Ideal shocks in 2‐layer flow Part II: Under a passive layer

In this second part of the study, ideal shock theory in two‐layer stratified flow is extended to include a third passive layer (i.e., a two and a half layer system). With the presence of a passive layer, two linear wave modes and "viscous tail modes" exist, complicating the solubility conditions and uniqueness proofs for two layer shocks. It is found however, that shocks can be unambiguously classified as external or internal based on the states of criticality that they connect. The steepening condition, while still necessary, provides a less restrictive constraint than it did with a rigid lid. Thus, we have to rely more on solutions to the full viscous shock equations to establish shock existence. The detailed structure, momentum exchange, and Bernoulli loss in a viscous shock are examined using an analytical weak shock solution and a set of numerical solutions for shocks with finite amplitudes. A shock regime diagram ( F ₁ by F ₂) is constructed based on the numerical integration of the full viscous shock equations. For strong external jumps, a cusp region (i.e., in the sense of catastrophe theory) is identified on the regime diagram. For pre‐shock states within the cusp, three end states are possible and two of these are realizable. The cusp has several physical implications. It indicates that an equal distribution of dissipation between the two layers in shocks is mathematically possible but physically inaccessible. It also allows hysteresis in time varying flows, and promotes the occurrence of double shocks (i.e., closely spaced shocks of different character). The results are compared with classical shock solutions and a set of time dependent numerical experiments.     

Bottom friction and its effects on periodic long wave propagation

A new set of Boussinesq-type equations describing the free surface evolution and the corresponding depth-integrated horizontal velocity is derived with the bottom boundary layer effects included. Inside the boundary layer the eddy viscosity gradient model is employed to characterize Reynolds stresses and the eddy viscosity is further approximated as a linear function of the distance measured from the seafloor. Boundary-layer velocities are coupled with the irrotational velocity in the core region through boundary conditions. The leading order boundary layer effects on wave propagation appear in the depth-integrated continuity equation to account for the velocity deficit inside the boundary layer. This formulation is different from the conventional approach in which a bottom stress term is inserted in the momentum equation. An iterative scheme is developed to solve the new model equations for the free surface elevation, depth-integrated velocity, the bottom stress, the boundary layer thickness and the magnitude of the turbulent eddy viscosity. A numerical example for the evolution of periodic waves propagating in one-dimensional channel is discussed to illustrate the numerical procedure and physics involved. The differences between the conventional approach and the present formulation are discussed in terms of the bottom frictional stress and the free surface profiles.     

Shock-capturing Boussinesq-type model for nearshore wave processes

This paper describes the formulation and validation of a nearshore wave model for tropical coastal environment. The governing Boussinesq-type equations include the conservative form of the nonlinear shallow-water equations for shock capturing. A Riemann solver supplies the inter-cell flux and bathymetry source term, while a Godunov-type scheme integrates the evolution variables in time. The model handles wave breaking through momentum conservation with energy dissipation based on an eddy viscosity concept. The computed results show very good agreement with laboratory data for wave propagation over a submerged bar, wave breaking and runup on plane beaches as well as wave transformation over fringing reefs. The model accurately describes transition between supercritical and subcritical flows as well as development of dispersive waves in the processes.     

Some features of the dynamic structure of a deep estuary

A boundary layer formulation for the dynamic structure of a deep estuary is developed. Cross-stream averages are used, but the boundary layer structure is shown to depend on the cross-stream geostrophic constraint. A similarity transformation and a weighted residual method are used to derive an approximate solution for the velocity and salinity structure of the upper layer. This solution indicates that, in the central regime of the estuary, outflow extends through the entire halocline. Inflow takes place in a much less stratified lower layer, and mass exchange between the layers is by upwelling. This structure is modified in the outer regime of the estuary, where mixing between the layers develops, and in the inner regime, where a sharp halocline develops and where the dynamics are dominated by river runoff. The implications of the dynamics for the flushing process and for pollutant movement and dispersion are discussed.     

A time‐split, forward‐backward numerical model for solving a nonhydrostatic and compressible system of equations

In this paper, we present a numerical procedure for solving a 2‐dimensional, compressible, and nonhydrostatic system of equations. A forward‐backward integration scheme is applied to treat high‐frequency and internal gravity waves explicitly. The numerical procedure is shown to be neutral in time as long as a Courant–Friedrichs–Lewy criterion is met. Compared to the leap‐frog‐scheme most models use, this method involves only two time steps, which requires less memory and is also free from unstable computational modes. Hence, a time‐filter is not needed. Advection and diffusion terms are calculated with a time step longer than sound‐wave related terms, so that extensive computer time can be saved. In addition, a new numerical procedure for the free‐slip bottom boundary condition is developed to avoid using inaccurate one‐sided finite difference of pressure in the surface horizontal momentum equation when the terrain effect is considered. We have demonstrated the accuracy and stability of this new model in both linear and nonlinear situations. In linear mountain wave simulations, the model results match the corresponding analytical solution very closely for all three cases presented in this paper. The analytical streamlines for uniform flow over a narrow mountain range were obtained through numerical integration of Queney's mathematical solution. It was found Queney's original diagram is not very accurate. The diagram had to be redrawn before it was used to verify our model results. For nonlinear tests, we simulated the famous 1972 Boulder windstorm and a bubble convection in an isentropic enviroment. Although there are no analytical solutions for the two nonlinear tests, the model results are shown to be very robust in terms of spatial resolution, lateral boundary conditions, and the use of the time-split scheme.     

Three-parameter model of turbulence for the atmospheric boundary layer over an urbanized surface

A modified three-parameter model of turbulence for a thermally stratified atmospheric boundary layer (ABL) is presented. The model is based on tensor-invariant parametrizations for the pressure-strain and pressure-temperature correlations that are more complete than the parametrizations used in the Mellor-Yamada model of level 3.0. The turbulent momentum and heat fluxes are calculated with explicit algebraic models obtained with the aid of symbol algebra from the transport equations for momentum and heat fluxes in the approximation of weakly equilibrium turbulence. The turbulent transport of heat and momentum fluxes is assumed to be negligibly small in this approximation. The three-parameter E ? ε ? 2> model of thermally stratified turbulence is employed to obtain closed-form algebraic expressions for the fluxes. A computational test of a 24-h ABL evolution is implemented for an idealized two-dimensional region. Comparison of the computed results with the available observational data and other numerical models shows that the proposed model is able to reproduce both the most important structural features of the turbulence in an urban canopy layer near the urbanized ABL surface and the effect of urban roughness on a global structure of the fields of wind and temperature over a city. The results of the computational test for the new model indicate that the motion of air in the urban canopy layer is strongly influenced by mechanical factors (buildings) and thermal stratification.     

A comparison of two formulations of barotropic–baroclinic splitting for layered models of ocean circulation

In numerical models of ocean circulation, it is widespread practice to split the fast and slow motions into barotropic and baroclinic subsystems, respectively. In the case of the baroclinic equations, the dependent variables can either be (1) slowly-varying baroclinic quantities, obtained from splitting the original flow variables into barotropic and baroclinic components, or (2) the original unsplit variables, which can vary on both the fast and slow time scales. In the second case, the variables in each layer are adjusted after each (long) baroclinic time step to ensure compatibility with the results produced from the barotropic equations. The second approach can be applied to the layer thickness equation to ensure exact conservation of mass within each layer. In the case of the momentum equations, the second approach amounts to replacing unresolved fast portions of Coriolis and pressure forcing with time averages of well-resolved forcing from the barotropic system. In this study, both approaches for the momentum equations are evaluated, in several test problems, by comparing to analytical solutions or to solutions computed with an unsplit code that uses short time steps. The two methods give very similar results in some simple problems for which analytical solutions are known. However, in some eddying double-gyre simulations, the formulation with unsplit variables requires a significant reduction in the baroclinic time step in order to avoid numerical difficulties that include grid noise and inaccurate representation of the flow field. In contrast, the formulation with split variables does not display such difficulties, and in those same examples it can be used with zero explicit horizontal viscosity. All of these computations employ a two-level time-stepping method that was previously developed by the author.     

Rotational instability of a horizontal shear flow in a stratified rotating fluid

The rotational instability of a thermally stratified, viscous, conducting, rotating fluid is investigated by means of linearized perturbation equations. It is assumed that the basic horizontal flow is vertically uniform and that the horizontal shear is confined in a thin layer. By solving a simplified boundary value problem as a model of rotational instability in the sea, we have shown that the vertical wave length of the neutral disturbance is of order 10 times as large as the laminar Ekman layer thickness, and that this scale is proportional to (L/N)^1/3, whereL is the width of the shear layer andN is the Brunt-Vaisala frequency.     

A Three-Dimensional Prediction Method for Thermal Diffusion

A three-dimensional,first order turbulence closure,thermal diffusion model is described inthis paper.The governing equations consist of an equation of continuity,three components of momentum,conservation equations for salt,temperature and subgridscale energy,and an equation of state.In the mod-el,according to the hypothesis of Kolmogorov and Prandtl,the viscosity coefficient of turbulent flow ofhomogeneous fluid is related to the local turbulent energy,and the horizontal and vertical exchangecoefficients of mass,heat and momentum are computed with the introduction of subgridscale turbulenceenergy.The governing equations are solved by finite difference techniques.This model is applied to theJiaozhou bay to predict thermal pollution by the Huangdao power plant.An instantaneous tidal currentfield is computed,then the distribution of temperature increment is predicted,and finally the effect of windstress on thermal discharge is discussed.     

An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator

A finite-volume formulation is presented that solves the three-dimensional, nonhydrostatic Navier–Stokes equations with the Boussinesq approximation on an unstructured, staggered, z-level grid, with the goal of simulating nonhydrostatic processes in the coastal ocean with grid resolutions of tens of meters. In particular, the code has been developed to simulate the nonlinear, nonhydrostatic internal wave field in the littoral ocean. The method is based on the formulation developed by Casulli, in that the free-surface and 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 with the second-order Adams–Bashforth method, while the pressure-correction method is employed for the nonhydrostatic pressure in order to achieve overall second-order temporal accuracy. Advection of momentum is accomplished with an Eulerian discretization which conserves momentum in cells that do not contain the free surface, and scalar advection is discretized in a way that ensures consistency with continuity, thereby ensuring local and global mass conservation using a velocity field that conserves volume on a local and global basis. The nonhydrostatic pressure field is solved efficiently using a block-Jacobi preconditioner, and while stability is limited by the internal gravity wave speed and vertical advection of momentum, applications requiring relatively small time steps due to accuracy or stability constraints are run efficiently on parallel computers, since the present formulation is written entirely with the message-passing interface (MPI). The ParMETIS libraries are employed in order to achieve a load-balanced parallel partitioning that minimizes interprocessor communication, and the grid is reordered to optimize per-processor performance by limiting cache misses while accessing arrays in memory. Test cases demonstrate the ability of the code to efficiently and accurately compute the nonhydrostatic lock exchange and internal waves in idealized as well as real domains, and we evaluate the parallel efficiency of the code using up to 32 processors.     

Formulation of a continuously stratified sea model with three-dimensional representation of the upper layer

A two-layered model is considered in which the upper layer is continuously stratified and the lower layer is homogeneous. The system is driven by atmospheric forces. Bottom stress and topography are included in the model. The linear three-dimensional hydrodynamical equations are used to describe the system. Taking the eddy viscosity in the upper layer as inversely proportional to the static stability, the dependent variables are expanded in terms of continuous functions in the vertical (eigenfunctions). Using this method it is possible to compute currents and internal displacements at any depth in the upper layer. The three-dimensional structure of the lower layer is not considered in this model. The equations describing the lower layer are integrated over depth to give depth mean currents. Using a staggered finite-difference grid in the horizontal and a forward time-stepping procedure, numerical test experiments are carried out for a cross section and for a closed rectangular basin.     

Estimating the characteristics of the vertical turbulent viscosity in the upper 200-m layer of the Black Sea

The influence of high vertical velocity gradients in the Black Sea Rim Current on the intensity of the vertical turbulent exchange is demonstrated on the basis of numerical modeling based on CTD data. The vertical turbulent exchange is confirmed by the anomalous distribution of the hydrochemical parameters in the redox layer. A system of equations for the kinetic energy of the turbulence and dissipation rate (k-? model) is used for the calculation of the coefficient of the vertical turbulent viscosity (diffusivity).     

Shocks in a coastal boundary current

We studied shocks in a coastal boundary current with zero potential vorticity. By coastal boundary current, we mean a semigeostrophic light fluid flow over an infinitely deep dense fluid and along a coast on its right hand side, with its lower interface exposed to the ocean surface at some finite distance from the coast. The shocks are assumed to conserve mass and momentum. It is found that the shocks can be classified into two categories, coastal shocks and frontal shocks, by the signs of the upper layer flux relative to the shocks. Coastal shocks, for which the relative upper layer flux is negative, always propagate downstream. The upper layer at the coast is thicker on the upstream sides of coastal shocks than on the downstream sides. Frontal shocks, for which the relative upper layer flux is positive, propagate upstream as well as downstream. In most cases, the current is wider on the downstream sides of frontal shocks than on the upstream sides. However, under the circumstances that the current is nearly separated from the coast, the current is wider on the upstream sides of frontal shocks. Coastal and frontal shocks both dissipate energy of the current. We also demonstrate that special shocks with no light fluid on the downstream sides cannot exist irrespective of the potential vorticity distribution.     

Infrared atmospheric sounding interferometer data information content; instrument characterization and the impact of a priori information

The use of linear estimation for the study of the information content of a given satellite radiance data set for temperature and humidity profile retrievals is first reviewed. A particular formulation of the retrieval approach is then used to obtain an intrinsic characterisation of the Infrared Atmospheric Sounding Interferometer (IASI) data set, in terms of accuracy versus vertical resolution of retrieved profiles. The performance of the IASI instrument alone is analysed and compared to that of the currently‐used HIRS‐TOVS. The problem is then regularized by addition of a priori independent information to the initial data set. The potential use of IASI data for some particular choices of the a priori information associated with practical problems such as profile inversion or data assimilation for weather forecasting is analysed. The approach is finally used to derive an "empirical" objective framework to define the vertical discretization adapted to these problems.     

AN ESTIMATION OF TURBULENCE STRESSES IN TIDAL CURRENTS OF HANGZHOU BAY

Based on the momentum equations, the turbulence stresses and eddy viscosities along five sections in Hangzhou Bay are estimated by using the observed data of tidal currents and tides. The coefficient of bottom stresses obtained from the calculation is 0.67 × 10-3 on the average and the vertical profiles of the amplitudes of turbulence stresses are almost linear and slight concave downwards, and the phases are deferred continuously from sea-bed. The coefficient of vertical eddy viscosity reaches its maximum at the layer below the mid-depth with a value of about 60 cm2S-I on the average.     

A proposed adiabatic formulation of 3‐dimensional global atmospheric models based on potential vorticity

A 2‐time‐level finite difference atmospheric general circulation model based on the semi‐Lagrangian advection of pseudo potential vorticity (which becomes potential vorticity in that part of the domain where the hybrid vertical coordinate becomes isentropic) has been formulated. At low levels, the hybrid vertical coordinate is terrain following. The problem of isentropic potential vorticity possibly becoming ill‐defined in the regions of planetary boundary layer is thus circumvented. The divergence equation is a companion to the (pseudo) potential vorticity equation and the model is thus called a PV‐D model. Many features of a previously developed shallow water PV‐D model are carried over: a modification of the PV equation needed to give computational stability of long Rossby waves; a semi‐Lagrangian semi‐implicit treatment of both the linear and the nonlinear terms; the use of an unstaggered grid in the horizontal; the use of a nonlinear multigrid technique to solve the nonlinear implicit equations. A linear numerical stability analysis of the model's gravity–inertia waves indicates that the potential temperature needs to be separated into horizontal mean and perturbation parts. This allows an implicit treatment of the vertical advection associated with the mean in the thermodynamic equation. Numerical experiments with developing baroclinic waves have been carried out and give realistic results.     

Modeling wave�Ccurrent bottom boundary layers beneath shoaling and breaking waves

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-ε model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.     

Computation of Viscous Uniform and Shear Flow over A Circular Cylinder by A Finite Element Method

The incompressible viscous uniform and shear flow past a circular cylinder is studied. The two-dimensional Navier-Stokes equations are solved by a finite element method. The governing equations are discretized by a weighted residual method in space. The stable three-step scheme is applied to the momentum equations in the time integration. The numerical model is firstly applied to the computation of the lid-driven cavity flow for its validation. The computed results agree well with the measured data and other numerical results. Then, it is used to simulate the viscous uniform and shear flow over a circular cylinder for Reynolds numbers from lO0 to lO00. The transient time interval before the vortex shedding occurs is shortened considerably by introduction of artificial perturbation. The computed Strouhal number, drag and lift coefficients agree well with the experimental data. The computation shows that the finite element model can be successfully applied to the viscous flow problem.     

Viscosity and nonlinearity effects on the forces and waves generated by a floating twin hull under heave oscillation

Nonlinear hydrodynamics of a twin rectangular hull under heave oscillation is analyzed using numerical methods. Two-dimensional nonlinear time-domain solutions to both inviscid and viscous problems are obtained and the results are compared with linear, inviscid frequency-domain results obtained in [26] to quantify nonlinear and viscous effects. Finite-difference methods based on boundary-fitted coordinates are used for solving the governing equations in the time domain [2]. A primitive-variables based projection method [6] is used for the viscous analysis and a mixed Eulerian–Lagrangian formulation [11] for inviscid analysis. The algorithms are validated and the order of accuracy determined by comparing the results obtained from the present algorithm with the experimental results of Vugt [22] for a heaving rectangle in the free surface. The present study on the twin-hull hydrodynamics shows that at large and non-resonant regular frequencies, and small amplitude of body oscillation, the fluid viscosity does not significantly affect the wave motion and the radiation forces. At low frequencies however the viscosity effect is found to be significant even for small amplitude of body oscillation. In particular, the hydrodynamic force obtained from the nonlinear viscous analysis is found to be closer to the linear inviscid force than the nonlinear inviscid force to the linear inviscid force, the reason for which is attributed to the wave dampening effect of viscosity. Since the wave lengths generated at smaller frequencies of oscillation are longer and therefore the waves could have a more significant effect on the dynamic pressure on the bottom of the hulls which contribute to the heave force, the correlation between the heave force and the wave elevation is found to be larger at smaller frequencies. Because of nonlinearity, the wave radiation and wave damping force remained nonzero even at and around the resonant frequencies – with the resonant frequencies as determined in [26] using linear potential flow theory. As to be expected, the nonlinear effect on the wave force is found to be significant at all frequencies for large amplitude of oscillation compared to the hull draft. The effect of viscosity on the force, by flow separation, is also found to be significant for large amplitude of body oscillation.     

A model for the propagation of nonlinear surface waves over viscous muds

The effect of a thin viscous fluid–mud layer on nearshore nonlinear wave–wave interactions is studied using a parabolic frequency-domain nonlinear wave model, modified to incorporate a bottom dissipation mechanism based on a viscous boundary layer approach. The boundary-layer formulation allows for explicit calculation of the mud-induced wave damping rate. The model performed well in tests based on laboratory data. Numerical tests show that damping of high frequency waves occurs, mediated by “difference” nonlinear interactions. Simulations of 2-dimensional wave propagation over a mud “patch” of finite extent show that the wave dissipation causes significant downwave diffraction effects.