Tobias Grafke

A. Svirsky, T. Grafke, and A. Frishman

Abstract

Pipe flow is a canonical example of a subcritical flow, the transition to turbulence requiring a finite perturbation. The Reynolds number (Re) serves as the control parameter for this transition, going from the ordered (laminar) to the chaotic (turbulent) phase with increasing Re. Just above the critical Re, where turbulence can be sustained indefinitely, turbulence spreads via the self-replication of localized turbulent structures called puffs. To reveal the workings behind this process, we consider transitions between one and two-puff states, which dynamically are transitions between two distinct chaotic saddles. We use direct numerical simulations to explore the phase space boundary between these saddles, adapting a bisection algorithm to identify an attracting state on the boundary, termed an edge state. At Re=2200, we also examine spontaneous transitions between the two saddles, demonstrating the relevance of the found edge state to puff self-replication. Our analysis reveals that the process of self-replication follows a previously proposed splitting mechanism, with the found edge state as its tipping point. Additionally, we report results for lower values of Re, where the bisection algorithm yields a different type of edge state. As we cannot directly observe splits at this Re, the self-replication mechanism here remains an open question. Our analysis suggests how this question could be addressed in future studies, and paves the way to probing the turbulence proliferation mechanism in other subcritical flows.

arXiv

Jelle Soons, Tobias Grafke, and Henk A. Dijkstra, J. Fluid Mech. 1009 (2025), A53

Abstract

There is a reasonable possibility that the present-day Atlantic Meridional Overturning Circulation is in a bi-stable regime and hence it is relevant to compute probabilities and pathways of noise-induced transitions between the stable equilibrium states. Here, the most probable transition pathway of a noise-induced collapse of the northern overturning circulation in a spatially-continuous two-dimensional model with surface temperature and stochastic salinity forcings is directly computed using Large Deviation Theory (LDT). This pathway reveals the fluid dynamical mechanisms of such a collapse. Paradoxically it starts off with a strengthening of the northern overturning circulation before a short but strong salinity pulse induces a second overturning cell. The increased atmospheric energy input of this two-cell configuration cannot be mixed away quickly enough, leading to the collapse of the northern overturning cell and finally resulting in a southern overturning circulation. Additionally, the approach allows us to compare the probability of this collapse under different parameters in the deterministic part of the salinity surface forcing, which quantifies the increase in collapse probability as the bifurcation point of the system is approached.

doi:10.1017/jfm.2025.248

arXiv

Jelle Soons, Tobias Grafke, René M. van Westen, and Henk A. Dijkstra

Abstract

Recently the global average temperature has temporarily exceeded the 1.5°C goal of the Paris Agreement, and so an overshoot of various climate tipping elements becomes increasingly likely. In this study we analyze the physical processes of an overshoot of the Atlantic Meridional Overturning Circulation (AMOC), one of the major tipping elements, using a conceptual box model. Here either the atmospheric temperature above the North Atlantic, or the freshwater forcing into the North Atlantic overshoot their respective critical boundaries. In both cases a higher forcing rate can prevent a collapse of the AMOC, since a higher rate of forcing causes initially a fresher North Atlantic, which in turn results in a higher northward transport by the subtropical gyre supplementing the salinity loss in time. For small exceedance amplitudes the AMOC is still resilient as the forcing rates can be low and so other state variables outside of the North Atlantic can adjust. Contrarily, for larger overshoots the trajectories are dynamically similar and we find a lower limit in volume and exceedance time for respectively freshwater and temperature forcing in order to prevent a collapse. Moreover, for a large overshoot an increased air-sea temperature coupling has a destabilizing effect, while the reverse holds for an overshoot close to the tipping point. The understanding of the physics of the AMOC overshoot behavior is important for interpreting results of Earth System Models and for evaluating the effects of mitigation and intervention strategies.

arXiv

T. Schorlepp, and T. Grafke

Abstract

We show how to efficiently compute asymptotically sharp estimates of extreme event probabilities in stochastic differential equations (SDEs) with small multiplicative Brownian noise. The underlying approximation is known as sharp large deviation theory or precise Laplace asymptotics in mathematics, the second-order reliability method (SORM) in reliability engineering, and the instanton or optimal fluctuation method with 1-loop corrections in physics. It is based on approximating the tail probability in question with the most probable realization of the stochastic process, and local perturbations around this realization. We first recall and contextualize the relevant classical theoretical result on precise Laplace asymptotics of diffusion processes [Ben Arous (1988), Stochastics, 25(3), 125-153], and then show how to compute the involved infinite-dimensional quantities — operator traces and Carleman-Fredholm determinants — numerically in a way that is scalable with respect to the time discretization and remains feasible in high spatial dimensions. Using tools from automatic differentiation, we achieve a straightforward black-box numerical computation of the SORM estimates in JAX. The method is illustrated in examples of SDEs and stochastic partial differential equations, including a two-dimensional random advection-diffusion model of a passive scalar. We thereby demonstrate that it is possible to obtain efficient and accurate SORM estimates for very high-dimensional problems, as long as the infinite-dimensional structure of the problem is correctly taken into account. Our JAX implementation of the method is made publicly available.

arXiv

Summary: Earth's climate is a highly complex, non-equilibrium and chaotic stochastic system. In this project, we attempt to classify its chaotic attractors with methods from non-equilibrium statistical mechanics, large deviation theory and manifold learning. For example, due to the ice albedo feedback, the climate is known to exist in two locally stable states, the current (warm) climate, and a "snowball" state, where the globe is covered in ice. Some models even suggest additional metastable climate states, such as the slushball Earth. Similarly, the currently active Atlantic Meridional Overturning Circulation (AMOC, colloquially known as Gulf Steram), transports hot water to northern Europe, but is suspected to be only metastable: If perturbed the right way, the climate would persist with the AMOC non-existent. Transitions between these climate states, and their local stability, can in principle be analyzed in light of the non-equilibrium quasipotential, characterizing the expected transition times and most likely escape paths out of the current climate state.

Relevant publications

1. G. Margazoglou, T. Grafke, A. Laio, and V. Lucarini, "Dynamical Landscape and Multistability of a Climate Model", Proc. R. Soc. A 447 (2021), 2250 (link)

2. Jelle Soons, Tobias Grafke, Henk A. Dijkstra, "Optimal Transition Paths for AMOC Collapse and Recovery in a Stochastic Box Model", J Physical Oceanography 54 (2024), 2537 (link)

3. Jelle Soons, Tobias Grafke, Henk A. Dijkstra, "Most Likely Noise-Induced Overturning Circulation Collapse in a 2D Boussinesq Fluid Model", J Fluid Mech 1009 (2025), A53 (link)

4. Jelle Soons, Tobias Grafke, René M. van Westen, and Henk A. Dijkstra, "Physics of an AMOC Overshoot in a Box Model", ArXiv (2025) (link)

Jelle Soons, Tobias Grafke, Henk A. Dijkstra

Abstract

The present-day Atlantic Meridional Overturning Circulation (AMOC) is considered to be in a bi-stable regime and hence it is important to determine probabilities and pathways for noise-induced transitions between its equilibrium states. Here, using Large Deviation Theory (LDT), the most probable transition pathways for the collapse and recovery of the AMOC are computed in a stochastic box model of the World Ocean. This allows us to determine the physical mechanisms of noise-induced AMOC transitions. We show that the most likely path of an AMOC collapse starts paradoxically with a strengthening of the AMOC followed by an immediate drop within a couple of years due to a short but relatively strong freshwater pulse. The recovery on the other hand is a slow process, where the North Atlantic needs to be gradually salinified over a course of 20 years. The proposed method provides several benefits, including an estimate of probability ratios of collapse between various freshwater noise scenarios, showing that the AMOC is most vulnerable to freshwater forcing into the Atlantic thermocline region.

doi:10.1175/JPO-D-23-0234.1

arXiv

Reyk Börner, Ryan Deeley, Raphael Römer, Tobias Grafke, Valerio Lucarini, Ulrike Feudel

Abstract

In multistable dynamical systems driven by weak Gaussian noise, transitions between competing states are often assumed to pass via a saddle on the separating basin boundary. In contrast, we show that timescale separation can cause saddle avoidance in non-gradient systems. Using toy models from neuroscience and ecology, we study cases where sample transitions deviate strongly from the instanton predicted by Large Deviation Theory, even for weak finite noise. We attribute this to a flat quasipotential and propose an approach based on the Onsager-Machlup action to aptly predict transition paths.

doi:10.1103/PhysRevResearch.6.L042053

arXiv

J. Liu, J. E. Sprittles, T. Grafke, J. Stat. Mech (2024) 103206

Abstract

Thermally activated phenomena in physics and chemistry, such as conformational changes in biomolecules, liquid film rupture, or ferromagnetic field reversal, are often associated with exponentially long transition times described by Arrhenius' law. The associated subexponential prefactor, given by the Eyring-Kramers formula, has recently been rigorously derived for systems in detailed balance, resulting in a sharp limiting estimate for transition times and reaction rates. Unfortunately, this formula does not trivially apply to systems with conserved quantities, which are ubiquitous in the sciences: The associated zeromodes lead to divergences in the prefactor. We demonstrate how a generalised formula can be derived, and show its applicability to a wide range of systems, including stochastic partial differential equations from fluctuating hydrodynamics, with applications in rupture of nanofilm coatings and social segregation in socioeconomics.

doi:10.1088/1742-5468/ad8075

arXiv

Paolo Bernuzzi, Tobias Grafke

Abstract

Noise-induced transitions between multistable states happen in a multitude of systems, such as species extinction in biology, protein folding, or tipping points in climate science. Large deviation theory is the rigorous language to describe such transitions for non-equilibrium systems in the small noise limit. At its core, it requires the computation of the most likely transition pathway, solution to a PDE constrained optimization problem. Standard methods struggle to compute the minimiser in the particular coexistence of (1) multistability, i.e. coexistence of multiple long-lived states, and (2) degenerate noise, i.e. stochastic forcing acting only on a small subset of the system's degrees of freedom. In this paper, we demonstrate how to adapt existing methods to compute the large deviation minimiser in this setting by combining ideas from optimal control, large deviation theory, and numerical optimisation. We show the efficiency of the introduced method in various applications in biology, medicine, and fluid dynamics, including the transition to turbulence in subcritical pipe flow.

arXiv