Reduced Models of Point Vortex Systems in Quasigeostrophic Fluid Dynamics

We develop a nonequilibrium statistical mechanical description of the evolution of point vortex systems governed by either the Euler, single-layer quasigeostrophic or two-layer quasigeostrophic equations. Our approach is based on a recently proposed optimal closure procedure for deriving reduced models of Hamiltonian systems. In this theory the statistical evolution is kept within a parametric family of distributions based on the resolved variables chosen to describe the macrostate of the system. The approximate evolution is matched as closely as possible to the true evolution by minimizing the mean-squared residual in the Liouville equation, a metric which quantifies the information loss rate due to model reduction. The point vortex approximation of the fluid dynamics allows the optimal closure, which is formulated on phase space, to be transferred to physical space resulting in an exact mean-field theory for the continuum limit. The near-equilibrium linearization of this theory is used to model the intrinsic relaxation rates for isolated coherent vorticity structures. The equilibration of initially disturbed vorticity fields is captured by a reduced model that has few resolved variables and no adjustable parameters. For the Euler and single-layer quasigeostrophic equations, the theory is used to model the axisymmetrization of a deformed vorticity patch. In particular, the reduced model exhibits how the rate of symmetrization depends upon the energy and the Rossby deformation radius. For the two-layer equations the study focuses on the relaxation of baroclinic perturbations of stable barotropic structures and the transfer of available potential energy to kinetic energy. The model predicts the dependence of the barotropization rate on the energy and the internal Rossby deformation radius. Both axisymmetrization and barotropization are prominent features of the coherent vortex structures observed in direct numerical simulations of two-dimensional and quasigeostrophic turbulence. The reduced model is tested against the evolution of an ensemble of point vortex systems to validate its predictions. Therefore, the reduced model furnishes a mathematical theory of these fluid dynamical phenomena

Reduced Models of Point Vortex Systems

Nonequilibrium statistical models of point vortex systems are constructed using an optimal closure method, and these models are employed to approximate the relaxation toward equilibrium of systems governed by the two-dimensional Euler equations, as well as the quasi-geostrophic equations for either single-layer or two-layer flows. Optimal closure refers to a general method of reduction for Hamiltonian systems, in which macroscopic states are required to belong to a parametric family of distributions on phase space. In the case of point vortex ensembles, the macroscopic variables describe the spatially coarse-grained vorticity. Dynamical closure in terms of those macrostates is obtained by optimizing over paths in the parameter space of the reduced model subject to the constraints imposed by conserved quantities. This optimization minimizes a cost functional that quantifies the rate of information loss due to model reduction, meaning that an optimal path represents a macroscopic evolution that is most compatible with the microscopic dynamics in an information-theoretic sense. A near-equilibrium linearization of this method is used to derive dissipative equations for the low-order spatial moments of ensembles of point vortices in the plane. These severely reduced models describe the late-stage evolution of isolated coherent structures in two-dimensional and geostrophic turbulence. For single-layer dynamics, they approximate the relaxation of initially distorted structures toward axisymmetric equilibrium states. For two-layer dynamics, they predict the rate of energy transfer in baroclinically perturbed structures returning to stable barotropic states. Comparisons against direct numerical simulations of the fully-resolved many-vortex dynamics validate the predictive capacity of these reduced model

Linear solvers for power grid optimization problems: a review of GPU-accelerated linear solvers

The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite and become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frameworks based on sparse LU or LDL^T decompositions. We benchmark five well known direct linear solver packages using matrices extracted from power grid optimization problems. The achieved solution accuracy varies greatly among the packages. None of the tested packages delivers significant GPU acceleration for our test cases

Deranged sodium to sudden death

In February 2014, a group of scientists convened as part of the University of California Davis Cardiovascular Symposium to bring together experimental and mathematical modelling perspectives and discuss points of consensus and controversy on the topic of sodium in the heart. This paper summarizes the topics of presentation and discussion from the symposium, with a focus on the role of aberrant sodium channels and abnormal sodium homeostasis in cardiac arrhythmias and pharmacotherapy from the subcellular scale to the whole heart. Two following papers focus on Na⁺ channel structure, function and regulation, and Na⁺/Ca²⁺ exchange and Na⁺/K⁺ ATPase. The UC Davis Cardiovascular Symposium is a biannual event that aims to bring together leading experts in subfields of cardiovascular biomedicine to focus on topics of importance to the field. The focus on Na⁺ in the 2014 symposium stemmed from the multitude of recent studies that point to the importance of maintaining Na⁺ homeostasis in the heart, as disruption of homeostatic processes are increasingly identified in cardiac disease states. Understanding how disruption in cardiac Na⁺-based processes leads to derangement in multiple cardiac components at the level of the cell and to then connect these perturbations to emergent behaviour in the heart to cause disease is a critical area of research. The ubiquity of disruption of Na⁺ channels and Na⁺ homeostasis in cardiac disorders of excitability and mechanics emphasizes the importance of a fundamental understanding of the associated mechanisms and disease processes to ultimately reveal new targets for human therapy.Centro de Investigaciones Cardiovasculare

Contractile Function during Angiotensin-II Activation:Increased Nox2 Activity Modulates Cardiac Calcium Handling via Phospholamban Phosphorylation

AbstractBackgroundRenin-angiotensin system activation is a feature of many cardiovascular conditions. Activity of myocardial reduced nicotinamide adenine dinucleotide phosphate oxidase 2 (NADPH oxidase 2 or Nox2) is enhanced by angiotensin II (Ang II) and contributes to increased hypertrophy, fibrosis, and adverse remodeling. Recent studies found that Nox2-mediated reactive oxygen species production modulates physiological cardiomyocyte function.ObjectivesThis study sought to investigate the effects of cardiomyocyte Nox2 on contractile function during increased Ang II activation.MethodsWe generated a cardiomyocyte-targeted Nox2-transgenic mouse model and studied the effects of in vivo and ex vivo Ang II stimulation, as well as chronic aortic banding.ResultsChronic subpressor Ang II infusion induced greater cardiac hypertrophy in transgenic than wild-type mice but unexpectedly enhanced contractile function. Acute Ang II treatment also enhanced contractile function in transgenic hearts in vivo and transgenic cardiomyocytes ex vivo. Ang II–stimulated Nox2 activity increased sarcoplasmic reticulum (SR) Ca2+ uptake in transgenic mice, increased the Ca2+ transient and contractile amplitude, and accelerated cardiomyocyte contraction and relaxation. Elevated Nox2 activity increased phospholamban phosphorylation in both hearts and cardiomyocytes, related to inhibition of protein phosphatase 1 activity. In a model of aortic banding–induced chronic pressure overload, heart function was similarly depressed in transgenic and wild-type mice.ConclusionsWe identified a novel mechanism in which Nox2 modulates cardiomyocyte SR Ca2+ uptake and contractile function through redox-regulated changes in phospholamban phosphorylation. This mechanism can drive increased contractility in the short term in disease states characterized by enhanced renin-angiotensin system activation