Geophysical flows under location uncertainty,Part II Quasi-geostrophy and efficient ensemble spreading

Models under location uncertainty are derived assuming that a component of the velocity is uncorrelated in time. The material derivative is accordingly modified to include an advection correction, inhomogeneous and anisotropic diffusion terms and a multiplicative noise contribution. In this paper, simplified geophysical dynamics are derived from a Boussinesq model under location uncertainty. Invoking usual scaling approximations and a moderate influence of the subgrid terms, stochastic formulations are obtained for the stratified Quasi-Geostrophy and the Surface Quasi-Geostrophy models. Based on numerical simulations, benefits of the proposed stochastic formalism are demonstrated. A single realization of models under location uncertainty can restore small-scale structures. An ensemble of realizations further helps to assess model error prediction and outperforms perturbed deterministic models by one order of magnitude. Such a high uncertainty quantification skill is of primary interests for assimilation ensemble methods. MATLAB^® code examples are available online.     

Geophysical flows under location uncertainty,Part I Random transport and general models

A stochastic flow representation is considered with the Eulerian velocity decomposed between a smooth large scale component and a rough small-scale turbulent component. The latter is specified as a random field uncorrelated in time. Subsequently, the material derivative is modified and leads to a stochastic version of the material derivative to include a drift correction, an inhomogeneous and anisotropic diffusion, and a multiplicative noise. As derived, this stochastic transport exhibits a remarkable energy conservation property for any realizations. As demonstrated, this pivotal operator further provides elegant means to derive stochastic formulations of classical representations of geophysical flow dynamics.     

Stochastic particle based models for suspended particle movement in surface flows

Modeling of suspended sediment particle movement in surface water can be achieved by stochastic particle tracking model approaches.In this paper,different mathematical forms of particle tracking models are introduced to describe particle movement under various flow conditions,i.e.,the stochastic diffusion process,stochastic jump process,and stochastic jump diffusion process.While the stochastic diffusion process can be used to represent the stochastic movement of suspended particles in turbulent flows,the stochastic jump and the stochastic jump diffusion processes can be used to describe suspended particle movement in the occurrences of a sequence of extreme flows.An extreme flow herein is defined as a hydrologic flow event or a hydrodynamic flow phenomenon with a low probability of occurrence and a high impact on its ambient flow environment.In this paper,the suspended sediment particle is assumed to immediately follow the extreme flows in the jump process（i.e.the time lag between the flow particle and the sediment particle in extreme flows is considered negligible）.In the proposed particle tracking models,a random term mainly caused by fluid eddy motions is modeled as a Wiener process,while the random occurrences of a sequence of extreme flows can be modeled as a Poisson process.The frequency of occurrence of the extreme flows in the proposed particle tracking model can be explicitly accounted for by the Poisson process when evaluating particle movement.The ensemble mean and variance of particle trajectory can be obtained from the proposed stochastic models via simulations.The ensemble mean and variance of particle velocity are verified with available data.Applicability of the proposed stochastic particle tracking models for sediment transport modeling is also discussed.     

STOCHASTIC THEORY OF THE DISTRIBUTION OF SEDIMENT PARTICLES SUSPENDED IN A TURBULENT FLOW

The random motion of sediment particles suspended in a turbulent flow is studied by means of stochastic process. Results of analysis of particle's frequency response to the random force exerted on the particle due to fluid turbulence suggest that only the lower part of the whole frequency range of the eddy motion will govern the motion of the particle. The mean values of particle velocity and displacement in the vertical direc- tion are calculated and it is found that particle velocity vp- can be decomposed into a mean motion and a velocity fluctuation vp- , where is equal to the fall velocity in tranquil fluid. An Ito^ random differential equation for particle dis- placement Yp is developed, from which a Fokker-Planck equation for the probability density function p(y,t) is derived on the basis of the theory of Markov process, where y denotes the vertical coordinate. The vertical distribution of the particle is thus interrelated to the random motion of the particle. The an effect that a particle will be subject to in the neighborhood or the bottom boundary is taken into consideration and a corresponding Fokker-Planck equation is developed. Analytical solution of the Fok- ker-Planck equation including the lift force effect shows that probability density p(y,t) for the particle displacement has a maximum value at y = H where the perpen- dicular component of the lift force balances the particle gravity. This theoretical result agrees with experimental observations as reported in literature.     

Stochastic analysis of immiscible displacement of the fluids with arbitrary viscosities and its dependence on support scale of hydrological data

Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow.     

Numerical simulation of three‐dimensional flow hydraulics in a braided channel

Results are presented from a numerical simulation of three‐dimensional flow hydraulics around a mid‐channel bar carried out using the FLUENT/UNS computational fluid dynamics (CFD) software package. FLUENT/UNS solves the three‐dimensional Reynolds‐averaged form of the Navier–Stokes equations. Turbulence closure is achieved using a RNG k–ϵ model. Simulated flow velocities are compared with measured two‐dimensional velocities (downstream and cross‐stream) obtained using an electromagnetic current meter (ECM). The results of the simulation are qualitatively consistent with the flow structures observed in the field. Quantitative comparison of the simulated and measured velocity magnitudes indicates a strong positive correlation between the two (r=0·88) and a mean difference of 0·09 m s⁻¹. Deviations between simulated and measured velocities may be identified that are both random and systematic. The former may reflect a number of factors including subgrid‐scale natural spatial variability in flow velocities associated with local bed structures and measurement uncertainty resulting from problems of ECM orientation. Model mesh configuration, roughness parameterization and inlet boundary condition uncertainty may each contribute to systematic differences between simulated and measured flow velocities. These results illustrate the potential for using CFD software to simulate flow hydraulics in natural channels with complex configurations. They also highlight the need for detailed spatially distributed datasets of three‐dimensional flow variables to establish the accuracy and applicability of CFD software. Copyright © 1999 John Wiley & Sons, Ltd.     

Interface dynamics in randomly heterogeneous porous media

We consider the dynamics of a fluid interface in heterogeneous porous media, whose hydraulic properties are uncertain. Modeling hydraulic conductivity as a random field of given statistics allows us to predict the interface dynamics and to estimate the corresponding predictive uncertainty by means of statistical moments. The novelty of our approach to obtaining the interface statistics consists of dynamically mapping the Cartesian coordinate system onto a coordinate system associated with the moving front. This transforms a difficult problem of deriving closure relationships for highly nonlinear stochastic flows with free surfaces into a relatively simple problem of deriving stochastic closures for linear flows in domains with fixed boundaries. We derive a set of deterministic equations for the statistical moments of the interfacial dynamics, which hold in one and two spatial dimensions, and analyze their solutions for one-dimensional flow.     

Geophysical flows under location uncertainty,Part III SQG and frontal dynamics under strong turbulence conditions

Models under location uncertainty are derived assuming that a component of the velocity is uncorrelated in time. The material derivative is accordingly modified to include an advection correction, inhomogeneous and anisotropic diffusion terms and a multiplicative noise contribution. This change can be consistently applied to all fluid dynamics evolution laws. This paper continues to explore benefits of this framework and consequences of specific scaling assumptions. Starting from a Boussinesq model under location uncertainty, a model is developed to describe a mesoscale flow subject to a strong underlying submesoscale activity. Specifically, turbulent diffusion and rotation effects have similar orders of magnitude. As obtained, the geostrophic balance is modified and the Quasi-Geostrophic assumptions remarkably lead to a zero Potential Vorticity. The ensuing Surface Quasi-Geostrophic model provides a simple diagnosis of warm frontolysis and cold frontogenesis.     

Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.     

A review on stochastic differential equations for applications in hydrology

Fundamentals of the theory of stochastic calculus and stochastic differential equations (SDE's) which are finding increasing application in water resources engineering are reviewed. The basics of probability theory, mean square calculus and the Wiener, white Gaussian and compound Poisson processes are given in preparation for a discussion of the general Itô SDE with drift, diffusion and jump discontinuity terms driven by Gaussian white noise and compound Poissionian impulses. Also discussed are stochastic integration and the derivation of moment equations via the Itô differential rule. The lierature of SDE's is reviewed with an emphasis on the more accessible sources.     

Wavelet-vaguelette decomposition of spatiotemporal random fields

A wavelet-based orthogonal decomposition of the solution to stochastic differential/pseudodifferential equations of parabolic type is derived in the cases of random initial conditions and random forcing. The family of spatiotemporal models considered can represent anomalous diffusion processes when the spatial operator involved is a fractional or multifractional pseudodifferential operator. The results obtained are applied to the generation of the sample paths of Gaussian spatiotemporal random fields in the family studied.     

The effect of random recharge on uniform steady free-surface flow in heterogeneous porous formations

The effect of parametric uncertainty in recharge rate and spatial variability of hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for sloped mean uniform flow in a randomly heterogeneous porous medium under the influence of random recharge. We develop analytic solutions for the variance of free-surface position, head, and specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of head and specific discharge beneath the free-surface. Statistical moments are derived using a first-order approximation and then compared with their parallel in an unbounded medium. The effect of recharge mean and variability on the statistical moments is analyzed. Results can be applied to more complex flows, slowly varying in the mean.     

Improving rainfall–runoff modelling through the control of uncertainties under increasing climate variability in the Ouémé River basin (Benin,West Africa)

ABSTRACT

The objective of this paper is to understand how the natural dynamics of a time-varying catchment, i.e. the rainfall pattern, transforms the random component of rainfall and how this transformation influences the river discharge. To this end, this paper develops a rainfall–runoff modelling approach that aims to capture the multiple sources and types of uncertainty in a single framework. The main assumption is that hydrological systems are nonlinear dynamical systems which can be described by stochastic differential equations (SDE). The dynamics of the system is based on the least action principle (LAP) as derived from Noether’s theorem. The inflow process is considered as a sum of deterministic and random components. Using data from the Ouémé River basin (Benin, West Africa), the basic properties for the random component are considered and the triple relationship between the structure of the inflowing rainfall, the corresponding SDE that describes the river basin and the associated Fokker-Planck equations (FPE) is analysed.

EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR D. Gerten     

Ground water contaminant transport by nondivergence-free, unsteady and nonstationary velocity fields

Pore flow velocity is assumed to be a nondivergence-free, unsteady, and nonstationary random function of space and time for ground water contaminant transport in a heterogeneous medium. The laboratory-scale stochastic contaminant transport equation is up scaled to field scale by taking the ensemble average of the equation by using the cumulant expansion method. A new velocity correction, which is a function of mean pore flow velocity divergence, is obtained due to strict second order cumulant expansion (without omitting any term after the expansion). The field scale transport equations under the divergence-free pore flow velocity field assumption are also derived by simplifying the nondivergence-free field scale equation. The significance of the new velocity correction term is investigated on a two dimensional transport problem driven by a density dependent flow.     

An eddifying Stommel model: fast eddy effects in a two-box ocean

ABSTRACT

A system of stochastic differential equations is formulated describing the heat and salt content of a two-box ocean. Variability in the heat and salt content and in the thermohaline circulation between the boxes is driven by fast Gaussian atmospheric forcing and by ocean-intrinsic, eddy-driven variability. The eddy forcing of the slow dynamics takes the form of a colored, non-Gaussian noise. The qualitative effects of this non-Gaussianity are investigated by comparing to two approximate models: one that includes only the mean eddy effects (the “averaged model”), and one that includes an additional Gaussian white-noise approximation of the eddy effects (the “Gaussian model”). Both of these approximate models are derived using the methods of fast averaging and homogenisation. In the parameter regime where the dynamics has a single stable equilibrium the averaged model has too little variability. The Gaussian model has accurate second-order statistics, but incorrect skew and rare-event probabilities. In the parameter regime where the dynamics has two stable equilibria the eddy noise is much smaller than the atmospheric noise. The averaged, Gaussian, and non-Gaussian models all have similar stationary distributions, but the jump rates between equilibria are too small for the averaged and Gaussian models.     

Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition

During probabilistic analysis of flow and transport in porous media, the uncertainty due to spatial heterogeneity of governing parameters are often taken into account. The randomness in the source conditions also play a major role on the stochastic behavior in distribution of the dependent variable. The present paper is focused on studying the effect of both uncertainty in the governing system parameters as well as the input source conditions. Under such circumstances, a method is proposed which combines with stochastic finite element method (SFEM) and is illustrated for probabilistic analysis of concentration distribution in a 3-D heterogeneous porous media under the influence of random source condition. In the first step SFEM used for probabilistic solution due to spatial heterogeneity of governing parameters for a unit source pulse. Further, the results from the unit source pulse case have been used for the analysis of multiple pulse case using the numerical convolution when the source condition is a random process. The source condition is modeled as a discrete release of random amount of masses at fixed intervals of time. The mean and standard deviation of concentration is compared for the deterministic and the stochastic system scenarios as well as for different values of system parameters. The effect of uncertainty of source condition is also demonstrated in terms of mean and standard deviation of concentration at various locations in the domain.     

Regionalization of unit hydrograph Parameters: 2. Uncertainty analysis

Hydrologic model parameters obtained from regional regression equations are subject to uncertainty. Consequently, hydrologic model outputs based on the stochastic parameters are random. This paper presents a systematic analysis of uncertainty associated with the two parameters, N and K, in Nash's IUH model from different regional regression equations. The uncertainty features associated with N and K are further incorporated to assess the uncertainty of the resulting IUH. Numerical results indicate that uncertainty of N and K from the regional regression equations are too significant to be ignored.     

Regionalization of unit hydrograph Parameters: 2. Uncertainty analysis

Hydrologic model parameters obtained from regional regression equations are subject to uncertainty. Consequently, hydrologic model outputs based on the stochastic parameters are random. This paper presents a systematic analysis of uncertainty associated with the two parameters, N and K, in Nash's IUH model from different regional regression equations. The uncertainty features associated with N and K are further incorporated to assess the uncertainty of the resulting IUH. Numerical results indicate that uncertainty of N and K from the regional regression equations are too significant to be ignored.     

Stochastic delineation of capture zones: classical versus Bayesian approach

A Bayesian approach to characterize the predictive uncertainty in the delineation of time-related well capture zones in heterogeneous formations is presented and compared with the classical or non-Bayesian approach. The transmissivity field is modelled as a random space function and conditioned on distributed measurements of the transmissivity. In conventional geostatistical methods the mean value of the log transmissivity and the functional form of the covariance and its parameters are estimated from the available measurements, and then entered into the prediction equations as if they are the true values. However, this classical approach accounts only for the uncertainty that stems from the lack of ability to exactly predict the transmissivity at unmeasured locations. In reality, the number of measurements used to infer the statistical properties of the transmissvity field is often limited, which introduces error in the estimation of the structural parameters. The method presented accounts for the uncertainty that originates from the imperfect knowledge of the parameters by treating them as random variables. In particular, we use Bayesian methods of inference so as to make proper allowance for the uncertainty associated with estimating the unknown values of the parameters. The classical and Bayesian approach to stochastic capture zone delineation are detailed and applied to a hypothetical flow field. Two different sampling densities on a regular grid are considered to evaluate the effect of data density in both methods. Results indicate that the predictions of the Bayesian approach are more conservative.