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Dissipation of incident forced long waves in the surf zone—Implications for the concept of “bound” wave release at short wave breaking
Coastal Engineering 60 (2012) 276–285
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Dissipation of incident forced long waves in the surf zone—Implications for the concept of “bound” wave release at short wave breaking T.E. Baldock School of Civil Engineering, University of Queensland, St Lucia, QLD 4072, Australia
a r t i c l e
i n f o
Article history: Received 8 July 2011 Received in revised form 13 October 2011 Accepted 1 November 2011 Available online 30 November 2011 Keywords: Surf beat Long waves Bound wave release Breaking Long wave dissipation Surf beat similarity
a b s t r a c t A review of laboratory data sets on surf beat is presented, with a focus on the dissipation of long wave energy in the surf zone. It is frequently assumed that incident forced long waves, or “bound” long waves, are released from short wave groups when the short waves break, subsequently propagating to the shore as a free wave. Free long waves may additionally be generated by the moving short wave breakpoint. Convincing evidence of the release of forced long waves as a result of short wave breaking is lacking, while there appears to be strong evidence to the contrary from a range of recent laboratory experiments. The data from the laboratory experiments are also consistent with field observations of strong nearshore dissipation of long waves. These data are also consistent with Longuet-Higgins and Stewart (1962), who suggest that the forced long wave may reduce in amplitude following short wave breaking, not that it might be released as a free wave. In contrast, forced long waves can be progressively “released” from the groups when the short waves are in shallow water, since these conditions correspond to those where the forced long wave satisfies the free wave dispersion relationship. This frequently occurs prior to short wave breaking for mild wave conditions, but here it is shown that these conditions are not usually satisfied at the short wave breakpoint for storm conditions. Energy transfers between free and forced waves are also discussed with regard the data. A surf beat similarity parameter that incorporates both relative beach slope and short wave steepness is suggested, which distinguishes between different long wave forcing regimes inside the surf zone. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Wave groups approaching the shore force associated long waves, commonly termed bound long waves as they propagate with the wave groups rather than at the free wave celerity. During shoaling, a range of non-linear triad interactions occur between pairs of short wave harmonics and the forced long wave harmonic. These interactions can be resonant or non-resonant and have been widely studied, commencing with the work of Biésel and continued since by LonguetHiggins and Stewart (1960; 1962), Phillips (1977), Freilich and Guza (1984), Herbers et al. (1995), Elgar and Guza (1985), Elgar et al. (1992), Schaffer (1993), Janssen et al. (2003), Battjes et al. (2004) and others. Collectively, these describe the evolution of the non-linear system from a non-resonant state in deep water through to a system where the departure from resonance varies during shoaling, with corresponding variations in the growth rate and phase relationship of the forced wave away from the Longuet-Higgins and Stewart (1962) solution (Battjes et al., 2004; Janssen et al., 2003). Nielsen and Baldock (2010) extended the particular solution of Longuet-Higgins and Stewart (1962) for shallow water waves to include free waves that are generated and propagate away from the group during short wave
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shoaling, leading to similar phase differences from the LonguetHiggins and Stewart (1962) solution. Collectively, low frequency waves, whatever their origin or mode are also termed infragravity waves. It is widely assumed that the bound long waves are released during short wave breaking, subsequently propagating to the shore as a free wave, where they are reflected (see reviews in Baldock and Huntley, 2002; Battjes et al., 2004; Janssen et al., 2003). The emphasis is on the short wave breaking process being the cause or trigger for the release of the bound waves. Statements asserting this release of the forced wave at short wave breaking are often unattributed with no citation of convincing evidence, or loosely attributed to Longuet-Higgins and Stewart (1962), and a physical mechanism for this release by short wave breaking is not usually stated. However, Longuet-Higgins and Stewart (1962) do not mention bound wave release on short wave breaking and a different interpretation of their statements is that the forced long wave will in fact decay in amplitude following short wave breaking. This is in agreement with a number of recent laboratory data sets that show strong dissipation of forced waves following short wave breaking (Baldock and O'Hare, 2004; Baldock et al., 2000; Battjes et al., 2004; Dong et al., 2009a; Dong et al., 2009b) and recent RANS numerical model results (Lara et al., 2011). Similarly, it is also consistent with field data showing strong nearshore dissipation of long waves (Henderson and Bowen, 2002; Henderson et al., 2006;
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2. Existing data Baldock and O'Hare (2004) presented a number of laboratory data sets that illustrated the strong dissipation of incident forced waves once short wave breaking commenced. Two different forms of wave
interaction were studied. Firstly, the forced waves (sub- and superharmonics) generated by a long wave-short wave interaction, which was the original application of the theory developed by LonguetHiggins and Stewart (1960). Secondly, the forced waves generated by broad-banded bichromatic short wave groups. In these experiments, the waves propagate over horizontal section of the wave flume, followed by shoaling on a 1:10 sloping beach which commences at x = −8 m, with the still water shoreline at x = 0 m. Second order wave generation was used and surface elevation measurements were made at closely spaced intervals along the full length of the flume, together with the shoreline motion. Several examples are illustrated here, where, despite the steep absolute beach slope, the forced waves propagate in a mild slope regime, as later defined by Battjes et al. (2004), and which represents near resonant conditions and weak breakpoint forcing. Fig. 1 shows the cross-shore variation of the amplitudes of sub- and super- harmonics forced by the interaction of a low steepness free long wave (f= 0.2 Hz) and a steep free short wave (f= 1 Hz). Here, the beach slope and wave amplitudes yield a narrow surf zone and Iribarren numbers corresponding to spilling and plunging waves, with rapid dissipation of short wave energy. The short wave breakpoint varies
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Thomson et al., 2006), which is not due to frictional losses but is suggested to occur through non-linear interaction that transfer energy from the infragravity waves back to swell and sea. However, these data sets have not been used to discuss the implications of this strong dissipation for the concept of bound wave release by short wave breaking. This is the primary purpose of the present paper. Following Longuet-Higgins and Stewart (1960), general solutions for a range of interacting wave modes were identified, with later extensions which describe a range of dynamical processes and forms of resonant interactions. Yuen and Lake (1982) provide an extensive general review, with the theoretical work of Freilich and Guza (1984), Elgar and Guza (1985); Herbers et al. (1995), Battjes et al. (2004) and Janssen et al. (2006) particularly applicable for shoaling waves and for conditions seaward of the breakpoint. Comprehensive analyses of forced wave propagation under field conditions prior to short wave breaking are given by Elgar et al. (1992) and Herbers et al. (1995) and references therein. Elgar and Guza (1985) showed that the phase of the long waves often evolves from the deep water bound wave biphase of 180° toward 90° as the short waves shoal, consistent with bound waves becoming free prior to short wave breaking, or with the infragravity band becoming dominated by freely propagating waves as the wave field shoals prior to breaking. Similarly, Herbers et al. (1995) show that as waves shoal, the relationship between infragravity waves and swell changes from one appropriate for forced, bound waves in deep water to freely propagating waves in shallow water. The triad interactions between swell waves and the infragravity waves become close to resonant in shallow water and require distances of order of a few wavelengths to transfer significant energy between the harmonic components (Freilich and Guza, 1984; Herbers et al., 1995). After the short waves break, there is then typically a reduction in correlation between the short waves and incident long waves inside the surf zone (Roelvink and Stive, 1989), because the short wave groupiness is usually reduced significantly in a random wave conditions. Once the short waves start to break, additional free long waves can be generated by the oscillating short breakpoint (Schaffer, 1993; Symonds et al., 1982), which again has been identified in the laboratory (e.g. Baldock and Huntley, 2002). The relative importance of the different regimes of forced wave amplification during shoaling has been clearly identified by Schaffer (1993), Battjes et al. (2004) and Janssen et al. (2008), and is primarily related to the relative beach slope for the long waves. Although field data sets exist as noted above, these are more difficult to interpret and they lack the spatial resolution of the laboratory data sets. Thus, this paper focuses on reviewing laboratory data sets, and particularly those that allow direct identification of the forced wave behavior because there was either minimal long wave generation at the breakpoint, or little shoreline reflection of the long waves, or the incident and reflected long waves were separated by careful analysis. The purpose of this paper is to discuss these recent laboratory data with respect to the dissipation of long wave energy in the surf zone, and if this is consistent with the concept of bound wave release as a result of short wave breaking. The paper is organized as follows. Section 2 provides examples of existing data sets that illustrate the cross-shore variation of the incident forced wave amplitude. A discussion of the data in view of the original ideas presented in Longuet-Higgins and Stewart (1962) is given in Section 3. Section 4 provides a further discussion of energy transfers and proposes a similarity parameter that accounts for both the relative beach slope and the short wave steepness when considering the different surf beat forcing mechanisms. Final conclusions follow.
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(m) Fig. 1. Cross-shore variation in sub- and super-harmonic amplitudes forced by a long waveshort-wave interaction. From Baldock and O'Hare (2004). The short waves break in the region x = − 0.75 m to x = − 1.5 m. Beach slope 1:10. a) Sub-harmonic, fsub = 0.586 Hz. b) Super-harmonic, fsup = 1.95 Hz.
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depending on the short wave amplitude. In each case, rapid dissipation of the forced harmonic occurs immediately following short wave breaking, which occurs in water depths between 0.05 m and 0.15 m. Fig. 2
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(m) Fig. 2. Cross-shore variation in sub- and super-harmonic amplitudes for a broad-banded bichromatic wave group, B1060A. − □ − data, —— Incident bound wave (2nd order quasi- local solution). c. Cross-shore variation in primary and bound wave amplitudes for a broad-banded wave group, B6055C. ⋄ f1 = 0.879 Hz, □ f2 = 0.341 Hz, ∇ fsub = 0.537 Hz, × fsup = 1.22 Hz, —— fsub (radiated). From Baldock and O'Hare (2004). The short waves commence breaking at approximately x = −1.05 m. Beach slope 1:10.
shows further examples, in these cases from bichromatic wave groups. Again, very rapid dissipation of the forced harmonics occurs following short wave breaking at x ∼ −1 m. Fig. 2c shows a very significant change in the division of energy (strictly surface variance) between the short waves and the forced harmonics prior to short wave breaking, usually interpreted as an energy transfer. Indeed, the energy transfer is sufficient to significantly reduce the short wave amplitude during shoaling, with the energy transferred almost exclusively from the higher frequency short wave component. In fact, the energy transfer is such that the forced harmonics grow larger than the short wave component that forces them. There is no evidence of a transfer of long wave energy back to the short waves during the dissipation of the long waves, but this would be very hard to detect once the short waves themselves are breaking. These data were obtained without the need to separate the total wave motion into free and forced waves and incident and radiated harmonics in the surf zone since there is little reflected wave energy at the frequencies of interest and, consequently, the incident wave amplitude is easily identified. The lack of reflection is a direct consequence of the strong dissipation of the primary short waves and the forced harmonics prior to the waves reaching the shore. However, for different conditions, and in general, a standing long wave pattern is typically formed, which may comprise of an incident forced, or bound, long wave, an incident breakpoint generated free wave, and a reflected free wave. In these cases, it is more difficult to determine the fate of the incident bound long wave after short wave breaking. Baldock et al. (2000) and Baldock and Huntley (2002) assumed a priori that the incident bound long wave was strongly dissipated following short wave breaking, and then calculated the resulting cross-shore standing wave structure of the surf beat, and found good agreement. However, this provides only indirect evidence of dissipation of the forced wave. As noted by Battjes et al. (2004) and subsequently demonstrated by van Dongeren et al. (2007), it is possible that the dissipation occurs because the long waves break in very shallow water. However, in order for this to explain the data illustrated above, this would require that long wave breaking occurred when the long wave height to water depth ratio, γL = HL/d, was about equal to 0.2, instead of the usual value of γ ≈ 0.8–1. Consequently, this is not a likely mechanism for the long wave dissipation in these cases. Battjes et al. (2004) present data that show a similar dissipation of the incident long waves and again the dissipation commences following short wave breaking and at a location where γL ≈ 0.1–0.3. Examples are shown in Figs. 3 and 4. On this double-barred beach, some dissipation occurs after the initial short wave breakpoint, with further dissipation occurring after renewed short wave breaking on the inner bar. However, it should be noted that other frequency bands from the same data set (Battjes et al., 2004) do not show the same degree of change in the incident long wave amplitude following initial breaking, e.g. Fig. 4, but show quite strong dissipation after the second breakpoint. Van Dongeren et al. (2007) also showed that some of these long waves break in very shallow water (h ≈ 0.02 m), which limits the amplitude of the outgoing wave shown in these figures. For these data, the short waves are in relatively shallow water at the short wave breakpoint, kh≈ 0.22, and this point is discussed further below. However, it is quite clear that in some experiments the incident wave forced is progressively released from the short waves, as frequently suggested. A clear example is shown by forced (outside the breakpoint) long waves propagating unchanged through the surf zone and reflecting from the shoreline (e.g. Janssen et al., 2003). However, this release is not necessarily related to short wave breaking. An example is a data set from van Dongeren et al. (2007), where the incident forced wave again reduces rapidly in amplitude shortly after short wave breaking, but subsequently “re-shoals” and then breaks in very shallow water (Fig. 5). Further inspection of the data in Fig. 5 indicates that kh≈0.5 at the short wave break point (h=0.2 m); for these waves shallow water conditions are reached at h≈0.07 m. Dissipation of short wave energy is initially
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Fig. 3. See Fig. 4 for caption.
rapid in the region 0.15 mb hb 0.2 m, but this then ceases and the long wave “re-shoals”. The amplification of the long wave in the region 0.5 m b h b 0.15 m is greater than that indicated by Green’ law for free wave shoaling. In fact, assuming shoaling as a near resonant shallow water forced wave (proportional to h− 5/2) and accounting for the reduction in the short wave forcing terms (proportional to H2), gives an expected amplification factor of about 2.2, only slightly greater than the observed amplification factor of 1.7. Thus, it seems possible that this long wave remains a forced wave across most of the surf zone until shallow water conditions for the short waves are reached at h ≈0.07 m. Conversely, as noted in Section 2, data from Elgar and Guza (1985) and Herbers et al. (1995) is consistent with bound waves becoming free prior to short wave breaking, or with the infragravity band becoming dominated by freely propagating waves as the wave field shoals prior to breaking. Dong et al. (2009a) present data from an experiment which excluded reflection of the long waves and therefore also enables direct observation of the incident bound long wave amplitudes. The experiments were performed over a sloping bottom, followed by a shelf with a constant water depth of 0.1 m. Hence, no, or only, partial breaking of the short waves occurred, not the near full dissipation of short wave energy that usually
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occurs across a sloping beach. Fig. 6 shows examples of the cross-shore variation of the incident bound long wave for cases where the short waves do not break (cases BS310, BS610, BS330 and BS630) and where they do break. For the non-breaking short wave groups, the incident bound long waves shoal to the edge of the shelf break (x =22 m) and then remain constant in amplitude since the forcing is then constant in the constant depth region. For the remaining cases, where the short waves do break (before or at the shelf break at x= 22 m), the difference is striking, and again the incident long waves decay rapidly in amplitude after short wave breaking occurs. Note again that the depth on the shelf was 0.1 m and the incident short wave heights were about 0.1 m in height, so only partial breaking occurs and some forcing remains possible in the constant depth region. Dong et al. (2009b) present data from similar experiments to those described above, but this time with and without adverse and favourable currents, and example data are illustrated in Figs. 7 and 8. The opposing current increases the short wave steepness and promotes short wave breaking where it would otherwise not occur. Fully modulated wave groups and weakly modulated groups were used, leading to a variation in the rate of change of wave height for the two data sets and therefore the rate of energy dissipation shortly after short wave breaking. When significant amounts of short wave energy is dissipated (fully modulated wave groups, Fig. 8), the incident bound wave amplitude decays rapidly, whereas when the short wave energy dissipation is lower (weakly modulated wave groups, Fig. 7), the incident bound wave amplitude reduces more slowly or stays constant as the short waves continue to shoal. There is also strong evidence from field data for rapid dissipation of incident infragravity energy in the surf zone. Henderson and Bowen (2002) observed strong dissipation of surf beat energy inside the surf zone and modeled this in terms of dissipation by the bed shear stress. However, the roughness required to match model-data predictions was very large. Henderson et al. (2006) and Thomson et al. (2006) analysed and modeled further field data and showed a strong transfer of energy away from infragravity bands in the inner surf zone. This could not be attributed to either bottom friction or breaking of the long waves, and was attributed to non-linear interactions between the forced long wave and the sea and swell. Since the interaction of free long waves with sea and swell is very weak (Phillips, 1977, §3.6), this suggests that the infragravity waves remained coupled to the sea and swell in the surf zone, enabling a reverse energy transfer to occur. This reverse energy transfer is well known during the focusing of transient wave groups (Rapp and Melville, 1990). Fig. 9 shows an example from Rapp and Melville (1990) where weak spilling breaking occurs during focusing and Fig. 10 shows an example with strong breaking at the focal point. The non-breaking (not shown) and spilling breaker cases show a near completely reversible energy transfer to the forced harmonics, where the energy in the low frequency harmonics increases during focusing (focal point is at x − xb = 0), and then decreases again during defocusing, with the energy returned to the short waves. Similarly, the majority of the low frequency energy is transferred back away from the low frequency band even in the case of a deep water plunging breaker (Fig. 10). For example, comparing the low frequency energy at kc(x−xb) = ±15 shows that the gain in low frequency energy at x − xb = 0 is reversed during defocusing, even after breaking. 3. Interpretation of Longuet-Higgins and Stewart (1962)
Fig. 4. Example of incident (triangles) and outgoing (dots) long waves forced by random waves (Battjes et al., 2004). The initial short wave breakpoint is at x ≈ 21 m (h ≈ 0.15 m, outer bar), followed by a second breakpoint at 25 m (h = ≈ 0.1 m, inner bar). From Battjes et al. (2004). Reproduced/modified by permission of American Geophysical Union.
Many papers discuss the fate of the incident bound long wave after breaking, and typically state an explanation similar to the following: “…the effect of short wave breaking on these non-linear interactions is not understood, ….it has been hypothesised that the
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However, this is not stated in Longuet-Higgins and Stewart (1962). The relevant quotation is: “Since long waves are more readily reflected by non-uniformities in the transmitting medium than are shorter waves, it is reasonable to suppose that at some depth the long wave associated with the mass transport undergoes partial reflexion while the shorter waves are allowed to pass on and be destroyed in shallower water”. This implies reflection of any long waves before short wave breaking. There is no implication that the separation of the short wave and forced wave requires short wave breaking to occur. In Lara et al. (2011), such reflection of the long wave before breaking is visible at beach slope discontinuities (their Fig. 9). Further, in relation to the observations that the long waves radiated offshore from the surf zone were observed to be (and are still typically observed to be) linearly proportional to the incident swell height (as opposed to showing a quadratic dependency), Longuet-Higgins and Stewart (1962) stated: Fig. 5. Cross-shore variation of Hrms for a bichromatic wave group. Dash-dot line, total; dashed line, short waves; solid line, long waves. From van Dongeren et al. (2007). Reproduce/modified by permission of American Geophysical Union.
incident bound waves are not destroyed in the surf zone, but are released as free waves (Longuet-Higgins and Stewart, 1962). …. After reflection at the shoreline, the long waves travel in a seaward direction and may either propagate into deep water or remain refractively trapped to the coast ”.
“It would seem that the amplitude of the long waves is proportional to the square of the envelope of the incoming swell. On the other hand, if breaking has taken place before the point of reflexion, the higher waves at least will have been reduced in amplitude, and so one expects in fact a law of variation rather weaker than a2.” This implies that the bound wave amplitude is determined by the local forcing that continues after short wave breaking (which is governed by the short wave amplitude and groupiness remaining after short wave
Fig. 6. Examples of cross-shore variation of incident bound long wave amplitudes from Dong et al. (2009a). Circles, slope = 1:40, triangles, slope = 1:10. Curves are asymptotic solutions of Longuet-Higgins and Stewart (1962). Depth is constant for x > 22 m. From Dong et al. (2009a). The short waves do not break for cases BS310, BS610, BS330 and BS630, but do so for the other cases shown.
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Fig. 7. Examples of cross-shore variation of incident bound long wave amplitudes from Dong et al. (2009b). Data is for a weakly modulated bichromatic wave group propagating up a 1:60 slope with and without an adverse current. For the adverse current, short wave breaking occurred at x = 15 m and x = 12 m in a) and b), respectively. Note that Uo = 0 and −12 cm/s in b). Depth is constant for x > 21 m. From Dong et al. (2009b).
breaking), and therefore that the bound wave is not released at breaking, i.e. it remains a forced wave. Such conditions were investigated by Schaffer (1993). This is consistent with most of the data shown above, which shows rapid reduction in either the incident long waves after short wave breaking, or a much decreased amplitude for the long waves reflected from the shore. Further, on a very mildly sloping beach, Ruessink (1998) observed free long wave amplitudes to be proportional to the square root of the offshore wave height. Also consistent with this interpretation, Ruessink et al. (1998) observed a shoreward decrease in the bound and total infragravity energy at locations landward of the breakpoint and Battjes et al. (2004) likewise observed a decrease in the total incident low frequency wave amplitude after breaking. In addition to the continued forcing postulated in Fig. 5, a further example of this continuing forcing can be found in Baldock (2006), which considered the propagation of transient wave groups over a 1:10 sloping beach and through breaking. Since the wave group is transient, the motion can be followed nearly to the shore before reflection complicates the picture (Fig. 11). In this example, the long wave surface slopes are inversely related to the spatial gradients of the wave envelope, as expected for a forced wave, even well inside the breakpoint. Note also that there is no significant lag between the wave envelope and bound long wave. The bound wave amplitude continues to increase after short wave breaking in this case, i.e. the
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Fig. 8. Examples of cross-shore variation of incident bound long wave amplitudes from Dong et al. (2009b). Data is for a strongly modulated bichromatic wave group propagating up a 1:60 slope with and without an adverse current. For the adverse current, short wave breaking occurred at x=20 m in b). Depth is constant for x>21 m. From Dong et al. (2009b).
long wave remains forced. This is because even though the short waves reduce in amplitude, the radiation stress gradient, dSxx/dx (estimated from the spatial wave envelope), remains approximately constant as the envelope continues to focus and shorten. Since the depth is also reducing, the response of the system increases and the bound wave continues to grow despite short wave breaking. However, this is not usual and typically the radiation stress gradients will reduce rapidly in the surf zone as the groupiness is largely destroyed or even reversed. Lara et al. (2011) have recently performed RANS modeling for wave conditions very similar to the experiments of Baldock (2006), and extended the numerical experiments to a wider range of equivalent beach slopes and wave conditions. The RANS model shows very close agreement with the patterns of long wave propagation observed in the laboratory. Notably, a long wave crest generated around the breakpoint radiates shoreward, propagates through the surf zone, and reflects with minimal dissipation and without breaking. Conversely, the reflected component of the incident bound long wave is much weaker, and in some cases almost zero. Their model runs for a range of beach slopes are illustrated in Fig. 12. Here, the data shown is for a location just offshore of the breakpoint where the incident bound wave is close to the maximum amplitude. Added arrows indicate the position of the reflected component of the incident bound wave. This outgoing negative pulse lags the incident short wave group by 8–10s but is almost non-existent in some cases and always much smaller than the incident bound wave at the same location, even though the final part of the beach profile around the shoreline was
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Fig. 9. Reversibility of short wave–long wave energy transfers during spilling breaking of a focusing-defocusing transient wave packet. From Rapp and Melville (1990). The wave group is focusing from kc(x − xb) = − 22 toward kc(x − xb) = 0, and thereafter defocusing. Reproduced with permission of the Royal Society.
Fig. 10. Reversibility of short wave–long wave energy transfers during plunging breaking of a transient wave packet. From Rapp and Melville (1990). The wave group is focusing from kc(x − xb) = − 22 toward kc(x − xb) = 0, and thereafter defocusing. Reproduced with permission of the Royal Society.
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the same in all cases. This implies a strong reduction in the amplitude of the incident bound wave at some location after short wave breaking, which is consistent with the collated experimental data discussed above. It is noted again that the short waves break in an intermediate water depth. 4. Discussion and surf beat similarity It is clear from the work of Schaffer (1993) and Battjes et al. (2004) that the relative beach slope is very important in controlling the amplitude of the incident forced wave, such that short wave groups over steep slopes behave similarly to long wave groups over mild slopes. The magnitude of the breakpoint excursion (closely equivalent to the relative beach slope) is also important in controlling the effectiveness of the Symonds et al. (1982) breakpoint forcing model (Baldock and Huntley, 2002). Baldock and Huntley (2002) also proposed that the relative water depth at the breakpoint was also important in determining the behaviour of the incident forced long wave after breaking. They proposed that if the short waves are shallow water waves before or at the breakpoint, then the forced long wave will be released as it propagates shoreward, consistent
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with the observations of Elgar and Guza (1985), Elgar et al. (1992) and Herbers et al. (1995) and others. Conversely, if the short waves are not shallow water waves at the breakpoint or in the surf zone, then the forced wave may reduce in amplitude along with the reduction in radiation stress (short wave) forcing after breaking. This may involve a non-linear energy transfer back to the short waves, as suggested by van Dongeren et al. (1996), Henderson et al. (2006) and Thomson et al. (2006). This latter mechanism would also appear to require that the forced waves are not released as free waves, since energy transfers among free waves are very weak (Phillips, 1977, §3.6). However, the reduction in the amplitude of the incident forced wave inside the surf zone could equally be regarded as reduction in the response of the system as the forcing is reduced, as proposed by Longuet-Higgins and Stewart (1962), and as observed in the data of Rapp and Melville (1990). In this interpretation, the forced wave and short waves cannot be viewed separately, but both are part of the same non-linear dynamic system, i.e. the forced wave appears from a linear analysis of a slowly varying nonlinear system. Longuet-Higgins and Stewart (1960) also derived an energy equation for the interaction between the primary free waves, see also Phillips (1977). In this energy equation, the relevant work done is given by the term Sxx
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which is the rate at which the convergence of the ambient flow (∂U/∂x) does work against the radiation stress Sxx. Here, U represents a free longer period wave or current, and Sxx is the radiation stress from another free shorter period wave. This work appears as additional wave energy, which are the forced waves (both super- and sub-harmonics). This additional wave energy can be lost during the breaking of the shorter free waves, when it is no longer available to be fed back to the longer period free waves (Longuet-Higgins and Stewart, 1964). It is important to note here that U represents the velocity field of a free wave, not the velocity field for a forced wave generated by the interaction process or the radiation stress gradients. Phillips (1977, §3.6) also gives an analogous
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(m) Fig. 11. Snapshot of spatial structure of short wave envelope (solid line and right scale) and long wave (dashed line and left scale) for a transient wave group. In a) the crest of the envelope is just seaward of the breakpoint. In b) breaking has commenced and about half the short wave energy has been lost. Note different scales. From Baldock (2006) Reproduced with permission of the Royal Society.
Fig. 12. Wave group and long wave elevation just offshore of the short wave breakpoint for a transient wave group in Lara et al. (2011). Thin solid line: total free surface elevation; thick solid line: long wave elevation (right scale). a) slope= 1:20, b) equivalent slope =1:25, c) equivalent slope =1:50. After Lara et al. (2011); added arrows indicate the offshore propagating long wave trough corresponding to the reflected component of the incident bound long wave. Reproduced with permission of the Royal Society.
284
T.E. Baldock / Coastal Engineering 60 (2012) 276–285
equation for the work done on the longer period free waves (or current) by the shorter period waves, where the relevant term is U
∂Sxx ∂x
ð2Þ
The author's interpretation is that again U represents the velocity field in a free longer period wave, or mean current, that interacts with the short wave, not the velocity field in the forced wave that results from the interaction. However, Schaffer (1993), and others (e.g. van Dongeren et al., 1996; Battjes et al., 2004; Henderson et al., 2006) interpret this term differently, taking U to represent the velocity field of the forced long wave and the product to represent the work done on the forced long wave. Hence, van Dongeren et al. (1996) and Battjes et al. (2004) find that the incident short wave groups do work on both the incident forced long waves (bound wave) and on the outgoing radiated long waves, and the latter are universally regarded as free waves. The outgoing free long wave amplitude therefore was predicted to oscillate as it propagates offshore, and showed good comparisons with data obtained from a linear reflection analysis which showed this same oscillation. It is not clear to the author why an outgoing free wave should exchange energy with incident free short wave groups, since net energy transfers during free wave interactions are regarded as being very weak (Phillips, 1977, §3.6). It is relevant to note that Van Dongeren et al. (2007) found that the oscillations in the outgoing free long wave observed by Battjes et al. (2004) are at least in part spurious, and result from assumptions and simplifications in the reflection analysis. If this is the case, it is not clear if the energy exchange that results from applying Eq. (2) to the forced wave, and which shows the same oscillations, is also correct. 4.1. Surf beat similarity The relative water depth at the breakpoint can be parameterised in terms of the deep water short wave steepness (Baldock and O'Hare, 2004). Conditions where the short waves are in shallow water at the breakpoint can be written conveniently in terms of the short wave deep water wavelength, Lo, and the wave height offshore of the breakpoint, Ho, as: Ho ≤0:016γ Lo
ð3Þ
where γ is the breaker index. For γ= 0.8, this requires that Ho/Lo b 0.013, typical of most field data which is obtained under relatively mild wave conditions and the laboratory data presented by Janssen et al. (2003) and Battjes et al. (2004). However, storm waves of period 10 s and height greater than 2 m do not satisfy the shallow water condition at the breakpoint, for example. Hence, under storm conditions, steep short waves,
particularly the larger sets, will break before true shallow water conditions are reached. Therefore, since both relative beach slope (Battjes et al., 2004) and wave steepness appear important for surf beat generation (Baldock and Huntley, 2002), it may be possible to develop a surf beat similarity parameter that indicates the type of surf beat likely to dominate in different conditions. The author tentatively suggests a parameter of the form:
ξsurfbeat
h ¼ x ωlow
rffiffiffisffiffiffiffiffiffiffi g Hos h Los
ð4Þ
where h is the depth in the shoaling zone, ωlow is the long wave frequency and Hos and Los refer to the short wave conditions. This parameter is a product of the normalised beach slope for the long waves (Battjes et al., 2004) and the (square root of the) short wave steepness. The latter parameter is inspired by the surf similarity parameter (Battjes, 1974). Table 1 shows values of ξsurfbeat for a range of typical beach conditions and the data sets used in Baldock et al. (2000) and Battjes et al. (2004). Also shown is the normalised beach slope and its classification into mild and steep slope regimes, together with the approximate value of khb at the short wave breakpoint, assuming hb =Ho/γ, with γ= 0.8. Shallow water conditions for the short waves at the breakpoint correspond approximately to khb b 0.3. Storm waves typically have khb >0.3. For mild slope regimes, the bound wave is strongly amplified during propagation into shallow water (Battjes et al., 2004), and for mild wave conditions (low short wave steepness) the bound wave satisfies shallow water condition before the breakpoint is reached. Hence, the surf beat arises predominantly from the progressive release of the incident forced long wave during shoaling and breakpoint forcing is weak on the assumption that this is linearly proportional to wave height (Symonds et al., 1982). For a mild slope regime and steep wave conditions, so that short wave breaking occurs before shallow water conditions are reached, the bound wave may decay to a smaller amplitude inside surf zone and breakpoint forcing is again weak, so surf beat can be minimal. Breakpoint forcing is weak in the mild slope regime since the forcing (breaking) region becomes large in comparison to the wavelength of the long wave (Baldock and Huntley, 2002). For a steep slope regime and mild short waves, both bound wave amplification and breakpoint forcing are weak, so surf beat is again minimal. For a steep slope regime and steep waves, there is insufficient time for progressive release of the bound wave, the incident forced wave may decay inside the surf zone, but breakpoint surf beat is strong and dominant inside the surf zone. A complicated pattern thus emerges; in essence, long groups of swell waves forcing long waves over mild slopes have low ξsurfbeat and generate surf beat that arises from the incident forced wave; storm waves (short wave groups) forcing long waves over steep slopes have high ξsurfbeat and generate breakpoint forced long waves and a nodal structure that differs from the classical standing wave pattern
Table 1 ξsurfbeat values for a range of typical beach slopes and incident wave conditions. The definition of the normalised beach slope as a mild or steep slope regime follows values suggested by Battjes et al. (2004). 1,2Wave conditions in experiments reported by Battjes et al. (2004) and Baldock et al. (2000). β
flow (Hz)
Tshort (s)
Ho (m)
βnorm
Ho/Lo
Slope
khb at breakpoint
ξsurfbeat
Conditions
0.01 0.01 0.01 0.014 0.01 0.01 0.01 0.1 0.1 0.02 0.1 0.1 0.02
0.2 0.014 0.02 0.04 0.005 0.005 0.01 0.5 0.25 0.005 0.1 0.1 0.0083
3.44 10 7 3.44 15 15 7 1 1.67 12 1.67 1 10
0.11 1 3 0.11 1 6 1 0.1 0.1 6 0.1 0.1 5
0.06 0.20 0.08 0.31 0.58 0.23 0.29 0.18 0.36 0.47 0.91 0.91 0.8
0.006 0.006 0.039 0.006 0.003 0.017 0.013 0.064 0.023 0.027 0.023 0.064 0.032
Mild Steep Mild Steep Steep Steep Steep Mild Steep Steep Steep Steep Steep
0.22 0.22 0.58 0.22 0.15 0.37 0.32 0.77 0.44 0.47 0.44 0.77 0.52
0.00 0.02 0.02 0.02 0.03 0.03 0.03 0.05 0.06 0.08 0.14 0.23 0.14
Battjes1 Field Field Battjes1 Field Field Field Baldock2 Baldock2 US Pacific NW coast, Baldock2 Baldock2 Gold Coast, Australia
T.E. Baldock / Coastal Engineering 60 (2012) 276–285
(see Baldock and Huntley, 2002). It is also of interest that many natural beach slopes usually regarded as mild slopes are in fact in the steep slope regime for the long waves. Similarly, laboratory experiments over “steep” beaches with short wave groups can actually represent mild slope regime conditions for the forced waves (e.g. Fig. 2), and as pointed out by Schaffer (1993), short short-wave groups over steep slopes behave similarly to long short-wave groups over mild slopes. 5. Conclusions The usual assumption that incident bound long waves are released at short wave breaking should not be attributed to Longuet-Higgins and Stewart (1962). In fact, they stated that the forced wave could be reflected before short wave breaking or that the amplitude might reduce after short wave breaking, which implies that the bound wave release is not related to the onset of short wave breaking. This interpretation is consistent with many laboratory data sets and field observations that show strong dissipation of forced harmonics immediately after short wave breaking commences. Finally, the type of surf beat generated by incident short wave groups on a beach is dependent on both the shoaling regime (Battjes et al., 2004) and the short wave steepness. A tentative surf beat similarity parameter is suggested that distinguishes these regimes. Acknowledgements Permission to publish the figures is gratefully acknowledged. The author thanks Lara et al. for permission to use Fig. 15 prior to publication of their paper. References Baldock, T.E., 2006. Long wave generation by the shoaling and breaking of transient wave groups on a beach. Proceedings of the Royal Society A—Mathematical Physical and Engineering Sciences 462, 1853–1876. Baldock, T.E., Huntley, D.A., 2002. Long-wave forcing by the breaking of random gravity waves on a beach. Proceedings of the Royal Society of London, A. 458, 2177–2201. Baldock, T.E., O'Hare, T.J., 2004. Energy transfer and dissipation during surf beat conditions. Proc. 29th Int. Conf. Coastal Engineering. World Scientific. Baldock, T.E., Huntley, D.A., Bird, P.A.D., O'Hare, T., Bullock, G.N., 2000. Breakpoint generated surf beat induced by bichromatic wave groups. Coastal Engineering 39 (2–4), 213–242. Battjes, J.A., 1974. Surf similarity. Proceedings of the 14th International Conference on Coastal Engineering American Society of Civil Engineers, New York, pp. 466–480. Battjes, J.A., Bakkenes, H.J., Janssen, T.T., van Dongeren, A.R., 2004. Shoaling of subharmonic gravity waves. Journal of Geophysical Research-Oceans 109 (C2). Dong, G., et al., 2009a. Experimental study of long wave generation on sloping bottoms. Coastal Engineering 56 (1), 82–89.
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Dong, G.H., Ma, X.Z., Xu, J.W., Ma, Y.X., Wang, G., 2009b. Experimental study of the transformation of bound long waves over a mild slope with ambient currents. Coastal Engineering 56 (10), 1035–1042. Elgar, S., Guza, R.T., 1985. Observations of bispectra of shoaling surface gravity-waves. Journal of Fluid Mechanics 161, 425–448. Elgar, S., Herbers, T.H.C., Okihiro, M., Oltmanshay, J., Guza, R.T., 1992. Observations of infragravity waves. Journal of Geophysical Research-Oceans 97 (C10), 15573–15577. Freilich, M.H., Guza, R.T., 1984. Nonlinear effects on shoaling surface gravity-waves. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, 311(1515), pp. 1–41. Henderson, S.M., Bowen, A.J., 2002. Observations of surf beat forcing and dissipation. Journal of Geophysical Research-Oceans 107 (C11). Henderson, S.M., Guza, R.T., Elgar, S., Herbers, T.H.C., Bowen, A.J., 2006. Nonlinear generation and loss of infragravity wave energy. Journal of Geophysical Research-Oceans 111 (C12). Herbers, T.H.C., Elgar, S., Guza, R.T., 1995. Generation and propagation of infragravity waves. Journal of Geophysical Research-Oceans 100 (C12), 24863–24872. Janssen, T.T., Battjes, J.A., van Dongeren, A.R., 2003. Long waves induced by short-wave groups over a sloping bottom. Journal of Geophysical Research-Oceans 108 (C8). Janssen, T.T., Herbers, H.C., Battjes, J.A., 2006. Generalized evolution equations for nonlinear surface gravity waves over two-dimensional topography. Journal of Fluid Mechanics 552, 393–418. Lara, J.L., Ruju, A., Losada, I.J., 2011. Rans modelling of long waves induced by a transient wave group on a beach. Proceedings of the Royal Society A 467, 1215–1242. Longuet-Higgins, M.S., Stewart, R.W., 1960. Changes in the form of short gravity waves on long waves and tidal currents. Journal of Fluid Mechanics 8, 565–583. Longuet-Higgins, M.S., Stewart, R.W., 1962. Radiation stress and mass transport in gravity waves, with application to `surf beats’. Journal of Fluid Mechanics 13, 481–504. Longuet-Higgins, M.S., Stewart, R.W., 1964. Radiation stresses in water waves; a physical discussion with applications. Deep Sea Research 11, 529–562. Nielsen, P., Baldock, T.E., 2010. I-shaped surf beat understood in terms of transient forced long waves. Coastal Engineering 57 (1), 71–73. Phillips, O.M., 1977. The dynamics of the upper ocean. Cambridge monographs on mechanics and applied mathematics. Cambridge University Press, Cambridge ; New York, p. viii. 336 pp. Rapp, R.J., Melville, W.K., 1990. Laboratory measurements of deep-water breaking waves. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences 331 (1622), 735-&. Roelvink, J.A., Stive, M.J.F., 1989. Bar-generating cross-shore flow mechanisms on a beach. Journal of Geophysical Research 94 (C4), 4785–4800. Ruessink, B.G., 1998. The temporal and spatial variability of infragravity energy in a barred nearshore zone. Continental Shelf Research 18 (6), 585–605. Ruessink, B.G., Kleinhans, M.G., van den Beukel, P.G.L., 1998. Observations of swash under highly dissipative conditions. Journal of Geophysical Research-Oceans 103 (C2), 3111–3118. Schaffer, H.A., 1993. Infragravity waves induced by short-wave groups. Journal of Fluid Mechanics 247, 551–588. Symonds, G., Huntley, D.A., Bowen, A.J., 1982. Two-dimensional surf beat—long-wave generation by a time-varying breakpoint. Journal of Geophysical Research-Oceans and Atmospheres 87 (NC1), 492–498. Thomson, J., Elgar, S., Raubenheimer, B., Herbers, T.H.C., Guza, R.T., 2006. Tidal modulation of infragravity waves via nonlinear energy losses in the surfzone. Geophysical Research Letters 33 (5). van Dongeren, A.R., Svendsen, I.A., Sancho, F.E., 1996. Generation of infragravity waves. In: Edge, B. (Ed.), Proceedings of the 25th International Conference on Coastal Engineering. Am. Soc. of Civ. Eng, Reston, Va, pp. 1335–1348. van Dongeren, A., et al., 2007. Shoaling and shoreline dissipation of low-frequency waves. Journal of Geophysical Research-Oceans 112 (C2). Yuen, H.C., Lake, B.M., 1982. Non-linear dynamics of deep-water gravity-waves. Advances in Applied Mechanics 22, 67–229.
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Dissipation of incident forced long waves in the surf zone—Implications for the concept of “bound” wave release at short wave breaking T.E. Baldock School of Civil Engineering, University of Queensland, St Lucia, QLD 4072, Australia
a r t i c l e
i n f o
Article history: Received 8 July 2011 Received in revised form 13 October 2011 Accepted 1 November 2011 Available online 30 November 2011 Keywords: Surf beat Long waves Bound wave release Breaking Long wave dissipation Surf beat similarity
a b s t r a c t A review of laboratory data sets on surf beat is presented, with a focus on the dissipation of long wave energy in the surf zone. It is frequently assumed that incident forced long waves, or “bound” long waves, are released from short wave groups when the short waves break, subsequently propagating to the shore as a free wave. Free long waves may additionally be generated by the moving short wave breakpoint. Convincing evidence of the release of forced long waves as a result of short wave breaking is lacking, while there appears to be strong evidence to the contrary from a range of recent laboratory experiments. The data from the laboratory experiments are also consistent with field observations of strong nearshore dissipation of long waves. These data are also consistent with Longuet-Higgins and Stewart (1962), who suggest that the forced long wave may reduce in amplitude following short wave breaking, not that it might be released as a free wave. In contrast, forced long waves can be progressively “released” from the groups when the short waves are in shallow water, since these conditions correspond to those where the forced long wave satisfies the free wave dispersion relationship. This frequently occurs prior to short wave breaking for mild wave conditions, but here it is shown that these conditions are not usually satisfied at the short wave breakpoint for storm conditions. Energy transfers between free and forced waves are also discussed with regard the data. A surf beat similarity parameter that incorporates both relative beach slope and short wave steepness is suggested, which distinguishes between different long wave forcing regimes inside the surf zone. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Wave groups approaching the shore force associated long waves, commonly termed bound long waves as they propagate with the wave groups rather than at the free wave celerity. During shoaling, a range of non-linear triad interactions occur between pairs of short wave harmonics and the forced long wave harmonic. These interactions can be resonant or non-resonant and have been widely studied, commencing with the work of Biésel and continued since by LonguetHiggins and Stewart (1960; 1962), Phillips (1977), Freilich and Guza (1984), Herbers et al. (1995), Elgar and Guza (1985), Elgar et al. (1992), Schaffer (1993), Janssen et al. (2003), Battjes et al. (2004) and others. Collectively, these describe the evolution of the non-linear system from a non-resonant state in deep water through to a system where the departure from resonance varies during shoaling, with corresponding variations in the growth rate and phase relationship of the forced wave away from the Longuet-Higgins and Stewart (1962) solution (Battjes et al., 2004; Janssen et al., 2003). Nielsen and Baldock (2010) extended the particular solution of Longuet-Higgins and Stewart (1962) for shallow water waves to include free waves that are generated and propagate away from the group during short wave
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shoaling, leading to similar phase differences from the LonguetHiggins and Stewart (1962) solution. Collectively, low frequency waves, whatever their origin or mode are also termed infragravity waves. It is widely assumed that the bound long waves are released during short wave breaking, subsequently propagating to the shore as a free wave, where they are reflected (see reviews in Baldock and Huntley, 2002; Battjes et al., 2004; Janssen et al., 2003). The emphasis is on the short wave breaking process being the cause or trigger for the release of the bound waves. Statements asserting this release of the forced wave at short wave breaking are often unattributed with no citation of convincing evidence, or loosely attributed to Longuet-Higgins and Stewart (1962), and a physical mechanism for this release by short wave breaking is not usually stated. However, Longuet-Higgins and Stewart (1962) do not mention bound wave release on short wave breaking and a different interpretation of their statements is that the forced long wave will in fact decay in amplitude following short wave breaking. This is in agreement with a number of recent laboratory data sets that show strong dissipation of forced waves following short wave breaking (Baldock and O'Hare, 2004; Baldock et al., 2000; Battjes et al., 2004; Dong et al., 2009a; Dong et al., 2009b) and recent RANS numerical model results (Lara et al., 2011). Similarly, it is also consistent with field data showing strong nearshore dissipation of long waves (Henderson and Bowen, 2002; Henderson et al., 2006;
T.E. Baldock / Coastal Engineering 60 (2012) 276–285
2. Existing data Baldock and O'Hare (2004) presented a number of laboratory data sets that illustrated the strong dissipation of incident forced waves once short wave breaking commenced. Two different forms of wave
interaction were studied. Firstly, the forced waves (sub- and superharmonics) generated by a long wave-short wave interaction, which was the original application of the theory developed by LonguetHiggins and Stewart (1960). Secondly, the forced waves generated by broad-banded bichromatic short wave groups. In these experiments, the waves propagate over horizontal section of the wave flume, followed by shoaling on a 1:10 sloping beach which commences at x = −8 m, with the still water shoreline at x = 0 m. Second order wave generation was used and surface elevation measurements were made at closely spaced intervals along the full length of the flume, together with the shoreline motion. Several examples are illustrated here, where, despite the steep absolute beach slope, the forced waves propagate in a mild slope regime, as later defined by Battjes et al. (2004), and which represents near resonant conditions and weak breakpoint forcing. Fig. 1 shows the cross-shore variation of the amplitudes of sub- and super- harmonics forced by the interaction of a low steepness free long wave (f= 0.2 Hz) and a steep free short wave (f= 1 Hz). Here, the beach slope and wave amplitudes yield a narrow surf zone and Iribarren numbers corresponding to spilling and plunging waves, with rapid dissipation of short wave energy. The short wave breakpoint varies
a) Sub-harmonics, f=0.78Hz
Amplitude (mm)
15
10
5
0 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
(m)
b) Super-harmonics, f=1.17Hz 15
Amplitude (mm)
Thomson et al., 2006), which is not due to frictional losses but is suggested to occur through non-linear interaction that transfer energy from the infragravity waves back to swell and sea. However, these data sets have not been used to discuss the implications of this strong dissipation for the concept of bound wave release by short wave breaking. This is the primary purpose of the present paper. Following Longuet-Higgins and Stewart (1960), general solutions for a range of interacting wave modes were identified, with later extensions which describe a range of dynamical processes and forms of resonant interactions. Yuen and Lake (1982) provide an extensive general review, with the theoretical work of Freilich and Guza (1984), Elgar and Guza (1985); Herbers et al. (1995), Battjes et al. (2004) and Janssen et al. (2006) particularly applicable for shoaling waves and for conditions seaward of the breakpoint. Comprehensive analyses of forced wave propagation under field conditions prior to short wave breaking are given by Elgar et al. (1992) and Herbers et al. (1995) and references therein. Elgar and Guza (1985) showed that the phase of the long waves often evolves from the deep water bound wave biphase of 180° toward 90° as the short waves shoal, consistent with bound waves becoming free prior to short wave breaking, or with the infragravity band becoming dominated by freely propagating waves as the wave field shoals prior to breaking. Similarly, Herbers et al. (1995) show that as waves shoal, the relationship between infragravity waves and swell changes from one appropriate for forced, bound waves in deep water to freely propagating waves in shallow water. The triad interactions between swell waves and the infragravity waves become close to resonant in shallow water and require distances of order of a few wavelengths to transfer significant energy between the harmonic components (Freilich and Guza, 1984; Herbers et al., 1995). After the short waves break, there is then typically a reduction in correlation between the short waves and incident long waves inside the surf zone (Roelvink and Stive, 1989), because the short wave groupiness is usually reduced significantly in a random wave conditions. Once the short waves start to break, additional free long waves can be generated by the oscillating short breakpoint (Schaffer, 1993; Symonds et al., 1982), which again has been identified in the laboratory (e.g. Baldock and Huntley, 2002). The relative importance of the different regimes of forced wave amplification during shoaling has been clearly identified by Schaffer (1993), Battjes et al. (2004) and Janssen et al. (2008), and is primarily related to the relative beach slope for the long waves. Although field data sets exist as noted above, these are more difficult to interpret and they lack the spatial resolution of the laboratory data sets. Thus, this paper focuses on reviewing laboratory data sets, and particularly those that allow direct identification of the forced wave behavior because there was either minimal long wave generation at the breakpoint, or little shoreline reflection of the long waves, or the incident and reflected long waves were separated by careful analysis. The purpose of this paper is to discuss these recent laboratory data with respect to the dissipation of long wave energy in the surf zone, and if this is consistent with the concept of bound wave release as a result of short wave breaking. The paper is organized as follows. Section 2 provides examples of existing data sets that illustrate the cross-shore variation of the incident forced wave amplitude. A discussion of the data in view of the original ideas presented in Longuet-Higgins and Stewart (1962) is given in Section 3. Section 4 provides a further discussion of energy transfers and proposes a similarity parameter that accounts for both the relative beach slope and the short wave steepness when considering the different surf beat forcing mechanisms. Final conclusions follow.
277
10
5
0 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
(m) Fig. 1. Cross-shore variation in sub- and super-harmonic amplitudes forced by a long waveshort-wave interaction. From Baldock and O'Hare (2004). The short waves break in the region x = − 0.75 m to x = − 1.5 m. Beach slope 1:10. a) Sub-harmonic, fsub = 0.586 Hz. b) Super-harmonic, fsup = 1.95 Hz.
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T.E. Baldock / Coastal Engineering 60 (2012) 276–285
depending on the short wave amplitude. In each case, rapid dissipation of the forced harmonic occurs immediately following short wave breaking, which occurs in water depths between 0.05 m and 0.15 m. Fig. 2
a) Sub-harmonic, fsub=0.586Hz. 12
Amplitude (mm)
10 8
6 4 2 0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
(m)
b) Super-harmonic, fsup=1.95Hz. 12
Amplitude (mm)
10 8 6 4 2 0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
(m)
c) 35
Amplitude (mm)
30 25 20 15 10 5 0 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
(m) Fig. 2. Cross-shore variation in sub- and super-harmonic amplitudes for a broad-banded bichromatic wave group, B1060A. − □ − data, —— Incident bound wave (2nd order quasi- local solution). c. Cross-shore variation in primary and bound wave amplitudes for a broad-banded wave group, B6055C. ⋄ f1 = 0.879 Hz, □ f2 = 0.341 Hz, ∇ fsub = 0.537 Hz, × fsup = 1.22 Hz, —— fsub (radiated). From Baldock and O'Hare (2004). The short waves commence breaking at approximately x = −1.05 m. Beach slope 1:10.
shows further examples, in these cases from bichromatic wave groups. Again, very rapid dissipation of the forced harmonics occurs following short wave breaking at x ∼ −1 m. Fig. 2c shows a very significant change in the division of energy (strictly surface variance) between the short waves and the forced harmonics prior to short wave breaking, usually interpreted as an energy transfer. Indeed, the energy transfer is sufficient to significantly reduce the short wave amplitude during shoaling, with the energy transferred almost exclusively from the higher frequency short wave component. In fact, the energy transfer is such that the forced harmonics grow larger than the short wave component that forces them. There is no evidence of a transfer of long wave energy back to the short waves during the dissipation of the long waves, but this would be very hard to detect once the short waves themselves are breaking. These data were obtained without the need to separate the total wave motion into free and forced waves and incident and radiated harmonics in the surf zone since there is little reflected wave energy at the frequencies of interest and, consequently, the incident wave amplitude is easily identified. The lack of reflection is a direct consequence of the strong dissipation of the primary short waves and the forced harmonics prior to the waves reaching the shore. However, for different conditions, and in general, a standing long wave pattern is typically formed, which may comprise of an incident forced, or bound, long wave, an incident breakpoint generated free wave, and a reflected free wave. In these cases, it is more difficult to determine the fate of the incident bound long wave after short wave breaking. Baldock et al. (2000) and Baldock and Huntley (2002) assumed a priori that the incident bound long wave was strongly dissipated following short wave breaking, and then calculated the resulting cross-shore standing wave structure of the surf beat, and found good agreement. However, this provides only indirect evidence of dissipation of the forced wave. As noted by Battjes et al. (2004) and subsequently demonstrated by van Dongeren et al. (2007), it is possible that the dissipation occurs because the long waves break in very shallow water. However, in order for this to explain the data illustrated above, this would require that long wave breaking occurred when the long wave height to water depth ratio, γL = HL/d, was about equal to 0.2, instead of the usual value of γ ≈ 0.8–1. Consequently, this is not a likely mechanism for the long wave dissipation in these cases. Battjes et al. (2004) present data that show a similar dissipation of the incident long waves and again the dissipation commences following short wave breaking and at a location where γL ≈ 0.1–0.3. Examples are shown in Figs. 3 and 4. On this double-barred beach, some dissipation occurs after the initial short wave breakpoint, with further dissipation occurring after renewed short wave breaking on the inner bar. However, it should be noted that other frequency bands from the same data set (Battjes et al., 2004) do not show the same degree of change in the incident long wave amplitude following initial breaking, e.g. Fig. 4, but show quite strong dissipation after the second breakpoint. Van Dongeren et al. (2007) also showed that some of these long waves break in very shallow water (h ≈ 0.02 m), which limits the amplitude of the outgoing wave shown in these figures. For these data, the short waves are in relatively shallow water at the short wave breakpoint, kh≈ 0.22, and this point is discussed further below. However, it is quite clear that in some experiments the incident wave forced is progressively released from the short waves, as frequently suggested. A clear example is shown by forced (outside the breakpoint) long waves propagating unchanged through the surf zone and reflecting from the shoreline (e.g. Janssen et al., 2003). However, this release is not necessarily related to short wave breaking. An example is a data set from van Dongeren et al. (2007), where the incident forced wave again reduces rapidly in amplitude shortly after short wave breaking, but subsequently “re-shoals” and then breaks in very shallow water (Fig. 5). Further inspection of the data in Fig. 5 indicates that kh≈0.5 at the short wave break point (h=0.2 m); for these waves shallow water conditions are reached at h≈0.07 m. Dissipation of short wave energy is initially
T.E. Baldock / Coastal Engineering 60 (2012) 276–285
Fig. 3. See Fig. 4 for caption.
rapid in the region 0.15 mb hb 0.2 m, but this then ceases and the long wave “re-shoals”. The amplification of the long wave in the region 0.5 m b h b 0.15 m is greater than that indicated by Green’ law for free wave shoaling. In fact, assuming shoaling as a near resonant shallow water forced wave (proportional to h− 5/2) and accounting for the reduction in the short wave forcing terms (proportional to H2), gives an expected amplification factor of about 2.2, only slightly greater than the observed amplification factor of 1.7. Thus, it seems possible that this long wave remains a forced wave across most of the surf zone until shallow water conditions for the short waves are reached at h ≈0.07 m. Conversely, as noted in Section 2, data from Elgar and Guza (1985) and Herbers et al. (1995) is consistent with bound waves becoming free prior to short wave breaking, or with the infragravity band becoming dominated by freely propagating waves as the wave field shoals prior to breaking. Dong et al. (2009a) present data from an experiment which excluded reflection of the long waves and therefore also enables direct observation of the incident bound long wave amplitudes. The experiments were performed over a sloping bottom, followed by a shelf with a constant water depth of 0.1 m. Hence, no, or only, partial breaking of the short waves occurred, not the near full dissipation of short wave energy that usually
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occurs across a sloping beach. Fig. 6 shows examples of the cross-shore variation of the incident bound long wave for cases where the short waves do not break (cases BS310, BS610, BS330 and BS630) and where they do break. For the non-breaking short wave groups, the incident bound long waves shoal to the edge of the shelf break (x =22 m) and then remain constant in amplitude since the forcing is then constant in the constant depth region. For the remaining cases, where the short waves do break (before or at the shelf break at x= 22 m), the difference is striking, and again the incident long waves decay rapidly in amplitude after short wave breaking occurs. Note again that the depth on the shelf was 0.1 m and the incident short wave heights were about 0.1 m in height, so only partial breaking occurs and some forcing remains possible in the constant depth region. Dong et al. (2009b) present data from similar experiments to those described above, but this time with and without adverse and favourable currents, and example data are illustrated in Figs. 7 and 8. The opposing current increases the short wave steepness and promotes short wave breaking where it would otherwise not occur. Fully modulated wave groups and weakly modulated groups were used, leading to a variation in the rate of change of wave height for the two data sets and therefore the rate of energy dissipation shortly after short wave breaking. When significant amounts of short wave energy is dissipated (fully modulated wave groups, Fig. 8), the incident bound wave amplitude decays rapidly, whereas when the short wave energy dissipation is lower (weakly modulated wave groups, Fig. 7), the incident bound wave amplitude reduces more slowly or stays constant as the short waves continue to shoal. There is also strong evidence from field data for rapid dissipation of incident infragravity energy in the surf zone. Henderson and Bowen (2002) observed strong dissipation of surf beat energy inside the surf zone and modeled this in terms of dissipation by the bed shear stress. However, the roughness required to match model-data predictions was very large. Henderson et al. (2006) and Thomson et al. (2006) analysed and modeled further field data and showed a strong transfer of energy away from infragravity bands in the inner surf zone. This could not be attributed to either bottom friction or breaking of the long waves, and was attributed to non-linear interactions between the forced long wave and the sea and swell. Since the interaction of free long waves with sea and swell is very weak (Phillips, 1977, §3.6), this suggests that the infragravity waves remained coupled to the sea and swell in the surf zone, enabling a reverse energy transfer to occur. This reverse energy transfer is well known during the focusing of transient wave groups (Rapp and Melville, 1990). Fig. 9 shows an example from Rapp and Melville (1990) where weak spilling breaking occurs during focusing and Fig. 10 shows an example with strong breaking at the focal point. The non-breaking (not shown) and spilling breaker cases show a near completely reversible energy transfer to the forced harmonics, where the energy in the low frequency harmonics increases during focusing (focal point is at x − xb = 0), and then decreases again during defocusing, with the energy returned to the short waves. Similarly, the majority of the low frequency energy is transferred back away from the low frequency band even in the case of a deep water plunging breaker (Fig. 10). For example, comparing the low frequency energy at kc(x−xb) = ±15 shows that the gain in low frequency energy at x − xb = 0 is reversed during defocusing, even after breaking. 3. Interpretation of Longuet-Higgins and Stewart (1962)
Fig. 4. Example of incident (triangles) and outgoing (dots) long waves forced by random waves (Battjes et al., 2004). The initial short wave breakpoint is at x ≈ 21 m (h ≈ 0.15 m, outer bar), followed by a second breakpoint at 25 m (h = ≈ 0.1 m, inner bar). From Battjes et al. (2004). Reproduced/modified by permission of American Geophysical Union.
Many papers discuss the fate of the incident bound long wave after breaking, and typically state an explanation similar to the following: “…the effect of short wave breaking on these non-linear interactions is not understood, ….it has been hypothesised that the
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However, this is not stated in Longuet-Higgins and Stewart (1962). The relevant quotation is: “Since long waves are more readily reflected by non-uniformities in the transmitting medium than are shorter waves, it is reasonable to suppose that at some depth the long wave associated with the mass transport undergoes partial reflexion while the shorter waves are allowed to pass on and be destroyed in shallower water”. This implies reflection of any long waves before short wave breaking. There is no implication that the separation of the short wave and forced wave requires short wave breaking to occur. In Lara et al. (2011), such reflection of the long wave before breaking is visible at beach slope discontinuities (their Fig. 9). Further, in relation to the observations that the long waves radiated offshore from the surf zone were observed to be (and are still typically observed to be) linearly proportional to the incident swell height (as opposed to showing a quadratic dependency), Longuet-Higgins and Stewart (1962) stated: Fig. 5. Cross-shore variation of Hrms for a bichromatic wave group. Dash-dot line, total; dashed line, short waves; solid line, long waves. From van Dongeren et al. (2007). Reproduce/modified by permission of American Geophysical Union.
incident bound waves are not destroyed in the surf zone, but are released as free waves (Longuet-Higgins and Stewart, 1962). …. After reflection at the shoreline, the long waves travel in a seaward direction and may either propagate into deep water or remain refractively trapped to the coast ”.
“It would seem that the amplitude of the long waves is proportional to the square of the envelope of the incoming swell. On the other hand, if breaking has taken place before the point of reflexion, the higher waves at least will have been reduced in amplitude, and so one expects in fact a law of variation rather weaker than a2.” This implies that the bound wave amplitude is determined by the local forcing that continues after short wave breaking (which is governed by the short wave amplitude and groupiness remaining after short wave
Fig. 6. Examples of cross-shore variation of incident bound long wave amplitudes from Dong et al. (2009a). Circles, slope = 1:40, triangles, slope = 1:10. Curves are asymptotic solutions of Longuet-Higgins and Stewart (1962). Depth is constant for x > 22 m. From Dong et al. (2009a). The short waves do not break for cases BS310, BS610, BS330 and BS630, but do so for the other cases shown.
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a
3.0
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Fig. 7. Examples of cross-shore variation of incident bound long wave amplitudes from Dong et al. (2009b). Data is for a weakly modulated bichromatic wave group propagating up a 1:60 slope with and without an adverse current. For the adverse current, short wave breaking occurred at x = 15 m and x = 12 m in a) and b), respectively. Note that Uo = 0 and −12 cm/s in b). Depth is constant for x > 21 m. From Dong et al. (2009b).
breaking), and therefore that the bound wave is not released at breaking, i.e. it remains a forced wave. Such conditions were investigated by Schaffer (1993). This is consistent with most of the data shown above, which shows rapid reduction in either the incident long waves after short wave breaking, or a much decreased amplitude for the long waves reflected from the shore. Further, on a very mildly sloping beach, Ruessink (1998) observed free long wave amplitudes to be proportional to the square root of the offshore wave height. Also consistent with this interpretation, Ruessink et al. (1998) observed a shoreward decrease in the bound and total infragravity energy at locations landward of the breakpoint and Battjes et al. (2004) likewise observed a decrease in the total incident low frequency wave amplitude after breaking. In addition to the continued forcing postulated in Fig. 5, a further example of this continuing forcing can be found in Baldock (2006), which considered the propagation of transient wave groups over a 1:10 sloping beach and through breaking. Since the wave group is transient, the motion can be followed nearly to the shore before reflection complicates the picture (Fig. 11). In this example, the long wave surface slopes are inversely related to the spatial gradients of the wave envelope, as expected for a forced wave, even well inside the breakpoint. Note also that there is no significant lag between the wave envelope and bound long wave. The bound wave amplitude continues to increase after short wave breaking in this case, i.e. the
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Fig. 8. Examples of cross-shore variation of incident bound long wave amplitudes from Dong et al. (2009b). Data is for a strongly modulated bichromatic wave group propagating up a 1:60 slope with and without an adverse current. For the adverse current, short wave breaking occurred at x=20 m in b). Depth is constant for x>21 m. From Dong et al. (2009b).
long wave remains forced. This is because even though the short waves reduce in amplitude, the radiation stress gradient, dSxx/dx (estimated from the spatial wave envelope), remains approximately constant as the envelope continues to focus and shorten. Since the depth is also reducing, the response of the system increases and the bound wave continues to grow despite short wave breaking. However, this is not usual and typically the radiation stress gradients will reduce rapidly in the surf zone as the groupiness is largely destroyed or even reversed. Lara et al. (2011) have recently performed RANS modeling for wave conditions very similar to the experiments of Baldock (2006), and extended the numerical experiments to a wider range of equivalent beach slopes and wave conditions. The RANS model shows very close agreement with the patterns of long wave propagation observed in the laboratory. Notably, a long wave crest generated around the breakpoint radiates shoreward, propagates through the surf zone, and reflects with minimal dissipation and without breaking. Conversely, the reflected component of the incident bound long wave is much weaker, and in some cases almost zero. Their model runs for a range of beach slopes are illustrated in Fig. 12. Here, the data shown is for a location just offshore of the breakpoint where the incident bound wave is close to the maximum amplitude. Added arrows indicate the position of the reflected component of the incident bound wave. This outgoing negative pulse lags the incident short wave group by 8–10s but is almost non-existent in some cases and always much smaller than the incident bound wave at the same location, even though the final part of the beach profile around the shoreline was
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Fig. 9. Reversibility of short wave–long wave energy transfers during spilling breaking of a focusing-defocusing transient wave packet. From Rapp and Melville (1990). The wave group is focusing from kc(x − xb) = − 22 toward kc(x − xb) = 0, and thereafter defocusing. Reproduced with permission of the Royal Society.
Fig. 10. Reversibility of short wave–long wave energy transfers during plunging breaking of a transient wave packet. From Rapp and Melville (1990). The wave group is focusing from kc(x − xb) = − 22 toward kc(x − xb) = 0, and thereafter defocusing. Reproduced with permission of the Royal Society.
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the same in all cases. This implies a strong reduction in the amplitude of the incident bound wave at some location after short wave breaking, which is consistent with the collated experimental data discussed above. It is noted again that the short waves break in an intermediate water depth. 4. Discussion and surf beat similarity It is clear from the work of Schaffer (1993) and Battjes et al. (2004) that the relative beach slope is very important in controlling the amplitude of the incident forced wave, such that short wave groups over steep slopes behave similarly to long wave groups over mild slopes. The magnitude of the breakpoint excursion (closely equivalent to the relative beach slope) is also important in controlling the effectiveness of the Symonds et al. (1982) breakpoint forcing model (Baldock and Huntley, 2002). Baldock and Huntley (2002) also proposed that the relative water depth at the breakpoint was also important in determining the behaviour of the incident forced long wave after breaking. They proposed that if the short waves are shallow water waves before or at the breakpoint, then the forced long wave will be released as it propagates shoreward, consistent
a
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Elevation (mm)
Elevation (mm)
40
with the observations of Elgar and Guza (1985), Elgar et al. (1992) and Herbers et al. (1995) and others. Conversely, if the short waves are not shallow water waves at the breakpoint or in the surf zone, then the forced wave may reduce in amplitude along with the reduction in radiation stress (short wave) forcing after breaking. This may involve a non-linear energy transfer back to the short waves, as suggested by van Dongeren et al. (1996), Henderson et al. (2006) and Thomson et al. (2006). This latter mechanism would also appear to require that the forced waves are not released as free waves, since energy transfers among free waves are very weak (Phillips, 1977, §3.6). However, the reduction in the amplitude of the incident forced wave inside the surf zone could equally be regarded as reduction in the response of the system as the forcing is reduced, as proposed by Longuet-Higgins and Stewart (1962), and as observed in the data of Rapp and Melville (1990). In this interpretation, the forced wave and short waves cannot be viewed separately, but both are part of the same non-linear dynamic system, i.e. the forced wave appears from a linear analysis of a slowly varying nonlinear system. Longuet-Higgins and Stewart (1960) also derived an energy equation for the interaction between the primary free waves, see also Phillips (1977). In this energy equation, the relevant work done is given by the term Sxx
5
-5
283
∂U ∂x
ð1Þ
which is the rate at which the convergence of the ambient flow (∂U/∂x) does work against the radiation stress Sxx. Here, U represents a free longer period wave or current, and Sxx is the radiation stress from another free shorter period wave. This work appears as additional wave energy, which are the forced waves (both super- and sub-harmonics). This additional wave energy can be lost during the breaking of the shorter free waves, when it is no longer available to be fed back to the longer period free waves (Longuet-Higgins and Stewart, 1964). It is important to note here that U represents the velocity field of a free wave, not the velocity field for a forced wave generated by the interaction process or the radiation stress gradients. Phillips (1977, §3.6) also gives an analogous
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(m) Fig. 11. Snapshot of spatial structure of short wave envelope (solid line and right scale) and long wave (dashed line and left scale) for a transient wave group. In a) the crest of the envelope is just seaward of the breakpoint. In b) breaking has commenced and about half the short wave energy has been lost. Note different scales. From Baldock (2006) Reproduced with permission of the Royal Society.
Fig. 12. Wave group and long wave elevation just offshore of the short wave breakpoint for a transient wave group in Lara et al. (2011). Thin solid line: total free surface elevation; thick solid line: long wave elevation (right scale). a) slope= 1:20, b) equivalent slope =1:25, c) equivalent slope =1:50. After Lara et al. (2011); added arrows indicate the offshore propagating long wave trough corresponding to the reflected component of the incident bound long wave. Reproduced with permission of the Royal Society.
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equation for the work done on the longer period free waves (or current) by the shorter period waves, where the relevant term is U
∂Sxx ∂x
ð2Þ
The author's interpretation is that again U represents the velocity field in a free longer period wave, or mean current, that interacts with the short wave, not the velocity field in the forced wave that results from the interaction. However, Schaffer (1993), and others (e.g. van Dongeren et al., 1996; Battjes et al., 2004; Henderson et al., 2006) interpret this term differently, taking U to represent the velocity field of the forced long wave and the product to represent the work done on the forced long wave. Hence, van Dongeren et al. (1996) and Battjes et al. (2004) find that the incident short wave groups do work on both the incident forced long waves (bound wave) and on the outgoing radiated long waves, and the latter are universally regarded as free waves. The outgoing free long wave amplitude therefore was predicted to oscillate as it propagates offshore, and showed good comparisons with data obtained from a linear reflection analysis which showed this same oscillation. It is not clear to the author why an outgoing free wave should exchange energy with incident free short wave groups, since net energy transfers during free wave interactions are regarded as being very weak (Phillips, 1977, §3.6). It is relevant to note that Van Dongeren et al. (2007) found that the oscillations in the outgoing free long wave observed by Battjes et al. (2004) are at least in part spurious, and result from assumptions and simplifications in the reflection analysis. If this is the case, it is not clear if the energy exchange that results from applying Eq. (2) to the forced wave, and which shows the same oscillations, is also correct. 4.1. Surf beat similarity The relative water depth at the breakpoint can be parameterised in terms of the deep water short wave steepness (Baldock and O'Hare, 2004). Conditions where the short waves are in shallow water at the breakpoint can be written conveniently in terms of the short wave deep water wavelength, Lo, and the wave height offshore of the breakpoint, Ho, as: Ho ≤0:016γ Lo
ð3Þ
where γ is the breaker index. For γ= 0.8, this requires that Ho/Lo b 0.013, typical of most field data which is obtained under relatively mild wave conditions and the laboratory data presented by Janssen et al. (2003) and Battjes et al. (2004). However, storm waves of period 10 s and height greater than 2 m do not satisfy the shallow water condition at the breakpoint, for example. Hence, under storm conditions, steep short waves,
particularly the larger sets, will break before true shallow water conditions are reached. Therefore, since both relative beach slope (Battjes et al., 2004) and wave steepness appear important for surf beat generation (Baldock and Huntley, 2002), it may be possible to develop a surf beat similarity parameter that indicates the type of surf beat likely to dominate in different conditions. The author tentatively suggests a parameter of the form:
ξsurfbeat
h ¼ x ωlow
rffiffiffisffiffiffiffiffiffiffi g Hos h Los
ð4Þ
where h is the depth in the shoaling zone, ωlow is the long wave frequency and Hos and Los refer to the short wave conditions. This parameter is a product of the normalised beach slope for the long waves (Battjes et al., 2004) and the (square root of the) short wave steepness. The latter parameter is inspired by the surf similarity parameter (Battjes, 1974). Table 1 shows values of ξsurfbeat for a range of typical beach conditions and the data sets used in Baldock et al. (2000) and Battjes et al. (2004). Also shown is the normalised beach slope and its classification into mild and steep slope regimes, together with the approximate value of khb at the short wave breakpoint, assuming hb =Ho/γ, with γ= 0.8. Shallow water conditions for the short waves at the breakpoint correspond approximately to khb b 0.3. Storm waves typically have khb >0.3. For mild slope regimes, the bound wave is strongly amplified during propagation into shallow water (Battjes et al., 2004), and for mild wave conditions (low short wave steepness) the bound wave satisfies shallow water condition before the breakpoint is reached. Hence, the surf beat arises predominantly from the progressive release of the incident forced long wave during shoaling and breakpoint forcing is weak on the assumption that this is linearly proportional to wave height (Symonds et al., 1982). For a mild slope regime and steep wave conditions, so that short wave breaking occurs before shallow water conditions are reached, the bound wave may decay to a smaller amplitude inside surf zone and breakpoint forcing is again weak, so surf beat can be minimal. Breakpoint forcing is weak in the mild slope regime since the forcing (breaking) region becomes large in comparison to the wavelength of the long wave (Baldock and Huntley, 2002). For a steep slope regime and mild short waves, both bound wave amplification and breakpoint forcing are weak, so surf beat is again minimal. For a steep slope regime and steep waves, there is insufficient time for progressive release of the bound wave, the incident forced wave may decay inside the surf zone, but breakpoint surf beat is strong and dominant inside the surf zone. A complicated pattern thus emerges; in essence, long groups of swell waves forcing long waves over mild slopes have low ξsurfbeat and generate surf beat that arises from the incident forced wave; storm waves (short wave groups) forcing long waves over steep slopes have high ξsurfbeat and generate breakpoint forced long waves and a nodal structure that differs from the classical standing wave pattern
Table 1 ξsurfbeat values for a range of typical beach slopes and incident wave conditions. The definition of the normalised beach slope as a mild or steep slope regime follows values suggested by Battjes et al. (2004). 1,2Wave conditions in experiments reported by Battjes et al. (2004) and Baldock et al. (2000). β
flow (Hz)
Tshort (s)
Ho (m)
βnorm
Ho/Lo
Slope
khb at breakpoint
ξsurfbeat
Conditions
0.01 0.01 0.01 0.014 0.01 0.01 0.01 0.1 0.1 0.02 0.1 0.1 0.02
0.2 0.014 0.02 0.04 0.005 0.005 0.01 0.5 0.25 0.005 0.1 0.1 0.0083
3.44 10 7 3.44 15 15 7 1 1.67 12 1.67 1 10
0.11 1 3 0.11 1 6 1 0.1 0.1 6 0.1 0.1 5
0.06 0.20 0.08 0.31 0.58 0.23 0.29 0.18 0.36 0.47 0.91 0.91 0.8
0.006 0.006 0.039 0.006 0.003 0.017 0.013 0.064 0.023 0.027 0.023 0.064 0.032
Mild Steep Mild Steep Steep Steep Steep Mild Steep Steep Steep Steep Steep
0.22 0.22 0.58 0.22 0.15 0.37 0.32 0.77 0.44 0.47 0.44 0.77 0.52
0.00 0.02 0.02 0.02 0.03 0.03 0.03 0.05 0.06 0.08 0.14 0.23 0.14
Battjes1 Field Field Battjes1 Field Field Field Baldock2 Baldock2 US Pacific NW coast, Baldock2 Baldock2 Gold Coast, Australia
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(see Baldock and Huntley, 2002). It is also of interest that many natural beach slopes usually regarded as mild slopes are in fact in the steep slope regime for the long waves. Similarly, laboratory experiments over “steep” beaches with short wave groups can actually represent mild slope regime conditions for the forced waves (e.g. Fig. 2), and as pointed out by Schaffer (1993), short short-wave groups over steep slopes behave similarly to long short-wave groups over mild slopes. 5. Conclusions The usual assumption that incident bound long waves are released at short wave breaking should not be attributed to Longuet-Higgins and Stewart (1962). In fact, they stated that the forced wave could be reflected before short wave breaking or that the amplitude might reduce after short wave breaking, which implies that the bound wave release is not related to the onset of short wave breaking. This interpretation is consistent with many laboratory data sets and field observations that show strong dissipation of forced harmonics immediately after short wave breaking commences. Finally, the type of surf beat generated by incident short wave groups on a beach is dependent on both the shoaling regime (Battjes et al., 2004) and the short wave steepness. A tentative surf beat similarity parameter is suggested that distinguishes these regimes. Acknowledgements Permission to publish the figures is gratefully acknowledged. The author thanks Lara et al. for permission to use Fig. 15 prior to publication of their paper. References Baldock, T.E., 2006. Long wave generation by the shoaling and breaking of transient wave groups on a beach. Proceedings of the Royal Society A—Mathematical Physical and Engineering Sciences 462, 1853–1876. Baldock, T.E., Huntley, D.A., 2002. Long-wave forcing by the breaking of random gravity waves on a beach. Proceedings of the Royal Society of London, A. 458, 2177–2201. Baldock, T.E., O'Hare, T.J., 2004. Energy transfer and dissipation during surf beat conditions. Proc. 29th Int. Conf. Coastal Engineering. World Scientific. Baldock, T.E., Huntley, D.A., Bird, P.A.D., O'Hare, T., Bullock, G.N., 2000. Breakpoint generated surf beat induced by bichromatic wave groups. Coastal Engineering 39 (2–4), 213–242. Battjes, J.A., 1974. Surf similarity. Proceedings of the 14th International Conference on Coastal Engineering American Society of Civil Engineers, New York, pp. 466–480. Battjes, J.A., Bakkenes, H.J., Janssen, T.T., van Dongeren, A.R., 2004. Shoaling of subharmonic gravity waves. Journal of Geophysical Research-Oceans 109 (C2). Dong, G., et al., 2009a. Experimental study of long wave generation on sloping bottoms. Coastal Engineering 56 (1), 82–89.
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Dong, G.H., Ma, X.Z., Xu, J.W., Ma, Y.X., Wang, G., 2009b. Experimental study of the transformation of bound long waves over a mild slope with ambient currents. Coastal Engineering 56 (10), 1035–1042. Elgar, S., Guza, R.T., 1985. Observations of bispectra of shoaling surface gravity-waves. Journal of Fluid Mechanics 161, 425–448. Elgar, S., Herbers, T.H.C., Okihiro, M., Oltmanshay, J., Guza, R.T., 1992. Observations of infragravity waves. Journal of Geophysical Research-Oceans 97 (C10), 15573–15577. Freilich, M.H., Guza, R.T., 1984. Nonlinear effects on shoaling surface gravity-waves. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, 311(1515), pp. 1–41. Henderson, S.M., Bowen, A.J., 2002. Observations of surf beat forcing and dissipation. Journal of Geophysical Research-Oceans 107 (C11). Henderson, S.M., Guza, R.T., Elgar, S., Herbers, T.H.C., Bowen, A.J., 2006. Nonlinear generation and loss of infragravity wave energy. Journal of Geophysical Research-Oceans 111 (C12). Herbers, T.H.C., Elgar, S., Guza, R.T., 1995. Generation and propagation of infragravity waves. Journal of Geophysical Research-Oceans 100 (C12), 24863–24872. Janssen, T.T., Battjes, J.A., van Dongeren, A.R., 2003. Long waves induced by short-wave groups over a sloping bottom. Journal of Geophysical Research-Oceans 108 (C8). Janssen, T.T., Herbers, H.C., Battjes, J.A., 2006. Generalized evolution equations for nonlinear surface gravity waves over two-dimensional topography. Journal of Fluid Mechanics 552, 393–418. Lara, J.L., Ruju, A., Losada, I.J., 2011. Rans modelling of long waves induced by a transient wave group on a beach. Proceedings of the Royal Society A 467, 1215–1242. Longuet-Higgins, M.S., Stewart, R.W., 1960. Changes in the form of short gravity waves on long waves and tidal currents. Journal of Fluid Mechanics 8, 565–583. Longuet-Higgins, M.S., Stewart, R.W., 1962. Radiation stress and mass transport in gravity waves, with application to `surf beats’. Journal of Fluid Mechanics 13, 481–504. Longuet-Higgins, M.S., Stewart, R.W., 1964. Radiation stresses in water waves; a physical discussion with applications. Deep Sea Research 11, 529–562. Nielsen, P., Baldock, T.E., 2010. I-shaped surf beat understood in terms of transient forced long waves. Coastal Engineering 57 (1), 71–73. Phillips, O.M., 1977. The dynamics of the upper ocean. Cambridge monographs on mechanics and applied mathematics. Cambridge University Press, Cambridge ; New York, p. viii. 336 pp. Rapp, R.J., Melville, W.K., 1990. Laboratory measurements of deep-water breaking waves. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences 331 (1622), 735-&. Roelvink, J.A., Stive, M.J.F., 1989. Bar-generating cross-shore flow mechanisms on a beach. Journal of Geophysical Research 94 (C4), 4785–4800. Ruessink, B.G., 1998. The temporal and spatial variability of infragravity energy in a barred nearshore zone. Continental Shelf Research 18 (6), 585–605. Ruessink, B.G., Kleinhans, M.G., van den Beukel, P.G.L., 1998. Observations of swash under highly dissipative conditions. Journal of Geophysical Research-Oceans 103 (C2), 3111–3118. Schaffer, H.A., 1993. Infragravity waves induced by short-wave groups. Journal of Fluid Mechanics 247, 551–588. Symonds, G., Huntley, D.A., Bowen, A.J., 1982. Two-dimensional surf beat—long-wave generation by a time-varying breakpoint. Journal of Geophysical Research-Oceans and Atmospheres 87 (NC1), 492–498. Thomson, J., Elgar, S., Raubenheimer, B., Herbers, T.H.C., Guza, R.T., 2006. Tidal modulation of infragravity waves via nonlinear energy losses in the surfzone. Geophysical Research Letters 33 (5). van Dongeren, A.R., Svendsen, I.A., Sancho, F.E., 1996. Generation of infragravity waves. In: Edge, B. (Ed.), Proceedings of the 25th International Conference on Coastal Engineering. Am. Soc. of Civ. Eng, Reston, Va, pp. 1335–1348. van Dongeren, A., et al., 2007. Shoaling and shoreline dissipation of low-frequency waves. Journal of Geophysical Research-Oceans 112 (C2). Yuen, H.C., Lake, B.M., 1982. Non-linear dynamics of deep-water gravity-waves. Advances in Applied Mechanics 22, 67–229.