Velocity potential formulations of highly accurate Boussinesq-type models

The highly accurate Boussinesq-type equations of Madsen et al. (Madsen, P.A., Bingham, H.B., Schäffer, H.A., 2003. Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: Derivation and analysis. Proc. R. Soc. Lond. A 459, 1075–1104; Madsen, P.A., Fuhrman, D.R., Wang, B., 2006. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Eng. 53, 487–504); Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective structures. Coast. Eng. 53, 929–945) are re-derived in a more general framework which establishes the correct relationship between the model in a velocity formulation and a velocity potential formulation. Although most work with this model has used the velocity formulation, the potential formulation is of interest because it reduces the computational effort by approximately a factor of two and facilitates a coupling to other potential flow solvers. A new shoaling enhancement operator is introduced to derive new models (in both formulations) with a velocity profile which is always consistent with the kinematic bottom boundary condition. The true behaviour of the velocity potential formulation with respect to linear shoaling is given for the first time, correcting errors made by Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective structures. Coast. Eng. 53, 929–945). An exact infinite series solution for the potential is obtained via a Taylor expansion about an arbitrary vertical position z = zˆ. For practical implementation however, the solution is expanded based on a slow variation of zˆ and terms are retained to first-order. With shoaling enhancement, the new models obtain a comparable accuracy in linear shoaling to the original velocity formulation. General consistency relations are also derived which are convenient for verifying that the differential operators satisfy a potential flow and/or conserve mass up to the order of truncation of the model. The performance of the new formulation is validated using computations of linear and nonlinear shoaling problems. The behaviour on a rapidly varying bathymetry is also checked using linear wave reflection from a shelf and Bragg scattering from an undulating bottom. Although the new models perform equally well for Bragg scattering they fail earlier than the existing model for reflection/transmission problems in very deep water.     

Tsunami generation,propagation, and run-up with a high-order Boussinesq model

In this work we extend a high-order Boussinesq-type (finite difference) model, capable of simulating waves out to wavenumber times depth kh < 25, to include a moving sea-bed, for the simulation of earthquake- and landslide-induced tsunamis. The extension is straight forward, requiring only an additional term within the kinematic bottom condition. As first test cases we simulate linear and nonlinear surface waves generated from both positive and negative impulsive bottom movements. The computed results compare well against earlier theoretical, numerical, and experimental values. Additionally, we show that the long-time (fully nonlinear) evolution of waves resulting from an upthrusted bottom can eventually result in true solitary waves, consistent with theoretical predictions. It is stressed, however, that the nonlinearity used far exceeds that typical of geophysical tsunamis in the open ocean. The Boussinesq-type model is then used to simulate numerous tsunami-type events generated from submerged landslides, in both one and two horizontal dimensions. The results again compare well against previous experiments and/or numerical simulations. The new extension compliments recently developed run-up capabilities within this approach, and as demonstrated, the model can therefore treat tsunami events from their initial generation, through their later propagation, and final run-up phases. The developed model is shown to maintain reasonable computational efficiency, and is therefore attractive for the simulation of such events, especially in cases where dispersion is important.     

A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations

A numerical scheme for solving the class of extended Boussinesq equations is presented. Unlike previous schemes, where the governing equations are integrated through time using a fourth-order method, a second-order Godunov-type scheme is used thus saving storage and computational resources. The spatial derivatives are discretised using a combination of finite-volume and finite-difference methods. A fourth-order MUSCL reconstruction technique is used to compute the values at the cell interfaces for use in the local Riemann problems, whilst the bed source and dispersion terms are discretised using centred finite-differences of up to fourth-order accuracy. Numerical results show that the class of extended Boussinesq equations can be accurately solved without the need for a fourth-order time discretisation, thus improving the computational speed of Boussinesq-type numerical models. The numerical scheme has been applied to model a number of standard test cases for the extended Boussinesq equations and comparisons made to physical wave flume experiments.     

Rossby waves with linear topography in barotropic fluids

Rossby waves are the most important waves in the atmosphere and ocean, and are parts of a large-scale system in fluid. The theory and observation show that, they satisfy quasi-geostrophic and quasi-static equilibrium approximations. In this paper, solitary Rossby waves induced by linear topography in barotropic fluids with a shear flow are studied. In order to simplify the problem, the topography is taken as a linear function of latitude variable y, then employing a weakly nonlinear method and a perturbation method, a KdV （Korteweg-de Vries） equation describing evolution of the amplitude of solitary Rossby waves induced by linear topography is derived. The results show that the variation of linear topography can induce the solitary Rossby waves in barotropic fluids with a shear flow, and extend the classical geophysical theory of fluid dynamics.     

Singularly perturbed solution of coupled model in atmosphere-ocean for global climate

A box model of the interhemispheric thermohaline circulation （THC） in atmosphere-ocean for global climate is considered. By using the multi-scales method, the asymptotic solution of a simplified weakly nonlinear model is discussed. Firstly, by introducing first scale, the zeroth order approximate solution of the model is obtained. Secondly, by using the multi-scales, the first order approximate equation of the model is found. Finally, second order approximate equation is formed to eliminate the secular terms, and a uniformly valid asymptotic expansion of solution is decided. The multi-scales solving method is an analytic method which can be used to analyze operation sequentially. And then we can also study the diversified qualitative and quantitative behaviors for corresponding physical quantities. This paper aims at providing a valid method for solving a box model of the nonlinear equation.     

全日潮海域风暴潮增水中全日扰动和半日扰动分析

风暴潮增水是风暴潮与天文潮相互作用理论研究的基本内容,也是风暴潮预报中的重要问题。最大余水位的产生机制对于提高预报精度及海岸带防护有着重要意义。为研究全日潮海域风暴潮增水中的全日扰动和半日扰动,对Horsburgh与Wilson的风暴潮余水位模型进行了改进和扩展,建立了包括多个分潮的余水位分解方法并将其应用于防城港站,对台风"启德"和"山神"影响下的潮位过程进行了分析。结果显示,建立的余水位的分解方法对于全日分潮和半日分潮有良好的适用性。由于高频分潮产生机制的复杂性,该方法对高频分潮应用尚需进一步研究。在全日潮的防城港海域,全日扰动与半日扰动具有相同的量级,二者的和约占总增水的15%~19%。台风过程不同,相位变化项和局地变化项对增水的贡献有较大差异。     

礁坪上波浪传播的数值模拟

本文基于具备间断捕捉能力的二阶全非线性Boussinesq数值模型,对规则波和随机波在礁坪地形上的传播变形进行了数值模拟。该模型采用高阶有限体积法和有限差分方法求解守恒格式的控制方程,将波浪破碎视为间断,同时采用静态重构技术处理了海岸动边界问题。重点针对礁坪上波浪传播过程中的波高空间分布和沿程衰减,礁坪上的平均水位变化,以及波浪能量频谱的移动和空间差异等典型水动力现象开展数值计算。将数值结果与实验结果对比,两者吻合情况良好,验证了模型具有良好的稳定性,具备模拟破碎波浪和海-岸动边界的能力,能较为准确地模拟波浪在礁坪地形上的传播过程中发生的各种水动力现象。     

地震波的场方程矩阵和能量的正定二次型及其意义

场方程是能够表述半空间地震波场的整体特征.波动方程、速度方程和能量方程.通过分析可知每个场方程都具有各自的“场方程矩阵”.能量方程能够对所有场方程矩阵进行综合和贯通,给出了能量方程以“弹性矩阵”为核心的普适性表达形式.最后,运用矩阵的正定二次型理论阐述了“能量矩阵与弹性矩阵”之间一致的对称性和正定性.能量矩阵蕴含的动态力的平衡关系、速度的时间_空间分布和能量的传播及变化的物理意义,能够从能量矩阵的正定二次型特性表述出来.本文研究分析问题的方法完全适用于复杂介质模型,相关的认识和结论可以拓展到均匀黏弹性各向同性介质、均匀弹性各向异性介质、均匀黏弹性各向异性介质以及比奥饱和流体介质.     

泉州湾赤潮藻类优势种细胞密度回归方程研究

于2006年5-11月对东海泉州湾赤潮监控区四个监测站位开展赤潮常规监测。根据监测结果,分别以各站位23项水质理化生物环境因子指标为自变量,相应赤潮藻类优势种的细胞密度为因变量,进行多元逐步回归分析,建立了各站位优势种中肋骨条藻(Skeletonema costatum)、太平洋海链藻(Thalassiosira pacifica)、微小原甲藻(Prorocentrum minimum)、尖刺拟菱形藻(Pseudo-nitzschia pungens)、丹麦细柱藻(Leptocylindrus danicus)和旋链角毛藻(Chaetoceros curvisetus)等的细胞密度多参量回归方程。结果表明,所有回归方程的复相关系数都接近于1,方差分析的结果均为回归极显著,表明所建立的回归方程可作为相应赤潮优势种细胞密度预报方程的高度有效性,将对今后泉州湾的赤潮预报提供良好的指导作用。     

New finite-gap solutions for the coupled Burgers equations engendered by the Neumann systems

On the tangent bundle TS^N-1 of the unit sphere S^N-1, this paper reduces the coupled Burgers equations to two Neumann systems by using the nonlinearization of the Lax pair, whose Liouville integrability is displayed in the scheme of the r-matrix technique. Based on the Lax matrix of the Neumann systems, the Abel--Jacobi coordinates are appropriately chosen to straighten out the restricted Neumann flows on the complex torus, from which the new finite-gap solutions expressed by Riemann theta functions for the coupled Burgers equations are given in view of the Jacobi inversion.