Beach Cusp Formation
THE STRUCTURING OF MOVING FLUIDS
Beach Cusps and Fluid Salients
(The Structuring of Moving Fluids) [2]
Michael A. Gorycki, Ph.D. (Revised June 5, 2018)
(Please ignore the earlier version of this essay posted July 2008)
ABSTRACT
Popular thought proposes that the formation of a beach cusp series results from the operation of one or both of two theories: standing edge wave, and self-organization. As will be discussed here, each presents a problem in its implementation. A third theory, fluid salient formation, offers a simple, problem-free explanation for cusp formation. It is supported by the observations of other workers, demonstrable laboratory mechanisms, and it also provides a sound mechanism for the structuring of moving fluids responsible for a number of disparate natural and laboratory phenomena. The mechanism appears in a variety of guises, depending on the conditions of its development. In the case of beach cusp formation, the broad leading edge of a sheet or wedge of moving fluid is impeded by a planar substrate and, as it overrolls and thins, it extends laterally (axially). To relieve the resulting compression that occurs everywhere along the axis, a series of evenly spaced salients, with intervening zones of retarded-reversed flow, form along the beach face at right angles to that edge. It is these parabola-shaped salients, separated by zones of retarded-reversed flow, produced as a wave of a suitable energy ascends the beach face, that apparently are responsible for the initiation, maturation, and maintenance of a stable, evenly spaced beach cusp series.
The fluid salient mechanism also allows for a better understanding of the reported even spacing and stability of rip currents, cusp/rip relationships, near shore circulation cells and associated giant cusps. It also explains a number of observations and experimental results, both in the field and the laboratory. These phenomena will also be discussed in this essay.
INTRODUCTION.
In my first website, [1] http://www.geocities.ws/magsalients/ or www.lchr.org/a/36/rp/, and others ([3-8},
[1] Fluid Salients and the Structuring of Moving Fluids
[2] Fluid Salients and the Formation of Beach Cusps
[3] Fluid Salients and Spiral Galaxies and Other Vortices
[4] Fluid Salients and Round Phenomena
[5] Fluid Salients and Linear Structures
[6] Fluid Salients and Stream Meandering
[7] Fluid Salients and the Jet Stream
[8] Fluid Salients and Planar Structures
I discuss a variety of natural or laboratory phenomena that can be attributed to what I call the fluid salient mechanism. Early work (Gorycki, 1973a) was only concerned with the cause of beach cusp formation. The literature is replete with a variety of observations related to cusp phenomena. For the convenience of discussion, I have grouped these works under several headings. They are:
Beach Cusps
Beach Cusps and Rip Currents
Beach Cusps and Near Shore Circulation Cells
Beach Cusps and Standing Edge Waves
Beach Cusps and the Self-Organization Model
Beach Cusps
Palmer first described beach cusps in 1834, but they have been a mystery for as long as people have observed the action of waves on the beach face. The cause of their periodicity has provided a kind of holy grail that has been sought after by beach morphologists for almost two centuries. As Johnson (1919, p. 457) states, “Among the minor forms of the shore zone none has proven more puzzling than the cuspate deposits of beach material built by wave action along the foreshore. Sand, gravel, or coarse cobblestones are heaped together in rather uniformly spaced ridges which trend at right angles to the sea margin, tapering out to a point near the water's edge. These 'beach cusps' have attracted the attention of many students.”
Much has been written about field and laboratory observations concerning beach cusp series, and research has generated a number of theories. Often, however, papers either avoid discussion of a mechanism that clearly provides for the even spacing of beach cusps, or suggest that further analysis is necessary to solve the problem of their periodicity. Cusps have been considered the result of hydrodynamic instabilities in the wave swash. These “hydrodynamic instabilities”, or “perturbations”, are terms often used to explain the cause of several evenly spaced structural phenomena produced by moving fluids, such as stream meanders, the lobe and cleft structure of gravity currents, the Hele-Shaw cell, or Langmuir circulation. However, a consideration of the uniformity of spacing and appearance in a cusp series would argue strongly against hydrodynamic instabilities or perturbations and argue for an active structuring which I contend is the fluid salient mechanism. Also, a number of physical models used to describe the uniformity in cusp spacing are presented here. Additionally, there are a number of other natural phenomena ([1], [3]-[8]) that exhibit multiple periodicities also suggesting something other than “instability”.
Once, having observed that the leading edge of a small amount of water, sloshed across the bottom of a tilted tray (approximately 45 cm in length), forms a series of salients separated by nascent zones of retarded-reversed flow, it immediately became clear that this could be the mechanism for the formation of evenly spaced beach cusps. I reasoned that an incoming wave on the beach face appears to overroll due to the wave orbit. Gravity would serve to reduce the diameter of the overroll and cause it to extend along its axis. Since it is everywhere attempting to lengthen, it can accommodate this situation only by forming evenly spaced fluid salients that run up the beach face. These are separated by zones of retarded-reversed flow. The salients and intervening retarded zones somewhat resemble a sine wave. The fluid salients would form the beach cusps, and the zones of retarded flow would form the intervening bays. Other workers (Simpson, 1972) have noticed similar lobes (and clefts) in density currents, but have offered no explanation for their formation or their even spacing (see “Density Currents” [8]). The fluid salient mechanism, observable in the laboratory (see below), easily explains the natural phenomenon observed in the field. As will be discussed here, the two popular current theories for the formation of beach cusp series; standing edge wave, and self-organization, each seek to apply a known mechanism to reveal the cause of cusp periodicity. However, each presents problems in its application.
As an aside, it would be remiss not to suggest that waves that are capable of forming beach cusps, in their repetition and at their maximum extent, might be considered to be standing waves of brief duration; just as the density current shown in Fig. 3 (shown below and in Fig. 6 [8]) would be a standing wave if we briefly travel with it downslope at the same speed. A stable meandering stream on the stream plate [6] could also be considered a standing wave if its flow is constant, as are the standing waves seen in vortices (see my argument about standing waves in an aside in Water Vortices, (Axial Flow) (Overrolling) [3]). Overrolling is also in evidence for all these phenomena, a component of the fluid salient mechanism.
To enhance the demonstration of the fluid salient mechanism observed in the tilted tray, a large rocking trough coated with high-gloss enamel paint was constructed to present a similar structuring on a larger scale (Fig. 1).
Fig. 1. The center section of a wedge of tinted water is shown overriding a 30 cm wide portion of the floor of a large rocking trough 75 cm wide. Evenly spaced, parabolic water salients with intervening zones of retarded-reversed flow are caused by the forward velocity and speed of overrolling. The further the water’s edge moves downslope, the larger and fewer the salients become due to cannibalization. The salients here are approximately 6.4 cm wide (compare with Fig. 4, where they are 3-5 cm wide and more numerous). No influence of the sidewalls on the salients is apparent. Scale line is 10 cm long; the arrow shows the direction of water motion [1].
Another physical model employs a thin rubber cylinder rolled between two pieces of plate glass (Fig. 2);
Fig. 2. A physical model of salients and zones of retarded-reversed flow formed by rolling a thin, straight rubber cylinder squeezed between two glass plates. Note the similar size of the evenly spaced salients, which point in the direction of motion of the upper plate (up), and the zones of retarded-reversed flow (which point down). Pressure on the rolling cylinder causes it to thin and generate axial extension everywhere along its length. This extension is relieved by the simultaneous production of salients which overroll forward and out, and the zones of retarded-reversed flow which overroll forward and in. This is a static demonstration, so the speed of forward motion of the upper plate is not a consideration here. The flattened cylinder is about 1 mm wide. Motion of the upper glass plate is perpendicular to the length of the cylinder, but if the upper plate is moved at an acute angle to the perpendicular, a similar pattern is also produced. The former situation suggests cusp formation produced by waves moving directly toward the shoreline, the latter by waves approaching the shoreline at an angle but which are also capable of producing a cusp series. If the upper plate moves parallel to the cylinder’s length, a similar pattern develops, but it is the result of a different internal distortion of the cylinder and is akin to fluid structuring responsible for stream meandering (Gorycki 1973b) [1]. (See also Figs. 8-9, “Physical Models of Meandering“ [6]). In the essay on meandering streams [6], the zones of retarded-reversed flow are shown to be reverse whorls of current that complement the reverse helical flow seen at each bend in the stream (see Figs. 11a and 11b in [6]).
others include gravity currents (Fig. 3), and the structuring of sand by water salients (see Fig. 4 here) (Gorycki, 1973a).
Fig. 3. Evenly spaced fluid salients in a gravity current. It is comprised of a tinted saturated salt solution moving downslope on a tilted plate glass surface beneath a layer of standing fresh water. Fresh water enters the salt flow both along its upper surface and within the retarded-reversed zones (clefts or tunnels) between the salients. Intermixing of current and standing water is apparent as the current slows, thins, and becomes less dense and lighter in color as it moves to the right. The bar is 1 cm long [1]. A simpler version of this structuring involves merely sloshing fresh water back and forth in a flat-bottomed tray.
After further experimentation, and a review of published research and field studies, I still conclude that the fluid salient mechanism offers the most simple and viable explanation for the formation of beach cusps, (Gorycki, 1973a). Interestingly, this mechanism, as treated in an early website [1], also seems responsible for a disparate variety of seemingly unrelated natural and laboratory phenomena involving the structuring of moving fluids. By definition, we may describe fluid salients as evenly spaced, similar projections produced by the extension/compression of an overrolling or shearing fluid moving at a critical speed and have intervening zones of retarded-reversed flow in the ratio of 1:1. They can have various shapes and form under various conditions (see “Introduction” [8]), and [1-8].
This concept of a combination of simultaneous extension/compression in a body does not seem to be a well-known mechanism, but it is a necessary feature in many of the phenomena discussed in these essays.
As an aside, the rubber cylinder model can be replaced by a 20 cm length of 6 mm diameter soft Tygon® tubing. The bends in the rubber cylinder can be recreated, on a more visible scale, if the hand-held Tygon® tubing is manipulated to simulate the effects of the two pieces of plate glass as they slide past each other.
In my paper (Gorycki, 1973a), I suggest that as waves initially encounter the beach face, they can develop evenly spaced salients separated by zones of retarded-reversed flow. This structuring apparently is due to overrolling of the plunging wave orbit, with thinning and axial extension of a portion of its mass in a direction parallel to the shoreline as the wave begins to “feel bottom”. Since the lateral extension of every portion of the wave everywhere meets with identical extension along the same axis, the resultant extension/compression all along the wave (assisted by gravity) consequently leads to the production of evenly spaced fluid salients separated by zones of retarded-reversed flow.
It should be noted that waves are often parallel as the beach face is approached. It is accepted that this is due to refraction, and I suggest that this mechanism would aid as a precursor to the formation of fluid salients. As an aside, I would like to offer a physical model that may give a clearer picture of the waves’ parallel approach. It involves placing a 15 cm length of a single strand of copper wire, having a diameter of 250 microns or less, between two hard, flat, pieces of plywood that measure 2 x 10 x 20 cm. The wire may initially have a number of bends, but if the blocks are pressed and rubbed together at right angles to the wire’s length, the wire will roll parallel to the movement between the two blocks of wood, loose its bends, and quickly become remarkably straight.
I suggest the blocks of wood symbolize the nearshore zone and the forward motion of the water caused by wind action, with gravity being represented the pressure between the blocks. As the wave contacts the nearshore zone, gravity decreases the diameter of the overrolling wave orbit. Like the copper wire, the rolling orbit would tend to cause the wave to become straight and aligned parallel to the shore. It would also, by diminishing any irregularities, reciprocally smooth the foreshore and possibly promote the development of the fluid salient mechanism and the subsequent formation of beach cusps.
Salients formed by a single wave might then deposit some sand at evenly spaced locations on the beach face to initiate cusp formation. More importantly, each salient would then spread bilaterally from each cusp (due to overrolling), and join with salients on each side, spreading from the adjacent cusps.
The salients on the beach face would then return to the sea as backwash and thus erode the bays that develop between cusps. The sediment derived from erosion of the bays would then begin to form submarine deltas seaward of the bays. The evenly spaced, stabilizing deltas and associated backwash would then inhibit and slow portions of the next approaching wave of similar strength. Since this next wave is already experiencing axial extension/compression, it readily becomes segmented so that unretarded portions will approach the beach face as salients already centered on the cusps. Because of the action of a single wave, an evenly spaced series of incipient cusps, bays, and stabilizing deltas are initiated and accentuated by continued action of a sequence of waves, and the alignment of salients to cusp locations. Uniform salient spacing would be a more or less fixed function of wave train velocity, amplitude, sediment particle size, water and sediment density, viscosity, etc (Gorycki, 1973a). However, the active operation of the fluid salient mechanism would be responsible for the initial even spacing of the cusp series throughout, and it also provides for their maturity, stability, and maintenance.
Not all wave regimes produce cusps. Sufficient lateral extension/compression, based on overrolling and speed of the uprush, generates the distortion resulting in salient formation. In the rubber cylinder model sufficient distortion must be generated by pressure to produce the salients as the cylinder is rolled, thinned and extended, but, in this case, without any velocity requirements. Importantly, Evans (1938) provides further evidence of fluid salients for cusp and bay initiation by describing the formation of a series of beach cusps as the result of the action of a single wave. This abrupt structuring of the beach face is also suggested by the image depicted below (Fig. 4).
Fig. 4. One pass of fluid salients on the rocking trough floor. Strewn sand is swept both forward and laterally into parabolic streaks (Gorycki, 1973a). On the beach face, the apices of the parabolas represent salients transporting sand shoreward to aid in cusp formation. The elongate limbs of the parabolas, formed in the retarded-reversed portions of the salients by the lateral motion of the water would, in nature, provide excavated sand carried seaward by backwash to deposit as submarine deltas in the bays. The purpose of this model is to show that a single wave moving across a surface is capable of forming salients and producing an arrangement of evenly spaced sedimentary structures. Scale line represents 6 cm; arrow shows direction of water motion. Salients are approximately 3-5 cm wide (compare with Fig. 1). This pattern (on a solid surface) is also similar to Langmuir circulation produced upon open water, the result of fluid salient structuring of the wind [1]. This physical model conveniently provides an extending beach face over which the swash of a single wave can traverse an extended distance under the extended energy provided by the controlled tilt of the trough. This allows salients and cusps to form at various even spacings depending on the extent of the traverse and the energy expended. This photo appears to be the first to show fairly evenly spaced redistribution of sand (allowing for cannibalization of fluid salients (see Fig. 1) in the process of beach cusp formation in the laboratory.
It is obvious that a single, persistent wave train of similar waves approaching parallel to the shore would be most likely to generate and maintain a beach cusp series. A single wave train approaching at a large angle to the shore would not be conducive to development of the axial compression required over a length of beach face for the operation of the fluid salient mechanism, and would thus inhibit beach cusp formation. If two wave trains, approaching parallel to the shoreline each exhibited a different wavelength or amplitude, they would collectively produce waves of irregular height, strength, and periodicity that could adversely affect the formation or maintenance of a cusp series. However, even such a random sea might also induce and maintain a beach cusp series. Uniform wave height and strength is conducive, but may not be essential, to cusp development and maintenance. In my paper on the formation of beach cusps (Gorycki, 1973a), I suggested that if wave conditions change, the normal variability of wave size could be accommodated by established beach cusps in that smaller than average waves would have little effect on established cusp systems. In the case of larger than average size waves, particularly if associated with a rising tide, incoming waves would escape the retarding effects of the submarine deltas and tend to passively and unimpededly surge directly into the bays between the cusps (Gorycki, 1973a). Some of the backwash of water within a bay would then tend to split symmetrically toward the cusps on either side, possibly adding sediment to them and would also diminish incoming wave energy centered on the cusps. This concept is supported by Coco et al.,(2004), who found that the height and cross-shore extent of beach cusps are tidally modulated, having cusps waning during rising tide and waxing during falling tide. However the cause of cusp spacing is not discussed in that paper. A prolonged or significant change in the wave regime would eventually destroy a cusp series, including the submarine deltas, or at least alter cusp spacing. In addition, local variation in the slope of a beach face would vary the axial extension of the overrolling waves and thus vary the local spacing of the cusps.
As an aside, there is some confusion in the literature with regard to water traveling across the beach face (or foreshore portion) that is normally exposed to the action of the swash. Swash (or uprush) is water moving up the beach face as the result of breaking waves. Backrush is the broad seaward return of swash on the beach face. The swash zone is that portion of the beach face alternately covered by uprush and backrush. Undertow, as its name implies, should refer to the return flow of submerged backrush on the beach face. Backwash is the short-term, periodic return flow of water in the bays (often called runout) derived from each salient (often called wave run up) centered on the cusp. Rip currents, on the other hand, are relatively narrow, widely spaced, sustained, linear currents of return flow to the sea derived from longshore currents on the beach face and should not be confused with backrush.
It should also be mentioned that, usually, waves advancing on the beach face can be comprised of a number of smaller, irregular salients not having the fluid salient structuring. These, and the parent wave, are incapable of producing a beach cusp series because they do not repeat at the same locations (Fig. 5).
Fig. 5. Irregular wave on the beach face, composed of a number of smaller, disorganized salients. This is the usual situation where beach cusps are not being produced.
Beach Cusps and Rip Currents
Rip currents receive a great deal of attention because they are dangerous and powerful enough to drag a swimmer out to sea. Beyond the surf zone, they generally operate at the sea’s surface, but can affect bottom sediment if water depth is not too great. Depending on the beach environment, rip currents (sensu stricto) are of long duration, usually spaced far apart, and can extend for hundreds of yards out to sea. At times, they appear to be evenly spaced, but Turner et al., (2007) maintain that their spacing is, in fact, irregular. Some workers have applied the term “rip” to the closely spaced backwash associated with beach cusp series. As such, use of the term “rip” might be confusing, and so, they are discussed below.
Rip currents are generated in several ways on a beach and a general understanding of their formation is appropriate at this point. The simplest situation involves waves which are parallel to the shore and which move directly toward the beach. As powerful waves break on the beach face, a portion of their mass (the set-up) can be supported above the general level of the sea by subsequent waves breaking in the surf zone as well as by strong onshore winds; conditions often caused by the onshore portion of hurricanes as they approach land. The unstable condition of the set-up is relieved by the development of rip currents at the shoreline, which return water to the sea. These are initiated by the formation of longshore currents, which are usually divergent and bi-directional, travel parallel and close to the shoreline, and are above but close to sea level. Depending on local conditions at the shore, each longshore current attains a certain size and mass, and has zero velocity at its point of divergence, which is located somewhere between their associated rip current pair.
When a pair of adjacent longshore currents converge, one might assume that they have met at a point conducive to promote a mutual seaward return. Again, their strengths, location, and spacing are a function of the amount of water pushed ashore. The converging longshore currents then merge to form a feeder current that returns water directly seaward, extending beyond the breaker zone, to produce the rip current. I would propose that if there is a suggestion of even spacing to a group of rip currents, it might merely be the result of persistent wave conditions operating on a uniform beach face over a lengthy section of the shoreline.
Further seaward, a rip current will meet opposing wave motion (and the general mass of the sea) to form a diffuse rip head (Shepard and Inman, 1951). I suggest that sediment plumes seen in rip currents and heads may form offshore submarine deltas, possibly added to by material from near shore rip channels gouged out of the sea bottom. These deltas, in combination with the rip heads and rip currents will oppose (in their vicinities) those portions of subsequent incoming waves that will break early at some distance from the beach. It is these portions of early breaking waves that cause the remainder of each incoming wave to be broken (“salient-like”) into lengthy, unhindered portions that reach the beach face, add to the set-up, and strengthen the diverging longshore currents (possibly symmetrically) so that the established cycle may continue. In time, the channels and deltas formed will become more defined and stabilized with regard to their locations on the beach. It has been observed that once established, some rip currents “tend to recur always in the same place” (see Wikipedia, Rip Current).
It is commonly suggested that a swimmer can free himself from a rip current by moving a short distance parallel to the shore. He then will be in a salient-like portion of the incoming waves. As a consequence, he will then have a better chance of reaching the beach face safely.
If waves approach at an angle to the shore, the longshore currents can be unidirectional, feeding the rip currents from only one side. Another variation of rip current formation involves the presence of an offshore bar, created in the surf zone, which acts to contain and channel the waters of the longshore currents. Water, which has overridden the bar shoreward, returns to the sea as a rip current, traveling through a breach in the offshore bar. Again, the development further seaward of a submarine delta and the action of the sediment plume, rip channel, current, and head would act to impede incoming waves in those regions. Perhaps, the unhindered portions of incoming waves could also promote the production of the offshore bar resulting in the passive production of a “breach” in those regions where rip currents exist.
McKenzie (1958) noted that during high-energy wave conditions rip currents were few (widely spaced) but strong, while during mild wave conditions rips were weaker and more numerous (closely spaced). This suggests that during high-energy conditions the more dynamic longshore currents must produce stronger (but fewer) feeder currents able to rush seaward. Under mild wave conditions, less dynamic longshore currents produce weaker, more closely spaced feeder currents that are still able to overcome the onshore advance of the set-up.
Bagnold, 1940; Russell and McIntire, 1965; Dean and Maurmeyer, 1980; Masselink et al., 1997; and Bird 2000, consider the return flow (backwash) in the bays between a beach cusp series to be “mini-rip systems”. I would suggest that these periodic return flows between beach cusps not be called rip currents or “mini-rip systems” so as not to be confused with the true rip currents already described. The backwash currents (retarded-reversed flow) of a beach cusp series are both short-term and recurrent, and no longshore currents operate. The backwash currents of a beach cusp series are the product of the run up of fluid salients from each wave of a size capable of cusp maintenance (Gorycki, 1973a). This contention agrees with the observation of Masselink (1999) that cusp spacing is strongly related to the horizontal swash excursion (Fig. 4.). That is, waves of equal strength engender, develop, and maintain a cusp series of a certain spacing. This is suggested by the rocking trough experiments (Gorycki, 1973a) where fluid salients are initially small and become larger as they traverse the trough (see Figs. 1 and 4 here). This backwash (retarded-reversed flow) derives from the confluence and return to the sea of what is the combined halves of fluid salients that fan out from adjacent cusps. This return flow between the cusps, and also the offshore submarine deltas produced by them, will impede (and lower) portions of the next incoming wave and will help to evenly subdivide it so that those portions which do reach the beach face are relatively higher salients aligned with the cusps. Since the backwash of a cusp series lacks longshore currents, these so-called “mini-rip currents” are short lived, and do not have well-developed (very elongate) channels, aligned with associated submarine deltas. In addition, the spacing between cusps is not very much greater than the zones of retarded-reversed flow (the width of the bays) (Gorycki, 1973a) which lie between the cusps. That is, the spacings between the backwash (bays) is similar to that between cusps, whereas rip currents are usually spaced far apart, are of long duration, exhibit longshore currents, and can extend far out to sea.
Masselink and Pattiaratchi (1998) describe three circulation patterns for swash water under fair weather conditions. Their oscillatory swash motion, which has little or no effect on established cusps (Gorycki, 1973a), represents weak, transitory, passive wave conditions. Their horn convergent swash motion, involving horn overtopping and erosion, represents what I describe as larger than average waves approaching the beach face (Gorycki, 1973a) and which should also be considered passive. One might consider this second situation as also representing the effect of morphology on flow, but I agree with Masselink and Pattiaratchi that this type “of swash flow is not capable of maintaining and reinforcing existing beach cusp morphology” They consider their horn divergent swash motion, for waves that are intermediate in strength, to be at least capable of maintaining and reinforcing existing beach cusp morphology. However, they admit that the “relationship between swash circulation and beach cusp morphology is not thoroughly understood, nor are the implications of the different types of swash flow for beach cusp development.” They feel that horn divergent swash motion somehow “concentrates the wave uprush on the horns and channels the backwash through the embayments.” This situation suggests a dynamic, active role of the waves, portions of which (fluid salients) are directed toward the cusps and which consequently subdivide passively to return, under the effects of gravity, as backwash through the bays. I would suggest that the fluid salient mechanism not only maintains and reinforces beach cusp series, but also is responsible for Masselink and Pattiaratchi’s “morphology” (i.e., even spacing) (Gorycki, 1973a). When we consider the dynamics of the rubber cylinder physical model (Fig. 2) for fluid salients and beach cusps we note that the advancing salients, representing swash flow onto the developing cusps, overroll outward. These form the intervening zones of retarded flow that converge inward, where the “flows meet to form a concentrated backwash” and concludes the horn divergent swash motion of Masselink and Pattiaratchi (1998).
Beach Cusps and Near Shore Circulation Cells
Evenly spaced giant cusp “series” have been described (Shepard, 1952; 1963) and measured as being from 150 m to 1000 m apart, with most spaced between 500 m and 600 m (Dolan, 1971). In addition, the giant cusps project an average of 15 m to 25 m seaward from the embayments. For a modest situation involving a 25 m cusp length and a 500 m cusp spacing, the ratio for giant cusps is 1:20, which is still many times greater than the roughly 1:2 cusp length to cusp spacing ratio that is commonly observed for a beach cusp series.
Shepard and Inman, (1951) describe near shore circulation cells with rip currents (fed by longshore currents) being aligned with cusps. In addition, Komar’s (1971) study of near shore circulation cells also describes field observations showing that rip currents, again, are generally aligned with giant cusps. Unfortunately, both Shepard and Inman’s (1951), and Komar's (1971) suggestions that the action of numerous paired near shore circulation cells are responsible for the even spacing of giant cusp “series” do not provide an explanation for the production, morphology, or the uniform size or spacing of the cells. Whether cusps are present or not, I would suggest (as mentioned earlier) that large scale (widely spaced), and what may only appear to be evenly spaced rip currents (Turner et al., 2007), may be the result of persistent wave conditions operating on a uniform beach face over a lengthy section of the shoreline. Again, these rip currents should not be confused with the more closely spaced zones of retarded-reversed flow (backwash) associated with a typical beach cusp series. It is also important to differentiate between beach cusp series and sequences of giant cusps in that they each have a different mechanism of origin and differing morphologies.
As stated earlier, during beach cusp formation, the zones of retarded-reversed flow (return flow, or backwash in the bays), and submarine deltas are located midway between the cusps. This backwash and associated submarine deltas act to subdivide the mass of each subsequent incoming wave into salients centered on the cusps of a series. Some sediment could be added to the cusps by incoming salients, but erosion of the bays by backwash provided by the salients splitting at the cusps and combining at the bays would be more effective to excavate the bays. Backwash thus actively aids in the passive development of the cusps, vigorously forms submarine deltas, and forcefully subdivides incoming waves. However, its action is intermittent and dependent on the arrival of each new wave.
To understand the formation of a sequence of giant cusps aligned with rip currents, I would again emphasize that rip currents are persistent components of beach dynamics, supported by lengthy portions of each incoming wave recurring over an extensive section of beach face. Since large volumes of incoming water persistently support the longshore currents that then supply feeder currents for the rips, any transported sediment excavated from the beach face would be moved toward the rips where the converging currents could then deposit it to form cusps. The rip currents would then move seaward, possibly erode rip channels offshore, and deposit submarine deltas further out. The combination of persistent rips, heads, and associated submarine deltas would then serve to subdivide incoming waves to maintain the mechanism. The formation of seemingly evenly spaced giant cusp “series”, aligned with rip currents, could result from strong, uniform onshore winds producing a powerful, uniform wave regime. If this causes a voluminous set-up and lateral migration of water acting on a lengthy, uniform beach face, it would produce rip currents of any seemingly uniform great spacing Turner et al., (2007) that would subdivide each incoming wave, and thus obviate the concept of near shore circulation cells. The disparity in surf dynamics between giant cusps and beach cusp series suggests that they should not be confused or even compared. Giant cusps result from the active deposition of sediment due to the mutual approach of longshore currents. Beach cusp series form rather passively due mainly to the erosion of the bays by the mutual approach of split fluid salients from adjacent cusps.
Komar (1971), states that, “Both cusp-rip current relationships appear to occur in nature.”, but finds that rips aligned with cusps is the more likely situation (as already described here for giant cusps (Shepard and Inman, 1951)). Komar proposes that they should be called “rip cusps” as in his Fig. 1, which shows longshore currents converging toward the cusps, resulting in widely spaced cusps. He also adds, “It is possible that in certain circumstances, such as on a steeper beach face, the rips will hollow out embayments leaving cusps midway between the rips.” (see his Fig. 2). For these, Komar proposes that the cusps are at “positions of zero transport”, passively produced as the bays are excavated. He feels that this, his “envisioned cuspate shoreline” seems reasonable but that it does not conform to observations in wave basin experiments and that it is “uncertain whether such a development occurs on natural beaches.” In this case, Komar may simply be describing typical closely spaced beach cusps, with salients aligned with cusps (at “positions of zero transport”), and “rips” (backwash) aligned with bays. The point to be made here is that active beach cusp formation requires that salients be aligned with cusps, and that return flow (backwash) be between cusps. Cusps aligned with Komar’s so-called “rips” (return flow) suggests either that fluid salients are passively aligned with bays in a beach cusp series (because of larger than typical size waves), or that true rip currents (including longshore currents) are active and responsible for widely spaced giant cusp production. Importantly, Komar does mention that some cusps he describes in the field and in his laboratory studies should be classified as beach cusps, based on their more closely spaced “rip” currents and small cusp spacing, but still feels that since some are “associated with rip currents” they should be called “rip cusps”. I contend that if the term rip cusp is to stand, then longshore currents should also be present. Otherwise, the term backwash (zones of retarded-reversed flow) should be used, particularly if the “rip currents” also are of short duration, dependent on the mass of each incoming wave, that cusps be closely spaced (1:2 ratio of cusp length to cusp spacing), and that widely spaced giant cusps not be considered members of a series.
Additionally, Komar (1976) states, “When wave crests are parallel or nearly parallel to the shoreline, the near shore currents are dominated by a cell circulation with seaward-flowing rip currents. This cell circulation is produced by longshore variations in wave breaker heights, which in turn produce longshore variations in the wave set-up. The set-up will raise the water in the near shore to high levels shoreward from positions of large breakers than shoreward of smaller breakers. Water will then flow alongshore toward locations of small breakers and set-up, converging and turning seaward as a rip current. The rip currents transport sand offshore to beyond the breaker zone, hollowing out embayments in the process. A series of rip currents can thereby produce a series of embayments separated by cuspate projections.” In response, I contend that: Komar’s small breaker portions, his small setup, eroded embayments, rip currents, and sand transported off shore beyond the breaker zone, (possibly to form submarine deltas), are components of what I have been calling zones of retarded-reversed flow for beach cusp formation, but which are too widely spaced for a beach cusp series. Importantly, these “small breaker portions” are aligned with rip currents, which locally inhibit wave approach, thus promoting broad areas of “large breakers” between the rips, similar to salients involved in beach cusp formation but which, again, are too large for that designation. This structuring of incoming waves, which is determined by the location of rip currents rather than “longshore variations in wave breaker heights”, would be responsible for the so-called “cell circulation”. That is, incoming waves can be structured both by widely spaced rip currents to produce so-called “near shore circulation cells” and an associated giant cusp “series”, or by the closely spaced backwash of fluid salients associated with the production of a typical beach cusp series. Again, the widely spaced “near shore circulation cells” may simply be the result of persistent wave conditions operating on a uniform beach face over a lengthy section of the shoreline.
In Komar's (1971) discussion of his laboratory experiments he states, “In all cases, it is found that cusps develop in the lee of the rip currents.” In that paper, the first of his wave basin experiments produced four cusps aligned with rip currents. Subsurface channels, also aligned with the cusps, were excavated by the “rips”, and extended away from the shoreline. Unfortunately, these experimental runs had to be of short duration. I suggest that here wave action was strong and that erosion of the bays by fluid salients as “horn convergent swash” (Masselink and Pattiaratchi, 1998) formed cusps (either actively or passively), and the seaward return flow (aligned with the cusps) was strong enough to excavate the subsurface channels.
The second of Komar's wave basin experiments at another laboratory initially produced “envisioned” cusps with rip currents in between. These cusps were of short duration, followed by development of three permanent cusps aligned with the rip currents (his Fig. 3, see below). Here, a large central cusp formed shoreward of a strong central rip and with a smaller cusp on either side, each aligned with a weaker rip. Of interest at this point is that these later cusps extend below the surface into deeper water as ridges rather than as channels seen in the first experiments. Komar suggests this configuration is an equilibrium condition because once these permanent cusps formed, all cell circulation, sediment transport, and longshore and rip currents ceased to exist. Komar also suggests that this equilibrium may be the reason why rip currents (and also their accompanying longshore currents) are not necessarily present with cusps in nature.
Several problems are suggested by Komar's second laboratory experiments. He finds that he initially produced rip currents midway between the cusps. I would prefer to say that the early cusps were aligned with fluid salients, and that the rip currents were really intervening zones of retarded-reversed flow. The cusps are spaced only about 5 m apart, which seems to suggest insufficient space for longshore currents to develop to produce rip currents. Longshore currents would require a proportionately much greater distance between cusps of the size produced, and Komar himself admits that his experimental forms look more like the typical natural beach cusp series that he observed in the field. In the first phase of these experiments, there might initially have been fluid salient deposition of some cusp material on the beach face with zones of retarded-reversed flow returning “seaward” as backwash with subsequent erosion of bays, possibly with limited deposition of submarine deltas between the cusps. The temporary cusps that formed passively midway between the “rip currents” are what I described above (Gorycki, 1973a) for the production of a beach cusp series, but with the emphasis on the structuring of the incoming waves forming evenly spaced fluid salients aligned with the cusps. The return flow or backwash (the so-called “rips” between cusps) would then cause erosion and development of the embayments between cusps and would tend to structure later waves so that salients would again be aligned with the cusps. Since the submarine deltas are just developing, they would initially add little to the retarding effect of the backwash.
Komar shows the beach configuration after two hours of operation of his second experiment (his Fig. 3, again see below). I would suggest that the initial experimental configuration of the beach (possibly due to its original slope) might have produced a typical beach cusp series due to “horn divergent swash” (Masselink and Pattiaratchi, 1998) centered on the early cusps. However, a change in the beach face slope due to continued strong wave activity would cause “horn convergent swash” to act on the early cusps. This would cause them to be eroded to form the new bays, with their sediment transported down the beach face to deeper water to help form the new submarine ridges aligned with the “rip currents”. These ridges, in combination with backwash, would thus continue to align incoming fluid salients with the locations of the early, short-lived cusps. Continued wave action then resulted not only in subsequent erosion of the early cusps, but the formation of the bays shown (at the former cusp locations), with concomitant formation of later cusps (at the former locations of the bays and now aligned with the rips) both on the beach and continuing as the ridges (stippled) below the water surface.
As stated previously, Komar suggests an equilibrium condition prevailing in the experiments in which these permanent cusps persist over hours of continuous wave action, and that all cell circulation, sediment transport, and longshore and rip currents would then cease to exist. To explain this situation, as I suggested earlier, a normal variability of wave size could be accommodated by natural beach cusp systems (Gorycki, 1973a). The wave basin experiments described here seem to offer no chance for the expression of the effects of a spectrum of wave sizes as seen in nature. The point to be made here is that, as these second experiments progress, the shoreline becomes more complex. There is the development of cusps, bays, submarine deltas, submerged cusp extensions (submarine ridges) and other changes in the bottom topography and beach face, including the erosional transition from a steep to a more gentle beach face, which may also have promoted the erosion of the early cusps. Consequently, there is also a gradual lengthening of the shoreline (within the limits set by the experiment), and a greater distance from wave generator to eroding shoreline. All these changes would eventually have a weakening effect on the identical, mechanically produced waves in the system. As described earlier, smaller than average waves (oscillatory of Masselink and Pattiaratchi, 1998) would have little effect on established natural cusp systems. Therefore, the gradual sapping of uniform wave energy against a more complex shoreline, would result in Komar's “equilibrium condition”, and would then have an overlooked, even detrimental effect on the experiments and the resulting observations and conclusions.
As an aside, I would also like to point out that the production of only three cusps at the end of Komar's second wave basin experiments (Fig. 6)
Fig. 6. Komar’s Fig. 3 depicting a large central cusp and two smaller cusps. It shows horn convergent swash eroding the two bays and moving sediment toward the three cusps and seaward as submarine deltas. A wall effect has both structured and limited wave activity on both sides of the tank.
strongly suggests a wall effect on the dynamics of the water motion. That is, a larger central cusp associated with a pair of adjacent, weaker cusps indicates the basin's sidewalls may artificially aid in structuring and weakening nearby portions of the incoming waves. I would prefer to see four or more identical cusps being produced in any experimental situation. In the case of my tilting trough experiments, depending on length of traverse, anywhere from more than twenty-five smaller fluid salients to seven large (due to cannibalization) were routinely produced with each flow of water across the length of the trough's bottom (Figs. 4 and 1, respectively). There was no suggestion of a wall effect in the central region of the trough. Similarly, Faller's (1978) wind and wave tank employed to produce Langmuir circulation cells (Fig. 7)
Fig. 7. Faller’s Fig. 2 depicting Langmuir circulation. Left and right dashed lines indicate side walls of tank. Arrows indicate streaks from individual dye crystals. Shaded area indicates bands of convergence of dye at the bottom of the tank. The two light areas represent fluid salients. Note wall effect causing frictional on both sides of the system. Compare with Fig. 6, with one of the images inverted so that air and water flow are in the same direction.
experimentally generates a regular pattern of crossed waves created by a double wedge that oscillates vertically in the water at a resonant frequency. An exhaust fan at the far end of the tank also draws a light wind over the waves as they move along the length of a long, narrow tank. A transparent windshield, which lies close to the water surface, induces the moving air to act on the waves. In 20 seconds, the combination of waves and wind causes: 1) scattered floats to align into two lanes parallel to the tanks walls, and 2) two fluid salients of clear water, moving with the trend of the wind, to displace dye at the tank's bottom. Faller suggests that this combination of events indicates proof of the production of Langmuir circulations, with two pair of longitudinal roll vortices operating in the water of the tank. However, Langmuir's (1938) own careful observations provide evidence that suggests it is the formation of fluid salients in the invisible, moving wind above a water surface, which is the mechanism responsible for the appearance of the so-called Langmuir circulation cells seen in the water. These assumed cylindrical cells, only sketched by later workers showing alternating rotation, do not comply with Langmuir’s own detailed description of unexpectedly shallow, near surface circulation patterns, which he observed and puzzled over during his experiments on Lake George [1]. Describing “a series of right and left helical vortices in the water having horizontal axes parallel to the wind.”, in the waters of Lake George, Langmuir (1938) perceptively states that the “longitudinal and transverse velocities of the water in the vortices have their maximum values at the surface and gradually decrease to zero at the thermocline. Thus the vortices are unsymmetrical with respect to depth, being increasingly diffuse at greater depths.” Langmuir, speaking of the force of the wind, found that on “quiet days”, the streaks were 5 to 10 meters apart, and on others, the streaks had spacings of 15 to 25 m, but never held that the spacings were even. A similar Langmuir-like pattern appears in the fluid salients of moving water as it flows over a flat, solid substrate as seen above (Figs. 1, 2 and 4) or in seif dunes in desert regions. Also, passive ablative snow striations appear to also have a slightly variable spacing due to the inconstancy of the air moving along them (see Fig. 14, [8].
In Faller's experiment, I would suggest that there is also a wall effect (as described here for Komar's (1971) experimental study of rip currents (see Fig. 6)) and that the symmetry of the oscillating wedge is reflected in the paired water salients and lines of floats. I would prefer to see the production of four or more water salients with superimposed float alignments and waves created by a simple paddle to obviate any suggestion of the influence of a wall or wedge effect. Further, just the flow of air over the water in a wider tank, without the operation of a paddle, might prove to be an alternative more in keeping with nature.
Beach Cusps and Standing Edge Waves
Some researchers have suggested that standing edge waves are responsible for the initiation of beach cusps (see Beach Cusps at Wikipedia). Briefly stated, this rather elaborate theory holds that edge waves can become set up perpendicular to the shoreline. That is, if two edge waves approach each other from opposite directions, they produce a standing edge wave trapped near the shoreline. This results in an interaction between the regularly arriving incoming waves (parallel to the shore) and the standing edge waves. If the standing edge waves happen to possess a wave period twice that of the incoming wave they are considered subharmonic, and more common. The interaction of these subharmonic standing waves with the incoming wave is thought to produce a regularly spaced series of peaks and troughs along that wave and that these are responsible for the initiation and even spacing of a beach cusp series. That is, if the incoming wave collides with a peak, there is an increase in height and greater erosion resulting in the production of an embayment; if with a trough, a decrease in height and a lessened erosive ability to the wave resulting in a horn. However, this dynamic is contrary to commonly observed cusp formation where the heightened portions of the incoming wave are directed toward the cusps and the lowered portions are aligned with the bays (Gorycki, 1973a).
I also find it difficult to reconcile the geometry of the paired, mutually approaching edge waves of unexplained origin, and indeterminate yet equal strength, with a wave period twice that of the incoming wave, interacting with the incoming wave to produce peaks or troughs for the initiation of a uniform series of cusps. This would require a complicated, yet uniform structuring over what could be a considerable length of shoreline (Fig. 8),
Fig. 8. Beach cusps at El Segundo, Calif. Note their uniform aspect over an extended section of the beach face. From; Beaches, by Willard Bascom, Scientific American, August, 1960.
and capable of producing beach cusp series capable of ranging from the very large to the very small.
Also as mentioned earlier, Evans (1938) described the formation of a series of beach cusps as the result of the action of a single wave. This action is photographically depicted here in Fig. 4.
Another problem with the edge wave mechanism is that it accounts only for the establishment of a nascent beach cusp series, and not their continued growth. That is, as the cusps increase in size, the amplitude of the edge waves must decrease. Once the nascent cusp pattern has emerged, and the edge waves are considered to cease to perform, it would then require the prompt action of a second mechanism to cause cusp maturation; the self-organization model (Coco et al., 2003). By comparison, the fluid salient mechanism is a simpler, easily demonstrated, and more readily understood explanation for beach cusps. It is based on the production of incipient cusps centered on the fluid salient structuring of incoming waves. The cusp series is bolstered by return flow from the bays and action of submarine deltas, which would be responsible for the maturation, and maintenance, of a beach cusp series of any evenly spaced periodicity (based on wave size), along an extended portion of a beach face exhibiting uniform waves.
Komar (1971) describes the waves at the points of cusps (his Fig. 6) as being appreciably smaller than waves present in the embayments on either side throughout his experiments, and that they remain so even after equilibrium had been achieved. He suggests that this indicates the presence of edge waves that are instrumental in producing the cell circulation with the rip currents developing in the positions of the lowest breakers. I, again, would suggest that in these experiments, deposited sediment, continuing from the cusps and becoming ridges below the water surface, would effectively serve to diminish any wave activity locally approaching the cusps, leaving larger wave portions to enter, and erode, the bays. Masselink (1999) rules out the edge wave mechanism of beach cusp formation of Guza and Inman (1975) because he could find no relationship between cusp spacing and beach face gradient. In addition, Inman and Guza (1982) conclude, “that swash cusps are formed by the swash and backwash acting directly on the beach face” They rely on edge waves “only to provide small periodic perturbations on an originally uniform beach” but felt the edge waves “need not persist for the development of mature cusp morphology.” That is, as cusp development proceeds, the amplitudes of edge waves correspondingly decrease. Inman and Guza’s edge waves, which produce “small periodic perturbations on an originally uniform beach”, might simply be a description of fluid salients produced within each wave as it interacts with the beach face.
Interestingly, Masselink and Pattiaratchi (1998) state, “Wave and current spectra collected within 10 m from the shoreline between the developing cusp horn and embayment did not reveal significant amounts of sub-harmonic energy during or just prior to cusp formation.” Coco et al., (2003) also report that standing edge waves have not been conclusively reported in laboratory or field measurements during beach cusp formation. In addition, Werner and Fink (1993) find that since “subharmonic edge waves decay strongly within one incident wavelength of the shore, they are difficult to detect. Standing subharmonic edge waves have not been observed unambiguously in conjunction with beach cusp formation in the field”
In a number of theoretical discussions, edge waves have been treated and variously described as:
1) “In fluid dynamics, an edge wave is a surface gravity wave fixed by refraction against a rigid boundary, often a shoaling beach. Progressive edge waves travel along this boundary, varying sinusoidally along it and diminishing exponentially in the offshore direction.” <A>
2) “ocean waves traveling parallel to a shore with crests normal to the shoreline, and having heights that diminish rapidly seaward and are negligible at a distance of one wavelength offshore” (Beer, 1997, p.75-76),
3) “often difficult to visualize, are coastally trapped, i.e., their amplitude is maximal at the shoreline and decays rapidly offshore, produce on the beach beautiful run-up patterns (highest points reached by a wave on the beach” <B>.
4) “being produced perpendicular to normally incident waves and which can produce nodal and antinodal points which are responsible only for cusp initiation” <C>.
5) and “generated by nonlinear interactions with incident wind waves.” (Bowen and Guza, 1978).
Edge waves, because of their theoretical limitations and varied descriptions, some suggestive of fluid salients, and some accompanied only by supportive sketches, do not seem a viable explanation for beach cusp periodicity. This is a case of unsuccessfully trying to employ a known, but unsuitable mechanism to explain a known phenomenon. Instead, one has an easier time discovering a new mechanism (if possible) and selecting a known phenomenon which it can successfully explain.
As an aside, edge waves become a problem in the manufacture of metal foils. Waves form on the edges of the sheet as a foil is rolled because the edges suffer a more intense compression and thinning compared to the rest of the sheet. A more familiar version of edge waves can be produced by tearing a heavy (5-mil) sheet of polyethylene plastic as one would normally tear a sheet of paper. To start the tear, it is best to initiate it by first making a small cut with a scissors. Interestingly, the sheet not only exhibits a uniform series of large edge waves (primary salients) a few mm from the edge, but also a uniform series of smaller, secondary edge waves at the torn edge, (Fig. 9). Both sets are caused by tension, extension, and thinning along the tear as it is forming.
Fig. 9. Primary edge waves approximately 3 mm in wavelength near the torn edge of a 5-mil thick polyethylene plastic sheet. Secondary edge waves at the edge of the sheet are also visible and are about 1 mm apart. The waves form due to compensatory lateral compression after initial tensional thinning of the sheet as the edge is being torn. Evenly spaced salients similarly form in fluids as the result of extension/compression at the leading edge. The fluid salient mechanism often presents as a series of primary, secondary, tertiary and even higher orders of waves in other phenomena.
The point to be made here is that the edge waves produced by rolling aluminum foil or those seen here in Fig. 9 have nothing to do with standing edge wave theory, but are obviously produced by an (axial) extension and thinning of material near its edge or along the tear. The edge waves might be reduced or eliminated by drastically stretching the entire plastic film in a direction parallel to its edge. Just as has been described here for the formation of salients using the various physical models (Figs. 1-4), a uniformly repeated wave pattern (as with fluid salients on the beach face) is created by extension/compression of material along an axis.
As a consequence of this discussion, I suggest that edge waves, when discussed or actually photographically depicted as being responsible for cusp formation on the beach face (or as “mini-rips” Masselink and Pattiaratchi (1998)), are in fact fluid salients which are generated by axial extension of individual waves as the waves are interacting with the beach face (Gorycki, 1973a). Salients, moving toward the shore, provide the mechanism for the initiation, production, maintenance and, especially, the even spacing of beach cusps.
Beach Cusps and the Self-Organization Model
Werner and Fink (1993) describe a computer simulation of flow and sediment transport in the swash zone that couples local flow acceleration and alongshore surface gradient. They present a simulated cusped beach developed after 250 computer generated swash cycles. Their image is not dissimilar to my sediment-strewn rocking trough pattern after one swash cycle (see Fig. 4). My observation reiterates Evans’ (1938) account of the formation of cusps on a beach as the result of the action of a single wave. Although cusps usually take time to mature in the field it has been noted that they can, “form a regular pattern almost instantly and they all appear at the same time.” (see Beach Cusps at Wikipedia). Werner and Fink, however, find that current observational data cannot determine whether their self-organization model or the standing edge wave model is responsible for the formation of beach cusps. They simply imply a passive role to swash stating that, “On a cuspate beach, runup is deflected by horns toward bays and from there flows seaward as runout.”
Coco et al., (2003) suggest that the formation and development of beach cusp morphology is associated with waves that approach the shoreline normally (Longuet-Higgins and Parkin, 1962; Sallenger, 1979; Guza and Bowen, 1981). Others take a less severe view. Evans (1938), Kuenen (1948), Guilcher (1950), and Rudowski (1964) find that asymmetrical cusps can be formed by waves that approach the beach at an angle. To corroborate this, the rubber cylinder model, already described here, can produce salients even when the upper glass plate is moved at an acute angle to the axis of the cylinder (Fig. 2). This observation suggests that waves approaching other than strictly parallel to the shore are capable of producing cusp-generating salients.
Masselink (1999) finds the strong relationship between cusp spacing and horizontal swash excursion to provide some support for the self-organization model of beach cusp formation. His observation supports my early conclusion that, “the increase in salient size and spacing with distance traveled in the experimental situation might suggest that the further swash extends up the beach face, the greater the intercusp spacing”, (Gorycki, 1973a). Since Masselink (1999) also could not find any correlation between cusp spacing and the gradient of the beach face, it implies that for any reasonable slope, the larger the wave (and the greater the swash excursion), the greater the cusp spacing. I suggest that other parameters which could be examined would be water viscosity, mass (wave size) (Gorycki, 1973a), as well as wave type (surging, non-breaking) and velocity. The momentum of any particular incoming wave would also be critical to the formation and spacing of the fluid salient mechanism since a slowly moving wave (traversing a broad, gently sloping beach face) would only be conducive to the production of the irregular salients (Fig. 5), produced by wavelets, commonly seen on any beach face (Gorycki, 1973a). Coco et al., (2003) also support the concept of a linear relationship between cusp spacing and swash excursion (Masselink, 1999).
Continuing, Coco et al., (2003) find that their field observations of swash flow patterns and morphology changes are in accord with the self-organization hypothesis. “Three signatures of this positive feedback during the growth of beach cusps are that, swash flow increasingly is affected by morphology, swash flow increasingly is diverted from horns to bays, and, both deposition on horns and erosion in bays increase.” They agree with Masselink et al., (1997) and state: “The initial growth in relief of a beach cusp is caused by the effect of morphology on flow. Specifically, swash is diverted from incipient horns to incipient bays, leaving residual deposition on horns and leading to enhanced erosion in bays.” Again, here is an emphasis on a passive role played by swash (Werner and Fink (1993)). However, if swash is diverted from incipient horns, and relief is caused by the effect of morphology on flow, how do the incipient horns come into being in the first place, so that it can have an effect on the swash, and lead to erosion of the bays? A simpler explanation holds that fluid salients initially approach what will be cusp locations where they may deposit some sediment, and then they diverge to erode the bays. The process is repeated, with the salients in register with the cusps because of the spacing of the previous backwash with the bays and the newly formed deltas. As I have shown; a moving sheet of water (fluid) is structured into evenly spaced salients (Figs. 1 and 3), the rubber cylinder physical model reveals that process (Fig. 2), sediment on a planar surface forms periodicities by a structured moving fluid (Fig. 4), and salients can therefore form on a sandy beach face that is otherwise featureless.
Additionally, Coco et al., (2003) also find three signatures of self-organization in beach cusp formation: “First, time lags between swash front motions in beach cusp bays and horns increase with increasing relief, representing the effect of morphology on flow. Second, differential erosion between bays and horns initially increases with increasing time lag, representing the effect of flow on morphology change because positive feedback causes growth of beach cusps. Third, after initial growth, differential erosion decreases with increasing time lag, representing the onset of negative feedback that stabilizes beach cusps.”
I interpret the first signature of this quote as having fluid salients initiating cusps that require longer periods of swash time as they mature which means that flow affects morphology. This is in keeping with the second signature, that relief increases with time lags and that, again, flow effects morphology causing “residual deposition on horns and leading to enhanced erosion in bays.” (see Masselink et al., (1997) above). I suggest that this reinforces and condenses my description of cusp formation as being the result of the fluid salient mechanism (Gorycki, 1973a). However, both signatures of Coco et al., (2003) do not explain the cause or the even spacing of the cusps. As an aside, this active fluid salient structuring also holds for the formation of the curved bend of a stream in Einstein’s famous tea cup paper (see The Coriolis Effect [6]) where the reversing helicoidal flow of the stream is passively formed by the supposed preexisting banks of the stream.
The third signature, “negative feedback”, recalls Komar’s (1971) “equilibrium condition”. Earlier, I described this “condition” as resulting from the inability of available energy of salients to cause further maturation of the cusps as the result the evolution of a more complex (mature) shoreline under constant wave energy. That is, at this point, flow and morphology are in equilibrium, each effecting the other (including cusp location and spacing), but not effecting structural change in the cusps (this will be discussed later).
Also, according to the self-organization theory (see Beach Cusps at Wikipedia), any surface areas lower than the average elevation on a flat beach face will attract and accelerate water resulting in further erosion and the development of bays. Consequently, areas not reduced will have higher relief capable of slowing water motion, thus causing deposition of sediment and the production of horns. As cusps and bays begin to form from slight irregularities on a smooth beach face, they would initially have a random array. This array is reworked by the swash so that the “uniform spacing of cusps is caused by the communication of surface gradients along the beach by the smoothing of the beach surface as the beach tries to rearrange itself to reduce variations in the plane.” The problem with this explanation is, how is the formation of a uniform, mature array of cusps and bays achieved from slight (random?) irregularities on a smooth beach face caused by the smoothing of that beach surface? That is, would not the random array of initial cusps and bays be smoothed as the beach “tries to rearrange itself to reduce variations in the plane.”? Another problem with this explanation of cusp production is that it is a time-consuming process, This, while it has been noted by others in the field that cusps can form a regular pattern almost instantly, and they all appear at the same time (again, see Beach Cusps at Wikipedia).
One could simply say that the “smoothing” of the beach surface to produce a “uniform spacing of cusps” could be that of fluid salients acting on the beach face.
In addition, if cusp spacing is determined, even predicted, by the value of the swash excursion, the question still remains; how does a vigorous, yet “passive” swash excursion relate to the production and maturation of an incipient series of evenly spaced cusps and bays? How would that maturing series affect later swash, or, how would that swash affect the series? Would not an incipient cusp series be destroyed by a duplicate, yet unstructured, featureless swash? Also, if the extent of swash excursion is responsible for cusp spacing, and if wave energy is responsible for the swash, then we might want to look seaward for an active mechanism, contained in the wave and swash, of a fixed spacing and strength, that would be responsible for the spacing, fixed locations, and eventual maturation of a cusp series. Again, morphology would have to be affected by flow.
Masselink et al., (1997) also describe the destruction, during a small storm, of cusps on the lower portion of a microtidal beach face in Perth and the reappearance of a cusp series “under the influence of declining wave conditions”. Interestingly, they find the cusps redeveloped at the same locations and with the same dimensions as the subtle remnants of the cusp series on the upper beach face and feel this observation supports the self-organization model of Werner and Fink (1993). Here again, a passive role is attributed to the swash, because Masselink et al., (1997) suggest that the cusp re-formation was “controlled more by the antecedent morphology than the hydrodynamic conditions. This indicates that positive feedback between swash hydrodynamics and beach face morphology, necessary to form beach cusps, does not require a large variation in relief.” However, they do not explain what the positive feedback is which limits cusp spacing and locations to pre-set conditions. That is, if wave energy is declining after a storm, why does a nascent cusp series with a larger spacing not appear before the cusps that have the original, pre-storm spacing and locations? Should not larger waves have a larger swash excursion, and therefore at least temporarily initiate cusps with a greater intercusp spacing and incompatible locations? Again, why do cusps only form at the same locations with the same spacings?
Coco et al., (2003), working at Duck, North Carolina, also discuss the re-formation of beach cusps after a number of existing beach cusps were first leveled by a bulldozer or where a portion of the beach face was initially smoothed by a storm <D>.
They show three surveys of measured beach morphology wherein cusps also reappear but do not remark on why the cusps are in the same former locations with the original spacing.
Difficulties arise if we consider how cusps can re-form as at Perth or Duck after the cusps were leveled after a storm. Is it that the developing cusp series, which has a fixed spacing at fixed locations, can ignore the greater limits of the storm’s swash excursion as it lessens through time? Said differently, how is it possible that the swash excursion (a function of diminishing wave strength through time, as wave conditions decline to some former strength) can reestablish only the re-forming cusps with their former spacing and at their former locations? At first glance, the fluid salient mechanism does not offer an explanation either. Nevertheless, by looking “deeper” into the conditions on these beaches, I find that the observations on the re-formation of cusps at fixed locations by Masselink et al., (1997) at Perth, and Coco et al., (2003) at Duck, do strongly support the operation of the fluid salient mechanism. That is, if pre-storm submarine deltas persist at Perth and Duck, they would be found seaward of the pre-storm bay locations. Consequently, these deltas would inhibit uprush into these bay locations. Also, any active or passive backwash at these locations, as it returns to the sea, would generate intervening salients that would produce and enhance cusps on the beach face at their former pre-storm positions as wave strength lessens (Gorycki, 1973a), and the deltas again become operational.
Submarine deltas, offshore dynamics and the structuring of incoming waves tend to be ignored in many discussions of beach cusp formation. Storms tend to raise sea level temporarily, and this may allow submarine deltas to both escape destruction and limit their effectiveness until some semblance of former conditions returns. The deltas, produced when the early cusp series first formed (before the cusps were leveled by erosion at Perth and Duck) would act to inhibit wave action at those locations especially under “declining wave conditions” and would eventually reassert their structuring of incoming waves to slow those portions centered on the bays. They would thus register the intervening portions of the waves (salients) with the remnants of cusps, no matter how subtle, at locations on the upper portion of the beach face. This would occur even though the cusps on the lower portion of the beach face were completely destroyed and had no influence on the incoming swash. Salients would then approach the pre-storm cusp locations to continue the cycle, and reestablish and enhance the cusp series. The salients would diverge at the former locations of cusps, possibly adding sediment to them, join with swash from adjacent salients, return seaward on the planar beach face as zones of retarded-reversed (return) flow, and further excavate the bays. They could also add material to enhance the already present submarine deltas, and, as backwash, impede and subdivide the incoming wave action in those areas as the return flow in the bays between the re-forming cusps (Gorycki, 1973a). A simple check on this assumption would be to examine the offshore areas between the cusp locations for the presence of submarine deltas before and during cusp reappearance.
Conditions on the bulldozed beach face would return to their former state simply by the reestablishment of the prior fluid salient locations and structuring of the routine incoming waves. One might also assume that persistent submarine deltas may support the spacing of a cusp series long after wave strength may have somewhat increased or decreased from the optimum for that series.
It should be noted that the Perth observations and the Duck experiments do not include a comprehensive surveying of the submerged portion of the beach. Only storm smoothing or bulldozing of the beach face and surveying as cusps developed. The beach face is defined as the, “section of the beach normally exposed to the action of the wave uprush” (Glossary of Geology and Related Sciences, 1957, p. 27). In addition, only three cusps were bulldozed at Duck, and the cusps on either side, in addition to their probable array of submarine deltas, might also have served to help reform cusps, to a limited extent, by helping to register three salient locations with those of the missing cusps. If the self-organization model were in effect for the larger storm smoothed beach face, one might expect that reforming cusps may not necessarily be in register with their former locations along the beach. Nor would they necessarily have the same spacing (and positioning on the beach face slope) throughout their re-formational history unless persistent submarine deltas were present.
One could now suggest that the presence of submarine deltas might represent the effects of morphology on flow, but the deltas are the result of incipient erosion of the bays formed by the early erosive action of fluid salients. They would serve to aid in the structuring of incoming waves, but their origin and function would be essentially that of flow on morphology. It should also be mentioned that analysis of nine years of video imagery from Duck, North Carolina revealed that a “peculiar suggestion of hysteresis within the cusp spacing time series was observed and may suggest that existing theories of cusp formation need to be reformulated.” (again, see Beach Cusps at Wikipedia).
The fluid salient mechanism has much to commend it. This concept includes not only the cusp-initiating aspect of the evenly spaced incoming salients but the evenly spaced backwash and the enhancement and action of stabilizing submarine deltas, so obvious in structuring the next incoming wave. It seems simpler to conclude that cusp morphology (with an established, initially fixed spacing) becomes increasingly enhanced from incipient to mature (including to the equilibrium state) and that this is due to the effect of flow on morphology, rather than by a progressive rearrangement of a random array of beach face material caused by the effect of morphology on flow. Additionally, the fluid salient mechanism appears to function in a variety of natural phenomena [1], and is also displayed in a variety of physical models (see Figs. 1 to 4, 9 in [1]). In addition, new or incipient cusp series can form a regular pattern exhibiting their mature spacings. Their development would merely represent their maturation over time under the influence of the fluid salient structured waves.
In addition, the three signatures of positive feedback and those of self-organization in the theory of Coco et al., (2003) would appear to be merely a description of the process of the maturation of beach cusps. That is, they would not be the cause of cusp formation or their even spacing, but seem, rather, to be the effect, or consequence, of the operation of the fluid salient mechanism as cusps form and mature wherein the evenly spaced cusp morphology, increasingly affected by swash, is enhanced from nascent to mature, resulting in increased time lags. Thus, suitable waves approaching the beach face would already be predisposed to have the same salient spacing and locations as the cusps already produced by the salients and zones of retarded-reversed flow of the swash of previous waves. A further increase in time lag leading to the onset of “negative feedback” that stabilizes beach cusps would, again, relate back to Komar's (1971) “equilibrium condition”. This condition would be due to the development of steeper (cusp) and more elongate (bay) slopes, a gradual lengthening of the shoreline due to cusp and bay formation on the beach face, and the growth of submarine deltas. These would all serve to dissipate the energy of suitable, uniform incoming waves, so that the beach face becomes stabilized for a certain wave strength. The material eroded from the bays would be deposited as submarine deltas that, in combination with the enhanced shoreline morphology and backwash, would also serve to dissipate the energy of incoming waves at those positions. This would be the cause of a waning of wave energy, and not the “differential erosion between bays and horns”, the so-called “negative feedback” of Coco et al., (2003) on the evolved beach face, or Komar’s (1971) “equilibrium condition”. In addition, as stated here, the gradual waning of wave energy and water depth during the period after a storm would reactivate the effects of submarine deltas on incoming waves and returning backwash and thus aid in reestablishing and stabilizing a previous cusp series.
To reiterate, no other mechanism provides a clearer explanation as to the genesis of a beach cusp series, that is, their initial growth on a planar surface, and more importantly, their spacing. The concepts of standing edge waves, “positive feedback”, and the “effect of morphology on flow” are all obviated if the fluid salient mechanism is endorsed. That is, if the swash were already structured in the form of fluid salients for waves of a certain size and energy, the initiation of a beach cusp series and its consequent maturation and equilibrium condition would follow directly. The fluid salient mechanism holds that it is structured flow that affects morphology under which the three signatures of positive feedback are recognized. This would be the result of a structured flow regime acting to shape the beach face rather than the developing cusp morphology structuring the flow upon which it supposedly depends. I would conclude that morphology is controlled by flow and not flow by morphology. Here, swash, in the form of fluid salients, plays an active role as it approaches horns and diverges toward bays.
Further, Masselink, et al., (1997) state, “Theories of cusp formation must be able to predict the conditions under which beach cusps occur on natural beaches and account for their observed rhythmicity.” While the fluid salient mechanism may be considered by some to be merely intuitive or descriptive, I feel there are no obvious drawbacks to its existence and operation, and that is not the case for the standing edge wave and self-organization theories. The fluid salient mechanism initiates and then maintains spaced cusp formation through maturity and can eventually reach equilibrium status for waves of a suitable, routine strength. Since it is wave structuring that operates to shape the beach face, many of the observed phenomena relating to cusp management can be accommodated. In addition, the mechanism may permit the numerous contradictory statements of others relating to beach cusp formation to be reconciled (Russell and McIntire, 1965), (Gorycki, 1973a). It provides an explanation for cusp periodicity as well as redevelopment in situ after storm destruction by focusing our attention on the role played by submarine deltas, the effects of backwash, structuring, and measurable parameters of incoming waves. Look to these rather than any effects due to beach sediment, swash infiltration, erosion versus deposition, breached beach ridges, topography, weather, etc. I have also provided simple physical models that can be examined as evidence for the fluid salient mechanism as related to beach cusp formation. Variations of the mechanism can also be seen in other phenomena [1] and [3-8]. To disregard the physical models presented here (Figs. 1-4) as evidence of the fluid salient mechanism would then require an alternate explanation for their existence and operation. Future data collection and quantification to allow prediction of cusp formation and spacing should be directed to the observation, examination, and quantification of the fluid salient mechanism in the field or as laboratory phenomenon. It is simpler to conceive of a structured surf, aided by backwash and submarine deltas, progressively shaping the beach face to form a maturing cusp series with a uniform spacing fixed through time.
CONCLUSIONS
The operation of the fluid salient mechanism suggests that structuring of the incoming wave is the primary morphological determinant in beach cusp formation. Therefore, sediment only plays a passive role in the development of beach cusps. This is made obvious by the numerous references to the mechanism given here to explain the disparate observations of others. The role attributed to standing edge waves in simply initiating cusp periodicity seems tenuous at best. The self-organization model of Coco et al., (2003), and the effects of swash patterns on the morphological evolution of beach cusps (Masselink and Pattiaratchi, 1998), are detailed, descriptive, and supported by a variety of field data. However, they do not resolve the problem of the periodicity of cusp spacing except to relate it to swash excursion and the connotation it suggests with the fluid salient mechanism. The evidence given for the fluid salient mechanism makes its operation appear obvious. The importance of submarine deltas and zones of retarded-reversed flow as backwash returns to the sea should also be acknowledged as to the effects they have on salient spacing and cusp location and morphology. Observing incoming waves on the beach face using drone technology might reveal the structuring of swash with or without the presence of cusps. So would ultrasound echo surveys to determine the presence and location of submarine deltas and the role that they may play under various weather conditions.
The analysis by Coco et al., (1999) of field and laboratory data collected over the past 50 years suggests a possible link between both edge waves and swash-sediment feedback for beach cusp formation (also see Coco et al., (2008)). These theories can now be replaced by a single theory that not only provides for cusp series initiation, uniform spacing, development and eventual equilibrium, but for a number of other periodic structures in nature and in the laboratory (Gorycki, 1973a) [1]. Consequently, I feel that the formation of beach cusp series is the result of the operation of the fluid salient mechanism. The various details that arise by considering the effects of the fluid salient mechanism on the disparate variety of phenomena that exhibit periodicities should merely be considered commentary.
FINAL COMMENTS
My original paper, discussing the mechanism for beach cusp spacing (Gorycki, 1973a) was precipitated merely by noting the uniformly spaced scalloped leading edge of a sheet of water traversing the bottom of a small tray as the tray is tilted. I called this structuring sheetflood, defined by McGee (1897) who noted that “pure water”, flowing over an indestructible surface, tends to divide into parallel streams (Glossary of Geology and Related Sciences, 1957, p. 263). However, I now feel it is just another version of the fluid salient mechanism that can operate in a number of disparate phenomena and under a variety of guises [1]. Likewise, a paper describing the mechanism for stream meandering (Gorycki, 1973b) involved use of the term “hydraulic drag”. Again, I find that this term merely describes another version of the fluid salient mechanism. This interpretation of the meander mechanism is described in [6].
The present web site (please ignore its earlier version posted July 2008) is one of a series presenting further comments and information as described in a monograph in my original website [1].
This web site is the second of a series [2-8] presenting further comments on my original web site [1].
Questions, comments and criticism are welcomed and may be addressed to me at: gorycki@yahoo.com
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