A Large Drop in Atmospheric 14C/12C and Reduced Melting in the Younger Dryas, Documented with 230Th Ages of Corals

Paired carbon-14 (14C) and thorium-230(230Th) ages were determined on fossil corals from the Huon Peninsula, Papua New Guinea. The ages were used to calibrate part of the 14C time scale and to estimate rates of sea-level rise during the last deglaciation. An abrupt offset between the 14C and 230Th ages suggests that the atmospheric 14C/12C ratio dropped by 15 percent during the latter part of and after the Younger Dryas (YD). This prominent drop coincides with greatly reduced rates of sea-level rise. Reduction of melting because of cooler conditions during the YD may have caused an increase in the rate of ocean ventilation, which caused the atmospheric 14C/12C ratio to fall. The record of sea-level rise also shows that globally averaged rates of melting were relatively high at the beginning of the YD. Thus, these measurements satisfy one of the conditions required by the hypothesis that the diversion of meltwater from the Mississippi to the St. Lawrence River triggered the YD event.

1 mM ascorbic acid in all buffers. 15. Changes in the plasma membrane potential were monitored by measurement of changes in the intensity of fluorescence emission of the dye 3,3' dipropylthiadicarbocyanine iodide [DiS-C3- (5)], which is sensitive to membrane potential (12). The assay medium was 20 mM MES (pH 5.5), 370 mM sorbitol, 20 mM KCI, 1 mM CaCI2, 90 mM sucrose, and 105 protoplasts per milliliter. The dye DiS-C3- (5) was added to the assay medium to a final concentration of 2 FM, and the suspension was incubated at 250C for 1 hour (in the experiments monitoring circadian rhythmicity) or for 5 min (in the experiments monitoring light sensitivity). As an antioxidant, 0.2 mM DTT was included to protect the dye during the hour-long incubations. Excitation of the dye was at 620 nm and emission was measured at 668 nm. 16. To demonstrate that the failure of added K+ to depolarize unresponsive protoplasts did not result from a loss of membrane integrity, we incubated the protoplasts with valinomycin and 200 mM K+. Depolarization induced by the ionophore plus K+ demonstrated that the protoplasts were intact (12). 17. White light was provided with Sylvania GTE fluorescent bulbs (32 W, 41 00K). Photon fluence rate, as measured with an L1-170 Quantum Meter (U-Corp.) or an 818 series photodetector (Newport Corp.), was 20 gmol M-2 s-1. 18. After 21 hours the protoplasts began to lose viability, preventing observations of further cycles. 19. We provided red and far-red light pulses by filtering the output of a fiber-optic lamp (Reichart Scientific Instruments, Buffalo, NY) through a heat-reflecting filter (Corion #HR750-F1-L255) and appropriate interference filters. The red filter was Oriel Optics no. 53960 and the far-red filter was Oriel Optics no. 53985. Peak transmittance was at 660 nm for red and 725 nm for far-red, with half-bandwidths of 10 nm in each case. The fluence rate was 80 gmol M-2 s-1 for red pulses and 40 Smol M-2 s-1 for far-red pulses. 20. Three-minute irradiation with white light was performed as in (17). Similar irradiation with red light was as in (19) except that the fluence rate was 3 to 4 gmol m-2 s-1. Irradiation with high-intensity red light could not be continued beyond 60 s because the potential-sensitive dye was bleached by continued illumination, even in the absence of protoplasts. Blue light irradiation was as described previously (12) at a fluence rate of 3 to 4 Smol M-2 s-1. The blue filters that were used transmitted significantly above 730 nm, in the absorbance range of the far-red-absorbing form of phytochrome. However, the blue filters were always used in combination with the heat reflecting filters, and transmittance by the combined filters was <1% at all wavelengths from 600 to 800 nm (red and far-red range). 21. We thank the University of Connecticut Biotechnology Center for the use of a fluorescence spectrophotometer. Supported by National Science Paired carbon-14 (1' C) and thorium-230(230Th) ages were determined on fossil corals from the Huon Peninsula, Papua New Guinea. The ages were used to calibrate part of the 14C time scale and to estimate rates of sea-level rise during the last deglaciation. An abrupt offset between the 14C and 23'Th ages suggests that the atmospheric 14C/12C ratio dropped by 15 percent during the latter part of and after the Younger Dryas (YD). This prominent drop coincides with greatly reduced rates of sea-level rise. Reduction of melting because of cooler conditions during the YD may have caused an increase in the rate of ocean ventilation, which caused the atmospheric 14C/12C ratio to fall. The record of sea-level rise also shows that globally averaged rates of melting were relatively high at the beginning of the YD. Thus, these measurements satisfy one of the conditions required by the hypothesis that the diversion of meltwater from the Mississippi to the St. Lawrence River triggered the YD event. spheric 14C/'2C ratio, but detailed records were not possible because alpha-counting measurements of 234U and 230Th were not sufficiently precise. In recent years, the development of thermal ionization mass spectrometric (TIMS) methods for the measurement of 234U (6) and 23o-fh (7) resulted in large increases in analytical precision and sensitivity. By the application of TIMS measurements to fossil corals, Edwards and colleagues (7) obtained 230Ml ages that were more precise than 14C ages. This breakthrough made it possible to calibrate the 14C chronometer with combined 23/>Th age determinations and '4C/'2C analyses. Fairbanks (8) recovered the first sequence of appropriate age for the extension of the atmospheric 14C/12C record by underwater drilling off the coast of Barbados. This sequence showed large offsets between 14C and 230Th ages (9), which suggests that the atmospheric 14C/12C ratio was high about 20,000 years ago. The subsequent drop in the atmospheric 14C/12C ratio was originally attributed to an increase in the terrestrial magnetic-field strength. However, Stuiver and co-workers (10) have raised questions about the Barbados data because this record disagrees with 14C/'2C records from some varved lake sediments.
The same Barbados corals show a record of sea-level rise during the last deglaciation (8). Their depth and age relationships suggest that the rise was characterized by two meltwater pulses separated by a period of reduced melting (PRM). The timing of the PRM has implications for the cause of the Younger Dryas (YD), an interval of time during deglaciation when the climate in northern Europe returned to near-glacial conditions. However, Broecker (11) has questioned the record because the inferred meltwater pulses occurred at core breaks.
In this report, we present a new coral record derived from a single drill core from the Huon Peninsula, Papua New Guinea. This record has similarities to the Barbados record but also important differences, which have implications regarding the calibration of the "4C time scale, the cause of a large decrease in the atmospheric '4C/12C ratio, the rate of sea-level rise during deglaciation, and the cause of the YD. Most of the New Guinea samples are from a drill site 5 km southeast of the village of Sialum (12). The top of the core is 6 m above sea level, and the base is 46 m below sea level. When this record is corrected for tectonic uplift (13), the core represents material that grew while sea level was rising from 71 to 8 m below the present sea level. Several common shallow-water coral genera, mostly Porites sp., were analyzed. Additional, previously reported 14C ages showed that the core ranges in age from 7,000 to 11,000 l4C years old (12,14).
A key aspect of the procedure in both laboratories is the addition of a selective dissolution step, which removes potential contamination. Diagenetic processes can add significant amounts of carbon with high 14C/12C ratios to corals (17,20,21), but selective dissolution techniques can remove such natural contaminants (17). Selective dissolution was not used in the original Barbados '4C calibration (9) but was applied later (22). The 14C/'2C values for selectively dissolved samples from the latter data set are generally lower than the original ones (21 ), which suggests that the latter data are more reliable. Consequently fur-ther reference to the Barbados 14C data in this report is restricted to the selectively dissolved samples in (22).
We were also concerned about potential inaccuracies in 230Th ages. Corals with ages that are known independently, from annual   (7), ellipse on left from (9)]. Compared with these two examples, our 8234U errors are smaller, an important factor in the identification of material that may have exchanged small amounts of uranium. All points plot within the range of surface seawater, consistent with closed-system evolution and accurate ages. Possible glacial-interglacial variations in marine uranium isotopic composition are less than 2 per mil. Numbers adjacent to ellipses represent depths in the core (in meters below the surface at the drilling site). The 230Th ages increase with depth, consistent with accurate ages. where X's are the respective decay constants (49), 8234Um is the measured 8234U value (27), [230Th/238U] is the 230Th/28U activity ratio, and T is the age. §Conventional 14C age calculated with the 5568 half-life. The magnitude of the reservoir effect (407 + 52 years, 49.4 + 6.6 per mil) was calculated from sample HOB-EE-5b as described in the text; this value has been subtracted from each age. The error in the reservoir effect is included in the quoted error for each age, except for HOB-EE-5b.
1ISee (30). SCIENCE * VOL. 260 * 14 MAY 1993 1M~9 growth bands and 14C dating, demonstrate that '30Th ages are accurate for carefully selected samples younger than 8000 years old (7,9). However, some corals older than several tens of thousands of years have initial 234U/238U ratios greater than modem marine values. Thus, exchange of uranium may have occurred after skeletal growth (7,23,24). To increase sensitivity in detecting uranium of diagenetic origin, we modified (25) procedures (6,7) to increase further the precision with which the 234U/238U ratio can be measured. Measurements (6,28); however, precisions are higher, which is important for the detection of samples that may have exchanged small amounts of uranium (Fig. 1).
Initial 8234U values (27) for two young Huon Peninsula corals collected from surface exposures (HOB-EE-5 and KAI-EE-1 in Table 1) agree with the value for a young (132 ± 3 years) sample from Barbados (24). The agreement of the initial values from different oceans suggests that spacial variations in the 234U/238U ratio of open-ocean surface seawater are not resolvable. The mean of 149.7 ± 1.5 per mil (2o) for the two Huon Peninsula samples and one Barbados sample gives a precise 8234U value for surface seawater. Initial 8234U values of all other samples ( Fig. 1 and Table 1) are indistinguishable, at a precision of ± 1 per mil, from this value. Thus, the data show no evidence for diagenetic exchange of uranium. The initial 234U/238U values show that the marine 234U/238U ratio varied by no more than 2 per mil in the last 13,000 years. This interval is just the period over which one would expect possible glacialinterglacial shifts in uranium isotopic composition of the sort that were recently documented for the 87Sr/66Sr ratio (29). Because none were observed, any shifts must have been smaller than 2 per mil.
The difference between the '4C/12C ratio of the atmosphere and that of the surface seawater (the reservoir effect) near the drill site was established by analysis of a surface sample (HOB-EE-5b in Table 1).
The Al4C value (30,31) at the time the coral grew was calculated from the measured '4C/12C ratio and the 230Th age. The difference between the known atmospheric 14C/'2C ratio (2,32) at the time of growth and the initial value for the coral is 49.4 ± 6.6 (2uf) per mil, equivalent to a difference in radiocarbon age of 407 ± 52 years. This value is consistent with previous estimates for the Huon Peninsula reservoir effect (12) as well as the estimate from KAI-EE-1, another coral that grew within the period for which the atmospheric '4C/'2C ratio is known from dendrochronology (Table 1). This correction has been subtracted from all of our 14C ages.
For drill-core samples, 14C ages range between 6530 ± 110 (2cr) and 10,970 ± 110 years B.P. (14) for coral that grew while the sea level rose from 71 to 8 m below the present sea level (13) (Fig. 2). The ages are similar to those previously reported for different samples from the same core (12). The rise in sea level inferred from our 14C curve is also similar to the value from the Barbados curve (8,12).
However, the Huon curve shows no evidence for slower sea-level rise during and before the YD, as does the Barbados 14C curve. The 230Th ages for core samples  13,129 + 84 years B.P. for coral that grew while the sea level rose from 71 to 14 m below its present height (13) (Fig. 2). The ages increase progressively with depth, as do the accurate values, and no genera-related discrepancies in age were noted. In contrast to the '4C curve, the 230Th curve shows dramatic evidence for slower sea-level rise between 12,332 ± 39 and 11,045 ± 57 years B.P. (14) (between 10,200 ± 130 and 9,990 ± 90 14C years B.P.). The rate of upward reef growth, and the presumed rate of sea-level rise, decreased from as rapid as 16 m per thousand years before this interval to 2 m per thousand years during the interval and then increased immediately after the interval to rates as high as 28 m per thousand years (Fig. 3). This pattern supports the idea that deglaciation was punctuated by at least one PRM (8,33). Unlike that of the Barbados curve, the PRM recorded in the Huon Peninsula core was shorter, younger, and characterized by a lower melt rate. The discrepancies may reflect differences in sample density, potential artifacts associated with core breaks, potential downslope movement of corals, or corals that grew at significant but variable depths below sea level. The relatively high melt rate for the PRM recorded at Barbados is in part an artifact because the original Barbados meltwater curve (8) was based on 14C dates. The "4C time scale is now known to be highly compressed at about 10,000 "4C 14C Age (ka) 12 (Fig. 3), it still gives a relatively high melt rate for the PRM. More refined studies may ultimately resolve these discrepancies. In support of the Huon Peninsula record, the most prominent feature in the sea-level curve is the PBM, which coincides with the most prominent feature in the '4C/'2C curve. We defend this relationship as more than coincidental.
Knowledge of the temporal relationship between the PRM and the YD helps to distinguish between hypotheses regarding the cause of the YD. It has been hypothesized that diversion of meltwater from the Mississippi into the St. Lawrence River caused the YD by lowering the salinity of North Atlantic surface water (34). Lowdensity surface water hindered the convective sinking of North Atlantic Deep Water (NADW). Because NADW sinking is responsible for meridonal heat transport to the North Atlantic, a lower rate of NADW formation resulted in cooler conditions during the YD. The Barbados chronology does not support this idea because it implies that the PRM started before and continued throughout the YD (8,35). A low melt rate that began before the YD would not have allowed formation of the low salinity cap required by the meltwater-diversion hypothesis. In contrast, our data (Figs. 2 and 3) suggest that the rise in sea level slowed after the onset of the YD and increased again several hundred years after its end. This discrepancy between the Barbados and Huon Peninsula data is critical. Because our data indicate high melt rates at the start of the YD, they suggest that meltwater input or diversion is a plausible trigger for the YD. Our data show that the melt rate fell to its lowest value during the YD. Therefore, one must determine another mechanism beside the continued input of low-density meltwater to explain the cool climate toward the end of the YD. The reduced melting after the onset of the YD would appear to be a response to cooling during the YD rather than a phenomenon that is necessarily related to the cause of the YD . The middle horizontal bar is the best estimate of the timing; with 6 errors taken into consid-13 11 9 7 eration, the upper hori-2Th or dendrochronology age (ka) zontal bar is the oldest possible timing and the lower bar is the youngest possible timing. The vertical bars represent corresponding estimates of the timing of the YD in 14C years, on the basis of the ice-core data and our calibration. The best and the earliest possible estimates agree with other radiocarbon-based estimates of the timing of the YD (35), suggesting that the calibration is accurate. The latest possible estimate for the start of the YD is later than other radiocarbon-based estimates, which suggests that the latest possible ice-core estimate is too young. The Papua New Guinea data define a 1280-year interval between 12.3 and 1 1.0 ka over which 14C ages changed by only 210 years. This period was coincident with the period of reduced melting (Figs. 2 and 3). A decrease in atmospheric 14C/12C (Fig. 5) caused the compression of the 14C time scale during this period. 230Th or dendrochronology age (ka) 2 and 3). The Papua New Guinea data show a rapid decrease in A14C between 12.3 and 1 1.0 ka, which is the most prominent feature yet identified in records of atmospheric A14C and is synchronous with the PRM. This decrease represents almost two-thirds of the total decrease in A14C between 19 and 11 ka. A large portion of the A14C decrease may have been caused by changes in the cycling of carbon in the ocean. SCIENCE * VOL. 260 * 14 MAY 1993 eral inferences. At 13,130 years B.P., '4C ages are 2,160 years younger than 23JO"h ages, corresponding to an atmospheric A'4C of 249 + 18 per mil (30). At 12,330 years B.P., the offset is about the same, 2,130 years (A14C of 247 + 20 per mil). However, by 11,050 years B.P., the offset has diminished to 1,060 years (A14C of 97 + 14 per mil), a value similar to that observed for the next several thousand years (2,3). Over a period of 1,280 years between 12,330 and 11,050 years B.P., radiocarbon ages only diminish by about 210 years. Within error, the Huon Peninsula data from 13 to 11 ka are consistent with the Barbados data (22) (Figs. 4 and 5), which indicates that coral analyses are reproducible from two localities with different diagenetic environments and potentially different reservoir effects. Because of their smaller analytical errors and higher sample density, the Huon Peninsula data suggest that there was a large A'4C decrease (150 + 24 per mil) within a short time interval (1280 ± 70 years). The decrease and the PRM were synchronous (Fig. 5), suggesting that the decrease was related to global melt rates. The decrease is the most prominent feature in the record of atmospheric 14C/12C variations over the last 20,000 years and represents almost twothirds of the total lowering of atmospheric A14C between 19 and 11 ka (Fig. 5) (2,3,9). However, the most prominent decrease in the A14C record (Fig. 5) does not correspond to a comparably prominent increase in the magnetic record, which suggests that changes in the terrestrial magnetic field did not cause most of the decrease. Furthermore, the large average rate of the atmospheric A14C decrease (117 ± 20 per mil per thousand years) is about the rate at which the average A'4C of the oceanatmosphere system would decrease if 14C production stopped completely. Thus, the rapidity of the decrease implicates a mechanism that does not involve production changes alone. 966 Because the ocean contains 95% of the exchangeable carbon in the ocean, atmosphere, and biosphere, nonproduction-related changes in atmospheric A&4C are most probably related to changes in the ocean. One manifestation of a perturbation in the carbon cycle is the rise in the atmospheric CO2 concentration during the last deglaciation, which is documented by measurements of CO2 in ice cores (40). Box models indicate that such a rise in CO2 could be accompanied by a decrease in atmospheric A'4C of 25 to 75 per mil (9,41). Although dating uncertainties preclude detailed correlation with the decrease in 14C/'2C that is inferred from the Huon Peninsula corals, the ice-core data show that the change in CO2 may not have been unidirectional during deglaciation and that some of the net glacial-interglacial rise in CO2 concentration could have taken place at about the YD interval. Thus, some but not all of the A'4C decrease may have resulted from ocean-circulation changes associated with an atmospheric CO2 increase.
An increase in the mixing rate of the ocean causes the average A&4C of the ocean to rise and the average A14C of the combined atmosphere and biosphere to fall. Box models indicate that an increase by a factor of 2 in the ventilation rate produces a decrease in atmospheric A'4C, at steady state, of several tens of per mil (42). Our melt rates (Fig. 3) suggest the presence of a mechanism that changes the ventilation rate. As melt rates decreased during the YD, North Atlantic surface-water salinity increased, causing an increase in NADW production. If global deep-ocean circulation paralleled NADW production, the rate of ocean ventilation increased and atmospheric A&4C decreased. When melt rates increased after the YD, the ventilation rate decreased, arresting the A&4C decrease. The coincidence of reduced global melting and the A'4C decrease (Fig. 5) supports this scenario. We suggest that this changing partitioning of "4C between the atmosphere and the ocean was superimposed on the gradual lowering of average A14C that was caused by the increasing intensity of the Earth's magneticdipole field strength between 20 and 10 ka (Fig. 5).
Previous studies have reconstructed certain aspects of past oceanic circulation.
Combined measurements of '4C/12C in planktonic and benthic foraminifera from the same deep-sea sediment strata (42,43) indicate that the ventilation rate was lower in the glacial ocean than in today's ocean. The ventilation rate about 10,000 14C years ago may have been somewhat higher than the present rate (42). However, more detailed work on the interval between 13 and 11 ka may be required to test whether the A"4C decrease was caused by a change in the SCIENCE * VOL. 260 * 14 MAY 1993 ventilation rate. Certainly the possibility of a ventilation-rate change by a factor of 2 or more at about the time of the YD cannot be ruled out. Significant changes in ocean circulation, which largely involved changes in the rate of formation of NADW and lower North Atlantic Deep Water (LNADW) took place (44 46) before and during the YD. At least some of the changes appear to relate directly to climate in the vicinity of the North Atlantic. In some form and at some rate, formation of NADW continued before and throughout the YD (45,46). Sarthein and colleagues (46) demonstrated that the rate ofNADW formation before the YD was slower than it is now and faster than the present rate at some time during the YD, which is consistent with our inferences from the "4C record. Although their study did not focus specifically on the YD interval, Duplessy and co-workers (47) showed that ventilation rates for short periods during deglaciation were two to three times higher than present rates. The inferred high rate of NADW formation at the end of the YD appears to conflict with cadmium data (48). However, this difference can be reconciled if NADW was more shallow at the end of the YD than at present height or if the ventilation rate of the ocean is not proportional to the rate of NADW production.
Meltwater input or diversion was a plausible trigger for the YD. During the YD, glacial melting slowed and atmospheric A&4C dropped dramatically, for approximately 1280 years, until several hundred years after the end of the YD. We infer from the abrupt decrease in atmospheric A"4C and its coincidence with slower rates of melting that the ventilation rate of the ocean was higher during this 1280-year interval, which is consistent with rates of NADW formation inferred from one other line of evidence (46). There is some uncertainty in the precise temporal relation between this interval and the YD (35) (Fig.   5). With errors taken into consideration, the PRM. may have overlapped with the end of the YD by as much as 900 years or as little as 400 years. In either case, despite low globally averaged rates of meltwater input, apparently high rates of NADW formation, and rapid ocean ventilation, the climate in the North Atlantic region remained cool during the late YD. One explanation for this apparent discrepancy includes the possibility (45) that the North Atlantic climate during the YD was controlled not by the global circulation rate or even by the rate of formation of NADW but by the rate of formation of LNADW. The rate of LNADW formation, in turn, may have been controlled not by globally averaged rates of melting but by meltwater discharge from the Fennoscandian ice sheet. A second possibility is that NADW formed and released heat at lower latitudes during the late YD. less than 200 years to 2100 years for coral that is 20,000 years old. The average A14C of the Barbados samples with 230Th ages between 15,000 and 25,000 years is 430 per mil without selective dissolution and 350 per mil with selective disso-lution [determined by the averaging of points in tables 1A and 1B in (22)]. This corresponds to a difference in 14C age of 460 years. If one chooses a pair of analyses for a single Barbados sample in this age range, the A14C value determined with selective dissolution (22) is typically lower than the value determined without selective dissolution (9); however, the two values typically agree within 2a analytical errors. 22. E. Bard, M. Arnold, R. G. Fairbanks, B. Hamelin, Radiocarbon 35, 191 (1993 The value of 234U/238U at secular equilibrium is 2318/X234 = 5.472 x 10-5, where X represents each respective decay constant (see Table 1) (49).
Initial 8234U values are equal to (8234Uured) Beach Cusps as Self-Organized Patterns B. T. Werner and T. M. Fink Computer simulations of flow and sediment transport in the swash zone on a beach demonstrate that a model that couples local flow acceleration and alongshore surface gradient is sufficient to produce uniformly spaced beach cusps. The characteristics of the simulated cusps and the conditions under which they form are in reasonable agreement with observations of natural cusps. The self-organization mechanism in the model is incompatible with an accepted model in which standing alongshore waves drive the regular pattern of erosion and deposition that gives rise to beach cusps. Because the models make similar predictions, it is concluded that currently available observational data are insufficient for discrimination between them.
Beach cusps are uniformly spaced, arcuate scallops in sediment that form at the shoreward edge of the episodically exposed portion of a beach known as the swash zone. Beach cusps and other regular topographic features in the nearshore, including ripples, megaripples, sand bars, ridges and runnels, sand waves, and swash marks (1), have attracted investigation owing to their beauty, their effect on sediment transport, and their uniformity in the presence of complex interactions between waves, currents, and sediment. According to a widely accepted hypothesis, the form and spacing of beach cusps reflect a pattern of alongshore standing waves on the beach (2)(3)(4). An alternative, physically incompatible model ascribes cusp formation to feedback between swash flow and beach morphology. This model has been discounted partially because of the difficulty of understanding how local interactions between fluid and sediment lead to globally uniform patterns (4). Using computer simulations, we show that uniform beach cusps can develop by local flow-morphology feedback, examine the implications of this self-organization model, and demonstrate that the predictions of the standing wave and self-organization models are sufficiently similar that current data and observations cannot distinguish unambiguously between them.
Beach cusps form on oceanic and lacustrine beaches and in sediments ranging 968 from fine sand to cobbles. They result from a varying combination of accretion at cusp horns and erosion in the interjacent cusp bays (1). Cusps are typically spaced 10 cm to 60 m apart (1). Beach cusp formation is favored under conditions of surging, nonbreaking waves of near-normal incidence on steep (usually coarse-grained) beaches (1,5). Swash runs up a beach without cusps as a thinning, often irregular sheet. On a cuspate beach, runup is deflected by horns toward bays and from there flows seaward as runout.
In the self-organization model, incipient topographic depressions in a beach are amplified by attracting and accelerating water flow, thereby enhancing erosion (Fig. 1A) (5). Dean and Maurmeyer (6) related welldeveloped beach cusp spacing to the swash excursion, the horizontal distance between the highest and lowest positions of the swash front on a beach, by considering the kinematical trajectory of elastic particles sliding on the surface of a cusp.
In contrast, according to the standingwave model, the pattern of erosion and deposition of sediment leading to beach cusps originates with the flow patterns of alongshore standing waves (2-4, 7, 8), generally taken to be subharmonic edge waves (Fig. 1B), which are trapped waves with half the frequency of incident waves. The standing wave model successfully predicts beach cusp spacing as a function of incident wave frequency and beach slope within about a factor of 2 (4,9) (compatible with uncertainties in model input parameters). In addition, standing waves induce cusp formation in sand at the swash zone margin on a laboratory concrete beach SCIENCE * VOL. 260 * 14 MAY 1993 (3). However, because 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 (10).
Our simulation algorithm uses a simplified model of coupled flow, sediment transport, and morphology change with the goal of simulating the evolution of cusps from arbitrary initial morphology. In contrast, earlier models considered wave and fluid flow interactions with a plane (3, 7) or a regular cuspate beach (6,8) only. We are able to investigate the development of uniform cusps, rather than postulating their uniformity and general form, and to simulate explicitly conditions under which cusps do not form.
Our model treats only the kinematics in gravity of swash flow over a beach (6), without directly considering the hydrodynamics of swash. This approximation is supported by an inviscid calculation (11) and laboratory measurements (12) on the motion of the swash front (the leading edge of the swash). Flow is simulated by cubical water particles, representing the swash front, that move according to Newton's laws in gravity (the kinematical equations) and are constrained to remain on the surface of the beach. For example, water particles move on parabolas in the swash zone on a plane beach (Fig. 2A). In addition to the gravitational force, water particles are repelled from regions of high water particle concentration and are attracted toward regions of low concentration in a simplified approximation to flow caused by pressure gradients induced by the sea surface.
The simulated beach is composed of thin, square slabs of sediment stacked on a rectangular grid of square cells with periodic