The evolution of steep waves in the open ocean is nonlinear. In narrow-banded but directionally spread seas, this nonlinearity does not produce significant extra elevation but does lead to a large change in the shape of the wave group, causing, relative to linear evolution, contraction in the mean wave direction and lateral expansion. We use the nonlinear Schrödinger equation (NLSE) to derive an approximate analytical relationship for these changes in group shape. This shows excellent agreement with the numerical results both for the NLSE and for the full water wave equations. We also consider the application of scaling laws from the NLSE in terms of wave steepness and bandwidth to solutions of the full water wave equations. We investigate these numerically. While some aspects of water wave evolution do not scale, the major changes that a wave group undergoes as it evolves scale very well.
The dynamics of extreme ocean waves is important to engineers who are designing for the maritime environment and is of major interest to those investigating ‘freak’ or ‘rogue’ waves. Much work has been devoted to predicting the amplitude of 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 & Trulsen 2007). Rather less work has gone into investigating the local shape of extreme waves. This is important as it influences the forces on maritime structures and is fundamental to understanding the nonlinear evolution of steep waves.
The starting point for this paper is the expected shape of a large wave event in a random 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 evolution might be expected to produce significant modifications to this shape. These may be investigated using a group which would, under linear evolution, form a NewWave group when all the wave components are in phase, or ‘focused’. Instead of using a linear model of evolution, we investigate how this shape is modified by nonlinear interactions both analytically and via numerical simulations.
In uni-directional (long-crested) seas, nonlinear changes will cause an increase in the amplitude relative to linear evolution, as well as a contraction of the length of the wave group in the mean wave direction (Baldock et al. 1996; Taylor & Vijfvinkel 1998; Adcock & Taylor 2009b), although these changes are inhibited by 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 of directionally spread (short-crested) wave groups has been investigated in both physical 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 through simple bound harmonics, there are large changes to the local aspect ratio of the wave group. An example of these changes is shown in figures 1 and 2 (see §3b for details of these results). This broadening along the crest direction is in agreement with some accounts of encounters with freak waves as ‘walls of water’ (Lawton 2001) and with the field observations of Monaldo (2000) and Krogstad et al. (2006).
In this paper, we start by deriving analytical relationships for the nonlinear changes to the aspect ratio of a localized wave group as it focuses on an otherwise quiescent ocean, using the approach used for uni-directional wave groups in Adcock & Taylor (2009b). We compare these results with numerical models and find excellent agreement. In the second half of the paper, we consider an approximate scaling behaviour for deep water waves based on the exact scaling laws of the nonlinear Schrödinger equation (NLSE). This is investigated numerically. We find that the NLSE (and its conserved quantities) is very useful for 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 a significant difference to the size of the wave crest but do not modify the underlying dynamics of the wave group. The second-order correction to the wave group shown in figure 1b is approximately 2 m. Once the freely propagating waves at an extreme event have been determined, it is straightforward to then calculate the bound waves using the interaction kernel (e.g. Forristall 2000).
2. Analytical results
The hyperbolic two-dimensional NLSE (equation 2.1) is the simplest nonlinear model for the evolution of water waves on deep water. This equation has been used extensively in the analysis of extreme wave events (Zakharov & Shabat 1972; Alber 1978; Janssen 2004; Zakharov & Ostrovsky 2009), 2.1 where u is the complex wave envelope, and ω0 and k0 are the frequency and wavenumber of the carrier wave which is travelling in the x-direction. This may be non-dimensionalized using the transformations T=ω0t, , Y =2k0y and to give 2.2
The NLSE does not capture the full dynamics of the nonlinear water wave problem. Various more complex equations have been derived which include additional terms (Davey & Stewartson 1974; Dysthe 1979; Trulsen & Dysthe 1996). 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) the NLSE has captured the overall change in aspect ratio (figure 2b) but not the full complexities of the nonlinear evolution (figure 2c) such as the shift in the position of the peak (see Lo & Mei 1987; Gibson & Swan 2007). In this section, we use the NLSE to seek an approximate analytical solution for the evolution of a Gaussian wave group on deep water following the approach of Adcock & Taylor (2009b). The Gaussian wave group which focuses and then de-focuses is an exact solution to the linear part of the NLSE, and a simple generalization of the one-dimensional result in Kinsman (1965), 2.3 Here, the linear Gaussian group is described by three parameters: the amplitude measure 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 which is closely related to the shape of the NewWave group in space—a group with a narrow bandwidth will give a longer spatial wave group and vice versa.
For the nonlinear problem, we make the assumption that the solution remains Gaussian in form but that the parameters vary slowly in time. We then use the conservation laws of the NLSE to identify the changes in these parameters as the group focuses and de-focuses. In this context the two-dimensional NLSE has two known (non-trivial) and useful conserved quantities, 2.4 and 2.5 We now introduce a nonlinear time scale τ. It would be possible to use a different nonlinear time scale in the X- and Y -directions, but for the analysis we present here this would not modify the results so we will keep just one time scale for simplicity. We substitute equation (2.3) into the conserved quantities to give 2.6 and 2.7 We evaluate these at τ=0 (focus) and (fully dispersed spatially), and equate these so as to generate two equations relating the shape of the group at focus to that when it is completely dispersed. This is also equivalent to equating the parameters for a group which has undergone linear evolution to one under nonlinear evolution, since the parameters describing the Gaussian do not change in the linear model, 2.8 and 2.9 This gives us two equations but three parameters (A, SX and SY). No additional useful conserved quantity is known for the two-dimensional NLSE, which only has a finite number of conserved quantities (Sulem & Sulem 1999). We stress the importance of the word ‘useful’ in the previous sentence. As well as the conserved quantities I2 (energy) and I4 (the Hamiltonian), only two others are known for the two-dimensional NLSE. These are 2.10 and 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 the group when the maximum of the envelope is located. When translated into the NLSE envelope, this implies that the U(X,Y,T=0) is symmetric in space. For a purely real and symmetric function, I3X for the one-dimensional NLSE and both I3X and I3Y for the two-dimensional NLSE are identically zero at focus. So at all other times the relevant I3 forms must also be identically zero.
When we substitute the assumed Gaussian form at focus into the conserved quantities, both I3 versions in two dimensions are again identically zero. Then, using the assumed Gaussian form as a slowly varying model for the nonlinear evolution, we find that both I3 variants are identically zero. This is independent of values for the parameters A, SX and SY, and, most importantly, time τ. So although I3X and I3Y are conserved, they yield no usable information.
The I3 forms would only be useful for constraining an approximation if they have a non-zero value when the group is most compact; but then the group is not perfectly symmetric and the focus is non-optimal. Possibly the I3 forms might then be useful for investigating the degree of non-optimality in focusing via the incorporation of a phase, , but in this work we explicitly assume that the focusing is optimal.
There is a second reason why we might not expect I3 to yield useful information for modelling the nonlinear evolution of a wave group. For arbitrary spatial and temporal structure within the group, both I2 and I3 are quadratic in the wave amplitude (and of course I2 is conserved independent of the nonlinearity of the evolution). Now consider nonlinear evolution in the one-dimensional NLSE. The parameters in the Gaussian approximation are (A,S,T), being evaluated at . The combination of I2 and I3, both being quadratic in A, would yield an equation relating to Sf independent of the wave group amplitude, A. But wave group evolution as we study it here is an inherently nonlinear effect—it simply does not occur for linear groups. In contrast I4 has both quadratic and quartic powers of the wave amplitude. It is an inherently nonlinear equation in amplitude, so it yields useful information to constrain our Gaussian approximation in both one and two dimensions.
An attempt was made to derive and use an additional approximately conserved quantity in Adcock (2009) by making the obvious generalization to two dimensions of the one-dimensional form for the next even-order conserved quantity (I6), but that analysis produced results inferior to those presented in this paper.
We choose to remove one of the free parameters by assuming that A remains constant so that . We justify this by considering the fully nonlinear numerical potential flow model analysis of Gibbs & Taylor (2005), which shows that the amplitude is within 5 per cent of the linear value, after removing the predominantly second-order bound wave structure. In an NLSE numerical model the difference is slightly larger, but this change is small compared with the change in bandwidth (Adcock 2009). We can now solve for the changes to the group shape, first for the waveform when it is dispersed given the properties at focus, 2.12 and 2.13 And the waveform at focus in terms of the parameters when the group is fully dispersed 2.14 and 2.15
While it is useful to have a closed-form solution for the evolution in each direction, the general behaviour of the groups becomes rather clearer if we take Taylor expansions in amplitude/bandwidth ratio. For the changes to the group shape as it focuses these are 2.16 and 2.17
Thus the bandwidth in the mean wave direction is increased at focus, so the group contacts in the mean wave direction. And in the along crest orthogonal direction the physical length of the crest at focus is increased by wave–wave interactions 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 3 as a function of the group properties at infinity. Marked on the figure with a square is the location of the case shown in figures 1 and 2, the analytical results predicting a change in the aspect ratio of approximately 3.6. Also marked with crosses 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-directional limit differs from the one-dimensional analysis in Adcock & Taylor (2009b), although the results are close in two dimensions. This discrepancy is to be expected since we have assumed that A remains constant, which is not the case for the 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. Physically this implies that a large wave group will not tend to last for more than a few wave periods. While there is no absolute limit found here for the two-dimensional problem 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 to contract. Thus we draw the general conclusion that the nonlinear physics will inhibit a large number of consecutive waves, moving in the same direction, which are taller than a background wave field.
A practical implication for offshore design is that the local directional spreading will be reduced, increasing the inline velocity and acceleration. This can be approximated by noting that , where theta is the r.m.s. directional spreading and is related to the inline wave kinematic factor used in offshore design, f, by 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 scheme in space to solve the NLSE. We found that Δx∼0.6/k0 and Δt∼0.5/ω0 gave accurate converged results. We start with a wave group at focus and allow it to disperse. When the spectrum stops changing and the evolution of the group becomes essentially linear we fit a Gaussian to the two-dimensional spectrum. We show a comparison of these against the analytical results in figure 4. These show excellent agreement with the analytical results above for a wide range of initial bandwidths, confirming at least that our analytical model works well for wave groups in the NLSE.
(b) Comparison with potential flow results
Gibbs & Taylor (2005) carried out fully nonlinear simulations of focusing wave groups in deep water, using the scheme developed by Bateman et al. (2001). The simulations used a Gaussian wave packet as the initial conditions with a spectrum based on a JONSWAP spectrum with γ=3.3 and with a r.m.s. directional spreading of 15° with a peak wavenumber of kp=0.0279 m−1, implying a peak period of 12 s—these are taken to be representative of a severe winter storm in the northern North Sea (Gibbs 2004). This gives a Gaussian group with sx=0.0046 m−1 and sy=0.0073 m−1. This, when non-dimensionalized as above, gives . The most nonlinear case run, ak0=0.33, equates to . The simulations were started 20 periods before linear focus and run for a number of different amplitudes. This is sufficiently long before focus that the 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- and even-order harmonics to be separated (Adcock & Taylor 2009a), and the effect of bound harmonics may simply be removed leaving mainly freely propagating waves. The results presented here are based on the maximum amplitude of the freely propagating waves recorded. The physical shape of these wave groups is shown in both figures 1 and 2.
As noted in §1, very little extra elevation was observed when compared with linear evolution. However, there was a significant change in the group shape owing to the nonlinear evolution. These changes may be quantified by fitting a Gaussian to the wave group along the x- and y-directions. Note that the values presented here are different from those in Gibbs & Taylor (2005), who fitted a Gaussian only to the peak of the group whereas here we are fitting the whole group. We show a comparison between the analytical theory and the numerical results in figure 5.
The agreement is excellent. For the steepest (most nonlinear) case the group contacts by more than 2× along the mean wave direction and expands along the crest by a factor of 2, giving a change in aspect ratio of approximately 5 compared with linear evolution. Given the approximations contained in reducing the full water wave equations to the NLSE, and the further approximations required to fit 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 show excellent agreement with fully nonlinear numerical simulations for a particular set of bandwidths representative of severe conditions in the North Sea. In this section, we examine the limitations of this observation through attempting to scale the full water wave equations.
The classic gravity wave problem (in the absence of surface tension) exhibits Froude 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 the group shape. However, the NLSE has a simple scaling law (Sulem & Sulem 1999), 4.1 4.2 4.3 and 4.4 where λ is the scaling parameter. Under this scaling, the solutions of the NLSE are invariant, whereas this scaling will not apply exactly to the full water wave equations. The NLSE model does not represent the permanent third-order interactions of Phillips (1960) and higher order interactions but does model the degenerate interactions of Phillips (1960) and the near-resonant interactions of Benjamin & Feir (1967). Thus, we would not expect all details of wave focusing to scale—however, the leading-order changes to group shape might be expected to scale reasonably well. We note that uni-directional scaling of the full water wave equations was briefly considered in Henderson et al. (1999), but we are not aware of any previous investigations of scaling for the two-dimensional surface problem.
The NLSE scaling laws will be applied to the full water wave equations using the following scalings: 4.5 4.6 4.7 and 4.8 The subscript s implies the scaled case. σ is the r.m.s. spreading of the wave group and for a simple Gaussian group is proportional to sy. The time the evolution starts, relative to linear focus at t=0, is ti. The peak period remains unchanged since this scaling can be viewed as an envelope scaling leaving the wiggles unaltered.
In this investigation we use as our unscaled case the same wave group as described in §3, i.e. a Gaussian group with sx=0.0046 m−1, σ=15° and kp=0.0279 m−1 implying Tp=12 s. All results presented are re-scaled back for comparison with the unscaled results. Likewise all results have been ‘linearized’ as above—however, we note that for cases λ>1 the spectrum is broad and thus the linearization process will be imperfect. It should also be noted that for λ>1 the cases were extremely steep and thus very demanding on the numerical code. Relatively small errors of approximatley 1 per cent in the total energy were observed for the steepest case presented here over the whole duration of the computation, whereas this error for smaller waves was typically several orders of magnitude smaller. Nonetheless, we believe the computations to be sufficiently accurate to draw firm conclusions.
(a) Local changes to group shape
We first consider the local changes to the shape of the wave group around the crest rather than the large scale changes to the whole wave group considered in §§2 and 3. These nonlinear changes to the shape of the wave group were investigated in 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 of the 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 ak and different scalings, re-scaled to λ=1. The same general behaviour is observed for all scalings, although there is some variation in the magnitude of the changes. These results suggest that groups with larger amplitude and larger bandwidth will contract more than those with smaller amplitudes and bandwidths. This is noteworthy as it suggests that, while the change in aspect ratio is primarily driven by nonlinearity which is captured by the NLSE, this process is further accentuated 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 when λ=1 at linear focus. This evolution is shown in figure 7. We again observe that there is reasonable agreement between the differently scaled cases.
The changes to the group in the lateral direction can be analysed by considering the local spreading angle introduced in Gibbs & Taylor (2005). This is defined by analogy to the inline wave kinematics factor used in offshore structural design 4.9 where uinline and utotal are the magnitude of the velocity components resolved in the mean wave direction and all the wave components, respectively. This parameter is based on the net horizontal displacement of fluid particles on the free surface as they perform the circular orbits of linear wave theory. Thus, we define the local spreading angle as 4.10 which gives a measure of how long-crested the wave group is at a particular spatial location. This amplitude and inline amplitude are found from 4.11 and 4.12 The envelope may then be found from , where is the Hilbert transform of the signal, obtained by incrementing the phase Φm,n by π/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, with slightly smaller changes in the wave groups which are smaller with narrower bandwidth. In figure 9, we track these changes with time for a particular scaled steepness; again we see good agreement for the changes in group structure up to focus.
(b) Large scale changes to group shape
We now consider the large scale changes to the wave group. We choose the same reference case described above with, for λ=1, ak=0.3. In figure 10, we compare 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 the mean wave direction and lateral expansion of similar magnitude. The changes to the wave group are strongest for the steeper cases (larger λ), as above. There is both a curvature to the wave group and a shift in the position of the peak observable in all cases which is stronger for larger λ. We would not expect this to scale as it is not a property of the NLSE evolution (see figure 2). We also observe that for λ=1.4 there is evidence of the group splitting along the mean wave direction. For highly nonlinear wave groups, the NLSE can split along the mean wave direction (Adcock & Taylor 2011). However, this is not the cause here as the wave groups are not sufficiently nonlinear. It is important to note that the convergence of the group at focus arising from matches the predictions of the NLSE scaling quite well. In contrast, the subsequent evolution is much less well modelled. However, here we are mostly interested in the formation of extreme events (i.e. pre-focus) where the scaling works quite well.
The amplitude spectra for the different scalings are presented at focus in figure 11. For reference we also show the constant spectrum under linear evolution, the predicted spectrum given by equations (2.14) and (2.15), and the spectrum found using the NLSE numerical model. Near the spectral peak we see an increase in the bandwidth in the kx direction and a decrease in the ky direction relative to the linear spectrum. The fully nonlinear cases appear to scale very well, as would be expected from the previous analysis. We also see the well-known down-shift in the spectral peak. However, there is clearly a higher transfer of energy to higher wavenumbers for larger λ whereas the transfers to small wave number are larger for small λ.
Groups of large waves will form owing to the random nature of wave fields in the open ocean. These wave groups will be modified by nonlinear interactions between the wave components. When the waves have a realistic directional spreading these interactions do not cause much, if any, extra amplitude but do cause large changes to the shape of the wave group and therefore substantially modify the local wave dynamics. Large numbers of consecutive large waves are inhibited, and there is a 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 relatively simple NLSE model of wave evolution.
In this paper, we have derived an approximate analytical relationship for the change in bandwidth of the wave group as it focuses, using the conservation laws of the NLSE. Despite the approximate nature of this solution, and of the NLSE itself as a model for deep water waves, this relationship shows excellent agreement with 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 and investigated this numerically. We find that, while NLSE scaling does not capture all 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 the water depth is reduced, this also implies that the shape changes of wave groups as they focus will be reduced in finite depth and absent for kd=1.363, where the coefficient of the cubic term in the NLSE vanishes (Johnson 1997, p. 304).
- Received January 17, 2012.
- Accepted March 28, 2012.
- This journal is © 2012 The Royal Society