Shoreface profile
Definition of Shoreface:
The shoreface is the nearshore zone of the inner continental shelf that is bounded landward by the mean lowwater line and that extends seaward to where the influence of wave action on crossshore 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 crossshore coastal depth profile of the shoreface.
This is the common definition for Shoreface profile, other definitions can be discussed in the article

This article focuses on the shoreface of sandy coasts. The shoreface of rocky shores is described in the article Rocky shore morphology and the shoreface of mud coasts in the article Coastal mud belt.
Contents
 1 Closure depth
 2 Morphodynamic feedback
 3 Bedforms and sediment sorting
 4 Beach classification; dissipative and reflective
 5 Difference between upper and lower shoreface
 6 Crossshore sediment transport
 7 Equilibrium shoreface profile (sandy beach)
 8 Analytical equilibrium models
 9 Related articles
 10 Further reading
 11 Table Symbols
 12 Appendix: Wave skewness
 13 References
Closure depth
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^{[1]}). 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 socalled outer closure depth [math]h_{out}[/math]; beyond this depth the seabed is hardly influenced by waves and waveinduced 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 Shallowwater wave theory). The outer closure depth [math]h_{out}[/math] has been empirically determined by Hallermeier (1981^{[2]}), yielding the formula (see also Closure depth)
[math]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)[/math]
where [math]H_s, T_s[/math] are the average significant wave height and period, [math]\sigma[/math] is the wave height standard deviation, [math]d_{50}[/math] the medium sediment grainsize and [math]g[/math] the gravitational acceleration. The second approximation in Eq. (1) follows from the assumption of a Rayleigh wave distribution (see Statistical description of wave parameters). Typical values of the outer closure depth are in the range 15 – 40 m.
The definition of the outer closure depth implies that wavedriven sediment transport across this boundary is very small on average. It does not rule out significant sediment transport across this boundary due to e.g. tidal currents, winddriven currents, ocean currents or internal waves. For example, the occurrence of shorefaceconnected ridges (see Sand ridges in shelf seas) on the lower shoreface can be theoretically related to sediment transport by strong winddriven currents. Other definitions of the outer boundary of the lower shoreface can also be found in the literature, for example the socalled Depth of Transport. This is the depth at which the total bed shear stress does not exceed the threshold of sediment transport under extreme storms^{[3]}. The Depth of Transport is typically further seaward than the outer closure depth Eq. (1).
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 [math]h_{in}[/math] between upper and lower shoreface was also given by Hallermeier (1978^{[4]}),
[math]h_{in} \approx 2.28 H_{12h/y} 68.5 \Large\frac{ H_{12h/y}^2}{g T_{12h/y}^2}\normalsize , \qquad (2)[/math]
where [math]H_{12h/y}[/math] 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 [math]T_{12h/y}[/math] the associated wave period. Later refinements of this formula are discussed in the article Closure depth. Typical values of [math]h_{in}[/math] 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 dune foot. 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 ^{[5]}^{[6]}^{[7]} (Fig. 1). Another indicator of the transition between upper and lower shoreface is a break in the trend of the sediment grainsize with depth^{[8]}. 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 threedimensional patterns are ubiquitous (see Rhythmic shoreline features and Fig. 3).
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 shorenormal direction) the relevant seabed bathymetry is the crossshore shoreface profile.
The shoreface profile influences crossshore sediment transport, but the inverse is also true: changes of the shoreface profile result from gradients in the crossshore 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, whilst 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^{[10]}^{[11]}^{[12]}. An illustration is presented in Fig. 2, which shows the recovery after strong perturbation of a shoreface profile that is initially close to equilibrium.
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 nonuniform coasts is a vanishing divergence of the longterm average total sediment transport vector (crossshore 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^{[13]}^{[14]}. 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 quasicyclical 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 ^{[15]}. 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 waveinduced 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 armoring' (see Seabed armoring).
The largest bedforms on the shoreface, such as shorefaceconnected ridges (scale of several kilometers) and shoreoblique 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 socalled nearshore sandbars or breaker bars (scale also of the order of 100 meters, Fig. 3) arise from morphodynamic coupling with the wave breaking process, through the resulting flow structure at the breaker bar and interaction with crossshore and longshore currents ^{[16]}^{[17]}  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 rippleinduced nearbed flow circulation^{[18]}, vortex shedding at ripple crests^{[19]} and nonlinear rippleripple interactions^{[20]}. 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 crossshore sediment transport on the shoreface, see section #Onshore waveinduced sediment transport.
Beach classification; dissipative and reflective
The beach profile in the upper shoreface zone can be highly variable. The profile tends to steepen in periods of lowenergy waves (especially longperiod swell waves) and to flatten under highenergy 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. Coarsegrained beaches typically have steeper slopes than finegrained beaches^{[21]}. Gently sloping beaches dissipate almost completely the energy of incident waves, whereas steepsloping beaches tend to reflect (partly) incident waves. For distinguishing between reflective and dissipative beaches the Deanparameter [math]\Omega[/math] is often used ^{[22]}^{[23]}:
[math]\Omega = \Large \frac{H_b}{wT_p} \normalsize , \qquad(3)[/math]
where [math]H_b[/math] is the significant wave height at breaking, [math]T_p[/math] is the peak spectral wave period and [math]w[/math] is the mean sediment fall velocity.
Wright and Short (1984) ^{[28]} analysed beach profiles in Australia by evaluating the parameter [math]\Omega[/math], considering the local longterm average wave climate. They found that beach types could be classified as follows :
 reflective beaches correspond to [math]\Omega \lt 1 [/math];
 intermediate beaches correspond to [math]1\lt \Omega \lt 6 [/math];
 dissipative beaches correspond to [math]\Omega \gt 6 [/math].
The tidal range [math]H_{tide}[/math] 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 [math]\Omega[/math] and small tides.
Several field studies^{[24]}^{[25]}^{[26]} have been performed to relate beach profiles to different combinations of the Dean parameter [math]\Omega[/math] and the relative tidal range [math]RTR=H_{tide}/H_b[/math]. Distinction is made between different beach morphologies: reflective beaches with steep graveled slopes, reflective/intermediate beaches with subtidal bars, intermediate beaches with rip cells and lowtide terrace or lowtide bar, dissipative beaches without significant bar formations, dissipative beaches with multiple intertidal bars and dissipative beaches with tidal flats, see Fig. 4. The figure only gives a rough indication of beach morphologies, because the natural variability is not fully captured by the two dimensionless parameters [math]\Omega[/math] and [math]RTR[/math]. Other factors play a role, in particular the temporal variability in morphology and environmental conditions and the role of geology^{[24]}.
Another parameter that is often used for analysing the beach response to variations in the wave climate is the socalled surf similarity parameter that relates the wave steepness (wave height to wave length ratio) to the surf zone slope.
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. In contrast, wave energy dissipation on the lower shoreface is small and hardly affects the wave energy flux toward the coast, [math]F^{(x)} = c_g H_s^2 \cos \alpha[/math], which can therefore be considered constant (Green's law; see Table Symbols for the symbols). Assuming approximate validity of linear wave theory, [math]c_g \approx c \approx \sqrt{gh}[/math], the height of waves propagating to shallower waters increases as [math]H_s \propto h^{1/4}[/math]. While shoaling, these waves develop wave skewness and asymmetry on the lower shoreface (see Appendix). Wave skewness and asymmetry generate net sediment transport (see next paragraph) and therefore play an important role in shaping the crossshore profile.
Sediment transport rates on the lower shoreface are much smaller than on the upper shoreface, 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 dayyear, compared to decademillennium time scales on the lower shoreface^{[1]}^{[29]}. Besides, waves are not the only sediment transport agent on the lower shoreface; tidal currents and winddriven 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 crossshore salinity gradient, fine sediment can be trapped on the lower shoreface and even on the upper shoreface, as explained in the article Coastal mud belt. However, on exposed coasts, energetic waves will transport fine sediment from the upper to 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^{[29]}. 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^{[30]}.
Crossshore sediment transport
Direct observation of net crossshore sediment transport and its separate components in the field is difficult. This is particularly true for the surf zone, where the temporal and spatial variability of transport processes is very high. The hydrodynamic conditions for the installation and operation of measuring devices are harsh, especially in periods where sediment transport is most significant. Bottommounted pressure sensors are most often employed for the determination of wave characteristics. This can easily yield erroneous (underestimated) values if the nonlinearity of wave dynamics is not accounted for^{[31]}. Acoustic and optical sensors are sensitive to air bubbles. Their use for determining suspended sediment concentrations and grainsizes is hampered as breaking waves entrain large volumes of air^{[32]}. 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 crossshore sediment transport. Mathematical models are often used as interpretation tools.
Sediment transport on the shoreface depends on many processes. The focus is on processes relevant to crossshore 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.
A distinction is often made between bedload transport and suspended load transport. Bedload transport represents the motion of sediment particles that are dragged by the flow along the seabed, either by rolling or successive jumps (saltation) or by entrainment of a thin seabed top layer (socalled sheet flow layer, of typically 1030 grains thick). Suspended load represents the transport of sediment particles that have been brought into suspension by turbulent fluid motions. An important difference between the two transport modes is the phase between the load of the transported sediment and the instantaneous flow velocity, especially in the case of waveinduced flow. Bedload responds almost instantaneously to changes in the wave orbital velocity; increase (decrease) of velocity implies immediate increase (decrease) of bedload. This implies that an onshore (offshore ) flow asymmetry (socalled velocity skewness) with higher peak velocity in onshore (offshore) flow induces a net bedload transport in onshore (offshore) direction. Suspended load generally displays a fast response to an increase in flow velocity; sediment particles can be lifted quickly from the seabed and brought into suspension when the flow velocity increases. However, unlike bedload, the response of the suspended sediment concentration to a decrease in flow velocity depends on the settling velocity of sediment particles, which induces a response delay that can be substantial compared to the wave period. Asymmetries in onshore (offshore) wave orbital flow are therefore less decisive for the net direction of suspended load than for bedload. In the surf zone where the waves break, suspended load transport dominates. Flume experiments suggest that bedload transport dominates in situations where unbroken waves approach the shore, e.g. on the lower shoreface and on steep, reflective beaches^{[33]}.
As explained below, onshore transport on the shoreface is mainly related to bedload transport and offshore transport to suspended load transport. As bedload transport involves mainly the coarse sediment fraction whilst suspended load transport involves mainly the fine sediment fraction, it follows that under average conditions coarser sediment is preferentially transported onshore and finer sediment offshore^{[34]}.
Onshore waveinduced sediment transport
Wave transformation in shallow water is a major factor for onshore sediment transport ^{[35]}. 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 ^{[36]}^{[37]}^{[38]}, see also Shallowwater wave theory. Wave skewness and asymmetry are primarily effective for inducing onshore bedload transport rather than suspended load transport, since bedload transport responds almost instantaneously to changes in nearbed velocities. 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, fine sands) 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^{[39]}^{[40]}^{[36]}^{[41]}, see Fig. 5. Although strongly skewed waves promote onshore ripple migration, the net sediment transport can be directed offshore. Onshore transport is stimulated by acceleration skewness^{[42]}. It creates a sedimentladen vortex on the stoss side of bed ripples, which travels toward the onshore direction as it is carried by the onshore ambient flow^{[43]}. Wave breaking that generates turbulent fluid motions and sediment resuspension under the wave crest also contributes to onshore transport, especially in the case of plunging waves^{[44]}. But at the same time, wave breaking removes most of the wave skewness^{[45]}, 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 can provide a further contribution to onshore transport ^{[46]}, 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^{[47]}^{[48]} (see also Sediment transport formulas for the coastal environment). Due to the high sediment concentration in the streaming layer (and in spite of the small vertical extent of the layer and the low flow velocity), the contribution of wave driven nearbed streaming to onshore transport is significant, especially in the shoaling zone and surf zone prior the plunging point^{[49]}.
Offshore waveinduced sediment transport
Offshore transport is mainly due to socalled undertow: the compensating return current for the net onshore mass transport in the upper water column between wave trough and wave crest ^{[50]}^{[51]}^{[52]}. The offshore suspended sediment transport by the undertow current is particularly strong just landward of the location where waves are breaking (plunging waves) and where much sediment is brought into suspension by wave energy dissipation and by the longshore current^{[53]}^{[54]}. Although the undertow velocity is smaller than the amplitude of the wave orbital velocity, it produces substantial offshore transport across the shoreface, in particular in the surf zone, where it often dominates the onshore transport related to waveasymmetry effects. The undertow current is strongest near the bottom in the surf zone, whereas it peaks near the surface on the lower shoreface. On the lower shoreface it is largely offset by onshoredirected Stokes drift and therefore less effective for offshore transport^{[55]}^{[56]}.
Bound infragravity waves may also contribute to net offshore sediment transport^{[57]}, 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^{[58]}. However, small transport contributions cannot be ignored because the large transport components related to wave skewness / asymmetry and undertow largely cancel each other out^{[59]}. For this reason, the downslope transport component must also be considered, see next section.
Downslope sediment transport
Several mechanisms can induce downslope sand transport. We do not consider here downslope turbidity currents or mud avalanching that may occur at the outer edge of deltas loaded with large amounts of fluvial sediment.
One mode of downslope transport consists of sand particles rolling down the shoreface slope. However, modeling suggests that this does not contribute significantly to downslope transport for fine to medium sandy beaches^{[1]}.
A second transport mode is socalled wavesupported gravity flow (WSGF). Observations on the Dutch coast show that wave stirring of the seabed under storm conditions can generate and sustain a highdensity layer of suspended fine sand (thickness 1520 cm, concentration exceeding 40 g/l) flowing down the shoreface slope (1/100  1/200) with a velocity of a few cm/s ^{[60]}. Modeling suggests that the highdensity sediment layer is kept in suspension by the combined action of waveinduced turbulent stresses and hindered settling. The wavesupported gravity flow was observed during a storm with nearbed wave orbital velocities of about 0.5 m/s, lasted for more than a day and far exceeded in volume other crossshore sand transport processes. However, the frequency of occurrence of WSGF events is not well known. Wavesupported gravity flows have also been observed on the shoreface of muddy coasts^{[61]}.
A third mechanism of downslope transport is related to the wave orbital excursion that moves sediment particles between zones of nearshore high bed shear stress and offshore low bed shear stress. Since resuspension after settling takes more time lower on the shoreface than higher on the shoreface, the downslope distance traveled by settling sediment particles is on average longer than the onshore distance. The average difference between offshore and onshore travel distance depends not only on the orbital excursion and shoreface slope, but also on the particle settling velocity. During the offshore excursion, sediment particles with a lower settling velocity settle later, thus lower, on the shoreface where the bed shear stresses are lower and the resuspension time longer. Fine sediment particles thus experience greater downslope transport than coarse particles. The slope effect requires a correction to the net sediment transport over a flat bottom, which is an increasing function of the product of bed slope [math]\tan \beta[/math] and wave orbital excursion, and a decreasing function of the fall velocity [math]w[/math]. Empirical formulas for suspended sediment transport assume in general that the slope effect can be represented by a correction factor to the instantaneous waveinduced transport components of the form^{[62]}
[math](1 \gamma_s \Large \frac{u}{w} \normalsize ) \, \tan \beta, \qquad (4)[/math]
where [math]u[/math] is the instantaneous wave orbital velocity, [math]w[/math] the sediment fall velocity and [math]\gamma_s[/math] an adjustable parameter depending, among other things, on the resuspension processes. This is consistent with laboratory experiments, in which an approximate correspondence was found between coastal profiles and an adjusted Dean parameter^{[63]}
[math]\Omega_{\beta} = \Large \frac{H_s \tan \beta}{wT_p} \normalsize . \qquad(5)[/math]
Experiments by Hattori and Kawamata (1980^{[64]}) with different profile slopes show that onshore transport dominates for [math]\Omega_{\beta} \lt 0.08[/math] and offshore transport dominates for [math]\Omega_{\beta} \gt 0.08[/math]. An equilibrium slope corresponds to (order of magnitude)
[math]\Omega_{\beta} \sim 0.08. \qquad(6)[/math]
Comparison with Eq. (3) shows that reflective beaches have a steep slope and dissipative beaches a gentle slope. The presence of the fall velocity [math]w[/math] in the denominator of the downslope correction term in Eq. (4) also implies that net waveinduced 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 waveinduced onshore transport (sum of all waveinduced onshore and offshore transport components including undertow) in situations close to equilibrium^{[65]}. This implies that the equilibrium shoreface slope [math]\tan \beta [/math] will be steeper (larger [math]\tan \beta [/math]) for a coarser seabed (large fall velocity [math]w[/math]) than for a finer sediment bed (small fall velocity [math]w[/math], smaller [math]\tan \beta [/math]). This is expressed by the correction factor (Eq. 4) and is in general agreement with field observations. An analysis of longterm shoreface profile observations at the Gold Coast (Australia) suggests that the bedslope 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 [math]\tan \beta_{eq}[/math] by the correction factor [math](1 – \tan \beta / \tan \beta_{eq})[/math] ^{[66]}, 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 Delft3D and XBEACH ^{[67]}, make use of approximate empirical formulations for several processes contributing to sediment transport and involve parameters that need to be tuned. Empirical formulas for waveinduced 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 sitespecific 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 certain (long) period to an equilibrium profile, as pointed out before. 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 ^{[68]} and the US Atlantic and Gulf coasts ^{[69]} Bruun (1954) and Dean (1977) derived an equilibrium profile (often called the Dean/Bruun profile) of the form (Fig. 6)
[math]h(x)=A \; x^{2/3}, \qquad(7)[/math]
where [math]h[/math] is the water depth [m], [math]x[/math] the crossshore distance [m] ([math]x=0[/math] at the shoreline) and [math]A[/math] a coefficient depending on the sediment fall velocity [math]w[/math] [m/s]. Dean (1987^{[70]}) empirically determined [math]A \approx 0.5 \; w^{0.44} [/math] [m^{1/3}]. Another estimate was given by Kriebel et al. (1991^{[71]}) [math]A \approx 2.25 \; (w^2/g)^{1/3}[/math] [m^{1/3}]. In Eq. (7) it is implicitly assumed that the fall velocity is uniform across the shoreface profile.
Dean ^{[69]} 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) ^{[72]},
[math]h(x)=B \; (1 – e^{kx}) , \qquad(8)[/math]
where [math]k[/math] is an empirical coefficient in the range [math]500 \lt k^{1} \lt 2000 [/math] [m] and where the coefficient [math]B[/math] is related to the beach slope [math]\tan \beta=Bk[/math] at the shoreline [math]x=0[/math].
The Bruun/Dean profile provides in general a reasonable description of observed longterm averaged profiles of micromesotidal sandy coasts^{[73]}. This holds in particular for the upper shoreface where breaking waves are the dominant driving agent for sediment transport. The 2/3 power law is generally less adequate for describing the profile of the lower shoreface. This is not surprising, as sediment transport here is mainly driven by nonbreaking waves. In a simplified ideal situation, based on realistic assumptions, a theoretical profile was derived through application of the principle of minimum energy dissipation^{[74]}^{[75]}. The theoretical profile also follows a power law (see Fig. 7):
[math]h(x) = \Big[ h_b^{1/n} + (h_c^{1/n}  h_b^{1/n}) \Large\frac{x  x_b}{x_c  x_b}\normalsize \Big]^n , \quad x_b \lt x \lt x_c , \qquad (9)[/math]
where [math]h_b=h(x_b)[/math] is the depth at the transition from upper to lower shoreface and [math]x_b[/math] the corresponding location, and [math]h_c=h(x_c)[/math] is the closure depth and [math]x_c[/math] the corresponding location. The exponent [math]n[/math] has a value in the range 0.3  0.4, depending on the formulation of the bottom friction^{[75]}. Inman et al. (1993) ^{[76]} found best fits with exponents ranging between 0.3 and 0.45 from an analysis of beach profiles for the coasts of south California, North Carolina and the Nile delta.
For gravel beaches other crossshore profiles are found, see Gravel Beaches.
Analytical equilibrium models
Because of the earlier mentioned difficulties to derive model estimates of the equilibrium shoreface profile, taking wave climate statistics fully into account, 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 shallowwater wave theory. An analytical expression for the equilibrium shoreface slope can be derived by requiring that the total waveintegrated crossshore sediment flux equals zero throughout the shoreface zone ^{[65]} ^{[1]}^{[77]}. 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 waveinduced sediment transport, which assumes zero phase difference between waveorbital velocity and sediment load, and which ignores many other processes described in the previous paragraphs. It has been shown^{[78]} 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^{[1]}, it is plausible that observed shoreface profiles generally correspond to transient profiles ^{[77]}.
Related articles
 Nearshore sandbars
 Bruun rule
 Active coastal zone
 Shoreline retreat and recovery
 Coastal Hydrodynamics And Transport Processes
 Modelling coastal hydrodynamics
 Coastal and marine sediments
 Gravel Beaches
Further reading
 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.
Table Symbols
Variable  Description  Variable  Description 

[math]c[/math]  wave propagation speed ([math][m/s][/math])  [math]c_g[/math]  wave group propagation speed ([math][m/s][/math]) 
[math]d_{50}[/math]  mean sediment grainsize ([math][m][/math])  [math]g[/math]  gravitational acceleration ([math]\approx 9.8 \; m/s^2[/math] ) 
[math]h[/math]  water depth ([math][m][/math])  [math]h_b[/math]  water depth at breaking location ([math][m][/math]) 
[math]H_b[/math]  significant wave height at breaking location ([math][m][/math])  [math]H_s[/math]  significant wave height ([math][m][/math]) 
[math]k[/math]  wave number ([math][m^{1}][/math])  [math]t[/math]  time ([math][s][/math]) 
[math]T_p[/math]  spectral peak wave period ([math][s][/math])  [math]u[/math]  velocity ([math][m/s][/math]) 
[math]w[/math]  sediment fall velocity ([math][m/s][/math])  [math]x[/math]  distance from shoreline (seaward >0, [math][m][/math]) 
[math]\alpha[/math]  wave incidence angle  [math]\tan \beta[/math]  shoreface slope 
[math]\theta[/math]  wave phase [math]\theta = \omega t  kx[/math]  [math]\rho[/math]  seawater density ([math][kg/m^3][/math]) 
[math]\omega[/math]  radial wave frequency ([math][s^{1}][/math])  [math]\Omega[/math]  Dean parameter ([math]= \Large \frac{H_b}{wT_p} \normalsize [/math]) 
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 sinusshaped 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
[math]u(\theta) = U_1 \cos(\theta) + U_2 \cos(2\theta) + V_2 \sin(2\theta) , \qquad (A1)[/math]
where [math]\theta = \omega t  kx[/math] is the wave phase, with [math]t=[/math]time, [math]x=[/math]the distance travelled in the wave propagation direction, [math]\omega =[/math]the wave angular frequency, [math]k=\omega / c=[/math]the wave number and [math]c=[/math]the wave propagation speed. The [math]U_2[/math] component represents the velocity skewness [math]S_u[/math] and the component [math]V_2[/math] represents the wave asymmetry or acceleration skewness [math]A_u[/math] (Fig. A1).
We define [math] \lt …\gt \equiv \Large\frac{1}{2 \pi}\normalsize \int_0^{2 \pi} … d \theta [/math] and assume [math]U_2,V_2 \lt \lt U_1 .[/math]
The velocity and acceleration skewness are defined as
[math]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)[/math]
(the Hilbert transform [math]\mathcal{H}[/math] applied to sinusoidal functions gives: [math]\mathcal{H}cos(x)=sin(x), \; \mathcal{H}sin(x)=cos(x), \; \mathcal{H}cos(2x)=sin(2x), \; \mathcal{H}sin(2x)=cos(2x)[/math], etc).
By analysing a large number of field observations of the wave orbital velocity on the shoreface, Ruessink et al. (2012^{[81]}) found empirical evidence that the velocity and acceleration skewness mainly depend on the Ursell number
[math]U_r = \Large\frac{3 g H}{8 h^2 \omega^2}\normalsize , \qquad (A3)[/math]
where [math]g \approx [/math]9.8 m s^{2} is the gravitational acceleration, [math]H[/math] the significant wave height and [math]h[/math] the local depth. They established the empirical expressions
[math] (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)[/math]
from which [math]S_u[/math] and [math]A_u[/math] can be solved. Besides the Ursell number, also other factors influencing the velocity and acceleration skewness have been identified, such as the offshore wave steepness, spectral band width and beach slope^{[82]}.
Abreu et al. (2010^{[37]}) derived a more general expression than Eq. (A1) for the wave orbital velocity, that includes velocity skewness and acceleration skewness,
[math]u(\theta)=U_1 \Large\frac{\sqrt{1r^2}}{1+\sqrt{1r^2}} \, \frac{r \sin{\phi}+(1+\sqrt{1r^2}) \sin(\theta)}{1  r \cos(\theta +\phi)}\normalsize . \qquad (A5)[/math]
The parameters [math]r[/math] and [math]\phi[/math] are related to the velocity skewness and acceleration skewness by
[math]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)[/math]
A predictive model for waveinduced crossshore sediment transport can be obtained by using the expression (A5) for the wave orbital velocity in formulas for crossshore 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 crossshore sediment transport discussed above have to be considered as well.
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} Stive, M.J.F. and De Vriend, H.J. 1995. Modelling shoreface profile evolution. Mar. Geol. 126: 235–248 Cite error: Invalid
<ref>
tag; name "S" defined multiple times with different content  ↑ Hallermeier, R.J. 1981. A profile zonation for seasonal sand beaches from wave climate. Coast. Eng. 4: 253–277
 ↑ Valiente, N.G., Masselink, G., Scott, T., Conley, D. and McCarroll, R.J. 2019. Role of waves and tides on depth of closure and potential for headland bypassing. Mar. Geol. 407: 60–75
 ↑ Hallermeier, R.J. 1978. Uses for a calculated limit depth to beach erosion. In: 16th Coastal Engineering Conference, American Society of Civil Engineers, pp. 1493 1512
 ↑ Wright, L.D., Chappel, J., Thom, B.G., Bradshow, M.P. and Cowell, P. 1979. Morphodynamics of reflective and dissipative beach and inshore systems: southeastern Australia. Mar.Geol. 32: 105140
 ↑ Niedoroda, A.W., Swift, D.J., Hopkins, T.S. and Ma, C.M. 1984. Shoreface morphodynamics on wavedominated coasts. Mar. Geol. 60: 331–354
 ↑ Larson, M., Kraus, N.C. and Wise, R.A. 1999. Equilibrium beach profiles under breaking and nonbreaking waves. Coast. Eng. 36: 59–85
 ↑ Aragones, L., Pagan, J.I., Lopez, I. and Serra, J.C. 2018. Depth of closure: New calculation method based on sediment data. International Journal of Sediment Research 33: 198207
 ↑ Patterson, D.C. and Nielsen, P. 2016. Depth, bed slope and wave climate dependence of long term average sand transport across the lower shoreface. Coastal Engineering 117: 113–125
 ↑ Wright, L.D., Short, A.D. and Green, M.O. 1985. Shortterm changes in the morphodynamic states of beaches and surf zones: an empirical predictive model. Marine Geology 62: 339—364
 ↑ Ludka, B. C., Guza, R. T. O’Reilly, W. C. and Yates M. L. 2015. Field evidence of beach profile evolution toward equilibrium, J. Geophys. Res. Oceans 120: 7574–7597
 ↑ Eichentopf, S., Van der Zanden, J., Caceres, I. and Alsina, J.M. 2019. Beach profile evolution towards equilibrium from varying initial morphologies. J. Mar. Sci. Eng. 7, 406
 ↑ Cornaglia, P. 1889. Delle Spiaggie. Accademia Nazionale dei Lincei, Atti. Cl. Sci. Fis., Mat. e Nat. Mem. 5: 284304
 ↑ Johnson, D.W. 1919. Shore processes and shoreline development. Prentice Hall, NY, 584 pp.
 ↑ Van Rijn, L.C. 1998. Principles of Coastal Morphology. Aqua Publications, The Netherlands (aqua publications.nl.
 ↑ Reniers, A.J.H.M., Thornton, E.B., Stanton, T.P. and Roelvink, J.A. 2004. Vertical flow structure during Sandy Duck: observations and modelling. Coast.Eng. 51: 237260
 ↑ Jacobsen, N.G. and Fredsoe, J. 2014. Formation and development of a breaker bar under regular waves. Part 2: Sediment transport and morphology. Coastal Eng. 88: 5568
 ↑ Sleath, J.F.A. 1984. Sea bed mechanics. Wiley, New York.
 ↑ Fredsøe, J., Andersen, K.H. and Sumer, M.B. 1999. Wave plus current over a ripplecovered bed. Coastal Eng. 38: 177221
 ↑ Marieu, V., Bonneton, P., Foster, D.L. and Ardhuin, F. 2008. Modeling of vortex ripple morphodynamics. J. Geophys. Res. 113, C09007, doi:10.1029/2007JC004659
 ↑ Bujan, N., Cox, R. and Masselink, R. 2019. From fine sand to boulders: Examining the relationship between beachface slope and sediment size. Marine Geology 417, 106012
 ↑ Gourlay, M. R. 1968. Beach and dune erosion tests. Rep. m935/m936, Delft Hydraul. Lab., Delft
 ↑ Dean, R.G. 1973. Heuristic models of sand transport in the surf zone. Proc. Conf. Eng. Dynamics in the Surf Zone, Sydney: 208214.
 ↑ ^{24.0} ^{24.1} ^{24.2} Scott, T., Masselink, G. and Russell, P. 2011. Morphodynamic characteristics and classification of beaches in England and Wales. Mar. Geol. 286: 1–20
 ↑ ^{25.0} ^{25.1} Short, A.D. and Jackson, D.W.T. 2013. Beach morphodynamics. In: Shroder, J.F. (Ed.), Treatise in Geomorphology, vol. 10. Academic Press, San Diego
 ↑ ^{26.0} ^{26.1} Jackson, D.W.T., Short, A.D., Loureiro, C. and Cooper, J.A.G. 2022. Beach morphodynamic classification using highresolution nearshore bathymetry and processbased wave modelling. Estuarine, Coastal and Shelf Science 268, 107812
 ↑ Li, Y., Zhang, C., Zhao, S., Qi, H., Cai, F. and Zheng, J. 2024. Equilibrium configurations of sandymuddy transitional beaches on South China coasts: Role of waves in formation of sandmud transition boundary. Coastal Engineering 187 (2024) 104401
 ↑ Wright, L.D. and Short, A.D. 1984. Morphodynamic variability of surf zones and beaches: a synthesis. Mar.Geol. 56: 93118
 ↑ ^{29.0} ^{29.1} Anthony, E.J. and Aagaard, T. 2020. The lower shoreface: morphodynamics and sediment connectivity with the upper shoreface and beach. Earth Sci. Rev. 210: 103334
 ↑ Short, A.D. 2013. Australia’s temperate carbonate coast: Sources, depositional environments and implications. In: Martini, I.P., Wanless, H.R. (Eds.), Sedimentary Coastal Zones from High to Low Latitudes: Similarities and Differences. London, Special Publications, Geological Society
 ↑ Martins, K., Bonneton, P., Mouragues, A. and Castelle, B. 2020. Nonhydrostatic, nonlinear processes in the surf zone. Journal of Geophysical Research: Oceans 125, e2019JC015521
 ↑ Aagaard, T. and Hughes, M.G. 2022. Sediment Transport. Chapter 8.6, Treatise on Geomorphology 2nd edition. Elsevier
 ↑ van der Zanden, J., van der A, D. A., Hurther, D., Cáceres, I., O'Donoghue, T., Hulscher, S. J. M. H. and Ribberink, J. S. 2017. Bedload and suspended load contributions to breaker bar morphodynamics. Coastal Engineering 129: 74–92
 ↑ Hassan, W.N. and Ribberink, J.S. 2005. Transport processes of uniform and mixed sands in oscillatory sheet flow. Coastal Engineering 52: 745– 770
 ↑ Elgar, S., Gallagher, E.L. and Guza, R.T. 2001.Nearshore sandbar migration. J.Geophys.Res. 106: 11,62311,727
 ↑ ^{36.0} ^{36.1} Nielsen, P. 1992. Coastal bottom boundary layers and sediment transport. In: Advanced Series on Ocean Engineering, IV. World Scientific.
 ↑ ^{37.0} ^{37.1} Abreu, T., Silva, P.A., Sancho, F. and Temperville, A. 2010. Analytical approximate wave form for asymmetric waves. Coastal Eng. 57: 656667
 ↑ Van der A, D.A., O’Donoghue, T., Davies, A.G. and Ribberink, J.S. 2011. Experimental study of the turbulent boundary layer in accelerationskewed oscillatory flow. J. of Fluid Mech. 684: 251283
 ↑ Vincent, C.E. and Green. M.O. 1990. Field measurements of the suspended sand concentration profiles and fluxes and of the resuspension coefficient over a rippled bed. J.Geophys.Res. 95: 1159111601
 ↑ Ribberink, J.S. and AlSalem, A.A. 1995. Sheet flow and suspension of sand in oscillatory boundary layer. Coastal Eng. 25: 205225
 ↑ Ruessink, B.G., Michallet, H., Abreu, T., Sancho, F., van der A, D.A., van der Werf, J.J. and Silva, P.A., 2011. Observations of velocities, sand concentrations, and fluxes under velocityasymmetric oscillatory flows. J. Geophys. Res. 116, C03004, doi:10.1029/2010JC006443
 ↑ Dubarbier, B., Castelle, B., Marieu, V. and Ruessink, G. 2015. Processbased modelling of crossshore sandbar behavior. Coastal Engineering 95: 35–50
 ↑ SalimiTarazouj, A., TianJian Hsu, TJ., Traykovski, P. and Chauchat, J. 2024. Investigating wave shape effect on sediment transport over migrating ripples using an Eulerian twophase model. Coastal Engineering 189, 104470
 ↑ Aagaard, T., Hughes, M.G. and Ruessink, B.G. 2018. Field observations of turbulence, sand suspension and crossshore transport under spilling and plunging breakers. Journal of Geophysical Research  Earth Surface 123: 2844–2862
 ↑ Guza, R.T. and Thornton, E.B. 1985. Velocity moments in the nearshore. J. Waterways Port Coastal Ocean Div. ASCE III: 235256
 ↑ LonguetHiggins, M.S. 1953. Mass transport in water waves. Royal Soc. London, Phil.Trans. 245A: 535581
 ↑ Trowbridge, J.H. and Madsen, O.S. 1984. Turbulent wave boundary layers: 2. Secondorder theory and mass transport. J.Geophys.Res. 89: 79998007
 ↑ Kranenburg, W. M., Ribberink, J. S., Uittenbogaard, R. E. and Hulscher, S. J. M. H. 2012. Net currents in the wave bottom boundary layer: on wave shape streaming and progressive wave streaming. J. Geophys. Res. 117(F03005), doi:10.1029/2011JF002070
 ↑ Zheng, P., Gumbira, G., Li, M., van der Zanden, J., van der A, D., van der Werf, J., Chen, X. and Tang, X. 2023. Development, calibration and validation of a phaseaveraged model for crossshore sediment transport and morphodynamics on a barred beach. Continental Shelf Research 258, 104989
 ↑ Stive, M.J.F. and Wind, H.G. 1986. Crossshore mean flow in the surf zone. Coastal Eng. 10: 325340
 ↑ Cox, D.T. and Kobayashi, N. 1998. Application of an undertow model to irregular waves on plane and barred beaches. J.Coast.Res. 14: 13141324
 ↑ Gallagher, E.L., Elgar, S. and Guza, R.T. 1998. Observations of sand bar evolution on a natural beach. J. Geophys. Res. 103: 3203–3215
 ↑ Thornton, E.B., Humiston, R.T. and Birkemeier, W. 1996. Bar/trough generation on a natural beach. J. Geophys. Res. 101: 219712110
 ↑ Reniers, A.J.H.M., Thornton, E.B., Stanton, T.P. and Roelvink, J.A. 2004. Vertical flow structure during Sandy Duck: observations and modelling. Coast.Eng. 51: 237260
 ↑ Lentz, S.J. and Fewings, M. 2012. The wind and wavedriven innershelf circulation. Annual Review of Marine Science 4: 317–341
 ↑ Larsen, B.E. and Fuhrman, D.R. 2023. Simulation of crossshore breaker bar development utilizing a stabilized twoequation turbulence model. Coastal Engineering 180, 104269
 ↑ Roelvink, J.A. and Stive, M.J.F. 1989. Bar generating crossshore flow mechanisms on a beach. J. Geophys. Res. 94: 47854800
 ↑ Aagaard, T., Greenwood, B. and Hughes, M. 2013. Sand transport on dissipative, intermediate and reflective beaches. Earth Sci. Rev. 124: 32–50
 ↑ Ruessink, B.G., Houwman, K.T. and Hoekstra, P. 1998. The systematic contribution of transporting mechanisms to the crossshore sediment transport in water depths of 3 to 9 m. Mar. Geol. 152: 295–324
 ↑ Flores, R. P., Rijnsburger, S., Meirelles, S., HornerDevine, A. R., Souza, A. J., Pietrzak, J. D., Henriquez, M. and Reniers, A. 2018. Wave Generation of GravityDriven Sediment Flows on a Predominantly Sandy Seabed. Geophysical Research Letters 45: 76347645
 ↑ Peng, Y., Yu, Q., Du, Z., Wang, L, Wang, Y and Gao, S. 2022, Gravitydriven sediment flows on the shallow sea floor of a muddy open coast. Marine Geology 445, 106759
 ↑ Bagnold, R.A. 1963. Mechanics of marine sedimentation. In: The sea, vol.3. Ed. M.N.Hill, WileyInterscience: 507528
 ↑ Bayle, P.M., Beuzen, T., Blenkinsopp, C.E., Baldock, T.E. and Turner, I.L. 2021. A new approach for scaling beach profile evolution and sediment transport rates in distorted laboratory models. Coastal Engineering 163, 103794
 ↑ Hattori, M. and Kawamata, R. 1980. Onshore–offshore transport and beach profile change. In: Proc. 17th ASCE Coastal Engineering Conference, 2, pp. 1175–1193
 ↑ ^{65.0} ^{65.1} Bowen, A.J. 1980. Simple models of nearshore sedimentation; beach profiles and longshore bars. In: The coastline of Canada, Ed. by S.B. McCann, Geological Survey of Canada, Ottawa, pp. 111
 ↑ Patterson, D.C. and Nielsen, P. 2016. Depth, bed slope and wave climate dependence of long term average sand transport across the lower shoreface. Coastal Engineering 117: 113–125
 ↑ https://oss.deltares.nl/web/xbeach/home
 ↑ Bruun, P. 1954. Coast erosion and the development of beach profiles. Beach Erosion Board, US Army Corps of Eng., Tech.Mem. 44: 179
 ↑ ^{69.0} ^{69.1} Dean, R.G. 1977. Equilibrium beach profiles: US Atlantic coast and Gulf coasts. Ocean Eng.Tech.Rep. 12, Univ. of Delaware, Newark, 45 pp.
 ↑ Dean, R. 1987. Coastal sediment processes: Toward coastal engineering solutions. Spec. Conf. Understanding Coastal Sediment Processes. N.C. Kraus, ed. ASCE, New York, pp. 124
 ↑ Kriebel, D.L., Kraus, N.C. and Larson, M. 1991. Engineering Methods for Predicting Beach Profile Response. Procs. Conf. Quantitative Approaches to Coastal Sediment Processes. N.C. Kraus, K.J. Gingerich, D.L. Kriebel eds. ASCE, New York, pp. 557571
 ↑ Bodge, K.R. 1992. Representing equilibrium beach profiles with an exponential expression. J. Coastal Research 8: 4755.
 ↑ Bernabeu, A.M., Medina, R. and Vidal, C. 2003. A morphological model of the beach profile integrating wave and tidal influences. Mar.Geol. 197: 95116
 ↑ Larson, M., Kraus, N. C. and Wise, R. A. 1999. Equilibrium beach profiles under breaking and nonbreaking waves. Coastal Engineering 36: 59–85
 ↑ ^{75.0} ^{75.1} Maldonado, S. 2020. Do beach profiles under nonbreaking waves minimize energy dissipation? Journal of Geophysical Research: Oceans 125, e2019JC015876
 ↑ Inman, D.L., Elwany, M.H. and Jenkins, S.A. 1993. Shorerise and barberm profiles on ocean beaches. J.Geophy.Res. 98: 1818118199
 ↑ ^{77.0} ^{77.1} Ortiz, A.C. and Ashton, A.D. 2016. Exploring shoreface dynamics and a mechanistic explanation for a morphodynamic depth of closure. J. Geophys. Res. Earth Surf 121: 442–464 doi:10.1002/2015JF003699
 ↑ Aagaard, T. and Huges, M.G. 2017. Equilibrium shoreface profiles: A sediment transport approach. Marine Geology 390: 321–330
 ↑ Van der A, D.A., O’Donoghue, T. and Ribberink, J.S. 2010. Measurements of sheet flow transport in accelerationskewed oscillatory flow and comparison with practical formulations. Coastal Engineering 57: 331–342
 ↑ Calantoni, J. and Puleo, J.A. 2006. Role of pressure gradients in sheet flow of coarse sediments under sawtooth waves. Journal of Geophysical Research 111, 2005JC002875
 ↑ Ruessink, B.G., Ramaekers, G. and van Rijn, L.C. 2012. On the parameterization of the freestream nonlinear wave orbital motion in nearshore morphodynamic models. Coastal Engineering 65: 56–63
 ↑ Rocha, M.V.L., Michallet, H. and Silva, P.A. 2017. Improving the parameterization of wave nonlinearities – The importance of wave steepness, spectral bandwidth and beach slope. Coastal Engineering 121: 77–89
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