doi: 10.1098/rspa.2012.0029 , 2704-2721 first published online 25 April 2012 468 2012 Proc. R. Soc. A Thomas A. A. Adcock, Richard H. Gibbs and Paul H. Taylor directionally spread wave groups on deep water The nonlinear evolution and approximate scaling of References html#ref-list-1 http://rspa.royalsocietypublishing.org/content/468/2145/2704.full. This article cites 34 articles, 8 of which can be accessed free Subject collections (45 articles) wave motion (20 articles) ocean engineering (84 articles) fluid mechanics Articles on similar topics can be found in the following collections Email alerting service here the box at the top right-hand corner of the article or click Receive free email alerts when new articles cite this article - sign up in http://rspa.royalsocietypublishing.org/subscriptions go to: Proc. R. Soc. A To subscribe to on August 19, 2012 rspa.royalsocietypublishing.org Downloaded from
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doi: 10.1098/rspa.2012.0029, 2704-2721 first published online 25 April 2012468 2012 Proc. R. Soc. A
Thomas A. A. Adcock, Richard H. Gibbs and Paul H. Taylor directionally spread wave groups on deep waterThe nonlinear evolution and approximate scaling of
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Proc. R. Soc. A (2012) 468, 2704–2721doi:10.1098/rspa.2012.0029
Published online 25 April 2012
The nonlinear evolution and approximatescaling of directionally spread wave groups
on deep waterBY THOMAS A. A. ADCOCK*, RICHARD H. GIBBS AND PAUL H. TAYLOR
Department of Engineering Science, University of Oxford, Parks Road,Oxford OX1 3PJ, UK
The evolution of steep waves in the open ocean is nonlinear. In narrow-banded butdirectionally spread seas, this nonlinearity does not produce significant extra elevationbut does lead to a large change in the shape of the wave group, causing, relative to linearevolution, contraction in the mean wave direction and lateral expansion. We use thenonlinear Schrödinger equation (NLSE) to derive an approximate analytical relationshipfor these changes in group shape. This shows excellent agreement with the numericalresults both for the NLSE and for the full water wave equations. We also consider theapplication of scaling laws from the NLSE in terms of wave steepness and bandwidthto solutions of the full water wave equations. We investigate these numerically. Whilesome aspects of water wave evolution do not scale, the major changes that a wave groupundergoes as it evolves scale very well.
The dynamics of extreme ocean waves is important to engineers who are designingfor the maritime environment and is of major interest to those investigating‘freak’ or ‘rogue’ waves. Much work has been devoted to predicting the amplitudeof extreme waves and the effect of nonlinear wave physics on estimating this(e.g. Socquet-Juglard et al. 2005; Onorato et al. 2006, 2009; Gramstad & Trulsen2007). Rather less work has gone into investigating the local shape of extremewaves. This is important as it influences the forces on maritime structures and isfundamental to understanding the nonlinear evolution of steep waves.
The starting point for this paper is the expected shape of a large wave event in arandom sea-state assuming linear evolution. Lindgren (1970) and Boccotti (1983)developed the theory which was applied by Tromans et al. (1991), Phillips et al.(1993) and Jonathan & Taylor (1997), demonstrating that, under linear evolution,the shape of the largest waves tends to the auto-correlation function, the so-called‘NewWave’. However, in extreme wave events, the nonlinearity of the evolutionmight be expected to produce significant modifications to this shape. These*Author for correspondence ([email protected]).
Figure 1. Wave groups at focus for (a) linear evolution and (b) fully nonlinear evolution. Wavegroups had a linear amplitude of 10.7 m. Other details are given in §3.
may be investigated using a group which would, under linear evolution, form aNewWave group when all the wave components are in phase, or ‘focused’. Insteadof using a linear model of evolution, we investigate how this shape is modified bynonlinear interactions both analytically and via numerical simulations.
In uni-directional (long-crested) seas, nonlinear changes will cause an increasein the amplitude relative to linear evolution, as well as a contraction of the lengthof the wave group in the mean wave direction (Baldock et al. 1996; Taylor &Vijfvinkel 1998; Adcock & Taylor 2009b), although these changes are inhibitedby finite water depth (Adcock & Yan 2010; Katsardi & Swan 2011).
In the open ocean, waves are not uni-directional (Forristall & Ewans 1998)and this makes a large difference in their nonlinear evolution. The evolution ofdirectionally spread (short-crested) wave groups has been investigated in bothphysical wave tanks (Johannessen & Swan 2001) and numerical wave tanks(Gibbs & Taylor 2005; Gibson & Swan 2007). These studies conclude that,while there is no extra elevation relative to linear amplitude other than throughsimple bound harmonics, there are large changes to the local aspect ratio of thewave group. An example of these changes is shown in figures 1 and 2 (see §3bfor details of these results). This broadening along the crest direction is inagreement with some accounts of encounters with freak waves as ‘walls of water’(Lawton 2001) and with the field observations of Monaldo (2000) and Krogstadet al. (2006).
In this paper, we start by deriving analytical relationships for the nonlinearchanges to the aspect ratio of a localized wave group as it focuses on an otherwisequiescent ocean, using the approach used for uni-directional wave groups inAdcock & Taylor (2009b). We compare these results with numerical modelsand find excellent agreement. In the second half of the paper, we consideran approximate scaling behaviour for deep water waves based on the exactscaling laws of the nonlinear Schrödinger equation (NLSE). This is investigatednumerically. We find that the NLSE (and its conserved quantities) is very usefulfor capturing the dominant nonlinear changes to a wave group as it focuses.
We do not analyse the second-order bound waves in this paper. These make asignificant difference to the size of the wave crest but do not modify the underlyingdynamics of the wave group. The second-order correction to the wave group
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Figure 2. The envelope of a wave group under different forms of evolution. (a) Linear evolution,(b) NLSE and (c) fully nonlinear. The simulations were started 20 periods before linear focus withak = 0.3, sx = 0.0046 m−1 and sy = 0.0073 m−1. Contours are at 1 m intervals.
shown in figure 1b is approximately 2 m. Once the freely propagating waves at anextreme event have been determined, it is straightforward to then calculate thebound waves using the interaction kernel (e.g. Forristall 2000).
2. Analytical results
The hyperbolic two-dimensional NLSE (equation 2.1) is the simplest nonlinearmodel for the evolution of water waves on deep water. This equation has beenused extensively in the analysis of extreme wave events (Zakharov & Shabat 1972;Alber 1978; Janssen 2004; Zakharov & Ostrovsky 2009),
ivuvt
= u0
8k20
v2uvx2
− u0
4k20
v2uvy2
+ u0k20
2|u|2u, (2.1)
where u is the complex wave envelope, and u0 and k0 are the frequency andwavenumber of the carrier wave which is travelling in the x-direction. This may benon-dimensionalized using the transformations T = u0t, X = 2
√2k0x , Y = 2k0y
and U = (ko/√
2)u to give
ivUvT
= v2UvX 2
− v2UvY 2
+ |U |2U . (2.2)
The NLSE does not capture the full dynamics of the nonlinear water waveproblem. Various more complex equations have been derived which includeadditional terms (Davey & Stewartson 1974; Dysthe 1979; Trulsen & Dysthe1996). The limitations of the NLSE are explored in Henderson et al. (1999).These can also be seen in figure 2—relative to linear evolution (figure 2a) theNLSE has captured the overall change in aspect ratio (figure 2b) but not the fullcomplexities of the nonlinear evolution (figure 2c) such as the shift in the positionof the peak (see Lo & Mei 1987; Gibson & Swan 2007). In this section, we use theNLSE to seek an approximate analytical solution for the evolution of a Gaussianwave group on deep water following the approach of Adcock & Taylor (2009b).
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The Gaussian wave group which focuses and then de-focuses is an exact solutionto the linear part of the NLSE, and a simple generalization of the one-dimensionalresult in Kinsman (1965),
U = A√1 + 2iS2
XT√
1 + 2iS2Y T
× Exp[−1
2
(S2
XX 2
1 + 4S4XT 2
+ S2Y Y 2
1 + 4S4Y T 2
)+ i
2
(S4
Y Y 2T1 + 4S4
Y T 2− S4
XX 2T1 + 4S4
XT 2
)].
(2.3)
Here, the linear Gaussian group is described by three parameters: the amplitudemeasure A, which is the actual amplitude of the group when it is at focus (T = 0),and the bandwidths in the mean wave direction and lateral direction SX and SY .The bandwidth as it is used here is a measure of the width of the spectrum whichis closely related to the shape of the NewWave group in space—a group with anarrow bandwidth will give a longer spatial wave group and vice versa.
For the nonlinear problem, we make the assumption that the solution remainsGaussian in form but that the parameters vary slowly in time. We then use theconservation laws of the NLSE to identify the changes in these parameters as thegroup focuses and de-focuses. In this context the two-dimensional NLSE has twoknown (non-trivial) and useful conserved quantities,
I2 =∫∞
−∞
∫∞
−∞|U |2 dX dY (2.4)
and
I4 =∫∞
−∞
∫∞
−∞
(|UX |2 − |UY |2 − 1
2|U |4
)dX dY . (2.5)
We now introduce a nonlinear time scale t. It would be possible to use a differentnonlinear time scale in the X - and Y -directions, but for the analysis we presenthere this would not modify the results so we will keep just one time scale forsimplicity. We substitute equation (2.3) into the conserved quantities to give
I2Gaussian = A2
SXSY(2.6)
and
I4Gaussian = pA2(S2X − S2
Y )2SXSY
− pA4
4SXSY (1 + 4S2X t2)1/2(1 + 4S2
Y t2)1/2. (2.7)
We evaluate these at t = 0 (focus) and ∞ (fully dispersed spatially), and equatethese so as to generate two equations relating the shape of the group at focusto that when it is completely dispersed. This is also equivalent to equatingthe parameters for a group which has undergone linear evolution to one under
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nonlinear evolution, since the parameters describing the Gaussian do not changein the linear model,
A2f
SXf SYf= A2∞
SX∞SY∞(2.8)
andpA2
f (S2Xf − S2
Yf )
2SXf SYf− pA4
f
4SXf SYf= pA2∞(S2
X∞ − S2Y∞)
2SX∞SY∞. (2.9)
This gives us two equations but three parameters (A, SX and SY ). No additionaluseful conserved quantity is known for the two-dimensional NLSE, which onlyhas a finite number of conserved quantities (Sulem & Sulem 1999). We stress theimportance of the word ‘useful’ in the previous sentence. As well as the conservedquantities I2 (energy) and I4 (the Hamiltonian), only two others are known forthe two-dimensional NLSE. These are
I3X =∫∞
−∞
∫∞
−∞i(UXU ∗ − UU ∗
X ) dX dY (2.10)
and
I3Y =∫∞
−∞
∫∞
−∞i(UY U ∗ − UU ∗
Y ) dX dY . (2.11)
In some applications of the NLSE, these are described as momentum equations.There are several reasons why these equations are not useful for constraining
our Gaussian approximation. Firstly, for any perfectly focused wave at focus,the wave is symmetric in space in both directions around the central point of thegroup when the maximum of the envelope is located. When translated into theNLSE envelope, this implies that the U (X , Y , T = 0) is symmetric in space. Fora purely real and symmetric function, I3X for the one-dimensional NLSE andboth I3X and I3Y for the two-dimensional NLSE are identically zero at focus. Soat all other times the relevant I3 forms must also be identically zero.
When we substitute the assumed Gaussian form at focus into the conservedquantities, both I3 versions in two dimensions are again identically zero. Then,using the assumed Gaussian form as a slowly varying model for the nonlinearevolution, we find that both I3 variants are identically zero. This is independentof values for the parameters A, SX and SY , and, most importantly, time t. Soalthough I3X and I3Y are conserved, they yield no usable information.
The I3 forms would only be useful for constraining an approximation if theyhave a non-zero value when the group is most compact; but then the group is notperfectly symmetric and the focus is non-optimal. Possibly the I3 forms mightthen be useful for investigating the degree of non-optimality in focusing via theincorporation of a phase, exp(if(X , Y , T )), but in this work we explicitly assumethat the focusing is optimal.
There is a second reason why we might not expect I3 to yield useful informationfor modelling the nonlinear evolution of a wave group. For arbitrary spatial andtemporal structure within the group, both I2 and I3 are quadratic in the waveamplitude (and of course I2 is conserved independent of the nonlinearity ofthe evolution). Now consider nonlinear evolution in the one-dimensional NLSE.The parameters in the Gaussian approximation are (A, S , T ), being evaluated atT = (0, ∞). The combination of I2 and I3, both being quadratic in A, would
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yield an equation relating S∞ to Sf independent of the wave group amplitude,A. But wave group evolution as we study it here is an inherently nonlineareffect—it simply does not occur for linear groups. In contrast I4 has bothquadratic and quartic powers of the wave amplitude. It is an inherently nonlinearequation in amplitude, so it yields useful information to constrain our Gaussianapproximation in both one and two dimensions.
An attempt was made to derive and use an additional approximatelyconserved quantity in Adcock (2009) by making the obvious generalization totwo dimensions of the one-dimensional form for the next even-order conservedquantity (I6), but that analysis produced results inferior to those presented inthis paper.
We choose to remove one of the free parameters by assuming that A remainsconstant so that Af = A∞. We justify this by considering the fully nonlinearnumerical potential flow model analysis of Gibbs & Taylor (2005), which showsthat the amplitude is within 5 per cent of the linear value, after removing thepredominantly second-order bound wave structure. In an NLSE numerical modelthe difference is slightly larger, but this change is small compared with the changein bandwidth (Adcock 2009). We can now solve for the changes to the groupshape, first for the waveform when it is dispersed given the properties at focus,
(SX∞SXf
)2
= 14
⎛⎜⎝2 −
(A
SXf
)2
− 2(
SYf
SXf
)2
+√√√√(
2 −(
ASXf
)2
− 2(
SYf
SXf
)2)2
+ 16(
SYf
SXf
)2
⎞⎟⎠ (2.12)
and
(SY∞SYf
)2
= 14
⎛⎜⎝2 +
(A
SYf
)2
− 2(
SXf
SYf
)2
+
+√√√√(
2 +(
ASYf
)2
− 2(
SYf
SXf
)2)2
+ 16(
SXf
SYf
)2
⎞⎟⎠ . (2.13)
And the waveform at focus in terms of the parameters when the group is fullydispersed
(SXf
SX∞
)2
= 14
⎛⎜⎝2 +
(A
SX∞
)2
− 2(
SY∞SX∞
)2
+√√√√(
2 +(
ASX∞
)2
− 2(
SY∞SX∞
)2)2
+ 16(
SY∞SX∞
)2
⎞⎟⎠ (2.14)
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While it is useful to have a closed-form solution for the evolution in eachdirection, the general behaviour of the groups becomes rather clearer if we takeTaylor expansions in amplitude/bandwidth ratio. For the changes to the groupshape as it focuses these are(
SXf
SX∞
)= 1 + 1
4((SY∞/SX∞)2 + 1)
(A
SX∞
)2
+ O
((A
SX∞
)4)
(2.16)
and (SYf
SY∞
)= 1 − 1
4((SX∞/SY∞)2 + 1)
(A
SY∞
)2
+ O
((A
SY∞
)4)
. (2.17)
Thus the bandwidth in the mean wave direction is increased at focus, so thegroup contacts in the mean wave direction. And in the along crest orthogonaldirection the physical length of the crest at focus is increased by wave–waveinteractions as the group focuses in from the initially dispersed state.
To consider the implications of these results we consider the aspect ratio (R =SX/SY ). We plot the change to the aspect ratio as the group focuses in figure 3as a function of the group properties at infinity. Marked on the figure with asquare is the location of the case shown in figures 1 and 2, the analytical resultspredicting a change in the aspect ratio of approximately 3.6. Also marked withcrosses in figure 3 are the initial conditions considered by Gibbs & Taylor (2005),whose results are compared with these predictions in §3b.
There is an obvious limitation to these results in that in the uni-directionallimit (SY → 0) differs from the one-dimensional analysis in Adcock & Taylor(2009b), although the results are close in two dimensions. This discrepancy is to beexpected since we have assumed that A remains constant, which is not the case forthe uni-directional result. We also observe that there is no ‘limiting nonlinearity’for a focused group, as was found for uni-directional waves in Adcock & Taylor(2009b). In this it was found that for a group focusing Af/SXf < 21/4. Physicallythis implies that a large wave group will not tend to last for more than a fewwave periods. While there is no absolute limit found here for the two-dimensionalproblem we note that, if a series of large waves are formed on a linear basis,this would imply a small SX , which would mean that the group would tend tocontract. Thus we draw the general conclusion that the nonlinear physics willinhibit a large number of consecutive waves, moving in the same direction, whichare taller than a background wave field.
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Figure 3. Change predicted by NLSE analytics to the aspect ratio of the group as it focuses.Contours show (Sxf /Syf )/(Sx∞/Sy∞). Also shown (crosses and squares) are the locations of theinitial conditions used by Gibbs & Taylor (2005) (see §3b) and the initial conditions of the caseanalysed in §4 (square).
A practical implication for offshore design is that the local directional spreadingwill be reduced, increasing the inline velocity and acceleration. This can beapproximated by noting that sy ≈ kp cos(q), where theta is the r.m.s. directionalspreading and is related to the inline wave kinematic factor used in offshoredesign, f , by
f = cos(q). (2.18)
3. Comparison with numerical results
(a) Comparison with NLSE
The analytical results may be compared with the results of numerical simulations.We use a fourth-order Runge–Kutta scheme in time and a pseudo-spectral schemein space to solve the NLSE. We found that Dx ∼ 0.6/k0 and Dt ∼ 0.5/u0 gaveaccurate converged results. We start with a wave group at focus and allow itto disperse. When the spectrum stops changing and the evolution of the groupbecomes essentially linear we fit a Gaussian to the two-dimensional spectrum.We show a comparison of these against the analytical results in figure 4. Theseshow excellent agreement with the analytical results above for a wide range ofinitial bandwidths, confirming at least that our analytical model works well forwave groups in the NLSE.
(b) Comparison with potential flow results
Gibbs & Taylor (2005) carried out fully nonlinear simulations of focusing wavegroups in deep water, using the scheme developed by Bateman et al. (2001).The simulations used a Gaussian wave packet as the initial conditions witha spectrum based on a JONSWAP spectrum with g = 3.3 and with a r.m.s.
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Figure 4. The group shape at infinity for various initially focused Gaussian groups. Solid linesrepresent the analytical predictions and crosses the numerical results. (a) Rf = 0.387, (b) Rf = 0.775,(c) Rf = 1.16 and (d) Rf = 1.94.
directional spreading of 15◦ with a peak wavenumber of kp = 0.0279 m−1, implyinga peak period of 12 s—these are taken to be representative of a severe winterstorm in the northern North Sea (Gibbs 2004). This gives a Gaussian groupwith sx = 0.0046 m−1 and sy = 0.0073 m−1. This, when non-dimensionalized asabove, gives SX∞/SY∞ = 0.45. The most nonlinear case run, ak0 = 0.33, equatesto A∞/SX∞ = 4.0. The simulations were started 20 periods before linear focus andrun for a number of different amplitudes. This is sufficiently long before focus thatthe resulting evolution is very similar to that of a group converging from infinity.Both ‘crest’ and ‘trough’ focused runs were carried out, allowing the odd- andeven-order harmonics to be separated (Adcock & Taylor 2009a), and the effectof bound harmonics may simply be removed leaving mainly freely propagatingwaves. The results presented here are based on the maximum amplitude of thefreely propagating waves recorded. The physical shape of these wave groups isshown in both figures 1 and 2.
As noted in §1, very little extra elevation was observed when compared withlinear evolution. However, there was a significant change in the group shape owingto the nonlinear evolution. These changes may be quantified by fitting a Gaussianto the wave group along the x- and y-directions. Note that the values presentedhere are different from those in Gibbs & Taylor (2005), who fitted a Gaussian onlyto the peak of the group whereas here we are fitting the whole group. We show acomparison between the analytical theory and the numerical results in figure 5.
The agreement is excellent. For the steepest (most nonlinear) case the groupcontacts by more than 2× along the mean wave direction and expands along thecrest by a factor of 2, giving a change in aspect ratio of approximately 5 comparedwith linear evolution. Given the approximations contained in reducing the fullwater wave equations to the NLSE, and the further approximations required tofit a Gaussian model to the NLSE, such good agreement is rather surprising.
4. NLSE scaling applied to fully water wave problem
Section 3 showed that simple analytical results derived from the NLSE showexcellent agreement with fully nonlinear numerical simulations for a particular
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Figure 5. Comparison of predicted change to group shape with the fully nonlinear results of Gibbs &Taylor (2005). Open circles, sxf /sx∞—fully nonlinear; cross symbols, syf /sy∞—fully nonlinear;plus symbols, (sxf /syf )/(sx∞/sy∞)—fully nonlinear; dashed line, sxf /sx∞—analytic; dotted line,syf /sy∞—analytic; solid line, (sxf /syf )/(sx∞/sy∞)—analytic.
set of bandwidths representative of severe conditions in the North Sea. In thissection, we examine the limitations of this observation through attempting toscale the full water wave equations.
The classic gravity wave problem (in the absence of surface tension) exhibitsFroude number scaling, as well as a number of others (Benjamin & Olver 1982).However, these are not useful for exploring the limitations of the changes to thegroup shape. However, the NLSE has a simple scaling law (Sulem & Sulem 1999),
U → lU , (4.1)
X → l−1X , (4.2)
Y → l−1Y (4.3)
and T → l−2T , (4.4)
where l is the scaling parameter. Under this scaling, the solutions of the NLSEare invariant, whereas this scaling will not apply exactly to the full waterwave equations. The NLSE model does not represent the permanent third-orderinteractions of Phillips (1960) and higher order interactions but does model thedegenerate interactions of Phillips (1960) and the near-resonant interactions ofBenjamin & Feir (1967). Thus, we would not expect all details of wave focusing toscale—however, the leading-order changes to group shape might be expected toscale reasonably well. We note that uni-directional scaling of the full water waveequations was briefly considered in Henderson et al. (1999), but we are not awareof any previous investigations of scaling for the two-dimensional surface problem.
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The NLSE scaling laws will be applied to the full water wave equations usingthe following scalings:
as → la, (4.5)
sxs → lsx , (4.6)
ss → ls (4.7)
and tis → l−2ti . (4.8)
The subscript s implies the scaled case. s is the r.m.s. spreading of the wavegroup and for a simple Gaussian group is proportional to sy . The time theevolution starts, relative to linear focus at t = 0, is ti . The peak period remainsunchanged since this scaling can be viewed as an envelope scaling leaving thewiggles unaltered.
In this investigation we use as our unscaled case the same wave groupas described in §3, i.e. a Gaussian group with sx = 0.0046 m−1, s = 15◦ andkp = 0.0279 m−1 implying Tp = 12 s. All results presented are re-scaled back forcomparison with the unscaled results. Likewise all results have been ‘linearized’as above—however, we note that for cases l > 1 the spectrum is broad and thusthe linearization process will be imperfect. It should also be noted that for l > 1the cases were extremely steep and thus very demanding on the numerical code.Relatively small errors of approximatley 1 per cent in the total energy wereobserved for the steepest case presented here over the whole duration of thecomputation, whereas this error for smaller waves was typically several orders ofmagnitude smaller. Nonetheless, we believe the computations to be sufficientlyaccurate to draw firm conclusions.
(a) Local changes to group shape
We first consider the local changes to the shape of the wave group around thecrest rather than the large scale changes to the whole wave group considered in §§2and 3. These nonlinear changes to the shape of the wave group were investigatedin Gibbs & Taylor (2005). They found that around the main crest (or trough)the contraction was even greater than for the whole wave group.
The ‘local’ bandwidth, sxl , is found by fitting the locally parabolic variation ofthe envelope of a Gaussian to the crest of the linearized wave group. In figure 6,we plot the maximum contraction of the group for different values of linear akand different scalings, re-scaled to l = 1. The same general behaviour is observedfor all scalings, although there is some variation in the magnitude of the changes.These results suggest that groups with larger amplitude and larger bandwidthwill contract more than those with smaller amplitudes and bandwidths. Thisis noteworthy as it suggests that, while the change in aspect ratio is primarilydriven by nonlinearity which is captured by the NLSE, this process is furtheraccentuated by other nonlinear processes.
We may also track the changes to the localized bandwidth during the evolution.We examine a highly nonlinear case, where the steepness would be ak = 0.3 whenl = 1 at linear focus. This evolution is shown in figure 7. We again observe thatthere is reasonable agreement between the differently scaled cases.
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Figure 6. Maximum localized contraction of the wave group relative to linear evolution for differentsteepness and scalings (open circle with solid line, l = 0.6; open square with solid line, l = 0.8; crosssymbol with solid line, l = 1; plus symbol with solid line, l = 1.2; and open diamond with solidline, l = 1.4).
–20 –15 –10 –5 0 5 10 15 200
0.005
0.010
0.015
0.020
time from linear focus (periods)
s xl
Figure 7. Change in localized bandwidth of the group with time for different scalings. Linear ak =0.3 (thick solid grey line, l = 1—linear; thin dashed line, l = 0.6; dot-dashed line, l = 0.8; verticalsmall bars, l = 1; thick dashed black line, l = 1.2; and thick solid black line, l = 1.4).
The changes to the group in the lateral direction can be analysed by consideringthe local spreading angle introduced in Gibbs & Taylor (2005). This is defined byanalogy to the inline wave kinematics factor used in offshore structural design
f = cos q = uinline
utotal, (4.9)
where uinline and utotal are the magnitude of the velocity components resolvedin the mean wave direction and all the wave components, respectively. Thisparameter is based on the net horizontal displacement of fluid particles on thefree surface as they perform the circular orbits of linear wave theory. Thus, wedefine the local spreading angle as
q = cos−1(
hinline,env
henv
), (4.10)
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Figure 8. Variation of local spreading angle with steepness and scaling (open circle with solid line,l = 0.6; open square with solid line, l = 0.8; cross symbol with solid line, l = 1; plus symbol withsolid line, l = 1.2; and open diamond with solid line, l = 1.4).
–20 –15 –10 –5 0 5 10 15 202
4
6
8
10
12
14
16
time from linear focus (periods)
loca
l spr
eadi
ng a
ngle
(°)
Figure 9. Change in local spreading angle with time for different scalings. Linear ak = 0.3 (thicksolid grey line, l = 1—linear; thin dashed line, l = 0.6; dot-dashed line, l = 0.8; vertical small bars,l = 1; thick dashed line, l = 1.2; and thick solid black line, l = 1.4).
which gives a measure of how long-crested the wave group is at a particular spatiallocation. This amplitude and inline amplitude are found from
h(x , y) =M∑
m=0
N∑n=0
amn cos(kxmx + kyny + Fmn) (4.11)
and
hinline(x , y) =M∑
m=0
N∑n=0
amn cos(kxmx + kyny + Fmn)kxm√
k2xm
+ k2yn
. (4.12)
The envelope may then be found from henv = √h2 + h2, where h is the Hilbert
transform of the signal, obtained by incrementing the phase Fm,n by p/2.Figure 8 shows the spreading angle at focus for different steepness and scalings.
Again we see that there is good agreement between the different scalings, withslightly smaller changes in the wave groups which are smaller with narrower
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Figure 10. Amplitude contours of the envelope of the locally linearized surface elevation. On allgraphs the horizontal axis is the mean wave direction in the frame of reference of the linear groupvelocity and both dimensions are in units of metres. Contours are given at 1 m intervals. (a) l = 0.6,(b) l = 1.0 and (c) l = 1.4. Rows are (from top to bottom): t = −6.9Tp, t = −2.8Tp, t = 1.3Tp andt = 5.2Tp, where time is relative to linear focus, based on the l = 1 case.
bandwidth. In figure 9, we track these changes with time for a particular scaledsteepness; again we see good agreement for the changes in group structureup to focus.
(b) Large scale changes to group shape
We now consider the large scale changes to the wave group. We choose the samereference case described above with, for l = 1, ak = 0.3. In figure 10, we compare
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Figure 11. Amplitude spectra of wave groups at focus. (a) Linear evolution, (b) prediction usingpredicted values from §2, (c) NLSE evolution, (d) fully nonlinear l = 0.6, (e) fully nonlinear l = 1and (f ) fully nonlinear l = 1.4. The contours are given at 0.0025, 0.005, 0.0075, 0.01, 0.02, 0.04,0.06, 0.08 and 0.09 m.
wave group envelope of different scaled cases, re-scaled back to the reference case.Again the general behaviour is very similar in all cases with a contraction in themean wave direction and lateral expansion of similar magnitude. The changes tothe wave group are strongest for the steeper cases (larger l), as above. Thereis both a curvature to the wave group and a shift in the position of the peakobservable in all cases which is stronger for larger l. We would not expect thisto scale as it is not a property of the NLSE evolution (see figure 2). We alsoobserve that for l = 1.4 there is evidence of the group splitting along the meanwave direction. For highly nonlinear wave groups, the NLSE can split along themean wave direction (Adcock & Taylor 2011). However, this is not the cause hereas the wave groups are not sufficiently nonlinear. It is important to note that theconvergence of the group at focus arising from −∞ matches the predictions of theNLSE scaling quite well. In contrast, the subsequent evolution is much less wellmodelled. However, here we are mostly interested in the formation of extremeevents (i.e. pre-focus) where the scaling works quite well.
The amplitude spectra for the different scalings are presented at focusin figure 11. For reference we also show the constant spectrum under linearevolution, the predicted spectrum given by equations (2.14) and (2.15), and thespectrum found using the NLSE numerical model. Near the spectral peak wesee an increase in the bandwidth in the kx direction and a decrease in the kydirection relative to the linear spectrum. The fully nonlinear cases appear toscale very well, as would be expected from the previous analysis. We also see thewell-known down-shift in the spectral peak. However, there is clearly a highertransfer of energy to higher wavenumbers for larger l whereas the transfers tosmall wave number are larger for small l.
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Groups of large waves will form owing to the random nature of wave fields in theopen ocean. These wave groups will be modified by nonlinear interactions betweenthe wave components. When the waves have a realistic directional spreading theseinteractions do not cause much, if any, extra amplitude but do cause large changesto the shape of the wave group and therefore substantially modify the local wavedynamics. Large numbers of consecutive large waves are inhibited, and there isa reduction in the directional spreading of the waves as they form a long-crested‘wall of water’. These changes are captured reasonably well by the relativelysimple NLSE model of wave evolution.
In this paper, we have derived an approximate analytical relationship for thechange in bandwidth of the wave group as it focuses, using the conservation lawsof the NLSE. Despite the approximate nature of this solution, and of the NLSEitself as a model for deep water waves, this relationship shows excellent agreementwith numerical simulations using a fully nonlinear potential flow model for sea-state parameters representative of large waves in a large storm in the North Sea.We have also applied the scaling laws of NLSE to the full water wave problem andinvestigated this numerically. We find that, while NLSE scaling does not captureall interactions, the dominant changes to the wave group scale reasonably well.Since there is a reduction in the strength of the nonlinear term in the NLSE as thewater depth is reduced, this also implies that the shape changes of wave groupsas they focus will be reduced in finite depth and absent for kd = 1.363, where thecoefficient of the cubic term in the NLSE vanishes (Johnson 1997, p. 304).
References
Adcock, T. A. A. 2009 Aspects of wave dynamics and statistics on the open ocean. DPhil thesis,University of Oxford, UK.
Adcock, T. A. A. & Taylor, P. H. 2009a Estimating ocean wave directional spreading from anEulerian surface elevation time history. Proc. R. Soc. A 465, 3361–3381. (doi:10.1098/rspa.2009.0031)
Adcock, T. A. A. & Taylor, P. H. 2009b Focusing of unidirectional wave groups on deep water:an approximate nonlinear Schrödinger equation-based model. Proc. R. Soc. A 465, 3083–3102.(doi:10.1098/rspa.2009.0224)
Adcock, T. A. A. & Taylor, P. H. 2011 Energy input amplifies the non-linear dynamics of deepwater wave groups. Int. J. Offshore Polar Eng. 21, 8–12.
Adcock, T. A. A. & Yan, S. 2010 The focusing of uni-directional Gaussian wave-groups in finitedepth: an approximate NLSE based approach. In Proc. 29th Int. Conf. Ocean, Offshore andArctic Engineering (OMAE), Shanghai, China, 6–11 June 2010, p. 20993.
Alber, I. E. 1978 The effects of randomness on the stability of two-dimensional surface wavetrains.Proc. R. Soc. Lond. A 363, 525–546. (doi:10.1098/rspa.1978.0181)
Baldock, T. E., Swan, C. & Taylor, P. H. 1996 A laboratory study of nonlinear surface waves onwater. Phil. Trans. R. Soc. Lond. A 354, 649–676. (doi:10.1098/rsta.1996.0022)
Bateman, W. J. D., Swan, C. & Taylor, P. H. 2001 On the efficient numerical simulationof directionally spread surface water waves. J. Comput. Phys. 174, 277–305. (doi:10.1006/jcph.2001.6906)
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory.J. Fluid Mech. 27, 417–430. (doi:10.1017/S002211206700045X)
Benjamin, T. B. & Olver, P. J. 1982 Hamiltonian structure, symmetries and conservation laws forwater waves. J. Fluid Mech. 125, 137–185. (doi:10.1017/S0022112082003292)
Proc. R. Soc. A (2012)
on August 19, 2012rspa.royalsocietypublishing.orgDownloaded from
Boccotti, P. 1983 Some new results on statistical properties of wind waves. Appl. Ocean Res. 5,134–140. (doi:10.1016/0141-1187 (83)90067-6)
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc.Lond. A 338, 101–110. (doi:10.1098/rspa.1974.0076)
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for applicationto deep water waves. Proc. R. Soc. Lond. A 369, 105–114. (doi:10.1098/rspa.1979.0154)
Forristall, G. Z. 2000 Wave crest distributions: observations and second-order theory. J. Phys.Oceanogr. 30. (doi:10.1175/1520-0485 (2000)030<1931:WCDOAS>2.0.CO;2)
Forristall, G. Z. & Ewans, K. C. 1998 Worldwide measurements of directional wavespreading. J. Atmos. Ocean. Technol. 15, 440–469. (doi:10.1175/1520-0426(1998)015<0440:WMODWS>2.0.CO;2)
Gibbs, R. H. 2004 Walls of water on the open ocean. DPhil thesis, University of Oxford, UK.Gibbs, R. H. & Taylor, P. H. 2005 Formation of wall of water in ‘fully’ nonlinear simulations. Appl.
Ocean Res. 27, 142–157. (doi:10.1016/j.apor.2005.11.009)Gibson, R. S. & Swan, C. 2007 The evolution of large ocean waves: the role of local and rapid
spectral changes. Proc. R. Soc. A 463, 21–48. (doi:10.1098/rspa.2006.1729)Gramstad, O. & Trulsen, K. 2007 Influence of crest and group length on the occurrence of freak
waves. J. Fluid Mech. 582, 463–472. (doi:10.1017/S0022112007006507)Henderson, K. L., Peregrine, D. H. & Dold, J. W. 1999 Unsteady water wave modulations: fully
nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29,341–361. (doi:10.1016/S0165-2125 (98)00045-6)
Janssen, P. A. E. M. 2004 The interaction of ocean waves and wind. Cambridge, UK: CambridgeUniversity Press.
Johannessen, T. B. & Swan, C. 2001 A laboratory study of the focusing of transient anddirectionally spread surface water waves. Proc. R. Soc. Lond. A 457, 971–1006. (doi:10.1098/rspa.2000.0702)
Johnson, R. S. 1997 A modern introduction to the mathematical theory of water waves. Cambridge,UK: Cambridge University Press.
Jonathan, P. & Taylor, P. H. 1997 On irregular, nonlinear waves in a spread sea. J. Offshore Mech.Arctic Eng. 119, 37–41. (doi:10.1115/1.2829043)
Katsardi, V. & Swan, C. 2011 The evolution of large non-breaking waves in intermediate andshallow water. I. Numerical calculations of uni-directional seas. Proc. R. Soc. A 463, 778–805.(doi:10.1098/rspa.2010.0280)
Kinsman, B. 1965 Wind waves, their generation and propagation on the ocean surface. EnglewoodCliffs, NJ: Prentice-Hall.
Krogstad, H. E., Magnusson, A.-K. & Donelan, M. A. 2006 Wavelet and local directional analysisof ocean waves. Int. J. Offshore Polar Eng. 16, 97–103.
Lawton, G. 2001 Monsters of the deep (the perfect wave). New Sci. 2297, 28–32.Lindgren, G. 1970 Some properties of a normal process near a local maximum. Ann. Math. Statist.
41, 1870–1883. (doi:10.1214/aoms/1177696688)Lo, E. Y. & Mei, C. C. 1987 Slow evolution of nonlinear deep water waves in two horizontal
directions: a numerical study. Wave Motion 9, 245–259. (doi:10.1016/0165-2125 (87)90014-X)Monaldo, F. M. 2000 Measurements of wave coherence properties using space borne synthetic
aperature radar. Mar. Struct. 13, 349–366. (doi:10.1016/S0951-8339 (00)00011-3)Onorato, M., Osborne, A., Serio, M., Cavaleri, L., Brandini, C. & Stansberg, C. 2006 Extreme
waves, modulational instability and second order theory: wave flume experiments on irregularwaves. Eur. J. Mech. B/Fluids 25, 586–601. (doi:10.1016/j.euromechflu.2006.01.002)
Onorato, M., et al. 2009 Statistical properties of directional ocean waves: the role of the modul-ational instability in the formation of extreme events. Phys. Rev. Lett. 102, 114502. (doi:10.1103/PhysRevLett.102.114502)
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. Theelementary interactions. J. Fluid Mech. 9, 193–217. (doi:10.1017/S0022112060001043)
Phillips, O. M., Gu, D. & Donelan, M. 1993 Expected structure of extreme waves in a gaussian sea.Part I. Theory and SWADE buoy measurements. J. Phys. Oceanogr. 23, 992–1000. (doi:10.1175/1520-0485(1993)023)
Proc. R. Soc. A (2012)
on August 19, 2012rspa.royalsocietypublishing.orgDownloaded from
Socquet-Juglard, H., Dysthe, K., Trulsen, K., Krogstad, H. E. & Liu, J. 2005 Probabilitydistributions of surface gravity waves during spectral changes. J. Fluid Mech. 542, 195–216.(doi:10.1017/S0022112005006312)
Sulem, C. & Sulem, P. L. 1999 The nonlinear Schrödinger eqaution: self-focusing and wave collapse.Applied Mathematical Sciences, vol. 139. New York, NY: Springer.
Taylor, P. H. & Vijfvinkel, E. 1998 Focussed wave groups on deep and shallow water. In Proc.1998 Int. DTRC Symposium, Houston, Texas, 30 April–1 May 1998, pp. 420–427. Reston, VA:ASCE.
Tromans, P. S., Anaturk, A. & Hagemeijer, P. 1991 A new model for the kinematics of large oceanwaves– application as a design wave. In Proc. 1st Int. Conf. Offshore Mech. and Polar Engng(ISOPE), Edinburgh, UK, 11–16 August, vol. 3 (eds J. S. Chung, B. J. Natvig, K. Kaneko & A.J. Ferrante), pp. 64–71. Golden, CO: International Society of Offshore and Polar Engineers.
Trulsen, K. & Dysthe, K. B. 1996 A modified nonlinear Schrödinger equation for broader bandwidthgravity waves on deep water. Wave Motion 24, 281–289. (doi:10.1016/S0165-2125 (96)00020-0)
Zakharov, V. E. & Ostrovsky, L. A. 2009 Modulation instability: the beginning. Physica D 238,540–548. (doi:10.1016/j.physd.2008.12.002)
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. J. Exp. Theor. Phys. 34, 62–69.
Proc. R. Soc. A (2012)
on August 19, 2012rspa.royalsocietypublishing.orgDownloaded from
