Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation

Transcription

1 Wave Motion 29 (1999) Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation K.L. Henderson 1,, D.H. Peregrine, J.W. Dold 2 School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Received 10 November 1997; received in revised form 20 May 1998; accepted 8 September 1998 Abstract The time evolution of a uniform wave train with a small modulation which grows is computed with a fully nonlinear irrotational flow solver. Many numerical runs have been performed varying the initial steepness of the wave train and the number of waves in the imposed modulation. It is observed that the energy becomes focussed into a short group of steep waves which either contains a wave which becomes too steep and therefore breaks or otherwise having reached a maximum modulation then recedes until an almost regular wave train is recovered. This latter case typically occurs over a few hundred time periods. We have also carried out some much longer computations, over several thousands of time periods in which several steep wave events occur. Several features of these modulations are consistent with analytic solutions for modulations using weakly nonlinear theory, which leads to the nonlinear Schrödinger equation. The steeper events are shorter in both space and time than the lower events. Solutions of the nonlinear Schrödinger equation can be transformed from one steepness to another by suitable scaling of the length and time variables. We use this scaling on the modulations and find excellent agreement particularly for waves that do not grow too steep. Hence the number of waves in the initial modulation becomes an almost redundant parameter and allows wider use of each computation. A potentially useful property of the nonlinear Schrödinger equation is that there are explicit solutions which correspond to the growth and decay of an isolated steep wave event. We have also investigated how changing the phase of the initial modulation effects the first steep wave event that occurs. c 1999 Elsevier Science B.V. All rights reserved. 1. Introduction The stability and evolution of a uniform wave train on deep water has been a subject of much interest over the years. It is well known that a periodic uniform deep water wave train suffers an instability known as the Benjamin Feir instability [1,2], which appears in the form of growing modulations. Unlike other instabilities it occurs for wave trains of steepness well below the maximum steepness of steady waves at which crests approach 120 and so may be especially relevant to the evolution of ocean waves. The wave groups that form are a useful model for the sea surface. Corresponding author. Fax: ; karen.henderson@uwe.ac.uk 1 Present address: Department of Mathematical Sciences, University of the West of England, Bristol BS16 IQY, UK. 2 Present address: Department of Mathematics, UMIST, Manchester M60 1QD, UK /99/$ see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S (98)

2 342 K.L. Henderson et al. / Wave Motion 29 (1999) From our deterministic computations full hydrodynamic details of such wave groups are available and hence may be used for further studies such as evaluating the forces on off-shore structures. The main limitation is that this study is restricted to two-dimensional motions. However, we show that the slowly varying modulation approximation which leads to the nonlinear Schrödinger equation is surprisingly effective. This effectiveness of the nonlinear Schrödinger equation can be of value for applications such as specifying wave groups in studies of the hydrodynamic forces on marine structures and also suggests that it may also be particularly useful for three-dimensional waves where computation of fully accurate solutions is impractical for more than a few wave periods. The analysis of Benjamin and Feir was for a linear perturbation to the weakly nonlinear solutions and predicted exponential growth over a finite band of perturbation frequencies. They conclude that the most unstable modes correspond to a pair of side-band modes which emerge distinctly if disturbances in the initial state have an appropriate bandwidth. When the side-band amplitudes become approximately equal the wave profile corresponds to a uniform modulation of the primary wave train. The qualitative nature of these theoretical results was confirmed by their experiments. Further experiments have been performed to look at the evolution of nonlinear wave trains on deep water; Lake et al. [3], Melville [4] and Su et al. [5]. They obtained good quantitative agreement for the initial growth of disturbance as predicted by Benjamin and Feir [2]. However, rather than a disintegration of the wave train, they observed that further evolution is characterised by a spreading out of the energy over many spectral components followed by demodulation and return of the energy to the initial spectral components. Over a longer timescale the modulation increases and decreases periodically and the wave train exhibits the Fermi-Pasta-Ulam type of recurrence phenomenon. An interesting result of these experiments is that for sufficiently steep initial waves the recurrence to a uniform wave train is accompanied by a decrease in frequency. Recently Trulsen and Dysthe [6] have published results on frequency down-shifting in three-dimensional wave trains in a deep basin. We have not observed frequency down-shifting numerically for inviscid waves except for a restricted time as mentioned in Section 3.3. Further stability analyses have been carried out by Longuet-Higgins [7] and McLean [8,9], which have confirmed the Benjamin Feir instability for a range of initial wave train steepnesses. There is another important instability of regular waves, the Tanaka [10,11] instability which occurs at the crest of steep waves. This is a relatively local response by the wave crest to disturbances, that affects only the steepest waves, that is ak > 0.43, H/L > The evolution of this instability in the ideal situation may take one of two courses, either breaking or adjustment to a lower wave; but for practical purposes it can be considered to lead to wave breaking since in the ocean environment there are many disturbances available to trigger the instability in this direction. Recent studies [12,13] clearly show that the Tanaka instability is not related to the wave train but is essentially related to the wave crest and hence is of equal significance for steep waves within wave groups. However, it is perhaps the Benjamin Feir instability which is more relevant to the evolution of ocean waves as it occurs for wave trains of steepness much less than the maximum steepness of steady waves. In considering the evolution of nonlinear deep water waves analytically, Lake et al. [3], Stiassnie and Kroszynski [14] and Lo and Mei [15] have used approximate equations. To third order in the initial wave steepness the slowly varying modulation amplitude satisfies the nonlinear Schrödinger equation [16 18]. The weakly nonlinear theory predicts strong modulation and recurrence of uniform waves; however, it cannot describe the wave breaking that occurs at high initial wave steepness. In this paper we use an efficient and accurate numerical code [19 21] to calculate the time evolution of small amplitude modulations on two-dimensional periodic deep water waves. It solves the fully nonlinear inviscid irrotational flow in a spatially periodic domain. Laplace s equation is solved using boundary integrals with the advantage that only a point discretisation of the surface is required. Boundary conditions are used in a Lagrangian form and the wave profile may be followed through to the initial overturning at breaking. We have completed many numerical runs varying the initial steepness and number of waves in the original wave train. The early evolution is characterised by a growth in modulation which either contains a steep wave which subsequently breaks or provided the initial steepness is not too great, demodulation occurs and a nearly uniform wave train recurs. We have also carried out some much longer runs over which several peak growths in the modulation are recorded. Dold and Peregrine [22]

3 K.L. Henderson et al. / Wave Motion 29 (1999) showed how modulations grow to fully modulated waves for a range of initial perturbations to a uniform wave train. This paper reports work which focusses on the evolution of such modulations with a brief report extending the Dold and Peregrine results. Banner and Tian [23,24] report results from the same Dold and Peregrine program and study the details of waves in a group as they approach breaking. Here we are mainly concerned in the wave evolution over long time periods and hence frequently keep the initial wave conditions such that the waves do not break. Some of the features of the modulations are consistent with the weakly nonlinear theory, in that for higher initial steepnesses the modulations are larger and occur more quickly than for lower initial steepnesses. By using appropriate scalings for the nonlinear Schrödinger equation we compare several computations and find good agreement particularly at lower steepnesses. This result is powerful in that, provided the initial steepness of the wave train is not too high, it renders the number of waves in the original wave train an almost redundant parameter, and allows more use to be made of each computation. 2. Water wave modulation In the next two sections we describe the numerical code and results of the evolution of small amplitude modulations on a uniform wave train Numerical code The numerical code used for the calculations was developed by Dold and Peregrine [19,20] and a detailed description of the method can be found in [21]. The code is based on a Cauchy theorem boundary integral for the evaluation of multiple time derivatives of the surface motion, thus the entire motion of an inviscid, incompressible, irrotational fluid may be modelled using a point discretisation of the surface. Cauchy s integral theorem is used iteratively to solve Laplace s equation for successive time derivatives of the surface motion and time-stepping is performed using truncated Taylor series. Thus the code is computationally fast and efficient and a detailed investigation into the efficiency and accuracy of the code was performed by Dold [21]. Time-stepping in the code is adjusted to ensure overall conservation of energy. Boundary conditions are chosen to make the spatial domain periodic. This domain may contain many waves. In order to run the code an initial wave surface together with the appropriate velocity potential on that surface must be supplied. This gives enormous freedom in the choice of initial conditions. In this paper we describe the case which is easy to specify yet appears to lead to natural waves: that is a small modulation of a uniform wave train. Given some suitable initial conditions the program can either track the surface profile of the fluid for many thousands of linear time periods, with a slight smoothing imposed as described in [21], or up to the initial overturning stage of wave breaking. For a particular computation the main parameters of the system are the steepness of the initial wave train and the number of these primary waves in one spatial period of the imposed initial modulation. The steepness of the initial wave train is measured in the form ak where a and k are the dimensional amplitude and wave number of the waves. We adopt length and timescales which set the dimensionless wavelength of each initial wave to 2π and scale gravity to 1. Another way of thinking of initial steepness is the ratio of the overall height of the wave divided by its wavelength, this quantity H/Lis related to initial steepness by the formula H/L = ak/π. Initial steepnesses considered in this work are mostly in the range 0.05 <ak<0.12, i.e < H/L < If a comparison is made with actively generated wind wave fields the energy per unit areas are equivalent to values of the wave steepness, ak, in the region where breaking occurs for all but the shortest wave groups. The initial conditions used throughout this work for the surface elevation η and corresponding velocity potential φ on the surface are made up in the following three stages: (i) Once the initial steepness ak is specified we compute an accurate steady wave form [25,26] on deep water which to first order in wave steepness would have the form η = ak cos(x)

4 344 K.L. Henderson et al. / Wave Motion 29 (1999) for the surface elevation and φ = ωak sin(x) for the velocity potential. We choose to take 16 discretisation points per wave since tests indicate that this affords a high level of accuracy whilst keeping running times reasonably low [21]. (ii) The number of waves in the initial modulation is specified and the computational region is then taken to contain n waves, where typically 4 <n<16. (iii) To this wave train perturbations of the form and ( n + m η p = εak cos x + θ n ) + εak cos ( n m n ) x + θ = 2εak cos(x + θ)cos ( ) n 1/2 ( ) ( ) n + m n 1/2 ( ) n m φ p = εak sin x + θ + εak sin x + θ n + m n n m n ( mx ) n are added to the surface elevation η and the velocity potential φ, respectively. This gives a modulation length of 2πn/m where m was taken to be 1 or 2. We will use the quantity l = n/m to denote the number of waves in the modulation. For the majority of runs the phase shift θ was taken to be π/4 as it has been indicated to give the most rapid growth of the modulation [27]; the effect of changing the phase is discussed later in the paper. The value of ε was typically taken to be 0.1, though in other studies a value of 0.05 has also been considered [22]. The periodic nature of the computational domain means that the effect of the above initial conditions is to give a weak periodic modulation on the wave train. The number of waves in one modulation must be consistent with the periodic nature of the spatial domain. Thus with one modulation in the domain up to 25 waves have been included, if two modulations are in that same domain then there are 12( 2 1 ) waves in each modulation. For five modulations in the same domain the modulation length is five waves. Of course, this latter case is much quicker to calculate by using a spatial domain of five waves ab initio. Once the initial conditions have been established the computation is followed in time until a specified end point is reached, in some cases illustrated in this paper this is over thousands of periods. The program stops as the numerical approximations fail if the surface curvature becomes too high as in the tip of an overturning jet in wave breaking. For waves of high initial steepness breaking typically occurs within 100 periods. For longer runs, over several hundred linear time periods, a slight level of smoothing is required to eliminate a very slowly developing steep wave instability as described in [21], while maintaining a high level of accuracy in the simulations Results For the first part of our results we have continued the work of Dold and Peregrine [22]. They, using the initial conditions described in Section 2.1, numerically followed the evolution of a modulated wave train for up to 200 linear time periods. For the number of waves in one modulation in the range 2( 2 1 ) l 10 they indicated the values of the initial steepness, ak, for which: (i) waves grew to breaking, (ii) the modulation grows and uniform waves recur, and (iii) there is no growth of modulation. In this paper we have extended the range of the parameter space in investigating the first evolution of the modulation. We report results from many computations with differing values of the two main parameters of the system, the initial steepness ak and the number of waves in the modulation l. The computation was continued until either a nearly uniform wave train was recovered or until wave breaking occurred and the computation broke down. A summary of these results is illustrated in Fig. 1, where open circles indicate the modulation grows and uniform waves recur, the crosses indicate the waves grew to breaking and the filled circles indicate no growth of modulation was observed. Some of the smaller modulations of 2( 2 1 ) and three

5 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 1. Summary of wave modulation computations. Crosses indicate waves grew to breaking, open circles indicate a growth in modulation with a recurrence of uniform waves, filled circles indicate no growth in modulation was observed. waves are too short to be unstable. Details of an instability of just two waves and its evolution can be found in [7,28]. Dold and Peregrine [22] considered the particular case of an initial five-wave modulation with steepness ak = 0.11 and followed its evolution over 200 linear time periods. This is a representative example of cases where the small initial modulation evolved into a steep, short wave group and back to a recurrence of a near uniform wave train. They also reported that: (i) for times of a few periods, linear theory gives a fair indication of wave behaviour, even for the steepest non-breaking waves, but the wave shape is nonlinear and (ii) for long time intervals, the weakly nonlinear NLS equation gives a qualitatively correct evolution, with quantitative discrepancies which are consistent with Dysthe s [17] higher order approximation. The results of a computation with an initial steepness of ak = 0.092,H/L = and nine waves in the modulation is illustrated in Fig. 2. The surface and modulation profiles are plotted every 10 linear time periods against x c g t, where c g = 2 1 is the non-dimensional linear group velocity of waves on deep water. The dotted lines represent the wave surface and the shaded regions enclosed between the solid lines represent the modulation envelope. In plotting the surface and modulation out at intervals of 2n periods the effects of the linear phase velocity and the linear group velocity lead to a first-order superposition of the wave and modulation positions and one can clearly see the evolution of the modulation. The main features of the evolution of the modulation illustrated in Fig. 2 are discussed in the following paragraph. It can be seen that the modulation grows developing a short group of steep waves of about wavelengths, a steep wave event (SWE), whilst either side of this group the amplitude of the remaining waves is small relative to initial values. In fact the modulation has near-zero minima either side of this short group. It can also be seen that over the time range of maximum growth the modulation travels slightly to the right indicating that the group is moving faster than the linear group velocity. Around the time of maximum modulation a wave crest is lost, so at T = 110 periods

6 346 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 2. Surface and modulation profiles plotted against x c g t for a nine wave modulation, initial steepness ak = at times of 0, 10, 20,...,160. The dotted lines represent the wave surface and the shaded regions enclosed between the solid lines represent the modulation envelope. Vertical exaggeration is five times. there are eight waves in one spatial period of the computation; however, as the surface evolves the initial number of nine waves is recovered and thus no permanent frequency down-shifting is observed. A fairly regular wave train is recovered at T = 160 periods with a change in phase relative to the initial profile. These features can also be seen in Fig. 3 in which the surface height for every second, linear time period is contoured in the (x c g t,t) plane. The loss and subsequent recovery of one wave crest can clearly be seen. The more detailed structure of the steep wave event is displayed in Fig. 4 in which we plot the wave profile against x from T/2π = at intervals of T /2π = 1/8. This picture clearly shows the effects of group velocity, that is that individual waves travel with speed c whilst the wave group, for waves on deep water, travels with speed c g = c/2. Even though during the SWE the waves are highly nonlinear the linear relationship for the velocity is still a good approximation over several time periods. A consequence of the group velocity being half the phase velocity is that if the surface elevation in the ocean is measured by a wave gauge at a fixed point in space then twice as many waves will be observed as that occur at a fixed point in time. To illustrate this a time-trace is plotted in Fig. 5 at the spatial position x = 5.77 over a time period of T/2π = It can be seen that the group has an approximate length of 3, twice as long as the groups shown in the spatial domain, as in Fig. 4. To further illustrate the group moving through the field we plot contours of the surface height in space time in Fig. 6. This figure clearly shows the enhanced phase velocity of the steeper waves. The evolution of a modulation that evolves to breaking follows a similar pattern, until a few periods before breaking, to the evolution structure of a modulation that grows and then decays. This is illustrated in Fig. 7 which is a similar picture to Fig. 2 but with an initial steepness of ak = 0.093, i.e. H/L = It can be seen that for the first 100 time periods the surface and modulation profile is very similar to the case of ak = However, in

7 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 3. Contour plot of the surface height, y, plotted against x c g t at every second time period for a nine wave modulation and initial steepness ak = Fig. 4. Surface profiles between T = 108 and T = 113 periods at time steps δt = 1/8. Initial steepness ak = 0.092,H/L = , modulation of nine waves. Vertical exaggeration is three times.

8 348 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 5. Time trace of surface height, y, atx = 5.77 for a nine wave modulation, initial steepness ak = Fig. 8(a) we plot the wave surface for an initial steepness of at T = 109 and it can be seen that an extreme event has occurred, the same wave is drawn to scale in Fig. 8(b). This steep wave has a sharp crest and the computation breaks down shortly after. Some much longer numerical runs have been computed for a range of initial steepnesses and number of waves in the modulation. The results of one such computation are illustrated in Figs. 9 and 10. Fig. 9 shows a time series, at frequent time intervals, of the maximum surface height for the case of an initial 14-wave modulation with initial steepness of The graph shows a thick line since there is a relatively rapid modulation, of period approximately 2, in the maximum height. The higher edge of the shaded area occurs when a wave crest is at the maximum of the modulation. The lower part of the shaded area shows the case when a wave trough is at the maximum of a modulation, if there is only one modulation. The range between trough and crest lines also indicates the spatial length of each modulation. In this example there are four particularly steep wave events in a time of 3500 wave periods, and some lesser events. The thickness of the line depicting this time series can be almost completely reduced by taking time series every two periods. That is just the time it takes for one crest to replace another in the modulation. Now, with much smoother data a better picture of the modulation can be obtained. The whole wave profile is stored every two periods, then an estimate is made of the modulation that gives the waves envelope. This modulation is then contoured in the modified space time, (x 2 1 t,t), plane to give Fig. 10. Fig. 10 uses the same computation as Fig. 9, but now we can see the location of each SWE event and see that in some instances two smaller ones are occurring at the same time. An idea of the sparsity of the maximum SWEs can immediately be obtained from Fig. 10. For example suppose the initial waves have a 10 s period so that their wavelength is approximately 160 m. The spatial domain is then 2.2 km and the duration of Fig. 10 is almost 10 h, yet only four very steep SWEs have occurred.

9 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 6. Contour plot of surface height, y, over the time range of maximum modulation for a nine wave modulation, initial steepness ak = Fig. 7. Surface and modulation profiles plotted against x c g t for a nine wave modulation, initial steepness ak = at times of 0, 10, 20,...,100. The dotted lines represent the wave surface and the shaded regions enclosed between the solid lines represent the modulation envelope. Vertical exaggeration is five times.

10 350 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 8. (a) Wave profile at T = 109 periods of a nine wave modulation, initial steepness ak = Vertical exaggeration is five times. (b) Same wave profile as that in (a) drawn to scale. Fig. 9. Time series of the maximum waveheight within the computational domain, for initial steepness ak = 0.06,H/L = 0.019, initial modulation = 14 wavelengths. 3. Nonlinear Schrödinger equation Several features of these modulations are consistent with analytic solutions of the nonlinear Schrödinger equation (NLS) which is the simplest approximate equation for weakly nonlinear modulations, e.g. see [29]. The steeper events are shorter in both space and time than the lower events Introduction The velocity potential of a two-dimensional modulated water wave train on deep water can be described, to first order by the expression φ(x,y,t) = A(x, t)e ky e i(kx ωt) + complex conjugate, (1) where A(x, t) is the slowly varying complex modulation amplitude, ω the frequency and k is the wave number of the modulated wave, or carrier wave which has an amplitude of s = 2k A /ω. Then to first order in the length of

11 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 10. Modulation amplitude contours for the same run as Fig. 9 (white indicates > 0.17, black < 0.02). the modulation, i.e. for very long modulations the modulation amplitude A satisfies the equation A t + c g A x = 0, (2) where c g = ω (k) is the linear group velocity of the modulated wave. That is long modulations of a uniform wave train travel at the group velocity. The next approximation, which is correct to third order in small wave steepness and long modulation length, takes into account the weakly nonlinear dispersive effects and the gradient of the modulation. This gives the equation for the complex modulation amplitude as 2iω(A t + c g A x ) c 2 g A xx = 4k 4 A 2 A. (3) This equation for A can be transformed to the self-focussing nonlinear Schrödinger equation iq T + q XX + 2 q 2 q = 0 (4)

12 352 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 11. Scaled modulation histories. (a) Initial steepness: ak = 0.07,H/L = , modulation of 12 waves; (b) initial steepness: ak = 0.056,H/L= , modulation of 15 waves. by the transformation T = 1 2 ωt, X = kx 1 2k 2 2 ωt, q = A, (5) ω where denotes complex conjugate [16 18]. Solutions of the NLS equation can be transformed from one steepness to another by the scaling q = q/q 0, x = q 0 X, t = q0 2 T. (6) As a result we make the hypothesis that for waves that do not grow too steep (a criterion to be made specific by considering computational results), the modulation growth should scale as solutions of the NLS equation and the actual number of waves in the initial modulation becomes an almost redundant parameter. To evaluate this hypothesis several computations have been run with appropriate matching of the number of waves with the initial wave steepness so that if this hypothesis held then the same modulation pattern would emerge. Fig. 11 shows two results similar to Fig. 9 since the length, time and steepness scales have been suitably adjusted. Although there are

13 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 12. As Fig. 11. Initial steepness ak = 0.084,H/L= , modulation of 10 waves. some discrepancies the major features of the wave evolution are closely comparable to a scaled time of almost 1000, i.e. over 2000 periods for Fig. 11(a) and over 3000 periods for Fig. 11(b). The NLS equation is formulated including terms of order s 2 in wave steepness, thus the timescale for the evolution of the modulation could be expected to vary as 1/s 2, e.g. 200/π wave periods in the case of s = 0.1. What we find from our computations is that there is good qualitative agreement over a much longer time scale than is expected, particularly when the initial waves are not too steep. In Fig. 12 the length, time and steepness scales have been suitably adjusted to be comparable with the plots in Fig. 11. In this case the initial steepness was ak = and the results are qualitatively similar to a scaled time of 400. Inspection of these results and those of other similar computations gives an indication that this approach holds for waves which grow no steeper than ak = 0.3,H/L = Once waves become steeper than this critical steepness the modulation regime appears to take on a different character, see the latter portion of Fig. 12 and see the irregular behaviour of Chereskin and Mollo-Christensen [30]. This more irregular behaviour is partly because there are often more than one or two wave groups within the calculations. However, we note that these solutions are like the homoclinic orbit solution of Ablowitz and Herbst [31] (see Section 3.3). They show how although the NLS equation is integrable, and hence does not have chaotic solutions, the perturbations introduced by numerical approximations may lead to the numerical version of this solution being chaotic. It would therefore not be surprising if the stronger nonlinear effects that arise for the steeper waves may also bring the possibility of chaotic solutions. Even these very long numerical computations are far too short to give any definite indication about chaos Changing the phase The numerical results presented so far have all been for an initial phase of the modulation relative to the carrier wave of 45. We now investigate how changing the initial phase effects the growth of the modulation. Many numerical runs have been carried out with an initial modulation of 12 waves, initial steepness ak = 0.07, modulation amplitude ε = 0.1, varying the value of the initial phase, θ, of the modulation. Some of the results are presented in Fig. 13 in which we plot a time series of the maximum surface height for values of the phase shift θ = 300, 285, 270,...,15, 30, 45. If the initial number of waves in the initial modulation is even then the results for phase θ and phase θ will be identical, so computations were only made for θ = 0, 15, 30,..., 150, 165. Fig. 13 shows that changing the phase has a dramatic effect on the time and size of the SWEs that occur over the first 1000 linear time periods. Note we are considering differences where 5 corresponds to a shift of the modulation by only 72 1 of a wavelength of the underlying carrier wave. The value

14 354 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 13. Time series of the maximum surface height for a 12-wave modulation of initial steepness ak = 0.07, for values of the phase shift θ = 300, 285, 270,...,15, 30, 45. of the phase for which the most rapid growth occurs is around θ = 30 with the first SWE occurring at about 200 linear time periods and the second SWE occurring at about 550 linear time periods. However, as we move away from this initial phase the position in time of both the first and second SWEs is later. At the initial phase of θ = 120 it can be seen that the first SWE occurs close to 350 linear time periods and the second at 900 linear time periods. In addition we can see that the two SWEs are much smaller than for other values of the phase. The initial conditions used in the numerical code are equivalent to an initial condition of q = h 2[1 + 2εe iθ cos(x/l)]

15 K.L. Henderson et al. / Wave Motion 29 (1999) Table 1 The maximum surface elevation during the first SWE, y max and the time, measured in linear time periods, T ymax at which it occurs for an initial 12-wave modulation with steepness ak = 0.07 and phase shift, θ. θ( ) T ymax y max in the NLS equation, where l = n/m represents the number of waves in the modulation. We can investigate the early development of this solution by performing a linear analysis on the solution q(x,t) = be i2b2t (1 + ɛ(x, T )), where b = h 2 and q(x,0) = q, assuming ε 1. Following the methods of Ablowitz and Herbst [31] we find that the early development of the modulation is governed by the equation ɛ(x,t ) = 2εbl {(l 2 + i ) sin(β θ)e T + (l 2 i ) sin(β + θ)e T } cos(x/l), (7) where = (4l 2 b 2 1)/l 2 and tan(β) = (4l 2 b 2 1). Thus there is an initial condition for which there is no initial growth in the modulation; the expression multiplying the e t term must be equal to 0 which implies that θ = β. Recalling that b = h 2, where h = ak is the initial steepness, we have that β = tan 1 { 8l 2 (ak) 2 1} and for an initial 12-wave modulation of steepness ak = 0.07 this gives β = Further time series of the maximum surface elevation are plotted in Fig. 14 for an initial 12-wave modulation with steepness ak = 0.07 with values of the phase close to β. We can see that for values of θ very close to β the SWE occurs at around 550 linear time periods which is much later than for other values of θ, see Fig. 13. In addition the SWE is much larger for θ = 114.9, 115, 115.1, than for θ = 113, 114, 116, 117. The largest SWE occurs for θ = rather than for θ = β, probably due to nonlinear effects becoming important. It can also be seen that the profiles are symmetrical about θ = 115, in that there is qualitative agreement between the times series for θ = and θ = 115.5,θ = 116 and θ = 114,θ = 117 and θ = 113,as is indicated from the linear analysis (7). Returning to the linear solution (7), it could be argued that in order for the first SWE to occur most rapidly, the expression multiplying the e t should be maximised. This gives a value for the phase angle of θ 1 = β Thus for an initial modulation containing 12 waves of steepness ak = 0.07,θ 1 = This result is confirmed in Table 1. Here we present the maximum surface height attained during the first SWE, y max and the time at which it is attained, measured in linear time periods, T ymax for an initial 12-wave modulation with steepness ak = 0.07 for phase angles θ = 60, 45, 35, 30, 27.5, 25, 22.5, 20, 15, 0. It can be seen that the SWE occurs most quickly for values of θ between 15 and 35 and that the maximum height attained is largest in the middle of this range. As the phase moves out of this range the SWE occurs later and is smaller in amplitude.

16 356 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 14. Time series of the maximum surface height for a 12-wave modulation of initial steepness ak = 0.07, for values of the phase shift θ = 113, 114, 114.5, 114.9, 115, 115.1, 115.2, 115.5, 116, Soliton solutions One of the potentially useful analogies with the NLS equation is that there are explicit solutions, in particular the Ma soliton [32], which corresponds to the periodic growth and decay of an isolated SWE. Peregrine [18] took the analytic solution for the Ma soliton and performed a double Taylor Series expansion about the amplitude peak (which occurs at x = t = 0) to arrive at the soliton q = e 2it ( 1 4(1 + 4it ) 1 + 4x t 2 ). (8) Formally this is the nonlinear sum of a finite plane wave of unit scaled amplitude and a soliton of zero amplitude. The maximum amplitude, of three times that of the uniform plane wave, is isolated in both space and time. The simplicity of the analytical expression in Eq. (8) makes it the most convenient approximation to one of the SWE. This soliton grows to three times the initial uniform wave amplitude. In considering the results of our numerical

17 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 15. (a) A for the isolated Ma soliton in solid line scaled such that the amplitude is the same as A estimated from an initial 14-wave modulation, ak = 0.06 at T = 247.6, shown in dashed line. The velocity potential for the isolated Ma soliton is the dotted line and the velocity potential from the water wave modulation in long dashes. (b) T = (c) T = 252 (d) T = 255 where the isolated Ma soliton is taken to travel with the group velocity calculated to third order and the wave itself travels with the phase velocity calculated to third order. Vertical exaggeration is 35 times. computations in the region for which the NLS scalings gives good agreement, we observe that three times the original wave amplitude appears to be a good measure of the steepest SWEs, see Figs. 9 and 11. Considering in particular the results in Fig. 9 of a 14-wave modulation of initial steepness ak = 0.06, we find that the largest SWE occurs at T = π (taken to be the time at which the maximum wave elevation occurs). An estimate of the modulation amplitude A at this time was made and the isolated Ma soliton fitted around these data, using the one free parameter to match the maximum height of A at the amplitude peak. These results are shown in Fig. 15(a), where the amplitude of the isolated Ma soliton is drawn in a solid line and the water wave modulation amplitude as

18 358 K.L. Henderson et al. / Wave Motion 29 (1999) a dashed line. Also illustrated in this figure is the velocity potential φ from the water wave modulation computation, drawn as a long dashed line and the velocity potential of the isolated Ma soliton, drawn as a dotted line, calculated from Eq. (1) and centred to get the best fit in the SWE. It can be seen that there is excellent agreement between the isolated Ma soliton and the water wave modulation particularly close to the amplitude peak; however, the isolated Ma soliton has two zeros either side of the peak whilst the modulation has two non-zero minima. Further from the amplitude peak the isolated Ma soliton has a higher value than the modulation amplitude. Turning to the comparison between the two velocity potentials, there is excellent agreement in the phase and amplitude close to the amplitude peak and reasonable agreement beyond the modulation amplitude minima. However, there is a difference between the two around the amplitude minima. In this region the isolated Ma soliton has two crests where the water wave modulation solution has only one, representing a temporary down-shift in frequency. In Figs. 15(b) (d) the same quantities as in Fig. 15(a) are plotted at times of T = π, T = 252 2π and T = 255 2π, respectively. The isolated Ma soliton is calculated from Eq. (8) with appropriate values of t and is taken to travel at the group velocity taken to be half the phase velocity calculated to third order. The velocity potential is again calculated from Eq. (1) with the wave itself travelling at the phase velocity c = 1 + (ak) 2. Again there is good qualitative agreement between the isolated Ma soliton and the modulation at the three times; however, it can be seen that the modulation has moved slightly ahead of the isolated Ma soliton at T = 255 2π. In comparing the velocity potentials over the three cases although there is good agreement away from the amplitude peak, it can be seen that the numerical steep water wave travels faster than the wave enclosed by the isolated Ma soliton and this effect is most noticeable in Fig. 15(d) at the furthest time from the amplitude peak. However, this good agreement over several time periods is encouraging and it has led us to look at other analytical solutions of the NLS equation satisfying periodic boundary conditions. Ablowitz and Herbst [31] have developed an analytical solution of the NLS equation which is periodic in space. It can be written as u(x, T ) = a o exp(2iao 2 T)1 + 2 cos(x/k)exp( T + 2iφ + γ)+ A 12exp(2 T + 4iφ + 2γ), (9) cos(x/k)exp( T + γ)+ A 12 exp(2 T + 2γ) which they call a homoclinic orbit, where k 1 = 2a o sin φ, =±k 1 4a 2 0 k 2,A 12 = sec 2 φ. Choosing T such that ε 0 = exp( T + γ)is small and second order terms in ε 0 are neglected, the solution (9) can be linearised to give an initial condition of ( u(x, 0) = a o 1 + ε ) o a 0 k cos(x/k)e i(β+π). Thus comparison with Eq. (7) shows that the homoclinic orbit identified by Ablowitz and Herbst is only appropriate to water waves when θ = β + π or, if there is an even number of waves in the modulation, when θ = β. For an initial modulation of 12 waves, steepness ak = 0.07, phase angle θ = β = 114.9, we find that the largest SWE occurs at T = π. An estimate of the modulation amplitude A at this time was made and the Ablowitz and Herbst solution calculated such that it has maximum amplitude (this occurs when exp( T +γ)= cos φ). These results are shown in Fig. 16, where the amplitude of the Ablowitz and Herbst solution is drawn in a solid line and the water wave modulation amplitude as a dashed line. Also illustrated in this figure is the velocity potential φ from the water wave modulation computation, drawn as a long dashed line and the velocity potential of the Ablowitz and Herbst solution, drawn as a dotted line, calculated from Eq. (1) and centred to get the best fit in the SWE. It can be seen that the Ablowitz and Herbst solution is noticeably lower than the water wave estimate of the modulation amplitude and consequently there is not such good agreement with the velocity potential profiles. We see that over a long time period corresponding to the growth of a SWE some discrepancies develop. The qualitative picture is still good, and as illustrated with the isolated Ma soliton if the match is made with the maximum height then for moderate time intervals agreement is usefully close.

19 K.L. Henderson et al. / Wave Motion 29 (1999) Fig. 16. A for the Ablowitz and Herbst solution in solid line, A estimated from an initial 12-wave modulation, ak = 0.07,θ = at T = , shown in dashed line. The velocity potential for the Ablowitz and Herbst solution is the dotted line and the velocity potential from the water wave modulation in long dashes. Vertical exaggeration is 35 times Higher-order approximations for modulations The NLS equation is derived by using approximations for small wave steepnesses and long modulation scales. Further approximations can be made and the next order equations were obtained by Dysthe [17]. A new feature arises at this approximation: the waves are influenced by small currents that are set up by the modulation gradients. They have a velocity potential which appears in the Dysthe equation: 2iω(A t + c g A x ) ω2 4k 2 A xx 4k 4 A 2 A = ik 3 (6A 2 A x 6AA A x 2 A 2 A x ) + iω2 8k 3 A xxx + 2ωkA( x i z ). The first group of terms on the right-hand side, in brackets, are nonlinear terms. It can be seen, by considering A = Re iθ, that the first two terms have a steepness dependent effect on phase but no effect on amplitude. The third term is easily seen to give an increase of 2 1 c gk 2 s 2 to the group velocity, recalling that the amplitude of the modulation is given by s = 2k A /ω. This corresponds to half of the Stokes correction to the phase velocity and is clearly seen in the computations. The third-derivative term is the next order improvement in approximation to the linear dispersion, and the final term gives the interaction with the induced current. This current comes from solving Laplace s equation for with the boundary condition z = 2k 2 ( A 2 ) x /ω at the mean free surface. There is a corresponding deviation of the mean surface equal to (1/g) t, but this is too small to be significant at this approximation. Note that we are now including terms as small as s 3 in wave steepness. These might be expected to be important only on a timescale of 1/s 3, e.g. (2000/π)T in the case of s = 0.1. Solutions of the Dysthe equations have been studied by Lo and Mei [15] and show differences from the NLS equation. The strongest is the group velocity increase already noted; however, the group velocity/phase velocity ratio appears to be unchanged as is also found from computations of the steepest SWE. Lo and Mei [15] show a steepening of the modulation at the front of wave groups; however, this does not appear in the evolving wave groups that are characteristic of our accurate computations. This is probably because these wave groups evolve too rapidly, on the shorter timescale of the NLS equation. This evolution on the faster NLS timescale could be because these groups are not sufficiently close to the steadily propagating solutions for envelope soliton solutions, hence giving little opportunity for the nonlinear contribution to group velocity to have a significant effect on the modulation shape. 4. Conclusions We have carried out many computations following the nonlinear evolution of a modulated wave train varying the initial wave steepness and the number of waves in the modulation. We have extended the work of Dold and

20 360 K.L. Henderson et al. / Wave Motion 29 (1999) Peregrine [22] in characterising the growth of the modulation over these two parameter ranges. They fall into two main categories; either the group contains a sharp-crested wave which breaks or otherwise having formed a short group of steep waves at the peak modulation, demodulation occurs and we find a recurrence to a near uniform wave train results. During the stages of peak modulation the linear group velocity is still a good approximation over several periods. We have also completed some much longer runs, over thousands of time periods. The initial wave train soon develops a strong modulation and then shows moderately regular variation with times of strong modulation. In comparing numerical runs of different initial steepnesses there are similarities to analytic solutions of the weakly nonlinear theory, in that modulations of steeper initial wave trains are both shorter in space and time. A scaling in space and time allows the initial steepness to be scaled out of the nonlinear Schrödinger equation. We compare appropriately scaled numerical computations and find excellent agreement over a much longer timescale than is expected particularly for lower initial steepnesses. This is a significant result and allows greater use to be made of each computation, reducing the parameter space to include only the steepness of the initial wave train. Another important result of the favourable comparison with weakly nonlinear theory is that the nonlinear Schrödinger equation has analytic solutions, in particular the isolated Ma soliton which closely corresponds to a short group of steep waves. The longer numerical computations that have been performed provide a realistic model of the ocean surface. Several of the SWEs that form could be used as design waves in modelling structures that need to withstand the full range of wave and current conditions that are expected for its lifetime in that position. Although the computational method uses a boundary integral technique, for any chosen time, a further simple computation gives any subsurface fluid properties that are required. As experiments and theory show, the long time evolutions shown here are vulnerable to three-dimensional instabilities, especially for the steeper waves. Nonetheless, the results and comparisons with the nonlinear Schrödinger solutions and scaling can be of value for practical application and give encouragement for the use of the nonlinear Schrödinger equation in three-dimensional modelling, particularly for cases where unsteady evolution means that the waves are not close to the steadily propagating solutions which give an opportunity for higher-order effects to be significant. The major higher-order effect showing in our computations is the enhanced group velocity which is described by the Dysthe equation. However, although the groups travel faster than the linear group velocity, they do not survive long enough for it to have much effect on their shape. Acknowledgements Support from the Marine Technology Directorate of EPSRC and from industrial contributors to the programme on Uncertainties in Loads of Offshore Structures under contract GR/J23624 is gratefully acknowledged. Dr. K.L. Henderson gratefully acknowledges the support of the Nuffield Foundation. References [1] T.B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. Roy. Soc. A 299 (1967) [2] T.B. Benjamin, J.E. Feir, The disintegration of wave trains on deep water. Part 1. Theory, J. Fluid Mech. 27 (1967) [3] B.M. Lake, H.C. Yuen, H. Rungaldier, W.E. Ferguson, Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train, J. Fluid Mech. 83 (1977) [4] W.K. Melville, The instability and breaking of deep-water waves, J. Fluid Mech. 115 (1982) [5] M.-Y. Su, M. Bergin, P. Marler, R. Myrick, Experiments on nonlinear instabilities and evolution of steep gravity-wave trains, J. Fluid Mech. 124 (1982) [6] K. Trulsen, K.B. Dysthe, Frequency downshift in three-dimensional wave trains in a deep basin, J. Fluid Mech. 352 (1997) [7] M.S. Longuet-Higgins, The instabilities of gravity waves of finite amplitude in deep water. II Subharmonics, Proc. Roy. Soc. London A 360 (1978) [8] J.W. McLean, Instabilities of finite-amplitude water waves, J. Fluid Mech. 114 (1982)

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