A range of planform morphologies emerge along sandy coastlines as a function of offshore wave climate. It has been implicitly assumed that the morphological response time is rapid compared to the timescales of wave climate change, meaning that coastal morphologies simply reflect the extant wave climate. This assumption has been explored by focussing on the response of two distinctive morphological coastlines – flying spits and cuspate capes – to changing wave climates, using a coastline evolution model. Results indicate that antecedent conditions are important in determining the evolution of morphologies, and that sandy coastlines can demonstrate hysteresis behaviour. In particular, antecedent morphology is particularly important in the evolution of flying spits, with characteristic timescales of morphological adjustment on the order of centuries for large spits. Characteristic timescales vary with the square of aspect ratios of capes and spits; for spits, these timescales are an order of magnitude longer than for capes (centuries vs. decades). When wave climates change more slowly than the relevant characteristic timescales, coastlines are able to adjust in a quasi-equilibrium manner. Our results have important implications for the management of sandy coastlines where decisions may be implicitly and incorrectly based on the assumption that present-day coastlines are in equilibrium with current conditions.
Many recent studies demonstrate how distinctive rhythmic planform coastline shapes on scales of kilometres to hundreds of kilometres (Fig. 1) can develop on wave-dominated sandy coasts due to morphodynamic interactions with wave climates (Ashton and Murray, 2006a, b; Ashton et al., 2001; Falqués and Calvete, 2005; Falqués et al., 2000; Kaergaard and Fredsoe, 2013a, b, c; van den Berg et al., 2012; Hurst et al., 2015; Nienhuis et al., 2013). Typically, previous studies assume (implicitly or explicitly) that coastline shapes chiefly reflect the contemporary wave climate (generally with specific regard to the angular distribution of wave angles), adjusting very rapidly to any changes. That is, intrinsic timescales for change in morphology are shorter than the timescales characterizing shifts in wave climate. Such rapid morphological change, relative to the rate of change in the wave climate, reflects a quasi- or dynamic-equilibrium morphological response to changing climate forcing: put simply, the rate of change in the morphology is much faster than the rate of change in the wave climate.
Diverse large-scale morphologies developed on open sandy coastlines
under the influence of dominantly high-angle offshore waves (
Changes in storm patterns in future climates will likely yield different wave climates (WASA Group, 1998; Hemer et al., 2013; Storlazzi et al., 2015; Wolf and Woolf, 2006) with concomitant changes in coastline behaviour. Moore et al. (2013) demonstrated temporal shifts in coastline shape on the Carolina capes caused by observed shifts in wave climate, supporting the prediction that even slight differences in wave climate can be associated with different morphologies (cf. Ashton and Murray, 2006a, b). Similarly, Allard et al. (2008) related spit growth and morphological change to wave climate variations on the Arçay Spit, France. Changes in wave climates will be manifest in altered patterns of shoreline change, including zones of enhanced erosion and/or accretion (Barkwith et al., 2014; McNamara et al., 2011; Slott et al., 2006). However, the assumption of quasi-equilibrium coastline change, with shoreline shape reflecting the present wave climate, has not been previously examined.
In an analogous study, Nienhuis et al. (2013) have shown how inherited delta morphology can influence the evolution of the coastline when sediment supply is perturbed: the shape of a delta can reflect not just present or recent sediment supply, but can depend strongly on the long-term history of sediment input. Might the influence of former morphology on coastline evolution also be true for sandy coastlines when wave climates are perturbed?
In this paper, we present the results of a study exploring the two key questions that arise from the above summary, specifically, the influence of former morphology of sandy coastlines on their subsequent evolution as wave climates change with regard to approaching wave angles (focussing on rates and spatial scales of response), and the degree to which coastlines can respond to changing wave climate in a quasi-equilibrium manner.
To address these questions, we have used the one-line Coastline Evolution Model (CEM; Ashton and Murray, 2006a) to examine (a) whether, and under what conditions, coastline shapes exhibit quasi-equilibrium responses to changes in the angular distribution of approaching waves; and (b) whether there are conditions under which coastlines might retain long-term memory of previous wave-climate forcing. Such hysteresis would greatly complicate forecasting and identification of coastline behaviours to be expected with changing wave climate.
We focussed on two end-member coastal morphologies, flying spits and cuspate capes (Fig. 1). These morphologies emerge as self-organized structures in response to high-angle wave instability that results when waves approach the coast at highly oblique angles, causing shoreline undulations to grow (Ashton et al., 2001; Falqués and Calvete, 2005). Their value from an academic perspective notwithstanding, our interest in these particular morphologies extends beyond their fascinating self-organized complexity. These morphologies are themselves important in many coastal regions because they often shelter fragile but highly diverse and dynamic shallow marine and estuarine ecosystems upon which human and animal communities depend, and also shelter socially and economically important coastal infrastructure. The fate of capes and spits under changing climate is thus of material concern to humanity and needs to be better understood to aid the development of appropriately informed coastal management policies.
In our experiments, we generated spit and cape coastlines using appropriate wave angle distributions (Ashton and Murray, 2006a). We then changed the wave angle distributions to be more diffusive in character, such that complex, high-amplitude perturbations (as exemplified here by spits and capes) are smoothed. Changes were made either instantaneously or gradually over a period of time. Outputs from the modelling were used to quantify how coastal morphologies responded to these changes.
In this initial investigation, we have focussed specifically on changes in the distributions of approaching wave angle, as these distributions fundamentally control gross coastal morphology at and above kilometre scales. Changes in other wave properties (height, period) control the rates at which changes occur, so are important in coastline evolution. However, we wished only to study the effects of changing wave angle distribution. The decision to move linearly from an unstable to diffusive wave angle distribution was motivated by wishing to understand the degree of stability of complex morphologies under changing wave angle distributions, hence the simplicity of our experiments. More complex experiments involving nonlinear or oscillating changes in wave climate were not warranted at this stage, but could be explored in future work. The timescales for change in those experiments where the wave angle distribution was changed gradually were to some extent arbitrary, but guided by timescales over which climate is known to be changing (decades–centuries) under the influence of anthropogenic activity.
Gradients in alongshore sediment flux, generated by wave-driven currents,
cause shoreline erosion or progradation on sandy coastlines. Assuming
conservation of mass in the shoreface, the temporal change in coastline
location in relation to flux gradients is given by
Critically,
The angle dependency of
For a nearly straight coastline, coastline evolution can be described by a
linear diffusion equation, where the diffusivity is either positive (stable)
or negative (unstable) (Ashton and Murray, 2006a, their Eq. 8). Every approaching wave condition
contributes to sediment transport and the consequent evolution of the
coastline. The overall effect of a wave climate on a coastline can be
determined from the net diffusivity,
For each wave at each location along a coastline, individual alongshore
sediment flux values are calculated using Eq. (2), and individual diffusivity
(
The CEM uses Eqs. (1) and (2) to explore coastline planform behaviour
numerically. The details of the theory and its implementation in the CEM are
discussed extensively elsewhere (Ashton et al., 2001; Ashton and Murray,
2006a, b), so here we summarize the model domain and setup. In this study,
the model domain consisted of
Wave angle distributions define the relative influences on alongshore
transport of all the waves approaching from angles falling within each angle
bin (Ashton and Murray, 2006b). Observed (or hindcast) wave records can be
transformed into such angular distributions, additionally weighting each wave
by height and period (
Because the principle interest of this study is in the effect of changing
wave angle distribution on planform change, the effects of variations in wave
period
Using linear wave theory, each offshore wave is refracted progressively over shore-parallel contours until depth-limited breaking occurs (e.g. see Hurst et al., 2015, Appendix A). At this point, the standard breaking-wave version of the CERC equation (e.g. Komar, 1998) is used to calculate the sediment flux as a function of the angle between the locally determined coastline orientation (Fig. 2a) and the wave angle at breaking and the breaking wave height. The coastline position is evolved based on the calculated gradients in flux, assuming conservation of mass in the shoreface (Eq. 1).
Net flux and diffusivity data generated by the CEM can be used to explore the
sensitivity of a coastline to change under existing and modified wave
climates, and the processes by which any change would occur, as discussed in
Sect. 2.2. In this study, the CEM was used to capture the
Snapshots of simulated coastline morphologies evolved under changing
wave climate. See Section 3.1.3 onwards for discussion.
We set up experiments by growing either flying spits or cuspate capes (“capes” from here on) from an initially straight coast (with small white noise perturbations) over an initial fixed period of time. In most experiments, this initial period was 250 model years. This time frame allows these morphologies to attain length scales commensurate with those observed along real coastlines. We then subjected these model coastlines either to a gradual change in wave climate, or an instantaneous change. In experiments with gradual wave climate change, the initial wave climate was evolved linearly towards the new state over an arbitrary period of 100 years. These experiments were used to explore the influence of pre-existing morphology on the nature and rate of response of a coastline to changing wave climate.
In experiments involving instantaneous change, the wave climate is transformed to the target state immediately following the period of initial growth; in these experiments, we also used initial periods of 50 and 125 years to provide additional data that we could use to determine characteristic timescales for change. This allowed us to explore the possibility of scaling relationships between time and the rate of change of length scales, and the degree to which a quasi-equilibrium response in morphology is possible for given rates of wave climate change.
In both cases, coastline morphology is in dynamic equilibrium with the
initial wave climate just before the wave-climate transition begins. As the
wave climate changes, the coastline progressively approaches a new
morphological state, settling into dynamic equilibrium with the final wave
climate. We characterized coastline morphology using the aspect ratio
(cross-shore extent
Model simulations were driven by wave approach angles drawn from a
probability density function (PDF) defined by two parameters (Ashton and
Murray, 2006a):
Ashton and Murray (2006a, their Fig. 9) mapped different coastline shapes
that emerge for different values of
Model capes are generated over 250 years with
Examples of the changes that occur in the planform morphology of our
experimental coastlines during the model runs are shown in Fig. 3; note that
these data are for experiments in which the wave climate is changed gradually
over 100 years from the initial
In Figs. 4 and 5 we plot the evolution of aspect ratio, wavelength and amplitude of coastal features (capes and/or spits) during the experiments. The data have been smoothed using a seven-point moving average window to aid clarity. Animations of the model simulations from which the results discussed in this study were derived are also included in the Supplement.
Cape morphology adjusts mostly in the 100-year period over which the wave
climate changes (Fig. 4ai). Cape amplitude declines through erosion of the
cape tips (Fig. 3aii, first and second panels). Rapidly declining aspect
ratio (mean amplitude
Time series of the evolution of aspect ratio
(amplitude
Time series of the evolution of aspect ratio
(amplitude
Flying spits change much more slowly, exhibiting pronounced long-term memory
(Fig. 4bi). The coast shape in dynamic equilibrium with
The results show that for both capes and spits, the scale of both wavelength
and amplitude of coastline features is larger than would be expected had the
coastlines formed under the final wave climates (
To derive characteristic timescales for change in morphology and to examine the relationship between temporal and spatial scales, further model experiments were run with initial wave climate conditions lasting 50, 125 and 250 years, followed by an instantaneous change in wave climate. The rates of morphological change in capes and spits are shown in Fig. 5a and b respectively, with reference to initial conditions run for 250 years.
Using the change in aspect ratio as the metric, we determined characteristic
timescales for morphological change (i.e. first and second
Based on the results from the instantaneous change experiments, we can distinguish two modes in which coastlines can adjust to changing wave climate, exemplified by the cape and spit experiments, respectively: cape morphologies adjust to a zero net flux condition, whilst spit morphologies adjust to a condition in which there is a constant down-drift translation of the feature.
To understand how capes adjust, we begin by considering capes in dynamic
equilibrium with a symmetric wave climate dominated by high-angle waves
(
For a dynamic equilibrium to develop under a symmetric wave climate – a
state in which the coastline adjusts very rapidly to small changes in wave
climate at short temporal and small spatial scales – all local coastlines
must adjust to orientations that produce little or no net sediment flux under
the local wave climate. As high-angle waves become more dominant in the wave
climate, the local coastlines adjusted to zero net flux lie at progressively
greater angles, relative to the regional coastline orientation, towards the
cape tips. Consequently, cape amplitudes increase with larger
Examples of these characteristics are shown in Fig. 6a, c for a section of
coastline with two sample capes. Figure 6a shows a cape pair from part of a
model cape coastline grown for 250 years under a symmetric wave climate with
As the wave climate shifts to one dominated by waves approaching more
directly onshore (e.g.
Detailed mapping of potential net flux and stability index across
example coastline features for different wave climates impacting on static
(unevolving) coastlines. Net flux data for
For spits, both the mode of emergence of the steady-state pattern and the mode of subsequent adjustment under a changing wave climate are more complicated. For an asymmetric wave climate, under which there is translation of finite-amplitude coastline features (Ashton and Murray, 2006a), a dynamic equilibrium is indicated by constant alongshore translation of the spits.
The tips of spits tend to propagate in the direction parallel to the
shoreline orientation that produces the maximum sediment flux for the given
regional wave climate (Ashton and Murray, 2006a; Ashton et al., 2016).
However, the flanks and tips of spits also experience alongshore translation
caused by erosion at their updrift ends. The updrift portion of each spit is
shadowed by its updrift neighbour. Given that flying spit growth requires a
wave climate in which the dominant wave angles are high and from the updrift
direction, this shadowing is greatest at the updrift end of each spit. The
shadowing decreases progressively downdrift, the spit coastline becoming more
exposed to the waves approaching from the dominant direction. Gradients in
net sediment flux arising from down-drift decline in wave-shadowing effects
tend to produce erosion, resulting in seaward concavity in the coastline
(Fig. 6b, d). However, the development of concavity is limited because the
increasing curvature of the coastline tends to result in accretion as the
change in coastline geometry interacts with the locally experienced wave
climate. Thus, the erosion tends to be balanced by coastline flattening
induced by the local wave climates that result from wave shadowing. The
concavity tends to be flattened and the locus of erosion propagates downdrift
to the spit flank and tip. As long as different parts of the spit translate
alongshore at different rates, the shape evolves (Ashton et al., 2016).
However, when the shape and the gradients in net sediment flux adjust such
that each segment of the spit translates alongshore at the same rate (given
by the local cross-shore erosion rate
When the wave climate shifts to one still asymmetric in character, but with
an increased proportion of low-angle waves (e.g. see wave rose PDFs in Fig. 3
for
Thus, for flying spits, the flux gradients scale approximately with the maximum net sediment flux divided by the total spit wavelength, rather than the small fraction of cape wavelength represented by cape tips. These two modes of adjustment – one in which gradients in net sediment flux occur over some small fraction of the wavelength, and one in which they occur over the whole wavelength – do not apply exclusively to capes and spits however. Once an adjusting flying spit reconnects with the down-drift coastline (Fig. 3bii, second panel), gradients in sediment flux occurring over a small fraction of the wavelength begin to exert an influence (Nienhuis et al., 2013). Furthermore, capes formed by slightly asymmetric wave climates (Ashton and Murray, 2006a), which produce a net alongshore sediment flux, also migrate alongshore and are, therefore, also subject to the constant alongshore-translation condition.
The experiments reported here involve only two types of coastline morphology
and two types of wave climate change. This limited exploration motivates a
wider, more systematic investigation of the responses of a broad range of
morphologies to changes in wave climate. Although beyond the scope of this
initial study, experiments like those depicted in Fig. 2c could be conducted
for morphologies within the (
The results reported here provide
These critical timescales depend not only on coastline morphology and the scale of coastline features (see also Hurst et al., 2015), but also on the characteristic wave heights and shoreface depths, which influence rates of coastline change in quantifiable ways. The timescales predicted by model experiments, such as those we present here, will be altered quantitatively when different wave heights, shoreface depths or alongshore-sediment flux relationships (or, indeed, empirical coefficient values) are used. However, such quantitative changes will not affect the relevance of comparing timescales for morphological response and wave climate change to understand or forecast whether coastlines will exhibit a quasi-equilibrium response or be influenced by a “memory” induced by preceding morphological states.
These results also have implications for management of potentially fragile sandy coasts. Management policies and plans are commonly underpinned by predictions of future shoreline erosion (or accretion) rates along developed coastlines that are based on shoreline change observed over previous (usually very few) decades. Observations accumulated over such relatively short timescales may not be sufficient to discern the true direction of morphological change, since these are of roughly the same order as the characteristic timescales indicative of limits to the potential for equilibrium morphological response, as calculated above. Furthermore, change in environmental drivers (weather patterns, storm frequency, etc.) may be poorly understood. Thus, wider analysis of environmental conditions and coastline response is likely needed for more informed decision making. Indeed, coastlines deemed to be under threat from climate change effects, and therefore requiring socio-economic as well as environmental management, should benefit from long-term monitoring of weather and geomorphology to understand what kind of intervention might be necessary, and to help preclude costly, non-beneficial or even damaging actions.
Our analysis has purposefully not considered changes in cross-shore sediment flux resulting from erosion related to sea-level rise (e.g. Moore et al., 2010; Wolinsky and Murray, 2009). However, the changes in coastline shape we have addressed can be superimposed on shoreline change associated with sea-level rise (e.g. Bruun, 1962; Walkden and Hall, 2005, 2011). Increases in the rate of sea-level rise generally cause increases in shoreline erosion, but the response to sea-level rise is approximately uniform along a sandy coastline at the scales of interest (e.g. Moore et al., 2010). In contrast, planform changes in coastline position arising from alongshore gradients in sediment flux indicate heterogeneous erosion and accretion along a shore: spatial patterns and magnitudes of accentuated shoreline erosion could be different in the future, compared to the recent past. This could result from quasi-equilibrium style adjustment to future changes in wave climate, analogous to that which Moore et al. (2013) found for cuspate capes over recent decades. However, our results illuminate the possibility that even without wave climate change occurring in the present or near future, coastline shape could be in the midst of a long-term readjustment to changes in wave climate that occurred in the past. Given the diffusive scaling of coastline response timescales with length scales, large coastal features inherited from the early Holocene may not yet have adjusted to current wave conditions. In such a case, zones of accentuated erosion would tend to shift in location and intensity over time, without any warning from identifiable changes in forcing. This possibility motivates future work to develop metrics in terms of alongshore patterns of local wave climates (e.g. Ashton and Murray, 2006b) or sediment fluxes that could identify coastlines that are in or out of (quasi-) equilibrium with the present directional wave climate.
Finally, the results of these model experiments might also have implications for geologic and paleo-climate interpretations. Our modelling results indicate that flying spits adjusting to changes in the wave angle distribution in a wave climate can leave behind records of the adjustment in the form of complex arrays of lagoons enclosed by beach ridges (e.g. see Fig. 3bii, second panel; 3biii, first and second panels). In natural settings, the lagoons, which can potentially extend far enough landward to be preserved, would fill with fine sediment over time. However, the morphological arrangement of such lagoons and associated bounding beach ridges, preserved in the geological record, could indicate the effects of changing wave climate on a spit coastline. Such complicated coastal plain deposits can form in other ways (including reworking of a relict, potentially crenulated coastline present at the beginning of the current sea-level high stand, and/or episodic fluvial or coastal sediment delivery, Nienhuis et al., 2015), but being aware of the possible influence of wave climate change on the morphology and structure of coastal hinterlands could inform reconstructions of coastal geographies and paleo-wave climates.
We have explored the degree to which complex sandy coastlines can adjust in a quasi-equilibrium manner to changing wave climate, and the degree to which past wave climates, as manifest in existing morphology, might influence subsequent coastline evolution during wave climate change. We conducted numerical modelling experiments in which capes and spits grown under some initial condition were subjected subsequently to a different wave climate. We find that the resulting evolution of complex coastlines (capes and spits) is influenced by previous morphology for a significant period of time – in some cases rather longer than indicative characteristic timescales of morphological change (few decades to few centuries) after wave climate change has occurred. Characteristic timescales for change show that spits respond at rates approximately an order of magnitude more slowly than capes, and these timescales depend on the spatial scale of the coastline feature (a diffusive scaling). In particular, quasi-equilibrium response cannot be assumed: such behaviour will depend on the rate of wave climate change, and how it compares with the characteristic timescale for morphological adjustment for a given coastline morphology. Significant changes in wave climate on decadal to centennial scales may result in hysteresis in the response of the coastline morphology. Thus, modern coastlines may be out of morphological equilibrium with respect to current wave conditions, reflecting instead the wave climates to which they have been subject in the past, and the resulting antecedent coastline morphology. The preservation of beach land forms in the coastal hinterlands behind modern sandy coasts may reflect this history and provide insights into past paleo-wave climate and coastal geography.
The Coastline Evolution Model (CEM) can be downloaded from
the Community Surface Dynamics Modeling System model
repository (
Christopher W. Thomas and A. Brad Murray planned the experimental design and undertook the modelling and research, and also prepared the manuscript, supported by Andrew D. Ashton and Martin D. Hurst. The original ideas for this work, and their subsequent development, grew out of initial discussions between Christopher W. Thomas and Andrew K. A. P. Barkwith; Andrew K. A. P. Barkwith subsequently provided further ideas, advice and support for the later modelling. Michael A. Ellis supported and helped shape the research through discussion and further ideas. All have contributed to the manuscript through additional or modified text and comments.
Christopher W. Thomas gratefully acknowledges support from the British Geological Survey and the Nicholas School of the Environment, Duke University, North Carolina, for a short sabbatical at Duke University in 2015, during which this work was largely undertaken. Eli Lazarus and an anonymous referee are thanked for constructive and insightful reviews. This work was funded by NERC national capability core funding to the British Geological Survey. The Coastline Evolution Model (CEM) can be downloaded from the Community Surface Dynamics Modeling System model repository. Edited by: P. Passalacqua Reviewed by: E. D. Lazarus and one anonymous referee