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

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

Incident type 2 diabetes attributable to suboptimal diet in 184 countries

The global burden of diet-attributable type 2 diabetes (T2D) is not well established. This risk assessment model estimated T2D incidence among adults attributable to direct and body weight-mediated effects of 11 dietary factors in 184 countries in 1990 and 2018. In 2018, suboptimal intake of these dietary factors was estimated to be attributable to 14.1 million (95% uncertainty interval (UI), 13.814.4 million) incident T2D cases, representing 70.3% (68.871.8%) of new cases globally. Largest T2D burdens were attributable to insufficient whole-grain intake (26.1% (25.027.1%)), excess refined rice and wheat intake (24.6% (22.327.2%)) and excess processed meat intake (20.3% (18.323.5%)). Across regions, highest proportional burdens were in central and eastern Europe and central Asia (85.6% (83.487.7%)) and Latin America and the Caribbean (81.8% (80.183.4%)); and lowest proportional burdens were in South Asia (55.4% (52.160.7%)). Proportions of diet-attributable T2D were generally larger in men than in women and were inversely correlated with age. Diet-attributable T2D was generally larger among urban versus rural residents and higher versus lower educated individuals, except in high-income countries, central and eastern Europe and central Asia, where burdens were larger in rural residents and in lower educated individuals. Compared with 1990, global diet-attributable T2D increased by 2.6 absolute percentage points (8.6 million more cases) in 2018, with variation in these trends by world region and dietary factor. These findings inform nutritional priorities and clinical and public health planning to improve dietary quality and reduce T2D globally. (c) 2023, The Author(s)

Children's and adolescents' rising animal-source food intakes in 1990-2018 were impacted by age, region, parental education and urbanicity

Animal-source foods (ASF) provide nutrition for children and adolescents physical and cognitive development. Here, we use data from the Global Dietary Database and Bayesian hierarchical models to quantify global, regional and national ASF intakes between 1990 and 2018 by age group across 185 countries, representing 93% of the worlds child population. Mean ASF intake was 1.9 servings per day, representing 16% of children consuming at least three daily servings. Intake was similar between boys and girls, but higher among urban children with educated parents. Consumption varied by age from 0.6 at <1 year to 2.5 servings per day at 1519 years. Between 1990 and 2018, mean ASF intake increased by 0.5 servings per week, with increases in all regions except sub-Saharan Africa. In 2018, total ASF consumption was highest in Russia, Brazil, Mexico and Turkey, and lowest in Uganda, India, Kenya and Bangladesh. These findings can inform policy to address malnutrition through targeted ASF consumption programmes. (c) 2023, The Author(s)

Repositioning of the global epicentre of non-optimal cholesterol

High blood cholesterol is typically considered a feature of wealthy western countries1,2. However, dietary and behavioural determinants of blood cholesterol are changing rapidly throughout the world3 and countries are using lipid-lowering medications at varying rates. These changes can have distinct effects on the levels of high-density lipoprotein (HDL) cholesterol and non-HDL cholesterol, which have different effects on human health4,5. However, the trends of HDL and non-HDL cholesterol levels over time have not been previously reported in a global analysis. Here we pooled 1,127 population-based studies that measured blood lipids in 102.6 million individuals aged 18 years and older to estimate trends from 1980 to 2018 in mean total, non-HDL and HDL cholesterol levels for 200 countries. Globally, there was little change in total or non-HDL cholesterol from 1980 to 2018. This was a net effect of increases in low- and middle-income countries, especially in east and southeast Asia, and decreases in high-income western countries, especially those in northwestern Europe, and in central and eastern Europe. As a result, countries with the highest level of non-HDL cholesterol—which is a marker of cardiovascular risk—changed from those in western Europe such as Belgium, Finland, Greenland, Iceland, Norway, Sweden, Switzerland and Malta in 1980 to those in Asia and the Pacific, such as Tokelau, Malaysia, The Philippines and Thailand. In 2017, high non-HDL cholesterol was responsible for an estimated 3.9 million (95% credible interval 3.7 million–4.2 million) worldwide deaths, half of which occurred in east, southeast and south Asia. The global repositioning of lipid-related risk, with non-optimal cholesterol shifting from a distinct feature of high-income countries in northwestern Europe, north America and Australasia to one that affects countries in east and southeast Asia and Oceania should motivate the use of population-based policies and personal interventions to improve nutrition and enhance access to treatment throughout the world

Diminishing benefits of urban living for children and adolescents’ growth and development

Optimal growth and development in childhood and adolescence is crucial for lifelong health and well-being1–6. Here we used data from 2,325 population-based studies, with measurements of height and weight from 71 million participants, to report the height and body-mass index (BMI) of children and adolescents aged 5–19 years on the basis of rural and urban place of residence in 200 countries and territories from 1990 to 2020. In 1990, children and adolescents residing in cities were taller than their rural counterparts in all but a few high-income&nbsp;countries. By 2020, the urban height advantage became smaller in most countries, and in many high-income western countries it reversed into a small urban-based disadvantage. The exception was for boys in most countries in sub-Saharan Africa and in some countries in Oceania, south Asia and the region of central Asia, Middle East and north Africa. In these countries, successive cohorts of boys from rural places either did not gain height or possibly became shorter, and hence fell further behind their urban peers. The difference between the age-standardized mean BMI of children in urban and rural areas was &lt;1.1 kg m–2 in the vast majority of&nbsp;countries. Within this small range, BMI increased slightly more in cities than in rural areas, except in south Asia, sub-Saharan Africa and some countries in central and eastern Europe. Our results show that in much of the world, the growth and developmental advantages of living in cities have diminished in the twenty-first century, whereas in much of sub-Saharan Africa they have amplified