Mathematical Analysis of a Model of River Channel Formation

The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin.     

具有精确色散性的非线性波浪水平二维数学模型

建立具有色散性的水平二维非线性波浪方程,方程的非线性近似到了三阶。方程以波面升高和自由表面速度势表达的微分-积分型数学方程,给出方程的数值求解方法和算例,对方程积分项的处理给出了计算方法。计算结果与Boussinesq方程模型和缓坡方程模型的对应计算结果进行了对比。     

Lambert problem solution in the hill model of motion

The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described. This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can be found.     

Boussinesq-type modelling using an unstructured finite element technique

A model for solving the two-dimensional enhanced Boussinesq equations is presented. The model equations are discretised in space using an unstructured finite element technique. The standard Galerkin method with mixed interpolation is applied. The time discretisation is performed using an explicit three-step Taylor–Galerkin method. The model is extended to the surf and swash zone by inclusion of wave breaking and a moving boundary at the shoreline. Breaking is treated by an existing surface roller model, but a new procedure for the detection of the roller thickness is devised. The model is verified using four test cases and the results are compared with experimental data and results from an existing finite difference Boussinesq model.     

A mass balance approach to estimate the dilution and removal of the pollutants in stream water polluted by acid mine drainage

 A few simple mass balance equations were developed to simultaneously estimate how much the pollutants from acid mine drainage (AMD) in stream water are diluted and removed during their migration. The application of the equations requires knowledge of the variations in the concentrations of the dissolved pollutants and the stoichiometry of the precipitation reaction of the pollutants when none of the pollutant shows a conservative behavior along the stream path. The calculation should be restricted to the pollutants showing much higher concentrations in the polluted main stream water than in the combining or diluting water of the same target area. The mass balance equations were applied to estimate the dilution factor and precipitation fractions of pollutants in Imgok Creek such as Fe, SO₄ and Al from the AMD of Yeongdong mine. The results show that the estimation, especially for SO₄ and Al, significantly depends on the kinds of the precipitates. When FeOHSO₄ and AlOHSO₄ are assumed to precipitate, the maximum removal fractions of SO₄ and Al by precipitation are respectively 34% and 46% of the original input, which is much higher than the values estimated when SO₄ is considered to be perfectly conservative. It indicates that the stoichiometry of precipitation reaction is very important in the interpretation of the pollutant dilution and migration and assessment of environmental impacts of AMD. The applicability of the mass balance equations may still need to be verified. However, examining the calculated dilution factor and precipitation fractions with the equations can provide invaluable information on not only the behavior but also unexpected input of the pollutants in the stream water polluted by AMD and other point sources. Received: 12 November 1997 · Accepted: 30 March 1998     

On the role of the Asian monsoon in the angular momentum and kinetic energy balances of the tropics

The balance conditions of relative angular momentum and time-mean kinetic energy and their annual variations are studied for the Northern Hemisphere tropical belt. The belt is divided into two roughly equal size parts, the monsoon and the extramonsoon regions. The data used consist of all available daily rawinsonde reports from the world areological network for the two 5-year periods 1958–63 and 1968–73.In winter, the trade winds in the monsoon and extramonsoon regions are both sources of westerly relative angular momentum for the middle latitude circulation. However, it is found that the angular momentum gained in the extramonsoon region of the Tropics is mostly destroyed by a net southward flow of mass in that region, and becomes regenerated in the monsoon region by a net northward flow of mass there. This excess of angular momentum together with the angular momentum picked up locally in the monsoon region is almost all exported across its northern boundary. It is further found that in winter the Tropics are also an important source of mean kinetic energy for middle latitudes. Again almost all export of kinetic energy was found to take place across the northern boundary of the monsoon sector. Most of this energy must be generated through the pressure gradient term inside the monsoon region itself, the transformation from transient eddy kinetic energy being very small. The proper evaluation of the pressure gradient appears to be the main stumbling block in the present study, preventing us from estimating the generation and thereby, as a residual, the frictional dissipation in the two regions.In summer, the extramonsoon region remains a source of angular momentum, but the monsoon region with its surface westerlies acts as a sink, leading to a sharp reduction (and even a midsummer reversal) of the export into middle latitudes. Also the export of mean kinetic energy almost vanishes in summer, except for a small southward transfer across the equator. The calculations for two 5-year periods give very similar estimates and thereby show the reliability of the results.Parts of this paper were presented at the International Symposium on Monsoons, March 7–12, 1977 in New Delhi, India.     

Dynamic model of mesoscale eddies. Eddy parameterization for coarse resolution ocean circulation models

In the framework of the eddy dynamic model developed in two previous papers (Dubovikov, M.S., Dynamical model of mesoscale eddies, Geophys. Astophys. Fluid Dyn., 2003, 97, 311–358; Canuto, V.M. and Dubovikov, M.S., Modeling mesoscale eddies, Ocean Modelling, 2004, 8, 1–30 referred as I–II), we compute the contribution of unresolved mesoscale eddies to the large-scale dynamic equations of the ocean. In isopycnal coordinates, in addition to the bolus velocity discussed in I–II, the mesoscale contribution to the large scale momentum equation is derived. Its form is quite different from the traditional down-gradient parameterization. The model solutions in isopycnal coordinates are transformed to level coordinates to parameterize the eddy contributions to the corresponding large scale density and momentum equations. In the former, the contributions due to the eddy induced velocity and to the residual density flux across mean isopycnals (so called Σ-term) are derived, both contributions being shown to be of the same order. As for the large scale momentum equation, as well as in isopycnal coordinates, the eddy contribution has a form which is quite different from the down-gradient expression.     

浅水方程数值计算方法的研究

求解以水位为变量的连续方程,并根据Navier-Stokes方程压力修正算法的基本思想,建立了浅水方程的水位修正算法,放宽了对离散时间步长的限制.通过对离散方程系数矩阵的重新构造,建立了高分辨率有限元格式,该格式既具有较高的离散精度又避免了数值解的伪振荡.对动量方程的阻力项做负坡线性化处理,提高了露滩计算的稳定性.数值模拟结果与解析解吻合良好,表明所建立的数值计算方法是正确的和可靠的.     

关于波浪Boussinesq方程的研究

对有关波浪 Boussinesq方程的研究成果进行了系统的归纳总结和评述 ,以期对本学科的发展起到一定的引导和促进作用     

浅述多元数据处理的一种简化拟合形式及算法剖析

在实际工程试验与研究中 ,多元数据处理经常遇到 ,根据 Weierstrass定理 ,采用多项式拟合是一种常见的数据处理方法。对于多元多项式的系数求解问题 ,可以归结为超定方程组的求解问题。该文以超定方程组的 SVD求解算法为核心 ,阐述了在工程实际中关于多元数据处理的一种应用方法 ,并尝试给出多元多项式拟合的一种简化方法