Initial Tests of a Possible Explanation For Alongshore Sandwaves on the Dutch Coast Andrew Ashton1, A. Brad Murray1, and G.B. Ruessink2 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, [email protected] 2 Department of Physical Geography, Faculty of Geographical Sciences, Institute for Marine and Atmospheric Sciences Utrecht, Utrecht University, P.O. Box 80.115, 3508 TC Utrecht, Netherlands. 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 A slocations h t o n et al. of the wave gauges used to estimate wave climates. From Ruessink and Jeuken [2]. computing a linear temporal trend of dunefoot position at each measurement location, and removing this trend, producing a residual time series xres, for each location. Then, the average of xres for each year was calculated along each coastline section. This alongshore average, representing the alongshore-homogenous shoreline behavior (e.g. alongshore-uniform response to hetero storms), is then removed. The result, xres , reflects alongshore-heterogeneous deviations from hetero the temporal trend. Fig. 2, which shows time stacks of xres , reveals organized, alongshore- migrating 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, Qs, involves the breaking wave height, Hb, and the relative angle between breaking wave crests and the shoreline (φb -θ ): 5/2 Qs = K Hb cos(φb -θ ) sin(φb -θ ) (1) where K is an empirical parameter, θ is the shoreline orientation, and φb is the orientation of wave crests. Assuming shore-parallel depth contours, this relation can be recast as (using the 1/5 approximation cos (φb - θ ) ≈ 1): 12/5 6/5 Qs = K’ Ho cos (φo -θ ) sin(φo -θ ) (2) where the subscript o denotes deep-water values. Alongshore sediment transport is maximized for (φo -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 (φo - Delta Coast Holland Coast Wadden C. 12 34 56 78 1950 1900 Time (yr) (a) 1850 31.3 6.4 17.5 1.5 118 103 60 51 26 20 0.2 41 52 9 16 100 (b) 50 St. dev. (m) 0 31.3 6.4 17.5 1.5 118 103 60 51 26 20 0.2 41 52 9 16 Distance (km) Fig. 2. Time-space diagram of how far shoreward or seaward each shoreline position is hetero compared to an alongshore and temporal average, xres , with positive values (seaward deviations) shown in dark gray and negative values in light gray. From Ruessink & Jeuken [2]. 3 A s h t o n et al. 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: 12/5 1/5 2 2 2 2 ∂η/∂t = K2H0 {cos (φ0 -θ)[ cos (φ0 -θ) - (6/5)sin (φ0 -θ) ]}∂ η/∂x , (4) where K2 is a constant. For given deep-water wave characteristics (H0 and φ0), (4) takes the form 2 2 of a diffusion equation, ∂η/∂t = Diff∂ η/∂x , where Diff is the diffusivity. When the angle between approaching wave crests and the shoreline (φ0 -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 Flux OCEAN φ Convergence SHORE θ Inflection η Points x Shoreline at t : Shoreline at t : 0 1 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.