PAUL D. KOMAR Department of Oceanography, Oregon State University, Corvallis, Oregon 97331

Nearshore Cell Circulation and the Formation of Giant Cusps

ABSTRACT 150 to 1,000 m, with most between 500 and Large cusps are common along many shore- 600 m, and the cusps projected on the average lines, sometimes isolated but at times forming some 15 to 25 m seaward from the embay- a rhythmic series of such forms with a uniform ments. Such features are therefore considerably spacing. The role of the cell circulation, rip larger than "beach cusps" as that term is gener- currents and associated longshore currents, in ally applied. producing such cusps is examined. A series of Dolan (1971) suggested that the origin of laboratory wave basin experiments is presented these giant cusp series may somehow be similar in which the cell circulation of water modifies to the development of meanders in a river. an initially smooth and straight sand beach. In Bruun (1954) indicated that there may be a all cases, it is found that cusps develop in the lee correspondence between them and the similar of the rip currents so that a series of cusps is migrating bars found in rivers. Bruun also in- formed with the same spacing as the rip cur- dicated that there may be some connection with rents. These are compared with giant cusps and the occurrence of rip currents but he did not beach cusps observed on natural beaches. An pursue this suggestion. Shepard (1963, p. 195), equilibrium cusp development is found in the from his observations of the beach at La Jolla, experiments in which, having produced the California, indicated that deposition in the lee cusps, all cell circulation and other longshore of a rip current may produce a giant cusp. currents suddenly cease to exist; the rip cur- The purpose of this study is to examine the rents disappear. It is demonstrated that the role of the cell circulation, rip currents and as- equilibrium state consists of a balance between sociated longshore currents, in producing such the forces that tend to drive a longshore current series of large cusps along the shore. In addi- from an oblique wave approach to the flanks of tion to the observations of Shepard at La Jolla, the cusps, and the forces that normally produce the possible significance of the cell circulation the cell circulation. Cusps, having been pro- is suggested by the longshore rhythmic nature duced by rip currents, can therefore be ob- of the rip currents whose spacings are similar in served on beaches although the rips are no range to the observed spacings of "sand longer present. waves." The currents of the cell circulation can be expected to produce a local longshore sand transport and thereby modify the shoreline into INTRODUCTION a series of cusps (Komar, 1969). Pertinent Sand beaches are seldom straight, but rather laboratory and field observations are presented. commonly contain crescentic seaward projec- tions, sometimes isolated but more often as a NEARSHORE CELL CIRCULATION rhythmic series of such forms with a fairly uni- Rip currents are commonly observed in the form spacing. These features have acquired the nearshore region forming plumes of turbulent descriptive names "sand waves" and "shoreline and often turbid water extending seaward from rhythms" (Bruun, 1954; Hom-ma and Sonu, the beach to beyond the breaker zone. The rip 1962; Bakker, 1968; Zenkovich, 1967; Dolan, currents are fed by a system of longshore cur- 1971). Shepard (1952; 1963, p. 195) classifies rents, illustrated in Figure 1 by the vectors such features as "giant cusps." The most recent within the surf zone. These increase in magni- study by Dolan (1971) contains particularly ex- tude from zero midway between the rips to a cellent examples of this feature from the North maximum where the water flow turns seaward Carolina coast. His measurements yielded spac- to enter the rips. Water removed from the surf ings between successive cusps ranging from zone by the rip currents is replenished by a slow

Geological Society of America Bulletin, v. 82, p. 2643-2650, 5 figs., September 1971 2643

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drift of water shoreward through the breaker line, and um is the bottom orbital velocity eval- zone. Such a current system consisting of two uated at the breaker zone (Table 1 contains a rip currents, associated longshore currents, and complete list of the notation). The rate of long- a drift through the breaker zone defines a near- shore sediment transport is expressed as an im- shore circulation cell (Shepard and Inman, mersed weight transport rate, If, which is 1951). The entire circulation pattern along a related to the more familiar volume transport length of beach consists of many individual rate, Sf, by cells. Although long recognized, it is only re-

cently, with the study of Bowen (1969) and (2) If=(ps — p)ga.'S£ Bowen and Inman (1969), that we have begun to understand the generation of this nearshore where ps and p are respectively the sand and circulation pattern. water densities and a' is the correction factor This paper is concerned with the sand trans- for pore-space and may be taken as 0.6.ty is the port produced by the currents of the cell circu- volume discharge rate of sediment in the long- lation system and the shoreline modifications shore direction on the beach and may be mea- this transport produces. To examine this, we sured in units of ft3/sec, cmVsec; If places the can apply the sand transport relationship devel- discharge rate on an immersed weight basis. oped by Inman and Bagnold (1963). Assuming Komar and Inman (1970) made simultaneous that the orbital motion of the waves provides measurements of the sand transport rate and the bottom stress to support and suspend the the wave and current parameters which demon- sediment so that any longshore current v can strate the validity of equation (1) with a value transport the sediment, they obtain K' — 0.28 for the dimensionless coefficient. As suggested by Komar (1969), equation (1) may be used in the context where the current lf= K' (D v in the relationship is the longshore current of the cell circulation to examine the sediment where E is the wave energy density, Cn is the transport due to these currents. In this applica- wave group velocity so that ECn is the wave tion, the current v in the relationship will be the energy flux (the subscript b indicates that it is vectors depicted in Figure 1. Let us consider evaluated at the breaker zone), ah is the angle what changes we might expect in an initially the breaking wave crest makes with the shore- straight shoreline. If we could theoretically pre-

' * '>. " \ *\ f "V ' RIP ,' / ' HEAD

' r

MASS TRANSPORT II II

LONGSHORE CURRENTS

: : --^'<'-:-.;:r.^:'-/ ^:vv^y;-."-':. BEACH ;;'•:•: .r;.V;vV;.:;-;v: •.->'. :y.v/^.-'-v

Figure 1. The nearshore cell circulation system of rip the vectors (modified from Shepard and Inman, 1951). currents and associated longshore currents indicated by

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diet the magnitudes of the cell current v, we TABLE 1. NOTATION could develop a mathematical model of the correction factor for sediment pore space (=0.6) shoreline modifications, using equation (1) as wave group velocity our basis. Unfortunately, we are at present una- wave energy ble to make such predictions. In addition, it is acceleration due to gravity apparent that the problem is somewhat more wave height complex in that once the shoreline is modified, still water depth the waves may strike the shore at an angle pro- total immersed weight longshore sand transport rate ducing a wave swash sand transport enhancing dimensionless proportionality coefficient that given by equation (1). Although we cannot longshore bulk volume sand transport rate give a sophisticated treatment of the shoreline maximum bottom wave orbital velocity changes, we can see the basic alterations that longshore current velocity would be expected under the currents of the offshore coordinate axis; positive onshore cell circulation. It is apparent from equation (1) longshore coordinate axis that the direction of the sediment transport will angle between the breaking wave crest and shoreline be that of the directions of the current vectors angle of beach face slope in Figure 1. The longshore currents will cause = H/(fT+ h) a general drift of sediment along the shore to- wave set-up in nearshore zone due to wave presence ward the rip currents, and because the transport density of sea water rate is governed principally by the magnitude density of sediment grains of the longshore current v, the transport will initially be greatest near the rip currents and decrease to zero at the center of the cell where the wave basin experiments. It is uncertain v = 0. Assuming that all this transported sedi- whether such a development occurs on natural ment is carried out the rip currents to deep beaches. water, it might be expected that cusps would develop at the cell centers midway between the LABORATORY WAVE BASIN rip currents where the transport is zero. The OBSERVATIONS initially straight shoreline would then be The controlled conditions of the laboratory modified to a series of cusps as depicted in Fig- wave basin provide the means to examine the ure 2, the rips occurring in the embayments. modifying effects of the nearshore circulation Although the above envisioned shoreline on the shoreline configuration. Experimental development seems reasonable, we shall see investigations have been conducted at two insti- that it does not conform to our observations in tutions for this purpose.

POSITIONS OF ZERO TRANSPORT

Figure 2. The envisioned cuspate shoreline pro- cell circulation. Cusps occupy the positions of zero sand duced by the sand transport under the currents of the transport and the rip currents occur in embayments.

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METERS UPPER SWASH LIMIT

Figure 3. The configuration of the granular coal of wave swash. The stippled areas denote the positions beach brought about by cell circulation after two hours and respective sizes of the three rip currents that pro- of operation in the wave tank. Contours are in inches duced the cusps. Table 2 gives the wave and beach char- below the still water level (0). The heavy contour repre- acteristics. sents the scarp notched into the beach by the upper limit

The initial observations were obtained in granular coal beach, smoothed to an initially 1968 in the 15.2- by 18.3-m wave basin of the straight shoreline, was used. Due to poorly Scripps Institution of Oceanography, La Jolla, defined beach limits it was not possible to con- California. The wave basin arrangement was trol the spacings of the rip currents. In the ex- exactly the same as that used by Bowen and periments, cusps first developed midway Inman (1969) in their study of rip currents, between the rip currents as envisioned above, except that a fine-grained quartz sand beach but they never grew very large and in a few replaced the solid, uniformly sloping beach. minutes entirely disappeared. They apparently The sand was initially smoothed to a flat, uni- were destroyed by the wave swash striking their formly sloping beach face. A resonant wave flanks at an angle producing a transport in the period of 3.24 sec produced four strong rip same direction as that caused by the cell circula- currents as predicted by Bowen and Inman tion. As in the runs at Scripps Institution of (1969). A cuspate shoreline quickly developed Oceanography, permanent cusps developed in under the cell circulation but with the cusps the lee of the rip currents, a larger cusp forming forming in the lee of the rip currents, not mid- in the lee of the stronger central rip current. way between the rips as envisioned above. Be- The shoreline configuration after two hours of low the water surface rip channels developed wave action in the first run is shown in Figure under each rip current so that the subsurface 3. We shall see later in the paper that this confi- contours were the reverse of the shoreline guration represents a steady-state equilibrium configuration. The two runs were continued for condition in which there are no longshore cur- 10 and 15 min, insufficient time for any equilib- rents and therefore no longshore sediment rium to be established. This series of experi- transport. The pertinent wave and beach data ments was discontinued due to the scheduling for this run are given in Table 2. of the wave basin for other studies. Additional pertinent experiments were con- TABLE 2. WAVE DATA FOR THE BEACH ducted in the large 16.2- by 58.0-m wave basin CONFIGURATION IN FIGURE 3 of the Hydraulics Research Station, Walling- ford, England. The original purpose of the ex- Wave period 1.46 sec periments was to investigate the effects of Nominal wave height 5.6 cm offshore topography on the shoreline configu- Water depth at generator 30.5 cm Specific gravity of granular coal 1.35 ration but it quickly became apparent that the Median grain size of crushed coal 0.8 mm cell circulation was playing the dominant role Initial beach slope 1:12 in producing the shoreline modifications. A

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Although additional wave basin studies may across the beach seaward of the cusps, similar to prove otherwise, we see that the laboratory ob- those observed in the basin experiments. servations to date do not conform with the In July 1970, the formation of such cusps was shoreline modifications envisioned earlier in observed on a small scale at Blue Stone Beach, the paper. Instead of obtaining permanent east of Crail, Fife, Scotland. A series of six rip cusps midway between the rip currents, the currents with an average spacing of 5.8 m was cusps formed in the lee of the rip currents. Ap- readily observable due to the presence of mi- parently not all the sand is carried out the rip nute bits of kelp and sea grass which collected currents to deep water as we had assumed. In- in the rips (Fig. 4). Any material placed in the stead, a back eddy forms shoreward of the rip swash zone slowly migrated alongshore to the which permits deposition of sand and the for- rip currents. At the time of observation the mation of a lee-rip cusp. Much of the sand is waves were low, the maximum breaker heights indeed carried out the rip as supposed, forming being approximately 6 cm. There was a pro- a submerged delta under the head of the rip nounced grouping of the waves and whenever current. The critical factor in the shoreline a series of larger waves arrived at the beach, the development may be then how much sand is currents accelerated and the widths of the rip deposited in the cusp versus how much is car- currents narrowed due to nonlinear effects. At ried out the rip. It is possible that under certain times of low waves the longshore currents were circumstances eddies are not formed shoreward slow and the rip currents diffused outward and of the rips and that the shoreline development increased in width. In the lee of each rip current is more like that envisioned in Figure 2. To a definite cusp had formed (Fig. 4). The cusps completely resolve this question, considerably were composed of coarse shell debris and grit more study of the sand transport under the cell size rock fragments, noticeably coarser than the circulation currents is required, in the field as medium-grained sand of the remainder of the well as in the laboratory wave basin. beach. The positions of the rip currents had apparently remained stationary for some time FIELD OBSERVATIONS as the coarse cusp material was strung out down The limited field observations indicate that the beach face by the retreating tide. both patterns of cusps (sand waves) and related At several other locations, cusps have been rip currents may be found. The relationship observed while studying other aspects of beach between cusps and rip currents diagrammed in processes. These cusps had spacings which Figure 2 would best explain the presence of closely correspond to rip current spacings com- deep-water channels found offshore of the em- monly measured at those particular locations. bay ments between the sand waves (Hom-ma Many of these cusps presumably owe their and Sonu, 1962; Dolan, 1971). These channels development to the cell circulation and rip cur- gap the offshore bar just as rip channels com- rents. Since their formation was not actually monly do, and suggest the presence of rip cur- observed, this development cannot be certain rents seaward of the embay ments, cor- and provides no information as to the relation- responding to Figure 2. However, these studies ship between the rips and the cusps. These do not confirm the presence of rip currents. cusps are "solitary" in that there is no true em- The relationship between rips and cusps with bayment, but instead they are separated by a a cusp in the lee of each rip, suggested by the stretch of straight beach. This suggests that they wave basin experiments, is definitely observed were formed in the lee of rip currents. on natural beaches. A good example is the cusp A more systematic field study would proba- described by Shepard (1963, p. 195) at Scripps bly demonstrate that cusps are commonly as- Beach, La Jolla, California. At that location, the sociated with rip currents. Such a study is rip currents are positioned by the effects of off- required to resolve the question concerning the shore submarine canyons on wave refraction relationship between the cusps and the rip cur- and remain approximately stable in position rents. most of the time. The high tide summer shore- Although this paper is concerned primarily line configuration and relationship to the rip with the formation of "giant cusps" or "sand currents is very similar to that found in the waves," it is apparent that some beach cusps wave basin experiments. At intermediate and could develop in this manner from more closely low tides, the configuration is greatly modified spaced rip currents. Probably most geomor- due to the presence of rip channels running phologists would consider the cusps described

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Figure 4. Cusp of coarse material forming in the lee The rip current is dark due to the presence of minute of a rip current at Blue Stone Beach, Crail, Fife, Scotland. bits of kelp and sea grass. from Scotland, with an average spacing of only pletely disappeared. Since there were no 5.8 m, to classify as "beach cusps" — "one of longshore currents, there was no sediment a series of low mounds of beach material sepa- transport alongshore and the cusps remained rated by crescentic-shaped troughs spaced at stable over hours of continued wave action. A more or less regular intervals along the beach stable state of equilibrium had been achieved in face" (Coastal Engineering Research Center, spite of waves striking the flanks of the cusps at 1966). The cusps developed in the laboratory appreciable angles, a condition which normally experiments would also strictly classify as beach generates longshore currents and produces a cusps. Certainly all beach cusps are not gener- longshore transport of sand. The observed ated in this way so we might call those defi- equilibrium state is diagrammed in Figure 5. nitely associated with rip currents "rip cusps" The waves at the point of the cusp were ob- to distinguish them from the more general and served to be appreciably smaller than those pre- inclusive term "beach cusps." sent in the embayments to either side throughout the entire experiment, remaining EQUILIBRIUM CUSPATE so even after equilibrium had been achieved. CONFIGURATION This indicates the presence of edge waves The wave basin experiments at the Hydrau- which are instrumental in producing the cell lics Research Station have produced some inter- circulation as demonstrated by Bowen and In- esting and unexpected results concerning the man (1969), the rip currents developing in the equilibrium condition or configuration that positions of lowest breakers. The continued ex- may be achieved when the cell circulation istence of the wave height variations even after modifies the shoreline. Having produced the the cell circulation ceased, points to the con- shoreline configuration shown in Figure 3, with tinued presence of the driving mechanism for a large cusp in the lee of the strong rip current the cell circulation (Bowen, 1969). It would and smaller cusps by the weaker rips, all cell appear from this that the two mechanisms or circulation and other longshore currents sud- forces which drive (1) the nearshore cell circu- denly ceased to exist; the rip currents com- lation, and (2) longshore currents from an

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FORMER POSITION OF RIP CURRENT

WAVE CRESTS SMALL BREAKERS

LARGE BREAKERS

Figure 5. Diagram of the equilibrium condition ob- waves striking obliquely to the flanks of the cusp and the served in the wave basin experiment of Figure 3 with the largest breakers occurring in the embayments. oblique wave approach, in this particular equi- the wave set-up, the water level rise TJ above librium state exactly opposed and balanced one the still-water depth /; due to the presence of another so that once the cusps had developed the waves (see a complete discussion in Bowen sufficiently, both currents ceased to exist. The and others, 1968), plus the term for the long- significance of this equilibrium condition is that shore variation in the longshore directed excess it is possible for cusps to have been produced wave momentum flux (the radiation stress as by rip currents, though the circulation is no defined by Longuet-Higgins and Stewart, longer present or is present in a much weak- 1964). Together these terms ordinarily drive ened state at the time of cusp observation. the cell circulation from within the surf zone This equilibrium balance can be placed on a and produce the rip currents (Bowen, 1969). mathematical basis by balancing the forces The right side of equation (3) is the force that tending to drive the cell circulation, as given by provides a thrust in the longshore direction and Bowen (1969), against the force tending to normally drives a steady longshore current produce a longshore current under the oblique from an oblique wave approach (Longuet-Hig- wave approach (Longuet-Higgins, 1970). Us- gins, 1970). Equation (3) therefore balances ing the coordinates of Figure 5 and the notation the forces tending to produce the two current defined in Table 1, the basic equilibrium condi- systems and should define the observed equilib- tion is1 rium condition. Simple algebraic manipulation of equation (3) leads to the expression «T(iJ + A)^M — i , _. , ,aw dy (3) a ^° "' fly h) (4) sin a cos a. h) dx 16' Jtan ft sin a cos a The left side of the equation contains, respec- for the equilibrium condition. The ratio y = tively, the term for the longshore variation in H/( TJ + h) is found to be essentially constant 1A more complete derivation of the equilibrium condition (Bowen and others, 1968). We see from the can be obtained from the author. relationship that for equilibrium the larger the

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longshore variation in the wave height a/// dy, Waden Isle of Vlieland: Shore and Beach, v. 36, the greater must be the angle of wave ap- p. 4-14. proach to the cusp flanks—the greater the devel- Bowen, A. J., 1969, Rip currents, 1, theoretical opment of the projecting cusps. Additional investigations: Jour. Geophys. Res., v. 74, p. 5467-5478. laboratory experiments are being conducted in Bowen, A. J., and Inman, D. L., 1969, Rip currents, which the equilibrium equation (4) will be ex- 2, laboratory and field observations: Jour. Geo- amined directly. phys. Res., v. 74, p. 5479-5490. Bowen, A. J., Inman, D. L., and Simmons, V. P., CONCLUSIONS 1968, Wave "set-down" and "set-up": Jour. Rip currents and associated longshore cur- Geophys. Res., v. 73, no. 8, p. 2569-2577. rents may produce cusps on the shoreline, the Bruun, P., 1954, Migrating sand waves and sand laboratory wave basin experiments indicating humps, with special reference to investigations that the cusps will develop in the lee of the rips. carried out on the Danish North Sea Coast: 5th Conf. Coastal Engr. Proc., p. 269-295. It is possible that in certain circumstances, such Coastal Engineering Research Center, 1966, Shore as on a steeper beach face, the rips will hollow protection, planning and design: U. S. Army, out embayments leaving cusps midway be- Corps of Engineers, Tech. Rept. no. 4. tween the rips. Both cusp-rip current relation- Dolan, R., 1971, Coastal landforms: crescentic and ships appear to occur in nature. rhythmic: Geol. Soc. America Bull., v. 82, p. Rip currents can produce either isolated 177-180. cusps, such as observed by Shepard (1963, p. Hom-ma, M., and Sonu, C., 1962, Rhythmic pat- terns of longshore bars related to sediment char- 195), or can form a rhythmic series where the acteristics: 8th Conf. Coastal Engr. Proc., p. cusp spacing corresponds to the spacings of the 248-278. rip currents. Rip currents are known to slowly Inman, D. L., and Bagnold, R. A., 1963, Littoral migrate alongshore (Komar and Inman, 1970) processes, in Hill, M. N., ed., The sea, v. 3: The so that the cusps may likewise migrate. earth beneath the sea: New York, Interscience, The wave basin experiments indicate the p. 529-553. possibility of an equilibrium cuspate configura- Komar, P. D., 1969, The longshore transport of tion in which no longshore currents or rip cur- sand on beaches [Ph.D. thesis]: San Diego, rents exist. Therefore it is possible for cusps to Univ. California, 143 p. Komar, P. D., and Inman, D. L., 1970, Longshore have been produced by rip currents, though the sand transport on beaches: Jour. Geophys. Re- currents are no longer present at the time of search, v. 75, no. 30, p. 5914-5927. cusp observation. Longuet-Higgins, M. S., 1970, Longshore currents ACKNOWLEDGMENTS generated by obliquely incident sea waves: Jour. Geophys. Research, v. 75, no. 33, p. 6778- This study was initiated while at the Hydrau- 6801. lics Research Station, Wallingford, Berkshire, Longuet-Higgins, M. S., and Stewart, R. W., 1964, England, and was sponsored by a NATO post- Radiation stresses in water waves; a physical dis- doctoral fellowship. It was completed while cussion, with applications: Deep-Sea Research, supported by Sea Grant Project R/O-1 with v. 77, p. 529-562. Shepard, F. P., 1952, Revised nomenclature for Oregon State University. depositional coastal features: Am. Assoc. Pe- I am grateful to those at the Hydraulics Re- troleum Geologists Bull., v. 36, no. 10, p. 1902- search Station who took an interest in the study: 1912. R.C.H. Russell, Director, W. A. Price and D. 1963, Submarine geology, 2d ed.: New York, H. Willis, who conducted the wave basin ex- Harper and Row, 557 p. periments described, and C. L. Abernethy, Shepard, F. P., and Inman, D. L., 1951, Nearshore whose discussions were most helpful. W. H. circulation: 1st Conf. Coastal Engr. Proc., p. 50- Quinn and B. T. Malfait, Oregon State Univer- 59. Zenkovich, V. P., 1967, Processes of coastal devel- sity, and Robert Dolan, University of Virginia, opment: Edinburgh—London, Oliver & Boyd, critically read the manuscript. 738 p. REFERENCES CITED Bakker, W. T., 1968, A mathematical theory about sand waves and its applications on the Dutch MANUSCRIPT RECEIVED BY THE SOCIETY APRIL 5, 1971

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