Dune erosion
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 socalled 'collision regime', where waves hit the dune without exceeding the dune crest.
Contents
Introduction
Coastal dunes have developed naturally along many shorelines worldwide during the last millennia when sealevel 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 lowlying 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 m^{3}/m, causing the dune foot to recede from 80100 m at some places^{[2]}.
The actual dune loss depends on many factors^{[3]}:
 beach geometry (prestorm): beach slope and width;
 grain size;
 dune geometry (prestorm): dune face slope, dune crest elevation;
 nearshore morphology: nearshore slope, ripchannel 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 Poststorm dune recovery. Because recovery of the beachdune 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]}.
Figure 2 shows as an example a model result of the relationship between dune foot retreat [math]RD[/math] (expressed as the distance between prestorm and poststorm 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
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 socalled 'Sand Engine' on the Dutch coast^{[10]}, which contributed to the reinforcement of the single protecting dune row through winddriven 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 crossshore 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]}.
The sand eroded from the dune is removed further down by the backwash and the seaward undercurrent ('undertow') under the breaking waves. 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 far offshore^{[6]}. The slope of the crossshore 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 crossshore 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 crossshore 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 crossshore direction. The same volume of sand which is eroded from the dunes and the very upper part of a crossshore profile is accumulated lower in the crossshore 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 meters^{3}/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 poststorm 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^{[13]}. It is based on empirical poststorm equilibrium profiles established by laboratory experiments and validated by field data of the Dutch coast ^{[14]}. The poststorm 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:
 the storm duration is sufficient for establishment of a poststorm equilibrium profile [math]y(x)[/math];
 the poststorm equilibrium profile does not (strongly) depend on the initial coastal profile [math]y_0(x)[/math];
 eroded dune sediment is deposited within a zone delimited by a storm closure depth [math]y_{max}[/math];
 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 poststorm equilibrium profile [math]y(x)[/math] and the storm closure depth [math]y_{max}[/math] are also indicated. The crossshore distance over which sand is redistributed (the storm active zone) is given by [math]x_{max}[/math].
The empirical formulas representing the poststorm 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 longterm 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 processbased, but highly idealized analytic model was developed by Larson et al. (2004^{[15]}) 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 runup 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{(Rz_D)^2}{T}\normalsize , \qquad (1)[/math]
where the constant [math]C_s[/math] depends on sitespecific 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 NarrabeenCollaroy 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^{[16]}. 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^{[15]}) 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 runup 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 runup component^{[17]}. A more elaborate version of the analytic dune erosion model includes the effects of overwash, windblown sand transport, and barberm material exchange (Larson et al., 2016^{[18]}).
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 longterm dune foot retreat can be estimated by making assumptions about poststorm dune recovery. The values for dune foot retreat obtained in this way by Ranasinghe et al. (2012^{[19]}) and Le Cozannet et al. (2019^{[20]}) 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)^{[21]} 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 crossshore coastal profile, including dune collapse by colliding waves. The model has been extensively tested in smallscale and largescale 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 setup, [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 runup 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^{[22]}. 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 processbased models that are capable to simulate the effect of such features, also including the effect of storm sequences. An example is the widely used depthaveraged wave spectral XBeach model ^{[23]}. This opensource processbased 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 setup 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 2014version 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 ^{[24]}. XBeach simulations further illustrated that the observed alongshore variation in dune erosion was steered primarily by the prestorm 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 poststorm 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 storminduced erosion during a long overwash/inundationdominant wave event when current highresolution input data is available^{[25]}.
Poststorm 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 ^{[26]}^{[27]}^{[28]}. 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 fairweather conditions. Aeolian sand transport to the dune starts once the upperbeach 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. Poststorm recovery of vegetation can take two to eight years depending on the extent that the roots and rhizomes are impacted^{[29]}^{[30]}.
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 decadeaveraged 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.
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^{[32]}.
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, socalled 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^{[33]}). 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 sealevel rise on dune erosion
The XBeach model predicts an approximately linear relationship between dune erosion volume and sea level rise ^{[34]}. This is primarily due to the higher water level in front of the dune rather than to changes in the significant shortwave or infragravity wave height. Changes in the offshore angle of wave incidence also affect the dune erosion volume. A 30° shift from shorenormal influences the dune erosion volume to the same extent as a 0.4 m sealevel 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^{[34]}.
XBeach simulations further show that the effectiveness of coastal sand nourishment to mitigate the impact of sealevel 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^{[34]}.
Appendix: Analytical dune erosion model
The analytic model of Larson et al. (2004^{[15]}) 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 runup [math]R[/math] are given in the article Wave runup. However, the 2% wave runup may not be the most pertinent parameter; less extreme thresholds may be more effective for dune erosion^{[3]}.
Wave runup 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^{[17]}.
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^{[15]}) 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=(Dz_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{(Rz_D)^2}{Dz_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 runup remains constant too (the slope [math]\beta[/math] was assumed constant). For storms of limited duration the runup [math]R[/math] can be assumed constant. The differential equation can then be solved, with solution
[math]t = \tau (DR)\Large(\frac{1}{Rz_D(t)}  \frac{1}{Rz_D(0)})\normalsize +\tau \ln\Large\frac{Rz_D(0)}{Rz_D(t)}\normalsize [/math], from which [math]z_D(t)[/math] and [math]V(t)[/math] can be derived.
Related articles
 Natural causes of coastal erosion
 Dune stabilisation
 Light revetments builtin into artificial dunes
 Shoreface profile
 Bruun rule
 Active coastal zone
 Dune development
 Shoreline retreat and recovery
 Swash zone dynamics
 Shore protection vegetation
 Dynamics, threats and management of dunes
 Risk and coastal zone policy: example from the Netherlands
References
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<ref>
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