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Formation of beach cusps in a wave tank
Coastal Engineering, 9 (1985) 81--98
81
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
FORMATION OF BEACH CUSPS IN A WAVE TANK
A. KANEKO
Research Institute for Applied Mechanics, Kyushu University 87, Kasuga, Fuhuoka 816 (Japan) (Received October 3, 1983; accepted for publication March 5, 1984)
ABSTRACT Kaneko, A., 1985. Formation of beach cusps in a wave tank. Coastal Eng., 9: 81--98. Beach cusps with a longshore spacing of 20 to 150 cm have been built by the continuous action of incident waves on a steep laboratory beach floor covered uniformly with a thin bed of glass beads. Breaking of incident waves was observed to induce vortices on the bed by interacting with swash motion along the beach face. Beach cusps formed when the value of a dimensionless parameter Hb/SgTi 2 became smaller than 0.042; H b is the breaking height of the incident waves, Ti their period, s the beach slope and g the acceleration due to gravity. This critical value occurred at a nearly central part of the generation region 0.003 < Hb/SgTi 2 < 0.068 for plunging breakers presented by Galvin (1968). Breaking-wave-induced vortices rather than breaker types controlled the movement of bed material in the nearshore zone. Most of the measured spacings of beach cusps, including previous observations, were in good agreement with half a wavelength of the zero-mode subharmonic edge wave, which is generated on the beach by the refraction of incident waves and has twice the period of the waves. The role of edge waves at each stage of cusp formation still remains as an important problem to be clarified.
INTRODUCTION
Beach cusps, prominent shoreline features with quasi-uniform longshore spacings, have attracted the interest of marine scientists and engineers for a long time (Komar, 1976). Although numerous mechanisms have been proposed for beach~usp formation, recent field observations provide evidence that edge waves, waves with longshore periodicity generated on a beach by refraction of incident waves, are the most plausible cause (Huntley and Bowen, 1973; Komar, 1973; Guza and Inman, 1975; Sallenger, 1979; Guza and Bowen, 1981; Inman and Guza, 1982). The better understanding of beach cusps, which are one of the most difficult nearshore problems, may be a key for establishing nearshore sediment dynamics. In this study, beach cusps are formed on a laboratory beach floor uniformly covered by a thin bed of glass beads. Flows induced thereon by breaking of incident waves are observed using a visualization technique and are related to bed profiles generated. The formation region and the
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© 1985 Elsevier Science Publishers B.V.
82
spacing of beach cusps are examined and compared with the data from previous observations. EXPERIMENTS
Experiments were carried out using a wave tank 250 cm long, 150 cm wide and 30 cm deep, equipped with a beach floor 200 cm long (Fig. 1). The slope (s) of the beach floor ranged from 0.081 to 0.141. Incident waves on the beach were formed by oscillating a wave-paddle-type generator with a m o t o r ~ r a n k system. The periods (Ti) of incident waves were adjustable in the range of 0.5 to 2.5 s. The side wails of the tank were made of transparent plastic to enable observation of bed profiles and flows on the beach floor. Glass beads of mean diameter (D) 0.028 cm and density (Ps) 2.43 g cm -3 were used as an erodible bed material instead of natural sands. When observing sedimentary processes on the beach floor, its surface was covered initially by a thin bed of the glass beads. Under such a condition, bed deformation cannot grow sufficiently to m o d i f y the initial condition of incident waves. The surface of the beach floor was also painted black to show clearly the movement of glass beads against the dark background. Flow visualization was made by tracing the paths of polystyrene beads of D = 0.15 cm and Ps = 1.04 g cm -3, which were suspended in the water from the bed by the action of breaking waves. The direction of mass transport flow close to the bed can also be estimated by observing the movem e n t of glass beads. The bed surface and a vertical slice of the flow field were illuminated with a 1-kW light projector. Photographs were taken by a 35-mm camera from the positions shown in Fig. 1. (o)
(e)
(e) Fig. 1. Sketch of experimental set-up: (a) motor; (b) wave generator; (c) model beach; (d) projector; (e) camera.
FORMATION
P R O C E S S O F CUSPS
W h e n surface waves climb on a beach of steep slope, the wave steepness increases continuously with the growth of the wave-height and the sharpened
83
wave surfaces break d o w n rapidly at a short distance from the shore. We shall define the breaking point as the position where the height of waves climbing on a beach becomes maximum and the plunging point as the position where the forward face of breaking waves impinges on the water surface ahead. For convenience, we shall use the terms offshore and nearshore zone, respectively, for the seaward and the shoreward sides of a breaking point. The nearshore zone is further divided into breaker and swash zones at a plunging point. It should be noted that no surf zone exists on a steep beach. In all photographs presented below, the right-hand direction indicates shoreward. Figure 2 shows the formation process of beach cusps obtained with the experimental conditions of s = 0.081, T i = 1.15 s, H b = 3.5 cm and h b = 4.0 cm, where H b is the height of breaking waves and h b the water depth at a breaking point. Bed features were photographed over a side wall o f the tank after the wave generator was stopped temporarily at each stage of cusp formation. Figure 2a shows the initial state of the glass-bead bed, a b o u t 0.2 cm thick over the nearshore zone of 49 cm wide. The dotted line indicates the shoreline for still water, and a bar at t o p right is 10 cm long. Shortly after breaking, waves begin to attack the beach, glass beads near the plunging point disappear and ripple marks form at the breaker zone (Fig. 2b). As time proceeds, glass beads in the ripple marks gradually move shoreward, forming a sedimentary zone parallel to the shoreline just at the seaward side of the plunging point (Figs. 2c and d). During this stage, the amounts o f glass beads piled up on the beach face increase successively. Such a transport process of bed materials means that the direction of mass transport flow along the bed is shoreward at the whole nearshore zone. The sedimentary zone near the plunging point begins to diminish with the disappearance of offshore glass beads, and the ridge of glass beads on the beach face is suddenly cut by a series o f narrow channels equi-spaced in the longshore direction (Fig. 2e). Finally a clear cusp-like feature forms (Fig. 2f). The run-up length of water had a m a x i m u m at the central point of the ridges and minimum at the channels. Therefore, water running up on the ridges turns back along the channels, forming strong rip currents. The ridge-channel system and the water circulation pattern have often been observed in wave tanks (Tamai, 1974; Matsunaga and Honji, 1981). In the field, Sallenger (1979) observed that a similar ridge-channel system generated in flood-tide changes to ordinary beach cusps after the m o u t h of the channels was eroded widely on ebb tide. It will, therefore, be reasonable to call such a ridge-channel system the beach cusp. Kaneko (1983) showed through a numerical analysis that horizontal circulation systems are induced by the periodic undulation of swash m o t i o n due to edge waves. This circulation may also play a role in the initiation of beach cusps. Figure 3 shows the formation process of bed profiles obtained at the condition of s = 0.081, T i = 0.62 s , H b = 3.3 cm and h b = 4.3 cm. Photographs were taken from the same direction as in Fig. 2 after the
O0
85
F i g . 2. F o r m a t i o n p r o c e s s o f b e a c h c u s p s ; s = 0 . 0 8 1 , T l = 1 . 1 5 s, H b = 3 . 5 c m a n d h b = 4 . 0 c m . ( a ) ~ = 0 s; ( b ) t = 3 1 s ; ( c ) t ffi 8 7 s; ( d ) t = 2 2 3 s; ( e ) t ffi 4 8 8 s; (f) t ffi 7 8 4 s.
86
Fig. 3. F o r m a t i o n
process of a longshore bar; s ffi 0.081, T i = 0.62 s, H b -- 3.3 c m t -~ 5 1 s ; ( c ) t = 3 1 1 s.
h b = 4 . 3 c m . ( a ) t -- 0 s ; ( b )
and
87 wave generator was temporarily stopped. Compared with the previous experiment, T i is remarkably reduced. Figure 3a shows the initial state o f the glass-bead bed, a b o u t 0.2 cm thick over the nearshore zone of 54 cm width. The d o t t e d line indicates the shoreline for still water, and a bar at t o p right is 10 cm. As soon as breaking waves begin to attack the bed, glass beads in the swash zone move rapidly seaward and those in the breaker zone move gradually shoreward, forming ripple marks (Fig. 3b). As a result, a sedimentary zone of glass beads forms along the plunging point in an equilibrium state (Fig. 3c). We shall next discuss the result of further experiments carried out to clarify the cause of the transition of the bed profiles from Figs. 2 to 3. Figure 4 shows the results obtained at the condition of s = 0.081, T i = 1.68 s, H b = 3.7 cm and h b = 3.9 cm under which beach cusps form. Photographs were taken horizontally through a side wall of the tank. The vertical slice at a distance of 10 cm from the side wall was projected. Figure 4a shows the profile of the water surface at the instance when a wave climbing on the beach breaks d o w n just beneath a horizontal scale of 4 cm. Figure 4b shows the anticlockwise-rotating vortex induced in the same position immediately after the breaking wave impinged on the forward water surface. This vortex, called the backwash vortex, was first found b y Matsunaga and Honji (1980) to form on a steep beach in a wave tank. The backwash from water running up on the beach face is responsible for this vortex formation. It is well known that the plunge of breaking waves into water induces clockwise-rotating vortices, called the plunge vortex (Miller, 1976). This experiment has a large swash length since the period o f breaking waves is longer than that of swash motion. Therefore, the backwash vortex can develop to overcome the plunge vortex. Figure 4c shows the sedimentary bed profile formed along a plunging point at the stage similar to Fig. 2d. This bed profile makes a step-like feature composed of a steep seaward face and a nearly horizontal top and will be called the longshore step in this paper. Although this step diminishes in an equilibrium state, as seen in Figs. 2e and f, on real beaches with abundant offshore sands the longshore step may continue to form. This kind of bed profile has often been observed on steep beaches (Bagnold, 1940; Horikawa, 1978; Wright et al., 1979). In comparison with Figs. 4b and c, the backwash vortex and the longshore step are almost at the same place. It is suggested that the backwash vortex is the cause of the step formation. A flow field induced b y the vortex also provides an explanation for the shoreward transport o f bed materials over the nearshore zone in Fig. 2. In the offshore, the wave-induced mass transport flow of Longuet-Higgins (1953) m a y play a role in transporting bed materials shoreward. Taking into account these observations, we can sketch the overall pattern of mass transport flow forming a longshore step and beach cusps, as in Fig. 5. Figure 6 shows the results obtained at almost the same conditions (s -- 0.081, T i = 0.62 s, H b = 3.3 cm and h b = 4.3 cm) as those for Fig. 3. The m e t h o d Of photography
88
Fig. 4. F l o w and b e d profile b e n e a t h t h e breaking wave f o r m i n g beach c u s p s ; s = 0.081, T i ffi 1.68 s, H b = 3.7 c m and h b = 3.9 cm. (a) Profile o f t h e breaking wave. (b) Anticlockwise v o r t e x i n d u c e d at t h e p l u n g i n g p o i n t . (c) L o n g s h o r e s t e p built along t h e plunging p o i n t .
89
and light projection was the same as that for Fig. 4. Figure 6a shows the profile of a water surface just before a breaking wave occurs beneath a horizontal scale o f 4 cm. An instantaneous streamline pattern also becomes visible in the wave b y tracing the path of polystyrene beads suspended in Backwash
vortex
Fig. 5. Sketch of mass transport flow forming a longshore step and beach cusps. water. The streamline pattern is analogous to that in the standing wave of constant water depth (Lamb, 1932). Breaking waves on a steep beach seem to be considerably influenced b y reflection from the beach face. A similar streamline pattern could also be seen to some extent in the observations by Wiegel (1964) and Wang et al. (1982). Figures 6b and c show the flow fields near a plunging point photographed at a short time interval immediately after the impingement of the breaking wave was complete. A short time elapses between Figs. 6b and c. In both cases, flows near the plunging point construct a pair of vortices, a clockwise, shoreward vortex and an anticlockwise, seaward one. A sedimentary bed profile built on the bed between these vortices is shown in Fig. 6d. In contrast to Fig. 4c, the bed profile has a steeper shoreward than seaward face. This property is representative of the longshore bar, which has been observed on real beaches. It will, therefore, be appropriate to call such a bed profile the longshore bar. A steeper shoreward face of the bar implies that the shoreward vortex is more intense than the seaward one. When the period of breaking waves is shorter than that o f swash m o t i o n along the beach face, the swash length becomes small. In this case, the backwash vortex cannot overcome the plunge vortex. In Figs. 6b and c, the clockwise and the anticlockwise vortices m a y be regarded as the plunge and the weakened backwash vortices, respectively. Taking these observations into account, we can also sketch the overall pattern of mass transport flow forming a longshore bar, as in Fig. 7.
C~
91 6dckwosh vortex
Plunge
vortex
Fig. 7. Sketch of mass transport flow forming a longshore bar.
FORMATION
R E G I O N O F CUSPS
In the present experiment, bed profiles generated from a thin glass-bead bed on a laboratory beach floor. Because of the small a m o u n t of glass beads used, the resulting bed deformation cannot develop sufficiently to change the initial conditions of incident waves, the height of breaking waves and the breaking point. Under such a condition, the formation region o f beach cusps m a y be discussed in relation to breaker types on a rigid plane beach. Galvin (1968) reported from wave-tank experiments that breaker types are controlled b y a dimensionless parameter Hb/sgTi 2. The values of Hb/gTi 2 obtained from the present experiment and the field observation of Longuet-Higgins and Parkin {1962) are plotted in Fig. 8 against those of s. Most of the plotted data for beach cusps are seen to collapse in the region:
Hb/sgTi 2 < 0.042
(1)
the b o u n d a r y of which is drawn with a solid line in Fig. 8. According to Galvin's study, the generation regions for surging, plunging and spilling breakers are respectively, Hb/sgTi 2 < 0.003, 0.003 < Hb/sgTi 2 < 0.068 and 0.068 < Hb/sgTi 2. The corresponding boundaries for the breaker types are also drawn with d o t t e d lines in Fig. 8. It should be noted that the critical line for eq. (1) passes through a nearly central part o f the generation region for plunging breakers. This means that the cusp formation has only a weak correlation with breaker type. Attempts to relate breaker types to the offshore wave parameter Ho/Lo, the ratio o f the height of deep water waves
Fig. 6. F l o w and bed profile beneath the breaking wave f o r m i n g a longshore bar; s = 0 . 0 8 1 , T i = 0 . 6 0 s, H b ffi3.2 c m and h b = 3.9 cm. (a) Profile o f the breaking wave; (b) V o r t e x pair induced at the plunging p o i n t ; (c) V o r t e x pair at a slightly later stage than b. (d) L o n g s h o r e bar built along the plunging point.
92
to their wavelength, have also been made (Galvin, 1968; Battjes, 1974). In this study, L0 is determined from the dispersion relation of deep-water waves a s :
Lo = gTi2/2~
(2)
and Ho, from the empirical relation of Komar and Gaughan (1973) connecting H b with H0 and L0, as: H0 = 2.05 (Hb/Lo)
(3)
1/4
The values of HolLo for the present experiment are also plotted in Fig. 9 against those o f s. Using eqs. (2) and (3), we can rewrite eq. (1) as:
Ho/Lo < 0.39
(4)
s s/+
The boundary of this region is drawn with a solid line in Fig. 9. Using offshore parameters, Galvin (1968) also gave the generation regions Ho/s2Lo < 0.09, 0.09 < Ho/s2Lo < 4.8 and 4.8 < Ho/s2Lo for surging, plunging and spilling breakers. The inverse of the square root of Ho/s2Lo is known as a dimensionless parameter controlling nearshore curreat systems and called the surf similarity parameter by Battjes (1974). The boundaries for breaker types are drawn with dotted lines in Fig. 9. Similarly to Fig. 8, the boundary of the cusp region is at a nearly central part of the region for plunging breakers. From experiments using a large wave tank, Tamai
cusp o Author • Longuet-Higglns & 0Y Parkin (1962) 4, bar • Author // ~Ob. ~
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• bar
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,.O0~
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/o X 8
~_-
,,;.~
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Fig. 8. The values of H b / s g T i ~ plotted against those of s. Fig. 9. The values of H o / L o plotted against those of s.
ii II
ii
0
/
Ho. ~o" 0"09s2
/
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93
(1974) gave the following empirical relation as a condition for cusp formation: H o / L o < 0.04
(5)
The boundary of this region is also drawn in Fig. 9. All the data for beach cusps in the present experiment satisfy the above condition. SPACING O F CUSPS
We shall here discuss the spacing of beach cusps compared with the wavelength of the edge waves excited on a beach. According to Ursell's t h e o r y (1952), the wavelength (Le) of edge waves can be expressed as: L e = g T e 2 sin [ ( 2 n + l ) s ] / 2 ~
(6)
where T e and n are the period and the offshore modal number of the edge waves, respectively. Edge waves on a beach may be classified into the two types of subharmonic and synchronous depending on whether the period of edge waves is twice or the same as that of incident waves on the beach. Guza and Davis (1974) predicted theoretically that the subharmonic edge wave of m o d e n = 0 is most easily excited b y incident waves. Huntley and Bowen (1973) and Guza and Bowen (1981) also found that in the p o w e r spectra of nearshore currents, the zero m o d e subharmonic edge wave exists on real beaches. When the longshore periodicity of small amplitude due to edge waves is superimposed on the uniform swash m o t i o n due to incident waves, the profiles of run-up and run
i!
•-.
Te/8
i=.
run-down
Te/2
5Te/8 I
Le
Le/2
0
I
Le
Le/2
3Te/8
run - up
(Q)
7 Te / 8
i i
r u n - up
3 Te/2
(b)
Fig. 10. Sketch of run-up and run-down profiles in plan view. The phases of oscillation are inscribed near the corresponding profiles. (a) Subharrnonic edge wave. (b) Synchronous edge wave.
95
the periodicity of swash motion, the spacing of beach cusps may be expected to be half a wavelength of subharmonic edge waves or a wavelength of synchronous edge waves. Using eq. (6) and T i instead of Te, we can express the calculated spacing (Lcal) o f cusps as: L c a 1 = L e / 2 = g T i 2 (sin s)/lr
(7)
for the zero m o d e subharmonic edge wave and: L c a 1 = L e = g T i 2 (sin s ) / 2 ~
(8)
for the zero m o d e synchronous edge wave. Figure 11 shows the equilibrium form of beach cusps with various spacings obtained in the present experiment. In all the photographs, a bar at t o p right is 10 cm. Similarly to Fig. 2, these beach cusps have regular ridgechannel systems. Using the width (B) of the wave tank and the total num1
TABLE
N u m e r i c a l d a t a o f b e a c h cusps Exp.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
s
0.081 0.081 0.081 0.081 0.081 0.081 0.081 0.107 0.107 0.107 0.107 0.107 0.107 0.107 0.118 0.141 0.141
Ti
Hb
hb
Hb/gTi 2
gTi2 sin s/Ir
Lob s
(s)
(cm)
(cm)
(× 10 -3)
(cm)
(cm)
2.38 1.72 1.71 1.58 1.27 1.15 0.85 1.11 1.06 1.01 0.98 0.94 0.88 0.76 1.38 1.38 0.98
3.4 3.1 2.7 3.3 4.0 3.5 3.5 3.5 3.2 3.3 3.4 3.0 3.0 3.5 3.6 2.6 4.2
3.1 3.1 2.8 3.3 3.5 4.0 3.0 3.5 3.3 3.0 2.9 2.6 2.8 3.1 3.6 2.8 3.2
0.61 1.08 0.94 1.35 2.53 2.70 4.94 2.90 2.93 3.30 3.61 3.50 4.04 6.18 1.93 1.39 4.46
143 74 74 63 41 33 18 41 37 34 32 29 26 19 70 83 42
150 75 50 75 50 38 30 30 38 25 30 30 25 21 75 75 38
Fig. 11. Equilibrium f o r m s of beach cusps with various spacings. (a) L o b s = 1 5 0 c m ; s = 0.081, T i = 2.38 s, H b = 3.4 c m a n d h b = 3.1 c m . (b) L o b s = 3 8 c m ; s = 0.141, T i = 0.98 s, H b = 4.2 c m a n d h b = 3 . 2 c m . (C) L o bs = 21cm;s = 0.107, T i = 0.76 s , H b = 3.5 c m a n d h b = 3.1 cm.
96 bet (N) o f the ridges, we can obtain the observational spacing (Lobs) of cusps as: Lob s = B / ( N
-
-
1)
(9)
The numerical data of beach cusps for the present experiment are presented in Table 1. The value of Lob s is in good agreement with that of Lca1. Although subharmonic edge waves have the swash profile as sketched in Fig. 10a, such a longshore undulation of swash motion was visible only in the experiment for Fig. l l a with the largest cusp spacing of 150 cm. In other cases, edge waves excited may be t o o weak to be f o u n d in swash motion. The values of Lob s for the laboratory experiments of Ann (1979), Tamai (1980) and the author, and the field observations of Longuet-Higgins and Parkin (1962), Komar (1973), Sallenger (1979), Guza and Bowen (1981) and Takeda and 8unamura (1982) are plotted in Fig. 12 against that of gTi 2 (sin s)/~. The value of Lcal for subharmonic and synchronous edge waves are also drawn with solid and d o t t e d lines, respectively. All the laboratory data and most of the field data have a best fit with the solid line. However, there are some data of Longuet-Higgins and Parkin, Komar and Sallenger which show a scatter around the d o t t e d line. It should be noted that the data of Longuet-Higgins and Parkin collapse in the neighb o r h o o d of the line H b / s g T i 2 = 0.003 in Fig. 8. A critical value of H b / sgTi 2 beneath which subharmonic edge waves are replaced b y synchronous edge waves m a y exist. I
~ .....
subharmonic synchronous
60~,/~ c~ ~ / 6 //
edge wave edge wave
c~ I I0 I
0
//
// 0
e ° / / e o ~ _
/
e / / i I 0 0 O0 0 0/// • // Field data /' 0 L onguet- Hl~Jlns • /// a Park In(1962) // • KomorU973) / e sanenger ¢1979) /~ e • Guza a
-i0 0 •
• • /
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/// /
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Ill
Bowen(19Sl) ~ Tanaka S Sunomura (1982) Laboratory data
o Ann tl979) o Tamai (1980) • Author
//" / L¢ / i 0 -I
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,011
iOo
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I
I
I
I I I1~
gT~sin s/IT (cnl)
I
I
I
!
I 1
I0 I
Fig. 12. The values of Lobs plotted against those of gT i2 (sin s)/~r.
97 CONCLUSIONS Th e f o r m a t i o n process o f beach cusps has been observed in detail on a l a b o r a t o r y beach. Flows forming the beach cusps are examined and their characteristics are c o m p a r e d with the result o f the previous studies. Main results obtained are summarized as follows: (1) Beach cusps form at the beach face when bed materials in a nearshore zone move shoreward forming a longshore step along the plunging point. (2) When bed materials, b o t h in the breaker zone and in the swash zone, move to the plunging point, beach cusps cannot develop and a longshore bar forms along this point. (3) Vortices induced on t he bed by breaking waves c o n t r o l t he movem e n t o f bed materials at a nearshore zone. (4) The transition of bed profiles from the beach cusp t o the longshore bar occurs at H b / s g T i 2 = 0.042 or H o l L o = 0.39 s s/4 which corresponds to a nearly central part of t h e generation region for plunging breakers on a rigid plane beach. (5) Most o f the observational spacings of beach cusps agree well with half a wavelength of t he zero m o d e subharmonic edge wave excited on the beach. A relatively large scatter of the field data m ay be explained consistently by considering the existence of beach cusps generated by the zero-mode s y n chr onous edge wave. ACKNOWLEDGMENTS I would like to express m y sincere thanks to Prof. H. Honji and Drs. A. Masuda and N. Matsunaga f or useful suggestions and t o Mr. Y. Shiraishi for technical assistance.
REFERENCES
Ann, K., 1979. Experimental study on the formation of rythmic shorelines. Master's Thesis, University of Tokyo, Tokyo, 135 pp. (unpubl., in Japanese). Bagnold, M.R.A., 1940. Beach formation by waves: some model-experiments in a wave tank. J. Inst. Civ. Eng., 15: 27--52. Battjes, J.A., 1974. Surf similarity. Proc. 14th Conf. Coastal Eng., Copenhagen, pp. 466--480. Galvin, C.J., 1968. Breaker type classification on three laboratory beaches. J. Geophys. Res., 73: 3651--3659. Guza, R.T. and Bowen, A.J., 1981. On the amplitude of beach cusps. J. Geophys. Res., 86: 4125--4132. Guza, R.T. and Davis, R.E~ 1974. Excitation of edge waves by waves incident on a beach. J. Geophys. Res., 79: 1285--1291. Guza, R.T. and Inman, D.L., 1975. Edge waves and beach cusps. J. Geophys. Res., 80:
2997--3012.
98 Horikawa, K., 1978. Coastal Engineering: An Introduction to Ocean Engineering. Univ. T o k y o Press and A. Halstead Press, Tokyo, 402 pp. Huntley, D.A. and Bowen, A.J., 1973. Field observations of edge waves. Nature, 243: 160--162. Inman, D.L. and Guza, R.T., 1982. The origin of swash cusps on beaches. Mar. Geol., 49: 133--148. Kaneko, A., 1983. A numerical experiment on nearshore circulation in standing edge waves. Coastal Eng., 7: 271--284. Komar, P.D., 1973. Observations of beach cusps at Mono Lake, California. Geol. Soc. Am. Bull., 84: 3593--3600. Komar, P.D., 1976. Beach Processes and Sedimentation. Prentice-Hall, EnglewoodCliffs, N.J., 427 pp. Komar, P.D. and Gaughan, M.K., 1972. Airy wave theory and breaker height prediction. Proc. 13th Conf. Coastal Eng., Vancouver, pp. 405--418. Lamb, H., 1932. Hydrodynamics. Camb. Univ. Press, Cambridge, 738 pp. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. London, Ser. A, 245: 535--581. Longuet-Higgins, M.S. and Parkin, D.W., 1962. Sea waves and beach cusps. Geogr. J., 128: 194--201. Matsunaga, N. and Honji, H., 1980. The backwash vortex. J. Fluid Mech., 99: 813-815. Matsunaga, N. and Honji, H., 1981. A petaline sand pattern along the shoreline. Rep. Res. Inst. Appl. Mech., 29: 1--12. Miller, R.L., 1976. Role of vortices in surf zone prediction: sedimentation and wave forces. In: R.A. Davis Jr. and R.L. Ethington (Editors), Beach and Nearshore Sedimentation. Spec. Publ. Soc. Econ. Paleontol. Mineral., Tulsa, 24: 92--114. Sallenger, A.H., 1979. Beach-cusp formation. Mar. Geol., 29: 23--37. Tamai, S., 1974. Study on the formation of beach cusps. Proc. 21st Japanese Conf. Coastal Eng., Sendai, pp. 115--120 (in Japanese). Tamai, S., 1980. Study on the characteristics of beach cusps and the prediction of shoreline changes. Doctor's Dissertation, K y o t o University, Kyoto, 197 pp. (in Japanese). Takeda, I. and Sunamura, T., 1982. F o r m a t i o n and wavelength of beach cusps. Proc. 29th Japanese Conf. Coastal Eng., Sendai, pp. 319--322 (in Japanese). Ursell, F., 1952. Edge waves on a sloping beach. Proc. R. Soc, London, Ser. A, 214: 79--97. Wang, H., Sunamura, T. and Hwang, P.A., 1982. Drift velocity at the wave breaking point. Coastal Eng., 6 : 121--150. Wiegel, R.L., 1964. Oceanographical Engineering. Prentice-Hall, Englewood-Cliffs, N.J., 532 pp. Wright, L.D., Chappell, J., Thorn, B.G., Bradshaw, M.P. and Cowell, P., 1979. Morphodynamics of reflective and dissipative beach and inshore systems: Southeastern Australia. Mar. Geol., 32: 105--140.
81
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
FORMATION OF BEACH CUSPS IN A WAVE TANK
A. KANEKO
Research Institute for Applied Mechanics, Kyushu University 87, Kasuga, Fuhuoka 816 (Japan) (Received October 3, 1983; accepted for publication March 5, 1984)
ABSTRACT Kaneko, A., 1985. Formation of beach cusps in a wave tank. Coastal Eng., 9: 81--98. Beach cusps with a longshore spacing of 20 to 150 cm have been built by the continuous action of incident waves on a steep laboratory beach floor covered uniformly with a thin bed of glass beads. Breaking of incident waves was observed to induce vortices on the bed by interacting with swash motion along the beach face. Beach cusps formed when the value of a dimensionless parameter Hb/SgTi 2 became smaller than 0.042; H b is the breaking height of the incident waves, Ti their period, s the beach slope and g the acceleration due to gravity. This critical value occurred at a nearly central part of the generation region 0.003 < Hb/SgTi 2 < 0.068 for plunging breakers presented by Galvin (1968). Breaking-wave-induced vortices rather than breaker types controlled the movement of bed material in the nearshore zone. Most of the measured spacings of beach cusps, including previous observations, were in good agreement with half a wavelength of the zero-mode subharmonic edge wave, which is generated on the beach by the refraction of incident waves and has twice the period of the waves. The role of edge waves at each stage of cusp formation still remains as an important problem to be clarified.
INTRODUCTION
Beach cusps, prominent shoreline features with quasi-uniform longshore spacings, have attracted the interest of marine scientists and engineers for a long time (Komar, 1976). Although numerous mechanisms have been proposed for beach~usp formation, recent field observations provide evidence that edge waves, waves with longshore periodicity generated on a beach by refraction of incident waves, are the most plausible cause (Huntley and Bowen, 1973; Komar, 1973; Guza and Inman, 1975; Sallenger, 1979; Guza and Bowen, 1981; Inman and Guza, 1982). The better understanding of beach cusps, which are one of the most difficult nearshore problems, may be a key for establishing nearshore sediment dynamics. In this study, beach cusps are formed on a laboratory beach floor uniformly covered by a thin bed of glass beads. Flows induced thereon by breaking of incident waves are observed using a visualization technique and are related to bed profiles generated. The formation region and the
0378-3839/85/$03.30
© 1985 Elsevier Science Publishers B.V.
82
spacing of beach cusps are examined and compared with the data from previous observations. EXPERIMENTS
Experiments were carried out using a wave tank 250 cm long, 150 cm wide and 30 cm deep, equipped with a beach floor 200 cm long (Fig. 1). The slope (s) of the beach floor ranged from 0.081 to 0.141. Incident waves on the beach were formed by oscillating a wave-paddle-type generator with a m o t o r ~ r a n k system. The periods (Ti) of incident waves were adjustable in the range of 0.5 to 2.5 s. The side wails of the tank were made of transparent plastic to enable observation of bed profiles and flows on the beach floor. Glass beads of mean diameter (D) 0.028 cm and density (Ps) 2.43 g cm -3 were used as an erodible bed material instead of natural sands. When observing sedimentary processes on the beach floor, its surface was covered initially by a thin bed of the glass beads. Under such a condition, bed deformation cannot grow sufficiently to m o d i f y the initial condition of incident waves. The surface of the beach floor was also painted black to show clearly the movement of glass beads against the dark background. Flow visualization was made by tracing the paths of polystyrene beads of D = 0.15 cm and Ps = 1.04 g cm -3, which were suspended in the water from the bed by the action of breaking waves. The direction of mass transport flow close to the bed can also be estimated by observing the movem e n t of glass beads. The bed surface and a vertical slice of the flow field were illuminated with a 1-kW light projector. Photographs were taken by a 35-mm camera from the positions shown in Fig. 1. (o)
(e)
(e) Fig. 1. Sketch of experimental set-up: (a) motor; (b) wave generator; (c) model beach; (d) projector; (e) camera.
FORMATION
P R O C E S S O F CUSPS
W h e n surface waves climb on a beach of steep slope, the wave steepness increases continuously with the growth of the wave-height and the sharpened
83
wave surfaces break d o w n rapidly at a short distance from the shore. We shall define the breaking point as the position where the height of waves climbing on a beach becomes maximum and the plunging point as the position where the forward face of breaking waves impinges on the water surface ahead. For convenience, we shall use the terms offshore and nearshore zone, respectively, for the seaward and the shoreward sides of a breaking point. The nearshore zone is further divided into breaker and swash zones at a plunging point. It should be noted that no surf zone exists on a steep beach. In all photographs presented below, the right-hand direction indicates shoreward. Figure 2 shows the formation process of beach cusps obtained with the experimental conditions of s = 0.081, T i = 1.15 s, H b = 3.5 cm and h b = 4.0 cm, where H b is the height of breaking waves and h b the water depth at a breaking point. Bed features were photographed over a side wall o f the tank after the wave generator was stopped temporarily at each stage of cusp formation. Figure 2a shows the initial state of the glass-bead bed, a b o u t 0.2 cm thick over the nearshore zone of 49 cm wide. The dotted line indicates the shoreline for still water, and a bar at t o p right is 10 cm long. Shortly after breaking, waves begin to attack the beach, glass beads near the plunging point disappear and ripple marks form at the breaker zone (Fig. 2b). As time proceeds, glass beads in the ripple marks gradually move shoreward, forming a sedimentary zone parallel to the shoreline just at the seaward side of the plunging point (Figs. 2c and d). During this stage, the amounts o f glass beads piled up on the beach face increase successively. Such a transport process of bed materials means that the direction of mass transport flow along the bed is shoreward at the whole nearshore zone. The sedimentary zone near the plunging point begins to diminish with the disappearance of offshore glass beads, and the ridge of glass beads on the beach face is suddenly cut by a series o f narrow channels equi-spaced in the longshore direction (Fig. 2e). Finally a clear cusp-like feature forms (Fig. 2f). The run-up length of water had a m a x i m u m at the central point of the ridges and minimum at the channels. Therefore, water running up on the ridges turns back along the channels, forming strong rip currents. The ridge-channel system and the water circulation pattern have often been observed in wave tanks (Tamai, 1974; Matsunaga and Honji, 1981). In the field, Sallenger (1979) observed that a similar ridge-channel system generated in flood-tide changes to ordinary beach cusps after the m o u t h of the channels was eroded widely on ebb tide. It will, therefore, be reasonable to call such a ridge-channel system the beach cusp. Kaneko (1983) showed through a numerical analysis that horizontal circulation systems are induced by the periodic undulation of swash m o t i o n due to edge waves. This circulation may also play a role in the initiation of beach cusps. Figure 3 shows the formation process of bed profiles obtained at the condition of s = 0.081, T i = 0.62 s , H b = 3.3 cm and h b = 4.3 cm. Photographs were taken from the same direction as in Fig. 2 after the
O0
85
F i g . 2. F o r m a t i o n p r o c e s s o f b e a c h c u s p s ; s = 0 . 0 8 1 , T l = 1 . 1 5 s, H b = 3 . 5 c m a n d h b = 4 . 0 c m . ( a ) ~ = 0 s; ( b ) t = 3 1 s ; ( c ) t ffi 8 7 s; ( d ) t = 2 2 3 s; ( e ) t ffi 4 8 8 s; (f) t ffi 7 8 4 s.
86
Fig. 3. F o r m a t i o n
process of a longshore bar; s ffi 0.081, T i = 0.62 s, H b -- 3.3 c m t -~ 5 1 s ; ( c ) t = 3 1 1 s.
h b = 4 . 3 c m . ( a ) t -- 0 s ; ( b )
and
87 wave generator was temporarily stopped. Compared with the previous experiment, T i is remarkably reduced. Figure 3a shows the initial state o f the glass-bead bed, a b o u t 0.2 cm thick over the nearshore zone of 54 cm width. The d o t t e d line indicates the shoreline for still water, and a bar at t o p right is 10 cm. As soon as breaking waves begin to attack the bed, glass beads in the swash zone move rapidly seaward and those in the breaker zone move gradually shoreward, forming ripple marks (Fig. 3b). As a result, a sedimentary zone of glass beads forms along the plunging point in an equilibrium state (Fig. 3c). We shall next discuss the result of further experiments carried out to clarify the cause of the transition of the bed profiles from Figs. 2 to 3. Figure 4 shows the results obtained at the condition of s = 0.081, T i = 1.68 s, H b = 3.7 cm and h b = 3.9 cm under which beach cusps form. Photographs were taken horizontally through a side wall of the tank. The vertical slice at a distance of 10 cm from the side wall was projected. Figure 4a shows the profile of the water surface at the instance when a wave climbing on the beach breaks d o w n just beneath a horizontal scale of 4 cm. Figure 4b shows the anticlockwise-rotating vortex induced in the same position immediately after the breaking wave impinged on the forward water surface. This vortex, called the backwash vortex, was first found b y Matsunaga and Honji (1980) to form on a steep beach in a wave tank. The backwash from water running up on the beach face is responsible for this vortex formation. It is well known that the plunge of breaking waves into water induces clockwise-rotating vortices, called the plunge vortex (Miller, 1976). This experiment has a large swash length since the period o f breaking waves is longer than that of swash motion. Therefore, the backwash vortex can develop to overcome the plunge vortex. Figure 4c shows the sedimentary bed profile formed along a plunging point at the stage similar to Fig. 2d. This bed profile makes a step-like feature composed of a steep seaward face and a nearly horizontal top and will be called the longshore step in this paper. Although this step diminishes in an equilibrium state, as seen in Figs. 2e and f, on real beaches with abundant offshore sands the longshore step may continue to form. This kind of bed profile has often been observed on steep beaches (Bagnold, 1940; Horikawa, 1978; Wright et al., 1979). In comparison with Figs. 4b and c, the backwash vortex and the longshore step are almost at the same place. It is suggested that the backwash vortex is the cause of the step formation. A flow field induced b y the vortex also provides an explanation for the shoreward transport o f bed materials over the nearshore zone in Fig. 2. In the offshore, the wave-induced mass transport flow of Longuet-Higgins (1953) m a y play a role in transporting bed materials shoreward. Taking into account these observations, we can sketch the overall pattern of mass transport flow forming a longshore step and beach cusps, as in Fig. 5. Figure 6 shows the results obtained at almost the same conditions (s -- 0.081, T i = 0.62 s, H b = 3.3 cm and h b = 4.3 cm) as those for Fig. 3. The m e t h o d Of photography
88
Fig. 4. F l o w and b e d profile b e n e a t h t h e breaking wave f o r m i n g beach c u s p s ; s = 0.081, T i ffi 1.68 s, H b = 3.7 c m and h b = 3.9 cm. (a) Profile o f t h e breaking wave. (b) Anticlockwise v o r t e x i n d u c e d at t h e p l u n g i n g p o i n t . (c) L o n g s h o r e s t e p built along t h e plunging p o i n t .
89
and light projection was the same as that for Fig. 4. Figure 6a shows the profile of a water surface just before a breaking wave occurs beneath a horizontal scale o f 4 cm. An instantaneous streamline pattern also becomes visible in the wave b y tracing the path of polystyrene beads suspended in Backwash
vortex
Fig. 5. Sketch of mass transport flow forming a longshore step and beach cusps. water. The streamline pattern is analogous to that in the standing wave of constant water depth (Lamb, 1932). Breaking waves on a steep beach seem to be considerably influenced b y reflection from the beach face. A similar streamline pattern could also be seen to some extent in the observations by Wiegel (1964) and Wang et al. (1982). Figures 6b and c show the flow fields near a plunging point photographed at a short time interval immediately after the impingement of the breaking wave was complete. A short time elapses between Figs. 6b and c. In both cases, flows near the plunging point construct a pair of vortices, a clockwise, shoreward vortex and an anticlockwise, seaward one. A sedimentary bed profile built on the bed between these vortices is shown in Fig. 6d. In contrast to Fig. 4c, the bed profile has a steeper shoreward than seaward face. This property is representative of the longshore bar, which has been observed on real beaches. It will, therefore, be appropriate to call such a bed profile the longshore bar. A steeper shoreward face of the bar implies that the shoreward vortex is more intense than the seaward one. When the period of breaking waves is shorter than that o f swash m o t i o n along the beach face, the swash length becomes small. In this case, the backwash vortex cannot overcome the plunge vortex. In Figs. 6b and c, the clockwise and the anticlockwise vortices m a y be regarded as the plunge and the weakened backwash vortices, respectively. Taking these observations into account, we can also sketch the overall pattern of mass transport flow forming a longshore bar, as in Fig. 7.
C~
91 6dckwosh vortex
Plunge
vortex
Fig. 7. Sketch of mass transport flow forming a longshore bar.
FORMATION
R E G I O N O F CUSPS
In the present experiment, bed profiles generated from a thin glass-bead bed on a laboratory beach floor. Because of the small a m o u n t of glass beads used, the resulting bed deformation cannot develop sufficiently to change the initial conditions of incident waves, the height of breaking waves and the breaking point. Under such a condition, the formation region o f beach cusps m a y be discussed in relation to breaker types on a rigid plane beach. Galvin (1968) reported from wave-tank experiments that breaker types are controlled b y a dimensionless parameter Hb/sgTi 2. The values of Hb/gTi 2 obtained from the present experiment and the field observation of Longuet-Higgins and Parkin {1962) are plotted in Fig. 8 against those of s. Most of the plotted data for beach cusps are seen to collapse in the region:
Hb/sgTi 2 < 0.042
(1)
the b o u n d a r y of which is drawn with a solid line in Fig. 8. According to Galvin's study, the generation regions for surging, plunging and spilling breakers are respectively, Hb/sgTi 2 < 0.003, 0.003 < Hb/sgTi 2 < 0.068 and 0.068 < Hb/sgTi 2. The corresponding boundaries for the breaker types are also drawn with d o t t e d lines in Fig. 8. It should be noted that the critical line for eq. (1) passes through a nearly central part o f the generation region for plunging breakers. This means that the cusp formation has only a weak correlation with breaker type. Attempts to relate breaker types to the offshore wave parameter Ho/Lo, the ratio o f the height of deep water waves
Fig. 6. F l o w and bed profile beneath the breaking wave f o r m i n g a longshore bar; s = 0 . 0 8 1 , T i = 0 . 6 0 s, H b ffi3.2 c m and h b = 3.9 cm. (a) Profile o f the breaking wave; (b) V o r t e x pair induced at the plunging p o i n t ; (c) V o r t e x pair at a slightly later stage than b. (d) L o n g s h o r e bar built along the plunging point.
92
to their wavelength, have also been made (Galvin, 1968; Battjes, 1974). In this study, L0 is determined from the dispersion relation of deep-water waves a s :
Lo = gTi2/2~
(2)
and Ho, from the empirical relation of Komar and Gaughan (1973) connecting H b with H0 and L0, as: H0 = 2.05 (Hb/Lo)
(3)
1/4
The values of HolLo for the present experiment are also plotted in Fig. 9 against those o f s. Using eqs. (2) and (3), we can rewrite eq. (1) as:
Ho/Lo < 0.39
(4)
s s/+
The boundary of this region is drawn with a solid line in Fig. 9. Using offshore parameters, Galvin (1968) also gave the generation regions Ho/s2Lo < 0.09, 0.09 < Ho/s2Lo < 4.8 and 4.8 < Ho/s2Lo for surging, plunging and spilling breakers. The inverse of the square root of Ho/s2Lo is known as a dimensionless parameter controlling nearshore curreat systems and called the surf similarity parameter by Battjes (1974). The boundaries for breaker types are drawn with dotted lines in Fig. 9. Similarly to Fig. 8, the boundary of the cusp region is at a nearly central part of the region for plunging breakers. From experiments using a large wave tank, Tamai
cusp o Author • Longuet-Higglns & 0Y Parkin (1962) 4, bar • Author // ~Ob. ~
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Fig. 8. The values of H b / s g T i ~ plotted against those of s. Fig. 9. The values of H o / L o plotted against those of s.
ii II
ii
0
/
Ho. ~o" 0"09s2
/
/
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93
(1974) gave the following empirical relation as a condition for cusp formation: H o / L o < 0.04
(5)
The boundary of this region is also drawn in Fig. 9. All the data for beach cusps in the present experiment satisfy the above condition. SPACING O F CUSPS
We shall here discuss the spacing of beach cusps compared with the wavelength of the edge waves excited on a beach. According to Ursell's t h e o r y (1952), the wavelength (Le) of edge waves can be expressed as: L e = g T e 2 sin [ ( 2 n + l ) s ] / 2 ~
(6)
where T e and n are the period and the offshore modal number of the edge waves, respectively. Edge waves on a beach may be classified into the two types of subharmonic and synchronous depending on whether the period of edge waves is twice or the same as that of incident waves on the beach. Guza and Davis (1974) predicted theoretically that the subharmonic edge wave of m o d e n = 0 is most easily excited b y incident waves. Huntley and Bowen (1973) and Guza and Bowen (1981) also found that in the p o w e r spectra of nearshore currents, the zero m o d e subharmonic edge wave exists on real beaches. When the longshore periodicity of small amplitude due to edge waves is superimposed on the uniform swash m o t i o n due to incident waves, the profiles of run-up and run
i!
•-.
Te/8
i=.
run-down
Te/2
5Te/8 I
Le
Le/2
0
I
Le
Le/2
3Te/8
run - up
(Q)
7 Te / 8
i i
r u n - up
3 Te/2
(b)
Fig. 10. Sketch of run-up and run-down profiles in plan view. The phases of oscillation are inscribed near the corresponding profiles. (a) Subharrnonic edge wave. (b) Synchronous edge wave.
95
the periodicity of swash motion, the spacing of beach cusps may be expected to be half a wavelength of subharmonic edge waves or a wavelength of synchronous edge waves. Using eq. (6) and T i instead of Te, we can express the calculated spacing (Lcal) o f cusps as: L c a 1 = L e / 2 = g T i 2 (sin s)/lr
(7)
for the zero m o d e subharmonic edge wave and: L c a 1 = L e = g T i 2 (sin s ) / 2 ~
(8)
for the zero m o d e synchronous edge wave. Figure 11 shows the equilibrium form of beach cusps with various spacings obtained in the present experiment. In all the photographs, a bar at t o p right is 10 cm. Similarly to Fig. 2, these beach cusps have regular ridgechannel systems. Using the width (B) of the wave tank and the total num1
TABLE
N u m e r i c a l d a t a o f b e a c h cusps Exp.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
s
0.081 0.081 0.081 0.081 0.081 0.081 0.081 0.107 0.107 0.107 0.107 0.107 0.107 0.107 0.118 0.141 0.141
Ti
Hb
hb
Hb/gTi 2
gTi2 sin s/Ir
Lob s
(s)
(cm)
(cm)
(× 10 -3)
(cm)
(cm)
2.38 1.72 1.71 1.58 1.27 1.15 0.85 1.11 1.06 1.01 0.98 0.94 0.88 0.76 1.38 1.38 0.98
3.4 3.1 2.7 3.3 4.0 3.5 3.5 3.5 3.2 3.3 3.4 3.0 3.0 3.5 3.6 2.6 4.2
3.1 3.1 2.8 3.3 3.5 4.0 3.0 3.5 3.3 3.0 2.9 2.6 2.8 3.1 3.6 2.8 3.2
0.61 1.08 0.94 1.35 2.53 2.70 4.94 2.90 2.93 3.30 3.61 3.50 4.04 6.18 1.93 1.39 4.46
143 74 74 63 41 33 18 41 37 34 32 29 26 19 70 83 42
150 75 50 75 50 38 30 30 38 25 30 30 25 21 75 75 38
Fig. 11. Equilibrium f o r m s of beach cusps with various spacings. (a) L o b s = 1 5 0 c m ; s = 0.081, T i = 2.38 s, H b = 3.4 c m a n d h b = 3.1 c m . (b) L o b s = 3 8 c m ; s = 0.141, T i = 0.98 s, H b = 4.2 c m a n d h b = 3 . 2 c m . (C) L o bs = 21cm;s = 0.107, T i = 0.76 s , H b = 3.5 c m a n d h b = 3.1 cm.
96 bet (N) o f the ridges, we can obtain the observational spacing (Lobs) of cusps as: Lob s = B / ( N
-
-
1)
(9)
The numerical data of beach cusps for the present experiment are presented in Table 1. The value of Lob s is in good agreement with that of Lca1. Although subharmonic edge waves have the swash profile as sketched in Fig. 10a, such a longshore undulation of swash motion was visible only in the experiment for Fig. l l a with the largest cusp spacing of 150 cm. In other cases, edge waves excited may be t o o weak to be f o u n d in swash motion. The values of Lob s for the laboratory experiments of Ann (1979), Tamai (1980) and the author, and the field observations of Longuet-Higgins and Parkin (1962), Komar (1973), Sallenger (1979), Guza and Bowen (1981) and Takeda and 8unamura (1982) are plotted in Fig. 12 against that of gTi 2 (sin s)/~. The value of Lcal for subharmonic and synchronous edge waves are also drawn with solid and d o t t e d lines, respectively. All the laboratory data and most of the field data have a best fit with the solid line. However, there are some data of Longuet-Higgins and Parkin, Komar and Sallenger which show a scatter around the d o t t e d line. It should be noted that the data of Longuet-Higgins and Parkin collapse in the neighb o r h o o d of the line H b / s g T i 2 = 0.003 in Fig. 8. A critical value of H b / sgTi 2 beneath which subharmonic edge waves are replaced b y synchronous edge waves m a y exist. I
~ .....
subharmonic synchronous
60~,/~ c~ ~ / 6 //
edge wave edge wave
c~ I I0 I
0
//
// 0
e ° / / e o ~ _
/
e / / i I 0 0 O0 0 0/// • // Field data /' 0 L onguet- Hl~Jlns • /// a Park In(1962) // • KomorU973) / e sanenger ¢1979) /~ e • Guza a
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Bowen(19Sl) ~ Tanaka S Sunomura (1982) Laboratory data
o Ann tl979) o Tamai (1980) • Author
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iOo
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I
I
I
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I
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I0 I
Fig. 12. The values of Lobs plotted against those of gT i2 (sin s)/~r.
97 CONCLUSIONS Th e f o r m a t i o n process o f beach cusps has been observed in detail on a l a b o r a t o r y beach. Flows forming the beach cusps are examined and their characteristics are c o m p a r e d with the result o f the previous studies. Main results obtained are summarized as follows: (1) Beach cusps form at the beach face when bed materials in a nearshore zone move shoreward forming a longshore step along the plunging point. (2) When bed materials, b o t h in the breaker zone and in the swash zone, move to the plunging point, beach cusps cannot develop and a longshore bar forms along this point. (3) Vortices induced on t he bed by breaking waves c o n t r o l t he movem e n t o f bed materials at a nearshore zone. (4) The transition of bed profiles from the beach cusp t o the longshore bar occurs at H b / s g T i 2 = 0.042 or H o l L o = 0.39 s s/4 which corresponds to a nearly central part of t h e generation region for plunging breakers on a rigid plane beach. (5) Most o f the observational spacings of beach cusps agree well with half a wavelength of t he zero m o d e subharmonic edge wave excited on the beach. A relatively large scatter of the field data m ay be explained consistently by considering the existence of beach cusps generated by the zero-mode s y n chr onous edge wave. ACKNOWLEDGMENTS I would like to express m y sincere thanks to Prof. H. Honji and Drs. A. Masuda and N. Matsunaga f or useful suggestions and t o Mr. Y. Shiraishi for technical assistance.
REFERENCES
Ann, K., 1979. Experimental study on the formation of rythmic shorelines. Master's Thesis, University of Tokyo, Tokyo, 135 pp. (unpubl., in Japanese). Bagnold, M.R.A., 1940. Beach formation by waves: some model-experiments in a wave tank. J. Inst. Civ. Eng., 15: 27--52. Battjes, J.A., 1974. Surf similarity. Proc. 14th Conf. Coastal Eng., Copenhagen, pp. 466--480. Galvin, C.J., 1968. Breaker type classification on three laboratory beaches. J. Geophys. Res., 73: 3651--3659. Guza, R.T. and Bowen, A.J., 1981. On the amplitude of beach cusps. J. Geophys. Res., 86: 4125--4132. Guza, R.T. and Davis, R.E~ 1974. Excitation of edge waves by waves incident on a beach. J. Geophys. Res., 79: 1285--1291. Guza, R.T. and Inman, D.L., 1975. Edge waves and beach cusps. J. Geophys. Res., 80:
2997--3012.
98 Horikawa, K., 1978. Coastal Engineering: An Introduction to Ocean Engineering. Univ. T o k y o Press and A. Halstead Press, Tokyo, 402 pp. Huntley, D.A. and Bowen, A.J., 1973. Field observations of edge waves. Nature, 243: 160--162. Inman, D.L. and Guza, R.T., 1982. The origin of swash cusps on beaches. Mar. Geol., 49: 133--148. Kaneko, A., 1983. A numerical experiment on nearshore circulation in standing edge waves. Coastal Eng., 7: 271--284. Komar, P.D., 1973. Observations of beach cusps at Mono Lake, California. Geol. Soc. Am. Bull., 84: 3593--3600. Komar, P.D., 1976. Beach Processes and Sedimentation. Prentice-Hall, EnglewoodCliffs, N.J., 427 pp. Komar, P.D. and Gaughan, M.K., 1972. Airy wave theory and breaker height prediction. Proc. 13th Conf. Coastal Eng., Vancouver, pp. 405--418. Lamb, H., 1932. Hydrodynamics. Camb. Univ. Press, Cambridge, 738 pp. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. London, Ser. A, 245: 535--581. Longuet-Higgins, M.S. and Parkin, D.W., 1962. Sea waves and beach cusps. Geogr. J., 128: 194--201. Matsunaga, N. and Honji, H., 1980. The backwash vortex. J. Fluid Mech., 99: 813-815. Matsunaga, N. and Honji, H., 1981. A petaline sand pattern along the shoreline. Rep. Res. Inst. Appl. Mech., 29: 1--12. Miller, R.L., 1976. Role of vortices in surf zone prediction: sedimentation and wave forces. In: R.A. Davis Jr. and R.L. Ethington (Editors), Beach and Nearshore Sedimentation. Spec. Publ. Soc. Econ. Paleontol. Mineral., Tulsa, 24: 92--114. Sallenger, A.H., 1979. Beach-cusp formation. Mar. Geol., 29: 23--37. Tamai, S., 1974. Study on the formation of beach cusps. Proc. 21st Japanese Conf. Coastal Eng., Sendai, pp. 115--120 (in Japanese). Tamai, S., 1980. Study on the characteristics of beach cusps and the prediction of shoreline changes. Doctor's Dissertation, K y o t o University, Kyoto, 197 pp. (in Japanese). Takeda, I. and Sunamura, T., 1982. F o r m a t i o n and wavelength of beach cusps. Proc. 29th Japanese Conf. Coastal Eng., Sendai, pp. 319--322 (in Japanese). Ursell, F., 1952. Edge waves on a sloping beach. Proc. R. Soc, London, Ser. A, 214: 79--97. Wang, H., Sunamura, T. and Hwang, P.A., 1982. Drift velocity at the wave breaking point. Coastal Eng., 6 : 121--150. Wiegel, R.L., 1964. Oceanographical Engineering. Prentice-Hall, Englewood-Cliffs, N.J., 532 pp. Wright, L.D., Chappell, J., Thorn, B.G., Bradshaw, M.P. and Cowell, P., 1979. Morphodynamics of reflective and dissipative beach and inshore systems: Southeastern Australia. Mar. Geol., 32: 105--140.