A connection between the Lagrangian stochastic–convective and cumulant expansion approaches for describing solute transport in heterogeneous porous media

The equation describing the ensemble-average solute concentration in a heterogeneous porous media can be developed from the Lagrangian (stochastic–convective) approach and from a method that uses a renormalized cumulant expansion. These two approaches are compared for the case of steady flow, and it is shown that they are related. The cumulant expansion approach can be interpreted as a series expansion of the convolution path integral that defines the ensemble-average concentration in the Lagrangian approach. The two methods can be used independently to develop the classical form for the convection–dispersion equation, and are shown to lead to identical transport equations under certain simplifying assumptions. In the development of such transport equations, the cumulant expansion does not require a priori the assumption of any particular distribution for the Lagrangian displacements or velocity field, and does not require one to approximate trajectories with their ensemble-average. In order to obtain a second-order equation, the cumulant expansion method does require truncation of a series, but this truncation is done rationally by the development of a constraint in terms of parameters of the transport field. This constraint is less demanding than requiring that the distribution for the Lagrangian displacements be strictly Gaussian, and it indicates under what velocity field conditions a second-order transport equation is a reasonable approximation.     

Generalized LSG models with spatially varying Coriolis parameter

In this paper, we derive and study approximate balance models for nearly geostrophic shallow water flow where the Coriolis parameter is permitted to vary across the domain as long as it remains nondegenerate. This situation includes, for example, the β-plane approximation to the shallow water equations at mid-latitudes. Our approach is based on changing configuration space coordinates in the underlying variational principle in such a way that consistent asymptotics in the transformed Lagrangian leads to a degenerate Lagrangian structure. In this paper, we restrict our attention to first-order models. We show that the resulting models can be formulated in terms of an advected potential vorticity with a nonlinear vorticity inversion relation. We study the associated solvability conditions and identify a subfamily of models for which these conditions are satisfied without additional restrictions on the data. Finally, we provide the link between our framework and the theory of constrained Hamiltonian systems.     

Arbitrary-order Taylor series expansion-based viscoacoustic wavefield simulation in 3D vertical transversely isotropic media

Taking the anisotropy of velocity and attenuation into account, we investigate the wavefield simulation of viscoacoustic waves in 3D vertical transversely isotropic attenuating media. The viscoacoustic wave equations with the decoupled amplitude attenuation and phase dispersion are derived from the fractional Laplacian operator and using the acoustic approximation. With respect to the spatially variable fractional Laplacian operator in the formulation, we develop an effective algorithm to realize the viscoacoustic wavefield extrapolation by using the arbitrary-order Taylor series expansion. Based on the approximation, the mixed-domain fractional Laplacian operators are decoupled from the wavenumbers and fractional orders. Thus, the viscoacoustic wave propagation can be conveniently implemented by using a generalized pseudospectral method. In addition, we perform the accuracy and efficiency analyses among first-, second- and third-order Taylor series expansion pseudospectral methods with different quality factors. Considering both the accuracy and computational cost, the second-order Taylor series expansion pseudospectral method can generally satisfy the requirements for most attenuating media. Numerical modelling examples not only illustrate that our decoupled viscoacoustic wave equations can effectively describe the attenuating property of the medium, but also demonstrate the accuracy and the high robustness of our proposed schemes.     

图法及其在Toeplitz矩阵分解中的应用

实现了基于图法的稀疏正定系统的求解，并在此基础上实现了具有Toeplitz结构的大型稀疏矩阵的快速LU分解，在基于波边方程的地震数据处理如地震波场模拟和叠前深度偏移等隐式方法中，拉普拉氏算子或亥姆霍兹算子的快速分解是这些方法能否实现的关键，在螺旋边界条件下，这些算子的表示矩阵是具有Toeplitz结构的正定厄密矩阵，可以通过本文方法实现快速分解。     

Analytical solution for the tidally induced Lagrangian residual current in a narrow bay

In a weakly nonlinear tidal system, the depth-averaged equations for the first-order Lagrangian residual velocity (LRV) are deduced systematically. For the case of a narrow bay, the equations are solved analytically and the results for a specific bottom profile are discussed in detail. According to the pattern of the first-order LRV, the bay can be divided into three parts, namely an inner part, a transitional zone, and an outer part. For the given depth profile, the streamline of the first-order LRV for a shorter bay is a part of that for a longer bay. The first-order LRV depends on a nondimensional parameter that combines the influences of the bottom friction coefficient, the tidal period and the averaged water depth. The form of the bottom friction also has a significant influence on the first-order LRV. The second-order LRV, i.e., the Lagrangian drift, is analytically solved and shows dependence on the initial tidal phase. The LRV differs from the Eulerian residual transport velocity both quantitatively and qualitatively. It is demonstrated that the residual currents obtained according to other definitions may cause misunderstanding of the mass transport in water exchange applications.     

3D analytical solution to the tidally induced Lagrangian residual current equations in a narrow bay

The 3D first-order Lagrangian residual velocity (LRV) equation is established, and its analytical solution is obtained in a narrow bay. The results show clearly the 3D structure of the first-order LRV. When the exponential bottom profile is assumed, the upper half layer of the water flows in through the deep channel from the open boundary directly to the head of the bay. Then the water will return to the area surrounding the lower half of the inflow area. The downwelling area is located mainly at the deep channel, while the upwelling area occupies both sides of the bay. The inter-tidal water transport, obtained by integrating the 3D first-order LRV through the water column, has a pattern similar to the previous study in which the 2D depth-averaged Lagrangian residual current equations were solved. The inter-tidal water transport is used to analyze the water exchange, and it is found that the water exchange at different cross sections increases smoothly with the distance between the cross sections and the head of the bay until about one wavelength. It is also found that the pattern of the breadth-averaged Lagrangian residual current varies with the length of the bay if a non-flat bottom profile is used. The depth-integrated LRV and the breadth-averaged LRV are mainly determined by the different terms of the tidal body force, with the former determined by the bottom friction related term and the latter by the eddy viscosity related term. When the bay is longer than one wavelength, different results in the outer bay can be observed.     

Large displacement behaviour of a structural model with foundation uplift under impulsive and earthquake excitations

This paper considers the dynamical behaviour of a structural model with foundation uplift. The equations of motion of the system considered are derived for large displacements thus allowing for the eventual overturning of the system. The transition conditions between successive phases of motion, derived in terms of the specific Lagrangian co‐ordinates used in the formulation of the equations of motion, present innovative aspects which resolve some previously inexplicable behaviour in the structural response reported in the literature. The dynamical behaviour of the model is considered under impulsive and long‐duration ground motions. The minimum horizontal acceleration impulses for the uplift and the overturning of the system are evaluated in analytical form. The sensitivity of the model to uplifting and to overturning under impulsive excitations is established as a function of few significant structural parameters. Numerical applications have been performed changing either the structural parameters or the loading parameter, in order to analyse several dynamical behaviours and also to validate the analytical results. For earthquake ground motions the results, reported in the form of response spectra, show that linearized models generally underestimate, sometimes significantly, the structural response. Copyright © 2003 John Wiley & Sons, Ltd.     

Note on Coriolis-Stokes force and energy

In this study, we consider the origin of the Coriolis-Stokes (CS) force in the wave-averaged momentum and energy equations and make a short analysis of possible energy input to the ocean circulation (i.e., Eulerian mean velocity) from the CS force. Essentially, we find that the CS force appears naturally when considering vertically integrated quantities and that the CS force will not provide any energy input into the system for this case. However, by including the “Hasselmann force”, we show some inconsistencies regarding the vertical structure of the CS force in the Eulerian framework and find that there is a distinct vertical structure of the energy input and that the net input strongly depends on whether the wave zone is included in the analysis or not. We therefore question the introduction of the “Hasselmann force” into the system of equations, as the CS force appears naturally in the vertically integrated equations or when Lagrangian vertical coordinates are used.     

Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay,China

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.

     

Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay, China

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.     

Fractional calculus in hydrologic modeling: A numerical perspective

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.     

Inferring groundwater system dynamics from hydrological time-series data

Abstract

The problem of identifying and reproducing the hydrological behaviour of groundwater systems can often be set in terms of ordinary differential equations relating the inputs and outputs of their physical components under simplifying assumptions. Conceptual linear and nonlinear models described as ordinary differential equations are widely used in hydrology and can be found in several studies. Groundwater systems can be described conceptually as an interlinked reservoir model structured as a series of nonlinear tanks, so that the groundwater table can be schematized as the water level in one of the interconnected tanks. In this work, we propose a methodology for inferring the dynamics of a groundwater system response to rainfall, based on recorded time series data. The use of evolutionary techniques to infer differential equations from data in order to obtain their intrinsic phenomenological dynamics has been investigated recently by a few authors and is referred to as evolutionary modelling. A strategy named Evolutionary Polynomial Regression (EPR) has been applied to a real hydrogeological system, the shallow unconfined aquifer of Brindisi, southern Italy, for which 528 recorded monthly data over a 44-year period are available. The EPR returns a set of non-dominated models, as ordinary differential equations, reproducing the system dynamics. The choice of the representative model can be made both on the basis of its performance against a test data set and based on its incorporation of terms that actually entail physical meaning with respect to the conceptualization of the system.

Citation Doglioni, A., Mancarella, D., Simeone, V. & Giustolisi, O. (2010) Inferring groundwater system dynamics from hydrological time-series data. Hydrol. Sci. J. 55(4), 593–608.     

Analytical expressions of conditional mean,covariance, and sample functions in geostatistics

This work presents analytical expressions for the best estimate, conditional covariance function, and conditional realizations of a function from sparse observations. In contrast to the prevalent approach in kriging where the best estimates at every point are determined from the solution of a system of linear equations, the best-estimate function can be represented analytically in terms of basis functions, whose number depends on the observations. This approach is computationally superior when graphing a function estimate and is also valuable in understanding what the solution should look like. For example, one can immediately see that all singularities in the best-estimate function are at observation points.     

On the exact shape of the horizontal profile of a topographically rectified tidal flow

Abstract

Starting from the nonlinear shallow water equations of a homogeneous rotating fluid we derive the equation describing the evolution of vorticity by a fluctuating bottom topography of small amplitude, using a multiple scale expansion in a small parameter, which is the topographic length scale relative to the tidal wave length. The exact response functions of residual vorticity for a sinusoidal bottom topography are compared with those obtained by a primitive perturbation series and by harmonic truncation, showing the former to be invalid for small topographic length scales and the latter to be only a fair approximation for vorticity produced by planetary vortex stretching. In deriving the exact shape of the horizontal residual velocity profile at a step-like break in the bottom topography, it is shown that the Lagrangian profile only exists in a strip having the width of the amplitude of the tidal excursion at both sides of the break, and that it vanishes outside that interval. Moreover, in the limit of small amplitude topography at least, it vanishes altogether for the generation mechanism by means of planetary vortex stretching. The Eulerian profile is shown to extend over twice the interval of the Lagrangian profile both for production by vortex stretching and by differential bottom friction. These finite intervals over which the residual velocity profiles exist for a step-like topography are not reproduced by harmonic truncation of the basic equation. This method gives exponentially decaying profiles, indicating spurious horizontal diffusion of vorticity. In terms of orders of magnitude, the method of harmonic truncation is reliable for residual velocity produced by vortex stretching but it overestimates the residual velocity produced by differential bottom friction by a factor 2.     

Particle tracking in the vicinity of Helgoland,North Sea: a model comparison

Station Helgoland Roads in the south-eastern North Sea (German Bight) hosts one of the richest long-term time series of marine observations. Hydrodynamic transport simulations can help understand variability in the local data brought about by intermittent changes of water masses. The objective of our study is to estimate to which extent the outcome of such transport simulations depends on the choice of a specific hydrodynamic model. Our basic experiment consists of 3,377 Lagrangian simulations in time-reversed mode initialized every 7 h within the period Feb 2002–Oct 2004. Fifty-day backward simulations were performed based on hourly current fields from four different hydrodynamic models that are all well established but differ with regard to spatial resolution, dimensionality (2D or 3D), the origin of atmospheric forcing data, treatment of boundary conditions, presence or absence of baroclinic terms, and the numerical scheme. The particle-tracking algorithm is 2D; fields from 3D models were averaged vertically. Drift simulations were evaluated quantitatively in terms of the fraction of released particles that crossed each cell of a network of receptor regions centred at the island of Helgoland. We found substantial systematic differences between drift simulations based on each of the four hydrodynamic models. Sensitivity studies with regard to spatial resolution and the effects of baroclinic processes suggest that differences in model output cannot unambiguously be assigned to certain model properties or restrictions. Therefore, multi-model simulations are needed for a proper identification of uncertainties in long-term Lagrangian drift simulations.     

Fractional crystallization in open systems of variable composition: Correlations between element concentrations

Fractional crystallization in open systems is modelled in a general mathematical form. The estimation of element concentrations in liquid and solid phases is done by means of a system of coupled nonlinear first order differential equations. With special, physically founded assumptions on distribution coefficients and source terms in the equations several special cases can be derived which are known in the literature.As an application, correlations between solid-phase concentrations of major elements, trace elements and isotopes are derived which can be used for the interpretation of geological processes. Especially the changes of strontium and rubidium concentrations of magmatic rocks are discussed in the progression of magma differentiation. In a second example the evolution of trace elements in the West-Erzgebirge (G.D.R.) as an open crystallizing magmatic system is discussed.     

On the theory of the axially-symmetric,time-average,state of the atmosphere

Summary A possible formal approach to a closed steady-state theory of the mean axially-symmetric variables is outlined. The approach involves alternating iterative solutions of the energy and momentum equations. In these equations the effects of transient eddy phenomena of all frequencies are assumed to be parameterized in terms of the mean symmetric variables.     

A unifying framework for watershed thermodynamics: balance equations for mass,momentum, energy and entropy,and the second law of thermodynamics

The basic aim of this paper is to formulate rigorous conservation equations for mass, momentum, energy and entropy for a watershed organized around the channel network. The approach adopted is based on the subdivision of the whole watershed into smaller discrete units, called representative elementary watersheds (REW), and the formulation of conservation equations for these REWs. The REW as a spatial domain is divided into five different subregions: (1) unsaturated zone; (2) saturated zone; (3) concentrated overland flow; (4) saturated overland flow; and (5) channel reach. These subregions all occupy separate volumina. Within the REW, the subregions interact with each other, with the atmosphere on top and with the groundwater or impermeable strata at the bottom, and are characterized by typical flow time scales.The balance equations are derived for water, solid and air phases in the unsaturated zone, water and solid phases in the saturated zone and only the water phase in the two overland flow zones and the channel. In this way REW-scale balance equations, and respective exchange terms for mass, momentum, energy and entropy between neighbouring subregions and phases, are obtained. Averaging of the balance equations over time allows to keep the theory general such that the hydrologic system can be studied over a range of time scales. Finally, the entropy inequality for the entire watershed as an ensemble of subregions is derived as constraint-type relationship for the development of constitutive relationships, which are necessary for the closure of the problem. The exploitation of the second law and the derivation of constitutive equations for specific types of watersheds will be the subject of a subsequent paper.     

Eddy diffusivity derived from drifter data for dispersion model applications

Ocean transport and dispersion processes are at the present time simulated using Lagrangian stochastic models coupled with Eulerian circulation models that are supplying analyses and forecasts of the ocean currents at unprecedented time and space resolution. Using the Lagrangian approach, each particle displacement is described by an average motion and a fluctuating part. The first one represents the advection associated with the Eulerian current field of the circulation models while the second one describes the sub-grid scale diffusion. The focus of this study is to quantify the sub-grid scale diffusion of the Lagrangian models written in terms of a horizontal eddy diffusivity. Using a large database of drifters released in different regions of the Mediterranean Sea, the Lagrangian sub-grid scale diffusion has been computed, by considering different regimes when averaging statistical quantities. In addition, the real drifters have been simulated using a trajectory model forced by OGCM currents, focusing on how the Lagrangian properties are reproduced by the simulated trajectories.