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.     

Testing Accuracy of Finite Element and Random Walk Schemes in Prediction of Pollutant Dispersion in Coastal Waters

Distribution of pollutants in coastal waters is usually represented by depth averaged twodimensional convection-dispersion equation. Under very specific conditions this equation can be solved analytically. Although such a solution is restricted to simplified situations it provides a very useful case for testing the performance of various numerical solution techniques currently available for the simulation of convective-dispersion of pollutants in natural water systems. In this paper the analytical solution of the convective dispersion equation is used as a benchmark against which the accuracy of other techniques are assessed. These assessments are based on quantitative comparisons between the results of the solution of two-dimensional convection-dispersion equation by the deterministic finite element and stochastic random walk methods. Both Eulerian and Lagrangian frameworks are employed to obtain the finite element solution of the convection-dispersion problem. It has been shown that the Lagrange–Galerkin finite element scheme yields the most accurate results for the case under study. However, computational costs of the Lagrange–Galerkin method can be relatively high and under certain conditions it may be reasonable to use a less accurate but cost effective random walk scheme to make water quality management decisions.     

Spreading of oil on water in the surface-tension regime

The third stage of oil spreading on water, in which surface-tension force promotes spreading against the resisting viscous effect, is investigated using a similarity solution in combination with an integral boundary-layer technique to solve the unidirectional oil-spreading dynamics problem in the last stage of spreading. The thin layer is assumed to be supplied by oil from a bulk boundary. Using a constitutive equation for oil-film surface tension versus oil-film thickness, analytical solutions near the bulk boundary and near the edge are developed. Using the asymptotic solutions to initiate integration, the differential equations for the oil thickness, oil velocity, and boundary-layer profiles are integrated starting from the leading edge and bulk boundary, which after matching provide a complete solution. The results for the spreading-law prefactors are found to differ by about 10 % from published theoretical results using the same constitutive equation. Using an empirical constitutive equation for oil-film surface tension versus distance from the bulk boundary leads to a spreading-law prefactor that is in excellent agreement with the published experimental result and published theoretical work providing and using the same empirical constitutive equation.     

Turbulent secondary flows in wall turbulence: vortex forcing,scaling arguments,and similarity solution

Spanwise surface heterogeneity beneath high-Reynolds number, fully-rough wall turbulence is known to induce a mean secondary flow in the form of counter-rotating streamwise vortices—this arrangement is prevalent, for example, in open-channel flows relevant to hydraulic engineering. These counter-rotating vortices flank regions of predominant excess(deficit) in mean streamwise velocity and downwelling(upwelling) in mean vertical velocity. The secondary flows have been definitively attributed to the lower surface conditions, and are now known to be a manifestation of Prandtl’s secondary flow of the second kind—driven and sustained by spatial heterogeneity of components of the turbulent (Reynolds averaged) stress tensor (Anderson et al. J Fluid Mech 768:316–347, 2015). The spacing between adjacent surface heterogeneities serves as a control on the spatial extent of the counter-rotating cells, while their intensity is controlled by the spanwise gradient in imposed drag (where larger gradients associated with more dramatic transitions in roughness induce stronger cells). In this work, we have performed an order of magnitude analysis of the mean (Reynolds averaged) transport equation for streamwise vorticity, which has revealed the scaling dependence of streamwise circulation intensity upon characteristics of the problem. The scaling arguments are supported by a recent numerical parametric study on the effect of spacing. Then, we demonstrate that mean streamwise velocity can be predicted a priori via a similarity solution to the mean streamwise vorticity transport equation. A vortex forcing term has been used to represent the effects of spanwise topographic heterogeneity within the flow. Efficacy of the vortex forcing term was established with a series of large-eddy simulation cases wherein vortex forcing model parameters were altered to capture different values of spanwise spacing, all of which demonstrate that the model can impose the effects of spanwise topographic heterogeneity (absent the need to actually model roughness elements); these results also justify use of the vortex forcing model in the similarity solution.     

Measuring sustainable governance in the European Union

This paper examines the problem of measuring sustainable governance in the European Union (EU-27) through the use of duality and the Slutsky equation. The proposed methodology is based on the application of a three-dimensional optimisation model, where the arguments of the objective (sustainable social welfare) function are economic goods that contribute to sustainable economic growth; environmental goods that provide for sustainable environmental protection; and social goods through which sustainable social development is achieved. The dual problem, formulated through this three-dimensional theoretical model, is solved to find the optimal solution, indicating a certain sustainability level. We suggest that this solution can be used for calculating the value of what we define here as the sustainable governmental policy indicator, which is considered to provide quantitative measurement of government policies on sustainable development within the context of ‘good governance’. Furthermore, it is suggested that the Slutsky equation can be used as a reliable method for long-term monitoring and planning of national as well as international good governance with regard to sustainable development policies. In its empirical part, the paper applies the theoretical model in an analysis of the sustainable development indicators (as set out in the Sustainable Development Strategy (SDS) of the EU) in Bulgaria for the period 2000–2010 and compares them to those of the EU-27.     

Assessment of the effects of a port expansion on algae appearance in a costal bay through mathematical modelling. Application to San Lorenzo Bay (North Spain)

In the last years several episodes of algae appearance affecting bathing areas have been observed in San Lorenzo Bay (north of Spain). The analysis of the collected algae revealed that they might come from near intertidal or shallow subtidal zones due to eutrophication processes or through drift algae movement by the action of marine currents. In the vicinity of this area, the expansion of the Port of Gijón (now under construction) supposes a significant modification of the coastal geometry. The magnitude of such an expansion could cause changes in the patterns of currents in the bay, with the consequent alteration of the observed algal appearance phenomena. A mathematical modelling study to evaluate the risk of generation of eutrophication processes in the San Lorenzo Bay area and the transport of drift algae from near sea bed areas was developed. This study required the use of different hydrodynamic models in order to characterize the currents caused by tides, winds and waves. The eutrophication processes in the bay were analyzed with a depth-averaged two-dimensional eutrophication model which deals with eight water quality variables. Calibration of model parameters with the observed data from a field survey was performed. A reasonable agreement with the field measurements was achieved. Model results showed that the maximum phytoplankton concentrations were below eutrophic conditions. Although, the port expansion has led to an increment of phytoplankton concentrations, chlorophyll a levels were not representative of eutrophic conditions. To analyse the transport of drift algae, a methodology based on the utilization of a two-dimensional model which solves the depth-averaged advection-diffusion equation considering seaweed as a conservative tracer was developed and applied. Numerical modelling allowed the identification of the coastal areas that seems to be the source of the seaweed found on the beach. It was also proven that port expansion does not significantly affect drift algae transport in the area.     

Theoretical modeling of suspended grain-size distribution in fluvial environment by stratification and secondary current approaches

Here we propose a theoretical model to compute the suspended grain-size distribution in fluvial environment. We derive the model based on the Reynolds averaged Navier–Stokes equation and the continuity equation of sediment phase. The model includes the effects of secondary current and stratification which are the cause of complex interaction between turbulence and grain-size distribution in the sediment-laden flow. Due to an immense importance of particle–particle and particle-turbulence interactions near the channel bed, we include their impacts in the boundary condition of the model. The present model has noteworthy contribution to demonstrate the phenomena of suspended grain-size distribution in the real world. Reported experimental data in literature shows well agreement with the numerical solution computed from the suggested model. The better computational accuracy of the present model is ascertained when the upper bound of calculated error between observed experimental data and computed values is found to be lowest for our model in comparison to a large number of existing models developed from different mathematical viewpoints.     

Anti-diffusion and source identification with the ‘RAW’ scheme: A particle-based censored random walk

This paper presents a novel methodology for time reversal in advective-diffusive pollutant transport in groundwater systems and other environmental flow systems (specifically: time reversal of diffusive terms). The method developed in this paper extends previous particle-based approaches like the Reversed Time Particle Tracking Method of Bagtzoglou [6]. The reversal of the ‘diffusive’ and/or ‘macrodispersive’ component of pollutant migration is especially under focus here. The basis of the proposed scheme for anti-diffusion is a continuous time, censored, non-local random walk capable of tracking groundwater solute concentration profiles over time while conserving the (reverse) Fickian properties of the anti-diffusing particle cloud in terms of moments. This scheme is an alternative to the direct solution of the eulerian concentration-based diffusion PDE, which is notoriously unstable in reverse time. Our analysis leads to the conclusion that an adaptive time stepping scheme—with decreasing time step—is necessary in order to maintain a constant amount of anti-diffusion (the reverse form of Fick’s law). Specifically, we study the relations between the following parameters: time step evolution vs. time (or vs. number of steps); variance evolution (decrease rate); total time (or number of steps) required to reach a fully anti-diffused solution. The proposed approach is shown to be quite efficient; typically, for every ten time steps, one to two orders of magnitude reduction of the dispersion width of the plume can be attained. Furthermore, the method is shown to be asymptotically exact for reverse Fickian diffusion. The method is applied with success to several situations involving the diffusive transport of a conservative solute in the following cases: (i) Single source recovery in one-dimensional space with constant diffusion parameters (this example serves as a validation test for the theory); (ii) Single source recovery in two-dimensional space with constant isotropic diffusion (this example also serves as a test for the theory); (iii) Multiple source recovery in two-dimensional space, assuming isotropic diffusion. It is expected that the methodology tested in this paper is applicable more generally to complex environmental pollution problems involving multiple sources, anisotropic hydrodynamic dispersion, and space-time variable advection-diffusion flow systems; the modeling of reverse diffusion/dispersion in such systems is currently under investigation.     

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.     

酸沉降临界负荷计算模式的修正

酸沉降临界负荷是酸沉降水平的函数,不同酸沉降水平下,矿物风化速率有数量级差别,在因此,酸沉降临界负荷的计算应先使用判别方程计算平衡ＰＨ值,然后确定采用何种方法计算,在欧洲可以直接采用动态模式计算临界负荷,是因为其酸沉水平能满足判别方程成立,但在中国大多数情况下不成立。     

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.     

A note on the development of the thermocline in temperate lakes

An exact solution has been obtained for a non-linear partial differential equation governing the vertical transport of heat in a lake. The equation, derived and analyzed numerically by Sundaram and Rehm (1971, 1973) is based on a balance between thermal buoyany forces and wind induced turbulence. The exact solution displays the qualitative features of the numerical solution and its representation of the seasonal descent of the thermocline is accurate.     

Plankton biodiversity along a gradient of productivity and its mediation by macrophytes

We studied the effect of aquatic vegetation on the process of species sorting and community assembly of three functional groups of plankton organisms (phytoplankton, seston-feeding zooplankton, and substrate-dwelling zooplankton) along a primary productivity gradient. We performed an outdoor cattle tank experiment (n = 60) making an orthogonal combination of a primary productivity gradient (four nutrient addition levels: 0, 10, 100, and 1000 microg P/L; N/P ratio: 16) with a vegetation gradient (no macrophytes, artificial macrophytes, and real Elodea nuttallii). We used artificial plants to evaluate the mere effects of plant physical structure independently from other plant effects, such as competition for nutrients or allelopathy. The tanks were inoculated with species-rich mixtures of phytoplankton and zooplankton. Both productivity and macrophytes affected community structure and diversity of the three functional groups. Taxon richness declined with increasing plankton productivity in each functional group according to a nested subset pattern. We found no evidence for unimodal diversity-productivity relationships. The proportional abundance of Daphnia and of colonial Scenedesmus increased strongly with productivity. GLM analyses suggest that the decline in richness of seston feeders was due to competitive exclusion by Daphnia at high productivity. The decline in richness of phytoplankton was probably caused by high Daphnia grazing. However, partial analyses indicate that these explanations do not entirely explain the patterns. Possibly, environmental deterioration associated with high productivity (e.g., high pH) was also responsible for the observed richness decline. Macrophytes had positive effects on the taxon richness of all three functional plankton groups and interacted with the initial productivity gradient in determining their communities. Macrophytes affected the composition and diversity of the three functional groups both by their physical structure and through other mechanisms. Part of the macrophyte effect may be indirect via a reduction of phytoplankton production. Our results also indirectly suggest that the often reported unimodal relationship between primary productivity and diversity in nature may be partially mediated by the tendency of submerged macrophytes to be most abundant at intermediate productivity levels.     

Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers

All numerical codes developed to solve the advection–diffusion-reaction (ADR) equation need to be verified before they are moved to the operational phase. In this paper, we initially provide four new one-dimensional analytical solutions designed to help code verification; these solutions are able to handle the challenges of the scalar transport equation including nonlinearity and spatiotemporal variability of the velocity and dispersion coefficient, and of the source term. Then, we present a solution of Burgers’ equation in a novel setup. Proposed solutions satisfy the continuity of mass for the ambient flow, which is a crucial factor for coupled hydrodynamics-transport solvers. By the end of the paper, we solve hypothetical test problems for each of the solutions numerically, and we use the derived analytical solutions for code verification. Finally, we provide assessments of results accuracy based on well-known model skill metrics.     

模拟酸雨下H+-Ca2+在红壤表面的动力学特征

研究模拟酸雨下H Ca2 在红壤表面的反应动力学吸附特征 ,结果表明 :Ca2 的吸附动力学能很好地用一级动力学方程、抛物线扩散方程和Elovich方程 (R2 >0 97)拟合 .随溶液酸度的增加 ,Ca2 的最大吸附量显著降低 ,Ca2 吸附反应速率控制步骤是离子在颗粒内的扩散 ,扩散速率常数随酸度的增加而下降 .酸溶液pH值为 4 5和 5 6时 ,有H 的释放 ,H 的表观释放量能用一级动力学方程和扩散方程拟合 ,Elovich方程不宜拟合pH 5 6酸处理的动力学数据 ;酸溶液pH值为 3 5和 2 5时 ,有H 的消耗 ,H 消耗总量看作表观消耗量 ,用一级动力学方程、Elovich方程和扩散方程进行拟合 .比较相关系数和标准误差的大小 ,除pH值为 2 5的酸处理用扩散方程拟合结果较差外 ,这 3种动力学模型进行拟合均得到较满意的结果     

Interactions between morphological and physiological plasticity optimize energy acquisition in corals

Morphological plasticity in response to environmental heterogeneity may be performance enhancing or may simply result from an intrinsic instability in morphology during development. Although patterns of morphological change are well documented for numerous taxa, it is often unclear whether this plasticity enhances the performance of organisms in the habitat to which they have acclimatized. Reef-building corals are an ideal model system in which to investigate this question. We here develop a three-dimensional geometric model and present a comprehensive photosynthesis data set with experimentally calibrated photosynthesis models that predicts energy acquisition by foliose corals as a function of colony shape. This allows us to assess the extent to which changes in colony morphology along an environmental gradient track the predicted optimal colony morphologies. Our results provide strong evidence that phenotypic plasticity in foliose corals optimizes photosynthetic energy acquisition and is not simply a mechanism to increase light capture. We show that the optimal morphology is constrained at the boundaries of the environmental gradient, with non-optimal morphologies in these habitats having greatly reduced energy acquisition. However, at the center of the environmental gradient, flexibility in photophysiology allows energy acquisition to be very similar for multiple morphologies. Our results highlight the importance of phenotypic plasticity at multiple scales. Variation in overall morphology is important at niche boundaries at which conditions are consistently more stressful, whereas physiological flexibility is important in intermediate and less predictable habitats in which a rapid and reversible response to environmental fluctuations is required.     

新生MnO2对甲基橙废水的脱色特性研究

以化学法合成的新生MnO2作吸附剂,对水中甲基橙染料进行吸附脱色研究,探讨了影响吸附的因素和吸附机理。结果表明：新生Mn2对甲基橙的吸附符合Langmuir吸附等温式,吸附速率大,吸附前溶液pH值是影响染料脱色效果的最主要因素,吸附后溶液pH和温度影响较小。在实验条件下可使甲基橙脱色率达99％。     

Self-aeration in the rapidly- and gradually-varying flow regions of steep smooth and stepped spillways

In high-velocity chute flows, free-surface aeration is often observed. The phenomenon is called self-aeration or white waters. When the turbulent shear stresses next to the free-surface are large enough, air bubbles are entrained throughout the entire air–water column. A rapidly-varied flow region is observed immediately downstream of the inception point of free-surface aeration. An analytical solution of the air diffusion equation is proposed and the results compare well with new experimental data. Both experiments and theory indicate that the flow bulking spans over approximately 3–4 step cavities downstream of the inception point of free-surface aeration on a stepped chute. Further downstream the void fraction distributions follow closely earlier solutions of the air diffusion equation. The application of the diffusion equation solution to prototype and laboratory data shows air bubble diffusivities typically larger than the momentum transfer coefficient. The result highlights however a marked decrease in the ratio of air bubble diffusivity to eddy viscosity with increasing Reynolds number. The finding might indicate some limitation of laboratory investigations of air bubble diffusion process in self-aerated flows and of their extrapolation to full-scale prototype applications.     

Predicting reproduction rate of varroa

A single equation is derived to predict population-density effects on the reproduction rate of the honey bee parasite Varroa destructor Anderson and Trueman. This equation provides a simpler alternative to the approach currently used in the literature, and additionally corrects an anomaly in that approach. The method is then extended to the case of co-existing haplotypes of Varroa. It thus derives an equation used without proof for modelling biocontrol of Varroa, and examines the error caused by an approximation necessary for a closed form solution. Additionally a varroa population model incorporating the derived equation is described.     

Negative effects of nitrogen deposition on Swiss butterflies

Nitrogen (N) deposition from agriculture and combustion of fossil fuels is a major threat to plant diversity, but its effects on organisms at higher trophic levels are unclear. We investigated how N deposition may affect species richness and abundance (number of individuals per species) in butterflies. We reviewed the peer-reviewed literature on variables used to explain spatial variation in butterfly species richness and found that vegetation variables appeared to be as important as climate and habitat variables in explaining butterfly species richness. It thus seemed likely that increased N deposition could indirectly affect butterfly communities via its influence on plant communities. To test this prediction, we analyzed data from the Swiss biodiversity monitoring program for vascular plants and butterflies in 383 study sites of 1 km² that are evenly distributed throughout Switzerland. The area has a modeled N deposition gradient of 2–44 kg N ha⁻¹ year⁻¹. We used traditional linear models and structural equation models to infer the drivers of the spatial variation in butterfly species richness across Switzerland. High N deposition was consistently linked to low butterfly diversity, suggesting a net loss of butterfly diversity through increased N deposition. We hypothesize that at low elevations, N deposition may contribute to a reduction in butterfly species richness via microclimatic cooling due to increased plant biomass. At higher elevations, negative effects of N deposition on butterfly species richness may also be mediated by reduced plant species richness. In most butterfly species, abundance was negatively related to N deposition, but the strongest negative effects were found for species of conservation concern. We conclude that in addition to factors such as intensified agriculture, habitat fragmentation, and climate change, N deposition is likely to play a key role in negatively affecting butterfly diversity and abundance.