Effects Of Bottom Slope Research Articles

Analytical and numerical modeling of sandbanks dynamics

Linear and nonlinear behavior of large‐scale underwater bedform patterns like sandbanks are studied using linear stability analysis and numerical modeling. The model is based on depth‐integrated hydrodynamics equations with a quadratic bottom friction law and a bed load sediment transport model including a bottom slope effect. First, the linear stability analysis of a flat erodible bottom subject to a steady current is performed. The direction of the most amplified bedform is controlled by the current, whereas its wavelength is selected by the gravity driven sediment flow. The instability proved to be due to the velocity component parallel to the mean flow, whereas the transverse velocity and the bottom slope effect are damping processes. The growth rate is then related to the spatial phase‐lag between flow velocity and bathymetry. In order to validate the numerical model in this linear regime, the growth rate and the phase celerity of an infinitesimal bottom perturbation are computed as a function of its wavevector. A good agreement with linear stability results is found. Second, using the property that the growth rates for block‐function and steady current are the same, the nonlinear behavior of the instability is investigated for a simple tidal current. The saturation height of the theoretically most amplified mode is estimated using the numerical model. The analysis of the sediment fluxes gradients shows that the saturation is mostly due to the hydrodynamic processes, although the saturation height slightly depends on the value of the bottom slope effect coefficient. Third, a Landau equation, whose coefficients are computed from the previous results, is used to predict the temporal evolution of the bedform amplitude from the initial infinitesimal perturbation to the saturation. A comparison with the characteristics of continental shelf sandbanks shows that the model gives a reasonable estimation of the temporal dynamics of these large‐scale bedforms. However, the saturation height seems to be slightly overestimated. The limitations of the method are also discussed.

A Proposal of New Breaker Height Formula

A new breaker height formula is developed based on a re-analysis of available laboratory data. The analysis shows that the breaking wave steepness is mainly governed by its deepwater wave steepness. The power form could be used to fit the relationship between breaking wave steepness and its deepwater wave steepness. This finding is new. Although the breaker height has been a subject of study for a century, no researcher has tried to relate the breaking wave steepness to the deepwater wave steepness. Bottom slope effect is included explicitly into the present formula. The present formula and existing formulas are examined and compared with a large number and wide range of published laboratory data (695 cases collected from 26 sources). Overall, the present formula gives very good predictions over a wide range of experimental conditions and is better than existing formulas.

Convection in Small Basins

Abstract Convection in the world's oceans often occurs in small, semienclosed basins where bottom slopes and nearby continental shelf breaks are commonplace. The evolution of convectively generated heat anomalies in such settings is studied using quasigeostrophic finite-difference and point vortex models. The displayed behaviors divide essentially into two categories: whole fluid column convection, in which bottom-slope effects are felt immediately, and partial fluid column convection, in which the topographic effects can be delayed. In both cases, topography significantly modifies the evolution of convective patches from that occurring over flat bottoms. Vertical walls induce strong self-propagation mechanisms that accelerate alongslope heat transport, while the continental shelf slope is repulsive and rejects lower-layer anticyclones. These anomalies are then “stranded,” being too far offshore to interact with the shelf break and having lost their heton partner in the interaction. Weaker deep ocean topo...

Baroclinic instability of density current along a sloping bottom and the associated transport process

Numerical experiments with a three‐dimensional nonhydrostatic ocean model have been carried out to investigate the dynamic processes of downslope density current and associated transport of dense water, focusing on the effects of bottom slope for the range of slope inclination S from 0 to 0.04. Bottom slope has two effects on baroclinic instability of the flow. One is the stabilizing effect due to the topographic β effect and to offshoreward increase in water depth. The other is the destabilizing effect due to steepened inclination of isopycnal surface, which increases available potential energy. Since the destabilizing effect over the upper slope overcomes the stabilizing one in the early stage, unstable waves develop more rapidly with increasing S. Nonlinear energy conversion due to Reynolds stresses also affects the flow stability and then the growth rate of instability increases linearly with S in this stage. After the unstable waves have finite amplitudes in the mature stage the stabilizing effect of bottom slope becomes dominant, especially over the lower slope. Thus vigorous unstable eddies are confined to the upper slope region in steep slope cases, while they cover the whole slope region in gentle slope cases, and this difference in the eddy field causes a drastic change in downslope transport of dense water. In gentle slope cases, efficiency of the transport becomes higher with increasing S since the destabilizing effect overcomes the stabilizing one as a whole. On the contrary, the efficiency becomes lower with increasing S in steep slope cases since the stabilizing effect surpasses the destabilizing one. As a result, the most effective transport is realized in the case with S = 0.005. Although the transport due to eddies in steep slope cases becomes weak over the lower slope, it is still larger than that due to the bottom Ekman current since the eddy transport occurs in the bottom layer with a thickness of 100∼200 m, which is 4∼8 times the Ekman layer thickness.

Channel Routing in Open-Channel Flows with Surges

Using numerical models for the purpose of channel-routing calculation has been well accepted in engineering practice. However, most traditional models fail to predict the transcritical flows because of numerical instability. This paper presents two high-resolution, shock-capturing schemes for the simulation of 1D, rapidly varied open-channel flows. The present schemes incorporate the method of characteristics to deal with the unsteady boundary conditions. Also, the Strang-type splitting operator is used to include the effects of bottom slope and friction terms. To assess the performance of the proposed algorithms, several steady and unsteady problems are simulated to verify the accuracy and robustness in capturing strong shocks in open-channel flows. Furthermore, the results of dynamic flood routing and steady routing are compared to demonstrate the risk of using steady routing for flood mitigation.

Verification and Modification of Breaker Height Formulas

This study is undertaken to find out the most reliable breaker height formulas that predict well for a wide range of hydraulic conditions. The applicability of 24 existing formulas, for computing breaking wave heights, is examined by using wide range and large amount of published laboratory data (574 cases collected from 24 sources). It is found that most formulas predict well for the breaking waves on the gentle slope (0 < m ≤ 0.07), but the prediction is unsatisfactory for the breaking waves on the steep slope (0.1 < m ≤ 0.44). Three formulas are selected and are modified by including the new form of bottom slope effect into the formulas. The new breaker height formulas predict well for wide range of wave and bottom slope conditions.

Effects of ambient stratification and shelfbreak topography on offshore transport of dense water on continental shelves

The transport of dense water offshore from an idealized coastal polynya is examined using a primitive equation numerical model. First, the prior scaling of the equilibrium density of water produced within the polynya is extended to include the effects of linear bottom slope. Next, ambient linear stratification is added. The equilibrium density is modified slightly by the density at the bottom at the initial time. Next, shelfbreak topography is included with both weak and strong ambient stratification. For the case of weak stratification, dense water carried to the shelfbreak within eddies moves rapidly downslope and is trapped near the bottom. The along‐shelf scale of the dense water cascading down the slope is set by the diameter of the dense water eddies that form along the periphery of the coastal polynya. For the case of strong stratification, dense water is carried offshore across the shelfbreak at an intermediate depth, injecting water originally within the polynya across the shelf and over the continental slope. A bottom‐trapped current flowing near the shelfbreak in the direction of coastal‐trapped wave propagation also forms. These results suggest that water formed within coastal polynyas over Arctic shelves may be capable of ventilating the continental slope regions and possibly central basins of the Arctic Ocean via eddy fluxes.

Experiments to dambreak wave: Sloping channel

Dambreak flows in a smooth, rectangular channel are considered. In all experiments, the abrupt rupture on an absolutely dry tailwater channel was modelled. The effect of bottom slope up to 50% is analyzed, and the results are compared with those pertaining for the horizontal channel. The positive and negative wave fronts are determined based on the characteristic equations. Also, the drying front is computed and the effect of viscosity is analyzed. The experiments are in agreement with the observations collected with a specially designed Video system. The positive wave front is found to tend towards a pseudo-uniform flow for large locations downstream of the dam section. The characteristics of the dambreak waves are then described, including the maximum wave height, and a generalized wave profile. The decay of maximum wave height is related to a combined parameter that includes the position, the bottom slope and the initial water depth. Further, the velocities in the wave body are determined and the velocity distribution is discussed. The discharge as the product of depth times velocity is demonstrated to be nearly independent of the bottom slope. With these results, a rather complete description of the dambreak wave can be furnished.

Drawdown curve and incipient aeration for chute flow

Turbulent rough flows in spillway chutes are considered in the first part. The drawdown curve is demonstrated to be identical with those of flows in the turbulent smooth regime. It is further shown that all drawdown curves can be expressed with a normalized streamwise coordinate that involves the bottom slope and the uniform and critical flow depths. An explicit solution is presented. The computational results are verified with model observations. Also, previous derivations are generalized and a universal drawdown curve for chute flow is introduced. The results are readily applicable for design. In the second part of this paper, incipient aeration on a spillway chute is investigated, based on the data of Bauer for the boundary-layer thickness. Predictions for the flow depth and location of incipient aeration are in agreement with observations. The effects of bottom slope and relative roughness are demonstrated to be comparably small, whereas the critical depth has a major influence on both flow depth and location of incipient aeration. Typical examples illustrate the computational procedure.Key words: aeration, chute, drawdown, spillway, supercritical flow.

Supercritical Flow in Chute Contraction

Contractions with supercritical flow generate shockwaves requiring significantly larger wall heights than predicted by one-dimensional flow equations. Three distinct waves in a straight-walled contraction may be identified: wave 1 downstream from the contraction point, wave 2 in the chute axis, and wall wave 3 beyond the contraction end. The characteristics of these three waves were experimentally determined, based on the concept of the shock number. The effects of contraction ratio and bottom slope were investigated. Means to reduce shock waves in a contraction were also studied. A shock diffractor was introduced as a simple element that reduces shocks significantly. A design procedure is presented, and the optimum diffractor geometry with its optimum location is specified. Corresponding wave heights without and with diffractors were studied; a typical reduction of wave height with diffractors was 20-40%. A discussion of off-design flow and choking flow is also included. Based on the present results, a complete hydraulic design of straight-walled chute contractions is possible.

Modelling shoreface profile evolution

Current knowledge of hydro-, sediment and morpho-dynamics in the shoreface environment is insufficient to undertake shoreface-profile evolution modelling on the basis of first physical principles. We propose a simple, panel-type model to map observed behaviour. The internal dynamics are determined by slope-dependent, wave-induced cross-shoreface transports, while the external driving factors are lateral sediment supply and sea-level rise. This model concept is tested with reasonable success against the observed behaviour of the Central Holland Coast, considering two hindcast periods, one covering the evolution over the last century, the other the Subboreal/Subatlantic evolution.A limitation of this model is that the cross-shoreface dynamics are solely steered by the variations of shoaling, short waves. Since a variety of other wave and current dynamics may be expected to be present in the coastal boundary layer, it may well be that the effects of the mechanisms and conditions which are not represented are hidden in the coefficients of the sediment-transport formula. This limits the accuracy of the coefficients as used, and our findings should be considered as an-order-of-magnitude estimate only. Indeed, behaviour-oriented modelling implies that generalization of results to arbitrary situations and conditions is not straightforward. Yet, we expect that some of the conclusions are more generally applicable.This concerns the substantiation of the assumption that the upper shoreface responds on a much smaller time scale than the lower shoreface, and the idea that the shoreface profile is not always and everywhere in equilibrium with its forcing. A worthwhile observation from the Holland Coast application is, that the bottom slope effect on the transport is only important at geological time scales. The profile evolution at the engineering time scales (say 10 to 100 years) is effectively quasi-static, in that there is no feedback between the long-term averaged transport and the state of the profile. This implies that at these smaller scales the profile changes can be predicted on the basis of a static sediment balance. This does not mean that the gravitational downslope transport is unimportant as a physical phenomenon in coastal profile evolution: It is only unimportant if a highly aggregated model like this is applied at relatively short time scales.

斜面上に表層放流される平面密度噴流の拡がり

This paper presents experimental study on the behavior of buoyant surface jet discharged on the sloping bottom. The important resuts of this study are concluded as follows.(A) Smaller bottom slope prompts jet type spreading rather than buoyant speading. (B) Smaller bottom slope prompts meandering of jet axis and cause large scale coherent motion. (C) bottom slope does not affect the temperature distributions even though the effects of bottom slope are very strong against spreadings.

Measurements of the effects of bottom slope on sound propagation in shallow water

Laboratory scale measurements have been made of sound propagation in shallow water over a sloping bottom. The measurements were made in a large test pool in a water layer over a model of the ocean floor that consisted of an epoxy layer on a concrete base. The properties of the model were consistent with those found in typical ocean sediments and substrates. The results presented in this paper will demonstrate downslope- and across-slope propagation for several different bottom angles, and comparisons will be made with measurements in a uniform depth water column over the same bottom. Important differences were found even in the special case of across slope propagation where the water depth was constant with range. Shadow zones and modal absorption will be demonstrated. [Work supported by ONR.]

CLOSED-FORM SOLUTIONS FOR THE PROBABILITY DENSITY OF WAVE HEIGHT IN THE SURF ZONE

By invoking the assumption that in the surf zone, random waves behave as a collection of individual regular waves, two closed-form solutions for the probability density function of wave height on planar beaches are derived. The first uses shallow water linear theory for wave shoaling, assumes a uniform incipient condition, and prescribes breaking with a regular wave model that includes both bottom slope and wave steepness effects on the rate of decay. In the second model, the shallow water assumption is removed, and a distribution in wave period (incipient condition) is included. Preliminary results indicate that the models exhibit much of the behavior noted for random wave transformation reported in the literature, including bottom slope and wave steepness effects on the shape of the probability density function.

2DH mathematical modelling of morphological evolutions in shallow water

A class of compound mathematical models of transient morphological evolutions in shallow water is discussed, based on commonly applied horizontally two-dimensional formulations of the water and sediment motion in the coastal zone. Three major aspects of the composition of such models are considered, viz. the specific adequacy of the constituent models, the balance of the total model and the possibility of spurious interactions. The lacunae in the knowledge and the tools needed to judge a proposed model at these points are indicated. Examples of spurious interactions are described in further detail and techniques to trace and avoid them are given. These examples, concerning morphological evolutions without the influence of waves, show that the advective acceleration terms in the flow model play an essential part in the bottom evolution and its interaction with the current and that incorporation of the bottom slope effects on the sediment transport is indispensable in models with a local transport formulation.

2DH COMPUTATION OF TRANSIENT SEA BED EVOLUTIONS

The interaction between the constituent models for waves, currents, sediment transport and bottom level changes in a class of compound 2DH mathematical models of transient sea bed evolutions in the coastal zone is investigated mathematically, using the theory of characteristics and a harmonic analysis technique. This leads to contrary indications as to the importance of including diffraction in the wave model and to the unambiguous conclusion that the bottom slope effect on the sediment transport is indespensable for the mathematical system to be inherently stable. This conclusion is shown to imply that depth-integrated current model are insufficient to describe the flow in this type of morphological computations for the nearshore zone or complex coastal areas.

Ebb-tidal jets: A model of suspended sediment and mass transport at tidal inlets

Mass transport by turbulent jets issuing from tidal inlets is investigated through a model that includes lateral mixing and entrainment, bottom friction, bathymetric changes, settling rate of particles (size), possible deposition/erosion at the bottom and ambient currents and concentrations. The bottom frictional jet becomes diluted more slowly than a classical jet. A non-vanishing concentration may result offshore and a maximum may occur in the core. The concentration of a jet on a sloping bottom decreases more rapidly due to increased dilution by entraiment. The effects of bottom friction and bottom slope compete in determining the jet concentration. Deposition to the bottom occurs within the jet mainly on both sides of the centre-line, and at lower rates on the centre-line. Erosion or deposition may occur at the jet core depending on the inlet flow conditions. In the case of erosion at the core, the material extracted is deposited on the margins and the offshore areas. Sorting of the sediments is expected, with coarser materials mainly deposited in the marginal areas, while the finer sediments are more uniformly distributed and jetted further off-shore. The main features of the model are verified through a limited set of observations. The qualitative agreement is enhanced for micro- and meso-tidal inlets that are dominated by tidal hydraulics.

Effect of Local Bottom Slope and Roughness Coefficient Variations in Overland Flows

The investigation analyses the effects of local bottom slope and roughness coefficient variations in overland flow over a plane by using the kinematic wave theory. Results are compared with the usual approach, and it is found that certain configurations of surface topography and roughness characteristics yield results, that deviate from the usual predictions.

Deep-ocean ray transmissions over convergent and divergent sloping bottoms

Surface-reflected/bottom-reflected transmissions over a slowly sloping bottom in a deep ocean are investigated using ray theory. For convenience, sound speed is taken to be bilinear with depth in the water, while the bottom structure is assumed to be uniform. The sound source and receiver are located on or near the ocean boundaries, and the effects of bottom slope on ray geometry, per-ray travel time, and incoherent total-field amplitude are examined. For transmission ranges of tens of km, and for typical deep-ocean slope inclinations, travel time may change by about 1.0 s, relative to its value for a horizontal bottom. The time difference between ray arrivals decreases (increases) in the divergent (convergent) channel. Principal effects of bottom slope on total-field amplitude arise primarily through bottom-loss modifications, rather than through spreading loss. An inclination-angle magnitude of only 0.25° is shown to cause a decrease of about 8 dB (increase of about 4 dB) in the incoherent amplitude in the convergent (divergent) channel, for ranges of about 100 km. Changing bottomed receiver location in a convergent channel, in order to improve source detectability, is shown in an example to be much more effective than in the case of a horizontal bottom. In a corresponding divergent channel, however, the strength of the acoustic reception is much less sensitive to variations in receiver location.

Coastal-Trapped Waves in a Continuously Stratified Ocean

The theory of coastal-trapped waves is extended to include the general features of continuous density stratification, variable bottom topography, and a finite coastal wall. In the limit of a vanishing coastal wall, topographic Rossby waves are the only class of sub-inertial frequency, trapped wave motion. The stratification effect on topographic Rossby waves depends on both the local baroclinic radius of deformation and the characteristic offshore length scale of the wave motion. For intermediate density stratification, long waves are nearly depth-independent in the shelf region, and are bottom-trapped in the slope region. The topographic Rossby waves reduce to the barotropic shelf waves and the bottom-trapped waves in the limits of small and large density stratification, respectively. In the general case of comparable influences from the coastal wall and bottom slope effects, baroclinic Kelvin waves and topographic Rossby waves are eigenmodes of the system. The eigenfunctions are modified from the elementary cases, which can be discerned by their structures along the coastal and bottom bound-aries. In particular, a resonance condition is suggested, i.e., the properties of a wavemode vary with the wavenumber and stratification. For example, mode 1 is a topographic Rossby wave for small wavenumbers and it is a baroclinic Kelvin wave for large wavenumbers. Also, the high-frequency cutoff found in the barotropic theories is lost.