Jets, stickiness, and anomalous transport

Turbulent coherent structures and early life below the Kolmogorov scale

Major evolutionary transitions, including the emergence of life, likely occurred in aqueous environments. While the role of water’s chemistry in early life is well studied, the effects of water’s ability to manipulate population structure are less clear. Population structure is known to be critical, as effective replicators must be insulated from parasites. Here, we propose that turbulent coherent structures, long-lasting flow patterns which trap particles, may serve many of the properties associated with compartments — collocalization, division, and merging — which are commonly thought to play a key role in the origins of life and other evolutionary transitions. We substantiate this idea by simulating multiple proposed metabolisms for early life in a simple model of a turbulent flow, and find that balancing the turnover times of biological particles and coherent structures can indeed enhance the likelihood of these metabolisms overcoming extinction either via parasitism or via a lack of metabolic support. Our results suggest that group selection models may be applicable with fewer physical and chemical constraints than previously thought, and apply much more widely in aqueous environments.

Models of the origin of life generally require a mechanism to structure emerging populations. Here, Krieger et al. develop spatial models showing that coherent structures arising in turbulent flows in aquatic environments could have provided compartments that facilitated the origin of life.

Time Recurrence Analysis of a Near Singular Billiard

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.

Polynomial dispersion of trajectories in sticky dynamics

Hamiltonian chaotic dynamics is, in general, not ergodic and the boundaries of the ergodic or quasiergodic area (stochastic sea, stochastic layers, stochastic webs, etc.) are sticky, i.e., trajectories can spend an arbitrarily long time in the vicinity of the boundaries with a nonexponentially small probability. Segments of trajectories imposed by the stickiness are called flights. The flights have polynomial dispersion that can lead to non-Gaussian statistics of displacements and to anomalous transport in phase space. In particular, the presence of flights influences the distribution of Poincaré recurrences. We use the distribution function of (l,t;epsilon, epsilon0) -separation of trajectories that at time instant t and trajectory length l are separated for the first time by epsilon<<1, being initially at a distance epsilon0 <<epsilon. The connection of this function, called the complexity function [Afraimovich and Zaslavsky, Chaos 13, 519 (2003)], with the distribution of Poincaré recurrences is established for three cases: (i) for the case of superdiffusion in standard and web maps for which the stickiness is defined by the boundaries of hierarchical sets of islands; (ii) for the case of the Sinai billiard with infinite horizon, where the stickiness is defined by zero-measure slits in the phase space; (iii) for the square billiard with a slit (bar-in-square billiard) where the Lyapunov exponent is zero and the stickiness is defined by the vicinity of the trajectory to the closest periodic trajectories obtained from the Diophantine approximation. Finally, the powerwise asymptotics of the Poincaré recurrences can be connected, in some cases, to the anomalous transport exponent.

Anomalous transport in Charney-Hasegawa-Mima flows

The transport properties of particles evolving in a system governed by the Charney-Hasegawa-Mima equation are investigated. Transport is found to be anomalous with a nonlinear evolution of the second moments with time. The origin of this anomaly is traced back to the presence of chaotic jets within the flow. All characteristic transport exponents have a similar value around mu = 1.75, which is also the one found for simple point vortex flows in the literature, indicating some kind of universality. Moreover, the law gamma = mu + 1 linking the trapping-time exponent within jets to the transport exponent is confirmed, and an accumulation toward zero of the spectrum of the finite-time Lyapunov exponent is observed. The localization of a jet is performed, and its structure is analyzed. It is clearly shown that despite a regular coarse-grained picture of the jet, the motion within the jet appears as chaotic, but that chaos is bounded on successive small scales.

Wave chaos and mode-medium resonances at long-range sound propagation in the ocean

We study how the chaotic ray motion manifests itself at a finite wavelength at long-range sound propagation in the ocean. The problem is investigated using a model of an underwater acoustic waveguide with a periodic range dependence. It is assumed that the sound propagation is governed by the parabolic equation, similar to the Schrodinger equation. When investigating the sound energy distribution in the time-depth plane, it has been found that the coexistence of chaotic and regular rays can cause a "focusing" of acoustic energy within a small temporal interval. It has been shown that this effect is a manifestation of the so-called stickiness, that is, the presence of such parts of the chaotic trajectory where the latter exhibit an almost regular behavior. Another issue considered in this paper is the range variation of the modal structure of the wave field. In a numerical simulation, it has been shown that the energy distribution over normal modes exhibits surprising periodicity. This occurs even for a mode formed by contributions from predominantly chaotic rays. The phenomenon is interpreted from the viewpoint of mode-medium resonance. For some modes, the following effect has been observed. Although an initially excited mode due to scattering at the inhomogeneity breaks up into a group of modes its amplitude at some range points almost restores the starting value. At these ranges, almost all acoustic energy gathers again in the initial mode and the coarse-grained Wigner function concentrates within a comparatively small area of the phase plane.

Topological instability along filamented invariant surfaces

In dynamical systems with a zero Lyapunov exponent, weak mixing can be governed by a specific topological structure of some surfaces that are invariant with respect to particle dynamics. In particular, when the genus of the invariant surfaces is more than one, they may have weak mixing and the corresponding fractional kinetics. This possibility is demonstrated by using a typical example from plasma physics, a three-dimensional resistive pressure-gradient-driven turbulence model. In a toroidal geometry and with a low-pressure gradient, this model shows the emergence of quasicoherent structures. In this situation, the isosurfaces of the velocity stream function have a web structure with filamentary surfaces emerging from the outer region of the torus and covering the inner region. The filamentary surfaces can result in stochastic jets of particles that cause a "topological instability." In such a situation, particle transport along the surfaces is of the anomalous superdiffusion type.

Space-time complexity in Hamiltonian dynamics

New notions of the complexity function C(epsilon;t,s) and entropy function S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with "flights," trappings, weak mixing, etc. The important part of the new notions is the first appearance of epsilon-separation of initially close trajectories. The complexity function is similar to the propagator p(t(0),x(0);t,x) with a replacement of x by the natural lengths s of trajectories, and its introduction does not assume of the space-time independence in the process of evolution of the system. A special stress is done on the choice of variables and the replacement t-->eta=ln t, s-->xi=ln s makes it possible to consider time-algebraic and space-algebraic complexity and some mixed cases. It is shown that for typical cases the entropy function S(epsilon;xi,eta) possesses invariants (alpha,beta) that describe the fractal dimensions of the space-time structures of trajectories. The invariants (alpha,beta) can be linked to the transport properties of the system, from one side, and to the Riemann invariants for simple waves, from the other side. This analog provides a new meaning for the transport exponent mu that can be considered as the speed of a Riemann wave in the log-phase space of the log-space-time variables. Some other applications of new notions are considered and numerical examples are presented.

Rain, power laws, and advection

Localized rain events have been found to follow power-law size and duration distributions over several decades, suggesting parallels between precipitation and seismic activity [O. Peters, Phys. Rev. Lett. 88, 018701 (2002)]]. Similar power laws are generated by treating rain as a passive tracer undergoing advection in a velocity field generated by a two-dimensional system of point vortices.