Difference between revisions of "Dune erosion"

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{{Definition|title=Dune erosion
 
{{Definition|title=Dune erosion
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|definition= [[Dune]] erosion involves that, during a severe [[storm surge]], sediments from the mainland and upper parts of the [[beach]] are eroded and settled at deeper water within a short time period; this is a typical cross-shore sediment transport process.}}
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|definition= Sand loss from a dune under wave attack, mainly by notching, avalanching and slumping processes.}}
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This article deals with dune erosion by storms and provides some simple rules from which retreat of the dune front can be estimated. The focus is on the so-called 'collision regime', where waves hit the dune without exceeding the dune crest. 
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==Introduction==
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[[Image:DuneScarpEgmond.jpg|400px|thumbnail|right|Fig. 1. Post-storm eroded dune, Egmond aan Zee (Netherlands NL).]]
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Coastal dunes have developed naturally along many shorelines worldwide during the last millennia when sea-level rise slowed down. Wave action and onshore winds are the main agents for coastal dune development and sufficient sand supply to the coast is a primary condition<ref>Anthony, E.J., Mrani-Alaoui. M. and Héquette, A. 2010. Shoreface sand supply and mid- to late Holocene aeolian dune formation on the storm-dominated macrotidal coast of the southern North Sea. Marine Geology 276: 100–104</ref>, see also the article [[Dune development]]. The coastal dune belt in many cases protects low-lying hinterland from flooding by the sea. Dune erosion therefore can be a serious threat.
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A coastal dune can suffer large losses when attacked by storm waves. The front dune can be taken away over several tens of meters, leaving a steep dune scarp, see Fig. 1. For the Dutch coast it has been estimated that under exceptional circumstances (extreme storms of very long duration, which may occur with a yearly probability of 1/100,000) the dune volume loss can amount to 800 m<sup>3</sup>/m, causing the dune foot to recede from 80-100 m at some places<ref>Den Heijer, C. 2013. The role of bathymetry, wave obliquity and coastal curvature in dune erosion prediction. PhD Thesis, Delft University</ref>.
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The actual dune loss depends on many factors<ref name=C23>Conlin, M.P., Cohn, N. and Adams, P.N. 2023. Total water level controls on the trajectory of dune toe retreat. Geomorphology 438, 108826</ref>:
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*beach geometry (pre-storm): beach slope and width;
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*grain size;
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*dune geometry (pre-storm): dune face slope, dune crest elevation;
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*nearshore morphology: nearshore slope, rip-channel embayments, [[nearshore sandbars]], subtidal morphology;
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*beach and dune vegetation; presence of (embryo-)foredunes;
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*presence of hard engineered structures;
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*alongshore variability in wave energy and dune height.
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The dune toe elevation (beach width<math>\times</math>slope) appears to be a major factor; observations show that vulnerability to erosion decreases with increasing dune toe elevation<ref name=S18>Splinter, K.D., Kearney, E.T. and Turner, I.L.  2018. Drivers of alongshore variable dune erosion during a storm event: Observations and modelling. Coastal Engineering 131: 31–41</ref>. Dune sand losses have also been positively correlated with the steepness of the dune face<ref name=BC>Burvingt, O. and Castelle, B. 2023. Storm response and multi-annual recovery of eight coastal dunes spread along the Atlantic coast of Europe. Geomorphology 435, 108735</ref>. There is instead strong evidence that dunes are less impacted by storms when situated on dissipative flat beaches than on reflective steep beaches<ref name=MH>Masselink, G. and van Heteren, S. 2014.  Response of wave-dominated and mixed-energy barriers to storms. Marine Geology 352 (2014) 321–347</ref>.
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Dune loss also depends on storm characteristics: maximum sustained wind speed and duration; maximum water level and simultaneous wave height, period and incidence angle.
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If at least part of the dune belt survives the storm, it can recover by natural processes – similar to the processes that created the original dune. See section [[#Post-storm dune recovery|Post-storm dune recovery]]. Because recovery of the beach-dune profile (especially the dune profile) takes much longer than storm erosion, the dune system is particularly vulnerable to storm clusters (quick succession of storms)<ref>Karunarathna, H., Pender, D., Ranasinghe, R., Short, A. D. and Reeve, D. E. 2014. The effects of storm clustering on beach profile variability. Mar. Geol. 348: 103–112</ref>. The probability that extreme dune recession is caused by a storm cluster is greater than the probability that this retreat is caused by an individual storm<ref>Baldock, T.E., Gravois, U., Callaghan, D.P. and Nichol, S. 2021. Methodology for estimating return intervals for storm demand and dune recession by clustered and non-clustered morphological events. Coastal Engineering 168, 103924</ref>.
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[[Image:imageDu002.JPG|350px|thumbnail|left|Fig. 2. Storm-induced dune foot retreat as a function of frequency of exceedance for the Dutch coast.]]
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Figure 2 shows as an example a model result of the relationship between dune foot retreat <math>RD</math> (expressed as the distance between pre-storm and post-storm dune foot locations) and exceedance frequency (the probability that in a particular year a storm occurs that produces greater erosion) for a location along the Dutch coast. According to this figure, which is based on the [[#Storm profile method|storm profile method]], there is a  10<sup>-5</sup> probability per year that dune erosion will exceed 85 m.
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To establish such a relationship, knowledge is required of extreme storm conditions with a very low probability. For these storms, the joint probability of extreme surge height, surge duration and corresponding wave conditions has to be determined<ref>Galiatsatou, P. and Prinos, P. 2016. Joint probability analysis of extreme wave heights and storm surges in the Aegean sea in a changing climate. In: FLOODrisk 2016-3rd European Conference on Flood Risk Management</ref>. Dune retreat estimates with an exceedance probability of 10<sup>-5</sup> are subject to great uncertainty, as reliable measurements of extreme water levels and wave conditions are only available over relatively short periods of the order of 100 years. This issue is further discussed in section [[#Dune failure probability|Dune failure probability]].
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==Impact of dune erosion==
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[[Image:BadhotelSchiermonnikoog_1924.jpg|350px|thumbnail|right|Fig. 3. Badhotel at Schiermonnikoog (NL) after the 1924 storm.]]
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Buildings on the front dune situated close to the dune foot are at risk under severe storms, see Fig. 3. A sound estimate of potential dune erosion is required when issuing building permits (see [[Setback area]]). In cases where the hinterland is situated below sea level, the dune belt serves as sea defence. Breach of the dune belt may have catastrophic consequences. Dune and beach monitoring and dune management are of crucial importance. In some cases, where the dune belt consists of a single dune row, dune reinforcement or shore nourishment may be needed. An example is the so-called 'Sand Engine' on the Dutch coast<ref>Luijendijk, A.P., Ranasinghe, R., de Schipper, M.A., Huisman, B.A., Swinkels, C.M., Walstra, D,J,R. and Stive, M.J.F. 2017. The initial morphological response of the Sand Engine: A process-based modelling study. Coastal Engineering 119: 1–14</ref>, which contributed to the reinforcement of the single protecting dune row through wind-driven sand transport from the widened beach<ref>Gerdes, E., Koning, R., Kort, M. and Soomers, H. 2021. Policy evaluation Sand Engine 2021. Rebel Economics & Transactions, report commissioned by Rijkswaterstaat, Netherlands</ref>.
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To ensure safety, several methods have been developed for estimating dune loss during exceptional storm conditions. 
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==Brief explanation of dune erosion processes==
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The initial cross-shore beach profile, considered to be in a more or less dynamic equilibrium condition with normally occurring hydrodynamic conditions, will be reshaped during a severe storm surge. Offshore directed sediment transport will occur, including sand eroded from the dune.
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Infiltration of swash into the dune via capillary action acts as a destabilizer by increasing the effective weight of dune sediment, thereby decreasing slope stability. Wave attack at the dune foot steepens the dune profile which may collapse by avalanching (Fig.4). Wave attack may also create a notch at the dune foot leading to mass failure: collapse of a dune slab, initiated by tensile cracking at the top surface of the dune followed by shear failure along an internal failure plane or overturning due to the weight of the overhang <ref>Erikson, Li, H., Larson, M. and Hanson, H. 2007. Laboratory investigation of beach scarp and dune recession due to notching and subsequent failure. Marine Geology 245: 1–19.</ref>, see Fig. 5.
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Dune scarping can occur over the entire dune face, but for high dunes the height of the dune scarp will generally not reach the dune crest<ref name=BC/>.
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{| border="0"
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|-
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| valign="top"|
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[[File: DuneFailureMechanisms.jpg|450px|thumbnail|left|Fig. 4. Dune failure mechanisms.]]
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| valign="top"|
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[[File: DuneErosionEgmond.jpg|450px|thumbnail|left|Fig. 5. Dune scarp after collapse. The people standing on the dune scarp have not read this article.]]
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|}
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A slab that slides down the dune scarp will temporarily defend the dune face, until waves and currents mobilize the sediment from the slump deposit and transport it offshore. After the sediment is transported away, the dune face is left exposed again, leaving room for a new slumping event. The total impact of a storm on the dune strongly depends on how fast the sediment is transported offshore during the storm.
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The sand eroded from the dune face is removed further down by the [[Swash|backwash]] and the seaward undercurrent ('[[undertow]]') under the breaking waves. Turbulence generated by wave bores is an important driver of offshore suspended sediment transport in the surf zone<ref>van Wiegen, P.P.J., de Vries, S. and Reniers, A.J.H.M. 2024. Field observations of wave-averaged suspended sediment concentrations in the inner surf zone with varying storm conditions. Marine Geology 473, 107302</ref>.
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Most of the sand is deposited on the lower beach or upper [[shoreface]]. In cases where strong [[rip current]]s are present or a strong ebb current from nearby inlets, part of the sand can even be transported to the lower shoreface<ref name=MH/>. The slope of the cross-shore profile gradually decreases, and consequently the rate of dune erosion will decrease with time during the storm surge. Usually, the limited storm surge duration does not allow for the development of a new equilibrium profile. The shape of the cross-shore profile after the storm surge, which is a transient state between the initial profile and the storm equilibrium profile, is often called 'storm erosion profile'.
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==Quantification of dune erosion==
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Although 3D effects are generally important in the dune erosion process, often a 1D or 2DV approach is adopted, ignoring alongshore variability. In this case the dune erosion process is considered as a typically offshore directed cross-shore sediment transport problem. Sand from the dunes is transported to deeper water and settles there.
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A first approach consists of assuming a closed sediment balance in cross-shore direction. The same volume of sand which is eroded from the dunes and the very upper part of a cross-shore profile is accumulated lower in the cross-shore profile. [Because of differences in porosity of the eroded dune material (often loosely packed) and the settled material (often a bit more densely packed), the volume balance is not always strictly closed.]
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During severe storm surges with greatly increased water levels (a water level increase of a few meters is possible along some coasts), huge volumes of sand from the dunes are transported in offshore direction, on the order of hundred or several hundred meters<sup>3</sup>/m <ref name=BC/>.
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===Storm profile method===
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A quick first order estimate of storm dune erosion can be obtained by assuming that the storm erosion profile is close to the storm equilibrium profile. This assumption is unrealistic for storms of short duration and strongly overestimates storm dune erosion in this case. However, for storms of very long duration, which produce great dune losses with low exceedance probability, the assumption of a post-storm equilibrium profile is not unreasonable. The method is still used in the Netherlands, besides other more advanced methods, to provide an upper bound of possible storm dune erosion<ref>van Wiechen, P.P.J., de Vries, S., Reniers, A.J.H.M. and Aarninkhof, S.G.J. 2023.  Dune erosion during storm surges: A review of the observations, physics and modelling of the collision regime. Coastal Engineering 186, 104383</ref>. It is based on empirical post-storm equilibrium profiles established by laboratory experiments and validated by field data of the Dutch coast <ref>van Gent, M.R.A., van Thiel de Vries, J.S.M., Coeveld, E.M., de Vroeg, J.H. and van de Graaff, J. 2008. Large-scale dune erosion tests to study the influence of wave periods Coastal Engineering 55: 1041–1051</ref>. The post-storm equilibrium profile was established for different values of the significant wave height <math>H</math>, peak wave period <math>T</math> and mean fall velocity of dune sand <math>w</math>. Major assumption are:
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# the storm duration is sufficient for establishment of a post-storm equilibrium profile <math>y(x)</math>;
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# the post-storm equilibrium profile does not (strongly) depend on the initial coastal profile <math>y_0(x)</math>;
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# eroded dune sediment is deposited within a zone delimited by a storm closure depth <math>y_{max}</math>;
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# the eroded dune scarp has a slope 1:1 and the slope at the toe of the sand deposit is 1:12 (these are less crucial assumptions).
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With these assumptions the eroded dune volume is given by the beach volume between  <math>y(x)</math> and <math>y_0(x)</math>, such that the total sand volume landward of the storm closure depth <math>y_{max}</math> is preserved. The procedure is explained in Fig. 6, where the empirical parameterized functions for the post-storm equilibrium profile <math>y(x)</math> and the storm closure depth <math>y_{max}</math> are also indicated. The cross-shore distance over which sand is redistributed (the storm active zone) is given by <math>x_{max}</math>.
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[[Image:ErosionProfile.jpg|700px|thumbnail|center|Fig. 6. Storm profile method for estimating dune erosion as given by the beach volume between  <math>y(x)</math> and <math>y_0(x)</math> for a storm of long duration. Underlying assumptions are explained in the text. ]]
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The empirical formulas representing the post-storm profile show that the slope is smaller (i.e. increasing dune erosion) as the wave period <math>T</math> is longer and the fall velocity smaller. 
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The storm profile method is based on the equilibrium profile concept that also underlies the [[Bruun rule]]. However, whereas the Bruun rule refers to the average coastal profile that settles over the long term, the storm profile method refers to storm conditions of limited but sufficient duration so that the most heavily reworked part of the beach and the foredune assumes an approximate equilibrium profile. The Bruun rule assumes that the sand is redistributed over the entire [[active coastal zone]] <math>x_{Bruun}</math> (from foredune to closure depth), which is wider than the storm active zone <math>x_{max}</math>.
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A drawback of the storm profile method is that hardly any physics is involved. The development with time of the storm profile is unknown. Effects of varying water levels and varying wave characteristics during the storm surge cannot be accounted for.
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Only in exceptional cases do storm surge conditions last long enough to induce profile changes that are close to equilibrium. The empirical profile formulas established from long-term hydraulic model experiments should therefore be considered an upper limit of dune erosion that can occur in naturally occurring extreme storms.
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===Analytic dune erosion model===
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A process-based, but highly idealized analytic model was developed by Larson et al. (2004<ref name=L4>Larson, M., Erikson, L. and Hanson, H. 2004. An analytical model to predict dune erosion due to wave impact. Coastal Engineering 51: 675– 696</ref>) for estimating the time evolution of dune erosion. This model is based on the assumption that the eroded dune volume <math>\Delta V</math> in a time interval <math>\Delta t</math> is proportional to the force <math>F</math> exerted by [[Swash zone dynamics|swash waves]] hitting the dune foot. By expressing the force <math>F</math> as a function of the dune foot height <math>z_D</math>,  the maximum water level <math>R</math> including [[wave run-up]] and the wave period <math>T</math>, the following equation was obtained for the loss rate <math>dV/dt</math> of dune volume (see the appendix and Fig. B1):
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<math>\Large\frac{dV}{dt}\normalsize=-4C_s\Large\frac{(R-z_D)^2}{T}\normalsize , \qquad (1)</math>   
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where the constant <math>C_s</math> depends on site-specific characteristics (such as sediment grainsize distribution, beach and foreshore morphology, vegetation, slope of the dune face) and has to be calibrated with field data. As shown in the appendix, the rate of dune erosion according to equation (1) is an increasing function of the wave height, the wave period and the beach slope. This is in line with the dune erosion estimate following from the storm profile of Fig. 6. Equation (1) also shows that dune erosion strongly depends on the height <math>z_D</math> of the dune foot (thus on the beach steepness and width). This is consistent with observations at Narrabeen-Collaroy Beach (East Australia, near Sidney), which showed that alongshore variability in dune erosion was highly correlated with alongshore variability in <math>z_D</math><ref name=S18>Splinter, K.D., Kearney, E.T. and Turner, I.L. 2018.  Drivers of alongshore variable dune erosion during a storm event: Observations and modelling. Coastal Engineering 131: 31–41</ref>. A strong relationship between dune erosion volumes and dune foot height was also found in wave tank experiments<ref>Palmsten, M.L. and Holman, R.A. 2012. Laboratory investigation of dune erosion using stereo video. Coastal Engineering 60: 123–135</ref>. Assuming that the beach slope <math>\beta</math> remains constant during dune erosion (i.e., dune retreat <math>\Delta x</math> equal to <math>\Delta z_D / \tan \beta</math>), Larson et al. (2004<ref name=L4/>) found reasonable agreement with observed dune erosion rates when using values of <math>C_s</math>  in the range 10<sup>-3</sup> - 2.10<sup>-3</sup>.
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The analytic model does not include the process of mass failure due to wetting and steepening of the dune face. Dune retreat will thus be underpredicted in situations where this process plays an important role. Ignoring the distinction between the wave run-up and storm level components in the total water level <math>R</math> is another simplification. Observations of dune erosion on the US northwest coast show that the influence of the storm level component is more important for dune erosion than the wave run-up component<ref name=C19>Cohn, N., Ruggiero, P., García-Medina, G., Anderson, D., Serafin, K.A. and Biel, R. 2019. Environmental and morphologic controls on wave-induced dune response. Geomorphology 329: 108–128</ref>. A more elaborate version of the analytic dune erosion model includes the effects of [[overwash]], wind-blown sand transport, and bar-berm material exchange (Larson et al., 2016<ref>Larson, M., Palalane, J., Fredriksson, F. and Hanson, H. 2016. Simulating cross-shore material exchange at decadal scale. Theory and model component validation. Coastal Engineering 116: 57–66 </ref>).
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The maximum water level <math>R</math> is subject to an increasing trend as a result of climate change (including sea level rise). The dune erosion model can therefore be used to estimate the impact of climate change on dune erosion. The associated long-term dune foot retreat can be estimated by making assumptions about post-storm dune recovery. The values for dune foot retreat obtained in this way by Ranasinghe et al. (2012<ref>Ranasinghe, R., Callaghan, D. and Stive, M. J. 2012. Estimating coastal recession due to sea level rise: beyond the Bruun rule. Climatic Change 110: 561–574</ref>) and Le Cozannet et al. (2019<ref>Le Cozannet, G., Bulteau, T., Castelle, B., Ranasinghe, R., Wöppelmann, G., Rohmer, J., Bernon, N., Idier, D.,  Louisor, J. and Salas-y-Mélia, D. 2019. Quantifying uncertainties of sandy shoreline change projections as sea level rises. Nature Scientific Reports 9:42</ref>) are higher than the values of coastal retreat obtained with the [[Bruun rule]].
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===DUNERULE===
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Based on a large number of numerical model runs, Van Rijn (2009)<ref>van Rijn, L.C. 2009. Prediction of dune erosion due to storms. Coastal Engineering 56: 441-457</ref> derived a simple formula for estimating the volume of dune erosion as a function of various parameters representing beach characteristics and hydraulic storm surge conditions. The underlying numerical model, CROSMOR, simulates the major wave transformation and sand transport processes in the surf zone responsible for the evolution of the cross-shore coastal profile, including dune collapse by colliding waves. The model has been extensively tested in small-scale and large-scale laboratory facilities and in natural settings. The DUNERULE formula reads
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<math>V(t) = V_5\Large (\frac{t}{t_{ref}})^{\alpha_t} \, (\frac{h}{h_{ref}})^{\alpha_h}  \, (\frac{H}{H_{ref}})^{\alpha_H}  \, (\frac{T}{T_{ref}})^{\alpha_T}  \, (\frac{d_{50}}{d_{50,ref}})^{\alpha_d}  \, (\frac{\tan \beta}{\tan \beta_{ref}})^{\alpha_{\beta}}\normalsize  \,  f(\theta) . \qquad (2)  </math>.
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The reference values correspond to a heavy storm at the Dutch coast:
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<math> t_{ref}= 18,000 \, s ; \; h_{ref}= 5 \, m ; \; H_{ref}= 7.6 \, m ; \; T_{ref}= 12 \, s ; \; \tan \beta_{ref}=  0.022; \; d_{50,ref}= 225 \, 10^{-6} \, m  ,</math>
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where <math>t</math> is the storm duration, <math>h</math> is the average storm surge level including wave set-up, <math>H</math> the storm significant wave height, <math>T</math> the peak wave period, <math>d_{50}</math> the mean sediment grain size, <math>\tan \beta</math> the average beach slope between the -3 m depth contour and the dune toe (+3 m) and <math>\theta</math> the wave incidence angle [<sup>o</sup>].
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The coefficients fitted to the CROSMOR simulations are:
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<math>V_5=170 \, m^3; \quad \alpha_t=0.5, \, t<18,000; \; \alpha_t=0.2, \, t>18,000; \quad \alpha_h=1.3, \, h<5 ; \alpha_h=0.5, \, h>5; \quad \alpha_H=0.5; \quad \alpha_T=0.5; \quad \alpha_d=1.3; \quad \alpha_{\beta}=0.3 \;</math> and <math>f(\theta)=(1+0.01 \, \theta)^{0.5} .</math>
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The main advantage of the DUNERULE formula is its simplicity. A constant beach slope is assumed; the effect of slope time dependence is approximated through the exponent <math>\alpha_t</math>. The formula implicitly takes wave run-up into account. One drawback of DUNERULE is that it does not account for fluctuating storm surge levels, which always occur along meso- and macrotidal coasts in case of long duration storms. Comparison with the analytical dune erosion model shows that DUNERULE produces much larger erosion volumes in case of short duration storms. The erosion volumes become similar for long duration storms, assuming that tidal water level variations can be ignored. Another drawback is the restriction to isolated storms; DUNERULE cannot be applied to storm clusters<ref>Baldock, T.E., Gravois, U., Callaghan, D.P., Davies, G., Nichol, S. 2021. Methodology for estimating return intervals for storm demand and dune recession by clustered and non-clustered morphological events. Coastal Engineering 168, September 2021, 103924</ref>. In cases of varying storm surge levels and storm clusters, the use of numerical models is required.
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===XBeach===
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The dune erosion models considered in the preceding sections represent strong simplifications that ignore specific hydrodynamic processes and bathymetric features that influence the local impact of storms, such as wave spectrum, wave deformation and refraction on morphological structures in the nearshore zone, longshore sand transport, beach morphology and vegetation and dune [[overwash]]. More generally applicable and accurate dune erosion forecasts therefore require numerical process-based models that are capable to simulate the effect of such features, also including the effect of storm sequences. An example is the widely used depth-averaged wave spectral XBeach model <ref>https://xbeach.readthedocs.io/en/latest/user_manual.html</ref>.  This open-source process-based numerical model was originally developed to simulate hydrodynamic and morphodynamic processes and impacts on sandy coasts with a domain size of kilometers and on the time scale of storms and has since been extended to wider applications. The model simulates the hydrodynamic processes of short [[wave transformation]] ([[refraction]], [[shoaling]] and breaking), long wave transformation (generation, propagation and dissipation of [[infragravity waves]]), [[wave set-up]] and unsteady currents, [[overwash]] and inundation. It includes the effect of wave skewness and asymmetry, the dynamics of nearshore sandbars, as well as the effects of vegetation and hard structures. The morphodynamic processes include bed load and suspended sediment transport, bed update and breaching. The model has been validated with a series of analytical, laboratory and field test cases using a standard set of parameter settings. The avalanching of sandy material from the dune face to the foreshore during storm conditions is taken into account when updating the bed levels. This is modeled through introduction of a critical bed slope (default critical slope of 1 for dry zones and 0.3 for wet zones). When the critical slope is exceeded, material is exchanged between adjacent cells to the amount needed to bring the slope back to the critical slope.  
  
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The basis of this article is especially written for the Coastal Wiki by the main author referred to at the bottom of this page.
 
  
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This article is under construction; not yet finished.
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Comparison with observed dune erosion events at the Dutch coast shows that XBeach estimates the magnitude and pattern of alongshore variations in erosion volume reasonably well. The 2014-version of XBeach overpredicted the erosion volume in the region where a dune scarp developed and underestimated the erosion volume where the whole dune face collapsed in a series of slumps <ref>De Winter, R.C,. Gongriep, F. and Ruessink, B.G. 2014. Observations and modeling of alongshore variability in dune erosion at Egmond aan Zee, the Netherlands. Coastal Engineering 99: 167-175.</ref>.  XBeach simulations further illustrated that the observed alongshore variation in dune erosion was steered primarily by the pre-storm dune topography i.e., the presence of embryonic dune fields and the steepness of the dune face. The importance of alongshore variability in intertidal beach topography was found to be secondary, but not negligible during the initial stage of the storm, when the surge level was still low. The 2019 version of XBeach was used to simulate the failure process of an experimental unvegetated dune on the German Baltic Sea coast. The post-storm dune breach was simulated well, but dune erosion was overestimated with respect to temporal profile evolution. It was concluded that the XBeach model is able to predict with sufficient accuracy the evolution of storm-induced erosion during a long [[overwash]]/inundation-dominant wave event when current high-resolution input data is available<ref>Schweiger, C., Kaehler, C., Koldrack, N. and Schuettrumpf, H. 2020. Spatial and temporal evaluation of storm-induced erosion modelling based on a two-dimensional field case including an artificial unvegetated research dune. Coastal Engineering 161, 103752</ref>.
  
  
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==Dunes as sea defense==
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==Post-storm dune recovery==
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River and sea dikes are good examples of structures to protect low-lying areas from flooding. Also a dune area (dune row) serves that aim in some cases. E.g. the safety of large parts of The Netherlands, often with ground levels even below Mean Sea Level (MSL), relies on dikes, but also on dunes for their protection against flooding.
 
  
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Visiting the beach and the coastal zone in e.g. The Netherlands under normal weather conditions would easily give the impression that the dunes are certainly strong enough to properly protect the hinterland. However, during a severe storm surge, with under design conditions water levels at sea which are approximately 5 - 6 m above MSL and together with the much more severe wave conditions than normal (cf. wave heights Hs ≈ 7 - 9 m and peak periods Tp ≈ 12 -18 s), the dunes will be eroded in a very short period of time.
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After being eroded under storm conditions, the dune front can recover naturally without human intervention. This has been observed for many sandy coasts with either a gently sloping or a steep sloping shoreface <ref>Vousdoukas, M.I., Pedro, L., Almeida, M. and Ferreira, O. 2012. Beach erosion and recovery during consecutive storms at a steep-sloping, meso-tidal beach. Earth Surf. Process. Landf. 37: 583–593</ref><ref>Scott. T., Masselink, G., O'Hare, T., Saulter, A., Poate, T,. Russell. P., Davidson, M. and Conley, D. 2016. The extreme 2013/2014 winter storms: Beach recovery along the southwest coast of England. Marine Geology 382: 224–241</ref><ref name=Ph>Phillips, M.S., Harley, M.D., Turner, I.L., Splinter, K.D. and Cox , R.J. 2017. Shoreline recovery on wave-dominated sandy coastlines: the role of sandbar morphodynamics and nearshore wave parameters. Marine Geology 385: 146–159</ref>. Strong beach accretion often already occurs during the phase of waning storm intensity. Shoreline recovery is dealt with in the article [[Shoreline retreat and recovery]]. It is a much faster process than dune recovery. Fast beach recovery is promoted by landward migration and welding of nearshore (intertidal) sandbars to the beach during fair-weather conditions. Aeolian sand transport to the dune starts once the upper-beach and any [[overwash|washover]] deposits become dry with the development of embryo dunes on the backshore. The expansion of the backshore is required to increase the fetch length, which controls the amount of sediment exchanged from the beach to the dune. Dune recovery also depends on vegetation recolonization. Post-storm recovery of vegetation can take two to eight years depending on the extent that the roots and rhizomes are impacted<ref name=Su>Suanez, S., Cariolet, J-M., Cancouët, R., Ardhuin, F. and Delacourt, C. 2012. Dune recovery after storm erosion on a high-energy beach: Vougot Beach, Brittany (France). Geomorphology 139-140: 16–33</ref><ref>Houser, C., Wernette, P., Rentschlar, E., Jones, H., Hammond, B. and Trimble, S. 2015. Post storm beach and dune recovery: implications for barrier island resilience. Geomorphology 234: 54–63</ref>.  
  
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Existing design rules in The Netherlands yield erosion rates of 80 - 100 m of the dunes during design storm conditions. (The rates of 80 - 100 m are given as an order of magnitude value only to facilitate the further discussion; the actual erosion rates under design conditions depend on the specific local conditions; e.g. shape of initial cross-shore profile and particle size of the dune material.) It must be realized that because of the specific Dutch conditions, the design conditions in The Netherlands are very strict.
+
An example of natural dune recovery is shown in Fig. 7a for a naturally retreating coast (Walcheren, the Netherlands). The extreme storm surge of 1953 caused a 30 m retreat of the dune foot. Afterwards the dune foot advanced slightly over a period of several years, until it reached about 15 years later the position corresponding to the retreating trend prior to the storm surge. The ongoing coastal retreat was finally halted in 1990, when the Dutch government decided to stop coastal retreat along the entire coastline with sand nourishments.  
  
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===Rather wide dune area is sea defense===
+
Figs. 7b and 7b show examples of dune recovery at two locations (Penhale, UK, and Lacanau, France) on the western European coast, which were severely impacted by the 2014 winter storm. The decade-averaged shoreline retreat in both locations is small, no more than a few meters per decade. The foredune in Penhale is higher than in Lacanau, more densely vegetated (with marram grass) and more alongshore uniform. The 2014 winter storm caused a dune foot retreat of almost 20 m in both locations. The retreat was fairly uniform along the coast at Penhale but showed pronounced erosion hotspots at Lacanau. At Penhale, the dune foot position partially recovered within a few years, but followed afterwards a trend of retreat. At Lacanau, the initial dune foot position was not recovered after 6 years, although a trend towards full recovery was observed. This example illustrates that dune recovery trajectories can differ considerably between sites with only moderately different characteristics.
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Often the dune areas in The Netherlands are wide enough to accommodate 80 - 100 m of dune erosion during a single severe storm surge. In some cases, however, the row of dunes is rather slender; a careful judgement has to be passed whether the dunes provide the required rate of protection. Is a break-through expected under design conditions?, and if yes: what reinforcement is necessary to fulfil the requirements?
 
  
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So far in the discussion the safety problem of people living behind the dunes was raised as the main issue. In the Sections 'Large scale safety problem' and 'Small scale safety problem'of this article it will be shown that more issues related to dune erosion are relevant to a coastal zone manager. Aspects like the safety of single houses and hotels in the erosion zone are dealt with.
 
  
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Also topics like how to deal with structural erosion and global sea level rise are relevant topics for a coastal zone manager. They are briefly discussed in this article with the present Dutch policy and insights as starting points.
+
[[File:DuneFootRecovery.jpg|thumb|center|700px|Fig. 7. Examples of dune foot retreat and recovery. 7a. Variation of the dune foot position in a beach transect of the island Walcheren (Netherlands) over the period 1900-2010 (solid white line, based on annual surveys of Rijkswaterstaat <ref>Jeuken, C., Ruessink, G. and Marchand, M. 2001. Ruimtelijke en temporele aspecten van de duinvoetdynamiek. Report WL/Delft Hydraulics Z2838 (in Dutch)</ref>). The trend is indicated by the dashed line, showing ongoing retreat of the dune foot from 1900-1990 and stabilization afterwards. 7b. Dune foot position at Penhale, UK, before and after the 2014 winter storm. After initial partial recovery the dune foot retreats towards its post-storm position. 7c. Longshore-averaged dune foot position at Lacanau, France, before and after the 2014 winter storm. The recovery of the pre-storm dune foot position is slow, and is variable along the coast. Figures 7b and 7c are redrawn from Burvingth and Castelle, 2023<ref name=BC/>.]]
  
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From the discussions it will become clear that for many reasons a proper insight in the rates of dune erosion as a function of the boundary conditions is necessary. Section 'Quantification of rate of dune erosion' will deal with the dune erosion process and the methods to quantify the rates of erosion during a severe storm surge.
 
  
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===Brief explanation of dune erosion process===
+
==Dune failure probability==
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Figure 1 shows schematically what happens with a (in this case: wide enough) dune during a severe storm surge.  
+
The numerical model XBEACH and the simplified dune erosion models discussed earlier provide estimates for dune retreat under given storm conditions. A highly relevant question is related to the probability of dune failure, i.e., the probability that dune retreat at a certain location exceeds a given value, leading to dune collapse and flood damage to settlements situated behind the dune. To answer this question, knowledge of the probability of extreme storm conditions occurring at this location is required. If the time interval between storms leading to dune failure is much shorter than the length of the available storm data record, the failure probability can be found by estimating the dune retreat for all extreme storm conditions occurring in the dataset. However, dune reinforcement programs usually require knowledge of the statistical return period of possible dune failure that far exceeds the length of available storm records. In this case, the available storm record must be extended by statistical methods to a time span much longer than this return period. This can be done by a method described below<ref>Li, F., van Gelder, P.H.A.J.M., Vrijling, J.K., Callaghan, D.P., Jongejan, R.B. and Ranasinghe, R. 2014. Probabilistic estimation of coastal dune erosion and recession by statistical simulation of storm events. Applied Ocean Research 47: 53–62</ref>.
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[[Image:imageDu001.JPG|350px|thumbnail|right|Figure 1 Dune erosion due to storm surge]]
 
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The initial cross-shore profile, which might be considered to be in a more or less dynamic equilibrium condition with the normal occurring boundary conditions, will be reshaped during the severe storm surge. The  much higher water levels and the much higher wave heights and peak periods call for a quite different shape of an equilibrium profile than the shape of the initial profile. Offshore directed sediment transports will occur, especially with sediments from the dunes. Sand is eroded from the dunes and is settled at the foreshore again. During these reshaping processes the slopes of the cross-shore profile gradually decrease, and consequently it can be understood that the rate of dune erosion will decrease with time during the storm surge. It is, however, not expected that a real equilibrium profile will develop during the storm surge. The time available during the storm is too short to achieve such a real equilibrium profile, but the developments are in the direction of achieving equilibrium. The shape of a cross-shore profile as encountered after the storm surge is often called 'erosion profile'.
 
  
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Right after a severe storm surge, when the boundary conditions are normal again, the shape of the (erosion) profile does not fit with these normal conditions. Onshore directed sediment transports will occur; wind will blow sand from the beach to the dunes; the 'old' situation will gradually be restored. Dune erosion because of severe storm surges is thus a temporary and a reversible process.
+
Storm conditions can be characterized by the parameters: maximum significant wave height, wave period, storm surge water level, wave direction, and storm duration. If these parameters are uncorrelated, the parameters corresponding to extreme storms with a given return period exceeding the data record can be estimated by fitting an appropriate cumulative distribution function (marginal cumulative distribution function) to the available storm dataset for each parameter separately. The overall probability is then given by the product of the separate probabilities. However, storm parameters are typically correlated. To account for these correlations, so-called copulas are constructed. [https://en.wikipedia.org/wiki/Copula_(probability_theory) Copulas] are multivariate cumulative distribution functions for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Choosing the appropriate copula and determining the copula parameters from the data is often not simple and requires acquaintance with copula theory (see for example McNeil et al. 2005<ref>McNeil, A., Frey, R. and Embrechts, P. 2005. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press</ref>). From the fitted copulas, random samples can be constructed of extreme storm surge conditions with known probabilities. By estimating dune retreat for these random samples (Monte Carlo method) a relation can be established between calamitous dune retreat (dune failure) and the corresponding probability.
  
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===Large scale safety problem===
 
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If, like in some cases, the dunes are really slender and landward of the row of dunes the ground levels are even below MSL (like in some places in The Netherlands) a large scale safety problem might occur. A break-through of the dunes causes flooding and will cause loss of lives and results in a lot of damage in the densely populated low-lying areas of The Netherlands.
 
−
Given the dimensions of a row of dunes and the specific design conditions, the Coastal Zone Manager has to judge whether the dunes are 'safe' or not. 'Safe' is in this respect in fact a relative notion. 'Safe' means that the strength (width, height) of the dunes fulfils the requirements. Absolute safety does not exist in many cases; often a set of even more extreme boundary conditions is conceivable which will result in a break-through of a row of dunes which was judged just 'safe' enough. The chance that such a set of boundary conditions will occur, is then, however, apparently smaller than has been agreed for the design conditions.
 
−
For the judgement of the safety of a row of dunes, a proper computation model is necessary and (a set of) design conditions. Section 'Quantification of rate of dune erosion' deals with these computation models.
 
  
−
For the further discussion in the present section it is good to realize that because of the many (stochastic) parameters which ultimately determine the rate of dune erosion, a probabilistic approach seems to be appropriate (see e.g. Van de Graaff, 1986). Stochastic parameters like: maximum storm surge level, wave height, wave period, storm duration, accuracy of computation model, particle size of dune sand and shape of initial cross-shore profile, all do play a role in a probabilistic approach.
+
==Impact of sea-level rise on dune erosion==
  
−
In a probabilistic approach an acceptable chance of failure must be the starting point instead of a single set of design conditions. Because of the high importance of the low-lying hinterland, a probability of failure for dunes of 10-5 per year (return period 100,000 years) has been agreed in The Netherlands for the most important parts of the country. This chance seems rather small, but compared to other threats (e.g. accidents with nuclear power plants, air planes or large industrial plants) the probability of failure is not that small taking into account the number of human lives and the high investments that are at stake.
+
The XBeach model predicts an approximately linear relationship between dune erosion volume and sea level rise <ref name=WR>De Winter, R,C. and Ruessink, B.G. 2017. Sensitivity analysis of climate change impacts on dune erosion: case study for the Dutch Holland coast. Climatic Change (2017) 141:685–701 DOI 10.1007/s10584-017-1922-3</ref>. This is primarily due to the higher water level in front of the dune rather than to changes in the significant short-wave or [[Infragravity waves|infragravity]] wave height. Changes in the offshore angle of wave incidence also affect the dune erosion volume. A 30° shift from shore-normal influences the dune erosion volume to the same extent as a 0.4 m sea-level rise. This increase in dune erosion volume is related to strong alongshore currents, generated as a result of the obliquity of the waves, that enhance stirring and hence offshore transport<ref name=WR/>.  
  
−
The probability of failure of 10-5 per year holds for the most important parts of The Netherlands; in some more rural areas larger probabilities of failure are the (legal) norm.
+
XBeach simulations further show that the effectiveness of coastal sand nourishment to mitigate the impact of sea-level rise on dune erosion depends on the location in the profile where this sand is added. The reduction of dune erosion volume relative to the total added nourishment volume never exceeds 30% in the case of shore or beach nourishment. This suggests that directly increasing the volume of sand in the dunes can be more efficient than beach or shoreface nourishment<ref name=WR/>.
  
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Figure 2 shows a schematic plot of the relationship between the rate of erosion RD (the distance RD is in Figure 1 the distance between the initial edge of the dune and the edge of the dune after the storm surge) and the frequency of exceedance. According to Figure 2 there is a chance of 10-5 per year that a rate of erosion of 85 m is reached or will be surpassed.[[Image:imageDu002.JPG|350px|thumbnail|right|Figure 2 Rate of erosion as a function of frequency of exceedance]]
 
  
 +
==Appendix: Analytical dune erosion model==
  
−
The safety problem of large parts of The Netherlands (as far as the problem depends on the protection by dunes (large scale problem)), seems thus solvable if a proper insight is available in the possible occurring conditions in the (very) small chances range. However, this also introduces many uncertainties. E.g. reliable measurements of (maximum) water levels of the sea in measuring stations near the coast (cf. in ports) are available since approximately 1850. So for only 150 years. With such an in fact restricted data set, it is hard to estimate water levels which are associated with frequencies of exceedance in the range of 10-3 to 10-5 per year.
+
[[File:DuneRetreatScheme.jpg|thumb|400px|left|Fig. B1. Simplified representation of dune retreat and notations used in the appendix.]]
  
−
Data sets of reliable measurements of wave characteristics cover even shorter periods of time. Nevertheless, these kinds of uncertainties have been taken into account to arrive at figures like Figure 2.
+
The analytic model of Larson et al. (2004<ref name=L4/>) is a crude simplification of the dune erosion process. However, it does allow an estimate of the order of magnitude of the eroded dune volume under severe storms for different beach morphologies. The model is based on the following assumptions and simplifications:
−
Based on a legal norm, with the help of a proper computation method, and for a given situation (cross-section of cross-shore profile and dune area) one is next able to judge whether the cross-section meets the requirements; i.e. whether the cross-section is safe enough or not. If not: iteratively an improvement scheme for the dune area can be developed; e.g. a strengthening of the dunes at the landward side.
+
* The eroded dune volume <math>\Delta V</math> in a time interval <math>\Delta t</math> is proportional to the force <math>F</math> exerted by [[Swash zone dynamics|swash waves]] hitting the dune foot, <math>\Delta V \propto F \Delta t .</math>
 +
* The force <math>F \Delta t </math> is proportional to the number of swash bores <math>n</math> hitting the dune foot (<math>n=\Delta t/T</math>) and the force exerted by each individual swash bore, <math>f=m \Large\frac{du}{dt}\normalsize \propto \Large\frac{mu}{T}\normalsize </math>, where <math>T</math> is the wave period, <math>m</math> the mass of the [[Swash zone dynamics|swash bore]] per unit dune width and <math>u</math> the velocity of the swash bore when hitting the dune foot.
 +
* The bore height <math>h</math> is related to the bore velocity by <math>u \propto \sqrt{gh}</math>, where <math>g</math> is the gravitational acceleration. The bore mass is thus proportional to <math>m \propto huT \propto u^3 T</math>, yielding <math>F = n \,f  / \Delta t \propto u^4 /T</math>.  
 +
* The bore velocity at the dune foot can be related to the [[Swash zone dynamics|swash runup]] <math>R</math>. A ballistic swash excursion model (neglecting friction) gives an estimate for this relation, <math>R=z_D+gu^2/2</math>, where <math>z_D</math> is the height of the dune foot compared to the beach level where waves collapse on the beach. Using this relation, the force <math> F </math> can be expressed as a function of the wave runup <math> R </math>, <math>F \propto (R – z_D)^2 / T</math>.
 +
* Substitution with <math>\Delta V / \Delta t \approx dV/dt </math> gives Eq. (1), where <math>C_s</math> is a proportionality constant, depending on characteristics of the site, such as sediment grainsize, wave incidence angle, shoreface profile, etc. The value of <math>C_s</math> should be calibrated with local field data.
 +
* Empirical expressions for the wave run-up <math>R</math> are given in the article [[Wave run-up]]. However, the 2% wave run-up may not be the most pertinent parameter; less extreme thresholds may be more effective for dune erosion<ref name=C23/>.
 +
Wave run-up increases with increasing offshore wave height <math>H</math>, increasing wave period <math>T</math> and increasing beach slope <math>\tan \beta</math>. From Eq. (1) it then appears that the rate of dune erosion increases as the wave height <math>H</math> increases, the wave period <math>T</math> increases, the beach slope <math>\beta</math> increases and the beach width <math>l</math> decreases. Greater dune erosion for steeper beaches under the same wave conditions is consistent with observations<ref name=C19>Cohn, N., Ruggiero, P., García-Medina, G., Anderson, D., Serafin, K.A. and Biel, R. 2019. Environmental and morphologic controls on wave-induced dune response. Geomorphology 329: 108–128</ref>.  
  
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===Small scale safety problem===
+
The dune foot height <math>z_D</math> (relative to the beach level of wave collapse <math>z_C</math>) and the beach slope <math>\beta</math> are time dependent. In order to solve Eq. (1) a prescription must be given how the eroded dune material is distributed over the beach – i.e. a model for the dependence of <math>z_D</math> and <math>\beta</math> on <math>dV/dt</math>. Based on experiments in large wave flumes, Larson et al. (2004<ref name=L4/>) assumed that the beach slope <math>\beta</math> remains constant during dune erosion. The dune retreat <math>\Delta x</math> was therefore set equal to <math>\Delta z_D / \tan \beta</math>, assuming that the major part of the dune volume loss is removed to the lower beach (below the level of wave collapse). Field observations of dune erosion events in different sites along coasts in the USA, Australia and France show that the upper beach slope can vary considerably depending on site characteristics and storm intensity<ref name=C23/>. These observations emphasize the tentative nature of the analytical model.
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If the width of the row of dunes is rather large (or at least large enough to meet the legal safety standard), the safety problem for the hinterland is not a real issue (any more). That holds for many stretches of the Dutch and other coasts bordering, sometimes, hostile seas.
 
  
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If these stretches concern bare dunes, loss of dune area will occur during a severe storm surge and some damage to e.g. nature will occur. Sooner or later the dunes will naturally repaired.
+
Considering a dune height <math>D</math> above the level of wave collapse <math>z_C</math> (the scarp height), the loss of dune volume is given by <math>\Delta V=-(D-z_D) \Delta x = -(D- z_D) \Delta z_D / \tan \beta</math>. Combining this result with Eq. (1) yields a differential equation for the dune foot height <math>z_D</math>:
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However, along the Dutch coast (like along many coasts elsewhere) at some places coastal villages and holiday resorts do exist with roads, houses and hotels built very close to the seaward brink of the dunes. And even if the dunes are safe enough to protect the hinterland, the (required or wanted) safety of single houses and hotels built in the zone prone to erosion during a severe storm surge, might be an issue. This safety problem might be classified as a small scale safety problem.
 
  
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It is to the direct interest of a large part of the Dutch population that the large scale safety problem is properly dealt with. Legal norms are available. The responsibilities of the various parties (Water Boards, Provinces, Central Government) have been legally embodied.
+
<math>\Large\frac{d}{dt}\normalsize z_D = \Large\frac{1}{\tau} \frac{(R-z_D)^2}{D-z_D}\normalsize , </math>
 +
where <math>\tau=T/(4 C_s \tan \beta)</math>.  
  
−
The small scale safety problem directly regards usually only a rather restricted part of the population. E.g. owners of houses and hotels built (or to be built in future) in the zone prone to erosion during a severe storm surge, but also governmental agencies are directly involved as owners of infrastructure like roads, promenades and car parks.
+
If it is further assumed that the level of wave collapse <math>z_C</math> does not change during the storm, then the wave run-up remains constant too (the slope <math>\beta</math> was assumed constant). For storms of limited duration the run-up <math>R</math> can be assumed constant. The differential equation can then be solved, with solution
  
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The government must also formulate and maintain a set of rules to avoid unbridled developments in the zone prone to erosion during a severe storm surge.
+
<math>t = \tau (D-R)\Large(\frac{1}{R-z_D(t)} - \frac{1}{R-z_D(0)})\normalsize +\tau \ln\Large\frac{R-z_D(0)}{R-z_D(t)}\normalsize  </math>, from which <math>z_D(t)</math> and <math>V(t)</math> can be derived.
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It must be realized, however, that within the zone which is 'necessary' during the design conditions for the safety of the hinterland, a distinction can be made in chances that a property will be lost. Close to the seaward brink of the dunes (say RD = 20 m according to Figure 2) the chance of loss of property is larger than at the landward side of the potential erosion zone (say RD = 60 m).
 
  
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The coastal zone very close to the brink of the dunes has high socio-economic potentials; many people would like to build houses and hotels with 'sea view' in this zone or would buy existing buildings. Owners of such properties are primarily responsible for possible damage to their properties by a severe storm surge. From this point of view the role of a Coastal Zone Manager would be a limited one in this respect. It is conceivable, however, that 'society' calls for some regulation. Too often (say: on an average of  every 10 years) loss of many properties in the coastal zone might be unwanted. A lot of commotion has to be expected; owners of properties who at once lost 'everything' might be considered as 'poor and innocent' fellow-citizens. At the other hand there seems to be no reason to avoid damage in the zone prone to erosion during a severe storm surge to chances comparable with the chance of failure of the dunes as sea defence. To find an acceptable compromise between these limits is a very difficult task for the responsible Coastal Zone Manager. Many Managers 'struggle' with this issue. E.g. financial, legal and insurance aspects might be helpful to be considered while developing a proper policy.
 
  
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So far the discussion referred to the erosion of the dunes due to a single storm surge (episodic effect). However, seen over a long period of time, the position of sandy coasts often show some distinct tendencies. It refers to either accreting, or eroding coasts (structural erosion), although also a more or less stable position with time of a coast might occur. Especially structural eroding coasts seriously complicate the coastal zone management task in dune areas where coastal villages or holiday resorts do exist. If the structural erosion is 'accepted', the zone prone to erosion during design conditions during severe storm surges is then continuously shifting in landward direction. It is good to realize that the effect of structural erosion (e.g. a few m per year), is quite different from the effect of a really serious storm surge: up to tens of meters per (really severe) event.
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==Related articles==
 +
* [[Natural causes of coastal erosion]]
 +
* [[Dune stabilisation]]
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* [[Light revetments built-in into artificial dunes]]
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* [[Shoreface profile]]
 +
* [[Bruun rule]]
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* [[Active coastal zone]]
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* [[Dune development]]
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* [[Shoreline retreat and recovery]]
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* [[Swash zone dynamics]]
 +
* [[Shore protection vegetation]]
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* [[Dynamics, threats and management of dunes]]
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* [[Risk and coastal zone policy: example from the Netherlands]]
  
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If structural erosion is not accepted, proper protection measures must be applied.
 
  
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===Quantification of rate of dune erosion===
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==References==
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Both for the ''small scale'' safety problem, but especially for the ''large scale'' safety problem a reliable quantification method for the rate of dune retreat during severe storm surge conditions is necessary. Rather much effort has spent on this specific quantification topic in research programs.
+
<references/>
  
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Although 3D effects are undoubtedly important in the dune erosion process, for the time being often a 2D approach is adopted. In that case the dune erosion process can be considered as a typical offshore directed cross-shore sediment transport problem. Sand from the dunes is transported to deeper water and is settled there.
 
  
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To a first approach a closed sediment balance in cross-shore direction can be assumed. The same volume of sand which is eroded from the dunes and the very upper part of a cross-shore profile (in m3/m) is accumulated elsewhere in the cross-shore profile. [Because of differences in porosity of the eroded dune material (often relatively loose packed) and of the settled material (often slightly denser packed), the sediment balance is not always strictly closed.]
 
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During really severe storm surges, with a serious increase of the water level (compare Fig.3.14; an increase of the water level with approximately 2.8 m was measured in that case) huge volumes of sand from the dunes are transported in offshore direction. And because dune erosion is a rather short lasting process, some computation methods take only offshore directed transports into account.
 
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A rather straightforward computation procedure uses the concept of a so-called 'erosion profile' (see also Section 7.4.2). The upper part of the shape of a cross-shore profile right after a (rather severe) storm surge is thought to be known in such a method. In the procedure still in use in The Netherlands, the shape of an erosion profile with the maximum storm surge level as reference, depends on the occurring wave height and the particle size of the dune/beach material. Once the characteristics of the erosion profile are known, the relevant dune retreat is easily to be determined with a closed sediment balance method. This concept forms the heart in a further probabilistic method taking into account the stochastic character of several dune erosion determining parameters. (E.g. water level, wave height, wave period, shape of initial cross-shore profile, estimated accuracy of computation method, duration of storm surge.) In TAW (1984) / CUR (1989) the approach followed in The Netherlands is described. In Van de Graaff (1986) some background information has been presented.
 
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A serious drawback of such a straightforward computation method with a 'known' profile is that hardly any physics is involved. E.g. only the 'end' profile after the storm surge is thought to be known; the development with time is unknown. Effects of varying water levels and varying wave characteristics during the storm surge cannot be accounted for.
 
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But also dune erosion during a severe storm surge is a real cross-shore sediment transport process.
 
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Based on theoretical and a lot of experimental work, Steetzel has developed the so-called DUROSTA computation model in which at many positions in a cross-shore profile actual sediment transports are calculated with a S = v.c concept.  [Steetzel (1993)]. Although in the mathematical descriptions of the cross-shore sediment transport the so-called wave related transport (see Section 5.3) is neglected, only the current related transport is taken into account, the results of the model compared with e.g. the 'reality' of large scale model tests in the Delta Flume of Delft Hydraulics are rather good. (See Fig.7.10.)
 
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During storm surge conditions the rather high breaking waves cause a considerable return flow (v in S = v.c) in the lower part of a water column in the surf zone. The fierce wave conditions result also in high sediment concentrations within the water column (c in S = v.c). Gradients in calculated sediment transport rates allow next for a bottom up-date. And so on. With the DUROSTA model the development with time of a bottom profile during a storm surge can be calculated and studied.
 
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Also with computation models like UNIBEST-TC (developed by Delft Hydraulics) in principle dune erosion computations can be made. With the present day (mid 2006) versions of UNIBEST-TC the results are not yet satisfying.
 
  
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==See also==
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{{2Authors
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* [[Types and background of coastal erosion]]: article on the background of erosion, dune erosion and [[structural erosion]] (''Jan van de Graaff is also planning to write a separate article on dune erosion'').
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|AuthorID1=11540
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* [[Natural Causes of Coastal Erosion]]: Effects of e.g. transport gradient, loss of sand, protruding areas, marine deposit shorelines, down stream erosion, sea level rise, subsidence and natural variation on coastal erosion.
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|AuthorFullName1= Jan van de Graaff
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|AuthorName1= Jan van de Graaff
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|AuthorID2=120
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|AuthorFullName2=Job Dronkers
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|AuthorName2=Dronkers J
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}}
  
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{{author
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[[Category:Coastal protection]]
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|AuthorID=11540
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[[Category:Physical coastal and marine processes]]
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|AuthorName= Jan van de Graaff
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[[Category:Beaches]]
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|AuthorFullName= Jan van de Graaff}}
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[[Category:Climate change, impacts and adaptation]]
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[[Category:Theme 8]]
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[[Category:Sea level rise]]

Latest revision as of 21:23, 3 August 2024

Definition of Dune erosion:
Sand loss from a dune under wave attack, mainly by notching, avalanching and slumping processes.
This is the common definition for Dune erosion, other definitions can be discussed in the article


This article deals with dune erosion by storms and provides some simple rules from which retreat of the dune front can be estimated. The focus is on the so-called 'collision regime', where waves hit the dune without exceeding the dune crest.


Introduction

Fig. 1. Post-storm eroded dune, Egmond aan Zee (Netherlands NL).

Coastal dunes have developed naturally along many shorelines worldwide during the last millennia when sea-level rise slowed down. Wave action and onshore winds are the main agents for coastal dune development and sufficient sand supply to the coast is a primary condition[1], see also the article Dune development. The coastal dune belt in many cases protects low-lying hinterland from flooding by the sea. Dune erosion therefore can be a serious threat.

A coastal dune can suffer large losses when attacked by storm waves. The front dune can be taken away over several tens of meters, leaving a steep dune scarp, see Fig. 1. For the Dutch coast it has been estimated that under exceptional circumstances (extreme storms of very long duration, which may occur with a yearly probability of 1/100,000) the dune volume loss can amount to 800 m3/m, causing the dune foot to recede from 80-100 m at some places[2].

The actual dune loss depends on many factors[3]:

  • beach geometry (pre-storm): beach slope and width;
  • grain size;
  • dune geometry (pre-storm): dune face slope, dune crest elevation;
  • nearshore morphology: nearshore slope, rip-channel embayments, nearshore sandbars, subtidal morphology;
  • beach and dune vegetation; presence of (embryo-)foredunes;
  • presence of hard engineered structures;
  • alongshore variability in wave energy and dune height.

The dune toe elevation (beach width[math]\times[/math]slope) appears to be a major factor; observations show that vulnerability to erosion decreases with increasing dune toe elevation[4]. Dune sand losses have also been positively correlated with the steepness of the dune face[5]. There is instead strong evidence that dunes are less impacted by storms when situated on dissipative flat beaches than on reflective steep beaches[6].

Dune loss also depends on storm characteristics: maximum sustained wind speed and duration; maximum water level and simultaneous wave height, period and incidence angle.

If at least part of the dune belt survives the storm, it can recover by natural processes – similar to the processes that created the original dune. See section Post-storm dune recovery. Because recovery of the beach-dune profile (especially the dune profile) takes much longer than storm erosion, the dune system is particularly vulnerable to storm clusters (quick succession of storms)[7]. The probability that extreme dune recession is caused by a storm cluster is greater than the probability that this retreat is caused by an individual storm[8].


Fig. 2. Storm-induced dune foot retreat as a function of frequency of exceedance for the Dutch coast.

Figure 2 shows as an example a model result of the relationship between dune foot retreat [math]RD[/math] (expressed as the distance between pre-storm and post-storm dune foot locations) and exceedance frequency (the probability that in a particular year a storm occurs that produces greater erosion) for a location along the Dutch coast. According to this figure, which is based on the storm profile method, there is a 10-5 probability per year that dune erosion will exceed 85 m. To establish such a relationship, knowledge is required of extreme storm conditions with a very low probability. For these storms, the joint probability of extreme surge height, surge duration and corresponding wave conditions has to be determined[9]. Dune retreat estimates with an exceedance probability of 10-5 are subject to great uncertainty, as reliable measurements of extreme water levels and wave conditions are only available over relatively short periods of the order of 100 years. This issue is further discussed in section Dune failure probability.

Impact of dune erosion

Fig. 3. Badhotel at Schiermonnikoog (NL) after the 1924 storm.


Buildings on the front dune situated close to the dune foot are at risk under severe storms, see Fig. 3. A sound estimate of potential dune erosion is required when issuing building permits (see Setback area). In cases where the hinterland is situated below sea level, the dune belt serves as sea defence. Breach of the dune belt may have catastrophic consequences. Dune and beach monitoring and dune management are of crucial importance. In some cases, where the dune belt consists of a single dune row, dune reinforcement or shore nourishment may be needed. An example is the so-called 'Sand Engine' on the Dutch coast[10], which contributed to the reinforcement of the single protecting dune row through wind-driven sand transport from the widened beach[11].

To ensure safety, several methods have been developed for estimating dune loss during exceptional storm conditions.

Brief explanation of dune erosion processes

The initial cross-shore beach profile, considered to be in a more or less dynamic equilibrium condition with normally occurring hydrodynamic conditions, will be reshaped during a severe storm surge. Offshore directed sediment transport will occur, including sand eroded from the dune.

Infiltration of swash into the dune via capillary action acts as a destabilizer by increasing the effective weight of dune sediment, thereby decreasing slope stability. Wave attack at the dune foot steepens the dune profile which may collapse by avalanching (Fig.4). Wave attack may also create a notch at the dune foot leading to mass failure: collapse of a dune slab, initiated by tensile cracking at the top surface of the dune followed by shear failure along an internal failure plane or overturning due to the weight of the overhang [12], see Fig. 5. Dune scarping can occur over the entire dune face, but for high dunes the height of the dune scarp will generally not reach the dune crest[5].


Fig. 4. Dune failure mechanisms.
Fig. 5. Dune scarp after collapse. The people standing on the dune scarp have not read this article.


A slab that slides down the dune scarp will temporarily defend the dune face, until waves and currents mobilize the sediment from the slump deposit and transport it offshore. After the sediment is transported away, the dune face is left exposed again, leaving room for a new slumping event. The total impact of a storm on the dune strongly depends on how fast the sediment is transported offshore during the storm.

The sand eroded from the dune face is removed further down by the backwash and the seaward undercurrent ('undertow') under the breaking waves. Turbulence generated by wave bores is an important driver of offshore suspended sediment transport in the surf zone[13]. Most of the sand is deposited on the lower beach or upper shoreface. In cases where strong rip currents are present or a strong ebb current from nearby inlets, part of the sand can even be transported to the lower shoreface[6]. The slope of the cross-shore profile gradually decreases, and consequently the rate of dune erosion will decrease with time during the storm surge. Usually, the limited storm surge duration does not allow for the development of a new equilibrium profile. The shape of the cross-shore profile after the storm surge, which is a transient state between the initial profile and the storm equilibrium profile, is often called 'storm erosion profile'.


Quantification of dune erosion

Although 3D effects are generally important in the dune erosion process, often a 1D or 2DV approach is adopted, ignoring alongshore variability. In this case the dune erosion process is considered as a typically offshore directed cross-shore sediment transport problem. Sand from the dunes is transported to deeper water and settles there.

A first approach consists of assuming a closed sediment balance in cross-shore direction. The same volume of sand which is eroded from the dunes and the very upper part of a cross-shore profile is accumulated lower in the cross-shore profile. [Because of differences in porosity of the eroded dune material (often loosely packed) and the settled material (often a bit more densely packed), the volume balance is not always strictly closed.]

During severe storm surges with greatly increased water levels (a water level increase of a few meters is possible along some coasts), huge volumes of sand from the dunes are transported in offshore direction, on the order of hundred or several hundred meters3/m [5].


Storm profile method

A quick first order estimate of storm dune erosion can be obtained by assuming that the storm erosion profile is close to the storm equilibrium profile. This assumption is unrealistic for storms of short duration and strongly overestimates storm dune erosion in this case. However, for storms of very long duration, which produce great dune losses with low exceedance probability, the assumption of a post-storm equilibrium profile is not unreasonable. The method is still used in the Netherlands, besides other more advanced methods, to provide an upper bound of possible storm dune erosion[14]. It is based on empirical post-storm equilibrium profiles established by laboratory experiments and validated by field data of the Dutch coast [15]. The post-storm equilibrium profile was established for different values of the significant wave height [math]H[/math], peak wave period [math]T[/math] and mean fall velocity of dune sand [math]w[/math]. Major assumption are:

  1. the storm duration is sufficient for establishment of a post-storm equilibrium profile [math]y(x)[/math];
  2. the post-storm equilibrium profile does not (strongly) depend on the initial coastal profile [math]y_0(x)[/math];
  3. eroded dune sediment is deposited within a zone delimited by a storm closure depth [math]y_{max}[/math];
  4. the eroded dune scarp has a slope 1:1 and the slope at the toe of the sand deposit is 1:12 (these are less crucial assumptions).

With these assumptions the eroded dune volume is given by the beach volume between [math]y(x)[/math] and [math]y_0(x)[/math], such that the total sand volume landward of the storm closure depth [math]y_{max}[/math] is preserved. The procedure is explained in Fig. 6, where the empirical parameterized functions for the post-storm equilibrium profile [math]y(x)[/math] and the storm closure depth [math]y_{max}[/math] are also indicated. The cross-shore distance over which sand is redistributed (the storm active zone) is given by [math]x_{max}[/math].


Fig. 6. Storm profile method for estimating dune erosion as given by the beach volume between [math]y(x)[/math] and [math]y_0(x)[/math] for a storm of long duration. Underlying assumptions are explained in the text.


The empirical formulas representing the post-storm profile show that the slope is smaller (i.e. increasing dune erosion) as the wave period [math]T[/math] is longer and the fall velocity smaller.

The storm profile method is based on the equilibrium profile concept that also underlies the Bruun rule. However, whereas the Bruun rule refers to the average coastal profile that settles over the long term, the storm profile method refers to storm conditions of limited but sufficient duration so that the most heavily reworked part of the beach and the foredune assumes an approximate equilibrium profile. The Bruun rule assumes that the sand is redistributed over the entire active coastal zone [math]x_{Bruun}[/math] (from foredune to closure depth), which is wider than the storm active zone [math]x_{max}[/math]. A drawback of the storm profile method is that hardly any physics is involved. The development with time of the storm profile is unknown. Effects of varying water levels and varying wave characteristics during the storm surge cannot be accounted for.

Only in exceptional cases do storm surge conditions last long enough to induce profile changes that are close to equilibrium. The empirical profile formulas established from long-term hydraulic model experiments should therefore be considered an upper limit of dune erosion that can occur in naturally occurring extreme storms.


Analytic dune erosion model

A process-based, but highly idealized analytic model was developed by Larson et al. (2004[16]) for estimating the time evolution of dune erosion. This model is based on the assumption that the eroded dune volume [math]\Delta V[/math] in a time interval [math]\Delta t[/math] is proportional to the force [math]F[/math] exerted by swash waves hitting the dune foot. By expressing the force [math]F[/math] as a function of the dune foot height [math]z_D[/math], the maximum water level [math]R[/math] including wave run-up and the wave period [math]T[/math], the following equation was obtained for the loss rate [math]dV/dt[/math] of dune volume (see the appendix and Fig. B1):

[math]\Large\frac{dV}{dt}\normalsize=-4C_s\Large\frac{(R-z_D)^2}{T}\normalsize , \qquad (1)[/math]

where the constant [math]C_s[/math] depends on site-specific characteristics (such as sediment grainsize distribution, beach and foreshore morphology, vegetation, slope of the dune face) and has to be calibrated with field data. As shown in the appendix, the rate of dune erosion according to equation (1) is an increasing function of the wave height, the wave period and the beach slope. This is in line with the dune erosion estimate following from the storm profile of Fig. 6. Equation (1) also shows that dune erosion strongly depends on the height [math]z_D[/math] of the dune foot (thus on the beach steepness and width). This is consistent with observations at Narrabeen-Collaroy Beach (East Australia, near Sidney), which showed that alongshore variability in dune erosion was highly correlated with alongshore variability in [math]z_D[/math][4]. A strong relationship between dune erosion volumes and dune foot height was also found in wave tank experiments[17]. Assuming that the beach slope [math]\beta[/math] remains constant during dune erosion (i.e., dune retreat [math]\Delta x[/math] equal to [math]\Delta z_D / \tan \beta[/math]), Larson et al. (2004[16]) found reasonable agreement with observed dune erosion rates when using values of [math]C_s[/math] in the range 10-3 - 2.10-3.

The analytic model does not include the process of mass failure due to wetting and steepening of the dune face. Dune retreat will thus be underpredicted in situations where this process plays an important role. Ignoring the distinction between the wave run-up and storm level components in the total water level [math]R[/math] is another simplification. Observations of dune erosion on the US northwest coast show that the influence of the storm level component is more important for dune erosion than the wave run-up component[18]. A more elaborate version of the analytic dune erosion model includes the effects of overwash, wind-blown sand transport, and bar-berm material exchange (Larson et al., 2016[19]).

The maximum water level [math]R[/math] is subject to an increasing trend as a result of climate change (including sea level rise). The dune erosion model can therefore be used to estimate the impact of climate change on dune erosion. The associated long-term dune foot retreat can be estimated by making assumptions about post-storm dune recovery. The values for dune foot retreat obtained in this way by Ranasinghe et al. (2012[20]) and Le Cozannet et al. (2019[21]) are higher than the values of coastal retreat obtained with the Bruun rule.

DUNERULE

Based on a large number of numerical model runs, Van Rijn (2009)[22] derived a simple formula for estimating the volume of dune erosion as a function of various parameters representing beach characteristics and hydraulic storm surge conditions. The underlying numerical model, CROSMOR, simulates the major wave transformation and sand transport processes in the surf zone responsible for the evolution of the cross-shore coastal profile, including dune collapse by colliding waves. The model has been extensively tested in small-scale and large-scale laboratory facilities and in natural settings. The DUNERULE formula reads

[math]V(t) = V_5\Large (\frac{t}{t_{ref}})^{\alpha_t} \, (\frac{h}{h_{ref}})^{\alpha_h} \, (\frac{H}{H_{ref}})^{\alpha_H} \, (\frac{T}{T_{ref}})^{\alpha_T} \, (\frac{d_{50}}{d_{50,ref}})^{\alpha_d} \, (\frac{\tan \beta}{\tan \beta_{ref}})^{\alpha_{\beta}}\normalsize \, f(\theta) . \qquad (2) [/math].

The reference values correspond to a heavy storm at the Dutch coast: [math] t_{ref}= 18,000 \, s ; \; h_{ref}= 5 \, m ; \; H_{ref}= 7.6 \, m ; \; T_{ref}= 12 \, s ; \; \tan \beta_{ref}= 0.022; \; d_{50,ref}= 225 \, 10^{-6} \, m ,[/math]

where [math]t[/math] is the storm duration, [math]h[/math] is the average storm surge level including wave set-up, [math]H[/math] the storm significant wave height, [math]T[/math] the peak wave period, [math]d_{50}[/math] the mean sediment grain size, [math]\tan \beta[/math] the average beach slope between the -3 m depth contour and the dune toe (+3 m) and [math]\theta[/math] the wave incidence angle [o].

The coefficients fitted to the CROSMOR simulations are:

[math]V_5=170 \, m^3; \quad \alpha_t=0.5, \, t\lt 18,000; \; \alpha_t=0.2, \, t\gt 18,000; \quad \alpha_h=1.3, \, h\lt 5 ; \alpha_h=0.5, \, h\gt 5; \quad \alpha_H=0.5; \quad \alpha_T=0.5; \quad \alpha_d=1.3; \quad \alpha_{\beta}=0.3 \;[/math] and [math]f(\theta)=(1+0.01 \, \theta)^{0.5} .[/math]

The main advantage of the DUNERULE formula is its simplicity. A constant beach slope is assumed; the effect of slope time dependence is approximated through the exponent [math]\alpha_t[/math]. The formula implicitly takes wave run-up into account. One drawback of DUNERULE is that it does not account for fluctuating storm surge levels, which always occur along meso- and macrotidal coasts in case of long duration storms. Comparison with the analytical dune erosion model shows that DUNERULE produces much larger erosion volumes in case of short duration storms. The erosion volumes become similar for long duration storms, assuming that tidal water level variations can be ignored. Another drawback is the restriction to isolated storms; DUNERULE cannot be applied to storm clusters[23]. In cases of varying storm surge levels and storm clusters, the use of numerical models is required.


XBeach

The dune erosion models considered in the preceding sections represent strong simplifications that ignore specific hydrodynamic processes and bathymetric features that influence the local impact of storms, such as wave spectrum, wave deformation and refraction on morphological structures in the nearshore zone, longshore sand transport, beach morphology and vegetation and dune overwash. More generally applicable and accurate dune erosion forecasts therefore require numerical process-based models that are capable to simulate the effect of such features, also including the effect of storm sequences. An example is the widely used depth-averaged wave spectral XBeach model [24]. This open-source process-based numerical model was originally developed to simulate hydrodynamic and morphodynamic processes and impacts on sandy coasts with a domain size of kilometers and on the time scale of storms and has since been extended to wider applications. The model simulates the hydrodynamic processes of short wave transformation (refraction, shoaling and breaking), long wave transformation (generation, propagation and dissipation of infragravity waves), wave set-up and unsteady currents, overwash and inundation. It includes the effect of wave skewness and asymmetry, the dynamics of nearshore sandbars, as well as the effects of vegetation and hard structures. The morphodynamic processes include bed load and suspended sediment transport, bed update and breaching. The model has been validated with a series of analytical, laboratory and field test cases using a standard set of parameter settings. The avalanching of sandy material from the dune face to the foreshore during storm conditions is taken into account when updating the bed levels. This is modeled through introduction of a critical bed slope (default critical slope of 1 for dry zones and 0.3 for wet zones). When the critical slope is exceeded, material is exchanged between adjacent cells to the amount needed to bring the slope back to the critical slope.


Comparison with observed dune erosion events at the Dutch coast shows that XBeach estimates the magnitude and pattern of alongshore variations in erosion volume reasonably well. The 2014-version of XBeach overpredicted the erosion volume in the region where a dune scarp developed and underestimated the erosion volume where the whole dune face collapsed in a series of slumps [25]. XBeach simulations further illustrated that the observed alongshore variation in dune erosion was steered primarily by the pre-storm dune topography i.e., the presence of embryonic dune fields and the steepness of the dune face. The importance of alongshore variability in intertidal beach topography was found to be secondary, but not negligible during the initial stage of the storm, when the surge level was still low. The 2019 version of XBeach was used to simulate the failure process of an experimental unvegetated dune on the German Baltic Sea coast. The post-storm dune breach was simulated well, but dune erosion was overestimated with respect to temporal profile evolution. It was concluded that the XBeach model is able to predict with sufficient accuracy the evolution of storm-induced erosion during a long overwash/inundation-dominant wave event when current high-resolution input data is available[26].


Post-storm dune recovery

After being eroded under storm conditions, the dune front can recover naturally without human intervention. This has been observed for many sandy coasts with either a gently sloping or a steep sloping shoreface [27][28][29]. Strong beach accretion often already occurs during the phase of waning storm intensity. Shoreline recovery is dealt with in the article Shoreline retreat and recovery. It is a much faster process than dune recovery. Fast beach recovery is promoted by landward migration and welding of nearshore (intertidal) sandbars to the beach during fair-weather conditions. Aeolian sand transport to the dune starts once the upper-beach and any washover deposits become dry with the development of embryo dunes on the backshore. The expansion of the backshore is required to increase the fetch length, which controls the amount of sediment exchanged from the beach to the dune. Dune recovery also depends on vegetation recolonization. Post-storm recovery of vegetation can take two to eight years depending on the extent that the roots and rhizomes are impacted[30][31].

An example of natural dune recovery is shown in Fig. 7a for a naturally retreating coast (Walcheren, the Netherlands). The extreme storm surge of 1953 caused a 30 m retreat of the dune foot. Afterwards the dune foot advanced slightly over a period of several years, until it reached about 15 years later the position corresponding to the retreating trend prior to the storm surge. The ongoing coastal retreat was finally halted in 1990, when the Dutch government decided to stop coastal retreat along the entire coastline with sand nourishments.

Figs. 7b and 7b show examples of dune recovery at two locations (Penhale, UK, and Lacanau, France) on the western European coast, which were severely impacted by the 2014 winter storm. The decade-averaged shoreline retreat in both locations is small, no more than a few meters per decade. The foredune in Penhale is higher than in Lacanau, more densely vegetated (with marram grass) and more alongshore uniform. The 2014 winter storm caused a dune foot retreat of almost 20 m in both locations. The retreat was fairly uniform along the coast at Penhale but showed pronounced erosion hotspots at Lacanau. At Penhale, the dune foot position partially recovered within a few years, but followed afterwards a trend of retreat. At Lacanau, the initial dune foot position was not recovered after 6 years, although a trend towards full recovery was observed. This example illustrates that dune recovery trajectories can differ considerably between sites with only moderately different characteristics.


Fig. 7. Examples of dune foot retreat and recovery. 7a. Variation of the dune foot position in a beach transect of the island Walcheren (Netherlands) over the period 1900-2010 (solid white line, based on annual surveys of Rijkswaterstaat [32]). The trend is indicated by the dashed line, showing ongoing retreat of the dune foot from 1900-1990 and stabilization afterwards. 7b. Dune foot position at Penhale, UK, before and after the 2014 winter storm. After initial partial recovery the dune foot retreats towards its post-storm position. 7c. Longshore-averaged dune foot position at Lacanau, France, before and after the 2014 winter storm. The recovery of the pre-storm dune foot position is slow, and is variable along the coast. Figures 7b and 7c are redrawn from Burvingth and Castelle, 2023[5].


Dune failure probability

The numerical model XBEACH and the simplified dune erosion models discussed earlier provide estimates for dune retreat under given storm conditions. A highly relevant question is related to the probability of dune failure, i.e., the probability that dune retreat at a certain location exceeds a given value, leading to dune collapse and flood damage to settlements situated behind the dune. To answer this question, knowledge of the probability of extreme storm conditions occurring at this location is required. If the time interval between storms leading to dune failure is much shorter than the length of the available storm data record, the failure probability can be found by estimating the dune retreat for all extreme storm conditions occurring in the dataset. However, dune reinforcement programs usually require knowledge of the statistical return period of possible dune failure that far exceeds the length of available storm records. In this case, the available storm record must be extended by statistical methods to a time span much longer than this return period. This can be done by a method described below[33].

Storm conditions can be characterized by the parameters: maximum significant wave height, wave period, storm surge water level, wave direction, and storm duration. If these parameters are uncorrelated, the parameters corresponding to extreme storms with a given return period exceeding the data record can be estimated by fitting an appropriate cumulative distribution function (marginal cumulative distribution function) to the available storm dataset for each parameter separately. The overall probability is then given by the product of the separate probabilities. However, storm parameters are typically correlated. To account for these correlations, so-called copulas are constructed. Copulas are multivariate cumulative distribution functions for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Choosing the appropriate copula and determining the copula parameters from the data is often not simple and requires acquaintance with copula theory (see for example McNeil et al. 2005[34]). From the fitted copulas, random samples can be constructed of extreme storm surge conditions with known probabilities. By estimating dune retreat for these random samples (Monte Carlo method) a relation can be established between calamitous dune retreat (dune failure) and the corresponding probability.


Impact of sea-level rise on dune erosion

The XBeach model predicts an approximately linear relationship between dune erosion volume and sea level rise [35]. This is primarily due to the higher water level in front of the dune rather than to changes in the significant short-wave or infragravity wave height. Changes in the offshore angle of wave incidence also affect the dune erosion volume. A 30° shift from shore-normal influences the dune erosion volume to the same extent as a 0.4 m sea-level rise. This increase in dune erosion volume is related to strong alongshore currents, generated as a result of the obliquity of the waves, that enhance stirring and hence offshore transport[35].

XBeach simulations further show that the effectiveness of coastal sand nourishment to mitigate the impact of sea-level rise on dune erosion depends on the location in the profile where this sand is added. The reduction of dune erosion volume relative to the total added nourishment volume never exceeds 30% in the case of shore or beach nourishment. This suggests that directly increasing the volume of sand in the dunes can be more efficient than beach or shoreface nourishment[35].


Appendix: Analytical dune erosion model

Fig. B1. Simplified representation of dune retreat and notations used in the appendix.

The analytic model of Larson et al. (2004[16]) is a crude simplification of the dune erosion process. However, it does allow an estimate of the order of magnitude of the eroded dune volume under severe storms for different beach morphologies. The model is based on the following assumptions and simplifications:

  • The eroded dune volume [math]\Delta V[/math] in a time interval [math]\Delta t[/math] is proportional to the force [math]F[/math] exerted by swash waves hitting the dune foot, [math]\Delta V \propto F \Delta t .[/math]
  • The force [math]F \Delta t [/math] is proportional to the number of swash bores [math]n[/math] hitting the dune foot ([math]n=\Delta t/T[/math]) and the force exerted by each individual swash bore, [math]f=m \Large\frac{du}{dt}\normalsize \propto \Large\frac{mu}{T}\normalsize [/math], where [math]T[/math] is the wave period, [math]m[/math] the mass of the swash bore per unit dune width and [math]u[/math] the velocity of the swash bore when hitting the dune foot.
  • The bore height [math]h[/math] is related to the bore velocity by [math]u \propto \sqrt{gh}[/math], where [math]g[/math] is the gravitational acceleration. The bore mass is thus proportional to [math]m \propto huT \propto u^3 T[/math], yielding [math]F = n \,f / \Delta t \propto u^4 /T[/math].
  • The bore velocity at the dune foot can be related to the swash runup [math]R[/math]. A ballistic swash excursion model (neglecting friction) gives an estimate for this relation, [math]R=z_D+gu^2/2[/math], where [math]z_D[/math] is the height of the dune foot compared to the beach level where waves collapse on the beach. Using this relation, the force [math] F [/math] can be expressed as a function of the wave runup [math] R [/math], [math]F \propto (R – z_D)^2 / T[/math].
  • Substitution with [math]\Delta V / \Delta t \approx dV/dt [/math] gives Eq. (1), where [math]C_s[/math] is a proportionality constant, depending on characteristics of the site, such as sediment grainsize, wave incidence angle, shoreface profile, etc. The value of [math]C_s[/math] should be calibrated with local field data.
  • Empirical expressions for the wave run-up [math]R[/math] are given in the article Wave run-up. However, the 2% wave run-up may not be the most pertinent parameter; less extreme thresholds may be more effective for dune erosion[3].

Wave run-up increases with increasing offshore wave height [math]H[/math], increasing wave period [math]T[/math] and increasing beach slope [math]\tan \beta[/math]. From Eq. (1) it then appears that the rate of dune erosion increases as the wave height [math]H[/math] increases, the wave period [math]T[/math] increases, the beach slope [math]\beta[/math] increases and the beach width [math]l[/math] decreases. Greater dune erosion for steeper beaches under the same wave conditions is consistent with observations[18].

The dune foot height [math]z_D[/math] (relative to the beach level of wave collapse [math]z_C[/math]) and the beach slope [math]\beta[/math] are time dependent. In order to solve Eq. (1) a prescription must be given how the eroded dune material is distributed over the beach – i.e. a model for the dependence of [math]z_D[/math] and [math]\beta[/math] on [math]dV/dt[/math]. Based on experiments in large wave flumes, Larson et al. (2004[16]) assumed that the beach slope [math]\beta[/math] remains constant during dune erosion. The dune retreat [math]\Delta x[/math] was therefore set equal to [math]\Delta z_D / \tan \beta[/math], assuming that the major part of the dune volume loss is removed to the lower beach (below the level of wave collapse). Field observations of dune erosion events in different sites along coasts in the USA, Australia and France show that the upper beach slope can vary considerably depending on site characteristics and storm intensity[3]. These observations emphasize the tentative nature of the analytical model.

Considering a dune height [math]D[/math] above the level of wave collapse [math]z_C[/math] (the scarp height), the loss of dune volume is given by [math]\Delta V=-(D-z_D) \Delta x = -(D- z_D) \Delta z_D / \tan \beta[/math]. Combining this result with Eq. (1) yields a differential equation for the dune foot height [math]z_D[/math]:

[math]\Large\frac{d}{dt}\normalsize z_D = \Large\frac{1}{\tau} \frac{(R-z_D)^2}{D-z_D}\normalsize , [/math] where [math]\tau=T/(4 C_s \tan \beta)[/math].

If it is further assumed that the level of wave collapse [math]z_C[/math] does not change during the storm, then the wave run-up remains constant too (the slope [math]\beta[/math] was assumed constant). For storms of limited duration the run-up [math]R[/math] can be assumed constant. The differential equation can then be solved, with solution

[math]t = \tau (D-R)\Large(\frac{1}{R-z_D(t)} - \frac{1}{R-z_D(0)})\normalsize +\tau \ln\Large\frac{R-z_D(0)}{R-z_D(t)}\normalsize [/math], from which [math]z_D(t)[/math] and [math]V(t)[/math] can be derived.


Related articles


References

  1. ↑ Anthony, E.J., Mrani-Alaoui. M. and Héquette, A. 2010. Shoreface sand supply and mid- to late Holocene aeolian dune formation on the storm-dominated macrotidal coast of the southern North Sea. Marine Geology 276: 100–104
  2. ↑ Den Heijer, C. 2013. The role of bathymetry, wave obliquity and coastal curvature in dune erosion prediction. PhD Thesis, Delft University
  3. ↑ 3.0 3.1 3.2 Conlin, M.P., Cohn, N. and Adams, P.N. 2023. Total water level controls on the trajectory of dune toe retreat. Geomorphology 438, 108826
  4. ↑ 4.0 4.1 Splinter, K.D., Kearney, E.T. and Turner, I.L. 2018. Drivers of alongshore variable dune erosion during a storm event: Observations and modelling. Coastal Engineering 131: 31–41 Cite error: Invalid <ref> tag; name "S18" defined multiple times with different content
  5. ↑ 5.0 5.1 5.2 5.3 Burvingt, O. and Castelle, B. 2023. Storm response and multi-annual recovery of eight coastal dunes spread along the Atlantic coast of Europe. Geomorphology 435, 108735
  6. ↑ 6.0 6.1 Masselink, G. and van Heteren, S. 2014. Response of wave-dominated and mixed-energy barriers to storms. Marine Geology 352 (2014) 321–347
  7. ↑ Karunarathna, H., Pender, D., Ranasinghe, R., Short, A. D. and Reeve, D. E. 2014. The effects of storm clustering on beach profile variability. Mar. Geol. 348: 103–112
  8. ↑ Baldock, T.E., Gravois, U., Callaghan, D.P. and Nichol, S. 2021. Methodology for estimating return intervals for storm demand and dune recession by clustered and non-clustered morphological events. Coastal Engineering 168, 103924
  9. ↑ Galiatsatou, P. and Prinos, P. 2016. Joint probability analysis of extreme wave heights and storm surges in the Aegean sea in a changing climate. In: FLOODrisk 2016-3rd European Conference on Flood Risk Management
  10. ↑ Luijendijk, A.P., Ranasinghe, R., de Schipper, M.A., Huisman, B.A., Swinkels, C.M., Walstra, D,J,R. and Stive, M.J.F. 2017. The initial morphological response of the Sand Engine: A process-based modelling study. Coastal Engineering 119: 1–14
  11. ↑ Gerdes, E., Koning, R., Kort, M. and Soomers, H. 2021. Policy evaluation Sand Engine 2021. Rebel Economics & Transactions, report commissioned by Rijkswaterstaat, Netherlands
  12. ↑ Erikson, Li, H., Larson, M. and Hanson, H. 2007. Laboratory investigation of beach scarp and dune recession due to notching and subsequent failure. Marine Geology 245: 1–19.
  13. ↑ van Wiegen, P.P.J., de Vries, S. and Reniers, A.J.H.M. 2024. Field observations of wave-averaged suspended sediment concentrations in the inner surf zone with varying storm conditions. Marine Geology 473, 107302
  14. ↑ van Wiechen, P.P.J., de Vries, S., Reniers, A.J.H.M. and Aarninkhof, S.G.J. 2023. Dune erosion during storm surges: A review of the observations, physics and modelling of the collision regime. Coastal Engineering 186, 104383
  15. ↑ van Gent, M.R.A., van Thiel de Vries, J.S.M., Coeveld, E.M., de Vroeg, J.H. and van de Graaff, J. 2008. Large-scale dune erosion tests to study the influence of wave periods Coastal Engineering 55: 1041–1051
  16. ↑ 16.0 16.1 16.2 16.3 Larson, M., Erikson, L. and Hanson, H. 2004. An analytical model to predict dune erosion due to wave impact. Coastal Engineering 51: 675– 696
  17. ↑ Palmsten, M.L. and Holman, R.A. 2012. Laboratory investigation of dune erosion using stereo video. Coastal Engineering 60: 123–135
  18. ↑ 18.0 18.1 Cohn, N., Ruggiero, P., García-Medina, G., Anderson, D., Serafin, K.A. and Biel, R. 2019. Environmental and morphologic controls on wave-induced dune response. Geomorphology 329: 108–128
  19. ↑ Larson, M., Palalane, J., Fredriksson, F. and Hanson, H. 2016. Simulating cross-shore material exchange at decadal scale. Theory and model component validation. Coastal Engineering 116: 57–66
  20. ↑ Ranasinghe, R., Callaghan, D. and Stive, M. J. 2012. Estimating coastal recession due to sea level rise: beyond the Bruun rule. Climatic Change 110: 561–574
  21. ↑ Le Cozannet, G., Bulteau, T., Castelle, B., Ranasinghe, R., Wöppelmann, G., Rohmer, J., Bernon, N., Idier, D., Louisor, J. and Salas-y-Mélia, D. 2019. Quantifying uncertainties of sandy shoreline change projections as sea level rises. Nature Scientific Reports 9:42
  22. ↑ van Rijn, L.C. 2009. Prediction of dune erosion due to storms. Coastal Engineering 56: 441-457
  23. ↑ Baldock, T.E., Gravois, U., Callaghan, D.P., Davies, G., Nichol, S. 2021. Methodology for estimating return intervals for storm demand and dune recession by clustered and non-clustered morphological events. Coastal Engineering 168, September 2021, 103924
  24. ↑ https://xbeach.readthedocs.io/en/latest/user_manual.html
  25. ↑ De Winter, R.C,. Gongriep, F. and Ruessink, B.G. 2014. Observations and modeling of alongshore variability in dune erosion at Egmond aan Zee, the Netherlands. Coastal Engineering 99: 167-175.
  26. ↑ Schweiger, C., Kaehler, C., Koldrack, N. and Schuettrumpf, H. 2020. Spatial and temporal evaluation of storm-induced erosion modelling based on a two-dimensional field case including an artificial unvegetated research dune. Coastal Engineering 161, 103752
  27. ↑ Vousdoukas, M.I., Pedro, L., Almeida, M. and Ferreira, O. 2012. Beach erosion and recovery during consecutive storms at a steep-sloping, meso-tidal beach. Earth Surf. Process. Landf. 37: 583–593
  28. ↑ Scott. T., Masselink, G., O'Hare, T., Saulter, A., Poate, T,. Russell. P., Davidson, M. and Conley, D. 2016. The extreme 2013/2014 winter storms: Beach recovery along the southwest coast of England. Marine Geology 382: 224–241
  29. ↑ Phillips, M.S., Harley, M.D., Turner, I.L., Splinter, K.D. and Cox , R.J. 2017. Shoreline recovery on wave-dominated sandy coastlines: the role of sandbar morphodynamics and nearshore wave parameters. Marine Geology 385: 146–159
  30. ↑ Suanez, S., Cariolet, J-M., Cancouët, R., Ardhuin, F. and Delacourt, C. 2012. Dune recovery after storm erosion on a high-energy beach: Vougot Beach, Brittany (France). Geomorphology 139-140: 16–33
  31. ↑ Houser, C., Wernette, P., Rentschlar, E., Jones, H., Hammond, B. and Trimble, S. 2015. Post storm beach and dune recovery: implications for barrier island resilience. Geomorphology 234: 54–63
  32. ↑ Jeuken, C., Ruessink, G. and Marchand, M. 2001. Ruimtelijke en temporele aspecten van de duinvoetdynamiek. Report WL/Delft Hydraulics Z2838 (in Dutch)
  33. ↑ Li, F., van Gelder, P.H.A.J.M., Vrijling, J.K., Callaghan, D.P., Jongejan, R.B. and Ranasinghe, R. 2014. Probabilistic estimation of coastal dune erosion and recession by statistical simulation of storm events. Applied Ocean Research 47: 53–62
  34. ↑ McNeil, A., Frey, R. and Embrechts, P. 2005. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press
  35. ↑ 35.0 35.1 35.2 De Winter, R,C. and Ruessink, B.G. 2017. Sensitivity analysis of climate change impacts on dune erosion: case study for the Dutch Holland coast. Climatic Change (2017) 141:685–701 DOI 10.1007/s10584-017-1922-3


The main authors of this article are Jan van de Graaff and Job Dronkers
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