Surface-water quality and flow Modeling Interest Group

Modeling Sand Bank Formation Around Tidal Headlands

¹ U.S. Geological Survey, Woods Hole, MA, U.S.A.

Please direct correspondence to:
  Richard P. Signell
  USGS, Woods Hole Field Center
  384 Woods Hole Road
  Woods Hole, MA 02543-1598
  Internet: rsignell@usgs.gov
  Phone: (508) 457-2229
  FAX: (508) 457-2309

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Citation:
Signell, R.P. and Harris, C.K., 2000, Modeling sand bank formation around tidal headlands, in Estuarine and Coastal Modeling, 6th Int. Conf., ASCE, New Orleans, LA, November 3-5, 1999. Editors: Malcolm L. Spaulding and Alan F. Blumberg.

Contents

Abstract
Introduction
Modeling Approach
Suspended Load Simulation
Bedload Simulation
Discussion
Conclusions
Acknowledgements
References

Abstract

Introduction

An equally strong anticyclonic residual eddy was found on the western side of the headland, but the sand bank associated with this eddy was much smaller. Pingree argued that the asymmetry in sand bank formation was due to the Earth's rotation. In the cyclonic eddy, the Coriolis force acting outward requires a larger inward directed pressure gradient to drive the centripetal acceleration (Figure 1). Conversely, in the anticyclonic eddy, the Coriolis force is directed toward the center of the eddy, thus requiring a smaller pressure gradient. Thus, near the bottom, where the pressure gradient balances stress divergence, more sand would be directed toward the center of the cyclonic eddy than toward the center of the anticyclonic eddy. In fact, if the residual vorticity of the anticylonic eddy were less than the planetary vorticity f, the pressure gradient at the bottom would be directed outward and no sand bank would form. This mechanism for generation of asymmetric banks around tidal headlands has remained popular (e.g. Dyer and Huntley, 1999).

Figure 1. Cartoon of the "tidal stirring" hypothesis for sandbank generation around tidal headlands, modified from Pingree (1979). Reversing tidal flows generate symmetric residual eddies on either side of the headland. The inward-directed pressure gradient PG, however, is larger in the cyclonic eddy because it must balance the Coriolis force C in addition to the centrifugal force CF. In the anticyclonic eddy, the Coriolis force helps balance the centrifugal force and thus the pressure gradient is smaller. The hypothesis states that this asymmetry in pressure gradient would result in asymmetry in sandbank generation, since sand moving down-pressure gradient near the bottom would feel a stronger force in the cyclonic eddy. The larger stippled region inside the cyclonic eddy indicates this pictorially.

Here we use idealized sediment transport simulations to explore sand bank formation around headlands by suspended and bed load and to test the "tidal stirring" hypothesis. We initialize the bed with a finite depth layer of uniform sediment, run a periodic tidal model with limiting cases of pure suspended load and pure bed load, then examine the resulting erosional and depositional patterns in relation to the tidal residual eddies.

Modeling Approach

For our idealized headland model domain, we utilize the base case of Signell and Geyer (1991) because the hydrodynamics have been thoroughly studied and it is a close match to the situation near the Shambles in terms of tidal flows, water depths and the scale of the headland. The domain is 50 x 25 km, with an 8 km width, 8 km amplitude Gaussian shaped headland located in the middle of the southern wall (Figure 2). The water depth is 20 m, except near the southern boundary, where it decreases to a minimum of 2 m over a distance of 3 km. The model grid spacing is variable, with 1-2 km spacing near the open boundaries, and 0.3-0.4 km spacing near the headland.

Figure 2. The headland model domain, grid and bathymetry used for this study. The tide is forced at the western boundary and a gravity wave radiation condition is applied at the eastern boundary.

There are open boundaries at the west and east and a free-slip wall at the north. The tide is specified as a periodic normal velocity at the western open boundary, and a gravity radiation condition is applied at the eastern open boundary to allow the tidal wave to leave the domain. In the base case, an M₂ (12.42 hour period) tidal current with amplitude of 0.5 m s^-1 is specified. Although Signell and Geyer (1991) used a curvilinear grid, we found that our Cartesian grid results closely match theirs. Table 1 summarizes other model characteristics and selected parameters.

Table 1. Adjustable model parameters. The FORTRAN variable names are shown in parentheses for the benefit of POM/ECOM users.

To model erosion and deposition due to bed load transport of sediment, we use the method of Van Rijn (1984a), where the rate is proportional to the excess shear stress to the 2.1 power. The convergence and divergence of bedload transport determines the erosion and deposition rate of the sediment from the bed.

To model erosion and deposition due to suspended load transport of sediment, we use a concentration-based approach, where the sediment is treated as a 3D tracer field with a vertical fall velocity w_s. We consider only a single size class of noncohesive sediment, with constant settling w_s. The advection of suspended sediment is computed using a high-order advection scheme (Margolin and Smolarkiewicz, 1989), and vertical diffusion of sediment is determined from the Mellor and Yamada (1982) level 2.5 turbulence closure model with the Galperin et al. (1988) extensions.

The bottom boundary condition for suspended sediment concentration is specified as a sediment flux between the bed and the water column. This sediment flux depends on excess shear stress, fall velocity, and sediment concentration in the lowest layer.  If the bottom shear velocity u_* is less than the threshold for movement u_*crit or less than the setting velocity w_s then only settling occurs and the flux to the bed is w_s*C₁ where C₁ is the concentration in the lowest layer. If u_* exceeds w_s and u_*crit, the sediment concentration in the lowest layer is assumed to be in equilibrium (upward diffusion balancing downward settling). The equilibrium horizontal sediment flux in the lower layer is then computed in a manner similar to Van Rijn (1984b), essentially integrating the product of the Rouse profile and the law-of-the-wall velocity profile over the thickness of the lowest sigma layer. The vertical flux at the bed is then specified as the difference between this equilibrium flux and the flux currently computed by the model (C₁*U₁), where U₁ is the velocity in the lowest layer and C₁ is the concentration in the lowest layer obtained from the previous time step. If the model flux is lower than equilibrium, erosion occurs so that the equilibrium flux is matched. If the model flux is higher than equilibrium, deposition occurs. Thus it is possible to have deposition even when u_* exceeds the critical movement and suspension thresholds. The cumulative impact of the vertical fluxes to and from the bed is tracked through erosional and depositional changes to the bed thickness. For simplicity, no waves, active layers, or multiple size classes were included.

The model is spun up from a state of rest for four tidal cycles before the sediment transport is initiated. The initial sediment bed is given a constant thickness of 3 m. As material is eroded and deposited, the bed thickness changes, and these modifications to the bed are allowed to change the overall depth in the model so that the developing bed features may affect the hydrodynamics. The present model configuration does not allow for formation of spits, islands, or other changes to the coastline because it cannot handle drying of previously wet grid cells.

Suspended Load Simulation

Insight into suspended sediment transport for this case can be gained by looking at snapshots of the tidal flow; suspended sediment; shear velocity magnitude; erosion rate; and net change in the bed thickness at key points during the 1^st tidal cycle after sediment transport is initiated (Figures 3 and 4). Just after maximum ebb (Figure 3), the westward flow is accelerating as it approaches the tip of the headland and then separates at the headland tip, forming a cyclonic eddy on the westward side of the headland. The shear stress is sufficient to resuspend material over a large region surrounding the headland, and the rate of erosion is particularly strong at the headland tip where maximum shear stress occurs. Deposition is starting to occur downstream of the headland tip, where the flow is decelerating and the sediment capacity of the lower layer is therefore decreasing. The bed thickness is just starting to decrease at the headland tip and increase where the tidal flows decelerate to the west of the headland tip.

Figure 3. Suspended load sediment transport characteristics near maximum ebb (1.3 hours after sediment transport was initiated): (a) Depth-averaged sediment concentration and near-bed (1 m) velocity vectors; (b) Shear velocity magnitude; (c) Erosion rate; (d) Net change in bed thickness. At maximum flood and ebb, high sediment concentrations are found near the tip of the headland, where strong shear velocities cause rapid erosion of the bed. Eroded material is advected away from the headland tip.

Figure 4. Suspended load sediment transport characteristics near slack water before flood (3.8 hours after sediment transport was initiated): (a) Depth-averaged sediment concentration and near-bed (1 m) velocity vectors; (b) Shear velocity magnitude; (c) Erosion rate; (d) Net change in bed thickness. Just 2.5 hours after the snapshot shown in Figure 3, the sediment resuspended during maximum ebb is being wrapped around a transient eddy. The shear velocities are weak, and the material is rapidly depositing due to the decreased carrying capacity of the flow. This erosion at maximum flood and ebb and deposition at slack before ebb and flood results in sand bank generation on both sides of the headland.

At slack before flood (Figure 4), the cyclonic eddy is fully developed, shear stresses have decreased, so the sediment that was eroded from the headland tip during ebb is continuing to fall out of suspension, building the bed thickness on the western side. At the headland tip, however, the flood tide is already beginning and this eastward flow is starting to pick up sediment. This sediment is transported and deposited in the anticyclonic eddy that forms on the eastward side of the headland, mirroring the process occurring during the ebb tide. An animation of suspended load transport over the tidal cycle adds considerable insight into the nature of the resuspension and transport process.

The mean erosion/deposition rate over the first two tidal cycles indicates that the coarse silt is being eroded from the headland and deposited in banks forming near the centers of the residual eddies (Figure 5a). The deposition rates, however, are nearly identical on either side of the headland, indicating nearly symmetric sand bank formation. While the tidal stirring hypothesis predicts that the bank on the left hand side of the headland should grow much faster (in the Northern Hemisphere), the simulation actually shows slightly greater growth of the sandbank to the right of the headland. Investigation revealed that this is the slight frictional damping of the tide that occurs as it propagates eastward from the western open boundary.

Figure 5. Net change in bed thickness over the 1^st two tidal cycles following sediment transport initiation and near-bottom (1 m) residual currents: (a) Coarse silt (suspended load) case in the northern hemisphere; (b) Coarse silt (suspended load) case in the southern hemisphere; (c) Medium sand (bedload) case in the northern hemisphere; (d) Medium sand (bedload) case in the southern hemisphere. The nearly symmetric sand bank formation in both the suspended load and bedload cases, and in both the northern and southern hemisphere indicates that the Earth's rotation is not a significant factor in the generation of these banks. The slightly increased rate of growth of the bank to the right of the headland is due to slight decay of the forced tidal wave as it propagates eastward from the western boundary, which generates slightly more resuspension on the eastward flowing phase of the tide.

As a further check as to the negligible influence of the Earth's rotation, the test case was run a second time using a Coriolis frequency f = -1x10^-4 s^-1, valid for the Southern Hemisphere. Figure 5b confirms that the development of the banks is not being significantly impacted by the Earth's rotation.

If we now focus on the long-term development of the sand banks instead of the first few tidal cycle patterns of erosion and deposition, we find that the source supply of sediment plays an important role in sand bank evolution. Within 1.5 years, the initial 3 m of sediment is completely removed from a roughly 25-km² region off the tip of the headland. As the sediment source near the headland tip is depleted, each bank rapidly shifts shoreward from the center of the residual eddy to a more stable position along the shoreward side of the residual eddy (Figure 6a). These banks continue to build in nearly the same locations, reaching heights of 4-5 m in 15 years when the simulation died (Figure 6b). Unfortunately, the banks were still slowly growing when the simulation died, and therefore the equilibrium condition (if one exists for this case) could not be established. At the time when the simulation died, the 4-5 m banks had only a modest impact on the structure and magnitude of the residual flow. An animation of the long-term development of the sandbanks clearly shows the rapid depletion of bed material near the tip and subsequent slow evolution of the banks.

Figure 6. Long-term changes in bed thickness and near-bottom (1 m) residual currents for the coarse silt (suspended load) case: (a) after 1.5 years; (b) after 15 years. Although the banks initially form near the centers of the residual eddies, they migrate away from the headland tip as the sediment supply is exhausted. After 1.5 years, all of the original 3 m of sediment has been eroded from a roughly 25-km² area in the vicinity of the headland tip, and the banks are found immediately adjacent to the erosional region. After 15 years the banks are nearly in the same location, but the region from which the material has been completely removed has grown to more than 50 km². The banks are nearly 5 m high, but their growth rate has slowed dramatically due to the lack of available sediment from the erosional area in the vicinity of the headland.

Bedload Simulation

Snapshots of the bedload transport rate, the shear velocity, the erosion rate and the net change to the bed again illustrate the mechanism of sand bank formation by bedload transport (Figure 7). At maximum flood, as in the suspended load case, erosion rates are highest near the tip of the headland where the flow speeds are greatest. Unlike the suspended load case, however, the erosion rate depends only on the divergence of the transport rate. While there is a region of deposition to the east of the headland near where the sandbank formed in the suspended load case, there is also a small region of strong deposition just offshore of the headland tip. There are regions of strong erosion along the upstream side of the headland and in the vicinity of the headland tip immediately adjacent to the small depositional zone. An animation of bedload transport over the tidal cycle clearly shows migration of the zone of deposition from the tip of the headland downstream as the flood tide progresses. Most of the deposition and erosion occurs during maximum flood and ebb, when the bottom stress gradients are strongest.

Figure 7. Bedload sediment transport characteristics at maximum flood: (a) Bedload transport rate (cm²/s); (b) Shear velocity magnitude and near-bed (1 m) velocity vectors; (c) Erosion rate (mm/hour); (d) Net change in bed thickness (mm). Strong bedload transport and shear velocities are found upstream and offshore as the flow separates from the headland tip. This causes rapid erosion on either side of the tip, but strong convergence of sand directly off the headland tip. As the shear velocities decrease downstream from the headland tip, bedload convergence of sand results in rapid deposition and sandbank generation.

The mean erosion/deposition rate for the bedload case over the first two tidal cycles indicates that medium sand will be both eroded and deposited from the headland tip and will also be deposited in banks forming near the centers of the residual eddies, just as in the suspended load case (Figure 5c). The bank positions are slightly closer to the headland tip than they were for the suspended load case since there is no water column advection of sediment in this case. The banks are again nearly symmetric and the effects of the Earth's rotation are negligible (Figure 5d).

The long-term evolution of the bedload generated sand banks could not be determined since the strong deposition near the headland tip caused the run to terminate within a few weeks. Future work is needed to determine if the inclusion of wave effects might redistribute the sand resulting from this strong deposition or whether this process might actually build spits, a phenomenon that the current model is not capable of representing because water cells cannot dry.

Discussion

If the teacup effect does not explain why the asymmetric sandbanks observed off of Portland Bill, what does? It is possible that meteorological conditions provide the explanation. Pingree (1979) shows that the response to a strong wind event from the west should result in preferential deposition on the Shambles. It is also possible that the sediment supply is such that it favors asymmetric development. Ultimately, the supply of sand needs to be linked to the wave-dominated nearshore zone, which is not considered here. Still more mechanisms for asymmetric sand bank formation could be the presence of persistent non-tidal residual flows caused by winds or density gradients or strong ebb or flood dominance in the large-scale tidal flow.

These simulations have explored sandbank formation only in a small part of parameter space. Previous work has shown that the nature of the flow around tidal headlands depends on the ratios of three important length scales: the tidal excursion, the length scale of the headland, and the frictional spin-down scale (Signell, 1989; Signell and Geyer, 1991). Flow separation occurs when the tidal excursion is at least as large as the headland length scale and at least as small at the frictional length scale. When the flow separates, then the ratio of the tidal excursion to the frictional length scale determines whether eddies formed on one phase of the tide can maintain their vorticity long enough to interact with eddies formed on the following phase of the tide. In the case explored here, which represents conditions commonly found in nature, all three length scales are about the same magnitude, so flow separation occurs, but eddies that form on one phase of the tide are spun down by friction before they can interact with eddies formed on the following phase of the tide. Since these parameters control the nature of the circulation, it is expected that they would also affect the structure of the resulting sandbanks. The length scale over which sediment falls from suspension and the length scale over which the critical suspension threshold is exceeded are important additional parameters that will affect the characteristics of sand bank generation. A more thorough study is necessary to map out the types of sandbanks that form in this parameter space.

The fact that all the relevant length scales for our test case have about the same magnitude provides an explanation for why the sandbanks are observed to initially form near the centers of the tidal residual eddies, an observation that would appear consistent with the teacup stirring mechanism. A more appropriate explanation, however, is that both vorticity and suspended load are generated near the headland tip, and are advected together downstream. The length scale that they are advected is proportional to the tidal excursion. If the sediment falls out of suspension over the same length scale that the vorticity decays, then the sandbank will coincide with the residual eddy. For the bedload case, the length scale of the headland dictates the length scale of shear stress variability, and if this scale is comparable to the tidal excursion and the frictional length scale, the bedload-generated bank will also coincide with the residual eddy.

Conclusions

While the suspended load and bedload cases both predict sandbank formation on either side of the headland, the suspended load case produced pure erosion in the vicinity of the headland tip, while the bedload case indicates a small region of strong deposition just off the headland tip. Future work is required to determine if this spit building mechanism would be significant in the presence of waves.

These simulations have demonstrated the great utility of numerical models in exploring the complex, sometimes nonintuitive aspects of sediment transport, even in fairly simple hydrodynamic situations. A thorough parameter study varying the tidal excursion, frictional length scale and sediment characteristics would likely yield additional insightful results, as would a study examining the interaction and sorting of sediment for mixed grain size beds.

Acknowledgements

References

Dyer, K.R. and D.A. Huntley, 1999. The origin, classification and modeling of sand banks and ridges, Cont. Shelf Res., 19, 1285-1330.

Galperin, B., Kantha, L.H., Hassid, S. and Rosati A., 1988. A quasi-equilibrium turbulent energy model for geophysical flows, J. Atmosph. Sci., 45, 55-62.

Margolin, L.G. and Smolarkiewicz, P.K., 1989. Antidiffusive velocities for multipass donor cell advection, Technical Report UCID 21866, Lawrence Livermore National Laboratory, Livermore, CA, 44 pp.

Mellor, G. and Yamada, T., 1982. Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys. Space Phys., 20, 851-875.

Pingree, R.D., 1979. The formation of the Shambles and other banks by tidal stirring of the seas, J. Mar. Biol. Ass. U.K., 58, 211-226.

Signell, R.P., 1989. Tidal dynamics and dispersion around coastal headlands, Ph.D thesis, MIT/WHOI Joint Program in Physical Oceanography, Woods Hole, MA, 159 pp.

Signell, R.P. & Geyer, W.R., 1991. Transient eddy formation around headlands, J. Geophys. Res., 96 (C2), 2561-2575.

Van Rijn, L.C., 1984a. Sediment transport, part I: Bedload transport, J. Hydraulic Eng., 110(10), 1431-1456.

Van Rijn, L.C., 1984b. Sediment transport, part II: Suspended load transport, J. Hydraulic Eng., 110(11), 1613-1641.

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