ON LONG-WAVE BREAKING Ⅳ. NUMERICAL RESULTS

ImODUcrIONBreakingwavesaretheagentSformanyimPohantupperoceanproassesinvolvingtransferofhobontalmornentumfromwindwivestosurfaceimtS.AfaIniliarandspancularpropertyoflongwivespropagatinginaninfiniteorhaif4nfinitechannelwithbottomsl0peisthemeCanismofwavebrmking.Alltheusualtheories(srnallamPlitudeapproxthetionsofAiryandStokes,noTilinearshallowwitertheory,K0rteweg-DeVireSequattonsforsolitaryandcnoidalWhves)areessentiallyapproxirnations,validonlyWhenthefluidadetionissuffidenhysmallcomPatalto…     

On Long-waves breaking II. in a half infinite channel with bottom slope

The propagation of long-waves, such as tidal waves from the coastal ocean into shallow estuarine waters, often produces asymmetries of velocity and water level in time series resulting in long-wave breaking. Tian (1994) studied the mechanism of long-wave breaking in an infinite channel with constant depth, consideringnth power bottom friction. This study is for the case of a half infinite channel with bottom slope, taking linear bottom friction into account. The wave breaking time and wave breaking location are estimated and the criteria for long-wave breaking in this particular case are obtained. The results obtained can also be easily applied to the case considering wind stress.     

ON LONG-WAVES BREAKING Ⅱ.IN A HALF INFINITE CHANNEL WITH BOTTOM SLOPE

The propagation of long-waves, such as tidal waves from the coastal oceam into shallow estuarine waters,often produces asymmetries of veolcity and water level in time series resulting in long-wave breaking.Tian (1994) studied the mechanism of long-wave breaking in an infinite channel with constant depth,considering nth power bottom friction. This study is for the case of a half infinite channel with bottomslope, taking linear bottom friction into account. The wave breaking time and wave breaking location areestimated and the criteria for long-wave breaking in this particular case are obtained. The results obtainedcan also be e asily applied to the case considering wind stress.     

Study of long-wave breaking

This paper reports study on the mechanism of long-wave breaking in a one-dimensional open chan-nel,without bottom friction,or with linear bottom friction.In general,the wave breaking phenomenon is related to the notion of"blow-up of solution"and can be studied by using the blow-up theory of par-tial differential equations(PDE).A pair of Riemann invariants are introduced to transform the governingequations into ordinary differential equations(ODE)along the characteristic directions of the governingequations.From the properties of the solutions of the ODE,the time and location of wave breakingare estimated and the criterion of long-wave breaking for the case with linear bottom friction is obtained.     

Determination of fractional energy loss of waves in nearshore waters using an improved high-order Boussinesq-type model

Fractional energy losses of waves due to wave breaking when passing over a submerged bar are studied systematically using a modified numerical code that is based on the high-order Boussinesq-type equations. The model is first tested by the additional experimental data, and the model’s capability of simulating the wave transformation over both gentle slope and steep slope is demonstrated. Then, the model’s breaking index is replaced and tested. The new breaking index, which is optimized from the several breaking indices, is not sensitive to the spatial grid length and includes the bottom slopes. Numerical tests show that the modified model with the new breaking index is more stable and efficient for the shallow-water wave breaking. Finally, the modified model is used to study the fractional energy losses for the regular waves propagating and breaking over a submerged bar. Our results have revealed that how the nonlinearity and the dispersion of the incident waves as well as the dimensionless bar height (normalized by water depth) dominate the fractional energy losses. It is also found that the bar slope (limited to gentle slopes that less than 1:10) and the dimensionless bar length (normalized by incident wave length) have negligible effects on the fractional energy losses.     

A VOF-based numerical model for breaking waves in surf zone

This paper introduces a numerical model for studying the evolution of a periodic wave train, shoaling, and breaking in surf zone. The model can solve the Reynolds averaged Navier-Stokes (RANS) equations for a mean flow, and (he k-s equations for turbulence kinetic energy k and turbulence dissipation rate e. To track a free surface, the volume of fluid (VOF) function, satisfying the advection equation was introduced. In the numerical treatment, third-order upwind difference scheme was applied to the convection terms of the RANS equations in order to reduce the effect of numerical viscosity. The shoaling and breaking processes of a periodic wave train on gently sloping beaches were modeled. The computed wave heights of a sloping beach and the distribution of breaking wave pressure on a vertical wall were compared with laboratory data.     

Large eddy simulation of the rotation effect on the ocean turbulence kinetic energy budget in the surface mixed layer

A non-hydrostatic, Boussinesq, and three-dimensional large eddy simulation(LES) model was used to study the impact of the Earth's rotation on turbulence and the redistribution of energy in turbulence kinetic energy(TKE) budget. A set of numerical simulations was conducted,(1) with and without rotation,(2) at different latitudes(10°N, 30°N, 45°N, 60°N, and 80°N),(3) with wave breaking and with Langmuir circulation, and(4) under different wind speeds(5, 10, 20, and 30 m/s). The results show that eddy viscosity decreases when rotation is included, indicating that rotation weakens the turbulence strength. The TKE budget become tight with rotation and the effects of rotation grow with latitude. However, rotation become less important under Langmuir circulation since the transport term is strong in the vertical direction. Finally, simulations were conducted based on field data from the Boundary Layer and Air-Sea Transfer Low Wind(CBLAST-Low) experiment. The results, although more complex, are consistent with the results obtained from earlier simulations using ideal numerical conditions.     

Analysis on velocity distribution and displacement along the profile of a slope using both empirical and analytical methods

Assessing the slope deformation is significant for landslide prediction. Many researchers have studied the slope displacement based on field data from the inclinometer in combination with complicated numerical analysis. They found that there was a shear zone above the slip surface, and they usually focused on the distribution of velocity and displacement within the shear zone. In this paper, two simple methods are proposed to analyze the distribution of displacement and velocity along the whole profile of a slope from the slip surface to the slope surface during slow movement. In the empirical method, the slope soil above the shear zone is assumed as a rigid body. Dual or triple piecewise fitting functions are empirically proposed for the distribution of velocity along the profile of a slope. In the analytical method, the slope soil is not assumed as a rigid body but as a deformable material. Continuous functions of the velocity and displacement along the profile of a slope are directly obtained by solving the Newton's equation of motion associated with the Bingham model. Using the two proposed methods respectively, the displacement and velocity along the slope profiles of three slopes are determined. A reasonable agreement between the measured data and the calculated results of the two proposed methods has been reached. In comparison with the empirical method, the analytical method would be more beneficial for slope deformation analysis in slope engineering, because the parameters are material constants in the analytical solution independent of time t, and the nonlinear viscosity of the soil can be considered.     

A report on Finite Element model design

Laplace's equation has been solved, discussed and applied by many mathematicians and physicians. Because of the significance and applicability of this equation in the field of physcial science, its approximate solution by Finite Element Method and comparison between computations and analytical results are presented.     

地下水溶质运移数值法和解析法预测结果对比分析—以沾化电厂为例

数值法和解析法是当前解决地下水流和溶质运移问题最常用的两种方法,虽然数值法以其广泛的适用性和较高的仿真性等优点取得了越来越普遍的应用,但解析法也以其简单易用等特点一直成为首选方法之一。该文以沾化电厂地下水环评项目为例,分别采用两种方法对污染物在长期和短期渗漏情况下对地下水所造成的污染情况进行了预测,并在对两种方法预测结果的差异进行分析的基础上,论述了解析法在该区的适宜性,为解析法在该区的应用提供了参考。     

毛乌素湖盆滩地湖水与地下水交互作用分析

湖水与地下水交互作用对于水资源合理开发与利用有着重要意义。基于温度示踪的原理，采用解析法、数值法2种方法，分析了湖床底部埋深0~0.4 m湖水与浅层地下水交互关系，并与水动力学方法进行了对比。结果表明，2018年5月20日至28日，湖水与地下水之间的垂向渗流速度为2×10-7~1×10-6 m/s，且在埋深0.4 m时大于埋深0.2 m处。降水会对解析法的结果造成一定影响，0.4 m处受到降雨影响表现为一定的滞后性。无降雨干扰情况下，数值法与水动力学方法估算结果较为吻合，且3种方法的计算结果处于同一数量级。同时，湖床沉积物体积热容和孔隙度2种参数对计算结果影响较大。在半干旱地区湖水与地下水交互研究中，数据较完备时，数值模拟法是更好的选择。      

Determination of fractional energy loss of waves in nearshore waters using an improved high-order Boussinesq-type model

Fractional energy losses of waves due to wave breaking when passing over a submerged bar are studied systematically using a modified numerical code that is based on the high-order Boussinesq-type equations.The model is first tested by the additional experimental data,and the model's capability of simulating the wave transformation over both gentle slope and steep slope is demonstrated.Then,the model's breaking index is replaced and tested.The new breaking index,which is optimized from the several breaking i...     

多级次地下水流系统的最小能耗率原理初探

20世纪60年代初期，Tóth基于定水头上边界条件推导出解析解，得出多级次地下水流系统，是水文地质学里程碑式的突破，成功地解决了一系列理论和实际问题。但Tóth解析解存在的缺陷也长期沿袭:单纯重视数学模拟而忽视物理机制；将地形控制地下水位看成是普适性规律；忽视给定水头上边界数学模拟的失真。这些缺陷，尤其是忽视物理机制探究，不仅妨碍Tóth理论自身发展，而且导致地下水流系统理论尚未被国际水文地质界普遍接受。参照河流动力学中应用的最小能耗率原理，类比提出地下水流最小能耗率的表达式。基于已有的通量上边界地下水流模式数值模拟结果，进一步探究物理机制，归纳得出地下水流系统遵循最小能耗率原理的结论。      

求解二阶线性常微分方程的一个显式差分格式

在求解常微分方程的方法中,有限差分法是使用最广泛的方法之一。考虑一个二阶线性常微分方程的边值问题,利用有限差分法,建立了一个具有二阶精度的显式差分格式。首先,通过讨论该显式差分格式的系数矩阵,证明了该显式差分格式解的存在性。然后,通过定义的3种不同范数之间的关系,证明了显式差分格式解的收敛性和稳定性。最后,通过计算机编程对实例的计算,验证了该显式差分格式的数值结果具有二阶精度,并且该显式格式数值结果绘制的图形稳定、光滑,与解析结果吻合较好。     

A COMBINED REFRACTION-DIFFRACTION-DISSIPATION MODEL OF WAVE PROPAGATION

A numerical model based on the mild-slope equation of water wave propagation over complicated bathymetry,taking into account the combined effects of refraction,diffraction and dissipation due to wavebreaking is presented.Wave breaking is simulated by modifying the wave height probability density func-tion and the wave energy dissipation mechanism is parameterized according to that of the hydraulic jumpformulation.Solutions of the wave height,phase function,and the wave direction at every grid point areobtained by finite difference approximation of the governing equations,using Gauss-Seidel Iterative Method(GSIM)row by row.Its computational convenience allows it to be applied to large coast regions tostudy the wave transformation problem.Several case studies have been made and the results compare verywell with the experiment data and other model solutions.The capability and utility of the model forreal coast areas are illustrated by application to a shallow bay of northeast Australia.     

基于谱-有限元法的SNERI地球模型固体潮形变计算

基于谱-有限元法计算一个球型、非自转、完全弹性、各向同性(SNERI)的地球固体潮形变,其中地球固体部分潮汐形变的弱解用哈密顿变分原理给出,液核部分的弱解采用静态中性分层的流体近似。计算过程中把SNERI地球进行等间距球层剖分,球面上对解函数和试探函数采用球谐展开,径向上采用线性插值。比较数值计算结果与同质地球模型的解析解结果得出,1 km径向等距剖分即可获得10~(-8)精度量级的低阶Love数。基于PREM地球模型的计算结果表明,谱-有限元法计算的固体潮2～3阶Love数与Runge-Kutta法的计算值差异在10~(-4)量级;与武汉台超导重力仪8个主潮波的实测重力潮汐因子相比,本方法计算的理论重力潮汐因子相差平均约0.15%。研究结果说明,谱-有限元法具有较好的收敛性与较高的计算精度,比传统Runge-Kutta法更适用于高精度地计算复杂地球模型的固体潮形变。     

求解二阶线性常微分方程的一个显式差分格式

在求解常微分方程的方法中,有限差分法是使用最广泛的方法之一。考虑一个二阶线性常微分方程的边值问题,利用有限差分法,建立了一个具有二阶精度的显式差分格式。首先,通过讨论该显式差分格式的系数矩阵,证明了该显式差分格式解的存在性。然后,通过定义的3种不同范数之间的关系,证明了显式差分格式解的收敛性和稳定性。最后,通过计算机编程对实例的计算,验证了该显式差分格式的数值结果具有二阶精度,并且该显式格式数值结果绘制的图形稳定、光滑,与解析结果吻合较好。     

航空重力梯度数据向下延拓的正则化方法及参数选取研究

围绕航空重力梯度测量数据处理的向下延拓问题,利用不同的正则化算法和参数选取方法对向下延拓的数学模型分别求解。通过梯度数值的模拟,联合EGM2008位系数计算的大地水准面上的重力场参数,对向下延拓结果进行检核。模拟数值计算结果表明,截断奇异值正则化（TSVD）方法结合广义交叉验证（GCV）参数选取方法能够实现稳定的重力场信号向下延拓,精度和稳定性都优于其他几种方法的组合。     

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