Effect of wave–bedform feedbacks on the formation of, and grain sorting over shoreface-connected sand ridges
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DOI: 10.1007/s10236-009-0210-9
- Cite this article as:
- Vis-Star, N.C., de Swart, H.E. & Calvete, D. Ocean Dynamics (2009) 59: 731. doi:10.1007/s10236-009-0210-9
Abstract
The influence of wave–bedform feedbacks on both the initial formation of shoreface-connected sand ridges (sfcr) and on grain size sorting over these ridges on micro-tidal inner shelves is studied. Also, the effect of sediment sorting on the growth and the migration of sfcr is investigated. This is done by applying a linear stability analysis to an idealized process-based morphodynamic model, which simulates the initial growth of sfcr due to the positive coupling between waves, currents, and an erodible bed. The sediment consists of sand grains with two different sizes. New elements with respect to earlier studies on grain sorting over sfcr are that wave-topography interactions are explicitly accounted for, entrainment of sediment depends on bottom roughness, and transport of suspended sediment involves settling lag effects. The results of the model indicate that sediment sorting causes a reduction of the growth rate and migration speed of sfcr, whereas the wavelength is only slightly affected. In the case where the entrainment of suspended sediment depends on bottom roughness, the coarsest sediment is found in the troughs; otherwise, the finest sediment occurs in the troughs. Compared to previous work, modeled maximum variations in the mean grain size over the topography are in better agreement with field observations. Settling lag effects are important for the damping of high-wavenumber mode instabilities such that a preferred wavelength of the bedforms is obtained.
Keywords
Sand ridges Stability analysis Storm-driven current Hiding Long Island shelf1 Introduction
Shoreface-connected sand ridges (hereafter abbreviated as sfcr) are rhythmic bottom patterns that are found on sandy, storm-dominated inner shelves in water depths of 5–30 m. During storms, the water motion is characterized by high waves (wave heights of 2–4 m) and a mean storm-driven alongshore flow of up to 0.5 m s^{ − 1}. Field observations (Swift et al. 1978; Antia 1996; Van de Meene and Van Rijn 2000; Schwab et al. 2000, and references therein) reveal that crests of sfcr have a length between 10 and 25 km, their heights are 1–6 m, and distances between successive crests are 2–6 km. Sfcr are attached to the shoreface and their seaward ends are located further upstream (with respect to the direction of the storm-driven flow). Angles between the crests and the coastline are in the range 20–50°. Sfcr evolve on a time scale of centuries, and they migrate in the direction of the storm-driven flow with 1–50 m year^{ − 1}, depending on the measuring period. Furthermore, field data indicate a persistent pattern of grain sorting over sfcr. In many cases, the sediment at the seabed is coarsest on the landward flanks and finest on the seaward flanks. This phase shift between the mean grain size and topography variations is documented for sfcr on the Mid Atlantic shelf (e.g., Swift et al. 1972; Schwab et al. 2000), the inner shelf of Argentina (Parker et al. 1982), and the German Bight (Antia 1996).
Both observations (Swift and Field 1981) and model studies (Trowbridge 1995; Calvete et al. 2001) have indicated that sfcr can grow due to positive feedbacks between the water motion and the erodible sandy bed. During storms, bottom stresses exerted by the waves are strong enough to entrain the sediment, which is subsequently transported by the storm-driven flow. The ridges induce spatial variations in the flow and in the sediment transport. This results in net deposition of sediment over the crests, whilst net erosion occurs in the troughs. As a result, the ridges grow. Their migration is due to the fact that the maximum deposition occurs slightly downstream of their crests. The models investigate the formation of sfcr by calculating the growth of arbitrary bottom perturbations with small amplitudes. Calvete et al. (2001) demonstrated that their fastest growing bottom perturbations have characteristics that agree quite well with those of observed sfcr.
One limitation of these models is that they consider sediment with a uniform grain size, which is not in accordance with field data. Walgreen et al. (2003) analyzed a model that considers the effect of sediment sorting on the formation of sfcr. Results of this model indicate that the phase shift between bed topography and mean grain size is due to the selective transport via suspended load of grains with different sizes. Furthermore, sediment sorting has a net stabilizing effect on the growth of sfcr, whereas their migration speed becomes larger and their wavelength is only slightly affected. Process-based models have also been developed to investigate grain sorting over other coastal bedforms, such as ripples (Foti and Blondeaux 1995), sorted bedforms on the inner shelf (Murray and Thieler 2004; Coco et al. 2007a, b; Huntley et al. 2008), tidal sand banks (Walgreen et al. 2004; Roos et al. 2007), and tidal sand waves (Roos et al. 2008; Van Oyen and Blondeaux 2009); see also the review by Holland and Elmore (2008).
The models on sfcr discussed above employ a strongly simplified description of waves in calculating stirring of sediment. In two recent studies by Lane and Restrepo (2007) and Vis-Star et al. (2007), a more sophisticated description of waves is applied, based on linear wave theory, which enables to include explicit feedbacks between waves and sfcr. The latter results in additional transport (by the storm-driven flow) of sediment being entrained by the wave orbital motion, which is generated by the bedforms. In the latter study, it is demonstrated that the spatial variations in this sediment transport are such that both the growth and downstream migration of sfcr are enhanced. Lane and Restrepo (2007), on the other hand, report that they do not obtain growth of bedforms if they include wave–bedform interactions. This seems to be a consequence of their assumption that sediment transport is proportional to the mass transport velocity (which accounts for currents and wave-induced Stokes drift), rather than being proportional to the near-bed current.
In this paper, the model of Vis-Star et al. (2007) is extended such that it enables the exploration of wave–bedform feedbacks in a model for bimodal sediment mixtures. This is done by adopting and extending the sediment sorting module for a bimodal sediment mixture of Walgreen et al. (2003). The first aim of this work is to investigate whether wave–bedform interactions play an important role in the initial growth of sfcr, as well as in the initial onset of grain sorting over sfcr. Secondly, the influence of sediment sorting on the characteristics of sfcr is explored. Several new aspects are included in the sediment module. First, the longshore-averaged value of the mean grain size is allowed to vary in the cross-shore direction. Second, motivated by the work of Murray and Thieler (2004), Coco et al. (2007a, b), and Huntley et al. (2008), settling lag and roughness-induced turbulence are accounted for when computing suspended load sediment transport.
In Section 2, a detailed discussion of the process-based model is given, followed by a description of the solution method in Section 3. Section 4 presents the results, which are discussed in Section 5. Finally, the conclusions are drawn in Section 6.
2 Model formulation
2.1 Shelf geometry
2.2 Hydrodynamics
2.2.1 Waves
Linear wave theory is used to describe the properties of offshore waves approaching the shoreface. For the waves and water depths considered here (the inner shelf region), this approximation is reasonable. The equations used are similar to those in the study by Vis-Star et al. (2007). Rayleigh distributed random waves are considered, characterized by a narrow band of frequencies and orientations. The narrow spectrum is centered around a peak frequency, wavenumber, and wave orientation.
The rigid-lid approximation is used, i.e., the effects of the free surface on the local water depth are neglected. Thus, the local water depth \(\tilde{D}=z_{\rm s}-z_{\rm b}\) is approximated by D = − z _{b}. Here, z _{s} and z _{b} are the free surface elevation and the bottom depth both measured with respect to the undisturbed water level z = 0. Refraction of waves by currents is not taken into account.
2.2.2 Currents
2.3 Sediment characteristics
2.4 Sediment dynamics
The boundary conditions for Eqs. 13 and 17 are that the bed level is fixed at x = 0 and far offshore and that \(\mathcal{C}_i\) is bounded far offshore.
3 Solution method
The initial growth of morphodynamic features is investigated by applying a linear stability analysis. It is a similar analysis to that employed in the early study by Trowbridge (1995), and it is based on the hypothesis that sfcr form as free morphodynamic instabilities of a basic state. As the initial growth is considered, equations are linearized. The basic state is discussed in Section 3.1 and the linearization procedure is outlined in Section 3.2.
3.1 Basic state
The cross-shore and alongshore momentum balance in the basic state describe a longshore current V(x), which is driven by an alongshore wind stress τ _{sy }. Due to Coriolis effects, it induces a mean sea level Z(x). The magnitude of V is inversely related to the local wave orbital velocity due to frictional effects. There is no cross-shore velocity in the basic state, i.e., U = 0.
3.2 Linear stability analysis
Solutions of this eigenvalue problem were obtained numerically using a spectral collocation method. This method involves an expansion of variables in Chebyshev polynomials, and it is subsequently imposed that the equations are exactly obeyed at N collocation points (for details, see Boyd 2001).
4 Results
4.1 Parameter values: default case
Parameter values are representative for the Long Island inner shelf and are partly taken from Figueiredo et al. (1982) and Schwab et al. (2000). Long Island is located along the Atlantic coast of North America at a latitude of ∼40° N, for which the Coriolis parameter is f = 1×10^{ − 4} s^{ − 1}. Going seaward, the depth of the inner shelf increases from H _{0} = 14 m to H _{s} = 20 m, and the inner shelf width is L _{s} = 5.5 km. Therefore, the transverse bottom slope is \(\beta=(H_{\rm s}-H_0)/L_{\rm s} \sim 1.1\times10^{-3}\). Typical values for the offshore root-mean-square wave height, offshore angle of wave incidence, wave period, and alongshore wind stress are H _{rms,s} = 1.5 m, \(\Theta_{{\rm s}}=-20^\circ\), T = 11 s, and τ _{sy } = − 0.4 N m^{ − 2} (southward), respectively. In the default experiment, a uniform fraction of fine and coarse sand is used: F _{1} = 0.7 and F _{2} = 0.3. Furthermore, a mean grain size d _{m} = 0.35 mm (Φ_{m} = 1.5) is adopted with a sorting parameter Υ _{s} = 0.5. For the exponent in the transport capacity function for bedload, c _{b} = 0.75 is used. In Eqs. 18–20, e _{ h } = 0.2, e _{ w } = 1.1, and e _{ f } = 0 are chosen, such that in Eq. 28a, for the sediment concentration in the basic state, the exponent c _{s} = − 1.1. Note that roughness-induced turbulence effects are neglected for the moment. Values of the other parameters are: r = 2.0×10^{ − 3}, \(\nu_{\rm b}=5.6\times10^{-5}\) s^{2} m^{ − 1}, λ _{b} = 0.65, λ _{s} = 0.30 s m^{ − 1}, w _{s*} = 0.04 m s^{ − 1}, \(E_{0*}=1.3\times10^{-4}\), \(\delta_*=0.20, c_{f*}=3.5\cdot 10^{-3}\), and p = 0.4. In the computations, 160 collocation points were used.
4.2 Basic state and linear stability analysis: default case
In this section, solutions of the morphodynamic (eigenvalue) problem are presented for the parameter values specified in the previous section. The characteristics of the basic state were already discussed in Vis-Star et al. (2007). In particular, the wave orbital velocity U _{ w } decreases monotonically with increasing depth (U _{ w }(x = 0) = 0.54 ms^{ − 1}, U _{ w }(x = L _{s}) = 0.40 ms^{ − 1}). The longshore current V points in the negative y direction and its magnitude increases with increasing depth (V(x = 0) = − 0.36 ms^{ − 1}, V(x = L _{s}) = − 0.48 ms^{ − 1}) since it experiences less bottom friction for larger depth.
The ratio of the maximum variation in the perturbed fraction of fine grains and the maximum variation in the bottom topography is [f _{1}′]/[h′] ∼0.56 /H _{0} . Thus, in the case of a ridge of 2 m in height, the variation in the fraction of fine grains will be ∼0.1. For diameters of the fine and coarse grains of 0.28 and 0.59 mm, respectively (as in the default case), this corresponds to a total variation in the mean grain size of approximately 0.05 mm.
4.3 Sensitivity to settling lag effects
Experiments reveal that settling lag effects are crucial for the damping of high wavenumber perturbations. Furthermore, growth rates become considerably smaller when settling effects are included, whereas migration speeds are hardly affected. The mean grain size and sorting pattern is not affected by settling lag effects. On the other hand, the ratio of the maximum variation in the perturbed mean grain size/sediment sorting and the maximum variation in the bottom topography slightly decreases when settling lag effects become more important.
4.4 Sensitivity to sediment characteristics
Experiments were conducted to explore the sensitivity of model results to different values of the standard deviation of the sediment mixture. For a fixed fraction of the fine and coarse sediment in the mixture and for a fixed mean grain size (default values), the standard deviation of the mixture was varied between zero and one. Thus, in the case where Υ _{s} = 0.0, sediment is uniform with a grain size d _{1} = d _{2} = 0.35 mm, whereas for, e.g., Υ _{s} = 1.0, sediment is nonuniform with d _{1} = 0.22 mm and d _{2} = 1.0 mm.
Bedform characteristics (longshore spacing, cross-shore extent, growth rate, e-folding growth time, migration speed, H _{0} [f _{1}′]/[h′], and H _{0} [υ _{s}′]/[h′]) of the initially fastest growing mode as a function of the mean grain diameter d _{m0} of the sediment mixture
d _{m0} (mm) |
0.20 |
0.30 |
0.40 |
0.50 |
0.60 |
---|---|---|---|---|---|
λ _{ p } (km) |
– |
3.5 |
2.1 |
1.6 |
1.2 |
cross-shore extent (km) |
– |
1.2 |
0.7 |
0.5 |
0.4 |
σ _{ r } (0.01 year^{ − 1}) |
– |
0.45 |
0.61 |
0.63 |
0.63 |
T _{ g } (years) |
– |
223 |
164 |
159 |
159 |
V _{m} (m year^{ − 1}) |
– |
− 62 |
− 38 |
− 27 |
− 20 |
H _{0} [f _{1}′]/[h′] |
– |
0.58 |
0.52 |
0.44 |
0.35 |
H _{0} [v _{s}′]/[h′] |
– |
0.28 |
0.24 |
0.21 |
0.17 |
In the previous experiments, it was assumed that the basic state properties of the sediment mixture (grain size distribution function, mean grain size, and standard deviation) have a fixed value in space. However, observations indicate that, often, sediment becomes finer as it goes offshore. To investigate the effect of an x-dependent sediment distribution, a run was performed for a grain size fraction, which is linear dependent on x: F _{1} = 0.5 at the shoreface to F _{1} = 0.8 at the transition to the outer shelf. The grain size of both the coarse and fine fractions are the same as in the default experiment. Thus, the mean grain size and the sorting parameter are also dependent on the cross-shore position. Results (not shown) indicate that bedform characteristics are hardly affected by including x-dependent sediment properties. The sorting pattern will change as soon as F _{1} < F _{2} in that part of the domain where sfcr are situated: the sediment is finer and more poorly sorted in the troughs, whilst it is coarser and better sorted on the crests.
4.5 Sensitivity to hiding and roughness-induced turbulence
A maximum growth rate of about 0.66×10^{ − 2} year^{ − 1} is found for k = k _{ p } ∼2.4 km^{ − 1}, which corresponds to a longshore wavelength of the bedforms of ∼2.7 km. This initially most preferred mode migrates with a speed of ∼65 m year^{ − 1} in the downstream (southward) direction. Taking into account roughness-induced turbulence effects thus leads to an increase in growth rates and a slight increase in migration speeds. The bottom pattern of the most preferred mode is shown in Fig. 10c and shows up-current-oriented sfcr that extend approximately 1 km offshore. The contour lines in this figure indicate the perturbations in the distribution of the mean grain size. Clearly, including roughness-induced turbulence effects leads to a change in the phase difference between the ridge topography and grain size distribution: the finest sediment is found slightly up-current of the crests instead of the troughs. The ratio of the maximum variation in f _{1}′ and the maximum variation in h′ increases to become [f _{1}′]/[h′] ∼0.84/H _{0}.
4.6 Sensitivity to other model parameters
The dependence of results on wave height, wave period, and angle of wave incidence at the offshore boundary was investigated, as well as the dependence on the water depth H _{s} at the outer shelf. The resulting trends in the growth rate, migration speed, and shape of bedforms are similar to those reported by Vis-Star et al. (2007). The phase shift between maxima in f _{1}′ and h′ is very robust (180°) under changes in offshore wave properties
Finally, the dependence of model results on the bed slope parameter λ _{s} in Eq. 16 was investigated. It turns out that, with decreasing values of λ _{s}→0, the sfcr become shorter and they grow faster. In the limit, λ _{s}→0, i.e., suspended load transport is not affected by the presence of bed slopes, the ridges grow a factor of 2 larger than in the default case, and their wavelengths are 20% smaller.
5 Discussion
5.1 Physical interpretation
A physical interpretation of the results presented so far will be given in this section. The growth of sfcr in the present model is due to three different mechanisms. The first mechanism was already described by Trowbridge (1995), in which the offshore deflection of the storm-driven flow over sfcr for a transversely sloping bottom of the inner shelf is essential. The second is the one described by Calvete et al. (2001), which involves both the offshore deflection of the current over sfcr and cross-shore gradients in the depth-averaged volumetric suspended sediment concentration in the basic state. The third mechanism was found most recently and described by Vis-Star et al. (2007) as the wave–bedform feedback mechanism. The up-current-oriented sfcr affect the wave field in such a way that wave rays converge on the upstream sides of sfcr. As a result, the perturbed wave energy in these areas increases, and stirring of sediment by waves is enhanced. Subsequently, the storm-driven flow transports the additional sediment as suspended load downstream. The third mechanism is more effective than the first two in the case of obliquely incident waves and even active in the absence of a transversely sloping bed. Both growth and migration of sfcr are controlled by suspended load transport of sediment in the case of the wave–bedform feedback mechanism. Below, an explanation is given of (1) the damping of high wavenumber perturbations due to settling lag effects, (2) the distribution of the mean grain size and standard deviation over sfcr, and (3) the sensitivity of model results to changes in characteristics of the sediment mixture.
5.1.1 Damping of high wavenumber perturbations due to settling lag effects
5.1.2 Distribution of the mean grain size and standard deviation over sfcr
In Eq. 42 above, V < 0 (southward) and the cross-shore gradient in the depth-averaged volumetric suspended sediment concentration C _{*}/H is negative, i.e., d/dx (C _{*}/H) < 0. In the default case, e _{ h } = 0 and c _{s} < 0; hence, in the basic state, the suspended sediment concentration of fine sediment is larger than that of coarse sediment (G _{s1} > G _{s2}). This implies that parameter M _{1} is positive. Furthermore, (M _{2} + M _{1} M _{3}) is positive for default parameter values.
The distribution of the mean grain size over the bedforms is dependent on the relative magnitude of the two terms on the left-hand side of Eq. 42. In previous work, only the term related to u′ was present, as interactions between bedforms and waves were ignored. In that case, the distribution of the fraction of fine grains is related to the perturbed cross-shore velocity. Trowbridge (1995) already showed that u′ > 0 if h′ > 0. Hence, \(\partial f_1'/\partial y \propto -u' \propto -h'\). As a consequence, the distribution of the mean grain size for suspended load is 90° out of phase with the topography such that the finer sand is found on the seaward (down-current) flank of sfcr. In this paper, wave–bedform interactions are included and appear to play a dominant role in both the evolution of the bottom and the mean grain size. Thus, the term proportional to \(\partial u_w'/\partial y\) dominates over the term proportional to u′ in Eq. 42. It follows immediately that f _{1}′ ∝ − u _{ w }′. The study by Vis-Star et al. (2007) revealed that the pattern of the perturbed wave orbital velocity is slightly shifted up-current with respect to that of the bottom. Thus, an approximate 180° phase shift exists between the pattern of the mean grain size for suspended load and the topography. According to Eq. 32, υ _{s}′ ∝ − f _{1}′ for F _{1} > F _{2}, and thus, the perturbed sorting of sediment is in phase with the topography. This explains the results shown in Section 4.2, where the finer and better-sorted sediment is found in the troughs and the coarser and more poorly sorted sediment on the crests.
As was shown in Section 4.5, including roughness-induced turbulence effects leads to a change in the phase difference between the perturbed mean grain size and the bedforms. Roughness-induced turbulence effects cause c _{s} to become positive, which implies that G _{s1} < G _{s2} and, thus, M _{1} < 0. The sign of (M _{2} + M _{1} M _{3}) does not change and is still positive. Therefore, the balance in Eq. 42 becomes f _{1}′ ∝ u _{ w }′ ∝ h′ due to the wave–bedform feedback mechanism. This explains the 0° phase shift between the bottom and mean grain size pattern, as obtained in Section 4.5. The finer (coarser) sediment, which is now located on the crests (in the troughs), is still better (more poorly) sorted, as υ _{s}′ ∝ − f _{1}′ still holds.
5.1.3 Sensitivity results to changes in sediment characteristics
For a bimodal sediment mixture compared to uniform sediment, both the growth rate and migration speed of sfcr decrease if the standard deviation is increased (see Fig. 7). The latter is caused by a reduction of the entrainment of both fine and coarse sediment due to a better packing of the sediment for a more poorly sorted sediment mixture (which is represented by the straining factor \(\lambda_{\mbox{\scriptsize{E}}}\) in the entrainment of sediment). In case that wave–bedform feedbacks are important, both growth and migration are controlled by suspended load transport and, thus, reduce with an increase in the standard deviation of the sediment mixture. Note that the decrease in migration speed with increase in standard deviation of the mixture is in contrast with previous results (Walgreen et al. 2003), which is due to the fact that, in their case, the migration speed was determined by bedload transport due to the neglect of wave–bedform interactions.
Figure 9 reveals that growth rates and migration speeds become smaller if the fraction F _{1} of fine sediment is increased. This is because a larger F _{1}, while keeping the standard deviation of the mixture fixed, implies a larger difference between the grain sizes of the coarse and fine sediment, and thus, hiding effects are more effective. Finally, the finding that growth of sfcr requires a minimum mean grain size of the sediment mixture (Table 1) is due to the fact that a smaller mean grain size causes settling lag effects in the concentration equation to become larger with respect to deposition and erosion terms. Thus, if the mean grain size is decreased, a more effective damping of all bottom perturbations occurs.
5.2 Comparison with observations
Most field data on sfcr, including sfcr on the Long Island shelf, reveal a phase difference of approximately 90° between the mean grain size and the topography: the coarsest sand appears on the landward (up-current) flank and the finest sediment on the seaward (down-current) flank. However, the modeled phase shift between the pattern of the mean grain size and the bottom topography for the default case is 180°: troughs (crests) consist of the finest (coarsest) sediment. The latter is attributed to the importance of wave–bedform interactions. The 180° phase difference between f _{1}′ and h′ is quite robust. It only changes when the hiding coefficient for suspended load transport c _{s} becomes positive, which is the case when the grain size dependency of the friction coefficient is taken into account. In that case, the mean grain size and topography are approximately in phase, with the finest sediment on the crests and coarsest sediment in the troughs. Notice that, for some of the sfcr, a phase shift between the mean grain size and topography is observed that is different from 90°. According to Hoogendoorn and Dalrymple (1986), the finest sediment is observed at the base of the downstream flanks of the Canadian sfcr, which seems to indicate a phase shift close to 180°. Figueiredo (1980) reports for the ridges on the inner shelf of southern Brazil that the coarse sediment is found in the troughs and the medium to fine sand appears on the ridge crests, thus implying a 0° phase shift. Grain size variations are also observed over other type of bedforms. Miselis and McNinch (2006) report for nearshore oblique bars of North Carolina that the coarser sediment almost always appears in the troughs. Both 0° and 180° phase shifts between mean grain size and topography are observed for tidal sand waves in the southern North Sea (Roos et al. 2007).
A considerable improvement with respect to Walgreen et al. (2003) is that the modeled maximum variation in the fraction of fine and coarse sediment over sfcr has increased. The latter is contributed to the fact that interactions between waves and the topography are included. According to model results in case c _{s} < 0, variations in the mean grain size d _{m} are ∼0.02 [h′] mm. For heights of sfcr h′∼1 − 6 m (observed values), the mean grain size would vary between 0.02 mm and 0.12 mm. These values become slightly higher for simulations with c _{s} > 0: grain size variations are 0.03–0.18 mm. Observations (e.g., Schwab et al. 2000, for the Long Island shelf) suggest that the variation in the mean grain size is in the order of 0.25–0.40 mm. Inspection of grain size variations on other shelves, as presented in Swift et al. (1978) and Stubblefield and Swift (1981), reveals grain size variations in the range of 0.05–0.65 mm. Thus, modeled grain size variations are slightly smaller than observed, but seem to be the right order of magnitude.
In literature, mostly qualitative information is given about variations in the standard deviation of sand over sfcr. For sfcr in general, including the sfcr on the Long Island shelf, sediment is best sorted on the crest and most poorly sorted in the troughs (Swift et al. 1972, 1978; Schwab et al. 2000). Data from sfcr in the German Bight (Antia 1993) indicate a better degree of sorting on the downstream flank. In the model, F _{1} > F _{2}, which implies that υ _{s}′ ∝ − f _{1}′ (see Eq. 32). Thus, the finer sediment (f _{1}′ > 0) is better sorted (υ _{s}′ < 0). Therefore, model results for which the friction coefficient is grain-size-dependent (c _{s} > 0) are in best agreement with the data for Long Island.
5.3 Model simplifications
The model used in this paper is based on several simplifying assumptions. First, a linear stability analysis is employed, which only yields information about the initial growth of and grain sorting over sfcr. If wave–bedform feedbacks are taken into account, both growth and migration of sfcr are still controlled by suspended load transport. However, in the case of finite-amplitude bedforms, exchanges of sediment between the active layer and the underlying substrate will occur, and they will probably have profound implications for the dynamics of the ridges.
In recent years, studies by Murray and Thieler (2004), Coco et al. (2007a, b), and Huntley et al. (2008) have focused on investigating so-called “sorted bedforms,” which are particularly found on sediment-starved shoreface and inner shelf environments. The sorted bedforms are found in very poorly sorted sediment mixtures, and a robust phase shift of 90° between f _{1}′ and h′ is observed. Inspired by their work, the effect of including the term \(L_{a0}\frac{\partial f_1'}{\partial t}\), which was excluded on the left-hand side of Eq. 34b, was investigated. This term defines the adjustment time scale of the grain size distribution to bottom changes, which, in general, will be very small for sfcr. For the default model setting, the thickness of the active layer in the basic state is \(L_{a0}=d_{\rm m} 2^{\Upsilon_{\rm s}}\sim 5\times10^{-4}\) m. In additional simulations, the latter was increased to a value of ∼3 m, which is the value used for sorted bedforms. The distribution of the mean grain size and standard deviation over sfcr was not affected. Only increasing L _{ a0} up to about 10 m would change the phase shift between h′ and f _{1}′ from 180° to 0°. However, this is certainly not realistic for sfcr.
6 Conclusions
The main objective of the present study was to obtain a better understanding of the initial formation of sfcr and the corresponding grain size distribution. For this, a model was developed and analyzed, which consists of the depth-averaged shallow water equations, a sediment transport formulation, and mass balance of sediment. A new aspect in the hydrodynamic module (with respect to previous studies on sfcr) is that the behavior of waves is described by equations, which follow from physical principles, rather than by parameterizations. The inclusion of feedbacks between the growing bedforms and the waves is important. The sediment is represented by two grain size classes. The important new aspects in the sediment transport module are that the entrainment of suspended sediment depends on bottom roughness, and settling lag effects are included in the sediment concentration equation.
Default model experiments for a setting that resembles the Long Island micro-tidal inner shelf show that the growth and migration of sfcr stabilizes for a bimodal sediment mixture compared to uniform sediment. Furthermore, results reveal the presence of the finer sand approximately in the troughs and the coarser sand slightly up-current of the crests. In the case where roughness-induced turbulence effects are taken into account in the suspended sediment transport, the trend is the opposite: the coarser sand is located in the troughs and the finer sand on the crests. Both results are not in agreement with field data for the Long Island sfcr in which the coarsest sand is found on the landward flank and the finest sand on the seaward flank. However, field data on, e.g., Canadian and Brazilian sfcr, and also on some other types of bedforms, reveal that, often, coarse sediment is observed in the troughs. An interesting model result is that the modeled maximum variation in the probabilities of fine and coarse sediment over the sfcr has the right order of magnitude (10^{ − 1} mm). The inclusion of wave–bedform interactions is crucial here.
A physical analysis has revealed that the phase shift between the mean grain size and the bottom topography obtained by the model is due to the wave–bedform feedback mechanism. It causes convergence of wave rays at the upstream sides of sfcr and, thus, results in enhanced stirring of sediment by waves at these locations. In the case where roughness-induced turbulence effects are neglected, the additional entrainment of grains is more effective for grains of size smaller than the mean compared to grains of size larger than the mean. Thus, more fine than coarse sand is eroded from the crests, and therefore, the crests become coarser. On the other hand, more fine than coarse sand is deposited in the troughs where wave energy is reduced, which results in troughs consisting of finer sand. In the case where the friction coefficient is dependent on the physical roughness of the seabed, results are fundamentally different. Due to a change in grain size and ripple size from the coarse to fine domains, the physical roughness of the seabed is larger (smaller) in areas where the sediment is relatively coarse (fine). Including this effect, the entrainment of grains of size larger than the mean is enhanced compared to the entrainment of grains of size smaller than the mean: troughs become coarser and crests become finer. Furthermore, the effects of settling lag are crucial to cause damping of small-scale perturbations.
Acknowledgements
The work of N.C. Vis-Star is supported by “Stichting voor Fundamenteel Onderzoek der Materie” (FOM), which is supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO). The work of D. Calvete has been partially funded by the Ministerio de Ciencia Tecnología of Spain through the “Ramón y Cajal” contract.
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