A unified formulation for the three-dimensional shallow water equations using orthogonal co-ordinates: theory and application

In this paper, the formulations of the primitive equations for shallow water flow in various horizontal co-ordinate systems and the associated finite difference grid options used in shallow water flow modelling are reviewed. It is observed that horizontal co-ordinate transformations do not affect the chosen co-ordinate system and representation in the vertical, and are the same for the three- and two-dimensional cases. A systematic derivation of the equations in tensor notation is presented, resulting in a unified formulation for the shallow water equations that covers all orthogonal horizontal grid types of practical interest. This includes spherical curvilinear orthogonal co-ordinate systems on the globe. Computational efficiency can be achieved in a single computer code. Furthermore, a single numerical algorithmic code implementation satisfies. All co-ordinate system specific metrics are determined as part of a computer-aided model grid design, which supports all four orthogonal grid types. Existing intuitive grid design and visual interpretation is conserved by appropriate conformal mappings, which conserve spherical orthogonality in planar representation. A spherical curvilinear co-ordinate solution of wind driven steady channel flow applying a strongly distorted grid is shown to give good agreement with a regular spherical co-ordinate model approach and the solution based on a β-plane approximation. Especially designed spherical curvilinear boundary fitted model grids are shown for typhoon surge propagation in the South China Sea and for ocean-driven flows through Malacca Straits. By using spherical curvilinear grids the number of grid points in these single model grid applications is reduced by a factor of 50–100 in comparison with regular spherical grids that have the same horizontal resolution in the area of interest. The spherical curvilinear approach combines the advantages of the various grid approaches, while the overall computational effort remains acceptable for very large model domains.     

The contribution of Very Long Baseline Interferometry to ITRF2005

The contribution of the International VLBI Service for Geodesy and Astrometry (IVS) to the ITRF2005 (International Terrestrial Reference Frame 2005) has been computed by the IVS Analysis Coordinator’s office at the Geodetic Institute of the University of Bonn, Germany. For this purpose the IVS Analysis Centres (ACs) provided datum-free normal equation matrices in Solution INdependent EXchange (SINEX) format for each 24 h observing session to be combined on a session-by-session basis by a stacking procedure. In this process, common sets of parameters, transformed to identical reference epochs and a prioris, and especially representative relative weights have been taken into account for each session. In order to assess the quality of the combined IVS files, Earth orientation parameters (EOPs) and scaling factors have been derived from the combined normal equation matrices. The agreement of the EOPs of the combined normal equation matrices with those of the individual ACs in terms of weighted root mean square (WRMS) is in the range of 50–60 μas for the two polar motion components and about 3 μs for UT1−UTC. External comparisons with International GNSS Serive (IGS) polar motion components is at the level of 130–170 μas and 21 μs/day for length of day (LOD). The scale of the terrestrial reference frame realized through the IVS SINEX files agrees with ITRF2000 at the level of 0.2 ppb.     

具有守恒特性的二维溃坝洪水演进数值模型

基于结构网格,采用有限体积法建立了二维水动力学模型,模拟溃坝洪水在复杂实际地形条件下的流动过程。该模型采用中心迎风格式求解界面通量,并结合对界面变量的线性重构,使其具有空间上的二阶精度。分别采用中心差分方法和半隐式方法对底床坡度项和摩擦阻力项进行离散,保证了模型的和谐性和稳定性。对于复杂地形条件下溃坝洪水的模拟,负水深的产生是影响模型稳定的关键因素。当库朗特数小于0.25时,模型能够保证任何时刻的计算水深都是非负的,而无需对负水深单元进行特殊处理。因此,相比于现有的大部分溃坝洪水模型,该模型具有更强的鲁棒性和稳定性。     

Climate,conflict and forced migration

Despite the lack of robust empirical evidence, a growing number of media reports attempt to link climate change to the ongoing violent conflicts in Syria and other parts of the world, as well as to the migration crisis in Europe. Exploiting bilateral data on asylum seeking applications for 157 countries over the period 2006–2015, we assess the determinants of refugee flows using a gravity model which accounts for endogenous selection in order to examine the causal link between climate, conflict and forced migration. Our results indicate that climatic conditions, by affecting drought severity and the likelihood of armed conflict, played a significant role as an explanatory factor for asylum seeking in the period 2011–2015. The effect of climate on conflict occurrence is particularly relevant for countries in Western Asia in the period 2010–2012 during when many countries were undergoing political transformation. This finding suggests that the impact of climate on conflict and asylum seeking flows is limited to specific time period and contexts.     

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Balancing the source terms in a SPH model for solving the shallow water equations

A shallow flow generally features complex hydrodynamics induced by complicated domain topography and geometry. A numerical scheme with well-balanced flux and source term gradients is therefore essential before a shallow flow model can be applied to simulate real-world problems. The issue of source term balancing has been exhaustively investigated in grid-based numerical approaches, e.g. discontinuous Galerkin finite element methods and finite volume Godunov-type methods. In recent years, a relatively new computational method, smooth particle hydrodynamics (SPH), has started to gain popularity in solving the shallow water equations (SWEs). However, the well-balanced problem has not been fully investigated and resolved in the context of SPH. This work aims to discuss the well-balanced problem caused by a standard SPH discretization to the SWEs with slope source terms and derive a corrected SPH algorithm that is able to preserve the solution of lake at rest. In order to enhance the shock capturing capability of the resulting SPH model, the Monotone Upwind-centered Scheme for Conservation Laws (MUSCL) is also explored and applied to enable Riemann solver based artificial viscosity. The new SPH model is validated against several idealized benchmark tests and a real-world dam-break case and promising results are obtained.     

The evolution of sub-mesoscale,coherent vortices on the β-plane

Abstract

An investigation is made of the evolution of small-scale, axisymmetric vortices in a stratified fluid with spatially variable Coriolis parameter. The criteria for smadness are a horizontal scale less than or equal to the first internal radius of deformation and a vertical scale less than or equal to that of the ambient stratification. These circumstances match those of Sub-mesoscale, Coherent Vortices frequently observed in the oceans. The dynamical model is the balance equations, which include various effects of finite Rossby number. The principal topics are the regime of nearly uniform propagation, the development of an equilibrium ratio of vertical and horizontal scales (i.e., Burger number selection), and the occurrence of various types of instability for vortices with extremes in amplitude or shape.     

应用二阶完全非线性Boussinesq方程模拟破碎波浪

建立了基于高阶Boussinesq水波方程的一维波浪破碎数值模型。基于一组具有二阶完全非线性特征的Boussinesq水波方程,建立了交错网格下的高精度差分格式,推导了适用于该组方程的永形波解析解,其和松弛造波技术相结合实现了数值波浪水槽中(强)非线性波浪的无反射入射。通过模拟封闭容器内水体晃动问题对数值格式进行了验证,通过模拟孤立波在斜坡海岸上的浅化过程说明了将方程从弱非线性发展到完全非线性的必要性。采用涡粘方法处理波浪破碎,利用物理模型实验数据,分析了模型中各波浪破碎参数对数值结果的影响并对参数进行了率定。应用该模型对规则波在斜坡海岸上的传播、变浅以及破碎过程进行了数值模拟研究,数值结果同实验数据吻合良好,验证了模型的有效性。