Extreme events and instantons in Lagrangian passive scalar turbulence models

M. Alqahtani, L. Grigorio, and T. Grafke, Phys. Rev. E 106 (2022) 015101

Abstract

The advection and mixing of a scalar quantity by fluid flow is an important problem in engineering and natural sciences. If the fluid is turbulent, the statistics of the passive scalar exhibit complex behavior. This paper is concerned with two Lagrangian scalar turbulence models based on the recent fluid deformation model that can be shown to reproduce the statistics of passive scalar turbulence for a range of Reynolds numbers. For these models, we demonstrate how events of extreme passive scalar gradients can be recovered by computing the instanton, i.e., the saddle-point configuration of the associated stochastic field theory. It allows us to both reproduce the heavy-tailed statistics associated with passive scalar turbulence, and recover the most likely mechanism leading to such extreme events. We further demonstrate that events of large negative strain in these models undergo spontaneous symmetry breaking.

doi:10.1103/PhysRevE.106.015101

arXiv

Enormous 'rogue waves' can appear out of nowhere. Math is revealing their secrets.

Once considered a maritime myth, these towering waves can pose serious risks to ships in the open sea. Now scientists are developing ways to predict them before they strike.

Dynamical mechanism of turbulence proliferation

A. Frishman and T.Grafke

Abstract

The subcritical transition to turbulence, as occurs in pipe flow, is believed to generically be a phase transition in the directed percolation universality class. At its heart is a balance between the decay rate and proliferation rate of localized turbulent structures, called puffs in pipe flow. Here we propose the first-ever dynamical mechanism for puff proliferation---the process by which a puff splits into two. In the first stage of our mechanism, a puff expands into a slug. In the second stage, a laminar gap is formed within the turbulent core. The notion of a split-edge state, mediating the transition from a single puff to a two puff state, is introduced and its form is predicted. The role of fluctuations in the two stages of the transition, and how splits could be suppressed with increasing Reynolds number, are discussed. Using numerical simulations, the mechanism is validated within the stochastic Barkley model. Concrete predictions to test the proposed mechanism in pipe and other wall bounded flows, and implications for the universality of the directed percolation picture, are discussed.

arXiv

Spontaneous Symmetry Breaking for Extreme Vorticity and Strain in the 3D Navier-Stokes Equations

T. Schorlepp, T. Grafke, S. May, and R. Grauer, Phil Trans Roy Soc A 380 (2022), 2226

Abstract

We investigate the spatio-temporal structure of the most likely configurations realising extremely high vorticity or strain in the stochastically forced 3D incompressible Navier-Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realising an extreme constraint as solution of a large optimisation problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realisations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry broken vortex-sheet configuration.

doi:10.1098/rsta.2021.0051

arXiv

Dynamical landscape of transitional pipe flow

A. Frishman and T.Grafke, Phys. Rev. E 105 (2022), 045108

Abstract

The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them generally. Specifically, we suggest the existence of an antipuff and a gap-edge---states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as antipuffs nucleating and decaying through the gap edge.

doi:10.1103/PhysRevE.105.045108

arXiv

Dynamical Landscape and Multistability of a Climate Model

G. Margazoglou, T. Grafke, A. Laio, and V. Lucarini, Proc. R. Soc. A 447 (2021) 2250

Abstract

We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model and analyze their interplay. First, drawing from the theory of quasipotentials, and viewing the state space as an energy landscape with valleys and mountain ridges, we infer the relative likelihood of the identified multistable climate states, and investigate the most likely transition trajectories as well as the expected transition times between them. Second, harnessing techniques from data science, specifically manifold learning, we characterize the data landscape of the simulation output to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate stable state in one of the two climate models we consider. The combination of our approaches allows to identify how the negative feedback of ocean heat transport and entropy production via the hydrological cycle drastically change the topography of the dynamical landscape of Earth's climate.

doi:10.1098/rspa.2021.0019

arXiv

Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems

T. Schorlepp, T. Grafke, and R. Grauer, J. Phys. A: Math. Theor. 54 (2021) 235003

Abstract

In recent years, instanton calculus has successfully been employed to estimate tail probabilities of rare events in various stochastic dynamical systems. Without further corrections, however, these estimates can only capture the exponential scaling. In this paper, we derive a general, closed form expression for the leading prefactor contribution of the fluctuations around the instanton trajectory for the computation of probability density functions of general observables. The key technique is applying the Gel'fand-Yaglom recursive evaluation method to the suitably discretized Gaussian path integral of the fluctuations, in order to obtain matrix evolution equations that yield the fluctuation determinant. We demonstrate agreement between these predictions and direct sampling for examples motivated from turbulence theory.

doi:10.1088/1751-8121/abfb26

arXiv

A new stochastic framework for ship capsizing

M.L. Bujorianu, R.S. MacKay, T. Grafke, S. Naik, E. Boulougouris

Abstract

We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extensi on of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs.

arXiv

Numerics and analysis of Cahn-Hilliard critical points

T. Grafke, S. Scholtes, A. Wagner, M. Westdickenberg

Abstract

We explore recent progress and open questions concerning local minima and saddle points of the Cahn-Hilliard energy in $$d\ge 2$$ and the critical parameter regime of large system size and mean value close to $$-1$$. We employ the String Method of E, Ren, and Vanden-Eijnden — a numerical algorithm for computing transition pathways in complex systems — in $$d=2$$ to gain additional insight into the properties of the minima and saddle point. Motivated by the numerical observations, we adapt a method of Caffarelli and Spruck to study convexity of level sets in $$d\ge2$$.

arXiv

Instantons for rare events in heavy-tailed distributions

M. Alqahtani, and T. Grafke, J. Phys. A: Math. Theor. 54 (2021), 175001

Abstract

Large deviation theory and instanton calculus for stochastic systems is widely used to gain insight into the evolution and probability of rare events. At its core lies the realization that rare events are, under the right circumstances, dominated by their least unlikely realization. Their computation through a saddle-point approximation of the path integral for the corresponding stochastic field theory then reduces an inefficient stochastic sampling problem into a deterministic optimization problem: finding the path of smallest action, the instanton. In the presence of heavy tails, though, standard algorithms to compute the instanton critically fail to converge. The reason for this failure is the divergence of the scaled cumulant generating function (CGF) due to a non-convex large deviation rate function. We propose a solution to this problem by "convexifying" the rate function through nonlinear reparametrization of the observable, which allows us to compute instantons even in the presence of super-exponential or algebraic tail decay. The approach is generalizable to other situations where the existence of the CGF is required, such as exponential tilting in importance sampling for Monte-Carlo algorithms. We demonstrate the proposed formalism by applying it to rare events in several stochastic systems with heavy tails, including extreme power spikes in fiber optics induced by soliton formation.

doi:10.1088/1751-8121/abe67b

arXiv