Flood routing based on diffusion wave equation using mixing cell method

A one-dimensional non-linear diffusion wave equation is derived from the Saint Venant equations with neglect of the inertia terms. This non-linear equation has no general analytical solution. Numerical schemes are therefore employed to discretize the space and time axes and convert the differential equation to difference form. In this study, the mixing cell method is used to convert the diffusion wave equation to difference form, in which the difference term can be eliminated by selecting an optimal space step size Δx when time step size Δt is given. When the time step size Δt→0, the space step size Δx=Q/(2S₀BC]_k) where Q is discharge, S₀ is bed slope, B is channel width and C_k is kinematic wave celerity, which is the same as the characteristic length proposed by Kalinin and Milyukov. The results of application to two cases show that the mixing cell and linear channel flow routing methods produce hydrographs that are in agreement with the observed flood hydrographs. © 1997 John Wiley & Sons, Ltd.     

Analysis of key parameters in a diffusion type beach profile evolution model

Diffusion type formulations are commonly used in beach profile evolution models. The practical idea behind that is to map the behaviour of the beach profile onto a simple mathematical model that exhibits the same behaviour under defined operating conditions. The success of this approach is based on the accurate determination of key parameters in the diffusion model that govern its behaviour, using observed beach behaviour in the field. In order to determine these parameters, i.e. diffusion coefficient and a time and space varying source function, we used observations of historic beach profiles at Milford-on-Sea beach in Christchurch Bay, Dorset, United Kingdom. The relationship between the diffusion coefficient and Dean's equilibrium profile was investigated, leading to a new interpretation of the diffusion coefficient in terms of the sediment characteristics. The analysis also shows the significance of the diffusion process in the medium to long term evolution of the beach profile. A canonical correlation analysis (CCA) was undertaken in order to identify patterns of behaviour between wave conditions and source terms, and the possible correlations between them. The analysis provides strong evidence of a useful link between the source term in the simple dynamical equation and the distribution of wave steepness.     

Distribution of uranium in intrusive bodies due to combined migration and diffusion

Uranium is enriched in the border zones of magmatic bodies and the enrichment is believed to be caused by the migration of hydrous solutions which carry that element along intergranular paths towards the contact zone with the wall rock. We propose that the contact zone is a geochemical barrier at which the uranium, present in the solution, would be deposited if it were not for diffusion away from the increased concentration at the margins.The two particle flows, the one caused by migration and the other caused by diffusion, can be described by a differential equation, whose solution is the concentration of uranium as a function of time, diffusion coefficient and velocity of migration.The distribution of uranium in two intrusive bodies, the Mont Blanc granite (Swiss Alps) and a pluton in the Dshetui-Oguz massif (U.S.S.R.), gives the following parameters: duration of process 0.3–1 m.y., diffusion coefficient 4 × 10^?4 to 5 × 10^?4 cm²/s, and velocity of migration 0.1–0.3 cm/year.The combined process of migration and diffusion is assumed to be an important mechanism for controlling the distribution of uranium throughout the earth's crust and for its change in geological time.     

The coupling of quasi-linear pitch angle and energy diffusion

Wave–particle interactions are described in the quasi-linear formulation by bounce-averaged diffusion coefficients for equatorial pitch angle α₀ and for momentum p, along with a mixed or cross diffusion coefficient D_α0p. Because the cross terms complicate the associated time-dependent diffusion equation, and because D_α0p has a somewhat unfamiliar character (e.g., it may be negative), it has frequently been omitted in numerical simulations of multidimensional diffusion. Generally, D_α0p becomes increasingly important for small values of f_pe/f_ce, as does D_pp, and for small widths of the wave frequency and wavenormal angle distributions. Here we use very simple models of diffusion to investigate the effect of the cross terms, and to demonstrate numerical problems associated with them.     

Inclusion of adsorption in the modelling of early diagenesis

A general one-dimensional diagenetic equation is derived which expresses the effect of the following processes upon the depth and time distribution of a dissolved species in a modern subaqueous sediment: adsorptive or ion exchange equilibrium between pore water and grain surfaces; slower chemical or radiogenic reactions in the pore solution and on the grain surfaces; diffusion in solution and on grain surfaces; advection of grains due to depositional burial; and advection of water due both to burial and to flow past the grains. It is shown that the effects of equilibrium adsorption can be expressed in terms of a lowered diffusion coefficient only in the special case where there is negligible flow of water relative to grains, adsorption is described by a simple linear isotherm, and there is nochemical reaction occurring in the sediment (i.e. only radioactive decay or no reaction at all). Consideration of a commonly used special case of the general equation indicates that if equilibrium adsorption is strong, the depositional term in the resulting diagenetic equation may become quantitatively as important as the terms for diffusion and chemical reaction. Thus, when modelling diagenesis, deposition often cannot be ignored. A coupled pair of differential equations are also derived which express the steady state depth distribution for two ions exhibiting equilibrium ion exchange while undergoing chemical reaction, diffusion, and deposition.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

A finite volume upwind scheme for the solution of the linear advection–diffusion equation with sharp gradients in multiple dimensions

A finite volume upwind numerical scheme for the solution of the linear advection equation in multiple dimensions on Cartesian grids is presented. The small-stencil, Modified Discontinuous Profile Method (MDPM) uses a sub-cell piecewise constant reconstruction and additional information at the cell interfaces, rather than a spatial extension of the stencil as in usual methods. This paper presents the MDPM profile reconstruction method in one dimension and its generalization and algorithm to two- and three-dimensional problems. The method is extended to the advection–diffusion equation in multiple dimensions. The MDPM is tested against the MUSCL scheme on two- and three-dimensional test cases. It is shown to give high-quality results for sharp gradients problems, although some scattering appears. For smooth gradients, extreme values are best preserved with the MDPM than with the MUSCL scheme, while the MDPM does not maintain the smoothness of the original shape as well as the MUSCL scheme. However the MDPM is proved to be more efficient on coarse grids in terms of error and CPU time, while on fine grids the MUSCL scheme provides a better accuracy at a lower CPU.     

Development of longitudinal profiles of alluvial channels in response to base-level lowering

Degradation of alluvial channels in cohesive sediments was studied in 15 m and 20 m long flumes with a slope of 0°01 cm/cm. Degradation was initiated by lowering base level to a fixed position, and the development of the longitudinal profile of the channel is analysed through a model formulated as a heat (diffusion) equation. It is based on the equation of sediment continuity, combined with an assumption regarding sediment transport, namely that sediment discharge is linearly proportional to the channel slope. In accordance with the boundary and initial conditions imposed by the experimental setup and procedure, the basic equation is amenable to an analytical solution, which defines bed elevation at any distance and time, as a function of the amount of base-level lowering and a ‘diffusion’ coefficient. Additional problems arising from bank erosion and channel armouring are also treated successfully within the framework of the same model. The results show that in homogeneous alluvial sediments, not subject to armouring, the ultimate result of base-level lowering by a certain amount is degradation all along the channel by the same amount. The main impact of erosion is felt in the early stages after initiation of the process, and mainly near the mouth. The rate of degradation at any station along the channel reaches a peak and then slowly decreases with time, and the peak rate is attenuated with distance from the outlet. The model permits the prediction of intermediate stages of profile development at any distance from the outlet and at different times.     

A new mixing equation for bed material composition in bed form dominant conditions

Vertical mixing beneath the bed surface and its effect on sediment composition has long been underestimated. This paper proposes a mixing equation to illustrate the temporal and spatial variations of bed material composition for bed form dominant rivers. A continuous mass conservation equation for elevation-specific nonuniform bed material composition is initially presented. A vertical mixing equation is finally derived as a diffusion-type partial differential equation with a source term charact...     

Impact of historic land‐use change on sediment delivery to a Chesapeake Bay subestuarine delta

Historic land use in the Chesapeake Bay drainage basin induced large fluxes of fluvial sediment to subestuarine tributaries. Stratigraphic and palaeoecologic analyses of deltaic deposits may be used to infer changes on the landscape, but are not sufficient to quantify past sediment supply. When viewed as an inverse boundary‐value problem, reconstruction of the sediment supply function may be achieved by combining deltaic sedimentation chronologies with an equation governing delta progradation. We propose that the diffusion equation is appropriate for simulating delta progradation and obtaining the sediment supply function provided a suitable diffusion constant (D) can be determined. Three new methods for estimating D are presented for the case of estuarine deltas. When the inverse boundary‐value technique was applied to Otter Point Creek, a tidal freshwater delta at the head of Bush River in upper Chesapeake Bay, D values ranged from 3763 to 6199 m² a^?1. Delta growth simulations showed a 1740–1760 initial pulse, a 1760–1780 erosive/redistributive interval, a 1780–1920 growth period, and a 1920‐present erosive/redistributive era. Coupling of simulated delta elevations with an empirical plant habitat predictive equation allowed for comparison of predicted versus actual relative habitat areas. Also, the model yielded reconstructed watershed erosion rates and stream suspended sediment concentrations that could be useful for development of water quality regulations. Copyright © 2001 John Wiley & Sons, Ltd.     

Implication of the flow resistance formulae on the prediction of flood wave propagation

ABSTRACT

This study examines the difference in the predictions of flood wave propagation in open channels depending on the flow resistance formulae, such as the Chézy and Manning’s equation. The celerity and diffusion coefficient are functions of the channel geometry, slope, roughness as well as the resistance formulae. The results suggest that substituting the Chézy equation with Manning’s equation results in different characteristics of flood propagation, which are consistent regardless of the cross-sectional geometry except for a circular cross-section: increasing celerity and decreasing diffusion coefficient. The celerity is more sensitive to the selection of resistance formulae than the diffusion coefficient. Geometry has a greater effect on the celerity and diffusion coefficient, and consequently on the resulting hydrographs. Manning’s equation results in a larger difference in celerity and diffusion coefficient compared to Chézy equation regardless of the water depth. Overall, this study shows that the selection of resistance formulae is important in terms of the resulting hydrographs and peak flow.

EDITOR Z.W. Kundzewicz ASSOCIATE EDITOR not assigned     

Estimates of lifetimes against pitch angle diffusion

We consider timescales on which particle distributions respond to pitch angle diffusion. On the longest timescale, the distribution decays at a single rate independent of equatorial pitch angle α₀, even though the diffusion coefficient, and the distribution itself, may vary greatly with α₀. We derive a simple integral expression to approximate this decay rate and show that it gives good agreement with the full expression. The roles of both the minimum and loss cone values of the diffusion coefficient are demonstrated and clarified.     

Erosion processes and landform evolution on agricultural land — new perspectives from caesium-137 measurements and topographic-based erosion modelling

Despite growing interest in soil erosion on agricultural land, relatively little attention has been paid to the influence of erosion processes on the pattern of contemporary landform evolution. This in part reflects the problems associated with up-scaling the results of short-term process studies to temporal and spatial scales relevant to the study of landform evolution. This paper presents a new approach to examining the influence of erosion processes on landform evolution on agricultural land which employs: caesium-137 (¹³⁷Cs) measurements to provide medium-term (c. 40 years) estimates of rates of landform change; experimental data and a topographic-based model to simulate soil redistribution by tillage; a mass-balance model of ¹³⁷Cs redistribution to separate the water erosion and tillage components of the ¹³⁷Cs ‘signatures’; and field observations of water erosion for validation. This approach is used to examine the relative importance of water erosion and tillage processes for contemporary landform evolution at contrasting sites near Leuven, in Belgium, and near Yanan, in Shaanxi Province, China. This application of the approach provides good agreement between the derived water erosion rates and field observations, and hitherto unobtainable insights into medium-term patterns and rates of contemporary landform evolution. At Huldenberg in Belgium, despite rill incision of slope concavities and ephemeral gully incision of the valley floor, contemporary landform evolution is dominated by infilling of slope and valley concavities (rates >0.5 mm a⁻¹) and gradual lowering of slope angles as a result of tillage. In contrast, at Ansai (near Yanan) the slope is characterized by increase in slope angle over most of the length, recession of the steepest section at a rate >5 mm a⁻¹ and by increasing planform curvature. At this site, contemporary landform evolution is dominated by water erosion. The constraints on the approach are examined, with particular attention being given to limitations on extrapolation of the results and to the sensitivity of the models to parameter variation. © 1997 by John Wiley & Sons, Ltd.     

新疆天山地区地貌分形与多重分形特征研究

利用投影覆盖法和投影覆盖概率对新疆天山地区不同构造地貌类型进行了分形与多重分形特征的研究。结果表明：在所研究的标度范围内，不同地貌区均表现出明显的多度域分形，分维值总体上呈现出高山区＞中低山区＞盆地区特点，多重分形谱Dq的形态和值域范围也表现出不同特征。研究认为，地貌表面的分维值与地貌形成的内外力地质作用方式和强度密切相关，并提出5-6km的尺度可作为地貌学研究中宏观与微观作用的分界点。     

On the using cumulant expansion method and van Kampen’s lemma for stochastic differential equations with forcing

Second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is obtained by using the cumulant expansion ensemble averaging method and by taking the time dependent sure part of the multiplicative operator into account. It is shown that the satisfaction of the commutativity and the reversibility requirements proposed earlier for linear stochastic differential equations without forcing are necessary for the linear stochastic differential equations with forcing when the cumulant expansion ensemble averaging method is used. It is shown that the applicability of the operator equality, which is used for the separation of operators in the literature, is also subjected to the satisfaction of the commutativity and the reversibility requirements. The van Kampen’s lemma, which is proposed for the analysis of nonlinear stochastic differential equations, is modified in order to make the probability density function obtained through the lemma depend on the forcing terms too. The second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is also obtained by using the modified van Kampen’s lemma in order to validate the correctness of the modified lemma. Second-order exact ensemble averaged equation for one dimensional convection diffusion equation with reaction and source is obtained by using the cumulant expansion ensemble averaging method. It is shown that the van Kampen’s lemma can yield the cumulant expansion ensemble averaging result for linear stochastic differential equations when the lemma is applied to the interaction representation of the governing differential equation. It is found that the ensemble averaged equations given for one the dimensional convection diffusion equation with reaction and source in the literature obtained by applying the lemma to the original differential equation are restricted with small sure part of multiplicative operator. Second-order exact differential equations for the evolution of the probability density function for the one dimensional convection diffusion equation with reaction and source and one dimensional nonlinear overland flow equation with source are obtained by using the modified van Kampen’s lemma. The equation for the evolution of the probability density function for one dimensional nonlinear overland flow equation with source given in the literature is found to be not second-order exact. It is found that the differential equations for the evolution of the probability density functions for various hydrological processes given in the literature are not second-order exact. The significance of the new terms found due to the second-order exact ensemble averaging performed on the one dimensional convection diffusion equation with reaction and source and during the application of the van Kampen’s lemma to the one dimensional nonlinear overland flow equation with source is investigated.     

Constrains on gravity wave induced diffusion in the middle atmosphere

A review of the important constraints on gravity wave induced diffusion of chemical tracers, heat, and momentum is given. Ground-based microwave spectroscopy measurements of H₂O and CO and rocket-based mass spectrometer measurements of Ar constrain the eddy diffusion coefficient for constituent transport (K _zz ) to be (1–3)×10⁵ cm²s^–1 in the upper mesosphere. Atomic oxygen data also limitsK _zz to a comparable value at the mesopause. From the energy balance of the upper mesosphere the eddy diffusion coefficient for heat transport (D _H ) is, at most 6×10⁵ cm²s^–1 at the mesopause and decreasing substantially with decreasing altitude. The available evidence for mean wind deceleration and the corresponding eddy diffusion coefficient for momentum stresses (D _M ) suggests that it is at least 1×10⁶ cm²s^–1, in the upper mesosphere. Consequently the eddy Prandtl number for macroscopic scale lengths is >3.     

砂岩核磁共振响应模拟及受限扩散

本文运用随机游走方法模拟了砂岩储层中流体的核磁共振(NMR)响应及其受限扩散现象.通过改变数字岩心的分辨率模拟生成不同孔隙尺寸的砂岩,研究了不同孔隙尺寸砂岩饱含水时流体扩散系数随扩散时间的变化关系,同时模拟了砂岩饱和单相流体和两相流体的NMR响应;研究了流体的受限扩散系数与横向弛豫时间T2的关系,分析了表面弛豫率和胶结指数对润湿相流体受限扩散系数线位置的影响,并将其用于解释砂岩储层的D-T2分布.结果表明:孔隙流体的扩散系数会随扩散时间的增加而逐渐减小并趋于定值.随着岩石孔隙尺寸的减小,受限扩散现象越明显,受限扩散对岩石NMR响应的影响也越大.润湿相流体受限扩散系数线的位置受岩石胶结指数和表面弛豫率的影响较大.由于润湿相流体扩散系数减小,导致D-T2分布中润湿相流体信号偏离其自由扩散系数线,需要利用流体的受限扩散系数线准确识别D-T2分布中的润湿相流体.     

Studies in diffusion,IV. Pressure dependence of Ar diffusion in phlogopite mica

Radiogenic Ar diffusion from phlogopite mica has been measured at 900° and 1080°C at 15 kbars pressure, using a Griggs-type hot creep tester operated at hydrostatic pressure, and followed by standard mass spectrometric analysis. The diffusion coefficients fall within the uncertainty of the diffusion results reported by Giletti (1974) at 1 kbar at 900° and on extrapolation of that Arrhenius plot to 1080°C. That linear plot is given by D₀ = 0.75 and Q = 57.9kcal/g-atom Ar. In the observed temperature range, the effect of pressure up to 15 kbars is not significant. A second Ar diffusion, which employed Ca(OH)₂-CaO as a water buffer, was run at 1 atm H₂O pressure and 550°C for 781 days. The resulting diffusion coefficient agrees, within the uncertainty, with the extrapolation of the earlier curve. This result, and both earlier ones at 600°, are similar in that they fall below the curve. Consequently, despite the agreement within the uncertainty, it is possible that diffusion of Ar from phlogopite below 650°C is somewhat slower than given by the above equation parameters.