Low-dimensional models for turbulent plane Couette flow in a minimal flow unit

We model turbulent plane Couette flow in the minimal flow unit (MFU) – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence – by expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier–Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on ‘uncoupled’ eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier–Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the ‘backbone’ for MFU turbulence is a periodic orbit, and support the roll–streak–breakdown–roll reformation picture of shear-driven turbulence

R-dimensional ESPRIT-type algorithms for strictly second-order non-circular sources and their performance analysis

High-resolution parameter estimation algorithms designed to exploit the prior knowledge about incident signals from strictly second-order (SO) non-circular (NC) sources allow for a lower estimation error and can resolve twice as many sources. In this paper, we derive the R-D NC Standard ESPRIT and the R-D NC Unitary ESPRIT algorithms that provide a significantly better performance compared to their original versions for arbitrary source signals. They are applicable to shift-invariant R-D antenna arrays and do not require a centrosymmetric array structure. Moreover, we present a first-order asymptotic performance analysis of the proposed algorithms, which is based on the error in the signal subspace estimate arising from the noise perturbation. The derived expressions for the resulting parameter estimation error are explicit in the noise realizations and asymptotic in the effective signal-to-noise ratio (SNR), i.e., the results become exact for either high SNRs or a large sample size. We also provide mean squared error (MSE) expressions, where only the assumptions of a zero mean and finite SO moments of the noise are required, but no assumptions about its statistics are necessary. As a main result, we analytically prove that the asymptotic performance of both R-D NC ESPRIT-type algorithms is identical in the high effective SNR regime. Finally, a case study shows that no improvement from strictly non-circular sources can be achieved in the special case of a single source.Comment: accepted at IEEE Transactions on Signal Processing, 15 pages, 6 figure

Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection

By utilizing diffusion maps embedding and transition matrix analysis we investigate sparse temperature measurement time-series data from Rayleigh--B\'enard convection experiments in a cylindrical container of aspect ratio $\Gamma=D/L=0.5$ between its diameter ($D$) and height ($L$). We consider the two cases of a cylinder at rest and rotating around its cylinder axis. We find that the relative amplitude of the large-scale circulation (LSC) and its orientation inside the container at different points in time are associated to prominent geometric features in the embedding space spanned by the two dominant diffusion-maps eigenvectors. From this two-dimensional embedding we can measure azimuthal drift and diffusion rates, as well as coherence times of the LSC. In addition, we can distinguish from the data clearly the single roll state (SRS), when a single roll extends through the whole cell, from the double roll state (DRS), when two counter-rotating rolls are on top of each other. Based on this embedding we also build a transition matrix (a discrete transfer operator), whose eigenvectors and eigenvalues reveal typical time scales for the stability of the SRS and DRS as well as for the azimuthal drift velocity of the flow structures inside the cylinder. Thus, the combination of nonlinear dimension reduction and dynamical systems tools enables to gain insight into turbulent flows without relying on model assumptions

Weak turbulence theory for rotating magnetohydrodynamics and planetary dynamos

A weak turbulence theory is derived for magnetohydrodynamics under rapid rotation and in the presence of a large-scale magnetic field. The angular velocity $\Omega_0$ is assumed to be uniform and parallel to the constant Alfv\'en speed ${\bf b_0}$. Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length $d \equiv b_0/\Omega_0$. In the large-scale limit ($kd \to 0$; $k$ being the wave number), the left- and right-handed waves tend respectively to the inertial and magnetostrophic waves whereas in the small-scale limit ($kd \to + \infty$) pure Alfv\'en waves are recovered. By using a complex helicity decomposition, the asymptotic weak turbulence equations are derived which describe the long-time behavior of weakly dispersive interacting waves {\it via} three-wave interaction processes. It is shown that the nonlinear dynamics is mainly anisotropic with a stronger transfer perpendicular ($\perp$) than parallel ($\parallel$) to the rotating axis. The general theory may converge to pure weak inertial/magnetostrophic or Alfv\'en wave turbulence when the large or small-scales limits are taken respectively. Inertial wave turbulence is asymptotically dominated by the kinetic energy/helicity whereas the magnetostrophic wave turbulence is dominated by the magnetic energy/helicity. For both regimes a family of exact solutions are found for the spectra which do not correspond necessarily to a maximal helicity state. It is shown that the hybrid helicity exhibits a cascade whose direction may vary according to the scale $k_f$ at which the helicity flux is injected with an inverse cascade if $k_fd < 1$ and a direct cascade otherwise. The theory is relevant for the magnetostrophic dynamo whose main applications are the Earth and giant planets for which a small ($\sim 10^{-6}$) Rossby number is expected.Comment: 4 figures, 33 page

A different view on the vector-valued empirical mode decomposition (VEMD)

The empirical mode decomposition (EMD) has achieved its reputation by providing a multi-scale time-frequency representation of nonlinear and/or nonstationary signals. To extend this method to vector-valued signals (VvS) in multidimensional (multi-D) space, a multivariate EMD (MEMD) has been designed recently, which employs an ensemble projection to extract local extremum locations (LELs) of the given VvS with respect to different projection directions. This idea successfully overcomes the problems of locally defining extrema of VvS. Different from the MEMD, where vector-valued envelopes (VvEs) are interpolated based on LELs extracted from the 1-D projected signal, the vector-valued EMD (VEMD) proposed in this paper employs a novel back projection method to interpolate the VvEs from 1-D envelopes in the projected space. Considering typical 4-D coordinates (3-D location and time), we show by numerical simulations that the VEMD outperforms state-of-art methods.Comment: 7th International Congress on Image and Signal Processing (CISP

Data-driven multivariate and multiscale methods for brain computer interface

This thesis focuses on the development of data-driven multivariate and multiscale methods for brain computer interface (BCI) systems. The electroencephalogram (EEG), the most convenient means to measure neurophysiological activity due to its noninvasive nature, is mainly considered. The nonlinearity and nonstationarity inherent in EEG and its multichannel recording nature require a new set of data-driven multivariate techniques to estimate more accurately features for enhanced BCI operation. Also, a long term goal is to enable an alternative EEG recording strategy for achieving long-term and portable monitoring. Empirical mode decomposition (EMD) and local mean decomposition (LMD), fully data-driven adaptive tools, are considered to decompose the nonlinear and nonstationary EEG signal into a set of components which are highly localised in time and frequency. It is shown that the complex and multivariate extensions of EMD, which can exploit common oscillatory modes within multivariate (multichannel) data, can be used to accurately estimate and compare the amplitude and phase information among multiple sources, a key for the feature extraction of BCI system. A complex extension of local mean decomposition is also introduced and its operation is illustrated on two channel neuronal spike streams. Common spatial pattern (CSP), a standard feature extraction technique for BCI application, is also extended to complex domain using the augmented complex statistics. Depending on the circularity/noncircularity of a complex signal, one of the complex CSP algorithms can be chosen to produce the best classification performance between two different EEG classes. Using these complex and multivariate algorithms, two cognitive brain studies are investigated for more natural and intuitive design of advanced BCI systems. Firstly, a Yarbus-style auditory selective attention experiment is introduced to measure the user attention to a sound source among a mixture of sound stimuli, which is aimed at improving the usefulness of hearing instruments such as hearing aid. Secondly, emotion experiments elicited by taste and taste recall are examined to determine the pleasure and displeasure of a food for the implementation of affective computing. The separation between two emotional responses is examined using real and complex-valued common spatial pattern methods. Finally, we introduce a novel approach to brain monitoring based on EEG recordings from within the ear canal, embedded on a custom made hearing aid earplug. The new platform promises the possibility of both short- and long-term continuous use for standard brain monitoring and interfacing applications

Spin-SILC: CMB polarisation component separation with spin wavelets

We present Spin-SILC, a new foreground component separation method that accurately extracts the cosmic microwave background (CMB) polarisation $E$ and $B$ modes from raw multifrequency Stokes $Q$ and $U$ measurements of the microwave sky. Spin-SILC is an internal linear combination method that uses spin wavelets to analyse the spin-2 polarisation signal $P = Q + iU$. The wavelets are additionally directional (non-axisymmetric). This allows different morphologies of signals to be separated and therefore the cleaning algorithm is localised using an additional domain of information. The advantage of spin wavelets over standard scalar wavelets is to simultaneously and self-consistently probe scales and directions in the polarisation signal $P = Q + iU$ and in the underlying $E$ and $B$ modes, therefore providing the ability to perform component separation and $E$-$B$ decomposition concurrently for the first time. We test Spin-SILC on full-mission Planck simulations and data and show the capacity to correctly recover the underlying cosmological $E$ and $B$ modes. We also demonstrate a strong consistency of our CMB maps with those derived from existing component separation methods. Spin-SILC can be combined with the pseudo- and pure $E$-$B$ spin wavelet estimators presented in a companion paper to reliably extract the cosmological signal in the presence of complicated sky cuts and noise. Therefore, it will provide a computationally-efficient method to accurately extract the CMB $E$ and $B$ modes for future polarisation experiments.Comment: 13 pages, 9 figures. Minor changes to match version published in MNRAS. Map products available at http://www.silc-cmb.org. Companion paper: arXiv:1605.01414 "Wavelet reconstruction of pure E and B modes for CMB polarisation and cosmic shear analyses" (B. Leistedt et al.