Solute Transport in Open-Channel Networks in Unsteady Flow Regime

In this paper we propose a robust algorithm to evaluate solute transport in open-channel networks with transient storage under an unsteady flow regime. In the proposed approach, through the integration of junction equations into the model and solving them explicitly, the analysis of solute transport problems in open-channel networks is simplified significantly. Furthermore, when coupled with a transient hydrodynamic open-channel network model for flow simulation, the proposed model can be utilized in the solution of solute transport problems under unsteady flow regimes. In the proposed model, the governing equations are written in a conservative form and are solved using a fractional-step algorithm, which is based on a relaxation and central difference scheme. The proposed algorithm is robust and accurate even for advection dominant cases. A pure advection with discontinuities, a field application and solute transport in an open-channel network in an unsteady flow regime are included, to demonstrate the performance of the proposed algorithm.     

Suspended particulate matter dynamics in a particle framework

Suspended particulate matter (SPM) dynamics in ocean models are usually treated with an advection–diffusion equation for one or more sediment size classes coupled to the hydrodynamical part of the model. Numerical solution of these additional partial differential equations unavoidably introduces numerical diffusion, i.e. in the case of sharp gradients the possible occurrence of artificial oscillations and non-positivity. A Lagrangian particle-tracking model has been developed to simulate short-term SPM dynamics. Modelling individual sediment particles allows a straightforward physical interpretation of the processes. The tracking of large numbers of individual and independent particles (up to 25 million in total in a single sediment class) can be achieved on high performance computer clusters, due to efficient parallelisation of particle tracking. The movement of the particles is described by a stochastic differential equation, which is consistent with the advection–diffusion equation. Here, the concentration profile is represented by a set of independent moving particles, which are advected according to the 3D velocity field, while the diffusive displacements of the particles are sampled from a random distribution, which is related to the eddy diffusivity field. To account for erosion a new parameterisation is proposed. Three numerical particle tracking schemes (EULER, MILSTEIN and HEUN) are presented and validated in idealised test cases. Finally, the particle tracking algorithms are applied to a realistic scenario, a severe winter storm in the East Frisian Wadden Sea (southern North Sea). The comparison with observations and an Eulerian SPM transport model seems to indicate a somewhat better fidelity of the Lagrangian approach.     

A universal approach to estimate biomass and carbon stock in tropical forests using generic allometric models

Allometric equations allow aboveground tree biomass and carbon stock to be estimated from tree size. The allometric scaling theory suggests the existence of a universal power-law relationship between tree biomass and tree diameter with a fixed scaling exponent close to 8/3. In addition, generic empirical models, like Chave's or Brown's models, have been proposed for tropical forests in America and Asia. These generic models have been used to estimate forest biomass and carbon worldwide. However, tree allometry depends on environmental and genetic factors that vary from region to region. Consequently, theoretical models that include too few ecological explicative variables or empirical generic models that have been calibrated at particular sites are unlikely to yield accurate tree biomass estimates at other sites. In this study, we based our analysis on a destructive sample of 481 trees in Madagascar spiny dry and moist forests characterized by a high rate of endemism (> 95%). We show that, among the available generic allometric models, Chave's model including diameter, height, and wood specific gravity as explicative variables for a particular forest type (dry, moist, or wet tropical forest) was the only one that gave accurate tree biomass estimates for Madagascar (R2 > 83%, bias < 6%), with estimates comparable to those obtained with regional allometric models. When biomass allometric models are not available for a given forest site, this result shows that a simple height-diameter allometry is needed to accurately estimate biomass and carbon stock from plot inventories.     

The Time Invariance Principle,the absence of ecological chaos,and a fundamental pitfall of discrete modeling

This paper is to show that most discrete models used for population dynamics in ecology are inherently pathological that their predications cannot be independently verified by experiments because they violate a fundamental principle of physics. The result is used to tackle an on-going controversy regarding ecological chaos. Another implication of the result is that all dynamical systems must be modeled by differential equations. As a result it suggests that researches based on discrete modeling must be closely scrutinized and the teaching of calculus and differential equations must be emphasized for students of biology.     

Prediction of mass and momentum transport in turbulent plane wall jets over smooth and transitionally rough surfaces

This paper is concerned with the prediction of mass and momentum transport in turbulent wall jets developing over smooth and transitionally rough plane walls. The ability to accurately predict the resulting wall shear stresses and vertical profiles of the Reynolds stresses in these flows is prerequisite to the accurate prediction of bed scour, sediment re-suspension and transport by turbulent diffusion. The computations were performed by solving the Reynolds-averaged forms of the equations describing conservation of mass, momentum and concentration. The unknown correlations that arise from the averaging process (the Reynolds stresses in the case of the momentum equation, and the turbulent mass fluxes in the case of concentration) were obtained from the solution of modeled differential equations that describe their conservation. Since these models are somewhat more complex than those typically used in practice, their benefits are demonstrated by comparisons with results obtained from simpler, eddy-viscosity based closures. Comparisons with experimental data show that results of acceptable accuracy can be obtained only by using the appropriate combination of models for the turbulent fluxes of mass and momentum that properly account for the reduction of the Reynolds stresses due to wall damping effects, and for the modification of the mass transfer rates due to interactions with the mean rates of strain.     

A resource-based approach to modelling the dynamics of interacting populations

The procedure for modelling the growth of single-species populations [Sakanoue, S., 2007. Extended logistic model for growth of single-species populations. Ecol. Model. 205, 159–168] is improved to be applicable to the study of the dynamics of interacting populations. The improved procedure is based on three assumptions: resource availability changes with population size as a variable, resource supply to populations and population demand for resources are defined as functions of resource availability and population size, and the variables of resource availability and population size shift in the supply function attracted to the demand function. These assumptions are organized into three equations. The equations can generate the dynamics models of plant, herbivore, and detritivore populations, and their own resources. The models can be used to describe prey–predator dynamics. They naturally contain nonlinear terms for the predator’s numerical and functional responses. Depending on the terms, the fluctuations in resource availability and population size stabilize. The three equations can also generate the dynamics models of different populations consuming the same resources. The analysis of zero isoclines of the models shows that a superior population can be simply defined as one with a higher intrinsic rate of natural increase, that a stable coexistence may be realized with the intraspecific interference or the interspecific facilitation of superiors, and that the interspecific interference or the intraspecific facilitation of inferiors may make the coexistence unstable and the inferiors winners depending on their initial population size.     

A bio-inspired computing model as a new tool for modeling ecosystems: The avian scavengers as a case study

The models used for ecosystems modeling are generally based on differential equations. However, in recent years new computational models based on biological processes, or bioinspired models, have arisen, among which are P systems. These are inspired by the functions of cells and present important advantages with respect to traditional models, such as a high computational efficiency, modularity and their ability to work in parallel. They are simple, individual-based models that use biological parameters that can be obtained experimentally. In this work, we present the framework for a model based on P systems applied to the study of an ecosystem in which three avian scavengers (predators) interact with 10 wild and domestic ungulates (preys). The computation time for 100 repetitions, corresponding to 14 simulation years each, with an initial population composed of 385,422 individuals, was 30 min. Our results suggest that the model presented, based on P systems, correctly simulates the population dynamics in the period of time analyzed. We discuss the usefulness of this tool in simulating complex ecosystems dynamics to aid managers, conservationists and policy-makers in making appropriate decisions for the improvement of management and conservation programs.     

Dressler’s theory for curved topography flows: iterative derivation,transcritical flow solutions and higher-order wave-type equations

The Dressler equations are a system of two non-linear partial differential equations for shallow fluid flows over curved topography. The theory originated from an asymptotic stretching method formulating the equations of motion in terrain-fitted curvilinear coordinates. Apparently, these equations failed to produce a transcritical flow profile changing from sub- to supercritical flow conditions. Further, wave-like motions over a flat bottom are excluded because the bed-normal velocity component is not accounted for. However, the theory was found relevant for several environmental flow problems including density currents over mountains and valleys, seepage flow in hillslope hydrology, the development of antidunes, the formation of geological deposits from hyper-concentrated flows, and shallow-water flow modeling in hydraulics. In this work, Dressler’s theory is developed in an alternative way by a systematic iteration of the stream and potential functions in terrain-fitted coordinates. The first iteration was found to be the former Dressler’s theory, whereas a second iteration of the governing equations results in velocity components generalizing Dressler’s theory to wave-like motion. Dressler’s first-order theory produces a transcritical flow solution over topography only if the total head is fixed by a minimum value of the specific energy at the transition point. However, the theory deviates from measurements under subcritical flow conditions, given that the bed-normal velocity component is significant. A second iteration to the velocity field was used to produce a second-order differential equation that resembles the cnoidal-wave theory. It accurately produces flow over an obstacle including the critical point and the minimum specific energy as part of the numerical solution. The new cnoidal-wave model compares well with the theory of a Cosserat surface for directed fluid sheets, whereas the Saint-Venant theory appears to be poor.     

Solving multidimensional bioeconomic problems with singular-perturbation reduction methods: Application to managing pest resistance to pesticidal crops

We investigate the application of ‘singular-perturbation’ reduction methods from the dynamical-systems literature to solve continuous-time multidimensional bioeconomic models resulting from integrating economics with increasingly complex biological structures. These methods reduce multidimensional solution space to the lower-dimensional subspace confining long-term dynamics. They arise naturally in problems with state variables evolving on widely disparate time scales. In particular, we demonstrate how the methods reduce the solution space of a linear-control specification—characterized by two state variables adjusting at widely disparate rates—to a single differential equation in the slow variable. All other system variables are determined by algebraic equations. We apply singular-perturbation methods to investigate the optimal management of pest resistance to pesticidal crops. The pest population evolves on a fast-time scale, while the population's genetic composition evolves on a slow-time scale. In comparison with past work, we can more fully characterize the continuous-time dynamics associated with a complex genetic specification.     

A new model for soil–water drainage problems

Seepage flow is an agent related to the transport and dispersion of contamination in groundwater. Steady two-dimensional seepage flow is governed by Laplace’s equation, for which several solution techniques are available. Because computations are complex from a practical point of view, simplified models encompass the Dupuit-Forchheimer approach assuming a horizontal flow. However this approach is inaccurate in seepage problems involving steep drawdowns. In this research, a new theoretical model for 2D seepage flow is proposed based on Fawer’s theory for curved flows Castro-Orgaz (Environ Fluid Mech 10(3):2971–2310, 2010), from which a second-order equation results describing the seepage surface. From this development, a numerical solution for the rectangular dam problem based on the second-order model is presented, whereas a simple first-order equation is found to describe flow to drains under a uniform rainfall. The results of this new model are compared with the full 2D solution of Laplace’s equation for typical test cases, resulting in an excellent agreement.     

Simulation of dam/levee-break hydrodynamics with a three-dimensional implicit unstructured-mesh finite element model

Dam failures usually cause huge economic and life losses , especially in urban areas where there is a high concentration of inhabitants and economic actors. In order to understand the physical mechanisms of the formation and development of dam-break flooding, lots of efforts have been put into different types of modelling techniques. However, most of existing models are 1D (one-dimensional) or 2D models based on the shallow water equations. In this paper, we present a 3D numerical modelling investigation of dam-break flow hydrodynamics in an open L-shape channel. A newly developed 3D unstructured mesh finite element model is used here. An absorption-like term is introduced to the Navier–Stokes equations in order to control the conditioning of the matrix equation in the numerical solution process and thus improve the stability. A wetting and drying algorithm is used here to allow the free surface height to be treated with a high level of implicitness and stability. The 3D model has been validated by comparing the results with the published experimental data. Good agreement has been achieved at six selected locations. This study shows that the 3D unstructured mesh model is capable of capturing the 3D hydraulic aspects and complicated local flows around structures in simulation of dam-break flows.     

CERES — A model of forest stand biomass dynamics for predicting trace contaminant,nutrient, and water effects. I. Model description

CERES is a forest stand growth model which incorporates sugar transport in order to predict both short-term effects and long-term accumulation of trace contaminants and/or nutrients when coupled with the soil chemistry model (SCHEM), and models of solute uptake (DIFMAS and DRYADS) of the Unified Transport Model, UTM. An important feature of CERES is its ability to interface with the soil-plant-atmosphere water model (PROSPER) as a means of both predicting and studying the effects of plant water status on growth and solute transport.CERES considers the biomass dynamics of plants, standing dead and litter with plants divided into leaves, stems, roots, and fruits. The plant parts are divided further into sugar substrate, storage, and in the case of stems and roots, heartwood components.Each ecosystem component is described by a mass balance equation written as a first-order ordinary differential equation.     

Production, consumption, and general equilibrium with physical constraints

This paper analyzes the consequences of integrating the conservation laws of mass and energy into the microeconomic models of production, consumption, and general equilibrium. We show that abstract models and especially general equilibrium theory are consistent with these physical constraints, but most applied and environmental economic models are not. We analyze the consequences of physical conservation laws for substitution possibilities and show that these constraints limit the number of independent substitution processes but not the value of the substitution elasticities. Finally, we propose a method for integrating physical constraints into static microeconomic models with a minimum of changes.     

Quantification of advective solute travel times and mass transport through hydrological catchments

This study has investigated and outlined the possible quantification and mapping of the distributions of advective solute travel times through hydrological catchments. These distributions are essential for understanding how local water flow and solute transport and attenuation processes affect the catchment-scale transport of solute, for instance with regard to biogeochemical cycling, contamination persistence and water quality. The spatial and statistical distributions of advective travel times have been quantified based on reported hydrological flow and mass-transport modeling results for two coastal Swedish catchments. The results show that the combined travel time distributions for the groundwater-stream network continuum in these catchments depend largely on the groundwater system and model representation, in particular regarding the spatial variability of groundwater hydraulic parameters (conductivity, porosity and gradient), and the possible contributions of slower/deeper groundwater flow components. Model assumptions about the spatial variability of groundwater hydraulic properties can thus greatly affect model results of catchment-scale solute spreading. The importance of advective travel time variability for the total mass delivery of naturally attenuated solute (tracer, nutrient, pollutant) from a catchment to its downstream water recipient depends on the product of catchment-average physical travel time and attenuation rate.     

Analytical solution of two-dimensional advection–dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method

In the present study analytical solutions of a two-dimensional advection–dispersion equation (ADE) with spatially and temporally dependent longitudinal and lateral components of the dispersion coefficient and velocity are obtained using Green’s Function Method (GFM). These solutions describe solute transport in infinite horizontal groundwater flow, assimilating the spatio-temporal dependence of transport properties, dependence of dispersion coefficient on velocity, and the particulate heterogeneity of the aquifer. The solution is obtained in the general form of temporal dependence and the source term, from which solutions for instantaneous and continuous point sources are derived. The spatial dependence of groundwater velocity is considered non-homogeneous linear, whereas the dispersion coefficient is considered proportional to the square of spatial dependence of velocity. An asymptotically increasing temporal function is considered to illustrate the proposed solutions. The solutions are validated with the existing solutions derived from the proposed solutions in three special cases. The effect of spatially/temporally dependent heterogeneity on the solute transport is also demonstrated. To use the GFM, the ADE with spatio-temporally dependent coefficients is reduced to a dispersion equation with constant coefficients in terms of new position variables introduced through properly developed coordinate transformation equations. Also, a new time variable is introduced through a known transformation.     

Models for the physiological effects of short O3 exposures on plants

Some published effects of ozone on plant photosynthesis and evaporation are detailed, and attempts were made to develop explanatory models of increasing complexity. Stomatal regulation, which keeps the CO₂ concentration inside the lead constant, is assumed. The O₃ concentration inside the leaf is assumed to be neglible, so O₃ uptake, CO₂ uptake and stomatal conductance must be proportional. The suppression of photosynthesis is assumed to be proportional to the integrated effective O₃ uptake. For a first model, a differential equation is derived from these assumptions and a simple analytical solution is found. A second model includes a threshold O₃ flux. The repair process is discussed and three models are investigated with a constant repair rate, a repair rate dependent on the stage of injury, and with a repair rate dependent on the photosynthetic rate. A comparison of the analytical solutions of the appropriate differential equations with literature data shows that the model with repair dependent on photosytheesis is the most successful one, but a constant repair rate also gives a fair approximation. The model with repair dependent on photosynthesis implies the existence of a threshold level for the suppression of photosynthesis which separates reversible and irreversible O₃ effects. With the assumption of an inhomogeneous leaf a model to predict visible leaf injury is derived. The possible existence of a maximum leaf injury index for one exposure concentration is explained from the properties of the models. The restrictions and possible extensions of the models are discussed. The concept of a threshold O₃ flux and of a critical suppression of photosynthesis are shown to give possible explanations of antagonistic and synergistic effects of O₃, SO₂ and NO₂.     

Secondary flow within a river bed and contaminant transport

We have developed a numerical method to simulate the transport of non-sorbing contaminants within the sediment layer of a stream and the leaching of these contaminants in the steam. Typical stream bottom surfaces are uneven with triangularly shaped undulation forms. The flow of the water above such triangular surfaces causes external pressure changes that result in a “pumping effect” and a secondary flow within the sediment. The latter causes a significant contaminant advection within the sediment layer. The flow field in the porous sediment layer is obtained by solving numerically Darcy’s equations. The unsteady mass transfer equation is solved by using a finite-difference method with an up-wind scheme. The effects of parameters, such as channel slope, hydraulic head and dispersion, are studied by quantitatively comparing the numerical results of the total mass flow rate from the contaminant source, the concentration front propagation, and the contaminant mass flow rate into the water column. The “pumping effect,” increases the flow in the vertical direction and, thus, enhances the vertical advective mass transport of the contaminant. This bedform-shape induced flow is largely responsible for the mass transfer of contaminants into the water column. The numerical results also show that the mechanical dispersion inside the sediment bed will significantly increase the contaminant mass flow rate from the source.     

A mathematical model for type II profile of concentration distribution in turbulent flows

This paper presents a mathematical model to investigate type II profile of suspension concentration distribution (i.e., the concentration profile where the maximum concentration appears at some distance above the bed surface) in a steady, uniform turbulent flow through open-channels. Starting from the mass and momentum conservation equations of two-phase flow, a theoretical model has been derived. The distribution equation is derived considering the effects of fluid lift force, drag force, particle inertia, particle–particle interactions, particle velocity fluctuations and drift diffusion. The equation is solved numerically and is compared with available experimental data as well as with other models existing in the literature. Good agreement between the observed value and computed result, and minimum error in comparison to other models indicate that the present model can be applied in predicting particle concentration distribution for type II profile for a wide range of flow conditions. The proposed model is also able to show the transition from type I profile to type II profile.     

k-ε Predictions of the initial mixing of desalination discharges

Predictions from a k-ε model are compared with recently acquired experimental data from inclined negatively buoyant discharges. The k-ε model is part of a standard computational fluid dynamics package (CFX). Two approaches are taken when implementing the model. One involves using an essentially standard form of the model to predict flow behaviour. The other approach involves calibrating the model, through adjustment of the turbulent Schmidt number in the tracer transport equation, to achieve reasonable predictions for positively buoyant vertical discharges and then applying it to inclined negatively buoyant discharges. While the calibrated approach improves the predictions of some bulk parameters (notably the tracer spread and dilution) when compared to predictions from the standard model, the overall effect on the quality of the predictions is small. Comparisons with experimental data indicate that predictions from both the standard and calibrated simulations compare favourably with trajectory data, but integrated dilution predictions at the centreline maximum height are conservative (mean-integrated concentrations are over-predicted). The standard and calibrated k-ε predictions confirm the importance of buoyant instabilities on the lower (inner) side of the flow, the effects of which are clearly evident in the mean concentration profiles. However, these simulations have a tendency to overestimate the influence of stabilizing density gradients on the upper (outer) side of the flow and are unable to effectively predict the cross-sectional distribution of a tracer. In contrast to a previous study, the above comparisons indicate that predictions of bulk parameters from such models can be poor and indeed are no better than those obtained from relatively simple analytical solutions.