Initial Tests of a Possible Explanation For Alongshore Sandwaves on the Dutch Coast

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1 Initial Tests of a Possible Explanation For Alongshore Sandwaves on the Dutch Coast Andrew Ashton 1, A. Brad Murray 1, and G.B. Ruessink 2 ABSTRACT: Long-term measurements of the shoreline position along the Dutch coast have revealed alongshore-heterogeneous shoreline changes with different characteristics on different coastline sections, ranging from migrating zones of erosion and accretion alongshore sandwaves to less coherent changes. The regional shoreline orientations, and therefore local wave climates, vary among the coastline sections, which is consistent with a potential explanation involving an instability in shoreline shape driven by alongshore sediment transport. A simple numerical model that treats alongshore transport as a function of the relative angle between wave crests and shoreline orientation produces behaviors, including sandwaves, that depend on the wave climate. Using measured wave climates off of the Dutch coast, relative to the regional coastline orientations, as input to the model, we explore this possible connection. The model predicts trends in wavelengths and alongshore-translation characteristics that roughly match the observations in some respects, suggesting that simple properties of alongshore sediment transport could explain much of the observed behavior. The match is not complete, however, suggesting that other processes (including dynamic inlet/delta behaviors) also play a role. 1. INTRODUCTION Analyses of long-term shoreline and dune-foot positions show alternating zones of erosion and accretion that migrate to the north and east along the Dutch coastlines [1, 2]. Such alongshore sandwaves have also been documented on other coasts, and are sometimes referred to as a type of erosional hot spot [3-6]. Annual observations extending up to 150 years along the Dutch coastline reveal sandwaves with exceptionally large time and space scales, with periods on the order of decades and wavelengths on the order of kilometers. Different explanations of this phenomenon have been suggested. Alongshore variations in the configuration of alongshore bars [7], which cause variations in the wave energy reaching shore, could be responsible for the pattern of shoreline change. However, Ruessink and Jeuken [2] pointed out that the characteristic time scales for the configuration of alongshore bars is much shorter than that of the sandwaves. The sandwaves might represent slugs of sediment input into the system that are translating downdrift (in the direction of net wave-driven alongshore transport) [2 and references therein]. For example, abrupt changes in tidal-channel position at an inlet could change the sediment flux into the downdrift shoreline segment [e.g. 8], acting as a temporally varying boundary condition for that segment. This mechanism could explain how perturbations are added to a shoreline, but not how a perturbation would translate in the downdrift direction. Alternatively, Ruessink and Jeuken [2] pointed out that an instability in shoreline shape [9] could produce such alongshore-inhomogeneous behavior, independent of the pattern of sediment flux into the shoreline segment. An investigation of this instability has shown that when waves 1 Division of Earth and Ocean Sciences, Nicholas School of the Environment and Earth Sciences/Center for Nonlinear and Complex Systems, Duke University, P.O. Box 90227, Durham, NC 27708, USA, andrew.ashton@duke.edu 2 Department of Physical Geography, Faculty of Geographical Sciences, Institute for Marine and Atmospheric Sciences Utrecht, Utrecht University, P.O. Box , 3508 TC Utrecht, Netherlands.

2 approach shore from high angles (angles between deep-water wave crests and the regional shoreline trend that are greater than the one that maximizes alongshore flux), any perturbations in the plan-view shoreline shape will grow. In addition, high-angle waves can cause a shoreline bump to translate alongshore [9] (as opposed to the typically symmetrical diffusion that occurs under the influence of low-angle waves). Ruessink and Jeuken [2] noted that changes in sandwave characteristics (degree of organization, wavelength, and rates of alongshore translation) correlate with changes in shoreline orientation. If the deep-water wave climate is approximately constant along a coastline, different regional shoreline orientations produce different distributions of high- and low-angle waves. Therefore, this variation of sandwave behavior with shoreline orientation could be consistent with the instability-driven explanation for Dutch sandwaves [2]. Here we use deep-water wave data and a simple shoreline-shape model [9] to test whether alongshore transport treated as a function of the relative angles of wave approach could produce variations in sandwave behavior as a function of regional shoreline orientation that are consistent with the observations along the Dutch Coast. 2. OBSERVATIONS Fig. 1 shows the Dutch coastline. Ruessink and Jeuken [2] analyzed long-term measurements of shoreline and dunefoot positions in the numbered sections. The analyzed shoreline sections represent regions with reasonably continuous annual measurements where human manipulation has been minimal. We will concentrate on the dunefoot data; the dunefoot location tends to follow the shoreline position, but with less short-term variation [2]. The analysis involves first ELD YM6 Fig. 1. Location of the study areas. Section boundaries shown as filled circles and numbers of kilometers from arbitrary reference locations. 2 Also shown are the approximate locations of the wave gauges used to estimate wave climates. From Ruessink and Jeuken [2].

3 computing a linear temporal trend of dunefoot position at each measurement location, and removing this trend, producing a residual time series x res, for each location. Then, the average of x res for each year was calculated along each coastline section. This alongshore average, representing the alongshore-homogenous shoreline behavior (e.g. alongshore-uniform response to storms), is then removed. The result, x res hetero, reflects alongshore-heterogeneous deviations from the temporal trend. Fig. 2, which shows time stacks of x res hetero, reveals organized, alongshoremigrating zones of seaward/shoreward deviations from average values which can be interpreted as migrating alongshore sandwaves. 3. INSTABILITY IN SHORELINE SHAPE A common formula [10, 11] for alongshore sediment flux, Q s, involves the breaking wave height, H b, and the relative angle between breaking wave crests and the shoreline (φ b -θ ): Q s = K H b 5/2 cos(φ b -θ ) sin(φ b -θ ) (1) where K is an empirical parameter, θ is the shoreline orientation, and φ b is the orientation of wave crests. Assuming shore-parallel depth contours, this relation can be recast as (using the approximation cos 1/5 (φ b - θ ) 1): Qs = K Ho 12/5 cos 6/5 (φo -θ ) sin(φo -θ ) (2) where the subscript o denotes deep-water values. Alongshore sediment transport is maximized for (φ o -θ) of approximately 43 degrees. Deigaard et al. [12], using a process-based model with a detailed treatment of sediment transport, found a maximum in sediment transport for (φ o - Delta Coast 1 2 Holland Coast Wadden C Time (yr) (a) (b) St. dev. (m) Distance (km) Fig. 2. Time-space diagram of how far shoreward or seaward each shoreline position is compared to an alongshore and temporal average, x res hetero, with positive values (seaward deviations) shown in dark gray and negative values in light gray. From Ruessink & Jeuken [2]. 3

4 θ) slightly greater than 45 degrees. Assuming that the rate of sediment exchange between the nearshore and deeper water is negligible compared to the alongshore flux, nearshore sediment is conserved, which can be expressed as: η/ t = -(1/D) Qs/ x, (3) where η is the position of the shoreline (Fig. 3), and D is the depth to which accumulation or erosion extends, i.e. the depth of the shoreface. Combining (2) and (3), and assuming small deviations from a straight shoreline, so that θ tan(θ) = η/ x, gives: η/ t = K 2 H 0 12/5 {cos 1/5 (φ 0 -θ)[ cos 2 (φ 0 -θ) - (6/5)sin 2 (φ 0 -θ) ]} 2 η/ x 2, (4) where K 2 is a constant. For given deep-water wave characteristics (H 0 and φ 0 ), (4) takes the form of a diffusion equation, η/ t = D iff 2 η/ x 2, where D iff is the diffusivity. When the angle between approaching wave crests and the shoreline (φ 0 -θ) is less than the maximizing angle, the term in the square brackets is positive, which makes (4) a diffusion equation with a positive diffusivity, which will smooth perturbations on a straight shoreline[13-15]. However, when waves approach at an angle greater than the maximizing one ( high-angle waves), (4) in effect has a negative diffusivity, and a straight shoreline is unstable [9]. In the high-angle case, moving alongshore in the transport direction between the inflection points of an infinitesimal-amplitude perturbation, the relative angle diverges from the value that maximizes the transport. Thus, along the crest of the perturbation, the alongshore sediment flux converges, causing accretion of sediment (Fig. 3). Wave Crests φ Flux Convergence OCEAN η x θ Inflection Points Shoreline at t0: Shoreline at t1: SHORE Fig. 3. Schematic plan view, showing the relative magnitude of the alongshore sediment flux (indicated by the arrows) and the consequent zones of flux divergence and convergence (erosion and accretion) on a perturbation to a straight shoreline subject to high-angle waves. (When alongshore flux is cast as a function of breaking-wave quantities, and the breaking-wave height is held constant, the alongshore flux is maximized for a breaking-wave angle of 45. Because of nearshore refraction, ocean waves essentially always break at much lower angles. However, because of the stretching of wave crests with refraction, breaking wave height decreases as the deep-water wave angle increases; breaking-wave height and breaking-wave angle are not independent. Analysis shows that alongshore transport will be maximized when ocean swells shoaling over shore-parallel contours (with a period of 10 s and a height of 2 m) break at 11, not 45. In this case, waves breaking at > 11 will cause perturbations on a straight shoreline to 4

5 grow.) For the case of high-angle waves approaching from a constant direction, as the amplitude of the feature increases, the angle between the deep-water wave crests and shoreline at the inflection point on the updrift side of a feature will approach, but not increase beyond, the angle that maximizes sediment transport. However, the inevitably continued erosion updrift of this point will cause the inflection point to migrate continually toward the crest. This erosion of the updrift flank and accumulation at the crest and downdrift of it will cause a shoreline bump to translate downdrift. Such translation will not occur under the influence of low-angle waves, or when the shoreline orientation at the updrift inflection point is less than that required to produce the maximum alongshore transport (for a given deep-water wave angle). If the relative angle between the waves and the regional shoreline trend is only slightly greater than the transport-maximizing angle, subtle, small-amplitude shoreline features should result. 4. NUMERICAL MODELING 4.1. Methods A simple numerical model has been developed to investigate the effects of the shoreline instability over an extended domain and with temporally varying wave angles [9]. Similar in concept to other one contour line numerical models [16-18], this model discreteizes sediment transport in the alongshore direction and assumes the conservation of nearshore sediment (equations (1) and (2)). This model, however, can handle an arbitrarily complex shoreline and uses a numerically stable solution during periods of high-angle wave approach. Deep-water wave variables are refracted to a depth-limited breaking point assuming shore-parallel contours and sediment flux is determined by equation (1) [9]. For the simulations performed here, initial conditions consist of a straight shoreline with white noise random perturbations. Every simulated day, a new deep-water wave approach angle (relative to the overall shoreline trend) is randomly selected from a probability distribution function (PDF). To represent the approximate local wave climate for the sections of the Dutch Coast analyzed by Ruessink and Jeuken [2], we are basing the PDFs on wave data gathered off of the Holland and Wadden Coasts (Fig. 1), relative to the orientation of the coastline in each section. For sections 3 and 5, we are using the Ijmuiden Munitiestortplaats platform, in 21 meters depth off of section 4. For sections 6-8 we are using the Eierlandse Gat platform, in 26 meters depth west of section 7. (We are not treating sections 1, 2 and 4 because the coastline orientation varies too much in those sections to meaningfully define a single orientation/local wave climate. Fig. 3a shows an example of the input wave climate relative to the orientation of the section 3 coastline. The probability in each bin, P i, is based on the average H o within that bin; P i is weighted by H o 12/5, the wave part of the alongshore-transport formula for deep-water waves (2). Summing the alongshore sediment transport given in (2) for all of the measured values and dividing by the total alongshore sediment transport (absolute value of the term in (2)), gives a relative measure of the net sediment transport, which we will call the Net Transport Index. 5

6 Fig 4. Wave climate for section 3 with a shore-parallel orientation of 40 degrees E of N. Data taken from Ijmuiden Munitiestortplaats station (YM6) from a. PDF of wave height contributions to alongshore sediment transport (obtained by summing H o 12/5 within each 7.5 degree bin). b. Graph of relative diffusivities (D iff ) calculated by multiplying the wave contribution to alongshore sediment transport by the factor in the brackets {} in (4). Fig. 3b shows the diffusivities produced by the waves in each angle bin, D iff,i, based on (4). (In other words, this graph is produced by multiplying the factor in the {} brackets in (4) by H o 12/5.) Summing the relative diffusivities in each bin provides a measure of whether perturbations will grow (positive sum) or be smoothed (negative sum) in the long term. We express this as an Instability Index, ΣD iff,i /Σ D iff,i, where the summation is over all the bins. Another measure of the local wave climate we will use involves the directional asymmetry of the high-angle waves, the High Angle Asymmetry Index, Σ (sign of wave angle)*d iff,i /Σ D iff,i, where the sum is only over the high-angle bins (D iff > 0). Table 1. Wave-climate parameters for coastline sections in Fig. 1. Negative values for the High-Angle Asymmetry Index and the Net Transport Index represent the North/East direction. Orientation (degrees E of N) Section Instability Index Asymmetry Index Net Transport Index Table 1 lists the statistics for each shoreline section. In general, two peaks in approaching wave energy, from the north and west, affect the Dutch coast. Within each section, the instability index is negative, indicating a predominance of diffusive, low-angle waves. However, high-angle waves affect all of the shoreline segments, sometimes representing almost half of the shoreline diffusivity, such as at sections 3 and 4, where the effect of low- and high-angle waves is nearly 6

7 balanced. Over all of the sections, the net sediment transport is much less than the gross sediment transport, in many sections less than 1% of the gross transport Model results Fig. 5 shows, for each section, the model results plotted in the same way as the observations are in Fig. 2 ( 2), and the shoreline shapes at each time. When the local wave climate is weighted relatively heavily toward low-angle waves (Instability Index below approximately 0.1), the shoreline evolution is dominated by diffusion. For example, in the section 7 and 8 results, the shoreline relaxes monotonically, toward longer-wavelength undulations. The space-time diagrams of these results show relatively long wavelengths and apparently organized, coherent behavior. The apparent polarity switches in these plots, however, are an artifact of the analysis technique applied to a diffusing shape (in which the long-term change at each point is not linear in time). In contrast, when the local wave climate is near the crux of instability (Instability Index near 0), as it is in our estimates for sections 3 and 5, shoreline undulations can maintain an approximately constant amplitude for years. During times of high-angle waves, new, short-wavelength perturbations can grow to significant amplitudes. However, these relatively short-wavelength bumps are diffused by subsequent low-angle waves more rapidly than the persistent undulations. The transient short-wavelength features give the time-space diagram a less-coherent appearance. However, in the simulations, the undulations always diffuse away eventually (perhaps starting with a time period of more-than-usual low-angle waves), as the smoother curves in the shorelineshape plots (Fig. 5) show. Because the high-angle waves are fairly symmetric in the section 5 wave-climate estimate (low High-Angle Asymmetry Index), the undulations do not migrate very consistently. The shorelineshape plots for section 5 shows little net migration; some of the apparent migration in the spacetime diagram is similar to the polarity shift in section 7. When the high-angle waves are more asymmetric, as in the section-3 estimate, undulations translate more consistently along the coastline. The shoreline-shape plot for section 3 shows more variable undulation positions. (Shoreline-shape plots for a shoreline orientation show undulations with amplitudes on the order of 100 m and relatively rapid translation.) 4.3. Comparisons Between Model Results and Observations Some of the trends in the model results also hold for the observations. In both model results and observations, the time-space diagrams for sections 3-5 show an apparently smaller wavelength than do those for sections 7 and 8 (Figs. 2 and 5). (In the model results, section 6 exhibits the same final wavelength as in sections 7 and 8. This 10-km wavelength is likely influenced by the alongshore-periodic boundary conditions and the size of the domain in these model runs, which will not allow a wavelength between 10 and 20 kilometers to develop. Section 6 is between the two wave platforms we used, and if we use data from the southern platform, the results look very similar to those for section 5.) 7

8 Fig. 5. Model results: Space-time plots of x res hetero, and shoreline shapes, for approximate wave climates for shoreline sections shown in Fig. 1. Section 8 simulations show essentially identical behavior to that in section 7. Boundary conditions are periodic in the alongshore direction. 8

9 Model predictions for trends in alongshore translation rates are only partially successful. In both model results and observations, features move to the NE in section 3, move less consistently in section 5, and are stationary in section 6. However, in the observations, sections 7 and 8 exhibit relatively rapid migrations, while in the model, shoreline undulations are stationary in these sections. 5. DISCUSSION We speculate that the biggest discrepancy between model results and observations, the translations evident in the observations of section 7 and 8, might arise from inlet effects, which are not considered in the model. Under the influence of diffusive wave climates, the shoreline undulations in the model have very low aspect ratios amplitudes of less than 10 m over 10-km wavelengths (Fig. 5). These undulations are apparently too subtle to be subject to the finiteamplitude effects that cause translation ( 3) under the influence of high-angle waves. However, sections 7 and 8 are both flanked by inlets, and abrupt shifts in inlet behavior could provide perturbations to the downdrift beaches [e.g. 8] that have more significant amplitudes. The relatively high observed standard deviations in x res hetero in sections adjacent to inlets, including sections 7 and 8 (Fig. 2), could be indications of the influence of dynamic inlet/delta behaviors. If significant slugs of sediment were being sporadically delivered to sections 7 and 8 from the updrift inlets, the high asymmetry in the high-angle waves in these sections (Table 1) would tend to cause the resulting shoreline bumps to translate to the east as they diffuse. Work is underway to test the effect of adding spatially coherent, relatively large bumps to the model shoreline. Work is also underway to test the effect of dynamically adding spatially independent perturbations to the model shoreline. Such added perturbations would represent the effects of smaller scale processes such as the development of beach cusps and the formation and movement of alongshore bars that prevent a shoreline from becoming perfectly straight as they tend towards in the simulations. This highly simplified, exploratory model is not designed to reproduce the details of observations from a specific region, and our purpose here is not to simulate historical coastline changes with a high degree of quantitative accuracy. Our model omits many factors and processes that affect shoreline behaviors, such as shoreface lithology, off-shore bathymetry that concentrates wave energy, and the possibly large variations in the empirical coefficient K in (1) and (2). Rather than testing quantitative model predictions, which can vary with model parameters and the details of the ways model processes are represented formally, we have focused on the trends predicted by the model [19, 20]. 6.CONCLUSIONS The trends in the wavelength and alongshore translation of alongshore-heterogeneous shoreline behaviors displayed in time-space diagrams (Fig. 5) are robust results of the numerical model, which treats only alongshore sediment flux as a function of relative wave angles. The characteristics of observed sandwaves show some of the same trends as a function of the regional wave climate (Fig. 2), suggesting that simple properties of alongshore transport provide a plausible explanation for some (though not all) aspects of sandwave behavior on the Dutch Coast. 9

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