Truncation errors of selected finite difference methods for two-dimensional advection-diffusion equation with mixed derivatives

The spread of a passive contaminant in an open-channel reach is considered with use of a two-dimensional advection-diffusion equation with the included off-diagonal dispersion coefficients. This paper presents the calculation of truncation errors, namely numerical diffusion and numerical dispersion for various finite difference schemes. The accuracy of the considered finite-difference approximations is analysed by deriving and studying the relevant modified partial differential equation.     

Lagrangian modelling of multi-dimensional advection-diffusion with space-varying diffusivities: theory and idealized test cases

To efficiently simulate the advection-diffusion processes along and across density surfaces, we need to deal with a diffusivity tensor containing off-diagonal elements (Redi, J Phys Oceanogr, 12:1154–1158, 1982). In the present paper, the Lagrangian model, in case of a space-varying diffusivity tensor, is developed. This random walk model is applied for two idealized test cases for which the analytical solutions are known. Results of the testing show that the Lagrangian approach provides accurate and effective solutions of advection-diffusion problems for general diffusivity tensor.     

Nonlocal transport models for capturing solute transport in one-dimensional sand columns: Model review,applicability, limitations and improvement

Modelling pollutant transport in water is one of the core tasks of computational hydrology, and various physical models including especially the widely used nonlocal transport models have been developed and applied in the last three decades. No studies, however, have been conducted to systematically assess the applicability, limitations and improvement of these nonlocal transport models. To fill this knowledge gap, this study reviewed, tested and improved the state-of-the-art nonlocal transport models, including their physical background, mathematical formula and especially the capability to quantify conservative tracers moving in one-dimensional sand columns, which represents perhaps the simplest real-world application. Applications showed that, surprisingly, neither the popular time-nonlocal transport models (including the multi-rate mass transfer model, the continuous time random walk framework and the time fractional advection-dispersion equation), nor the spatiotemporally nonlocal transport model (ST-fADE) can accurately fit passive tracers moving through a 15-m-long heterogeneous sand column documented in literature, if a constant dispersion coefficient or dispersivity is used. This is because pollutant transport in heterogeneous media can be scale-dependent (represented by a dispersion coefficient or dispersivity increasing with spatiotemporal scales), non-Fickian (where plume variance increases nonlinearly in time) and/or pre-asymptotic (with transition between non-Fickian and Fickian transport). These different properties cannot be simultaneously and accurately modelled by any of the transport models reviewed by this study. To bypass this limitation, five possible corrections were proposed, and two of them were tested successfully, including a time fractional and space Hausdorff fractal model which minimizes the scale-dependency of the dispersion coefficient in the non-Euclidean space, and a two-region time fractional advection-dispersion equation which accounts for the spatial mixing of solute particles from different mobile domains. Therefore, more efforts are still needed to accurately model transport in non-ideal porous media, and the five model corrections proposed by this study may shed light on these indispensable modelling efforts.     

Stochastic analysis of biodegradation fronts in one-dimensional heterogeneous porous media

We consider a one-dimensional model biodegradation system consisting of two reaction–advection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (pdf) and the traveling front profiles. The travel time pdf does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles are expressed analytically or semi-analytically as functions of the travel time. The problem of nonlinear transport by a random velocity reduces to two subproblems: one being nonlinear transport by a known (unit) velocity, and the other being linear (advective) transport by a random velocity. The approach is illustrated through some examples where the randomness in velocity stems from the spatial variability of porosity.     

SIMULATION OF POLLUTANTS IN RIVER SYSTEMS USING FINITE DIFFERENCE METHOD

1 INTRODUCTION Numerical computation provides an easily extended and user-friendly environment with computer aided programming for the simulation of pollutants in river systems. In the most of water quality assessment and monitoring problems during water pollution control and environmental impact assessment studies of river systems, mathematical modeling has been playing a key role for the last two decades. The majority of existing water quality models are the mechanistic and are based o…     

The impact of boundary on the fractional advection–dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the Caputo derivatives

The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl⁻ in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation.     

A process-based model for sediment transport under various wave and current conditions

The purpose of this study is to investigate the capability of a newly developed process-based model for sediment transport under a wide variety of wave and current conditions.The model is based on the first-order boundary layer equation and the sediment advection-diffusion equation.In particular,a modified low Reynolds number k-e model is coupled to provide the turbulence closure.Detailed model verifications have been performed by simulating a number of laboratory experiments,covering a considerable range of hydrodynamic conditions such as sinusoidal waves,asymmetric waves and wave-current interactions.The model provides satisfactory numerical results which agree well with the measured results,including the time-averaged/dependent sediment concentration profiles and sediment flux profiles,as well as the time series of concentration at given elevations.The observed influences of wave orbital velocity amplitude,wave period and sediment grain size are correctly reproduced,indicating that the fundamental physical mechanisms of those processes are properly represented in the model.It is revealed that the present model is capable of predicting sediment transport under a wide range of wave and current conditions,and can be used to further study the morphodynamic processes in real coastal regions.     

Multi-block lattice Boltzmann simulations of solute transport in shallow water flows

We report a two-dimensional multi-block lattice Boltzmann model for solute transport in shallow water flows, which is developed based on the advection–diffusion equation for mass transport and the shallow water equations for the flows. A weighting factor is included in the centered scheme for improved accuracy. The model is firstly verified by simulating three benchmark tests: wind-driven circulation in a dish-shaped lake, jet-forced flow in a circular basin, and flow formed by two parallel streams containing different uniform concentrations at the same constant velocity; and then it is applied to a practical wind-induced flow, Baiyangdian Lake, which is characterized by irregular geometries and complex bathymetries. The numerical results have shown that the model is able to produce accurate and detailed results for both water flows and solute transport, which is attractive, especially for flows in narrow zones of practical terrains and certain areas with largely varying pollutant concentrations.     

鄱阳湖二维水动力和水质耦合数值模拟

针对大型通江湖泊水位变化剧烈,地形起伏多变,岸线复杂,湖泊内部窄小洪道与大面洲滩和洼地连接复杂,湖泊洲滩出露、淹没频繁交替等湖泊水情和地貌特征,基于二维浅水方程和对流扩散方程组构建了鄱阳湖二维水动力和水质耦合模拟模型.模型采用非结构网格有限体积法进行离散,以HLLC算法计算单元界面的水量、动量和物质输运通量.水陆边界通...     

轴对称各向异性介质波动方程深度偏移的对称非平稳相移方法

由所建立的三维qP波相速度表示式出发,导出并解析求解各向异性介质中的频散方程,得到三维各向异性介质中的相移算子,进而将以相移算子为基础的对称非平稳相移方法推广到各向异性介质,发展了一个三维各向异性介质的深度偏移方法. 文中使用的各向异性介质的速度模型与现行的各向异性构造的速度估计方法一致,将各向同性、弱各向异性及强各向异性统一在一个模型中. 所建立的各向异性介质对称非平稳相移波场延拓算子可以同时适应速度及各向异性参数横向变化;文中给出的算例虽然是针对二维VTI介质的,但所提出的算法同样适用于三维TI介质.     

基于四维变分同化法的巢湖流域南淝河水质模拟

针对河流模拟中未知不确定性源对模拟精度的影响,以巢湖流域南淝河为研究对象,建立了基于四维变分同化方法的南淝河干流水质模型,研究了含未知污染源的南淝河水质过程模拟.模型以未知污染负荷的动态变化过程为控制变量,通过同化沿河不同断面的逐日水质监测数据,识别不同河段的逐日入河污染负荷过程来实现水质过程的模拟,改变了常规模型模拟需提前预知并输入污染负荷的应用前提.模拟结果表明,采用四维变分同化方法的水质模拟结果有明显改进,重点河段水质模拟的纳什效率系数从小于0提高到0.5以上.识别的入河污染过程与降雨过程波动总体一致,证实南淝河的入河污染与降雨过程密切;同时,模型也可识别异常的入河负荷,提高模型对水环境问题的诊断分析能力.该方法可推广应用于复杂河流系统,为巢湖等流域污染来源定量解析、水质预测预警及污染管控提供支持.     

关于动力Alfven波的色散与朗道阻尼

根据完全动力论理论,证明了可由简化漂移动力论方程求得动力Ａｌｆｖｅｎ波色散方程,即横向用磁流体力学方程,纵向用Ｖｌａｓｏｖ方程,在初级近似下由此推导出适合于日冕和太阳风的色散关系和Ｌａｎｄａｕ阻尼,得到在性质上与ＭＨＤ　Ａｌｆｖｅｎ波完全不同的动力Ａｌｆｖｅｎ波,太阳风中Ａｌｆｖｅｎ湍流很容易由动力Ａｌｆｖｅｎ波演化而来。提出由动力Ａｌｆｒｅｎ波构筑太阳风高速流模型将更符合观测结果。     

Analytical and experimental study on dissolved pollutant wash‐off over impervious surfaces

Non‐point source pollution in the impervious surface of city, which including dissolved and particulate pollutants, is a significant source of water pollution. Simple first‐order decay models can generally simulate the cumulative wash‐off process of the particulate pollutants. There is inadequate knowledge as to whether or not they are suitable for dissolved pollutants. This study presents a mathematical wash‐off model for dissolved pollutants, which combines analytical equations for overland flows and the exponential equation for the pollutant wash‐off. A series of laboratory experiments have been conducted to verify this wash‐off model. It shows that the pollutant concentration and pollutant transport rate can be predicted well by the newly developed equations. It is found that the pollutant concentration monotonically decreases to zero as the accumulated pollutants are washed off, whereas the pollutant transport rate first increases to the maximum value and then decreases to zero. The maximum pollutant transport rate is found to increase with the decrease of the arrival time of the maximum value. The difference between the simplified exponential model and the amended wash‐off equation depends on the initial residual percentage (P_c), but the present equation generally provides a more accurate representation of the wash‐off process of dissolved pollutants.     

Suspended load in flows on erodible bed

Steady state suspended-load of sediment transported in flow over erodible beds usually is treated by the advection-diffusion approach, though in recent years, it is being treated as a two-phase flow phenomenon incorporating kinetics of sediment particles. Among the advection-diffusion approaches, Rouse＇s equation is the well-known, although a number of researchers in later periods have attempted to improve it by modifying the mixing length concept taking into account other aspects. In this paper, the advection-diffusion approach and associated logarithmic law of flow velocity are revisited. It is concluded from the logarithmic law that the Reynolds shear stress is a linear function of height above the bed, which reduces to bed shear stress in the case of a long horizontal channel. As a consequence, it is shown that the volumetric concentration of sediment is best approximated by the sum of two power laws of height above the bed. An equation is derived for the suspended-load transport rate in terms of elementary functions.     

MODELING OF SEDIMENT AND NONPOINT SOURCE POLLUTANT YIELD

1 INTRODUCT1ONWaer bodies commonly receive both point soure and nonpoint source pollutans. NonPoint pollution ischaracterized by occasional high loads and high concotions (PfatO, et al, l990; Shen, et al., l992;Novotuy and Olem,l994; Li and Shen, l996). It is very difficult to measure, to colltrOl and to trCat. FornonPoint source pollution control, the most productive research area is probably the develOPmnt andtesting of better watershed water quality models (Haith, l998). Measureme…     

太湖富营养化条件下影响蓝藻水华的主导气象因子

利用2004-2018年卫星遥感解译的太湖蓝藻水华信息构建蓝藻综合指数,采用随机森林机器学习算法分析同期气象因子与蓝藻水华综合指数的关系,定量评估影响蓝藻水华的主要气象因子特征变量的重要性度量和贡献率.结果表明,在光、温、水、风等主要气象要素中,气温对蓝藻水华综合指数起着主导的作用,其次是风速和降水,日照时间的影响或可忽略.其中气温条件中重要性度量最大的是年平均气温,其次是冬、春季节的平均气温;风速因子中影响较大的是7月份的平均风速;水分条件中主导因子是9月累计降水量.优选的随机森林模型模拟值与实际蓝藻水华综合指数的变化趋势基本一致,拟合优度为0.91,通过0.01显著性检验,随机森林模型模拟效果较好.用随机森林模型模拟值对太湖蓝藻水华分等级评估,模型模拟精度达到了86.7%,其中5个重度等级年份模型模拟结果完全一致,中度等级的6个年份模型模拟值有5年与之相符,中度以上等级的模拟精度达90.9%,模型能够反映气象因子对蓝藻水华综合指数的综合影响,对中、重度蓝藻水华的模拟效果更好.随机森林模型有助于理解富营养化状态下影响蓝藻水华的主导气象因子,利用气象因子的可预测性可以促进蓝藻水华预测预警能力的提升.     

三维VTI介质中波动方程深度偏移的最优分裂Fourier方法

从含Thomsen各向异性参数的qP波相速度表示式出发,建立并求解三维VTI介质中的频散方程,得到三维VTI介质中的相移算子,进而将以相移算子为基础的最优分裂Fourier方法推广到三维VTI介质,发展了一个三维VTI介质的深度偏移方法.文中使用的各向异性介质的速度模型与现行的各向异性构造的速度估计方法一致,将各向同性、弱各向异性及强各向异性统一在一个模型中.文中提出的偏移算法对相移法引入了高阶校正项来补偿介质横向变化的影响,使该方法可应用于横向非均匀VTI介质的陡角度成像,文中给出的偏移脉冲响应很好地证明了这一点.     

Adjoint method for the optimum planning of industrial pollutant sources

The deterioration of air quality is threatening the life and health of people. Scientists in China and other countries have done a great deal of research work on the details of air pollution and the methods of preven-tion and control during the past decades. Up to now, most of the achievements are concentrated on the techniques of controlling pollutant sources and the programs of reduction, which focus on the improve-ment of air quality and the restoration of environment. The techniques of con…