Stochastic Lagrangian Models for Turbulent Dispersion in Atmospheric Boundary Layer

A one-particle 3D stochastic Lagrangian model in for transport of particles in horizontally-homogeneous atmospheric surface laeyr with arbitrary one-point probability density function of Eulerian velocity uctuations is suggested. A uniquely defined Lagrangian stochastic model in the class of well-mixed models is constructed from physically plausible assumptions: (i) in the neutrally stratified horizontally homogeneous surface layer, the vertical motion is mainly controlled by eddies whose size is of order of the current height; and (ii), the streamwise drift term is independent of the crosswind velocity. Numerical simulations for neutral stratification have shown a good agreement of our model with the well known Thomson's model, with Flesch & Wilson's model, and with experimental measurements as well. However there is a discrepancy of these results with the results obtained by Reynolds' model

Probability of error deviations for the dependent sampling Monte Carlo methods: Exponential bounds in the uniform norm

Under Bernstein's condition a non-asymptotic exponential estimation for the probability of deviations of a sum of independent random fields in uniform norm is proposed. Application of this result to the problem of the error estimation for the dependent sampling Monte Carlo method is presented. It is shown that in the domain of moderately large deviations the suggested estimations have optimal asymptotics

Stochastic flow simulation and particle transport in a 2D layer of random porous medium

A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed

One-Particle Stochastic Lagrangian Model for Turbulent Dispersion in Horizontally Homogeneous Turbulence

A one-particle stochastic Lagrangian model in 2D and 3D dimensions is constructed for transport of particles in horizontaly homogeneous turbulent flows with arbitrary one-point probability density function. It is shown that in the case of anisotropic turbulence with gaussian pdf, this model essentially differs from the known Thomson's model. The results of calculations according to our model in the case of neutrally stratified atmospheric surface layer agree satisfactorily with the measurements known from the literature

Convergence of Fourier-wavelet models for Gaussian random processes

Mean square convergence and convergence in probability of Fourier-Wavelet Models (FWM) of stationary Gaussian Random processes in the metric of Banach space of continuously differentiable functions and in Sobolev space are studied. Sufficient conditions for the convergence formulated in the frame of spectral functions are given. It is shown that the given rates of convergence of FWM in the mean square obtained in the Nikolski\u{i}-Besov classes cannot be improved

Stochastic spectral and Fourier-wavelet methods for vector Gaussian random fields

Randomized Spectral Models (RSM) and Randomized Fourier-Wavelet Models (FWM) for simulation of homogeneous Gaussian random fields based on spectral representations and plane wave decomposition of random fields are developed. Extensions of FWM to vector random processes are constructed. Convergence of the constructed Fourier-Wavelet models (in the sense of finite-dimensional distributions) under some general conditions on the spectral tensor is given. A comparative analysis of RSM and FWM is made by calculating Eulerian and Lagrangian statistical characteristics of a 3D isotropic incompressible random field through an ensemble and space averagin

A Lagrangian stochastic model for the transport in statistically homogeneous porous media

A new type of stochastic simulation models is developed for solving transport problems in saturated porous media which is based on a generalized Langevin stochastic differential equation. A detailed derivation of the model is presented in the case when the hydraulic conductivity is assumed to be a random field with a lognormal distribution, being statistically isotropic in space. To construct a model consistent with this statistical information, we use the well-mixed condition which relates the structure of the Langevin equation and the probability density function of the Eulerian velocity field. Numerical simulations of various statistical characteristics like the mean displacement, the displacement covariance tensor and the Lagrangian correlation function are presented. These results are compared against the conventional random displacement method

Stochastic simulation of flows and particle transport in porous tubes

A Monte Carlo method is developed for stochastic simulation of flows and particle transport in tubes filled with a porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modelled in a tube with prescribed boundary conditions. Numerical experiments are carried out by solving the random Darcy equation for each sample of the hydraulic conductivity by a SOR iteration method, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, diffusion coefficients, the mean and variance of Lagrangian trajectories, and discuss a ''stagnation" effect which was found in our simulations

Extensions of multiscale Gaussian random field simulation algorithms

We analyze and compare the efficiency and accuracy of two simulation methods for homogeneous random fields with multiscale resolution. We consider in particular the Fourier-wavelet method and three variants of the Randomization method: (A) without any stratified sampling of wavenumber space, (B) with stratified sampling of wavenumbers with equal energy subdivision, (C) stratified sampling with a logarithmically uniform subdivision. We focus on fractal Gaussian random fields with Kolmogorov-type spectra. As noted in previous work [3,6], variants (A) and (B) of the Randomization method are only able to generate a self-similar structure function over three to four decades with reasonable computational effort. By contrast, variant (C), suggested by [34,22], along with the Fourier-wavelet method developed by [6], is able to reproduce accurate self-similar scaling of the structure function over a number of decades increasing linearly with computational effort (for our examples we will show that nine decades can be reproduced). We provide some conceptual and numerical comparison of the various cost contributions to each random field simulation method (overhead, cost per realization, cost per evaluation). When evaluating ensemble averaged quantities like the correlation and structure functions, as well as some multi-point statistical characteristics, the Randomization method can provide good accuracy with considerably less cost than the Fourier-wavelet method. The Fourier-wavelet method, however, has better ergodic properties, and hence becomes more efficient for the computation of spatial (rather than ensemble) averages which may be important in simulating the solutions to partial differential equations with random field coefficients

Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models

Forward and backward stochastic Lagrangian trajectory simulation methods for calculation of the mean concentration of scalars and their fluxes for sources arbitrarily distributed in space and time are constructed and justified. Generally, absorption of scalars by medium is taken into account. A special case of the source structure, when the scalar is generated by a plane source, say, located close to the ground, is treated. This practically interesting particular case is known in the literature as the footprint problem