A semi-implicit three-dimensional numerical model for non-hydrostatic pressure free-surface flows on an unstructured,sigma grid

A semi-implicit 3-D numerical formulation for solving non-hydrostatic pressure free-surface flows on an unstructured,sigma grid is proposed.Pressure-splitting and 9 semi-implicit methods are inherited and reformed from Casulli’s z-coordinate formulation.The non-orthogonal sigma-coordinate transformation leads to additional terms. The resulting linear system for the non-hydrostatic correction is diagonally dominant but unsymmetric,and it is solved by the BiCGstab method.In contrast with z-coordinate non-hydrostatic models,the new model fits vertical boundaries much better,which is important for the long-time simulation of sediment transport and riverbed deformation.A lock-exchange density flow is computed to determine whether the new scheme is able to simulate non-hydrostatic free-surface flows.The new model is further verified using the field data of a natural river bend of the lower Yangtze River.Good agreement between simulations and earlier research results,field data is obtained, indicating that the new model is applicable to hydraulic projects in real rivers.     

A wetting and drying algorithm with a combined pressure/free-surface formulation for non-hydrostatic models

A wetting and drying method for free-surface problems for the three-dimensional, non-hydrostatic Navier–Stokes equations is proposed. The key idea is to use a horizontally fixed mesh and to apply different boundary conditions on the free-surface in wet and dry zones. In wet areas a combined pressure/free-surface kinematic boundary condition is applied, while in dry areas a positive water level and a no-normal flow boundary condition are enforced. In addition, vertical mesh movement is performed to accurately represent the free-surface motion. Non-physical flow in the remaining thin layer in dry areas is naturally prevented if a Manning–Strickler bottom drag is used. The treatment of the wetting and drying processes applied through the boundary condition yields great flexibility to the discretisation used. Specifically, a fully unstructured mesh with any finite element choice and implicit time discretisation method can be applied. The resulting method is mass conservative, stable and accurate. It is implemented within Fluidity-ICOM [1] and verified against several idealized test cases and a laboratory experiment of the Okushiri tsunami.     

A 3D unstructured non-hydrostatic ocean model for internal waves

A 3D non-hydrostatic model is developed to compute internal waves. A novel grid arrangement is incorporated in the model. This not only ensures the homogenous Dirichlet boundary condition for the non-hydrostatic pressure can be precisely and easily imposed but also renders the model relatively simple in its discretized form. The Perot scheme is employed to discretize horizontal advection terms in the horizontal momentum equations, which is based on staggered grids and has the conservative property. Based on previous water wave models, the main works of the present paper are to (1) utilize a semi-implicit, fractional step algorithm to solve the Navier-Stokes equations (NSE); (2) develop a second-order flux-limiter method satisfying the max–min property; (3) incorporate a density equation, which is solved by a high-resolution finite volume method ensuring mass conservation and max–min property based on a vertical boundary-fitted coordinate system; and (4) validate the developed model by using four tests including two internal seiche waves, lock-exchange flow, and internal solitary wave breaking. Comparisons of numerical results with analytical solutions or experimental data or other model results show reasonably good agreement, demonstrating the model’s capability to resolve internal waves relating to complex non-hydrostatic phenomena.     

Non-hydrostatic and non-linear contributions to the internal wave energy flux in sill regions

A three-dimensional non-linear, non-hydrostatic model in cross-sectional form is used to determine the factors influencing the relative importance of the linear, non-hydrostatic and non-linear contributions to the internal wave energy flux in sill regions due to tidal forcing. The importance of the free surface elevation term is also considered. Idealised topography representing the sill at the entrance to Loch Etive, the site of a recent measurement programme, is used. Calculations show that the non-linear terms in the energy flux become increasingly important as the sill Froude Number (F _s) increases and the sill aspect ratio is increased. The vertical profile of the stratification, in particular its value close to the sill crest where internal waves are generated, has a significant influence on unsteady lee wave and mixed tidal–lee wave generation and the non-linear contribution to the energy flux. Calculations show that as F _s increases, the energy flux due to the non-linear and non-hydrostatic terms increases more rapidly than the linear term. The importance of the non-linear terms in the energy flux also increases as the sill aspect ratio is increased. Increasing the buoyancy frequency reduces the contribution of the non-hydrostatic and non-linear terms to the total energy flux. Also, as the buoyancy frequency is increased, this reduces unsteady lee wave and mixed tidal–lee wave generation. In essence, these calculations show that the energy flux due to the non-hydrostatic and non-linear terms is appreciable in sill regions.     

On the importance of non-hydrostatic processes in determining tidally induced mixing in sill regions

The importance of using a non-hydrostatic model to compute tidally induced mixing and flow in the region of a sill is examined using idealized topography representing the sill at the entrance to Loch Etive. This site is chosen since detailed measurements were recently made there. Calculations are performed with and without the inclusion of non-hydrostatic dynamics using a vertical slice model for a range of sill widths corresponding to typical sill regions. Initial non-hydrostatic calculations showed that the model could reproduce the observed flow characteristics in the region. However, when calculations were performed using the model in hydrostatic form, the significant artificial convective mixing that occurred in order to remove density inversions led to excessively high vertical mixing. This influenced the computed temperature field and the intensity of the current jet that separated from the sill on its lee side. In addition it affected the magnitude and spatial characteristics of the lee waves generated on the lee side of the sill. Calculations with a range of sill widths, showed that as the sill width decreased the difference between the solution computed with the non-hydrostatic and hydrostatic model increased.     

VTI介质起伏地表地震波场模拟

起伏地表下地震波场模拟有助于解释主动源和被动源地震探测中穿过山脉和盆地的测线所获得的资料.然而传统的有限差分法处理起伏的自由边界比较困难,为了克服这一困难,我们将笛卡尔坐标系的各向异性介质弹性波方程和自由边界条件变换到曲线坐标系中,采用一种稳定的、显式的二阶精度有限差分方法离散(曲线坐标系)VTI介质中的弹性波方程;对...     

Numerical studies of flow over a sill: sensitivity of the non-hydrostatic effects to the grid size

A non-hydrostatic terrain-following model in cross sectional form is applied to study the processes in the lee of a sill in an idealized stratified fjord during super-critical tidal inflow. A sequence of numerical studies with horizontal grid sizes in the range from 100 to 1.5625 m are performed. All experiments are repeated using both hydrostatic and non-hydrostatic versions of the model, allowing a systematic study of possible non-hydrostatic pressure effects and also of the sensitivity of these effects to the horizontal grid size. The length scales and periods of the internal waves in the lee of the sill are gradually reduced and the amplitudes of these waves are increased as the grid size is reduced from 100 down to 12.5 m. With a further reduction in grid size, more short time and space scale motions become superimposed on the internal waves. Associated with the internal wave activity, there is a deeper separation point that is fairly robust to all parameters investigated. Another separation point nearer to the top of the sill appears in the numerical results from the high-resolution studies with the non-hydrostatic model. Associated with this shallower separation point, an overturning vortex appears in the same set of numerical solutions. This vortex grows in strength with reduced grid size in the non-hydrostatic experiments. The effects of the non-hydrostatic pressure on the velocity and temperature fields grow with reduced grid size. In the experiments with horizontal grid sizes equal to 100 or 50 m, the non-hydrostatic pressure effects are small. For smaller grid sizes, the time mean velocity and temperature fields are also clearly affected by the non-hydrostatic pressure adjustments.     

Free-surface implementation in a mesh-free finite-difference method for elastic wave propagation in the frequency domain

We present an original implementation of the free-surface boundary condition in a mesh-free finite-difference method for simulating elastic wave propagation in the frequency domain. For elastic wave modelling in the frequency domain, the treatment of free surfaces is a key issue which requires special consideration. In the present study, the free-surface boundary condition is directly implemented at node positions located on the free-surface. Flexible nature of the mesh-free method for nodal distribution enables us to introduce topography into numerical models in an efficient manner. We investigate the accuracy of the proposed implementation by comparing numerical results with an analytical solution. The results show that the proposed method can calculate surface wave propagation even for an inclined free surface with substantial accuracy. Next, we calculate surface wave propagation in a model with a topographic surface using our method, and compare the numerical result with that using the finite-element method. The comparison shows the excellent agreement with each other. Finally, we apply our method to the SEG foothill model to investigate the effectiveness of the proposed method. Since the mesh-free method has high flexibility of nodal distribution, the proposed implementation would deal with models of topographic surface with sufficient accuracy and efficiency.     

临界窗口外S波分裂的初步研究

本文在详细讨论了自由界面对入射Ｓ波的影响的基础上，首次提出一种对临界窗口之外的Ｓ波分裂资料的处理方法。首先对径向和竖向记录资料进行振幅和相位矫正，进而利用最大特征值和波形识别算子法（刘希强，１９９２）对资料进行分析。利用本文提出的方法对天水爆破资料进行了分析，结果表明，爆破资料显示了比较明显的Ｓ波震相，且产生了明显的Ｓ波分裂。     

非对称型强飓风中的准平衡流特征分析Ⅱ:波动结构

利用非对称波分量分解和小波分析的方法,对准平衡动力模型下非对称强飓风内中尺度波动的空间结构和时间序列特征进行分析.结果表明,平衡流场中1波扰动占主要地位且具有涡旋Rossby波的典型结构特征,准平衡流各波数下扰动的空间分布反映了中尺度波动的混合性质;模式大气和准平衡垂直运动的全局功率谱中,超过信度检验的强波动信号中不仅包含分别表征重力波和涡旋Rossby波的高频和低频波动信号,还存在表征具有物理性质不可分特性的混合涡旋Rossby-重力波的中频波动.混合波的出现建立了不同频段波动之间的能量交换通道,其信号的变化对飓风系统的强弱变化具有一定的指示作用.非平衡垂直运动的波动强信号则主要集中在高频和低频区域,反映了在飓风强度变化情况下,与高频重力波有关的快波调整过程所引起的垂直扰动的振荡和传播.强垂直风切变对飓风内中尺度波动的切向和径向传播具有重要影响,当环境垂直风切变很强时,准平衡1波扰动在径向和切向方向上均呈"驻波"形态,随着环境垂直风切变的减弱,1波扰动以混合波波速逆基本气流传播.     

The effect of random recharge on uniform steady free-surface flow in heterogeneous porous formations

The effect of parametric uncertainty in recharge rate and spatial variability of hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for sloped mean uniform flow in a randomly heterogeneous porous medium under the influence of random recharge. We develop analytic solutions for the variance of free-surface position, head, and specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of head and specific discharge beneath the free-surface. Statistical moments are derived using a first-order approximation and then compared with their parallel in an unbounded medium. The effect of recharge mean and variability on the statistical moments is analyzed. Results can be applied to more complex flows, slowly varying in the mean.     

Three-dimensional numerical modelling of wind-driven circulation in a homogeneous lake

A three-dimensional numerical model has been developed to study wind-induced circulation patterns in a shallow homogeneous lake with a complex bathymetry. The governing equations are the unsteady Reynolds-Averaged Navier–Stokes equations in which the non-hydrostatic pressure distribution has been included. The model was tested against analytical solutions and laboratory data for wind-induced currents and then applied to Esthwaite Water, a small lake in Cumbria, UK. The model was used to study the main model parameters and to generate typical circulation patterns for a variety of conditions in the lake. Simulations showed that a non-hydrostatic pressure distribution did not have any noticeable influence on the overall circulation pattern in the lake. However, comparisons with field data at some measurement stations in the near-shore region with sharply varying bottom topography showed that the hydrodynamic pressure component had some influence on the vertical velocity profile.     

GRAPES全球非静力大气模式的正规模分析

为分析数值模式动力学框架中不同波动的特性及对数值天气预报模式计算稳定性的影响,文章对GRAPES（Global/Regional Assimilation and PrEdiction System）全球非静力大气模式进行了正规模分析.首先,建立了静止大气状态下模式的线性化系统,并在适当的边界条件下将线性化系统分解成具有垂直与水平结构方程的本征值、本征函数耦合问题.然后在等温大气条件下,利用耦合问题的数值结果分析得出：GRAPES非静力模式除了有几乎和对应的静力模式一致的向东、向西传播的重力惯性波及向西传播的Rossby波外,还有一个向东、向西传播的声惯性波;特别是,只有当纵横比较大时,非静力模式对重力惯性波才会有显著影响.     

2D finite-difference viscoelastic wave modelling including surface topography

We have pursued two-dimensional (2D) finite-difference (FD) modelling of seismic scattering from free-surface topography. Exact free-surface boundary conditions for the particle velocities have been derived for arbitrary 2D topographies. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank–Nicholson method in time.
In order to demonstrate the capabilities of the surface topography modelling technique, we simulate incident point sources with a sinusoidal topography in seismic media of increasing complexities. We present results using parameters typical of exploration surveys with topography and heterogeneous media. Topography on homogeneous media is shown to generate significant scattering. We show additional effects of layering in the medium, with and without randomization, using a von Kármán realization of apparent anisotropy. Synthetic snapshots and seismograms indicate that prominent surface topography can cause back-scattering, wave conversions and complex wave patterns which are usually discussed in terms of inter-crust heterogeneities.     

A 3-D phase-averaged model for shallow-water flow with waves in vegetated water

A 3-D shallow-water flow model has been developed to simulate the flow in coastal vegetated waters with short waves. The model adopts the 3-D phase-averaged shallow-water flow equations with radiation stresses induced by short waves. It solves the governing equations using an implicit finite volume method based on quadtree rectangular mesh in the horizontal plane and stretching mesh in the vertical direction. The flow model is coupled with a spectral wave deformation model called CMS-Wave. The wave model solves the spectral wave-action balance equation and provides wave characteristics to the flow model. The model considers the effects of vegetation on currents and waves by including the drag and inertia forces of vegetation in the momentum equations and the wave energy loss due to vegetation resistance in the wave-action balance equation. The model has been tested using several sets of laboratory experiments, including steady flows in a straight channel with submerged vegetation and in a compound channel with vegetated floodplain and random waves through a vegetated channel and on a vegetated beach slope. The calculated water levels, current velocities, and wave heights are in general good agreement with the measured data.     

Finite-difference modeling of surface waves in poroelastic media and stress mirror conditions

During seismic wave propagation on a free surface, a strong material contrast boundary develops in response to interference by P- and S- waves to create a surfacewave phenomenon. To accurately determine the effects of this interface on surface-wave propagation, the boundary conditions must be accurately modeled. In this paper, we present a numerical approach based on the dynamic poroelasticity for a space–time-domain staggeredgrid finite-difference simulation in porous media that contain a free-surface boundary. We propose a generalized stess mirror formulation of the free-surface boundary for solids and fluids in porous media for the grid mesh on which lays the free-surface plane. Its analog is that used for elastic media, which is suitable for precise and stable Rayleigh-type surface-wave modeling. The results of our analysis of first kind of Rayleigh (R1) waves obtained by this model demonstrate that the discretization of the mesh in a similar way to that for elastic media can realize stable numerical solutions with acceptable precision. We present numerical examples demonstrating the efficiency and accuracy of our proposed method.     

High-resolution modelling of a large-scale river plume

Evolution of a large-scale river plume is studied numerically using the Massachusetts Institute of Technology general circulation model. The model parameters were set close to those observed in the area of the Columbia River mouth. The fine-resolution grid along with the non-hydrostatic dispersion included in the model allowed for the reproduction of detailed inner plume structure, as well as a system of internal waves radiated from the plume’s boundary. It was found that not only first-mode but also second- and third-mode internal waves are radiated from the plume at the latest stages of its relaxation when the velocity of the front propagation drops below an appropriate wave phase speed of internal baroclinic mode. The model output shows that the amplitude of these high-mode waves is of the same order as the leading first-mode waves, which in combination with the specific vertical structure (location of the maximum structure function beyond the pycnocline layer) creates favourable conditions for the generation of shear instabilities. High-resolution model output also reveals evidence of a fine internal structure of the plume characterized by the presence of secondary fronts inside the plume and secondary internal wave systems propagated radially from the lift-off area to the outer boundary. These structures intensify the mixing processes within the propagating plume with predominance of the entrainment mechanism developing on the lower boundary between the plume’s body and underlying waters. The scheme of horizontal circulation in the plume was reproduced by the methodology of Lagrange drifters released near the mouth at different depths.

     

High-resolution modelling of a large-scale river plume

Evolution of a large-scale river plume is studied numerically using the Massachusetts Institute of Technology general circulation model. The model parameters were set close to those observed in the area of the Columbia River mouth. The fine-resolution grid along with the non-hydrostatic dispersion included in the model allowed for the reproduction of detailed inner plume structure, as well as a system of internal waves radiated from the plume’s boundary. It was found that not only first-mode but also second- and third-mode internal waves are radiated from the plume at the latest stages of its relaxation when the velocity of the front propagation drops below an appropriate wave phase speed of internal baroclinic mode. The model output shows that the amplitude of these high-mode waves is of the same order as the leading first-mode waves, which in combination with the specific vertical structure (location of the maximum structure function beyond the pycnocline layer) creates favourable conditions for the generation of shear instabilities. High-resolution model output also reveals evidence of a fine internal structure of the plume characterized by the presence of secondary fronts inside the plume and secondary internal wave systems propagated radially from the lift-off area to the outer boundary. These structures intensify the mixing processes within the propagating plume with predominance of the entrainment mechanism developing on the lower boundary between the plume’s body and underlying waters. The scheme of horizontal circulation in the plume was reproduced by the methodology of Lagrange drifters released near the mouth at different depths.     

Hydro-elastic wave proliferation over an impermeable seabed with bottom deformation

ABSTRACT

A hydro-elastic frame has been considered to investigate the proliferation of waves over small base deformation on an infinitely extended flexible seabed. The flexible base surface is assumed as a thin elastic plate of very small thickness and it depends on the Euler–Bernoulli beam equation. For any particular frequency, there are two different modes of time-harmonic propagating wave exists rather than one mode of propagating wave along the positive horizontal direction. The waves with smaller wavenumber spread along the free-surface of the sea (say, free-surface mode) and the waves with higher wavenumber spread along the flexible base surface (say, flexural mode). A simplified perturbation approach is utilised to bring down the entire equations which govern the original boundary value problem (bvp) to a less complex bvp for the first-order velocity potential function. The first-order potential function along with the first-order reflection and transmission coefficients for both modes are calculated by a procedure based upon Fourier transform approach. A shape of sinusoidal swells flexible base surface is taken as an example to approve the scientific results. It is observed that when the train of normal incident propagating wave spreads over base distortion because of either the free-surface unsettling influence or the flexural wave movement in the sea, the reflected and transmitted energy are always feasible to be exchanged from one particular wave mode to another wave mode. Furthermore, we notice that the realistic changes in the flexural rigidity behaviour on the flexible base surface of the sea have a significant effect on the problem of water wave proliferation over small base deformation. Moreover, the energy conservation equation is derived with the help of the Green's integral theorem. The results for the values of reflection and transmission coefficients obtained for both the free-surface unsettling influence as well as flexural wave movement in the fluid are found to satisfy the energy conservation equation almost accurately.     

On the solution of transient free-surface flow problems in porous media by the finite element method

A numerical procedure is presented to deal with solution of transient free-surface flows in porous media. The governing boundary-value problem for the piezometric potential is solved by the finite element method. The initial-value problem which describes the transient motion of the free-surface is solved by the method of quasi-linearization. The numerical scheme has been applied to isotropic and anisotropic earth dam problem and also to a ditch drainage problem. Excellent agreements have been reached when compared with known solutions. This computational procedure is shown to be stable and suitable for this class of problems with the aid of a digital computer.