On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

The relationship between Koopman mode decomposition, resolvent mode decomposition and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalised to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatio-temporal Koopman operator, which has a travelling wave interpretation. The relationship leads to a generalisation of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known

A foundation for analytical developments in the logarithmic region of turbulent channels

An analytical framework for studying the logarithmic region of turbulent channels is formulated. We build on recent findings (Moarref et al., J. Fluid Mech., 734, 2013) that the velocity fluctuations in the logarithmic region can be decomposed into a weighted sum of geometrically self-similar resolvent modes. The resolvent modes and the weights represent the linear amplification mechanisms and the scaling influence of the nonlinear interactions in the Navier-Stokes equations (NSE), respectively (McKeon & Sharma, J. Fluid Mech., 658, 2010). Originating from the NSE, this framework provides an analytical support for Townsend's attached-eddy model. Our main result is that self-similarity enables order reduction in modeling the logarithmic region by establishing a quantitative link between the self-similar structures and the velocity spectra. Specifically, the energy intensities, the Reynolds stresses, and the energy budget are expressed in terms of the resolvent modes with speeds corresponding to the top of the logarithmic region. The weights of the triad modes -the modes that directly interact via the quadratic nonlinearity in the NSE- are coupled via the interaction coefficients that depend solely on the resolvent modes (McKeon et al., Phys. Fluids, 25, 2013). We use the hierarchies of self-similar modes in the logarithmic region to extend the notion of triad modes to triad hierarchies. It is shown that the interaction coefficients for the triad modes that belong to a triad hierarchy follow an exponential function. The combination of these findings can be used to better understand the dynamics and interaction of flow structures in the logarithmic region. The compatibility of the proposed model with theoretical and experimental results is further discussed.Comment: Submitted to J. Fluid Mec

Basis for finding exact coherent states

One of the outstanding problems in the dynamical systems approach to turbulence is to find a sufficient number of invariant solutions to characterize the underlying dynamics of turbulence [Annu. Rev. Fluid Mech. 44, 203 (2012)]. As a practical matter, the solutions can be difficult to find. To improve this situation, we show how to find periodic orbits and equilibria in plane Couette flow by projecting pseudorecurrent segments of turbulent trajectories onto the left-singular vectors of the Navier-Stokes equations linearized about the relevant mean flow (resolvent modes). The projections are, subsequently, used to initiate Newton-Krylov-hookstep searches, and new (relative) periodic orbits and equilibria are discovered. We call the process project-then-search and validate the process by first applying it to previously known fixed point and periodic solutions. Along the way, we find new branches of equilibria, which include bifurcations from previously known branches, and new periodic orbits that closely shadow turbulent trajectories in state space

Periodic Shadowing Sensitivity Analysis of Chaotic Systems

The sensitivity of long-time averages of a hyperbolic chaotic system to parameter perturbations can be determined using the shadowing direction, the uniformly-bounded-in-time solution of the sensitivity equations. Although its existence is formally guaranteed for certain systems, methods to determine it are hardly available. One practical approach is the Least-Squares Shadowing (LSS) algorithm (Q Wang, SIAM J Numer Anal 52, 156, 2014), whereby the shadowing direction is approximated by the solution of the sensitivity equations with the least square average norm. Here, we present an alternative, potentially simpler shadowing-based algorithm, termed periodic shadowing. The key idea is to obtain a bounded solution of the sensitivity equations by complementing it with periodic boundary conditions in time. We show that this is not only justifiable when the reference trajectory is itself periodic, but also possible and effective for chaotic trajectories. Our error analysis shows that periodic shadowing has the same convergence rates as LSS when the time span $T$ is increased: the sensitivity error first decays as $1/T$ and then, asymptotically as $1/\sqrt{T}$. We demonstrate the approach on the Lorenz equations, and also show that, as $T$ tends to infinity, periodic shadowing sensitivities converge to the same value obtained from long unstable periodic orbits (D Lasagna, SIAM J Appl Dyn Syst 17, 1, 2018) for which there is no shadowing error. Finally, finite-difference approximations of the sensitivity are also examined, and we show that subtle non-hyperbolicity features of the Lorenz system introduce a small, yet systematic, bias

Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known

Worldwide trends in diabetes since 1980: a pooled analysis of 751 population-based studies with 4.4 million participants

BACKGROUND: One of the global targets for non-communicable diseases is to halt, by 2025, the rise in the age-standardised adult prevalence of diabetes at its 2010 levels. We aimed to estimate worldwide trends in diabetes, how likely it is for countries to achieve the global target, and how changes in prevalence, together with population growth and ageing, are affecting the number of adults with diabetes. METHODS: We pooled data from population-based studies that had collected data on diabetes through measurement of its biomarkers. We used a Bayesian hierarchical model to estimate trends in diabetes prevalence—defined as fasting plasma glucose of 7·0 mmol/L or higher, or history of diagnosis with diabetes, or use of insulin or oral hypoglycaemic drugs—in 200 countries and territories in 21 regions, by sex and from 1980 to 2014. We also calculated the posterior probability of meeting the global diabetes target if post-2000 trends continue. FINDINGS: We used data from 751 studies including 4 372 000 adults from 146 of the 200 countries we make estimates for. Global age-standardised diabetes prevalence increased from 4·3% (95% credible interval 2·4–7·0) in 1980 to 9·0% (7·2–11·1) in 2014 in men, and from 5·0% (2·9–7·9) to 7·9% (6·4–9·7) in women. The number of adults with diabetes in the world increased from 108 million in 1980 to 422 million in 2014 (28·5% due to the rise in prevalence, 39·7% due to population growth and ageing, and 31·8% due to interaction of these two factors). Age-standardised adult diabetes prevalence in 2014 was lowest in northwestern Europe, and highest in Polynesia and Micronesia, at nearly 25%, followed by Melanesia and the Middle East and north Africa. Between 1980 and 2014 there was little change in age-standardised diabetes prevalence in adult women in continental western Europe, although crude prevalence rose because of ageing of the population. By contrast, age-standardised adult prevalence rose by 15 percentage points in men and women in Polynesia and Micronesia. In 2014, American Samoa had the highest national prevalence of diabetes (>30% in both sexes), with age-standardised adult prevalence also higher than 25% in some other islands in Polynesia and Micronesia. If post-2000 trends continue, the probability of meeting the global target of halting the rise in the prevalence of diabetes by 2025 at the 2010 level worldwide is lower than 1% for men and is 1% for women. Only nine countries for men and 29 countries for women, mostly in western Europe, have a 50% or higher probability of meeting the global target. INTERPRETATION: Since 1980, age-standardised diabetes prevalence in adults has increased, or at best remained unchanged, in every country. Together with population growth and ageing, this rise has led to a near quadrupling of the number of adults with diabetes worldwide. The burden of diabetes, both in terms of prevalence and number of adults affected, has increased faster in low-income and middle-income countries than in high-income countries. FUNDING: Wellcome Trust