Undulating Bed Research Articles

Experiment on the Hydrodynamic Characteristics of Plunging Flows in Bedrock Canyon Bends

AbstractBedrock river incision drives the evolution of the Earth's landscape and is influenced by river hydraulic power. However, the relationship between plunging flows and the curvature of bedrock canyons is poorly understood, which encouraged us to explore how the bend's curvature and undulating beds affect plunging flows. A generalized physical model was built to investigate the hydrodynamic characteristics of plunging flows in a constant curvature flume with plain and undulating beds. Our experimental findings demonstrated that plunging flows were related to secondary circulation, topography, and width‐to‐depth ratios. Plunging flows occurred when secondary circulation reached its peak, and as secondary circulation vanished, the intensity of plunging flows decreased. The undulating bed topography in the bedrock bends suppressed secondary circulation and the development of plunging flows. Bed topography may be a dominant factor in plunging flows in bedrock bends. The potential erosion area in bedrock bends was related to the intensity of plunging flows, which caused velocity inversion to increase the shear stress near the bed. With higher discharges and undulating beds, the intensity of the transverse shear stress of the riverbed near the center was greater. The potential erosion area in bedrock bends was concentrated at the center of the cross‐section around the bend apex upstream. Our experimental results can improve incision models in terms of the distribution of shear stress and flow structure in bedrock bends.

Width undulation drives flow convergence routing in five flashy ephemeral river types across a dry summer subtropical region

AbstractDuring the last decade, meter‐resolution topo‐bathymetric digital elevation models (DEMs) have become increasingly utilized within fluvial geomorphology, but most meter‐scale geomorphic analyses are done on just one to a few rivers. While such analyses have contributed greatly to our collective understanding of river discharge‐topography interactions, which is applicable in both river restoration design and environmental flow regulation contexts, their generalizability across a range of river types remains largely unevaluated. This study assessed the dominance of a single hydro‐morphodynamic mechanism, flow convergence routing, in 35 ephemeral rivers divided among five river types in California's South Coast region by answering five questions. Geomorphic covariance structure (GCS) analysis was performed on longitudinal standardized width and standardized, detrended bed elevation spatial series from meter‐resolution DEMs. All river types had coherent, multi‐scalar structures of longitudinal fluvial topography, implicating a process‐morphology link. GCS metrics revealed that landform patterning was consistent with the requirements of the morphodynamic mechanism of flow convergence routing. Thus, that process was found to be a broadly relevant channel altering mechanism among sites, but its relationship with water stage differed between river types. Specifically, river types in unconfined valleys exhibited a strong bankfull width control over base flow bed undulations, with no obvious flood‐stage control over bankfull landform patterning. River types in partially confined valleys also exhibited strong bankfull width control over base flow bed undulations, but their bankfull landform patterns appear to have coalesced with coherent width and bed elevation undulations during flood flows. Finally, metrics for confined river types showed that it takes higher magnitude, less frequent floods to set their coherent width and bed elevation undulations, but even these channels do exhibit flow convergence routing when given enough discharge for sufficient duration.

Investigating the Linkage Between Spiral Trough Morphology and Cloud Coverage on the Martian North Polar Layered Deposits

AbstractThe martian North Polar layered deposits (NPLD) are composed of alternating water‐ice and dust‐rich layers resulting from atmospheric deposition and are key to understanding Mars climate cycles. Within these deposits are spiral troughs whose migration affects deposition signals. To understand the relationship between NPLD stratigraphy and Martian climate, we must identify modern‐day drivers of NPLD ice migration. Prevailing theory posits migration driven by upstream‐migrating bed undulations bounded by hydraulic jumps, caused by katabatic winds flowing over trough walls with asymmetric cross‐sectional relief. This is supported by trough‐parallel clouds, whose formation has been attributed to hydraulic jumps. We present a cloud atlas across the Martian north pole using ∼13,800 THEMIS images spanning ∼18 Earth years. We find trough‐parallel clouds in ∼400 images, with regions nearer to the pole having higher cloud frequency. We compare spiral trough geometry to our cloud atlas and find regions with trough‐parallel clouds often correlate with metrics associated with modern‐day sublimation‐deposition cycles (i.e., relief and asymmetry), but not always. In some regions, troughs with morphologies conducive to cloud formation have no clouds. Overall, trough geometry varies greatly across the deposits, both within and between troughs, suggesting localized differences in deposition relative to migration, varying katabatic wind intensities, differing past climatic states influencing the troughs, varying trough initiation properties, or the possibility of additional mechanisms for trough initiation and migration (e.g., in situ trough erosion). Understanding what controls trough shape variability across the NPLD and how these controls change through time and space is key when interpreting Martian paleoclimate.

Velocity Field due to a Vertical Deformation of the Bottom of a Laminar Free-Surface Fluid Flow

This article investigates the velocity field of a free-surface flow subjected to harmonic deformation of the channel bottom, progressing asymptotically from a flat initial state to a maximum amplitude. Assuming a uniform main flow with the primary velocity component transverse to the bed undulation, analytical solutions are obtained for the three velocity components and free surface distortion using the method of perturbations. The perturbation components of the velocity field, streamlines, and surface deformation depend on a dimensionless parameter that reflects the fluid inertia induced by bed deformation relative to viscous resistance. When viscous effects dominate, a monotonic decay of the perturbations from the bed to the free surface is observed. In contrast, when inertia dominates, the perturbations can exhibit an oscillatory behavior and introduce circulation cells in the plane normal to the main flow. The interplay between inertia and viscosity reveals scenarios where surface and bed deformations are either in or out of phase, influencing vertical velocity components. Figures illustrate these phenomena, providing insights into the complex dynamics of free-surface flows with harmonic bed deformation in the direction normal to the main flow, and amplitude growing with time. The results are limited to small deformations of the channel bottom, as imposed by the linearization of the momentum equations. Even so, to the best of the authors’ knowledge, this problem has not been addressed before.

A hybrid boundary element method based model for wave interaction with submerged viscoelastic plates with an arbitrary bottom profile in frequency and time domain

This study investigates the scattering of surface ocean waves by a submerged viscoelastic plate placed over the variable bottom topography under the assumptions of linear theory. The solution is derived using the hybrid boundary element method. The boundary element method is faster and easier to use when compared to the most widely used analytical method, such as the eigenfunction expansion method. Moreover, the application of analytical methods is restricted to structures with regular geometries and flat sea bottom. However, the present solution technique works for the plate at any angle placed over the variable bottom topography. Furthermore, as a particular case, the scattering problem is analyzed when a rigid wall is placed downstream. Energy balance relations are derived to check the accuracy of the computed numerical results. The effect of sinusoidally varying bottom topography, damping parameter, and plate edge conditions on the Bragg resonance phenomenon is analyzed. Initially, the solutions are presented in the frequency domain using the hybrid boundary element method and then extended to the time domain using the Fourier transform. It is observed that when the edges of the submerged plate are fixed, the Bragg resonance occurs at lower values of the frequency parameter. However, the Bragg resonance occurs around the primary Bragg value when the plate has free edges. For certain incident wave frequencies, the viscoelastic plate that completely covers the undulating bed dissipates a greater amount of wave energy than when the plate only partially covers the seabed.

Identifying key channel variability functions controlling ecohydraulic conditions using synthetic channel archetypes

AbstractGeometric modelling of river channel topography is a method of design synthesis wherein specific 2D geometric elements of river topography, such as the bed profile, cross‐sectional shape and channel planform contours, are expressed mathematically in isolation and then combined to produce a 3D heightmap. We utilized the geometric modelling framework to synthesize channel terrains that reveal flow–form–function linkages to investigate what roles variability in bed roughness, thalweg elevation and channel width play in defining hydraulic and ecohydraulic conditions of a channel reach from baseflow to bankfull discharge. To achieve a robust inquiry for a range of settings, this study developed four distinct synthetic channel terrain models for each of three stream reaches of different channel types in the South Fork Eel River in northern coastal California, USA. To test the process‐based effects of these diverse terrain synthesis options, we compared the resulting hydraulic patterns and preferred habitat availability for fry/juvenile steelhead trout and coho salmon over a range of discharges. Among thalweg bed undulation, width variation and bed roughness, we found thalweg bed undulation was the key factor affecting the channel ecohydraulic response at baseflow condition. At bankfull condition, thalweg bed elevation had the largest effect in high‐order mainstem streams identified by gravel‐cobble dominated high width‐to‐depth, riffle‐pool sequences, width variation had the largest effect in mid‐order confined channels with gravel‐cobble, high width‐to‐depth ratio with expansions/contractions and bed roughness in low‐order streams with low width‐to‐depth ratio, high‐gradient, cobble‐boulder and step‐pool/cascade channels.

Performance of conservative and non-conservative two-dimensional shallow water models in wavy riverbed

The conservativeness, stability, and robustness of the numerical models applied in braided river systems is primarily influenced by morphological complexities adverse, flow pattern and the channel roughness. Two-dimensional flow modelling over an undulating bed profile is critical on account of the existence of subsequent wet-dry regions in the flow domain. This work provides the applicability of the conservative and non-conservative form of two-dimensional unsteady flow equations in braided river system. Both models are solved by TVD McCormack predictor corrector finite difference method and applied in the Brahmaputra River near Majuli Island, Assam India. The computed outputs are compared with the field observed results for validation. Results indicate a satisfactory agreement between the computed and measured value. Two statistical indices are used in the model performance evaluation.

Modeling Optimal Canal Conveyance Capacity for the Ahero Irrigation Scheme using the Hydrologic Engineering Centre River Analysis System (HEC-RAS)

Irrigation plays a critical role in addressing food security as envisaged in Kenya’s development blueprint, the Big Four Agenda. However, the performance of any open channel irrigation system is a function of its canal conveyance efficiency, among other factors. To overcome challenges with irrigation water conveyance at the Ahero Irrigation Scheme, a Hydrologic Engineering Centre River Analysis System (HEC-RAS) model was used to simulate the flow characteristics at the tail-end section of the canal network, covering a total length of 2.6 km. The study also consisted of a comparative review of an FAO-CROPWAT model estimation water requirement for rice. The manual estimation of the canal capacity in its unmaintained state revealed a discharge capacity of 0.228 m3/s, which was significantly lower than the minimum crop water demand requirement estimation of 0.3166 m3/s (a 28% water deficit). The simulated characteristics projected an optimal flow capacity of 0.583 m3/s. The study recommends canal maintenance (levelling bed undulations, dredging, and smooth concrete lining) to attain the optimal flow capacity at the tail end of the network.

Propagation, mixing, and turbulence characteristics of saline and turbidity currents over rough and permeable/impermeable beds

A series of constant-flux saline and turbidity current experiments were carried out, focusing on the coupling impact of bed roughness and permeability on current propagation, mixing, and turbulence characteristics. The distinct current propagation phases on RI (rough and impermeable) and RP (rough and permeable) beds are identified, respectively. Experimental results revealed that the intermittently undulating bed surface breaks the strict no-slip boundary, thus, increasing local current velocity near the bed, while its roughness reduces the current peak profile velocity. Interbed pores induced vertical fluid exchange, which synchronously decreases the current peak profile velocity and local velocity near the bed, causes the density profile to no longer follow a monotonous variation trend along with water depth. The larger bed surface roughness or the interbed porosity leads to the smaller upper TKE (turbulent kinetic energy) peak. The lower TKE peak is inversely proportional to the bed surface roughness of the RI beds, while it is proportional to the porosity of the RP bed. A rough bed surface intensifies the asymmetry of the mean velocity distribution around peak velocity resulting in a transfer barrier of turbulent momentum triggered by the interbed pores. On the RP bed, the cross-correlation function based on two-point statistics captures the spikes associated with pore-scale eddies locally, but under the RI condition, it only obtains the logical timescale characterizing the largest eddies of the current. The sediment deposition makes the turbidity current easier to separate from the RP and RI bed than the saline type, causing a consequence of growing the current height.

An augmented lagrangian algorithm for recovery of ice thickness in unidirectional flow using the shallow ice approximation

A key parameter in ice flow modelling is basal slipping at the ice-bed interface as it can have a large effect on the resultant ice thickness. Unfortunately, its contribution to surface observations can be hard to distinguish from that of bed undulations. Therefore, inferring the ice thickness from surface measurements is an interesting and non-trivial inverse problem. This paper presents a method for recovering dually the ice thickness, in turn the bed topography, and the basal slip using only surface elevation and speed measurements. The unidirectional shallow ice approximation is first implemented to model steady state ice flow for given bedrock and basal slip profiles. This surface is then taken as synthetic observed data. To invert the data, an optimisation approach is taken, employing a basic Uzawa type algorithm to minimise the augmented Lagrangian objective functional. This optimisation recovers the diffusion coefficient which gives the best fit to observations. Combining this recovered diffusion with observed surface velocity, a simple Newton’s method is used to recover both the ice thickness and basal slip. The method was successful in each test case and this implies that it should be possible to recover both of these parameters in two-dimensional cases also.

The Causes of Debris-Covered Glacier Thinning: Evidence for the Importance of Ice Dynamics From Kennicott Glacier, Alaska

The cause of debris-covered glacier thinning remains controversial. One hypothesis asserts that melt hotspots (ice cliffs, ponds, or thin debris) increase thinning, while the other posits that declining ice flow leads to dynamic thinning under thick debris. Alaska’s Kennicott Glacier is ideal for testing these hypotheses, as ice cliffs within the debris-covered tongue are abundant and surface velocities decline rapidly downglacier. To explore the cause of patterns in melt hotspots, ice flow, and thinning, we consider their evolution over several decades. We compile a wide range of ice dynamical and mass balance datasets which we cross-correlate and analyze in a step-by-step fashion. We show that an undulating bed that deepens upglacier controls ice flow in the lower 8.5 km of Kennicott Glacier. The imposed velocity pattern strongly affects debris thickness, which in turn leads to annual melt rates that decline towards the terminus. Ice cliff abundance correlates highly with the rate of surface compression, while pond occurrence is strongly negatively correlated with driving stress. A new positive feedback is identified between ice cliffs, streams and surface topography that leads to chaotic topography. As the glacier thinned between 1991 and 2015, surface melt in the study area decreased, despite generally rising air temperatures. Four additional feedbacks relating glacier thinning to melt changes are evident: the debris feedback (negative), the ice cliff feedback (negative), the pond feedback (positive), and the relief feedback (positive). The debris and ice cliff feedbacks, which are tied to the change in surface velocity in time, likely reduced melt rates in time. We show this using a new method to invert for debris thickness change and englacial debris content (∼0.017% by volume) while also revealing that declining speeds and compressive flow led to debris thickening. The expansion of debris on the glacier surface follows changes in flow direction. Ultimately, glacier thinning upvalley from the continuously debris-covered portion of Kennicott Glacier, caused by mass balance changes, led to the reduction of flow into the study area. This caused ice emergence rates to decline rapidly leading to the occurrence of maximum, glacier-wide thinning under thick, insulating debris.

Formation of undulating topography and gravel beds at the bases of incised valleys: Last Glacial Maximum examples beneath the lowlands facing Tokyo Bay

Recent studies using well density distributed borehole logs have revealed undulating topography at the bases of incised valleys formed during the Last Glacial Maximum (LGM). In this study, from analysis of 4702 borehole logs, undulating topography forming a series of pits 1–2 km long, < 1 km wide, 5–10 m deep, and spaced at 1–2-km intervals was discovered at the bases of LGM incised valleys beneath the Tama River Lowland on the west coast of Tokyo Bay. This undulating topography can be attributed to scouring at braided river channel confluences. In the study area, single borehole logs are available within each 187 m × 187 m grid cell, and the logs sample both the bottom and marginal portions of the scouring, which suggests that this undulating topography is not an artifact of erroneous values arising from mathematical interpolation. The morphologies and incision depths of two incised valleys in the study area show a cover effect of the gravel bed at the base of the post-LGM incised-valley fills. The basal age of this basal gravel bed (BG) is confirmed at < 30 ka because the LGM incised valleys dissect the MIS 3 Tachikawa buried terrace overlain by the AT tephra dated 30.0 ka. This means that the BG, which represents braided-river sediments, is interpreted as resulting from the LGM sea-level lowstand after 30 ka.

Experimental investigation of the morphodynamic response of riffles and pools to unsteady flow and increased sediment supply

AbstractGravel‐bed rivers characteristically exhibit shallow riffles in wide sections and deeper pools where the channel becomes constricted and narrow. While rivers can adjust to changing flow and sediment supply through some combination of adjustments of channel slope, bed‐surface sorting, and channel shape, the degree to which riffle‐pools may adopt these changes in response to changing flows and sediment supplies remains unclear. This article presents results from a flume experiment investigating how constant‐ and variable‐width channels adjust their morphology in response to changing flow and increased sediment supply. Two flume geometries were used: (1) constant‐width and (2) variable‐width, characterized by a sinusoidal pattern with a mean width equal to that of the first channel. The variable‐width channel developed bed undulations in phase with the width, representing riffle‐pools. The experiment consisted of three phases for each flume geometry: (1) steady flow, constant sediment supply; (2) unsteady flow, constant sediment supply; and (3) unsteady flow, doubled sediment supply. Unsteady flow was implemented in the form of repeated symmetrical stepped hydrographs, with a mean discharge equal to that in the steady flow phase. In all phases the bed and sediment supply were composed of a sand/gravel mixture ranging from 1 to 8 mm. In both the straight and variable‐width channels, transitioning from steady flow to repeated hydrographs did not result in significant changes in bed morphology. The two channel geometries had different responses to increased sediment supply: the slope of the straight channel increased nearly 40%, while the variable‐width channel reduced the relief between bars and pools and decreased the variability in cross‐sectional elevation with a slight slope increase. Bar‐pool relief varied with repeat discharge hydrographs. Pool elevation changed twice the distance of bar elevations, emphasizing the relevance of pool scour for riffle‐pool self‐maintenance in channels with width variations.

Hyporheic exchanges due to channel bed and width undulations

Riffle-pool sequences are fundamental, ubiquitous morphological features of alluvial rivers that are thoroughly studied in general and commonly incorporated into river restoration projects. Most previous investigations on the effect of riffle-pool sequences on hyporheic exchanges focused on solely bed undulation, because that is widely thought to be the defining feature of riffle-pool sequences. However, riffle- pool sequences also have significant width undulations that are vital to riffle-pool sequences morphodynamics, yet relatively few studies exist on the effect of such undulations on hyporheic exchanges. Thus, in this study based on laboratory experiments and numerical simulations, we investigate the effect of bed and width undulations on hyporheic zone characteristics for various ratio of width amplitude to bed amplitude. The variation of hyporheic exchange characteristics (i.e. hyporheic exchange flux and residence time) for different riffle-pool designs were also assessed by creating prescribed topography of synthetic river valleys. The results showed that due to pressure variation along width undulations, the upwelling and downwelling patterns can also be observed for only width undulations in rivers, in the absence of bed undulations, and as width undulation amplitude decreases, the normalized hyporheic exchanges (Q*HZ) increase and normalized median residence time (RT*) decreases. Also, simultaneous channel bed and width undulations result in higher Q*HZ especially when a pool is located in an expansion (aka “oversized” cross-section) and a riffle in a constriction (“nozzle”). Our results suggest that river restoration project that artificially construct wide pools and constricted riffles will achieve maximum Q*HZ and RT*, though they will not be geomorphically self-sustainable.

Investigation of water wave scattering by an elastic sea-bed of varying depth in two superposed fluids covered by an ice-sheet

A hydroelastic model is taken up in this work to investigate the propagation of normal incident waves over a small undulation located on an elastic sea-bed in a fluid flow for two superposed fluids where a floating ice-sheet covers the upper fluid. Here, both the floating ice-sheet and elastic sea-bed are approximated as thin elastic plates which follow the Euler–Bernoulli beam equation, familiar for such studies. Due to its minimal contribution, the surface tension at the interface of the fluid layers is not accounted for. The behavior of the roots of the associated complex dispersion relation is investigated through the application of a counting argument along with a contour plot. The time-harmonic waves, for a given frequency, are observed to propagate with exactly two wavenumbers such that the one corresponding to the smaller wavenumber propagates along the ice-sheet while the other corresponding to the higher wavenumber along the interface. By assuming the fluid to be incompressible and inviscid following an irrotational motion, and then applying a perturbation technique, the first-order contribution to the velocity potentials in each layer is evaluated with the help of Fourier transform. Following that, the associated reflection and transmission coefficients are computed by carrying out appropriate integration for integrals containing a shape function depicting the sea-bed undulation. In order to validate the present model developed in this work, one obtained result for reflection coefficient is validated against an available result which points towards a good agreement. This validation establishes the model and the method followed in this work to be effective and allows subsequent investigation for this problem to be taken up. For verification of the theory developed here, one specific undulating bed topography is selected to assess the reflection as well as transmission coefficients which are depicted by graphical representations to examine and establish the energy conversion between the flexural wave mode and the interfacial wave mode. Present investigation demonstrates that the reflection and transmission of the waves exhibit an oscillatory behavior, the reason for which is the outcome of multiple interaction of the incident wave with the elastic bottom undulation, interface and the ice-sheet. It further shows that flexural wave moment is significantly affected as compared to the interfacial wave influence. It is noted that significant variation in the elastic specification of the ice-sheet and the elastic sea-bed has a considerable influence for wave scattering by the undulating elastic sea-bed. It is observed that, when water waves propagate over a deformable bottom, the wave characteristics undergo a significant change. This occurs due to the elasticity of the sea-bed which immensely influences the kinematics and dynamics of the wave over and above providing the demonstration of the elastic behavior of the soil located below the elastic sea-bed. The approach adopted here is likely to be of immense significance in the design and construction of various types of floating breakwaters and very large floating structures (VLFS) for the purpose of various scientific activities mainly in polar regions. Further, the solutions of problems like these dealing with a deformable sea-bed will be beneficial in providing the required background to the researchers with respect to the incorporation of the characteristics of the infinite depth soil located below the sea-bed.

Effect of undulating bottom on wave interaction with a floating flexible plate coupled with a flexible porous barrier

In the present study, under the assumption of small amplitude water wave theory and structural response, effect of bed undulation on the wave interaction with a combination of flexible porous barrier and a flexible floating plate, is studied. The flexible porous barrier is modelled using the porous wave maker-theory while the elastic floating plate is modelled using thin plate theory. The physical problem is handled for solution using eigenfunction expansion method by matching pressure and velocity at interface boundaries while finite difference method is used to deal with modified mild-slope equation. In the present study, two types of plate configurations are namely (a) finite plate and (b) semi-infinite plate. To understand the role of undulating seabed, wave and structural parameters, in attenuating wave force and plate deflection, numerical results are computed and compared with available literature. It is found that full wave reflection occurs for certain critical angle and wave force acting on the barrier vanishes for the same critical angle for the semi-infinite plate case while the wave reflection tends to attain maximum value at critical angle for the finite case and these maximum value increases with the increase of plate length. It is observed that less wave reflection occurs for sloping bed profile compare to rest of the bed profile. The study reveals that positioning a barrier between plate and undulated region, helps in the reduction of plate deflection significantly. As special case, wave interaction problem with a finite floating elastic plate is studied experimentally and the experimental result has shown good agreement with the numerical result. The findings of the present study are likely to be of immense help in the design of various types of marine structures for protecting very large floating structures (VLFS). The present theory can be extended to handle large class of acoustic wave interaction problems with flexible porous structures.

Mathematical modeling of oscillating water column wave energy converter devices over the undulated sea bed

The present study deals with the hydrodynamic performance of an OWC (oscillating water column) wave energy converter device placed on an undulated bed. The bed undulation is finitely extended and beyond that, the water bed is uniform in nature. The mathematical problem is studied in the two-dimensional Cartesian coordinate system under the linearized water wave theory. To solve the associated boundary value problem, a coupled eigenfunction expansion - boundary element method is used. The numerical convergence of the coupled eigenfunction expansion - boundary element method is provided. The effects of the width of the OWC device chamber, front wall submergence depth and inclination with the vertical axis, and the bed undulations on the hydrodynamic efficiency of the OWC device is discussed in detail.

Propagation of Oblique Waves Over a Small Undulating Elastic Bottom Topography in a Two-Layer Fluid Flowing Through a Channel

A hydrodynamic model, with the incorporation of elasticity, is considered to study oblique incident waves propagating over a small undulation on an elastic bed in a two-layer fluid, where the upper layer fluid is bounded above by a rigid lid, which is an approximation to the free surface. Following the Euler–Bernoulli beam equation, the elastic bed is approximated as a thin elastic plate. The surface tension at the interface of the layers is completely ignored since its contribution will be minimal. The behavior and properties of the roots of the dispersion relation are analyzed using counting argument and contour plot. In this case, time-harmonic waves propagate with only one wavenumber. Considering an irrotational motion in an incompressible and inviscid fluid, and applying perturbation technique, the first-order corrections to the velocity potentials are evaluated by an appropriate application of Fourier transform and, subsequently, the corresponding reflection and transmission coefficients are computed through integrals containing a shape function which depicts the bottom undulation. To validate the theory developed, two particular undulating bottom topography are taken up as examples in order to evaluate the hydrodynamic coefficients which are represented through graphs. It is noted that the reflection coefficient shows an oscillatory pattern, when the wavenumber of the undulating bed takes a value almost double the wavenumber of the incident wave. When the ratio of the wavenumber of the undulating bed and the wavenumber of the incident wave approaches 2, the theory predicts existence of resonance between the undulating elastic bed and the interface of the layers. It is worthwhile to note that reflected wave energy of the incident wave is higher if the number of ripples of the bottom undulation is large. It is further noted that the reflected energy increases in response to an increase in the values of the elastic parameters of the ocean bed. The results for a patch of sinusoidal ripples having different wavenumbers are found to be similar. Further, the reflected energy decreases for an increase in the angle of incidence. It is noted that reasonable changes in elasticity, ripple wavenumber and the number of the ripples of the bed have a remarkable effect when the propagating wave passing through a channel encounters a small bottom undulation. When results are compared with the available results, good agreement is observed.

A simple and efficient numerical model for simulating one dimensional dam break flows

A semi implicit finite difference scheme is presented in this paper for simulation of dam break flow over undulated bed. For this purpose, the one dimensional (1D) governing equations of shallow water flow, in a deviatoric version is solved here. One of the main important features of the present model is that, no special treatment is required here to deal with the offending friction slope source term for very less water depth. Regarding the bed slope term also, simple central difference approximation is sufficient to handle undulated channel bed. To investigate the performance of the present model in diverse situations, it is used to replicate different dam break test problems including one of known analytical solution and two experimental dam break cases. A detail analysis of the results show that the model capable is of producing good quality results for varied conditions. A sensitivity analysis is carried out for the damping parameter and an optimal range is proposed for it.

Revisiting terrain-following Boussinesq equations on a highly variable periodic bed

On re-deriving the Boussinesq-type equations in a terrain-following coordinate system based on the conformal map for an arbitrarily (submerged) periodic bed, it is pointed out that some aspects of the existing approach need to be clarified, specifically the use of velocity potential gradient in the mapped plane in connection with the transformation of velocity vector, and the appropriate frequency–wavenumber relation that is from the solution to the linearized Boussinesq equations. It is shown that as the bed undulation height increases, this relationship increasingly departs from the flat-bottom dispersion relation that has previously been assumed for the terrain-following Boussinesq systems. Over a highly variable periodic bed, the waveforms of linear time harmonic waves already have features reminiscent of nonlinear waveforms, which should be distinguished from the subsequent nonlinear evolution.