# Shoreface profile

 Definition of Shoreface: The shoreface is the nearshore zone of the inner continental shelf that is bounded landward by the mean low-water line and that extends seaward to where the influence of wave action on cross-shore sediment transport is on average minor compared to other influences. A similar definition is: The shoreface is the zone seaward of the shoreline where offshore generated waves interact with the upward sloping seabed. This is the common definition for Shoreface, other definitions can be discussed in the article
 Definition of Shoreface profile: The shoreface profile, often called beach profile, is the cross-shore coastal depth profile of the shoreface. This is the common definition for Shoreface profile, other definitions can be discussed in the article

## Introduction Figure 1. Schematic representation of a shoreface profile. The vertical/horizontal scale ratio is greatly exaggerated.

In the literature one can find various definitions of the shoreface. The definition given here, which is most commonly used, follows the definition given in Stive and De Vriend (1995). The shoreface can be divided in two zones, the upper shoreface and the lower shoreface. The upper shoreface is the zone where under average conditions waves break and the lower shoreface the zone where waves shoal. The lower part of the shoreface extends to the so-called outer closure depth $h_{out}$; beyond this depth the seabed is hardly influenced by waves and wave-induced sediment transport is (on average) insignificant. The lower part of the shoreface is also called shoaling zone; wave shoaling is the process of wave amplification when waves travel from deep to shallow water before breaking (see Shallow-water wave theory). The outer closure depth $h_{out}$ has been empirically determined by Hallermeier (1981), yielding the formula (see also Closure depth)

$h_{out} \approx (H_s – 0.3 \sigma) \, T_s \sqrt{\large\frac{g}{5000 d_{50}}\normalsize} \approx 0.013 H_s T_s \sqrt{\large\frac{g}{d_{50}}\normalsize} , \qquad (1)$

where $H_s, T_s$ are the average significant wave height and period, $\sigma$ is the wave height standard deviation, $d_{50}$ the medium sediment grainsize and $g$ the gravitational acceleration. The second approximation in Eq. (1) follows from the assumption of a Rayleigh wave distribution (see Statistical description of wave parameters). Other criteria for the outer closure depth can be derived from the condition that the wave-induced bed shear stress should not exceed the threshold of sediment transport under extreme storms. Typical values of the outer closure depth are in the range 15 – 40 m.

Most energy is dissipated in the upper part of the shoreface, where wave overturning and breaking occurs. The depth where waves start breaking depends on the wave climate and is therefore highly variable. An often used definition of the transition depth $h_{in}$ between upper and lower shoreface was also given by Hallermeier (1978),

$h_{in} \approx 2.28 H_{12h/y}- 68.5 \Large\frac{ H_{12h/y}^2}{g T_{12h/y}^2}\normalsize , \qquad (2)$

where $H_{12h/y}$ is the effective wave height just seaward of the breaker zone that is exceeded on average for 12 hours per year (i.e. the significant wave height with a probability of yearly exceedance of 0.137%) and $T_{12h/y}$ the associated wave period. Later refinements of this formula are discussed in the article Closure depth. Typical values of $h_{in}$ are in the range 3 – 10 m. The width of the upper shoreface typically ranges between about 100 - 1000 meters and the average slope between 1/10 and 1/100. The upper shoreface is sometimes also called breaker zone or surf zone, although onshore surfing wave bores usually only occur in the shoreward part of the upper shoreface. It is therefore preferable to use the designation upper shoreface as a fixed geographical indication, and the designations surf zone or breaker zone as a physical indication of the zone where broken waves surf onshore. This latter zone is narrow and close to the shoreline in a gentle wave climate and can be very wide under storm conditions, extending from the seaward boundary of the upper shoreface to the dunefoot. Other subdivisions of the coastal profile include the active coastal zone, the beach face or foreshore and the backshore, see Definitions of coastal terms.

The division between upper and lower shoreface profiles is often marked by an inflexion point, a bar or a terrace, indicating transition of morphodynamics  (Fig. 1). Another indicator of the transition between upper and lower shoreface is a break in the trend of the sediment grainsize with depth. The shoreface is bordered on the landward side by the beach face. This is the zone where under average conditions waves collapse and run up the beach (see Swash zone dynamics). The upper shoreface profile is often undulated due to the presence of one or several bars, see Nearshore sandbars. These bars have mainly a shore parallel orientation, but more complex three-dimensional patterns are ubiquitous (see Rhythmic shoreline features and Fig. 2).

## Morphodynamic feedback

Sediment transport on the shoreface depends on the local wave climate. By 'wave climate' is meant the description of incoming waves in terms of a statistical distribution of incident wave directions, wave heights and wave periods (see Statistical description of wave parameters). Sediment transport also depends on the characteristics of bed sediments and on the seabed bathymetry. For small wave incidence angles (close to the shore-normal direction) the relevant seabed bathymetry is the cross-shore shoreface profile.

The shoreface profile influences cross-shore sediment transport, but the inverse is also true: changes of the shoreface profile result from gradients in the cross-shore sediment transport. This is called morphodynamic feedback: for a given incident wave field the shoreface profile determines the transformation of the wave field on the shoreface and therefore the resulting sediment transport, whereas the resulting sediment transport determines the shoreface profile. As a result of this morphodynamic feedback, the shoreface profile tends to an equilibrium shape for a given stable wave climate. A dynamic equilibrium is reached when onshore directed sediment transport and offshore directed transport are equal on average over a period exceeding the morphodynamic adaptation time. This holds for alongshore uniform coasts; the equilibrium requirement for non-uniform coasts is a vanishing divergence of the long-term average total sediment transport vector (cross-shore and alongshore). It was realised already long ago that the shoreface of a sedimentary coast should not be regarded as a geologically inherited feature, but as the result of natural adaptation to modern hydrodynamic conditions and sediment supply. In real field situations, a coast never assumes its equilibrium profile; it is always in a transient state due to fluctuations in the wave climate and the mean water level. The coastal profile deviates from the equilibrium profile, even in the hypothetical case of a stationary wave climate, due to the cyclical or quasi-cyclical migration of seabed structures along the profile (see Nearshore sandbars).

Knowing how to determine the equilibrium shoreface profile has great practical interest. It provides quantitative insight in the response of the shoreface to changes in the local wave climate by human and natural causes; such insight is important for testing and improving the effectiveness of measures to combat coastal erosion and shoreline retreat.

## Bedforms and sediment sorting

Morphodynamic feedback involves not only shoreface profile development, but also sediment grainsize and bed roughness. Grainsize and bed roughness strongly influence sediment transport . Inversely, sediment transport strongly influences the grainsize distribution of bed sediments. Bed roughness depends on the presence of bedforms (ripples, dunes), which in turn depend on sediment transport and sediment grainsize. Consequently, shoreface profile development, wave transformation on the shoreface, sediment transport, grainsize sorting and bedform emergence are all dynamically coupled processes.

Grainsize sorting is primarily due to differences in transport modes of different grainsize fractions. The finer fractions are more easily brought and maintained in suspension than the coarser fractions. They are transported further away from their initial location and tend to settle on the lower shoreface zone where wave-induced bed shear stresses are lower than on the upper shoreface. The coarse fraction is transported partly as suspended load and partly as bedload in the wave boundary layer. Wave transformation in shallow water leading to wave skewness (stronger onshore than offshore wave orbital velocity) and wave asymmetry (stronger onshore than offshore wave orbital acceleration) favour net onshore transport in the wave boundary layer. Fine sediments are resuspended from the upper shoreface due to undertow currents, longshore currents and turbulent shear stresses generated by breaking waves. Sediments on the upper shoreface are therefore coarser than on the lower shoreface. Fine sediment particles can still be present on the upper shoreface if they are 'hidden' by overlying coarser sediment particles – a phenomenon called 'bed armouring'. Figure 2. The barred coast of North Holland, The Netherlands. https://beeldbank.rws.nl, Rijkswaterstaat / Rens Jacobs

The largest bedforms on the shoreface, such as shoreface-connected ridges (scale of several kilometers) and shore-oblique bars (scale of the order of hundred meters) are mainly related to morfodynamic feedback processes that interact with longshore transport, see the articles Sand ridges in shelf seas and Rhythmic shoreline features. Other large bedforms, the so-called nearshore sandbars or breaker bars (scale also of the order of 100 meters) arise from morphodynamic coupling with the wave breaking process, through the resulting flow structure at the breaker bar and interaction with cross-shore and longshore currents  - see Nearshore sandbars for a more detailed discussion. Bedforms at much smaller scales – ripples with wavelengths of the order of 0.3 - 1 m – are formed in the less exposed parts of the shoreface under low to moderate wave conditions. They are often found on the lower shoreface and in the troughs between nearshore breaker bars. Morphodynamic feedback processes, leading to regular ripple patterns, involve ripple-induced near-bed flow circulation, vortex shedding at ripple crests and non-linear ripple-ripple interactions. For a more detailed discussion, see the articles Wave ripples and Wave ripple formation. The time lag of sediment suspension and settling related to vortex shedding at the ripple crests strongly influences the direction of residual cross-shore sediment transport on the shoreface, see section #Onshore wave-induced sediment transport.

## Dissipative and reflective beaches

The beach profile in the upper shoreface zone can be highly variable. The profile tends to steepen in periods of low-energy waves (especially long-period swell waves) and to flatten under high-energy storm waves (see Active coastal zone). Sediment characteristics, which largely depend on the supply source (see Coastal and marine sediments), also play an important role. Coarse-grained beaches typically have much steeper slopes than fine-grained beaches. Gently sloping beaches dissipate almost completely the energy of incident waves, whereas steep-sloping beaches tend to reflect (partly) incident waves. For distinguishing between reflective and dissipative beaches the Dean-parameter $\Omega$ is often used :

$\Omega = \Large \frac{H_s}{wT_p} \normalsize , \qquad(3)$

where $H_s$ is the offshore significant wave height before breaking, $T_p$ is the peak spectral wave period and $w$ is the mean sediment fall velocity. Wright and Short (1984)  analysed beach profiles in Australia by evaluating the parameter $\Omega$, considering the local long-term average wave climate. They found that beach types could be classified as follows :

• reflective beaches correspond to $\Omega \lt 1$;
• intermediate beaches correspond to $1\lt \Omega \lt 6$;
• dissipative beaches correspond to $\Omega \gt 6$.

The tidal range also influences the beach profile, because the surf zone is shifted across the shoreface with tidal level. For large tides (tidal range much larger than wave height) the surf zone covers during the tidal cycle a much larger area than for small tides. Beaches with large tides therefore have smaller upper shoreface slopes compared to beaches with the same $\Omega$ and small tides.

Another parameter that is often used for analysing the beach response to variations in the wave climate is the so-called surf similarity parameter that relates the wave steepness (wave height to wave length ratio) to the surf zone slope.

## Cross-shore sediment transport

Direct observation of net cross-shore sediment transport and its separate components in the field is very difficult, in particular in the surf zone. Experimental data seldom yield conclusive results, for several reasons: the hydrodynamic conditions for the installation and operation of measuring devices are harsh, especially in periods where sediment transport is strongest, the temporal and spatial variability of the transport processes is very high, and sediment concentration sensors cannot be easily calibrated because of their sensitivity to air bubbles (especially acoustic backscatter sensors) and grainsize (especially optical backscatter sensors). For these reasons, most knowledge is derived from laboratory experiments. Knowledge from the field is often obtained indirectly, through interpretation of the observed erosion and accretion patterns produced by cross-shore sediment transport. Mathematical models are often used as interpretation tools.

Sediment transport on the shoreface depends on many processes. We focus on processes relevant for cross-shore transport. It should be realised, however, that longshore transport processes can also play an important role, for example by influencing the sediment balance of the shoreface or by generating various kinds of bedforms, see for example the articles: Natural causes of coastal erosion, Littoral drift and shoreline modelling and Rhythmic shoreline features. In the following we consider sandy beaches, with grain sizes typically between 0.2 and 0.5 mm (fine to medium sand). Coarse sedimentary beaches are dealt with in the article Gravel Beaches.

### Onshore wave-induced sediment transport

Wave transformation in shallow water is a major factor for onshore sediment transport . Due to shoaling effects, starting on the lower shoreface, incident waves become increasingly skewed (asymmetry in onshore / offshore peak flow velocity) and asymmetric (asymmetry in onshore / offshore flow acceleration, also called acceleration skewness), see the Appendix for details. These asymmetries in the wave orbital velocity produce shear stresses on the seabed which are stronger in onshore direction than in offshore direction , see also Shallow-water wave theory. Wave skewness and asymmetry are most pronounced on gently sloping dissipative coasts; they hardly develop on steep reflective coastal profiles.

Phase lags between bed shear stress and suspended sediment concentration determine to which degree wave skewness induces onshore transport. In some situations (rippled bed, sheet flow) a large phase lag can reverse the net direction of suspended sediment transport (A large time lag implies that sediment particles that are stirred up by turbulent eddies during onshore orbital motion are transported offshore after reversal of the wave orbital velocity ). Onshore transport is stimulated by acceleration skewness and by wave breaking that generates turbulent fluid motions and sediment resuspension under the wave crest, especially for plunging waves. But at the same time, wave breaking removes most of the wave skewness, thus reducing the associated onshore sediment transport component in the surf zone.

The development of forward streaming at the top of the wave boundary layer also contributes to onshore transport , but this only holds for smooth beds and not in cases where the seabed is rippled, in cases of strongly skewed waves or for sheet flow conditions  (see also Sediment transport formulas for the coastal environment). Another, but generally lesser contribution to onshore sediment transport is further due to net mass transport in the upper part of the water column between wave trough and wave crest (in-phase variation of wave water level and wave orbital velocity).

### Offshore wave-induced sediment transport

Offshore transport is due to so-called undertow: the compensating return current for the net onshore mass transport in the upper water column between wave trough and wave crest . The offshore undertow sand transport is particularly strong just landward of the location where waves are breaking and where much sediment is brought into suspension by wave energy dissipation and by the longshore current. Although the undertow current is smaller than the wave orbital velocities, it produces substantial offshore transport across the shoreface, and in particular in the surf zone, where it can dominate the onshore transport related to wave-asymmetry effects. Bound infragravity waves may also contribute to net offshore sediment transport, as discussed in the article Infragravity waves. The transport component related to infragravity waves can be quite substantial on dissipative beaches, but the net contribution is subdued after wave breaking because the phase relationship with incident waves is lost and the net sand transport can be either onshore or offshore directed. However, small transport contributions cannot be ignored because the large transport components related to wave skewness / asymmetry and undertow largely cancel each other out. For this reason, the downslope transport component must also be considered, see next section.

### Downslope sediment transport

Downslope transport occurs when sediment particles roll down the shoreface slope. However, this is not the primary cause of downslope sediment transport for fine to medium sandy beaches  (we do not consider here downslope turbidity currents or mud avalanching occurring at the outer edge of river deltas that receive large amounts of sediment) . Gravity-induced sediment transport is generally a minor effect for two reasons: (1) the shoreface slope is small and (2) sediment particles are mainly transported in suspension, more than as bedload, especially on the upper shoreface. When a sediment particle is lifted from the bed it travels in onshore or in offshore direction depending on the phase of the wave orbital motion. When it settles higher on the shoreface it is more easily resuspended than when it settles lower on the shoreface, because it is subject to greater bed shear stresses in the former case. This implies that settling high on the shoreface will occur more frequently for coarser sediment particles with high fall velocity, while finer sediment particles with small fall velocity will preferentially settle lower on the shoreface. The slope effect thus requires a correction to the net sediment transport over a flat bottom, which is an increasing function of the bed slope $\tan \beta$ and a decreasing function of the fall velocity $w$. Formulas for suspended sediment transport assume in general that the slope effect can be represented by a correction factor to the instantaneous wave-induced transport components of the form 

$(1- \alpha \, \tan \beta \Large \frac{u }{w} \normalsize ) , \qquad(4)$

where $u$ is the instantaneous wave orbital velocity, $w$ the sediment fall velocity and $\alpha$ an efficiency parameter. This is consistent with laboratory experiments, in which an approximate correspondence was found between coastal profiles and an adjusted Dean parameter

$\Omega_{\beta} = \Large \frac{H_s \tan \beta}{wT_p} \normalsize . \qquad(5)$

Experiments by Hattori and Kawamata (1980) with different profile slopes show that onshore transport dominates for $\Omega_{\beta} \lt 0.08$ and offshore transport dominates for $\Omega_{\beta} \gt 0.08$. An equilibrium slope corresponds to (order of magnitude)

$\Omega_{\beta} \sim 0.08. \qquad(6)$

Comparison with Eq. (3) shows that reflective beaches have a steep slope and dissipative beaches a gentle slope. The presence of the fall velocity $w$ in the denominator of the downslope correction term in Eq. (4) also implies that net wave-induced transport can be downslope (offshore) for fine sediment even if upslope transport (onshore) prevails for coarse sediment.

Although not proven, it is often assumed that the slope effect is a major factor compensating for net wave-induced onshore transport (sum of wave-induced onshore and offshore transport components) in situations close to equilibrium. This implies that the equilibrium shoreface slope $\tan \beta$ will be steeper (larger $\tan \beta$) for a coarser seabed (large fall velocity $w$) than for a finer sediment bed (small fall velocity $w$, smaller $\tan \beta$). This is expressed by the correction factor (4) and is in general agreement with field observations. An analysis of long-term shoreface profile observations at the Gold Coast (Australia) suggests that the bed-slope effect on the total net sediment transport averaged over the wave period (or, more generally, the wave climate) can be related to the equilibrium bed slope $\tan \beta_{eq}$ by the correction factor $(1 – \tan \beta / \tan \beta_{eq})$ , which is equivalent to Eq. (4).

### Shoreface modelling

The processes involved in sediment transport on the shoreface, as described above, are complicated and not yet fully understood. Even the most sophisticated numerical morphodynamic models, like XBEACH , make use of approximate empirical formulations for several processes contributing to sediment transport and involve parameters that need to be tuned. Empirical formulas for wave-induced sediment transport on the shoreface that are often used in practice are given in the articles Sediment transport formulas for the coastal environment and Sand transport. These formulas are based on laboratory and field measurements for a large range of conditions. Rough estimates can be obtained with the empirical default parameters given in these articles. For more accurate site-specific estimates it is necessary to tune these parameters to field data. A more detailed discussion of shoreface modelling is presented in the article Modelling coastal hydrodynamics.

## Equilibrium shoreface profile (sandy beach)

For a given stable wave climate the shoreface profile tends after a long period to an equilibrium profile. This equilibrium profile can be computed by applying sediment transport formulas to an initial profile for a period comprising all wave conditions according to local wave climate statistics. This requires heavy calculations that are moreover subject to large uncertainty margins. Therefore, equilibrium profiles derived from field observations are often used. By analyzing a large number of beach profiles of the Californian and Danish coasts  and the US Atlantic and Gulf coasts  Bruun (1954) and Dean (1977) derived an equilibrium profile (often called the Dean/Bruun profile) of the form (Fig. 3) Figure 3. The Bruun/Dean profile (Eq. 7) and the equivalent Bodge profile (Eq. 8).

$h(x)=A \; x^{2/3}, \qquad(7)$

where $h$ is the water depth [m], $x$ the cross-shore distance [m] ($x=0$ at the shoreline) and $A$ a coefficient depending on the sediment fall velocity $w$ [m/s]. Dean (1987) empirically determined $A \approx 0.5 \; w^{0.44}$ [m1/3]. Another estimate was given by Kriebel et al. (1991) $A \approx 2.25 \; (w^2/g)^{1/3}$ [m1/3]. In Eq. (7) it is implicitly assumed that the fall velocity is uniform across the shoreface profile.

Dean  showed that the exponent 2/3 is consistent with the assumption of constant wave energy dissipation per unit volume throughout the surf zone.

Because the Bruun/Dean profile has an infinite slope at the shoreline, an alternative form was proposed by Bodge (1992) ,

$h(x)=B \; (1 – e^{-kx}) , \qquad(8)$

where $k$ is an empirical coefficient in the range $500 \lt k^{-1} \lt 2000$ [m] and where the coefficient $B$ is related to the beach slope $\tan \beta=Bk$ at the shoreline $x=0$.

From an analysis of terrace-shaped shoreface profiles in northern Spain, Bernabeu et al. (2003)  found that the Bruun/Dean profile was adequate only for the upper shoreface; for the lower shoreface they proposed the form

$h(x)=A' \; (x-x_0)^{2/3} , \qquad(9)$

where $x_0$ is determined such that the two profiles (7) and (9) match at the inflexion point between the upper and lower shoreface.

Inman et al. (1993)  used a similar procedure for the analysis of beach profiles for the coasts of south California, North Carolina and the Nile delta. They found best fits with exponents ranging between 0.3 and 0.45 instead of 2/3, which is more consistent with uniform wave energy dissipation in the surf zone per unit surface rather than per unit volume.

The cross-shore profile of gravel beaches is not the same as for sandy beaches, see: Gravel Beaches.

## Difference between upper and lower shoreface

The upper shoreface is the zone where the energy of incident waves is converted into water level gradients, currents, seabed agitation and sediment transport. This goes with strong morphodynamic activity manifested in profile change, generation of bedforms and shoreline advance or retreat. Waves also play an important role on the lower shoreface in shaping the cross-shore profile while wave interaction with the seabed generates wave skewness and asymmetry and resulting net sediment transport. But transport rates on the lower shoreface are typically small, implying a long adaptation time scale to changes in sea level and wave climate. On the upper shoreface these time scales are on the order of day-year, compared to decade-millennium time scales on the lower shoreface. Besides, waves are not the only sediment transport agent on the lower shoreface; tidal currents and wind-driven currents are important as well. The strongest currents are associated with extreme events (storm surges, cyclones), but persistent wind stress over longer periods can produce up or downwelling and associated up or downslope sediment transport. At places where river plumes flowing along the coast induce a cross-shore salinity gradient, fine sediment can be trapped in the nearshore zone, as explained in the article Coastal mud belt. However, fine sediment is more generally transported from the upper to the lower shoreface, as the medium sediment grainsize is typically larger on the upper shoreface than on the lower shoreface. The lower shoreface can also receive sediment directly from nearby river deltas. When sediment sources are absent or sparse the seabed sediment of the lower shoreface has mainly a relict origin with possibly rocky outcrops, indurated clay or coarse terrigenous deposits. Biotic activity on the lower shoreface can be intense with a high production of calcareous bioclasts. The lower shoreface can therefore act as a source of calcareous sand to the upper shoreface and the beach.

## Analytical equilibrium models

Because of the earlier mentioned difficulties to derive model estimates of the equilibrium shoreface profile, taking into account the full wave climate, analytical models have been developed in which the wave climate is replaced by a single representative monochromatic incident wave. In these models an alongshore uniform shoreface is assumed with perpendicular wave incidence, and a simple wave transformation model is used based on depth saturation of broken waves (wave height limited by depth) and shallow-water wave theory. An analytical expression for the equilibrium shoreface slope can be derived by requiring that the total wave-integrated cross-shore sediment flux equals zero throughout the shoreface zone  . Although these models reproduce certain properties also observed in the field, such as increasing profile steepness with increasing grain size and with increasing wave period, a more detailed comparison of computed profiles with observations exhibits large discrepancies. This is perhaps not surprising in view of the simplicity of the model used for wave-induced sediment transport, which assumes zero phase difference between wave-orbital velocity and sediment load, and which ignores many other processes described in the previous paragraphs. It has been shown that some discrepancies can be repaired by using expressions for the sediment transport components derived from field observations.

Analytical models also ignore the influence of extreme storms, which can flatten the profile out to great depths within a short period. The profile adaptation time scale for the entire shoreface being quite long, it is plausible that observed shoreface profiles generally correspond to transient profiles .

## Related articles

Nearshore sandbars
Bruun rule
Active coastal zone
Coastal Hydrodynamics And Transport Processes
Modelling coastal hydrodynamics
Coastal and marine sediments
Gravel Beaches

Komar, P.D. 1998. Beach processes and sedimentation. Prentice Hall, London, pp. 544.
Woodroffe, C.D. 2002. Coasts, form, processes and evolution. Cambridge Univ.Press, 623 pp.
Dronkers, J. 2017. Dynamics of Coastal Systems. World Scientific Publ. Co. 753 pp.

## Appendix: Wave skewness

When a sinusoidal wave travels into shallow water the wave shape changes. The equations describing wave motion in deep water are almost linear; the solution is a simple sinus-shaped wave. The nonlinear terms in the wave equations become important in shallow water. These nonlinear terms introduce higher harmonic wave components in the solution. Initially, the first harmonic components grow the fastest. The wave orbital velocity can then be written as

$u(\theta) = U_1 \cos(\theta) + U_2 \cos(2\theta) + V_2 \sin(2\theta) , \qquad (A1)$

where $\theta = \omega t - kx$ is the wave phase, with $t=$time, $x=$the distance travelled in the wave propagation direction, $\omega =$the wave angular frequency, $k=\omega / c=$the wave number and $c=$the wave propagation speed. The $U_2$ component represents the velocity skewness $S_u$ and the component $V_2$ represents the wave asymmetry or acceleration skewness $A_u$ (Fig. A1). Fig. A1. Wave orbital velocity as function of the phase $\theta$. Left panel: Velocity skewness. Right panel: Acceleration skewness. Blue curves: Eq. (A1). Red curves: Eq. (A5).

We define $\lt …\gt \equiv \Large\frac{1}{2 \pi}\normalsize \int_0^{2 \pi} … d \theta$ and assume $|U_2|,|V_2| \lt \lt |U_1| .$

The velocity and acceleration skewness are defined as

$S_u \equiv \Large\frac{\lt u^3\gt }{\lt u^2\gt ^{3/2}}\normalsize \approx \Large\frac{3 U_2}{\sqrt{2} U_1}\normalsize, \quad A_u \equiv \Large\frac{ \lt (\mathcal{H}u)^3 \gt }{\lt (\mathcal{H}u)^2\gt ^{3/2} }\normalsize \approx \Large\frac{3 V_2}{\sqrt(2) U_1}\normalsize . \qquad (A2)$

(the Hilbert transform $\mathcal{H}$ applied to sinusoidal functions gives: $\mathcal{H}cos(x)=sin(x), \quad \mathcal{H}cos(2x)=sin(2x), \quad \mathcal{H}sin(2x)=-cos(2x)$).

By analysing a large number of field observations of the wave orbital velocity on the shoreface, Ruessink et al. (2012) found empirical evidence that the velocity and acceleration skewness mainly depend on the Ursell number

$U_r = \Large\frac{3 g H}{8 h^2 \omega^2}\normalsize , \qquad (A3)$

where $g \approx$9.8 m s-2 is the gravitational acceleration, $H$ the significant wave height and $h$ the local depth. They established the empirical expressions

$(S_u^2+A_u^2)^{-1/2} = 1.17 \, (1 + 0.205 \, U_r^{-3.37}), \quad \Large\frac{A_u}{S_u}\normalsize = -\cot \Big(\Large\frac{\pi}{2}\normalsize \tanh \big(0.815 U_r^{-0.672}\big)\Big) \normalsize, \qquad (A4)$

from which $S_u$ and $A_u$ can be solved.

Abreu et al. (2010) derived a more general expression than Eq. (A1) for the wave orbital velocity, that includes velocity skewness and acceleration skewness,

$u(\theta)=U_1 \Large\frac{\sqrt{1-r^2}}{1+\sqrt{1-r^2}} \, \frac{r \sin{\phi}+(1+\sqrt{1-r^2}) \sin(\theta)}{1 - r \cos(\theta +\phi)}\normalsize . \qquad (A5)$

The parameters $r$ and $\phi$ are related to the velocity skewness and acceleration skewness by

$r = \Large\frac{2 \sqrt{p+1}}{p+2}\normalsize, \quad p = \Large\frac{9}{2 (S_u^2+A_u^2)}\normalsize , \quad \phi = -\Large\frac{\pi}{2}\normalsize -\arctan(A_u/S_u) . \qquad (A6)$

A predictive model for wave-induced cross-shore sediment transport can be obtained by using the expression (A5) for the wave orbital velocity in formulas for cross-shore sediment transport, for example the formulas presented in the article Sediment transport formulas for the coastal environment. This yields only the transport contribution of the oscillatory wave component. Other components of cross-shore sediment transport discussed above have to be considered as well.