A new approach to the local time stepping problem for scalar transport

A new scalar transport method is proposed to reduce computational time when a large number of scalars are transported in coupled hydrodynamic-ecosystem models. The new Local Mass Transport (LMT) method confines subtime transport computations to regions where the local Courant–Freidrichs–Lewy (CFL) number exceeds a given numerical stability criteria for a global (large) time step, but the method does not require either contiguous regions or special boundary algorithms between regions as used in previous Local Time Stepping (LTS) approaches. The new method uses conservative transport of mass rather than dissolved concentration. This approach allows different faces of a single grid cell to use different subtime steps. The new LMT method is further extended to include background filtering (LMTB) so that scalars below a pre-defined background concentration are ignored in transport calculations. This new approach can further reduce computational time where large regions are at or below an irrelevant background concentration. Both LMT and LMTB methods can be more computationally efficient than global subtime stepping.     

Fourier filtering and coefficient tapering at the North Pole in OGCMs

This article considers how some of the measures used to overcome numerical problems near the North Pole affect the ocean solution and computational time step limits. The distortion of the flow and tracer contours produced by a polar island is obviated by implementing a prognostic calculation for a composite polar grid cell, as has been done at NCAR. The severe limitation on time steps caused by small zonal grid spacing near the pole is usually overcome by Fourier filtering, sometimes supplemented by the downward tapering of mixing coefficients as the pole is approached; however, filtering can be expensive, and both measures adversely affect the solution. Fourier filtering produces noise, which manifests itself in such effects as spurious static instabilities and vertical motions; this noise can be due to the separate and different filtering of internal and external momentum modes and tracers, differences in the truncation at different latitudes, and differences in the lengths of filtering rows, horizontally and vertically. Tapering has the effect of concentrating tracer gradients and velocities near the pole, resulting in some deformation of fields. In equilibrium ocean models, these effects are static and localised in the polar region, but with time-varying forcings or coupling to atmosphere and sea ice it is possible that they may seriously affect the global solution. The marginal stability curve in momentum and tracer time-step space should have asymptotes defined by diffusive, viscous, and internal gravity wave stability criteria; at large tracer time steps, tracer advection stability may become limiting. Tests with various time-step combinations and a flat-bottomed Arctic Ocean have confirmed the applicability of these limits and the predicted effects of filtering and tapering on them. They have also shown that the need for tapering is obviated by substituting a truncation which maintains a constant time step limit rather than a constant minimum wave number over the filtering range.     

A 3-D model based on Navier–Stokes equations for regular and irregular water wave propagation

A spatial fixed σ-coordinate is used to transform the Navier–Stokes equations from the sea bed to the still water level. In the fixed σ-coordinate system only a very small number of vertical grid points are required for the numerical model. The time step for using the spatial fixed σ-coordinate is efficiently larger than that of using a time dependent σ-coordinate, as there is substantial truncation error involved in the time dependent σ-coordinate transformation. There is no need to carry out the σ-coordinate transformation at each time step, which can reduce computational times. It is important that wave breaking can be potentially modeled in the fixed σ-coordinate system, but in a time-dependent σ-coordinate system the wave breaking cannot be modeled. A projection method is used to separate advection and diffusion terms from the pressure terms in Navier–Stokes equations. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. In order to reduce computational time of solving the hydrodynamic pressure equation, at every time step the initial pressure is extrapolated in time domain using computed pressures from previous time steps, and then corrected in spatial domain using a multigrid method. For each time step, only a few of iterations (typically six iterations) are required for solving the pressure equation. The model is tested against available experimental data for regular and irregular waves and good agreement between calculation results and the measured data has been achieved.     

A reduced grid for a parallel global ocean general circulation model

A limitation of many global climate models with explicit finite-difference numerics is the timestep restriction caused by the decrease in cell size associated with the convergence of meridians near the poles. To keep the longitudinal width of model cells as uniform as possible, we apply a “reduced” grid to a three-dimensional primitive equation ocean-climate model. With this grid the number of cells in the longitudinal direction is reduced at high latitudes. The grid consists of subgrids which interact at interfaces along their northern and southern boundaries, where the resolution changes by a factor of three. We extend the finite-difference techniques to these interfaces, focusing on the conservation required to perform long time integrations, while preserving the staggered spatial arrangement of variables and the numerics used on subgrids. The common alternative used to reduce the timestep restriction caused by the spherical grid is the filtering of high-frequency modes from the high-latitude solution. The reduced grid allows an increased timestep while eliminating the need for filtering and reduces execution time per model step by roughly 20%. We implement the reduced grid model for parallel computer architectures with two-dimensional domain decomposition and message passing, with speedup results similar to those of the original model. We present results of model runs showing small effects on the solution and sizable improvements to the execution time.     

Moving shoreline boundary condition for nearshore models

The paper develops and analyzes two fully nonlinear boundary conditions that incorporate the motion of the shoreline in nonlinear time domain nearshore models. A moving shoreline essentially means the computational domain is changing with the solution of the flow. The problem is solved in two steps. The first is to establish an equation that determines the motion of the shoreline based on the local momentum balance. The second is to develop and implement into a shoreline model the capability of accommodating a changing computational domain. The two models represent two different ways of addressing this step: one is to track the position of the shoreline in a fixed grid by establishing a special shoreline point which generally is not a fixed grid point. The second is by a coordinate transformation that maps the changing domain onto a fixed domain and solves the basic equations in the mapped domain. The two shoreline conditions are tested against three known solution for nonlinear shoreline motion. Two are the 1-D solutions to the nonlinear shallow water (NSW) equations by Carrier and Greenspan [J. Fluid Mech. 4 (1958) 97], one representing the response to a transient change in the offshore water level, the other the motion due to a periodic standing wave, both on slopes steep enough to allow full reflection. The third is the 2-D horizontal (2DH) computational solution by Zelt [Coast. Eng. 15 (1991) 205] for the run-up of a solitary wave on a cusped beach. In all cases, both models are shown to behave well and give high accuracy results for suitably chosen grid and time spacings.     

数值模式与统计模型相耦合的近岸海浪预报方法

针对数值模式和统计模型预报近岸海浪存在的局限性,构建了数值模式和统计模型相耦合的近岸海浪预报框架,在模式计算格点和近岸预报目标点之间定义一个海浪能量密度谱传递系数,通过经验正交函数分解和卡尔曼滤波方法建立传递系数的统计预报模型并与数值模式进行耦合。经过对近岸波浪观测站1a的预报试验表明:该方法能够提高近岸海浪有效波高预报精度,有效波高的均方根误差降低了约0.16m,平均相对误差降低约9%。进一步试验和分析发现,该方法的预报有效时间小于24h,将海浪能量密度谱经过分解后得到的基本模态反映了近岸波侯的主要特征,海浪能量密度谱传递系数的变化体现了波侯的季节变化特点。     

Numerical modeling of long waves in shallow water using Incremental Differential Quadrature Method

Incremental Differential Quadrature Method (IDQM) as a rapid and accurate method for numerical simulation of Nonlinear Shallow Water (NLSW) waves is employed. To the best of authors’ knowledge, this is the first endeavor to exploit DQM in coastal hydraulics. The one-dimensional NLSW equations and related boundary conditions are discretized in space and temporal directions by DQM rules and the resulting system of equations are used to compute the state variables in the entire computational domain. It was found that the splitting of total simulation time into a number of smaller time increments, could significantly enhance the performance of the proposed method. Furthermore, results of this study show two main advantages for IDQM compared with other conventional methods, namely; unconditional stability and minimal computational effort. Indeed, using IDQM, one can use a few grid points (in spatial or time direction) without imposing any stability condition on the time step to obtain an accurate convergent solution.     

Modeling Black Sea circulation with high resolution in the coastal zone

We present a numerical model of Black Sea circulation based on primitive equations with improved spatial resolution in the coastal zone. The model equations are formulated in a two-pole orthogonal coordinate system with arbitrary locations of the poles and a vertical σ coordinate. Increased horizontal resolution is gained by displacing the pole into the vicinity of the separated subdomain. The problem is solved over a grid with a variable step. The northern coordinate pole is displaced to the vicinity of Gelendzhik; the grid step varies from 150 m in the coastal zone to 4.6 km in the main basin. We simulated the fields of currents, sea level, temperature, and salinity under the given atmospheric forcing in 2007. The model is capable of reproducing the large-scale Black Sea circulation and submesoscale variations in the coastal currents.     

完全非线性深水波的数值模拟

基于势流理论,并结合深水波质点运动从水面向下呈e指数衰减的特性,建立了完全非线性数值变深水槽模型,通过实时模拟活塞式造波机运动来产生波浪.采用时域高阶边界元法进行模拟,利用混合欧拉-拉格朗日方法和四阶Runge-Kutta方法追踪流体瞬时水面,应用镜像格林函数消除了水槽两个侧面的积分,在水槽末端布置人工阻尼层来消除反射...     

An Efficient Model for Transient Surface Waves in Both Finite and Infinite Water Depth

A numerical model is developed to simulate fully nonlinear extreme waves in finite and infinite water-depth wave tanks. A semi-mixed Eulerian-Lagrangian formulation is adopted and a higher-order boundary element method in conjunction with an image Green function is used for the fluid domain. The boundary values on the free surface are updated at each time step by a fourth-order Runga-Kutta time-marching scheme at each time step. Input wave characteristics are specified at the upstream boundary by an appropriate wave theory. At the downstream boundary, an artificial damping zone is used to prevent wave reflection back into the computational domain. Using the image Green function in the whole fluid domain, the integrations on the two lateral walls and bottom are excluded. The simulation results on extreme wave elevations in finite and infinite water-depths are compared with experimental results and second-order analytical solutions respectively. The wave kinematics is also discussed in the present study.     

Implementation of method of lines to predict water levels due to a storm along the coastal region of Bangladesh

In this study, the method of lines (MOL) has been applied to solve two-dimensional vertically integrated shallow water equations in Cartesian coordinates for the prediction of water levels due to a storm surge along the coast of Bangladesh. In doing so, the partial derivatives with respect to the space variables were discretized by the finite difference (central) method to obtain a system of ordinary differential equations (ODEs) with time as independent variable. The classical fourth-order Runge–Kutta method was used to solve the obtained system of the ODEs. We used a nested finite difference scheme, where a high resolution fine grid model (FGM) capable of incorporating all major islands along the coastal region of Bangladesh was nested into a coarse grid model (CGM) covering up to 15°N latitude of the Bay of Bengal. The boundaries of the coast and islands were approximated through proper stair step. Appropriate tidal condition over the model domain was generated by forcing the sea level to be oscillatory with the constituent M ₂ along the southern open boundary of the CGM omitting wind stress. Along the northeast corner of the FGM, the Meghna River discharge was taken into account. The developed model was applied to estimate water levels along the coast of Bangladesh due to the interaction of tide and surge associated with the April 1991 storm. We also computed our results employing the standard finite difference method (FDM). Simulated results show the MOL performs well in comparison with the FDM with regard to CPU time and stability, and ensures conformity with observations.     

Numerical and Experimental Investigation of Interactions Between Free-Surface Waves and A Floating Breakwater with Cylindrical-Dual/Rectangular-Single Pontoon

This paper investigates the hydrodynamic performance of a cylindrical-dual or rectangular-single pontoon floating breakwater using the numerical method and experimental study. The numerical simulation work is based on the multi-physics computational fluid dynamics (CFD) code and an innovative full-structured dynamic grid method applied to update the three-degree-of-freedom (3-DOF) rigid structure motions. As a time-marching scheme, the trapezoid analogue integral method is used to update the time integration combined with remeshing at each time step. The application of full-structured mesh elements can prevent grids distortion or deformation caused by large-scale movement and improve the stability of calculation. In movable regions, each moving zone is specified with particular motion modes (sway, heave and roll). A series of experimental studies are carried out to validate the performance of the floating body and verify the accuracy of the proposed numerical model. The results are systematically assessed in terms of wave coefficients, mooring line forces, velocity streamlines and the 3-DOF motions of the floating breakwater. When compared with the wave coefficient solutions, excellent agreements are achieved between the computed and experimental data, except in the vicinity of resonant frequency. The velocity streamlines and wave profile movement in the fluid field can also be reproduced using this numerical model.     

1994年发生在台湾海峡的一次地震海啸的数值模拟

建立了一个地震海啸数值模式,模式包含越洋海啸传播部分和近岸海啸变形部分,在越洋海啸传播部分中采用线性浅水方程,使用蛙跃格式求解,并且选择合适的空间步长与时间步长,使差分格式中产生的数值频散与包辛尼斯克方程中的物理频散一致,这样在不影响海啸数值计算精度的前提下,节省了计算机的机时与内存.在近岸海啸变形部分的计算中,考虑了非线性对流项与海底摩擦项.同时该模式采用了多重网格嵌套技术,提高了所关心地区的计算精度.利用这个地震海啸模式模拟了1994年发生在台湾海峡的一次地震海啸,结果与观测记录较吻合.这个模型已用于我国沿海核电站可能最大地震海啸的数值计算.     

Tidal and surge modelling using differential quadrature: A case study in the Bristol Channel

The Incremental Differential Quadrature Method (IDQM) was applied to a tidal and surge model of the Bristol Channel, UK. The method is considered as an alternative new numerical technique in the field of marine hydraulics and its performance was examined through this case study. For validation of the simulated results, tide gauge data along the Bristol Channel was used. Another well known 1D model (MIKE11) and a quasi-3D model (POLCOMS) provided more insight into the flow pattern of the estuary and additional validation data. Similar to MIKE11, IDQM is unconditionally stable and so time steps of around 45 min achieved good results for the Bristol Channel, whereas for methods which are restricted to the CFL criterion (e.g. explicit finite differencing schemes), the time step is limited to around 3 min. Since there is no stability constraint in IDQM, the time step must be selected with reference to accuracy rather than stability. The usefulness of IDQM was also demonstrated by using a small number of grid points (11 along the 110 km length of the Bristol Channel) to produce accurate results. Based on the results of this case study, it is concluded that IDQM can be successfully implemented for 1D modelling of tidal elevations and surges in non-prismatic irregular channels.     

长江河口盐水入侵对大通枯季径流量变化的响应时间

应用河口海岸三维数值模式, 计算区域包括大通至长江河口及其邻近海域, 设计高分辨率网格, 数值模拟和分析不同潮型下长江河口盐水入侵对大通径流量变化的响应时间。计算结果表明, 不同潮型期间大通径流量的增加, 河口盐度响应的时间在4.0～6.2 d之间, 但小潮期的响应时间明显长于其他潮型期的响应时间。本文给出了长江河口盐水入侵对大通枯季径流量变化的响应时间, 可为河口水文、泥沙和环境等研究中取何时径流量提供了依据。     

Diffusion reduction in an arbitrary scale third generation wind wave model

The numerical schemes for the geographic propagation of random, short-crested, wind-generated waves in third-generation wave models are either unconditionally stable or only conditionally stable. Having an unconditionally stable scheme gives greater freedom in choosing the time step (for given space steps). The third-generation wave model SWAN (“Simulated WAves Nearshore”, Booij et al., 1999) has been implemented with this type of scheme. This model uses a first order, upwind, implicit numerical scheme for geographic propagation. The scheme can be employed for both stationary (typically small scale) and nonstationary (i.e. time-stepping) computations. Though robust, this first order scheme is very diffusive. This degrades the accuracy of the model in a number of situations, including most model applications at larger scales. The authors reduce the diffusiveness of the model by replacing the existing numerical scheme with two alternative higher order schemes, a scheme that is intended for stationary, small-scale computations, and a scheme that is most appropriate for nonstationary computations. Examples representative of both large-scale and small-scale applications are presented. The alternative schemes are shown to be much less diffusive than the original scheme while retaining the implicit character of the particular SWAN set-up. The additional computational burden of the stationary alternative scheme is negligible, and the expense of the nonstationary alternative scheme is comparable to those used by other third generation wave models. To further accommodate large-scale applications of SWAN, the model is reformulated in terms of spherical coordinates rather than the original Cartesian coordinates. Thus the modified model can calculate wave energy propagation accurately and efficiently at any scale varying from laboratory dimensions (spatial scale O(10 m) with resolution O(0.1 m)), to near-shore coastal dimension (spatial scale O(10 km) with resolution O(100 m)) to oceanic dimensions (spatial scale O(10 000 km) with resolution O(100 km).     

Numerical modeling of nonlinear resonance of semi-enclosed water bodies: Description and experimental validation

The purpose of this work is to a present a numerical model to solve a set of modified Boussinesq equations to analyse nonlinear resonance of semi-enclosed water bodies. The equations are solved on a finite element unstructured grid in order to achieve an optimal mesh resolution with the local geometry. The model is able to simulate long time lapses and realistic forcing in real bathymetries with a reasonable computational cost. To validate the numerical results, a set of experiments was carried out in a physical model of two adjacent elongated basins. Comparisons between numerical and experimental results for different geometries and nonlinear conditions show that the model is able to simulate with an excellent agreement the transient nonlinear resonant process.     

A coupled modeling scheme for longshore sediment transport of wave-dominated coasts—A case study from the southern Baltic Sea

A modeling scheme based on dynamic coupling of a high-resolution 1D cross-shore model to a 2DH area model is developed to calculate the total longshore sediment transport (LST) rate in wave-dominated coasts. The purpose of this coupling strategy aims at resolving the LST with a high-resolution (both temporally and spatially) inside the surf-zone and with a coarser spatial resolution seaward of the surf-zone. The 2DH area model operates on a fixed pre-designed regional grid (parent grid) and the 1D cross-shore model is dynamically coupled to the boundary of the parent grid with a time-varying domain, starting from the first wave breaking point and ending at the maximum wave set-up point. The time-varying domain is generated in the 1D model by resolving the landward wave propagation from the offshore conditions provided by the 2DH area model at every time step. With a high-resolution cell size the 1D model resolves the wave propagation processes and resulting LST along the profile. The coupled model is applied to study the LST in the Pomeranian Bight at the southern Baltic Sea. Simulation results are compared with three other different hierarchical modeling methods (from empirical formulas such as CERC and Kamphuis to a 2DH area simulation). The comparative study indicates that the dynamically coupled model can be a reliable tool in practical applications, especially for the areas where hydrodynamics is controlled by complex bathymetry (e.g., multiple longshore bars) or morphologically induced circulation patterns.     

一个两时间层分裂显格式海洋环流模式(MASNUM)及其检验

A two-time-level, three-dimensional numerical ocean circulation model（named MASNUM） was established with a two-level, single-step Eulerian forward-backward time-differencing scheme. A mathematical model of large-scale oceanic motions was based on the terrain-following coordinated, Boussinesq, Reynolds-averaged primitive equations of ocean dynamics. A simple but very practical Eulerian forward-backward method was adopted to replace the most preferred leapfrog scheme as the time-differencing method for both barotropic and baroclinic modes. The forward-backward method is of second-order of accuracy, computationally efficient by requiring only one function evaluation per time step, and free of the computational mode inherent in the three-level schemes. This method is superior to the leapfrog scheme in that the maximum time step of stability is twice as large as that of the leapfrog scheme in staggered meshes thus the computational efficiency could be doubled. A spatial smoothing method was introduced to control the nonlinear instability in the numerical integration. An ideal numerical experiment simulating the propagation of the equatorial Rossby soliton was performed to test the amplitude and phase error of this new model. The performance of this circulation model was further verified with a regional（northwest Pacific） and a quasi-global（global ocean simulation with the Arctic Ocean excluded） simulation experiments. These two numerical experiments show fairly good agreement with the observations. The maximum time step of stability in these two experiments were also investigated and compared between this model and that model which adopts the leapfrog scheme.     

闽江感潮河段潮汐-洪水相互作用数值模拟

本文分析了闽江感潮河段洪水、潮汐特征,利用高精度GIS数据建立了基于非结构三角网的高分辨率洪-潮耦合模型,在闽江口重点区域的网格分辨率达到50~100m。选取竹岐断面作为径流边界并基于"2006.6.6"洪水过程设计了3组数值实验,模拟结果表明:相比于只考虑洪水或者潮汐,在耦合洪水和潮汐后,各代表站的模拟值与实测值更为吻合;在30年一遇洪水的作用下,闽江感潮河段各断面的原有潮汐特征都不同程度地被洪水信号所影响,其中,文山里和解放大桥站表现出明显的洪水特征,而峡南、白岩潭和琯头站则表现出洪、潮混合特征;从峡南到琯头对应河段在高潮时段流速减小而低潮时段则流速增大,说明该河段存在很明显的洪-潮相互作用。