Brief Communication: Dimensionality Reduction in Total Dynamic Mode Decomposition Using A Simple Geometric Method

Dynamic mode decomposition (DMD) and its variants have emerged as popular methods for the post-processing of fluid dynamics' simulations in order to visualize dominant coherent structures and to reduce the practical degrees of freedom to a restricted set of ``modes''. In this brief communication we provide a geometric method for choosing the number of modes for the Total DMD technique and test its efficacy using a synthetic example (to examine the effect of noise) and a cylinder wake case.Comment: 8 pages, 3 figure

Gradual multifractal reconstruction of time-series: Formulation of the method and an application to the coupling between stock market indices and their Hoelder exponents

A technique termed gradual multifractal reconstruction (GMR) is formulated. A continuum is defined from a signal that preserves the pointwise Hoelder exponent (multifractal) structure of a signal but randomises the locations of the original data values with respect to this (φ=0), to the original signal itself(φ=1). We demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dual-tree complex wavelet transform. That is, the φ=0 end of the continuum is realised using the recently proposed iterated, amplitude adjusted wavelet transform algorithm (Keylock, 2017) that fully randomises the wavelet phases. This is extended to the GMR formulation by selective phase randomisation depending on whether or not the wavelet coefficient amplitudes exceeds a threshold criterion. An econophysics application of the technique is presented. The relation between the normalised log-returns and their Hoelder exponents for the daily returns of eight financial indices are compared. One particularly noticeable result is the change for the two American indices (NASDAQ 100 and S & P 500) from a non-significant to a strongly significant (as determined using GMR) cross-correlation between the returns and their Hoelder exponents from before the 2008 crash to afterwards. This is also reflected in the skewness of the phase difference distributions, which exhibit a geographical structure, with Asian markets not exhibiting significant skewness in contrast to those from elsewhere globally

Multifractal surrogate-data generation algorithm that preserves pointwise Hölder regularity structure, with initial applications to turbulence

An algorithm is described that can generate random variants of a time series while preserving the probability distribution of original values and the pointwise Hoelder regularity. Thus, it preserves the multifractal properties of the data. Our algorithm is similar in principle to well-known algorithms based on the preservation of the Fourier amplitude spectrum and original values of a time series. However, it is underpinned by a dual-tree complex wavelet transform rather than a Fourier transform. Our method, which we term the iterated amplitude adjusted wavelet transform can be used to generate bootstrapped versions of multifractal data, and because it preserves the pointwise Hoelder regularity but not the local Hoelder regularity, it can be used to test hypotheses concerning the presence of oscillating singularities in a time series, an important feature of turbulence and econophysics data. Because the locations of the data values are randomized with respect to the multifractal structure, hypotheses about their mutual coupling can be tested, which is important for the velocity-intermittency structure of turbulence and self-regulating processes

The Schur decomposition of the velocity gradient tensor for turbulent flows

The velocity gradient tensor for turbulent flow contains crucial information on the topology of turbulence, vortex stretching and the dissipation of energy. A Schur decomposition of the velocity gradient tensor (VGT) is introduced to supplement the standard decomposition into rotation and strain tensors. Thus, the normal parts of the tensor (represented by the eigenvalues) are separated explicitly from non-normality. Using a direct numerical simulation of homogeneous isotropic turbulence, it is shown that the norm of the non-normal part of the tensor is of a similar magnitude to the normal part. It is common to examine the second and third invariants of the characteristic equation of the tensor simultaneously (the diagram). With the Schur approach, the discriminant function separating real and complex eigenvalues of the VGT has an explicit form in terms of strain and enstrophy: where eigenvalues are all real, enstrophy arises from the non-normal term only. Re-deriving the evolution equations for enstrophy and total strain highlights the production of non-normality and interaction production (normal straining of non-normality). These cancel when considering the evolution of the VGT in terms of its eigenvalues but are important for the full dynamics. Their properties as a function of location in space are characterized. The Schur framework is then used to explain two properties of the VGT: the preference to form disc-like rather than rod-like flow structures, and the vorticity vector and strain alignments. In both cases, non-normality is critical for explaining behaviour in vortical regions

Hypothesis testing for nonlinear phenomena in the geosciences using synthetic, surrogate data

©2018. The Authors. Studying nonlinear and potentially chaotic phenomena in geophysics from measured signals is problematic when system noise interferes with the dynamic processes that one is trying to infer. In such circumstances, a framework for statistical hypothesis testing is necessary but the nonlinear nature of the phenomena studied makes the formulation of standard hypothesis tests, such as analysis of variance, problematic as they are based on underlying linear, Gaussian assumptions. One approach to this problem is the method of surrogate data, which is the technique explained in this paper. In particular, we focus on (i) hypothesis testing for nonlinearity by generating linearized surrogates as a null hypothesis, (ii) a variant of this that is perhaps more appropriate for image data where structural nonlinearities are common and should be retained in the surrogates, and (iii) gradual reconstruction where we systematically constrain the surrogates until there is no significant difference between data and surrogates and use this to understand geophysical processes. In addition to time series of sunspot activity, solutions to the Lorenz equations, and spatial maps of enstrophy in a turbulent channel flow, two examples are considered in detail. The first concerns gradual wavelet reconstruction testing of the significance of a specific vortical flow structure from turbulence time series acquired at a point. In the second, the degree of nonlinearity in the spatial profiles of river curvature is shown to be affected by the occurrence of meander cutoff processes but in a more complex fashion than previously envisaged

Large eddy simulation of the velocity-intermittency structure for flow over a field of symmetric dunes

Owing to their frequent occurrence in the natural environment, there has been significant interest in refining our understanding of flow over dunes and other bedforms. Recent work in this area has focused, in particular, on their shear-layer characteristics and the manner by which flow structures are generated. However, field-based studies, are reliant on single-, or multi-point measurements, rather than delimiting flow structures from the velocity gradient tensor, as is possible in numerical work. Here, we extract pointwise time series from a well-resolved large eddy simulation as a means to connect these two approaches. The at-a-point analysis technique is termed the velocity-intermittency quadrant method and relates the fluctuating, longitudinal velocity, u 1 ′ (t) , to its fluctuating pointwise Hölder regularity, . Despite the difference in boundary conditions, our results agree very well with previous experiments that show the importance, in the region above the dunes, of a quadrant 3 (u 1 ′ 0) dominant near the wall and quadrant 2 dominant close to the lower part of the separated shear layer. These results are consistent with a near-wall advection of vorticity into a region downstream of a temporarily foreshortened reattachment region, and the entrainment of slow moving and quiescent fluid into a faster, more turbulent shear layer. A comparison of instantaneous vorticity fields to the velocity-intermittency analysis shows how the pointwise results reflect larger-scale organisation of the flow. We illustrate this using results from two instantaneous datasets. In the former, extreme velocity-intermittency events corresponding to a foreshortened recirculation region (and high pressures on the stoss slope of the dune immediately downstream) arise, and the development of intense flow structures occurs as a consequence. In the other case, development of a 'skimming flow' with relatively little exchange between the inner and outer regions results in exceedances because of the coherence associated with this high velocity, high turbulence outer region. Thus, our results shed further light on the characteristics of dune flow in the near-wall region and, importantly for field-based research, show that useful information on flow structure can be obtained from single-point single velocity component measurements

A joint velocity-intermittency analysis reveals similarity in the vertical structure of atmospheric and hydrospheric canopy turbulence

Turbulent fow through and over vegetation continues to draw signifcant research attention given its relevance to a plethora of applications in earth and environmental science. Canopy fows are characterized by three-dimensional coherent vortical motions not directly accessible from single-point measurements, which pose a challenge to formalizing links between vegetation structure and turbulent motion. A joint velocity-intermittency technique is applied to velocity data collected within and above aquatic vegetation in a hydraulic fume and above a forested canopy. The approach reveals behavior that provides greater insight into canopy fow dynamics than may be inferred from the vertical profles of mean velocity, turbulence intensity and Reynolds stresses, which are the quantities usually studied. There is a remarkable similarity in the structure of such fows between the forest canopy and the fume study despite large diferences in morphology and stem rigidity. In particular, these results determine an outer fow type arising above 1.5 canopy heights, while turbulent in-rushing events are most signifcant at the zero-plane displacement. The approach also implies ways in which improved models for canopy turbulence may be developed

A gene trap transposon eliminates haematopoietic expression of zebrafish Gfi1aa, but does not interfere with haematopoiesis

A transposon-mediated gene trap screen identified the zebrafish line qmc551 that expresses a GFP reporter in primitive erythrocytes and also in haemogenic endothelial cells, which give rise to haematopoietic stem and progenitor cells (HSPCs) that seed sites of larval and adult haematopoiesis. The transposon that mediates this GFP expression is located in intron 1 of the gfi1aa gene, one of three zebrafish paralogs that encode transcriptional repressors homologous to mammalian Gfi1 and Gfi1b proteins. In qmc551 transgenics, GFP expression is under the control of the endogenous gfi1aa promoter, recapitulates early gfi1aa expression and allows live observation of gfi1aa promoter activity. While the transposon integration interferes with the expression of gfi1aa mRNA in haematopoietic cells, homozygous qmc551 fish are viable and fertile, and display normal primitive and definitive haematopoiesis. Retained expression of Gfi1b in primitive erythrocytes and upregulation of Gfi1ab at the onset of definitive haematopoiesis in homozygous qmc551 carriers, are sufficient to allow normal haematopoiesis. This finding contradicts previously published morpholino data that suggested an essential role for zebrafish Gfi1aa in primitive erythropoiesi

Studying turbulence structure near the wall in hydrodynamic flows: An approach based on the Schur decomposition of the velocity gradient tensor

The Schur decomposition of the velocity gradient tensor (VGT) is an alternative to the classical Cauchy-Stokes decomposition into rotation rate and strain rate components. Recently, there have been several strands of work that have employed this decomposition to examine the physics of turbulence dynamics, including approaches that combine the Schur and Cauchy-Stokes formalisms. These are briefly reviewed before the latter approach is set out. This partitions the rotation rate and strain rate tensors into normal/local and non-normal/non-local contributions. We then study the relation between the VGT dynamics and ejection-sweep events in a channel flow boundary-layer. We show that the sweeps in particular exhibit novel behaviour compared with either the other quadrants, or the flow in general, with a much-reduced contribution to the dynamics from the non-normal terms above the viscous sub-layer. In particular, the reduction in the production term that is the interaction between the non-normality and the normal straining reduces in the log-layer as a consequence of an absence of alignment between the non-normal vorticity and the strain rate eigenvectors. There have been early forays into using the Schur transform approach for subgrid-scale modelling in large-eddy simulation (LES) and this would appear to be an exciting way forward