Divergence Of Velocity Field Research Articles

Automated detection of exploding granules with SDO/HMI data

Exploding granules on the solar surface play a major role in the dynamics of the outer part of the convection zone, especially in the diffusion of the magnetic field. We aim to develop an automated procedure able to investigate the location and evolution of exploding granules over the solar surface and to get rid of visual detection. We used sequences of observations of intensity and Doppler velocity, as well as magnetograms, provided by the Helioseismic and Magnetic Imager aboard the Solar Dynamics Observatory. The automated detection of the exploding granules was performed by applying criteria on either three or two parameters: the granule area, the amplitude of the velocity field divergence, and, at the disc centre, the radial Doppler velocity. Our analyses show that granule area and divergence amplitudes are sufficient to detect the largest exploding granules; thus, we can automatically detect them, not only at the disc centre, but across the whole solar surface. Using a 24-hour-long observation sequence, we have demonstrated the important contribution of the most dynamic exploding granules in the diffusion of the magnetic field in the quiet Sun. Indeed, we have shown that the most intense exploding granules are sufficient to build a large part of the photospheric network. We have also applied our procedure on Hinode observations to locate the exploding granules relative to trees of fragmenting granules (TFGs). We conclude that, during a first phase of about 300 minutes after the birth of a TFG, exploding granules are preferentially located on its edge. Finally, we also show that the distribution of exploding granules is homogeneous (at the level of our measurement errors) over the solar surface without a significant dependency on latitude.

Enhanced schemes for resolution of the continuity equation in projection-based SPH

To the best knowledge of the authors, the numerical resolution of the continuity equation, i.e., the simultaneous achievement of accurate and physically consistent velocity divergence and density fields, has not been investigated in the context of projection-based particle methods. In this study, based on the Velocity-divergence Error Mitigating (VEM)/Volume Conservation Shifting (VCS) schemes developed for the WCSPH method, which improve the numerical resolution of the continuity equation, VEM/VCS schemes are proposed for projection-based particle methods. Furthermore, two other novel schemes are introduced. First, a Dynamic VEM (DVEM) scheme is proposed, where VEM is dynamically activated only when the velocity divergence instantaneously diverge from the reference solution at individual particles. The second scheme is referred to as Enhanced VCS by reference density (EVCS), where precise imposition of dynamic free surface boundary condition is considered. Several benchmark tests with different features have been conducted, and it is shown that the proposed schemes enhance the numerical resolution of the continuity equation for incompressible flows, i.e., the divergence free velocity and the constant density conditions. Application of DVEM and EVCS has provided enhanced resolution of the continuity equation along with clear reductions in the numerical energy dissipation associated with the approximated additional numerical terms.

Heterogenous atrial conduction in patients with hypertrophic cardiomyopathy and atrial fibrillation

Abstract Background/Introduction Hypertrophic cardiomyopathy (HCM) is characterised by myocardial thickening, fibrosis, and disarray, affecting both ventricles and atria. This pathology extends to atrial tissue, potentially influencing conductivity and contributing to arrhythmias, notably atrial fibrillation (AF). Pulmonary vein isolation to treat AF in HCM patients yields less satisfactory results compared to other patient groups. Improved understanding of the pathophysiology of AF in HCM patients could improve treatment stratification and delivery. Purpose This study aims to analyse atrial conduction properties in HCM patients with AF, compared to age, body mass index and sex-matched control patients. Methods Pre-ablation electroanatomic mapping data from 10 HCM patients and a matched control group of patients with structurally normal hearts undergoing AF ablation was retrospectively analysed. All patients underwent pre-ablation electroanatomic mapping during proximal coronary sinus pacing. This data was analysed using OpenEP software to quantify voltage mapping, conduction velocity and wave collision points. Conduction velocity was calculated using both plane fitting and triangulation methods. Wave collisions were identified by calculating the percentage of points with conduction velocity field divergence values less than -1.5. Results HCM patients exhibited significantly reduced mean left atrial bipolar voltage and greater low voltage burden <0.5mV than matched control patients (1.43mV vs. 2.14mV, p=0.0177; and 38.28% vs. 15.68%, p<0.0001), (Figure 1A and B). Wave collision points were significantly more prevalent in the HCM group (5.4% vs. 1.6%, p=0.0004), indicating more heterogeneous conduction (Figure 1C). Figure 2 shows example of divergence map with local electrograms around wave collision points. Both methods of quantifying conduction velocity showed comparable results. While average conduction velocity showed no statistically significant difference (plane fitting method - 0.61m/s in HCM group vs. 0.64m/s in control group, p = 0.465; triangulation method - 0.61m/s in HCM group, 0.64m/s in control group, p=0.393), a trend toward lower velocities in HCM patients was observed (Figure 1D and 1E). Conclusion HCM patients demonstrate significantly lower bipolar voltage and significantly greater low voltage areas than matched control patients. Atrial conduction in patients with HCM is significantly more heterogenous than matched control patients. These findings highlight marked conduction abnormalities in HCM patients which are likely to contribute to AF pathogenesis.

Enhanced resolution of the continuity equation in explicit weakly compressible SPH simulations of incompressible free‐surface fluid flows

This paper is dedicated to improved resolution of the continuity equation and thus more precise satisfaction of incompressibility conditions in explicit weakly compressible SPH (Smoothed Particle Hydrodynamics) simulations of incompressible free-surface fluid flows. In the explicit weakly compressible SPH methods, including δ-SPH with a widely adopted particle shifting scheme (or the so-called δ-plus-SPH), two incompressibility conditions corresponding to the invariant density condition and divergence-free velocity condition are not well resolved. To achieve enhanced resolution of the continuity equation as well as satisfaction of both incompressibility conditions, this paper presents two novel schemes, namely, Velocity-divergence Error Mitigating (VEM) and Volume Conservation Shifting (VCS) schemes. The VEM scheme corresponds to an additional term in the momentum equation that mitigates the velocity divergence error at every computational time step. The VEM term is derived by linking the instantaneous velocity divergence error to the undesired density time variations and accordingly to an error mitigating pressure pVEM obtained through an Equation of State (EOS). The corresponding pressure gradient term or acceleration leads to attenuation of numerical noise in the velocity divergence field. The VCS acts as a distinct particle shifting scheme to enforce local and thus global volume conservation. This distinct shifting is obtained through calculation of the gradient of a corresponding pressure, namely, pVCS, obtained through efficient solution of a Poisson Pressure Equation (PPE) derived from the concept of projection-based particle methods. By adopting both proposed schemes, the explicit SPH method is shown to provide enhanced satisfaction of both incompressibility conditions, i.e., divergence-free velocity and invariant density conditions, in simulation of incompressible free-surface fluid flows.

Spatial Scales and Time Variation of Solar Subsurface Convection

Spectral analysis of the spatial structure of solar subphotospheric convection is carried out for subsurface flow maps constructed using the time–distance helioseismological technique. The source data are obtained from the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory from 2010 May to 2020 September. A spherical harmonic transform is applied to the horizontal velocity divergence field at depths from 0 to 19 Mm. The range of flow scales is fairly broad in the shallow layers and narrows as the depth increases. The horizontal flow scales increase rapidly with depth, from supergranulation to giant-cell values, and indicate the existence of large-scale convective motions in the near-surface shear layer. The results can naturally be interpreted in terms of a superposition of differently scaled flows localized at different depth intervals. There is some tendency toward the emergence of meridionally elongated (banana-shaped) convection structures in the deep layers. The total power of the convective flows is anticorrelated with the sunspot number variation over the solar activity cycle in shallow subsurface layers, and positively correlated at larger depths, which is suggestive of the depth redistribution of the convective flow energy due to the action of magnetic fields.

Divergence of Velocity Fields in Electrochemical Systems

The passage of current through electrochemical systems results in the development of concentration gradients in the electrolytic phase that can be modeled using concentrated solution theory. Application of this theory requires knowledge of three concentration-dependent transport coefficients, which are often taken to be conductivity, diffusion coefficient, and the cation transference number with respect to the solvent velocity. The governing diffusion equation for molar concentration contains two additional terms—the thermodynamic factor which is related to activity of the electrolytic species and the solvent velocity. The main advance in this paper is the derivation of an expression for the divergence of the solvent velocity. Solving this equation requires knowledge of the partial molar volume of the electrolyte. Analogous expressions are derived for the mass average and molar average velocities. These velocities occur naturally in the diffusion equation if concentration is expressed as weight fraction and mole fraction of the electrolytic species, respectively.

Superclusters from velocity divergence fields

ABSTRACT Superclusters are a convenient way to partition and characterize the large-scale structure of the Universe. In this Letter, we explore the advantages of defining superclusters as watershed basins in the divergence velocity field. We apply this definition on diverse data sets generated from linear theory and N-body simulations, with different grid sizes, smoothing scales, and types of tracers. From this framework emerges a linear scaling relation between the average supercluster size and the autocorrelation length in the divergence field, a result that holds for one order of magnitude from 10 up to 100 Mpc h−1. These results suggest that the divergence-based definition provides a robust context to quantitatively compare results across different observational or computational frameworks. Through its connection with linear theory, it can also facilitate the exploration of how supercluster properties depend on cosmological parameters, paving the way to use superclusters as cosmological probes.

Development of computationally efficient augmented Lagrangian SPH for incompressible flows and its quantitative comparison with WCSPH simulating flow past a circular cylinder

SummaryIn Lagrangian particle‐based methods such as smoothed particle hydrodynamics (SPH), computing totally divergence‐free velocity field in a flow domain with the smallest error possible is the most critical issue, which might be achieved through solving pressure Poisson equation implicitly with higher particle resolutions. However, implicit solutions are computationally expensive and may be particularly challenging in the solution of multiphase flows with highly nonlinear deformations as well as fluid‐structure interaction problems. Augmented Lagrangian SPH (ALSPH) method is a new alternative algorithm as a prevalent pressure solver where the divergence‐free velocity field is achieved by iterative calculation of velocity and pressure fields. This study investigates the performance of the ALSPH technique by solving a challenging flow problem such as two‐dimensional flow around a cylinder within the Reynolds number range of 50 to 500 in terms of improved robustness, accuracy, and computational efficiency. The same flow conditions are also simulated using the conventional weakly compressible SPH (WCSPH) method. The results of ALSPH and WCSPH solutions are not only compared in terms of numerical validation/verification studies, but also rigorous investigations are performed for all related physical flow characteristics, namely, hydrodynamic coefficients, frequency domain analyses, and velocity divergence fields.

Study on the Structure of a Horizontal Shear Line over the Tibetan Plateau Based on CRA-Interim Datasets and Its Comparison with ERA-Interim Datasets

The CRA-Interim trial production of the global atmospheric reanalysis for 10 years from 2007 to 2016 was carried out by the China Meteorological Administration in 2017. The structural characteristics of the horizontal shear line over the Tibetan Plateau (TPHSL) based on the CRA-Interim datasets are examined by objectively identifying the shear line, and are compared with the analysis results of the European Centre for Medium-Range Weather Forecasts reanalysis data (ERA-Interim). The case occurred at 18UTC on July 5, 2016. The results show that both of the ERA-Interim and CRA-Interim datasets can well reveal the circulation background and the dynamic and thermal structure characteristics of TPHSL, and they have shown some similar features. The middle and high latitudes at 500 hPa are characterized by the circulation situation oftwo troughs and two ridges, and at 200 hPa, the TPHSL is located in the northeast quadrant of the South Asian High Pressure (SAHP). The TPHSL locates in the positive vorticity zone and passes through the positive vorticity center corresponding to the ascending motion. Near the TPHSL, the contours of pseudo-equivalent potential temperature (θse) tend to be intensive, with a high-value center on the south side of the TPHSL. The TPHSL can extend to 460 hPa and vertically inclines northward. There is a positive vorticity zone near the TPHSL which is also characterized by the northward inclination with the height, the ascending motion near the TPHSL can extend to 300 hPa, and the atmospheric layer above the TPHSL is stable. However, the intensities of the TPHSL's structure characteristics analyzed with the two datasets are different, revealing the relatively strong intensity of geopotential height field, vertical velocity field, vorticity field and divergence field from the CRA-Interim datasets. In addition, the vertical profiles of the dynamic and water vapor thermal physical quantities of the two datasets are also consistent in the east and west part of the TPHSL. In summary, the reliable and usable CRA-Interim datasets show excellent properties in the analysis on the structural characteristics of a horizontal shear line over the Tibetan Plateau.

Advanced strain and mass transfer analysis in crustal-scale oroclinal buckling and detachment folding analogue models

The PIV (particle image velocimetry) method became a standard tool for the calculation of displacement fields in physical geodynamic models. For understanding the deformation dynamics of geodynamic models, in our study, we implemented several post-processing algorithms on the derived displacement field and calculated the velocity and strain(-rate) components, such as the divergence of the velocity field, vorticity and shear strain-rate.In the model of oroclinal buckling, we focused on strain analysis of the upper crust and correlated the shear strain-rate, vorticity and divergence anomalies with visual deformation patterns in the upper crust. The divergence of velocity fields in these models correspond to the pop-up and pop-down belts oriented along the axial trace of the oroclinal bends. High shear strain-rate domains correlate with horizontal, isovolumic shear zones alongside these belts, while vorticity shows rotational trend of fold axial traces of the pop-up and pop-down belts, around orocline inflection.In another series of models, we simulated the development of melt-cored crustal scale detachment folds and employed the same set of parameters to investigate the ductile deformation visible in side-view of the model domain. We developed a method that allows tracing the divergence in subcells locked on target subdomains. We tracked and quantified melt flow between the melt source area at the bottom of the model and progressively developing folds. This mass transfer analysis revealed polyphase fold evolution, where initial fold perturbations quickly amplify as the melt accumulates in the triangular hinge sector below and between the rotating fold limbs. While the early amplification leads to decompression driving the melt into the hinge zone area between the limbs, the fold lock-up stage and continued attenuation of the vertical limbs is associated with melt expulsion from the fold interlimb domain back into the source layer, where it can be transferred laterally to the foreland.

Exact solution of the compressible Euler–Helmholtz equation and the millennium prize problem generalization

SummaryFor the first time the exact strong solution of the three-dimensional (3D) compressible Euler–Helmholtz (EH) vortex equation in unbounded space is obtained. It corresponds to the general solution obtained in Euler variables for the 3D Riemann–Hopf (RH) equation and has a singularity in finite time. The exact closed description for the evolution of all one and multi point moments of the hydrodynamic fields and their spectra is now possible and thus, the mysterious turbulence problem has the exact solution for this case. As an example, for the first time the two-dimensional (2D) energy spectrum in the inertial scale interval is obtained directly from the exact solution of compressible 2D Navier–Stokes (NS) equation. For this purpose, a smooth continuation at all times for these exact solutions to the EH and RH equations is obtained by introducing a super-threshold homogeneous friction or any even extremely low effective viscosity. A new, smooth for all time, analytic solutions to the 1D, 2D and 3D NS equations is obtained. This solution coincides with the above-mentioned modified by viscosity solutions to the EH and RH equations, when also taking into account a linear relation between the pressure and the velocity field divergence, which is well known for the systems far from the equilibrium with relatively large second viscosity. This gives the positive answer to the generalization of the millennium prize problem (www.claymath.org) for the case of the compressible NS equation. Indeed, earlier only a negative answer to the problem of the existence of smooth solutions for any finite time has been a priori considered for the compressible case with the finite initial energy of the smooth velocity field in the unbounded space.

Accurate fitting functions for peculiar velocity spectra in standard and massive-neutrino cosmologies

We estimate the velocity field in a large set of N-body simulations including massive neutrino particles, and measure the auto-power spectrum of the velocity divergence field as well as the cross-power spectrum between the cold dark matter density and the velocity divergence. We perform these measurements at four different redshifts and within four different cosmological scenarios, covering a wide range in neutrino masses. We find that the nonlinear correction to the velocity power spectra largely depends on the degree of nonlinear evolution with no specific dependence on the value of neutrino mass. We provide a fitting formula based on the value of the rms of the matter fluctuations in spheres of 8 h−1 Mpc, describing the nonlinear corrections with 3% accuracy on scales below k = 0.7 h Mpc−1.

Application of the Entropically Damped Artificial Compressibility model to direct numerical simulation of turbulent channel flow

The feasibility of the Entropically Damped Artificial Compressibility (EDAC) model for simulation of wall-bounded turbulence is examined. A favourable feature of EDAC is the purely parabolic type of governing equations resulting from the introduction of a damping term to the pressure evolution equation. This significantly reduces noise in the velocity divergence field when central schemes are applied for the spatial discretisation. The method of lines is used for the solution of the EDAC equations. A conservative finite difference method of purely 2nd- and mixed 4th/2nd-order is applied. The explicit 4th-order, six-stage low-storage Runge–Kutta method is used for time advancement. As expected, the mixed scheme is superior to the 2nd-order one in terms of both accuracy and computational efficiency.For the first time, the EDAC model is assessed for the direct numerical simulation of wall-bounded turbulent flow requiring a non-uniform mesh. As a particular test case we have chosen the channel flow at the friction Reynolds numbers Reτ=180 and 395. A very good agreement with the reference data is obtained, as documented by the chosen one-point velocity statistics: the mean and r.m.s. profiles and the budgets of the Reynolds stresses. The use of the explicit time discretisation and local (rather than compact) spatial schemes results in easily and efficiently parallelisable solution algorithm. We achieved very high computational performance on a desktop computer when solving the EDAC equations using a code dedicated for the graphics processing unit (GPU). The aspects of the GPU implementation are discussed.

A lattice Boltzmann model for thermal compressible flows at low Mach numbers beyond the Boussinesq approximation

A modified weakly compressible lattice Boltzmann model is developed for simulating low Mach number thermal compressible flows. Using the low Mach number approximation (LMNA) and splitting the pressure into two parts, namely thermodynamic and hydrodynamic pressure, the Navier–Stokes equations at low Mach number limits with variable density are recovered from the kinetic model. The fluid density is governed by the chosen equation of state. In order to satisfy the continuity equation at large temperature variations, which leads to large density differences, a suitable compensation for the finite divergence of the velocity field is introduced via a forcing term. This relationship correlates the total derivatives of the temperature and thermodynamic pressure to the velocity field divergence. The present model is successfully validated against well-known benchmarks. Numerical experiments are conducted to predict the critical Rayleigh number in a compressible Rayleigh–Benard convection test case, where the Boussinesq approximation is no longer valid due to the large temperature differences across the domain. The results are in good agreement with theory and previously published works. Moreover, two-dimensional natural convection heat transfer in a square cavity with large temperature differences is simulated and the results agree with available literature data. One interesting aspect of the presented algorithm is its capability to be extended to two-phase liquid-gas flows to model the gas phase as thermally compressible for low Mach numbers flows.

Static and dynamic properties of smoothed dissipative particle dynamics

In this paper, static and dynamic properties of the smoothed dissipative particle dynamics (SDPD) method are investigated. We study the effect of method parameters on SDPD fluid properties, such as structure, speed of sound, and transport coefficients, and show that a proper choice of parameters leads to a well-behaved and accurate fluid model. In particular, the speed of sound, the radial distribution function (RDF), shear-thinning of viscosity, the mean-squared displacement (〈R2〉∝t), and the Schmidt number (Sc∼O(103)−O(104)) can be controlled, such that the model exhibits a fluid-like behavior for a wide range of temperatures in simulations. Furthermore, in addition to the consideration of fluid density variations for fluid compressibility, a more challenging test of incompressibility is performed by considering the Poisson ratio and divergence of velocity field in an elongational flow. Finally, as an example of complex-fluid flow, we present the applicability and validity of the SDPD method with an appropriate choice of parameters for the simulation of cellular blood flow in irregular geometries. In conclusion, the results demonstrate that the SDPD method is able to approximate well a nearly incompressible fluid behavior, which includes hydrodynamic interactions and consistent thermal fluctuations, thereby providing, a powerful approach for simulations of complex mesoscopic systems.

Blowup criteria for full compressible Navier-Stokes equations with vacuum state

This paper deals with the global strong solution to the three-dimensional (3D) full compressible Navier-Stokes systems with vacuum. The authors provide a sufficient condition which requires that the Sobolev norm of the temperature and some norm of the divergence of the velocity are bounded, for the global regularity of strong solution to the 3D compressible Navier-Stokes equations. This result indicates that the divergence of velocity fields plays a dominant role in the blowup mechanism for the full compressible Navier-Stokes equations in three dimensions.

Two is better than one: joint statistics of density and velocity in concentric spheres as a cosmological probe

The analytical formalism to obtain the probability distribution functions (PDFs) of spherically-averaged cosmic densities and velocity divergences in the mildly non-linear regime is presented. A large-deviation principle is applied to those cosmic fields assuming their most likely dynamics in spheres is set by the spherical collapse model. We validate our analytical results using state-of-the-art dark matter simulations with a phase-space resolved velocity field finding a 2% percent level agreement for a wide range of velocity divergences and densities in the mildly nonlinear regime (~10Mpc/h at redshift zero), usually inaccessible to perturbation theory. From the joint PDF of densities and velocity divergences measured in two concentric spheres, we extract with the same accuracy velocity profiles and conditional velocity PDF subject to a given over/under-density which are of interest to understand the non-linear evolution of velocity flows. Both PDFs are used to build a simple but accurate maximum likelihood estimators for the redshift evolution of the variance of both the density and velocity divergence fields, which have smaller relative errors than their sample variances when non-linearities appear. Given the dependence of the velocity divergence on the growth rate, there is a significant gain in using the full knowledge of both PDFs to derive constraints on the equation of state of dark energy. Thanks to the insensitivity of the velocity divergence to bias, its PDF can be used to obtain unbiased constraints on the growth of structures ($\sigma_8$,f) or it can be combined with the galaxy density PDF to extract bias parameters.

Monitoring crustal movements in northern Tianshan Mountain based on GPS technology

Monitoring results for the period 2009–2014 by ten standard GPS stations allowed to determine the crustal movements of seismically active region in south-eastern Kazakhstan. Maps of movement velocity were made in geocentric coordinate system and in reference system of the Eurasian continent. GPS points displacements reflect the features of modern deformation processes that are notable in the high seismic activity region. The structure of the velocity field divergence qualitatively confirms major deformation in the sublatitudinal direction which is parallel to the main ridge of the northern Tianshan Mountain. The epicenters of earthquakes are in agreement with the border areas of compression–tension, as well as the allocated areas of multidirectional rotary motion. The conclusion is that GPS monitoring of the movements of the Earth's crust can be used to evaluate the territory's stress–strain state for the purpose of seismic zoning and seismic risk assessment.

Fast large scale structure perturbation theory using one-dimensional fast Fourier transforms

The usual fluid equations describing the large-scale evolution of mass density in the universe can be written as local in the density, velocity divergence, and velocity potential fields. As a result, the perturbative expansion in small density fluctuations, usually written in terms of convolutions in Fourier space, can be written as a series of products of these fields evaluated at the same location in configuration space. Based on this, we establish a new method to numerically evaluate the 1-loop power spectrum (i.e., Fourier transform of the 2-point correlation function) with one-dimensional Fast Fourier Transforms. This is exact and a few orders of magnitude faster than previously used numerical approaches. Numerical results of the new method are in excellent agreement with the standard quadrature integration method. This fast model evaluation can in principle be extended to higher loop order where existing codes become painfully slow. Our approach follows by writing higher order corrections to the 2-point correlation function as, e.g., the correlation between two second-order fields or the correlation between a linear and a third-order field. These are then decomposed into products of correlations of linear fields and derivatives of linear fields. The method can also be viewed as evaluating three-dimensional Fourier space convolutions using products in configuration space, which may also be useful in other contexts where similar integrals appear.

Convergence of a Finite Difference Scheme for Two-Dimensional Incompressible Magnetohydrodynamics

We investigate a finite difference scheme for the two-dimensional, incompressible magnetohydrodynamics equations that was introduced in [J.-G. Liu and W.-C. Wang, J. Comput. Phys., 174 (2001), pp. 12--37]. It uses central difference and averaging operators on a staggered grid and was shown not only to keep the magnetic and the velocity field divergence free but, moreover, to preserve the energy exactly. We extend Liu and Wang's result by a discrete $H^1$ bound, or more precisely, we show that the discrete solution for the velocity and the magnetic field is bounded in $L^\infty(0,\infty;H^1(\Omega))$ and $L^2(0,T;H^2(\Omega))$. This bound allows us to prove the convergence of the scheme. The convergence is strong in $L^2$ and weak in $H^1$.