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 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.     

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 non-negative and high-resolution finite volume method for the depth-integrated solute transport equation using an unstructured triangular mesh

This paper proposes a new high-resolution finite volume method for solving the two-dimensional (2D) solute transport equation using an unstructured mesh. A new simple r-factor algorithm is introduced into the Total Variation Diminishing flux limiter to achieve a more efficient yet accurate high-resolution scheme for solving the advection term. To avoid the physically-meaningless negative solutions resulted from using the Green–Gauss theorem, a nonlinear two-point flux approximation scheme is adopted to deal with the anisotropic diffusion term. The developed method can be readily coupled with a two-dimensional finite-volume-based flow models under unstructured triangular mesh. By integrating with the ELCIRC flow model, the proposed method was verified using three idealized benchmark cases (i.e., advection of a circle-shaped solute field, advection in a cyclogenesis flow field and transport of a initially square-shaped solute plume), and further applied to simulate the non-reactive solute transport process driven by irregular tides in the Deep Bay, eastern Pearl River Estuary of China. These cases are also simulated by models using other existing methods, including different r-factor for advection term and the Green–Gauss theorem for diffusion term. The comparison between the results from the new method and those from other existing methods demonstrated the new method could describe advection induced concentration shock and discontinuities, and anisotropic diffusion at high resolution without providing spurious oscillations and negative values.     

The wave-induced solute flux from submerged sediment

The issue of the transport of dissolved nutrients and contaminants between the sediment in the bottom of a lake or reservoir and the body of water above it is an important one for many reasons. In particular the biological and chemical condition of the body of water is intricately linked to these mass transport processes. As the review by Boudreau (Rev Geophys 38(3):389–416, 2000) clearly demonstrates those transport processes are very complex involving mechanisms as diverse as the wave-induced flux between the sediment and the overlying water and the effect of burrowing animals on the transport within the sediment as well as basic diffusion mechanisms. The present paper focuses on one facet of these transport processes; we re-examine the balance of diffusion and wave-induced advection and demonstrate that the wave-induced flux of a solute from submerged sediment is not necessarily purely diffusive as suggested by Harrison et al. (J Geophys Res 88:7617–7622, 1983) but can be dominated by a mean or time-averaged flux induced by the advective fluid motion into and out of the sediment caused by the fluctuating pressure waves associated with wave motion. Indeed along the subtidal shoreline where the fluctuating bottom pressures are greatest, wave-induced advection will dominate the mean, time-averaged transport of solute into or out of the sediment as suggested in the work of Riedl et al. (Mar Biol 13:210–221, 1972). However, the present calculations also indicate that this advective flux decreases rapidly with increasing depth so that further away from the shoreline the advective flux becomes negligible relative to the diffusive flux and therefore the latter dominates in deeper water.     

A σ-coordinate transport model coupled with rotational boussinesq-type equations

This paper describes a σ-coordinate scalar transport model coupled with a Boussinesq-type hydrodynamic model. The Boussinesq model has the ability to calculate both three-dimensional velocity distributions and the water surface motion. To capture ‘dispersion’ processes in open channel flow, horizontal vorticity effects induced by a bottom shear stress are included in the Boussinesq model. Thus, a reasonable representation of vertical flow structure can be captured in shallow and wavy flow fields. To solve the coupled Boussinesq and scalar transport system, a finite-volume method, based on a Godunov-type scheme with the HLL Riemann solver, is employed. Basic advection and advection–diffusion numerical tests in a non-rectangular domain were carried out and the computed results show good agreement with analytic solutions. With quantitative comparisons of dispersion experiments in an open channel, it is verified that the proposed coupled model is appropriate for both near and far field scalar transport predictions. From numerical simulations in the surf zone, physically reasonable results showing expected vertical variation are obtained.     

Impact of aquacultures on the marine ecosystem: Modelling benthic carbon loading over variable depth

In order to be used within Environmental Impact Assessment study, we have developed a three-dimensional particle tracking model for prediction of benthic carbon loading (BCL) caused by fish farms. The model is based on stochastic differential equations for particle transport consistent with the well-known semi-empirical advection/diffusion equation. It requires only easily obtainable input data in the form of measured current record, the source location and a specification of local bathymetry. The model accounts for advection by long-term residual and tidal currents, turbulent diffusion, realistic bathymetry and variations in daily (monthly or yearly) emissions from fish farm.Here, we concentrate on the changes in sedimentation pattern caused by various bathymetric shapes. Examination of idealized cases reveals where and why we can expect the worst impact on benthic communities. For future reference, these results will be included into guidelines for fish farming.     

Solute transport in a semi-infinite homogeneous aquifer with a fixed point source concentration

This study derives an analytical solution of the advection–dispersion (AD) equation commonly used to describe the transport of pollutants in a semi-infinite homogeneous aquifer. When an extra constant source term is added to the AD equation, it changes the solution of the equation. The AD equation is solved analytically using Laplace transform. Also, the equation is solved numerically using an explicit finite difference method and its stability condition is presented with the aid of matrix method. For the solution of the AD equation the following considerations are made: (1) The dispersion and velocity are considered as time-dependent; (2) dispersion is expressed as directly proportional to the square of velocity; (3) there is also diffusion; (4) there is some initial concentration and the aquifer domain is, therefore, not pollutant-free; (5) There is a time-dependent exponentially decreasing input source; and (6) the concentration gradient is assumed to be zero at the exit boundary. It is found that the contaminant concentration decreases with time contrary to what happens when the extra term is not included.     

Modelling the biochemical oxidation processes in artificial conditions of waste treatment plants

For natural water bodies there are many models of biochemical oxidation of organic pollutants, from simple to complicated ones. For the artificial conditions of waste water treatment plants there are numerous models of oxidation processes too. The authors believe that a better understanding of these processes in nature would be gained if the oxidation processes in more simple artificial conditions were understood.An attempt has been made to explain the variety of types of models of waste treatment in an activated sludge system or a biofilm reactor by the diffusion mechanism of oxidation of single substrates on the one hand, and by superposition of the kinetics of oxidation of single substrates in measurements of the dynamics of oxidation of a complex pollutant on the other hand.The conditions in which zero- and first-order models are valid in the oxidation of single substrates are derived. Respective formulae for rate constants are given. Monod's model is valid in a broader range of substrate concentration variations. An approximate formula for the half-saturation constant is proposed. Satisfactory agreement with experimental data for glucose and ammonia substrates is shown.The formal character of models being used at present is claimed for multicomponent substrates. The accuracy of the calculation is largely governed by the number of model coefficients. Similar results obtained by Monod's and Grau's models, often used in practice, are emphasized.     

Models predicting the fate of pollutants in rivers

The spill of 2,4,‐D in the Rhine river was used to show the evolution from simple to sophisticated models. The first simulation was done with an analytical solution of the dispersion‐advection equation without elimination. Elimination was introduced in a second simulation. The third simulation was carried out with a numerical model. This included elimination and variable dispersion. The lack of data limited the use of very detailed models.     

Beyond eddy diffusivity: an alternative model for turbulent dispersion

Turbulent dispersion proceeds not only much faster but also in a qualitatively different manner than molecular diffusion. Yet, the majority of hydraulic, oceanic and atmospheric models rely on the concept of an eddy diffusivity. It is shown here that an alternative model can be developed to exhibit observed behavior. The new term in the diffusion equation, which is non-local, may be interpreted in terms of the probability density function (pdf) of the turbulent velocity. Different assumptions about this distribution lead to a family of models, one of which is the model proposed here and another, the classical Fickian model of diffusion. A connection is also made with models using fractional calculus.     

Total phosphorus accident pollution and emergency response study based on geographic information system in Three Gorges Reservoir area

• A new algorithm of two-dimensional water quantity and the quality model was built. • The migration and diffusion of TP was simulated. • The emergency measures for sudden water pollution accidents was proposed. In recent years, sudden water pollution accidents in China’s rivers have become more frequent, resulting in considerable effects on environmental safety. Therefore, it is necessary to simulate and predict pollution accidents. Simulation and prediction provide strong support for emergency disposal and disaster reduction. This paper describes a new two-dimensional water quantity and the quality model that incorporates a digital elevation model into the geographic information system. The model is used to simulate sudden water pollution accidents in the main stream of the Yangtze River and Jialing River in the Chongqing section of the Three Gorges Reservoir area. The sectional velocity distribution and concentration change of total phosphorus are then analyzed under four hydrological situations. The results show that the proposed model accurately simulates and predicts the concentration change and migration process of total phosphorus under sudden water pollution accidents. The speed of migration and diffusion of pollutants is found to be greatest in the flood season, followed by the water storage period, drawdown season, and dry season, in that order. The selection of an appropriate water scheduling scheme can reduce the peak concentration of river pollutants. This study enables the impact of pollutants on the ecological environment of river water to be alleviated, and provides a scientific basis for the emergency response to sudden water pollution accidents in the Three Gorges Reservoir area.     

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.     

A comparison of three Eulerian numerical methods for fractional-order transport models

Tracer transport in complex systems like turbulent flows or heterogeneous porous media is now more and more regarded as a non-local process that can hardly be represented by second-order diffusion models. In this work, we consider diffusion models that assume that tracer particles follow a heavy-tail Lévy distribution, which allows for large displacements. We show that such an assumption leads to a fractional-order diffusion operator in the governing equation for tracer concentration. A comparison of three Eulerian numerical methods to discretize that equation is then performed. These consist of the finite difference, finite element and spectral element methods. We suggest that non-local methods, like the spectral element method, are better suited to transport models with fractional-order diffusion operators.     

3D numerical modelling of turbidity currents

During floods, the density of river water usually increases due to a subsequent increase in the concentration of the suspended sediment that the river carries, causing the river to plunge underneath the free surface of a receiving water basin and form a turbidity current that continues to flow along the bottom. The study and understanding of such complex phenomena is of great importance, as they constitute one of the major mechanisms for suspended sediment transport from rivers into oceans, lakes or reservoirs. Unlike most of the previous numerical investigations on turbidity currents, in this paper, a 3D numerical model that simulates the dynamics and flow structure of turbidity currents, through a multiphase flow approach is proposed, using the commercial CFD code FLUENT. A series of numerical simulations that reproduce particular published laboratory flows are presented. The detailed qualitative and quantitative comparison of numerical with laboratory results indicates that apart from the global flow structure, the proposed numerical approach efficiently predicts various important aspects of turbidity current flows, such as the effect of suspended sediment mixture composition in the temporal and spatial evolution of the simulated currents, the interaction of turbidity currents with loose sediment bottom layers and the formation of internal hydraulic jumps. Furthermore, various extreme cases among the numerical runs considered are further analyzed, in order to identify the importance of various controlling flow parameters.     

Local calibration of coliforms parameters of water quality problem at Igapó I Lake, Londrina, Paraná, Brazil

This article presents results concerning the local calibration of the transport parameters (longitudinal and transversal diffusions and decay coefficient) for a two-dimensional problem of water quality at Igapó I Lake, located in Londrina, Paraná, Brazil, using fecal coliforms as an indicator of water quality. The simulation of fecal coliforms concentrations all over the water body is conducted by means of a structured discretization of the geometry of Igapó I Lake, together with the finite difference and finite element methods. By using the velocity field, modeled by the Navier-Stokes and Poisson equations, the flow of fecal coliforms is described by means of a transport model, which considers advective and diffusive processes, as well as a process of fecal coliforms decay. In the checkpoint, the longitudinal and transversal diffusion coefficients and the coliforms decay coefficient that best fitted the value of the fecal coliforms concentration were D_x = D_y = 0.001 m²/h and k = 0.5 d⁻¹ = 0.02083 h⁻¹. A qualitative and quantitative analysis of the numerical simulations conducted in function of the diffusion coefficients and of the coliforms decay parameter provided a better understanding of the local water quality at Igapó I Lake.     

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.     

Solution of fully-coupled shallow water equations and contaminant transport using a primitive-variable Riemann method

A Riemann-solver scheme, using primitive variables rather than conserved variables, is configured and tuned for the solution of the fully-coupled two-dimensional shallow water and contaminant transport equations. This scheme is based on the unstructured finite volume discretization using primitive-variable Roe-flux approximation with an entropy fix. The primitive-variable flux associated with the exact source-term balancing is well-behaved and well-balanced for both still-water and dry regions with arbitrary bed topography. Second-order accuracy is used in space and time. The present study uses a nonlinear implicit scheme based on Newton-iterative algorithm for the time integration. In order to show the accuracy of the scheme, numerical results are verified by different test cases for contaminant advection and diffusion. A scenario of contaminant transport in a complex geometry with wet and dry elements is also simulated to demonstrate that the present work can be implemented on practical applications involving flooding and contaminant transport.     

Geochemistry of burrow waters vented by a bioturbating shrimp in Bermudian sediments

Waters collected at various times from the burrows of Callianassa spp. shrimp are characterized chemically by concentrations of nutrients, sulfide and organic carbon, intermediate between those of pore waters and overlying waters of Coot Pond, Bermuda. Although the shrimp vigorously flush their burrows, irrigation events are insufficently frequent to maintain a burrow water composition completely like that of oxygenated overlying waters; in instances where departure is extensive, models describing the diffusion of interstitial-water solutes into the water column via burrows would have to take this into account. Measured oxidation potentials are substantially reduced with respect to those of the water column, and relatively high levels of reduced arsenic (As III) are encountered. Burrow irrigation provides a mechanism for the rapid, conservative advection of solutes across the sediment-water interface and therefore is at least partially responsible for the concentration and speciation variations typical of coastal seawater. Of the solutes tested, only total dissolved inorganic arsenic occurs at seawater concentration in the burrows, implying that some specialized mechanism intercepts the diffusion of pore water arsenic into the burrows.