Experimental research on unstable movements of sandbars under wave actions

Jing YIN*, Zhi-li ZOU, Ping DONG, Hai-fei ZHANG, Guo-qiang WU, Yi-nan PAN

1. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, P. R. China

2. Department of Civil Engineering, School of Engineering, University of Dundee, Dundee DD1 4HN, UK

Experimental research on unstable movements of sandbars under wave actions

Jing YIN*1, Zhi-li ZOU1, Ping DONG2, Hai-fei ZHANG1, Guo-qiang WU1, Yi-nan PAN1

1. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, P. R. China

2. Department of Civil Engineering, School of Engineering, University of Dundee, Dundee DD1 4HN, UK

This study is motivated by recognition of complex sandbar evolution patterns under wave actions inside the surf zone. A series of physical model experiments were conducted in a wave flume to investigate sandbar migration under various wave conditions, including wave groups, regular waves, and random waves. It was observed that under certain wave conditions sandbars move alternately shoreward and seaward rather than continuously in the same direction. The measurements show that the unstable movement of sandbars is closely related to the amplitude modulation of waves. Smaller amplitude modulation tends to produce more intense unstable bar movements. Data analysis further shows that the sandbar migration does not seem to be a passive response of the sea bed to wave forcing, but is most likely caused by the feedback interaction between waves and bed topography.

unstable movement of sandbar; sediment transport; forced response; wave action

1 Introduction

The formation and migration of sandbars are commonly observed in the surf zone. Recent studies indicate that sandbars are not stable but in continuous migration during storms (Hoefel and Elgar 2003). Such migration may cause submarine pipelines to be exposed and erosion of the nourished beach. Therefore, understanding and prediction of sandbar formation and migration is one of the key problems in coastal morphodynamic research and is essential for developing effective soft engineering measures to prevent beach erosion.

In the past few decades morphological research has largely focused on equilibrium beach profiles formed under the action of waves. Many field observations of alongshore uniform equilibrium profiles worldwide can be found in the literature (Dean 1977; Elgar et al. 2001; Plant et al. 2001). It is generally accepted that barred profiles are generated by large storms while planar beach profiles are caused by shoreward sediment transport in the milder wave conditions in between the storms.

On the mechanisms of sandbar formation, there currently exist two kinds of theories. Oneis the forced response mechanism (Yu and Mei 2000), which states that the formation of sandbars is a passive response to the hydrodynamic forcing, and thus the beach profile has the same length scale as that of the hydrodynamic forcing when the profile reaches equilibrium. The other theory is the self-organization mechanism (Falques et al. 2000), which does not consider the beach profile a stable profile, but a continually evolving one.

Many experimental studies have also been carried out to study the mechanisms of wave-sand interactions in the surf zone. Kraus and Smith (1994) studied several cross-shore sediment transport processes at intermediate scales with the SUPERTANK experiment. Their data were used to validate a second generation shoreline model (Larson and Kraus 1989). The experiments of Dally and Barkaszi (1994) concerned sediment suspension at the micro-scale under wave breaking and during boundary layer processes. In the SISTEX99 project, Ribberink et al. (2000) observed wave-induced flows, turbulent velocities, and sediment response at high frequencies, including suspension and sheet flow transport. Dong et al. (2009) performed an experimental study of long wave generation on sloping bottoms. Recently, Baldock et al. (2010) conducted a new experiment with a special focus on the influence of free long waves, bound long waves, and wave groups on sediment transport in the surf and swash zones. Beetham and Kench (2011) quantified infragravity wave transformation through field measurements, and identified the long wave motion in a significant morphodynamic process.

The present study investigated the development and evolution of sandbars on an initially planar beach profile in a laboratory wave flume under different wave actions, including wave groups, regular waves, and random waves. All these waves can be considered wave groups with different amplitude modulations: regular waves can be considered a special wave group with an amplitude modulation coefficient equal to zero, while random waves can be considered a more general wave group with spatially and temporally varying amplitude modulations. This feature was used to investigate the correlation between sandbar movements and the intensity of waves, i.e., the variation of wave heights. Different beach slopes (1:20 and 1:10) were considered in order to determine gravity’s effect on sandbar migrations. In order to show more characteristics of sandbar migrations under these wave conditions, sandbar migration velocities were also measured and analyzed.

2 Experimental setup

2.1 Wave flume and beaches

The laboratory experiment was performed in a wave flume at the State Key Laboratory of Coastal and Offshore Engineering at Dalian University of Technology (Yin et al.2008). The flume is 56 m long, 0.7 m wide, and 0.7 m deep. A piston-type wave generator was set at one end of the flume. From the wave generator to the toe of the beach was the horizontal bottom with an undisturbed water depth of 0.45 m. A sandy beach with a slope of 1:20 or 1:10 was installed. The experiment set-up is shown in Fig. 1. The beach profile was formed from natural sand with a median grain sized50=0.224 mm and the sorting coefficientSc=1.551.

Fig. 1 Experimental setup

Along the flume, seven capacity-type wave gauges were used to measure wave heights. The beach profile was recorded by marking the profile on the glass wall of the flume every 30 min or 45 min, and the first marking was performed 15 min after wave generation. The marked lines were measured manually. The time intervals above were used to record beach profiles and sandbar migrations without remarkable oscillations, such as those in random wave and wave group cases with larger amplitude modulation. To measure the sandbar migrations with intense bar crest oscillation in the regular wave case, shorter time intervals were adopted. After each run of tests finished, the beach profile was remolded to the initial planar slope for the next run testing.

2.2 Wave conditions

Wave conditions, including wave groups with different amplitude modulations, regular waves, and random waves, were considered in this experiment. The random waves were generated using the JONSWAP spectrum (the p eak enhancement factorγ0=3.3). The wave group was generated according to the following equations:

whereηis the wave surface elevation;tis time;Tandaare the period and amplitude of short waves that form the wave group;Tgis the period of the wave group:Tg=nT;nis the number of short waves in the wave group; andδis the amplitude modulation coefficient of the wave group. Eq. (1) was used to determine the control signal of a wave paddle by setting the translation velocity of the wave paddle to be equal to the horizontal fluid particle velocity corresponding to the wave surface elevation given by Eq. (1). For all the wave group cases,T=1.2 s andn=15. Three values of the amplitude modulation coefficient,δ= 0.25, 0.5, and 1.0, were chosen to examine the effect of wave amplitude modulation on sandbar evolution. Wave groups are simplified (idealized) models of random waves. Different modulated wave groups were used to provide a transition from regular waves (zero modulation) to random waves (varied modulation). Typical time series of the wave surface elevation for these controlled wave groups are shown in Fig. 2. The test conditions are summarized in Table 1. The slope toe was set as the coordinate origin, and the shoreward direction was chosen as the positive direction of thexaxis.

Fig. 2 Time series of wave groups with different modulation parametersδ(H1/3= 0.09 m)

Table 1 Test conditions of experiments

The effects of second-order long waves, including bound and free second-order long waves as well as second-order long waves produced using only the first-order wave theory in wave generation, were not considered in the experiments, as these long waves, compared with short waves, are not the dominant factors in the development and evolution of a sandbar (Roelvink and Stive 1989; Baldock et al.2010).

The difficulties in generating long-duration time series of waves in a flume are well known, especially for regular waves. But this was not a serious problem in our tests, since the sandy beach at the end of the wave flume can dissipate almost all of the wave energy through wave breaking and wave running-up. It is observed that the wave reflection from the sandy beach is very small, and the secondary reflection can be neglected. This can be verified by a recorded time series of the wave surface elevation, which shows an almost uniform wave height over the whole experiment, as shown in Fig. 3. The time series shown in Fig. 3includes observations made every 3 h, which last for 5 min. The mean wave heights of the three records are 10.80 cm, 10.99 cm, and 10.26 cm, respectively, and the differences between them are small, less than 4.0%.

Fig. 3 Time series of wave surface elevation for Test R7 at different times

3 Evolution of sandbars under wave actions

3.1 Regular waves

Regular waves can be seen as a special wave group with an amplitude modulation coefficient equal to zero. Figs. 4 and 5 are examples of beach profile evolution for two beaches, with slopes of 1:20 and 1:10, respectively. Figs. 6 and 7 show the crest displacements of sandbars for two kinds of slopes under the regular wave conditions. Regular waves with various wave heights were used to investigate the correlation between the bar movement and wave height. It can be seen that the sandbar movements all contain shoreward-seaward oscillation patterns.

Fig. 4 Beach profile evolution for Test R3

Fig. 5 Beach profile evolution for Test R7

In the regular wave case the initial position of a sandbar only relies on the wave breaking point, which is dependent on the incident wave height. But after the formation of a sandbar, the water depth around the bar region changes, leading to the variation of the wave breaking pointand subsequent changes of bar features. It is known that the interaction between waves and the beach profile results in the unstable movement of the sandbar, and the intense unstable movements of a sandbar under regular waves are mainly dependent on the perturbation of the wave envelop caused by the wave-beach profile interaction. In this case, the beach slope also has a great effect. For the steeper slope of 1:10, the critical wave height that separates shoreward and seaward bar migrations was found to be= 0.145 m, but for the milder slope 1:20, no such condition could be found. This is presumably due to the fact that the wave height needed to produce the seaward bar migration is too large to be generated in the wave flume.

Fig. 6 Displacement of bar crest for tests R1-R6 on beach with slope of 1:20

Fig. 7 Displacement of bar crest for tests R7-R10 on beach with slope of 1:10

Because of strong oscillation of sandbars caused by regular waves, we used the results under the action of regular waves to examine the features of oscillation of sandbars. The bar displacement was recorded with shorter time intervals for determination of the bar migration velocity, as shown in Fig. 8, where three runs of Test R7 are presented. The solid dots in Fig. 8 correspond to the results of bar crest displacements of the bar profile, which are shown in Fig. 7, and the crosses are used for the measurement of the bar migration velocity. The latter has denser distribution and displays the details of higher frequency bar oscillations.

To further demonstrate this feature, Fig. 9 shows the energy spectrum of the bar oscillation, wherefis the frequency, andS(f) is the energy spectrum density. Since the time series (Fig. 8) is short, the Fourier transform can only generate the same number of energy spectrum density values as that of the measuring points in the time series, and the spectrum curves are very ragged. We adopted the maximum entropy method (Burg 1975) to obtain the energy spectrum estimation at a high resolution and to smooth the short-length data series. The peaks of the spectrum curves show the frequencies of the bar oscillation and correspond to the periods of 87 min, 76 min, and 69 min. The three runs under the same test conditions do not have similar forms of spectrum curves and major periods of oscillations. As shown in Fig. 9,the peak spectral periods of bar oscillations are several tens of minutes, so they are too large to be related to the oscillation periods of flow fluctuations in the flume, which are only tens of seconds for the wave-group period of long waves. This reflects the fact that the bar oscillation is not a forced response corresponding to hydrodynamic actions, but a self-organized process corresponding to bed instability. Therefore, it can be expected that the oscillation period can be predicted theoretically by the bed instability theory.

Fig. 8 Time series of bar movement displacements for Test R7

Fig. 9 Energy spectrum of bar oscillations for Test R7

3.2 Wave groups

Here we examine the effect of different amplitude modulations of wave groups on the bar evolution through varying mean wave heights and beach slopes. Figs. 10 and 11 show the evolution of beach profiles under the wave amplitude modulation coefficientδof 0.25 for the slopes of 1:20 and 1:10. The lines connecting the bar crests at successive times indicate the sandbar migration.

Fig. 10 Beach profile evolution for Test G3

Fig. 11 Beach profile evolution for Test G12

Figs. 12 and 13 show the displacements of the sandbar under the wave action of wave groups with amplitude modulation coefficientsδof 1.0, 0.5, and 0.25, and significant wave heightsH1/3of 0.09 m and 0.13 m for the slopes of 1:20 and 1:10, respectively. Three separate runs under identical test conditions were performed to demonstrate the repeatability of the experiments. It can be seen that after the formation of a sandbar, the bar has an overall trend of shoreward or seaward migration, but this is accompanied by slight seaward-shoreward alternating migration. It is noted that for the smaller wave height (H1/3= 0.09 m) the overall direction of bar migration is shoreward, while for the larger wave height (H1/3= 0.13 m) the overall direction is seaward, which is the case for both slopes.

Fig. 12 Displacements of bar crest for tests G1-G6 on beach with slope of 1:20

To understand this process, it is necessary to consider the two factors mainly affecting the propagation direction of the sandbar. One is the seaward bed load transport over the bar crest, which is related to the down-slope components, including gravity, net mean seaward flow, andundertow. The other is the shoreward bed load transport over the bar crest, associated with the asymmetry of maximum shoreward and seaward wave orbital velocities and the streaming flow in the wave bottom boundary layer. As the bar migration is caused by the net wave period and averaged sand transport, the overall direction of the bar migration depends on these two factors. The dominancy of the former leads to the seaward propagation of the bar, while the dominancy of the latter leads to the shoreward propagation.

Fig. 13 Displacements of bar crest for tests G7-G12 on beach with slope of 1:10

If the two factors balance each other, the bar is in the morphodynamic equilibrium state. This state can be disturbed by either a small seabed perturbation or a small flow velocity perturbation, and, as a consequence, a net shoreward migration, net seaward migration, or other more complicated forms of the bar migration may occur. Thus, a balance of the net shoreward and seaward sand transports should be considered the necessary condition for the bar oscillation.

Besides these factors, the suspended load also affects the bar migration, which has been investigated by measuring the flow velocity and suspended load concentration in the experiment (Wu 2011). The analyses show that the suspended load over the bar crest is transported mainly in the seaward direction, as it is dominated by the undertow and low frequency waves. The shoreward suspended load transport induced by the vertical asymmetry of short waves is negligible. This result shows that the contribution of the suspended load makes the bar move seaward.

The above analyses lead to the following conclusions on the mechanisms of bar migrations. First the seaward migration of the bar (as shown in Figs. 12 and 13 in the cases ofH1/3= 0.13 m) is related to the bed load and suspended load induced by the undertow and low frequency waves; that is, the seaward undertow dominates the wave asymmetry effect and results in a net seaward sand transport over the bar crest. Conversely, the shoreward migrationof the bar (as observed in Figs. 12 and 13 in the cases ofH1/3= 0.09 m) implies a dominant contribution of the asymmetry of short waves in the vertical direction, which means that the wave asymmetry and the streaming flow in the wave bottom boundary layer have greater effects on the bar migration than the gravity, undertow, and low frequency waves, and hence lead to a net shoreward sand transport over the bar crest.

However, when the net shoreward and seaward sand transports are nearly in balance, a small change in the beach profile will cause complex interactions between waves and the beach profile, which may lead to the bar oscillation. The instability mechanism is clearly supported by the experimental evidence. Assuming that the sandbar migration was forced by a wave group, the sandbar position would be ex pected to vary with forward and backward shifts of wave breaking points due to the wave height variation in the wave group, and an increase inδwould enlarge the variation range of the wave height, resulting in a larger variation range of wave breaking points and, correspondingly, a large oscillation of the sandbar. In fact, the opposite is seen in Figs. 12 and 13; that is, with the increase ofδ, the ranges of seaward and shoreward migrations of the sandbar tend to decrease instead: the smaller the value ofδis, the larger the oscillating frequency and amplitude become. As shown in Fig. 13, tests G9 and G12, which have a smaller amplitude modulation withδ= 0.25, show a larger bar oscillation, whereas tests G7 and G10, which have a larger amplitude modulation withδ=1.0, show no discernable oscillation. Tests G8 and G11 fall between the two extremes, and small bar oscillations can be identified in tests G7 and G10, indicating a stable bar profile. This correlation between the sandbar oscillation and wave amplitude modulation indicates that the sandbar migration is not merely a passive response of the sea bed to wave forcing, but is affected by the feedback interaction between waves and bed topography.

The instability feature of the sandbar is more apparent in the results of the regular wave case discussed in Section 3.1, where the oscillation of the sandbar is stronger than that under the wave group conditions. Relatively stable bar profiles are persistent, especially in random wave cases, which will be discussed below.

3.3 Random waves

A random wave can be considered a more general wave group with spatially and temporally varying amplitude modulation. Under this special amplitude modulation, the experimental results show that the bar profile is stable.

For the slope of 1:20 (Fig. 14), the sandbar generated in the wave flume is quite small, but for the slope of 1:10, an evident sandbar is produced, as shown in Fig. 15. Fig. 16 shows the movement displacement of the bar crest of three runs in different test conditions.

It can be seen from Figs. 14 and 15 that the bar moves continuously in one direction (shoreward for the slope of 1:20 and seaward for the slope of 1:10) with small oscillations, which means that the bar profile is stable with small topographic perturbations in the random wave case.

Fig. 14 Beach profile evolution for Test IR1

The migration speed of the sandbar decreases with the bar migration, although there may be a short retarding time during the process, and at the later duration, the bar migration almost stops and the bar reaches a steady location (Fig. 15). The three repeated runs for each test condition in Fig. 16 show similar trends. To check the stability of the final steady location of the bar migration, the test duration was extended from 370 min to 570 min for Test IR2. It was found that the bar location remained stable, and no marked sandbar movements took place during the extended time period, showing that the chosen test duration, 360 min, is sufficient for the sandbar to reach a steady location under these test conditions.

Fig. 15 Beach profile evolution for Test IR2

Fig. 16 Displacements of bar crest for tests IR2, IR3, and IR4 on beach with slope of 1:1.0

The overall shoreward or seaward migrations of the sandbar in random wave cases are similar to those in wave group cases with the amplitude modulation coefficientδ= 1.0, as shown in Figs. 12 (a) and (d) for the slope of 1: 20 and Figs. 13 (a) and (d) for the slope of 1:10. The seaward migration of the sandbar measured in this study is consistent with that of Wijnberg and Terwindt (1995).

4 Sandbar migration velocities

The above analyses show that there are two types of sandbar migrations: stable and unstable migrations. The stable bar migration has a unidirectional and decreasing migration velocity, while the unstable bar migration has a forward-backward oscillating migration velocity. The test conditions with the slope of 1:10 were chosen to discuss these two types of bar migration velocities. The migration velocity was calculated from the measured sandbar displacement divided by the corresponding time interval. The shoreward migration velocity was defined as positive and the seaward migration velocity as negative. The time interval was from 5 min to 15 min, and the change of the velocity direction was considered in the determination of the time interval. Thus, the time interval was not uniform and changed according to the bar movement: the faster the bar moved, the smaller the time interval adopted.

4.1 Regular waves

Fig. 17 shows the time series of sandbar migration velocities for Test R7 under the action of regular waves. It is a clear indication of the strong shoreward-seaward oscillation of the sandbar under the action of regular waves. Fig. 18 shows the energy spectra of bar migration velocities. It is seen that the peaks of the spectrum distribute evenly over the whole frequency range, meaning a wide-band distribution of the maximum velocities from the long period to the short period.

Fig. 17 Time series of sandbar migration velocities for Test R7

Table 2 gives the mean values of the positive velocityV+and negative velocityV?, denoted byand, respectively. The corresponding standard deviations are denoted byσ+andσ?and also given in the table. It is seen that the mean valuesandfor the three runs are about 0.8 cm/min and –1.0 cm/min, respectively. The latter (seaward migration) is slightly greater than the former (shoreward migration).

Fig. 18 Energy spectrum of bar migration velocities for Test R7

Table 2 Mean migration velocities of sandbar and standard deviations for Test R7

4.2 Wave groups

Fig. 19 shows the sandbar migration velocity for the slope of 1:10 under the action of wave groups with different amplitude modulation coefficientsδ(tests G10, G11, and G12). It is known from the previous results that for this kind of beach slope the bar migration is stable, i.e., the sandbar migration is continuously seaward until a final steady location is reached. This is reflected by the decrease of the negative migration velocities in Fig. 19. It is seen that the decreasing rate of the sandbar migration velocity is dependent on the amplitude modulation coefficientδ. The larger the amplitude modulation coefficient, the larger the velocity decrease rate. Thus, the larger wave amplitude modulation indicates a higher stability of the bar profile. In order to discuss the decreasing rate of the sandbar migration velocity quantitatively, the velocity is fitted with an exponential function as follows:

whereAvis a constant, andαis the velocity decrease rate. They are determined by the least square meth od:

whereViis the measured data of the sandbar migration velocity at timetiin three individual runs, andNis the number of measurements. Table 3 gives the results of the velocity decrease rate calculated with Eq. (3). The correlation coefficientρvis also g iven in the table. Since the sandbar oscillation accompanying the bar seaward migration is irregular, the correlationcoefficientρvprovides an effective indication of the degree of matching between the fitting curve and the measured data. It is seen that for the Test G10 with smaller sandbar oscillations,ρv= 0.851, and the correlation between the fitting curve and measured data is satisfactory, but for tests G11 and G12 with larger sandbar oscillations, the correlations are poor, with correlation coefficientsρvof 0.438 and 0.152, respectively. Fig. 19 shows the fitting curves with theAvandαvalues given in Table 3. The result thatαandρvdecrease withδfurther illustrates the presence of bed instability, since smallαand poorρvmean intense unstable movements of sandbars.

Fig. 19 Time series of bar migration velocities for tests G10, G11, and G12

Table 3 Analysis of deviation betweenVand

Table 3 Analysis of deviation betweenVand

TestvA αvρG10 –0.986 0.007 3 0.851 G11 –0.281 0.006 5 0.438 G12 –0.215 0.002 5 0.152

4.3 Random waves

Fig. 20 gives the sandbar migration velocity under the action of random waves in test IR2. In other test conditions, e.g., tests IR3 and IR4, with a slope of 1:10, the sandbar migrations have the similar movement trends to that in Test IR2, which are not shown here. As in the cases of wave groups, the sandbar migration velocity in Fig. 20 is always negative and decays to zero, and Eq. (2) can also be used to match the measured data. The values ofAvandαare given in Table 3. It is seen that under this random wave condition the velocity decrease rateαis 0.004 6,σvis 0.083, andρvis 0.721. The measured data and the fitted values have a significant correlation. The value ofαis between thoseof the two wave group cases whereδ= 0.25 and 0.5, and the random wave is considered to have spatially and temporally varying amplitudemodulations. This result also shows that in this random wave case the bar profile is stable.

Fig. 20 Time series of bar migration velocity for Test IR2

5 Conclusions

This study investigated the evolution features and mechanisms of sandbar migration in the surf zone through movable-bed physical model experiments. The main conclusions are as follows:

(1) Two kinds of sandbar migrations are identified in the surf zone: one is the stable migration which has a fast decaying migration velocity in the shoreward or seaward direction, and the other is the unstable migration which has a shoreward-seaward oscillation superimposing on the overall shoreward or seaward beach migration.

(2) The measurements show that the unstable migration of sandbars is closely related to the amplitude modulation of waves. Smaller amplitude modulation tends to produce more intense unstable bar movements under the action of waves. The correlation between the sandbar oscillation and wave amplitude modulation indicates that the sandbar migration is not merely a passive response of the sea bed to wave forcing, but is affected by the feedback interaction between waves and topography.

(3) The strongest bar oscillation was found in the case of regular waves with an average migration speed of sandbars being around 1.0 cm/min. The decreasing rate of the bar migration velocity in wave group cases is dependent on the wave amplitude modulation coefficientδ: the larger the amplitude modulation coefficient, the larger the decreasing rate of the bar migration velocity. This further supports the idea that bar oscillations are caused by bed instability. The decreasing rate of the bar migration velocity in random wave cases lies between those in wave group cases withδ= 0.25 and 0.5, which is an encouraging result as the wave height variation of the random waves is similar to those of the two kinds of wave groups.

(4) The experiments show that the overall direction of the bar migration is affected by the type of incident waves, the wave height, and the beach slope. The effects of these factors are different under different test conditions. In random wave cases, after a bar profile is generated, the bar always migrates seaward. In wave group cases, the bar migrates seaward under the condition of larger wave heights (H1/3= 0.13 m), and shoreward under small wave heights (H1/3= 0.09 m) for both beach slopes (1:10 and 1:20). But in regular wave cases, the beach slope has a much stronger effect: for the steeper slope 1:10, the critical wave height that separates shoreward and seaward bar migrations was found to be= 0.145 m, but for the milder slope 1:20, no suchcondition could be found. We conclude that the wave height needed to produce the seaward bar migration is too large to be generated in the present wave flume.

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(Edited by Ye SHI)

This work was supported by the National Natural Science Foundation of China (Grant No. 51079024).

*Corresponding author (e-mail:yinjing@mail.dlut.edu.cn)

Received Jul. 26, 2011; accepted Nov. 9, 2011