Session Y04: Nonlinear Dynamics: Turbulence (11:30am - 12:15pm CST)

Symmetry-breaking-induced rare fluctuations in a time-delay dynamic system

Inspired by the experimental and numerical findings, we study the dynamic instabilities of two coupled nonlinear time-delay differential equations that was used to describe the coherent oscillations between the top and bottom boundary layers (BLs) in turbulent Rayleigh-Bénard convection. By introducing two sensitivity parameters for the instabilities of the top and bottom BLs, we find three different types of solutions, namely, in-phase oscillations, period doubling and chaos. The chaos solution contains rare but large amplitude fluctuations. The statistical properties of these fluctuations are consistent with those observed in the experiment for the massive eruption of thermal plumes, which causes random reversals of the large-scale circulation in turbulent Rayleigh-Bénard convection. Our study thus provides new insights into the origin of rare massive eruptions and sudden changes in large-scale flow pattern that are often observed in closed thermal convection systems of geophysical and astrophysical scale.   

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

Time-delayed feedback control (Pyragas 1992 \textit{Phys. Letts. A} \textbf{170}(6) 421-428), is a method known to stabilise periodic orbits in chaotic dynamical systems. A system $\dot{\mathbf{x}}(t)=f(\mathbf{x})$ is supplemented with $G(\mathbf{x}(t)-\mathbf{x}(t-T))$ where $G$ is a `gain matrix' and $T$ a time delay. The form of the delay term is such that it will vanish for any orbit of period $T,$ making it an orbit of the uncontrolled system. This non-invasive feature makes the method attractive for stabilising exact coherent structures in fluid turbulence. Here we validate the method using the basic flow in Kolmogorov flow; a two-dimensional incompressible viscous flow with a sinusoidal body force. Linear predictions are well captured by direct numerical simulation. By applying an adaptive method to adjust the streamwise translation of the delay, a known travelling wave solution is able to be stabilised up to relatively high Reynolds number. Finally an adaptive method to converge the period $T$ is also presented to enable periodic orbits to be stabilised in a proof of concept study at low Reynolds numbers. These results demonstrate that unstable ECSs may be found by timestepping a modified set of equations, thus circumventing the usual convergence algorithms.   

Multiscale entropy analysis of forced Burgers turbulence

Scale invariance is one of the key characteristics of nonlinear fluid systems exhibiting multiscale interactions, which is often manifested by a power-law scaling in energy spectrum. The Fourier analysis is commonly employed, but its application to flows with strong inhomogeneities and spatial locality is limited due to the fundamental assumption on its basis. In this study, an information-based alternative is proposed, and its validity and physical significance are discussed. The multiscale entropy (MSE) was developed in physiological and biomedical contexts to diagnose irregularities in time-series data that sample the underlying dynamical systems, typically not governed by mathematical models. The standard MSE algorithm evaluates the average rate at which new information is created at different scales. Direct numerical simulations of Burgers turbulence (as a one-dimensional analogue of the three-dimensional Navier--Stokes turbulence) are performed and analyzed by evaluating the standard MSE. Results show that MSE provides consistent and physically meaningful descriptions of Burgers turbulence when compared to the Fourier analysis.   

Reduced order representations of turbulent Kolmogorov flow using machine learning

A long-standing challenge in low-order modelling is to design reduced representations of turbulent flows which are connected to the underlying dynamical system. In this work, we train a family of deep convolutional autoencoders to identify highly efficient low-dimensional representations of monochromatically forced, two-dimensional turbulence which are connected to the simple invariant solutions embedded in the turbulent attractor. We establish this connection by developing a technique we term ``latent Fourier analysis'' — a decomposition of the low-dimensional latent representation of vorticity into a set of orthogonal modes parameterised by latent wavenumbers. Individual latent Fourier modes decode into physically meaningful recurrent patterns. We show how projections onto latent Fourier modes can identify different dynamical regimes visited by the turbulence. Moreover, using latent Fourier modes to measure near recurrences along long turbulent orbits results in the identification of an order of magnitude more periodic orbits than are flagged using a Euclidean norm in physical space.   

Linearly forced fluid flow on a rotating sphere

We investigate generalized Navier–Stokes (GNS) equations that couple nonlinear advection with a generic linear instability. This analytically tractable minimal model for fluid flows driven by internal active stresses has recently been shown to permit exact solutions on a stationary two-dimensional sphere. Here, we extend the analysis to linearly driven flows on rotating spheres. We derive exact solutions of the GNS equations corresponding to time-independent zonal jets and superposed westward-propagating Rossby waves, qualitatively similar to those seen in planetary atmospheres. Direct numerical simulations with large rotation rates obtain statistically stationary states close to these exact solutions. The measured phase speeds of waves in the GNS simulations agree with analytical predictions for Rossby waves.