D. Calvete

Morphological development of rip channel systems: Normal and near-normal wave incidence

The process of formation of a rip channel/crescentic bar system on a straight, sandy coast is examined. A short review of earlier studies is presented. A morphodynamic stability model is then formulated. The resulting model includes a comprehensive treatment of shoaling and surf zone hydrodynamics, including wave refraction on depth and currents and waves. The sediment transport is modeled using a total load formula. This model is used to study the formation of rip currents and channels on a straight single-barred coast. It is found that this more comprehensive treatment of the dynamics reveals the basic rip cells predicted in earlier studies for normal incidence. Also as before, cell spacings (λ) scale with shore-to-bar crest distance (X b ), while growth rates decrease. The λ increases with offshore wave height (H) up to a saturation value; increasing H also increases instability. Experiments at off-normal wave incidence ( > 0) introduce obliquity into the evolving bed forms, as expected, and λ increases approximately linearly. the e-folding times also increase with . At normal incidence, λ increases weakly with wave period, but at oblique angles, λ decreases. Tests also reveal the presence of forced circulation cells nearer to the shoreline, which carve out bed forms there. The dynamics of these forced cells is illustrated and discussed along with the associated shoreline perturbation. Transverse bars are also discovered. Their dynamics are discussed. Model predictions are also compared with field observations. The relevance of the present approach to predictions of fully developed beach states is also discussed.

Large‐scale dynamics of sandy coastlines: Diffusivity and instability

The dynamics of small-amplitude perturbations of an otherwise rectilinear coastline due to the wave-driven alongshore sediment transport is examined at large time and length scales (years and kilometers). A linear stability analysis is performed by using an extended one-line shoreline model with two main improvements: (1) the curvature of the coastline features is accounted for and (2) the coastline features are assumed to extend offshore as a bathymetric perturbation up to a finite distance. For high incidence angles, instability is found in accordance with Ashton et al. (2001). However, it is seen that instability is inhibited by high waves with long periods and gently sloping shorefaces so that in this case the coastline may be stable for any angle. Similarly, there is no instability if the bathymetric perturbation is confined very close to the coast. It is found that the traditional linearized one-line model (Larson et al., 1987) tends to overpredict the coastline diffusivity. The overprediction is small for the conditions leading to a stable coastline and for moderate incidence angles but can be very dramatic for the conditions favoring instability. An interesting finding is that high-angle waves instability has a dominant wavelength at the linear regime, which is in the order of 4–15 km, one to two orders of magnitude larger than the length scale of surf zone rhythmic features. Intriguingly, this is roughly the same range of the wavelength of some observed shoreline sand waves and, in particular, those observed along the Dutch coast. A model application to this coast is presented.

Modelling the formation and the long‐term behavior of rip channel systems from the deformation of a longshore bar

A nonlinear numerical model based on a wave- and depth-averaged shallow water equation solver with wave driver, sediment transport, and bed updating is used to investigate the long-term evolution of rip channel systems appearing from the deformation of a longshore bar. Linear and nonlinear regimes in the morphological evolution have been studied. In the linear regime, a crescentic bar system emerges as a free instability. In the nonlinear regime, merging/splitting in bars and saturation of the growth are obtained. In spite of excluding undertow and wave-asymmetry sediment transport, the initial crescentic bar system reorganizes to form a large-scale and shore-attached transverse or oblique bar system, which is found to be a dynamical equilibrium state of the beach system. Thus the basic morphological transitions “Longshore Bar and Trough” → “Rhythmic Bar and Beach” → “Transverse Bar and Rip” described by earlier conceptual models are here reproduced. The study of the physical mechanisms allows us to understand the role of the different transport modes: The advective part induces the formation of crescentic bars and megacusps, and the bedslope transport damps the instability. Both terms contribute to the attachment of the megacusps to the crescentic bars. Depending on the wave forcing, the bar wavelength ranges between 180 and 250 m (165 and 320 m) in the linear (nonlinear) regime.

Generation and nonlinear evolution of shore-oblique/transverse sand bars

The coupling between topography, waves and currents in the surf zone may self-organize to produce the formation of shore-transverse or shore-oblique sand bars on an otherwise alongshore uniform beach. In the absence of shore-parallel bars, this has been shown by previous studies of linear stability analysis, but is now extended to the finite-amplitude regime. To this end, a nonlinear model coupling wave transformation and breaking, a shallow-water equations solver, sediment transport and bed updating is developed. The sediment flux consists of a stirring factor multiplied by the depth-averaged current plus a downslope correction. It is found that the cross-shore profile of the ratio of stirring factor to water depth together with the wave incidence angle primarily determine the shape and the type of bars, either transverse or oblique to the shore. In the latter case, they can open an acute angle against the current (up-current oriented) or with the current (down-current oriented). At the initial stages of development, both the intensity of the instability which is responsible for the formation of the bars and the damping due to downslope transport grow at a similar rate with bar amplitude, the former being somewhat stronger. As bars keep on growing, their finite-amplitude shape either enhances downslope transport or weakens the instability mechanism so that an equilibrium between both opposing tendencies occurs, leading to a final saturated amplitude. The overall shape of the saturated bars in plan view is similar to that of the small-amplitude ones. However, the final spacings may be up to a factor of 2 larger and final celerities can also be about a factor of 2 smaller or larger. In the case of alongshore migrating bars, the asymmetry of the longshore sections, the lee being steeper than the stoss, is well reproduced. Complex dynamics with merging and splitting of individual bars sometimes occur. Finally, in the case of shore-normal incidence the rip currents in the troughs between the bars are jet-like while the onshore return flow is wider and weaker as is observed in nature.

Modelling the formation of shoreface-connected sand ridges on storm-dominated inner shelves

A morphodynamic model is developed and analysed to gain fundamental understanding of the basic physical mechanisms responsible for the characteristics of shorefaceconnected sand ridges observed in some coastal seas. These alongshore rhythmic bed forms have a horizontal lengthscale of order 5 km and are related to the mean current along the coast: the seaward ends of their crests are shifted upstream with respect to where they are attached to the shoreface. The model is based on the two-dimensional shallow water equations and assumes that the sediment transport only takes place during storms. The flux consists of a suspended-load part and a bed-load part and accounts for the e ects of spatially non-uniform wave stirring as well as for the preferred downslope movement of sediment. The basic state of this model represents a steady longshore current, driven by wind and a pressure gradient. The dynamics of small perturbations to this state are controlled by a physical mechanism which is related to the transverse bottom slope. This causes a seaward deflection of the current over the ridges and the loss of sediment carrying capacity of the flow into deeper water. The orientation, spacing and shape of the modelled ridges agree well with eld observations. Suspended-load transport and spatially non-uniform wave stirring are necessary in order to obtain correct e-folding timescales and migration speeds. The ridge growth is only due to suspended-load transport whereas the migration is controlled by bed-load transport.

Effect of wave-topography interactions on the formation of sand ridges on the shelf

[1] The role of wave-topography interactions in the formation of sand ridges on microtidal inner shelves is investigated with an idealized morphodynamic model. The latter uses the two-dimensional shallow water equations to describe a storm-driven flow on an inner shelf with an erodible bottom and a transverse slope. Both bed load and suspended load sediment transport are included. New are the incorporation of a wave module based on physical principles and a critical shear-stress for erosion. A linear stability analysis is used to study the initial growth of bed forms, by analyzing the initial growth of small perturbations evolving on an alongshore uniform basic state, which describes a storm-driven flow on a microtidal inner shelf. Model simulations show that wave-topography interactions cause the ridges to become more trapped to the coast. Both growth and migration of the ridges are controlled by suspended load transport. The physical mechanism responsible for ridge growth is related to transport by the stormdriven current of sediment that is entrained due to wave orbital motions induced by bed forms. This new mechanism even acts in absence of a transverse bottom slope. The orientation, spacing and shape of the modeled ridges agree well with field observations from different shelves.

On beach cusp formation

A system of shallow water equations and a bed evolution equation are used to examine the evolution of perturbations on an erodible, initially plane beach subject to normal wave incidence. Both a permeable (under Darcys law) and an impermeable beach are considered. It is found that alongshore-periodic morphological features reminiscent of swash beach cusps form after a number of incident wave periods on both beaches. On the permeable (impermeable) beach these patterns are accretional (erosional). In both cases flow is ‘horn divergent’. Spacings of the cusps are consistent with observations, and are close to those provided by a standing synchronous linear edge wave. An analysis of the processes leading to bed change is presented. Two physical mechanisms are identified: concentration gradient and flow divergence, which are dominant in the lower and upper swash respectively, and their difference over a wave cycle leads to erosion or deposition on an impermeable beach. Infiltration enters this balance in the upper swash. A bed wave of elevation is shown to advance up the beach at the tip of the uprush, with a smaller wave of depression on the backwash. It is found that cusp horns can grow by a positive feedback mechanism stemming from decreased (increased) backwash on positive (negative) bed perturbations.

A nonlinear model study on the long-term behavior of shore face-connected sand ridges

A morphodynamic model is analyzed to gain further knowledge about the finite amplitude behavior of shore face–connected sand ridges observed on storm-dominated inner shelves. The present work elaborates on previous studies in which it was demonstrated that ridge formation may be due to an instability of a storm-driven current moving over a sandy inner shelf with a transverse slope. Here, the long-term evolution of the ridges is studied by performing a nonlinear stability analysis in which the physical variables are expanded in eigenmodes of the linear stability problem. New physical aspects are that the longshore pressure gradient in the momentum equations and settling lag in the suspended load sediment transport are incorporated. Furthermore, along-shelf uniform shelf modes and subharmonic eigenmodes are accounted for. The model shows that after the transient stage the competition between the modes results in saturation behavior that is dominated by a few modes only. The characteristic height of the final bed forms increases with increasing transverse slopes of the shelf, while the timescale of transient behavior decreases. The longshore uniform modes, pressure gradient, and settling lag effects only have a minor effect on the dynamics. A process analysis reveals that the mechanism responsible for the saturation behavior is the sediment transport related to the bottom slope and the effect of small-scale bed forms. Subharmonic modes significantly affect the transient behavior of the ridges and cause the final bed forms to have larger amplitudes and longer wavelengths.

Effect of grain size sorting on the formation of shoreface-connected sand ridges

Field data of shoreface-connected ridges show persistent spatial variations of mean grain size over the bed forms. In the shore-normal direction, the profiles of bottom topography and mean grain size are approximately 90° out of phase. To investigate the mechanisms responsible for the observed grain size distribution and the influence of sediment sorting on the temporal and spatial characteristics of shoreface-connected ridges, a model is developed and analyzed. A linear stability analysis of an alongshore uniform basic state (describing a storm-driven flow on a microtidal inner shelf) with respect to small bottom perturbations is carried out. The transport of nonuniform sediment is described by formulations for both bed load and suspended load, both of which account for dynamic hiding effects. A one-layer model for the bed evolution and a bottom friction term, which depends on the grain size, are used. The initial formation of the ridges is studied for a bimodal sediment mixture. The results of the model indicate that the phase shift between bed topography and mean grain size for shoreface-connected ridges is due to the selective transport via suspended load of grains with different sizes. A net stabilizing effect on the growth of bed forms and enhanced migration are predicted, caused by the bimodal character of the sediment. The wavelengths of the bed forms are only slightly affected. Including a tidal current or a grain size dependent formulation for the bottom friction has no effect on the results. A physical explanation for the model results is also given.

Growth of large-scale bed forms due to storm-driven and tidal currents: a model approach

An idealized morphodynamic model is used to gain further understanding about the formation and characteristics of shoreface-connected sand ridges and tidal sand banks on the continental shelf. The model consists of the 2D shallow water equations, supplemented with a sediment transport formulation and describes the initial feedback between currents and small amplitude bed forms. The behaviour of bed forms during both storm and fair weather conditions is analyzed. This is relevant in case of coastal seas characterized by tidal motion, where the latter causes continuous transport of sediment as bed load. The new aspects of this work are the incorporation of both steady and tidal currents (represented by an M2 and M4 component) in the external forcing, in combination with dominant suspended sediment transport during storms. The results indicate that the dynamics during storms and fair weather strongly differ, causing different types of bed forms to develop. Shoreface-connected sand ridges mainly form during storm conditions, whereas if fair weather conditions prevail the more offshore located tidal sand banks develop. Including the M4 tide changes the properties of the bed forms, such as growth rates and migration speeds, due to tidal asymmetry. Finally a probabilistic formulation of the storm and fair weather realization of the model is used to find conditions for which both types of large-scale bed forms occur simultaneously. These conditions turn out to be a low storm fraction and the presence strong tidal currents in combination with strong steady currents during storms.