Stability of Atmospheric Jets

Atmospheric flows on rotating planets are prone to form jets that are surprisingly stable in time. This behaviour can be predicted by simulating the planet's atmosphere numerically. In certain parameter regimes, the underlying dynamical system has multiple locally stable solutions, corresponding to atmosphere configurations with different numbers of jets. This project explores the mechanisms by which random turbulent fluctuations in the atmosphere drive the system to transition between these fixed points, effectively creating or destroying atmospheric jets in the process.

Rogue Waves and Large Deviations

Summary: A statistical theory of rogue waves is proposed and tested against experimental data collected in a long water tank where random waves with different degrees of nonlinearity are mechanically generated and free to propagate along the flume. Strong evidence is given that the rogue waves observed in the tank are hydrodynamic instantons, that is, saddle point configurations of the action associated with the stochastic model of the wave system. As shown here, these hydrodynamic instantons are complex spatio-temporal wave field configurations which can be defined using the mathematical framework of Large Deviation Theory and calculated via tailored numerical methods. These results indicate that the instantons describe equally well rogue waves that originate from a simple linear superposition mechanism (in weakly nonlinear conditions) or from a nonlinear focusing one (in strongly nonlinear conditions), paving the way for the development of a unified explanation to rogue wave formation.

Experimental Setup

The experimental data were recorded in the 270m long wave flume at Marintek (Norway), schematically represented in the schematic. At one end of the tank a plane-wave generator perturbs the water surface with a predefined random signal, here taken to be the JONSWAP spectrum, modelling a random sea state, with enhancement factor \(\gamma\). These perturbations create long-crested wave trains that propagate along the tank toward the opposite end, where they eventually break on a smooth beach that suppresses most of the reflections. The water surface \(\eta(x,t)\) is measured by probes placed at different distances from the wave maker (\(x\)-coordinate). The signal at the wave maker \(\eta(x=0,t) \equiv \eta_0(t)\) is prepared according to the stationary random-phase statistics with deterministic spectral amplitudes \(C(\omega_j)\). Experimental data were collected for three different regimes: quasi-linear (\(\gamma=1\), \(H_s=0.11\) m), intermediate (\(\gamma=3.3\), \(H_s=0.13\) m), and highly nonlinear (\(\gamma=6\), \(H_s=0.15\) m). Note that these three regimes have comparable significant wave heights \(H_s\), but the difference in their enhancement factors \(\gamma\) has significant dynamical consequences.

Extreme-event filtering: Extracting rogue waves from experimental data

To characterize the dynamics leading to extreme events of the water surface, we adopt the following procedure: at a fixed location \(x=L\) along the flume, we select small observation windows around all temporal maxima of \(\eta\) that exceed a threshold \(z\). The choice of the threshold \(z\) is meant to select extreme events with a similar probability for all sets: the values of \(z=H_s=4\sigma\) for the quasi-linear set, \(z=1.1\,H_s=4.4\sigma\) for the intermediate set and \(z=1.2\,H_s=4.8\sigma\) for the highly-nonlinear set, where the maximum of the surface elevation exceeds the threshold at the \(45\) m probe, \(\eta(x=45\) m\(,t)\ge z\). We track the wave packet backward in space and look at its shape at earlier points in the channel. This allows us to build a collection of extreme events and monitor their precursors.

Theoretical description of rogue waves via instantons of NLSE

To avoid solving fully nonlinear water wave equations that are complicated from both theoretical and computational viewpoints, it is customary to use simplified models such as the Nonlinear Schr\"odinger equation (NLSE). If we exclude very nonlinear initial data, it is known that NLSE captures the statistical properties of one dimensional wave propagation to a good degree of accuracy up to a certain time and it can be improved upon by using higher order envelope equations. Because of their simplicity, NLSE and extensions thereof have been successfully used to explain basics mechanisms such as the modulational instability in water waves. With the aim of capturing leading order effects, rather than describing the full wave dynamics, here we restrict ourselves to the NLSE as a prototype model for describing the nonlinear and dispersive waves in the wave flume.

In the limit of deep-water, small-steepness, and narrow-band properties, the evolution of the system is described, to leading order in nonlinearity and dispersion, by the one-dimensional NLSE:

$$ \frac{\partial \psi}{\partial x} + 2\frac{k_0}{\omega_0} \frac{\partial \psi}{\partial t} + {i} \frac{k_0}{\omega_0^2} \frac{\partial^2 \psi}{\partial t^2}+ 2 i {k_0^3} |\psi|^2\psi = 0\,. $$

The NLSE describes the change of the complex envelope \(\psi\equiv\psi(x,t)\) that relates to the surface elevation via the Stokes series truncated at second order:

$$ \eta = |\psi|\cos(\theta) + \tfrac12 k_0 |\psi|^2\cos(2\theta) + O(k_0^2 |\psi|^3)\,, $$

where \(\theta=k_0 x - \omega_0 t + \beta\) and \(\beta\) is the phase of \(\psi\). In this expression the second order term can be neglected when the field amplitude \(|\psi|\) is small — this is the case near the wave maker at \(x=0\), where we will specify initial conditions for the NLSE. However, this second order correction is important when \(|\psi|\) becomes large, i.e. when rogue waves develop.

The NLSE is written as an evolution equation in space (rather than in time) in order to facilitate the comparison with experimental data which are taken along the spatial extend of the flume. Consistent with the wave generator located at \(x=0\), we specify \(\psi(x=0,t)=\psi_0(t)\) as initial condition for the NLSE, which we take to be a Gaussian random field with a covariance whose Fourier transform is related to the JONSWAP spectrum.

Large Deviation Theory and Instanton Calculus

Our analytical and computational descriptions of rare events rely on instanton theory. Developed originally in the context of quantum field theory, at its core lies the realization that the evolution of any stochastic system, be it quantum and classical, reduces to a well-defined (semi-classical) limit in the presence of a small parameter. Concretely, the simultaneous evaluation of all possible realizations of the system subject to a given constraint results in a (classical or path-) integral whose integrand contains an action functional \(S(\psi)\). The dominating realization can then be obtained by approximating the integral by its saddle point approximation, using the solution to \(\delta S(\psi^*)/\delta \psi=0\). This critical point \(\psi^*\) of the action functional is called the instanton, and it yields the maximum likelihood realization of the event. This conclusion can also be justified mathematically within Large Deviation Theory.

Specifically, we are interested in the probability

$$ P_L(z) \equiv \mathbb{P}(\eta(L,0)\ge z) $$

i.e. the probability of the surface elevation at position \(L\) at an arbitrary time \(t=0\) exceeding a threshold \(z\). This probability can in principle be obtained by integrating the distribution of the initial conditions over the set

$$ \Lambda(z)=\{\psi_0: \eta(L,0))\ge z\},\label{eq:13} $$

i.e. the set of all initial conditions \(\psi_0\) at the wave maker \(x=0\) that exceed the threshold \(z\) further down the flume at \(x=L\). Since the initial field \(\psi_0(t)\) is Gaussian, the probability \(P_L(z)\) can therefore be formally written as the path integral

$$ P_L(z) = Z^{-1} \int_{\Lambda(z)} \exp(-\tfrac12 \|\psi_0\|^2_C)\,D[\psi_0]\,, $$

where \(Z\) is a normalization constant and the \(C\)-norm is induced by the energy spectrum (in this case, JONSWAP) of the initial condition. The set \(\Lambda(z)\) has a very complicated shape in general, that depends non-trivially on the nonlinear dynamics of the NLSE since it involves the field at \(x=L>0\) down the flume rather than \(x=0\). One way around this difficulty is to estimate the integral via Laplace's method. This strategy is the essence of Large deviation theory (LDT), or, equivalently, instanton calculus, and it is justified for large \(z\), when the probability of the set \(\Lambda(z)\) is dominated by a single \(\psi_0\) contributing most to the integral. The optimal condition leads to the constrained minimization problem

$$ \tfrac12\min_{\psi_0\in \Lambda(z)}\, \| \psi_0\|^2_C\equiv I_L(z)\,, $$

and gives the large deviation estimate for \(P_L(z)\) of,

$$ P_L(z) \asymp \exp\left(-I_L (z) \right)\,,$$

where the symbol \(\asymp\) means asymptotic logarithmic equivalence, i.e. the ratio of the logarithms of the two sides tends to 1 as \(z\to\infty\), or, in other words, the exponential portion of both sides scales in the same way with \(z\). Intuitively, this estimate says that, in the limit of extremely strong (and unlikely) waves, their probability is dominated by their least unlikely realization, the instanton.

Now, the stochastic sampling problem is replaced by a deterministic optimization problem, which we solve numerically. The trajectory initiated from the minimizer \(\psi_0^*\) of the action will be referred to as the instanton trajectory, and in the following we compare it to trajectories obtained from the experiment.

Experimental Rogue Waves compared to Instantons

In the figure above, numerically computed instanton realizations are depicted in solid black, while experimental averages and their standard deviation are in color. We compare the evolution of rogue waves observed in the experiment and averaged over many realizations to that of the instanton, both constrained at \(x=45\) m. In all cases the instanton tracks the dynamics of the averaged wave very closely during the whole evolution. Moreover, in the focusing region the standard deviation around the mean is small, especially toward the end of the evolution. This observation in itself is a statement that indeed all of the rogue waves such that \(\eta(L,0)\ge z\) resemble the instanton plus small random fluctuations. The instanton approximation shows excellent agreement not only across different degrees of nonlinearity (and therefore substantially different physical mechanisms), but also captures the behavior of precursors earlier along the channel.

It is worth stressing that the instanton approach captures both the linear and the fully nonlinear cases, unlike previous theories that could describe each of these regimes individually but not both. To make that point, in the next two sections we compare the predictions of our approach to those of the quasi-determinism and semi-classical theories that hold in the dispersive and nonlinear regimes, respectively.

Conclusion

Here we have proposed a unifying framework based on Large Deviation Theory and Instanton Calculus that is capable to describe with the same accuracy the shape of rogue waves that result either from a linear superposition or a nonlinear focusing mechanism. In the limit of large nonlinearity, the instantons closely resemble the Peregrine soliton to describe extreme events, but with the added bonus that our framework predicts their likelihood; in the limit of linear waves, the instanton reduces to the autocorrelation function. A smooth transition between the two limiting regimes is also observed, and these predictions are fully supported by experiments performed in a large wave tank with different degrees of nonlinearity. These results were obtained for one dimensional propagation, but there are no obstacles to apply the approach to two horizontal dimensions, which may finally explain the origin and shape of rogue waves in different setups, including the ocean.

Relevant publications

  1. G. Dematteis, T. Grafke, M. Onorato, and E. Vanden-Eijnden, "Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves", Phys. Rev. X 9 (2019), 041057

  2. G. Dematteis, T. Grafke, and E. Vanden-Eijnden, "Extreme event quantification in dynamical systems with random components", J. Uncertainty Quantification 7 (2019), 1029

  3. G. Dematteis, T. Grafke, and E. Vanden-Eijnden, "Rogue Waves and Large Deviations in Deep Sea", PNAS 115 (2018), 855-860

  4. T. Grafke, R. Grauer, and T. Schäfer, "The Instanton Method and its Numerical Implementation in Fluid Mechanics", J. Phys. A: Math. Theor. 48 (2015), 333001

Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves

G. Dematteis, T. Grafke, and E. Vanden-Eijnden, Phys. Rev. X 9 (2019), 041057

Abstract

A statistical theory of rogue waves is proposed and tested against experimental data collected in a long water tank where random waves with different degrees of nonlinearity are mechanically generated and free to propagate along the flume. Strong evidence is given that the rogue waves observed in the tank are hydrodynamic instantons, that is, saddle point configurations of the action associated with the stochastic model of the wave system. As shown here, these hydrodynamic instantons are complex spatio-temporal wave field configurations, which can be defined using the mathematical framework of Large Deviation Theory and calculated via tailored numerical methods. These results indicate that the instantons describe equally well rogue waves that originate from a simple linear superposition mechanism (in weakly nonlinear conditions) or from a nonlinear focusing one (in strongly nonlinear conditions), paving the way for the development of a unified explanation to rogue wave formation.


doi:10.1103/PhysRevX.9.041057

arXiv

Extreme event quantification in dynamical systems with random components

G. Dematteis, T. Grafke, and E. Vanden-Eijnden, J. Uncertainty Quantification 7 (3), (2019), 1029

Abstract

A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. Here this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system's parameters and/or its initial conditions. Specifically, it is established under which conditions such extreme events occur in a predictable way, as the minimizer of the LDT action functional. It is also shown how this minimization can be numerically performed in an efficient way using tools from optimal control. These findings are illustrated on the examples of a rod with random elasticity pulled by a time-dependent force, and the nonlinear Schrödinger equation (NLSE) with random initial conditions.


doi:10.1137/18M1211003

arXiv

Numerical computation of rare events via large deviation theory

T. Grafke, and E. Vanden-Eijnden, Chaos 29 (2019), 063118

Abstract

An overview of rare events algorithms based on large deviation theory (LDT) is presented. It covers a range of numerical schemes to compute the large deviation minimizer in various setups, and discusses best practices, common pitfalls, and implementation trade-offs. Generalizations, extensions, and improvements of the minimum action methods are proposed. These algorithms are tested on example problems which illustrate several common difficulties which arise e.g. when the forcing is degenerate or multiplicative, or the systems are infinite-dimensional. Generalizations to processes driven by non-Gaussian noises or random initial data and parameters are also discussed, along with the connection between the LDT-based approach reviewed here and other methods, such as stochastic field theory and optimal control. Finally, the integration of this approach in importance sampling methods using e.g. genealogical algorithms is explored.


doi:10.1063/1.5084025

arXiv

String Method for Generalized Gradient Flows: Computation of Rare Events in Reversible Stochastic Processes

T. Grafke, J. Stat. Mech 2019/4 (2019) 043206

Abstract

Rare transitions in stochastic processes often can be rigorously described via an underlying large deviation principle. Recent breakthroughs in the classification of reversible stochastic processes as gradient flows have led to a connection of large deviation principles to a generalized gradient structure. Here, we show that, as a consequence, metastable transitions in these reversible processes can be interpreted as heteroclinic orbits of the generalized gradient flow. As a consequence, this suggests a numerical algorithm to compute the transition trajectories in configuration space efficiently, based on the string method traditionally restricted only to gradient diffusions.


doi:10.1088/1742-5468/ab11db

arXiv

Rogue Waves and Large Deviations in Deep Sea

G. Dematteis, T. Grafke, and E. Vanden-Eijnden, Proc. Natl. Acad. Sci., 115 (2018), 855-860

Abstract

The appearance of rogue waves in deep sea is investigated using the modified nonlinear Schrödinger (MNLS) equation with random initial conditions that are assumed to be Gaussian distributed, with a spectrum approximating the JONSWAP spectrum obtained from observations of the North Sea. It is shown that by supplementing the incomplete information contained in the JONSWAP spectrum with the MNLS dynamics one can 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 hit small pockets of wave configurations hidden in the core of their distribution that trigger large disturbances of the surface height via modulational instability. 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 builds on tools from large deviation theory that reduce the calculation of the most likely rogue wave precursors to an optimization problem that can be solved efficiently.


doi:10.1073/pnas.1710670115

arXiv

Large Deviations for Rogue Waves

In this project, Rogue Waves in deep sea are investigated using the modified nonlinear Schrödinger (MNLS) equation in one spatial-dimension with random initial conditions. The initial conditions approximate realistic conditions of a uni-directional sea state, taken as the JONSWAP spectrum. 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. The Rogue Wave events encountered in numerical simulations, as well as in experiments in a 270m wave channel in Norway, agree with their most likely (instanton) configuration obtained from the theory.

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Spatiotemporal Self-Organization of Fluctuating Bacterial Colonies

T. Grafke, M. Cates, and E. Vanden-Eijnden, Phys. Rev. Lett., 119 (2017), 188003

Abstract

We model an enclosed system of bacteria, whose motility-induced phase separation is coupled to slow population dynamics. Without noise, the system shows both static phase separation and a limit cycle, in which a rising global population causes a dense bacterial colony to form, which then declines by local cell death, before dispersing to re-initiate the cycle. Adding fluctuations, we find that static colonies are now metastable, moving between spatial locations via rare and strongly nonequilibrium pathways, whereas the limit cycle becomes quasi-periodic such that after each redispersion event the next colony forms in a random location. These results, which resemble some aspects of the biofilm-planktonic life cycle, can be explained by combining tools from large deviation theory with a bifurcation analysis in which the global population density plays the role of control parameter.


doi:10.1103/PhysRevLett.119.188003

arXiv

Metastability in Active Matter: Motile Microorganisms

Active materials can self-organize in many more ways than their equilibrium counterparts. For example, self-propelled particles whose velocity decreases with their density can display motility-induced phase separation (MIPS), a phenomenon building on a positive feedback loop in which patterns emerge in locations where the particles slow down. Here, we investigate the effects of intrinsic fluctuations in the system's dynamics on MIPS. We show that these fluctuations can lead to transitions between metastable patterns. The pathway and rate of these transitions is analyzed within the realm of large deviation theory, and they are shown to proceed in a very different way than one would predict from arguments based on detailed-balance and microscopic reversibility.

Details...