Spatial and Temporal Extrapolation of Disdrometer Size Distributions Based on a Lagrangian Trajectory Model of Falling Rain

Methodologies to improve disdrometer processing, loosely based on mathematical techniques common to the field of particle flow and fluid mechanics, are examined and tested. The inclusion of advection and vertical wind field estimates appears to produce significantly improved results in a Lagrangian hydrometeor trajectory model, in spite of very strict assumptions of noninteracting hydrometeors, constant vertical air velocity, and time independent advection during a radar scan time interval. Wind field data can be extracted from each radar elevation scan by plotting and analyzing reflectivity contours over the disdrometer site and by collecting the radar radial velocity data to obtain estimates of advection. Specific regions of disdrometer spectra (drop size versus time) often exhibit strong gravitational sorting signatures, from which estimates of vertical velocity can be extracted. These independent wind field estimates can be used as initial conditions to the Lagrangian trajectory simulation of falling hydrometeors.Comment: 25 pages, 15 figures, 4 tables. Submitted to The Open Atmospheric Science Journal, http://www.bentham.org/open/toascj

Anomalous dispersion in correlated porous media: A coupled continuous time random walk approach

We study the causes of anomalous dispersion in Darcy-scale porous media characterized by spatially heterogeneous hydraulic properties. Spatial variability in hydraulic conductivity leads to spatial variability in the flow properties through Darcy's law and thus impacts on solute and particle transport. We consider purely advective transport in heterogeneity scenarios characterized by broad distributions of heterogeneity length scales and point values. Particle transport is characterized in terms of the stochastic properties of equidistantly sampled Lagrangian velocities, which are determined by the flow and conductivity statistics. The persistence length scales of flow and transport velocities are imprinted in the spatial disorder and reflect the distribution of heterogeneity length scales. Particle transitions over the velocity length scales are kinematically coupled with the transition time through velocity. We show that the average particle motion follows a coupled continuous time random walk (CTRW), which is fully parameterized by the distribution of flow velocities and the medium geometry in terms of the heterogeneity length scales. The coupled CTRW provides a systematic framework for the investigation of the origins of anomalous dispersion in terms of heterogeneity correlation and the distribution of heterogeneity point values. Broad distributions of heterogeneity point values and lengths scales may lead to very similar dispersion behaviors in terms of the spatial variance. Their mechanisms, however are very different, which manifests in the distributions of particle positions and arrival times, which plays a central role for the prediction of the fate of dissolved substances in heterogeneous natural and engineered porous materials

Imaging of a fluid injection process using geophysical data - A didactic example

In many subsurface industrial applications, fluids are injected into or withdrawn from a geologic formation. It is of practical interest to quantify precisely where, when, and by how much the injected fluid alters the state of the subsurface. Routine geophysical monitoring of such processes attempts to image the way that geophysical properties, such as seismic velocities or electrical conductivity, change through time and space and to then make qualitative inferences as to where the injected fluid has migrated. The more rigorous formulation of the time-lapse geophysical inverse problem forecasts how the subsurface evolves during the course of a fluid-injection application. Using time-lapse geophysical signals as the data to be matched, the model unknowns to be estimated are the multiphysics forward-modeling parameters controlling the fluid-injection process. Properly reproducing the geophysical signature of the flow process, subsequent simulations can predict the fluid migration and alteration in the subsurface. The dynamic nature of fluid-injection processes renders imaging problems more complex than conventional geophysical imaging for static targets. This work intents to clarify the related hydrogeophysical parameter estimation concepts

Cloud microphysical effects of turbulent mixing and entrainment

Turbulent mixing and entrainment at the boundary of a cloud is studied by means of direct numerical simulations that couple the Eulerian description of the turbulent velocity and water vapor fields with a Lagrangian ensemble of cloud water droplets that can grow and shrink by condensation and evaporation, respectively. The focus is on detailed analysis of the relaxation process of the droplet ensemble during the entrainment of subsaturated air, in particular the dependence on turbulence time scales, droplet number density, initial droplet radius and particle inertia. We find that the droplet evolution during the entrainment process is captured best by a phase relaxation time that is based on the droplet number density with respect to the entire simulation domain and the initial droplet radius. Even under conditions favoring homogeneous mixing, the probability density function of supersaturation at droplet locations exhibits initially strong negative skewness, consistent with droplets near the cloud boundary being suddenly mixed into clear air, but rapidly approaches a narrower, symmetric shape. The droplet size distribution, which is initialized as perfectly monodisperse, broadens and also becomes somewhat negatively skewed. Particle inertia and gravitational settling lead to a more rapid initial evaporation, but ultimately only to slight depletion of both tails of the droplet size distribution. The Reynolds number dependence of the mixing process remained weak over the parameter range studied, most probably due to the fact that the inhomogeneous mixing regime could not be fully accessed when phase relaxation times based on global number density are considered.Comment: 17 pages, 10 Postscript figures (figures 3,4,6,7,8 and 10 are in reduced quality), to appear in Theoretical Computational Fluid Dynamic

Secular Gravitational Instability of a Dust Layer in Shear Turbulence

We perform a linear stability analysis of a dust layer in a turbulent gas disk. Youdin (2011) investigated the secular gravitational instability of a dust layer using hydrodynamic equations with a turbulent diffusion term. We obtain essentially the same result independently of Youdin (2011). In the present analysis, we restrict the area of interest to small dust particles, while investigating the secular gravitational instability in a more rigorous manner. We discuss the time evolution of the dust surface density distribution using a stochastic model and derive the advection-diffusion equation. The validity of the analysis by Youdin (2011) is confirmed in the strong drag limit. We demonstrate quantitatively that the finite thickness of a dust layer weakens the secular gravitational instability and that the density-dependent diffusion coefficient changes the growth rate. We apply the obtained results to the turbulence driven by the shear instability and find that the secular gravitational instability is faster than the radial drift when the gas density is three times as large as that in the minimum-mass disk model. If the dust particles are larger than chondrules, the secular gravitational instability grows within the lifetime of a protoplanetary disk.Comment: 32 pages, 6 figures, accepted for publication in Ap

Dust settling in local simulations of turbulent protoplanetary disks

In this paper, we study the effect of MHD turbulence on the dynamics of dust particles in protoplanetary disks. We vary the size of the particles and relate the dust evolution to the turbulent velocity fluctuations. We performed numerical simulations using two Eulerian MHD codes, both based on finite difference techniques: ZEUS--3D and NIRVANA. These were local shearing box simulations incorporating vertical stratification. Both ideal and non ideal MHD simulations with midplane dead zones were carried out. The codes were extended to incorporate different models for the dust as an additional fluid component. Good agreement between results obtained using the different approaches was obtained. The simulations show that a thin layer of very small dust particles is diffusively spread over the full vertical extent of the disk. We show that a simple description obtained using the diffusion equation with a diffusion coefficient simply expressed in terms of the velocity correlations accurately matches the results. Dust settling starts to become apparent for particle sizes of the order of 1 to 10 centimeters for which the gas begins to decouple in a standard solar nebula model at 5.2 AU. However, for particles which are 10 centimeters in size, complete settling toward a very thin midplane layer is prevented by turbulent motions within the disk, even in the presence of a midplane dead zone of significant size. These results indicate that, when present, MHD turbulence affects dust dynamics in protoplanetary disks. We find that the evolution and settling of the dust can be accurately modelled using an advection diffusion equation that incorporates vertical settling. The value of the diffusion coefficient can be calculated from the turbulent velocity field when that is known for a time of several local orbits.Comment: 15 pages, 16 figures, accepted in Astronomy & Astrophysic

Chaotic Mixing in Three Dimensional Porous Media

Under steady flow conditions, the topological complexity inherent to all random 3D porous media imparts complicated flow and transport dynamics. It has been established that this complexity generates persistent chaotic advection via a three-dimensional (3D) fluid mechanical analogue of the baker's map which rapidly accelerates scalar mixing in the presence of molecular diffusion. Hence pore-scale fluid mixing is governed by the interplay between chaotic advection, molecular diffusion and the broad (power-law) distribution of fluid particle travel times which arise from the non-slip condition at pore walls. To understand and quantify mixing in 3D porous media, we consider these processes in a model 3D open porous network and develop a novel stretching continuous time random walk (CTRW) which provides analytic estimates of pore-scale mixing which compare well with direct numerical simulations. We find that chaotic advection inherent to 3D porous media imparts scalar mixing which scales exponentially with longitudinal advection, whereas the topological constraints associated with 2D porous media limits mixing to scale algebraically. These results decipher the role of wide transit time distributions and complex topologies on porous media mixing dynamics, and provide the building blocks for macroscopic models of dilution and mixing which resolve these mechanisms.Comment: 36 page