Tobias Grafke

Numerical simulations of possible finite time singularities in the incompressible Euler equations : Comparison of numerical methods

Abstract The numerical simulation of the 3D incompressible Euler equations is analyzed with respect to different integration methods. The numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods. Based on this comparison, a Kida–Pelz-like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given. This estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective.

Large Deviations in Fast–Slow Systems

The incidence of rare events in fast–slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior—such fluctuations are rare on short time-scales but become ubiquitous eventually. Classical results prove that this LDP involves an Hamilton–Jacobi equation whose Hamiltonian is related to the leading eigenvalue of the generator of the fast process, and is typically non-quadratic in the momenta—in other words, the LDP for the slow variables in fast–slow systems is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone. It is shown here that the eigenvalue problem for the Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables, and the fast variables enter quadratically the equation for the slow variables. These results are illustrated via examples, inspired by kinetic theories of turbulent flows and plasma, in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an SDE.

The instanton method and its numerical implementation in fluid mechanics

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations.

Instanton filtering for the stochastic Burgers equation

We address the question of 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 Phys. Rev. E 64 026306). 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.

Arclength Parametrized Hamilton's Equations for the Calculation of Instantons

A method is presented to compute minimizers (instantons) of action functionals using arclength parametrization of Hamiltons equations. This method can be interpreted as a local variant of the geometric minimum action method introduced to compute minimizers of the Freidlin--Wentzell action functional that arises in the context of large deviation theory for stochastic differential equations. The method is particularly well suited to calculate expectations dominated by noise-induced excursions from deterministically stable fixpoints. Its simplicity and computational efficiency are illustrated here using several examples: a finite-dimensional stochastic dynamical system (an Ornstein--Uhlenbeck model) and two models based on stochastic partial differential equations: the

Finite-time Euler singularities: A Lagrangian perspective

\phi^4

Rogue waves and large deviations in deep sea

-model and the stochastically driven Burgers equation.

Efficient Computation of Instantons for Multi-Dimensional Turbulent Flows with Large Scale Forcing

We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate for a finite-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, 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 and inject Lagrangian tracer particles to monitor geometric properties of vortex line segments. We then verify the assumptions made by analytical non-blowup criteria introduced by Deng et. al [Commun. PDE 31 (2006)] connecting vortex line geometry (curvature, spreading) to velocity increase to rule out singular behavior.

Turbulence properties and global regularity of a modified Navier–Stokes equation

Significance Quantifying the departure from Gaussianity of the wave-height distribution in the seas and thereby estimating the likelihood of appearance of rogue waves is a long-standing problem with important practical implications for boats and naval structures. Here, a procedure is introduced to identify ocean states that are precursors to rogue waves, which could permit their early detection. Our findings indicate that rogue waves obey a large deviation principle—i.e., they are dominated by single realizations—which our method calculates by solving an optimization problem. The method generalizes to estimate the probability of extreme events in other deterministic dynamical systems with random initial data and/or parameters, by using prior information about the nature of their statistics. The appearance of rogue waves in deep sea is investigated by using the modified nonlinear Schrödinger (MNLS) equation in one spatial dimension with random initial conditions that are assumed to be normally distributed, with a spectrum approximating realistic conditions of a unidirectional sea state. It is shown that one can use the incomplete information contained in this spectrum as prior and supplement this information with the MNLS dynamics to reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. Our results indicate that rogue waves occur when the system hits unlikely pockets of wave configurations that trigger large disturbances of the surface height. The rogue wave precursors in these pockets are wave patterns of regular height, but with a very specific shape that is identified explicitly, thereby allowing for early detection. The method proposed here combines Monte Carlo sampling with tools from large deviations theory that reduce the calculation of the most likely rogue wave precursors to an optimization problem that can be solved efficiently. This approach is transferable to other problems in which the system’s governing equations contain random initial conditions and/or parameters.

Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools

Extreme events play a crucial role in fluid turbulence. Inspired by methods from field theory, these extreme events, their evolution and probability can be computed with help of the instanton formalism as minimizers of a suitable action functional. Due to the high number of degrees of freedom in multi-dimensional fluid flows, traditional global minimization techniques quickly become prohibitive in their memory requirements. We outline a novel method for finding the minimizing trajectory in a wide class of problems that typically occurs in turbulence setups, where the underlying dynamical system is a non-gradient, non-linear partial differential equation, and the forcing is restricted to a limited length scale. We demonstrate the efficiency of the algorithm in terms of performance and memory by computing high resolution instanton field configurations corresponding to viscous shocks for 1D and 2D compressible flows.