Modulational Instabilities and Breaking Strength for Deep ... · role in air–sea interaction and...

Modulational Instabilities and Breaking Strength for Deep-Water Wave Groups

ALINA GALCHENKO, ALEXANDER V. BABANIN, DMITRY CHALIKOV, AND I. R. YOUNG

Swinburne University of Technology, Melbourne, Victoria, Australia

TAI-WEN HSU

National Cheng Kung University, Tainan, Taiwan

(Manuscript received 24 November 2009, in final form 8 June 2010)

ABSTRACT

Progression of nonlinear wave groups to breaking was studied numerically and experimentally. Evolution

of such wave group parameters as a function of distance to breaking and modulation depth—the height ratio

of the highest and the lowest waves in the group—was described. Numerical model results demonstrated

good agreement with experimental results in describing the behavior of the distance to breaking and

modulation depth as functions of initial wave steepness. It was shown that energy loss appears to be

a function of the modulation depth at the breaking onset. Energy loss grows with modulation depth up to

a certain threshold of the latter. It was also shown that breaking probability for wave groups with modu-

lation depth below 2.2 is very low.

1. Introduction

Wave breaking is a frequent event observed by ev-

eryone who has ever been to a sea. It plays an important

role in air–sea interaction and influences the dynamics

of the upper ocean and exchanges of energy, momen-

tum, and gases between the atmosphere and the ocean.

Breaking of waves limits the height of the waves, po-

tentially generates new waves, and transfers momentum

from the wind to surface currents. Wave breaking is of

great importance for maritime and coastal engineering

applications, because it imposes large forces on ships and

offshore structures. Thus, wave breaking plays a signifi-

cant role in many meteorological and oceanographical

processes. Understanding of this interesting phenome-

non, however, is still quite blurred. Over the last 30 yr,

wave breaking has been studied via experimental, the-

oretical, and numerical approaches, and significant prog-

ress in theoretical understanding of wave instabilities has

been made. Still, significant work is required to un-

derstand how breaking rates can be predicted and how

wave breaking can be described as a function of the

physical characteristics of real waves.

One of characteristics that influence the contribution

of breaking waves to all the processes mentioned above

is breaking severity or breaking strength. Breaking se-

verity, together with breaking probability, provides the

means of statistical description of the wave energy dis-

sipation (see, e.g., Babanin 2009). Breaking severity is

defined as the amount of energy lost in an individual

breaking event.

Measurements of the breaking severity in the open

sea are extremely complicated and need sophisticated

equipment. That is why this parameter is investigated in

laboratory experiments. A number of important labo-

ratory studies were performed over the last 20 yr, but

the means by which the wave breaking was achieved were

often quite different. For example, Rapp and Melville

(1990) and Meza et al. (2000) used superposition of linear

waves through frequency dispersion (linear focusing).

Pierson et al. (1992) used superposition of nonlinear

waves through amplitude dispersion. Melville (1982)

and Babanin et al. (2007, 2010) employed evolution of

nonlinear wave groups. Unfortunately, the research

works with the latter method were focused mostly on

questions other than breaking severity. The present study

is an attempt to fill this gap; here, the breaking severity

due to the modulational instability is analyzed. We gen-

erate modulated wave groups and then observe their

nonlinear evolution to breaking; in the course of this

Corresponding author address: Tai-Wen Hsu, National Cheng

Kung University, Tainan, Taiwan.

E-mail: [email protected]

OCTOBER 2010 G A L C H E N K O E T A L . 2313

DOI: 10.1175/2010JPO4405.1

� 2010 American Meteorological Society

evolution to the point of breaking, characteristics of the

initial groups significantly change.

In the present work, it is proposed that breaking se-

verity can depend on modulation depth. Modulation

depth is the ratio of heights of the highest and the lowest

waves in the group (Babanin et al. 2010). For the pre-

diction of the development of physical wave properties

of nonlinear wave trains, a fully nonlinear numerical

model (Chalikov and Sheinin 2005) is used. This model

allows one to qualitatively describe the behavior of the

breaking onset depending on initial steepness and other

initial parameters. In the model, instability modes, if

they were present and relevant, had to grow from back-

ground noise, in which case quantitative rates of their

growth can be quite different to the laboratory because

of the numerical background noise being suppressed in

the model, three-dimensional (3D) crests, and other fac-

tors (i.e., Babanin 2009; Babanin et al. 2010).

The subsequent laboratory experiment is based on the

information provided by this model. In the experiment,

we record time series of surface elevation before and

after the breaking. Based on these records, energy be-

fore and after the breaking is estimated and energy loss

is calculated.

In section 2, outcomes of the numerical simulations of

nonlinear wave groups are described. Section 3 of the

present paper is dedicated to the experimental results,

their comparison with numerical results presented in

section 2, and analysis of dependence of breaking se-

verity on the modulation depth of the group. In section 4,

the overall conclusions of this study are summarized

and the issue of the influence of wind on the group evo-

lution to breaking is briefly discussed.

2. Numerical simulation of the breaking onset

The present study is dedicated to the investigation of

properties of the breaking waves and wave groups and

to the evolution of the parameters responsible for the

breaking. As mentioned above, simulation of the evolu-

tion of waves to the point of breaking is done by means

of the fully nonlinear Chalikov–Sheinin (CS) model

(Chalikov and Sheinin 1996, 1998, 2005). The CS model

is designed for direct hydrodynamic modeling of 1D non-

linear gravity and gravity–capillary periodic waves based

on conformal mapping of a finite-depth water domain. The

classical complex variable method was originally de-

veloped by Stokes (1847), and for the stationary problem

it employs the velocity potential F and the streamfunction

C as independent variables. The nonstationary conformal

method was introduced by a number of scientists in the

1970s (e.g., Whitney 1971). In the CS model, the non-

stationary conformal mapping allows the principal

equations of a potential flow with a free surface to be

rewritten in a surface-following coordinate system. The

hydrodynamic system is represented by two relatively

simple evolutionary equations that can be solved numer-

ically in a straightforward manner.

In the surface-following coordinate system (j, z), the

potential wave equations appear to be as follows:

Fjj

1 Fzz

5 0, (1)

zt5�x

j§

t� z

jj

t, and (2)

ft5�j

tf

j� 1

2J�1(f2

j �F2z)� z� p

0, (3)

where x and z are Cartesian coordinates, and kinematic

condition (2) and Lagrange integral (3) are written for

the surface z 5 0 (so that z 5 h , h–surface elevation; for

details, see Chalikov and Sheinin 2005), p0 is surface

pressure, J is Jacobian of the transformation

J 5 x2j 1 z2

j 5 x2z 1 z2

z , (4)

jt and §t are linked through the relationship

zt5�(J�1F

z)

z50, (5)

and u 5 F (z 5 0). Boundary condition is assumed to be

vanishing of vertical velocity in depth,

Fz(j, z! �‘, t) 5 0. (6)

Wind forcing of the waves is introduced as follows:

According to the linear theory, Fourier components of

the surface pressure p0 are connected with the surface

elevation through the expression

pk

1 ip�k5 (b

k1 ib�k

)(hk

1 ih�k), (7)

where bk and b2k are the real and imaginary parts of b

function (e.g., Fourier coefficients at cosine and sine,

respectively). Traditionally, it is suggested that both co-

efficients are a function of nondimensional frequency

V 5 vkU, where U is a nondimensional wind velocity.

Approximation for functions bk and b2k is given in

Chalikov and Rainchik (2010). More details about the

CS numerical model can be found in Chalikov and

Sheinin (1996, 1998, 2005).

The CS model has some important characteristics. The

principal advantage of this scheme is that it demonstrates

much greater accuracy compared to most other existing

schemes. In addition, it has high computational efficiency.

The CS scheme preserves the integral invariants, which

2314 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 40

gives an opportunity to use it for long evolution processes.

This scheme is able to reproduce a nonlinear concentra-

tion of energy in physical space resulting in wave breaking

and potentially the appearance of freak waves. The CS

scheme demonstrates stability over thousands of periods

of the dominant wave.

In the present study, a wave group is generated as a

superposition of two waves, one primary wave and one

perturbation, with close wavenumbers. Initial wave pha-

ses were kept constant. Babanin et al. (2007) investi-

gated initially monochromatic wave trains, where the side

bands necessary for the Benjamin–Feir (BF) modulation

(Benjamin and Feir 1967) grew naturally from the back-

ground noise. These monochromatic waves experienced

self-modulation and developed into strongly modulated

wave groups. In the subsequent study, Babanin et al. (2010)

noticed that the depth of this modulation is essentially

affected by the wind and in turn influences the breaking

severity. In the present paper, to achieve different mod-

ulation depths and investigate this connection of wave

groups with the breaking strength but to avoid the com-

plicating action of the wind, we create wave groups: that

is, modulate the wave train initially, thus skipping the

process of self-modulation and therefore controlling the

parameters of the modulation. The initially generated

wave group is a superposition of two sinusoidal signals.

Here we should note that we did not intend to study BF

instability in the present paper but concentrate on the

dependence of the breaking severity S on the modulation

depth R. The sideband imposed was only needed to

control R: that is, to achieve different depths of the mod-

ulation at the breaking onset. Therefore, in our numerical

experiments, only one mode was imposed, and even that

was not necessarily one of the BF fastest growing modes.

Thus, the BF modes, if they were present and relevant, had

to grow from background noise.

The key parameters that define the behavior of the

resulting wave group are the initial steepness of the pri-

mary (carrier) wave «1; steepness of the sideband «2; and

bandwidth y,

y 52(k

2� k

1)

k2

1 k1

, (8)

where k1 and k2 represent the wavenumbers of the pri-

mary and secondary waves, respectively. In the CS model,

only relative rather than absolute values of these numbers

are important. Therefore, in the model these values are

always kept as integer values because of requirements of

the numerics. These integer numbers, however, do not

signify any particular dimensional values of wavenumbers,

and that is why the agreement between the numerical

results obtained using this model and experimental results

are only qualitative (see also a detailed discussion of dif-

ferences between physical and numerical modeling of

two-dimensional breaking in Babanin 2009).

Figure 1 shows the surface elevation Z of three mod-

ulated wave trains with the same steepness of the carrier

wave «1 5 0.23 and of the sideband «2 5 0.033. Steep-

nesses are «1 5 a1k1 and «2 5 a2k2, where ai is wave

amplitude. In the nondimensional model, however, the

wave amplitude is not an independent parameter and is

defined by a choice of the wave steepness and wave-

number. The bandwidth y is varied; as a result, these

wave groups have different numbers of waves in the

modulation N 5 1/y. In Fig. 1, this number is 16.5 in the

top panel, 6.9 in the middle panel, and 4.5 in the bottom

panel: the broader bandwidth corresponds to fewer waves

FIG. 1. Surface elevation of the waves with steepness «1 5 0.23 and «2 5 0.033 and different values of N: (top) k1 5 32,

k2 5 34, and N 5 16.5; (middle) k1 5 32, k2 5 37, and N 5 6.9; and (bottom) k1 5 32, k2 5 40, and N 5 4.5 for (left)

beginning of the run and (right) end of the run, one period before the breaking.


in the modulation. For further simulations, we use N 5

6.9, because this is closest to those occurring in natural

wave fields. In Fig. 1, snapshots of the computational

domain are shown for two moments: on the left side this

moment corresponds to the beginning of the run, and on

the right side it corresponds to one period before the

breaking. The total number of grid points n in the do-

main is: n 5 4M, where M is a truncation number of the

Fourier transform (spatial resolution) used in the pro-

gram. The nondimensional wavenumber of the primary

mode is 32, so there are 32 wave crests in the domain. If

the wavenumber was different, 16, for instance, the do-

main would include 16 crests.

A breaking criterion—criterion of terminating a run

in the numerical model—was defined by the first ap-

pearance of a nonsingle value of the surface in the in-

terval x 5 (0, L):

x(i 1 1) , x(i), i 5 1, 2, 3, . . . , Np� 1, (9)

where Np is the number of points on the wave profile

over its length L. It is important to emphasize that, after

the moment when the criterion (9) has been reached, the

solution never returns to stability: the volume of fluid

crossing the vertical x(i) experiences rapid growth. Up

to this moment, conservation of the sum of potential and

kinetic energy, horizontal momentum, and the volume is

excellent. When the surface becomes a nonsingle value

(at the initial stage of breaking), conservation of invariants

still holds, but later a sharp increase of energy occurs and

a further integration becomes useless. Usually it happens

just within one Runge–Kutta time step, and that is why it is

very difficult to find a moment to record a numerical wave

profile at the breaking point. The surface elevation re-

cords meaningful for the present paper (where the pre-

breaking parameters of the groups are estimated) are the

ones made at one period before breaking (Fig. 1, right).

Numerical simulations of fully nonlinear waves have

been conducted to predict the lifetime and modulation

depth of nonlinear wave trains. The lifetime tl of the

wave is the dimensionless time between the moment the

wave is created and the moment when it breaks: that is,

duration to the wave breaking in terms of the number of

wave periods. Lifetime tl was of interest for two reasons.

The practical reason was that, knowing how lifetime qual-

itatively depends on the initial parameters of the wave

group, we could estimate the point of breaking of the

wave and plan the position of gauges in the experiment.

The other reason is that the lifetime is directly related to

the breaking probability. Breaking probability is an im-

portant statistical characteristic of a wave field. Thus, an

ability to estimate and describe breaking probability and

severity allows an estimation of the energy dissipation.

The equations underpinning the CS model are written

in a nondimensional form with the following scales: length

L, time L1/2g21/2, and the velocity potential L3/2 g1/2.

Therefore, within the frame of the model, wave de-

velopment in time and in space is equivalent, and the

present term ‘‘lifetime’’ can be associated with the dis-

tance to breaking in terms of the wavelength and tank

length.

According to the numerical simulations, lifetime ap-

pears to be a function of the proportion of the ratio of

the initial primary and secondary wave steepnesses «1/«2

(Fig. 2), which is consistent with the experimental results

of Tulin and Waseda (1999). A logarithmic function pro-

vided a good fit to the observed data,

tl5 a ln(«

1/«

2) 1 b, (10)

where a and b appear to decrease linearly as a function

of «1 as follows:

a 5�90«11 35 and

b 5�80«11 15. (11)

Here and below, the choice of a fitting function was

justified by calculating correlation coefficients between

experimental data (xn, yn) and values of fitting functions

[xn, f(xn)].

The lifetime itself also decreases with «1 (Fig. 3).

Babanin et al. (2007) found that all waves ultimately

break when they reach a certain limiting steepness « 5

0.44. Here, steepness is defined as « 5 ak 5 pH/l, where

a is wave amplitude, H is wave height, k 5 2p/l is wave-

number, and l is wavelength (horizontal distance between

FIG. 2. Lifetime tl as a function of the ratio «1/«2 for «1 5 0.17

(circles), «1 5 0.19 (diamonds), «1 5 0.21 (crosses), and «1 5 0.24

(squares). Solid lines are fitted logarithmic functions.


two successive upcrossings of the mean water level). In

the CS numerical model, limiting steepness is lower due

to the strictly two-dimensional nature of the crest and

is only « ; 0.3 [see discussions in Babanin et al. (2007,

2010), whose experiments also showed that the higher the

initial steepness, the faster the wave can reach its lim-

iting steepness and break].

Because in our initially modulated wave trains the

lifetime also depends on «1/«2 (Fig. 2), values of lifetime

for «1/«2 5 7 in Fig. 3 are generally smaller than for «1/«2 5

30. Fitted lines show that the lifetime decreases with «1

exponentially,

tl5 A

1exp(A

2«

1) 1 B and 0 , «

1# 0.44. (12)

These exponential lines were forced to asymptote at the

point «1 5 0.44, which is the limiting steepness. Al-

though in the two-dimensional model the waves will

break within one period if « . 0.3, in the physical world

they develop a three-dimensional instability at the crest

at such steepness and continue on without breaking until

the steepness reaches « 5 0.44, which is therefore the

natural asymptote for dependences (12) (see, e.g., Babanin

2009).

Thus, we can control lifetime or distance to breaking

by varying the steepness of the primary wave and the

ratio of steepnesses of the primary and secondary waves.

Simulations have shown that it works both ways.

The wave modulation depth R is a height ratio of the

highest Hh and the lowest H‘ waves in the group (Babanin

et al. 2010),

R 5H

h

H‘

. (13)

As it has already been mentioned, we generated wave

groups that are modulated initially. Modulation depth

tends to grow with time: in Fig. 1, it can be clearly seen

that modulation depth at the moment of one period

before breaking is much higher than the initial modu-

lation depth.

Modulation depth R at the moment one period before

breaking is smaller if the initial primary wave steepness

«1 is larger (Fig. 4). The decrease is exponential,

R 5 C exp(«11D). (14)

Fitted exponential functions were forced to asymptote

at R 5 1, because R 5 1 signifies no modulation by

definition.

Unlike the lifetime above, the modulation depth does

not appear to depend on «1/«2 (Fig. 5), at least in the

range of «1/«2 shown. It does decrease at values of «1/«2

even higher than those plotted, but these high values are

achievable only numerically and not experimentally.

This decrease does not have a practical value here and

will be discussed in more detail in section 4. In Fig. 5, one

can notice that modulation depth for higher values of «1,

in most cases, is smaller than for lower values of «1. This

difference can be expected from Fig. 4 but is not as large

FIG. 3. Lifetime tl as a function of primary wave steepness «1

for «1/«2 5 7 (circles), «1/«2 5 16.5 (crosses), and «1/«2 5 30 (di-

amonds). Fitted exponential lines are y7 5 173.5 exp(28«1) 2 5

(solid), y16.5 5 228.45 exp(27«1) 2 10 (dotted), and y30 5 231 exp

(25.5«1) 2 20 (dashed).

FIG. 4. Modulation depth R as a function of primary wave

steepness «1 for «1/«2 5 7 (circles), «1/«2 5 11 (dots), «1/«2 5 16.5

(crosses), «1/«2 5 20 (stars), «1/«2 5 30 (diamonds), and «1/«2 5 36

(asterisks). Solid («1/«2 5 7, «1/«2 5 36), dotted («1/«2 5 16.5, «1/«2 5

30), and dashed («1/«2 5 11, «1/«2 5 20) lines are exponential

functions: z7 5 33 exp(214«1) 1 1, z11 5 216 exp(222«1) 1 1, z16 5

451 exp(225«1) 1 1, z20 5 341 exp(223«1) 1 1, z30 5 123 exp

(219«1) 1 1, and z36 5 110 exp(218«1) 1 1.


as shown in Fig. 5. The reason appears to rest with the

third parameter, which affects R at the breaking point,

and that is the lifetime of the wave. Modulation depth

grows in time, and for waves with higher initial primary

steepness it grows faster (Fig. 6). However, waves with

higher initial primary steepness also have a much shorter

lifetime, because they reach the limiting steepness quicker

(Figs. 3, 6). That is why modulation depth for waves with

higher initial steepness does not have enough time to grow

to the levels achievable for waves with lower steepness.

This is clearly shown in Fig. 6: the wave with «1 5 0.17 has

twice the lifetime as the wave with «1 5 0.25, and the

modulation depth before breaking is more than twice as

large.

Thus, the ability to control initial steepnesses of the

waves allowed us to investigate quite a wide range of

magnitudes of the modulation depth R. This is impor-

tant for experimentally studying the breaking severity as

a function of R. The breaking process as such cannot be

simulated by a potential wave model, and therefore the

study of the breaking strength, which was the main target

of this paper, was conducted in laboratory conditions.

3. The experiment

The laboratory experiment was conducted in the wave

flume of the Department of Hydraulic and Ocean Engi-

neering of the National Cheng Kung University, Taiwan.

This is a glass tank: 24 m long, 1.3 m high, and 1 m wide

(Fig. 7). A piston-type generator was mounted at one end

of the tank.

The programmable wavemaker was used to produce

wave conditions similar to the numerical experiments.

The wave paddle generated a wave train consisting of

seven to eight wave groups, and the groups were created

by superposing the primary and secondary wave com-

ponents. We have to emphasize again that this is not the

way to generate wave groups designed for studying the

modulational-instability evolution (e.g., Tulin and Waseda

1999; Stansberg 2005; Babanin et al. 2007, 2010; Toffoli

et al. 2008). As in the numerical simulations described

above, our experimental setup was intended for achieving

different depths of modulation at the point of breaking

onset, with the purpose of having a reasonably broad

range of variations of this depth R.

Surface elevations were measured and recorded by

means of an array of six capacitance-type wave gauges

(Fig. 7). They are 40-cm-long parallel-wire-capacity-type

wave gauges (WHP-400) manufactured by Procal Tech-

nology Com. Ltd., Taiwan. The measured voltage signal

is amplified through the amplifier. The analog signal is

converted into digital signal by a National Instruments

AD/DA platform (PCI-6014 and BNC-2090), and a vi-

sualization interface is established by LabView software.

The National Instrument PCI-6014 AD/DA trans-

formation card’s resolution is 16 bit; the resolution of the

40-cm-long wave gauge could reach 400/216 mm, about

FIG. 5. Modulation depth R as a function of ratio «1/«2 for

«1 5 0.17 (circles), «1 5 0.19 (diamonds), «1 5 0.21 (crosses), and

«1 5 0.24 (squares).

FIG. 6. Modulation depth R vs time to breaking for waves with

«1/«2 5 7, «1 5 0.17 (circles), «1 5 0.21 (squares), and «1 5 0.25

(stars).

FIG. 7. Experimental setup (proportions are arbitrary).


0.0061 mm. The system error of the wave gauge set in

the tank is below 0.5 mm.

An example of a modulated wave train is shown in

Fig. 8. It is clearly seen how the initially moderate group-

ing develops into strong sawtooth modulation consistent

both with the numerical simulations above and with an-

alytical expectations for the modulational-instability

evolution (Shemer and Dorfman 2008). This fact is also

in agreement with experimental results of Trulsen and

Stansberg (2001) and Stansberg (1995, 1998), who ob-

served and described nonlinear propagation of initially

bichromatic wave trains: asymmetric forward-leaning

evolution of initially symmetric groups and an increase

of their amplitudes, as well as exchange of energy be-

tween the spectral components.

The horizontal array of closely spaced wave probes was

designed so as to have wave-breaking events occurring

between the gauges: that is, to measure the surface el-

evation just prior to breaking and straight after. Gauge 1

placed at the distance of 2–3 wavelengths from the gen-

erator was used to record the initial conditions of the

wave trains: that is, after the piston-produced waves ad-

justed to the depth-distributed wave profile. Because the

distance to breaking could be approximately predicted

based on the numerical simulations with the CS model

and thus controlled, the experiment was divided into two

stages: first, the waves with relatively short lifetimes were

generated and recorded, and then the gauges were moved

further along the tank and waves with longer lifetimes

were generated and measured.

A single ‘‘breaking event,’’ in most situations, con-

sisted of two consecutive waves breaking within a group.

Such breaking behavior is well known (e.g., Donelan

et al. 1972, and many other studies) and the two breaking

waves are strongly coupled to an extent that it is often

not possible to separate them into two single breaking

events (Babanin 2009). If such double breaking happens

between, for example, gauges 2 and 4, then we consider

the record at gauge 2 as the record before breaking

and the record at gauge 4 as the record after breaking.

As the beginning of the breaking process, we assume the

moment when the fluid surface collapses at the crest

(this is the point where the CS model was stopped in

the numerical experiments). The next stage is termed

‘‘developing breaking’’ (Liu and Babanin 2004), and

the before-breaking gauge should be located before the

start of this stage in the first of the double breakers. The

after-breaking gauge should be located after the break-

ing ceased in the second double breaker. In the present

study, we did not separate plunging and spilling breakers

and concentrated on the dependence of breaking severity

on modulation depth.

The wavelengths l generated in this experiment var-

ied from 0.8 to 1.6 m with most measurements con-

ducted for waves with l 5 0.9 2 1.2 m. The water depth

in the tank was 1.2 m, and therefore the deep-water

conditions were satisfied (e.g., Young 1999). Experi-

mental results show good qualitative agreement with the

predictions of the CS model. As was expected (Fig. 2),

the distance to breaking grows logarithmically as a func-

tion of «1/«2 (Fig. 9). As was also expected (Fig. 4),

modulation depth exponentially decreases with primary

wave steepness «1 (Fig. 10). For the modulation depth R,

even quantitative agreement between the model and

experiment is good, and in both Figs. 4 and 10 the value

of R for «1 ’ 0.24 2 0.25 is approximately 2–2.5. In

Fig. 10, R decreases further with increasing «1.

FIG. 8. An example of surface elevation records on gauges (top)–

(bottom) 1–6: N 5 7, «1 5 0.23, and «2 5 0.086. FIG. 9. Dimensionless distance to breaking in terms of number of

wavelengths vs ratio of primary and secondary wave steepnesses

«1/«2, range of steepnesses is 0.2 , «1 , 0.23: experimental results.

Solid line is fitted logarithmic function, y 5 11 lnx 1 1.5.


Numerical modeling and even laboratory measure-

ments of breaking severity represent a considerable chal-

lenge and knowledge of this important property is sparse.

Rapp and Melville (1990) found that the loss of energy

ranged from 10% for single spilling events to 25% for

plunging breakers. Similar results were obtained by

Kway et al. (1997): they had energy loss varying from

14% to 22% for different plunging waves. These were

investigations of the breaking resulting from linear wave

superposition. Babanin (2009), however, demonstrated

that in the case of breaking brought about by the non-

linear evolution of wave trains, the breaking severity can

vary broadly from virtually 0% to 100%.

In this study, wave energy is estimated for one group as

E 5

ðT

0

h2(t) dt, (15)

where h(t) is the envelope of the group and T is the

period of the whole group. To find the envelopes of the

groups before and after the breaking, a method of

smoothing splines is used (Craven and Wahba 1979).

The severity coefficient is defined as follows:

S 5E

bb� E

ab

Ebb

, (16)

where Ebb is the energy before breaking and Eab is the

energy after breaking. Thus, the severity coefficient S is

a nondimensional parameter.

The decay due to wall friction was estimated for a

monochromatic train of nonbreaking waves. Average

energy of a wave in such a train decreases exponentially:

for example, for l 5 1 for our data on gauges 2–6 the

approximation is E(d) 5 142 exp(20.0003d) and for l 5

1.2 it is E(d) 5 125 exp(20.0002d), where d is the dis-

tance covered by the wave and E is the average energy of

the wave. Because the losses are small, this exponential

function can be easily replaced by a linear function.

Knowledge of energy of these nonbreaking waves at

each gauge allows the calculation of the coefficient of

energy loss due to decay, g 5 Ej11/Ej, where j is a gauge

number. It can be found that the average coefficient for

the waves with l 5 0.9 2 1 m is g 5 0.96; for l 5 1.2 m,

it is g 5 0.98. Gauges 2–6 were equally spaced.

As anticipated from Babanin et al. (2010), in Fig. 11

severity appears to be a function of modulation depth:

energy loss grows with modulation depth. Figure 11

contains 109 points, and the correlation coefficient be-

tween R and S is K 5 0.4410.13�0.17 (for 95% confidence

interval). Energy loss varies very significantly from 2%

to 35%. Note that this is energy loss from the group

where not every wave is breaking: that is, the severity of

individual breaking waves is much greater (see Babanin

2009). Here and in Figs. 12–14, straight lines are fittings

based on reduced major axis (geometric mean) regression

(e.g., Sinclair and Blackwell 2002). A linear function, with

which severity is approximated in Fig. 11, is

S 5 (0.16 6 0.03)R� (0.35 6 0.06). (17)

Here and in approximations in captions of Figs. 12 and

13, 95% confidence limits are shown in brackets.

The parameterization (17) is the main result of this

paper as it identifies and quantifies the function form of

FIG. 10. Modulation depth R vs primary wave steepness «1;

experimental data, waves with l 5 1. Solid line is exponential fit,

z 5 50 exp(218x) 1 1.6.

FIG. 11. Severity coefficient S [Eq. (16)] as a function of modu-

lation depth R [Eq. (13)] for all wavelengths and steepnesses. Solid

line is parameterization [Eq. (17)].


the wave-breaking severity, which is routinely treated as

a fixed fraction of the before-breaking wave energy. In

this regard, it should be stressed that our breaking was

achieved as a result of modulation instability. In the case

of linear-superposition breaking, which most of the

previous studies of the breaking severity have considered,

modulation depth prior to breaking may perhaps also

vary. The breaking dynamics of a steep wave created by

linear superposition, however, is different to the dynamics

of the breaking cause by nonlinear evolution (see Babanin

2009; Babanin et al. 2010), and therefore severity of the

linear-superposition breaking can be different as a func-

tion of the modulation depth or maybe even constant, as

was found in Rapp and Melville (1990).

The scatter of the severity coefficient in Fig. 11 is

significant. This points out that the breaking strength is a

complex property that possibly depends on many fac-

tors, rather than solely on the dynamics identified by the

modulation depth R at the breaking onset. Indeed, even

within the modulational-instability breaking, R can per-

haps vary depending on relative phase of the two mono-

chromatic trains generated as the initial condition.

Besides, the evolution of nonlinear wave groups re-

alized here is not a classical Benjamin–Feir scenario but

is rather evolution of the bichromatic trains. This is still

a Benjamin–Feir–like evolution as mentioned above in

the context of studies by Trulsen and Stansberg (2001)

and Stansberg (1995, 1998). As we know, the necessary

resonant sidebands will grow from the background noise

even if not imposed at all (e.g., Melville 1982; Babanin

et al. 2007, 2010). However, because the initial conditions,

FIG. 12. Severity coefficient S as a function of modulation depth

R for runs with (top) l 5 0.9 m, (middle) l 5 1 m, and (bottom)

l 5 1.2 m. Solid lines are linear fits S0.9 5 (0.22 6 0.07)R 2 (0.6 6

0.2), S1 5 (0.2 6 0.05)R 2 (0.4 6 0.1), and S1.2 5 (0.12 6 0.03) R2

(0.3 6 0.1), respectively.

FIG. 13. Severity coefficient S as a function of modulation depth R for different ranges of �1 5

«1 1 «2. Solid lines are linear fits: (top left) S1 5 (0.2 6 0.07)R 2 (0.41 6 0.14), (top right)

S2 5 (0.14 6 0.03)R 2 (0.31 6 0.09), and (bottom left) S3 5 (0.18 6 0.05)R 2 (0.4 6 0.15).


again, are different, they may affect the prebreaking

modulation-depth outcome and, correspondingly, the

scatter.

Other physical features could have contributed to the

scatter too. The surface tension, even in the case of purely

gravitational waves, can control development of the bulge

and/or jet at the breaking-onset wave crest or influence

such development and lead to generation of parasitic

capillary waves that would consume some energy and

thus affect the breaking severity (e.g., Qiao and Duncan

2001; Iafrati 2009). This can be a noticeable contribution

to the dissipation, in addition to the observed trend.

All these above-mentioned potential contributions

have not been accounted in this paper, where we con-

centrated on demonstrating the dependence of breaking

severity on one property only. However, these addi-

tional dependences are feasible and may have contrib-

uted to the scatter. Reduction of the scatter in such

a multiparameter problem is left to future studies.

It should be also kept in mind that measurements

of both R and S in the experiment contain some un-

certainties. One of the measurement errors of R is due to

the distance from the gauge to the beginning of break-

ing. Although the wave-breaking groups were selected

in such a way that the breaking onset is close to one of

the wave probes, obvious variability is unavoidable in such

a method. With respect to the energy lost, here we base

our estimate on measurements of the surface elevation

only: that is, Eqs. (15) and (16), where the changes of

kinetic energy are not accounted for explicitly, although

their proportion of contributions to the total energy loss

in a nonlinear wave system can be different to that of the

potential energy.

Another potential cause of experimental error is the

use of the smoothed envelopes in (15). Wave reflection

from the beach was minimized by applying only short

packets of seven to eight groups (Fig. 8), but such re-

flection could introduce some error for waves with lon-

ger lifetimes.

Therefore, the dependence (17) is approximate, but

the positive trend is statistically robust. We conducted

380 runs with seven to eight wave groups in each run. So,

even with all the errors mentioned above, the trends that

we observe are sufficiently reliable to conclude that en-

ergy loss grows with modulation depth up to a certain

threshold of the latter. In our experiments, this threshold

was different for different wavelengths, and it increases

for longer primary waves (Fig. 12). For waves with l 5

1 m (Fig. 12, middle), energy loss increases with R and the

correlation coefficient is K 5 0.610.2�0.3 (for 95% confi-

dence interval). The energy loss is approximately 35% for

R 5 3.7 and 2% for R 5 2 (here we have already excluded

energy dissipation due to wall friction as described above).

For waves with l 5 1.2 m (Fig. 12, bottom), S grows with

R with a correlation coefficient K 5 0.6410.16�0.24 . The high-

est energy loss is 29%, and that is for R 5 4.7. For R . 4.7,

the energy loss does not continue to grow (data points are

not shown). Waves with l 5 0.9 m experience growth of

energy loss up to R 5 3.3, and then S decreases (Fig. 12,

top). The highest value of R is 19% for R 5 3.3. The

correlation coefficient between S and R for the range of

growing R (Fig. 12, top) is K 5 0.5610.24�0.31 .

In Fig. 12, the data are separated into groups in terms

of wavelengths. However, wavelength is a dimensional

parameter and therefore cannot be used for general in-

vestigation of the functional dependence. Attempts to

sort the data by initial steepness of the carrier wave «1

and ratio of initial steepnesses of the carrier wave and

sideband «1/«2 were unsuccessful. Such separation de-

creases the correlation coefficients between R and S.

Figure 13 shows the severity coefficient versus mod-

ulation depth for data sorted by total steepness, a

nondimensional parameter « 5 «1 1 «2. The strongest

dependence is for medium values of total steepness « 5

0.35 2 0.4 (K 5 0.71; Table 1). For smaller values « 5

0.2 2 0.3 and « 5 0.3 2 0.35, saturation of the severity

values S at higher R can perhaps be observed, similar to

that discussed with respect to Fig. 12. Because it is not

quite as distinct, it was not quantified; as a result, the

respective correlation coefficients are lower (Table 1).

For higher total steepnesses, no dependence could be

determined because of the limited range of variation of

R and S.

Thus, the severity loss S was parameterized as a linear

function of modulation depth R, up to a certain thresh-

old of R as described above. To reduce the scatter of the

FIG. 14. Lifetime tl as a function of ratio «1/«2 for «1 5 0.17

(circles), «1 5 0.19 (diamonds), «1 5 0.21 (crosses), and «1 5 0.24

(squares). Solid line is a fitted exponential function Y 24 5 16

ln(«1/«2) 2 12; U/c 5 4.


parameterization and investigate variation and causes of

the threshold, further experimental investigations are

required.

4. Discussion and conclusions

This study is an experimental investigation of nonlinear

modulation and its relationship with breaking severity. In

accordance with previously known results for the modu-

lational instability (i.e., Tulin and Waseda 1999; Waseda

and Tulin 1999; Babanin et al. 2007, 2010), it was con-

firmed numerically (Fig. 2) and experimentally (Fig. 9)

that the distance to breaking for a modulated wave group

grows with the ratio of initial steepnesses «1/«2 and de-

creases with the initial primary wave steepness «1 (Fig. 3).

The modulation depth R [Eq. (13)], which is a primary

focus of this paper, has been previously suggested to be

connected with the wave-breaking strength S [Eq. (16);

Babanin 2009; Babanin et al. 2010]. In this study, it is

found to decrease with primary wave steepness (Figs. 4,

10). Most importantly, it has been shown that breaking

strength is a function of modulation depth that occurred at

the point of breaking onset and grows with the modulation

depth for values of R that are typical for ocean waves.

In Babanin et al. (2010), difference in modulation

depths was achieved by varying the wind forcing. Here,

to separate purely hydrodynamic effects from compli-

cating issues of air–sea interaction, the initial modula-

tion was imposed in the absence of winds and was allowed

to evolve in the course of propagation of nonlinear wave

groups. Investigation of the influence of the wind, how-

ever, is important in the considering application of these

results to real oceanic waves.

Therefore, the next step in studying the properties

of modulated wave groups, their evolution to breaking,

and their breaking severity should be to perform sim-

ilar experiments, but in the presence of wind. This will

be a separate study, but preliminary numerical exper-

iments with the CS model have been conducted and are

instructive in shaping future studies. The simulations

showed that weak and moderate wind forcing tends to

reduce lifetime (increase breaking probability; see also

Babanin et al. 2007), but for strong wind forcing this

reduction slows significantly. The logarithmic increase

of lifetime as a function of «1/«2 is distorted if moderate-

to-strong wind forcing is applied.

Figure 14 shows lifetime as a function of the ratio «1/«2

with moderately strong wind forcing U/c 5 4, where U

is the wind speed in the nondimensional CS model and

c is the phase speed of the primary wave. When values

of «1 are low, lifetime first rapidly grows and then rea-

ches saturation: that is, stays approximately constant

(Fig. 14). In our simulations, for «1 5 0.17 lifetime rea-

ches saturation, for example, when «1/«2 5 16.5 (Fig. 14);

however, for «1 5 0.24 it continues to grow and can be

approximated by a logarithmic function, just as in the

absence of wind (Figs. 2, 14). In general, wind increases

breaking probability and therefore reduces lifetime,

which is consistent with the previous observations. It is

interesting to notice that, in the presence of this strong

wind forcing, for high values of «1/«2, steeper waves have

smaller breaking probability (longer distance to break-

ing; Fig. 14).

Stronger wind always reduces modulation depth. This

has been found numerically for initially modulated wave

groups in the present research project (Fig. 15) and was

also observed experimentally in self-modulated wave

trains by Babanin et al. (2010). Reduction of the mod-

ulation depth signifies a slower development of the

modulational instability in presence of wind forcing,

which has been previously demonstrated theoretically

(Trulsen and Dysthe 1992).

It can be seen (Fig. 6) that initially steeper waves with

«1 5 0.25 have a cutoff for modulation depth at approx-

imately R 5 2.2. Looking at the dependence of severity

TABLE 1. Correlation coefficients K between modulation depth R

and severity coefficient S for data shown in Fig. 13.

Range «

Correlation coefficient K with lower

and upper bounds for a 95%

confidence interval No. of points

0.2–0.3 K 5 0.47, 0.07 , K , 0.73 24

0.3–0.35 K 5 0.52, 0.27 , K , 0.7 48

0.35–0.4 K 5 0.71, 0.42 , K , 0.87 23

0.4–0.6 K 5 0 12

FIG. 15. Modulation depth as a function of wind forcing for wave

trains with three different steepnesses of «1 5 0.17, «2 5 0.005

(asterisk); «1 5 0.19, «2 5 0.005 (diamond); and «1 5 0.21, «2 5

0.006 (circle).


on modulation depth (Figs. 11–13), one can see that the

average coordinate of the point where the severity co-

efficient approaches zero (i.e., there is no breaking) is

R ’ 2.2 (it is higher for waves with low initial steepness;

see Fig. 6). This makes it possible to conclude that R 5

2.2 represents a lower limit to modulation depth, below

which the probability of breaking in the absence of wind

significantly decreases. In the presence of wind, how-

ever, the modulation depth immediately before break-

ing can be as low as R ’ 1, the value that signifies no

modulation by definition (Fig. 15). This fact emphasizes

that the question of the actual role of the wind in wave

breaking is very important and is still in need of serious

investigation.

Wind forcing, as well as nonlinear modulation, is an

integral part of field conditions. This makes nonlinear

modulation in the presence of wind an important subject

for future research. Also of interest are the following

topics: severity loss for higher values of modulation

depth, development of new wave groups after breaking

and multiple breaking events, and breaking severity of

three-dimensional waves with continuous spectrum.

Acknowledgments. We thank Prof. Ou Shan-Hwei for

helpful discussion and Dr. Lai J.W. for his interest, sup-

port, and valuable assistance with the laboratory experi-

ment. Alex Babanin acknowledges financial support

through the ARC Grants LP0883888 and DP1093349 and

Alina Galchenko through her Ph.D. scholarship granted

by the Centre for Sustainable Infrastructure of Swin-

burne University.

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