Rectangular Basin Research Articles

Baroclinic Modes in a Two-Layer Basin

The objective of this study is to investigate the time-dependent circulation in a closed basin where the steady circulation is included and long Rossby wave speeds are consistent with observations. Specifically, the large-scale baroclinic eigenmodes of a two-layer rectangular basin forced by surface wind stress in the limit of small dissipation are examined. Low-frequency modes with small decay rates independent of friction result when the constraint of mass conservation is enforced. The magnitude of the wind stress is found to be critical to the eigenspectrum. For all forcing magnitudes, including forcings with closed geostrophic contours, oscillatory modes with decay rates independent of friction emerge. For forcings with closed geostrophic contours, two important classes of eigenmodes with comparable decay rates emerge: purely decaying modes confined to the region of closed contours, and basin-scale oscillatory modes. The purely decaying modes also exist without the constraint of total mass conservation but their decay rates depend on the magnitude of friction to leading order.

Influence of a step-like coastline on the basin scale vorticity budget of mid-latitude gyre models

Global vorticity budgets in C-grid shallow water (SW) and quasi-geostrophic (QG) models of winddriven ocean circulation with free-slip boundary conditions are considered. For both models, it is pointed out that the discretized vorticity equation is defined only over a subdomain that excludes boundary grid nodes. At finite resolution, this implies an advective flux of vorticity across the perimeter of the discretized vorticity domain. For rectangular basins where grid axes are aligned with the basin walls, this flux tends to zero as resolution is increased. We also consider the case in which the grid is rotated with respect to the basin, so that a step-like coastline results. Increased resolution then leads to more steps and, because the advective flux of vorticity out of the domain is particularly large at steps, it is no longer obvious that increased resolution should reduce the advective flux. Results are found to be sensitive to numerical details. In particular, we consider different formulations for the non-linear terms (for both the SW and QG models) and two formulations of the viscous stress tensor for the SW model [the conventional five-point Laplacian and the δ—ζ stress tensor suggested by Madec et al. (J. Phys. Oceanogr. 21, 1349—1371)]. For the SW model, the overall circulation and the behavior of the flux term are dependent on both the formulation of the viscous stress tensor and the non-linear terms. The best combination is found to be the δ—ζ tensor with an enstrophy-preserving advection scheme. With this combination, the circulation of the non-rotated basin is recovered in rotated basins and the advective flux tends to converge towards zero with increasing resolution. The poorest combination is the δ—ζ tensor with the conventional advective scheme. In this case, the advective flux term diverges with increasing resolution for some rotation angles and the model crashes for some others. For the QG model, the convergence order of the advective flux term of absolute vorticity is near unity (roughly the same as with the SW model). Most of the error (especially at high resolution) is related to errors in the β term (which is hidden in the advective contribution in the SW model). However, the overall circulation is less sensitive to the rotation of the grid with respect to the basin, especially when the Jacobian proposed by Arakawa (J. Comput. Phys. 1, 119—143) is used.

Current Structure and Behavior of the River Plume in Suo-Nada

The behavior of a river plume in Suo-Nada, Japan, has been studied using a primitive equation numerical model, the Princeton Ocean Model. Special attention has been paid to the current structure and behavior of the anticyclonic eddy (bulge) induced by high freshwater inflow changing on a timescale of one week. First, the freshwater is supplied from a river to a rectangular basin with a simple topography. When the river discharge subsides after reaching its peak value, the bulge propagates upstream (i.e., opposite to the direction of the Kelvin wave propagation). Next, the freshwater is supplied from eight major rivers to the basin with realistic topography. The less saline water mass in the southern part of Suo-Nada propagates to the west (i.e., upstream) after the river discharge subsides. This is consistent with an observed phenomenon, viz., that the less saline water mass appears in the western part of Suo-Nada, suggesting that the upstream propagation of the bulge is possible in the real ocean. Finally, the cause of the upstream propagation is considered. Onshore currents appear in the bottom layer beneath the bulge, propagating upstream. They produce an anticyclonic barotropic eddy due to the conservation of potential vorticity. The current component associated with the eddy crosses normally to the isohaline in the upper layer, and therefore transports the bulge upstream. No other current component (such as surface current velocity minus vertically-averaged value) is responsible for the upstream propagation of the bulge.

A study of the effects of grid non-orthogonality on the solution of shallow water equations in boundary-fitted coordinate systems

In the present study, an existing two-dimensional boundary-fitted model [J. Hydraul. Eng.-ASCE 122 (9) (1996) 512] is used to study the effect of grid non-orthogonality on the solution of shallow water equations using boundary-fitted grids. The linearized two-dimensional shallow water equations are expressed in terms of the grid angle and aspect ratio. The truncation errors of the finite difference approximations used in the solution of the governing equations are shown to be dependent on the grid angle and the aspect ratio. The coefficient of the truncation error was shown to increase, with the decrease in the grid angle. The RMS errors in model predicted surface elevations and velocities for the case of seiching in a rectangular basin are found to increase gradually, as the grid resolution decreases from 174 to 80 gridpoints per wavelength or as the grid angle decreases from 90° to 50° and increases rather sharply for a grid angle of 30° at grid resolutions less than 80 gridpoints per wavelength. The model predicted surface elevations for the case of tidal forcing in a rectangular basin are found to be insensitive to the grid angle at grid resolutions higher than 600 gridpoints per wavelength. The RMS error in the model predicted velocities is found to increase gradually as the grid angle decreases from 90° to 30° or as the grid resolution decreases from 1400 gridpoints per wavelength to 400 gridpoints per wavelength and increases sharply as the grid resolution decreases from 400 to 150 gridpoints per wavelength. Two-dimensional depth averaged hydrodynamic modeling of tidal circulation in Narragansett Bay, using three different boundary-fitted grids showed that the model predicted surface elevations are insensitive to the grid angle at grid resolutions as low as 200 gridpoints per wavelength. However, the model predicted velocities were found to increase as the grid resolution decreases from 600 to 200 gridpoints per wavelength. We conclude from this study that grid angle and grid resolution affects the accuracy of the model predicted currents and the numerical dispersion increases with the decrease in grid angle or grid resolution and these are in agreement with that reached by Sankaranarayanan and Spaulding [Dispersion and Stability Analyses of Shallow Water Equations in Boundary-fitted Coordinates, Department of Ocean Engineering, University of Rhode Island, 2001, p. 33] through a Fourier analysis of the discretized equations in boundary-fitted coordinates.

Three-dimensional hydrodynamic model, study cases for quarter annular and idealized ship channel problems

An extended version of the three-dimensional hydrodynamic model, ADCIRC 3D-DSS, was utilized to simulate both horizontal and vertical flows in a (quarter) annular harbor (QATP and ATP) and rectangular basin with an idealized ship channel (RBSC). Comparison of horizontal and vertical solutions to the analytical solution and results of other researchers are in good agreement. The vertical velocity solution is highly sensitive to the horizontal velocity solutions. The presence of the sidewall boundary may also affect the vertical solutions. Around the sloping bank of RBSC channel with one-third gradient, the vertical velocity becomes important. The maximum vertical velocity approaches ±50% of the sediment fall velocity of fine sand.

Baroclinic Rossby Waves in Irregular Basins*

The properties of baroclinic, quasigeostrophic Rossby basin waves are examined. Full analytical solutions are derived to elucidate the response in irregular basins, specifically in a (horizontally) tilted rectangular basin and in a circular one. When the basin is much larger than the (internal) deformation radius, the basin mode properties depend profoundly on whether one allows the streamfunction to oscillate at the boundary or not, as has been shown previously. With boundary oscillations, modes occur that have low frequencies and, with scale-selective dissipation, decay at a rate less than or equal to that of the imposed dissipation. These modes approximately satisfy the long-wave equation in the interior. Using both unforced and forced solutions, the variation of the response with basin geometry and dissipation is documented. The long-wave modes obtain with scale-selective dissipation, but also with damping that acts equally at all scales. One finds evidence of them as well in the forced response, even when the dissipation is weak and the corresponding free modes are apparently absent.

Long Rossby Wave Basin-Crossing Time and the Resonance of Low-Frequency Basin Modes

The ability of long-wave low-frequency basin modes to be resonantly excited depends on the efficiency with which energy fluxed onto the western boundary can be transmitted back to the eastern boundary. This efficiency is greatly reduced for basins in which the long Rossby wave basin-crossing time is latitude dependent. In the singular case where the basin-crossing time is independent of latitude, the amplitude of resonantly excited long-wave basin modes grows without bound except for the effects of friction. The speed of long Rossby waves is independent of latitude for quasigeostrophic dynamics, and the rectangular basin geometry often used for theoretical studies of the wind-driven ocean circulation is such a singular case for quasigeostrophic dynamics. For more realistic basin geometries, where only a fraction of the energy incident on the western boundary can be transmitted back to the eastern boundary, the modes have a finite decay rate that in the limit of weak friction is independent of the choice of frictional parameters. Explicit eigenmode computations for a basin geometry similar to the North Pacific but closed along the equator yield basin modes sufficiently weakly damped that they could be resonantly excited.

Long Rossby Wave Basin-Crossing Time and the Resonance of Low-Frequency Basin Modes

Abstract The ability of long-wave low-frequency basin modes to be resonantly excited depends on the efficiency with which energy fluxed onto the western boundary can be transmitted back to the eastern boundary. This efficiency is greatly reduced for basins in which the long Rossby wave basin-crossing time is latitude dependent. In the singular case where the basin-crossing time is independent of latitude, the amplitude of resonantly excited long-wave basin modes grows without bound except for the effects of friction. The speed of long Rossby waves is independent of latitude for quasigeostrophic dynamics, and the rectangular basin geometry often used for theoretical studies of the wind-driven ocean circulation is such a singular case for quasigeostrophic dynamics. For more realistic basin geometries, where only a fraction of the energy incident on the western boundary can be transmitted back to the eastern boundary, the modes have a finite decay rate that in the limit of weak friction is independent of the choice of frictional parameters. Explicit eigenmode computations for a basin geometry similar to the North Pacific but closed along the equator yield basin modes sufficiently weakly damped that they could be resonantly excited.

Intermediate modelling of tidal inlet systems: spatial asymmetries in flow and mean sediment transport

In order to identify mechanisms which determine the morphology of tidal inlet systems, experiments with a shallow water model and sediment transport model of intermediate complexity are carried out and interpreted. A highly schematised geometry is used, consisting of a rectangular outer area connected to a rectangular basin by a narrow strait. The bottom topography is either flat or constantly sloping in the landward direction and the bed is composed of fine sand. Forcing at the open boundary consists of prescribed water levels, which can represent a (shore-parallel) progressive tidal wave or a fully standing wave. The symmetry breaking effects of Coriolis force and a progressive Kelvin wave on the tidal motion, the mean flow field and the mean sediment transport field are investigated. Tidal ellipticity properties on the outer delta are also discussed. Experiments for a rectangular basin without outer area are carried out in order to compare the intermediate model results with those of a 1D idealised model. The overall agreement is satisfactory. Next the influence of earth rotation and of the progressive wave in the outer sea on mean sediment transport in the basin is investigated. It turns out that both effects, which cannot be included in the idealised model, lead to different patterns of erosion and deposition. In both sets of experiments the effects of a progressive Kelvin wave were found to be dominant over earth rotation effects in the flow and mean sediment transport fields.

Review of History of hydraulic research in India by G. S. Dhillon

The book attempts to cover only those studies in which a practicing engineer is involved in problem formulation and in the evaluation and application of the results. Such studies mainly have been conducted by a handful of government organizations. One such organization is the Irrigation and Power Research Institute, Amritsar. The author had a long association with this institute; consequently, he emphasized studies conducted in this institute. Most studies are related to the development of irrigation and hydropower. The title of the book should be ‘‘Development of Irrigation Structures in India.’’ The book contains 20 chapters. The first chapter, entitled ‘‘Growth of Hydraulic Research in India,’’ summarizes the chronology of the establishment of hydraulic research organizations during the British rule and after independence in 1947. The remaining chapters deal with the various aspects of river behavior and control, including research areas ~e.g., ice modeling, tidal power, and lock structures! in which no significant contribution was made by Indian research organizations. The history of hydraulics in India dates back 3000 B.C. and it is not covered in the book. In 1998 the reviewer led a group of graduate students from The Univ. of Iowa to India and visited some of the ancient sites, e.g.: ~1! the remains of an ancient city at Lothal, Gujarat, built about 4,500 years ago, which includes an intricate system of surface and underground drains and a large rectangular basin probably for docking trade ships; ~2! the remains of one of the finest of the Gujarati step wells, known as Adalaj Vav, built some 500 years ago; and ~3! a number of crossdrainage works on the Ganga Canal that are over 150 years old. There still is a scope for another book on the history of hydraulics that also covers the ancient period and the basic hydraulic research.

Green functions for initial free-surface flows due to 3D impulsive bottom deflections

The immediate impulsive flow of an incompressible fluid due to a concentrated flux through an otherwise impermeable boundary is investigated analytically in three dimensions. The flow is inviscid and irrotational, and obeys the equipotential condition at the free surface, which is initially horizontal. Various elementary bottom geometries are analyzed: rectangular basins, sloping beaches, semi-cylindrical and hemispherical basins. Special attention is paid to the case of impulsive free-surface flows generated on a uniform sloping beach. A general integral solution is presented and compared against a series solution found for a discrete set of angles. The results are relevant for the modeling of tsunami generation due to rapid bottom deflections.

Scaling Up of Plugflow-Dissolved Air Flotation

In a conventional rectangular flotation basin, acceptable separation efficiencies in excess of 90% can rarely be achieved with an operational surface loading in excess of 5 m/h, because of the great degree of dispersion, turbulence, and dead space present in the basin. A pilot-scale coaxial cylinder-type flotation column was built to fulfill the required plugflow performance, to reduce dead space, and to optimize the size. This coaxial plugflow flotation column was used in a pilot-scale flotation unit capable of handling 10 m3/h of wastewater, operating at a maximum surface loading of 12 m/h. Near-plugflow was established using baffles in the different sections. Flotation efficiencies in excess of 90% were obtained for industrial wastewater at high surface loadings (up to 12 m/h) and a low recycle ratio (9%). Efficiencies obtained in a traditional unit are significantly lower. Experience with the coaxial unit has enabled us to formulate scale-up guidelines for the plugflow dissolved air flotation, and the use of a modular design is required. The process can be described by a model using two parameters, i.e., the bubble-floc attachment efficiency, αpb, and the air-to-solid volume ratio in the agglomerate, β. Fitting experimental data and model predictions defines both model parameters, which are in line with the literature data and with values obtained in a laboratory-scale unit, thus justifying the possible use of small-scale experiments for scale-up purposes.

Barotropic response in a lake to wind-forcing

Abstract. We report results gained with a three-dimensional, semi-implicit, semi-spectral model of the shallow water equations on the rotating Earth that allowed one to compute the wind-induced motion in lakes. The barotropic response to unidirectional, uniform winds, Heaviside in time, is determined in a rectangular basin with constant depth, and in Lake Constance, for different values and vertical distributions of the vertical eddy viscosities. It is computationally demonstrated that both the transitory oscillating, as well as the steady state current distribution, depends strongly upon the absolute value and vertical shape of the vertical eddy viscosity. In particular, the excitation and attenuation in time of the inertial waves, the structure of the Ekman spiral, the thickness of the Ekman layer, and the exact distribution and magnitude of the upwelling and downwelling zones are all significantly affected by the eddy viscosities. Observations indicate that the eddy viscosities must be sufficiently small so that the oscillatory behaviour can be adequately modelled. Comparison of the measured current-time series at depth in one position of Lake Constance with those computed on the basis of the measured wind demonstrates fair agreement, including the rotation-induced inertial oscillation.Key words. Oceanography: general (limnology) – Oceanography: physical (Coriolis effects; general circulation)

Satellite altimetry observations of the Black Sea level

The paper is devoted to the application of satellite altimetry from the TOPEX/Poseidon and ERS‐1 missions to the Black Sea. We use the NASA Ocean Altimeter Pathfinder Project collinear data set and monthly climatic hydrography to restore the dynamical sea level, that which connected with the Black Sea circulation. Two realizations are created from collinear and grid data sets. The Black Sea hydrographic survey data collected by the Cooperative Marine Science Programme for the Black Sea are used for validation of both products. The estimated rms accuracy of the Black Sea dynamical level is about 3 cm. The grid data product is used for the analysis of seasonal and mesoscale variability of the Black Sea level. The observations show that the Black Sea circulation has strong seasonal variation. It attenuates in summer to autumn and intensifies in winter to spring. This variability is accompanied by a western‐phase propagation of the sea level. A simple two‐layer model of the wind‐induced circulation in the rectangular basin provides an interpretation of these results. It is shown that the seasonal variability of the geostrophic circulation is most likely produced by the annual changes of the wind stress curl. The western phase propagation is treated as the radiation of Rossby waves from the eastern coast of the basin. Typical periods of the mesoscale oscillations of the dynamical sea level are estimated from the TOPEX/Poseidon altimeter time series in different parts of the Black Sea by means of a spectral analysis. The strongest mesoscale oscillations have periods of about 120 days and are located in the southeastern part of the basin, where the Rim Current bifurcates, and off the Crimean peninsula, where the Rim Current meanders as a result of the local orography.

Multiple thermohaline states due to variable diffusivity in a hierarchy of simple models

The effect of variable vertical diffusivity is investigated in dynamically reduced models of the thermohaline circulation (THC) in a rectangular basin. In a simple box model, sufficiently strong variation of the diffusivity κ v with stability G can lead to the existence of two stable equilibria. Related behaviour is found in well-resolved frictional geostrophic (FG) models. A hierarchy of under-resolved FG models is constructed, the simplest of which is an 8-cell cube, to connect the two extremes of resolution. Multiple solutions in low-order models are found to correspond to the formation of high-gradient layers which are unlikely to be resolved by current ocean models. Physical arguments show that layering and multiple solutions require κ v to decrease more rapidly than 1/ G and sensitivity experiments suggest that, in addition, κ v must vary by a factor of 10–100. In two-hemisphere runs with salinity forcing included, the dependence of diffusivity on stratification is found to marginally favour equatorially symmetric states. Finally, such variation is shown to have a profound effect on the periodic, flush-collapse cycle under strong saline forcing; specifically, if diffusivity is taken to be a function of stratification rather than depth, regime transitions can occur much more easily. It will therefore be important for climate modelling to determine which is more realistic.

The effect of advective processes on the morphodynamic stability of short tidal embayments

Abstract An idealised model is used to gain understanding about the formation of channels and shoals in a rectangular basin with fixed sides and a sandy bottom. Forcing is due to a prescribed tidal constituent at the entrance. The basin length is assumed to be small compared to the tidal wavelength. The model consists of the 2D shallow water equations, a suspended load sediment transport formulation and a tidally averaged bottom evolution equation. Channels and shoals can develop as free instabilities in this system. Here the role of nonlinear tides and advective sediment fluxes in the formation process is studied. It is found that advective processes can have both a stabilising and destabilising effect. The tidal residual circulation plays a dominant role in the growth process. Moreover, the destabilising effect of the diffusive sediment transport is reduced by advection. With increasing influence of advective (with respect to diffusive) fluxes the preferred location of the bottom perturbations shifts towards the basin entrance. The model results are consistent with field data and the outcome of complex morphodynamic models.

Three-dimensional wind-induced baroclinic circulation in rectangular basins

We present results of various circulation scenarios for the wind-induced currents in two vertically stratified rectangular basins of constant depth with different sizes; these are obtained with the aid of a semi-spectral semi-implicit finite difference code developed in Haidvogel DB, Wilkin JL, Young R. J. Comput. Phys. 94 (1991) 151–185 and Wang Y, Hutter K. J. Comput. Phys. 139 (1998) 209–241. Our focus is to see whether the code allows reproduction of the many well-known processes exhibited in stratified waters of a lake basin on the rotating Earth. Often, the internal dynamics exhibits Kelvin- and Poincaré-type oscillations, whose periods depend upon the stratification and the geometry of the basin and which persist for a long time, the attenuation being the result of the turbulent dissipation mechanisms. It is shown that the numerical dissipation of our code can be sufficiently restricted that such wave dynamics obtained with it is realistically persistent for typical time scales of physical limnology. Direct responses to wind forcing and the oscillating behaviour after wind secession are studied and numerical results are illustrated for longitudinal and transverse winds, respectively. By solving the eigenvalue problem of the linearized shallow water equations of two-layered closed rectangular basins, the interpretation of the oscillations as Kelvin- and Poincaré-type waves is corroborated.

Barotropic wind-driven circulation patterns in a closed rectangular basin of variable depth influenced by a peninsula or an island

Abstract. We study how a coastal obstruction (peninsula or coastal island) affects the three-dimensional barotropic currents in an oblong rectangular basin with variable bathymetry across the basin width. The transverse depth profile is asymmetric and the peninsula or island lies in the middle of the long side of the rectangle. A semi-spectral model for the Boussinesq-approximated shallow water equations, developed in Haidvogel et al. and altered for semi-implicit numerical integration in time in Wang and Hutter, is used to find the steady barotropic state circulation pattern to external winds. The structural (qualitative) rearrangements and quanti2tative features of the current pattern are studied under four principal wind directions and different lengths of the peninsula and its inclination relative to the shore. The essentially non-linear relationships of the water flux between the two sub-basins (formed by the obstructing peninsula) and the corresponding cross-sectional area left open are found and analysed. It is further analysed whether the depth-integrated model, usually adopted by others, is meaningful when applied to the water exchange problems. The flow through the channel narrowing is quantitatively estimated and compared with the three-dimensional results. The dynamics of the vortex structure and the identification of the up-welling/down-welling zones around the obstruction are discussed in detail. The influence of the transformation of the peninsula into a coastal island on the global basin circulation is considered as are the currents in the channel. The geometric and physical reasons for the anisotropy of the current structure which prevail through all obtained solutions are also discussed.Key words: Oceanography: general (limnology; numerical modeling) - Oceanography: physical (currents)

Barotropic wind-driven circulation patterns in a closed rectangular basin of variable depth influenced by a peninsula or an island

We study how a coastal obstruction (peninsula or coastal island) affects the three-dimensional barotropic currents in an oblong rectangular basin with variable bathymetry across the basin width. The transverse depth profile is asymmetric and the peninsula or island lies in the middle of the long side of the rectangle. A semi-spectral model for the Boussinesq-approximated shallow water equations, developed in Haidvogel et al. and altered for semi-implicit numerical integration in time in Wang and Hutter, is used to find the steady barotropic state circulation pattern to external winds. The structural (qualitative) rearrangements and quanti2tative features of the current pattern are studied under four principal wind directions and different lengths of the peninsula and its inclination relative to the shore. The essentially non-linear relationships of the water flux between the two sub-basins (formed by the obstructing peninsula) and the corresponding cross-sectional area left open are found and analysed. It is further analysed whether the depth-integrated model, usually adopted by others, is meaningful when applied to the water exchange problems. The flow through the channel narrowing is quantitatively estimated and compared with the three-dimensional results. The dynamics of the vortex structure and the identification of the up-welling/down-welling zones around the obstruction are discussed in detail. The influence of the transformation of the peninsula into a coastal island on the global basin circulation is considered as are the currents in the channel. The geometric and physical reasons for the anisotropy of the current structure which prevail through all obtained solutions are also discussed.Key words: Oceanography: general (limnology; numerical modeling) - Oceanography: physical (currents)

Three-dimensional numerical simulation for tidal motion in Singapore's coastal waters

This paper describes the development and application of a three-dimensional (3-D) multi-level hydrodynamic model to tidal motions in Singapore's coastal waters. In this model, a two-step Euler predictor–corrector algorithm is introduced to predict and correct the free surface water elevation. The governing equations are explicitly and implicitly discretized into finite difference forms in a non-uniform staggered grid system. The velocity components in the x, y and z directions are directly obtained. The 3-D model is tested against analytical solutions for two standard cases in rectangular basins. The model results for the velocities and the free surface water elevations coincide with analytical results. The model is applied to the Singapore coastal region and its surrounding waters. In the first simulation, the tidal current behavior including current circulation, time history of sea levels, currents and discharges as well as patterns of residual currents without wind effects is predicted. The numerically predicted results show good agreement with field measurements. In the second simulation, the response of velocity fields to seasonal wind forcing is investigated.