THE FORMATION OF FREAK OCEAN WAVES William Bateman SUMMARY When sufficient energy within the ocean congregates at a particular point, resonant nonlinear effects lead to a series of changes in the distribution of energy that allows larger extremes to be created and also extreme crests that are locally more unidirectional in shape and often hundreds of metres wide. This has important consequences for mariners as it greatly increases the opportunity for an extreme wave encounter, with the position of the vessel or structure relative to the peak wave crest becoming increasingly irrelevant due to wide lateral extent of these waves. The scientific understanding behind how these events arise has improved in recent years with better numerical models and much greater availability of field data. There is now a growing understanding of the sea conditions that are most likely to promote these freak, or rogue, extremes, and therefore this provides an opportunity for seafarers to avoid these areas if they can recognise the signs. This paper attempts to identify some of these conditions by exploring the impact of third-order resonant interactions. NOMENCLATURE

k wave-number vector (k,l) k0 fundamental wave-number vector (k0,l0) kn wave-number in the principal horizontal

direction x, equal to kn = n ⋅ 2π /Lx = n ⋅ k0 ln wave-number in the principal horizontal

direction y, equal to ln = n ⋅ 2π /Ly = n ⋅ l0 ω wave frequency, T time Lx wave length in the x horizontal direction Ly wave length in the y horizontal direction ∇ differential operator, ∂ /∂x,∂ /∂y,∂ /∂z( ) φ velocity potential η surface elevation

1. INTRODUCTION Mariners have for many centuries passed around stories about giant walls of water that appeared unexpectedly out of the ocean. In 1995, the Master of the Queen Elizabeth 2 recalls seeing a ‘wall of water’ for a couple of minutes during a storm off Newfoundland. More recently, photos and videos have provided more evidence, while the measurements of a 18.5m high crest that struck the Statoil’s ‘Draupner’ gas platform on the 1st January 1995 provided the first conclusive proof that these freak events can and do occur. There are, however, two conventional problems with our understanding of these observed extremes: first they appear too often; and secondly they often appear to be the wrong shape. Statistically, the occasional occurrence of a particularly large wave is entirely natural. If the ocean is assumed to be a random ensemble of individual waves, then if you wait long enough, eventually this superposition will lead to something

large, and the longer you wait the larger the waves you can expect. A fit of the linear Rayleigh distribution to field measurements approximates the size and occurrence of crest heights well for the vast majority of events. However, very large extremes seem to be occurring far more frequently than the models predict. Therefore, waves that are expected to occur once every 1,000 or 10,000 years are actually arising every 100 or 1,000 years, respectively (Figure 1). While the measurement of waves, and particularly the measurement of extremes, is far from a precise science, the magnitude of this difference is difficult to ignore.

Figure 1: The normalised distribution of wave crest heights against annual probability of exceedance, for random waves measured in the MARIN basin. For the less frequent events, i.e. 1:100 year (10-2) or less, there is a distinct difference between the observed crest heights and the statistical extrapolations due to linear Rayleigh and Rayleigh + a second order correction [22].

Real seas are always directionally spread, such that wave energy moves through the ocean along a number of independent headings. An analysis of the field data recorded at the Tern Platform in the northern North Sea by Jonathan et al. [1] found the sea to be very broad-banded and strongly directional, while Tamuea et al. [2] demonstrated that a possible ‘freak’ event led to the loss of a fishing boat off the Japanese coast during the crossing of two strong storm surge systems. Directionality generally leads to a reduction in the length of extreme crests and also reduces nonlinear effects. Jonathan et al. demonstrated they could model all of their field measurements using only a second-order solution. Yet many freak waves are frequently described as ‘walls of water’ that are highly nonlinear with particularly long crests, perhaps 300-500m wide, which is inconsistent with a second-order solution. These differences between conventional wisdom and observations have prompted many questions, such as: Do observers in their panic over estimate the shapes of waves? Are wave gauges or satellites inaccurately measuring the seas? Is the measurement of directionality wrong? However, once you start to consider the higher order nonlinear interactions, in particular 3rd order resonant interactions, then many of the observed features can actually be explained by real physical processes within the sea that are all ignored by traditional numerical methodologies [5]. For the shipping industry, the frequency and shape of waves (height, width and slope) are all equally important considerations when determining the loads applied to a vessel and its response. Conversely, in the offshore industry the wave height is the driving consideration where fixed structures are typically designed so the decks avoid even the highest waves. The aim of the current work is to explore the characteristics of extreme waves of varying bandwidth and directional spread, and consider how the directionality of a sea can be reduced during the formation of a ‘freak’ wave event. 2. BACKGROUND 2.1 WAVE THEORY Water waves are caused by a movement of energy through the water, which creates a disturbance along the free surface due to the significant density difference between the water and air. With all ocean

waves, the energy alternates between kinetic (due to motion) and potential (due to height) energy, with gravity providing a restoring force that, like all natural systems, is attempting to reach an equilibrium with minimal energy; i.e. flat water. For the most part, ocean waves are so large that density differences in the water, viscous energy losses, and forces due to surface tension can be ignored. Although the additional energy input from wind is an important effect in the creation of extreme waves, it is a complicated process which is beyond the scope of the current investigation. Despite the surface shape of ocean waves being anything but sinusoidal, it is convenient to describe the ocean surface elevation and velocity potential using a summation of Fourier waves:

) cos(),,(,

, tlykxatyxji

ji ωη −+⋅= ∑ (1a)

where a is the amplitude of each wave component.

) sin())(cosh(

),,,(

,,

, tlykxhzkA

tzyx

jiji

ji ω

φ =

−+⋅+⋅∑

ki, j = ki2 + l j

2

(1b)

where and for a linear solution

)sinh(khkA αω

=

gk=2ω

k

.

For many seas, the wave amplitudes, a, are sufficiently small relative to the wave length, (L), that this formulation leads to a complete description of the ocean surface with k and ω related by the linear deep water dispersion relation, (2) This equation connects wave lengths to wave period such that:

=2π (3) L

ω =2πand T

. (4)

For rougher seas, the waves start to form a conidial shape, with peaked crests and shallower troughs, as shown in Figure 2.

114

Figure 2: Stokes wave profile, with peaked crests and shallower troughs. The blue dotted line is the linear solution; the green solid line is a 2nd order bound term; the red dotted line is the combined linear and 2nd order solution; the black solid line is a 5th order Stokes solution. Inset is a discrete amplitude spectrum of waves. In order to describe these waves it is necessary to consider further wave components, in addition to those given in (1a), which make the crests sharper and more peaks and troughs wider and flatter and also less shallow. These new terms are called ‘bound waves’ because they move with the same speed and direction as the underlying ‘free’ linear wave components. Fundamentally, they are nothing more than a correction to this original linear approximation that for other mathematical advantages assumes the waves are sinusoidal. These corrections occur at wave-numbers and frequencies that are approximate multiples of the original, yet the travel at the same speed as the parent wave, such that

knknc ωω

≈⋅⋅

= (5)

where n is nth nonlinear improvement to the underlying free wave. Thus, n=2 is a second order correction. For a regular, and therefore also unidirectional, single wave, Stokes [9] derived the necessary amplitudes for each correction up to the fifth order (n=5). In doing so he sought a solution that approximately satisfied the two key surface boundary conditions. The first is the kinematic boundary condition that relates the speed of the water particles at the surface with the motion of the surface itself, or equivalently, fluid particles on the free surface remain on the free surface (in the absence of wave breaking) ηt = φz −φxηx −φyηy (6)

The second is the dynamic boundary condition, from Bernoulli, which relates to the static convective and unsteady pressure terms,

φt = −12

(φx2 +φy

2 +φz2) − gη

k + = k1 + k2

ω + = ω1 + ω2

k− = k1 + k2

(7)

In a more realistic unsteady sea, which contains many free-waves of different frequencies, direction and phases, these boundary conditions still apply, only now many corrections are needed to bring sinusoidal waves into agreement with the true ocean. This leads to the deep water second order theory of Longuet-Higgins and Stewart [3], which was later extended to shallow water by Sharma and Dean [4]. In both cases, only a second-order improvement is calculated, which, in addition to the Stokes corrections above, also yields two new bound-wave components for every possible free-wave pair within the sea. These are often known as the frequency sum (+) and frequency difference (-) terms:

and (8) ω − = ω1 + ω2

The sum terms represent a concentration of energy towards the centre of the wave group, leading to a larger crest amplitude and steeper waves, which is balanced by a draining of energy both fore and aft of a wave group which is achieved by the different terms. The difference terms are often referred to as ‘set down’, since they locally lower the mean sea level beneath energetic groups. These second order corrections also lead to a greatly improved estimate of wave heights, increasing the crest elevation by up to 10-12% for steep seas. For strongly directional seas, second-order is often sufficient to describe even the largest crests, as confirmed by the analysis of Jonathan et al. [1]. 2.2 RESONANT INTERACTIONS In contrast to bound wave nonlinearities, there exists a series of higher order nonlinear interactions that can redistribute energy among the free-wave components within a spectrum. These changes are often called 'resonant' interactions, because of their reversible nature, which leads to energy oscillating between the underlying free-waves. Phillips [19] was the first to consider the transfer of energy among two pairs of wave components. He demonstrated that for energy transfers to occur, the following equations must be satisfied: 115

(9) 4231

4231

kkkk +=++=+ ωωωω

Each of these four waves must also satisfy the free-wave dispersion relationship given in (2): (10) nn gk=2ω To satisfy both (9) and (10) Phillips established the figure of eight resonance loop sketched in Figure 3, which includes a band of instability within which the conditions are approximately met. McLean [20] established that this was just the first of many loops, with higher order interactions, which involve more combinations of waves, becoming increasingly important with steeper waves.

Figure 3: The figure of eight resonance loop sketch by Phillips [19]. The vector k3=k1 is labelled 2k1. The conditions for energy movement require the energy from each wave to be in close proximity and also of a comparable phase. Achieving this condition is not easy and many energy transfers, that might otherwise have occurred, will fail because one or more of the four waves is in the wrong state. However, a particular case illustrated by the wave-number vectors in Figure 3 occurs when the k1 and k3 are identical in both magnitude and direction, i.e.

(11) 421

421

22

kkk +=+= ωωω

This leads to 3-wave resonant condition, which is substantially easier to form and therefore leads to more frequent and larger overall energy transfer. It is important to recognise that, unlike the bound waves, resonant interactions are a physical process (rather than a mathematical analysis) that take many wave periods to gradual transform the ocean energy distribution. 2.3 NUMERICAL SCHEME In order to study the formation of extreme waves the 3D numerical model of Bateman, Swan and Taylor [6], hereafter referred to as BST is used to simulate the frequency and directional focusing of

ocean energy. This model is a significant extension to the earlier work of Craig and Sulem [10] and has been carefully validated against experimental data of Baldock et al. [11], Johannessen and Swan [7], and repeatedly used in studies undertaken by both Gibson et al. [8] and Gibbs and Taylor [5]. The key approximations implicit to the BST model are that the fluid is incompressible and irrotational, and therefore inviscid. This means the main fluid domain conforms to the Laplace equation, ∇ . The kinematics throughout the domain can now be calculated directly from values around the boundary,

02 =φ

1 which in deep water reduces to only surface values of the velocity potential, φS(x,y,t), as

0

φz

. φ =z=−∞

Within the BST model, a Dirichlet_Neumann operator (G-Operator) is defined that converts values of φS into the vertical derivative of φ at the surface,

z=η= G(η)ϕ S

G(η) = Gi (η)i=0

m

∑

G0

(12) To create the G-Operator, a Fourier representation of φS and its vertical derivate ∂φS/∂z are expanded about still water level using a Taylor series. A comparison of terms of the same degree of homogeneity on both sides of (12) yield the necessary form of each order of the G-Operator within

(13)

the first three terms of G(η) being,

= D tanh(Dh)

1 D2 D tanh(Dh)ηD tanh(Dh)

G2 =12

(14a-c)

G = η −

η2D2D tanh(Dh) − ηD2ηD tanh(Dh)

−12

D tanh(Dh)η2D2

+D tanh(Dh)ηD tanh(Dh)ηD tanh(Dh)

D = −i

where h is the water depth, and d∂ / r is a complex radial derivative operator with

r = x2 + y2 . In deep water, tanh(Dh) reduces to the sign(D).

1 The Cauchy Integral Theorem is a classical mathematical approach that calculates the values at any point within the domain from those expressed around the boundary.

Within this G-Operator evaluation, multiplication by η must occur in physical space while the application of the D operator must be conducted in spectral domain. The transformation back and forth between physical and spectral domains is conducted efficiently using Fast Fourier Transforms (FFT), which also require that the wave domain is periodic in both x and y. As 80% of the computation effort is involved in performance of FFTs and their inverse, the computation time for the entire model is O(n log n), where n is the number of surface points. With this G-Operator in place, the time derivatives of η(x,y,t) and φS(x,y,t) can now be calculated using a combination of the total derivatives of φS with respect to each horizontal dimension and time,

φx z=η = dxφS − dxη⋅G(η)φ S (15a)

φy z=η

= dyφS − dyη⋅G(η)φ S (15b)

φt z=η = dtφ

S − dtη ⋅G(η)φ S (15c) where the horizontal total derivatives (dx, dy) of both φS and η can be calculated efficiently using simple Fourier techniques. Combining equations (15a-c) and (12) with the exact boundary conditions given in (6) and (7) yields the time derivatives of η(x,y,t) and φS(x,y,t). 2.4 INITIAL CONDITIONS In normal circumstances, particularly those concerning field data, an evolving wavefield is recorded as a time-history, η(t), at a single point fixed in space. As a result, it is characterised by a frequency spectrum, Sηη (ω) . In this paper, the main results are based on a fetch limited JONSWAP spectrum (Hasselman [16]), which is widely applied in offshore design calculations (see Figure 4a),

Sηη (ω) =αg 2

ω 5 exp −βω p

4

ω 4

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ γ

exp−(ω −ω p )2

2ω p2σ 2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ (16)

where wp is the peak frequency of the spectrum and

σ = 0.07, ω ≤ ω p

σ = 0.09, ω > ω p

Figure 4a: JONSWAP spectrum in frequency with γ=2, ωp=0.551, α=0.0081, β=1.25. With a peak enhancement factor of γ=1, the JONSWAP is identical to the fully developed sea states described by the Pierson Moskowitz spectrum, and is broadly identical to the Bretchsnider spectrum used in many marine applications. Although it is important to know the distribution of energy within the ocean, the next challenge is to identify what energy congregates during the formation of an extreme wave. Building on the work by Lindgen [13], Boccotti [14] and Tromans et al. [12] established a ‘NewWave’ theory that calculates the most probable shape of the highest wave in either space or time. This solution ultimately reduces to the autocorrelation of the underlying spectrum.2Thus, the expected or most probable shape of the water surface elevation, η, which is conditional on the presence of a large crest elevation, η=A, at some arbitrary time, t=τ, is

(17) η(t) A= ⋅ ρ(t)

where ρ(t)

ρ(t) =1

ση2

is the autocovariance function, ∞

Sηη (ω) cos(ωt) dω0∫

ση2 = Sηη (ω) dω

(18)

and the variance is defined by ∞

0∫

ai =

(19)

Adopting the temporal profile in (17), the amplitudes, ai required in (1), for each frequency components, ωI, may be identified with a Fourier transform, which is in essence a reversal of (18), such that,

A Sηη (ω i) ⋅ Δω (20) ση

2

2 A more extensive discussion of this technical can be found in Bateman [15], Gibbs and Taylor [5] and also Gibson and Swan [8].

To apply the wave model described above, both η(x,y,t) and φS(x,y,t) must be defined at some initial time, t=t0, at discrete spatial points evenly distributed over the entire computational domain 0<x<Lx and 0<y<Ly. The surface must also be periodic, therefore, in equation (1)

kn = n ⋅ k0 (21)

where k 0 =2πL

.

To satisfy (21) within (1), then Sηη (ω) may be converted into a wave-number spectrum, S . If it is assumed that all wave components are freely propagating and satisfied by the dispersion relationship of (2), then

ηη (k)

kSkS

∂∂

=ωωηηηη )()( (22)

with , so (20) is now redefined as ω2 = gk

ai =A

ση2 Sηη (ki) ⋅ Δk (23)

It is common practice in analytical and statistical descriptions of directionally spread wavefields to assume that a sea state may be represented by a uni- directional wavefield coupled with a frequency independent, directional weighting, function. Two widely applied examples of this weighting include the Mitsuyasu [17] spreading parameter, s, whereby

α(θ ) = β coss θ2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ (24)

and the application of a wrapped-normal distribution having a standard deviation of σθ so that

α(θ ) =β

σθexp −θ 2

2σθ2

⎛

⎝ ⎜

⎞

⎠ ⎟ (25)

where α defines the proportion of the total amplitude that propagates at an angle θ to the mean wave direction and β is a normalising coefficient dependent upon the model applied. To put these distributions in perspective, field data recorded at the Tern Platform suggests that the directional distribution associated with severe storms is typically represented by a wrapped-normal distribution with a standard deviation of σθ = 30° as shown in Figure 4b [1]. This also corresponds closely to a Mitsuyasu distribution with a spreading parameter of s=7. 118

Figure 4b: Wrapped-normal distribution, σθ = 30°

ai, = α(θ)

, with ideal analytical solution and actual spreading within a discrete 2D wave-number spectrum. To calculate the amplitudes in a directional spectrum with two wave-number coordinates k and l, then (23) is combined with (24) to give

Aj (26)

ση2 Sηη (ki, j ) ⋅ Δki, j

θ = tan−1 l j /ki

( ) where

ki, j = ki2 + l j

2 and

Δki, j = ko cos2 θ + l0 sin2 θ

Figure 5: Directional wave-number distribution, Tp=11.4s, γ=2.0, 30°. σ =θ A plot of ai,j is shown in Figure 5 for a typical North Sea condition. This also represents all the energy beneath the linearly focused wave event that is defined by NewWave. To start the BST model, the ai,j values are transformed into η(x,y,t) and φS(x,y,t), through the application of equations (1a-b). Unfortunately, at the focal time this wave is very nonlinear, while (1) is a linear solution. This is resolved by moving back in time until the surface is sufficiently dispersed that a linear solution becomes valid (Baldock et al. [11]). As a result, there are no large waves present within the computational domain. The approach first suggested by Johannessen [7] and later used in the numerical studies of [15,8,1] is to work back in time to a point

when the second-order correction to the surface elevation, η, (Dean and Sharma [4]) is less than 2% of the linear value.

The methodology underlying this process is highlighted in Figures 6 and 7. These calculations concern a JONSWAP spectrum with Tp=11.4s, γ=2.0, which at the focal location produces a NewWave profile with maximum elevation η=7.8m. Figure 6 provides a linear description of the maximum crest elevation occurring anywhere across the computational domain for times prior to the focal event, at t=0s. The directional simulation disperses quickly, so a start time of t=-200s for BST would seem appropriate. However, for the unidirectional case, it could be argued that the initial time should be chosen so as to minimise the maximum crest elevation, or at least select a time when the minimum value is asymptotically approached. This would be t=-2000s in Figure 6, which is very long time before the focal event. To resolve this, Johnnessen used a second-order initial conditions, which is valid so long as the correction is less than 2%, which occurs at t=-800s in Figure 7 for a unidirectional simulation.

Figure 6: Maximum surface elevation anywhere across the domain, from a JONSWAP spectrum with Tp=11.4s, γ=2.0 and σθ for unidirectional case and = 0° σθ = 30° for the directional case. Maximum surface elevation in both cases is η=7.8m.s

Figure 7: Magnitude of the second order correction. For a directional problem, there is little need for the second-order solution as directional dispersion rapidly reduces the surface to something that is

appropriately linear. In addition, the second-order method by Dean and Sharma [4] is actually incredible slow, taking longer to compute the velocity potential, φ, at one time, than the BST model takes to simulate 20 subsequent periods. For the current purposes a directional sea is create using an initial surface calculated using (1) with t=-200s, which forms the input to BST, see Figure 8a. The BST model then evolves the surface from this point forward without any further inputs to create the surfaces illustrated in Figure 8b-c

Figure 8a: Initial surface elevation at t=-200s, for Tp=11.4s, γ=2.0 and 30°. σ =θ

Figure 8b: Intermediate surface elevation at t=-100s.

Figure 8c: Focal point, and position of largest wave. 3. EXTREME WAVES In terms of crest height, the most extreme waves you can model for a specific sea condition will, by definition, be on the brink of breaking. Adding more energy to these waves will cause it to break

119

earlier in its evolution. However, near breaking waves will also exhibit the most nonlinear and interesting changes, the largest crest velocities and the largest vertical accelerations. As the BST model is a pseudospectral method with a surface elevation that must be a single valued function of the horizontal coordinate, then simulations are limited to the early stages of breaking before the surface becomes vertical. For this reason, the current work considers the simulation of waves at about 98% of the maximum numerical limit, by considering the maximum vertical acceleration in the Lagrangian frame of reference,

DwDt

=∂w∂t

+ u ∂w∂x

+ v ∂w∂y

+ w ∂w∂z

(27)

From an analysis of Johannessen’s [7] laboratory data concerning waves close to breaking, Bateman [15] established that when Dw/Dt≈-0.38g a real wave will break. This is well below the upper limit of -0.5g established by Longuet-Higgins [17], who looked at conditions within steady Stokes waves with a perfect 120° corner; but very consistent with his later work [18] for unsteady unidirectional waves groups that have accelerations between -0.388g in the crest and 0.315g in the trough. With the initial conditions established in Section 2, the BST method is now used to model the full nonlinear evolutions of the surface. A in equation (26), is gradually increased until the Dw/Dt≈-0.38g is achieved once at some point during the evolutions, typically near the linear focal time. Each of these cases uses a JONSWAP spectra (14) with the peak enhancement varied from γ=1 to 5 and a wrapped normal distribution (25) from σθ = 0° (unidirectional) and σθ = 60° . In the following subsection, these results are analysed. 3.1 MAXIMUM HEIGHT In Figure 9, the near breaking maximum surface elevations are plotted, with the largest waves for any peak enhancement (γ), occurring between σθ = 20° and 30°. Interestingly, this is consistent with conditions in the northern North Sea, where γ≈3.3 (c.f. [1]). 120

0 10 20 30 40 50 60

Spread, σθ (degrees)

Spectral Peakiness, γDeep Water Depth (h=∞)

1 (PM)23

10

11

12

13

14

ηm

ax (m

)

45

Figure 9: Maximum surface elevation, JONSWAP spectra (Tp=11.4s, α=0.0081, β=1.25). 3.2 PERMANENT ENERGY CHANGES Once you have eliminated additional inputs from wind and losses due to friction, the energy within the overall ocean must remain constant. This is particularly true of a model like BST, which ignores these features. This energy within a wave is proportional to the wave amplitude squared, i.e

∑∝i

iaE 2 (28)

where the ai comes from a Fourier transform of the surface elevation. While tracking E is useful to ensure stability of the numerical model, it adds little other value. However, the direct summation,

∑ ∝i

iaA (29)

is more informative because it reveals how the energy is distributed across different frequencies, although it does not tell us at which frequencies the energy resides. High values of A correspond to energy being evenly distributed across a lot of frequency components, while the lowest possible value of A occurs when all the energy is concentrated into a single frequency wave component.

-200 -150 -100 -50 0 50 100 150 200

Time (s)

0.8

1.0

1.2

1.4

1.6

A /

Ain

p

UnidirectionalLong Crested, 10 degIntermediate, 30 degShort Crested, 60 deg

UnidirectionalLong Crested, 10 degShort Crested, 60 deg

0

2

4

6

8

10

12

Max

imum

Ele

vatio

n, η

max

(m)

-200 -150 -100 -50 0 50 100 150 200

Time (s)

Figure 10: Changes in energy distribution, γ=3. In Figure 10, the relative change in the amplitude sum (A) relative to the initial input (Ainp) against time are shown for four cases, all with γ=3. In all these, the main wave focuses around t=0, but in the three directional cases there is a permanent change in A either side of the focal event. However, with the unidirectional simulation A gradually returns to its original value. The inclusion of directionality appears to tear apart the energy, making previous resonant energy movement permanent. In unidirectional seas, the 3rd or even 4th order resonant interactions are less effective because all the components lie in the same direction, so from Figure 3, k1≈k2≈k3≈k4. Although Benjamin and Feir [21] have demonstrated near resonant interaction that lead to instabilities of a regular wave train, this has not been demonstrated in broad-banded transient seas. This is confirmed in Figure 10, with the unidirectional A returning to unity at approximately t=300s. Despite these large changes, the physical difference between the unidirectional and long-crested ( °= 10θσ ) sea surface is surprisingly small. This is illustrated in Figure 11, which plots the maximum surface elevation at any point across the surface of each case. The amplitude sum (A) also has another important characteristic. It effectively represents the largest possible wave elevation that can be achieved within the given sea state. This would occur if all wave components were perfectly in phase, which is virtually impossible. Nevertheless, if A is permanently increased, then statistically this new sea state is more likely to create larger waves that the original lower value. Therefore, one question to consider is: Having formed one extreme event, should we expect to see others?

Figure 11: Maximum surface elevation, γ=3.

0 10 20 30 40 50 60

Spread, σθ (degrees)

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Am

ax /

Ain

p1 (PM)2345

Spectral Peakiness, γ

Deep Water Depth (h=∞)

°

Figure 12: Maximum change in the amplitude sum ratio, illustrating the degree of nonlinearity at the focal time. γ=3. Figure 12 summarises these results around the focal point with a plot of the maximum achieved ratio of A/Ainp. For the unidirectional cases, there is an increase of over 60% in the amplitude due almost entirely to the creation of bound wave energy. However, with a small amount of directionality, i.e.

= 10θσ , this ratio is increased by over 7% and can only come from 3rd (or higher) order resonant interactions. As directionality is increased, the bound terms reduce dramatically and as Jonathan et al. [1] found, a reasonably spread sea can be described adequately with a second-order solution (c.f. Figure 7). This is confirmed in Figure 12, where the ratio A/Ainp ratio is less than 20% once °> 30θσ . One remarkable feature in the formation of a focused wave crest is that global nonlinear behaviour is essentially identical to that of a focused trough, when the initial conditions are 121

inverted, i.e. π out of phase. As Gibbs and Taylor [5] explain, the formation of an extreme wave is dependent on the group properties of the wave field, such as amplitude and shape, rather than the absolute phasing of the waves within the group. To illustrate this, a Stokes-like perturbation expansion for a wave-crest group would be of the form,

∑ ∑∑ +++=i kji

kjiijkji

jiijicrest aaacaaca,,,

.....η (30)

where all the amplitudes, a, are positive and c are interaction coefficients. Repeating this for a trough-focused wave group where all the amplitudes are now negative, then only the odd-terms of expansion change sign,

∑ ∑∑ +−+−=i kji

kjiijkji

jiijitrough aaacaaca,,,

.....η (31)

Thus, the addition of (30) and (31) yields a surface due only to even order interactions, while a subtraction will yield only odd order interactions, i.e.

ηodd = (ηcrest − ηtrough ) / 2 (32)

-5.0e-4

-3.5e-4

-2.0e-4

-5.0e-5

1.0e-4

2.5e-4

4.0e-4

-0.04

-0.02

0.00

0.02

0.04

l (ra

d/m

)

0.00 0.02 0.04 0.06 0.08k (rad/m)

Focal Point

(2)

a)

(1)

(1)

-0.04

-0.02

0.00

0.02

0.04

l (ra

d/m

)

0.00 0.02 0.04 0.06 0.08k (rad/m)

Final (t=200s)b)

(3)

(4)

(2)

Amplitude (m)

Figure 13: Net energy transfers due to odd order interactions a) at the focal point, b) at the final dispersed solution after the nonlinear extreme. Red is a reduction in energy; blue is gained energy. Vertical line corresponds to the initial peak of the spectrum. γ=3,

°= 30θσ . With a Fourier transformation of (32) the amplitude spectrum will contain only the underlying free-waves and 3rd, 5th, 7th, etc order interactions. While 5th+ ordered interactions are included within these results, the scale of these terms drops exponentially with each additional order so that their contribution is minimal. In Figure 13, a sample of the net odd-order energy transfers are shown for: a) the focal time; and b) the fully dispersed sea after the focal event. 122

In a) energy is removed from the peak of the spectrum in regions (1) to (2), which is at a lower frequency with reduced directionality. In b) there is a partial reversal of some of these energy transfers; in particular region (1) is restored ('white' represents the norm), but we also see substantial growth in (4) with energy continually being removed from the inline spectral peak. Relating these changes back to Figure 3, the dominant shift is energy from wide angles to slightly lower and higher frequencies can only occur if k2 and k4 received the energy from k1, the central frequency. It is difficult to see how one resonant interaction can achieve both of these goals; instead, it is more likely that two separate processes are ongoing simultaneously. In the first, Figure 3 is rotated so k2 lies close to the mean wave direction and energy is, on average, drawn from wave-number k1 and k4 (c.f. Figure 14-a), i.e.

k 2 2 k1⇐ − k 4

k4

(33) In the second set of interactions, Figure 3 is rotated so k4 lies closer to the mean wave direction and energy is drawn from k1 and k2 (c.f. Figure 14-b),

2k1 k2⇐ − (34) What is important to note is that while the magnitude k2<k1 and k1<k4 , the changes in absolute wave number observed in Figure 13 are subtle although the changes in direction are large. To maintain a small change in wave-number, then all three vectors must lie relatively close to each other in order to satisfy the figure of eight resonance loop of Figure 3 (see sketch in Figure 14). Therefore to effect a large change in direction, a very large number of fairly similar resonant interactions occur (differing mainly by change in the mean direction), that cascade energy from wide angles toward the mean direction of the group.

Figure 14: Sketch of wave-number vector, with blue vectors receiving energy and red loosing energy. a) left diagram with energy moving to low frequency. b) right diagram has energy moving to higher frequency.

k2

k1 k4

k2

k1

k4

t=-100s t=-90s t=-80s t=-70s

t=-60s t=-50s t=-40s t=-30s

t=-20s t=-10s t=0s (*) t=10s

t=20s t=30s t=40s t=50s

t=60s t=70s t=80s t=90s

Figure 15:Net energy transfers due to odd order interactions. γ=3, σθ = 30° . (*) is the approximate position of the nonlinear extreme event. To some extent this can be seen in Figure 15, which plots the net change in odd-order interactions for a range of times. At the earlier times, most of the energy transfers occur close to the mean wave direction and change directionality of the group very little. However, as focusing continues, two distinct islands (c.f. region (1) in Figure 13) gradually form as energy is pumped into region (2). After the focal event, the spectrum changes only gradually, with most of the changes occurring in the first two wave periods (T=11.4s). In Figure 16, the changes in the angular distribution of amplitude are shown, which at the initial time is identical to Figure 4b. Figure 17a presents the same data, but from a plan view, while in Figure 17b only the odd-order contributions to the distribution are given. This last figure demonstrates the true impact of the 3rd- (or higher) order resonant interactions, and their impact on the underlying free-waves (1st order). Contrasting with Figure 17a, it is now clear that nearly all the reductions in directionality are being driven by 3rd order interactions. Nevertheless, there is a clear tendency for this sea state to become more unidirectional in nature, and therefore longer crested.

For completeness, Figure 18 plots the maximum reduction in directionality for each of the JONSWAP seas (c.f. Figure 9).

Figure 16:Variations in directional distribution of amplitude with time, γ=3, = 30° σθ

-80 -60 -40 -20 0 20 40 60 80

Angle (deg)

-200

-150

-100

-50

0

50

100Ti

me

(s)

0.00

0.05

0.11

0.16

0.21

All orders

Amplitude (m)

Figure 17a: Plan view of variations in directional distribution of amplitude with time, γ=3, 30°. σ =θ

-80 -60 -40 -20 0 20 40 60 80

Angle (deg)

-200

-150

-100

-50

0

50

100

Tim

e (s

)

0.00

0.05

0.11

0.16

0.21

Odd orders

Amplitude (m)

Figure 17b: Variations in directional distribution of amplitude due only to odd order interactions, γ=3, 30°. σ =θ

0 10 20 30 40 50 60

Spread, σθ (degrees)

0.7

0.8

0.9

1.0

θ e / 1

.645

σ θ2

1 (PM)2345

Spectral Peakiness, γ

Deep Water Depth (h=∞)

Figure 18: Reduction in directionality: Tp=11.4s, α=0.0081, β=1.25. 3.3 WALLS OF WATER To demonstrate the creation of a wall of water, the author has chosen to refer to work by Gibbs and Taylor [5], whose focus was on the changes in the surface properties rather than the changing nature of the underlying energy as the author has presented above. Key to their work was the video processing they performed on the results from the BST model to highlight the main wave group and its internal wave structure (see Figure 19). The success of this approach is an illustration of how graphical movies can often reveal processes and structure within the wave group that can not be ascertained from observations at distinct times.

Figure 19: Initial conditions with linear and second order correction for a Gaussian Wave group. Gibbs and Taylor [5].

The basis of their work was to consider a narrow-banded sea using a purely Gaussian energy distribution in both wave-number directions, such that,

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−= 2

2

2)(

exp)(w

p

kkk

kSηη (32)

where kp is the wave-number at the peak of the spectrum and ks is the spectral width. They then selected both of these parameters to fit to the shape of the JONSWAP spectra (16), with Tp=12s, such that ks = 0.004606 m-1 and kp = 16k0 = 0.02796 m-1

corresponds to the JONSWAP spectra with γ=3.3 (a value typical of the northern North Sea). Physically, this creates the wave surface, shown in Figure 20. The approach dramatically simplifies the sea state by removing the high frequency tail from the spectrum.

Gibbs and Taylor demonstrate that in the initial stages, the evolution and shape of the surface are broadly equivalent to a linearly focussed event. At about four wave periods prior to focusing, there is a sufficient concentration of energy to enable resonant interactions to take place. At this point, the nonlinear surface undergoes a rapid broadening toward the focal point, growing over 2.5 times wider and 20% higher than the linear equivalent. A maximum crest height of 12.8m is achieved at t=1.3Tp. There is also a major contraction from the front of the nonlinear group. Figure 19, demonstrates these features, which are also perfectly in line with the reduction in directionality first seen in Figure 18 at approximately t=-40s.

After the focal event, the disintegration of the surface due to directional dispersion leads to a very interesting transformation in the nonlinear case. As Figure 21 illustrates, the nonlinearities counter the directional dispersion and hold the group together, leading to several subsequent peaks with elevations of 12.6m, 12.0m, 11.2m, and so on. In contrast to a linear model, which unwinds in a reverse sequence to focusing and has a maximum wave elevation of 10.7m, the nonlinear simulation exceeded the linear crests height in ten separate wave periods, each with waves that are over 230m wide at a height of 10.7m. As the nonlinear group evolves from the focal point, it grows rapidly in the transverse direction yet it locally maintains a fairly planer front. This structure is very reminiscent of the ‘wall of water’ seen by mariners, and is over 500m wide in Figure 19e. These findings appear to be in complete agreement with the field measurement plotted in Figure 1, and explain why larger events are being measured far more often than they might otherwise (linearly) be expected.

4. CONCLUSIONS Real oceans are nonlinear, unsteady and directional. All these properties must be considered if you want to accurately approximate both the surface properties and the statistical rates of occurrence. The inclusion of directionality alongside third-order nonlinearities has been

emonstrated to cause significant

Figure 20: Nonlinear (left) aprofile, the nonlinear surfa

nd linear (right) surface prce is about 2.5 times wider

ofiles at the focal time. Relative to the linear and 20% higher, and has contraction in the

mean wave direction. Gibbs and Taylor [5].

d internal energy

also leading to a succession of decreasingly large, but nevertheless extreme, waves during the subsequent six or more wave periods. To contrast this with linear theory, the leading wave event can be over 20% higher, with a frontal profile that is

nter

vessel being struck by a wide wave is much higher than traditional statistics would estimate. This would explain the higher than events involving ships caught by extreme wave events.

movements, which not only lead to larger maximum waves before the onset of breaking, but also leads to the creation of walls of water, similar to those observed by mariners. Although the precise nature of individual energy movement is not known, there is little doubt that, with sufficient local energy density, these resonant interactions change the direction of dispersed energy that would otherwise travel through the wave group. This leads to more energy being aligned with the dominant wave direction, thus increasing the height of the main focal event and

over 200m wider the previously suggested. Furthermore, as third-order nonlinearities coudirectional dispersion, the succession of wave event after focussing each grow with width. Any one of these extreme waves would be reminiscent of the ‘walls of water’ reported by mariners. An important consideration in estimating the likelihood of encountering an extreme wave is to consider the relative sizes of both the observer and the wave. Ocean wave statistics traditionally calculate the frequency of events at a point; however most offshore structures and ships have plan areas covering many hundreds of square metres. Therefore, the larger the vessel (or structure), the greater the chance it encounters an extreme event. The same principle can also be applied to the ‘walls of waters’ modelled in this paper, relative to a single point observer. The observer can now be in a number of positions in the sea and still be struck by the same wave. Combining these points, the probability of a large

Figure 21: Nonlinear surface elevation at times after the peak event has occurred at t=0. (a) & (b) t=2.3Tp

(c) & (d) t=3.2Tp (e) & (f) t=4.2Tp. The left hand column gives the rear view of the wave group and the right hand column shows the fontal view. The frame of reference is moved at the linear group velocity.

5. ACKNOWLEDGEMENTS I would like to thank Paul Taylor and Richard Gibbs of Oxford University for use of their figures on the evolution of Gaussian wave groups. I would also like to thank MARIN for the data provided from the CresT JIP and plotted in Figure 1, and Andrew Gibbons for his technical proof reading. 6. REFERENCES 1. Jonathan P, Taylor PH and Tromans PS, ‘Storm waves in the northern north sea’. Proceedings of the seventh international conference behaviour of offshore structures (BOSS). Massachusetts Institute, USA, 12-15 July 1994, p.481-494. 2. Tamura H, Waseda Tand Miyazawa Y, ‘Numerical study of the sea state in the Kuroshio Extension region at the time of an accident’, Rogue Waves Conference, 2008. 3. Longuet-Higgins MS and Stewart RW, ‘Changes in the form of short gravity waves on long waves and tidal currents’, J.Fluid Mechanics, 1960, 8, p.565-583. 4. Dean RG and Sharma JN, ‘Simulation of Wave Systems due to Nonlinear Directional Spectra’, Int. Symp. On Hydrodynamics in Ocean Engineering, Trondheim, Norway, 1981, p.1211-1222. 5. Gibbs RH and Taylor PH, ‘Formation of walls of water in fully nonlinear simulations’, Applied Ocean Research 27 p.142-157, 2005. 6. Bateman WJDB, Swan C and Taylor PH, ‘On the efficient numerical simulation of directionally-spread surface water waves’, J.Comp Physics. 2001, 174, p.277-305. 7. Johannessen TB and Swan C, ‘A laboratory study of the focussing of transient and directionally spread surface water waves’, Proc. Roy. Soc Lond A 2001, 457, p.971-1006. 8. Gibson R, Swan C, ‘The evolution of large ocean waves: the role of local and rapid spectral changes’. Proc. Roy. Soc., A. 463 (2077), 2007, p.21-48. 9. Stokes CG, ‘On the theory of oscillatory waves’, Trans. Cambridge Phil. Soc., 1847, 8 p.441-445. 10. Craig W and Sulem C, ‘Numerical simulation of gravity waves’, J. Comp Physics, 1993, 108, p.73-83.

11. Baldock TE, Swan C, Taylor PH, ‘A laboratory study of nonlinear surface waves on water’, Phil Tans, R Soc Lond A 1996, 354, p.649-676. 12. Tromans PS, Anaturk A, Hagemeijer P, ‘A new model for the kinematics of large ocean waves – application as a design wave.’, In: Proceedings of the first international offshore and polar engineering conference (ISOPE), Edinburgh, UL, 3, 11-16 August 1991, p.64-71. 13.Lindgren G, ‘Some properties of a normal process near a local maximum’, Ann Math Stat, 1970, 41, p.1870-1883. 14. Boccotti P, ‘Some new results on statistical properties of wind waves’, Applied Ocean Research, 1983, 5, p.134-140. 15. Bateman WJD, ‘A numerical investigation of three dimensional extreme water waves’, PhD Thesis, Imperial College, University of London, UK, 2000. 16. Hasselman, K. ‘On the nonlinear energy transfer in a gravity-wave spectrum. Part 1’, General theory. J.Fluid Mechanics, 1962, p. 481-500. 17. Longuet-Higgins MS, ‘The deformation of steep surface waves on water. A numerical method of computation’, Proc. Roy. Soc. London, 1976, A 350, p.1-26. 18. Longuet-Higgins MS, ‘Eulerian and Lagrangian aspects of surface waves’, J.Fluid Mechanics, 1986, 173, p.683-707. 19. Phillips OM, 'On the dynamics of unsteady gravity waves of finite amplitude, part 1 the elementary interactions', J.Fluid Mechanics, 1960, 9, p.193-217. 20. McLean J, 'Instabilities of finite-amplitude water waves'. Journal of Fluid Mechanics, 1982, 114, p. 315–330. 21. Benjamin TB, Feir JE, 'The disintegration of wave trains on deep water, part 1', Theory. J Fluid Mechanics 1967;27:417–30. 22. Buchner B, Dijk R, Voogt A, ‘The Spatial Analysis of an Extreme Wave in a Model Basin’, OMAE 2007. 127