Beach Cusps

Introduction

Beach cusps are one of the most commonly observed shoreline features of sandy and gravel beaches. They appear at the edge of the so-called swash zone, the part of the beach where waves have collapsed into a thin bore that runs up and down the shore (see Swash zone dynamics). Beach cusps are characterized by highs and lows in elevation with a spacing that, depending on beach slope and incoming wave field characteristics, can vary from less than one meter to 50-100 meters (Figure 1). The highs in elevation are usually called “beach cusp horns” and are essentially a ridge of sediment that protrudes towards the sea with a right angle to the shore. The lows, usually defined as “beach cusp embayments” (or simply bays), tend to be much smoother in elevation and are characterized by a gentler cross-shore slope. Beach cusps appear under a wide variety of hydrodynamic conditions (Holland 1998), from calm to almost stormy conditions, and sediments sizes from fine sand to gravel. Usually, beach cusps are associated to steeper beaches with coarse-grained sediments and normally incident waves. Overall, beach cusps are probably one of the most intriguing and striking patterns in the nearshore.

Figure 1: Beach cusps (left panel, courtesy of M. Olabarrieta; right panel courtesy of Amaia Alegria).

Flow and sediment characteristics

When normally approaching waves run up the beach and encounter a beach cusp field, the flow diverges at the horns and converges into the bays (Figure 2, left panel) where the water is concentrated before running down the beach (Figure 2 right panel). Narrow, concentrated and offshore-directed flows in the embayments are a typical feature of beach cusps. Other types of circulation patterns around beach cusps have also been reported and for a thorough discussion the reader should refer to Masselink and Pattiaratchi (1998). From a sedimentological perspective, beach cusps have been observed to develop on gravel and sandy beaches. When sand and gravel are present or when the sand distribution covers fine and coarse sands, the coarser sediment fractions concentrate on the horns of the beach cusps.

Figure 2: Snapshots of flow circulation over beach cusps. Courtesy of Ron Ovenden.

Theories of beach cusp formation

Probably because of their impressive regularity, beach cusps have attracted the attention of many scientists and a variety of mechanisms have been suggested since the beginning of the past century (see Coco et al., 1999 for details and references to theories that have not survived field evidence or that lack generality). Over more recent times, two theories have emerged to explain the conditions that lead to the development of beach cusps and to predict their geometry (primarily the spacing between beach cusp horns). The two contrasting hypotheses suggest that beach cusps are the result of either (Theory 1) self-organization through sediment-flow feedbacks (Werner and Fink, 1993) or (Theory 2) a pattern in the hydrodynamics due to the presence of standing edge waves (Guza and Inman, 1975).

Self-organization

Figure 3: Beach cusp spacing as a function of swash excursion for measurements (field and laboratory) and numerical simulations.

This approach is based on the idea that feedbacks between flow, sediment transport and morphological change can lead to the development of regular features like beach cusps. The approach has already been successfully applied to a variety of other coastal bedforms ranging from ripples to sand waves, from sandbars to shoreface-connected ridges (see Stability models, Wave ripples, Rhythmic shoreline features, Sand ridges in shelf seas). In the case of beach cusp formation, the positive feedbacks that lead to bedform growth are actually quite simple and almost intuitive: once an area of lower elevations starts to develop at some location, the surrounding flow will start to be directed towards it, accelerate and enhance erosion.

Small depressions in the beach surface attract swash flow from the surrounding higher areas. The resulting concentration of return flow can enhance sediment removal from these depressions, causing them to deepen. Conversely, reduced sediment removal or increased deposition on the adjacent higher areas promotes the development of cusp horns. These interactions between beach morphology, swash circulation and sediment transport constitute a positive feedback that can amplify an initially irregular beach surface (see the Appendix). Opposite behaviour can be observed on depositional areas (the incipient beach cusp horns) where a slowing flow will cause further deposition of sediment and flow diversion.

As cusp relief increases, the horn-to-bay swash circulation changes and the sediment transport gradients responsible for cusp growth diminish. Eventually, a quasi-equilibrium cusp morphology may develop in which net erosion and deposition are small (Coco et al., 2003). This morphology can persist as long as hydrodynamic conditions remain favourable. Changes in wave conditions, particularly the development of strong alongshore currents associated with oblique wave incidence, can disrupt the swash circulation and cause the cusp pattern to weaken or disappear (Masselink and Pattiaratchi, 1998; Coco et al., 2003). Cusp disappearance can also result from storm erosion or progressive infilling of the embayments under accretionary conditions (Almar et al., 2008).

These feedbacks have been tested through numerical models (Werner and Fink, 1993; Coco et al., 2000) and tested against field observations. (Coco et al., 1999). They predict that the beach cusp spacing is linearly dependent on the swash excursion, consistent with the observations shown in Figure 3. The model simulating beach cusp formation is a radical simplification of the complicated processes that operate in the swash zone, and many of such complications are bypassed by using a cellular type of approach. Nevertheless, the model can simulate also conditions that, in agreement with observations, are not conducive to beach cusp formation or that lead to a less regular pattern growth. The model also simulates conditions that lead to beach cusp disappearance (e.g., the presence of a strong longshore component in the flow). The model and the role of positive and negative feedbacks have also been studied through field observations (known as DozerDuck) where a number of cusps were destroyed by a bulldozer and conditions monitored while cusps reformed (Coco et al., 2003 and 2004).

Edge waves

Edge waves are gravity waves trapped near the shoreline by refraction, with amplitudes that decrease offshore (see Edge waves). They can propagate along the coast or form standing patterns through the superposition of oppositely travelling edge waves, with a periodic sequence of high and low amplitudes in the alongshore. These standing edge-wave patterns are assumed to become imprinted on the shoreline giving rise to beach cusps. The mechanism leading to the appearance of edge waves is related to a transfer of energy from reflected incident waves normally approaching a planar beach (Guza and Davis, 1974) and allows prediction of beach cusp spacing as a function of incident wave period, beach slope and edge wave mode. Each mode reflects a different cross-shore structure in amplitude variation and mode zero is assumed to be the most likely to be excited (Guza and Davis, 1974).

Figure 4. Measured beach cusp spacing versus the spacing predicted assuming the presence of subharmonic (left panel) or synchronous (right panel) standing edge waves. Data from Coco et al. (1999). Dashed blue line indicates perfect agreement. Notice the logarithmic scale.

Two types of standing edge waves are considered likely to develop: synchronous and subharmonic, with frequencies the same or half the frequency of the incident wave field, respectively. The alongshore structure of a subharmonic standing edge wave, combined with incident waves, presents a number of nodes (where wave run-up is minimized and where horns are likely to develop) and antinodes (where wave run-up is maximized and bays are likely to be scoured). Edge wave predictions of beach cusp spacing compare reasonably well with a large number of field and laboratory observations (Figure 4, Coco et al., 1999). Although observed beach cusp spacing and edge wave predictions are somehow consistent with the subharmonic predictions (discrepancies are larger for synchronous predictions, see Figure 4), no subharmonic or synchronous standing edge wave was directly measured in any of the data used in Figure 4. Edge wave presence and alongshore structure was inferred from incident wave conditions (sometimes collected in deep water). Guza and Bowen (1981) showed that large-amplitude beach cusps detune edge wave motions rendering their detection complicated. Aside from the difficulty in measuring the presence of subharmonic standing edge waves, theoretical analysis shows that friction is likely to suppress subharmonic motions (Guza and Bowen, 1976).

Conclusion

Both standing edge waves and self-organization processes can generate beach cusps. Although the generation of a standing edge wave pattern is theoretically possible at shore-normal wave incidence, field evidence of longshore standing edge waves on unbounded beaches is scarce and largely indirect. Most field measurements with longshore arrays show predominantly progressive edge waves (Huntley et al., 1981). Standing edge waves have been documented on a pocket beach, where they result from resonance with the beach width (Özkan-Haller et al., 2001). Working on a gravel beach, Wang and Sherman (2016) detected a synchronous standing edge wave that should have generated cusp spacing of 130 m, while the observed spacing was 7.6 m, close to the spacing of 7.6 m predicted by the self-organization mechanism. The available evidence therefore suggests that self-organization through swash–sediment feedbacks is a more general mechanism for beach cusp formation, whereas formation controlled by standing edge waves may occur under more specific conditions (Almar et al., 2008).

Appendix: Dean-Maurmeyer cusp formation model

A simple quantitative model consistent with the self-organization concept was given by Dean and Maurmeyer (1980). The model considers swash flow driven by the collapsing wave (initial velocity [math]U_0[/math]) and by gravity, ignoring friction and infiltration.

The scouring of incipient beach cusp embayments is due to the diversion of the wave uprush (swash excursion) from the incipient horns to the incipient embayments where consequently the wave downrush velocity is greater than the uprush and accelerating. The inverse (minimum downrush) holds for the incipient horns that therefore will accrete. The Dean-Maurmeyer model is based on the assumption: The cusp planform that will develop after an initial perturbation will correspond to the situation where the embayment wave downrush is maximum.

Fig. A1. Schematic representation of a beach cusp with wave uprush and downrush. The beach profile is concave-up at the horn and convex-up in the embayment. For simplicity, the same mean cross-shore slope [math]\beta_x[/math] is assumed at the horn and in the embayment. The horn–embayment relief decreases from [math]Z[/math] at the lower end of the swash to approximately zero at the uprush limit; the effective longshore slope felt by a swash particle is therefore taken as the mean value [math]\overline{\beta}_y \approx \beta_y/2[/math] over the swash excursion, where [math]\beta_y = 2Z/L_y[/math].

The proportion of uprush that is diverted from the incipient horns to the incipient embayments is maximum when the lateral swash diversion (uprush and downrush) equals half the embayment width, i.e. half the beach cusp spacing [math]L_y[/math], see Fig. A1. The maximum diversion occurs at time [math]2T[/math] when the downrush leaves the embayment. In the simplest model of frictionless particle motion over a slope, the time [math]T[/math] depends on the square root of the ratio of distance to slope, for both the cross-shore and longshore directions (Fig. A1). The maximum downrush in the embayment thus corresponds to the situation where the ratio lateral uprush diversion to the mean longshore slope [math]L_y / (2 \overline{\beta}_y) = L_y / \beta_y[/math] equals four times the ratio cross-shore uprush (swash excursion) to the mean cross-shore slope, [math]L_y /\beta_y = 4 L_x / \beta_x[/math]. The slope [math]\beta_y[/math] times the beach cusp spacing [math]L_y[/math] is about twice the elevation difference [math]Z[/math] between horn and valley. Substitution of [math]\beta_y=2 \, Z/L_y[/math] yields the formula [math]L_y^2 \approx 8 Z L_x / \beta_x [/math]. Dean and Maurmeyer report measurements of the beach cusp parameter [math]\epsilon[/math], which approximately equals the ratio of the horn height [math]Z[/math] to twice the uprush height [math]2 \beta_x L_x[/math]. Substitution of [math]\beta_x=Z/(2 \epsilon L_x)[/math] gives [math]L_y/L_x \approx 4 \sqrt{\epsilon}[/math]. The measurements of [math]\epsilon[/math] fall in the range 0.07-0.29, yielding [math]L_y/L_x \sim 1.06 – 2.15[/math]. The resulting range is consistent with the observed approximately linear relationship between beach cusp spacing and swash excursion shown in Figure 3.

The above model is of course an oversimplification of the beach cusp development process. It does not take into account many factors that can have a significant influence, such as frictional momentum dissipation, oblique and varying wave incidence angles, infragravity waves, etc. However, the basic assumption that beach cusps develop by maximizing the scouring downrush in an incipient embayment along the shoreline is highly plausible and yields a result that is close to what is observed in the field.

Related articles

Swash zone dynamics

Edge wave

Wave run-up

Rhythmic shoreline features

Stability models

Infragravity waves

References

The main authors of this article are Giovanni Coco and Job Dronkers Please note that others may also have edited the contents of this article. Citation: Giovanni Coco; Job Dronkers; (2026): Beach Cusps. Available from http://www.coastalwiki.org/wiki/Beach_Cusps [accessed on 8-11-2026] For other articles by this author see Category:Articles by Giovanni Coco For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Giovanni Coco For an overview of contributions by this author see Special:Contributions/Dronkers J