Numerical simulation of wave-induced laminar boundary layers

The three-dimensional numerical model with σ-coordinate transformation in the vertical direction is applied to the simulation of surface water waves and wave-induced laminar boundary layers. Unlike most of the previous investigations that solved the simplified one-dimensional boundary layer equation of motion and neglected the interaction between boundary layer and outside flow, the present model solves the full Navier–Stokes equations (NSE) in the entire domain from bottom to free surface. A non-uniform mesh system is used in the vertical direction to resolve the thin boundary layer. Linear wave, Stokes wave, cnoidal wave and solitary wave are considered. The numerical results are compared to analytical solutions and available experimental data. The numerical results agree favorably to all of the experimental data. It is found that the analytical solutions are accurate for both linear wave and Stokes wave but inadequate for cnoidal wave or solitary wave. The possible reason is that the existing analytical solutions for cnoidal and solitary waves adopt the first-order approximation for free stream velocity and thus overestimate the near bottom velocity. Besides velocity, the present model also provides accurate results for wave-induced bed shear stress.     

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.     

Implementation of baroclinic terms in a three-dimensional hydrodynamic model and its application to Anzali Port, Iran

Baroclinic terms have been implemented in a three-dimensional fully hydrodynamic model developed by Badiei et al. [2008. A three-dimensional non-hydrostatic boundary fitted model for free surface flows. International Journal for Numerical Methods in Fluids, 56(6), 607-627] modifying its momentum equations to account for density gradients and utilizing the scalar (salinity, temperature, etc.) conservation equation (SCE) and a state equation for the calculation of density. In the solution of advection-diffusion terms of the governing Navier-Stokes equations (NSE) and SCE, a symmetric splitting method was applied to ensure the long-term stability of simulations. Correction terms proposed by Ruddic et al. (1995) were applied to SCE to ensure the conservation of the scalar quantity. In the presence of baroclinic terms, the zero gradient pressure in the vertical direction in the vicinity of surface and bottom boundaries assumed by Badiei et al. [2008. A three-dimensional non-hydrostatic boundary fitted model for free surface flows. International Journal for Numerical Methods in Fluids, 56(6), 607-627] created spurious currents. This problem was solved by assuming a hydrostatic pressure variation at those boundaries. The ability of extended model was validated by comparing its results with an experimental test case. The simulation of hydrodynamic and salt intrusion at Anzali Port located at the southern coasts of Caspian Sea in Iran was carried out by the model with both barotropic and baroclinic modes. The simulated results with baroclinic mode show a better agreement with measured data as compared to the results of barotropic mode that clearly demonstrate the significance of baroclinic terms in the simulation of cyclic intrusion of salt wedge into the Port Basin.     

A hybrid spectral/finite-difference large-eddy simulator of turbulent processes in the upper ocean

A three-dimensional numerical model for large-eddy simulation (LES) of oceanic turbulent processes is described. The numerical formulation comprises a spectral discretization in the horizontal directions and a high-order compact finite-difference discretization in the vertical direction. Time-stepping is accomplished via a second-order accurate fractional-step scheme. LES subgrid-scale (SGS) closure is given by a traditional Smagorinsky eddy-viscosity parametrization for which the model coefficient is derived following similarity theory in the near-surface region. Alternatively, LES closure is given by the dynamic Smagorinsky parametrization for which the model coefficient is computed dynamically as a function of the flow. Validation studies are presented demonstrating the temporal and spatial accuracy of the formulation for laminar flows with analytical solutions. Further validation studies are described involving direct numerical simulation (DNS) and LES of turbulent channel flow and LES of decaying isotropic turbulence. Sample flow problems include surface Ekman layers and wind-driven shallow water flows both with and without Langmuir circulation (LC), generated by wave effects parameterized via the well-known Craik–Leibovich (C–L) vortex force. In the case of the surface Ekman layers, the inner layer (where viscous effects are important) is not resolved and instead is parameterized with the Smagorinsky models previously described. The validity of the dynamic Smagorinsky model (DSM) for parameterizing the surface inner layer is assessed and a modification to the surface stress boundary condition based on log-layer behavior is introduced improving the performance of the DSM. Furthermore, in Ekman layers with wave effects, the implicit LES grid filter leads to LC subgrid-scales requiring ad hoc modeling via an explicit spatial filtering of the C–L force in place of a suitable SGS parameterization.     

Modeling the pathways and ages of inflowing salt- and freshwater in the Baltic Sea

A three-dimensional, eddy-permitting ocean circulation model with implemented bottom boundary layer model and flux-corrected transport scheme is used to calculate the pathways and ages of various water masses in the Baltic Sea. The agreement between simulated and observed temperature and salinity profiles of the period 1980–2004 is satisfactory. Especially the renewal of the deep water in the Baltic proper by gravity-driven dense bottom flows is better simulated than in previous versions of the model. Based upon these model results details of the mean circulation are analyzed. For instance, it is found that after the major Baltic inflow in January 2003 saline water passing the Słupsk Furrow flows directly towards northeast along the eastern slope of the Hoburg Channel. However, after the baroclinic summer inflow in August/September 2002 the deep water flow spreads along the southwestern slope of the Gdansk Basin. Further, the model results show that the patterns of mean vertical advective fluxes across the halocline that close the large-scale vertical circulation are rather patchy. Mainly within distinct areas are particles of the saline inflow water advected vertically from the deep water into the surface layer. To analyze the time scales of the circulation mean ages of various water masses are calculated. It is found that at the sea surface of the Bornholm Basin, Gotland Basin, Bothnian Sea, and Bothnian Bay the mean ages associated to inflowing water from Kattegat amount to 26–30, 28–34, 34–38, and 38–42 years, respectively. Largest mean sea surface ages of more than 30 years associated to the freshwater of the rivers are found in the central Gotland Basin and Belt Sea. At the bottom the mean ages are largest in the western Gotland Basin and amount to more than 36 years. In the Baltic proper vertical gradients of ages associated to the freshwater inflow are smaller than in the case of inflowing saltwater from Kattegat indicating an efficient recirculation of freshwater in the Baltic Sea.     

南海等密度面环流季节变化的数值模拟研究

In this study, we develop a variable-grid global ocean general circulation model(OGCM) with a fine grid(1/6)°covering the area from 20°S–50°N and from 99°–150°E, and use the model to investigate the isopycnal surface circulation in the South China Sea(SCS). The simulated results show four layer structures in vertical: the surface and subsurface circulation of the SCS are characterized by the monsoon driven circulation, with basin-scaled cyclonic gyre in winter and anti-cyclonic gyre in summer. The intermediate layer circulation is opposite to the upper layer, showing anti-cyclonic gyre in winter but cyclonic gyre in summer. The circulation in the deep layer is much weaker in spring and summer, with the maximum velocity speed below 0.6 cm/s. In fall and winter, the SCS deep layer circulation shows strong east boundary current along the west coast of Philippine with the velocity speed at 1.5 m/s, which flows southward in fall and northward in winter. The results have also revealed a fourlayer vertical structure of water exchange through the Luzon Strait. The dynamics of the intermediate and deep circulation are attributed to the monsoon driving and the Luzon Strait transport forcing.     

A higher-efficient three-dimensional numerical model for small amplitude free surface flows

A higher-efficient three-dimensional non-hydrostatic model is developed to simulate small amplitude free surface flows based on a staggered unstructured grid. In this model, a fractional step algorithm is adopted to solve the Navier-Stokes equations in two major steps. A top-layer pressure method is proposed to minimize the number of vertical layers and subsequently the computational cost. Three classical examples of small amplitude free surface flows are used to demonstrate the capability and efficiency of the model. The satisfactory results demonstrated the capability and efficiency of modelling a range of small amplitude free surface flows with only a small number of vertical layers.     

Critical Latitude in Tidal Dynamics Using the Kara Sea as an Example

It is well known that, within the linear nonviscous equations of tidal dynamics, the amplitudes of oscillations of the barotropic and baroclinic tidal velocity components unlimitedly increase when approaching the critical latitude. It is also known that the linear equations of tidal dynamics with a constant and specified vertical eddy viscosity indicate the occurrence of significant tidal velocity shears in the near-bottom layer, which are responsible for increasing the baroclinic tidal energy dissipation, the turbulent kinetic energy, and the thickness of the bottom boundary layer. The first circumstance—the growth of the amplitudes of oscillations of the barotropic and baroclinic tidal velocity components—is due to the elimination in the original equations of small terms, which are small everywhere except for the critical latitude zone. The second circumstance—the occurrence of significant tidal velocity shears—is due to the fact that internal tidal waves, which induce the dissipation of the baroclinic tidal energy and the diapycnal diffusion, are either not taken into account or described inadequately. It is suggested that diapycnal diffusion can lead to the degeneration (complete or partial) of tidal velocity shears, with all the ensuing consequences. The aforesaid is confirmed by simulation results obtained using the QUODDY-4 high-resolution three-dimensional finite-element hydrostatic model along the 66.25° E section, which passes in the Kara Sea across the critical latitude.     

Boundary layer approach in the modeling of breaking solitary wave runup

The boundary layer is very important in the relation between wave motion and bed stress, such as sediment transport. It is a known fact that bed stress behavior is highly influenced by the boundary layer beneath the waves. Specifically, the boundary layer underneath wave runup is difficult to assess and thus, it has not yet been widely discussed, although its importance is significant. In this study, the shallow water equation (SWE) prediction of wave motion is improved by being coupled with the k–ω model, as opposed to the conventional empirical method, to approximate bed stress. Subsequently, the First Order Center Scheme and Monotonic Upstream Scheme of Conservation Laws (FORCE MUSCL), which is a finite volume shock-capturing scheme, is applied to extend the SWE range for breaking wave simulation. The proposed simultaneous coupling method (SCM) assumes the depth-averaged velocity from the SWE is equivalent to free stream velocity. In turn, free stream velocity is used to calculate a pressure gradient, which is then used by the k–ω model to approximate bed stress. Finally, this approximation is applied to the momentum equation in the SWE. Two experimental cases will be used to verify the SCM by comparing runup height, surface fluctuation, bed stress, and turbulent intensity values. The SCM shows good comparison to experimental data for all before-mentioned parameters. Further analysis shows that the wave Reynolds number increases as the wave propagates and that the turbulence behavior in the boundary layer gradually changes, such as the increase of turbulent intensity.     

Effect of surface wave breaking on the surface boundary layer of temperature in the Yellow Sea in summer

Near-surface enhancement of turbulent mixing and vertical mixing coefficient for temperature owing to the effect of surface wave breaking is investigated using a two-dimensional (2-D) ocean circulation model with a tidal boundary condition in an idealized shelf sea. On the basis of the 2-D simulation, the effect of surface wave breaking on surface boundary layer deepening in the Yellow Sea in summer is studied utilizing a 3-D ocean circulation model. A well-mixed temperature surface layer in the Yellow Sea can be successfully reconstructed when the effect of surface wave breaking is considered. The diagnostic analysis of the turbulent kinetic energy equation shows that turbulent mixing is enhanced greatly in the Yellow Sea in summer by surface wave breaking. In addition, the diagnostic analysis of momentum budget and temperature budget also show that surface wave breaking has an evident contribution to the turbulent mixing in the surface boundary layer. We therefore conclude that surface wave breaking is an important factor in determining the depth of the surface boundary layer of temperature in the Yellow Sea in summer.     

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.     

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.     

A WDM method on a generalized curvilinear grid for calculation of storm surge flooding

In this paper, the wet-dry grid point method (WDM) with generalized curvilinear computation grids is introduced with the aim of presenting a kind of effective numerical model for real-time forecasting of storm surge flooding. To realize this general WDM method, generalized 2-D shallow sea dynamic equations in curvilinear co-ordinates are derived, and the contravariant components of the velocity vector are employed for easily realizing boundary conditions and making the wet-dry point judgement criterion more reasonable. As the generalized equations are not limited to a specific co-ordinate transformation, a self-adaptive grid generation method, which optimizes simultaneously grid smoothness, orthogonality and variation in cell volumes, is then proposed to meet the needs of WDM with varying spacing grids. The paper also provides a numerically stable difference scheme and this scheme is proved accurate by the verification calculation of observed storm surge.     

A Three-dimensional Tidal Model in Boundary-fitted Curvilinear Grids

In hydrodynamic models of marine and estuarine currents the use of boundary-fitted curvilinear grids not only makes the model grids fit to the coastline and bathymetry well, but also makes the kinetic boundary conditions simple and more accurate. Because of these advantages, a three-dimensional tidal model with boundary-fitted curvilinear grids has been developed to simulate both tide and current in estuarine and shelf water. The basic idea is to use a set of coupled σ-stretched and elliptic transformations to map the physical space into a corresponding transformed space such that all boundaries are coincident with co-ordinate lines and the transformed grids are rectangular. The hydrodynamic equations in the transformed space are solved in a rectangular mesh of the transformed grid system. The application of the model in the simulation of the M₂,S₂ , K₁and O₁tidal waves in the Bohai Sea, China, shows that the numerical results are in good agreement with the observations. The method of boundary-fitted curvilinear grids is effective for improving simulation accuracy of current in the estuarine and shallow seas, especially at the coastal regions where the current was usually impractical by a uniform Cartesian grid system with the shoreline and bathymetry represented by numerous stair-steps.     

PSME model of parametric excitation of two-layer liquid in a tank

Internal waves driven by external excitation constitute important phenomena that are often encountered in environmental fluid mechanics. In this study, a pseudospectral σ-transformation model is used to simulate parametric excitation of stratified liquid in a two-layer rectangular tank. The σ-transformation maps the physical domain including the liquid free surface, the interface between the liquid layers, and the bed, onto a pair of fixed rectangular computational domains corresponding to the two layers. The governing equation and boundary conditions are discretised using Chebyshev collocation formulae. The numerical model is verified for two analytical sloshing problems: horizontal excitation of constant density liquid in a rectangular tank, and vertical excitation of stratified liquid in a rectangular tank. A detailed analysis is provided of liquid motions in a shallow water tank due to excitations in the horizontal and the vertical directions. Also, the effect of pycnocline on the wave motions and patterns is studied. It is found that wave regimes and patterns are considerably influenced by the pycnocline, especially when the excitation frequency is large. The present study demonstrates that a pseudospectral σ-transformation is capable to model non-linear sloshing waves in a two-layer rectangular tank.     

Mass Transport in a Thin Layer of Bi-Viscous Mud Under Surface Waves

The mass transport in a thin layer of non-Newtonian bed mud under surface waves is examined with a two-fluid Stokes boundary layer model. The mud is assumed to be a bi-viscous fluid, which tends to resist motion for small-applied stresses, but flows readily when the yield stress is exceeded. Asymptotic expansions suitable for shallow lluid layers are applied, and the second-order solutions for the mass transport induced by surface progressive waves are obtained numeri-cally. It is found that the stronger the non-Newtonian behavior of the mud, the more pronounced intermittency of the flow. Consequently, the mass transport velocity is diminished in magnitude, and can even become negative (i. e. , oppo-site to wave propagation) for a certain range of yield stress.     

黄海冷水团的环流结构

运用定性分析和数值模拟，对黄海冷水团的环流结构进行了探讨，结果表明黄海冷水团的垂向环流结构为双环结构：跃层以上区为中心上升，边缘下降的弱环流;跃层以下区为中心下降，边缘上升的强环流；在冷水团的中心区域，流动很弱，且无穿越温跃层的垂向流动。同时也对以往有关黄海冷水团垂向环流结构的工作进行了讨论。     

Influence of the White Sea on tides in adjacent marginal seas of the North European Basin

An investigation into the interaction of surface M ₂ tides in the system of marginal seas of the North European Basin is carried out using the three-dimensional finite-element hydrostatic model QUODDY-4. Three numerical experiments are performed for this purpose. In the first (control), the model equations are solved in the system of the Norwegian, Greenland, Barents, and White seas; thereby the interaction of the tides in these seas is explicitly taken into account. In the second experiment, the White Sea is excluded from consideration and the no-flux condition is posed at the entrance to the sea. The third experiment uses an approach in which the observed tidal elevations that determine the existence of a finite horizontal transport of barotropic energy to the White Sea are specified at the open boundary of the White Sea. It is shown that changes in tidal dynamics represented by changes in the amplitudes and phases of tidal elevations and in the barotropic tidal velocity ellipse parameters are within the model noise in experiments 2 and 3 when compared with the control experiment. On the contrary, changes in energy characteristics (the horizontal wave transport, density, and dissipation rate of barotropic tidal energy) are equal to or greater (in order of magnitude) than the energy characteristics themselves.