Lennaert van Veen

The Significance of Simple Invariant Solutions in Turbulent Flows

Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. A significant advance in this respect has been the numerical discovery of simple invariant sets, such as nonlinear equilibria and periodic solutions, in well-resolved Navier-Stokes flows. This review describes some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows. It is shown that the near-wall regeneration cycle of coherent structures can be reproduced by such solutions. The typical similarity laws of turbulence, i.e., the Prandtl wall law and the Kolmogorov law for the viscous range, as well as the pattern and intensity of turbulence-driven secondary flow in a square duct can also be represented by these simple invariant solutions.

Periodic motion representing isotropic turbulence

Temporally periodic solutions are extracted numerically from forced box turbulence with high symmetry. Since they are unstable to small perturbations, they are not found by forward integration but can be captured by Newton–Raphson iterations. Several periodic flows of various periods are identified for the micro-scale Reynolds number Rλ between 50 and 67. The statistical properties of these periodic flows are compared with those of turbulent flow. It is found that the one with the longest period, which is two to three times the large-eddy-turnover time of turbulence, exhibits the same behaviour quantitatively as turbulent flow. In particular, we compare the energy spectrum, the Reynolds number dependence of the energy-dissipation rate, the pattern of the energy-cascade process, and the magnitude of the largest Lyapunov exponent. This periodic motion consists of high- and low-activity periods, which turbulence approaches, more often around its low-activity part, at the rate of once over a few eddy-turnover times. With reference to this periodic motion the Kaplan–York dimension and the Kolmogorov–Sinai entropy of the turbulence with high symmetry are estimated at Rλ = 67 to be 19.7 and 0.992, respectively. The significance of such periodic solutions, embedded in turbulence, for turbulence analysis is discussed.

Overturning and wind-driven circulation in a low-order ocean–atmosphere model

Abstract A low-order ocean–atmosphere model is presented which combines coupling through heat exchange at the interface and wind stress forcing. The coupling terms are derived from the boundary conditions and the forcing terms of the constituents. Both the ocean and the atmosphere model are based on Galerkin truncations of the basic fluid dynamical equations. Hence, the coupled model can readily be extended to include more physics and more detail. The model presented here is the simplest of a hierarchy of low-order ocean–atmosphere models. The behaviour of the coupled model is investigated by means of geometric singular perturbation theory and bifurcation analysis. Two ways are found in which the slow time scales can play a role in the coupled dynamics. In the first scenario, a limit cycle on the overturning time scale is created. The associated oscillatory behaviour is governed by internal ocean dynamics. In the second scenario, intermittent behaviour occurs between periodic and chaotic regimes in parameter space.

Active and passive ocean regimes in a low-order climate model

A low-order climate model is studied which combines the Lorenz-84 model for the atmosphere on a fast time scale and a box model for the ocean on a slow time scale. In this climate model, the ocean is forced strongly by the atmosphere. The feedback to the atmosphere is weak. The behaviour of the model is studied as a function of the feedback parameters. We find regions in parameter space with dominant atmospheric dynamics, i.e., a passive ocean, as well as regions with an active ocean, where the oceanic feedback is essential for the qualitative dynamics. The ocean is passive if the coupled system is fully chaotic. This is illustrated by comparing the Kaplan–Yorke dimension and the correlation dimension of the chaotic attractor to the values found in the uncoupled Lorenz-84 model. The active ocean behaviour occurs at parameter values between fully chaotic and stable periodic motion. Here, intermittency is observed. By means of bifurcation analysis of periodic orbits, the intermittent behaviour, and the rôle played by the ocean model, is clarified. A comparison of power spectra in the active ocean regime and the passive ocean regime clearly shows an increase of energy in the low frequency modes of the atmospheric variables. The results are discussed in terms of itinerancy and quasi-stationary states observed in realistic atmosphere and climate models.

On Matrix-Free Computation of 2D Unstable Manifolds

Recently, a flexible and stable algorithm was introduced for the computation of two-dimensional (2D) unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this manifold as the solutions of an appropriate boundary value problem (BVP). The BVP is underdetermined, and a one-parameter family of solutions can be found by means of arclength continuation. This family of orbits covers a piece of the manifold. The quality of this covering depends on the way the BVP is discretized, as do the tractability and accuracy of the computation. In this paper, we describe an implementation of the orbit continuation algorithm which relies on multiple shooting and Newton-Krylov continuation. We show that the number of time integrations necessary for each continuation step scales with the number of shooting intervals but not with the number of degrees of freedom of the dynamical system. The number of shooting intervals is chosen based on linear stability analysis to keep the conditioning of the BVP in check. We demonstrate our algorithm with two test systems: a low-order model of shear flow and a well-resolved simulation of turbulent plane Couette flow.

Metabifurcation analysis of a mean field model of the cortex

Mean field models (MFMs) of cortical tissue incorporate salient, average features of neural masses in order to model activity at the population level, thereby linking microscopic physiology to macroscopic observations, e.g., with the electroencephalogram (EEG). One of the common aspects of MFM descriptions is the presence of a high-dimensional parameter space capturing neurobiological attributes deemed relevant to the brain dynamics of interest. We study the physiological parameter space of a MFM of electrocortical activity and discover robust correlations between physiological attributes of the model cortex and its dynamical features. These correlations are revealed by the study of bifurcation plots, which show that the model responses to changes in inhibition belong to two archetypal categories or “families”. After investigating and characterizing them in depth, we discuss their essential differences in terms of four important aspects: power responses with respect to the modeled action of anesthetics, reaction to exogenous stimuli such as thalamic input, and distributions of model parameters and oscillatory repertoires when inhibition is enhanced. Furthermore, while the complexity of sustained periodic orbits differs significantly between families, we are able to show how metamorphoses between the families can be brought about by exogenous stimuli. We here unveil links between measurable physiological attributes of the brain and dynamical patterns that are not accessible by linear methods. They instead emerge when the nonlinear structure of parameter space is partitioned according to bifurcation responses. We call this general method “metabifurcation analysis”. The partitioning cannot be achieved by the investigation of only a small number of parameter sets and is instead the result of an automated bifurcation analysis of a representative sample of 73,454 physiologically admissible parameter sets. Our approach generalizes straightforwardly and is well suited to probing the dynamics of other models with large and complex parameter spaces.

Open-source tools for dynamical analysis of Liley's mean-field cortex model

Abstract Mean-field models of the mammalian cortex treat this part of the brain as a two-dimensional excitable medium. The electrical potentials, generated by the excitatory and inhibitory neuron populations, are described by nonlinear, coupled, partial differential equations that are known to generate complicated spatio-temporal behaviour. We focus on the model by Liley et al. (Network: Computation in Neural Systems 13 (2002) 67–113). Several reductions of this model have been studied in detail, but a direct analysis of its spatio-temporal dynamics has, to the best of our knowledge, never been attempted before. Here, we describe the implementation of implicit time-stepping of the model and the tangent linear model, and solving for equilibria and time-periodic solutions, using the open-source library PETSc. By using domain decomposition for parallelization, and iterative solving of linear problems, the code is capable of parsing some dynamics of a macroscopic slice of cortical tissue with a sub-millimetre resolution.

Sub critical transition to turbulence in three-dimensional Kolmogorov flow

We study Kolmogorov flow on a three dimensional, periodic domain with aspect ratios fixed to unity. Using an energy method, we give a concise proof of the linear stability of the laminar flow profile. Since turbulent motion is observed for high enough Reynolds numbers, we expect the domain of attraction of the laminar flow to be bounded by the stable manifolds of simple invariant solutions. We show one such edge state to be an equilibrium with a spatial structure reminiscent of that found in plane Couette flow, with stream wise rolls on the largest spatial scales. When tracking the edge state, we find an upper and a lower branch solution that join in a saddle node bifurcation at finite Reynolds number.

Algorithm 956: PAMPAC, A Parallel Adaptive Method for Pseudo-Arclength Continuation

Pseudo-arclength continuation is a well-established method for generating a numerical curve approximating the solution of an underdetermined system of nonlinear equations. It is an inherently sequential predictor-corrector method in which new approximate solutions are extrapolated from previously converged results and then iteratively refined. Convergence of the iterative corrections is guaranteed only for sufficiently small prediction steps. In high-dimensional systems, corrector steps are extremely costly to compute and the prediction step length must be adapted carefully to avoid failed steps or unnecessarily slow progress. We describe a parallel method for adapting the step length employing several predictor-corrector sequences of different step lengths computed concurrently. In addition, the algorithm permits intermediate results of correction sequences that have not converged to seed new predictions. This strategy results in an aggressive optimization of the step length at the cost of redundancy in the concurrent computation. We present two examples of convoluted solution curves of high-dimensional systems showing that speed-up by a factor of two can be attained on a multicore CPU while a factor of three is attainable on a small cluster.

PERIODIC MOTION IN HIGH-SYMMETRIC FLOW

We investigate unstable periodic motion embedded in isotropic turbulence with high symmetry. Several orbits of different period are continued from the regime of weak turbulence into developed turbulence. The orbits of short period diverge from the turbulent state as the Reynolds number increases but the orbit of longest period we analysed, about two to three eddy-turnover times, represents several average values of the turbulence well. In particular we measure the energy dissi- pation rate and the largest Lyapunov exponent as a function of the viscosity. At the largest micro-scale Reynolds number attained in the continuation we com- pare the energy spectra of periodic and turbulent motion. The results suggest that periodic motion of a sufficiently long period can represent turbulence in a statistical sense.