Singularities and Instantons

The question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time is addressed. Analytical considerations and numerical simulations suggest high-symmetry flows being promising candidates for a blowup of the Euler equations in finite time [Pelz, 2001]. We present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high resolution adaptively refined numerical simulations (effective resolution of up to 81923 mesh points) using Lagrangian tracer particles to verify the assumptions made by recent theorems by Deng et al. [2005,2006]. This approach connects vortex line geometry (curvature, spreading) and velocity increase with a non-blowup statement.

We then ask the question, which role these singular structures play in turbulence statistics. More than 15 years ago, for certain systems like the problem of passive advection and Burgers turbulence the door for attacking this issue was opened by getting access to the probability density function to rare and strong fluctuations by the instanton approach. We address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [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.