# Folding of the frozen-in-fluid di-vorticity field in two-dimensional hydrodynamic turbulence

## TL;DR

This paper investigates the behavior of di-vorticity in 2D turbulence, revealing exponential growth and thinning near quasi-shocks, with a power-law relation indicating folding of the divergence-free vector field.

## Contribution

It demonstrates the exponential increase and thinning of di-vorticity near quasi-shocks, and establishes a power-law relation between maximum di-vorticity and thickness, indicating folding behavior.

## Key findings

- Maximum di-vorticity grows exponentially over time.
- Thickness of the maximum area decreases exponentially.
- Power-law relation between di-vorticity and thickness with exponent ~2/3.

## Abstract

The vorticity rotor field ${\bf B}=\mbox{rot}\,\mathbf{\omega}$ (di-vorticity) for freely decaying two-dimensional hydrodynamic turbulence due to a tendency to breaking is concentrated in the vicinity of the lines corresponding to the position of the vorticity quasi-shocks. The maximum value of the di-vorticity $B_{max}$ at the stage of quasi-shocks formation increases exponentially in time, while the thickness $\ell(t)$ of the maximum area in the transverse direction to the vector ${\bf B}$ decreases in time also exponentially. It is numerically shown that $B_{max} (t)$ depends on the thickness according to the power law: $B_{max}(t)\sim \ell^{-\alpha}(t)$, where the exponent $\alpha\approx 2/3$. This behavior indicates in favor of folding for the divergence-free vector field of the di-vorticity.