Phase space analysis in complex systems - From quantum dynamics to turbulent flows

We address the long standing problem of how to describe, by means of discrete symbolic dynamics, the spatiotemporal chaos (or turbulence) in spatially extended, strongly nonlinear field theories. One way to capture the essential features of turbulent motions is offered by coupled map lattice models, in which the spacetime is discretized, with the dynamics of small-scale spatial structures modeled by maps attached to lattice sites. The discretization that we study, the "spatiotemporal cat," has a remarkable feature that its every solution is uniquely encoded by a linear transformation from the corresponding finite alphabet symbol lattice. A spatiotemporal window into system dynamics is provided by a finite block of symbols, and the central question is to determine the likelihood of a given block's occurrence. As spatiotemporal states that share the same sub-blocks shadow each other exponentially well within the corresponding spatiotemporal windows, the dynamical zeta functions are now sums over spacetime tori, rather than time-periodic orbits. In the spatiotemporal formulation of turbulence there is no evolution in time, there are only a repertoires of admissible spatiotemporal patterns. In other words: throw away your integrators, and look for guidance in clouds' repeating patterns.

For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. At this regard, we relate our results to the debate around the prominence of contigency vs. convergence in biological evolution. Finally, we propose a new method for constructing Melancholia states from direct numerical simulations, thus bypassing the need to use the edge-tracking algorithm.

I will present in the hopefully accessible way to the audience of various background the work accomplished together with Bruno Eckhardt on restricted dimensionality models of multielectron ionisation of atoms and molecules, starting with classical models and reaching full quantum mechanical simulations of dynamics.

There has been much recent progress with regards to characterizing self-similar behavior in wall turbulence in experiments, in simulation, and in the mean and instantaneous forms of the Navier-Stokes equations. Indeed, this is a topic about which Bruno thought deeply. Here, we identify commonalities and differences between these observations, and draw some conclusions concerning the requirements for self-similarity and self-sustaining processes in wall turbulence. Recent developments with respect to resolvent analysis are exploited to identify low-rank representations of these processes, their signatures and their limitations in physical and spectral space. We close with a discussion of some outstanding challenges related to the existence, self-sustenance and modeling of self-similar solutions and structures in the canonical flows.

Lobes and clefts are the primary structures formed at the fronts of sand storms and other meteorological gravity currents. Though the velocity field and the density field are coupled, the present numerical simulations and stability analyses of the gravity currents show that the local Rayleigh-Taylor instability (RTI) at the density interface can determine the position and the original spanwise wave number of the strongest perturbations. Consequently, a RTI model is proposed without considering any information of the flow field, and includes only the fluids’ properties (Grashof number and Prandtl number). The predictions of the RTI model, i.e. the original dominating spanwise wave number of the Boussinesq current substantially depends on the Prandtl number and has a 1/3 scaling law with the Grashof number, are confirmed by the three-dimensional numerical simulations and the experiments. Furthermore, by applying the turbulent eddy viscosity, the turbulent Schmidt number, and the sand storm density measured at Qingtu Lake in the RTI model, it is shown that the dominating spanwise wavelength predicted by the RTI model agrees qualitatively with the lobes’ scale observed at the front of sand storm.

Townsend's model of attached eddies for boundary layers is revisited within a quasi-linear approximation. The velocity field is decomposed into a mean profile and fluctuations. While the mean is obtained from the nonlinear equations, the fluctuations are modelled by replacing the nonlinear self-interaction terms with an eddy-viscosity-based turbulent diffusion and stochastic forcing. Under this particular approximation, the resulting fluctuation equations remain linear, and it allows one to use superposition of their solutions for the calculation of Reynolds stress as in the original attached model. By leveraging this feature, the stochastic forcing is determined self-consistently by solving an optimisation problem which minimises the difference between the Reynolds shear stresses from the mean and fluctuation equations, subject to a constraint that the averaged Reynolds shear-stress spectrum is sufficiently smooth in the spatial wavenumber space. The proposed quasi-linear approximation is subsequently applied to turbulent channel flow in a range of friction Reynolds number from Re_tau=500 to Re_tau=20,000. The best result is obtained when the Reynolds stress is calculated by considering two leading POD (proper orthogonal decomposition) modes, which further filters out the modelling artifact caused by the unphysical stochastic forcing. In this case, the resulting turbulence intensity profile and energy spectra exhibit exactly the same qualitative behaviour as DNS data throughout the entire wall-normal location, while reproducing the early theoretical predictions of the original attached eddy model within a controlled approximation to the Navier-Stokes equations. *This is the work done with Bruno just before he passed away.

In recent years detailed experiments combined with advances in dynamical systems theory and statistical mechanics have allowed much progress in our understanding of the transition to turbulence in linearly stable shear flows. I will summarize part of this work and highlight open questions and future directions.

In 1926 Einstein published a short paper explaining the meandering of rivers. He famously began the paper by discussing the secondary flow generated in a stirred tea cup -- the flow now widely known to be responsible for the collection of tea leaves at the center of a stirred cup of tea. In 2014, Luo and Hou presented detailed numerical evidence of a finite-time singularity in a rotating, incompressible, inviscid flow. The key driving mechanism of that singularity is the secondary tea-cup flow. The present work is not aimed at proving the existence of a singularity in this flow, nor is it aimed at generating more highly resolved numerical evidence for the singularity than already exists. Rather, I will assume that the flow simulated by Luo and Hou genuinely develops a singularity. My goal is to understand, from a fluid-mechanics perspective, why.

In the last few years classical set oriented numerical methods for the approximation of invariant sets of finite dimensional dynamical systems have been extended to the infinite dimensional setting. This novel approach is based on embedding techniques that allow to a compute one-to-one image of the invariant set in a finite dimensional space. After a brief review of the set-oriented continuation method for the computation of unstable manifolds I will apply this algorithm to a fluid dynamics problem: The transition to turbulence in parallel shear flows is connected with the existence of a lower-dimensional manifold in state space that distinguishes between the basin of attraction of the laminar fixed point and initial conditions leading to turbulent flow. Relative attractors on this manifold, so-called edge states have therefore at least one unstable direction. Here, we calculate and visualize the unstable manifold of such an edge state in channel flow.

Newtonian fluids are known to exhibit hydrodynamic instabilities and/or transition to turbulence at large enough Reynolds numbers. Recently it has been discovered that in simple shear flows (like pressure-driven flows in a pipe or between two plates) there exist the so-called coherent structures which organize the turbulent dynamics close to the laminar-turbulent transition. In that region, their dynamics are low-dimensional and can be described by a few (order ten) well-chosen degrees of freedom. On the other hand, complex fluids, in general, and polymer solutions, in particular, do not flow like Newtonian fluids. Their flows exhibit instabilities at very low Reynolds numbers which are driven not by inertia, but rather by anisotropic elastic stresses. Further increase of the flow rate results in a truly chaotic flow -- the so-called purely elastic turbulence. The mechanism of this new type of chaotic motion is poorly understood. In this talk I will discuss our recent attempts to generalise the Newtonian theory of the transition to turbulence to the purely elastic case. We identify the relevant coherent structures and construct a viscoelastic self-sustaining process that can organise flow dynamics close to the transition.

The “Edge” is the state space manifold of codimension 1 separating the basins of attraction of the laminar and the turbulent state. Its definition and identification are relatively straightforward in bistable systems, and less trivial in systems where turbulence is transient. It is of interest to understand what the notion of separatrix becomes in the presence of an additional linear instability of the laminar state, i.e. when turbulence is the only attractor of the system. Such a situation arises in many high-Reynolds number shear flows such as plane Poiseuille flow or the Blasius flow, or more simply in imperfect subcritical flows. In this talk I will review the status of edge tracking for the Blasius flow and the practical difficulties encountered in simulations. A new three-dimensional model will be used to illustrate the global bifurcation responsible for the gradual disappearance of the Edge as the Reynolds number is increased.

In nature and in engineering, periodic forcing leads often to nonlinear resonances which are challenging to model and predict. We show that the sloshing of water in a rectangular container driven by harmonic horizontal excitation is accurately described by the Duffing equation. The system exhibits a bent response curve with strong hysteresis (increasing with the driving amplitude), transitions occurring at $90^\circ$-phase-lag, as well as period-three-motion. At large driving amplitudes highly nonlinear effects like competition between states and wave breaking, cause deviations from the Duffing dynamics and call for the development of more advanced models.

The formation and breakup of drops in turbulent flows is of key importance in chemical, mechanical and aerospace engineering, but their physical mechanisms and time scales remain poorly understood. We investigate drop breakup in homogeneous isotropic turbulence with direct numerical simulations, thereby resolving all temporal and spatial scales. A new GPU code solving the Cahn–Hilliard–Navier–Stokes equations enables the simulation of thousands of independent cases and a detailed analysis of the breakup process. We find that drop breakup is a memoryless process, whose rate depends on the Weber number. Our results allow estimating the evolution of drop-size distributions in the dilute regime and can be used to parametrize population-balance equations.

TBA

Bruno Eckhardt had broad interests in the nonlinear dynamics of complex systems. This talk is comprised of two vignettes, one on nonlinear dynamics and turbulence in viscoelastic polymer solutions, and the other on applications of machine learning tools to nonlinear dynamics problems. One of my last conversations with Bruno was on the latter topic. Addition of polymers to a liquid can lead to dramatic reductions in energy dissipation during turbulent flow while having a negligible effect on laminar flow. We perform simulations of viscoelastic polymer solution flow in the so-called elastoinertial turbulence (EIT) regime. Localized polymer stretch fluctuations are observed, which are very similar to structures arising from linear stability (Tollmien-Schlichting modes) and resolvent analyses. Further simulations reveal the existence of a new family of attractors whose structure closely resembles the linear Tollmien-Schlichting (TS) mode and is nonlinearly self-sustained by viscoelasticity. These results suggest that, in the parameter range considered here, the bypass transition leading to EIT is mediated by nonlinear amplification and self-sustenance of perturbations that excite the Tollmien- Schlichting mode. Partial differential equations such as the Navier-Stokes equations are formally infinite dimensional, but the presence of dissipation leads to the expectation that the long-time dynamics collapse onto a finite dimensional invariant manifold, sometimes called an inertial manifold (IM). A data-driven framework is developed to represent chaotic dynamics on an IM, and applied to solutions of the Kuramoto-Sivashinsky equation (KSE). The dimension of the IM is determined by training NNs with various dimensions and observing that the approximation error drops abruptly once the dimension reaches that of the IM. Additional neural networks predict time-evolution on the IM, maintaining good agreement with exact results for several correlation times and yielding excellent agreement for the probability densities of important quantities. The formalism accounts for translation invariance and energy conservation, and substantially outperforms linear dimension reduction with PCA.

In this talk we will review some localization scenarios in canonical shear flows. The fundamental mechanism consists of subharmonic Hopf bifurcations by which travelling waves exhibit spatio-temporal modulation. The resulting Navier-Stokes solutions are relative periodic orbits that, sometimes, may exhibit streamwise localization. To monitor the localization, we perform arclength continuation of modulated waves in wavelength and Reynolds number. We will also study the role of recently found asymmetric Tollmien-Schlichting waves in this type of scenarios.

Over the last 10 years, one of Bruno's focus research areas was Taylor-Couette Turbulence. In this talk I will present new experimental, numerical, and theoretical results on this wonderderful system, including our results on wall roughness and on the profile of the velocity.

The hypothalamus controls a wide variety of physiological processes such as body temperature, appetite, sexual behavior, daily physiological cycles and emotional responses. The hypothalamus secrets powerful hormones into the body and engages in neuronal networks with other regions of the brain. Additionally, the hypothalamus encloses a few millimeter large, 100 microns wide cavity called ventricle that is filled with cerebrospinal fluid (CSF), a liquid, containing a wide variety of biologically active substances. Similar to neuronal and hormonal signals, the transport of CSF in the ventricle is highly selective occuring along specific, evolutionarily conserved, network-forming routes (Faubel et al., Science 2016). Such guided transport is achieved by bundles of beating cilia, tiny molecular machines that are tethered to the ventricular wall and propel CSF at rates in the range of 1mm/s. The directional guidance of ciliary transport relies on the location of cilia bundles as well as their direction of their beating. We will discuss the phenomenon of directed transport, will present the cellular structure of the tissues and will show that biomimetic lipid vesicles are selectively transported by the network of ciliary roads and locally deployed to selective sites. Finally, we show examples of the dynamic nature of the network. We hypothesize that the physiology of the hypothalamus is influence by this novel network transporting signaling factors.

We establish rigorous upper bounds on the time-averaged heat transport for a model of rotating Rayleigh-B\'enard convection between no-slip boundaries at infinite Prandtl number and with Ekman pumping. The analysis is based on the asymptotically reduced equations derived for rotationally constrained dynamics with no-slip boundaries, and hence includes a lower order correction that accounts for the Ekman layer and corresponding Ekman pumping into the bulk. Using the auxiliary functional method we find that, to leading order, the temporally averaged heat transport is bounded above as a function of the Rayleigh and Ekman numbers $\Ra$ and $\Ek$ according to $\Nu \leq 0.3704 \Ra^2 \Ek^2$. Dependent on the relative values of the thermal forcing represented by $\Ra$ and the effects of rotation represented by $\Ek$, this bound is both an improvement on earlier rigorous upper bounds, and provides a partial explanation of recent numerical and experimental results that were consistent yet surprising relative to the previously derived upper bound of $\Nu \lesssim \Ra^3 \Ek^4$.

For any quantity of interest in a system governed by nonlinear differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Upper bounds can be proved a priori using auxiliary functions, the best choice of which is a convex optimization. We show that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on maximal time averages. They also provide volumes in phase space where maximal trajectories must lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming which we illustrate using the Lorenz and Kuramoto-Sivashinsky equations. This is joint work with Ian Tobasco and David Goluskin, part of which appears in Physics Letters A 382, 382-386 (2018).

We propose a dynamical mechanism for a strictly finite prediction horizon, i.e. a scenatio of chaotic motion where asymptotically a more precise knowledge of the initial condition does note translate into a longer closeness of the forecast to the truth. For this, we propose a class of hierarchical dynamical systems which possess a scale dependent error growth rate in the form of a power law. Actually, this is motivated by and consistent with well known hierarchies of patterns in atmospheric dynamics. This scale dependent error growth rate in form of a power law translates in power law error growth over time instead of exponential error growth as in conventional chaotic systems. The consequence is a strictly finite prediction horizon, since in the limit of infinitesimal errors of initial conditions, the error growth rate diverges and hence additional accuracy is not translated into longer prediction times. By re-analyzing data of the National Center for Environmental Protect Global Forecast System, a weather prediction model, published by Harlim et al (2005 Phys. Rev. Lett. 94 228501) we show that such a power law error growth rate can indeed be found in numerical weather forecast models and estimate it average maximal prediction horizon to about 15 d.

Experiments and numerical simulations have shown that transitional turbulence in wallbounded shear flows is not statistically homogeneneous, but composed of long oblique turbulent bands, if the domains are sufficiently large to accomodate them. These bands have been observed in in plane Couette, plane Poiseuille flow, counter-rotating Taylor-Couette flow, torsional Couette flow, and annular pipe flow. At their upper Reynolds number threshold, they appear with a regular spatial wavelength as laminar regions carve out gaps in otherwise uniform turbulence. At the lower threshold, turbulent bands sparsely populate otherwise laminar domains and complete laminarization takes place via their disappearance. In consequence, the spatial domains in which transition must be studied must be much larger than the gap. We achieve this by simulating a modified version of plane Couette flow, in which the rigid walls are replaced by stress-free boundaries and the resolution is drastically truncated. This Model Waleffe Flow nonetheless displays the same sequence of transitional states as wall-bounded Couette flow, i.e. transient spots, regular turbulent-laminar bands, and uniform turbulence. This shows these phenomena are very robust and that the walls and their associated boundary layers do not play a role in producing them: the necessary ingredients are only shear and confinement. MWF is sufficiently economical to allow numerical simulations with very large horizontal dimensions (4096 x 4096 or 8192 x 2048 in gap units) and over very long times (10^6 advective time units). These simulations demonstrate that the transition to turbulence is continuous, via band extinction, and is in the universality class of directed percolation in two spatial dimensions.

Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the orthogonal, unitary, and symplectic one. If there is in addition a particle-antiparticle symmetry, the chiral variants of these ensembles appear. The list is completed by four more random matrix ensembles resulting in the ten-fold way. A microwave study of the chiral orthogonal, unitary, and symplectic ensembles is presented using a linear chain of evanescently coupled dielectric cylindrical resonators. A typical feature of these ensembles is a mirror symmetry of the spectrum with respect to an energy zero $E_0$. Close to $E_0$ the eigenvalues feel the neighborhood of their symmetry equivalent partners leading to a possible eigenvalue repulsion at $E_0$. In all cases the predicted repulsion behavior could be experimentally verified.

Cell polarization is a key mechanism of symmetry breaking in morphogenesis. It involves the coarsening of domains of membrane proteins that interact with cytosolic enzymes. The front propagation in classical phase separation is driven by thermodynamic relaxation subjected to thermal noise. In contrast, the domains on the cell surface are driven by catalytic biochemical reactions, and they are subjected to chemical noise. Hence, the front propagation is coupled to a dissipative chemical current. We will set up a stochastic-thermodynamic framework to describe the evolution of the domains boundaries in the classical hermodynamic and in the biochemical system. Based on the analytical solutions for the two models we will discuss differences and commonalities of the processes.

On the request of the organisers, I'll attempt to give an overview of Bruno's considerable work in fluid mechanics with a particular focus on his work in transition.

I will review recent results on Rayleigh-Taylor turbulent convection and mixing on the basis of direct numerical simulations. The effect of the density stratification in the evolution of the mixing process and the possibility to stabilize the flow will be in particular addressed.

Based on the example of a paradigmatic low-dimensional Hamiltonian system subjected to different scenarios of parameter drifts of non-negligible rates, we show that the dynamics of such systems can best be understood by following ensembles of initial conditions corresponding to tori of the initial system. When such ensembles are followed, torus-like objects, called snapshot tori are obtained which change their location and shape. In their center one finds a time-dependent, snapshot elliptic orbit. After some time, many of the tori break up and spread over large regions of the phase space, however, one may find some smaller tori, which remain closed curves throughout the whole scenario. We also show that the cause of torus break-up is the collision with a snapshot hyperbolic orbit and the surrounding chaotic sea, which forces the ensemble to adopt chaotic properties. Within this chaotic sea we demonstrate the existence of a snapshot horseshoe structure and a snapshot saddle. An easily visualizable condition for torus break-up is found in relation to a specific snapshot stable manifold. The average distance of nearby pairs of points initiated on an original torus at first hardly changes in time, but crosses over into an exponential growth when the snapshot torus breaks up. This new phase can be characterized by a novel type of finite time Lyapunov exponent which depends both on the torus and on the scenario followed.

We consider active particles with circular trajectories on a plane, subject to elastic collisions. We show that the chaotic state of this system is a long-living transient, after which a simple synchronized regime appears. We discuss the statistical properties of the transition.

The investigation of the propagation of light in mesoscopic, i.e. often micrometer-scale, systems is a rich subject providing insights ranging from quantum chaos in open systems to new schemes for realizing microlasers. The concept of quantum-classical, here wave-ray, correspondence, proves to be a useful tool to efficiently capture the properties of light confined by total internal reflection in optical microcavities. The phase-space view is of particular interest in these open systems as it allows a direct comparison of the Poincaré surface of section of billiards for lights and the Husimi function reflecting the resonance properties. It reveals how the unstable manifold determines the far-field emission properties of deformed microdisk lasers as well as semiclassical deviations from the ray-picture expectation in the reflection and refraction of light. For arrays of coupled microcavities and billiards with sources, the phase space picture is a powerful instrument in analyzing the dynamics of light out of equilibrium. We illustrate these effects and address how fundamental research can motivate innovative applications.

I will describe examples of recent work where massive simulations of turbulence have shed new light on the dynamics of turbulence.

Catastrophic events in Nature can be often triggered by small perturbations, with ``remote triggering" of earthquakes being an important example. Here we present a mechanism for the giant amplification of small perturbations that is expected to be generic in systems whose dynamics is not derivable from a Hamiltonian. We offer a general discussion of the typical instabilities involved (being oscillatory with an exponential increase of noise) and examine the normal forms that determine the relevant dynamics. The high sensitivity to external perturbations is explained for systems with and without dissipation. Numerical examples are provided using the dynamics of frictional granular matter. Finally we point out the relationship of the presently discussed phenomenon to the highly topical issue of ``exceptional points" in quantum models with non-Hermitian Hamiltonians.

In recent years it has been shown that the concept in the chemistry literature of a transition state through which a system has to pass in order to react can can made precise in terms of normally hyperbolic invariant manifold. The normally hyperbolic invariant manifold has stable and unstable manifolds that are codimension 1 separatrices that form the phase space conduits from reactants to products. In this talk we will review these phase space structures and their implications for the classical and quantum reaction dynamics.

Two-dimensional (2d) and quasi-2d flows occur at macro- and mesoscale in a variety of physical systems. Examples include stratified layers in Earth's atmosphere and the ocean, soap films and more recently also dense bacterial suspensions, where the collective motion of microswimmers induces patterns of mesoscale vortices. A characteristic feature of 2d turbulence is the occurrence of an inverse energy cascade. In absence of large-scale friction the inverse energy cascade results in the formation of large-scale coherent structures, so-called condensates. We here study the formation of the condensate as a function of the kind and amplitude of the forcing. Direct numerical simulations show that the condensate does not appear gradually but in a phase transition. For prescribed energy dissipation the transition is second order; for active matter, where the forcing is due to a small-scale instability, the transition is first order. The phase transition separates two markedly different types of 2d turbulence: in turbulence with a condensate, energy input is predominantly balanced by dissipation in the condensate and intermediate scales follow an inertial cascade; without a condensate dissipation is spread over the intermediate scales and the properties of the energy transfer are noticeably different and non-universal.