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Statistical properties of multiple bars
COASTAL ENGINEERING ELSEVIER
Coastal Engineering
3 1 (1997) 263-280
Statistical properties of multiple bars Z. Pruszak, G. R6iyiiski Polish Academy
of Sciences, Institute ofHydra-Engineering
*, R.B. Zeidler (IBW PAN), 7 Kdciersku,
’ 80-953 Gdansk,
Poland Received
14 May 1996; revised 7 February
1997; accepted 7 February
1997
Abstract A natural, mild-slope, multiple-bar shore with a sandy bed (D,, = 0.022 cm) at the Coastal Research Facility (CRF) Lubiatowo (Poland) is analysed. The bathymetric dataset consists of 81 cross-shore transects surveyed from 1964 to 1994. Correlations have been computed for couples of parameters of each bar and for the same parameters of different bars. The most significant correlations have been obtained for parameters of the innermost stable bar. Correlations of the same parameters of different bars decrease rapidly with the distance from shoreline. Further analysis has shown that water depths over consecutive bar crests increase similarly to the mean shore profile. Their change can be presented by a Dean-type function describing a new ‘bed equilibrium line’ for the envelope of bar crests. The line is directly coupled with breaker parameters. Hence a wave breaking criterion for multiple-bar profiles could be established. Wave breaking is put forth as a major factor of bar formation. 0 1997 Elsevier Science B.V. Keywords: analysis
Shore morphodynamics;
Equilibrium
profile; Multiple
longshore
bars; Bar geometry;
Correlation
1. Introduction A number of bed-form systems, which undergo spatial scales can be distinguished in the nearshore crucial role among them. The origin of such bars theories developed so far associate bar formation Dean et al., 1992) and various types of long waves
evolution in different temporal and zone, longshore sand bars playing a is still insufficiently explored. The with wave breaking (Dally, 1987; and their interactions (edge waves,
* Corresponding author. Fax: + 48-58-5242 11. ’ Associate Professor, Senior Research Associate, Professor and Head of Department, Polish Academy of Sciences’ Institute of Hydro-Engineering (IBW PAN). 0378.3839/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO378-3839(97)00010-O
respectively,
of
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31 (1997) 263-280
surf beats, infragravity waves) or relate bar emergence to components of the wave spectrum and nonlinear wave effects, together with energy exchange between those components (cf. Short, 1975; Salenger and Howd, 1989; Howd et al., 1991; BoczarKarakiewicz et al., 1995). Many recent studies on bars have concentrated on the assumption that bar location and characteristics depend primarily on the existence and location of breakers. Dally (1987) revealed no relationship between bar formation and surf beats, but he found very likely the linkage between bar formation and wave breaking. In addition, he noticed that bar formation did not necessarily need to be accompanied by spilling breaker, as it was also frequently observed in the case of plunging breakers. Similar findings were reported by Dean et al. (1992), who proved there was no relationship between bar formation and spatial distribution of nodes and antinodes of infragravity waves. They also suggested that wave breaking and convergence of cross-shore currents and sediment transport rates are crucial factors of bar formation. Hence bar shape may be assumed to depend on the stability of breaker location during bar formation. If the breaker stability is sufficient in time and space, the resulting bar has a conspicuous, regular shape. By contrast, unstable breakers generate flatter, less conspicuous bars. Bar emergence is thus the part of cross-shore evolution of the nearshore zone which might consist in erosion at locations after breaking and accretion before breaking. For multi-bar shores, breakers occur several times along the wave ray, so there are also several erosion and accretion subzones. Bars are more likely to appear on gentle nearshore slopes and fine, non-cohesive bed material, when the development of a multiple-bar profile is enhanced. Mildly sloping shores with fine sediment can produce up to 4 or 5 bars, although Dolan and Dean (1985) show that even a dozen or so small bars may sometimes appear. Quite obviously, an analysis of bar generation, evolution and characteristics can be more reliable if supported by prototype evidence (laboratory data may suffer from scale effects). Therefore the field data of CRF Lubiatowo collected from 1964 until 1994 has been dealt with in this study by means of statistical tools. Some common findings with Dolan and Dean (1985) may be sought, as the latter researchers found multiple bar systems in small water depths of the upper Chesapeake Bay and claimed similarities with the Baltic Sea (Gelding Bay) and also discussed multiple wave break hypothesis as a possible mechanism for the formation of multiple bars. Undoubtedly, both sites, Lubiatowo and Chesapeake Bay, are highly dissipative systems. Yet other similarities seem limited, not only because of the negligible tides in the Baltic Sea but mostly on account of the very conspicuous bars at CRF Lubiatowo, while the bar height identified by Dolan and Dean (1985) ranged from as little as 0.03 to 0.61 m. The statistical approach applied herein was also avoided for the Chesapeake Bay. Field conditions closest to CRF Lubiatowo are reported by Hands for Lake Michigan in 1980 (Dolan and Dean, 1985). In statistical terms, not only may multi-bar profiles, within a certain morphodynamic system, exhibit correlations between parameters of the same bar but also the same parameters of different bars may be correlated. The correlation analysis of a bar system allows one to determine the coupling of neighbourin, 0 bars, inquire whether a bar acts independently or is influenced by other bars, and monitor temporal changes in bar
Z. Pruszak et al./ Coastal Engineering 31 (1997) 263-280
265
characteristics. The problem is extremely interesting and somewhat novel, since multi-bar profiles are examined here, contrary to earlier studies on simpler one or two-bar profiles done by Larson and Kraus (1992).
2. Site description
and data
The analysed shore at CRF Lubiatowo, situated at the southern edge of the Baltic Sea (Fig. l), usually consists of multiple longshore bars (Figs. 1 and 2). Its mean slope is mild (m = tan a = l- 1.5%) and the sediment diameter equals D,, = 0.022 cm. The resultant long-term wave energy flux is directed obliquely to the shore, and so is the resulting longshore sediment transport. Multiple breaking of waves is often observed in the surf zone. On the basis of multi-yearly records and monitoring one can conclude that, for average storm conditions in this region, the significant wave outside the surf
55’ N
Fig.
I.
Study site location and bathymetry.
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1991.06.04 1991.10 29
h [ml
~B92.05.21
20
1992 10.22 19930720
-x-x-
Lubiatw
10
1991 _ 93
00
100
IS0
200
IS0
300
350
Fig. 2. Examples of multi-bar
400
450
profiles employed
SW
550
600
650
700
in analysis.
zone (h = 7 m) reaches H, = 2-2.5 m (or at most 3.5 m) and T- 5-7 s. The storm wave loses its energy upon shoreward transformations and at the depth h = 2-3 m, the mean wave height is e = 0.5- 1 m (up to 1.5 m), while the mean wave period becomes T = 4-5 s. In the close proximity of the shoreline (h = 1 m), the storm wave has a height and a period of 0.3-0.5 m (at most 0.6 m) and 4-5 s, respectively. Because of its semi-enclosed area, i.e. isolation of the Baltic Sea from the Atlantic, the influence of tides is negligible. The average surf scaling parameter E = [(H, w’>/( g . m*>] (where w = K2n))/(7-)I, g = acceleration due to gravity and m = average bed slope, b being an index standing for wave breaking) is in the range of 1000 < F < 3000 for the analysed region, thus falling in the category of highly dissipative shores. The seabed at Lubiatowo was first surveyed in 1964, well before the research facility was established. In the course of many years to date, the offshore range of the surveys varied and, on average, was equal to 800 m, thus sufficient to record all visible bars. Each bathymetric campaign from 1964 to 1994 contained several cross-shore profiles, spaced about 100 m. All measurements of depths greater than 1 m were carried out with an echosounder, while shallower depths were measured manually. The accuracy of the depth measurements is estimated at 5 to 10 cm, depending i.a. on weather conditions. The implications of that survey accuracy for the analysis of bar properties consists mainly in the scatter of statistical relationships. Four central transects spaced at 50-100 m (cf. Fig. 1) were eventually selected for further analysis of bar properties, in order to avoid longshore distorsions, such as longshore migration of beach cusps. All surveys were fixed at their mean shoreline (slightly different at each transect), treated as a common origin for all profiles analysed. The shoreline on a given day is identified as the mean water line on that day, sea level change reduction being included. The bathymetric database for the years 1964-1994 so generated and employed subsequently in the analysis consists of depth sets for 81 transects (Table 1). Sand bars display a substantial interannual variation and their geometry varies considerably; some bars may even be washed away for some time. This behaviour is attributed to the high variability of wave climate in the study area. It is worthwhile to
Z. Pruszuk
Table 1 Analysed bathymetric
et al./
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267
profiles
No.
Date of survey
Surveyed profile names
No. of stable bars (1.2, 3, 4)
1
1964.09.26 1966.09.30 1973,05.11 1974.05.15 1974.06.18 1974.08.09 1976.07.15 1980.07.09 1981.08.13 1981.09.14 1981.10.10 1987.05.06 1987.08.14 1987.09.22 1987.10.15 1988.04.28 1988.10.05 1989.05.24 1989.09.13 1990.06.05 1990.08.14 199 1.06.04 1991.10.29 1992.05.21 1992.10.22 1993.07.20 1993.09.30 1994.04.24
4.5.6 4, 5, 6 6
3 3 4 4 4 3 3 3 3 3 3 4 4 4 4 3 4 3 4 4 4 4 4 4 4 4 4 4
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
0.4, 5.6 0, 4, 5.6 4 0, 4 4,536 5 4, 5 5.6 4, 5, 6 576 4.5, 6 0.4, 5, 6 4.5.6 4,5,6 4,576 4, 5, 6 4, 5, 6 4, 5, 6 0 0, 4, 0.4, 074, 0,4, 0, 4, 0, 4,
576 5.6 5,6 5, 6 5, 6 5, 6
note, though, that despite all its shortcomings the analysed dataset is the only one for the Polish coast which contains so many cross-shore transects covering a fairly long time span for a single coastal unit.
3. Data analysis 3.1. Equilibrium
profile
The equilibrium Dean function): v’ = A . xr2/3
profile is usually
defined by the power function
(referred
to as the
(1) Two simple modifications of Eq. (1) have been attempted in this study in order to draw conclusions on the analytical form of the equilibrium profile itself as well as on sand bars emerging above that profile. In the first version (Fig. 3a), at the shoreline PO( X s, y,> the tangent of Eq. (1) is made equal to the bottom slope m = tan cx (taken as
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y’=m-x’
Fig. 3. Two modified versions of equilibrium
0.05, averaged from shoreline to midpoint one has X, = (8/27)(A/tan cuj3 and y = A
( x + xS)2’3
- A . 4/3
between
;
profile.
shoreline
and innermost
bar). Thus
= A . [[~+%kJl?j-(&i’)
C2)
In the second modification (Fig. 3b), the shoreline is at the origin of coordinates (x’, y’) and the equilibrium profile is assumed to be described by two equations, each valid in its own range: y’ = m x’ for x’ < x0
and
y’ = A . x”‘~
for x’ 2 = x0
(3) x0 can be determined at the intersection of both lines, which yields: x0 = ( A/m>3. The value of m = tan (Y is slightly different from its counterpart in the previous modification, it was found as 0.04, averaged from the shoreline to the innermost bar. In both versions the value of the coefficient A agreed with the grain size, as stipulated in the Dean formula. Eqs. (2) and (3) were least-square fitted to the data in order to compare the accuracy of the resultant equilibrium profiles. Since a slightly better fit was achieved with Eq. (31, the latter was employed subsequently in our identification of bars (‘riding’ on the equlibrium profile). Hence, equilibrium profiles were adapted in the form of Eq. (3) and were identified separately for each transect 0, 4, 5 and 6, as the mean profiles measured in all surveys. Bar parameters could then be calculated as specified in Fig. 4, upon intersection of the actual profile and the equilibrium profile for each transect and each bar, such as: L, = z= h, = xp =
bar length bar height water depth to bar crest distance from shoreline to the with equilibrium profile) xk = distance from shoreline to the x8 = distance from shoreline to bar x, = distance from shoreline to bar x,,~ =distance from geodetic baseline V = bar volume (above equilibrium
shoreward
extremity
seaward extremity crest centre of gravity to shoreline profile)
of bar (shoreward
of bar
intersection
Z. Prrrsruk et ul./ Cmstul
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600
400
800
269
x
[ml
Fig. 4. Bar parameters.
3.2. Variability of bar parameters
Six basic bar parameters L,, xg, xc, h,, z and V have been examined statistically. Data sets with those parameters were created for five bars (0, I, II, III, IV) emerging in the surveys. The minimum, average and maximum values of the analysed bar parameters are put together in Table 2. 3.2.1. Bar length Mean bar length depends on bar location in cross-shore profiles. It is short (25-30 m, max. 60 m> for the unstable (ephemeral), innermost bar 0 and the first stable bar I
Table 2 Minimum, Bar No.
average and maximum
No. of occurrences
Bar origin xp [ml
bar parameters,
Bar end xk
[ml
Bar length
L, [ml
Lubiatowo
1964- 1994
Bar height
Depth over bar crest
Location of crest
z [ml
h, [ml
xg
[ml
Location of centre of gravity x,
0
29
0 29.6 69.
30 56.3 76.3
0 26.6 60
0 0.35 0.80
I
73 100.4
50.9 134.1 195.7
69.2 33.7 240
0 0.55 72.7
0 1.38 1.28
0.1 0.6 1.7
Bar volume V [m3]
[ml 0 6.17 19.5
10 39 70
16.7 44.5 79.8
0.9 112.6 2.66
60 116.2 200
62.4 12.88 207
48.9
0
II
77
130.6 215 481.8
170 272.4 517.8
0 57.3 153.2
0 0.77 1.35
1.01 2.38 5.18
160 234.6 510
169.7 239.8 500.7
0 30.2 103.9
111
69
282.8 362 518.6
360 461.2 630
0 99.1 200
0 0.7 1 1.45
2.7 3.76 5.2
310 393.9 520
321.8 406.6 519.9
0 46.5 119
IV
55
379.2 534.5 676.9
478.3 738.6 850
10.8 204.3 409.8
0.07 0.64 1.23
3.8 5.04 7.3
400 586.3 770
444.9 627.8 776.1
0.97 76.1 139.5
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(30-35 m on average, max. 75 m>. The longest outermost bar IV is 6-8 times longer (200-210 m on average, max. 370 m>. The maximum standard deviation was also observed for the outermost bar, indicating high variability of its length. Thus, taking into account its high mean length and small height z, bar IV is a large, flat and irregular bed form, often without a distinct crest. Such shape can be attributed to spatially distributed wave breaking over this segment of the shore profile. 3.2.2. Distance from shoreline to bar crest and centre of gravity The spacing between the bar crest and the centre of gravity is usually small. The centre of gravity is situated several meters offshore of the bar crest for bars (0, I, II, III), with the maximum of 40 m for bar IV, showing the degree of bar asymmetry. As in the case of bar length, the maximum standard deviations were observed for the outermost bar and it again can be assigned to spatial distribution of wave breaking, especially during heavy storms. The spacing between bars grows in the offshore direction. Thus, the inner bars lie closer to each other and their spacing is equal to 75 m between bars 0 and I, 120 m between I and II and 200 m between III and IV. 3.2.3. Bar height and water depth to bar crest Bar height is defined as an elevation of crest above the equilibrium profile, i.e. the reference line given by Eq. (3). Maximum bar heights so defined may reach 1.5 m above that line but, if counted from trough to crest, they may even exceed 2.5-3 m. The height is greatest for the central bar II, the mean height of that bar is 40% greater than that of the inner bar I, 10% more than for the outer bar III and 20% more than the outermost bar IV. The standard deviation of bar height is also maximum for bar II. Thus, bar II is the most conspicuous bed form of a multi-bar profile and the one undergoing the strongest morphodynamic changes. It lies in the stable area of the most frequent wave breaking, which in turn is the most vital factor of cross-shore profile evolution. The mean water depth over consecutive bars h, bGrows with offshore distance. At the highly unstable ephemeral bar, the depth over its crest varies from 0.1 m (direct proximity of the shoreline) to 1.7 m. On the other hand, depth oscillations over the crest of the outermost bar vary from 3.8 m to 7.3 m. The latter maximum value corresponds to the existence of a flat bar, being a residual form of bar IV. That value is also close to the water depth where the analysed cross-shore profiles vary insignificantly during most of the year. 3.2.4. Bar volume Mean bar volume, determined by elevation of the bed profile above the equilibrium line, grows with the distance from shoreline. The volume of the outermost bar IV is ten times greater than that of bar 0, six times than for bar I and twice as much as the neighbouring bar III. It is clear that the volume strongly depends on geometrical parameters of bars, particularly their length. Bar IV also exhibits the strongest standard deviation of volume despite its remoteness from shoreline and the highest depth. This may indicate that sediment movement is not negligible in the outermost section of the profile.
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3.3. Corre2htion.s Since the total number of all transects analysed was 81, up to 81 values of a given parameter could be found for each bar, depending on its occurrence (all four bars did not show up on all dates). The correlations for bar parameters were then computed by the well-known formula:
where p, and p2 represent two bar parameters deviations. The covariance was defined as:
COV( PI9 P2) =
and cry,, aP, stand for their standard
tlif(
Pti_Ft)(P2i-~2)
1=
I
where the overbars denote mean values, i being the index of the given bar parameter, up to the maximum II = 81 transects (events). The analysis was carried out separately for couples of parameters belonging to the same bar and couples of the same parameter measured at different bars. The correlations calculated for different couples of bar parameters are generally low, with a few exceptions. High correlations of 0.7-0.95 were found for V CJ z and V - L, which is otherwise obvious. Notable are also the correlations obtained for V * xg and h, * xR belonging to bar I. The first correlation (i.e. V 0 x,) indicates that the more offshore bar I, the greater its volume. This can be caused by the exchange of sediment between bars I and 0 (and the beach as well). During storm events the sediment moves seawards and bar I can accumulate sand from bar 0, which is ephemeral and disappears at times. On the other hand, during calm periods the sediment moves shorewards and bar I may convey sand to bar 0, which emerges occasionally. The latter then stores temporarily the sediment migrating from beach towards the system of more stable bars. Correlations for other couples of parameters (still same bars) are much lower, being about 0.3-0.5 for bar I and only 0.0-0.3 for other bars. The correlations are all put together in Table 3. In order to see whether bars act independently or as a system, correlations of the same parameters L,, xg, xc, h,, z, V were computed for different bar couples. The symbols RO, RI, RII, RI11 and RIV henceforth denote the respective bars that were set in different combinations for which correlations were computed (Table 4).
Table 3 Correlations
between parameters
of the same bar
Bar No.
L, versus xg
z versus xy
v versus xg
V versus z
V versus h,
v versus L,
0
- 0.27 0.3 1 0.28 - 0.22 - 0.34
0.06 0.44 0.00 - 0.09 0.46
-0.18 0.49 0.19 - 0.07 0.11
0.59 0.92 0.79 0.82 0.53
- 0.50 - 0.26 - 0.21 -0.16 - 0.26
0.89 0.84 0.82 0.75 0.67
I II 111 IV
Z. Pruszuk et d./Coastd
272 Table 4 Correlations
the same parameters
of different
Parameters
Bar 0 versus bar I
Bar I versus bar II
-5
0.37
L, ” z X<
0.46 0.55 0.10 0.44
0.49 0.12 0.54 0.39 0.52
Engineering 31 (1997) 263-280
bars Bar II versus bar III
Bar 111 versus bar IV
Bar I versus bar III
Bar II versus bar IV
0.42 0.17 0.05 0.02 0.39
0.36 -0.18 - 0.20 -0.14 - 0.53
0.36 PO.15 -0.14 -0.13 0.40
0.06 0.13 -0.16 - 0.24 0.06
The analysis shows that the correlations are not high and decrease with the distance from shoreline. The highest values (although below 0.6) have been obtained for RO e RI (L,, x8, x,, V and for RI CJ RI1 ( xg, x,, Z, V. The correlations of x,. and x8 computed for all consecutive pairs of bars imply that the cross-shore migration of the inner bars 0 to II is somewhat correlated while the outer bars III and IV are not statistically coupled with the movement of the inner bar system. Thus it is only the inner bar subsystem that may be claimed to move as an entity either offshore or onshore. Smaller values of the correlation coefficients may partly stem from the high instability and mobility of bar 0 and a relatively high stability of bar IV. The latter is particularly vulnerable to substantial evolution at extreme events. Bar lengths are coupled only within RO @ RI, whilst bar heights and volumes are correlated significantly within the subsystem RI e RII. This may indicate once again that the inner bars 0, I and II are much more conjugated than the outer bars III and IV. It can also be concluded that bars III and IV change their size and volume independently of other bars. Crucial causes of their evolution may be either rare, extremely heavy storms or long-term shoreline changes and the respective disturbances of sediment transport. Therefore, the evolution of those outer bars is associated with greater spatial and temporal scales. It is worthwhile to note that, on the contrary, the inner bars 0, I and II are subject to the most rapid and dynamic bed changes due to coastal phenomena of smaller scales. Hence a system of multiple bars can be split up into two subsystems, inner and outer, controlled by coastal processes of different scales.
4. Empirical
relationships
In the search for empirical relationships between the parameters L,, xfi, h,, z, V, the latter have been normalised with respect to two characteristic morphodynamic lengths (Fig. 4): the depth ho = 8 m and the width xn = 950 m corresponding to the depth h,. The latter has been taken somewhat arbitrarily as the depth where the standard deviation of depth change equals 3% of that depth. h, should not be identified with the closure depth, although both quantities are numerically similar for the study site. The very reason h, and xn were introduced consisted in normalisation of morphodynamic quantities. The normalised parameters have been defined as & = [(L,)/(h,)],
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Kh,)/(h,)l, 5 = Kz)/(h,k 3 = [(x,)/(x,>l, V= [(V>/(lOO . h, . x,)1 and empirical relationships were then sought on the basis of the least-square fit for the following functions: 5 = fllr, >, fir = f. After testing several functional types, the linear function was chosen for all parameters because of its simplicity, a relatively good approximation of the field data and no clear advantage of more sophisticated functions. For & and L, the following relationships have been obtained:
h, =
4r=35.3
(5a)
L, = 0.295 . xg
(5b)
with the correlation coefficient r = 0.73 (Fig. 5). The greatest deviations appear for bars III and IV. Those bars are often greater and more irregular than inner bars, because the outer bars fairly often merge or bifurcate, a bar can have two crests or be so flat that it can hardly be distinguished. For z and z the equations have been sought separately shorewards and seawards of the trough between bars I and II: ~=0.62.~
fors
and
z = 0.0052. + 0.818
for x>
-z=
-0.03.~+0.1
x8 for x < 150 m
and
forsbO.158 z = - 0.000252.
150 m;
L,= 0.295.
r
x+,
oLr= 51.9 m.
for one has to realise This scatter can be (1) an adjustment of instability and high while wave breaking
r = 0.729
-
300
L[m1200
100
1 0
0 0
100
xg (6b)
with a rather low r = 0.38 (Fig. 6). However, this is not surprising that bar height is highly scattered over the whole range of 3. attributed to the simultaneous effect of at least two processes: seabed to a continuously evolving equilibrium profile and (2) variability of wave breaking over bars. The first process is slow
400
(6a)
200
300
400
500
600
x8 [ml Fig. 5. Bar length as linear function of distance to bar crest.
700
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$6<150 xg +0.818
s 2 150
oz
= 0.31
m r =0.38
1.6 0
= [ml
0
100
200
300
400 xg
Fig. 6. Maximum bar height as piecewise
500
600
700
[ml
linear function of distance to bar crest.
can vary fast during the same storm, at a given bar. Hence Fig. 6 can be used as an indirect proof of the feedback between bar formation and wave breaking. A relatively high correlation of r = 0.71 has resulted for the bar volume: -v = 0.000155~, V = 0.124. xg
(7a) (7b)
Like for bar length, greater distances
v = 0.124.
z+,
from shoreline
0” = 22.6 m31m,
correspond
to greater volumes
r = 0.711
160
"0
100
200
300
400
500
600
xg [ml Fig. 7. Bar volume as linear function of distance to bar crest.
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275
and high deviations appear around locations of bars III and IV (Fig. 71, the reason being the same as for bar lengths. A very weak, negative correlation r = -0.091 has been found for the regressional relationship between simultaneous cross-shore movement of the shoreline and the entire bar system (Fig. 8): ‘/,b
This evolve affected manner pivotal The
=
- 0.006 . xg + 52.7
(8)
weak correlation seems to suggest that the subsystems of shoreline and bars independently of each other and that beach erosion and accretion might be by longshore sediment transport to the extent previously unexpected and in a different from bar response patterns. Sometimes the shoreline can act as a point, when it does not move while bars do so. water depth to bar crest versus distance from shoreline can be described as
4 = 1.08 .s
(94
h, = 0.0091 . xg
(9b)
with a high correlation coefficient I = 0.94 (Fig. 9). This equation may be regarded as a new linear ‘bed equilibrium envelope’ linking bar crests, at which wave breaking is enforced. It is interesting to note that the bed gradient of the line given by Eq. (9b) is close to the empirical mean bed slope m = 0.01-0.015 and both slopes happen to decrease with increasing 5. This finding may be used as an indirect argument that wave energy dissipation, which is proved to be in balance with the slope of the Dean profile, occurs primarily as wave breaking at bar crests. The relationship h, = f( xp> can also be approximated by other functional types. If a
Fig. 8. Distance of shoreline to base as linear function of distance to bar crest.
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h, = 0.0091.
h,
31 (1997)
cshr = 0.53 m
263-280
r = 0.94
h [ml
0
200
100
300
400 x,
500
600
700
[ml
Fig. 9. Depth to bar crest as linear function of distance to bar crest.
power function is employed (Fig. lo), the fit is slightly better (r = 0.97) and the respective Eq. (10) reads: h = a . Xb = 0 036 . x”.777 r (10) R R . If in turn a curve analogical to the Dean function is employed, the fit is only a bit worse (r = 0.95) and the water depth over bar crest reads h r =A,
.x,T’~=O.O~.X;‘~
The coefficient
A * in Eq. (11) is not the common __._....._ h, = 0.069 xg?
parameter
appearing
shr = 0.46 m
r = 0.95
m
r = 0.97
h, = 0.036 x+OrnS oh, = 0.43
(‘1) in the Dean
a 0
7
,. :
6
< .._ Dean-type function
5 h,
[ml 4 3
0
0
100
200
300
400
500
600
700
xg [ml
Fig. 10. Depth to bar crest as power and Dean-type
function of distance to bar crest.
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277
curve but instead it stands for a new ‘bed equilibrium of bar crests’. Its value of 0.07 represents average conditions and it is believed to vary similarly to the coefficient A(r) in the time-dependent Dean function (Pruszak, 1993): h = A(t)
. x2’3 = 0.06 f 0.09 . x2’3
(12)
However, further research is needed to come up with a precise A,(t) in order to establish a formula resembling Eq. (121, which would describe an envelope of bar crests. The envelope would then represent the equilibrium between bars and breaking waves during their shoaling over the bar system. Field observations of waves provide background for such reasoning since waves lose their stability just about bar crests and there is a substantial feedback between waves, which stir up and shape the bed and bars, which in turn bring about wave transformation. By analogy to the long-term variation of the Dean profile, it can thus be postulated that the evolution and location of bars vary with time and depend on the equilibrium profile, as reflected by the parameter A(t) given by Pruszak (1993). If the two time-dependent parameters A(t) and A ~(t> are known, a more general formula can be obtained for bar height: (13) More light can be shed by observations and in-situ measurements carried out at CRF Lubiatowo, supported by numerical computations of wave transformation. From the data on the height of storm waves, breaking a couple of times over consecutive bars and generated by various wind climates, it can be concluded that the breaker index varies across shore, that is over bars in cross-shore profile. As an example, the height of waves breaking at five bars is shown in Table 5 for a westerly wind with a speed of 12 m/s. Similar tables and computations based on continuous monitoring of wave parameters at two points of the cross-shore profile were carried out at CRF Lubiatowo for 20 wind situations. An overwhelming majority of that database confirms the trend visible in Table 5. The mean values of the breaker index stemming from the twenty situations analysed are presented graphically in Fig. 11. The ratio (H/h,), is seen to decrease in the offshore direction, it is 0.70-0.75 for the first stable bar (I) but falls to 0.45-0.5 at the outermost bar IV. The breaker indices shown in Table 5 and Fig. 11 are generally smaller than those measured in laboratories or embayments and estuaries, cf. Dolan and Dean (1985). The discrepancy probably stems from a greater dynamics of the open sea and differences
Table 5 Variation
of breakino
wave height at multiole bars due to wind W 12 m/s
Bar No.
H, (m)
z7, Cm)
(H)/(h,),
0 1 2 3 4
0.5
0.6 1.38 2.38 3.76 5.04
0.75 0.72 0.635 0.50 0.38
1.o 1.5 1.8 1.9
2. Pruszuk
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0.40
+-
0.00
_ ~~~~-~ 0.20
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31 (1997)
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-~~~ 0 40
Fig. Il. Wave breaking
0.60
criterion for multi-bar
0.80
1 .oo
profiles.
between the breaking processes of spectral and monochromatic waves (prototype versus laboratory). Hence the following dimensionless formula was put forth for the empirical description of the observed relationships:
where: H is the rms wave height, h, is the water depth over bar crest, x8,, is the distance from shoreline to crest of ith bar (i = 0, 1, . . , p) and xx,,, is the distance from shoreline to crest of the outermost bar ( p = 4). The subscript b stands for wave breakin, 0 and i denotes consecutive bar crests. Several curve types were least-square fitted to match the field data (Fig. 11). The best fit (r = 0.95), has been achieved for the following exponential function: ( H/!z,),,~
= 0.76 . exp( - 0.5~~
(15)
Eq. (14) can be treated as a general form for the criterion of wave breaking at multiple bars, within the concept of ‘bed equilibrium of bar crest envelope’. The dynamics, evolution and interaction of multiple bars are coupled with wave transformation processes, deep-water wave parameters, shore morphology and a number of other factors. In the light of the data presented in this paper, the wave breaking
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phenomenon appears to prevail in the realm of coastal phenomena potentially controlling multiple bars. Hence from among the basic physical processes of bar formation and evolution, leading to a few bar formation models briefly referred to in the introduction i.e. wave breaking, nonlinear interactions and infragravity waves, the authors opt for the former, with a possible secondary effect of wave breaking-induced currents, at least under the conditions measured at CRF Lubiatowo.
5. Conclusions The study confirms earlier findings by Dean et al. (1992) that bar size and shape may depend upon the stability of wave breaking location during bar formation. If spatial and temporal stability of that location is sufficiently high, the bar has a conspicuous shape. Less mobile are bars controlled by waves with spatially and/or temporally distributed breaker locations. Only a few couples of a single bar parameters are significantly correlated (r 2 0.5), i.e. V - z, V - L,, V - xg, h, - x L, * xg. The strongest correlations have been computed for bar I; for other bars the’iorrelations, even if significant, are weaker. Bar I is clearly backfed with the unstable innermost bar 0, which in turn disappears either during calm periods, when it feeds the beach, or during heavy storms when bar I is nourished. Bar 0 is therefore a temporary bed form, typical of short time scales. It usually appears 30-40 m away from the shoreline for H, < 1 m, mainly when a heavy storm decays and shortly thereafter. Correlations of the same parameters belonging to different bars decrease rapidly with the offshore distance. The highest correlations of 0.4-0.5 have been found between bars 0 and I for L,, xg, x, and V and between bars I and II for x8, x,, z and V. All correlations for bar III and IV are negative, except for xR. Those bars change size and volume independently of other bars. The processes of their evolution are linked to rare extreme storms or to long-term coastal changes bringing about respective variations of sediment transport. Therefore, the evolution of bars III and IV should be associated with spatial and temporal large-scale phenomena. Bars 0, I and II undergo a fairly fast evolution. They are continuously influenced by short scale phenomena and are exposed to the most dynamic changes. The water depth over bar crests increases in a manner similar to the variation of the mean shore profile. Hence the offshore growth of that depth can be described by a Dean-type curve and a new ‘bed equilibrium envelope’ can be defined for bar crests. The envelope can be directly related to parameters of breaking waves, and a new wave breaking criterion for multiple bars could thus be formulated (cf. Eqs. (14) and (15). In the context of the data and findings presented in this paper, the wave breaking phenomenon prevails in the realm of coastal phenomena potentially controlling multiple bars. Hence from among the basic physical processes of bar formation and evolution, the authors choose wave breaking as the most crucial factor of bar formation, with a possible secondary effect of wave breaking-induced currents, at least under the conditions typical of sites similar to CRF Lubiatowo.
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Acknowledgements This paper is based on work in the PACE-project, in the framework of the EU-sponsored Marine Science and Technology Programme (MAST III), under contract No. MAS3-CT950002. The work was co-sponsored by the PAN Supported Programme IBW 2-1996. The authors hereby express their gratitude to the reviewers of the paper, for their thorough reading of its earlier version and many valuable suggestions on improvement of the contents.
References Boczar-Karakiewicz, B., Forbes, D.L., Drapeau, G., 1995. Nearshore bar development in Southern Gulf of St. Lawrence. .I. Waterway Port Coastal Ocean Eng. 121 (1). 49-60. Dally, W., 1987. Longshore bar formation: Surf beat or undertow? Proceedings of Coastal Sediments ‘87. ASCE, 1987, pp. 71-86. Dean, R., Srinivas, R., Parchure, T., 1992. Longshore bar generation mechanisms. Proceedings of the 23rd Coastal Engineering Conference. Venice, 1992, pp. 200 l-20 14. Dolan, T.J., Dean, R.G., 1985. Multiple longshore sand bars in the upper Chesapeake Bay. Estuarine Coastal Shelf Sci. 21, 727-743. Howd, P., Bowen, T., Holman, R., Oltman-Shay, J., 1991. Infragravity waves, longshore currents and linear sand bar formation. Proceedings of Coastal Sediments ‘91. ASCE, pp. 72-84. Larson, M., Kraus, N., 1992. Dynamics of longshore bars. Proceedings of the 23rd Coastal Engineering Conference. Venice, 1992, pp. 2219-2232. Pruszak, Z., 1993. The analysis of beach profile changes usin, 0 Dean’s method and empirical orthogonal functions. Coastal Eng. 19, 245-261. Salenger, A.H., Howd, P.A., 1989. Nearshore bars and the break point hypothesis. J. Coastal Eng. 12, 301-313. Short, A., 1975. Multiple offshore bars and standing waves. J. Geophys. Res. 80, 3838-3840.
Coastal Engineering
3 1 (1997) 263-280
Statistical properties of multiple bars Z. Pruszak, G. R6iyiiski Polish Academy
of Sciences, Institute ofHydra-Engineering
*, R.B. Zeidler (IBW PAN), 7 Kdciersku,
’ 80-953 Gdansk,
Poland Received
14 May 1996; revised 7 February
1997; accepted 7 February
1997
Abstract A natural, mild-slope, multiple-bar shore with a sandy bed (D,, = 0.022 cm) at the Coastal Research Facility (CRF) Lubiatowo (Poland) is analysed. The bathymetric dataset consists of 81 cross-shore transects surveyed from 1964 to 1994. Correlations have been computed for couples of parameters of each bar and for the same parameters of different bars. The most significant correlations have been obtained for parameters of the innermost stable bar. Correlations of the same parameters of different bars decrease rapidly with the distance from shoreline. Further analysis has shown that water depths over consecutive bar crests increase similarly to the mean shore profile. Their change can be presented by a Dean-type function describing a new ‘bed equilibrium line’ for the envelope of bar crests. The line is directly coupled with breaker parameters. Hence a wave breaking criterion for multiple-bar profiles could be established. Wave breaking is put forth as a major factor of bar formation. 0 1997 Elsevier Science B.V. Keywords: analysis
Shore morphodynamics;
Equilibrium
profile; Multiple
longshore
bars; Bar geometry;
Correlation
1. Introduction A number of bed-form systems, which undergo spatial scales can be distinguished in the nearshore crucial role among them. The origin of such bars theories developed so far associate bar formation Dean et al., 1992) and various types of long waves
evolution in different temporal and zone, longshore sand bars playing a is still insufficiently explored. The with wave breaking (Dally, 1987; and their interactions (edge waves,
* Corresponding author. Fax: + 48-58-5242 11. ’ Associate Professor, Senior Research Associate, Professor and Head of Department, Polish Academy of Sciences’ Institute of Hydro-Engineering (IBW PAN). 0378.3839/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO378-3839(97)00010-O
respectively,
of
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31 (1997) 263-280
surf beats, infragravity waves) or relate bar emergence to components of the wave spectrum and nonlinear wave effects, together with energy exchange between those components (cf. Short, 1975; Salenger and Howd, 1989; Howd et al., 1991; BoczarKarakiewicz et al., 1995). Many recent studies on bars have concentrated on the assumption that bar location and characteristics depend primarily on the existence and location of breakers. Dally (1987) revealed no relationship between bar formation and surf beats, but he found very likely the linkage between bar formation and wave breaking. In addition, he noticed that bar formation did not necessarily need to be accompanied by spilling breaker, as it was also frequently observed in the case of plunging breakers. Similar findings were reported by Dean et al. (1992), who proved there was no relationship between bar formation and spatial distribution of nodes and antinodes of infragravity waves. They also suggested that wave breaking and convergence of cross-shore currents and sediment transport rates are crucial factors of bar formation. Hence bar shape may be assumed to depend on the stability of breaker location during bar formation. If the breaker stability is sufficient in time and space, the resulting bar has a conspicuous, regular shape. By contrast, unstable breakers generate flatter, less conspicuous bars. Bar emergence is thus the part of cross-shore evolution of the nearshore zone which might consist in erosion at locations after breaking and accretion before breaking. For multi-bar shores, breakers occur several times along the wave ray, so there are also several erosion and accretion subzones. Bars are more likely to appear on gentle nearshore slopes and fine, non-cohesive bed material, when the development of a multiple-bar profile is enhanced. Mildly sloping shores with fine sediment can produce up to 4 or 5 bars, although Dolan and Dean (1985) show that even a dozen or so small bars may sometimes appear. Quite obviously, an analysis of bar generation, evolution and characteristics can be more reliable if supported by prototype evidence (laboratory data may suffer from scale effects). Therefore the field data of CRF Lubiatowo collected from 1964 until 1994 has been dealt with in this study by means of statistical tools. Some common findings with Dolan and Dean (1985) may be sought, as the latter researchers found multiple bar systems in small water depths of the upper Chesapeake Bay and claimed similarities with the Baltic Sea (Gelding Bay) and also discussed multiple wave break hypothesis as a possible mechanism for the formation of multiple bars. Undoubtedly, both sites, Lubiatowo and Chesapeake Bay, are highly dissipative systems. Yet other similarities seem limited, not only because of the negligible tides in the Baltic Sea but mostly on account of the very conspicuous bars at CRF Lubiatowo, while the bar height identified by Dolan and Dean (1985) ranged from as little as 0.03 to 0.61 m. The statistical approach applied herein was also avoided for the Chesapeake Bay. Field conditions closest to CRF Lubiatowo are reported by Hands for Lake Michigan in 1980 (Dolan and Dean, 1985). In statistical terms, not only may multi-bar profiles, within a certain morphodynamic system, exhibit correlations between parameters of the same bar but also the same parameters of different bars may be correlated. The correlation analysis of a bar system allows one to determine the coupling of neighbourin, 0 bars, inquire whether a bar acts independently or is influenced by other bars, and monitor temporal changes in bar
Z. Pruszak et al./ Coastal Engineering 31 (1997) 263-280
265
characteristics. The problem is extremely interesting and somewhat novel, since multi-bar profiles are examined here, contrary to earlier studies on simpler one or two-bar profiles done by Larson and Kraus (1992).
2. Site description
and data
The analysed shore at CRF Lubiatowo, situated at the southern edge of the Baltic Sea (Fig. l), usually consists of multiple longshore bars (Figs. 1 and 2). Its mean slope is mild (m = tan a = l- 1.5%) and the sediment diameter equals D,, = 0.022 cm. The resultant long-term wave energy flux is directed obliquely to the shore, and so is the resulting longshore sediment transport. Multiple breaking of waves is often observed in the surf zone. On the basis of multi-yearly records and monitoring one can conclude that, for average storm conditions in this region, the significant wave outside the surf
55’ N
Fig.
I.
Study site location and bathymetry.
2. Pruszuk
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Enginerring
31 (1997) 263-280
1991.06.04 1991.10 29
h [ml
~B92.05.21
20
1992 10.22 19930720
-x-x-
Lubiatw
10
1991 _ 93
00
100
IS0
200
IS0
300
350
Fig. 2. Examples of multi-bar
400
450
profiles employed
SW
550
600
650
700
in analysis.
zone (h = 7 m) reaches H, = 2-2.5 m (or at most 3.5 m) and T- 5-7 s. The storm wave loses its energy upon shoreward transformations and at the depth h = 2-3 m, the mean wave height is e = 0.5- 1 m (up to 1.5 m), while the mean wave period becomes T = 4-5 s. In the close proximity of the shoreline (h = 1 m), the storm wave has a height and a period of 0.3-0.5 m (at most 0.6 m) and 4-5 s, respectively. Because of its semi-enclosed area, i.e. isolation of the Baltic Sea from the Atlantic, the influence of tides is negligible. The average surf scaling parameter E = [(H, w’>/( g . m*>] (where w = K2n))/(7-)I, g = acceleration due to gravity and m = average bed slope, b being an index standing for wave breaking) is in the range of 1000 < F < 3000 for the analysed region, thus falling in the category of highly dissipative shores. The seabed at Lubiatowo was first surveyed in 1964, well before the research facility was established. In the course of many years to date, the offshore range of the surveys varied and, on average, was equal to 800 m, thus sufficient to record all visible bars. Each bathymetric campaign from 1964 to 1994 contained several cross-shore profiles, spaced about 100 m. All measurements of depths greater than 1 m were carried out with an echosounder, while shallower depths were measured manually. The accuracy of the depth measurements is estimated at 5 to 10 cm, depending i.a. on weather conditions. The implications of that survey accuracy for the analysis of bar properties consists mainly in the scatter of statistical relationships. Four central transects spaced at 50-100 m (cf. Fig. 1) were eventually selected for further analysis of bar properties, in order to avoid longshore distorsions, such as longshore migration of beach cusps. All surveys were fixed at their mean shoreline (slightly different at each transect), treated as a common origin for all profiles analysed. The shoreline on a given day is identified as the mean water line on that day, sea level change reduction being included. The bathymetric database for the years 1964-1994 so generated and employed subsequently in the analysis consists of depth sets for 81 transects (Table 1). Sand bars display a substantial interannual variation and their geometry varies considerably; some bars may even be washed away for some time. This behaviour is attributed to the high variability of wave climate in the study area. It is worthwhile to
Z. Pruszuk
Table 1 Analysed bathymetric
et al./
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267
profiles
No.
Date of survey
Surveyed profile names
No. of stable bars (1.2, 3, 4)
1
1964.09.26 1966.09.30 1973,05.11 1974.05.15 1974.06.18 1974.08.09 1976.07.15 1980.07.09 1981.08.13 1981.09.14 1981.10.10 1987.05.06 1987.08.14 1987.09.22 1987.10.15 1988.04.28 1988.10.05 1989.05.24 1989.09.13 1990.06.05 1990.08.14 199 1.06.04 1991.10.29 1992.05.21 1992.10.22 1993.07.20 1993.09.30 1994.04.24
4.5.6 4, 5, 6 6
3 3 4 4 4 3 3 3 3 3 3 4 4 4 4 3 4 3 4 4 4 4 4 4 4 4 4 4
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
0.4, 5.6 0, 4, 5.6 4 0, 4 4,536 5 4, 5 5.6 4, 5, 6 576 4.5, 6 0.4, 5, 6 4.5.6 4,5,6 4,576 4, 5, 6 4, 5, 6 4, 5, 6 0 0, 4, 0.4, 074, 0,4, 0, 4, 0, 4,
576 5.6 5,6 5, 6 5, 6 5, 6
note, though, that despite all its shortcomings the analysed dataset is the only one for the Polish coast which contains so many cross-shore transects covering a fairly long time span for a single coastal unit.
3. Data analysis 3.1. Equilibrium
profile
The equilibrium Dean function): v’ = A . xr2/3
profile is usually
defined by the power function
(referred
to as the
(1) Two simple modifications of Eq. (1) have been attempted in this study in order to draw conclusions on the analytical form of the equilibrium profile itself as well as on sand bars emerging above that profile. In the first version (Fig. 3a), at the shoreline PO( X s, y,> the tangent of Eq. (1) is made equal to the bottom slope m = tan cx (taken as
Z. Pruszuk et ul./Coustul
268
En+zeering
31 (1997) 263-280
y’=m-x’
Fig. 3. Two modified versions of equilibrium
0.05, averaged from shoreline to midpoint one has X, = (8/27)(A/tan cuj3 and y = A
( x + xS)2’3
- A . 4/3
between
;
profile.
shoreline
and innermost
bar). Thus
= A . [[~+%kJl?j-(&i’)
C2)
In the second modification (Fig. 3b), the shoreline is at the origin of coordinates (x’, y’) and the equilibrium profile is assumed to be described by two equations, each valid in its own range: y’ = m x’ for x’ < x0
and
y’ = A . x”‘~
for x’ 2 = x0
(3) x0 can be determined at the intersection of both lines, which yields: x0 = ( A/m>3. The value of m = tan (Y is slightly different from its counterpart in the previous modification, it was found as 0.04, averaged from the shoreline to the innermost bar. In both versions the value of the coefficient A agreed with the grain size, as stipulated in the Dean formula. Eqs. (2) and (3) were least-square fitted to the data in order to compare the accuracy of the resultant equilibrium profiles. Since a slightly better fit was achieved with Eq. (31, the latter was employed subsequently in our identification of bars (‘riding’ on the equlibrium profile). Hence, equilibrium profiles were adapted in the form of Eq. (3) and were identified separately for each transect 0, 4, 5 and 6, as the mean profiles measured in all surveys. Bar parameters could then be calculated as specified in Fig. 4, upon intersection of the actual profile and the equilibrium profile for each transect and each bar, such as: L, = z= h, = xp =
bar length bar height water depth to bar crest distance from shoreline to the with equilibrium profile) xk = distance from shoreline to the x8 = distance from shoreline to bar x, = distance from shoreline to bar x,,~ =distance from geodetic baseline V = bar volume (above equilibrium
shoreward
extremity
seaward extremity crest centre of gravity to shoreline profile)
of bar (shoreward
of bar
intersection
Z. Prrrsruk et ul./ Cmstul
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31 (1997) 263-280
600
400
800
269
x
[ml
Fig. 4. Bar parameters.
3.2. Variability of bar parameters
Six basic bar parameters L,, xg, xc, h,, z and V have been examined statistically. Data sets with those parameters were created for five bars (0, I, II, III, IV) emerging in the surveys. The minimum, average and maximum values of the analysed bar parameters are put together in Table 2. 3.2.1. Bar length Mean bar length depends on bar location in cross-shore profiles. It is short (25-30 m, max. 60 m> for the unstable (ephemeral), innermost bar 0 and the first stable bar I
Table 2 Minimum, Bar No.
average and maximum
No. of occurrences
Bar origin xp [ml
bar parameters,
Bar end xk
[ml
Bar length
L, [ml
Lubiatowo
1964- 1994
Bar height
Depth over bar crest
Location of crest
z [ml
h, [ml
xg
[ml
Location of centre of gravity x,
0
29
0 29.6 69.
30 56.3 76.3
0 26.6 60
0 0.35 0.80
I
73 100.4
50.9 134.1 195.7
69.2 33.7 240
0 0.55 72.7
0 1.38 1.28
0.1 0.6 1.7
Bar volume V [m3]
[ml 0 6.17 19.5
10 39 70
16.7 44.5 79.8
0.9 112.6 2.66
60 116.2 200
62.4 12.88 207
48.9
0
II
77
130.6 215 481.8
170 272.4 517.8
0 57.3 153.2
0 0.77 1.35
1.01 2.38 5.18
160 234.6 510
169.7 239.8 500.7
0 30.2 103.9
111
69
282.8 362 518.6
360 461.2 630
0 99.1 200
0 0.7 1 1.45
2.7 3.76 5.2
310 393.9 520
321.8 406.6 519.9
0 46.5 119
IV
55
379.2 534.5 676.9
478.3 738.6 850
10.8 204.3 409.8
0.07 0.64 1.23
3.8 5.04 7.3
400 586.3 770
444.9 627.8 776.1
0.97 76.1 139.5
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(30-35 m on average, max. 75 m>. The longest outermost bar IV is 6-8 times longer (200-210 m on average, max. 370 m>. The maximum standard deviation was also observed for the outermost bar, indicating high variability of its length. Thus, taking into account its high mean length and small height z, bar IV is a large, flat and irregular bed form, often without a distinct crest. Such shape can be attributed to spatially distributed wave breaking over this segment of the shore profile. 3.2.2. Distance from shoreline to bar crest and centre of gravity The spacing between the bar crest and the centre of gravity is usually small. The centre of gravity is situated several meters offshore of the bar crest for bars (0, I, II, III), with the maximum of 40 m for bar IV, showing the degree of bar asymmetry. As in the case of bar length, the maximum standard deviations were observed for the outermost bar and it again can be assigned to spatial distribution of wave breaking, especially during heavy storms. The spacing between bars grows in the offshore direction. Thus, the inner bars lie closer to each other and their spacing is equal to 75 m between bars 0 and I, 120 m between I and II and 200 m between III and IV. 3.2.3. Bar height and water depth to bar crest Bar height is defined as an elevation of crest above the equilibrium profile, i.e. the reference line given by Eq. (3). Maximum bar heights so defined may reach 1.5 m above that line but, if counted from trough to crest, they may even exceed 2.5-3 m. The height is greatest for the central bar II, the mean height of that bar is 40% greater than that of the inner bar I, 10% more than for the outer bar III and 20% more than the outermost bar IV. The standard deviation of bar height is also maximum for bar II. Thus, bar II is the most conspicuous bed form of a multi-bar profile and the one undergoing the strongest morphodynamic changes. It lies in the stable area of the most frequent wave breaking, which in turn is the most vital factor of cross-shore profile evolution. The mean water depth over consecutive bars h, bGrows with offshore distance. At the highly unstable ephemeral bar, the depth over its crest varies from 0.1 m (direct proximity of the shoreline) to 1.7 m. On the other hand, depth oscillations over the crest of the outermost bar vary from 3.8 m to 7.3 m. The latter maximum value corresponds to the existence of a flat bar, being a residual form of bar IV. That value is also close to the water depth where the analysed cross-shore profiles vary insignificantly during most of the year. 3.2.4. Bar volume Mean bar volume, determined by elevation of the bed profile above the equilibrium line, grows with the distance from shoreline. The volume of the outermost bar IV is ten times greater than that of bar 0, six times than for bar I and twice as much as the neighbouring bar III. It is clear that the volume strongly depends on geometrical parameters of bars, particularly their length. Bar IV also exhibits the strongest standard deviation of volume despite its remoteness from shoreline and the highest depth. This may indicate that sediment movement is not negligible in the outermost section of the profile.
2. Pruszak et d/Coastal
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271
3.3. Corre2htion.s Since the total number of all transects analysed was 81, up to 81 values of a given parameter could be found for each bar, depending on its occurrence (all four bars did not show up on all dates). The correlations for bar parameters were then computed by the well-known formula:
where p, and p2 represent two bar parameters deviations. The covariance was defined as:
COV( PI9 P2) =
and cry,, aP, stand for their standard
tlif(
Pti_Ft)(P2i-~2)
1=
I
where the overbars denote mean values, i being the index of the given bar parameter, up to the maximum II = 81 transects (events). The analysis was carried out separately for couples of parameters belonging to the same bar and couples of the same parameter measured at different bars. The correlations calculated for different couples of bar parameters are generally low, with a few exceptions. High correlations of 0.7-0.95 were found for V CJ z and V - L, which is otherwise obvious. Notable are also the correlations obtained for V * xg and h, * xR belonging to bar I. The first correlation (i.e. V 0 x,) indicates that the more offshore bar I, the greater its volume. This can be caused by the exchange of sediment between bars I and 0 (and the beach as well). During storm events the sediment moves seawards and bar I can accumulate sand from bar 0, which is ephemeral and disappears at times. On the other hand, during calm periods the sediment moves shorewards and bar I may convey sand to bar 0, which emerges occasionally. The latter then stores temporarily the sediment migrating from beach towards the system of more stable bars. Correlations for other couples of parameters (still same bars) are much lower, being about 0.3-0.5 for bar I and only 0.0-0.3 for other bars. The correlations are all put together in Table 3. In order to see whether bars act independently or as a system, correlations of the same parameters L,, xg, xc, h,, z, V were computed for different bar couples. The symbols RO, RI, RII, RI11 and RIV henceforth denote the respective bars that were set in different combinations for which correlations were computed (Table 4).
Table 3 Correlations
between parameters
of the same bar
Bar No.
L, versus xg
z versus xy
v versus xg
V versus z
V versus h,
v versus L,
0
- 0.27 0.3 1 0.28 - 0.22 - 0.34
0.06 0.44 0.00 - 0.09 0.46
-0.18 0.49 0.19 - 0.07 0.11
0.59 0.92 0.79 0.82 0.53
- 0.50 - 0.26 - 0.21 -0.16 - 0.26
0.89 0.84 0.82 0.75 0.67
I II 111 IV
Z. Pruszuk et d./Coastd
272 Table 4 Correlations
the same parameters
of different
Parameters
Bar 0 versus bar I
Bar I versus bar II
-5
0.37
L, ” z X<
0.46 0.55 0.10 0.44
0.49 0.12 0.54 0.39 0.52
Engineering 31 (1997) 263-280
bars Bar II versus bar III
Bar 111 versus bar IV
Bar I versus bar III
Bar II versus bar IV
0.42 0.17 0.05 0.02 0.39
0.36 -0.18 - 0.20 -0.14 - 0.53
0.36 PO.15 -0.14 -0.13 0.40
0.06 0.13 -0.16 - 0.24 0.06
The analysis shows that the correlations are not high and decrease with the distance from shoreline. The highest values (although below 0.6) have been obtained for RO e RI (L,, x8, x,, V and for RI CJ RI1 ( xg, x,, Z, V. The correlations of x,. and x8 computed for all consecutive pairs of bars imply that the cross-shore migration of the inner bars 0 to II is somewhat correlated while the outer bars III and IV are not statistically coupled with the movement of the inner bar system. Thus it is only the inner bar subsystem that may be claimed to move as an entity either offshore or onshore. Smaller values of the correlation coefficients may partly stem from the high instability and mobility of bar 0 and a relatively high stability of bar IV. The latter is particularly vulnerable to substantial evolution at extreme events. Bar lengths are coupled only within RO @ RI, whilst bar heights and volumes are correlated significantly within the subsystem RI e RII. This may indicate once again that the inner bars 0, I and II are much more conjugated than the outer bars III and IV. It can also be concluded that bars III and IV change their size and volume independently of other bars. Crucial causes of their evolution may be either rare, extremely heavy storms or long-term shoreline changes and the respective disturbances of sediment transport. Therefore, the evolution of those outer bars is associated with greater spatial and temporal scales. It is worthwhile to note that, on the contrary, the inner bars 0, I and II are subject to the most rapid and dynamic bed changes due to coastal phenomena of smaller scales. Hence a system of multiple bars can be split up into two subsystems, inner and outer, controlled by coastal processes of different scales.
4. Empirical
relationships
In the search for empirical relationships between the parameters L,, xfi, h,, z, V, the latter have been normalised with respect to two characteristic morphodynamic lengths (Fig. 4): the depth ho = 8 m and the width xn = 950 m corresponding to the depth h,. The latter has been taken somewhat arbitrarily as the depth where the standard deviation of depth change equals 3% of that depth. h, should not be identified with the closure depth, although both quantities are numerically similar for the study site. The very reason h, and xn were introduced consisted in normalisation of morphodynamic quantities. The normalised parameters have been defined as & = [(L,)/(h,)],
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Kh,)/(h,)l, 5 = Kz)/(h,k 3 = [(x,)/(x,>l, V= [(V>/(lOO . h, . x,)1 and empirical relationships were then sought on the basis of the least-square fit for the following functions: 5 = fllr, >, fir = f
h, =
4r=35.3
(5a)
L, = 0.295 . xg
(5b)
with the correlation coefficient r = 0.73 (Fig. 5). The greatest deviations appear for bars III and IV. Those bars are often greater and more irregular than inner bars, because the outer bars fairly often merge or bifurcate, a bar can have two crests or be so flat that it can hardly be distinguished. For z and z the equations have been sought separately shorewards and seawards of the trough between bars I and II: ~=0.62.~
fors
and
z = 0.0052. + 0.818
for x>
-z=
-0.03.~+0.1
x8 for x < 150 m
and
forsbO.158 z = - 0.000252.
150 m;
L,= 0.295.
r
x+,
oLr= 51.9 m.
for one has to realise This scatter can be (1) an adjustment of instability and high while wave breaking
r = 0.729
-
300
L[m1200
100
1 0
0 0
100
xg (6b)
with a rather low r = 0.38 (Fig. 6). However, this is not surprising that bar height is highly scattered over the whole range of 3. attributed to the simultaneous effect of at least two processes: seabed to a continuously evolving equilibrium profile and (2) variability of wave breaking over bars. The first process is slow
400
(6a)
200
300
400
500
600
x8 [ml Fig. 5. Bar length as linear function of distance to bar crest.
700
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<
-0.000252
31 (1997)
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$6<150 xg +0.818
s 2 150
oz
= 0.31
m r =0.38
1.6 0
= [ml
0
100
200
300
400 xg
Fig. 6. Maximum bar height as piecewise
500
600
700
[ml
linear function of distance to bar crest.
can vary fast during the same storm, at a given bar. Hence Fig. 6 can be used as an indirect proof of the feedback between bar formation and wave breaking. A relatively high correlation of r = 0.71 has resulted for the bar volume: -v = 0.000155~, V = 0.124. xg
(7a) (7b)
Like for bar length, greater distances
v = 0.124.
z+,
from shoreline
0” = 22.6 m31m,
correspond
to greater volumes
r = 0.711
160
"0
100
200
300
400
500
600
xg [ml Fig. 7. Bar volume as linear function of distance to bar crest.
700
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and high deviations appear around locations of bars III and IV (Fig. 71, the reason being the same as for bar lengths. A very weak, negative correlation r = -0.091 has been found for the regressional relationship between simultaneous cross-shore movement of the shoreline and the entire bar system (Fig. 8): ‘/,b
This evolve affected manner pivotal The
=
- 0.006 . xg + 52.7
(8)
weak correlation seems to suggest that the subsystems of shoreline and bars independently of each other and that beach erosion and accretion might be by longshore sediment transport to the extent previously unexpected and in a different from bar response patterns. Sometimes the shoreline can act as a point, when it does not move while bars do so. water depth to bar crest versus distance from shoreline can be described as
4 = 1.08 .s
(94
h, = 0.0091 . xg
(9b)
with a high correlation coefficient I = 0.94 (Fig. 9). This equation may be regarded as a new linear ‘bed equilibrium envelope’ linking bar crests, at which wave breaking is enforced. It is interesting to note that the bed gradient of the line given by Eq. (9b) is close to the empirical mean bed slope m = 0.01-0.015 and both slopes happen to decrease with increasing 5. This finding may be used as an indirect argument that wave energy dissipation, which is proved to be in balance with the slope of the Dean profile, occurs primarily as wave breaking at bar crests. The relationship h, = f( xp> can also be approximated by other functional types. If a
Fig. 8. Distance of shoreline to base as linear function of distance to bar crest.
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h, = 0.0091.
h,
31 (1997)
cshr = 0.53 m
263-280
r = 0.94
h [ml
0
200
100
300
400 x,
500
600
700
[ml
Fig. 9. Depth to bar crest as linear function of distance to bar crest.
power function is employed (Fig. lo), the fit is slightly better (r = 0.97) and the respective Eq. (10) reads: h = a . Xb = 0 036 . x”.777 r (10) R R . If in turn a curve analogical to the Dean function is employed, the fit is only a bit worse (r = 0.95) and the water depth over bar crest reads h r =A,
.x,T’~=O.O~.X;‘~
The coefficient
A * in Eq. (11) is not the common __._....._ h, = 0.069 xg?
parameter
appearing
shr = 0.46 m
r = 0.95
m
r = 0.97
h, = 0.036 x+OrnS oh, = 0.43
(‘1) in the Dean
a 0
7
,. :
6
< .._ Dean-type function
5 h,
[ml 4 3
0
0
100
200
300
400
500
600
700
xg [ml
Fig. 10. Depth to bar crest as power and Dean-type
function of distance to bar crest.
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curve but instead it stands for a new ‘bed equilibrium of bar crests’. Its value of 0.07 represents average conditions and it is believed to vary similarly to the coefficient A(r) in the time-dependent Dean function (Pruszak, 1993): h = A(t)
. x2’3 = 0.06 f 0.09 . x2’3
(12)
However, further research is needed to come up with a precise A,(t) in order to establish a formula resembling Eq. (121, which would describe an envelope of bar crests. The envelope would then represent the equilibrium between bars and breaking waves during their shoaling over the bar system. Field observations of waves provide background for such reasoning since waves lose their stability just about bar crests and there is a substantial feedback between waves, which stir up and shape the bed and bars, which in turn bring about wave transformation. By analogy to the long-term variation of the Dean profile, it can thus be postulated that the evolution and location of bars vary with time and depend on the equilibrium profile, as reflected by the parameter A(t) given by Pruszak (1993). If the two time-dependent parameters A(t) and A ~(t> are known, a more general formula can be obtained for bar height: (13) More light can be shed by observations and in-situ measurements carried out at CRF Lubiatowo, supported by numerical computations of wave transformation. From the data on the height of storm waves, breaking a couple of times over consecutive bars and generated by various wind climates, it can be concluded that the breaker index varies across shore, that is over bars in cross-shore profile. As an example, the height of waves breaking at five bars is shown in Table 5 for a westerly wind with a speed of 12 m/s. Similar tables and computations based on continuous monitoring of wave parameters at two points of the cross-shore profile were carried out at CRF Lubiatowo for 20 wind situations. An overwhelming majority of that database confirms the trend visible in Table 5. The mean values of the breaker index stemming from the twenty situations analysed are presented graphically in Fig. 11. The ratio (H/h,), is seen to decrease in the offshore direction, it is 0.70-0.75 for the first stable bar (I) but falls to 0.45-0.5 at the outermost bar IV. The breaker indices shown in Table 5 and Fig. 11 are generally smaller than those measured in laboratories or embayments and estuaries, cf. Dolan and Dean (1985). The discrepancy probably stems from a greater dynamics of the open sea and differences
Table 5 Variation
of breakino
wave height at multiole bars due to wind W 12 m/s
Bar No.
H, (m)
z7, Cm)
(H)/(h,),
0 1 2 3 4
0.5
0.6 1.38 2.38 3.76 5.04
0.75 0.72 0.635 0.50 0.38
1.o 1.5 1.8 1.9
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0.40
+-
0.00
_ ~~~~-~ 0.20
et ul./
Cocc.d
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31 (1997)
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-~~~ 0 40
Fig. Il. Wave breaking
0.60
criterion for multi-bar
0.80
1 .oo
profiles.
between the breaking processes of spectral and monochromatic waves (prototype versus laboratory). Hence the following dimensionless formula was put forth for the empirical description of the observed relationships:
where: H is the rms wave height, h, is the water depth over bar crest, x8,, is the distance from shoreline to crest of ith bar (i = 0, 1, . . , p) and xx,,, is the distance from shoreline to crest of the outermost bar ( p = 4). The subscript b stands for wave breakin, 0 and i denotes consecutive bar crests. Several curve types were least-square fitted to match the field data (Fig. 11). The best fit (r = 0.95), has been achieved for the following exponential function: ( H/!z,),,~
= 0.76 . exp( - 0.5~~
(15)
Eq. (14) can be treated as a general form for the criterion of wave breaking at multiple bars, within the concept of ‘bed equilibrium of bar crest envelope’. The dynamics, evolution and interaction of multiple bars are coupled with wave transformation processes, deep-water wave parameters, shore morphology and a number of other factors. In the light of the data presented in this paper, the wave breaking
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phenomenon appears to prevail in the realm of coastal phenomena potentially controlling multiple bars. Hence from among the basic physical processes of bar formation and evolution, leading to a few bar formation models briefly referred to in the introduction i.e. wave breaking, nonlinear interactions and infragravity waves, the authors opt for the former, with a possible secondary effect of wave breaking-induced currents, at least under the conditions measured at CRF Lubiatowo.
5. Conclusions The study confirms earlier findings by Dean et al. (1992) that bar size and shape may depend upon the stability of wave breaking location during bar formation. If spatial and temporal stability of that location is sufficiently high, the bar has a conspicuous shape. Less mobile are bars controlled by waves with spatially and/or temporally distributed breaker locations. Only a few couples of a single bar parameters are significantly correlated (r 2 0.5), i.e. V - z, V - L,, V - xg, h, - x L, * xg. The strongest correlations have been computed for bar I; for other bars the’iorrelations, even if significant, are weaker. Bar I is clearly backfed with the unstable innermost bar 0, which in turn disappears either during calm periods, when it feeds the beach, or during heavy storms when bar I is nourished. Bar 0 is therefore a temporary bed form, typical of short time scales. It usually appears 30-40 m away from the shoreline for H, < 1 m, mainly when a heavy storm decays and shortly thereafter. Correlations of the same parameters belonging to different bars decrease rapidly with the offshore distance. The highest correlations of 0.4-0.5 have been found between bars 0 and I for L,, xg, x, and V and between bars I and II for x8, x,, z and V. All correlations for bar III and IV are negative, except for xR. Those bars change size and volume independently of other bars. The processes of their evolution are linked to rare extreme storms or to long-term coastal changes bringing about respective variations of sediment transport. Therefore, the evolution of bars III and IV should be associated with spatial and temporal large-scale phenomena. Bars 0, I and II undergo a fairly fast evolution. They are continuously influenced by short scale phenomena and are exposed to the most dynamic changes. The water depth over bar crests increases in a manner similar to the variation of the mean shore profile. Hence the offshore growth of that depth can be described by a Dean-type curve and a new ‘bed equilibrium envelope’ can be defined for bar crests. The envelope can be directly related to parameters of breaking waves, and a new wave breaking criterion for multiple bars could thus be formulated (cf. Eqs. (14) and (15). In the context of the data and findings presented in this paper, the wave breaking phenomenon prevails in the realm of coastal phenomena potentially controlling multiple bars. Hence from among the basic physical processes of bar formation and evolution, the authors choose wave breaking as the most crucial factor of bar formation, with a possible secondary effect of wave breaking-induced currents, at least under the conditions typical of sites similar to CRF Lubiatowo.
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Acknowledgements This paper is based on work in the PACE-project, in the framework of the EU-sponsored Marine Science and Technology Programme (MAST III), under contract No. MAS3-CT950002. The work was co-sponsored by the PAN Supported Programme IBW 2-1996. The authors hereby express their gratitude to the reviewers of the paper, for their thorough reading of its earlier version and many valuable suggestions on improvement of the contents.
References Boczar-Karakiewicz, B., Forbes, D.L., Drapeau, G., 1995. Nearshore bar development in Southern Gulf of St. Lawrence. .I. Waterway Port Coastal Ocean Eng. 121 (1). 49-60. Dally, W., 1987. Longshore bar formation: Surf beat or undertow? Proceedings of Coastal Sediments ‘87. ASCE, 1987, pp. 71-86. Dean, R., Srinivas, R., Parchure, T., 1992. Longshore bar generation mechanisms. Proceedings of the 23rd Coastal Engineering Conference. Venice, 1992, pp. 200 l-20 14. Dolan, T.J., Dean, R.G., 1985. Multiple longshore sand bars in the upper Chesapeake Bay. Estuarine Coastal Shelf Sci. 21, 727-743. Howd, P., Bowen, T., Holman, R., Oltman-Shay, J., 1991. Infragravity waves, longshore currents and linear sand bar formation. Proceedings of Coastal Sediments ‘91. ASCE, pp. 72-84. Larson, M., Kraus, N., 1992. Dynamics of longshore bars. Proceedings of the 23rd Coastal Engineering Conference. Venice, 1992, pp. 2219-2232. Pruszak, Z., 1993. The analysis of beach profile changes usin, 0 Dean’s method and empirical orthogonal functions. Coastal Eng. 19, 245-261. Salenger, A.H., Howd, P.A., 1989. Nearshore bars and the break point hypothesis. J. Coastal Eng. 12, 301-313. Short, A., 1975. Multiple offshore bars and standing waves. J. Geophys. Res. 80, 3838-3840.