Spatiotemporal dynamics in 2D Kolmogorov flow over large domains

Kolmogorov flow in two dimensions - the two-dimensional Navier-Stokes equations with a sinusoidal body force - is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimicks the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn-Hilliard-type equation and as a result coarsening dynamics are observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially-localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) PDEs based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions - a kink and antikink - which connect two steady one-dimensional flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour

Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows

We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point-type introduced by Gibbon et al. (1999). By employing the method of mapping to regular systems, presented in Bustamante (2011) and extended to the symmetry-plane case by Mulungye et al. (2015), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^2.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t < T^* - 10^{-9000}$, where $T^*$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating point double precision ($\approx 10^{-16}$) creates a `machine-epsilon' barrier. Due to this limitation on the \emph{original} independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^*- t \approx 10^{-140}.

Chance in the Modern Synthesis

The modern synthesis in evolutionary biology is taken to be that period in which a consensus developed among biologists about the major causes of evolution, a consensus that informed research in evolutionary biology for at least a half century. As such, it is a particularly fruitful period to consider when reflecting on the meaning and role of chance in evolutionary explanation. Biologists of this period make reference to “chance” and loose cognates of “chance,” such as: “random,” “contingent,” “accidental,” “haphazard,” or “stochastic.” Of course, what an author might mean by “chance” in any specific context varies. In the following, we first offer a historiographical note on the synthesis. Second, we introduce five ways in which synthesis authors spoke about chance. We do not take these to be an exhaustive taxonomy of all possible ways in which chance meaningfully figures in explanations in evolutionary biology. These are simply five common uses of the term by biologists at this period. They will serve to organize our summary of the collected references to chance and the analysis and discussion of the following questions: • What did synthesis authors understand by chance? • How did these authors see chance operating in evolution? • Did their appeals to chance increase or decrease over time during the synthesis? That is, was there a “hardening” of the synthesis, as Gould claimed (1983)

Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow

Motivated by recent success in the dynamical systems approach to transitional flow, we study the efficiency and effectiveness of extracting simple invariant sets (recurrent flows) directly from chaotic/turbulent flows and the potential of these sets for providing predictions of certain statistics of the flow. Two-dimensional Kolmogorov flow (the 2D Navier-Stokes equations with a sinusoidal body force) is studied both over a square [0, 2{\pi}]2 torus and a rectangular torus extended in the forcing direction. In the former case, an order of magnitude more recurrent flows are found than previously (Chandler & Kerswell 2013) and shown to give improved predictions for the dissipation and energy pdfs of the chaos via periodic orbit theory. Over the extended torus at low forcing amplitudes, some extracted states mimick the statistics of the spatially-localised chaos present surprisingly well recalling the striking finding of Kawahara & Kida (2001) in low-Reynolds-number plane Couette flow. At higher forcing amplitudes, however, success is limited highlighting the increased dimensionality of the chaos and the need for larger data sets. Algorithmic developments to improve the extraction procedure are discussed

Stabilization of exact coherent structures in two-dimensional turbulence using time-delayed feedback

This work is supported by EPSRC New Investigator Award EP/S037055/1, “Stabilisation of exact coherent structures in fluid turbulence.”Time-delayed feedback control, attributed to Pyragas [Phys. Lett. A 170, 421 (1992)], is a method known to stabilize periodic orbits in low-dimensional chaotic dynamical systems. A system of the form ẋ (t) = f (x) has an additional term G(x(t - T) - x(t)) introduced where G is some "gain matrix" and T a time delay. The form of the delay term is such that it will vanish for any orbit of period T, therefore making it also an orbit of the uncontrolled system. This noninvasive feature makes the method attractive for stabilizing exact coherent structures in fluid turbulence. Here we begin by validating the method for the basic flow in Kolmogorov flow; a two-dimensional incompressible Navier-Stokes flow with a sinusoidal body force. The linear predictions for stabilization are well captured by direct numerical simulation. By applying an adaptive method to adjust the streamwise translation of the delay, a known traveling wave solution is able to be stabilized up to relatively high Reynolds number. We discover that the famous "odd-number" limitation of this time-delayed feedback method can be overcome in the fluid problem by using the symmetries of the system. This leads to the discovery of eight additional exact coherent structures which can be stabilized with this approach. This means that certain unstable exact coherent structures can be obtained by simply time stepping a modified set of equations, thus circumventing the usual convergence algorithms.Publisher PDFPeer reviewe

Optical Gain from InAs Nanocrystal Quantum Dots in a Polymer Matrix

We report on the first observation of optical gain from InAs nanocrystal quantum dots emitting at 1.55 microns based on a three-beam, time resolved pump-probe technique. The nanocrystals were embedded into a transparent polymer matrix platform suitable for the fabrication of integrated photonic devices.Comment: 8 pages, 3 figures. This second version is excactly the same as the first. It is resubmitted to correct some format errors appeared in the pdf file of the first versio