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Ontheshapeofnonlinearwavetrains
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ON THE SHAPE OF NONLINEAR WAVE TRAINS
A. Toffoli1, A. Babanin2, E. Bitner-Gregersen3, A. K. Magnusson4,J. Monbaliu1, J. Portilla1, G. Dumon5, M. Onorato6
Abstract: As the water depth decreases or the sea severity increases, non-linearity becomes more relevant and waves deviate from the Gaussian surface.Second–order wave theory is, in general, able to capture much of the nonlinear-ity. Measurements made in finite water depth, however, show that second–orderwave profiles fail to restore second–order effects at high nonlinear levels. Influ-ence of directional spreading and combined seas is also looked at.
INTRODUCTION
The theory associated with the Gaussian processes is well understood and commonly usedto derive the sea surface elevation at a fixed spatial location. As the water depth decreases orthe sea severity increases, nonlinearity becomes more relevant and waves deviate from theGaussian surface. The most obvious manifestation of such a nonlinearity is the sharpeningof the wave crests and flattening of the troughs, which has several consequences such as theunusually large dynamic structural responses (e.g. Pastoor et al., 2003).
For an actual sea state, the wave energy is distributed along both frequencies and direc-tions. Longuet-Higgins (1963) proved theoretically that the skewness (Eqn.1) of the freesurface elevation for a directional sea state in deep water depth is limited within a lowerbound corresponding to the superposition of two orthogonal long–crested seas and an upperbound corresponding to a single long–crested sea.
Prevosto (1998) studied the effects of directional spreading on the nonlinearity in bothdeep and intermediate water depth. The analysis, which was based on simulated profilesfrom a JONSWAP spectrum and the Mitsuyasu directional distribution, concluded that thehypothesis of unidirectionality remains conservative in deep water but not when the depthis sufficiently shallow to affect the nonlinear behavior. A real sea state is often, however, a
1Dept. Civil Engineering, K.U.Leuven, Kasteelpark Arenberg 40, 3001 Heverlee, Belgium, e-mails:[email protected], [email protected], [email protected]
2Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Hawthorn, Victoria3122, Australia, [email protected]
3Det Norske Veritas AS, N1322 Høvik, Norway, [email protected] Meteorological Institute, Alleten 70, 5007 Bergen, Norway, [email protected] and Maritime Affairs Administration, Ministry of the Flemish Community Vrijhavenstraat 3,
8400 Oostende, Belgium, [email protected] di Torino, via P. Giuria 1, 10125 Torino, Italy, [email protected]
1 Toffoli et al.
combination of two or more wave systems i.e. the wind sea and the swell. These wave trainsare characterized by a certain directional spreading and are free to travel along any directions.An analysis undertaken within the E.U. project MaxWave on ship accidents reported as beingdue to bad weather (Toffoli et al., 2004) showed, in that respect, that a relatively high numberof casualties might have occurred during sea state with wind sea and swell along almostperpendicular directions (i.e. crossing seas).
In the present study the nonlinearity of wave trains is classified by using a general steep-ness factor. By means of in situ measurements, nonlinear effects on wave shape character-istics (i.e. wave heights, crests, troughs, steepness, and horizontal asymmetry) are conse-quently looked at and compared with second-order wave theory at different degree of non-linearity. Particular emphasis is also given to the influence of different combinations of thewind sea and the swell. We begin with a general description of the nonlinear and shapeparameters that are used thereafter. The next section introduces wave data sets. Then wepresent results of combining measurements and numerical simulated data. We finally givesome conclusions.
WAVE PARAMETERS
Several definitions to describe an individual wave exist in literature. Throughout thispaper an individual wave at a fixed time or position is defined as the part of the record, whichfalls between two consecutive zero down–crossing. Specifically, the distance between thestill water level and the highest point of the wave is defined as the wave crest (HCr), whilethe distance between the still water level and the minimum value of the wave representsthe wave trough excursion (HTr). The wave height is given by the sum of crests and troughs(H = HCr+HTr). The average of the highest 1/3 of the waves is by definition the significantwave height (H1/3), which can also be calculated from the variance of the wave spectrum(m0): Hm0 = 4.01
√m0.
Each individual wave is not only described by its height, but also by its period. An averagedescription of the wave period is given herein by means of the peak period TP = 1/ωP ,where ωP is the frequency carrying the maximum amount of energy. The wavelength at anarbitrary depth can be approximately calculated as a function of the deep water wavelength(λ0 = f(T0)) and the water depth (d): λ = λ0
√
tanh(2πdλ−10 ) (e.g. WMO, 1998). The ratio
of the wave height to the wavelength is a measure of the wave steepness (ε).
Real wind waves tend to have crests greater than troughs (i.e. vertical asymmetry) and tobe tilted to the front side (i.e. horizontal asymmetry). A measure of the vertical asymmetryis given in Eqn.1:
β3 = E|x− µ|3σ−3 (1)
where µ and σ are the mean and standard deviation of the sample, and E|·| is the expectation.The skewness is equal to 0 for normal distributed wave elevations (linear waves), and greaterthan 0 for nonlinear waves.
Coefficients describing the horizontal asymmetry were introduced by e.g. Myrhaug andKjeldsen (1986). For the present work, the horizontal asymmetry (ς3) is calculated as theskewness of the 90–degree shift of the original signal, which is derived from the Hilberttransform. Although the first derivative would be a more appropriate definition of the 90–
2 Toffoli et al.
degree shift, for our purpose differences are marginal. Whereas a value of zero correspondto symmetric waves, positive asymmetry indicates wave fronts steeper than wave backs.
NONLINEAR FACTORS
Water waves are normally described in terms of a wave potential Φ(x, y, z, t) such thatthe water particle velocities are given as the spatial derivative of Φ. In case of uniform waterdepth, Φ is determined by the following boundary value problem (e.g. Dean and Dalrymple,2000):
∇2Φ = 0 (2)
∂Φ
∂t+
1
2(∇Φ)2 + gz = 0 at z = η(x, y, t) (3)
∂η
∂t+∂Φ
∂x
∂η
∂x+∂Φ
∂y
∂η
∂y− ∂Φ
∂z= 0 at z = η(x, y, t) (4)
∂Φ
∂z= 0 at z = −h (5)
The boundary conditions are not linear and a solution (Stokes wave) can be obtained from aperturbation expansion (Eqn.6).
Φ = Φ(1) + Φ(2) + . . . whereΦ(n+1)
Φ(n)= O(ε) (6)
ε is a parameter typically proportional to the wave steepness. If the wave slopes areinfinitesimal, the perturbations are negligible (linear problem). But if they are not very small,higher order approximation are needed. For deep water the steepness represents, therefore,a degree of nonlinearity.
In case of arbitrary depth (i.e. d/λ < 1/4, WMO, 1998) the perturbation parameter (ε)assumes a more general form (Whitham, 1974), which is expressed in Eqn.7 in terms ofwavelength:
µ2 =H
λ· coth(2π
λd) · (1 + 3
2 sinh2(2πλd)
) (7)
This general nonlinear factor (µ2) tends to the value of H/λ for deep water waves and toa form of the Ursell number (Ursell, 1952) for shallow water waves.
DATA SETS
In situ measurementsThe data used throughout this study were collected from two locations in the North Sea,
i.e., the Ekofisk field (ConocoPhillips) and the Belgian coast, and from Lake George, Aus-tralia.
Within the framework of the Monitoring Network Flemish Banks (operated by the Min-istry of the Flemish Community), wave surface profiles have been recorded by non–directionalWaveriders, and stored continuously at the sample frequency of 2Hz since January 2003.Within the same time period, additional two–dimensional spectra measured by directionalbuoys were, also, made available. Although measurement devices are deployed at several
3 Toffoli et al.
locations, data used for this study were limited only to location Bol van Heist (BVH). Thebathymetry varies due to the tide between 9.5 meters and 14.5 meters. The range of thenondimensional water depth is representative of intermediate water depth waves.
At the Ekofisk field observations have been stored in the environmental database of theNorwegian Meteorological Institute since 1980, and include wave profiles sampled at 2 Hz.The instruments comprise one Waverider buoy and two vertically pointing lasers. At thislocation the water depth reaches 74 meters and measurements are representative of deepwater. Data are limited to the period from January 2003 to June 2003.
An integrated set of measurements in the atmospheric, sub-surface boundary layers, andon the surface were carried out at the Lake George field experimental site from September1997 to August 2000 (Young et al., 2005). The lake bottom in the region of the observationsite is relatively flat. As the lake was drying out, following its natural cycle, the waterdepth gradually changed from 1.1 m in the beginning of the measurements down to 0.4 mby the end of the experiment in 2000. Under typical meteorological conditions the range ofnondimensional depth was mainly representative of intermediate depth wind sea. The surfacewave field was measured by means of a spatial array of 8 capacitance gauges sufficiently farfrom any disturbances.
Second–order wave profilesA second–order expansion of the sea surface can capture the effects of wave steepness,
water depth, and directional spreading up to the second order (e.g. Forristall, 2000). For thepurpose of the present work, second–order deep and intermediate water depth wave profileswere simulated under both directional and unidirectional conditions.
A second–order directional wave theory at an arbitrary water depth was derived by Sharmaand Dean (1981) as an extension of the theory developed by Longuet-Higgins (1963) for in-finite water depth.
A linear solution of the boundary value problem in Eqn.2 to 5 can be expressed as inEqn.8, where t is the time, and x is the position vector. ωi, εiθ, and kiθ are the angularfrequency, phase, and vector wave number respectively of the Fourier wave components ata given frequency and direction (i, θ). Note that frequency and wave number are related bythe linear dispersion relation, i.e., ωi = g|kiθ| tanh(|kiθ|d).
The second order correction to the linear solution is given by Sharma and Dean (1981),and is expressed in Eqn.9.
η(1)(x, y, t) =M
∑
θ=1
N∑
i=1
aiθ cos(kiθ · x− ωit+ εiθ) (8)
η(2)(x, y, t) =1
4
M∑
θ,ψ=1
N∑
i,j=1
aiθajψ{K−
iθjψ cos(ϕiθ − ϕjψ) +K+iθjψ cos(ϕiθ + ϕjψ)} (9)
where ϕiθ = kiθ · x − ωit + εiθ, and K+iθjψ and K−
iθjψ are the interaction kernels. The fullexpression for the interaction kernels can be found in Eqn. 34 to 37 of Sharma and Dean(1981), and also in Eqn. 12 to 19 of Forristall (2000).
4 Toffoli et al.
A second–order simulation of a given wave spectrum was performed with Eqn.8 and9 by assigning the wave amplitudes (ai =
√
2S(ω)∆ω for the unidirectional case, andaiθ =
√
2S(ω, θ)D(ω, θ)∆ω∆θ for the directional case), the wave frequencies, the wavenumbers, and a uniform distributed random phase εiθ on the interval [0, 2π]. For two–dimensional simulation, the use of the same spectral variance at each frequency does not in-clude all of the natural variability (Forristall, 2000). To restore the actual variability Prevostoand Forristall (2002) and Birknes and Bitner–Gregersen (2003) used a Rayleigh distributedamplitude for their simulations. If a directional spectrum is simulated, the addition of sev-eral wave components with different directions and same frequency produce automaticallythe natural variability of the amplitudes (Forristall, 2000). As suggested by Forristal (2000),the amplitude is randomized only for two-dimensional simulations for consistency.
A set of two-dimensional deep water wave simulations were performed by using a JON-SWAP spectrum with significant wave height Hm0 of 11 meters, peak period Tp of 12 sec-onds, and at water depth of 10000 meters (µ2
∼= ε ∼= 0.050). Intermediate water depth waveswere also obtained using a JONSWAP spectrum with the same Hm0 and Tp but at two dif-ferent depths: 40 meters (µ2 = 0.1, ε = 0.055), and 30 meters (µ2 = 0.15, ε = 0.060).Although a TMA spectrum (Bouws et al., 1985) is more appropriate to describe finite waterdepth, simulations of dimensionless heights and crests did not show any significant differ-ences in wave statistics, at least not within the limit of this study.
Three–dimensional simulation were performed to study the effects of different combina-tions of wind sea and swell for both deep and intermediate water depth. To this end, waveprofiles were generated by using the Torsethaugen (Torsethaugen, 1996) two–peak spectrumas a point spectrum and the Mytsuyasu (see e.g. Goda, 2000) directional distribution withHm0 = 6 meters, Tp = 8 seconds, and a depth of 10000 meters (µ2
∼= ε ∼= 0.06) as well as25 meters (µ2 = 0.084, and ε = 0.0625). The s coefficient of the directional function wasassumed to be equal to 10 for wind waves and 25 for swell (Goda, 2000). Different combi-nations were obtained considering the swell along the wind sea direction (following sea), aswell as along a 45 and 90 degree direction with respect to the wind sea (45o and crossing searespectively) (fig.1).
30
210
60
240
90
270
120
300
150
330
180 0
30
210
60
240
90
270
120
300
150
330
180 0
30
210
60
240
90
270
120
300
150
330
180 0
Fig. 1. Combined sea states: a) following sea; b) 45o sea, and c) crossing sea.
For two–dimesional simulation 2048 frequencies, 4096 time steps, and a sample fre-quency of 4 Hz (the latter is in agreement with Tayfun, 1993) were used. A total of 1000time series were consequently simulated (≈ 110000 waves). Three–dimensional simulations,on the other hand, were perfomed using 1024 frequencies, 15 directions within the interval[−π,+π], 2048 time steps, and a sample frequency of 4 Hz. 510 repetitions were made.
5 Toffoli et al.
EFFECT OF QUASI–LAGRANGIAN BEHAVIOR ON WAVE RECORDS
The wave records are collected with both fixed and floating instrumentations. The fixedsensors (Eulerian devices) are able to provide an absolute measure of the height of the seasurface at a fixed position, and give in principle an accurate profile of the evolution of thewave form. The representation of the sea surface using floating instruments differs from thatof fixed devices because of a quasi-Lagrangian behavior. Longuet–Higgins (1986) showedthat this yields deeper troughs and lower crests. In order to recover the actual Euleriantime history, a quasi-lagrangian correction method proposed by Magnusson et al (1999)was applied. The process assumes that the horizontal motion of the instrument is in onedirection; the buoy follows the fluid velocity at a depth of 0.2 m; and the stretching of themooring system is negligible.
For the deep water profiles collected at Ekofisk, wave crest statistics from both Eulerianand corrected Lagrangian measurements are comparable. In the case of intermediate waterdepth, nonlinear effects become stronger, and the deviation of wave crest statistics observedat Lake George confirm this. Unexpectedly, however, corrected wave profiles from BVHinstrumentation show lower crests (almost Rayleigh distributed) (fig.2a) and deeper troughs.Most likely the influence of the mooring system on measurements is not negligible becauseof a combined effect of the relatively strong tidal and wind induced currents.
0 0.5 1 1.5
10−6
10−4
10−2
100
HCr
/ Hm0
Pro
ba
bili
ty o
f E
xce
ed
an
ce
RayleighEkofisk − Fixed DeviveEkofisk − Floating DeviceBVH − Floating DeviceLake George − Fixed Device
(a)0 0.5 1 1.5 2 2.5
10−6
10−4
10−2
100
H / Hm0
Pro
ba
bili
ty o
f E
xce
ed
an
ce
RayleighEkofisk − Floating DeviceEkofisk − Fixed DeviceBVH − Floating DeviceLake George − Fixed Device2nd order model
(b)Fig. 2. Probability of exceedance of wave crests (a) and heights (b) for different in situmeasurements.
The quasi lagrangian motion produces an effect of the same magnitude on both crests andtroughs (Magnusson et al., 1999). Therefore it does not modify the wave heights. More-over, nonlinearity up to the second order does not affect the crest-to-trough wave height (e.g.Tayfun, 1980). A comparison of in situ and simulated wave heights collected from differ-ent devices and at different degrees of nonlinearity shows, in that respect, good agreement(fig.2b). Nonetheless, measurements taken from the Belgian coast show a deviation of waveheight statistics starting at ≈ 10−3. The probable effects of the mooring system yields anenlargement of the troughs greater than the cutting of the crests.
NONLINEAR EFFECTS – UNIDIRECTIONAL HYPOTHESIS
Classification of nonlinearityA classification of wave nonlinearity is given on the basis of the generalized nonlinear
factor (Eqn.7) as follow: C1 → 0.02 ≤ µ2 < 0.06; C2 → 0.06 ≤ µ2 < 0.10; C3 →
6 Toffoli et al.
0.10 ≤ µ2 < 0.15; C4 → 0.15 ≤ µ2 < 0.25. As observed from in situ data, class C1represents deep water waves, i.e., µ2
∼= ε. Note that µ2 is calculated using Hm0 and TP .
Deep water wavesDeep water waves usually show a weak nonlinearity, and hence a deviation from linear
theory can be observed (fig.3a). Much of the theoretical nonlinearity can, however, be cap-tured by calculations corrected to second order. Forristal (2000) and Prevosto and Forristal(2002) concluded that second–order simulations of wave crests agree well with measure-ments of high wave crests. According to these works, similar results were found comparingEkofisk field data (from the laser instrumentation) and simulated profiles (fig.3a).
0 0.5 1 1.5 2
10−6
10−4
10−2
100
HCr
/ Hm0
Pro
babi
lity
of E
xcee
danc
e
Rayleigh2nd order µ
2 = ε = 0.05
Ekofisk µ2 = 0.050
Lake George µ2 = 0.043
(a)0 0.5 1 1.5
10−6
10−4
10−2
100
HCr
/ Hm0
Exc
eeda
nce
Pro
babi
lity
Rayleigh2nd order: µ
2 = ε = 0.05
2nd order: µ2 = 0.10
2nd order: µ2 = 0.15
Lake George: µ2 = 0.087
Lake George: µ2 = 0.133
(b)Fig. 3. In situ measurements vs. 2nd order simulation: a) deep water; b) intermediatewater.
Intermediate water depth wavesAs the relative depth decreases, the nonlinearity becomes stronger and the factor µ2 starts
depending on both the steepness and the shallow water factor (kd). Moreover, due to thepresence of coth and sinh in Eqn.7, µ2 increases rapidly within the finite water regime. Theenhancement of nonlinear asymmetry can be seen in fig.4. Class C4 (highest nonlinearity)contains only 4% of the total number of waves. This should explain the rapid decay of itsdistribution for probability of exceedance lower than 10−2 (line of class C4 in fig.4a).
The class of lowest nonlinearity (C1) comprises waves around the deep and the interme-diate water depth regime, i.e., d/λ = 1/4 (WMO, 1998). Their vertical asymmetry can becaptured relatively well by second–order theory (deep water depth case). The ratio of theempirical skewness to the model skewness is relatively close to unity (Table 1). The discrep-ancy, which is observed mainly for low probability (fig.3a), can be explained by both a lackof data (only 1000 repetitions were performed), and finite water effects.
In case of intermediate water depth regime, at least according to our simulations andlimitations, the second–order theory fails to restore the measured nonlinearity, even thoughthe limit for its applicability is respected. Namely, as proposed by Dean and Dalrymple(2000), the magnitude of the Ursell number (Ursell, 1953) must be lower than ≈ 26, i.e.,λ2H/d3 < 8π2/3. For data, which fall in class C3, the Ursell number is on average equalto 8.5 << 26. Figure 3b shows, in that respect, the comparison between measures of classC2 and C3 (µ2 = 0.0875, and µ2 = 0.1325 respectively) and the two simulated data sets inintermediate water depth regime, which have µ2 = 0.10 and µ2 = 0.15 (edges of class C3).
7 Toffoli et al.
The deviation of in situ data is also evident from the ratio of the sample skewness (β3,field)to the model skewness (β3,2nd) at level µ2 = 0.10 as reported in Table 1.
0 0.5 1 1.5
10−6
10−4
10−2
100
HCr
/ Hm0
Pro
babi
lity
of E
xcee
danc
e
Rayleigh0.02<µ
2<0.06
0.06<µ2<0.10
0.10<µ2<0.15
0.15<µ2<0.25
(a)0 0.5 1 1.5
10−6
10−4
10−2
100
HTr
/ Hm0
Pro
babi
lity
of E
xcee
danc
e
Rayleigh0.02<µ
2<0.06
0.06<µ2<0.10
0.10<µ2<0.15
0.15<µ2<0.25
(b)Fig. 4. Probability of exceedance of Lake George’s in situ measurements: a) wavecrests; b) wave troughs.
Table 1. Vertical and horizontal asymmetryParameter C1 C2 C3 C4
β3,field/β3,2nd 1.10 1.42 2.47 2.78
ς3,field/ς3,2nd 1.1 1.0 1.0 0.7
Whereas enhancements of µ2 yields a sharpening of crests and flattening of troughs, i.e.,increasing of skewness, nonlinearity does not have any effect on the horizontal asymmetry, atleast within the limits of the present study. The horizontal asymmetry assumes a small con-stant value (almost symmetric profiles) for all simulated conditions. Assuming the secondorder simulated asymmetry as a normalizing factor, we observed that horizontal asymme-try (ς3) measured at the Lake George (wind–driven waves) slightly decreases towards largeclasses of nonlinearity (Table 1).
NONLINEAR EFFECTS – DIRECTIONAL HYPOTHESIS
A real sea state is characterized by a distribution of energy along both frequencies anddirections. One of the main effects that the directionality has on second–order theory indeep water is the reduction of the skewness (Longuet–Higgins, 1963). During following and45 degree sea states, which were simulated in this study, the directional spreading of thetwo spectra is similar (fig.1). Consequently, the reduction of skewness due to directionaldistribution is of the same magnitude (≈ 30%). Condition of crossing seas generates, onthe other hand, a larger directional spreading with a consequently larger reduction of thenonlinear effects (see Table 2).
Simulations performed under the hypothesis of intermediate water depth show that thedirectional spreading has only a limited effect on the vertical asymmetry. Table 2 showsin that respect that the skewness of the directional seas are almost similar to the one of theunidirectional case. Note that the skewness for directional seas (three–dimensional waves)is calculated with Eqn.1, which does not distinguish wave directionality, and hence resultsshould be used with care.
8 Toffoli et al.
Although nonlinearity (up to the second order) seems not to play any significant role interms of horizontal asymmetry, the latter appears to be weakly influenced by the directionalspreading. The ratio ς3,dir/ς3,uni is approximatelly equal to 1.05. Influence of different windsea and swell combinations was not observed.
According to second–order theory, wave statistics performed for different combined seas(i.e. following, 45–degree, and crossing seas) does not change (fig.5). Although their limi-tation, also wave heights measured from BVH confirms this result.
Table 2. Skewness during combined seas.β3,dir/β3,uni Following 45
o Crossing
d/λ > 1/4 0.72 0.74 0.66
d/λ < 1/4 0.98 0.98 0.99
0 0.5 1 1.5
10−4
10−3
10−2
10−1
100
HCr
/ Hm0
Exceedance P
robabili
ty
RayleighFollowing
45o
Crossing
(a)0 0.5 1 1.5 2 2.5
10−4
10−3
10−2
10−1
100
H / Hm0
Pro
ba
bili
ty o
f E
xce
ed
an
ce
RayleighFollowing
45o
CrossingBVH − Following
BVH − 45o
BVH − Crossing
(b)
Fig. 5. Wave statistics for combined seas: a)crests; b) heights.
CONCLUSIONS
Deep and intermediate water depth wave measurements were used to study wave shapecharacteristics at different levels of nonlinearity. A comparison with second–order theorywas also performed.
As expected, in an intermediate water depth regime, when the finite water depth effectsbecome stronger, nonlinearity enhances rapidly. Whereas second–order 2D simulated pro-files describes well the nonlinear effects in deep water, they fail to restore high degrees ofnonlinearity in finite water depths, even though the limit of the theory was not overtaken.
It is observed that directional spreading yields a reduction of the vertical asymmetry ofdeep water depth waves. In case of intermediate water depth, however, a limited influenceof the spreading was observed.
Horizontal asymmetry is not captured by the second–order theory. However, according tomeasurements taken at Lake George, nonlinear effects does not seem to have any significanteffects on horizontal wave asymmetry.
ACKNOWLEDGEMENTS
This work was carried out in the framework of the project F.W.O. G.0228.02 and G.0477.04.The wave profiles at the EKOFISK field were gathered by the Phillips Petroleum Company
9 Toffoli et al.
of Norway, and made available by Norwegian Meteorological Institute. BeGrid computerfacilities are also acknowledged.
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