Random Advection Research Articles

From Zero-Mode Intermittency to Hidden Symmetry in Random Scalar Advection

From Zero-Mode Intermittency to Hidden Symmetry in Random Scalar Advection

Dynamics for a diffusive prey–predator model with advection and free boundaries

This article deals with a free boundary problem of the Lotka–Volterra type prey–predator model with advection in one space dimension. The model considered here may be applied to describe the expanding of an invasive or new predator species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundaries representing expanding fronts of predator species. The main purpose of this article is to understand the influence of the advection environment on the dynamics of the model. We provide sufficient conditions for spreading and vanishing of the predator species, and we find a sharp threshold between the spreading and vanishing concerning this model. Moreover, for the case of successful spreading for the predator, we give estimates of asymptotic spreading speeds and nonlocal long‐time behavior of the prey and the predator.

Dynamics and asymptotic profiles on an age-structured SIS epidemic model with random diffusion and advection

Dynamics and asymptotic profiles on an age-structured SIS epidemic model with random diffusion and advection

Lattice models of random advection and diffusion and their statistics.

We study in detail a one-dimensional lattice model of a continuum, conserved field (mass) that is transferred deterministically between neighboring random sites. The model belongs to a wider class of lattice models capturing the joint effect of random advection and diffusion and encompassing as specific cases some models studied in the literature, such as those of Kang-Redner, Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setup comes from a straightforward interpretation of the advection of particles in one-dimensional turbulence, but it is also related to a problem of synchronization of dynamical systems driven by common noise. For finite lattices, we study both the coalescence of an initially spread field (interpreted as roughening), and the statistical steady-state properties. We distinguish two main size-dependent regimes, depending on the strength of the diffusion term and on the lattice size. Using numerical simulations and a mean-field approach, we study the statistics of the field. For weak diffusion, we unveil a characteristic hierarchical structure of the field. We also connect the model and the iterated function systems concept.

A Wavenumber–Frequency Spectrum Model for Sheared Convective Atmospheric Boundary Layer Flows

Abstract Parameterizing turbulence in the atmospheric boundary layer as a function of space and time is essential for weather and climate models. Here, we explore a model for wavenumber–frequency spectra based on a linear random advection approach to characterize sheared convective atmospheric boundary layer flows. Building on previous works, we obtain the wavenumber–frequency spectrum as a product of the wavenumber spectrum and a Gaussian frequency distribution, whose mean and variance are given by the mean advection and random sweeping velocities, respectively. The applicability of the model is tested with direct numerical simulation data in the mixed layer and the entrainment zone for the streamwise and vertical velocity components and buoyancy. To obtain a fully analytical model, we propose using a von Kármán wavenumber spectrum parameterized by the characteristic variances and integral length scales. These parameters are height dependent and vary considerably with the relative balance of buoyancy and shear forces. The introduced analytical model relies on fitting parameters obtained from numerical data in the relevant range of scales. The comparison of the von Kármán–based spectra for velocity and buoyancy to simulation results shows that the main features of the measured spectra are captured by the model.

Persisting asymmetry in the probability distribution function for a random advection–diffusion equation in impermeable channels

In this paper, we study the effect of impermeable boundaries on the symmetry properties of a random passive scalar field advected by random flows. We focus on a broad class of nonlinear shear flows multiplied by a stationary, Ornstein–Uhlenbeck (OU) time varying process, including some of their limiting cases, such as Gaussian white noise or plug flows. For the former case with linear shear, recent studies (Camassa et al., 2019) numerically demonstrated that the decaying passive scalar’s long time limiting probability distribution function (PDF) could be negatively skewed in the presence of impermeable channel boundaries, in contrast to rigorous results in free space which established the limiting PDF is positively skewed (McLaughlin and Majda, 1996). Here, the role of boundaries in setting the long time limiting skewness of the PDF is established rigorously for the above class using the long time asymptotic expansion of the N-point correlator of the random field obtained from the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in Camassa et al. (2019). Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Péclet number. We demonstrate the convergence of the Ornstein–Uhlenbeck case to the white noise case in the limit γ→∞ of the OU damping parameter, which generalizes the results for free space in Resnick (1996) to the channel domain problem. We show that the long time limit of the first three moments depends explicitly on the value of γ, which is in contrast to the conclusion in Vanden Eijnden (2001) for the limiting PDF in free space. To find a benchmark for theoretical analysis, we derive the exact formula of the N-point correlator for a flow with no spatial dependence and Gaussian temporal fluctuation, generalizing the results of Bronski et al. (2007). The long time analysis of this formula is consistent with our theory for a general shear flow. All results are verified by Monte-Carlo simulations.

The Transport Equation for the Dispersal of Passive Tracers in a Nonuniform Turbulent Fluid: Numerical Simulations

Abstract The random advection of passive additives in a turbulent fluid plays an important role in solar physics, astrophysics, and atmospheric sciences. We concern ourselves here with the case where the fluctuations are not statistically homogeneous in space, and, hence, where the transport coefficients vary with position. Using a numerical model in which the fluid turbulence is defined kinematically, we show that the evolution of the distribution of passive tracers in the fluid is not always governed by the ordinary diffusion equation. We find it is governed by a more general transport equation whose form depends on the nature of the turbulence, particularly on its compressibility, or divergence. The more general transport equation resembles the ordinary diffusion equation, but the transport coefficient appears in two places and is raised to a power that depends on the divergence of the fluid velocity. If the flow has zero divergence, the case for incompressible turbulence, the resulting transport equation is found to be the regular diffusion equation.

Transport under advective trapping

Abstract

Eulerian space-time correlation of strong magnetohydrodynamic turbulence

In this work the scale-dependent temporal decorrelation of incompressible MHD turbulence is investigated by means of high-resolution numerical simulations and phenomenological modeling. The simulations show that counterpropagating Alfv\'enic fluctuations at each scale decorrelate at the same rate in both balanced and imbalanced turbulence, and consistent with random advection of small-scale eddies by the velocity of the energy containing scales.

Brownian fluctuations of flame fronts with small random advection

We study the effect of small random advection in two models in turbulent combustion. Assuming that the velocity field decorrelates sufficiently fast, we (i) identify the order of the fluctuations of the front with respect to the size of the advection; and (ii) characterize them by the solution of a Hamilton–Jacobi equation forced by white noise. In the simplest case, the result yields, for both models, a front with Brownian fluctuations of the same scale as the size of the advection. That the fluctuations are the same for both models is somewhat surprising, in view of known differences between the two models.

A lattice Boltzmann study on Brownian diffusion and friction of a particle in a confined multicomponent fluid

We study the diffusivity of a small particle immersed in a square box filled with a non-ideal multicomponent fluid in the presence of thermal fluctuations. Our approach is based on the numerical integration of fluctuating lattice Boltzmann models (LBM) for multicomponent mixtures. At changing the wettability on the particle's surface, we measure the mean square displacement (MSD) and compare with the prediction of the Stokes-Einstein theory. Two main set-ups are tested, involving periodic boundary conditions and wall boundary conditions realized on the computational box. We find that full periodic boundary conditions give rise to random advection after millions of lattice Boltzmann time steps, while this effect is mitigated in the presence of wall boundary conditions. The matching with the Stokes-Einstein relation is therefore guaranteed when we use the appropriate frictional properties measured in the presence of confinement. Our results will help the exploration of nanoscale applications with multicomponent fluids using LBM in the presence of thermal fluctuations.

Dispersion of particles in two-dimensional circular vortices

The spreading of n particles simultaneously released is modeled numerically and analytically using a Langevin’s equation. In the numerical experiments, particles are dispersed utilizing the random walk technique plus advection caused by a vortex with rigid-body rotation (RV) or irrotational (IV). In each vortex, the dispersion is described analyzing the statistical behavior of the particles’ position as a function of time t. In the RV case, analytical expressions for the statistics indicate (like the numerical experiments) that the dispersion or variance is linear in t and the diffusion coefficient D depends on the angular speed Ω of the vortex. If the time scale of the random walk is much smaller than the rotation period (i.e., Δt ≪ 2π/Ω), then D decreases lightly if Ω increases relative to a pure two-dimensional (2D) random walk. In the experiments of the IV in which all particles start at the same point, we observe an asymmetric dispersion (the cloud becomes a comet and then a spiral), which leads to a rapid growth in the variance proportional to t3 in the first time steps. This anomalous dispersion occurs while the particles distribute around the origin and the spiral reaches to form a belt. Afterward, the variance grows linearly in t as in standard dispersion, but D is in general larger than the case without the vortex, even if initially the particles formed a ring around the origin. D tends to increase with the circulation for more intense vortices.

Extending the study on the linear advection equation subject to stochastic velocity field and initial condition

In this paper we extend the study on the linear advection equation with independent stochastic velocity and initial condition performed in Dorini and Cunha (2011). By using both existing and novel results on the stochastic chain rule, we solve the random linear advection equation in the mean square sense. We provide a new expression for the probability density function of the solution stochastic process, which can be computed as accurately as wanted via Monte Carlo simulations, and which does not require the specific probability distribution of the integral of the velocity. This allows us to solve the non-Gaussian velocity case, which was not treated in the aforementioned contribution. Several numerical results illustrate the computations of the probability density function by using our approach. On the other hand, we derive a theoretical partial differential equation for the probability density function of the solution stochastic process. Finally, a shorter and easier derivation of the joint probability density function of the response process at two spatial points is obtained by applying conditional expectations appropriately.

Large-Scale Pattern Formation in the Presence of Small-Scale Random Advection.

Despite the presence of strong fluctuations, many turbulent systems such as Rayleigh-Bénard convection and Taylor-Couette flow display self-organized large-scale flow patterns. How do small-scale turbulent fluctuations impact the emergence and stability of such large-scale flow patterns? Here, we approach this question conceptually by investigating a class of pattern forming systems in the presence of random advection by a Kraichnan-Kazantsev velocity field. Combining tools from pattern formation with statistical theory and simulations, we show that random advection shifts the onset and the wave number of emergent patterns. As a simple model for pattern formation in convection, the effects are demonstrated with a generalized Swift-Hohenberg equation including random advection. We also discuss the implications of our results for the large-scale flow of turbulent Rayleigh-Bénard convection.

Log-Correlated Large-Deviation Statistics Governing Huygens Fronts in Turbulence

Analyses have disagreed on whether the velocity $$u_T$$ of bulk advancement of a Huygens front in turbulence vanishes or remains finite in the limit of vanishing local front propagation speed $$u_0$$ . Here, a connection to the large-deviation statistics of log-correlated random processes enables a definitive determination of the correct small- $$u_0$$ asymptotics. This result reconciles several theoretical and phenomenological perspectives with the conclusion that $$u_T$$ remains finite for vanishing $$u_0$$ , which implies a propagation anomaly akin to the energy-dissipation anomaly in the limit of vanishing viscosity. Various leading-order structural properties such as a novel $$u_0$$ dependence of a bulk length scale associated with front geometry are predicted in this limit. The analysis involves a formal analogy to random advection of diffusive scalars.

A two-species weak competition system of reaction-diffusion-advection with double free boundaries

<p style='text-indent:20px;'>In this paper, we investigate a two-species weak competition system of reaction-diffusion-advection with double free boundaries that represent the expanding front in a one-dimensional habitat, where a combination of random movement and advection is adopted by two competing species. The main goal is to understand the effect of small advection environment and dynamics of the two species through double free boundaries. We provide a spreading-vanishing dichotomy, which means that both of the two species either spread to the entire space successfully and survive in the new environment as time goes to infinity, or vanish and become extinct in the long run. Furthermore, if the spreading or vanishing of the two species occurs, some sufficient conditions via the initial data are established. When spreading of the two species happens, the long time behavior of solutions and estimates of spreading speed of both free boundaries are obtained.

Fluctuations of Random Semilinear Advection Equations

We consider a semilinear advection equation driven by a highly oscillatory space-time Gaussian random field, with the randomness affecting both the drift and the nonlinearity. In the linear setting, classical results show that the characteristics converge in distribution to a homogenized Brownian motion, hence, the pointwise law of the solution is close to a functional of the Brownian motion. Our main result is that the nonlinearity plays the role of a random diffeomorphism, and the pointwise limiting distribution is obtained by applying the diffeomorphism to the limit in the linear setting.

Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources

This paper mainly study the dynamics of a Lotka-Volterra reaction-diffusion-advection model for two competing species which disperse by both random diffusion and advection along environmental gradient. In this model, the species are assumed to be identical except spatial resource distribution: heterogeneity vs homogeneity. It is shown that the species with heterogeneous resources distribution is always in a better position, that is, it can always invade when rare. The ratio of advection strength and diffusion rate of the species with heterogeneous distribution plays a crucial role in the dynamics behavior of the system. Some conditions of invasion, driving extinction, and coexistence are given in term of this ratio and the diffusion rate of its competitor.

A Duality-Based Optimization Approach for Model Adaptivity in Heterogeneous Multiscale Problems

This paper introduces a novel framework for model adaptivity in the context of heterogeneous multiscale problems. The framework is based on the idea to interpret model adaptivity as a minimization problem of local error indicators that are derived in the general context of the dual weighted residual (DWR) method. Based on the optimization approach a postprocessing strategy is formulated that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. This allows for the systematic, “goal-oriented” tuning of effective models} with respect to a quantity of interest. The framework is tested numerically on elliptic diffusion problems with different types of heterogeneous, random coefficients, as well as an advection-diffusion problem with a strong microscopic, random advection field.

Stochastic consistency and stochastic stability in mean square sense for Cauchy advection problem

This work deals with the construction of finite difference solutions of random advection Cauchy type partial differential equation containing uncertainty through the coefficient of the velocity. Under appropriate hypothesis on the velocity random variable, we establish that the constructed random finite difference solution is mean square consistent and mean square stable over the whole real line. In addition, the main statistical functions, such as the mean, of the approximate solution stochastic process generated by truncation of the exact finite difference solution are given. Finally, we apply the proposed technique to several illustrative examples which show our discussing for the mean square stability.