LUBIN, P., and CHANSON, H. (2017). "Are breaking waves, bores, surges and jumps the same flow?" Environmental
Fluid Mechanics, Vol. 17, No. 1, pp. 47-77, (DOI: 10.1007/s10652-016-9475-y) (ISSN 1567-7419 [Print] 1573-1510
[Online]).
1
Are breaking waves, bores, surges and jumps the same flow?
by
Pierre LUBIN (
1
) (
3
) and Hubert CHANSON (
2
)
(
1
) Université de Bordeaux, I2M, CNRS UMR 5295, 16 avenue Pey-Berland, 33607 Pessac, France
(
2
) The University of Queensland, School of Civil Engineering, Brisbane QLD 4072, Australia
(
3
) Corresponding author, Email: p.lubin@i2m.u-bordeaux1.fr
Abstract
The flow structure in the aerated region of the roller generated by breaking waves remains a great challenge to study,
with large quantities of entrained air and turbulence interactions making it very difficult to investigate in details. A
number of analogies were proposed between breaking waves in deep or shallow water, tidal bores and hydraulic jumps.
Many numerical models used to simulate waves in the surf zone do not implicitly simulate the breaking process of the
waves, but are required to parameterise the wave-breaking effects, thus relying on experimental data. Analogies are
also assumed to quantify the roller dynamics and the energy dissipation. The scope of this paper is to review the
different analogies proposed in the literature and to discuss current practices. A thorough survey is offered and a
discussion is developed an aimed at improving the use of possible breaking proxies. The most recent data are revisited
and scrutinised for the use of most advanced numerical models to educe the surf zone hydrodynamics. In particular, the
roller dynamics and geometrical characteristics are discussed. An open discussion is proposed to explore the actual
practices and propose perspectives based on the most appropriate analogy, namely the tidal bore.
Keywords: Breaking waves, Breaking bores, Hydraulic jumps, Air bubble entrainment, Flow singularity, Tidal bores.
1. INTRODUCTION
Surface wave breaking, occurring in the open ocean or the coastal zone, is a complex and challenging two-phase flow
phenomenon which plays an important role in numerous processes, including air–sea transfer of gas, momentum and
energy, and in a number of technical applications such as acoustic underwater communications and optical properties of
the water column. The major visible feature during wave breaking is the large quantities of air entrained in the form of
bubble clouds and whitecaps, generally coined surface foam (Figures 1 and 2). The generation of bubble clouds has
been shown to induce energy dissipation and turbulent mixing, to contribute to heat exchange and enhance gas transfer
(Hwung et al., 1992; Wanninkhof et al., 2009). Bubble clouds have been shown to influence climate and intensification
of tropical cyclones (Véron, 2015), and cause the ocean ambient noise (Prosperetti, 1988). The breakup and evolution
of entrained air into numerous bubbles is a source of acoustic noise, which is important for naval hydrodynamics. The
hydrodynamic performance of ships is influenced by the wake modified by the air entrainment, and the sound generated
by the bubble clouds render the ships subject to detection. In hydraulic engineering, large spillways are often protected
from cavitation damage by controlling aeration (Russell and Sheehan, 1974; Falvey 1990).
Many numerical models (e.g. Boussinesq equations) used to study waves in the surf zone do not implicitly simulate the
breaking process of the waves (Christensen et al., 2002). The wave-breaking effects have to be parameterised by
incorporating additional terms in the mass and momentum equations (e.g. Musumeci et al. 2005; Cienfuegos et al.,
2010; Bjørkavåg and Kalisch, 2011; Tissier et al., 2012, Kazolea et., 2014). The challenge is to take the breaking
process into account to ensure an accurate description of the surf zone, including the wave height decay and the setup
development. The main consideration is to dissipate energy when wave breaking is likely to occur. Svendsen (1984a,
1984b) proposed the roller concept, in the form of a volume of water carried shoreward with the wave. Local roller
thickness and mean front slope of the breaker were used to quantify part of the local momentum deficit. But the vertical
surface roller of the breaking wave is only considered to play an important part in the momentum and energy
conservation. However, the energy flux and dissipation during wave breaking remain difficult to quantify. Most recent
modelling attempts are still struggling with the lack of physical knowledge of the finest details of the breaking
processes, which makes the task of parameterising breaking effects very difficult since no universal scaling laws for
physical variables have been proposed so far. Physical parameters, such as the height and length of the roller, have to
be quantified and criteria have to be defined with critical bounded values to estimate where the waves break and stop
breaking. Thus models still need calibration and further improvements (Brocchini, 2013).
The turbulent flow dynamics in bubble clouds is a very challenging numerical problem. Esmaeeli and Tryggvason
(1996) studied direct numerical simulations of buoyant bubbles in a two-dimensional periodic domain. They simulated
144 and 324 bubbles, showing that the work done by the buoyant bubbles increased the energy of flow structures much
larger than the bubbles. But 3D direct modeling of air bubble entrainment and evolution at the scale of the surf zone is
computationally unaffordable. Another way of tackling dispersed two-phase flows is using a continuum-mechanical
LUBIN, P., and CHANSON, H. (2017). "Are breaking waves, bores, surges and jumps the same flow?" Environmental
Fluid Mechanics, Vol. 17, No. 1, pp. 47-77, (DOI: 10.1007/s10652-016-9475-y) (ISSN 1567-7419 [Print] 1573-1510
[Online]).
2
approach (Drew, 1983). Two-fluid models are used to model the polydisperse two-fluid bubbly flow based on mixture
theory (Carrica et al. 1999; Moraga et al., 2008, Shi et al., 2010; Ma et al. 2011; Derakhti & Kirby, 2014). A first
attempt to use a continuum type model for studying bubbly flow under surface breaking waves was made by Shi et al.
(2010). They proposed a physically-based numerical model for prediction of air bubble population in a surf zone-scale
domain. The air entrainment was formulated by connecting the shear production at air–water interface and the bubble
number density with the bubble size spectra as observed by Deane and Stokes (2002). The model was initially fed with
the entrained bubbles and used to simulate the evolution of the bubble plumes. This approach requires much less spatial
and temporal resolution than needed to capture detailed air entrainment process in DNS simulations. The model results
revealed that bubbles larger than 1 mm provide a major contribution to void fraction, while smaller bubbles contribute
significantly to the cumulative interfacial area of the bubble cloud but do not contribute much to the total volume of air.
Discrepancies between observations and model behaviour were nevertheless reported. Based on the works of Ma et al.
(2011), Derakhti and Kirby (2014) used an Eulerian–Eulerian polydisperse two-fluid model in an LES framework.
Detailed overviews on methods and models for CFD of multiphase flows can be found in textbooks (Drew and
Passman, 1999; Crowe et al. 2011). More information about turbulence modelling in the framework of multiphase
flows is given by Labourasse et al. (2007) and Bombardelli (2012). Smoothed-particle hydrodynamics (SPH) is also a
mesh-free method which can be used to describe accurately the 3D surf zone hydrodynamics, as recently shown by
Farahani and Dalrymple (2014) who investigated some novel coherent turbulent vortical structures under broken
solitary waves. The state-of-the-art is detailed by Gomez-Gesteira et al. (2010) and Violeau and Rogers (2016), who
detailed a number of examples in which SPH simulations have been successfully used in fluid flow research and
hydraulic engineering.
Numerical models still rely on experimental data. Detailed information on the temporal and spatial variations of the
void fraction fields beneath breaking waves is required. Instantaneous void fraction and interfacial velocity data are
critically needed to calibrate and improve numerical models of the two-phase flow generated beneath plunging and
spilling breaking waves. Models for air-entrainment are critically dependent upon accurate estimates of the surface area
affected by wave breaking. Controlled laboratory experiments and accurate measurements of void fraction and bubble
size distributions beneath plunging and spilling breakers are still very challenging. When a wave breaks, the tip forms a
liquid jet which impinges on the front face of the wave and creates an air cavity which breaks into bubbles. The
characterisation of the bubble sizes resulting from the cavity collapse has to be measured and the trajectories of these
entrained bubbles are also critical information. The initial stages of the breaking of a wave generated a large amount of
bubbles production and to the distribution at greater depths. The bubble clouds will then form, grow and decay during
the propagation of the turbulent air/water mixing region forming the bore, the temporal variations of all bubble cloud
dimensions reflecting this evolution. The large volumes of air in bubbles rapidly evolve into a distribution of bubble
sizes which interacts with liquid turbulence and organised motions.
The motion of bubbles relative to the liquid causes velocity fluctuations in the water column and increases the energy
of liquid motion at the scales comparable with the bubble diameter (Derakhti and Kirby, 2014). Bubble plume
kinematics and dynamics, and the structure of the turbulent bubbly flow under breaking waves constitute critical
information to be taken into account for an accurate description of the wave breaking process (Melville 1996). While
the former can be studied experimentally, the liquid–bubble interactions, i.e. the effects of dispersed bubbles on
organised and turbulent motions, are still poorly understood.
When looking at a bore, whereas it has been generated by a stationary hydraulic jump, a surface wave breaking on the
ocean or in the surf zone, or a tidal bore propagating upstream a river, the question is: are we looking at the same flow?
Is there only one bore structure, or are there variations depen
ding on the initial conditions leading to its occurrence and
behaviour? To what extent can we compare the bores and use the quantities through similarity? It is the aim of this
contribution to contribute to the transfer of knowledge from detailed measurements realised in hydraulic jumps and
tidal bores, to the wave breaking investigation. The first part of the article is dedicated to the identification of the
knowledge gaps encountered when attempting to simulate numerically the hydrodynamics of breaking waves and a
review of the various analogies which have been proposed in the literature. The next part reports on the state-of-the-art
of the studies focusing on the void fraction and velocity analysis under breaking waves, tidal bores and hydraulic
jumps. Based on this survey, we attempt to identify and assess the quantities which can be considered for possible
analogies. The most recent data are revisited and scrutinised for the use of most advanced numerical models to educe
the surf zone hydrodynamics, highly influenced by the wave breaking process. An open discussion is proposed to
explore the actual practises and propose perspectives based on the most appropriate analogy, namely the tidal bore.
2. KNOWLDEGE GAPS FOR THE MODELLING OF THE SURF ZONE
HYDRODYNAMICS
2.2 Current state of practice in numerical modelling and limitations
Most numerical models only consider macro-scale roller properties. The roller formation and propagation have been
LUBIN, P., and CHANSON, H. (2017). "Are breaking waves, bores, surges and jumps the same flow?" Environmental
Fluid Mechanics, Vol. 17, No. 1, pp. 47-77, (DOI: 10.1007/s10652-016-9475-y) (ISSN 1567-7419 [Print] 1573-1510
[Online]).
3
shown to be a highly unsteady process, with air entrainment and turbulence generation. The most advanced models,
which are generally used to simulate non-linear wave transformations in coastal areas, are based either on the Non-
linear Shallow Water equations (NSW), the Boussinesq-type equations (BT), or some form of hybrid model. Extensive
developments and break-through progress have been made recently for a large variety of coastal engineering
applications (e.g. Tissier, 2012; Bacigaluppi et al., 2014; Brocchini, 2013; Kazolea, 2014). A key feature, the breaking
process, is however not explicitly simulated and missing in these models. Several approaches and parametrisations have
thus been proposed to introduce wave breaking in NSW and BT models.. Any such approach requires the quantification
of energy dissipation, dynamically activated when wave breaking is likely to occur. Some physically based criteria have
to be able to activate or deactivate these extra terms and simple expressions are generally favoured. Simple quantities
include geometrical aspects of the roller, including heights, lengths and angles, easily extracted from any visual
observations in laboratory and in the field, All these quantities cannot be estimated from a single experiment. Instead a
composite set of data and practices have been elaborated though time by looking at various analogue flows, and some
variations have been proposed in order to fill the gaps.
Practically, most numerical models need to evaluate:
1. a Froude number Fr characteristic of wave breaking, of when it occurs and stops (with Fr varying with water
depth). Currently, an accepted value for the transition between non breaking and breaking waves has been
identified in Froude number range between 1.3 and 1.6 (Okamoto and Basco, 2006). This is based upon the
analogy with non-breaking undular hydraulic jump and bore (Favre, 1935; Treske, 1994; Chanson and
Montes, 1995, Lennon and Hill 2006, Chanson and Koch 2008);
2. the roller height h
r
, derived from momentum considerations (see Appendix II);
3. the roller length L
r
, determined empirically. A common parameterisation is L
r
= 2.91h
r
(Haller and Catalan,
2009), although the re-analysis of large-scale experiments suggests L
r
/h
r
1 to 8 (Figure 5). In Figure 5,
steady breaker, stationary hydraulic jump and tidal bore data are compared;
4. the mean front slope angle ɸ (Schäffer et al., 1993), typically between 8° to 30° for the termination and
initiation of the breaking event respectively;
5. the roller celerity (or celerity of the breaking wave);
6. the energy dissipation in the roller region;
7. the bubble size distributions, often improperly estimated based upon Hinze's (1955) model developed in the
case of a single droplet under non-coalescecing conditions (!).
To estimate most of these quantities, flow analogies have been considered, but some limitations are clearly identified
and some modifications, based on new experimental data analysis, are proposed in the following sections.
2.2 Flow analogies or not?
A number of analogies were proposed between breaking waves, bores and jumps (Appendix I). Appendix I lists a
number of early seminal references and Figure 1 presents definition sketches. The steady breaker configuration was
proposed as a simplification of the spilling breaker (Banner and Phillips 1974, Banner and Melville 1976). Important
results were obtained (Duncan 1981, Banner and Peregrine 1993, Cointe and Tulin 1994, Lin and Rockwell 1995,
Dabiri and Gharib 1997), but there is still on-going argument about the validity of this analogy (Kiger & Duncan 2012).
Further links were developed between breaking waves and steady flow configurations. These encompassed
comparisons between plunging breakers and plunging jets (Cipriano and Blanchard 1981, Hubbard et al. 1987,
Chanson and Cumming 1994, Oguz et al. 1995, Chanson et al. 2002,2006, Salter et al. 2014), between spilling breakers
and stationary hydraulic jumps (Longuet-Higgins 1973, Peregrine and Svendsen 1978, Madsen 1981, Brocchini et al.
2001a,b), and between spilling breakers and translating hydraulic jumps (also called positive surges or tidal bores)
(Longuet-Higgins 1973, Peregrine and Svendsen 1978, Brocchini and Peregrine 2001b). In parallel, there have been
numerous discussions about the similarities and differences between stationary and translating hydraulic jumps (e.g.
Darcy and Bazin 1865, Stoker 1957, Tricker 1965, Lighthill 1978), although the open channel hydraulic literature
develops the same integral approach for both types (Henderson 1966, Lighthill 1978, Chanson 2004, 2012).
To date, the mechanistic connections between these flows are not well understood and have not always been successful.
Wave-plunging jet conditions appear to produce a qualitatively different type of impact, with almost no penetration into
the oncoming flow and a pronounced splash that cascades multiple times down the face of the wave. What is better
characterised however, is the volume of air trapped by the initial contact of the jet with the wave face, which has been
numerically simulated, and its shape has been successfully modelled, at least for a limited set of conditions (e.g. Lubin
and Glockner 2015). However, the processes that follow the initial contact are only known qualitatively for the majority
of the breaking conditions, and thus still require further study in order to acquire improved physical understanding.
Furthermore, wave breaking is a combination of transient processes which evolve within the breaking duration making
adequate physical understanding a challenging proposition. Overall, “In studying any turbulent flow it is very helpful if
it can be shown to be similar to other well known flows” (Peregrine and Svendsen,1978). Below, a number of seminal
flow configurations are explored and the relevance of flow comparisons is discussed.
LUBIN, P., and CHANSON, H. (2017). "Are breaking waves, bores, surges and jumps the same flow?" Environmental
Fluid Mechanics, Vol. 17, No. 1, pp. 47-77, (DOI: 10.1007/s10652-016-9475-y) (ISSN 1567-7419 [Print] 1573-1510
[Online]).
4
3. VOID FRACTION KINEMATICS
3.1 Breaking waves
Fûhrboter (1970) discussed the correlation between the turbulence generated in the surf zone and the amount of air
entrained during the breaking of the waves, as well as the sudden reduction of wave height and energy. He highlighted
the importance to study quantitatively the air entrainment process for a detailed comprehension of the surf zone
physics. Vagle and Farmer (1992) and recently Anguelova and Huq (2012) reviewed the different techniques used to
quantify the void fraction under breaking waves. Both works concluded that combined techniques were the best
approach. Indeed, the higher the concentrations of bubbles within bubble clouds, the more difficult it is to count and
measure individual bubbles.
Some studies have been conducted in field while others have been completed in physical wave tanks. Thorpe (1982)
studied wind-waves breaking and speculated that wind speed, salinity, and temperature were major factors, possibly
responsible for existing discrepancies that arised when comparing data from different sources. Monahan (1993)
proposed the terms Alpha-plume (high void fraction, short lifespan), Beta-plume, and Gamma-plume (low void
fraction, long lifespan) to describe the evolution of a bubble cloud, from its formation to its disappearance (e. g.
dissolution, degassing and advection). Most field studies confirmed that the Alpha-plumes consist of high void
fractions (10% or more) with large bubble sizes (radii ranging from tens of micrometers to millimeters). At the other
end of the process, the Gamma-plume were observed to be very low void fraction between 10
-5
to 10
-8
and containing
bubbles with radii on the order of O(10-100)m. The lifetime of a whole bubble cloud may be about a hundred of
seconds. The bubble clouds are also generally confined to the first few meters of the water column. For example,
Lamarre & Melville (1992) compared field and laboratory void fraction measurements obtained with an impedance
probe, and showed large void fraction values at shallow locations while lower void fraction values were found deeper.
Deane (1997) used acoustic and optical measurements of individual breaking waves in the surf zone, off La Jolla
Shores beach, California. Total void fractions of 0.3–0.4 were measured, consisting of bubbles with radius greater than
1 mm. Stokes and Dean (1999) observed that the time scale for the generation of clouds of submillimetric bubbles was
on the order of about 90 ms. Dahl & Jessup (1995) found comparable quantities in deep-ocean studies. Gemmrich and
Farmer (1999) measured void fraction values (e.g. 10
-2
at 0.25 m below the free-surface), associated with low
penetrating breaking events (spilling breakers), while they speculated that higher values of void fractions found deeper
would be associated with more energetic violent events (plunging breakers). Interestingly Gemmrich (2010) found
higher turbulence levels within the wave crest region of the breaking waves, suggesting that the bubble fragmentation
process is mainly driven by turbulence. Most studies reported that void fraction changes significantly during the
lifetime of the bubble cloud, from high void fractions in the first seconds of the breaking event to residual void
fractions persisting for long times. Most field studies consisted in wind-waves breaking observations, with only few
events giving data susceptible to be accurately analysed.
A lot of studies investigated the hydrodynamics in the surf zone, especially the general mechanisms involved during the
breaking process (Peregrine, 1983), the generation of turbulence (Battjes, 1988), and sediment transport. When waves
break, the flow suddenly exhibits a violent transition from irrotational to rotational motion over the entire water
column. Two main types of breaker types have been studied: (1) the spilling breakers, where white foam, consisting of
a turbulent air/water mixture, appears at the wave crest and spills down the front face of the propagating wave; and (2)
the plunging breakers, where the front face of the steepening wave overturns and impacts the forward face of the wave.
These two breaker types have been shown to have similar initial motions, but with different length scales (Basco,
1985). When approaching a beach, the waves change form due to the decrease in water depth. The forward face of the
wave steepens and the wave becomes asymmetric. Once the front face becomes almost vertical, a jet of liquid is
projected from the crest of the wave. The tongue of water thrown from the crest develops and free falls down forward
into a characteristic overturning motion, and eventually hits the water at the plunge point. Depending on the position of
the plunge point, different breaker types can be observed. If the plunge point is located very near to the crest of the
wave, the resulting splash is directed down the wave leading to a spilling breaker. Otherwise, if the jet is ejected farther
towards the lower part of the face of the steepening wave, the wave becomes a plunging breaker. The plunging jet
encloses an air pocket when it finally hits the wave face at the plunge point. The jet re-enters the water after impact,
forcing up a second jet, called splash-up. The early works of Miller (1976), Basco (1985), Jansen (1986) and Bonmarin
(1989) were dedicated to qualitative description of the dynamics of the breaking process, the air entrainment and the
evolution of the large-scale geometric properties of bubble plumes. The overturning process, subsequent overturning
motion and plunging jet impact were described, resulting in the identification and tracking of breaker vortices
trajectories. Some information about the evolution (size, shape and position) of the bubble plumes were also detailed.
The jet-splash cycles, occurring several times in a single breaker, have been shown to be responsible for the generation
of a sequence of large-scale vortices with a horizontal axis of rotation, some of these eddies have been shown to be co-
rotating vortices and some counter-rotating vortices depending on the splash-up mechanism (Miller, 1976; Bonmarin,
LUBIN, P., and CHANSON, H. (2017). "Are breaking waves, bores, surges and jumps the same flow?" Environmental
Fluid Mechanics, Vol. 17, No. 1, pp. 47-77, (DOI: 10.1007/s10652-016-9475-y) (ISSN 1567-7419 [Print] 1573-1510
[Online]).
5
1989). Nadaoka et al. (1989) detailed the flow field under a turbulent bore propagating towards the shoreline, resulting
from a spilling breaking wave. Large-scale horizontal eddies are present in the bore front, while behind the wave crest
the flow structure changes rapidly into obliquely downward stretched three-dimensional (3D) eddies, so-called
‘obliquely descending eddies’. Lin and Hwung (1992), Govender et al. (2002) and Kimmoun and Branger (2007) also
described the large motions of aerated regions under plunging breaking waves, with splash-ups and vortical structures.
Miller (1976) measured the average bubble concentration in plunging and spilling breakers and indicated a larger
bubble density presence in plunging breakers (about 31% in the late stage compared to 19% for spilling breakers); these
results were in agreement with earlier descriptions from Miller (1972). Lamarre and Melville (1991) concluded that a
large portion of the mechanical energy of the wave was lost in entraining the bubble clouds. High values of void
fractions (up to 100 %) were found next to the free-surface, and void fractions of at least 20% were observed for up to
half a wave period after the breaking occurrence. They later confirmed that air entrainment was closely related with the
energy dissipation of the breaking wave (Lamarre and Melville, 1994). Several other works provided more
comprehensive laboratory measurements of the void fraction in breaking waves, detailing the vertical and horizontal
distributions of void fraction. Cox and Shin (2003) measured the void fraction in the aerated region at a point using a
capacitance probe, and observed peak ensemble-averaged void fractions in the range of 15–20%. Surprisingly, they
measured higher void fractions under the spilling breakers than under the plunging breakers. The temporal variation of
void fraction, above and below the still water level, was analysed using three breaker types (spilling, spilling/plunging
and plunging). The temporal variation of void fraction above the still water normalized by the wave period and average
void fraction appears to be remarkably self-similar (independently of the breaker type). Hwung et al. (1992) found a
deeper penetration of air bubbles under plunging breaking waves and higher void fractions (18%), compared to the
spilling breaking waves (12%). Similarly Hoque and Aoki (2005), using a conductivity probe, measured maximum
void fractions of 20% and 16% beneath plunging and spilling breakers, respectively. Mori et al. (2007, 2008) obtained
void fractions of 19% beneath spilling breaking waves and 24% beneath plunging breaking waves, using dual-tip
resistivity void probe. Interestingly they also studied scale effects according to Froude similarity and using two
different scaled experiments. Void fractions were affected by the geometric scale, with larger quantities being found in
the larger experiment, while the bubble size spectra proved to be nearly independent. Kimmoun and Branger (2007)
estimated the evolution of void fractions using particle image velocimetry images and velocity measurements. They
reported large void fractions of up to 88% in the first splash-up location, decreasing slowly when the breaking wave
propagates towards the shore, with values between 20 and 30%. Much lower void fractions were found in other studies.
Kalvoda et al. (2003) investigated the geometric and kinematic characteristics of large air bubbles clouds produced by
spilling breaking wind waves. They observed that the lifetime of the bubble cloud was about 1.4 times the wave period,
with bubbles diameters in the range of 1.0–10 mm. They found a void fraction of about 0.4%. Leifer et al. (2006)
reported void fractions between 0.2% and 2.3% beneath breaking wind waves, using a video system to characterise the
bubble clouds. Blenkinsopp and Chaplin (2007) studied plunging, spilling/plunging and spilling breaking waves. They
calculated integral properties of the bubble clouds and splash-ups, such as areas and volumes of air entrained,
trajectories of centroids and energy dissipation, and showed remarkable similarity between plunging and spilling
breakers. Their data indicated that the evolution of the bubble clouds was subjected to scale effect. Rojas and Loewen
(2010) detailed the void fraction evolution in spilling and breaking breakers. They observed that beneath plunging
breaking breakers, the mean void fraction ranged between 1.2 to 37%, while beneath spilling breaking waves, the mean
void fraction ranged between 17 to 29%. They found that “an energetic spilling breaker may entrain approximately the
same volume of air as a steeper, larger-amplitude plunging breaker”. They identified and tracked successive bubble
clouds, detailed the void fractions at each step of the breaking events, and found that, beneath the spilling breaker, the
celerity of the bubble cloud compared with the phase speed. Beneath the plunging breaker, the celerity of the air cavity
was about 70% of the phase speed. This has to be compared to the celerity of the bubble cloud entrained by the
propagating splash-up which has been measured to be about 90% of the phase speed. Blenkinsopp and Chaplin (2007)
and Rojas and Loewen (2010) found that the volume of air entrained by the splash-up, observed during a plunging
breaking event, was greater than the volume of air entrained by the initial plunging jet (about 60% more). More
recently Anguelova and Huq (2012) used an imaging technique to quantify the phase dependent void fraction, and
measured values reaching 80–99% at the wave crest phase and decreasing to 20-30% at the trough phase. Lim et al.
(2015) confirmed these results in the case of a plunging breaker. They showed that the distribution of the turbulent
intensities matched the vorticity and void fraction fields. Nevertheless, some differences could be observed in the
experimental results for the peak values of void fraction, indicating a strong temporal and spatial variability in the
unsteady breaking waves (Lim et al., 2015). The difference in the locations of the measurements and the method used
to generate the breaking could also be responsible for the discrepancies. Some authors indicated that the mean void
fraction could be modelled by a linear function of time followed by an exponential decay. Hoque and Aoki (2005)
found that the void fraction distribution followed the analytical solution of an advection equation. This is not surprising
as there is a general consensus about in void fractions contours in the breaking waves, shapes and general kinematics of
the aerated regions. However, some differences can also be noted. Lamarre and Melville (1991) found that that the
temporal variation of the normalized void fraction in deep water breakers could be fitted by a power law t
-2.3
, while
