Short-time exponential growth of the integrated decorrelator

Prove that, at short times, the spatially integrated decorrelator \(\Phi(t)\) grows exponentially with a constant rate \(\lambda\), i.e., \(\frac{\dot{\Phi}}{\Phi} = \lambda\), in three-dimensional, forced, homogeneous and isotropic Navier–Stokes turbulence.

Background

The authors adapt decorrelators to paper chaos in turbulence and posit that, at short times, the integrated decorrelator exhibits exponential growth characterized by a Lyapunov exponent. They then confirm this behavior numerically using direct numerical simulations and shell-model calculations.

While the paper presents numerical evidence, the initial assertion is formulated as a conjecture, motivating a rigorous theoretical establishment of exponential decorrelator growth in the specified turbulent setting.