Rectangular Basin Research Articles

Two-Dimensional Simulation Model of Sediment Removal and Flow in Rectangular Sedimentation Basin

A steady, two-dimensional numerical model was created to study the hydrodynamics of a rectangular sedimentation basin under turbulent conditions. The strip integral method was used to formulate the flow equations, using a forward marching scheme for solving the governing partial differential equations of continuity, momentum, advection–diffusion, turbulent kinetic energy, and its dissipation. In this way the flow equations were converted to a set of ordinary differential equations (ODEs) in terms of the key physical parameters. These parameters, along with a set of shape functions, describe flow variables including the velocity, the concentration of suspended sediments, and both the kinetic energy and its dissipation rate. Four Gaussian distributions were investigated, one corresponding to each flow parameter. In order to calculate the turbulent shear stresses, a two-equation turbulence model (i.e., k-e model) was used. A fourth order Runge–Kutta method numerically integrates the set of ODEs. Simulation r...

A numerical study on the influence of geometry on the formation of sandbanks

Applying a two-dimensional, non-linear hydrodynamic numerical model in combination with a semiempirical equation for bedload sediment transport, the influence of geometry on the formation of sandbanks is investigated. In the first experiment, the formation of sandbanks in an ideal rectangular basin, resembling the Taylor’s problem, was calculated. Sandbanks occur in a small area at the closed boundary. Similar experiments were carried out for a range of wavelengths of the incident Kelvin wave. The results reveal that large wavelengths favor the generation of sandbanks. In subsequent calculations, the basin was modified by introducing new geographical features like bays and peninsulas. The numerical experiments show that geometry is a fundamental factor to determine the position where groups of sandbanks appear. The results suggest that in regions where the Kelvin wave is diffracted, the formation of sandbanks occurs. An experiment, in which we applied an ideal geometric configuration representing that of the Southern Bight of the North Sea, generated sandbank patterns resembling those observed in the region.

Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation

The wind-driven double-gyre circulation in a rectangular basin goes through several dynamical regimes as the amount of lateral friction is decreased. This paper studies the transition to irregular flow in the double-gyre circulation by applying dynamical systems methodology to a quasigeostrophic, equivalent-barotropic model with a 10-km resolution. The origin of the irregularities, in space and time, is the occurrence of homoclinic bifurcations that involve phase-space behavior far from stationary solutions. The connection between these homoclinic bifurcations and earlier transitions, which occur at larger lateral friction, is explained. The earlier transitions, such as pitchfork and asymmetric Hopf bifurcation, only involve the nonlinear saturation of linear instabilities, while the homoclinic bifurcations are associated with genuinely nonlinear behavior. The sequence of bifurcations—pitchfork, Hopf, and homoclinic—is independent of the lateral friction and may be described as the unfolding of a singularity that occurs in the frictionless, Hamiltonian limit of the governing equations. Two distinct chaotic regimes are identified: Lorenz chaos at relatively large lateral friction versus Shilnikov chaos at relatively small lateral friction. Both types of homoclinic bifurcations induce chaotic behavior of the recirculation gyres that is dominated by relaxation oscillations with a well-defined period. The relevance of these results to the mid-latitude oceans’ observed low-frequency variations is discussed. A previously documented 7-year peak in observed North-Atlantic variability is shown to exist across a hierarchy of models that share the gyre modes and homoclinic bifurcations discussed herein.

Response of a harbor with two connected basins to incoming long waves

The general features of the resonant response for a given basin, including the determination of the normal modes and relative amplification of incoming long waves can be easily determined by using a numerical model. However, when the relative role of the different geometric parameters want to be studied, the use of analytical solutions becomes extremely useful. An analytical model is developed to determine the coupled oscillations between two rectangular basins connected through a gap, when only one of them is opened to the ocean. This geometry resembles the topographic features of Ciutadella inlet and its adjacent sub-basin (Cala Busquets) in Menorca, Balearic Islands. The analytical solutions show a good agreement with numerical simulations and capture well the changes induced by some hypothetical modifications of Ciutadella–Cala Busquets system. The agreement shown justifies the use of the analytical model as a basis for the discussion on the influence of the geometrical parameters in the normal mode characteristics of the coupled system.

스펙트럴 방법에 의한 실해역파 재현 및 파 방향 해석

Real sea waves in a towing wave basin have been generated using random periodic motion of the segmented wave makers and the wave reflections of sidewalls. Theoretically, the real sea waves can be described by the superposition of many random oblique waves. This paper introduces numerical real sea wave generation in a rectangular wave basin using spectral method that uses a superposition of orthogonal functions which have to satisfy the Laplace equation. Oblique regular waves, long crested irregular waves and real sea waves were simulated and met the requirement of sidewall wave reflection and wave absorption. MLM (Maximum Likelihood Method) and Spatial Fourier Transform were used in order to obtain propagated wave direction characteristics. The estimated results proved the usefulness of the method and the performances showed reasonable directional patterns comparing with generating patterns.

The Effects of Wind Forcing and Background Mean Currents on the Latitudinal Structure of Equatorial Rossby Waves

Abstract The latitudinal structure of annual equatorial Rossby waves in the tropical Pacific Ocean based on sea surface height (SSH) and thermocline depth observations is equatorially asymmetric, which differs from the structure of the linear waves of classical theory that are often presumed to dominate the variability. The nature of this asymmetry is such that the northern SSH maximum (along 5.5°N) is roughly 2 times that of the southern maximum (along 6.5°S). In addition, the observed westward phase speeds are roughly 0.5 times the predicted speed of 90 cm s−1 and are also asymmetric with the northern phase speeds, about 25% faster than the southern phase speeds. One hypothesized mechanism for the observed annual equatorial Rossby wave amplitude asymmetry is modification of the meridional structure by the asymmetric meridional shears associated with the equatorial current system. Another hypothesis is the asymmetry of the annually varying wind forcing, which is stronger north of the equator. A reduced-gravity, nonlinear, β-plane model with rectangular basin geometry forced by idealized Quick Scatterometer (QuikSCAT) wind stress is used to test these two mechanisms. The model with an asymmetric background mean current system perturbed with symmetric annually varying winds consistently produces asymmetric Rossby waves with a northern maximum (4.7°N) that is 1.6 times the southern maximum (5.2°S) and westward phase speeds of approximately 53 ± 13 cm s−1 along both latitudes. Simulations with a symmetric background mean current system perturbed by asymmetric annually varying winds fail to produce the observed Rossby wave structure unless the perturbation winds become strong enough for nonlinear interactions to produce asymmetry in the background mean current system. The observed latitudinal asymmetry of the phase speed is found to be critically dependent on the inclusion of realistic coastline boundaries.

INCREASE IN GROUNDWATER LEVEL DUE TO ARTIFICIAL RECHARGE

ABSTRACT Artificial recharge of the ground-water table from rectangular basins has been the subject of extensive study in the last few decades. We develop a numerical model, based on the Finite Difference Method with an explicit scheme, to predict the transient position of the water table in response to recharge from a rectangular basin. The rate of recharge is varied by applying a time decay constant in order to simulate realistic infiltration behaviour. The model is validated through comparison with an existing analytical solution and is then applied to estimate the mounding below a proposed recharge basin at the Indian Institute of Technology Kanpur.

Separation of scales on a broad, shallow turbulent flow

Two different mechanisms acting on a broad, shallow turbulent flow whose energy spectrum is separated by a gap is investigated experimentally. The study concentrates on the spectral response evolution of the shallow flow approaching an opening induced by a lateral contraction, located halfway in streamwise direction of a large rectangular basin. The geometrical forcing, induced by the lateral contraction that breaks the plan-form symmetry of an otherwise broad and uniform shallow flow, produces an energy concentration in the range of low wavenumbers (large-scale) by an inverse cascade process. The geometrical forcing continues to pile up energy until scales the size of the entire flow geometry are strongly excited. The topographical forcing, induced by the presence of a fixed scour hole located at the contraction end that breaks an otherwise flat bottom, produces an energy concentration in the range of high wavenumbers (small-scale) by trapping some flow modes. The topographical forcing is only local, and has the effect of acting as an eddy viscosity on the large-scale turbulent motion. The problem presented is relevant to determine (he fate of suspended sediment downstream of the opening of relief bridges located on flat floodplains in periods of high waters.

On the three-dimensional structure of the inertial wave field in a rectangular basin with one sloping boundary

Inertial waves propagate obliquely through a rotating fluid with an angle with respect to the rotation axis that is determined by the ratio of wave frequency and rotation frequency. This constraint leads to wave focusing or defocusing upon reflection at a sloping wall. In an enclosed basin repeated reflection may lead to standing waves or wave attractors: limit cycles to which all energy converges. In a two-dimensional setting (including the rotation-axis), wave patterns can be predicted mathematically; in three dimensions this is not generally possible, but ray-tracing indicated that attractors can still be found due to refractive trapping in the along-channel direction. Two wave attractors and a standing wave were investigated experimentally in a rectangular basin with a sloping wall, using particle image velocimetry. Wave attractors and a standing wave were indeed observed, with inhomogeneous energy and phase distribution in the along-channel direction, different for the different attractors and the standing wave. This behaviour can be partly understood using ray-tracing. Scale estimates revealed that the width of the attractor is limited by nonlinear advection.

A comprehensive ocean prediction and analysis system based on the tangent linear and adjoint of a regional ocean model

The regional ocean modelling system (ROMS) is a new generation ocean general circulation model that is rapidly gaining favour in the ocean modelling community. The tangent linear and adjoint versions of ROMS have recently been developed, and a new suite of tools that utilize these models for a variety of applications are now available to the ocean modelling community. In this paper we will describe the tangent linear and adjoint components of ROMS, and present examples from the tools that are currently available to ROMS users. In particular we will consider the finite time eigenmodes and the adjoint finite time eigenmodes of the tangent linear propagator, the singular vectors of the propagator, and its forcing singular vectors and stochastic optimals. The pseudospectra of the tangent linear resolvent matrix are also considered. Examples of each type of calculation will be presented for a time evolving double gyre ocean circulation in a rectangular ocean basin.

Finite-Element Model for Modified Boussinesq Equations. II: Applications to Nonlinear Harbor Oscillations

The finite-element model for modified Boussinesq equations developed in Part I of this paper is used to study harbor resonance problems. The internal wavemaker technique is implemented on an unstructured finite-element mesh, and a sponge layer is used as an open boundary condition. Both linear and nonlinear harbor oscillations in rectangular and circular basins are examined, and experimental measurements and analytical solutions are used to validate the model. Finally, the wavelet transform is employed to analyze transient features of the nonlinear resonance problem.

Removing the separation singularity in a barotropic ocean with bottom friction

We consider western boundary outflow in a two-dimensional rectangular basin on a beta plane with bottom drag. The nature of the flow in the boundary layer is determined by a parameter λ which measures the nonlinearity of the flow. When λ is smaller than some critical value λ c, the boundary layer remains attached to the western wall, but as λ is increased above λ c, a separation bubble develops, and the point of separation moves south with increasing λ. If a classical boundary layer expansion as suggested by Prandtl, (Motion of Fluids with Very Little Viscosity, Technical Memo-452, NACA, Washington, DC, 1928) is attempted, a Goldstein separation singularity develops (Quart. J. Mech. Appl. Math. 1 (1948) 43). We show that if an alternative expansion is used inside the boundary layer, this singularity can be removed and the flow can be extended past separation.

A Tropical Ocean Recharge Mechanism for Climate Variability. Part II: A Unified Theory for Decadal and ENSO Modes

Decadal to interdecadal timescale variability in the Pacific region, commonly referred to as the Pacific decadal oscillation (PDO), is studied in this research using analytical and numerical models. A coupled analytical model is formulated to analyze the physical mechanism of both the PDO and ENSO. It has the equatorial β-plane dynamics of a reduced-gravity model coupled with the wind stress of fixed spatial patterns. The amplitude of the latter is proportional to the sea surface temperature (SST) anomaly in the eastern equatorial Pacific. The SST anomaly is governed by a simple thermal dynamic equation used for ENSO modeling. It is found that when a warm SST is coupled with cyclonic wind stress patterns in the eastern subtropical Pacific, an oscillation with a timescale of around 10–15 yr could be generated. In contrast, when a warm SST is coupled with only a westerly wind stress in the central equatorial Pacific, an ENSO-like oscillation could be generated with a timescale of around 3–5 yr. Thus the present research is potentially relevant to aspects of the PDO and the mechanism of the PDO may be understood as a weakly coupled decadal recharge oscillator similar to the recharge oscillator dynamics of ENSO. The sensitivity of these two kinds of coupled modes to different parameters is tested. Numerical integrations with the reduced-gravity shallow-water model in a rectangular basin and a similar coupled framework confirm the results of the analytical model.

Observations of inertial waves in a rectangular basin with one sloping boundary

Inertial waves in a homogeneous rotating fluid travel along rays that are inclined with respect to the rotation axis. The angle of inclination depends solely on the ratio of the wave frequency and twice the angular frequency. Because of this fixed angle, the waves can become focused when reflected at a sloping wall. In an infinitely long channel with a sloping wall, the repeated action of focusing may lead to the approach to a limit cycle, the so-called wave attractor, where the energy is concentrated. This effect is studied in the laboratory in a rectangular tank with one sloping wall, placed excentrically on a rotating table. The waves are excited by modulation of the background rotation. Several frequency ratios are used to study different wave attractors and one standing wave. The observations consist mainly of particle image velocimetry data in horizontal and vertical cross-sections in one half of the basin. The attractors are observed in the vertical cross-sections. They show continuous phase propagation, which distinguishes them from the standing wave where the phase changes at the same time over the whole cross-section. However, particle motion of inertial waves is three-dimensional and the actual basin is not an ideal twodimensional channel but is of finite length. This implies that the waves must adapt to the vertical endwalls, although a prediction of the nature of these adaptations and the structure of the three-dimensional wave field is at present lacking. For critical waves, whose rays are parallel to the slope, clear three-dimensional behaviour is observed. The location of most intense motion along this critical slope attractor changes in the horizontal direction and horizontal phase propagation is observed, with a wavelength between 1/5 and 1/4 of the basin length. For the other attractors there is little evidence of phase propagation in the horizontal direction. The motion along the attractor is however stronger near the vertical endwalls for attractors with wave rays of slopes close to 1 or larger. The standing wave and the other attractors are more clearly visible near 1/5 of the tank length.

Two-Dimensional Simulation Model for Contour Basin Layouts in Southeast Australia. I: Rectangular Basins

Contour basin irrigation layouts are used to irrigate rice and other cereal crops on heavy cracking soils in Southeast Australia. In this study, a physically based two-dimensional simulation model that incorporates all the features of contour basin irrigation systems is developed. The model’s governing equations are based on a zero-inertia approximation to the two-dimensional shallow water equations of motion. The equations of motion are transformed into a single nonlinear advection–diffusion equation in which the friction force is described by Manning’s formula. The empirical Kostiakov equation and the quasi-analytical Parlange equation are used to model the infiltration process. The governing equations are solved by using a split-operator approach. The numerical procedure described here is capable of modeling rectangular basins; a procedure for irregular shaped basins is presented in Paper II. The model was validated against field data collected on commercial lasered contour layouts.

Impact of the AD 79 explosive eruption on Pompeii, I. Relations amongst the depositional mechanisms of the pyroclastic products, the framework of the buildings and the associated destructive events

A quantitative and qualitative evaluation of the damage caused by the products of explosive eruptions to buildings provides an excellent contribution to the understanding of the various eruptive processes during such dramatic events. To this end, the impact of the products of the two main phases (pumice fallout and pyroclastic density currents) of the Vesuvius AD 79 explosive eruption onto the Pompeii buildings has been evaluated. Based on different sources of data, such as photographs and documents referring to the archaeological excavations of Pompeii, the stratigraphy of the pyroclastic deposits, and in situ inspection of the damage suffered by the buildings, the present study has enabled the reconstruction of the events that occurred inside the city when the eruption was in progress. In particular, we present new data related to the C.J. Polibius’ house, a large building located inside Pompeii. From a comparison of all of the above data sets, it has been possible to reconstruct, in considerable detail, the stratigraphy of the pyroclastic deposits accumulated in the city, to understand the direction of collapse of the destroyed walls, and to evaluate the stratigraphic level at which the walls collapsed. Finally, the distribution and style of the damage allow us to discuss how the emplacement mechanisms of the pyroclastic currents are influenced by their interaction with the urban centre. All the data suggest that both structure and shape of the town buildings affected the transport and deposition of the erupted products. For instance, sloping roofs ‘drained’ a huge amount of fall pumice into the ‘impluvia’ (a rectangular basin in the centre of the hall with the function to collect the rain water coming from a hole in the centre of the roof), thus producing anomalous deposit thicknesses. On the other hand, flat and low-sloping roofs collapsed under the weight of the pyroclastic material produced during the first phase of the eruption (pumice fall). In addition, it is evident that the walls that happened to be parallel to the direction of the pyroclastic density currents produced during the second eruptive phase were minimally damaged in comparison to those walls oriented perpendicular to the flow direction. We suggest that the lower depositional parts of the pyroclastic currents were partially blocked (locally reflected) and slowed down because of recurring encounters with the closely spaced walls within buildings. Locally, the percentage of demolished walls decreases down-current, which has been interpreted as a loss in kinetic energy within the depositional system of the flow. However, it seems that the upper transport system by-passed these obstacles, then supplied new pyroclasts to the depositional system that restored its physical characteristics and restored enough kinetic energy to demolish the next walls and buildings further along its path.

Transient and steady-state amplitudes of forced waves in rectangular basins

A weakly-nonlinear analysis of the transient evolution of two-dimensional, standing waves in a rectangular basin is presented. The waves are resonated by periodic oscillation along an axis aligned with the wavenumber vector. The amplitude of oscillation is assumed to be small with respect to the basin dimensions. The effects of detuning, viscous damping, and cubic nonlinearity are all simultaneously considered. Moreover, the analysis is formulated in water of general depth. Multiple-scales analysis is used in order to derive an evolution equation for the complex amplitude of the resonated wave. From this equation, the maximum transient and steady-state amplitudes of the wave are determined. It is shown that steady-state analysis will underestimate the maximum response of a basin set into motion from rest. Amplitude response diagrams demonstrate good agreement with previous experimental investigations. The analysis is invalid in the vicinity of the “critical depth” and in the shallow-water limit. A separate analysis, which incorporates weak dispersion, is presented in order to provide satisfactory results in shallow water.

Low-Frequency Variability in Shallow-Water Models of the Wind-Driven Ocean Circulation. Part I: Steady-State Solution*

Successive bifurcations—from steady states through periodic to aperiodic solutions—are studied in a shallowwater, reduced-gravity, 2‰-layer model of the midlatitude ocean circulation subject to time-independent wind stress. The bifurcation sequence is studied in detail for a rectangular basin with an idealized spatial pattern of wind stress. The aperiodic behavior is studied also in a North Atlantic‐shaped basin with realistic continental contours. The bifurcation sequence in the rectangular basin is studied in Part I, the present article. It follows essentially the one reported for single-layer quasigeostrophic and 1‰-layer shallow-water models. As the intensity of the north‐ south-symmetric, zonal wind stress is increased, the nearly symmetric double-gyre circulation is destabilized through a perturbed pitchfork bifurcation. The low-stress steady solution, with its nearly equal subtropical and subpolar gyres, is replaced by an approximately mirror-symmetric pair of stable equilibria. The two solution branches so obtained are named after the inertial recirculation cell that is stronger, subtropical or subpolar, respectively. This perturbed pitchfork bifurcation and the associated Hopf bifurcations are robust to changes in the interface friction between the two active layers and the thickness H2 of the lower active layer. They persist in the presence of asymmetries in the wind stress and of changes in the model’s spatial resolution and finitedifference scheme. Time-dependent model behavior in the rectangular basin, as well as in the more realistic, North Atlantic‐shaped one, is studied in Part II.

Low-Frequency Variability in Shallow-Water Models of the Wind-Driven Ocean Circulation. Part II: Time-Dependent Solutions*

The time-dependent wind-driven ocean circulation is investigated for both a rectangular and a North Atlantic‐ shaped basin. Multiple steady states in a 2‰-layer shallow-water model and their dependence on various parameters and other model properties were studied in Part I for the rectangular basin. As the wind stress on the rectangular basin is increased, each steady-state branch is destabilized by a Hopf bifurcation. The periodic solutions that arise off the subpolar branch have a robust subannual periodicity of 4‐5 months. For the subtropical branch, the period varies between sub- and interannual, depending on the inverse Froude number F2 defined with respect to the lower active layer’s thickness H2 .A sF2 is lowered, the perturbed-symmetric branch is destabilized baroclinically, before the perturbed pitchfork bifurcation examined in detail in Part I occurs. T ransition to aperiodic behavior arises at first by a homoclinic explosion off the isolated branch that exists only for sufficiently high wind stress. Subsequent global and local bifurcations all involve the subpolar branch, which alone exists in the limit of vanishing wind stress. Purely subpolar solutions vary on an interannual scale, whereas combined subpolar and subtropical solutions exhibit complex transitions affected by a second, subpolar homoclinic orbit. In the latter case, the timescale of the variability is interdecadal. The role of the global bifurcations in the interdecadal variability is investigated. Numerical simulations were carried out for the North Atlantic with earth topography5 minute (ETOPO-5) coastline geometry in the presence of realistic, as well as idealized, wind stress forcing. The simulations exhibit a realistic Gulf Stream at 20-km resolution and with realistic wind stress. The variability at 12-km resolution exhibits spectral peaks at 6 months, 16 months, and 6‐7 years. The subannual mode is strongest in the subtropical gyre; the interannual modes are both strongest in the subpolar gyre.

Deviation of Long-Period Tides from Equilibrium: Kinematics and Geostrophy

New empirical estimates of the long-period fortnightly (Mf ) tide obtained from TOPEX/Poseidon (T/P) altimeter data confirm significant basin-scale deviations from equilibrium. Elevations in the low-latitude Pacific have reduced amplitude and lag those in the Atlantic by 308 or more. These interbasin amplitude and phase variations are robust features that are reproduced by numerical solutions of the shallow-water equations, even for a constant-depth ocean with schematic interconnected rectangular basins. A simplified analytical model for cooscillating connected basins also reproduces the principal features observed in the empirical solutions. This simple model is largely kinematic. Zonally averaged elevations within a simple closed basin would be nearly in equilibrium with the gravitational potential, except for a constant offset required to conserve mass. With connected basins these offsets are mostly eliminated by interbasin mass flux. Because of rotation, this flux occurs mostly in a narrow boundary layer across the mouth and at the western edge of each basin, and geostrophic balance in this zone supports small residual offsets (and phase shifts) between basins. The simple model predicts that this effect should decrease roughly linearly with frequency, a result that is confirmed by numerical modeling and empirical T/P estimates of the monthly (Mm) tidal constituent. This model also explains some aspects of the anomalous nonisostatic response of the ocean to atmospheric pressure forcing at periods of around 5 days.