Instanton filtering for the stochastic Burgers equation (Vol. 44 No. 3)

Comparison of the filtered velocity field (new direct method, top) and the instanton field (Chernykh-Stepanov method, bottom) as a space-time contour plot

Extreme events in stochastic nonlinear systems play an essential role in nature. Characterizing their likelihood is a fundamental albeit challenging problem since the tails of the underlying probability distributions are usually non-Gaussian and governed by saddlepoints of the corresponding path integrals, so-called “instantons”.

Understanding intermittency in turbulent systems is still one of the open problems in classical physics. Since intermittency is governed by the non-Gaussianity of rare fluctuations, instantons might offer a way to better understand the behavior of turbulent systems. In the present work we concentrate on rare fluctuations in Burgers turbulence and we address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. This is of special importance since this demonstrates that instantons indeed form the skeleton of rare turbulent fluctuations. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [Phys. Rev. E 64, 026306 (2001)]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution, which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement.

T. Grafke, R. Grauer and T. Schäfer, ‘Instanton filtering for the stochastic Burgers equation’, J.Phys.A: Math. Theor., 46, 062002 (2013)