A three-dimensional semi-implicit unstructured grid finite volume ocean model

A new three-dimensional semi-implicit finite-volume ocean model has been developed for simulating the coastal ocean circulation, which is based on the staggered C -unstructured non-orthogonal grid in the horizontal direction and z -level grid in the vertical direction. The three-dimensional model is discretized by the semi-implicit finite-volume method, in that the free-surface and the vertical diffusion are semi-implicit, thereby removing stability limitations associated with the surface gravity wave and vertical diffusion terms. The remaining terms in the momentum equations are discretized explicitly by an integral method. The partial cell method is used for resolving topography, which enables the model to better represent irregular topography. The model has been tested against analytical cases for wind and tidal oscillation circulation, and is applied to simulating the tidal flow in the Bohai Sea. The results are in good agreement both with the analytical solutions and measurement results.     

A coastal ocean model of semi-implicit finite volume unstructured grid

A two-dimensional coastal ocean model based on unstructured C-grid is built, in which the momentum equation is discretized on the faces of each cell, and the continuity equation is discretized on the cell. The model is discretized by semi-implicit finite volume method, in that the free surface is semi-implicit and the bottom friction is implicit, thereby removing stability limitations associated with the surface gravity wave and friction. The remaining terms in the momentum equations are discretized explicitly by integral finite volume method and second-order Adams-Bashforth method. Tidal flow in the polar quadrant with known analytic solution is employed to test the proposed model. Finally, the performance of the present model to simulate tidal flow in a geometrically complex domain is examined by simulation of tidal currents in the Pearl River Estuary.     

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.     

Nonhydrostatic aspects of coastal upwelling meanders and filaments off eastern ocean boundaries

Coastal upwelling meanders and filaments are common features off eastern ocean boundaries. Their growth is reinvestigated herein using a nonhydrostatic three-dimensional model and a reduced-gravity model, with the objective of assessing contributions from two mechanisms that emerge in the nonhydrostatic regime. The first mechanism is caused by the vertical projection of the Coriolis force in the momentum equation. It is found that the vertical Coriolis force often acts as a restoring force against numerical damping off eastern ocean boundaries and thus enhances the growth of meanders and filaments. The second mechanism arises from unstable ocean stratification when the cold upwelled water intrudes seaward over the warm layer. The unstable stratification, albeit transient, further enhances the growth of meanders and filaments. It is concluded that although nonhydrostatic effects do not change our understanding of how meanders and filaments grow, the realism can be enhanced using a nonhydrostatic model insofar as meanders and filaments off eastern ocean boundaries are concerned.     

斜压海洋动力学的一种三维数值模式 Ⅰ.动力学方程数值格式

依据自由海面海洋动力学原始方程建立了一种三维有限差分数值模式,可用于潮波、风暴潮和海流的数值模拟和预报。运动方程和连续方程的数值格式采用内、外模态分离的技术。外模态采用交替方向隐格式,用于计算海面高度和垂直平均流速,时间步长不受Ｃｏｕｒａｎｔ－Ｆｒｉｄｅｒｉｃｈｓ－Ｌｅｗｙ条件限制;内模态采用半隐格式,用于计算海流的垂直２颁布,其时间步长可大于外模态时间步长。模式的计算程度比一般显式模式可快１０倍     

The role of nonhydrostatic dynamics in controlling development of a surface ocean front

Numerical studies of surface ocean fronts forced by inhomogeneous buoyancy loss show nonhydrostatic convective plumes coexisting with baroclinic eddies. The character of the vertical overturning depends sensitively on the treatment of the vertical momentum equation in the model. It is less well known how the frontal evolution over scales of O(10 km) is affected by these dynamics. Here, we compare highly resolved numerical experiments using nonhydrostatic and hydrostatic models and the convective-adjustment parametrization. The impact of nonhydrostatic processes on average cross-frontal transfer is weak compared to the effect of the O(1 km) scale baroclinic motions. For water-mass distribution and formation rate nonhydrostatic dynamics have similar influence to the baroclinic eddies although adequate resolution of the gradients in forcing fluxes is more important. The overall implication is that including nonhydrostatic surface frontal dynamics in ocean general circulation models will have only a minor effect on scales of O(1 km) and greater.     

Analysis of a simplification strategy in a nonhydrostatic model for surface and internal wave problems

This paper examines the simplification strategy of retaining only the nonhydrostatic effect of local acceleration in a three-dimensional fully nonhydrostatic model regarding the submesoscale wave phenomenon in the ocean.Elaborate scale analysis of the vertical component of the Reynold-averaged Navier–Stokes(RANS) equation was performed, confirming the rationalization of this simplification. Then, the simplification was implemented in a RANS equation-based nonhydrostatic model NHWAVE(nonhydrostat...     

Modeling vertical motion at ocean fronts: Are nonhydrostatic effects relevant at submesoscales?

Through a suite of three-dimensional, high-resolution numerical modeling experiments, we examine the role of nonhydrostatic effects on O(1 km) submesoscale processes at ocean fronts, with particular focus on the vertical velocity field. Several differences between nonhydrostatic and hydrostatic models are pointed out using a framework that enables precise comparison, but it is difficult to identify categorical differences between the model solutions at the grid resolutions afforded. The instantaneous vertical velocity structure is sensitive to the model choice and, even more so, to grid resolution, but the average vertical flux is similar in both hydrostatic and nonhydrostatic cases.When a frontal region with horizontal density gradients is perturbed by wind, a profusion of submesoscale, O(1 km), secondary circulation features develops in the upper 50 m. Narrow, elongated cells of intense up- and down-welling are found to occur close to the surface, overlying broader regions of weaker up- and down-welling associated with the mesoscale meanders of the baroclinically unstable front. The submesoscale down-welling is considerably stronger than up-welling and is concentrated in 1–2 km width filaments within which velocities can attain magnitudes as high as 200 m day⁻¹. The submesoscale features are found to be robust at horizontal grid resolutions varying between 1 and 0.25 km and exist even in the hydrostatic model. Submesoscale circulation is difficult to observe or resolve in coarser resolution circulation models, but is likely to play a significant role in the exchange of energy and properties between the surface ocean and thermocline. Possible mechanisms for the generation of these features are investigated in a follow-on paper.     

Resolving eddies by local mesh refinement

Nesting in large-scale ocean modeling is used for local refinement to resolve eddy dynamics that would not be accessible otherwise. Unstructured meshes offer this functionality too by adjusting their resolution according to some goal function. However, by locally refining the mesh one does not necessarily achieve the goal resolution, because the eddy dynamics, in particular the ability of eddies to release the available potential energy, also depend on the dynamics on the upstream coarse mesh. It is shown through a suite of experiments with a zonally re-entrant channel that baroclinic turbulence can be out from equilibrium in wide (compared to a typical eddy size) zones downstream into the refined area. This effect depends on whether or not the coarse part is eddy resolving, being much stronger if it is not. Biharmonic viscosity scaled with the cube of grid spacing is generally sufficient to control the smoothness of solutions on the variable mesh. However, noise in the vertical velocity field may be present at locations where the mesh is varied if momentum advection is implemented in the vector invariant form. Smoothness of vertical velocity is recovered if the flux form of momentum advection is used, suggesting that the noise originates from a variant of the Hollingsworth instability.     

Dispersion and stability analyses of the linearized two-dimensional shallow water equations in Cartesian coordinates

In the present study, a Fourier analysis is used to develop expressions for phase and group speeds for both continuous and discretized, linearized two-dimensional shallow water equations, in Cartesian coordinates. The phase and group speeds of the discrete equations, discretized using a three-point scheme of second order, five-point scheme of fourth order and a three-point compact scheme of fourth order in an Arakawa C grid, are calculated and compared with the corresponding values obtained for the continuous system. The three-point second-order scheme is found to be non-dispersive with grid resolutions greater than 30 grids per wavelength, while both the fourth-order schemes are non-dispersive with grid resolutions greater than six grids per wavelength. A von Neumann stability analysis of the two- and three-time-level temporal schemes showed that both schemes are stable. A wave deformation analysis of the two-time-level Crank–Nicolson scheme for one-dimensional and two-dimensional systems of shallow water equations shows that the scheme is non- dispersive, independent of the Courant number and grid resolution used. The phase error or the dispersion of the scheme decreases with a decrease in the time step or an increase in grid resolution.     

Study of internal wave generation by tide-topography interaction

The generation mechanism of internal waves by a relatively strong tidal flow over a sill is clarified analytically. Special attention is directed to the role of the tidal advection effect, which is examined by use of characteristics. An internal wave which propagates upstream is gradually formed through interference of infinitesimal amplitude internal waves (elementary waves) emanating from the sill at each instant of time. In the accelerating (or decelerating) stage of the tidal flow, the effective amplification of the internal wave takes place as the internal Froude number exceeds (or falls below) unity because during this period the internal wave slowly travels downstream (or upstream) while crossing over the sill where elementary waves are efficiently superimposed. In fact, the variability in the internal wave field actually observed in a realistic situation (Stellwagen Bank in Massachusetts Bay) is shown to be satisfactorily interpreted in terms of this mechanism. Furthermore, by using this analytical model, the relation between the strength of the tidal advection effect and the resulting internal waveform is clarified. This theory is easily extended to include a vertically sheared steady flow. In this case, although the fundamental generation mechanism is the same as above, the amplitude of the elementary wave varies with time depending on the relative direction of the tidal flow and steady shear flow, so that the internal wave field over the sill differs markedly between the ebb and flood tidal phases. As an example, the internal wave generation process over the sill in the Strait of Gibraltar is qualitatively discussed on the basis of this analytical model. The effect of vertical mixing caused by breaking of these large-amplitude internal waves on the coastal environment is also pointed out. In particular, a brief discussion is made on the control of water exchange by the fortnightly modulation of tidal mixing processes at the sills and constrictions in channels connecting freshwater sources with the ocean.     

Difference-frequency wave loads on a large body in multi-directional waves

The second-order difference-frequency wave forces on a large three-dimensional body in multi-directional waves are computed by the boundary integral equation method and the so-called FML formulation (assisting radiation potential method). Semi-analytic solutions for a bottom-mounted vertical circular cylinder are also developed to validate the numerical method. Difference-frequency wave loads on a bottom-mounted vertical cylinder and stationary four legs of the ISSC tension-leg platform (TLP) are presented for various combinations of incident wave frequencies and headings. These force quadratic transfer functions (QTF) can directly be used in studying slowly varying wave loads in irregular short-crested seas described by a particular directional spectrum. From our numerical results, it is seen that the slowly varying wave loads are in general very sensitive to the directional spreading function of the sea, and therefore wave directionality needs to be taken into account in relevant ocean engineering applications. It is also pointed out that the uni-directionality of the sea is not necessarily a conservative assumption when the second-order effects are concerned.     

An unrealistic high-salinity tongue simulated in the tropical Atlantic: another example illustrating the need for a more careful treatment of vertical discretizations in OGCMs

An unrealistically high-salinity maximum is found to develop in a high-resolution model of the north and equatorial Atlantic below the shallow halocline in the Gulf of Guinea. The spurious water mass with salinities too high by as much as 1 psu is formed when the vertical advection is treated by the standard central-differencing advection scheme. The problem is considerably reduced either by increasing the vertical resolution of the numerical grid, or by switching to a higher-order upwind-weighted scheme for vertical advection. This note stresses the need for a careful consideration of vertical discretization even in typical high-resolution ocean general circulation models (OGCMs). Particular attention is needed for studying heat and salt budgets or transports of biogeochemical tracers.     

Second-order moment advection scheme applied to Arctic Ocean simulation

We apply the second-order moment (SOM) advection scheme of (Prather, M.J. 1986. Numerical advection by conservation of second-order moments. J. Geophys. Res. 91, 6671–6681.) to the simulation of the large-scale circulation of the Arctic Ocean with a coupled ocean–sea-ice model. Compared to three other advection schemes commonly employed in ocean simulations (centred differences, flux corrected transport, and multidimensional positive definite advection transport), the SOM method helps preserve the vertical structure of Arctic water masses. The depth, thickness and hydrographic properties of the Arctic Surface Water and the Arctic Atlantic Layer are better represented with SOM than with any of the other three advection algorithms. We also present a convenient method for calculating the implicit numerical diffusivity of upstream based schemes, such as the SOM method, and discuss three approaches for improving the monotonicity properties of the SOM algorithm.     

Algorithm for non-hydrostatic dynamics in the Regional Oceanic Modeling System

A non-hydrostatic algorithm for the Regional Oceanic Modeling System (ROMS) is proposed. It is based on a decomposition technique for hydrostatic and non-hydrostatic pressure. The algorithm has a pressure-correction scheme with split-explicit time-stepping for baroclinic and barotropic vertical modes with a free surface. The algorithm implementation requires solving a Poisson equation for a non-hydrostatic pressure that has a non-symmetric matrix in discrete form. The efficiency of a different class of solvers and preconditioners were tested. The algorithm is successfully implemented with several examples where non-hydrostatic effects are important. These include standing external gravity waves; strongly nonlinear internal wave generation and transformation; stratified shear instability and its associated mixing; and nonlinear internal tidal generation over a ridge. The corresponding changes in the pre-processing and post-processing infrastructure in the existing hydrostatic ROMS code were performed to implement parallel elliptic solvers and a new set of dynamical equations.     

Simulation of nearshore wave processes by a depth-integrated non-hydrostatic finite element model

This paper presents CCHE2D-NHWAVE, a depth-integrated non-hydrostatic finite element model for simulating nearshore wave processes. The governing equations are a depth-integrated vertical momentum equation and the shallow water equations including extra non-hydrostatic pressure terms, which enable the model to simulate relatively short wave motions, where both frequency dispersion and nonlinear effects play important roles. A special type of finite element method, which was previously developed for a well-validated depth-integrated free surface flow model CCHE2D, is used to solve the governing equations on a partially staggered grid using a pressure projection method. To resolve discontinuous flows, involving breaking waves and hydraulic jumps, a momentum conservation advection scheme is developed based on the partially staggered grid. In addition, a simple and efficient wetting and drying algorithm is implemented to deal with the moving shoreline. The model is first verified by analytical solutions, and then validated by a series of laboratory experiments. The comparison shows that the developed wave model without the use of any empirical parameters is capable of accurately simulating a wide range of nearshore wave processes, including propagation, breaking, and run-up of nonlinear dispersive waves and transformation and inundation of tsunami waves.     

Internal wave excitation from a collapsing mixed region

The collapse of a uniform density fluid (a “mixed region”) into a surrounding ambient fluid with complex stratification is examined by way of laboratory experiments and fully nonlinear numerical simulations. The analysis focuses upon the consequent generation of internal gravity waves and their influence upon the evolution of the collapsing mixed region. In experiments and simulations for which the ambient fluid has uniform density over the vertical extent of the mixed region and is stratified below, we find the mixed region collapses to form an intrusive gravity current and internal waves are excited in the underlying stratified fluid. The amplitude of the waves is weak in the sense that the intrusion is not significantly affected by the waves. However, scaling the results to the surface mixed layer of the ocean we find that the momentum flux associated with the waves can be as large as 1 N/m². In simulations for which the ambient fluid is stratified everywhere, including over the vertical extent of the mixed region, we find that internal waves are excited with such large amplitude that the collapsing mixed region is distorted through strong interactions with the waves.     

无结构网格二维河口海岸水动力数值模式的建立及其应用

为完全拟合河口近海复杂岸线和工程结构以及有效局部加密,设计并建立了一个无结构三角形网格二维河口海岸水动力数值模式。空间离散主要基于有限体积法以保证守恒性,时间积分采用预估修正法以提高精度。水位在三角形网格中心通过连续方程求解;水平x方向和y方向的流速U和V均在网格边中点上通过动量方程求解。流速平流项的求解中采用了TVD格式。TVD流速平流通量为一个一阶迎风格式通量和一个二阶格式通量的组合,一阶格式通量和二阶格式通量根据流速的局部分布情况得出配比,最终组合得到TVD通量。TVD格式具有低耗散和无频散的优点,提高了模式的稳定性。应用实测资料验证建立的模式,结果显示水位、流速和流向的计算值与实测值均符合良好。     

Model of the Baltic Sea’s dynamics with nonhydrostatic effects on a triangular grid

A numerical model of the sea’s thermohydrodynamics with a finite-difference approximation of the equations of the nonhydrostatic dynamics on a grid with a triangular form of the horizontal section of its element is formulated. The slope of the lower side of the grid’s bottom cell is determined by the given linear profile of the bottom relief. Within the shallow-water approach, the dispersion relations of the B and C grids and the developed discrete model are compared; the results of tests for different approximations of the bottom relief for a rectangular basin are given and analyzed. The developed model of the thermohaline dynamics is used for studying the influence of the nonhydrostatic effects on the circulation of the Baltic Sea and a part of Vistula Bay. The comparison of the simulation results obtained according to the version with the full equation of the vertical momentum and to that using the hydrostatic approach shows the influence of the effects of the nonhydrostatic dynamics on the structure of the simulated fields even with small horizontal resolution (the step of the grid is 3.5 km). This is manifested in the strengthening of the field of the vorticity and the increasing of the sea level gradients and the velocities of the horizontal currents, whose growth reaches 1.5 cm/s.     

Generation of internal waves by barotropic tidal flow over a steep ridge

A three-dimensional nonhydrostatic numerical model is used to study the generation of internal waves by the barotropic tidal flow over a steep two-dimensional ridge in an ocean with strong upper-ocean stratification. The process is examined by varying topographic width, amplitude of the barotropic tide, and stratification at three ridge heights. The results show that a large amount of energy is converted from the barotropic tide to the baroclinic wave when the slope parameter, defined as the ratio of the maximum ridge slope to the maximum wave slope, is greater than 1. The energy flux of internal waves can be normalized by the vertical integral of the buoyancy frequency over the ridge depths and the kinetic energy of the barotropic tides in the water column. A relationship between the normalized energy flux and the slope parameter is derived. The normalized energy flux reaches a constant value independent of the slope parameter when the slope parameter is greater than 1.5. It is inferred that internal wave generation is most efficient at the presence of strong upper-ocean stratification over a steep, tall ridge. In the Luzon Strait, the strength of the shallow thermocline and the location of the Kuroshio front could affect generation of internal solitary waves in the northern South China Sea.