A mathematical model for the distribution and abundance of benthic algae in a Norwegian fjord

A mathematical model has been developed to study the distribution and abundance of benthic algae species in a Norwegian fjord. The main forcing functions are the physical factors which have well defined gradients in this area, but biotic interactions are also accounted for in the model.The distribution of algae along vertical transects in the intermediate fjord area has been simulated, and the observed distribution is reproduced fairly well. This indicates that the major factors determining the equilibrium elevation between intertidal and subtidal vegetations in this area are adequately represented in the model.Based on observations in a fjord branch with a high level of freshwater influx and polluted by heavy metals, assumptions are made of the combined influence of these contaminants on growth characteristics of the species considered. The resulting effects on community development and community patterns are simulated. For a certain level of contamination, the decrease in abundance, community development rate and species separation have been determined. It is also shown that the fluctuations in biomass during a year are more pronounced than under unaffected conditions.The relative importance of external growth regulating factors and self-shading is demonstrated for different degrees of community development, and we have indicated that this may be an important factor in the assessment of transient and long-term responses to “natural” and man-made influences on the shore-line vegetation.One of the main problems during model development was the formulation of the interaction function for factors influencing growth parameters. Several alternatives were examined.The differences between the most frequently used forms were small with respect to abundance and not very large with respect to possible changes in distribution. It was noted that even by including only the most limiting factor at each instant, (Liebig's form) a high percentage of the potentially limiting factors did actually limit growth during periods of the year.The model is intended as a working tool for the studies of changes in the shore-line vegetation caused by man-made interferences like heavy metal and oil pollution, and we believe that the results presented demonstrate that it can successfully be used for this purpose.     

A stochastic Lagrangian micromixing model for the dispersion of reactive scalars in turbulent flows: role of concentration fluctuations and improvements to the conserved scalar theory under non-homogeneous conditions

Several reaction schemes, based on the conserved scalar theory, are implemented within a stochastic Lagrangian micromixing model to simulate the dispersion of reactive scalars in turbulent flows. In particular, the formulation of the reaction-dominated limit (RDL) reaction scheme is here extended to improve the model performance under non-homogeneous conditions (NHRDL scheme). The validation of the stochastic model is obtained by comparison with the available measurements of reactive pollutant concentrations in a grid-generated turbulent flow. This test case describes the dispersion of two atmospheric reactant species (NO and O₃) and their reaction product (NO₂) in an unbounded turbulent flow. Model inter-comparisons are also assessed, by considering the results of state-of-the-art models for pollutant dispersion. The present validation shows that RDL reaction scheme provides a systematic overestimation (relative error of ca. 85% around the centreline) in computing the local reactant consumption/production rate, whereas the NHRDL scheme drastically reduces this gap (relative error lower than 5% around the centreline). In terms of NO₂ production (or reactant consumption), neglecting concentration fluctuations determines overestimations of the product mean of around 100% and a NO₂ local production of one order of magnitude higher than the reference simulation. In terms of standard deviations, the concentration fluctuations of both the passive and reactive scalars are generally of the same order of magnitude or up to 1 or 2 orders of magnitudes higher than the corresponding ensemble mean values, except for the background reactant close to the plume edges. The study highlights the importance of modelling pollutant reactions depending on the instantaneous instead of the mean concentrations of the reactants, thus quantifying the role of the turbulent fluctuations of concentration, in terms of scalar statistics (mean, standard deviation, intensity of fluctuations, skewness and kurtosis of concentration, segregation coefficient, simulated reaction rate). This stochastic particle method represents an efficient numerical technique to solve the convection–diffusion equation for reactive scalars and involves several application fields: micro-scale air quality (urban and street-canyon scales), accidental releases, impact of odours, water quality and fluid flow industrial processes (e.g. combustion).     

Prediction of pollutant dispersion in turbulent two-phase flows

With a growing awareness of water pollution problems, in recent years there has been a considerable increased effort in developing and applying numerical models to predict accurately the contaminant distributions, particularly in free surface flows. This numerical study presents a predictive hydrodynamic model in order to explore the dispersion phenomenon of a pollutant injected from time-dependent sources in a turbulent free surface flow. More precisely, we study the impact of pulsation on the dispersion of an injected material. The air/water interface was modeled with the volume of fluid method and sharpness of the free surface was assured by means of Geo-Reconstruct scheme. The numerical results showed that the pulsation played a dominant role at the early stage of the pollutant transport. It was also observed that the pulsation affected the distribution of the injected material especially near the front and that a major swirling action was developed compared to the constant-rate-injection case.     

A dynamic mathematical model for wastewater stabilization ponds

A dynamic mathematical model was developed to predict the effluent quality of facultative wastewater stabilization ponds. For a sound representation of sediment–water column, water column–atmosphere interactions and stratification due to variations in dissolved oxygen concentrations, a two-dimensional hydraulic model was employed considering dispersed flow and diffusion in horizontal and vertical directions, respectively. Resulting partial differential equation system was solved using finite difference methods and matrix manipulation techniques. The model has been calibrated and evaluated on the basis of collected data from a full-scale facultative stabilization pond in Selçuk, Izmir. Variations of COD, NH₄-N, PO₄-P, dissolved oxygen, bacteria and algae concentrations with time and the dimensions of the pond were estimated by using the dynamic model. The model can be used for design of new stabilization ponds and also, for improving the effluent quality of existing ponds.     

A mathematical model of PCB bioaccumulation in plankton

In batch experiments exposing individual plankton constituents to Aroclor 1254 PCB, the rate at which the organism approaches partitioning equilibrium appears to be partly size-dependent while the extent of PCB accumulation is species-specific. The sorptive desorptive kinetics of PCB in these experiments can be described mathematically by a first-order expression. Employing this expression in a model plankton food web permits examination of the role of feeding and sorptive processes in determining PCB body burden under various environmental conditions. When ingestion rates exceed desorption and excretion rates, a consuming organism accumulates PCB above levels predicted by equilibrium partitioning relationships. Feeding-induced oscillations in PCB body burden could thus obscure the reduction of soluble PCB concentration which determine a “baseline” PCB body burden. Unless referenced to a specific set of biological and environmental conditions, the importance of direct partitioning from water vs. food uptake appears to be a moot topic.     

Large-scale turbulent structure of uniform shallow free-surface flows

The hydrodynamics of super- and sub-critical shallow uniform free-surface flows are assessed using laboratory experiments aimed at identifying and quantifying flow structure at scales larger than the flow depth. In particular, we provide information on probability distributions of horizontal velocity components, their correlation functions, velocity spectra, and structure functions for the near-water-surface flow region. The data suggest that for the high Froude number flows the structure of the near-surface layer resembles that of two-dimensional turbulence with an inverse energy cascade. In contrast, although large-scale velocity fluctuations were also present in low Froude number flow its behaviour was different, with a direct energy cascade. Based on our results and some published data we suggest a physical explanation for the observed behaviours. The experiments support Jirka’s [Jirka GH (2001) J Hydraul Res 39(6):567–573] hypothesis that secondary instabilities of the base flow may generate large-scale two-dimensional eddies, even in the absence of transverse gradients in the time-averaged flow properties.     

A mathematical model for the uptake of heavy metals in benthic algae

A mathematical model for the uptake of heavy metals in the benthic algae Ascophyllum nodosum has been developed. The model allows assignment of age-dependent growth parameters which also may be a function of external factors. Mortality curves for A. nodosum corresponding to a highly polluted and a nearly unpolluted area have been used in the study of metal uptake. Uptake has been simulated for different values of growth parameters and different concentrations of heavy metals in seawater. Emphasis has been on the uptake of zinc. The simulations showed that the concentration of zinc in the algae changed much less (6%) than the biomass (30%) with changes in intrinsic growth rate, mortality and carrying capacity.The concentration in the algae was found to be an approximately linear function of the mean concentration in seawater up to about 100 ppb. At very high concentrations, associated with high mortality, the deviation from linearity may be significant (about 13% at 162 ppb at one locality). The calculation indicates further that variations in the zinc concentration in the seawater are considerably damped in the algae.Simulations have also been performed to study the effects of sampling different parts of the algae, and some initial simulations have been performed to study the effects of heavy metal exchange between algae and sea water.     

Velocity scales in steady-nonuniform turbulent flows with low relative submergence

Environmental Fluid Mechanics - The most employed velocity scale in open-channel flows is the shear velocity. Although it is effortlessly identified in uniform flow condition, its value becomes...     

Turbulent velocity profile in fully-developed open channel flows

The determination of velocity profile in turbulent narrow open channels is a difficult task due to the significant effects of the anisotropic turbulence that involve the Prandtl’s second type of secondary flow occurring in the cross section. With these currents the maximum velocity appears below the free surface that is called dip phenomenon. The well-known logarithmic law describes the velocity distribution in the inner region of the turbulent boundary layer but it is not adapted to define the velocity profile in the outer region of narrow channels. This paper relies on an analysis of the Navier–Stokes equations and yields a new formulation of the vertical velocity profile in the center region of steady, fully developed turbulent flows in open channels. This formulation is able to predict time averaged primary velocity in the outer region of the turbulent boundary layer for both narrow and wide open channels. The proposed law is based on the knowledge of the aspect ratio and involves a parameter C_Ar depending on the position of the maximum velocity (ξ_dip). ξ_dip may be derived, either from measurements or from an empirical equation given in this paper. A wide range of longitudinal velocity profile data for narrow open channels has been used for validating the model. The agreement between the measured and the computed velocities is rather good, despite the simplification used.     

A mathematical model and mesh-free numerical method for contact-line motion in lubrication theory

Environmental Fluid Mechanics - We introduce a mathematical model with a mesh-free numerical method to describe contact-line motion in lubrication theory. We show how the model resolves the...     

Detailed experimental study of hydrodynamic turbulent flows generated in vertical slot fishways

The kinematics of hydrodynamic turbulent flows developed in vertical slot fishways (VSF) was studied in detail in flow patterns not yet published to date for the purposes of modifying existing devices and to allow for the passage of all fishes, particularly the smaller species. A transparent device based on the typical prototype dimensions of VSF in France was constructed for the experiment. The velocity measurements were carried out by Particle Image Velocimetry (PIV). These measurements were used to determine the various kinematics parameters characterizing the flow. From the dimensions and slope of the fishway, two flow topologies highlighting the swirling pattern were proposed. The method of Proper Orthogonal Decomposition (POD) was used to undertake unsteady and energetic analyses to characterize the main phases of flow evolution that fish passing through the passage may encounter.     

Assessment of spatial distribution of submerged vegetation in the Orbetello lagoon by means of a mathematical model

Coastal lagoons are characterized by a constant threat of eutrophication and a critical coexistence of differing submerged vegetation forms. This paper investigates the competitive equilibrium of macroalgae and phanerogams in the Orbetello lagoon in relation to physico-chemical and environmental factors, including wind, nutrients in the water column, and sediment characteristics. A mathematical model describing the evolution of the submerged vegetation as a function of the abiotic parameters is used here in conjunction with specific experimental studies to explain the relationship between phanerogams (seagrasses) prairie expansion, water movements, and sediment characteristics. The combination of specific sediment sampling and mathematical modelling shows that water circulation and the state of the upper sediment are both dominant factors in determining the phanerogams distribution in the lagoon and the mutually exclusive growth of these groups in differing parts of the lagoon. Water currents control the distribution of floating macroalgae, resulting in an uneven accumulation of decomposing biomass and phanerogams seed dispersal. The oxygenation provided by the rooted phanerogams affects the sediment characteristics, making them suitable for further prairie expansion. In addition to sediment analysis the use of a mathematical model combining the hydrodynamics and the water quality of the lagoon provides a thorough explanation of the expansion of the rooted vegetation in critical areas. A further result of this research is the validation of the model, originally calibrated with the lagoon central stations’ data, with the newly acquired data from several other parts of the ecosystem. The model predictions are in good agreement with the field observations under a number of environmental conditions and explain the observed expansion trend of phanerogams, which are beneficial for the lagoon ecology, more thoroughly than by relying on the sediment observations alone.     

Stability,resonance and role of turbulent stresses in 1D alluvial flows

Environmental Fluid Mechanics - Linear stability analysis is used to investigate the behavior of small perturbations of a uniform flow in a straight channel with an erodible bed composed by a...     

Re-examining log law velocity profile in smooth open channel flows

Environmental Fluid Mechanics - A comprehensive investigation of velocity distribution is presented, and the log law is re-examined using experimental data from a smooth uniform open channel flow....     

A mathematical model for dispersal of an annual plant population with a seed bank

In order to describe the spatial dispersion of a population of annual plants with a seed bank, we have formulated a delay integrodifference equation model. The plant population growth is considered as being dependent on the density of seedlings that spring from seeds of the two previous years. We have analyzed the effects of the seed bank on the total population size, on the population spatial distribution as well as on the population velocity of invasion. It is showed that the seed bank can significantly alter the dynamics of annual plants, since it can have both a stabilizing and a destabilizing effect on the total population and can also homogenize the spatial distribution of the plants.     

A new model for the CBL growth based on the turbulent kinetic energy equation

Starting from the evolution equation for the turbulent energy density spectrum (EDS), we develop a new model for the growth of the Convective boundary layer (CBL). We apply dimensional analysis to parameterize the unknown inertial transport and convective source term in the dynamic equation for the three-dimensional (3-D) spectrum and solve the 3-D EDS equation. The one-dimensional vertical spectrum is derived from the 3-D spectrum, employing a weight function. This allows us to select the magnitude of the vertical spectral component for the construction of the growing 3-D EDS. Furthermore, we employ the vertical component of the energy spectrum to calculate the eddy diffusivity (required in dispersion models). Currently there are no available experimental data to directly verify our EDS model.     

Mixing at a sediment concentration interface in turbulent open channel flow

In this work we address the role of turbulence on mixing of clear layer of fluid with sediment-laden layer of fluid at a sediment concentration interface. This process can be conceived as the entrainment of sediment-free fluid into the sediment-laden layer, or alternatively, as the transport of sediment into the top sediment-free flow. This process is governed by four parameters—Reynolds number of the flow \(Re_\tau\), non-dimensional settling velocity of the sediment (proxy for sediment size) \(\tilde{V}\), Richardson number \(Ri_\tau\) and Schmidt number Sc. For this work we have performed direct numerical simulations for fixed Reynolds and Schmidt numbers while varying the values of Richardson number and particle settling velocity. In the simple model considered here, the flow’s momentum and turbulence pre-exists over the entire layer of fluid, while the sediment is initially confined to a layer close to the bed. Mixing of sediment-free fluid with the sediment-laden layer is associated primarily with upward transport of sediment and buoyancy. There is no simultaneous upward transport of fluid momentum and turbulence into the sediment-free fluid layer, which is already in motion and turbulent. The analysis performed shows that the ability of the flow to transport a given sediment size decreases with the distance from the bottom, and thus only fine enough sediment particles are transported across the sediment concentration interface. For these cases, the concentration profiles evolve to a final steady state in good agreement with the well-known Rouse profile. The approach towards the Rouse profile happens through a transient self-similar state. This behavior of the flow is not seen for larger particles. Detailed analysis of the three dimensional structure of the sediment concentration interface shows the mechanisms by which sediment particles are lifted up by tongues of sediment-laden fluid with positive correlation between vertical velocity and sediment concentration. Finally, the mixing ability of the flow is addressed by monitoring the time evolution of the center of mass of the sediment-laden layer and the vertical location of the sediment-free/sediment-laden interface.