Transient Growth in Stochastic Burgers Flows

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.

25 pages, 12 figures; to appear in "Discrete & Continuous Dynamical Systems -- B"

References in corpus (6)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Burgers Turbulence
• Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation
• Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows
• Optimal rate of convergence for stochastic Burgers-type equations
• The non-viscous Burgers equation associated with random positions in coordinate space: a threshold for blow up behaviour

Cited by in corpus (6)

• Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows
• Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics
• Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation
• Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation
• On vortex stretching for anti-parallel axisymmetric flows
• Singularity Formation in the Deterministic and Stochastic Fractional Burgers Equation