Random Velocity Field Research Articles

Clustering of passive tracers in a random acoustic velocity field

We consider the effects of passive tracer clustering (e.g., the magnetic field energy in stellar atmospheres) in a random acoustic velocity field. A method for numerical modeling of a two-dimensional random acoustic field is proposed. The field is described by a correlation tensor defined by traveling isotropic waves, taking into account dissipation. Two metrics for measuring the clustering effects are used: concentration and density. Using numerical modeling, we show that the tracer concentration is almost always clustered. The situation with the density is different; as the dissipation tends to zero, the time to reach the clustered states increases significantly. In addition, due to the tracer transport out of the density clustering regions, only a part of the tracer is clustered. For the presented analyses, we considered ensembles of Lagrangian particles and introduced and applied the statistical topography methodology.

Analysis of the spatiotemporal velocity of annual precipitation based on random field

. Changes in precipitation directly impact river runoff volume, subsequently influencing food production, and the security of downstream urban areas. In this study, we introduce a random velocity field (RVF) capable of performing multi-step predictions while providing interpretable insights into precipitation variations. The RVF leverages the gradient of a Gaussian random field to learn spatiotemporal velocity patterns and employs a predictive process to reduce dimensionality and enable multi-step forecasting. Bayesian parameter estimation is obtained using the Markov Chain Monte Carlo (MCMC) method. Our analysis reveals a noticeable shifting trend in annual precipitation based on diverse real datasets. This trend serves as a valuable foundation for further exploration of urban flood control and agricultural development strategies.

Clustering of Floating Tracers in a Random Velocity Field Modulated by an Ellipsoidal Vortex Flow

The influence of a background vortex flow on the clustering of floating tracers is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (1) a steady state, (2) oscillation and (3) rotation of the ellipsoidal vortex core. The latter two induce an unsteady velocity field for the tracer, thus leading to irregular (chaotic) tracer motion. Superimposing a stochastic divergent velocity field onto the deterministic vortex flow allows us to observe significantly different tracer evolution. An ellipsoidal vortex has ellipsoidal symmetry, and the tracer’s trajectories exhibit the same symmetry inside the vortex. Outside the vortex, the external deformation flow symmetry dominates. Diffusion scattering and chaotic advection give tracers the opportunity to leave the region of ellipsoidal symmetry and form a picture of shear flow symmetry. We use the method of characteristics to integrate the floating tracer density evolution equation and the Euler Ito scheme for obtaining the floating tracer trajectories with a random velocity field. The cluster area and cluster mass from the statistical topography are used as the quantitative diagnostics of a floating tracer’s clustering. For the case of a steady ellipsoidal vortex embedded into the deformation flow with a random velocity field component, we found that the clustering characteristics were weakened by the steady vortex. For the cases of an unsteady ellipsoidal vortex, we observed clustering in the floating tracer density field if the contribution of the divergent component was greater than or equal to that of the rotational (nondivergent) component. Even when the initial floating tracer patch was set on the boundary of the oscillating ellipsoidal vortex, we observed the formation of clusters. In the case of a rotating ellipsoidal vortex, we also observed pronounced clustering. Thus, we argue that unsteady ellipsoidal vortex regimes (oscillation and rotation), which induce chaotic motion of the nearby passive tracer’s trajectories, are still conducive to clustering of floating tracers observed in the density field, despite the intense deformation introduced by strain and shear.

Computing differential operators of the particle velocity in moving particle clouds using tessellations

We propose finite-time measures to compute the divergence, the curl and the velocity gradient tensor of the point particle velocity for two- and three-dimensional moving particle clouds. For this purpose, a tessellation of the particle positions is performed to assign a volume to each particle. We introduce a modified Voronoi tessellation which overcomes some drawbacks of the classical construction. Instead of the circumcenter we use the center of gravity of the Delaunay cell for defining the vertices. Considering then two subsequent time instants, the dynamics of the volume can be assessed. Determining the volume change of tessellation cells yields the divergence of the particle velocity. Reorganizing the various velocity coefficients allows computing the curl and even the velocity gradient tensor. The helicity of particle velocity can be likewise computed and swirling motion of particle clouds can be quantified. First we assess the numerical accuracy for randomly distributed particles. We find a strong Pearson correlation between the divergence computed with the modified tessellation, and the exact value. Moreover, we show that the proposed method converges with first order in space and time in two and three dimensions. Then we consider particles advected with random velocity fields with imposed power-law energy spectra. We study the number of particles necessary to guarantee a given precision. Finally, applications to fluid and inertial particles advected in three-dimensional fully developed isotropic turbulence show the utility of the approach for real world applications to quantify self-organization in particle clouds and their vortical or even swirling motion.

Numerical simulations of the random angular momentum in convection – II. Delayed explosions of red supergiants following ‘failed’ supernovae

ABSTRACT When collapse of the iron core in a massive red or yellow supergiant does not lead to an energetic supernova, a significant fraction of the convective hydrogen envelope will fall in towards the black hole formed from the collapsing core. The random velocity field in the convective envelope results in finite specific angular momentum in each infalling shell. Using 3D hydrodynamical simulations, we follow the infall of this material to small radii, resolving the circularization radii of the flow. We show that infall of the convective envelope leads to nearly complete envelope ejection in a ≳1048 erg explosion with outflow speeds of ≳200 km s−1. The light curve of such an explosion would show a characteristic, red plateau as the ejecta cools and a hydrogen recombination front recedes through the expanding ejecta. Adopting supernova IIp scalings, the event would have a plateau luminosity of ≳1040 erg s−1 and a duration of several hundreds of days. These events would appear quite similar to luminous red novae with red or yellow supergiant progenitors; some luminous red novae may, in fact, be signposts of black hole formation. The mechanism studied here produces more energetic explosions than the weak shock generated from radiation of neutrino energy during the protoneutron star phase. Because we cannot simulate all the way to the horizon, our results are likely lower limits on the energy and luminosity of transients produced during the collapse of a red or yellow supergiant to form a black hole.

On Clustering of Floating Tracers in Random Velocity Fields

AbstractIn this paper, we investigate the aggregation of a floating tracer into clusters. Motivated by observations of dense patches of buoyant material in the real ocean (e.g., microplastic pollutants, plankton, and sargassum), we develop an idealized model that can reproduce the clustering process. A stochastic, kinematic 2D velocity field is chosen to represent turbulent oceanic surface currents, with a weakly divergent component. Lagrangian particles are introduced and we track their concentrations. We differ from delta‐correlated fields used in previous studies by including finite time correlations. Clustering in these fields can be compared to the traditional setting, through global measures and cluster detection algorithms. The enhanced velocity fields can be deformed using various interpolation methods. We can then investigate the sensitivity of clustering to the representation of temporal/spatial velocity structure to inform future studies of this phenomenon. We find coherency of time‐correlated velocities leads to significantly faster rates of clustering, causing a larger number of longer lived/more populated clusters to form. Clustering is likely relevant to a host of biogeochemical processes of urgent interest, such as phytoplankton blooms and the ecological risk of microplastic pollutants. This work aims to establish an accurate basis for clustering simulations, to enable further exploration.

Cosmic web & caustic skeleton: non-linear constrained realizations — 2D case studies

The cosmic web consists of a complex configuration of voids, walls, filaments, and clusters, which formed under the gravitational collapse of Gaussian fluctuations. Understanding under what conditions these different structures emerge from simple initial conditions, and how different cosmological models influence their evolution, is central to the study of the large-scale structure. Here, we present a general formalism for setting up initial random density and velocity fields satisfying non-linear constraints for specialized N-body simulations. These allow us to link the non-linear conditions on the eigenvalue and eigenvector fields of the deformation tensor, as specified by caustic skeleton theory, to the current-day cosmic web. By extending constrained Gaussian random field theory, and the corresponding Hoffman-Ribak algorithm, to non-linear constraints, we probe the statistical properties of the progenitors of the walls, filaments, and clusters of the cosmic web. Applied to cosmological N-body simulations, the proposed techniques pave the way towards a systematic investigation of the evolution of the progenitors of the present-day walls, filaments, and clusters, and the embedded galaxies, putting flesh on the bones of the caustic skeleton. The developed non-linear constrained random field theory is valid for generic cosmological conditions. For ease of visualization, the case study presented here probes the two-dimensional caustic skeleton.

Clustering Data for Passive Tracers in Random Velocity Fields

Clustering Data for Passive Tracers in Random Velocity Fields

Deep learning reconstruction of the large-scale structure of the Universe from luminosity distance

ABSTRACT Supernovae Ia (SNe) can provide a unique window on the large-scale structure (LSS) of the Universe at redshifts where few other observations are available, by solving the inversion problem (IP) consisting in reconstructing the LSS from its effects on the observed luminosity distance. So far the IP was solved assuming some restrictions about space–time, such as spherical symmetry for example, while we obtain for the first time solutions of the IP problem for arbitrary space–time geometries using deep learning. The method is based on the use of convolutional neural networks (CNN) trained on simulated data. The training data set is obtained by first generating random density and velocity fields, and then computing their effects on the luminosity distance. The CNN, based on an appropriately modified version of U-Net to account for the tridimensionality of the data, is then trained to reconstruct the density and velocity fields from the luminosity distance. We find that the velocity field inversion is more accurate than the density field, because the effects of the velocity on the luminosity distance only depend on the source velocity, while in the case of the density it is an integrated effect along the line of sight, giving rise to more degeneracy in the solution of the IP. Improved versions of these neural networks, modified to accommodate the non-uniform distribution of the SNe, can be applied to observational data to reconstruct the LSS of the Universe at redshifts at which few other observations are available.

Universal and Non-Universal Features in the Random Shear Model

The stochastic transport of particles in a disordered two-dimensional layered medium, driven by correlated y-dependent random velocity fields is usually referred to as random shear model. This model exhibits a superdiffusive behavior in the x direction ascribable to the statistical properties of the disorder advection field. By introducing layered random amplitude with a power-law discrete spectrum, the analytical expressions for the space and time velocity correlation functions, together with those of the position moments, are derived by means of two distinct averaging procedures. In the case of quenched disorder, the average is performed over an ensemble of uniformly spaced initial conditions: albeit the strong sample-to-sample fluctuations, and universality appears in the time scaling of the even moments. Such universality is exhibited in the scaling of the moments averaged over the disorder configurations. The non-universal scaling form of the no-disorder symmetric or asymmetric advection fields is also derived.

Energetics and vortex structures near small-scale shear layers in turbulence

Vortices and kinetic energy distributions around small-scale shear layers are investigated with direct numerical simulations of isotropic turbulence. The shear layers are examined with the triple decomposition of a velocity gradient tensor. The shear layers subject to a biaxial strain appear near vortices with rotation, which induce energetic flow that contributes to the shear. A similar configuration of rotating motions near the shear layers is observed in a multi-scale random velocity field, which is free from the dynamics of turbulence. Therefore, the mechanism that sustains shearing motion is embedded as a kinematic nature in random velocity fields. However, the biaxial strain is absent near the shear layers in random velocity because rotating motions appear right next to the shear layers. When a random velocity field begins to evolve following the Navier–Stokes equations, the shear layers are immediately tilted to the nearby rotating motions. This misalignment is a key for the vortex to generate the compressive strain of the biaxial strain around the shear layer. As the configuration of shearing and rotating motions arises from the kinematic nature, the shear layers with the biaxial strain are formed within a few times the Kolmogorov timescale once the random velocity field begins to evolve. The analysis with high-pass filtered random velocity suggests that this shear layer evolution is caused by small-scale turbulent motions. These results indicate that the kinematic nature of shear and rotation in velocity fluctuations has a significant role in the formation of shear layers in turbulence.

The effects of driving time scales on coronal heating in a stratified atmosphere

Aims. We investigate the atmospheric response to coronal heating driven by random velocity fields with different characteristic time scales and amplitudes. Methods. We conducted a series of three-dimensional magnetohydrodynamic simulations of random driving imposed on a gravitationally stratified model of the solar atmosphere. In order to understand differences between alternating current (AC) and direct current (DC) heating, we considered the effects of changing the characteristic time scales of the imposed velocities. We also investigated the effects of the magnitude of the velocity driving. Results. In all cases, complex foot point motions lead to a proliferation of current sheets and energy dissipation throughout the coronal volume. For a given driving amplitude, DC driving typically leads to a greater rate of energy injection when compared to AC driving. This ultimately leads to the formation of larger currents, increased heating rates, and higher coronal temperatures in DC simulations. There is no difference in the spatial distribution of energy dissipation across simulations; however, energy release events in AC cases tend to be more frequent and last for less time than in DC cases. This results in more asymmetric temperature profiles for field lines heated by AC drivers. Higher velocity driving is associated with larger currents, higher temperatures, and the corona occupying a larger fraction of the simulation volume. In all cases, the majority of heating is associated with small energy release events, which occur much more frequently than larger events. Conclusions. When combined with observational results that highlight a greater abundance of oscillatory power in lower frequency modes, these findings suggest that energy release in the corona is more likely to be driven by longer time scale motions. In the corona, AC and DC driving occur concurrently and their effects remain difficult to isolate. The distribution of field line temperatures and the asymmetry of temperature profiles may reveal the frequency and longevity of energy release events and therefore the relative importance of AC and DC heating.

Diffusion in a Frozen Random Velocity Field

Particle diffusion in a frozen isotropic 2D random velocity field is studied by simulation, and the results are compared with the prediction of a simple model. The model accounts for the effects of particle trapping and infinite correlation time.

Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations

We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier–Stokes equations and 3D hyper-viscous Navier–Stokes equations in Td subjected to nondenegenerate Hσ-regular noise for any σ sufficiently large. That is, for all s>0 there is a deterministic exponential decay rate such that all mean-zero Hs passive scalars decay in H−s at this same rate with probability one. This is equivalent to what is known as quenched correlation decay for the Lagrangian flow in the dynamical systems literature. This is a follow-up to our previous work, which establishes a positive Lyapunov exponent for the Lagrangian flow—in general, almost-sure exponential mixing is much stronger than this. Our methods also apply to velocity fields evolving according to finite-dimensional models, for example, Galerkin truncations of Navier–Stokes or the Stokes equations with very degenerate forcing. For all 0≤k<∞, this exhibits many examples of CtkCx∞ random velocity fields that are almost-sure exponentially fast mixers.

Non-Gaussian limit of a tracer motion in an incompressible flow

We consider a massless tracer particle moving in a random, stationary, isotropic and divergence free velocity field. We identify a class of fields for which the limit of the laws of appropriately scaled tracer trajectory processes is a non-Gaussian Rosenb

A quenched local limit theorem for stochastic flows

We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by a Gaussian random velocity field that is white in time and smooth in space. We show that conditional on the velocity field, the quenched density of the particle after a long time can be approximated pointwise by the product of a deterministic Gaussian density and a spacetime-stationary random field U. If the velocity field is additionally assumed to be incompressible, then U≡1 almost surely and we obtain a local central limit theorem.

Statistics of local Reynolds number in box turbulence: ratio of inertial to viscous forces

In high-Reynolds-number turbulence the spatial distribution of velocity fluctuation at small scales is strongly non-uniform. In accordance with the non-uniformity, the distributions of the inertial and viscous forces are also non-uniform. According to direct numerical simulation (DNS) of forced turbulence of an incompressible fluid obeying the Navier–Stokes equation in a periodic box at the Taylor microscale Reynolds number $R_\lambda \approx 1100$ , the average $\langle R_{loc}\rangle$ over the space of the ‘local Reynolds number’ $R_ {loc}$ , which is defined as the ratio of inertial to viscous forces at each point in the flow, is much smaller than the conventional ‘Reynolds number’ given by $Re \equiv UL/\nu$ , where $U$ and $L$ are the characteristic velocity and length of the energy-containing eddies, and $\nu$ is the kinematic viscosity. While both conditional averages of the inertial and viscous forces for a given squared vorticity $\omega ^{2}$ increase with $\omega ^{2}$ at large $\omega ^{2}$ , the conditional average of $R_ {loc}$ is almost independent of $\omega ^{2}$ . A comparison of the DNS field with a random structureless velocity field suggests that the increase in the conditional average of $R_ {loc}$ with $\omega ^{2}$ at large $\omega ^{2}$ is suppressed by the Navier–Stokes dynamics. Something similar is also true for the conditional averages for a given local energy dissipation rate per unit mass. Certain features of intermittency effects such as that on the $Re$ dependence of $\langle R_{loc}\rangle$ are explained by a multi-fractal model by Dubrulle (J. Fluid Mech., vol. 867, 2019, P1).

Quasi-stationarity of scalar turbulent mixing statistics in a non-symmetric case

The existence of an asymptotic shape at large times of the probability density function (PDF) of a non-reacting scalar, mixed by a solenoidal turbulent velocity field and molecular diffusive transport, was investigated by Sinai and Yakhot [Y. G. Sinai and V. Yakhot, “Limiting probability distributions of a passive scalar in a random velocity field,” Phys. Rev. Lett. 63, 1962 (1989)]. The quasi-stationarity of the mixing statistics along the time evolution by Valiño et al. [“Quasistationary probability density functions in the turbulent mixing of a scalar field,” Phys. Rev. Lett. 72, 3518 (1994)] was an extension to symmetric scalar pdfs; analytic solutions for the scalar fluctuation dissipation rates, conditional upon the scalar value, and the pdf were obtained. This manuscript examines the generalization of the latter results to asymmetric scalar pdfs, further scrutinizes underlying mechanisms for a quasi-stationary statistics, and shows a Monte Carlo implementation which allows non-Gaussian relaxations to prescribed values of skweness and kurtosis.

The Propagation of a Chain Isothermal Flame in a Random Medium

The dependences of the turbulent flame velocity in a random flow on the turbulence intensity have been studied in terms of the Kolmogorov–Petrovsky–Piskunov nonlinearity model in the one-dimensional case. We show that for a static random medium the turbulent velocity decreases and tends to zero with increasing turbulence intensity. This is explained by the fact that the random flow velocity is the random potential gradient, leading to traps and diffusion retardation. Apparently, this is a general fact: a random gradient velocity field in the two- and three-dimensional cases must give the same effect. This may be true not only for the Kolmogorov–Petrovsky–Piskunov nonlinearity. We have shown analytically that the introduction of a random nonzero mean velocity field accelerating the flame motion enables the flame to overcome the barriers more easily. For a dynamic medium an initial increase of the velocity with growing turbulence is possible in some ranges of scales in space and time. We consider an example of a two-dimensional random divergence-free field for which our numerical computation has shown an increase of the turbulent flame velocity.

Comments on Wave-Like Propagation with Binary Disorder

In a recent paper (J Stat Phys 179:729–747, 2020) time evolution of the telegrapher’s equation subjected to a random velocity field was studied in the context of the effective medium approximation. In addition, the solution for the mean-value of the field was presented as a series expansion in Terwiel’s cumulants. Here we have proved that this expansion cuts if the random velocity field is emulated with a stationary exponential-correlated symmetric binary disorder. The present conclusion can also be used to tackle other wave-like models of propagation with Markovian binary disorder.