Prediction of average mixing depth of sediment in the surf zone

Marine Geology, 62 (1985) 1--12

1

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

PREDICTION OF AVERAGE MIXING DEPTH OF SEDIMENT IN THE SURF ZONE

TSUGUO SUNAMURA and NICHOLAS C. KRAUS*

Institute of Geoscience, University of Tsukuba, Ibaraki 305 (Japan) Nearshore Environment Research Center, Tokyo (Japan)

ABSTRACT Sunamura, T. and Kraus, N.C., 1985. Prediction of average mixing depth of sediment in the surf zone. Mar. Geol., 62: 1--12. A predictive model for the average depth of mixing of sediment in the surf zone is formulated on the basis of the wave-induced shear stress o n the bottom. The model is calibrated using field data obtained through tracer experiments performed on Pacific beaches of Japan. The result gives Z/D ffi 81.4 (*b -- ~I'c), where Z is the average depth of sediment mixing in the surf zone, D is the grain size of the sediment, ~b is the Shields parameter at the wave breaking point, and *c is the critical Shields number for oscillatory flow. The predicted mixing depth is found to increase in an approximately linear fashion with the breaking wave height for significant breaking wave heights up to about 1.5 m. For greater breaking wave heights, the rate of increase in the mixing depth decreases with increasing wave height. The mixing depth is also a strong function of the wave period in the larger wave height region. Finally, the predicted mixing depth is found to be a weakly increasing function of the sediment grain size.

INTRODUCTION

Intense vertical mixing of the bottom sediment by waves takes place in the surf zone. It is generally assumed that the depth of this mixing is equal to the thickness of the horizontal moving layer of sediment. Studies of the mixing depth are of twofold interest. One is directly related to the empirical estimation of the bulk longshore sediment transport rate, Q, using the relation Q = Va Z Xb, where V~_is the average longshore advection velocity of the sediment in the surf zone, Z is the average value of the depth of mixing and Xb is the width of the surf zone (e.g., Komar and Inman, 1970; Inman et al., 1980; Kraus et al., 1981, 1982). The other point of interest is closely related to the erosion of underwater bedrocks on rocky coasts where a veneer of sediment exists (e.g., King, 1972, p. 257; Sunamura, 1983). If sediment particles do not move, erosion of the bedrock will not occur; once particles move, the bedrock suffers abrasion (Sunamura, 1978). *Present address: Coastal Engineering Research Center, U.S. A r m y Engineer Waterways Experiment Station, P.O. Box 631, Vicksburg, MS 39180, U.S.A. 0025-3227/85/$03.30

© 1985 Elsevier Science Publishers B.V.

A limited number of studies, beginning with that of King (1951), have dealt with the topic of the depth of sediment mixing produced by wave action (Komar and Inman, 1970; Madsen, 1974; Gaughan, 1978; Greenwood and Hale, 1980; Inman et al., 1980; Kraus et al., 1981, 1982; Kraus, 1985). All of these concern work performed in the field, except Madsen's (1974) theoretical study. Kraus (1985) has given a comprehensive discussion of the defining procedure for the mixing depth and reviewed the literature. Komar (1983) provides a good summary of the field studies and emphasizes the need for further investigation on this topic. King (1951) found a correlation between the depth of mixing and the breaking wave height: a linear relation with a proportionality coefficient of about 0.03 was obtained from the rather widely scattered data. The data obtained by Komar and Inman (1970) show a similar tendency for the depth of mixing to increase with the breaking wave height. By means of a theoretical model relating soil failure to the wave-induced horizontal pressure gradient at the bed, Madsen (1974) deduced a linear relation between the mixing depth and the breaking wave height and estimated a proportionality coefficient of 0.11 for the region immediately under the breaking wave. Recently, based on extensive tracer experiments, Kraus et al. (1982) and Kraus {1985) have clearly demonstrated that the mixing depth can vary across the surf zone. The average surf-zone mixing depth was also found to be linearly related to the breaker height in these experiments. It has been suggested by both King {1951, 1972, p. 257) and Komar (1983) that the mixing depth should increase in some (unspecified) manner with the sediment grain size. However, no detailed examination of the grain-size dependence has been made. For a more rational treatment, the grain-size factor should be incorporated. In the present study we reanalyze the data of Kraus et al. (1982) using a model based on the shear stress exerted by waves on the b o t t o m . The Shields parameter emerges as the important governing quantity and a predictive equation is proposed in which the average mixing depth in the surf zone is given as a linear function of the Shields parameter. DATA USED IN THIS STUDY The depth to which sediment mixes in the surf zone was investigated in eight tracer experiments performed on four wave-dominated Pacific beaches of northeastern Honahu, Japan: Oarai, Ajigaura, Hirono and Shimokita (Table I; Kraus et al., 1982; Kraus, 1985). These beaches experience a mean tidal range of 1 m. Hirono beach is composed o f coarse sand and the other beaches are composed of fine to medium sands. The significant wave parameters were calculated from data instrumentally obtained during each experiment and statistically processed, with the exception of Ajigaura-78, for which the wave parameters were mainly measured visusally during the experiment.

TABLE I Experiment conditions and results Experiaent

8 e d i u e n t and bee©h c h a t a ¢ t e , l l t i c s Mean gzain lieershoze Date

1

AJIGAURA 70

12/14/78

0.23

2.6S

qua,iS

aand

1/50 - 1/70

2

AJIGAURA 79

8/31/79

0.27

2.65

qua,tz

oand

1/50 -

1/70

3

SHINOKITA

10/27/79

0.18

3.13

quartz lind heavy mine,als

1/40 -

1/60

4

HIBOMO - 1

11/13/80

0.59

2.67

q u a c t s sand & pebbles

1/10

He. Beach

Specific gravity

| r e e k i n g wave c h a r a c t e r i s t t e s H e i g h t 8 Period e Depth Type h

SiZe (mm)

slope

(¢!)

(s)

Average m i x i n g depth i n s u [ f zone

(cm)

Coipoeition

(elm) 06

9.0

90

P & S

].8

110

6.5

130

P & S

2.9

63

4.9

110 c

8 k P

2.3

161

S.7

100

P & C

3.7

S

HIBOUO - 2

11/14/80

0.59

2.67

quarts

sand

1/10

100

6.4

120

6

OAEAI 00

12/8/80

0.25

2.79

quartz

|and

1/50 -

1/?0

100

10.2

100

P i

S

2.8

7

OAItAI

01

6/27/81

0.24

2.79

q u e s t s sand

1/50 -

1/70

111

6.1

160

P & S

2.3

8

OARAI 62

8/26/82

0.25

2.79

quartz

1/30 -

1/40

80

7.5

100

P & S

1.0

a)

Significant

waves•

b)

e = Plunging;

S -

& pebbles

Splllinq;

sand

C - Collaplingp

¢)

Chenqd

to

81 cl

in

calculationl

P

-

3.0

nee

text

The experiments were carried out on a time scale long enough for the tracer mixing to attain equilibrium (Kraus et al., 1981), but short enough not to suffer from contamination produced b y changes in sand level due to: (1) changes in wave and tidal conditions (e.g., Inman and Filloux, 1960; Hattori, 1982); and (2) migration of large-scale bedforms such as inner bars (e.g., Hayes, 1972; Sunamura and Takeda, 1984). Native sand dyed with fluorescent color was injected along a straight line from the shoreline to the average breaker line. For all experiments, a large number of core samples were taken on a two-dimensional grid downcurrent of the injection line and spanning the surf zone {spatial sampling). In addition, core sampling was performed frequently on one or two lines crossing the surf zone in five of the experiments (temporal sampling). In these experiments sheet flow dominated with no ripples forming on the surf-zone b o t t o m . An objective determination of the mixing depth was made through systematic processing of the core samples. For details of the mixing depth determination, see Kraus (1985). Figure 1 shows two examples of cross-shore distributions of the mixing depth in the surf zone. The depth of mixing, Z, is plotted on the vertical axis. The horizontal axis indicates the offshore distance from the shoreline, normalized b y the surf-zone width, Xb. The notation "T-20 m " denotes a temporal sampling on a line located 20 m from the tracer injection line; similarly, the notation "S-120 min" denotes a spatial sampling performed 120 min after injection. The length of the bars at the plotted points is two standard deviations and the number of samples averaged at each point is given in parentheses below the bars. Considerable variation in local Z-values is found (Fig. 1). We plan to discuss the variation of Z across the surf zone in a future publication. An average value for the surf zone, denoted as Z,, was obtained for each ex-

p e r i m e n t (Table I), as illustrated in Fig. 1 b y the h o r i z o n t a l lines and numbers in brackets. ( T h e indicated values are averages f o r m e d f r o m all samples comprising the particular e x p e r i m e n t ; the distributions s h o w n are a subset o f t h e t o t a l samplings for t h e respective e x p e r i m e n t s . ) 8 7

AJIGAURA 79 T xb = 52.5 m

6

I I

T- 20m 5-120 min

T

I

~5

(6)

E4 N

i~:.,~

3 2

,

<'°>l <'°> a

x

;,>~

T

<'~'
oll

I

!1

,7 ,~ 7

i (~, ('~>J-J i (18)

, -L!30> iI1(~)

(26)

(27)

(~'

(14)

( i; No. of c o r e s

averaged

.2'5

1.2! DistQnce o f f s h o r e

OARAI Xb =

T

T

1

il

Y

u

82

S,60min

36 m

S-180 rain

l

-

?+ ~

T/ I

"2 N

X/X b

T

,

-

i

i

'

I

I

4>

~

1

'

4>

,

It

.4~ ]1 ,,Ji 7 ~] +T I/ i/ I (23>j. '~' II i T (24> / ',1 Jl IJ, (23~ / /

1 (33)

(31'

1 ¢27) 11

(11)

x (,,>cs,I ~(34) I J.

0

(34)

.is Distonce

I

(3,, 1

0)'!"

b

(12)

TM

(II)

(24) ): NO. o f c o r e s

.~ offshore

I

(24)

(,6 (21)

averaged

.7'5 ×/X b

Fig. 1. Example of on-offshore distributions of the mixing depth (a) for the Ajigaura beach experiment in 1979 and (b) for the Oarai beach experiment in 1982.

MODEL BASED ON THE SHEAR STRESS Waves in the surf z o n e e x e r t a strong shear stress o n t h e b o t t o m . This shear will act t o fluidize t h e u p p e r layer o f s e d i m e n t t o s o m e d e p t h and in t h a t state t h e s e d i m e n t grains can m o v e laterally and vertically. L e t us assume initially t h a t t h e stress e x e r t e d o n t h e b o t t o m , t o , s u p p o r t s an i m m e r s e d weight o f s e d i m e n t layer, ~, having u n i t surface area and

thickness ~. We then have: p. = r o

(I)

and: =

(Ps

-

p ) g ( 1 - e)~"

(2)

where Ps is the density of the sediment (assumed to be mainly quartz sand), e is the porosity of the sediment layer (e = 0.4), p is the density of sea water and g is the acceleration of gravity. From eqs. 1 and 2: ~o

=

(Ps - p ) g ( l

(3)

- e)

For an estimate o f ~, the quantity TO will be replaced by the shear stress at the wave breaking point, rb, which can be calculated directly from the data. Therefore: =

Tb (Ps -- p)g(1 -- e)

(4)

The quantity r b is given by: 1

Tb

2 PfwUb

(5)

where Ub is the maximum near-bottom horizontal orbital velocity of the breaking waves and fw is a wave friction factor (Jonsson, 1966) c o m m o n l y employed in wave and sediment transport studies. Applying linear wave theory: Ub

T sinh (21rhb/Lb)

(6)

where Hb is the breaker height, hb is the depth at the breaking point, L b is the wavelength at the breaking point and T is the wave period. The wave friction factor is obtained by iteratively solving the equation (Jonsson, 1966): 1 I ab 4~/fw + log 4~/f w = "0.08 + ]Og--r

(7)

where log is t o the base 10, r is a roughness length which we identify as the sediment grain diameter, D, for the case of a smooth bottom (no ripples), and ab is the horizontal semi-excursion distance of the wave orbit at the bottom. In linear wave theory ab is given by: ab

Hb 2 sinh (27rhb/Lb)

(8)

It is noted that fw is an implicit function of the wave period, entering through L b in eq. 8. Using the Oarai beach data (Table I) to arrive at a representative value of ~, direct application of the naive model expressed by eq. 4 yields ~ -~1 mm. This result is an order of magnitude less than the depth to which tracer is actually forced to mix under moderate wave conditions, typically 2--4 cm as shown in Fig. 1 and Table I. Because of fluid-to-grain interactions, however, the surface layer of the b o t t o m will be dilated (Reynolds, 1885; Bagnold, 1954; Bailard and Inman, 1979) to a greater depth, permitting free lateral and vertical movement of grains. The amount of dilation is quantitatively unknown, but nevertheless an underestimation by eq. 4 is to be expected. Based on the above development, we assume that the average surf-zone mixing depth, Z, is proportional to the b o t t o m shear stress exerted by breaking waves, rb. To this end, ~ in eq. 4 is replaced by Z and a nondimensional constant, k, is introduced to account for the effect of dilatancy. Then: 2 =k

Tb (Ps - p)g(1 - e)

(9)

Normalization of eq. 9 by means of the sediment grain size leads to: 2 - - = K ~b, D

K =

k 1-e

- (= constant)

(10)

where: ~b ~'b = (p, _ p)glJ

(11)

is the Shields parameter at the breaking point. It is well known that sediment will not begin to move until the Shields parameter exceeds a certain critical value, ~c. Therefore, eq. 10 is modified to become: (~b--q'c)

--=K'

D

(12)

where K' is a constant. The critical Shields number ~ c for oscillatory flow can be obtained graphically (Madsen and Grant, 1976, curve in Fig. 1) with knowledge of the nondimensional fall velocity of the sediment, S , , which is expressed by: D

S, = ~

[(pslp - 1)gD] ~/2

(13)

where v is the kinematic viscosity of the fluid (-~ 0.01 cm 2 s-l). The quantities ~I'b and x~c were calculated using the data set to determine

K'. Results of the intermediate calculations are given in Table II. Figure 2 shows the normalized mixing depth, Z/D, plotted against the effective Shields parameter, ~I'b -- q'c. The bars are one standard deviation in length. It was noted that the No. 3 data point (Shimokita beach) deviated considerably from the general trend of the data. Inspection of Table I and experience at the Shimokita experiment indicate that the breaking depth may have been missurveyed. Therefore, the breaking depth was estimated from the more reliable observed breaking wave height via the well-known empirical breaking criterion:

Hb/h b

=

0.78

(14)

TABLE II Calculated quantities associated with Fig. 2 Exp' t NO.

ub (cmls)

fw

1 2 3 4 5 6 7 8

159 145 105 177 140 155 139 122

0.0054 0.0062 0.0066 0.0067 0.0072 0.0054 0.0062 0.0062

~Ub

~)c

Z/D

1.813 1.456 0.904 1.051 0.692 1.436 1.391 1.008

0.045 0.042 0.051 0.033 0.033 0.043 0.044 0.043

165 107 128 63 51 112 96 76

2.0

I

1

T_i

1.5 A 0 0

3 1

/1

"

i.o a

T

tN .L

0.5

ZD= m.4(tYb-~) I

0

I

0.5

I

I

1.0

I

I

1.5

i

2.0

'¥b--'¥c

Fig. 2. Relation between normalized average mixing depth and the effective Shields parameter.

As a result of correcting the breaking depth, the No. 3 data point moved to the right, as shown in Fig. 2 by the dashed line. The correlation line in Fig. 2 was drawn based on the calculation incorporating this shifted data point. The line is expressed by:

2

- - = 81.4 (45 - - ~ c ) D

(15)

The model eq. 15 will be examined in the next section and the predictions compared to the empirical results. DISCUSSION For the surf zones of energetic coasts composed of fine to coarse sands, the value of ~b is much larger than that of ~I,c {Table II). Therefore, we make a negligibly small error by setting ~c = 0 in eq. 15. Then: 40.7 f w U ~ -- = 81.4 q2b = D (Ps/P - 1 ) g D

(16)

Applying eq. 14 and the shallow-water wave approximation s i n h ( 2 n h b / L b ) 2 u h b / L b and L b = T ( g h b ) 1/2, eq. 6 reduces to:

=

(17)

Ub = ( g l i b ~ 5 . 1 3 ) lp

From eqs. 16 and 17, we finally obtain: 2=

7.93 fw

Hb

(18)

(Ps/P - 1)

To the extent that fw can be assumed to be constant, eq. 18 indicates that 2 should be directly proportional to Hb. A linear dependence of 2 on

/[

, 1

~

s

IN 2

1

Z

A

o

,

,

,

J

,

,

,

,

so

i

,

~oo

Hb

~

= 0.027

~

,

J

1so

Hb

,

,

L

zoo

(cm)

Fig. 3. Average measured mixing depth as a function o f breaking wave height.

Hb was obtained empirically by King (1951) and Kraus et al. (1982). Figure 3 shows the relationship between Z and Hb for the data set in Table I. A somewhat wider scatter in the data points is seen in Fig. 3 as compared with Fig. 2. The straight line drawn through the origin gives the result (Kraus et al., 1982; Kraus, 1985): = 0.027 Hb

(19)

In general, however, eq. 18, or the more exact eq. 15, indicates that the mixing depth is a weak function of the grain size (which enters through fw) and a function of the wave period (which enters through fw and Ub), as well as the wave height. That is to say, the wave-induced stress on the b o t t o m varies with the b o t t o m roughness and the wave height and period. In order to examine these dependences, sample calculations were made with eq. 15 for the three grain sizes D = 0.02, 0.04 and 0.06 cm and the four wave periods of T = 4, 6, 8 and 12 s, for breaking wave heights up to 5 m. The results are given in Fig. 4, in which eq. 19 is also plotted. In this

12I

, , ~ ~

D=0.02 cm

12

10

D=O.O4cm

/

~.~

I0 12S

8

?

,~,

8

8S

U

~s

",--6

IN

24

~ 4 s

a o 12

l

,

|

1

.

i

,

i

,

i

,

i

2 3 Hb (m)

D=0.06cm

,

1

,

i

,

i

,

o

4

i

,

I

1

,

I

i

I

2

,

t

,

I

i

I

i

3

Hb (m)

4 i

i

I

i

,,'~iz,/

10 8 A

E

tJ

~6

IN

~4= 0

1

2 3 Hb (m)

C 4

5

Fig. 4. P r e d i c t e d values o f t h e average m i x i n g d e p t h as a f u n c t i o n o f wave wave p e r i o d f o r grain sizes o f (a) 0 . 0 2 cm, ( b ) 0 . 0 4 c m a n d (c) 0 . 0 6 cm.

heightand

10 figure, it was taken into account that a wave will break if its steepness exceeds a certain value. The Mitchell criterion, H o / L o = 0.142, was used, where H o / L o is the deep-water wave steepness (see, e.g., Komar, 1976, pp. 51--52). This limited the range of allowable breaking wave height for the 4-s waves. A number of interesting observations can be made from Fig. 4. First, the predicted ,~ is only a weak function of the grain size, as can be seen by comparing curves for the same wave period. The mixing depth increases with the grain size; this confirms the conclusions of King (1951, 1972, p. 257) and Komar (1983), arrived at on empirical grounds. Second, eq. 15 agrees well with the empirical results, eq. 19, for moderate wave heights (Hb less than approximately 1.5 m) and a wide range in wave period. However, a striking property of eq. 15 is that the rate of increase of Z falls off for larger wave heights, indicating that the rate of increase of the shear stress becomes smaller as the wave height increases. This is because although the quantity u~ in eq. 5 is approximately proportional to the wave height (eq. 17), the friction coefficient fw decreases with increase in the wave height. Finally, Fig. 4 shows that the stress-induced mixing depth is a fairly strong increasing function of the wave period for larger breaking wave heights. Typically, for a fixed Hb a noticeable increase in Z occurs for waves of periods varying between 4 to 8 s; the rate of increase of Z with T is much less for periods exceeding 8 s. This result indicates that the longshore sediment transport rate and the abrasion of bedrock will depend on the wave period at a fair level of significance. These results are expected to be valid for many typical surf zones. However, it should be cautioned that the model is applicable only where the wave-induced b o t t o m stress is the principle cause of mixing. Other factors could be equally or more important. For example, scouring and mixing under plunging breakers, turbulence caused by the collision of runup and rundown in the swash zone, pressure gradient at the b o t t o m (Madsen, 1974} and other factors may produce larger values than predicted here. CONCLUDING REMARKS A shear stress model was presented to predict the average depth of waveinduced sediment mixing in the surf zone. The model shows t h a t the normalized mixing depth is linearly related to the effective Shields parameter (eq. 15). Equation 15, which was calibrated with field data, agrees well with the empirical results (a linear relationship between 2 and Hb as expressed by eq. 19) for moderate wave heights and for a wide range in wave period on beaches composed of fine to coarse sands (Fig. 4). The predicted mixing depth is found to be a weakly increasing function of the grain size of the sediment for all wave conditions (height and period}, while it is a strongly increasing function of the wave period for larger wave heights

11 (Fig. 4). This finding indicates t h a t t h e wave p e r i o d is a p o t e n t i a l l y imp o r t a n t f a c t o r in empirical e s t i m a t i o n s o f t h e l o n g s h o r e s e d i m e n t t r a n s p o r t rate w h i c h involve t h e m i x i n g d e p t h . It is believed t h a t t h e p r e s e n t w o r k is t h e first t o find a clear c o r r e l a t i o n b e t w e e n field m e a s u r e m e n t s o f local s e d i m e n t m o v e m e n t a n d a t h e o r e t i c a l m o d e l based o n t h e h y d r o d y n a m i c s o f t h e surf zone. This p o i n t s o u t t h e e x t r e m e d i f f i c u l t y o f m a k i n g reliable s i m u l t a n e o u s m e a s u r e m e n t s o f t h e waves and s e d i m e n t m o v e m e n t in real surf zones, as well as s h o w i n g t h e general lack o f k n o w l e d g e o f t h e f l u i d / s e d i m e n t i n t e r a c t i o n in high waveenergy environments. REFERENCES

Bagnold, J.A., 1954. Experiments on a gravity-free dispersion of large solid spheres in a newtonian fluid under shear. Proc. R. Soc. London, Ser. A, 225: 49--63. Bailard, J.A. and Inman, D.L., 1979. A reexamination of Bagnold's granular-fluid model and bed load transport equation. J. Geophys. Res., 84, C12: 7827--7833. Gaughan, M.K., 1978. Depth of disturbance of sand in surf zones. Proc. 16th Coastal Eng. Conf., ASCE, pp. 1513--1530. Greenwood, B. and Hale, P.B., 1980. Depth of activity, sediment flux and morphological change in a barred beach environment. In: S.B. McCann (Editor), The Coastline of Canada. Geol. Surv. Can., pp. 89--109. Hattori, M., 1982. Field study on onshore-offshore sediment transport. Proc. 18th Coastal Eng. Conf., ASCE, pp. 923--940. Hayes, M.O., 1972. Forms of sediment accumulation in the beach zone. In: R.E. Meyer (Editor), Waves on Beaches. Academic Press, New York, N.Y., 297--356. Inman, D.L. and Filloux, J., 1960. Beach cycles related to tide and local wind regime. J. Geol., 68: 225--231. Inman, D.L., Zampol, J.A., White, T.E., Hanes, D.M., Waldorf, B.W. and Kastens, K.A., 1980. Field measurements of sand motion in the surf zone. Proc. 17th Coastal Eng. Conf., ASCE, pp. 1215--1234. Jonsson, I.G., 1966. Wave boundary layers and friction factors. Proc. 10th Coastal Eng. Conf., ASCE, pp. 127--148. King, C.A.M., 1951. Depth of disturbance of sand on sea beaches by waves. J. Sediment. Petrol., 21: 131--140. King, C.A.M., 1972. Beaches and Coasts (2nd ed.). Edward Arnold, London, 570 pp. Komar, P.D., 1976. Beach Processes and Sedimentation. Prentice-Hall, Englewood Cliffs, N.J., 429 pp. Komar, P.D., 1983. Nearshore currents and sand transport on beaches. In: B. Johns (Editor), Physical Oceanography of Coastal and Shelf Seas. Elsevier, Amsterdam, pp. 67--109. Komar, P.D. and Inman, D.L., 1970. Longshore sand transport on beaches. J. Geophys. Res., 75: 5914--5927. Kraus, N.C., 1985. Field experiments on vertical mixing of sand in the surf zone. J. Sediment. Petrol., 55(1 ). Kraus, N.C., Farinato, R.S. and Horikawa, K., 1981. Field experiments on longshore sand transport in the surf zone; time-dependent motion, on-offshore distribution and total transport rate. Coastal Eng. Jpn., 24: 171--194. Kraus, N.C., Isobe, I., Igarashi, H., S a ~ i , T.O. and Horikawa, K., 1982. Field experiments on longshore sand transport in the surf zone. Proc. 18th Coastal Eng. Conf., ASCE, pp. 969---988.

12 Madsen, O.S., 1974. Stability of a sand bed under breaking waves. Proc. 14th Coastal Eng. Conf., ASCE, pp. 776--794. Madsen, O.S. and Grant, W.D., 1976. Quantitative description of sediment transport by waves. Proc. 15th Coastal Eng. Conf., ASCE, pp. 1093--1112. Reynolds, O., 1885. On the dilatancy of media composed on rigid particles in contact. Philos. Mag., Ser. 5, 20: 469--481. Sunamura, T., 1978. A model of the development of continental shelves having erosional origin. Geol. Soc. Am. Bull., 89: 504--510. Sunamura, T., 1983. Processes of sea cliff and platform erosion. In: P.D. Komar (Editor), Handbook of Coastal Processes and Erosion. CRC Press, Boca Raton, Fla., pp. 233-265. Sunamura, T. and Takeda, I., 1984. Landward migration of inner bars. In: B. Greenwood and R.A. Davis, Jr. (Editors), Hydrodynamics and Sedimentation in Wave-Dominated Coastal Environments. Mar. Geol., 60: 63--78.