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Keywords:

  • solitary waves;
  • bar formation;
  • ripples;
  • sediment transport;
  • bed forms free surface interaction;
  • wave flume

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] The surface profile of water waves propagating in shoaling water approaches the solitary waveform before wave breaking. The effect of the high non-linearity of solitary waves may be very significant on bed forms induced in the nearshore zone. In this study, experiments on bed form generation beneath solitary waves are carried out in a 10-m-long flume used in resonant mode. Solitary waves are generated in shallow water on the background of a standing harmonic wave. One solitary wave (soliton) propagates in each direction of the flume on the time period of the flow, above an initially flat sandy bed. Ripples form rapidly on the bed and a strong interaction with the free surface occurs. The amplitude of the soliton and the phase shift between the soliton and the harmonic wave decrease with time, while the ripple amplitude increases. The amplitude of the harmonic wave is not affected by the ripples. The final ripple wavelength is about 1000 times the sand median diameter. Bars with superimposed ripples appear, with bar crests being positioned beneath the nodes of the standing wave, when bars form with crests beneath the antinodes of surface wave for standing waves without solitons. The Eulerian drift distribution in the flume is not affected by the propagation of solitons. We propose an energy balance for solitons propagating in shallow water above flat beds in which a term for the dissipation due to sand ripples is introduced, which defines a coefficient of interaction between solitons and ripples.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Waves, currents or combined wave-current flows over a bed of sand often generate bed forms patterns. Small-scale bed forms called ripples with a typical wavelength of about 10 cm may be observed both in steady [Engelund, 1970] and oscillatory flows [Sleath, 1984]. Yalin [1977] proposed a very simple formula for the estimation of the ripple wavelength in the case of a steady current. The ripples strongly influence the boundary layer structure and the turbulence intensity near the bed. In the case of waves, the steady streaming (wave-induced) current is significantly affected by these small bed structures [Marin, 2004]. Numerous studies have been carried out on ripple formation under waves. These studies have been performed in the field [see, e.g., Inman, 1957; Myrhaug et al., 1995; Hanes et al., 2001], in laboratory with an experimental approach [see, e.g., Kennedy and Falcon, 1965; Jarno-Druaux et al., 2004], and in a theoretical way [see, e.g., Fredsøe and Brøker, 1983; Blondeaux, 1990; Vittori and Blondeaux, 1990]. Larger-scale bed forms such as the offshore sand waves have wavelengths of several hundreds of meters, and their heights can rise up to 10 m. Large parts of shallow seas (as the North Sea) are covered with such bed features which are the object of intensive work [see, e.g., Németh et al., 2003]. In the nearshore zone, bars consisting of ridges of sediment running roughly parallel to the shore, are common features on sandy beaches. Nielsen [1979] has shown in a laboratory study that the bar formation may be explained by the steady drift which takes the form of closed recirculating cells in the case of standing waves [Longuet-Higgins, 1953]. The bar heights and spacings are about 1 m and of the order of 100 m, respectively. These structures provide a possible mechanism of natural beach protection from the energy of incident waves. O'Hare and Davies [1993] have shown the significance of the mode of sediment transport on the bar position under partially standing waves, the bars having spacing equal to half the surface wavelength. This spacing corresponds to the Bragg condition for which strong reflection of the incident waves may occur.

[3] It is well known that as an oscillatory wave moves into shoaling water, the wave amplitude becomes higher, the trough becomes flatter, and the surface profile approaches the solitary waveform before wave breaking [Munk, 1949]. The cnoidal wave theory approaches the solitary wave theory as the wavelength becomes very long. In this paper, solitary waves are generated in shallow water on the background of a standing harmonic wave, above an initially flat sandy bed. The interaction between the solitary waves and the bed is considered. The bed response may have practical importance in the nearshore zone, in particular for natural barred beaches. It has been shown in the case of partially standing waves that the bar position is a very significant parameter as far as the ability of bars to reflect incident wave energy is concerned [O'Hare and Davies, 1993]. The effect of solitary waves, highly nonlinear waves, on the bar position is considered in the present work. In nature, partially standing waves over bar systems are connected with long-period waves, such as leaky waves or edge waves [van Rijn, 1998]. In other respects, long waves such as tsunamis and waves resulting from large displacements of water caused by landslides and earthquakes often behave like solitary waves [Liu et al., 1991]. In particular, the run-up and shoreward inundation are often simulated using solitary waves [Lo and Shao, 2002]. To the authors' knowledge, the only study on the interaction between solitary waves and a sandy bed is a preliminary study carried out by the present authors [Marin et al., 2005]. Only one test has been carried out for this experimental study; this test has shown qualitatively a decrease of the solitary wave amplitude induced by the ripples which form on the sandy bed. The aim of the present work is to extend this preliminary study by a detailed analysis of sand bed forms induced by solitary waves in shallow water, and to develop an analytical model of the interaction between solitary waves and a sandy bed. The physical processes leading to the decrease of the solitary wave amplitude are investigated; this work may find applications in the protection of coastal areas subjected to tsunamis. Finally, the present tests are carried out in a wave flume used in resonant mode, and this study may also be relevant to the understanding of bed forms which are generated in natural resonators, such as bays and lakes.

[4] The first observations and experiments on solitary waves are attributed to Russell [1838, 1844]. Boussinesq [1871] performed pioneering theoretical studies of solitary waves. Korteweg and de Vries [1895] derived a model equation which describes the unidirectional propagation of long waves in water of relatively shallow depth. This equation is known as the Korteweg-de Vries equation or KdV equation. The damping of large-amplitude solitary waves has recently been studied in the framework of the extended KdV equation, that is the usual KdV equation supplemented with a cubic nonlinear term, by Grimshaw et al. [2003]. Recently, Ezersky et al. [2003] presented an original method to generate solitary waves in shallow water in a wave flume. This method is used in the present study, and contra-propagative solitary waves are excited in the flume. These solitary waves which preserve their shape while passing through each other, are known as solitons [Remoissenet, 1996]. Previous studies have been carried out on soliton excitation in resonators in systems whose physical nature is very different from that of water waves, for instance solitons of electromagnetic waves [Ostrovsky and Potapov, 1999], but which nevertheless possess similar dispersive and nonlinear characteristics as surface waves in shallow water. By analogy with these solitons of electromagnetic waves, Ezersky et al. [2004] have proposed equations for the amplitude and the phase of soliton in the case of surface waves propagating in shallow water above fixed flat beds. From these equations, we present in this paper a modeling of the interaction between solitary waves and a sandy bed, and we define a coefficient of interaction.

[5] In section 2, the experimental set-up and test conditions are presented. The section 3 which is split into four sections shows the results. Qualitative observations and the representation of solitary wave are described in the sections 3.1 and 3.2, respectively. In section 3.3, the ripple formation and the interaction with the free surface are considered. The section 3.4 is devoted to the bar formation. Velocity measurements have been carried out with a laser-Doppler anemometer to study the processes leading to the bars position. In section 4, a discussion is presented. The conclusions are stated in section 5.

2. Experimental Setup and Test Conditions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[6] The tests have been carried out in a 10-m-long and 0.49-m-wide wave flume at Le Havre University (France). The mean water depth was d = 0.26 m. Surface waves are produced by an oscillating paddle at one end of the flume (Figure 1); a near-perfect reflection takes place at the other end. The flume is used in resonant mode, without an absorbing beach. A resonant mode is established when an integer number of lengths of the wave equals the distance over which the wave is propagating as it reflects repeatedly forward and back [Sorensen, 1997]. The flume has resonant frequencies which are given by

  • equation image

where Lr is the length of the resonant cavity, g is the acceleration due to gravity, and n is equal to 0, 1, 2, etc., for the fundamental, first, and second harmonic modes, etc. In the present study, the frequency of the oscillating paddle f is chosen very close to the resonant frequency f1 = 0.165 Hz, corresponding to the mode for which n = 1. In this case, the wavelength Lh of the standing harmonic wave equals the effective flume length Lr (9.63 m; Figure 1). The ratio d/Lh = 0.027 lies in the range where the theory of very long shallow water waves is valid, and the harmonic wave speed is V0 = equation image. For values of the amplitude of horizontal displacement of the oscillating paddle averaged over depth ah lower than 2 cm, only standing harmonic waves are generated in the flume. For values of ah greater than 2 cm, solitary waves are excited on the background of the standing harmonic wave [Ezersky et al., 2004]; one solitary wave propagates in each direction of the flume on the time period of the flow. The generation of solitary waves in the flume results from the excitation of higher harmonics. Figure 2 depicts schematically the free surface elevation when the solitary waves are close to the ends of the flume. At the central part of the channel, the solitary waves collide, then continue their propagation until being reflected at the flume ends.

image

Figure 1. (a) Side and (b) top view of the flume.

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image

Figure 2. Sketch of solitary waves when they are close to the ends of the channel, with the free surface envelope of the standing wave (N, node of standing wave; A, antinode of standing wave).

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[7] The instantaneous free surface elevation is measured with one resistive probe located at the reflective end of the flume. The acquisition frequency is 100 Hz. The test conditions are shown in Table 1. In this table, As is the soliton amplitude, ϕs is the phase shift between the soliton and the harmonic wave, and A0 is the harmonic wave amplitude. The phase shift ϕs is given by the relation ϕs = Δτsω, where Δτs is the time interval between the passage of the peak of the soliton and the passage of the zero upcrossing of the harmonic wave (see Figure 6b in section 3.2), and ω = 2πf the angular frequency of the flow. In no test did wave breaking occur in the flume.

Table 1. Test Conditions
Test Numberf, Hzah, cmAs at the Beginning of the Test, mmequation images at the Beginning of the Test, radA0 at the Beginning of the Test, mm
  • a

    Test 7 was carried out without sand at the bottom.

10.170664.61.247.5
20.177680.70.443.9
30.17345.8
40.173567.40.954.1
50.173673.41.005.5
60.173776.00.896.4
7a0.173681.81.094.54

[8] For Tests 1 to 6, the bed was initially flat and covered by a 20-mm sand layer of median grain diameter D = 0.15 mm and relative density s = 2.65. The resonance frequencies defined by equation (1) depend on the mean water depth d, and are not affected by the bed morphology. An additional test (Test 7) was carried out without sand at the bottom to perform detailed velocity measurements. For this test, the frequency and the displacement amplitude of the oscillating paddle equaled those of Test 5. However, at the beginning of the test, the soliton amplitude As and the phase shift ϕs were slightly higher than for Test 5 (Table 1). This is due to the energy dissipation induced for Test 5 by the porosity of the sandy bed and by very small ripples which begin to appear during the time required for the solitons to establish in the flume (section 3.1).

[9] The fluid velocities were measured for Test 7 with a two-component, 4 W argon laser-Doppler anemometer operated in backward scatter mode. The measurement volume was 0.14 mm3. Fourteen vertical profiles of velocity measurements were carried out along the flume (Figure 1), from y = 0 to y = 18 cm, where y is the vertical position above the bed. The distance between two vertical profiles was 20 cm. At each measurement point, data acquisition was performed during 180 s with an acquisition frequency of 50 Hz.

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

3.1. Qualitative Observations

[10] At time t = 0, the wave generator is switched on and the bed is flat. Approximately 1 min is necessary for the appearance of solitons in the flume. Then the soliton amplitude grows until, after an average period of about 4 min, they are fully established. Let us consider Test 5 and the dimensionless time τ defined by

  • equation image

Figure 3a shows schematically the initial sand distribution along the channel, and Figure 3b shows the temporal evolution of the free surface elevation η at the reflective end of the flume when τ = 262 (ω = 1.09 rad/s), where the level 0 mm corresponds to the water level at rest. The peaks in the free surface elevation correspond to the passage of solitary waves. Small ripples form rapidly after a few minutes of surface wave excitation. These ripples spread all over the flume, except in the central part where the bed remains flat along a stretch of approximately 40 cm. This region corresponds to the zone of collision of the two solitary waves. Figure 4a shows a typical temporal evolution of the horizontal and vertical components of velocity, u and v, respectively, for the same test conditions but without sand at the bed (Test 7). This velocity measurement was carried out at x = 0.80 m where x is the distance measured in the horizontal direction, from the flume center toward the reflective end of the flume (Figure 1), and at y = 95 mm. Positive and negative peaks correspond to the passage of a solitary wave, propagating toward the reflective end of the flume and to the wave generator, respectively. Figure 4b depicts the temporal evolution of u and v in the middle part of the flume (x = 0) where the solitary waves collide, at y = 175 mm. At this height from the bed, the peak values of the vertical component of velocity exceed the peak values of the horizontal velocity. The contra-propagative solitary waves have horizontal velocities of opposite sign which give a resultant horizontal velocity close to zero in the collision zone. Near the bed in the central part of the flume (x = 0), fluid velocities are very small; the value of the bed shear stress is close to zero and below the critical value τcr for the initiation of sand movement. The threshold Shields parameter θcr for the incipient motion is defined as

  • equation image

where ρs and ρ are the density of sediment grains and water, respectively. The value of θcr can be estimated with the formula proposed by Soulsby and Whitehouse [1997],

  • equation image

where D* = [g(s − 1)/ν2]1/3D. This leads to a value of τcr equal to 0.16 N/m2 for present tests. The fluid velocities being very small close to the bed in the central part of the flume, the flow regime may be assumed to be laminar in this area where the bed is flat. Using the classical formulas for the bed friction under waves alone [Sleath, 1984], it is very easy to obtain an estimation of the amplitude Ucr of the horizontal component of velocity close to the bed in the central part of the flume, which would lead to a bed shear stress value equals to τcr. This value of Ucr is approximately 0.3 m/s; it is clear from Figure 4b that the maximum real value of u is far below this critical value. The bed is not affected by the surface waves in the central part of the flume. Moving away from the center, the ripple size increases. Figures 3c and 3d show respectively the distribution of sand along the channel and the temporal evolution of the free surface for a dimensionless time τ = 1.4 · 103 (ω = 1.09 rad/s; Test 5). A strong interaction between the bed and the free surface occurs. Compared to the initial stage of the test, the surface elevation is significantly lower (Figure 3b), which must be due to the dissipation at the now rippled bed. At τ = 9.8 · 103, when the ripples are bigger (Figure 3e), the peak value of the surface elevation has decreased further (Figure 3f). From this moment, the ripple size does not significantly increase any more. However, two sand accumulation zones will appear and the dynamics of bed evolution will be slow. These accumulation zones are clearly observable when τ = 6.1 · 104 (Figure 3g), and the value of the peak of the surface elevation has only very slightly decreased (Figure 3h) since the appearance of these accumulation zones. These ones are symmetrical in respect to the vertical plane oriented perpendicularly to the longitudinal axis of the flume and crossing the flume center (x = 0). They develop and form bars with crests located beneath the nodes of the harmonic surface wave (The position of the nodes is marked N in Figure 2). The bars have not been observed in the preliminary study of Marin et al. [2005], as they appear within longer timescales than the ripples. Figure 5 presents a series of four photographs which have been assembled to depict the sand distribution along 6 m in the central part of the flume, for τ = 3.13 · 105.

image

Figure 3. (a, c, e, g) Schemes of sand distribution in the flume and (b, d, f, h) temporal evolution of the free surface (Test 5) (ω = 1.09 rad/s); beginning of the test (Figures 3a and 3b), τ = 1.4 · 103 (Figures 3c and 3d), τ = 9.8 · 103 (Figures 3e and 3f), τ = 6.1 · 104 (Figures 3g and 3h).

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image

Figure 4. Temporal evolution of the horizontal and vertical components of velocity (Test 7); (a) x = 0.80 m; y = 95 mm, and (b) x = 0 m; y = 175 mm (collision zone of solitary waves).

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image

Figure 5. Sand distribution induced by solitary waves along 6 m of the flume (central part) (Test 5); τ = 3.13 · 105.

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[11] The temporal bed form evolution is qualitatively the same for the different tests. However, the energy dissipation at the bed due to ripples may lead to the vanishing of the solitary waves. This is the case for Tests 2 and 4. For Test 3, the amplitude of displacement ah of the oscillating paddle was too weak to get solitary waves during several minutes; these waves vanished very quickly after their appearance and only standing waves took place in the flume. The bed remained flat for this test.

3.2. Representation of Solitary Wave

[12] The solitons and the standing harmonic wave are extracted from the free surface elevation signal using a method proposed by Ezersky et al. [2003]. In this method, the position of the peaks of the free surface elevation are first detected (t = tmax; Figure 6a). Then the temporal series obtained with the resistive probe is replaced within a time window [t1, t2] by a linear dependence of the free surface on the time (t1 < tmax < t2; soliton cut off in Figure 6a). The last step consists in extracting the harmonic with the same frequency as the oscillating paddle frequency, and to subtract this harmonic from the measured signal. The temporal evolution of the free surface in Figure 6a is shown for Test 5 with the harmonic wave extracted from this signal, when the solitons are just established in the flume. Figure 6b depicts separately the sequence of solitons and the harmonic wave, with the theoretical sech-squared profile ηs of soliton,

  • equation image

where Vs is the propagation velocity of soliton: Vs = V0(1 + As/2d). The shape of the solitons generated in the wave flume is well represented by equation (5) (Figure 6b), except in regions where the amplitude is low. The profile of the soliton (equation (5)) which propagates from the oscillating paddle to the fixed end of the channel is a solution to the Korteweg-de Vries (KdV) equation which governs the displacement of the free surface η in shallow water [Remoissenet, 1996],

  • equation image
image

Figure 6. (a, c, d) Temporal evolution of the free surface and of the extracted harmonic wave, close to the reflective end of the flume (Test 5); τ = 294 (Figure 6a), τ = 1.7 · 103 (Figure 6c); τ = 7.2 · 103 (Figure 6d). (b) Sequence of solitons and harmonic wave (Test 5).

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[13] The soliton which propagates toward the oscillating paddle is solution of a symmetrical KdV equation. The KdV equation does not describe the interaction of contra-propagative waves. Neglecting this interaction, the free surface displacement consists of three components,

  • equation image

where k = 2π/Lh is the wave number of harmonic waves. Equation (7) can be applied for times t greater than the establishment time of soliton (about 4 min), and for −Lh/2 < x < Lh/2. Maxworthy [1976] has shown that the reflection at a vertical wall of one solitary wave is equivalent to a collision of two contra-propagative solitary waves of equal amplitude in an unbounded medium. Neglecting the interaction of contra-propagative waves, the free surface displacement close to the fixed reflective end of the flume (x = Lh/2), where the resistive probe is located, can be described by

  • equation image

The amplitude of the soliton measured by this probe is twice the amplitude of the propagative solitons (Figure 6b: peak of height 2 As). The temporal evolution of the free surface and of the extracted harmonic wave are presented in Figures 6c and 6d, for τ = 1.8 · 103 and τ = 7.2 · 103, respectively.

3.3. Ripple Formation and Interaction With the Free Surface

[14] The variation with τ of the ripple height h, horizontally averaged over the flume length, is shown in Figure 7a for Tests 5 and 6. The temporal variation of the ripple wavelength L, horizontally averaged over the flume length, is presented in Figure 7b for the same tests. The dimensions of the ripples forming on the bed increase for increasing values of τ (for τ < 104; Figure 7), when the soliton amplitude As decreases, as depicted in Figure 8. The test carried out without sand at the bottom (Test 7) has shown that the value of As does not vary with time as soon as the solitons are established in the flume, that is for values of τ greater than approximately 260. For the tests carried out with sand at the bottom, a strong interaction between the bed and the free surface occurs (Figures 7 and 8). However, the harmonic wave amplitude A0 is not affected by the ripples (Figure 8). The ripple wavelength (L ∼ 10 cm) is much smaller than the wavelength Lh of the standing surface harmonic wave (Lh = 9.63 m). The ripples correspond to small scale roughness for the harmonic wave. This scale difference (L/Lh ∼ 10−2) induces a negligible scattering of the harmonic waves by the ripples, and the harmonic wave amplitude is approximately constant during the ripple formation. The decay of the soliton amplitude As may result from the turbulence induced by the bed ripples, and from the scattering of the solitons by the ripples. The scale of the solitons (∼50 cm) is much smaller than the scale of the harmonic wave, and of the same order than the ripple scale. The scattering due to the bed morphology is then much greater for the soliton than for the harmonic wave. For Test 5, the amplitude of the soliton is only about half the initial value when τ = 7.8 · 103 (Figure 8a). The temporal evolution of As is shown for four tests in Figure 9. Let us consider Test 5 and Test 6. At each time, the value of As for Test 6 is greater than the value of As for Test 5; however, the dimensions of the ripples are very similar for both tests (Figure 7). These results show that the dimensions of the ripples do not depend on the soliton amplitude. The ripple wavelength at the equilibrium state, that is when L and h do not vary significantly any more with time, is about 1000 times the median diameter of sand. This matches the formula L = 1000D proposed by Yalin [1977] for unidirectional current ripples. The good agreement for the estimation of ripple wavelength, between this formula for the case of steady flows and present results for solitary waves excited in a wave flume used in resonant mode, shows that the instability mechanisms involved in ripple formation in the two cases may be related. The pulsating nature of the present flow does not appear to affect the ripple wavelengths in comparison with the case of a steady flow.

image

Figure 7. Variation of (a) the ripple height and (b) the ripple wavelength, horizontally averaged over the flume length, with the dimensionless time τ (Tests 5 and 6).

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image

Figure 8. Variation of the soliton amplitude As and of the harmonic wave amplitude A0 with the dimensionless time τ. (a) Test 5. (b) Test 6.

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image

Figure 9. Variation of the soliton amplitude As with the dimensionless time τ (Tests 1, 4, 5, 6).

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[15] For Test 4, the amplitude of the soliton rapidly decreases with time. For the same excitation frequency, the amplitude ah of horizontal displacement of the oscillating paddle is one cm weaker than for Test 5. This amplitude is too weak to prevent the decay of the soliton due to the high energy dissipation at the bottom. The decay of the soliton was even faster for Test 2 and 3 leaving only standing waves in the flume.

[16] The temporal evolution of the phase shift ϕs between the soliton and the harmonic wave is depicted in Figure 10. It is clear that sand ripples induce a decrease of the phase shift ϕs.

image

Figure 10. Variation of the phase shift ϕs between the soliton and the harmonic wave with the dimensionless time τ (Tests 1, 4, 5, 6).

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3.4. Bar Formation

[17] In the case of standing waves alone, Longuet-Higgins [1953] showed theoretically that the steady drift takes the form of closed recirculating cells with boundaries at a spacing of one-quarter the wavelength of the surface waves, as illustrated in Figure 11. Very close to the bed, that is, inside the Stokes boundary layer thickness δss = equation image,where ν is the kinematic viscosity), the fluid particle drift is oriented toward the node of the water surface profile and away from the antinodes. However, the drift is in the opposite direction further in the outer flow. These recirculating cells generate a sediment transport pattern resulting in bar formation, as shown by Nielsen [1979]. The bar crests are positioned beneath the antinodes of the surface elevation when the suspended load transport is dominant, and near the nodes (just up-wave) when the bed load transport is dominant, under purely standing waves as under partially standing waves [Carter et al., 1972; O'Hare and Davies, 1993]. In the present study, solitary waves are excited on the background of a standing harmonic wave, and large ripples form on the bed (section 3.3). These ripples generate vortices which lift into suspension a lot of sand, leading to a persistent layer of suspended material, and to a significant amount of suspended load transport. As mentioned in section 3.1, the bar crests are positioned beneath the nodes of the harmonic surface wave. This is an important difference compared to the case of standing waves alone for which the bar crests are located beneath the antinodes of the surface elevation when the suspended load transport dominates. An additional test was carried out with the same flow conditions than for Test 5, but with only a very thin layer of sediments (initially uniformly distributed along the flume), in order to prevent the ripples formation and the induced suspended load transport. Sediment accumulation was observed under the nodes of the harmonic surface wave, where the bar crests are positioned for Test 5. The mode of sediment transport does not seem to control the bar crests position when solitary waves, highly nonlinear waves, are generated on the background of a standing harmonic wave. Detailed velocity measurements have been carried out to analyze if the steady drift may explain the position of bars in the case of solitary waves. These measurements have been carried out without sand on the bed (Test 7) along 2.60 m of the wave flume, from the flume center (x = 0) where the antinode of the standing harmonic surface wave is located, to 20 cm downward of the node of the standing harmonic surface wave (x = 2.60 m), as illustrated in Figure 1b. Velocity measurements could not be carried out for greater values of x since the sidewalls of the flume are not transparent at the ends of the flume. Figure 12 shows the distribution of time-averaged velocity vectors. These time-averaged values have been obtained by a temporal averaging over 31 periods of the recorded velocities, the velocity measurements being performed during 180 s. A big clockwise recirculating cell can clearly be seen in Figure 12. Let us now focus our attention on the steady streaming close to the bed, where the vertical drift velocities are negligible. Figure 13 presents vertical profiles of equation image, the horizontal component of near-bed Eulerian drift, for y < 15 mm. In the central part of the flume (x = 0), the drift velocity is close to zero in the vicinity of the bed. For 0 < x < 2200 mm, the drift velocity is positive, that is, oriented toward the reflective fixed end of the channel, for y < 1.5 mm, that is inside the Stokes boundary layer, since δs = 1.4 mm for Test 7; the drift velocity is oriented in the opposite direction for greater values of y. Figure 13 shows that for x > 2200 mm, the drift velocity reverses close to the bottom and is directed toward the wave maker. Figures 12 and 13 show that the Eulerian drift distribution in the flume is approximately the same in the case of the solitary waves propagating on the background of a standing harmonic wave, as in the case of standing waves alone. The shift of the bar position obtained with the solitary waves has to be explained by other physical processes than the steady drift in the flume. Cooker et al. [1997] have investigated the collision of a solitary wave traveling over a horizontal bed with a vertical wall, using a boundary-integral method to compute the potential fluid flow described by the Euler equations. These authors have shown that the soliton amplitude increases more than twice at the instant of reflection at the wall. This reflection is equivalent to a collision of two contra-propagative solitons of equal amplitude [Maxworthy, 1976]. Ezersky et al. [2004] have shown that the velocity of the soliton is slightly higher after a collision with a contra-propagative soliton than before the collision. This slight increase of the soliton velocity in the center of the flume (zone of the collision) and at the ends of the flume (where the solitons are reflected) may explain the sand accumulation, and then the position of the bar crest, beneath the nodes of the standing harmonic wave. The interaction between the contra-propagative solitons has to be considered to get more insight into this point, but this is beyond the scope of this paper.

image

Figure 11. Sketch of steady drift for standing waves.

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image

Figure 12. Distribution of time-averaged velocity vectors (Test 7); [RIGHTWARDS ARROW]0.007 m s−1.

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image

Figure 13. Vertical profiles of the drift velocity equation image (Test 7).

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4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[18] Previous studies have been carried out on soliton excitation in resonators in systems whose physical nature is very different from that of surface waves, but which nevertheless possess similar dispersive and nonlinear characteristics as surface waves in shallow water. For instance, solitons of electromagnetic waves were observed on a section of an LC-line excited by sinusoidal voltage [Ostrovsky and Potapov, 1999]. An LC-line is a chain used for electromagnetic wave transmission which consists of a succession of capacities and inductances. Ezersky et al. [2004] have shown that the equations for the amplitude and phase of the soliton may be written, for surface waves propagating in shallow water above flat beds, by analogy with the solitons of electromagnetic waves. For present tests, the solitons propagate above a sandy bed and ripples form. The energy dissipation induced by the ripples has to be taken into account. The equations for the amplitude and phase of the soliton may be written, for surface waves propagating in shallow water above a mobile sandy bed, in the following form:

  • equation image
  • equation image

In these equations, Es is soliton energy (Es = equation imageηs2dx ∼ As3/2), δ is a coefficient describing the part of the damping of the soliton which is independent of the scale of the perturbations, α is a phenomenological coefficient for the dissipation of the soliton due to sand ripples, and Δ = 2π(f − f1) is the frequency mismatch. Equation (9) is an energy balance. The first term in the right side of Equation (9) describes the energy transfer from the harmonic wave to the soliton. The second term of this equation corresponds to the dissipation of the soliton. We suppose that the rate of soliton energy dissipation is proportional to the energy; this means that the amplitude of the soliton decays like an exponential function, as shown by Grimshaw et al. [2003] for large-amplitude solitary waves in the framework of the extended KdV equation. For a fixed flat bed, α = 0 holds; this case has been considered by Ezersky et al. [2004] who showed that that the value δ = 0.015 Hz allows to have a good agreement between experiment and theory. In the following, we will consider the same value for δ. The dissipation of the soliton due to sand ripples is supposed to be proportional to the ripple height h, as a linear function is the simplest way to propose a parameterization. The coefficient α, which has dimension m−1s−1, represents the percentage of soliton energy which is lost per time unit and per ripple height unit. Equation (10) is a kinematic condition. The first and second terms in the right side of equation (10) describe the dependence of the soliton velocity with respect to the harmonic wave velocity on the soliton amplitude As, and on the phase shift ϕs, respectively. The third term in the right side of equation (10) corresponds to the effect of the frequency mismatch between the oscillating paddle frequency and the resonance frequency on the phase shift ϕs. Introducing the dimensionless coefficients δ′ and α′,

  • equation image
  • equation image

equations (9) and (10) give the following relations for ϕs and As at steady state:

  • equation image
  • equation image

It is clear from equations (13) and (14) that the dissipation of the soliton due to sand ripples (related to the coefficient α′) induces a decrease of the phase shift ϕs and of the amplitude of the soliton As, as observed in the experiments. The value of the coefficient α′ may be estimated at 0.03 for present tests using equation (13) and the measured values of ϕs, A0 and h. Figure 14 presents the temporal variation of ϕs and As obtained from equations (13) and (14), with the experimental values for Tests 5 and 6. The experimental and theoretical values of the phase shift ϕs are in good agreement (Figures 14a and 14b). The decrease of the soliton amplitude is well reproduced by equation (14) (Figures 14c and 14d). However, there is a shift between the theoretical and experimental values of As; this may partly be due to the interaction of counterpropagating solitary waves which is not taken into account in the proposed formulation (equation (7)).

image

Figure 14. Comparison between experimental and theoretical (equations (13) and (14)) values of (a, b) ϕs and (c, d) As (Tests 5 and 6).

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5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[19] A laboratory study has been carried out on bed forms induced by solitary waves (solitons) propagating in shallow water. The solitons are generated on the background of a standing harmonic wave, in a wave flume used in resonant mode. One soliton propagates in each direction of the flume on the time period of the flow, above an initially flat sandy bed. Ripples form rapidly along the flume, except in the collision zone of the two solitary waves, in the central part of the flume. A strong interaction between the bed and the free surface occurs. The energy dissipation induced by the ripples leads to a decrease of the amplitude of the soliton, and to a decrease of the phase shift between the soliton and the harmonic wave. An analytical model of the interaction between the solitons and a sandy bed is proposed. The agreement between the experiments and the model is satisfactory. The final ripple wavelength is about 1000D where D is the median grain diameter, as in the case of steady flows. Bars on which ripples are superimposed appear, with crests located beneath the nodes of harmonic surface wave. In the case of standing waves alone and when the sediment transport is dominated by suspended load, bar crests form beneath the antinodes of surface elevation. This important difference may have practical consequence in the nearshore zone for the ability of bars to reflect incident wave energy. The shift of the bar position cannot be explained by the Eulerian drift distribution in the flume, which is the same for the standing waves alone and for the solitons propagating on the background of a standing harmonic wave. This shift may be explained by the interaction between the contra-propagative solitons. The study of this interaction requires further research.

Notation
ah

amplitude of horizontal displacement of the oscillating paddle averaged over depth, cm.

A0

amplitude of harmonic wave, mm.

As

amplitude of soliton, mm.

d

mean water depth, m.

D

median grain diameter, mm.

D*

dimensionless grain diameter: [g(s − 1)/ν2]1/3D.

Es

soliton energy: Es = equation imageηs2dx, m3.

f

oscillating paddle frequency, Hz.

fn

resonant frequency of the mode n for the resonant cavity, Hz.

g

acceleration due to gravity, m/s2.

h

ripple height, horizontally averaged over the flume length, mm.

k

wave number of harmonic waves, m−1.

L

ripple wavelength, horizontally averaged over the flume length, mm.

Lh

wavelength of the surface harmonic wave, m.

Lr

length of the resonant cavity, m.

s

sediment relative density.

t

time, s.

u

horizontal component of fluid velocity, m/s.

equation image

horizontal component of fluid Eulerian drift, m/s.

Ucr

critical value of the amplitude of u for which the bed shear stress value equals τcr, m/s.

v

vertical component of fluid velocity, m/s.

V0

speed of surface waves of small amplitude in shallow water, m/s.

Vs

propagation velocity of soliton, m/s.

x

distance measured in the horizontal direction, from the flume center toward the reflective end of the flume, mm.

y

distance from the bed measured in the vertical direction, mm.

α

coefficient of interaction between solitons and ripples, m−1s−1.

α′

dimensionless coefficient of interaction between solitons and ripples: α′ = αd3/2/equation image.

δ

coefficient describing exponential damping of soliton, Hz.

δ′

dimensionless coefficient describing exponential damping of soliton: δ′ = δequation image.

δs

Stokes boundary layer thickness, mm.

Δ

2π(f − fr), frequency mismatch, rad/s.

Δτs

time interval between the passage of the peak of soliton and the passage of the zero upcrossing of the harmonic wave, s.

η

free surface elevation, mm.

ηs

component of free surface elevation corresponding to soliton profile, mm.

θcr

threshold Shields parameter for the incipient motion.

ν

kinematic viscosity, m2/s.

ρ

density of water, kg/m3.

ρs

density of sediment grains, kg/m3.

τ

tω, dimensionless time.

τcr

threshold bed shear-stress for motion of sediment, N/m2.

ϕs

Δτsω, phase shift, rad.

ω

2πf, angular frequency of the flow, rad/s.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[20] The authors acknowledge I. Mutabazi for the invitation of A. B. Ezersky, who got a financial support from the French Ministry of Research for his stay at Le Havre University. We would also like to thank the reviewers for their helpful and interesting suggestions on how to improve the original manuscript.

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  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental Setup and Test Conditions
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
jgrf96-sup-0001-t01.txtplain text document0KTab-delimited Table 1.

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