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 'collision regime', where waves hit the dune without exceeding the dune crest (the so-called 'overwash regime').

Introduction

Fig. 1. Post-storm eroded dune, Egmond aan Zee.

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, 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.

The actual dune loss depends on many factors:

• beach geometry: beach slope and width;
• grain size;
• dune geometry: dune face slope, dune crest elevation;
• nearshore morphology: nearshore slope, rip-channel embayments, nearshore sandbars, subtidal morphology;
• beach and dune vegetation; presence of 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. However, there is also strong evidence that dunes are less impacted by storms when situated on dissipative flat beaches than on reflective steep beaches.

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.

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

As an example, Fig. 2 shows a model result of the relationship between dune retreat [math]RD[/math] (expressed as the distance between the initial dune foot location and the dune foot location after the storm surge) 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 1/100,000 probability per year that dune erosion will exceed 85 m. To establish such a plot, a proper insight is needed in the possibly occurring extreme storm conditions in the (very) small probability range. However, such insight is subject to great uncertainty, as reliable measurements of extreme water levels and wave conditions are available only over relatively short periods of the order of 100 years. With such a restricted data set, it is hard to make estimates of storm conditions with a yearly exceedance probability less than about 1/1000.

Impact of dune erosion

Fig. 3. Badhotel at Schiermonnikoog 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. 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. 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, which might be considered to be in a more or less dynamic equilibrium condition with normally occurring hydrodynamic conditions, will be reshaped during a severe storm surge. The much higher water levels and 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 sand eroded from the dunes.

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 , see Fig. 5.

Fig. 4. Dune failure mechanisms.

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

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, some of the sand can even be transported far offshore. 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 a great water level increase (a water level increase of a few metres is possible along some coasts) 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.

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. It is based on empirical post-storm equilibrium profiles established by laboratory experiments and validated by field data of the Dutch coast . 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 slope of the post-storm profile decreases (i.e. increasing dune erosion) as the wave period [math]T[/math] increases and the fall velocity decreases.

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 much 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 most naturally occurring storms.

Analytic dune erosion model

A simple analytic model was developed by Larson et al. (2004) 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 site-dependent constant [math]C_s[/math] has to be determined empirically. According to this equation, the rate of dune erosion is an increasing function of the wave height, the wave period and the beach slope (see appendix). This is consistent 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 (i.e. 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 strongly correlated with alongshore variability in [math]z_D[/math]. A strong relationship between dune erosion volumes and dune foot height was also found in wave tank experiments. 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) found reasonable agreement with observed dune erosion rates when using values of [math]C_s[/math] in the range 10^-3 - 2.10^-3.

Contrary to the erosion profile method, the analytic model only considers the maximum water level [math]R[/math] without distinction between the wave run-up and storm level components. Observations of dune erosion on the US northwest coast suggest that the influence of the storm level component is more important for dune erosion than the wave run-up component. 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).

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) and Le Cozannet et al. (2019) 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) 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]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. One drawback is that it does not account for fluctuating storm surge levels, which always occur along tidal coasts in case of long duration storms. The formula implicitly takes wave run-up into account. Another simplification is the constant beach slope; the effect of slope time dependence is approximated through the exponent [math]\alpha_t[/math]. 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.

XBeach

The dune erosion models considered in the preceding sections represent strong simplifications that ignore specific hydrodynamic 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. An example is the widely used XBeach model . 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 includes 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, as well as the dynamics of nearshore sandbars. The morphodynamic processes include bed load and suspended sediment transport, bed update and breaching. Effects of vegetation and hard structures have been included. 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 . 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. An application of the 2019 version of XBeach to a breach experiment at a large-scale sand dune in Germany showed that the post-storm beach profile was well simulated, but dune erosion was overestimated with respect to temporal profile evolution. It was concluded that the XBeach model is capable of accurately predicting the evolution of storm-induced erosion during a sufficiently long and overwash/inundation-dominant surge event when current high-resolution input data is available.

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 . Strong beach accretion often already occurs during the phase of waning storm intensity. After-storm dune recovery starts with the development of embryo dunes on the backshore stimulated by vegetation growth. An example of natural dune recovery is shown in Fig. 7 for a 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 by coastal sand nourishments. Recovery of the shoreline precedes recovery of the dune foot and is much faster. Fig. 8 shows after-storm recovery of the shoreline position at Narrabeen-Collaroy Beach in East Australia, near Sidney. The recovery period after each storm-induced shoreline retreat is only a few months at this location. Shoreline recovery at this location is significantly correlated with onshore migration of nearshore sandbars. It further appears that post-storm recovery of wide gently sloping beaches is promoted by infragravity swash.

Fig. 7. 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 ). The trend is indicated by the dashed line, showing ongoing retreat of the dune foot from 1900-1990. The large dune erosion by the exceptional 1953 storm surge has no influence on the long-term trend of dune foot retreat. Stabilization after 1990 is due to implementation of the current 'hold-the-line' coastal policy by sand nourishments.

Figure 8. Variation of the shoreline position in a transect of Narrabeen-Collaroy Beach (East Australia, near Sidney) over the period October 2004-May 2005. Strong shoreline retreat occurs during the storm periods indicated in light blue. After-storm recovery occurs during the periods in between. Figure redrafted after .

Dune failure probability

The numerical model XBEACH and the simplified dune erosion models discussed earlier provide estimates for dune retreat under given storm conditions. In practice, however, 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 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 frequency of storms leading to dune failure is much greater than the inverse duration 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 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.

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). 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 a linear relationship between dune erosion volume and sea level rise . This is primarily due to the higher water level in front of the dune and not 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.

The model simulations further show that the effectiveness of coastal sand nourishment to mitigate the impact of sea-level rise, strongly depends on the location in the profile where this sand is added. However, the reduction in dune erosion volume relative to the total added volume remains low (<0.3) for all mitigation options. This suggests that directly increasing the volume of sand in the dunes may be more efficient than beach or shoreface nourishment.

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) 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 coefficient [math]C_s[/math] should be determined empirically.
• Empirical expressions for the wave run-up [math]R[/math] are given in the article Wave run-up. 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.

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) 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). Considering a dune height [math]D[/math] above the level of wave collapse [math]z_C[/math], 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

• Natural causes of coastal erosion
• Dune stabilisation
• Light revetments built-in 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

The main authors of this article are Jan van de Graaff and Job Dronkers Please note that others may also have edited the contents of this article. Citation: Jan van de Graaff; Job Dronkers; (2022): Dune erosion. Available from http://www.coastalwiki.org/wiki/Dune_erosion [accessed on 25-11-2024] For other articles by this author see Category:Articles by Jan van de Graaff For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Jan van de Graaff For an overview of contributions by this author see Special:Contributions/Dronkers J