Transient Growth in Stochastic Burgers Flows

TL;DR

This paper investigates how stochastic forcing affects the maximum enstrophy growth in Burgers flows, finding that noise does not amplify extreme enstrophy beyond deterministic limits, with implications for understanding singularity formation.

Contribution

It demonstrates through numerical analysis that stochastic forcing does not increase enstrophy growth beyond deterministic bounds in Burgers flows.

Findings

Expected enstrophy scales similarly to deterministic case.

Stochastic excitation does not enhance extreme enstrophy growth.

Expected and actual enstrophy bracket the deterministic solution.

Abstract

This study considers the problem of the extreme behavior exhibited by solutions to Burgers equation subject to stochastic forcing. More specifically, we are interested in the maximum growth achieved by the "enstrophy" (the Sobolev $H^1$ seminorm of the solution) as a function of the initial enstrophy $\mathcal{E}_0$, in particular, whether in the stochastic setting this growth is different than in the deterministic case considered by Ayala \& Protas (2011). This problem is motivated by questions about the effect of noise on the possible singularity formation in hydrodynamic models. The main quantities of interest in the stochastic problem are the expected value of the enstrophy and the enstrophy of the expected value of the solution. The stochastic Burgers equation is solved numerically with a Monte Carlo sampling approach. By studying solutions obtained for a range of optimal initial data and different noise magnitudes, we reveal different solution behaviors and it is demonstrated that the two quantities always bracket the enstrophy of the deterministic solution. The key finding is that the expected values of the enstrophy exhibit the same power-law dependence on the initial enstrophy $\mathcal{E}_0$as reported in the deterministic case. This indicates that the stochastic excitation does not increase the extreme enstrophy growth beyond what is already observed in the deterministic case.