta$, the spatially integrated strain and viscous contributions to decorrelator growth, respectively.