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Field observations of beach cusps and swash motions
ELSEVIER
Marine Geology 134(1996) 77-93
Field observations of beach cusps and swash motions K.T. Holland a, R.A. Holman b a Naval Research Laboratory Code 7442, Bldg. 2437, Stennis Space Center, MS 39529, USA’ b College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331 USA’ Received 15 May 1995; accepted 25 March 1996
Abstract We tested the hypothesis of cusp formation by the longshore structure of synchronous or subharmonic, mode-zero edge waves using detailed field measurements of foreshore topography, incident waves and swash motions. Two distinct intervals of cusp formation were observed. Alongshore separated swash measurements were analyzed to detect structures at wavelengths and frequencies consistent with the measured cusp spacings. For both events, no statistical support for cusp generation by the traditional subharmonic or synchronous edge wave mechanisms was found. However, once cusps became well developed, there was evidence of interaction between fluids and topography at a wavelength equal to the cusp spacing, but at the period of the incident swash. We propose that this interaction could have been responsible for the generation of the second set of cusps with a spacing of half the pre-existing cusp wavelength.
1. Introduction
Rhythmic, foreshore topographic features known as beach cusps have attracted scientific speculation concerning their formation for nearly a century (Branner, 1900; Johnson, 1910; Evans, 1938; Kuenen, 1948; Guza and Inman, 1975; Werner and Fink, 1993 and many others). In addition to being visually striking (Fig. l), the nearly uniform spacing between alternating cusp horns and embayments stands as an excellent example of a simple structure in nature that has eluded scientific explanation. In the following, we test one of the more commonly accepted theories for cusp formation (the edge wave hypothesis) using detailed observations of swash motions ‘601-688-5320;(fax) 601-688-4476;[email protected]. ‘541-737-2914;(fax) 541-737-2064;[email protected]. 0025-3227/96/$15.000 1996Elsevier Science B.V. All rights reserved PII SOO25-3227(96)00025-4
during cusp formation. Our observations show no obvious indication of cusp development due to edge wave forcing. A few generalizations of when (not why) cusps occur are typically agreed upon. For example, evidence suggests that cusps form during still stands of water level of sufficient duration such that a perturbation in the beach topography (by any possible mechanism) can develop (Williams, 1973; Huntley and Bowen, 1978; Seymour and Aubrey, 1985). In addition, many field observations of cusps suggest that initiation is most favorable under reflective conditions of incident wave approach normal to the shoreline (Johnson, 1910; Longuet-Higgins and Parkin, 1962; Sallenger, 1979; Guza and Bowen, 1981) and characteristically following storm subsidence (Miller et al., 1989). There is less agreement on causative models of
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K. T. Holland, R. A. HolmanlMarine Geology 134 (1996) 77-93
Fig. 1. Video time exposure image showing rhythmic beach cusps at the experiment site in Duck, NC. The time exposure technique is useful in the identification of breaking wave patterns corresponding to changes in topography. Image sampled on August 28, 1993.
cusp formation. Beginning with Johnson (1910), a number of researchers have suggested that cusps develop because of feedback between the underlying topography and the fluid motions (see also Russell and McIntire, 1965; Werner and Fink, 1993). Although the specific mechanisms are undefined, the general explanation is that existing, irregularly spaced perturbations on the otherwise featureless longshore become more uniform due to the apparently regular succession of incoming swash fronts. Over time, the depressions are eroded into embayments until the point that the swash circulation pattern reaches an equilibrium with the topography. Although the necessary physical relationships between fluids and topography are only vaguely stated and hardly proven, the “feedback” circulation pattern has been commonly observed, especially for conditions where the dominant swash period and incident wave period are approximately equal (Bagnold, 1940; Dean and Maurmeyer, 1980). This pattern, shown in Fig. 2a, has a maxi-
mum (in elevation) wave runup on the cusp horns that diverges into the adjacent embayments. Landward flows from neighboring horns accumulate in the center of the embayment to create a spatially concentrated downwash that interferes destructively with the next incoming wave (Fig. 3). This flow pattern promotes cusp development by allowing landward sediment transport at the horns and erosion of the bays. However, as pointed out by Seymour and Aubrey (1985), the growth of a small depression by the feedback provided by this circulation pattern does not necessarily exclude the possibility of a rhythmic, initial perturbation caused by some other mechanism. So although feedback between fluids and topography almost certainly plays a role in cusp growth, it is unclear whether this mechanism is solely responsible for cusp formation. A competing theory, first proposed by Bowen and Inman (1969) is that beach cusps form in response to an alongshore differential in shoreline fluid motions caused by edge waves. Edge waves
K. T. Holland, R A. HolmanJMarine Geology 134 (1996) 77-93
19
Beach
Fig. 2. Circulation patterns of the feedback model (top) and the subharmonic edge wave model (bottom). The regular cusp spacing in the feedback model is hypothesized to result due to division of uprush at the horns and convergence as an intense backwash in embayments. Cusps are predicted to form in the edge wave model due to topographic changes in the beach face induced by an alongshore differential in edge wave amplitude.
are gravity waves trapped to the nearshore region that have a maximum amplitude at the shoreline. On a plane beach, edge waves exhibit the following relationship between wavelength, L e,and period,
T,(Eckart, 1951): L,= $ Tz(2n+l)tanP
(1)
80
K T. Holland, R. A. HolmanlMarine Geology 134 ( 1996) 77-93
Fig. 3. A video image from October 20 showing fluid interaction with well developed cusp topography. Fluids are concentrated in the embayments during downwash and destructively interfere with incoming swash bores.
where g is the acceleration due to gravity, n is the mode number, and p is the beach slope. For monochromatic incident waves, Guza and Inman (1975) found that the most easily excited edge wave, which results from triad interactions between frequencies, is a mode n =0 edge wave with a period twice that of the incident wave period, T, = 2Ti. This “subharmonic” edge wave, defined to be standing in the Iongshore, is hypothesized to result in a cusp wavelength, h,, of one-half the edge wave length. On a plane beach, the cusp length formed by a subharmonic edge wave is related to the incident wave period by: subharmonic:
1, = 2
= a Tf tan /I
(2)
The circulation pattern associated with the edge wave model is somewhat opposite to that of the feedback model in that maximum runup occurs at the antinodes of the edge wave (corresponding to embayments under erosive conditions) and flows toward neighboring horns (Fig. 2b). Kuenen (1948) noted the distinctive characteristic of this
circulation in that antinodes alternately correspond to the location of swash maxima and minima. To develop cusp topography, the swash surge transports sediment removed from the embayments toward the horns. Guza and Inman also mention the possible importance of “synchronous” edge waves of period equal to the incident wave period in cusp formation. In the laboratory, they found the synchronous mechanism to be a higher order, weaker resonance than the subharmonic type. Under the condition of incident waves of infinite wavelength interacting with edge waves of smaller amplitude and equal period, the corresponding plane beach relationship between cusp wavelength, edge wave length, and incident wave period is: synchronous: 1, = L e = & T z tan p Bowen (1972) explains that this relationship is independent of whether the edge wave is progressive or standing because the incident and edge
K. T. Holland, R. A. HolmanlMarine
Geology 134 (1996) 77-93
81
6 4
10
15
20
25
-----___ I
I
15
20
25
October 1994
Fig. 4. Significant wave height (top) and wave period (middle) measured in 8 m depth and tide level (bottom). Experiment interval indicated by dashed bar.
waves are phase locked. If however, the edge wave amplitude greatly exceeds that of the incident wave, as might be expected under storm conditions, the cusp spacing and edge wave length will no longer be equal. In this case, the edge wave must be standing for cusps to develop (as dominantly progressive motions exhibit no alongshore structure) and will result in an alongshore pertur-
bation of one-half the edge wave length via a mechanism analogous to the subharmonic model. To prevent confusion, we will restrict our use of the term synchronous to refer only to cusp formation conditions where cusp lengths and edge wave lengths are equal. Numerous authors have shown suggestions of edge waves association with observed beach cusp
82
K. T Holland, R A. HolmanlMarine Geology 134 (1996) 77-93
x [ml
x [ml
100 110 x [ml
120
Fig. 5. Topographic contours for October 15, 20, and 21. Offshore distances increase to the right (positive x). The foreshore in the region between the 1 and 2 m elevation contours was essentially featureless in the southern half of the study area on October 15. Cusps with a length of approximately 32 m appeared on October 17 and became well developed by October 20. Shorter scale cusps with spacings of approximately half that of the most well developed cusps were observed on October 21 (most obvious in the northern half of the region).
spacings using (2) and (3) (Huntley and Bowen, 1975, 1978; Sallenger, 1979; Guza and Bowen, 1981; Inman and Guza, 1982; Seymour and Aubrey, 1985). However, as pointed out by Inman and Guza (1982), several sources for potential error in the use of these equations exist. Foremost, field wave spectra are seldom monochromatic making identification of a specific incident wave period difficult. Since cusp wavelength scales as the square of the incident wave period, small errors in the incident period may result in relatively large errors in the predicted cusp wavelength. Additionally, the choice of beach slope is problematic in that natural beaches are rarely planar and the region over which a proper beach slope should be calculated is poorly defined. In practice, the relation between edge wave period and wavelength can be calculated numerically for natural topography (see Appendix) and often differs substantially
from the plane beach equations. The possibility of imprecision certainly does not negate the general agreement under a variety of conditions between observations and predictions using (2) and (3). However, the interpretation of these findings as conclusive proof of the edge wave hypothesis is quite possibly erroneous. A stronger confirmation of cusp formation by edge wave processes exists in the findings of Huntley and Bowen (1978). As part of a study of nearshore current velocities, these authors observed cusp formation during the measurement period of their experiment. Velocity spectra sampled approximately 10 m seaward of the shoreline showed energy at subharmonic frequencies preceding the formation of foreshore cusps. Although this subharmonic peak was not significant at the 95% confidence level, it persisted for several hours, was coherent between various sen-
83
K. T. Holland, R.A. HolmanjMarine Geology 134 (1996) 77-93
16 -
i 650
I
700
I
750
I
800 Alongshore
I
850 distance [m]
I
I
900
950
1000
Fig. 6. The measured l-m eontour in the study region as a function of cross-shore and alongshore distance stacked vertically by date (increasing down and denoted by the numbers). A quadratic alongshore trend has been removed. Vertical ticks are separated by 5 m.
sors, and exhibited a phase relationship between vertical and horizontal velocity components consistent with a mode zero, standing, subharmonic edge wave. Equally important was the fact that the mean spacing between cusps (12.7 m) matched the predicted wavelength of a cusp formed by a subharmonic edge wave with a period of around 14 seconds. What separates their study from other cusp-edge wave associations is the observation of energy at the appropriate frequency near the shoreline prior to cusp formation, rather than the more typical the assumption of edge wave presence through the use of the incident wave period. Yet Huntley and Bowen (1978) lacks the unambiguous identification of an alongshore wave pattern in the observed water motions corresponding to the measured cusp wavelength. It is possible that edge waves provide an initial longshore periodic perturbation in topography that further evolves due to feedback with the incident wave train. Indeed, we recognize that beach cusps may form by some, all, or none of
these models. Testable predictions involving the feedback theory of cusp development under natural conditions are still in development. However, we can formulate objective criteria for the verification of the edge wave hypothesis. Indications of beach cusp formation by edge waves include: ( 1) identification of a pronounced, and presumably significant, energy peak in a nearshore spectrum that precedes the formation of cusps; (2) correspondence between the calculated edge wave length, the period of the energy peak, and measured cusp spacing; and (3) observation of longshore magnitude and phase relationships in swash signals that correspond to the expected edge wave pattern. In this paper, we test the edge wave models following these criteria.
2. Field observations Detailed measurements of foreshore topography and swash elevations overlapping two sequences
K. T Holland, R. A. HolmanlMarine Geology 134 (1996) 77-93
84
Table 1 Predicted and observed cusp lengths Date 15 October, 16 October, 17 October, 18 October, 19 October, 20 October, 21 October,
1994 1994 1994 1994 1994 1994 1994
Z(s)
X,(m) [subharmonic]
j,(m) [synchronous]
x,(m)
10.9 kO.6 11.2+0.5 10.6kO.4 11.5kO.5 14.2k1.2 13.5kO.9 11.3kO.6
34.4 & 3.4 36.5k2.9 32.7k2.2 38.Ok2.8 55.1k6.9 50.8+5.0 36.7f3.6
17.2+ 1.9 18.4k1.6 16.3&-1.2 19.2& 1.6 29.7k3.5 26.9 _t 3.5 18.5k2.0
36 29 32 36 40 37 20
[observed]
Observed, average incident wave period, T,, predicted cusp spacings, A,, and observed cusp spacings, IC,.XCestimates calculated along the z = 1 m contour.
of cusp formation were made from October 15 to 21, 1994 as part of the Duck94 nearshore processes experiment at the US Army Corps of Engineers Field Research Facility (FRF) in Duck, NC. The Duck foreshore during this period was comprised of a medium sand (mean grain size of 1.5 phi) with a cross-shore slope of approximately 1:12. Relevant environmental data collected at the FRF (significant wave heights, peak periods, and tidal elevations) are shown in Fig. 4. Topography was sampled daily to a vertical accuracy of better than 2 cm using vehicles equipped with Global Positioning System (GPS) surveying instrumentation. Survey points were densely spaced (approximately every 3 m) within a 400 m longshore by 50 m cross-shore region. The established coordinate system was such that positive y designated northerly increases in the alongshore direction while positive x increased offshore. Swash time series were measured during daylight hours along 5 1 cross-shore transects spaced evenly within a subset of the above region. The total alongshore coverage was 250 m. Swash elevations were calculated from video recordings of the view from two cameras mounted approximately 45 m above mean sea level and positioned looking alongshore. Individual video frames were digitized to find the location of the swash edge, indicated by pixel intensity exceeding a time averaged threshold. The process was repeated every second for the length of standard video cassette tape (2 hours). The vertical resolution of this technique varies as a function of distance from the camera, but a typical vertical
resolution of 2 cm was obtained by projecting the average horizontal resolution to an elevation (for further details, see Lippmann and Holman (1989)). The beachface seaward of a set of extreme high tide cusps was to some extent flattened following extremely high wave conditions on the night of October 15, although, remnant alongshore irregularities from a time period preceding the experiment were still apparent. Foreshore cusps with a spacing (of approximately 38 m) mimicking that of the remnant cusps began to form throughout the study region on October 17 (which we will denote as event #l) and continued to develop through October 20 (Fig. 5). Since the resulting cusp lengths closely approximated the pre-existing, higher elevation cusp field, but had a slightly different alongshore phase, it is not certain whether the cusps associated with event #l developed independently of the remnant topography. To reduce subjectivity in cusp length measurements, we calculated average (alongshore) cusp spacings from the wavenumber (reciprocal wavelength) spectrum of the alongshore variation in horizontal position of the measured 1 m elevation contour (Fig. 6). The average cusp spacing, xc, was defined as the wavelength corresponding to the peak in spectral density. For event #l, the cusp spacing varied between 29 and 40 m (Table 1), with a mean cusp spacing during the last three days of the event (the portion with the most well developed cusps) of 38 m. We observed an approximate halving in the spacing of cusps on the lower beachface, a decrease to 20 m, on the morning of October 21, which will be denoted as event #2 (Figs. 5 and 6). The event #2
K. T Holland, R A. Holman/Marine
cusp structure is more clearly unrelated to the still remnant, high tide cusps.
3. Model tests Offshore wave spectra from an array of pressure sensors in 8 m depth from October 15 through 21 were examined to determine incident wave periods corresponding to both of the cusp formation events. In all cases, the wave spectra were narrowbanded, making identification of Ti (the incident wave period associated with the peak in spectral density) fairly simple. However, there was considerable variation in the peak period over time, therefore a daily mean period, Ti, and standard deviation, or, were calculated to display the range of possible periods. We also predicted cusp spacings, fi,, under the assumption of either synchronous or subharmonic mode zero edge waves with periods proportional to Ti. The relationship between edge wave period and wavelength (analogous to ( 1)) was found numerically for the alongshore averaged cross-shore profiles for each day (method fully explained in the Appendix). Under the hypothesis of edge wave forcing of cusp formation, comparison of these results (displayed in Table 1) with the average spacings observed for events #l and #2 (38 and 20 m, respectively), indicates that the alongshore wave structure most consistent with the measured cusp spacing and incident wave period during the first event was a subharmonic edge wave present before October 19 with a period of around 22 seconds and a wavelength of approximately 76 m. The most consistent structure for event #2 was a synchronous edge wave with an approximate period of 11 seconds and wavelength of 20 m occurring on October 21. Since the shorter wavelength cusps were first observed at 0700 EST on the morning of the October 21 and the change in incident wave period to 11 seconds did not occur until the late evening of October 20 (after 2300, Fig. 4), for this model to be correct, the cusp development must have occurred rapidly (i.e. within 8 hours). Synchronous and subharmonic edge wave periods responsible for cusp formation are typi-
Geology 134 (1996) 77-93
85
cally referenced in terms of the incident wave peak (because it represents the hypothesized forcing). As above, these mechanisms are usually tested by assuming an edge wave period related to the measured incident wave period at the time of the cusp observation, solving for a corresponding cusp wavelength and comparing the predicted wavelength with that observed. However, the usual test can also be reversed to predict the required edge wave period for the measured cusp length independently of Ti.Predicted edge wave periods, fek, corresponding to the mean cusp spacings of events #l and #2 determined in this manner are shown in Table 2. These results indicate that for event #l the subharmonic (L e = 2X,) and synchronous (L e = h,) edge wave periods corresponding to the measured cusp length of 38 m are approximately 23 and 16 s, respectively. For event #2 the same periods corresponding to a 20 m cusp spacing are 16 and 12 sec. We examined swash data runs between October 16 and 21 to verify the presence of edge wave energy at the appropriate periods for the two events. The runs, listed in Table 3, were selected as having swash ranges centered within the cusp region. [On October 18, human activity in the beachface region obstructed the camera view during the optimal sampling period which prevented accurate swash measurement. Results from the following run were substituted.] Autospectra and cross-spectra were calculated from demeaned and detrended time series. Seven ensembles of length 1024 s and 2 frequency bins were merged to yield 28 degrees of freedom and a frequency bandwidth of 0.002 Hz. The alongshore averaged swash spectrum for each run is given in Fig. 7. The approximate frequencies of subharmonic and synchronous edge waves consistent with the measured topography are indicated and the mean offshore wave period, Ti, is shown. The swash spectra show little evidence of dominant peaks in energy at frequencies associated with the predicted subharmonic. In fact, subharmonic structures associated with any higher frequency spectral peak are not obvious. For the runs on October 19, 20, and 21, there was an obvious peak in swash energy in close proximity to the predicted synchronous frequency. However
86
K. T. Holland, R. A. HolmanlMarine
a)
Geology 134 ( 1994) 77-93
October 16
October 17
lo*
7 1
l
t
-
10’
100
10“
c)
October 16
October 19
lo’~i
1
* +
10’ :
loo
e)
October 20 lo2
:
9
October 21
t 1’1
IO0 10’
x
t
?
;I:;-
10-l 0
0.05 0.1 Frequency [Hz]
0
0.05 0.1 Frequency [Hz]
Fig. 7. Alongshore averaged energy density as a function of frequency for the six swash runs (a-f). The 95% confidence interval for spectral estimates (assuming 28 degrees of freedom) is indicated by the vertical bar. Frequencies for hypothesized edge waves corresponding to the measured cusp spacings of the two events are marked by arrows (event #l arrows are black; event #2 arrows are gray). Subharmonic edge waves are indicated by the upward pointing arrows and synchronous edge waves are indicated by downward pointing arrows. The frequencies associated with the peak in offshore wave energy density are given by the asterisk.
as mentioned, given frequency spectra alone we cannot be certain that this energy corresponds to an appropriate edge wave structure. Measurements of the alongshore structure are required.
To test for evidence of edge waves of the appropriate length scales, we partitioned swash measurements into wave structures of a specified wavelength. Specifically, we modeled the observa-
K. T Holland, R. A. Holman/Marine Geology 134 (1996) 77-93
Table 2 Predicted edge wave periods, lengths, 1,
Event #l Event #2
Te, based on observed cusp
L(m)
PC(s) [subharmonic]
Fe(s) [synchronous]
36-40 20
22.3-23.6 16.4
1.5.6-16.4 11.7
Ott Ott Ott act Ott Ott
16 17 18 19 20 21
Time (EST)
Mean (m)
Maximum (m)
Minimum (m)
R,(m)
0648 0652 1034 0834 0830 0843
1.7 1.5 0.5 1.4 1.5 1.2
2.8 2.2 1.3 2.6 2.6 2.2
0.5 0.3 -0.3 0.1 0.2 0.4
2.9 2.4 2.0 3.2 2.8 2.8
Significant swash heights, R,, were defined as 4 x the measured, alongshore averaged, standard deviation in swash elevation.
tions, tj, following the procedure given by Haines (1987) as a linear combination of two complex modes propagating in opposite directions plus a residual _E _O=Ec(+e __ where I%= [exp
(iEy)
exp (-izp)]
(4)
E is a matrix
of two columns, each of which represents the complex form of an alongshore structure of wavelength, L., given at the alongshore locations contained in the vector, 21. The covariance matrix associated with the complex modal amplitudes, g, is given by (&i>=(E’E)-lE’(@‘)E(E’E)-l
diagonal elements represents the real valued amplitudes, a, and a2, of each wave. Squared coherence values, y2, indicate the degree of phase locking between the two modes and are given by:
(6)
Table 3 Statistics of swash motions Date
81
(5)
where ( ) indicates time averaging, ’ denotes transpose, and ( )-’ represents the matrix inverse. It can be shown (Haines, 1987) that this least squares solution is equivalent to fitting the crossspectral matrix, (00) at a given frequency where the solution A = (CM) is a 2 x 2, positive, semidefinite, Hermitian matrix. The square root of the
Perfectly phase locked modes indicative of a pure standing wave structure have a squared coherence of 1. The degrees of freedom of the observed crossspectral matrix can be used in the calculation of the significance level associated with the model. To identify edge wave structures corresponding to the expected edge wave periods listed in Table 2 (based on the cusp length observations), the observed cross-spectra were tested using the above model. We first looked for evidence of subharmonic, standing edge wave structures where the modeled edge wave length was defined at twice the observed cusp length. A frequency interval for edge waves consistent with the measured topography, l/r, +O.Ol Hz, was determined to account for possible errors in the dispersion relation resulting from inaccurate inputs (see Appendix). Squared coherence estimates above the 95% significance level (0.20 for 28 degrees of freedom) and within the appropriate frequency interval were interpreted as being suggestive of standing edge wave motions of the proper length scale. The results, shown in Fig. 8 and listed in Tables 4 and 5, indicate that for event #l, only one significant coherence value in the proper interval was observed (October 19). The amplitudes of the two associated edge waves were relatively small, approximately 1 cm. Also the period of the significant coherence did not coincide with an obvious energy peak in the spectrum. For event #2, large significant squared coherence values were observed near but certainly not centered within the expected frequency interval on October 20; however, the edge wave amplitudes were again small and the model was found to describe less than 20% of the observed variance. More importantly, there was no indication of this energy being associated with the first subharmonic of any dominant spectral peak. Considering these results, there was little evidence of cusp formation during either event due
K. T. Holland, R A. HolmanlMarine Geology 134 (1996) 77-93
88
October 16 L,: 76m II
b)
0.8
October 17 L,: 76m
r
+
0.6
I
+
October 18 L,: 76m
October 19 L,: 76m
4 0.8
r
+
0.8
0.4
October 20 L,+ 76m
e,
Lip----
V
October 20 L,: 40m
October 21 L,: 40m
I 0.8 0.6 0.4 0.2 n 0.05 Frequency
0.1 [Hz]
Frequency
[Hz]
Fig. 8. Squared coherence between oppositely propagating modes for subharmonic edge waves of wavelength L, as a function of frequency for the six swash runs. The 95% critical coherence level is indicated by the dashed line. Test results corresponding to the event #l cusp measurements are given in panels a-e. Event #2 results are displayed in panels f and g. Horizontal bars indicate the possible range of frequencies for edge waves consistent with the measured topography.
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Geology 134 (1996) 77-93
89
Table 4 Event #1 Edge wave characteristics Time
Date
T,
Subharmonic L e = 16 m Y2
Ott Ott Ott Ott Ott
16 17 18 19 20
0648 0652 1034 0834 0830
20.7
n/o n/o n/o 0.34 n/o
aI (cm)
1.3
Synchronous L e = 38 m a2 (cm)
1.2
T,(s)
Y’
al (cm)
a2 (cm)
a3 (cm)
14.7 15.2
nlo n/o n/o 0.58 0.62
2.9 3.5
1.3 2.0
7.8 1.4
T,, edge wave period corresponding to the maximum coherence level, y’, within the expected frequency interval. a,, a,, a3 represent the measured amplitudes of a progressive wave of wavelength L e propagating in the +y direction, the -y direction, and a normally incident wave, respectively. n/o = significant squared coherence value not observed.
Table 5 Event #2 Edge wave characteristics Date
Ott 20 Ott 21
Time
0830 0843
T, (s)
14.3 14.3
Y2
a1 (cm)
a2 (cm)
T, (s)
Y’
a1 (cm)
a2 (cm)
a3 (cm)
0.52 0.22
3.0 0.8
1.7 0.7
11.4
n/o 0.66
1.4
1.0
4.8
subharmonic edge waves. It is conceivable, however, that the high significant coherence levels on October 20 for event #2 occurred in association with a non-traditional mechanism of cusp formation involving swash energy at the incident peak. This possibility will be examined more fully in the discussion. We also used the edge wave model to test for synchronous edge waves with wavelengths equal to the measured cusp lengths. As described in the introduction, the synchronous mechanism requires that the edge wave structure (either standing or progressive) be phase locked to and of the same period as the incident wave field. For this reason, we added a third mode describing an alongshore homogeneous structure to the model described in (4). For event #l, evidence of synchronous, standing waves (indicated by high coherence values between the two edge wave forms and within the appropriate frequency interval) was only seen on October 19 and 20 (Table 4). Interestingly, these motions were also coherent with the incident wave indicating a standing phase locked structure. For event #2, there was evidence on October 21 of a similar coupling between two edge waves and the to standing,
Synchronous L e = 20 m
Subharmonic L c = 40 m
incident wave. However, in each of these cases, the edge wave amplitudes were exceedingly small compared to the significant swash height and were less than half the amplitude of the incident wave mode. In addition, no appropriate edge wave structures were evident prior to the formation of the cusp fields (i.e. before October 17 and 20). So it is unlikely that synchronous, standing wave energy formed the observed beach cusps. When we tested for individual progressive motions coupled to the incident wave field, there was only one example (the October 17 run) of a progressive wave significantly coherent with the incident wave field in which the wave propagating in the opposite direction was not significantly coupled to either the first edge wave or the incident wave (indicative of a synchronous, progressive wave). However, the amplitude of this wave was less than 1 cm and a corresponding spectral energy peak was not apparent, so we found no strong evidence of progressive, synchronous edge wave forcing either. In contrast to the lack of identifiable edge wave structures, we did find evidence in the October 20 swash data of the feedback circulation pattern.
90
K. T Holland, R. A. HolmanlMarine
-5’
’
780
I
I
800
820
1
840
Geology 134 (1996) 77-93
I
I
I
I
I
880
880
900
920
940
960
v[ml Fig. 9. Magnitude (top) and phase (middle) of the dominant empirical orthogonal function of the October 20, 0830 run at frequency 0.068 Hz versus (bottom) the cusp pattern shown by cross-shore and alongshore coordinates of the l-m contour (longshore trend removed). This mode explains 75% of variance of the observations at this frequency.
Fig. 9 shows the longshore magnitude and phase structure of the dominant, complex empirical orthogonal function (CEOF) of the cross-spectral matrix at the frequency corresponding to the peak of the spectrum (see Wallace and Dickinson, 1972; Holland et al., 1995). The CEOF method is somewhat similar to the least-squares model in that orthogonal wave structures are detected, although in this case without the constraint of an input wavelength. The observed magnitude and phase represent a strong interaction between the fluid motions and the existing cusp topography given that (local) maximum magnitudes correspond to the longshore locations of cusp horns. The phase relationships indicate that swash motions in the embayments lag motions at the horns by approximately 4 s at that frequency. These observations are identical to the feedback pattern in that the amplitude of swash bores is largest at the cusp horns and the phase at the horns leads the bays due to destructive interference in the embayments between the incoming bore and the preceding
downwash (Fig. 3). We emphasize that these topographic interaction results correspond to the timing of the most well developed cusp field and are therefore more indicative of cusp maintenance than cusp formation.
4. Discussion Our detailed measurements and careful testing showed little suggestion that either synchronous or subharmonic edge waves caused the observed cusp structures during this experiment. Obviously, it is much simpler to present convincing evidence in support of a specific model of cusp formation than it is to absolutely disprove the possibility of any particular mechanism. With only one observation of cusp development that obviously differed from the remnant topography (event #2), we lacked the necessary range of conditions to conclusively discount the feasibility of the edge wave mechanism.
K. T Holland, R A. HolmanlMarine
One possible explanation for the lack of prominent edge wave structures of the proper length scale is that edge wave induced topography may provide a negative feedback to the edge wave excitation process (Guza and Inman, 1975; Guza and Bowen, 1981). In other words, cusp growth may dampen edge wave excitation which makes the edge wave structure less obvious. In addition, the lack of obvious edge wave patterns could be due to a short lived initiation event that occurred apart from our sampling. For example, Evans (1945) and Komar (1973) observed cusp fields developing rapidly on the order of minutes. Regardless of the strength of our conclusions, we believe that the methods presented here stand as a considerable improvement over those previously utilized and that our findings emphasize the need for more cautious interpretation of prior results. We were intrigued by the relationship between the swash motions on October 20 and the cusps that developed on October 21. From these observations, there was evidence (Fig. 9) of a strong interaction between the well-developed event #l cusp field and the swash motions occurring within subharmonic frequency interval for event #2. We also note, for October 20, the approximate coincidence of the frequency associated with the maximum in swash spectral density, the frequency associated with the incident wave peak, and the predicted value of the event #2 subharmonic (Fig. 7e). Although we found little indication of this energy being associated with an edge wave form, we find interesting the possibility that energy standing in the alongshore at the wavelength of the topography could have been excited by topographic interaction at the frequency of the swash spectral peak. In fact, there was some evidence to suggest that this standing pattern did occur (Fig. 8f ). If this was indeed the case, then energy scattered from this interaction between swash and topography could exist as a free, standing edge wave given the proper dispersion relationship. As noted, the predicted event #2 subharmonic (h, =L,/2) frequency was close to the frequency of dominant interaction. We propose that a slight change in tidal level or incident wave period on
Geology 134 ( 1996) 77-93
91
the evening of October 20 caused a strong resonance that developed into a 40 m standing edge wave structure of larger amplitude than that measured earlier in the day, and that this structure in turn forced topography at one-half the edge wave length, corresponding to the cusps observed on October 21. This mechanism differs from the conventional Guza and Inman (1975) hypothesis in that the excitation of the edge wave did not result from fluid interactions at twice the incident wave period, but occurred in response to swash interaction with topography. In addition, this alternative mechanism of cusp formation is for the specific condition of a halving of the existing cusp spacing. We intend to pursue this possibility in future research.
5. Summary For two distinct intervals of cusp formation, we tested the hypothesis of cusp formation by synchronous or subharmonic mode zero edge waves in two ways. First, cusp wavelengths corresponding to both subharmonic and synchronous edge wave forcing were predicted given the observed incident wave period and compared to observed cusp spacings. These comparisons indicated that the most likely edge wave forcing (if any) was a subharmonic edge wave for the first event and a synchronous edge wave for the second event. However, the timing of the cusp events and the lack of identifiable subharmonic peaks in swash autospectra indicated that the presence of these particular wave structures was unlikely. Alternately, swash measurements separated in the longshore were analyzed for evidence of edge wave structures within an appropriate frequency interval irrespective of the incident wave period. For both events, the statistical connections between the measured swash motions and the predicted wave structures given the observed cusp wavelengths, were not significantly different from expectations for random noise. In other words, for these two events, there was no statistical support for cusp generation by the traditional subharmonic or synchronous edge wave mechanisms.
K T. Holland, R. A. HolmanJMarine Geology 134 (1996) 77-93
92
However, there were suggestions in the data of an unconventional mechanism for cusp generation. Close examination of the longshore structure of swash motions over well-developed cusps showed interaction at a wavelength equal to the cusp spacing but at the period of the incident swash. We propose that if this interaction occurred at a frequency satisfying the dispersion relation, free edge waves may have been resonantly forced, which, in turn, could explain the generation of cusps with a spacing of half the pre-existing cusp wavelength. This mechanism differs from the traditional hypothesis for the formation of cusps at half the edge wave length in that edge wave excitation at subharmonic frequencies is not required.
Acknowledgment
This research was supported by funding from the Office of Naval Research, Coastal Dynamics Program (N00014-9411196) and the US Geological Survey, National Coastal Geology Program (Cooperative agreement #1434-93-A-l 124).
Appendix The linear equation governing edge wave motions of the form, q(x,yJ)=agf(x) sin (ky-crt), as (htp&+tj
(
$ -k2h
>
=0
(AlI
where a is the edge wave amplitude, g is gravitational acceleration, f(x) represents a cross-shore structure, u =2x/T, k = 27t/L, h is the water depth, and x and y represent the offshore and alongshore spatial coordinates. If a plane beach topography of constant slope, g, is assumed, the dispersion relation c?=gk sin (2n+ l)S (equivalent to (1)) for edge wave motions can be derived analytically (Ursell, 1952). More often, however, we wish to find edge wave solutions on arbitrary topography, h(x, y). Typically these solutions are calculated numerically assuming h,=O (Holman and Bowen, 1979; Kirby et al., 1981; Howd et al., 1992). In the present study, numerical solutions for the relationship between o and k on arbitrary topography were found following the method of Holman and Bowen (1979). Equation al is expressed as two coupled first order differential equations, subject to the boundary conditions n(0) =flnite and n(co) = 0. These equations are solved using a Runge-Kutta technique
(Press et al., 1992) where h and n are expanded as h(x)=/31x+/&XZ+... ~(x)=~o(l+~,x+~L,xz+...)
(A2)
Substituting Eq. A2 into Al and equating coefficients of like powers of x yields the initial conditions at the shoreline rl(0)=ilO= given
(A3)
For any particular frequency, the corresponding value of k is found by maximizing the offshore distance at which the solution diverges from the deep water boundary condition. These values represent the o, k pairs satisfying the edge wave dispersion relation. Given that the choice of a shoreline location is integral to the solution results, a shoreline elevation of 1.5 m was selected as representative of the contour where most cusps were concentrated. Changes in a given frequency value resulting from +0.5 m change in the shoreline elevation were less than 0.005 Hz for frequencies between 0.05 and 0.15 Hz. In all cases, we solved only for mode zero. Multiple cross-shore transects were averaged to give a mean profile which was assumed homogeneous in the alongshore. However, it is obvious that this assumption is invalid when cusps are well-developed such that the effect of rhythmic variations in the alongshore topography would be to alter the dispersion relationships. Since the above method assumes alongshore inhomogeneities are negligible, we attempted to quantify the maximum error in the calculated dispersion relation, given the measured cusp field. Guza and Bowen (1981) present an analytic solution for the change in the edge wave frequency on an otherwise plane beach of slope g, with a superimposed longshore periodicity, h,, of half the edge wave length. This situation, h=hO+hI
where
ho = x tan /l and h, =f(x)
cos (2ky)
(A4)
results in a decrease of the natural frequency, oO, as given by: 1 dcma,aTk _~ tanp Z
(
2P 70+2)
>
where aT represents the cusp amplitude and cusp perturbations are defined to decay exponentially offshore at exp+‘kX.For a subharmonic edge wave existing on the well-developed cusp field on October 20 (cusp spacing= 38 m, estimated cusp amplitude = 34 cm and approximate cross-shore decay scale = 20 m), the solution indicates a maximum change in the natural edge wave frequency of about 15%.
K T. Holland, R. A. HolmanlMarine Geology 134 ( 1996) 77-93
References Bagnold, R.A., 1940. Beach formation by waves: Some model experiments in a wave tank. J. Inst. Civ. Eng., 15: 27-52. Bowen, A.J., 1972. Edge waves and the littoral environment. Proc. 13th Coastal Eng. Conf. ASCE, Vancouver, BC, pp. 1313-1320. Bowen, A.J. and Inman, D.L., 1969. Rip currents, 2. Laboratory and field observations. J. Geophys. Res., 74(23): 5479-5490. Branner, J.C., 1900. The origin of beach cusps. J. Geol., 8: 481-484. Dean, R.G. and Maurmeyer, E.M., 1980. Beach cusps at Point Reyes and Drakes Bay Beaches, California. In: B.L. Edge (Editor), Proc. 17th Coastal Eng. Conf. AXE, Sydney, pp. 863-884. Eckart, C., 1951. Surface waves on water of variable depth. Scripps Inst. Oceanogr. Wave Rep., 100, Ref. 51-12. Evans, O.F., 1938. The classification and origin of beach cusps. J. Geol., 46: 615-627. Evans, O.F., 1945. Further observations on the origin of beach cusps. J. Geol., 53: 403-404. Guza, R.T. and Bowen, A.J., 1981. On the amplitude of beach cusps. J. Geophys. Res., 86(C5): 4125-4132. Guza, R.T. and Inman, D.L., 1975. Edge waves and beach cusps. J. Geophys. Res., 80(21): 2997-3012. Haines, J.W., 1987. An investigation of low-frequency wave motions: A comparison of barred and planar beaches. Thesis. Dalhousie Univ., 257 pp. Holland, K.T., Raubenheimer, B., Guza, R.T. and Holman, R.A., 1995. Runup kinematics on a natural beach. J. Geophys. Res., lOO(C3): 49854993. Holman, R.A. and Bowen, A.J., 1979. Edge waves on complex beach profiles. J. Geophys. Res., 84(ClO): 6339-6346. Howd, P.A., Bowen, A.J. and Holman, R.A., 1992. Edge waves in the presence of strong longshore currents. J. Geophys. Res., 97(C7): 11,357-11,371. Huntley, D.A. and Bowen, A.J., 1975. Field observations of edge waves and their effect on beach material. J. Geol. Sot. London, 131: 69-81. Huntley, D.A. and Bowen, A.J., 1978. Beach cusps and edge
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waves. In: K.E. Nauman (Editor), Proc. 16th Coastal Eng. Conf. ASCE, Hamburg, pp. 137881393. Inman, D.L. and Guza, R.T., 1982. The origin of swash cusps on beaches. Mar. Geol., 49: 133-148. Johnson, D.W., 1910. Beach cusps. Geol. Sot. Am. Bull., 21: 599-624. Kirby, J.T., Dalrymple, R.A. and Liu, P.L.F., 1981. Modification of edge waves by barred beach topography. Coastal Eng., 5: 35549. Komar, P.D., 1973. Observations of beach cusps at Mono Lake, California. Geol. Sot. Am. Bull., 84(11): 3593--3600. Kuenen, P.H., 1948. The formation of beach cusps. J. Geol., 56: 34-40. Lippmann, T.C. and Holman, R.A., 1989. Quantification of sand bar morphology: A video technique based on wave dissipation. J. Geophys. Res., 94(Cl): 995-1011. Longuet-Higgins, M.S. and Parkin, D.W., 1962. Sea waves and beach cusps. Geogr. J., 128(2): 194-201. Miller, J.R., Miller, S.M.O., Torzynski, C.A. and Kochel, R.C., 1989. Beach cusp destruction, formation, and evolution during and subsequent to an extratropical storm, Duck, North Carolina. J. Geol., 97(6): 7499760. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P., 1992. Numerical Recipes in C: the art of scientific computing. Cambridge Univ. Press, 994 pp. Russell, R.J. and McIntire, W.G., 1965. Beach cusps. Geol. Sot. Am. Bull., 76(3): 307-320. Sallenger Jr., A.H., 1979. Beach-cusp formation. Mar. Geol., 29( l-4): 23-37. Seymour, R.J. and Aubrey, D.G., 1985. Rhythmic beach cusp formation: A conceptual synthesis. Mar. Geol., 65(3-4): 289-304. Ursell, F., 1952. Edge waves on a sloping beach. F’roc. R. Sot. Ser. A, 214: 79-97. Wallace, J.M. and Dickinson, R.E., 1972. Empirical orthogonal representation of time series in the frequency domain, Part I: Theoretical considerations. J. Appl. Meteorol., 11(6): 887-892. Werner, B.T. and Fink, T.M., 1993. Beach cusps as self-organized patterns. Science, 260: 968-971. Williams, A.T., 1973. The problem of beach cusp development. J. Sediment. Petrol.. 43(3): 857-866.
Marine Geology 134(1996) 77-93
Field observations of beach cusps and swash motions K.T. Holland a, R.A. Holman b a Naval Research Laboratory Code 7442, Bldg. 2437, Stennis Space Center, MS 39529, USA’ b College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331 USA’ Received 15 May 1995; accepted 25 March 1996
Abstract We tested the hypothesis of cusp formation by the longshore structure of synchronous or subharmonic, mode-zero edge waves using detailed field measurements of foreshore topography, incident waves and swash motions. Two distinct intervals of cusp formation were observed. Alongshore separated swash measurements were analyzed to detect structures at wavelengths and frequencies consistent with the measured cusp spacings. For both events, no statistical support for cusp generation by the traditional subharmonic or synchronous edge wave mechanisms was found. However, once cusps became well developed, there was evidence of interaction between fluids and topography at a wavelength equal to the cusp spacing, but at the period of the incident swash. We propose that this interaction could have been responsible for the generation of the second set of cusps with a spacing of half the pre-existing cusp wavelength.
1. Introduction
Rhythmic, foreshore topographic features known as beach cusps have attracted scientific speculation concerning their formation for nearly a century (Branner, 1900; Johnson, 1910; Evans, 1938; Kuenen, 1948; Guza and Inman, 1975; Werner and Fink, 1993 and many others). In addition to being visually striking (Fig. l), the nearly uniform spacing between alternating cusp horns and embayments stands as an excellent example of a simple structure in nature that has eluded scientific explanation. In the following, we test one of the more commonly accepted theories for cusp formation (the edge wave hypothesis) using detailed observations of swash motions ‘601-688-5320;(fax) 601-688-4476;[email protected]. ‘541-737-2914;(fax) 541-737-2064;[email protected]. 0025-3227/96/$15.000 1996Elsevier Science B.V. All rights reserved PII SOO25-3227(96)00025-4
during cusp formation. Our observations show no obvious indication of cusp development due to edge wave forcing. A few generalizations of when (not why) cusps occur are typically agreed upon. For example, evidence suggests that cusps form during still stands of water level of sufficient duration such that a perturbation in the beach topography (by any possible mechanism) can develop (Williams, 1973; Huntley and Bowen, 1978; Seymour and Aubrey, 1985). In addition, many field observations of cusps suggest that initiation is most favorable under reflective conditions of incident wave approach normal to the shoreline (Johnson, 1910; Longuet-Higgins and Parkin, 1962; Sallenger, 1979; Guza and Bowen, 1981) and characteristically following storm subsidence (Miller et al., 1989). There is less agreement on causative models of
78
K. T. Holland, R. A. HolmanlMarine Geology 134 (1996) 77-93
Fig. 1. Video time exposure image showing rhythmic beach cusps at the experiment site in Duck, NC. The time exposure technique is useful in the identification of breaking wave patterns corresponding to changes in topography. Image sampled on August 28, 1993.
cusp formation. Beginning with Johnson (1910), a number of researchers have suggested that cusps develop because of feedback between the underlying topography and the fluid motions (see also Russell and McIntire, 1965; Werner and Fink, 1993). Although the specific mechanisms are undefined, the general explanation is that existing, irregularly spaced perturbations on the otherwise featureless longshore become more uniform due to the apparently regular succession of incoming swash fronts. Over time, the depressions are eroded into embayments until the point that the swash circulation pattern reaches an equilibrium with the topography. Although the necessary physical relationships between fluids and topography are only vaguely stated and hardly proven, the “feedback” circulation pattern has been commonly observed, especially for conditions where the dominant swash period and incident wave period are approximately equal (Bagnold, 1940; Dean and Maurmeyer, 1980). This pattern, shown in Fig. 2a, has a maxi-
mum (in elevation) wave runup on the cusp horns that diverges into the adjacent embayments. Landward flows from neighboring horns accumulate in the center of the embayment to create a spatially concentrated downwash that interferes destructively with the next incoming wave (Fig. 3). This flow pattern promotes cusp development by allowing landward sediment transport at the horns and erosion of the bays. However, as pointed out by Seymour and Aubrey (1985), the growth of a small depression by the feedback provided by this circulation pattern does not necessarily exclude the possibility of a rhythmic, initial perturbation caused by some other mechanism. So although feedback between fluids and topography almost certainly plays a role in cusp growth, it is unclear whether this mechanism is solely responsible for cusp formation. A competing theory, first proposed by Bowen and Inman (1969) is that beach cusps form in response to an alongshore differential in shoreline fluid motions caused by edge waves. Edge waves
K. T. Holland, R A. HolmanJMarine Geology 134 (1996) 77-93
19
Beach
Fig. 2. Circulation patterns of the feedback model (top) and the subharmonic edge wave model (bottom). The regular cusp spacing in the feedback model is hypothesized to result due to division of uprush at the horns and convergence as an intense backwash in embayments. Cusps are predicted to form in the edge wave model due to topographic changes in the beach face induced by an alongshore differential in edge wave amplitude.
are gravity waves trapped to the nearshore region that have a maximum amplitude at the shoreline. On a plane beach, edge waves exhibit the following relationship between wavelength, L e,and period,
T,(Eckart, 1951): L,= $ Tz(2n+l)tanP
(1)
80
K T. Holland, R. A. HolmanlMarine Geology 134 ( 1996) 77-93
Fig. 3. A video image from October 20 showing fluid interaction with well developed cusp topography. Fluids are concentrated in the embayments during downwash and destructively interfere with incoming swash bores.
where g is the acceleration due to gravity, n is the mode number, and p is the beach slope. For monochromatic incident waves, Guza and Inman (1975) found that the most easily excited edge wave, which results from triad interactions between frequencies, is a mode n =0 edge wave with a period twice that of the incident wave period, T, = 2Ti. This “subharmonic” edge wave, defined to be standing in the Iongshore, is hypothesized to result in a cusp wavelength, h,, of one-half the edge wave length. On a plane beach, the cusp length formed by a subharmonic edge wave is related to the incident wave period by: subharmonic:
1, = 2
= a Tf tan /I
(2)
The circulation pattern associated with the edge wave model is somewhat opposite to that of the feedback model in that maximum runup occurs at the antinodes of the edge wave (corresponding to embayments under erosive conditions) and flows toward neighboring horns (Fig. 2b). Kuenen (1948) noted the distinctive characteristic of this
circulation in that antinodes alternately correspond to the location of swash maxima and minima. To develop cusp topography, the swash surge transports sediment removed from the embayments toward the horns. Guza and Inman also mention the possible importance of “synchronous” edge waves of period equal to the incident wave period in cusp formation. In the laboratory, they found the synchronous mechanism to be a higher order, weaker resonance than the subharmonic type. Under the condition of incident waves of infinite wavelength interacting with edge waves of smaller amplitude and equal period, the corresponding plane beach relationship between cusp wavelength, edge wave length, and incident wave period is: synchronous: 1, = L e = & T z tan p Bowen (1972) explains that this relationship is independent of whether the edge wave is progressive or standing because the incident and edge
K. T. Holland, R. A. HolmanlMarine
Geology 134 (1996) 77-93
81
6 4
10
15
20
25
-----___ I
I
15
20
25
October 1994
Fig. 4. Significant wave height (top) and wave period (middle) measured in 8 m depth and tide level (bottom). Experiment interval indicated by dashed bar.
waves are phase locked. If however, the edge wave amplitude greatly exceeds that of the incident wave, as might be expected under storm conditions, the cusp spacing and edge wave length will no longer be equal. In this case, the edge wave must be standing for cusps to develop (as dominantly progressive motions exhibit no alongshore structure) and will result in an alongshore pertur-
bation of one-half the edge wave length via a mechanism analogous to the subharmonic model. To prevent confusion, we will restrict our use of the term synchronous to refer only to cusp formation conditions where cusp lengths and edge wave lengths are equal. Numerous authors have shown suggestions of edge waves association with observed beach cusp
82
K. T Holland, R A. HolmanlMarine Geology 134 (1996) 77-93
x [ml
x [ml
100 110 x [ml
120
Fig. 5. Topographic contours for October 15, 20, and 21. Offshore distances increase to the right (positive x). The foreshore in the region between the 1 and 2 m elevation contours was essentially featureless in the southern half of the study area on October 15. Cusps with a length of approximately 32 m appeared on October 17 and became well developed by October 20. Shorter scale cusps with spacings of approximately half that of the most well developed cusps were observed on October 21 (most obvious in the northern half of the region).
spacings using (2) and (3) (Huntley and Bowen, 1975, 1978; Sallenger, 1979; Guza and Bowen, 1981; Inman and Guza, 1982; Seymour and Aubrey, 1985). However, as pointed out by Inman and Guza (1982), several sources for potential error in the use of these equations exist. Foremost, field wave spectra are seldom monochromatic making identification of a specific incident wave period difficult. Since cusp wavelength scales as the square of the incident wave period, small errors in the incident period may result in relatively large errors in the predicted cusp wavelength. Additionally, the choice of beach slope is problematic in that natural beaches are rarely planar and the region over which a proper beach slope should be calculated is poorly defined. In practice, the relation between edge wave period and wavelength can be calculated numerically for natural topography (see Appendix) and often differs substantially
from the plane beach equations. The possibility of imprecision certainly does not negate the general agreement under a variety of conditions between observations and predictions using (2) and (3). However, the interpretation of these findings as conclusive proof of the edge wave hypothesis is quite possibly erroneous. A stronger confirmation of cusp formation by edge wave processes exists in the findings of Huntley and Bowen (1978). As part of a study of nearshore current velocities, these authors observed cusp formation during the measurement period of their experiment. Velocity spectra sampled approximately 10 m seaward of the shoreline showed energy at subharmonic frequencies preceding the formation of foreshore cusps. Although this subharmonic peak was not significant at the 95% confidence level, it persisted for several hours, was coherent between various sen-
83
K. T. Holland, R.A. HolmanjMarine Geology 134 (1996) 77-93
16 -
i 650
I
700
I
750
I
800 Alongshore
I
850 distance [m]
I
I
900
950
1000
Fig. 6. The measured l-m eontour in the study region as a function of cross-shore and alongshore distance stacked vertically by date (increasing down and denoted by the numbers). A quadratic alongshore trend has been removed. Vertical ticks are separated by 5 m.
sors, and exhibited a phase relationship between vertical and horizontal velocity components consistent with a mode zero, standing, subharmonic edge wave. Equally important was the fact that the mean spacing between cusps (12.7 m) matched the predicted wavelength of a cusp formed by a subharmonic edge wave with a period of around 14 seconds. What separates their study from other cusp-edge wave associations is the observation of energy at the appropriate frequency near the shoreline prior to cusp formation, rather than the more typical the assumption of edge wave presence through the use of the incident wave period. Yet Huntley and Bowen (1978) lacks the unambiguous identification of an alongshore wave pattern in the observed water motions corresponding to the measured cusp wavelength. It is possible that edge waves provide an initial longshore periodic perturbation in topography that further evolves due to feedback with the incident wave train. Indeed, we recognize that beach cusps may form by some, all, or none of
these models. Testable predictions involving the feedback theory of cusp development under natural conditions are still in development. However, we can formulate objective criteria for the verification of the edge wave hypothesis. Indications of beach cusp formation by edge waves include: ( 1) identification of a pronounced, and presumably significant, energy peak in a nearshore spectrum that precedes the formation of cusps; (2) correspondence between the calculated edge wave length, the period of the energy peak, and measured cusp spacing; and (3) observation of longshore magnitude and phase relationships in swash signals that correspond to the expected edge wave pattern. In this paper, we test the edge wave models following these criteria.
2. Field observations Detailed measurements of foreshore topography and swash elevations overlapping two sequences
K. T Holland, R. A. HolmanlMarine Geology 134 (1996) 77-93
84
Table 1 Predicted and observed cusp lengths Date 15 October, 16 October, 17 October, 18 October, 19 October, 20 October, 21 October,
1994 1994 1994 1994 1994 1994 1994
Z(s)
X,(m) [subharmonic]
j,(m) [synchronous]
x,(m)
10.9 kO.6 11.2+0.5 10.6kO.4 11.5kO.5 14.2k1.2 13.5kO.9 11.3kO.6
34.4 & 3.4 36.5k2.9 32.7k2.2 38.Ok2.8 55.1k6.9 50.8+5.0 36.7f3.6
17.2+ 1.9 18.4k1.6 16.3&-1.2 19.2& 1.6 29.7k3.5 26.9 _t 3.5 18.5k2.0
36 29 32 36 40 37 20
[observed]
Observed, average incident wave period, T,, predicted cusp spacings, A,, and observed cusp spacings, IC,.XCestimates calculated along the z = 1 m contour.
of cusp formation were made from October 15 to 21, 1994 as part of the Duck94 nearshore processes experiment at the US Army Corps of Engineers Field Research Facility (FRF) in Duck, NC. The Duck foreshore during this period was comprised of a medium sand (mean grain size of 1.5 phi) with a cross-shore slope of approximately 1:12. Relevant environmental data collected at the FRF (significant wave heights, peak periods, and tidal elevations) are shown in Fig. 4. Topography was sampled daily to a vertical accuracy of better than 2 cm using vehicles equipped with Global Positioning System (GPS) surveying instrumentation. Survey points were densely spaced (approximately every 3 m) within a 400 m longshore by 50 m cross-shore region. The established coordinate system was such that positive y designated northerly increases in the alongshore direction while positive x increased offshore. Swash time series were measured during daylight hours along 5 1 cross-shore transects spaced evenly within a subset of the above region. The total alongshore coverage was 250 m. Swash elevations were calculated from video recordings of the view from two cameras mounted approximately 45 m above mean sea level and positioned looking alongshore. Individual video frames were digitized to find the location of the swash edge, indicated by pixel intensity exceeding a time averaged threshold. The process was repeated every second for the length of standard video cassette tape (2 hours). The vertical resolution of this technique varies as a function of distance from the camera, but a typical vertical
resolution of 2 cm was obtained by projecting the average horizontal resolution to an elevation (for further details, see Lippmann and Holman (1989)). The beachface seaward of a set of extreme high tide cusps was to some extent flattened following extremely high wave conditions on the night of October 15, although, remnant alongshore irregularities from a time period preceding the experiment were still apparent. Foreshore cusps with a spacing (of approximately 38 m) mimicking that of the remnant cusps began to form throughout the study region on October 17 (which we will denote as event #l) and continued to develop through October 20 (Fig. 5). Since the resulting cusp lengths closely approximated the pre-existing, higher elevation cusp field, but had a slightly different alongshore phase, it is not certain whether the cusps associated with event #l developed independently of the remnant topography. To reduce subjectivity in cusp length measurements, we calculated average (alongshore) cusp spacings from the wavenumber (reciprocal wavelength) spectrum of the alongshore variation in horizontal position of the measured 1 m elevation contour (Fig. 6). The average cusp spacing, xc, was defined as the wavelength corresponding to the peak in spectral density. For event #l, the cusp spacing varied between 29 and 40 m (Table 1), with a mean cusp spacing during the last three days of the event (the portion with the most well developed cusps) of 38 m. We observed an approximate halving in the spacing of cusps on the lower beachface, a decrease to 20 m, on the morning of October 21, which will be denoted as event #2 (Figs. 5 and 6). The event #2
K. T Holland, R A. Holman/Marine
cusp structure is more clearly unrelated to the still remnant, high tide cusps.
3. Model tests Offshore wave spectra from an array of pressure sensors in 8 m depth from October 15 through 21 were examined to determine incident wave periods corresponding to both of the cusp formation events. In all cases, the wave spectra were narrowbanded, making identification of Ti (the incident wave period associated with the peak in spectral density) fairly simple. However, there was considerable variation in the peak period over time, therefore a daily mean period, Ti, and standard deviation, or, were calculated to display the range of possible periods. We also predicted cusp spacings, fi,, under the assumption of either synchronous or subharmonic mode zero edge waves with periods proportional to Ti. The relationship between edge wave period and wavelength (analogous to ( 1)) was found numerically for the alongshore averaged cross-shore profiles for each day (method fully explained in the Appendix). Under the hypothesis of edge wave forcing of cusp formation, comparison of these results (displayed in Table 1) with the average spacings observed for events #l and #2 (38 and 20 m, respectively), indicates that the alongshore wave structure most consistent with the measured cusp spacing and incident wave period during the first event was a subharmonic edge wave present before October 19 with a period of around 22 seconds and a wavelength of approximately 76 m. The most consistent structure for event #2 was a synchronous edge wave with an approximate period of 11 seconds and wavelength of 20 m occurring on October 21. Since the shorter wavelength cusps were first observed at 0700 EST on the morning of the October 21 and the change in incident wave period to 11 seconds did not occur until the late evening of October 20 (after 2300, Fig. 4), for this model to be correct, the cusp development must have occurred rapidly (i.e. within 8 hours). Synchronous and subharmonic edge wave periods responsible for cusp formation are typi-
Geology 134 (1996) 77-93
85
cally referenced in terms of the incident wave peak (because it represents the hypothesized forcing). As above, these mechanisms are usually tested by assuming an edge wave period related to the measured incident wave period at the time of the cusp observation, solving for a corresponding cusp wavelength and comparing the predicted wavelength with that observed. However, the usual test can also be reversed to predict the required edge wave period for the measured cusp length independently of Ti.Predicted edge wave periods, fek, corresponding to the mean cusp spacings of events #l and #2 determined in this manner are shown in Table 2. These results indicate that for event #l the subharmonic (L e = 2X,) and synchronous (L e = h,) edge wave periods corresponding to the measured cusp length of 38 m are approximately 23 and 16 s, respectively. For event #2 the same periods corresponding to a 20 m cusp spacing are 16 and 12 sec. We examined swash data runs between October 16 and 21 to verify the presence of edge wave energy at the appropriate periods for the two events. The runs, listed in Table 3, were selected as having swash ranges centered within the cusp region. [On October 18, human activity in the beachface region obstructed the camera view during the optimal sampling period which prevented accurate swash measurement. Results from the following run were substituted.] Autospectra and cross-spectra were calculated from demeaned and detrended time series. Seven ensembles of length 1024 s and 2 frequency bins were merged to yield 28 degrees of freedom and a frequency bandwidth of 0.002 Hz. The alongshore averaged swash spectrum for each run is given in Fig. 7. The approximate frequencies of subharmonic and synchronous edge waves consistent with the measured topography are indicated and the mean offshore wave period, Ti, is shown. The swash spectra show little evidence of dominant peaks in energy at frequencies associated with the predicted subharmonic. In fact, subharmonic structures associated with any higher frequency spectral peak are not obvious. For the runs on October 19, 20, and 21, there was an obvious peak in swash energy in close proximity to the predicted synchronous frequency. However
86
K. T. Holland, R. A. HolmanlMarine
a)
Geology 134 ( 1994) 77-93
October 16
October 17
lo*
7 1
l
t
-
10’
100
10“
c)
October 16
October 19
lo’~i
1
* +
10’ :
loo
e)
October 20 lo2
:
9
October 21
t 1’1
IO0 10’
x
t
?
;I:;-
10-l 0
0.05 0.1 Frequency [Hz]
0
0.05 0.1 Frequency [Hz]
Fig. 7. Alongshore averaged energy density as a function of frequency for the six swash runs (a-f). The 95% confidence interval for spectral estimates (assuming 28 degrees of freedom) is indicated by the vertical bar. Frequencies for hypothesized edge waves corresponding to the measured cusp spacings of the two events are marked by arrows (event #l arrows are black; event #2 arrows are gray). Subharmonic edge waves are indicated by the upward pointing arrows and synchronous edge waves are indicated by downward pointing arrows. The frequencies associated with the peak in offshore wave energy density are given by the asterisk.
as mentioned, given frequency spectra alone we cannot be certain that this energy corresponds to an appropriate edge wave structure. Measurements of the alongshore structure are required.
To test for evidence of edge waves of the appropriate length scales, we partitioned swash measurements into wave structures of a specified wavelength. Specifically, we modeled the observa-
K. T Holland, R. A. Holman/Marine Geology 134 (1996) 77-93
Table 2 Predicted edge wave periods, lengths, 1,
Event #l Event #2
Te, based on observed cusp
L(m)
PC(s) [subharmonic]
Fe(s) [synchronous]
36-40 20
22.3-23.6 16.4
1.5.6-16.4 11.7
Ott Ott Ott act Ott Ott
16 17 18 19 20 21
Time (EST)
Mean (m)
Maximum (m)
Minimum (m)
R,(m)
0648 0652 1034 0834 0830 0843
1.7 1.5 0.5 1.4 1.5 1.2
2.8 2.2 1.3 2.6 2.6 2.2
0.5 0.3 -0.3 0.1 0.2 0.4
2.9 2.4 2.0 3.2 2.8 2.8
Significant swash heights, R,, were defined as 4 x the measured, alongshore averaged, standard deviation in swash elevation.
tions, tj, following the procedure given by Haines (1987) as a linear combination of two complex modes propagating in opposite directions plus a residual _E _O=Ec(+e __ where I%= [exp
(iEy)
exp (-izp)]
(4)
E is a matrix
of two columns, each of which represents the complex form of an alongshore structure of wavelength, L., given at the alongshore locations contained in the vector, 21. The covariance matrix associated with the complex modal amplitudes, g, is given by (&i>=(E’E)-lE’(@‘)E(E’E)-l
diagonal elements represents the real valued amplitudes, a, and a2, of each wave. Squared coherence values, y2, indicate the degree of phase locking between the two modes and are given by:
(6)
Table 3 Statistics of swash motions Date
81
(5)
where ( ) indicates time averaging, ’ denotes transpose, and ( )-’ represents the matrix inverse. It can be shown (Haines, 1987) that this least squares solution is equivalent to fitting the crossspectral matrix, (00) at a given frequency where the solution A = (CM) is a 2 x 2, positive, semidefinite, Hermitian matrix. The square root of the
Perfectly phase locked modes indicative of a pure standing wave structure have a squared coherence of 1. The degrees of freedom of the observed crossspectral matrix can be used in the calculation of the significance level associated with the model. To identify edge wave structures corresponding to the expected edge wave periods listed in Table 2 (based on the cusp length observations), the observed cross-spectra were tested using the above model. We first looked for evidence of subharmonic, standing edge wave structures where the modeled edge wave length was defined at twice the observed cusp length. A frequency interval for edge waves consistent with the measured topography, l/r, +O.Ol Hz, was determined to account for possible errors in the dispersion relation resulting from inaccurate inputs (see Appendix). Squared coherence estimates above the 95% significance level (0.20 for 28 degrees of freedom) and within the appropriate frequency interval were interpreted as being suggestive of standing edge wave motions of the proper length scale. The results, shown in Fig. 8 and listed in Tables 4 and 5, indicate that for event #l, only one significant coherence value in the proper interval was observed (October 19). The amplitudes of the two associated edge waves were relatively small, approximately 1 cm. Also the period of the significant coherence did not coincide with an obvious energy peak in the spectrum. For event #2, large significant squared coherence values were observed near but certainly not centered within the expected frequency interval on October 20; however, the edge wave amplitudes were again small and the model was found to describe less than 20% of the observed variance. More importantly, there was no indication of this energy being associated with the first subharmonic of any dominant spectral peak. Considering these results, there was little evidence of cusp formation during either event due
K. T. Holland, R A. HolmanlMarine Geology 134 (1996) 77-93
88
October 16 L,: 76m II
b)
0.8
October 17 L,: 76m
r
+
0.6
I
+
October 18 L,: 76m
October 19 L,: 76m
4 0.8
r
+
0.8
0.4
October 20 L,+ 76m
e,
Lip----
V
October 20 L,: 40m
October 21 L,: 40m
I 0.8 0.6 0.4 0.2 n 0.05 Frequency
0.1 [Hz]
Frequency
[Hz]
Fig. 8. Squared coherence between oppositely propagating modes for subharmonic edge waves of wavelength L, as a function of frequency for the six swash runs. The 95% critical coherence level is indicated by the dashed line. Test results corresponding to the event #l cusp measurements are given in panels a-e. Event #2 results are displayed in panels f and g. Horizontal bars indicate the possible range of frequencies for edge waves consistent with the measured topography.
K. T Holland, R A. HolmanlMarine
Geology 134 (1996) 77-93
89
Table 4 Event #1 Edge wave characteristics Time
Date
T,
Subharmonic L e = 16 m Y2
Ott Ott Ott Ott Ott
16 17 18 19 20
0648 0652 1034 0834 0830
20.7
n/o n/o n/o 0.34 n/o
aI (cm)
1.3
Synchronous L e = 38 m a2 (cm)
1.2
T,(s)
Y’
al (cm)
a2 (cm)
a3 (cm)
14.7 15.2
nlo n/o n/o 0.58 0.62
2.9 3.5
1.3 2.0
7.8 1.4
T,, edge wave period corresponding to the maximum coherence level, y’, within the expected frequency interval. a,, a,, a3 represent the measured amplitudes of a progressive wave of wavelength L e propagating in the +y direction, the -y direction, and a normally incident wave, respectively. n/o = significant squared coherence value not observed.
Table 5 Event #2 Edge wave characteristics Date
Ott 20 Ott 21
Time
0830 0843
T, (s)
14.3 14.3
Y2
a1 (cm)
a2 (cm)
T, (s)
Y’
a1 (cm)
a2 (cm)
a3 (cm)
0.52 0.22
3.0 0.8
1.7 0.7
11.4
n/o 0.66
1.4
1.0
4.8
subharmonic edge waves. It is conceivable, however, that the high significant coherence levels on October 20 for event #2 occurred in association with a non-traditional mechanism of cusp formation involving swash energy at the incident peak. This possibility will be examined more fully in the discussion. We also used the edge wave model to test for synchronous edge waves with wavelengths equal to the measured cusp lengths. As described in the introduction, the synchronous mechanism requires that the edge wave structure (either standing or progressive) be phase locked to and of the same period as the incident wave field. For this reason, we added a third mode describing an alongshore homogeneous structure to the model described in (4). For event #l, evidence of synchronous, standing waves (indicated by high coherence values between the two edge wave forms and within the appropriate frequency interval) was only seen on October 19 and 20 (Table 4). Interestingly, these motions were also coherent with the incident wave indicating a standing phase locked structure. For event #2, there was evidence on October 21 of a similar coupling between two edge waves and the to standing,
Synchronous L e = 20 m
Subharmonic L c = 40 m
incident wave. However, in each of these cases, the edge wave amplitudes were exceedingly small compared to the significant swash height and were less than half the amplitude of the incident wave mode. In addition, no appropriate edge wave structures were evident prior to the formation of the cusp fields (i.e. before October 17 and 20). So it is unlikely that synchronous, standing wave energy formed the observed beach cusps. When we tested for individual progressive motions coupled to the incident wave field, there was only one example (the October 17 run) of a progressive wave significantly coherent with the incident wave field in which the wave propagating in the opposite direction was not significantly coupled to either the first edge wave or the incident wave (indicative of a synchronous, progressive wave). However, the amplitude of this wave was less than 1 cm and a corresponding spectral energy peak was not apparent, so we found no strong evidence of progressive, synchronous edge wave forcing either. In contrast to the lack of identifiable edge wave structures, we did find evidence in the October 20 swash data of the feedback circulation pattern.
90
K. T Holland, R. A. HolmanlMarine
-5’
’
780
I
I
800
820
1
840
Geology 134 (1996) 77-93
I
I
I
I
I
880
880
900
920
940
960
v[ml Fig. 9. Magnitude (top) and phase (middle) of the dominant empirical orthogonal function of the October 20, 0830 run at frequency 0.068 Hz versus (bottom) the cusp pattern shown by cross-shore and alongshore coordinates of the l-m contour (longshore trend removed). This mode explains 75% of variance of the observations at this frequency.
Fig. 9 shows the longshore magnitude and phase structure of the dominant, complex empirical orthogonal function (CEOF) of the cross-spectral matrix at the frequency corresponding to the peak of the spectrum (see Wallace and Dickinson, 1972; Holland et al., 1995). The CEOF method is somewhat similar to the least-squares model in that orthogonal wave structures are detected, although in this case without the constraint of an input wavelength. The observed magnitude and phase represent a strong interaction between the fluid motions and the existing cusp topography given that (local) maximum magnitudes correspond to the longshore locations of cusp horns. The phase relationships indicate that swash motions in the embayments lag motions at the horns by approximately 4 s at that frequency. These observations are identical to the feedback pattern in that the amplitude of swash bores is largest at the cusp horns and the phase at the horns leads the bays due to destructive interference in the embayments between the incoming bore and the preceding
downwash (Fig. 3). We emphasize that these topographic interaction results correspond to the timing of the most well developed cusp field and are therefore more indicative of cusp maintenance than cusp formation.
4. Discussion Our detailed measurements and careful testing showed little suggestion that either synchronous or subharmonic edge waves caused the observed cusp structures during this experiment. Obviously, it is much simpler to present convincing evidence in support of a specific model of cusp formation than it is to absolutely disprove the possibility of any particular mechanism. With only one observation of cusp development that obviously differed from the remnant topography (event #2), we lacked the necessary range of conditions to conclusively discount the feasibility of the edge wave mechanism.
K. T Holland, R A. HolmanlMarine
One possible explanation for the lack of prominent edge wave structures of the proper length scale is that edge wave induced topography may provide a negative feedback to the edge wave excitation process (Guza and Inman, 1975; Guza and Bowen, 1981). In other words, cusp growth may dampen edge wave excitation which makes the edge wave structure less obvious. In addition, the lack of obvious edge wave patterns could be due to a short lived initiation event that occurred apart from our sampling. For example, Evans (1945) and Komar (1973) observed cusp fields developing rapidly on the order of minutes. Regardless of the strength of our conclusions, we believe that the methods presented here stand as a considerable improvement over those previously utilized and that our findings emphasize the need for more cautious interpretation of prior results. We were intrigued by the relationship between the swash motions on October 20 and the cusps that developed on October 21. From these observations, there was evidence (Fig. 9) of a strong interaction between the well-developed event #l cusp field and the swash motions occurring within subharmonic frequency interval for event #2. We also note, for October 20, the approximate coincidence of the frequency associated with the maximum in swash spectral density, the frequency associated with the incident wave peak, and the predicted value of the event #2 subharmonic (Fig. 7e). Although we found little indication of this energy being associated with an edge wave form, we find interesting the possibility that energy standing in the alongshore at the wavelength of the topography could have been excited by topographic interaction at the frequency of the swash spectral peak. In fact, there was some evidence to suggest that this standing pattern did occur (Fig. 8f ). If this was indeed the case, then energy scattered from this interaction between swash and topography could exist as a free, standing edge wave given the proper dispersion relationship. As noted, the predicted event #2 subharmonic (h, =L,/2) frequency was close to the frequency of dominant interaction. We propose that a slight change in tidal level or incident wave period on
Geology 134 ( 1996) 77-93
91
the evening of October 20 caused a strong resonance that developed into a 40 m standing edge wave structure of larger amplitude than that measured earlier in the day, and that this structure in turn forced topography at one-half the edge wave length, corresponding to the cusps observed on October 21. This mechanism differs from the conventional Guza and Inman (1975) hypothesis in that the excitation of the edge wave did not result from fluid interactions at twice the incident wave period, but occurred in response to swash interaction with topography. In addition, this alternative mechanism of cusp formation is for the specific condition of a halving of the existing cusp spacing. We intend to pursue this possibility in future research.
5. Summary For two distinct intervals of cusp formation, we tested the hypothesis of cusp formation by synchronous or subharmonic mode zero edge waves in two ways. First, cusp wavelengths corresponding to both subharmonic and synchronous edge wave forcing were predicted given the observed incident wave period and compared to observed cusp spacings. These comparisons indicated that the most likely edge wave forcing (if any) was a subharmonic edge wave for the first event and a synchronous edge wave for the second event. However, the timing of the cusp events and the lack of identifiable subharmonic peaks in swash autospectra indicated that the presence of these particular wave structures was unlikely. Alternately, swash measurements separated in the longshore were analyzed for evidence of edge wave structures within an appropriate frequency interval irrespective of the incident wave period. For both events, the statistical connections between the measured swash motions and the predicted wave structures given the observed cusp wavelengths, were not significantly different from expectations for random noise. In other words, for these two events, there was no statistical support for cusp generation by the traditional subharmonic or synchronous edge wave mechanisms.
K T. Holland, R. A. HolmanJMarine Geology 134 (1996) 77-93
92
However, there were suggestions in the data of an unconventional mechanism for cusp generation. Close examination of the longshore structure of swash motions over well-developed cusps showed interaction at a wavelength equal to the cusp spacing but at the period of the incident swash. We propose that if this interaction occurred at a frequency satisfying the dispersion relation, free edge waves may have been resonantly forced, which, in turn, could explain the generation of cusps with a spacing of half the pre-existing cusp wavelength. This mechanism differs from the traditional hypothesis for the formation of cusps at half the edge wave length in that edge wave excitation at subharmonic frequencies is not required.
Acknowledgment
This research was supported by funding from the Office of Naval Research, Coastal Dynamics Program (N00014-9411196) and the US Geological Survey, National Coastal Geology Program (Cooperative agreement #1434-93-A-l 124).
Appendix The linear equation governing edge wave motions of the form, q(x,yJ)=agf(x) sin (ky-crt), as (htp&+tj
(
$ -k2h
>
=0
(AlI
where a is the edge wave amplitude, g is gravitational acceleration, f(x) represents a cross-shore structure, u =2x/T, k = 27t/L, h is the water depth, and x and y represent the offshore and alongshore spatial coordinates. If a plane beach topography of constant slope, g, is assumed, the dispersion relation c?=gk sin (2n+ l)S (equivalent to (1)) for edge wave motions can be derived analytically (Ursell, 1952). More often, however, we wish to find edge wave solutions on arbitrary topography, h(x, y). Typically these solutions are calculated numerically assuming h,=O (Holman and Bowen, 1979; Kirby et al., 1981; Howd et al., 1992). In the present study, numerical solutions for the relationship between o and k on arbitrary topography were found following the method of Holman and Bowen (1979). Equation al is expressed as two coupled first order differential equations, subject to the boundary conditions n(0) =flnite and n(co) = 0. These equations are solved using a Runge-Kutta technique
(Press et al., 1992) where h and n are expanded as h(x)=/31x+/&XZ+... ~(x)=~o(l+~,x+~L,xz+...)
(A2)
Substituting Eq. A2 into Al and equating coefficients of like powers of x yields the initial conditions at the shoreline rl(0)=ilO= given
(A3)
For any particular frequency, the corresponding value of k is found by maximizing the offshore distance at which the solution diverges from the deep water boundary condition. These values represent the o, k pairs satisfying the edge wave dispersion relation. Given that the choice of a shoreline location is integral to the solution results, a shoreline elevation of 1.5 m was selected as representative of the contour where most cusps were concentrated. Changes in a given frequency value resulting from +0.5 m change in the shoreline elevation were less than 0.005 Hz for frequencies between 0.05 and 0.15 Hz. In all cases, we solved only for mode zero. Multiple cross-shore transects were averaged to give a mean profile which was assumed homogeneous in the alongshore. However, it is obvious that this assumption is invalid when cusps are well-developed such that the effect of rhythmic variations in the alongshore topography would be to alter the dispersion relationships. Since the above method assumes alongshore inhomogeneities are negligible, we attempted to quantify the maximum error in the calculated dispersion relation, given the measured cusp field. Guza and Bowen (1981) present an analytic solution for the change in the edge wave frequency on an otherwise plane beach of slope g, with a superimposed longshore periodicity, h,, of half the edge wave length. This situation, h=hO+hI
where
ho = x tan /l and h, =f(x)
cos (2ky)
(A4)
results in a decrease of the natural frequency, oO, as given by: 1 dcma,aTk _~ tanp Z
(
2P 70+2)
>
where aT represents the cusp amplitude and cusp perturbations are defined to decay exponentially offshore at exp+‘kX.For a subharmonic edge wave existing on the well-developed cusp field on October 20 (cusp spacing= 38 m, estimated cusp amplitude = 34 cm and approximate cross-shore decay scale = 20 m), the solution indicates a maximum change in the natural edge wave frequency of about 15%.
K T. Holland, R. A. HolmanlMarine Geology 134 ( 1996) 77-93
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waves. In: K.E. Nauman (Editor), Proc. 16th Coastal Eng. Conf. ASCE, Hamburg, pp. 137881393. Inman, D.L. and Guza, R.T., 1982. The origin of swash cusps on beaches. Mar. Geol., 49: 133-148. Johnson, D.W., 1910. Beach cusps. Geol. Sot. Am. Bull., 21: 599-624. Kirby, J.T., Dalrymple, R.A. and Liu, P.L.F., 1981. Modification of edge waves by barred beach topography. Coastal Eng., 5: 35549. Komar, P.D., 1973. Observations of beach cusps at Mono Lake, California. Geol. Sot. Am. Bull., 84(11): 3593--3600. Kuenen, P.H., 1948. The formation of beach cusps. J. Geol., 56: 34-40. Lippmann, T.C. and Holman, R.A., 1989. Quantification of sand bar morphology: A video technique based on wave dissipation. J. Geophys. Res., 94(Cl): 995-1011. Longuet-Higgins, M.S. and Parkin, D.W., 1962. Sea waves and beach cusps. Geogr. J., 128(2): 194-201. Miller, J.R., Miller, S.M.O., Torzynski, C.A. and Kochel, R.C., 1989. Beach cusp destruction, formation, and evolution during and subsequent to an extratropical storm, Duck, North Carolina. J. Geol., 97(6): 7499760. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P., 1992. Numerical Recipes in C: the art of scientific computing. Cambridge Univ. Press, 994 pp. Russell, R.J. and McIntire, W.G., 1965. Beach cusps. Geol. Sot. Am. Bull., 76(3): 307-320. Sallenger Jr., A.H., 1979. Beach-cusp formation. Mar. Geol., 29( l-4): 23-37. Seymour, R.J. and Aubrey, D.G., 1985. Rhythmic beach cusp formation: A conceptual synthesis. Mar. Geol., 65(3-4): 289-304. Ursell, F., 1952. Edge waves on a sloping beach. F’roc. R. Sot. Ser. A, 214: 79-97. Wallace, J.M. and Dickinson, R.E., 1972. Empirical orthogonal representation of time series in the frequency domain, Part I: Theoretical considerations. J. Appl. Meteorol., 11(6): 887-892. Werner, B.T. and Fink, T.M., 1993. Beach cusps as self-organized patterns. Science, 260: 968-971. Williams, A.T., 1973. The problem of beach cusp development. J. Sediment. Petrol.. 43(3): 857-866.