Wave Breaking

93

Wave Breaking

D.H. Peregrine

Dept. of Mathematics,Bristol University, University Walk, Bristol, BS8 IlW, England.

Introduction.

Water waves break as they advance over a gently sloping beach for all off-shore sea states except for the calmest conditions. Most theoretical, experimental and field studies have been directed to this type of wave breaking, yet waves can break with greater violence on steep slopes and close to coastal structures and c1iffs. Further, on those occasions when winds are actively generating waves, the offshore incident wave field already has a significant density of breaking waves before any effects of shoaling or coastal currents stimulate more breaking.

It is easy to appreciate why most attention has been given to waves breaking on gently sloping beaches. Comparison with experirnents shows that the assumption of slowly-varying properties for a periodic wave train can give a description of wave height variation, which together with a breaking criterion is adequate for many engineering purposes, though the effects of irregular and three-dimensional incident waves are not weil documented or analysed. Theoretical treatments for other bathymetry or steep incident waves have only recently become available for research purposes, and the interaction between theory and experiment which is so vital to greater understanding is only just developing.

Here, we first discuss the breaking criterion for classical slowly-varying wave theory. Current knowledge on the kinematics and dynamics of wave breaking is then briefly surveyed. Brief comments are given on the effects of subsurface structures and vertical walls on two-dimensional breaking, followed by some thoughts on the problems of sealing from experiments, or numerical modeis, to prototype.

Breaking criteria.

When wave refraction is calculated using a "slowly-varying wave train" theory some criterion is needed to determine when breaking will occur. Since a train waves of a given period on water with a given constant depth and given mean current have a maximum steepness, this maximum steepness is the most obvious criterion to use. However, as Longuet-Higgins and Fenton (1974) first demonstrated, lower waves close to the maximum have greater phase veloeities and energy den sities than the Iimiting wave of maximum steepness. As aresult, computations using "numerically exact" wave trains such as those of Stiassnie and Peregrine (1980) and Sakai & Battjes (1980) reach a limit at a lower wave. Further, Tanaka (1983,1985,1986) has shown, for both deep-water waves and solitary waves, that waves steeper than those with maximum energy density are unstable. Thus the steepness of the wave of maximum energy density gives a more logical choice of breaking criterion. In fact,at the present time the value of this steepness is not readily available, except for deep water waves and waves which are sufficiently long that the solitary-wave result is

relevant. In addition, there are several reasons for using any breaker criterion with care.

ft is unusual to use fully accurate wave train solutions in refraction computations. In fact I know of no fully consistent examples with better than third order accuracy other than those for the especially simple geometries where waves vary with only a single spatial coordinate. Often linear theory is used. Inaccuracies due to less-than-accurate wave-train properties are likely to be more significant than a 10% variation in choice of breaking criterion.

A breaking criterion only indicates where a slowly-varying wave train may commence evolution to breaking. The comparison between Hansen and Svendsen's (1979) experiments and Stiassnie and Peregrine's (1980) computations (figures 9 to 11 in the latter paper) shows the type of discrepancy that may be found. This is iIIustrated here in figure 1 which shows both the experimental and theoretical results. Alternatively, the numerical computations of the growth of Tanaka's instability (Tanaka, Dold, Lewy and Peregrine, 1987, and Jillians, 1989) give an indication of the time taken for a wave's full development to breaking. For waves in shallow water a crest can travel 8 times the water depth as breaking develops. This is consistent with the expertmental comparison mentioned above.

In addition there is the wide range of environmental effects which are usually neglected in practical wave modeIs. Some of the more significant are the variations between successive waves, three-dimensional effects and current shear. Modulations and irregularities in the incident waves can be modelled by using a statistical distribution of wave heights in those cases where the breaking behaviour is dominated by the shallow depth of water (Battjes & lanssen 1978).One effect of nonlinearity in three-dimensional waves is to strongly enhance diffraction of wave energy, "along crests," away from locally high waves. This has been demonstrated experimentally and theoretically. See Peregrine, Skyner, Stiassnie and Dold (1988) for a preliminary account of an experimental and theoretical comparison for waves that are focussed, and Peregrine (1988) for a survey of nonlinear effects in refraction.

It is now some time since Phillips and Banner (1974) and Banner and PhiJlips (1974) argued that the vorticity in the surface layer of the water, such as in a wind-drift layer can substantially affect wave breaking. The only experimental investigation, by Douglas and Weggel (1988), was more qualitative than quantitative, but it did show that strong effects are found for wave breaking on a beach. A following wind caused waves to break "early" at a significantly lower steepness. An adverse wind de1ayed breaking till a higher steepness. Surfers are weil aware of these differences but further quantification is needed. One theoretical result that may help is Teles da Silva and Peregrine's (1988) study of waves with constant vorticity. For waves in shallow water the variation of height of the Iimiting solitary wave with such constant shear is shown in figure 2. Perhaps the most useful result to highlight is for small shear, which we can measure as óu, which is the difference between the velocity at the surface and the bed in the sense that is positive if the surface velocity is greater

against

the wave direction. For current shear that is not too large a Iinear approximation based on the slope of the line in figure 2 at zero vorticity is probably quite adequate.The slope is just I.O. Thus for weak shears:

WAVE BREAKING 95

300

100 200

It* (rnrn)

Figure l(a). A cornparison of slowly-varying-wave -train theory using accurate wave-train solutions (Stiassnie & Perergine 1980) and experirnenta! results from Hansen&

Svendsen (1979):wave height against water depth for a plane beach of slope

1:34.26; wave period 1.67secs; theoretica! deep-water wave height for the

middle theoretical curve is 37.5mm. The theoretica! wavelength of the Iimiting wave is indicated.

mrn /

~~~nn

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'm

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mm",,,,,,,,nmn'

"'0"'"

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Figure l(b).Measured wave profiles,from a slowly-moving wave gauge,adjusted with the

calculated phase velocity to give equivalent spatia! profiles for the steeepest waves measured in the experiment iIIustrated in figure 1(a).

o

Figure 2. The height of the highest solitary waves on water with constant vorticity.

Data courtesy A.F.Teles da Silva.The vorticity

n=

~u/(gh)ln where

~u is the difference in velocity between the surface and the bed..where ~u

is positive if the surface velocity is greater against the wave's motion.

WAVE BREAKING 97 maxjmum wave hei~ht

h

= 0.83 + ~

(gh)l/2

whereh is the water depth and 0.83isthe maximum heightof an irrotational solitary

wave.

Mechanics of wave breaking.

"Wave breaking" occurs in a range of ways which are described by terms such as

plunging, spilling and surging. There isno clear-cut distinction between these types

of breakers, yet these descriptive terms continue to be used since no good quantitative

measure of breaking type has yet been developed. In only two cases can some details

of the flow be described as known.That is,in the initial stages of a plunging breaker

and in the quasi-sready flow of aspilling breaker or bore.

Waves that overturn and produce a plunging breaker can be modelled as an

inviscid, irrotational flow. Since the work of Longuet-Higgins and Cokelet (1976)

several computational methods have been developed, which describe the overturning

motion. In principle this can be followed until the jet plunges into the water in front

of the wave and causes a splash. In practice the jet is usually too thin to be well

resolved. However itsinitial development gives a good indication of the veloeities

involved. The most striking feature discovered from those computations, (peregrine

et al., 1980) is that the water rising up the front of the wave into the jet is subject to

large accelerations. Typical computed maxima are up to 5g. New et al. (1985) give

details of profiles, veloeities and accelerations of representative waves. An example

shown in figure 3has a maximum acceleratioin of 3g at the last time shown. The

sequence of profiles also gives an indication of the time and distance involved as a

wave develops towardsbreaking.

Large accelerations of fluid imply large pressure gradients. These are normal to

the water surface since the pressure of the atmosphere is almost constant along the

free surface. The large pressure gradients give extra "buoyancy" to objects in the

water which are not accelerating and may be a useful contribution to support of surf

boards. They also imply that the pressure a short distance behind the face of an

overtuming wave is relatively large. This pressure is that provided by the

near-hydrostatic vertical pressure gradient in the mass of water forming the main wave

crest, which for a typical propagating wave has a relatively gentle backslope. This

pressure and its high gradient may be an important factor to consider for some

floating objects, but for structures in the surf zone wave impact is likely to be of

greater concern.

The traditional criterion for wave breaking is that horizontal water veloeities in the

crest must exceed the speed ofthe crest. This appears self evident, but once detailed

flow fields are examined it is found that since the crest shape is changing there is

often no precisely relevant crest velocity. There is a range of veloeities which

roughly correspond to crest speed. and water veloeities usually exceed these by

appreciabie margins.

Despite the fact thatoverturning waves are unsteady two-dirnensional flows with a

free surface some progress has been made towards analytical description.

Longuet-Higgins (1980)suggests that asolution for a rotating hyperbola falling under gravity

could represent the motion at the tip of the jet. New (1983) found that the curve of the face of waves underneath a jet is often well described by an ellipse, and found unsteady solutions for flow around an elliptical free surface. More comprehensive, but approximate solutions have been described by Greenhow (1983) and Jillians (1988). One feature of all these analytical solutions is that they have several free parameters. Although plunging breakers often appear to have very similar shapes, details of computed solutions show that there is appreciabie variation from case to case. The initial direction and velocity of the jet can vary substantially as well as its size relative to the rest of the wave.

Observational papers, such as Myrhaug and Kjeldsen (1984) and Bonmarin and Ramamonjiarisoa (1985) for deep water waves, tend to concentrate on geometrical parameters such as asymrnetry which are hard to re1ate to dynamical aspects of a wave. Velocity measurements in the jet are difficult to make but some velocity measurements in breaking waves have been made, e.g. Hedges and Kirkgöz (1981), Easson et al. (1988), Skjelbreia (1987), Skyneret al. (1990).

Comparisons between experiments and fully nonlinear computed solutions have been made in a few cases. The most ambitious is Dommermuth and Yue's (1987) modelling of a wavetank with dispersive waves focussed in time and space to provide one larger breaking wave. Greenhow (1987) follows the experimental work by Lin et al. (1984), which describes the rise of water at a plate set in impulsive motion among other examples, with computations of the splash formed when a wedge hits the water surface. Skyner et al. (1990), note that from their computations, the actual details of wave breaking have a sensitive dependenee on the initial conditions but in Skyner & Greated (1992) they report a remarkably good agreement between theory and measurement,even in the jet. Computations of solitary waves meeting a submerged semicircular cylinder mentioned below are also in agreement with experiment (Cooker et al., 1990).

As soon as the jet of a breaking wave touches down, at the plunge point, the wave's dynamics change dramatically. The flow around the trapped air forms a vortex. The circulation around the vortex can be found in those computations which follow the falling jet to the plunge point. Other vorticity is generated in a shear layer between water from the jet and the water it is penetrating. A splash is created which can rise higher than the crest of the original wave. The sheet of water pushed up by the falling jet comes from the previously undisturbed water and falls back down repeating a cycle of vortex formation and splashing which can continue many times, although increasingly obscured by drops and bubbles. These splashing and vortical motions on the largest scales appear to be fully deterministic despite the growing level of turbulence. Initially the turbulence is associated with the shear layer, but the instabilities of the splashes and trapped air augment the turbulence. Eventually the flow becomes fully turbulent, either as a propagating spilling breaker, or a turbulent bore depending on the size of the breaking event, the wave, and the depth of water.

The study of this stage of wave breaking is still at the observational stage with contributions by Miller (1976), Peregrine (1983), Basco (1985), Jansen (1986), Nadaoka (1986)and Nadaoka et al. (1988). the latter paper contributes ana1ysis of the velocity , vorticity and rate of strain fie1ds. Tallent, Yamashita &Tsuchiya (1990) have been able to obtain particu1arly usefu1 pictures which identify the main bulk of

WA VE BREAKING

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water from regions with many bubbles or drops, and as aresult many cycles of splash

and vortex formation are found.

An attempt at theoretical one-dimensional modelling of splashes occuning in

shallow water is described in Peregrine (1981). A number of interesting features

emerge about theflowstructure and sketches are given, but it is concluded that since

the shear bet ween the falling jet and the water beneath it is not included in the model

more complicated flow patterns are likely.

The transition from wave overtuming through to some quasi-sready spilling breaker or bore is c1early seen in measurements of waves on beaches where the change of character is c1early identified by Svendsen et al. (1978) from their

substantial study of periodic waves. See Svendsen (1992 companion to this paper). It

is probably sensible to consider this transition zone sperately in wave modelling.

Quasi-steady breaking waves.

After the initial overtuming and splashing of wavebreaking, breaking often

continues. In the continuing breakers, the waves are usually quasi-sready in the sense

that they change their shape only slightly whilst propagating an 0(1) distance, e.g. a distance equal to the depth in shallow water, or perhaps wave height or crest width in

deeper water. Peregrine and Svendsen (1978) proposed a "spectrum" of quasi-sready

waves which they had found helpful in trying to interpret the flow in spilling breakers

and bores.

At one extreme of this spectrum is the spilling breaker with just a touch of

turbulence near its crest,though Duncan's (1981,1983) observations of waves behind

a hydrofoil suggest there is a minimum size for a steady spilling breaker relative to

the size of the wave crest. Moving across the spectrum leads to"spilling" regions

covering more of the face of the wave,ranging through to turbulent bores if the water

is sufficiently shallow. If a new reference frame is taken,looking at a steady wave on

a current rather than a propagating wave, the turbulent bore becomes equivalent to a hydraulic jump on a supercritical stream. Hydraulic jumps of differing fonn occur

according to the bed profile and the shear in the incoming flow. The simplest

extension of the breaking-wave spectrum is to consider jumps over beds of increasing

slope, right up to the case of a vertically falling stream: a waterfall.

As a result of studying this extended 'spectrum' of quasi-sready breaking waves in

the laboratory, with various flow visualization techniques, Peregrine and Svendsen

(1978) suggest that these flows may be best modelled by considering the whole

region of turbulence. Previously most attent ion had been directed to the "surface

roller"; that is, that part of the mean flow which forms a recirculating region

travelling with the wave. However, the turbulent veloeities are of the same order of

magnitude as the wave velocity. The fluid content of the "roller" is continually

mixing withthe rest of the turbulent fluidin the wave.These two views are illustrated

in figure 3.Experimental velocity measurements from deep water waves by Battjes

and Sakai (1981)andDuncan (1981) and for waves on beaches byStive (1980) and

Sakai et al. (1982) and Nadaoka et al. (1988) give support to this description. An important souree of the turbulence is at the "toe" of the roller where previously

WAVE BREAKING 101

-r\

"J

)

Figure 4. (a) Traditional view of a spilling breaker with a recirculating roller.

(b) An indication of the intense turbulence that occurs in a breaker, making it more appropriate to consider a turbulent region than a relatively

isolated roller.

undisturbed water meets the turbulent layer for the first time. This region has a similarity to the classical turbulent mixing layer.

Theoretical modelling hasmade some progress in recent years beyond the classical model of the hydraulic jump. Madsen and Svendsen (1982) and Svendsen and Madsen(1984) model hydraulic jumps and bores by describing two layers of water, one turbulent, the other irrotational, in a nonlinear shallow-water approximation with apparent success. Banner (1987)uses a"Iumped mass"approximation to a roller and obtains ordinary differential equations which give a rough simulation of his measurements of the unsteady response of a steady breaker to a disturbance. The work which has been carried furthest at present (Tulin and Cointe, 1986, and Cointe, 1987)uses modelling intermediate between the above mentioned. Linear wave theory is combined with a passive hydrostatic model of the roller, and results compare favourably with experiments. All these mode1s require refinement, but constitute a promising start to understanding a complex flow. Svendsen(l992, companion to this paper) describes further modelling of waves and associated flows in the surf zone. Wave breaking near structures.

Until recently information on wave breaking above bottom topography which varies rapidly, as for example on the steep underwater slope of a breakwater, could only come from experiments. Now, two-dimensional irrotational flow computations are permitting detailed study of some useful examples. Cooker et al. (1990) describe solitary waves meeting an underwater semi-eireular cylinder for the full parameter range of solitary wave height and size of cylinder. A surprising result was found: even the highest waves only break after passing the crest of the cylinder, some waves breaking beyond the far side of the cylinder. The absence of any backwash from a previous wave may explain why breaking does not occur as the waves meet the cylinder. Grilli and Svendsen (1990) gives examples of waves meeting a steep slope.

One of the more interesting of these computations is that of Cooker and Peregrine (1990a), describing a breaking wave slamming onto avertical wall. Contrary to expectations, theirrotational flow can describe very violent water motions. Astrong vertical jet can form without an actual impact taking place. The water at the wall suddenly rises extremely rapidly just as the vertical face of a wave appears about to hit the wall. Brief very high pressures on the wall are computed. From a practical point of view the Cooker and Peregrine (1990a) computations are particularly useful for comparison with simpier approximations and computations. In particular, the classical concept of pressure-impulse (Lamb, 1932, §11) can be used to give pressure fields,which except in the vicinity of the most violent motion,compare weil with the more detailed computations (Cooker and Peregrine 1990b). Pressure-impulse calculations show that wave shapes may be greatly simplified without significant loss of accuracy. Since the calculations only involve solving Laplace's equation there are many techniques and solutions available.

The wide-variation of peak impact pressures due to breaking waves has long been appreciated. Recent experimental work illustrates this weIl. Chan and MelviIle (1989) measure pressures on a vertical plate which is placed at different positions relative to a repeatable, single breaking wave. Small variations of plate position lead to substantial changes in peak pressure. Perhaps more closely related to coastal waves are experirnents such as thoseof Witte (1988) and Kirkghöz (1991) where

WAVE BREAKING 103

pressures are measured for periodic breaking waves meeting a wall. AIthough the

waves are nominally periodic, the pressures and its properties, such as rise time etc.,

show asubstantial scatter which is presented in statistical form. Most of the data are

closetoa log-normal distribution.

Measurements of the pressure field on a wall at impact show that it spreads right

down to the bed. the computations mentioned above show that the short-lived high

pressure fieldcanalso have substantial gradientsalong the bed near the wall. Cooker

and Peregrine (1992) calculate that for objects on the bed this pressure gradient can

be more significant than other hydrodynamic forces. This impact related force

increases with the size of the object more rapidly than fluid drag and thus may have

significanee foritems such asarmour units,which could be driven away from impact

areas by multiple impulsesduring a storm.

Sea

l

ing to prototype

With the increase of experimental and computational data concerning breaking wave impact on structures the problem of sealing up to prototype conditions becomes

more pressing. Simple plots of peak pressures against typical length-scales of

experiments show IittJe useful correlation.

My present views on thesubject are that measurements must be examined more

closely. Those papers where the time history of impact pressure is weIl recorded

show two distinct types of impact; one, a single intense, very short peak, the other a

strong, short peak followed by oscillations. There are two experiments (Chan and

Melville, 1988 and Arami and Hattori, 1989) where photos/video of the waves give a

cIear indication of the motion. The first more severe type of flow involves little or no

trapped air and is similar toCooker and Peregrine's (l990a) most intense example. In

the second oscillatory case, a significant amount of air is trapped near the wall.

Simple calculations involving thecompressibility of trapped air indicate oscillation

frequencies close to those observed (Topliss, Cooker and Peregrine, 1992). This

suggests that sealing for the two types of impact should be quite different with compressibility of trapped air to be allowed for in the second case.

The computations of Cooker and Peregrine (I990a) provide some guidance for the

most extreme examples. At first sight simple Froude sealing appears appropriate.

However, it is found that really high pressures only occur in a small region where the free surface is"focussed" before ajet forms. The smaller this focussing region the

higher the pressure from a given wave. Thereis a practicallower limit of size of such

a region given by the initial roughness of the water surface or of the wall. In

prototype conditions the surface roughness seems Iikely to play a dominant role in

reducing the maximum intensity of impact.

R

ef

e

r

en

ces

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WA VE BREAKING 105

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