Linking reduced breaking crest speeds to unsteady nonlinear water wave group behavior
Michael Banner, Xavier Barthelemy, Francesco Fedele, Michael Allis, Alvise Benetazzo, Frederic Dias, William Peirson
LLinking reduced breaking crest speeds to unsteady nonlinear water wave group behavior
M.L. Banner *, X. Barthelemy , F. Fedele , M. Allis , A. Benetazzo , F. Dias and W.L. Peirson . 1. School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia 2. Water Research Laboratory, School of Civil and Environmental Engineering, The University of New South Wales, Manly Vale, NSW 2093, Australia 3. School of Civil and Environmental Engineering, and School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. 4. Institute of Marine Sciences, National Research Council (CNR-ISMAR), Venice, Italy. 5. UCD School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland. Abstract
1 Observations show that maximally-steep breaking water wave crest speeds are much slower than 2 expected. We report a wave-crest slowdown mechanism generic to unsteady propagating deep water 3 wave groups. Our fully nonlinear computations show that just prior to reaching its maximum height, each 4 wave crest slows down significantly and either breaks at this reduced speed, or accelerates forward 5 unbroken. This finding is validated in our extensive laboratory and field observations. This behavior 6 appears to be generic to unsteady dispersive wave groups in other natural systems. 7 8
Introduction
9 Nonlinear wave groups occur in a wide range of natural systems, exhibiting complex behaviors 10 especially in focal zones where there is rapid wave energy concentration and possible ‘wave-breaking’. 11 The incompletely-understood interplay between dispersion, directionality and nonlinearity presents a 12 significant knowledge gap presently beyond analytical treatment. Here, we investigate maximally-steep, 13 deep-water wave group behaviour, but the findings appear relevant to dispersive nonlinear wave motion 14 in many other natural systems. 15 In the open ocean, wind forcing generates waves that can steepen and break conspicuously as 16 whitecaps, strongly affecting fundamental air-sea exchanges, including greenhouse gases. This has 17 stimulated recent interest in measuring whitecap properties spectrally. While accurately measuring 18 wavelengths of individual breakers is difficult, measuring whitecap speeds can provide a less direct but 19 more convenient method: since a whitecap remains attached to the underlying wave crest during active 20 breaking. The dispersion relation from Stokes’ classical deep water wave theory discussed below (Stokes 21 [1]) conventionally provides the wavelength from the observed whitecap speed (Phillips [2]). 22 Stokes’ theory was developed for a steady , uniform train of two-dimensional (2D) non-linear, deep-23 water waves of small-to-intermediate mean steepness ak (=2π×amplitude/wavelength), for which the 24 intrinsic wave speed c increases slowly with ak : 25 c = c [1+1/2 ( ak ) +higher order terms in ( ak )] (1) 26 where c is the wave speed for linear (infinitesimally-steep) waves. Extending (1) computationally to 27 maximally-steep, steady waves (Longuet-Higgins [3]), c approaches 1.1 c Thus, increased wave 28 steepness has long been associated with higher wave speeds. 29 Natural wind-waves comprise a spectrum of modes interacting on different scales, producing evolving 30 wave-group patterns rather than steady, uniform wavetrains (Longuet-Higgins [4]). Here we investigate 31 the ‘dominant’ waves, i.e. those with the largest spectral amplitudes after filtration of higher-wavenumber 32 components. Within a group, each advancing dominant wave gradually changes its height and shape, 33 characterized by slow forward and backward leaning of the crests (Tayfun [5]), also transiently becoming 34 the tallest wave. This tallest wave may break, or else decrease in height while advancing unbroken 35 towards the front of the group. 36 In this context, previous deep-water breaking wave laboratory studies (Rapp and Melville [6]; Stansell 37 and McFarlane [7]; Jessup and Phadnis [8]) suggest that breaking-crest speeds are typically O(20%) 38 lower than expected from linear-wave theory, contrary to the expectation from (1) that steeper breaking 39 waves should propagate faster. Understanding this paradoxical crest slowdown behaviour is central to 40 both refining present knowledge on water-wave propagation and dynamics, and optimal implementation 41 of Phillips’ spectral framework for breaking waves (Phillips [2]; Kleiss and Melville [9]; Gemmrich et al. 42 [10]).
43 Historically, an appreciable literature has developed on non-breaking, focusing, deep-water, nonlinear 44 wave packets. However, only the studies of Johannessen and Swan ([11], [12]) identified crest slowdown 45 at focus, reporting an O(10%) crest-speed slowdown relative to its linear-theory prediction. To understand 46 the underlying physics, the present study investigates how very steep unsteady, non-periodic, deep-water 47 wave groups propagate when frequently-assumed theoretical constraints are relaxed, including steady-48 state, spatially-uniform or slowly-varying, weakly-nonlinear wavetrain behavior. Our goal was to 49 investigate initial breaker speeds, hence it was crucial to track changes, up to the point of breaking 50 initiation, in dominant wave-crest speeds within evolving nonlinear wave groups. 51 Methodology and results
53 No presently-available analytic theory can predict the evolution of fully-nonlinear, deep-water wave 54 groups. Our primary research strategy utilized simulations from a fully-nonlinear, 3D numerical wave 55 code, validated against results from our innovative laboratory and ocean-wave observations. 56 Our simulations were generated using a numerical wave tank (Grilli et al. [13]). This boundary element 57 code simulates fully-nonlinear potential flow theory and is able to model extreme water waves to the 58 point of overturning. A programmable wave paddle produces a specific 2D or 3D chirped wave-group 59 structure comprising a prescribed number of carrier waves with given initial amplitudes, wavenumbers, 60 frequencies and phases. This shapes the spatial and temporal bandwidths characterizing the group 61 structure and its spectrum. For the simulations, including the 2D example below, the paddle followed the 62 displacement-motion equation (3) described in Song and Banner [14], with N=5, 7 and 9. We also 63 investigated corresponding laterally-converging 3D chirped packet cases with 10- and 25-wavelength 64 focal distances. In this study, breaking occurred predominantly as sequential spilling events with 65 occasional local plunging. The complementary wave-basin experiments described below also included 66 comparable bimodal, modulating nonlinear wave packets specified by equation (2) in [14]. The half-67 power bandwidths were O(8) times broader than investigated in [9]. 68 Figure 1(a) shows the complex growth behavior experienced by all dominant wave crests evolving 69 within a representative 2D nonlinear, non-breaking wave group. The initial steepest wave decays and is 70 replaced by the following growing wave, which grows modestly, then slows down and is replaced by the 71 annotated faster-growing crest, which evolves to its maximum height and decays. As each new crest 72 develops, it grows (A-B-C) then slows down and attenuates (C-D), then accelerates (D-E) back to its 73 original speed while advancing towards the front of the group. 74 Figure 1(b) shows the spatial wave profile in greater detail at the evolution times A-E in Figure 1(a). 75 The dominant wave grows asymmetrically, initially leaning forward as it steepens within the group. In the 76 absence of breaking, the steepest wave advances leaning forward, relaxing back to symmetry near its 77 maximum height (the focal point), then leans backwards past the maximum elevation. Forward-leaning 78 crests are accompanied by backward leaning troughs, and vice-versa. This leaning is a generic feature of 79 each crest in natural, unsteadily-evolving dispersive nonlinear water wave groups (Tayfun [5]). 80 Relative to the speed of a classical (symmetrical) Stokes wave, significant crest (and trough) speed 81 changes accompany the leaning, measured by tracking the (horizontal) speed of a given wave-crest profile 82 in space and time. The generic crest-speed slowdown is identified in Figure 1(c) by the steeper slope of 83 the displacement-time curve between B and D relative to the indicated linear wave trajectory (speed c ), 84 where c is the speed of the spectral peak determined from the computed wave-packet dispersion relation. 85 The actual speed reduction relative to c is 18%. This lasts about one wave period, with a spatial extent of 86 about one wavelength. Figure 1(d) shows a typical trajectory when crest speed is plotted against local 87 crest steepness s c = a c k c , where a c is the time-dependent crest height above mean-water level and the 88 corresponding local wavenumber k c is defined as divided by the local zero-crossing separation spanning 89 the given crest. Introducing s c was necessary to describe the complexity of unsteady nonlinear wave crest 90 behaviour, and was easily computed for Stokes waves for the crest-speed comparison shown. 91 The significant departure of the crest speed versus crest steepness trajectory for waves in unsteady 92 wave groups compared with the classical Stokes for steady wavetrain prediction underpins the central 93 findings in this study. In this example, the maximum crest steepness marginally precedes the slowest 94 crest speed, with the trajectory looping counter-clockwise about this point. This trajectory is not generic, 95 since other simulated cases and the experimental curve of Figure 2(b) below showed clockwise looping. 96 Further studies are needed to explain this effect. Also, as seen in Figure 1(d), the asymmetry of the 97 dominant wave shape near its maximum steepness results in different crest speeds for growing and 98 decaying crests of the same steepness. Note that the local peak in crest speed between A and B at s c ≈ x/L ~4, t/T ~13.5) when the detected crest location jumps abruptly from the receding crest to the newly-101 developing crest. 102 The above discussion was for 2D waves, but laterally-focused (3D) wave fronts in both our simulations 103 and wave-basin investigation (described below) show similar leaning and crest-slowdown behavior, with 104 the subsequent breaker-crest speed initiated at ~0.8 c
105 Relative to classical ocean-wave speeds, our model results for the speeds of the left-hand and right-106 hand zero-crossings spanning the tallest wave shows that their average remains close to the linear wave 107 speed c (Fig. 1(c)), with modest local fluctuations of (+7% to -1%). Hence, aside from the strong 108 unsteady leaning crest and trough motions, the waves propagate largely as expected from Stokes theory. 109
110 111 Figure 1. (a) space-time evolution diagram of a non-breaking 2D chirped wave group, moving toward the 112 right, showing the decay of the initial tallest crest, growth of the following tallest crest and complex 113 transitions of other developing crests. Wave properties at annotated times A-E are shown in panels (b), (c) 114 and (d). T and L are reference carrier-wave period and wavelength scales. (b) tallest crest shapes at 115 evolution times A-E, showing crest transition from forward-leaning through symmetry to backward-116 leaning (c) horizontal location of the tallest crest (solid line) versus time. The steeper slope between B 117 and D shows the crest-speed reduction relative to c (dotted line). Horizontal locations versus time of the 118 two adjacent zero-crossings (long-short dashed lines) are also shown. (d) trajectory of the corresponding 119 tallest crest speed c , normalized by c , against local crest steepness s c defined in the text. Stokes theory 120 prediction (1) is shown in terms of s c (dashed line) for comparison. The apparent crest-speed surge at 121 s c ≈ Breaking onset and speed
124 If the tallest wave in the group proceeds to break rather than recur, our simulations found that breaking 125 onset occurs when this wave attains maximum steepness and close to its minimum crest speed. This can 126 certainly explain why initial breaking wave crest speeds are observed to be O(80%) of the linear carrier-127 wave speed (Rapp and Melville [6]; Stansell and McFarlane [7]; Jessup and Phadnis [8]). This behavior 128 was found in all our simulations and verified in our laboratory measurements (see Figure 2(b)). 129
Is the crest slowdown a nonlinear effect?
131 Insight on this key question is available from previous linear and weakly-nonlinear theory. For 132 uniform, deep-water, linear gravity wavetrains, the carrier wave speed c follows from the dispersion 133 relation = ( gk ) and c = / k . However, narrow-band wave groups are characterized by non-134 uniformity in both space and time. Correct to O ( ), a local frequency can be defined (Chu and Mei [15]) 135 as 136 = ( gk ) - a xx /( ak ) (2) 137 where v is a characteristic spectral bandwidth, =dc g / dk , c g = d / dk is the linear group velocity and a ( x , t ) 138 is the wavetrain envelope that satisfies the linear Schrödinger equation (Mei [16]). The associated local 139 phase speed is given approximately by 140 c ≈ c - a xx /( ak ). (3) 141 Equations (2), (3) and the associated relationships above show that c varies along the group and in time. 142 Since <0, c attains its lowest value at the envelope maximum, where the largest crest occurs ( a xx <0). 143 The other crests and troughs in the group also experience similar local speed variations. 144 Furthermore, for dispersive, weakly-nonlinear unsteady wave groups, we find that the slowdown effect 145 due to dispersion is counterbalanced by the increase in phase speed due to nonlinearity [Equation (1)], 146 limiting the phase-velocity slowdown within the group (Fedele [17]). 147 Our focus on breaking-crest slowdown for large wave steepness approaching breaking onset is beyond 148 conventional analysis methodologies. We now validate our fully-nonlinear numerical simulation findings 149 on wave-crest slowdown against laboratory and open-ocean measurements. 150 151
Wave basin measurements
152 Complementary experiments were performed in a 27m x 7.75m wave basin with 0.55m water depth. 153 Wave groups were generated at one end of the basin by a computer-controlled wave-generator comprising 154 13 bottom-cantilevered, flexible-plate segments. Lateral focusing was achieved by suitably setting the 155 phase of each segment (Dalrymple [18]). A 95% absorbing beach minimized end reflections. Heights of 156 evolving wave groups matching the simulations were measured to within ±0.5mm by a traversable in-line 157 array of nine wave-wire probes spanning one wavelength. c was calculated using linear theory from the 158 spectrally-weighted wave frequency of the wave probe closest to the wave-generator. 159 Identified wave crests were tracked between the wave probe signals, their motion interpolated using 160 cubic splining and their crest speeds extracted. An overhead one-megapixel videocamera imaged the 161 breaking crests at 100 Hz. The imagery, corrected for lens and mounting distortion, was transformed onto 162 a regular grid, sequential leading-edge location and lateral extent data were extracted for each breaker and 163 their speeds determined. 164 Figure 2(a) shows surface profiles measured at evolution times A-E, with breaking initiation near C, for 165 a modulating 5-wave, bi-modal breaking case. Figure 2(b) shows its crest speed trajectory. Also shown is 166 the ensemble-mean trajectory for spilling-breaker speeds in the measured ensemble of 240 modulational 167 and chirped 2D and 3D cases. The measurement resolution enabled resolving the crest leaning and 168 slowing at the maximum surface elevation (C). Crest-speed oscillations observed for smaller-steepness 169 waves (e.g. at B), are the same crest-leanings, but occurring earlier as the crest moves through the wave 170 group. This figure confirms the reduced speeds of crests preceding breaking onset and the accompanying 171 generic breaker slowdown. 172 173 174 175 176 177 178 179 180 181 Figure 2. (a) measured surface profiles of a 5-wave, bi-modal wave packet, at times A-E, with breaking 182 initiation at C. (b) corresponding trajectory of normalized crest speed c/c of the tallest wave, against local 183 crest steepness s c . The ensemble-mean breaker speed trajectory is shown by the dot-dash line. The dashed 184 line shows Stokes’ prediction. Reference wave scales were L =1.09m, T =0.836 sec. 185 186 Open ocean observations
187 Our
Wave Acquisition Stereo System (WASS) was deployed at the
Acqua Alta oceanographic tower 16 188 km offshore from Venice in 17m water depth (Fedele et al. [19]; Benetazzo et al. [20]).
WASS cameras 189 were 2.5m apart, 12.5m above sea level at 70 o depression angle, providing a trapezoidal field-of-view 190 with sides increasing from 30m to 100m over a 100m fetch. The mean windspeed was 9.6 ms -1 with a 110 191 km fetch. The uni-modal wave spectrum had a significant wave height H s =1.09m and dominant period 192 T p =4.59s. Most observed crests were very steep, with sporadic spilling breaking. We describe results 193 using 21,000 frames captured at 10 Hz. 194 The speeds c of crests reaching maximum local steepness within the imaged area were estimated using 195 a crest-tracking methodology, as in the wave-basin measurements. The data were filtered above 1.5 Hz to 196 remove short riding waves. Sub-pixeling reduced quantization errors in estimating the local 3D crest 197 position from the surface-displacement time series spaced along the wave-propagation direction. The 198 local reference c was calculated from the peak frequency of the short-term Fourier spectrum of a time 199 series of duration D centered at the crest event, using D =120sec as a suitable record length and Doppler-200 corrected for the in-line 0.20 ms -1 mean current. We analyzed 200 dominant local wave crests with 201 elevations >0.3 H s and local crest steepness s c >0.3( s c ) max using the observed ( s c ) max =0.45, and determined 202 ~12,000 evolving crest speeds from a 60-point spatial grid, with 0.5m spacing along the wave-203 propagation direction. 204 205 206 207 208 209 210 211 212 213 Figure 3. Probability density function of normalized crest speed c/c for all crests transitioning through a 214 maximum local crest steepness, from a 35-minute WASS stereo-video sequence from an ocean tower. 215 Note the tall peak at c/c ~ 0.75. Local standard error bounds are indicated. 216 217 Values of D and were chosen so that the empirical probability density function (pdf) of c / c was 218 insensitive to changes in these parameters. Figure 3 shows the pdf, which peaks at close to 0.75 c . Values 219 for c/c > 1.5 (7% of the total ensemble) are outliers with >15% uncertainty in estimating c and crest 220 location. This figure highlights the observed crest slowdown, consistent with the nonlinear simulations 221 and experiments described above. 222 223 Discussion and conclusions
224 Our study provides fundamental new insights into the behavior of chirped, bi-modal and open-ocean 225 unsteady steep, deep-water nonlinear wave groups. We found that as carrier waves reach maximum 226 steepness, their crests decelerate strongly (O(20%)), which results from unsteady crest sloshing modes 227 arising from the complex interplay between nonlinearity and dispersion. This behaviour departs markedly 228 from the speed increase with wave steepness predicted by steady-wavetrain theory. 229 Our findings have significant, broader consequences. For ocean waves, they explain the puzzling 230 (O(20%)) reduced initial speed of breaking-wave crests, central to assimilating whitecap data accurately 231 into sea-state forecast models. Parameterizations of air-sea fluxes of momentum and energy, which 232 depend on the square and cube of the sea-surface velocity, may be modified appreciably. Atmospheric 233 and oceanic internal waves, (Helfrich and Melville [21]), should also experience similar effects to those 234 described here. As noted above, even weakly-nonlinear, unsteady dispersive water-wave groups described 235 by the nonlinear Schrödinger equation (NLSE) (Zakharov [22]) exhibit crest slowdown. The NLSE is 236 commonly used to describe wave phenomena in other natural systems (e.g. geophysical flows (Osborne 237 [23]), nonlinear optics (Kibler et al. [24], amongst others). Exploring implications of the present findings 238 should provide refined insights when the wave-group nonlinearity and bandwidth are beyond the validity 239 of the NLSE. 240 References
243 1. Stokes, G.G., 1847.
Trans. Cambridge Phil. Soc.
8, 441–455. 244 2 . Phillips, O .M., 1 9 8 5 .
J. Fluid Mech. , 1 5 6 , 5 0 5 -5 3 1 . 245 3. Longuet-Higgins, M.S., 1975.
Proc. Roy. Soc.
A 342, 157-174 246 4. Longuet-Higgins, M.S., 1984.
Phil. Trans. R. Soc. A J. Geophys. Res . 91 C6:7743–7752. 248 6. Rapp, R.J. and Melville, W.K., 1990.
Phil. Trans. R. Soc.
A 331, 735–800. 249 7. Stansell, P. and MacFarlane, C., 2002.
J. Phys. Oceanogr.
32, 1269–1283. 250 8. Jessup, A.T. and Phadnis, K.R., 2005.
Meas. Sci. Technol . 16, 1961-1969. 251 9. Kleiss, J.M. and Melville, W.K., 2010.
J. Phys. Oceanogr.
40, 2575–2604 252 10. Gemmrich, J.R., Zappa, C.J., Banner, M.L. and Morison, R.P. 2013.
J. Geophys. Res. Oceans,
253 In press, doi: 10.1002/jgrc.20334 254 11. Johannessen, T.B. and Swan, C., 2001.
Proc. Roy. Soc. Lond. A , 457, 971-1006.
255 12. Johannessen, T.B. and Swan, C., 2003.
Proc. R. Soc. Lond. A , 459, 1021-1052 256 1 3 . Grilli, S., Guyenne, P. and Dias, F., 2 0 0 1 .
Int. J. Num. Methods Fluids , 3 5 , 8 2 9 – 8 6 7 . 257 14. Song, J. and Banner, M.L., 2002.
J. Phys. Oceanogr . 32, 2541–2558
258 15. Chu, V.H. and Mei, C.C., 1970.
J. Fluid Mech ., 41, 873-887 259 16. Mei, C.C., 1983.
The Applied Dynamics of Ocean Surface Waves . Wiley-Interscience, 740pp. 260 17. Fedele, F., 2013. http://arxiv.org/abs/1309.0668 261 18. Dalrymple, R.A., 1989.
J. Hydraulic Res.
27, 23-34. 262 19. Fedele, F., Benetazzo, A., Gallego, G., Shih, P.-C., Yezzi, A., Barbariol and F., Ardhuin, F., 2013. In 263 press, http://dx.doi.org/10.1016/j.ocemod.2013.01.001 264 20. Benetazzo, A., Fedele, F., Gallego, G., Shih, P.C. and Yezzi, A., 2012.
Coastal Engineering , 64, 127-265 138. 266 21. Helfrich, K.R. and Melville, W.K., 2006.
Annu. Rev. Fluid Mech . 38:395–425. 267 22. Zakharov, V. E., 1968.
J. Appl. Mech. Tech. Phys . 9, 190-194 268
23. Osborne, A.R., 2010.
Nonlinear Ocean Waves and the Inverse Scattering Transform.
Academic Press, 269 ISBN: 978-0-12-528629-9, 949pp. 270 24. Kibler, B., Fatome, J., Finot, C., Millot, G., Genty, G., Wetzel B., Akhmediev, N., Dias F. and 271 Dudley, J.M., 2012.
Nature Scientific Reports
Acknowledgements
274 Financial support is gratefully acknowledged for XB, MA, MB and WP from the Australian Research 275 Council through their support of Discovery Projects DP0985602, DP120101701. Financial support for 276 MB is also gratefully acknowledged from the National Ocean Partnership Program, through the U.S. 277 Office of Naval Research (Grant N00014-10-1-0390). The WASS experiment at