Kinematics of breaking waves and associated suspended sediment in the nearshore zone

ContinentalShelfResearch,Vol. 13, No. 11, pp. 1219-1242,1993. Printedin Great Britain.

0278-4343/93 $6.00+ 0.00 PergamonPressLtd

Kinematics of breaking waves and associated suspended s e d i m e n t in the n e a r s h o r e z o n e YANLING

Yu,* RICHARD W .

STERNBERG* a n d REGINALD A . BEACHt

(Received 16 June 1992; accepted 20 May 1993) Abstract--This paper reports the results of a study to describe systematic variations in sea surface shape, horizontal and vertical velocity components, and nearbed suspended sediment concentrations as waves shoal, break and propagate shoreward as bores. Data were collected as part of the Duck '85 nearshore experiments carried out at the Army Research Facility at Duck, North Carolina in September 1985. The data set includes time series measurements of sea surface elevation, currents, and suspended sediment concentrations located at five positions across the surf zone. Additionally, a video recording of the sea surface was analyzed to classify surface wave types as unbroken, breaking (with or without associated form), or bores. Results show that the degree of asymmetry of wave shape increases as waves shoal and break. The cross-shore velocity varies systematically, being somewhat undulatory for unbroken waves and decomposing into large eddy structures upon breaking. The patterns of suspended sediment are strongly related to wave type. For unbroken waves, sediment suspensions can be high under the crest but occur within several centimeters of the seabed, on the order of the wave boundary layer thickness. For breaking waves, suspended sediment concentrations increase dramatically and sediment is mixed to higher levels in the water column. Sediment inversions and localized high concentration patches that appear to correspond to eddy motions impinging directly on the seabed are observed. For bores, overall sediment concentrations decrease as they propagate shoreward. The ratios of suspended load for the unbroken waves, breaking waves, and bores are 1: 2.9: 4 ~ 1.3 (averaging 2.3), respectively. Maximum suspended load occurs just shoreward of the breakpoint and most sediment settles out within 10-15 m of the plunge point.

INTRODUCTION

MUCH of our early knowledge of suspended sediment transport in the surf zone has been obtained from water-suspended sediment samples collected by divers, concurrently with wave and fluid flow measurements (e.g. KANA, 1979; INMAN et al., 1980; ZAMPOL and INMAN, 1989). These studies have provided fundamental information on the mean vertical and horizontal distributions of suspended sediment and additionally have shown that suspended sediment concentrations are strongly related to breaker type (spilling vs plunging) and location relative to wave breakpoint. Diver techniques have been invaluable for defining the gross relationships between waves and suspended sediment; however, the detailed relationships between fluid motion within a wave and sediment response could not be resolved by these techniques. *School of Oceanography, WB-10, University of Washington, Seattle, WA 98195, U.S.A. tCollege of Oceanography, Oregon State University, Corvallis, OR 97331, U.S.A. 1219

1220

YANLINGYV et al.

Since the development of the optical backscatter sensor (OBS; DOWNINGet al., 1981), it has been possible to monitor small-scale suspended sediment concentrations in the surf zone, i.e. on temporal and spatial scales commensurate with electromagnetic current meters [e.g. 10 Hz, 0(cm)]. Simultaneous OBS--current meter measurements have provided useful data regarding small-scale fluid-sediment interactions (e.g. JAFFE et al., 1985; STERNBERGet al., 1985, 1989; HANES and HUNTLEY,1986; BEACH and STERNBERG, 1988; HUNTLEY and HANES, 1987; however, the detailed relationships between wave shape, internal velocity structure and sediment response still are largely unknown because of the difficulty of differentiating wave types in a time series data record. For example, at a single location within the breaker zone, unbroken waves, spilling breakers, plunging breakers and bores all may occur over a short time period, due in part to variability of offshore wave characteristics (e.g. groupiness). To associate sediment response with wave type, each wave type must be identified with respect to its shape and velocity structure in a time series record before it can be correlated systematically to the suspended sediment distribution. In an attempt to identify the different wave types in a data record, a sediment transport experiment was included as part of the "Duck '85" field experiment (MAsoN et al., 1987), carried out at the U.S. Army Corps of Engineers Field Research Facility, Duck, NC, on 69 September 1985. The overall sediment transport experiment consisted of twice-daily time-series observations of waves, currents, and suspended sediment concentration profiles at five locations across the surf zone. During the final data run of the experiment, a video camera was used to record the surface wave patterns over the instrument arrays while the time series was being collected. The data run lasted about 44 min, while approximately 200 individual waves propagated shoreward. The initial analysis of this data set was carried out by BEACH (1989). The Duck '85 data set was used in the present study to describe the shape and internal velocity structure of the wave types observed in the surf zone (e.g. unbroken, breaking, bore) and to relate these types to the observed suspended sediment concentration. Specifically, measurements were used to: (1) describe the wave shape and internal velocity associated with the various wave types identified; (2) describe the sediment suspension patterns of the observed wave types; and (3) calculate the mean suspended sediment load associated with each wave type. BACKGROUND Investigations of wave shape and the internal velocity field in the breaker region have been carried out in numerous laboratory studies and in a limited number of field studies. M~LLER and ZEIGLER(1965), using acoustic and electromagnetic current meters and wave gauges across the surf zone, showed that various breaker types can occur in the same vicinity on the same beach because of the interactions of incoming waves, wave groups, and backwash. They also found that the internal velocity structure is strongly related to wave shape (asymmetry) and that breakers may be characterized by one or more vortex systems. In a laboratory experiment, MILLER (1976) found differences in the wave structure (shape and velocity) of plunging and spilling waves. A vortex motion was generated by the curling over of the breaking wave and vortices of plunging waves reached the bottom, while those of spilling waves were restricted to a near-surface region.

Kinematics of breaking waves

1221

NADAOKA (1986) and NADAOKAet al. (1988, 1989), using a laser doppler current meter, observed large-scale "obliquely descending eddies" associated with wave breaking. These eddies were believed to be responsible for the large-scale variation of Reynolds stress in the wave velocity field and the generation of entrained surface air bubbles, associated with upwelling of the sediment-fluid mixture in close proximity to the breakers [Fig. I(A) and (B)]. A similar eddy-like motion due to the separation of the wave boundary layer also was observed by MATSUNAGAet al. (1988). Observations of sediment suspension by wave action in the surf zone generally have been limited by instrumentation; however, several studies of different aspects of sediment movement have been reported. DOWNING (1983), using OBS sensors to document individual sediment suspension events, found that typical events could be separated into three phases: (1) the water column is relatively clear during initial offshore acceleration of flow prior to the passage of a wave; (2) high suspended sediment concentrations and gradients develop near the bed with sediment rapidly mixed into the water column immediately following the passage of the wave crest; and (3) suspended load settles with a uniform decrease of concentration at all levels following the passage of a wave. HANES and HUNTLEY (1986) used OBS sensors and electromagnetic current meters to investigate suspension events of sand related to individual waves and wave groups seaward of the surf zone at Pt. Sapin, Canada. The vertical transfer of sediment appeared to be more strongly related to the onshore directed phase of wave motion than to the offshore phase. Also, they found that the initiation of suspension may be determined more readily by fluid deceleration than by velocity. BEACH (1989) observed mean sediment suspension patterns among three types of waves--unbroken, breaking, and bores in the data set used in the present study. This experiment was conducted in North Carolina at the Army Corps of Engineers Field Research Facility during high tide on 9 September 1985. The beach is oriented N N W - S S E , has an average slope of 1.68 °, and consists of fine sand with a mean diameter of 0.17 mm. The waves approached shore-normal and had significant heights of 0.4-0.5 m and periods of 10-12 s (BEACH, 1989). Instrumentation consisted of five optical backscatter arrays, seven electromagnetic current meters, and five pressure transducers. Instruments were located at five positions on a shore-normal transect extending from approximately 20 m (position 1) to 50 m (position 5) seaward from the shoreline (Fig. 2). Measurements at each position included suspended sediment concentrations at five levels above sea bed, u and v velocity components, and sea surface variations (the pressure gauge at position 5 malfunctioned). In addition, vertical velocity components (w) were measured at one level at positions 3 and 4. The nominal elevations of current meter and pressure sensors are noted in Fig. 2; however, they vary _+1 cm due to changes in the level of the seabed. The elevations of the individual OBS sensors on each array are z = 4, 7, 13, 25 and 55 cm for AO, BO, and CO arrays and z = 4, 7, 10, 15 and 20 cm, respectively, for A M and BM arrays. Data were recorded for 44 min at a 10 Hz sampling rate while waves broke (primarily plunging) within the experimental section. During the data run 194 waves passed the experimental transect. Most waves broke between positions 3 and 4 (Fig. 2). There were also a number of unbroken waves and a few bores. The video recording during the data run provided a visual means of recognizing wave types.

1222

YANLINGYU et al.

A

~= 11,5 cm 8

I/(cm) 0 -4 i0

._ 2

4

6

IO(s)

8

40 20

w (ca/s)

0

-

-

~----

+.....

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. . . .

2

-

60 40 20 0 20

-

40

-

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u (ca/s)

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B Downward development of an eddy involving air bubbles

Upwell.Jng motion of fluid

~ . ! .,,

• j.,

.----

containing bottom sediments

(*!:tv"S I*-, ~

-

induced by buoyancy effect of air bubbles

-,_-~. !-~.~..:~.:.............,. .... . . . . . .

. •

.

".

.-

..

• ".'..

~-.;...../.......~:.-..

,

~

. ".....

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Fig. 1. (A) Results of a laboratory study of NADAOKAet al. (1989) showing surface elevation (r/), horizontal velocity (w), cross-shore velocity (u), and instantaneous Reynolds stress ( - u w ) . The arrows indicate the times when large-scale eddies descend through the velocity measuring point. Note that horizontal (u) and vertical (w) velocity symbols have been transposed. (B) Schematic illustration of eddy action on the bottom and succeeding upwelling of sediment-laden fluid (after NADAOKAet al., 1989).

Kinematics of breaking waves

1223

~50m

o,v,2oco, Pll5cm) ~ _ ~

-55c

AO ~ 4-55crnS[~

. . ~ U,V(45 cm) -(~-d~ P(44 cm) .~__~w(19cm)

ITION 5 \

\ POSITION4 ~40m DISTANCE OFFSHORE

~

BO

I

~

_

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"(P'I u,v(72 crn) . 1-~ p ((3 cm)

~P--'l_~.,,(19cr,)

" 2ocZ__L

\

\ POSITION 5

\

?ocm, \POS,T,ON,

Fig. 2. Instrument deployment at Duck, NC. AO, AM, BO, BM and CO refer to vertical arrays of optical backscatter sensors; u, v, w and P indicate cross-shore velocity, longshore velocity, vertical velocity and pressure sensor, respectively (after BEACH, 1989).

ANALYTICAL METHODS

Wave type The video recording was analyzed visually to estimate the type of wave occurring at each sensor location. Each wave was coded on the basis of its stage in breaking as it crossed each of the five sensor positions. The terminology, unbroken, breaking without foam, breaking withfoam and bore, follows the order of development as waves approach shore. Unbroken waves were distinguished by their undulatory shape and lack of foam at the crest. Breaking waves without foam signified that the breakpoint had occurred (foam in the crest region) but the wave was still in the process of collapsing. The category, breaking with foam, referred to waves that had completed the plunging process at, or close enough to, the sensor array so that water surrounding the plunge point was a mixture of foam and bubbles (and sediment) rising to the surface. The term bore was applied to turbulent bores that evolved from the breaking process and propagated shoreward from the point of collapse. Since shoaling waves occur as a continuum, and also the timing of a wave type observed at the water surface does not exactly match the timing of the signal received by the sensors near the seabed, the visual classification scheme is not precise and serves only as a guide to assist in sorting out the time series signals. To correlate wave type, internal velocity structure and sediment suspensions, it was necessary to derive a quantitative measure for classifying wave types.

1224

YANUN~Y u

et al.

Wave shape and internal velocity Various parameters to predict wave breaking have been proposed (e.g. GALVIN, 1968; BATI'JES, 1988). These parameters generally define breaker types as a function of wave steepness and beach slope. None, however, use wave asymmetry as a defining condition, although, according to MILLERand ZEIGLER(1965), this aspect is closely related to internal velocity structure. Thus, it would be desirable to compare the slopes of the leading and trailing edges of the wave crest as a measure of wave asymmetry; as shown below; however, the region behind the wave crest is undulating and a slope calculation is not possible. In lieu of this, we selected the maximum slope of the leading edge of the wave (AHmax/AX) as a meaningful parameter to describe the wave profile and therefore to distinguish wave type. AHmax/AX was computed from time-series pressure records (AH) and the wave celerity between sensors at positions 4 and 3. The use of pressure sensors to reconstruct the sea surface in a region of strong accelerations is subject to errors of approximately 5% in the surf zone and 17% near the break point (THORNTON and GUZA, 1989). Nevertheless, it is felt that AHmax/AX is useful for a first approximation in comparing differences in wave slopes during the shoaling-breaking process. By comparing AH/Ax of each individual wave with the visual classification identified in the video records, it was found that in general this parameter is a good representation of the various waves. The AHmax/AXparameterization shows the continuous evolution of the wave shoaling-breaking process better than visual observation of the video records. It is used, therefore, in the study as a quantitative descriptor of the wave shape during the shoaling-breaking process. In addition, the maximum onshore velocity (Urnax) and vertical velocity (Wmax) of each wave were used to represent the fluid flow conditions.

Reynolds stress and vorticity To be able to sense the onset, or nature of vortices, or eddy fluctuations associated with the various wave types, the instantaneous Reynolds stress, as represented by the product u' w', and the y-component of vorticity, ~y, have been estimated. For the data collected in this study (e.g. 10 Hz collection rate over a spectrum of wave periods), the moving average method of NAKAGAWA(1983) is used. It assumes that the unsteady flow component under waves is comprised of periodic (fi, ~) and turbulent flow (u',w') components. To determine the u'w' component, the fluid velocity is separated into mean and unsteady parts -

u = ff + h

(1)

where ff is mean flow and t~unsteady flow. The value of the unsteady flow for the u and w components is (h, ~) = (u - if, w - #).

(2)

The turbulent component (u'w') is estimated by removing the periodic component (ti, fi~) from t~:

(u', w') = (~ - ~, ~ where h in the x direction at a time t = i is defined by

~,)

(3)

Kinematics of breaking waves

1225

Ui = ~(u/_ 1 t, 1 + 2~/- + u~+,) A

(4)

and the periodic component in the z direction is calculated using the w components, respectively. Equation (4), used to estimate the periodic flow components, is a linear low-pass filter which passes frequencies near zero and cuts off at 5 Hz. There is certainly leakage of low frequency energy in the computed u' and w' values. However, according to HATrORI and AoNo (1985), a large portion of turbulent energy is transferred from large-scale eddies to frequencies between 1 and 10 Hz upon breaking; thus the u'w' estimates may be useful as a qualitative parameter to identify this process. Additional errors can occur in Reynolds stress calculations when using electromagnetic current meters. As discussed by HUNTtEY (1988), misalignment of sensor axes when measuring u and w velocity components can lead to enormous errors in Reynolds stress calculations ( > 10% error per degree of tilt). Although Reynolds stress calculations used in this study cannot be evaluated for errors (which may be very large), they are used only as a guide in identifying variations in wave conditions during wave breaking. While Reynolds stress estimates may be useful in identifying turbulent characteristics within waves, large-scale eddy motions can be approximated by a calculation of vorticity ~Y

Au

Aw

AZ

~f

(5)

In this study, the calculation of vorticity is limited by sensor placement. While the u velocity was measured at two elevations with 26 cm vertical spacing, the w velocity was measured at positions 3 and 4 across the surf zone, separated by 5.5 m. As a result, the Au/Az term can be calculated directly but an approximation must be made to compute the Aw/~x term in equation (5). Assuming that over the sampling inter4al (At = 0.1 s) the horizontal motion of a water parcel is small relative to its vertical motion, then A w / > ~ c can be approximated by: A ~ _ w(t) -- w ( t - 1)

2t2

~c

(6)

where A#/2t2 is the approximation of Aw/&r, ~c = u(t - 1) x 0.1 s, and t is time. To test the validity of this approximation, u and w velocities, plus an error function, are used in the wave equation. If A w / ~ c is defined as the theoretical term and A # .~ w(t) - w(t - 1)

Ax as the approximation of Aw/Ax, the ratio of A# law ~/ ~x

is about 50%. Realizing this limitation, the calculation of ~y is considered as an "estimated vorticity" term and is used only qualitatively in this paper to suggest the onset of vortex motion under breaking waves. Sediment suspension

OBS data were collected at a 10 Hz sampling rate and logged in parallel with current meter and pressure measurements. Calibration of OBS sensors was carried out with

1226

YANLING YU et al.

208

.

.

.

.

1o8

9

10

11

12

13

14 15 Tilde (minl

16

17

IB

19

20

Fig. 3. Time series as recorded at position 4, (A) cross-shore velocity, (B) longshore velocity, (C) sea level fluctuations, ( D - H ) suspended sediment concentration at 4, 7, 13, 25 and 55 cm above the bed, respectively (after BEACH, 1989).

bottom sediment from the study site. Suspended sediment concentration data are presented in two ways. Plots of sediment concentration within each wave type are based on the time-series 10 Hz data without filtering or averaging. Estimates of the total mass of suspended sediment per unit area of the seabed for each wave in the data set (i. e. Fig. l 5) is calculated by: Mt = f C( z) dz where C"is a 4-s mean concentration beginning from the wave crest and the integral was evaluated over the elevation range of the sensors. A 4-s average is used because it approximately corresponds to the suspended load during the time of onshore velocity. RESULTS

Data set An example of an 11 min segment of the time series collected at position 4 is shown in Fig. 3. The data include cross-shore and longshore velocity (panels A and B), sea level fluctuations (C), and susp_ended sediment concentration at five levels above the bed (DH). Mean water depth (h), cross-shore velocity (if), longshore velocity (~-), RMS wave height (Hrms), and the ratio of wave height to water depth (y), observed during the

Kinematics of breaking waves

1227

Table 1. Mean water depth (la), mean cross-shore (~) and longshore velocity (~) and rms wave height (H,~,~) of all waves by cross-shore position (BEACH,1989) f/

~*

~*

nrm s

Position

(cm)

(cm s-x)

(cm s-1)

(cm)

5t 4

147 126

37(20) 6(45)

-48

3

116

11(72)

42

2 1

110 104

2(20) -4(45) -8(19) -10(72) -5(19) -5(17) <1(20)

14(17) 18(20)

32 33

*Numbersin parentheses indicate instrument height above the bed. -~Wave height statistics not obtained due to failure of pressure sensor.

sampling period are listed in Table 1 (from BEACH,1989) for each of the five cross-shore positions. During the data run, mean sea level varied from about 1.5 m at position 5 to I m at the innermost position (Table 1). The mean cross-shore velocity at positions 2--4 were directed onshore (negative values) and varied between 5 and 10 cm s -1, whereas positions I and 5 had small offshore directed m e a n flows of < 1 - 2 cm s -1. The m e a n longshore current was to the north at all positions (positive values) and varied from a m a x i m u m of 37 cm s -1 at position 5 to a minimum of 6 cm s-1 at position 4 and then steadily increased towards shore with a secondary m a x i m u m of 18 cm s -1 occurring at position 1. The rms wave height calculated from the variance of sea surface fluctuations in the frequency band 0.3-0.03 Hz, according to the methods of THORNTON and GUZA (1982) and WRIGHT et al. (1982), decreased towards shore from a high of 48 cm at position 4 to 33 cm at position 1. A spectrum of sea surface fluctuation and cross-shore velocity from position 4 (Fig. 4) indicates that the predominant wave period during the data run was 12 s (0.083 Hz). The frequency of occurrence of each wave type propagating through the sensor array is summarized in Table 2 (from BEACH,1989). A systematic shoreward variation is observed. For example, at position 4 unbroken waves accounted for 67% of the total, while at position 1 only 28% of waves were unbroken. The percentage of bores increased from 1% at position 4 to 47% at position 1. Plunging waves account for 32% at position 4 and reached a m a x i m u m of 44% at position 3, then decreased towards position 1, implying that wave breaking occurred primarily around position 4.

DISCUSSION W a v e characteristics W a v e s h a p e . The analysis of wave conditions was carried out at position 4. Position 4 was selected because it represented the approximate boundary of the surf zone where a

1228

YANLING YO et el.

Period (s) 1000

100

10

I

ol

I

t~lO -'

0 0.001 Fig. 4.

I

~

~ 0.01 0.1 Frequency (HZ)

1

Cross-shore (u) and sea level (r/) spectra (h = 1.26 m) (after BEACH, 1989).

Table 2. Number o f waves and percentage o f wave type occurring at each sensor position. Position 5 is omitted from the analysis because the pressure time series was not obtained (after BEACH, 1989)

Type (%) Plunging Position 4 3 2 1

Waves (no.)

With foam

No foam

Total

Bores

Unbroken

212 226 184 187

18 29 10 10

14 15 22 14

32 44 32 24

1 6 41 47

67 49 27 28

mixture of wave types could be observed. Longshore currents were minimum (Table 1) and the infra gravity band (Fig. 4) had relatively low energy. Visual analyses of 194 waves propagating past position 4 indicated that AHmax/Ax for unbroken waves increased from a minimum value of 0.01 to approximately 0.2. AHmax/Ax for breaking waves varied from about 0.2 to 0.5 and then decreased to values of 0.3 for the bores propagated shoreward. The systematic variation of wave shape as a function of AHmax/Z~¢ for shoaling waves is illustrated in Fig. 5. The x-axis in this figure (and subsequent figures showing wave profiles) is the normalized distance x/L from the wave crest, where L is the wavelength. Values ofx/L were estimated from the ratio of At/Tor the fraction of the wave period (T) before ( - ) or after (+) the wave crest. The wave shape is rounded and the sea surface smoothly undulating (A) prior to breaking (Fig. 5). As the

Kinematics of breaking waves 200

1229

AHm :0.t3

Ax

100 - ' / 0.2

.0.2

0.4

0.6

0.8

0.6

0.8

AHm

i

---0.2

0

0.2

0.4

200

~Hm =0.34

zso

/

.0.2

AX

,.,..... ~

0

0.2

0.4

0.6

0.8

20O [

AHm ' = 0.28

tso

100 -0.2

0

0.2

0.4

0.6

0.8

200

E

AHm= 0 . 2 4 Ax

I00 .0.2

o

0.2

0.4

0.6

0.8

x/L

Fig. 5. Evolution of the shape of shoalingwavesfor variousvaluesof AHmax/Ax.Unbroken wave

(A); breaking wave (B); sequential bore development (C-E).

breakpoint is approached, the shape becomes increasingly asymmetrical and sharply peaked (B). Once breaking occurs, the wave peak collapses and secondary undulations develop behind the wave crest (C-E). Velocity structure. Three wave shapes (Figs 6-8) have been selected from the data set to illustrate the characteristics of unbroken waves, breaking waves and bores [from Fig. 5(A)

1230

YANLING Y u etal.

UNBROKEN WAVE AT POSITION 4

2OO

AHm /xX --0.15 150 -r. 100 i

i

-0.2

0

0.2

-0.2

i 0

i 0.2

0.4

0.6

0.8

0.4

0.6

~8

100 0

-lOO -2o0

~L Fig. 6.

Wave height (A) and horizontal velocity (B) for an unbroken wave at position 4. Solid line is velocity at 19 cm and dashed line at 45 cm above seabed.

to (C) respectively]. The breaking wave is just beginning to break and exhibits maximum steepness. The bore is an "initial bore", just evolving from a collapsed wave. The differences in the internal velocity field associated with each wave type are evident in the figures. The unbroken wave (AH,,ax/AX = 0.13) has a very smooth and relatively symmetric shape [Fig. 6(A)]. The u velocity tends to be symmetrical about the wave crest and the magnitudes of u at the two measurement levels exhibit a smooth variation (exceeding 100 cm s -1) and track each other closely [Fig. 6(B)] similar to a potential flow field associated with ideal waves. The breaking wave (without foam) (AHmax/AX = 0.50) shows a strong degree of asymmetry in wave shape and velocity [Fig. 7(A)], increased onshore flow (-163 cm s-l), and strong similarity in the magnitude of u velocities at the two measurement elevations [Fig. 7(B)]. The bore (Fig. 8) has a moderate slope ( A H m a x/AX of 0.34), and a second crest which follows immediately after the main one [Fig. 8(A)]. The horizontal velocity (u) is variable in speed and direction at the two measurement levels, shows strong vertical shear, and suggests major fluctuations in the flow field occurring on a time scale of a few seconds [Fig. 8(B)]. The magnitude of AHmax/~k~f for all the waves passing position 4 is illustrated in Fig. 9(A). In this plot, the envelope of data points for waves designated as unbroken, breaking (with or without foam), and bores is outlined to illustrate their differences. Substantial overlap of the visual classification fields occurs. The magnitude of the maximum crossshore velocity (Umax) [Fig. 9(A)] increases from about 20 to 180 cm s-1 (onshore direction) as unbroken waves shoal and begin to break (Aamax/Z~k.~ reaching 0.5). As breaking proceeds and bores begin to form, both AHmax/AX and Ureax decrease; however, Anma x/Ax remains higher for the broken waves and bores than for unbroken waves. The phase of Urnax relative to the wave crest also varies throughout the shoaling cycle [Fig. 9(B)]. For unbroken w a v e s (AHmax/,~ < 0 . 2 ) , the phase of//max is scattered and

1231

Kinematics of breaking waves B R E A K I N G W A V E A T POSITION 4

2OO

f~

i

AHm 0,50"

150 =

lO0 -0.2

i 0

0.2

-0.2

0

0.2

0.4

i 0.6

i 0.8

0.4

0.6

0.8

100

. 100,i

-200

x/L Fig. 7.

Wave height (A) and horizontal velocity (B) for a breaking wave at position 4. Solid line is velocity at 19 cm and dashed line at 45 cm above seabed.

B O R E A T POSITION 4

2OO

T

A

T

::/~Hm ~

=0.341

150 -e 100

21

" i

-0.2

0

-0.2

0

0.2

0.4

0.6

0.8

0.4

0.6

0.8

100

u -100 -200

i

0.2 x/L

Fig. 8.

Wave height (A) and horizontal velocity (B) for a bore at position 4. Solid line is velocity at 19 cm and dashed line at 45 cm above seabed.

1232

YANLING Y u et al.

A +

BREAKING WITH FOAM

l

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....... /..~.B

WITHOUT FOAM o -200

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:

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0

Time Lag of Urn to Wove Crest (see)

The relationship of maximum horizontal velocity (A) and of its time lag to the wave crest (B) relative to AHmax/~x. The (0) -- unbroken wave; (©) = plunging wave without foam; (+) = plunging wave with foam; and ( x ) = bore (at position 4). The envelope of data points for each wave type is outlined in (A).

Fig. 9.

varies from leading the w a v e crest by as m u c h as 1 s to following the w a v e crest by 1 s. For breaking w a v e s (AHmax/Z~ = 0.2--0.5), Umax lags the w a v e crest by 0.6 s. Only three bores occurred at position 4, so the relationship b e t w e e n AHmax/Ax and the time lag of Umax is not well defined•

1233

K i n e m a t i c s of b r e a k i n g w a v e s

A 20 ................................................................................................ i ........................................................ !

0

~20

i 0,

I

-02

0

02

I

I

06

08

20

0 -20 ............. i .................................................

i ........................... i ......................... i .......................... i ....

i -0.2

0

0.2

-0.2

0

0.2

i

i

i

0.4

0.6

0.8

0.4

0.6

0.8

2O

0

-20

x/L Fig. 10.

V e r t i c a l v e l o c i t y structure: u n b r o k e n w a v e ( A ) , b r e a k i n g w a v e (B) and b o r e (C). V e l o c i t y m e a s u r e d at z = 15 cm.

The vertical velocity structure for the three wave types are shown in Fig. 10. Previously, it had been suggested that breaking waves are related to the onset of eddy activity and energy redistribution (HuNTLEY, 1988; NADAOI(Aet al., 1988), as may be observed in the vertical velocity structure. The m a x i m u m vertical velocity (Wmax) of the unbroken wave [Fig. 10(A)] is seen as an irregular pattern with Wmaxreaching 7 cm s - I about 1 s before the wave crest. The breaking wave exhibits a strong, single p e a k with Wmax of 22 cm s -1, occurring immediately after the wave crest [Fig. 10(B)]. The vertical velocity under the bore [Fig. 10(C)] is characterized by major fluctuations of about a 2 s period and reaches a m a x i m u m of 22 cm s -1 about 2.3 s (x/L = 0.25) behind the wave crest. M a x i m u m fluctuations in the vertical velocity occur at times of m a x i m u m shear between the u velocities as shown in Fig. 8(B). The magnitude and phase of Wm~, as a function of hHmax/AX and wave type for all waves is shown in Fig. 11. The magnitude of Wmax[Fig. 11(A)] increases from 0 to a m a x i m u m of about 25 cm s -1 as hHmax/AX increases from 0 to about 0.5, the m a x i m u m condition of breaking. The phase of Wm~ [Fig. 11(B)] shows that it leads the wave crest for unbroken waves, occurs at the crest for breaking waves, and successively lags the wave crest as broken waves transform into bores propagating shoreward.

1234

et al.

YAr, rI.rN(; Yu

8"

A ÷

o

.

i

. . . . . . . . . . . . . *.+..-: . . . . . . . . -*.. . . . . . . . . ~. . . . . . . . . . . . . . . . . . . ÷

o

:

:

i :

i ~

:÷

~

0

•

O

,~

°i

*o

o:

: ÷ :*x

**

:

' .,.: / i * +

o

i . . . . . . . o - * - - . . . . ~:~-- - . . . * - . . . . ~ . . . . . . . :. . . . . . . . . . . . . . . . . . .

o

:

4.o

0

o

x:

o: o

. ~0

0+o + . , i ,

i"

+

÷ + " " + 0•'0 " i" o .2 . . . . . . . . . . . . . . . . :+'",--"'O-'O'"";,"? ......... +~........ !;'"'.;, ................................ ÷

. ..

.

° : ":.

•

! •

•°.0

I.:. i•

.

" o . " .~

0

+. . . . . . . . . . . . . . . . . . .

:

i :o

I1-

". ":: ~" .

:

i:

!

'

+~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

"

.!

1'5

20

w~

25

35

30

( c m s -~)

.8-

B +

6 ..................................

~.................. -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,

....................................................

~o + !÷ 4 ..................

. . . . . . . . . . . . . . . . . +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~.

÷ . . . . . . . . . . . . . ~. . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . . . . . :. . . . . . . . . . . . . . . . .

i

a .................................................. i ...........i~.~: .............. ::.................................................... •

i.

~"

::

o :';.'.

. :

•

• .•

.

" ~ •

• :° 0

*:

-6

-fl

-4

-2

0

T i m e Lag of w ~

Fig.

crest

11. (B)

The

relationship

in relation (+)

to =

of maximum

~Hm~/~x. plunging

The

wave

vertical

velocity

unbroken

(O)

=

with

foam;

8

to Wave Cresl

and

(A)

and

wave; (x)

=

(©)

bore

of its time =

lag relative

plunging

(at position

wave

to the wave

without

foam;

4).

Eddy motions. T o e x a m i n e further the turbulent eddy structures, R e y n o l d s stress (-u'w') and vorticity (~y) also w e r e calculated for the three w a v e types (Fig. 12). The u n b r o k e n w a v e s h o w e d relatively l o w values of -u'w' and Cy [Fig. 12(A) and (B)] although a m a x i m a in Cy occurred at x/L of - 0 . 2 and 0.5, corresponding to the position of Wmax [Fig. 10(A)] and the change of cross-shore velocity from an o n s h o r e direction to an offshore direction [Fig. 6(B)]. U n d e r the breaking w a v e [Fig. 12(C) and (D)], a large

1235

Kinematics of breaking waves

•

-2 -0.2

0

0.2

0,4

0.6

0.8

_:°t ...........!......................... i........................... J......................... !........................... ;....................... i,t

-0.2

0

0.2

0.4

0.6

0.8

~.

o

~.

2 ........... ! ......................... + .................... ! ..........................

fi

,

,-.

+

+

-2

I

- .....

+

+

+

.......... i .......................... i ........................... ! ........................... : ......................................................

i

-0.2

15,I

m

- .............................................

+

0

':

........... :, .............

++...................

'~ ........ +......

-0.2

:,

0.2

0.4

!

+................. f --!

+

0.6

+.................................. +

c

i .........

+I

0.8

.....................i +I . ~+

I

13

+................. +...........i~.............: ...........................+...........................+..........."I

0

0,.2

0.4

'

0.6

i

0.8

!°

E

2 ....................................... :.......................+.....................,;.........................:,...........................:...........

-0.2 15 l

0

0.2

0.4

0.6

0.8

:

.

~

~

~

.sl-..... ...........+:...........................;+....................+.................... ~...........................;...........................~..........

-m + ...........+i....................... ~ ................................................. ; -0.2

0

0.2

0,4

-.......................... ~ : ...........

0.6

0,8

x/L Fig. 12.

The variation of - u ' w ' and ~, respectively, under the unbroken wave (A, B); breaking wave (C, D); and bore (E, F).

f l u c t u a t i o n in - u ' w ' a n d ~y occurs n e a r t h e w a v e crest; a large s p i k e in ~y O c c u r s at x / L = 0 . 3 , t h e p o s i t i o n in t h e w a v e a s s o c i a t e d v~ith reversals in u [Fig. 7 ( B ) ] . A l s o , t h e u ' w ' signal is i n i t i a t e d a h e a d o f t h e w a v e crest similar t o ~y but r e a c h e s a m a x i m u m i n t e n s i t y f o l l o w i n g t h e m a x i m u m in vorticity.

1236

YANLINGYUet al.

The distribution of - u ' w ' and ey under the bore is particularly interesting. Two prominent fluctuations occur at about x/L = 0.05 and 0.2, respectively [Fig. 12(E) and (F)], and an additional spike in ey is seen at x/L = 0.4 where the cross-shore flow reverses [Fig. 8(B)]. The Reynolds stress and vorticity spikes are consistent with the suggestion of large-scale eddy motions and turbulence generation associated with collapsing waves. NADAOKA and KONDOH (1982) and NADAOKAet al. (1988) also observed, in both their laboratory and field experiments, this type of large-scale fluctuation in instantaneous Reynolds stress [e.g. Fig. I(A)]. In addition, the laboratory experiment of NADAOKAand KONDOH(1982) showed a higher turbulent energy around the wave plunging point than at the breaking point, which is also true for these data [i.e. the breaking wave shown in (C) and (D) is close to the breaking point while the bore shown in (E) and (F) is a recently collapsed wave closer to the plunge point]. Sediment suspensions Patterns. The patterns of suspended sediment can be observed as a series of snapshots along the axis of wave propagation showing the distribution of suspended sediment for various wave conditions. These patterns are illustrated first in relation to the three wave types used as examples throughout this paper, and second, in relation to a single wave as it breaks and propagates shoreward past each cross-shore instrument position. Figure 13 shows the sediment response to the three types of waves shown in Figs 6, 7, and 8, respectively. For unbroken waves [Fig. 13(A)], the suspended sediment distribution is restricted to a region very close to the seabed [on the order of the wave boundary layer, 0(cm)] and is generated under the shoreward moving part of the wave, i.e. from the wave crest to about 0.35 wavelengths seaward ( - 4 s in time). Background levels of suspended sediment in the water column are about 1 g 1-1, whereas the near-bed concentration under the crest reaches 14 g 1-1. The breaking wave pattern [Fig. 13(B)] shows an increased nearbed concentration of suspended sediment (--<30 g 1-1) and an increased horizontal distribution that begins slightly before the wave crest and extends over 70% of the wavelength (x/L = 0.7). The concentration patterns occurring at x/L > 0.4 reflect previously suspended sediment being carried offshore after the flow reverses [Fig. 7(B)]. Additionally, significant quantities of sediment have been transferred vertically to an elevation of about 24 cm. The suspended sediment associated with the bore [Fig. 13(C)] shows a mixed pattern; nearbed concentrations are not pronounced ( - 2 g 1-1) but are higher than background concentrations of 1 g 1-1 throughout the water column, and high concentration patches and inversions of suspended sediment are observed in the water column. OBS sensors are not strongly affected by entrained air bubbles (DowNING et al., 1981), and since the bore depicted in Fig. 13(C) has just evolved from the plunging process and is associated with major vortex motion [e.g. Figs 8(C), 10(C), 12(E) and (F)], the sediment pattern is interpreted to represent sediment-laden water parcels mixed upwards by the breaking process. The development of the suspended sediment concentration field as one wave propagates across the surf zone is shown in Fig. 14. Beginning at position 4, the wave is at the midbreaking stage with strong nearbed gradients under the crest and significant vertical mixing associated with the wave trough. As the wave propagates shoreward, the continued vertical mixing and a suggestion of vortex structure are seen to dominate (positions 3

1237

Kinematics of breaking waves Unbroken Wave -0.3 -0.2 -0.1 -0.0 o

~

~

I

0.1

0.2

I~..~l

0.3

0.4

LI

'

I

0.5 I

I

0.6 I

'

0.7 I

I

0.8 I

I

0.9

A

l"

45.5

45.5

34.9

~ 24-.3

24.3

.--~'E-13.6

13.6

f:

3.0 ~-, ~, -0.3 -0.2 -0.1 -0.0

0.1

0.2

I ~, ~"--..,.-,~, 0.3 0.4 0.5 0.6 0.7 x/L 14g/l

, ,

0.8

0.9

3.0

Breaking Wave ~"

-0.3 -0.2 -0.1 -0.0

~

~

4

5

.

'

0.1

0.2

0.3

' '

0.4

0.5

0.6

0.7

0.8

0.9

B

' ' ' ' ' ~

5

-

45.5

/ " / " O~

¢~ 34-.9

34.9

>~ 24-.3 /]

24.3

.~ 1,,3.6 o) m

13.6

-0.3-0.2-0.1

-0.0

0.1

0.2

0.3 X/L 50g

0.4

0.5

0.6

0.7

0.8

0.9

0.4- 0.5

0.6

0.7

0.8

0.9

3.0

13.

Bore •~

-0.3-0.2-0.1

-0.0

0.1

0.2

0.5

C

~j M-.9

34.9

e> 24-.3

24.3

,3.6

,3.6

.~.o 7- q-q ,~"~--0.3-0.2-0.1-0.0

c

3.0

0.1

0.2

0.3 0.4- 0.5 x/L 2g/J~

0.6

0.7

0.8

0.9

Fig. 13. Spatial distribution of sediment concentration (g 1-1) under the three types of waves (shown in Figs 9-11, respectively). The maximum near-bed sediment concentration under the crest region is shown below each panel.

and 2). Suspended sediment concentration steadily decreases in a shoreward direction, until at position 1 a nearbed concentration gradient of reduced magnitude is observed under the crest of the bore. The maximum suspended load occurs shoreward of the breakpoint at position 3 and continually decreases as the wave propagates toward shore.

Suspended load. The suspended load has been computed for every wave in the data set. As only three bores were identified at position 4, for the analysis of suspended sediment characteristics, the data from the numerous bores (n = 24) that occurred at position 3 are

1238

YANLING YU et al.

,76

4,6

Breaking Wave (Position 4) -0.3 --0.2 -0.1 -0.0 0.1 0.2 0.3 0.4 0.5 0.6

~o

0.7

o.a

0.9

32.6

32.a

~17.9

17.9

-r

3.0 - 0 . 3 - 0 . 2 -0.1

-0.0

0.1

0.2

0.3

Initial Bore

--0.3-0.2-0.1--0.0

v

0.1

0.2

0.4

0.5

0.6

0.7

0.8

0.9

0.6

0.7

0.8

0.9

3,0

(Position 3)

0.3

0.4

0.5

47.6

47.6

32.8

32.8

.~ 17.9

17.9

>=

-~ 3.0

3.0

•.r

- 0 . 3 -0.2 -0.1

-0.0

oE

--0.3-0.2-0.1-0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

O.S

0.7

0.8

0.9

Bore (Position 2) 0.1

0.2

0.3

0.4

0.5

~m

f~

•$

:z:

18.2

3.0

-0.3-0.2-0.1-0.0

0.1

-0.3-0.2-0.1-0.0

0.1

' 0.3

0.2

0.4

3.0

0.5

0,6

0.7

0.8

0.9

0.4

0.5

o.e

0,7

o.a

0.9

0.4

0.5

~ ' 0.6 0.7

0.B

0.9

Bore (Position 1) 0.2 .

.

.

0.3 .

.

7.6 .c -~

=

Fig. 14.

3.0 I ~_ -0.3-0.2-0.1-0.0

0.1

0.2

0.3 x/L

.

Distribution of sediment concentration field (g 1-1) under a single wave as it propagates shoreward past positions 4--1, respectively.

used. Using the time lag (At) of 1Vmax tO the wave crest to differentiate wave type, the relationship between the respective suspended load is shown in Fig. 15. Time lag of Wmax was selected as the independent variable in Fig. 15 because it is interpreted to be indicative of wave position relative to its break point (minus = prior to breaking, positive = broken). From these data a systematic variation in suspended load is observed (Table 3). For unbroken waves, suspended loads range from 16 to 127 mg cm-2 and average 38 mg cm -2. Breaking waves without foam range from 21 to 107 mg cm -2 with an average of 51 mg cm -2, while breakers associated with foam range from 20 to 443 mg cm -2 with an average of 110 mg cm -2. Sediment loads under bores (position 3) show a wide range of values as

1239

Kinematics of breaking waves 8 t~

x ¢q

...................................

. ...................................

! ...................................................................

.o

ix

t,o

o g !

Eo

; +

i ..................................

f it.

i

x

~t +

!

i

o~

i

i

•

' '

! • •

.

I:' •

~o

"-:

.

~

.

i

" i."

: -6

+ ................................................................

": +

g

-8

x

+i

! :..................................

c5

-~

:

+ i

x

*

"i x ,

x

! x

° +

Xx

...

°

"~_ .

i-

.P.;ol

! ~++"

.

x

x: x-

x x

x

i

,

i

i

-4

-2

2

4

Time Lag of Wmax to Wave Crest (sec)

Fig. 15. Total mass sediment load relative to the time lag of maximum vertical velocity to the wave crest. The (O) = unbroken wave; (O) --- plunging wave without foam; ( + ) = plunging wave with foam; and ( x ) = core (at position 3). The underlined bore symbols designate bores at position 4 (a third point is off scale).

they evolve from collapsed waves and propagate shoreward. Suspended sediment concentrations range from 17 to 336 mg cm -2 with average suspended loads of 87 mg cm -2. This average value represents a mean for all bores passing position 3 (24 waves); however, as shown in Fig. 15, the mean value steadily decreases from about 150 mg cm -2 for bores occurring near the breaker line (time lag - 0 s) to about 50 mg cm -2 for bores some distance shoreward of the breaker line (time lag - 6 s). Suspended loads appear to continually decrease towards shore (Fig. 14). The large suspended sediment concentration range observed in the region dominated by breakers with foam (20-443 mg cm -2) and initial bores (17-336 mg cm -2) is qualitatively consistent with suggestions from previous studies showing plunging waves impinging on the seabed and curling upward (MILLER, 1976) and entrained air bubbles and sediment upwelling in close proximity to breakers (NADAOr~A, 1986; NADAOr,A et al., 1988, 1989). The internal velocity structure observed in the present data set also supports the observed suspended sediment pattern. Upon breaking, the waves tend to form major vortices with associated strong nearbed shear in the u velocity and major oscillations in the w velocity. Sediment response under these conditions is one of rapid upward mixing and suspended sediment patchiness within the water column.

1240

YANLINGYU et al. Table 3. Mean, minimum and maximum values of suspended load for the various classes of waves and the ratio of mean concentration relative to unbroken waves. Numbers in parentheses are number of waves

Breaking Unbroken (114) Mean Min Max Ratio

38.3 mg c m 15.7 126.9 1.0

-2

without foam (41) 50.7 mg c m 20.7 106.9 1.3

-2

with foam (39) 109.8 mg c m 20.0 443.0 2.9

-2

Bore* (24) 86.9 mg c m 17.3 335.7 2.3

-2

*At location 3. The absolute values of suspended load depend on many factors related to wave conditions, beach slope, and grain characteristics on a particular beach. It may be more relevant to compare relative magnitudes of suspended load as a way of expressing crossshore variability (Table 3). Thus suspended load under breakers is as much as 2.9 times that of unbroken waves. Bores suspend on the average about 2.3 times as much as unbroken waves; however, this factor varies form a magnitude of 4 near the plunge point to 1.3 at a distance of 10-15 m shoreward. CONCLUSIONS The following conclusions can be drawn from this study. (1) A progressive changing shape and velocity structure is associated with shoaling waves. The onshore velocity under the wave crest increases as the wave approaches the breakpoint, and the vertical velocity changes significantly in magnitude and phase relative to the wave crest. In general, the time lag of the vertical velocity indicates the stage of a wave's development in the surf zone. In unbroken waves, the maximum vertical velocity within the wave leads the passage of the wave crest. At the breakpoint, the vertical velocity occurs under the crest and, after breaking, the maximum vertical velocity sequentially lags the wave crest as bores propagate shoreward. (2) Major eddies, or vortices, begin to form upon breaking. Initially, wave orbital motions (potential flow patterns) dominate the velocity structure and wave shape. However, as the wave breaking process progresses, orbital velocities are reduced and eddy motions and associated turbulence dominate the wave velocity field. These internal motions also are manifested in sea surface undulations on the trailing side of the bores. (3) The breakdown of the wave into eddy motions is observed in a variety of ways: systematic variations of Reynolds stress and estimated vorticity; strong vertical shear in the horizontal velocity; the strong oscillations of vertical velocity following the wave crest. The oscillatory motions described above have approximately 2 s periods ( T s = 10-12 s). (4) Field observations of the shape, internal velocity field and eddy structure of shoaling and breaking waves are qualitatively similar to waves produced in laboratory simulations. (5) Sediment suspension patterns are responses to various fluid motions within the wave. For unbroken waves and waves near the breaking point, sediment suspensions occur within several centimetres of the seabed (within the wave boundary layer). Nearbed concentrations increase as a result of increasing onshore flow as waves approach the

Kinematics of breaking waves

1241

breakpoint. For breaking waves, suspended sediment concentrations tend to increase dramatically and sediment is mixed to higher levels in the water column. Sediment inversions and localized high concentration patches of sediment also occur. This pattern appears to correspond to eddy motions which impinge directly on the seabed, produce intense turbulence, and entrain significant quantities of bubbles which all contribute to suspending and mixing sediment upward into the water column. As bores propagate shoreward, their overall sediment concentration decreases and occurs mainly under the bore crest. (6) The average suspended load varies as a function of wave type. For the conditions measured at Duck, NC, suspended sediment loads were 38 mg cm-2 for unbroken waves, 51 mg cm -2 under the initial break point (breakers without foam), and 110 mg cm -z under the plunge point (breakers associated with foam). The suspended load under bores averaged 87 mg 1-1 but steadily decreased from 150 mg cm -2 near the breakpoint to 50 mg cm -2 as bores propagated past position 3. The ratios of the suspended loads for unbroken waves, breaking waves, and bores propagating shoreward are 1:2.9:4 ~ 1.3 (averaging 2.3), respectively. The maximum suspended load occurs just shoreward of the breaker and the majority of the suspended load settles out of suspension within 10-15 m shoreward of the plunge point. Acknowledgements--The authors would like to thank Rob Holman, Rex Johnson and Gail Kineke for their assistance in the field; the crew at the USACOE Field Research Facility at Duck, NC, for their overall facilities support to carry out the field experiment and Dora P. Henry and Joan Oltman-Shay for their review of the manuscript. This research project was supported by the Office of Naval Research under Grant N00014-90-J-1189. This is Contribution No. 1936 from the School of Oceanography, University of Washington.

REFERENCES BEACH R. A. and R. W. STERNBERG (1988) Suspended sediment transport in the surf zone: response to cross-shore infragravity motion. Marine Geology, 80, 61-79. BEACH R. A. (1989) Suspended sediment transport in the surf zone. Dissertation to University of Washington, pp. 143. BAVI'JESJ. A. (1988) Surf-zone dynamics. Annual Review of Fluid Mechanics, 20,257-293. DOWNINGJ. P. (1983) Field studies of suspended sand transport, Twin Harbors beach, Washington. Dissertation to University of Washington, pp. 121. DOWNING J. P., R. W. STERNRERGand C. R. B. LISTER (1981) New instrumentation for the investigation of sediment suspension processes in the shallow marine environment. In: Sedimentary dynamics ofcontinential shelves, C. A. NIYrROUER,editor, pp. 19-34. Marine Geology, 42 (special issue). GALVINC. J. JR (1968) Breaker type classification on three laboratory beaches. Journal of Geophysical Research, 73, 3651-3659. HANES D. M. and D. A. HUNTEEY(1986) Continuous measurements of suspended sand concentration in a wave dominated nearshore environment. ContinentalShelf Research, 6,585-596. HAYrORI M. and T. AoNo (1985) Experimental study on turbulence structures under breaking waves. Coastal Engineering in Japan, 28, 97-116. HUNTLEY O. A. (1988) A modified internal dissipation method for estimating seabed stresses at low Reynolds numbers, with application to wave/current boundary layer measurements. Journal of Physical Oceanography, 18, 339-346. HUNTEEY D. A. and D. M. HANES (1987) Direct measurement of supended sediment transport. In: Coastal Sediments '87, N. C. KRAUS,editor. Proceedings of a Specialty Conference on Advances in Understanding of coastal Sediment Processes, May 1987, New Orleans, Louisiana, pp. 723-737. INMAN D. L., J. A. ZAMPOL,T. E. WHITE,D. M. HANES,B. W. WALDORFand K. A. KASTENS(1980) Field measurements of sand motion in the surf zone. Proceedings of the 17th Coastal Engineering Conference, pp. 1215-1234.

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