VOLUME 32 JOURNAL OF PHYSICAL MAY 2002

Experimental Investigation of Wave Breaking Criteria Based on Wave Phase Speeds

PAUL STANSELL Department of Civil and Offshore Engineering, Heriot-Watt University, Edinburgh, Scotland

COLIN MACFARLANE Department of Ship and Marine Technology, University of Strathclyde, Glasgow, Scotland

(Manuscript received 26 May 2000, in ®nal form 16 August 2001)

ABSTRACT Experiments were performed that test the kinematic breaking criterion, which states that the horizontal ¯uid particle velocity at the surface of a crest exceeds the local phase speed of the crest prior to breaking. Three different de®nitions of phase speed are used to calculate phase velocities of the wave crests from detailed surface elevation measurements. The ®rst de®nition, based on the equivalent linear wave, is constant over the wavelength of the wave. The second, based on partial Hilbert transforms of the surface elevation data, is local in space and time giving instantaneous values at all space and time measurements. The third, based on the speed of the position of the crest maximum, is local in time but not in space. Particle image velocimetry is used to obtain horizontal components of ¯uid particle velocity at the surface of crests of breaking and nonbreaking waves produced in a wave ¯ume using the chirp-pulse focusing technique. The ratios of these ¯uid particle speeds to crest phase speeds are calculated and are consistently less than unity. The speed ratio for the rounded crest of a plunging breaker is at most 0.81, implying that the kinematic breaking criterion is far from satis®ed. The ratio for the more sharply pointed crest of a spilling breaker is at most 0.95, implying that the kinematic breaking criterion is closer to being satis®ed. Thus, for the breaking waves in this study the kinematic breaking criterion is not satis®ed for any of the three de®nitions of phase speed and so it cannot be regarded as a universal predictor of wave breaking.

1. Introduction horizontal ¯uid particle velocity in the crest to the phase speed of the crest is greater than unity. The breaking of surface gravity waves is an important This hypothesis has been stated as early as Rankine phenomenon. It is one of the most signi®cant energy (1864), by Kinsman (1984, section 5.4), Kjeldsen redistribution terms in wave models. It contrib- (1990), Tulin and Li (1992), and as recently as Banner utes to wave damping, horizontal momentum transport and Peregrine (1993, p. 386) where they say that: from waves to currents, and sediment transport in coast- al waters. The prediction of hydrodynamic loads on The traditional [kinematic] criterion for wave breaking structures in the ocean are dependent on the velocity is that horizontal water velocities must exceed the speed and acceleration of the ¯uid particles, and in breaking of the crest. This appears self evident, but from detailed waves these are far greater than those found in non- ¯ow ®elds it is found that since the crest shape is chang- breaking waves of the same height (Melville and Rapp ing there is often no precisely relevant crest velocity. 1988). Yet, despite their importance, breaking waves in Rather, there is a range of velocities which roughly cor- irregular are still poorly understood. Many authors respond to the crest speed, and water velocities usually have proposed breaking criteria (by which we mean a exceed these by appreciable margins. parameter that can, in principle at least, be measured in the region of a wave crest and for which a threshold Nevertheless, it seems that this kinematic criterion for crossing would imply that the wave is breaking or about wave breaking has not been veri®ed, either experimen- to break). One criterion that is often suggested is that tally or computationally. Indeed, Melville and Rapp any surface will break if the ratio of the (1988) state in their introduction that what motivated their paper was the ``desire to identify breaking by searching for events in which the ¯uid velocity at the surface exceeded the phase velocity of the wave'' but Corresponding author address: Dr. Paul Stansell, Department of Civil and Offshore Engineering, Heriot-Watt University, Riccarton, in the body of the paper they ``did not pursue this com- Edinburgh EH14 4AS, Scotland. parison of phase and ¯uid velocities.'' E-mail: [email protected] There exists some confusion and contradiction con-

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and time. We achieve this in section 2 by proposing a particular de®nition of phase speed based on the partial Hilbert transforms of the surface elevation of a wave, ␩(x, t). The de®nition can be applied to obtain a number that can be said to represent the local phase speed of the wave. It can be applied to nonlinear wave forms that are evolving rapidly in space and time. It is truly local in the sense that the phase speed so calculated is a function of both space and time and so can yield different values of phase speed for different positions and times within the same wave. A discussion of related work is deferred until section 3. In section 4 we compare local phase speed values, nonlocal phase speed values obtained from the equivalent linear wave, and phase FIG. 1. Schematic diagram of the parameters used in the de®ning speed values from another de®nition local only in time, equation, Eq. (1), of the kinematic breaking criterion. The hatched with the horizontal ¯uid particle velocities measured at region on the (x, t)-coordinate plane, xd(t␷ ) Ͻ x Ͻ x␷ , td(x␷ ) Ͻ t Ͻ the surface of breaking and nonbreaking wave crests. t␷ , is the region in which the criterion is expected to hold at least one point. Finally, in sections 5 and 6 we offer overall discussions and conclusions. cerning this issue in the literature, where the universality of the kinematic breaking criterion is often assumed 2. Three de®nitions of phase speed correct without question. In this study, we set out to From the pro®le a train of irregular gravity waves investigate this claim directly. We pose the following evolving in space and time on the surface of a ¯uid it question: is it a necessary condition of wave breaking is not straightforward to de®ne the speed of any par- that, at or shortly before the time of breaking and within ticular wave at any particular arbitrary point in space the spatial extent of the crest, there exist horizontal ¯uid and time. Thus, we begin by proposing three possible particle velocities that exceed the phase speed of the de®nitions. crest? To express this question mathematically we de®ne Our ®rst de®nition of phase speed is that based on a the z direction to be that of the outward normal to the simple approach commonly used in ocean studies. We undisturbed ¯uid surface and the x direction to be in de®ne the nonlocal phase speed of an irregular surface the plane of the undisturbed ¯uid surface. The position wave as the speed of an equivalent linear wave. For a and time of the inception of a is de®ned linear wave consisting of a single Fourier component to be the position and time of the ®rst occurrence of a we can always ®nd a frame of reference in which the vertical tangent to the front face to the wave and is wave is stationary. From this observation we can de®ne denoted by (x , t ). The above statement of the kine- ␷ ␷ the phase speed of the wave, c , to be minus the speed matic breaking criterion can now, for a wave traveling p of the moving reference frame. Thus, for the linear wave in the positive x direction, be written as ␩(x, t) ϭ a cos(kx Ϫ ␻t), where k ϭ 2␲/␭ is the wave-

u[z ϭ ␩(x, t), x, t] Ͼ cp(x, t) number, ␻ ϭ 2␲ f is the angular frequency, ␭ is the wavelength, f ϭ 1/T is the frequency, and T is the

for xd(t␷␷) Ͻ x Ͻ x , td(x␷␷) Ͻ t Ͻ t , (1) period, we de®ne where u(z, x, t)isthex component of ¯uid particle ␻ velocity in the coordinate plane (z, x), ␩(x, t)isthez cp ϭ . (2) k coordinate of the ¯uid's free surface at (x, t), t is the time, xd(t) is the position of the ®rst zero up-crossing Application of this de®nition to an irregular wave (for behind the wave crest parameterized by time, td(x␷ )is which a stationary frame of reference does not exist) the time of the ®rst zero up-crossing behind the wave requires the approximation that the values of T and ␭ crest parameterized by position, and cp(x, t) is some are constant over the time and space scale of any par- appropriate de®nition of the wave's phase speed. A ticular wave of interest. To calculate the phase speed of schematic diagram illustrating this de®nition is shown the equivalent regular wave the required values of T and in Fig. 1. ␭ can be measured as the time and distance between As suggested by Banner and Peregrine (1993), ap- consecutive zero-downcrossings on the irregular wave plying the kinematic breaking criterion to an irregular pro®le's time series and space series. If, as is common, wave is complicated by the dif®culty in de®ning or mea- only the time series of the surface pro®le is known, then suring the phase speed of a wave that is not of permanent a relation, such as that for deep water, form. To overcome this problem we require a way of de®ning properties of the wave that are local in space ␻2 ϭ gk, (3)

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ץ -where g is the acceleration due to gravity, can be in ␻(x, t) ϵ ␾ (x, t), (11) t tץ .(voked to eliminate k from Eq. (2 Application of this simple de®nition of the phase speed to a wave that is about to break is not satisfactory where the phase function ␾t(x, t) is given by since, for example, the speed of the breaking crest must Ht[␩(x, t)] be greater than the speed of the trough preceding it if tan␾t(x, t) ϭ , (12) the front face of the wave is to become vertical. By this ␩(x, t) simple de®nition of cp, however, the phase speed is we can write ␻(x, t)as identical at all points between consecutive zero-down- ץץ crossings of a wave. Thus, for a breaking wave it is not 1 ␻(x, t) ϭ ␩ [␩] Ϫ [␩] ␩ . (13) acceptable to consider ␻ and k constant over the time 22 HttH tץ tץ␩ ϩ H t [␩]΂΃ and length scales of the wave as de®ned by the positions of the zero-downcrossings. From Eqs. (4), (10), and (13) a general expression for To overcome this problem we propose a second, in- a phase speed local in space and time is ץץstantaneous or local, de®nition of phase speed as the  ratio of an instantaneous angular frequency, ␻(x, t), with ␩ H [␩] Ϫ H [␩] ␩ t ץt ttץ ␩22ϩ H [␩]  an instantaneous wavenumber, k(x, t). That is c (x, t) ϭ x . (14) ץץ [p ␩22ϩ H [␩ ␻(x, t) t ␩ H [␩] Ϫ H [␩] ␩ xץx xxץc (x, t) ϵ . (4)  p k(x, t) Note that in the linear (small amplitude or Airy wave) The theory of analytic functions provides a convenient approximation Ht[␩] ϭϪHx[␩], which simpli®es the way to de®ne ␻(x, t) and k(x, t). For a two-dimensional expression for the local phase speed to wave (one time and one space dimension) denoted ␩(x, ץץ -t), we require the de®ning equations for the partial Hil ␩ H [␩] Ϫ H [␩] ␩ tץt ttץ -bert transforms of this arbitrary two-dimensional func tion with respect to x and t. These are, respectively [see, cp(x, t) ϭ . ץץ e.g., chapter 7 of Poularikas (1995)], H [␩] ␩ Ϫ ␩ H [␩] xץ xץtt 1 ϱ ␩(xЈ, t) Hx[␩(x, t)] ϭ PdxЈ, (5) Use of the Hilbert transform to de®ne the instanta- ␲ ͵ (x Ϫ xЈ) Ϫϱ neous frequency and wavenumber in this way is not 1 ϱ ␩(x, tЈ) new and is considered by some as controversial, par- H [␩(x, t)] ϭ PdtЈ, (6) ticularly for functions with wideband spectra. For such t ␲ ͵ (t Ϫ tЈ) Ϫϱ functions values of the frequency may be obtained that where P stands for the Cauchy principal value of the are zero or even negative. Such results are regarded by integral. Using Eq. (5), we de®ne the local wave- some as unphysical. They are associated with maxima number by in ␩(x, t) occurring below, or minima occurring above, the mean water level. These large ¯uctuations have been ץ k(x, t) ϵ ␾ (x, t), (7) observed and explained in some detail by Melville x x (1983, section 4.3). In this study we are only interestedץ in obtaining the instantaneous frequencies (and wave- where the phase function ␾ (x, t) is given by x numbers) in a relatively small time (or space) domain H [␩(x, t)] in close vicinity to the crest of the wave where prob- tan␾ (x, t) ϭ x . (8) x ␩(x, t) lematic zero and negative values of the local frequencies and wavenumbers are not present. Differentiating with respect to x gives The third working de®nition of phase speed that we use is de®ned as the speed of the position of the surface ץץ ␾ cos2␾ץ xxϭ ␩ [␩] Ϫ [␩] ␩ . (9) elevation maximum associated with the wave crest of 2 HxxH (x interest (see, e.g., Baldock et al. 1996). Letting xc(tץ xץx ␩ ΂΃ץ 2 2 2 2 denote the x coordinate of the surface maximum of the From Eq. (8) we see that cos ␾x ϭ ␩ /(␩ ϩ Hx[␩]), crest then its phase speed, local only in the time di- which along with Eqs. (7) and (9), gives the following mension, is de®ned to be expression for the local wavenumber

dxc(t) (cp(t) ϵ . (15 ץץ 1 k(x, t) ϭ ␩ H [␩] Ϫ H [␩] ␩ . (10) dt xץx xxץ [␩22ϩ H [␩ x ΂΃This de®nition of phase speed is only valid at the locus

In an analogous fashion, but starting from Eq. (6) and of the crest maximum given by [xc(t), t] and is therefore de®ning the local angular frequency by instantaneous only in time, not in space.

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Note the use of the arguments to distinguish the three the gradient of the phase function ␾t(x1, t), as given by de®nitions of phase speed given by Eqs. (2), (14), and Eq. (16), is ¯awed in that it is an approximation when (15). applied to the case of a ®nite amplitude wave. It is valid in the linear wave limit, that is to say, the limit of in- ®nitesimal wave amplitude. This is because for a wave 3. Survey of related work composed of a linear superposition of Airy waves In this section we review previous studies and show Ht[␩(x, t)] ϭϪHx[␩(x, t)]. [Further discussion on this how none of them answer the question associated with point is given in relation to Hwang et al. (1989, p. 22).] Eq. (1). We discuss three studies that used the Hilbert Melville (1983), however, does not compare his local transform of the surface elevation to de®nite local an- phase speed with ¯uid particle velocities. gular frequency and wavenumber for irregular waves. A numerical study that used the Hilbert transform of We also review some studies, both experimental and the surface elevation to de®ne local aspects of irregular computational, that compare measured ¯uid particle ve- breaking and nonbreaking waves was performed by locities in a breaking crest with some de®nition of phase Banner and Tian (1998). Boundary integral simulations speed. Inconsistency in the results is evident. It will be of fully nonlinear surface waves were used to assess the seen that each of these studies invoked linear theory at behavior of nondimensional relative rates of change of some point. As far as we are aware, of each of the studies local mean energy and momentum densities following that test the kinematic breaking criterion, ours is the their envelopes. The de®nition of these local variables required the calculation of the instantaneous wavelength only one in which measurements of cp(x, t) and u are made that avoid any linear approximations. This rep- and frequency of the wave, although, in their ®nal cal- resents an important aspect of this study which differ- culation, the authors used the frequency of a linear wave entiates our results from those of other studies, espe- as an approximation to the local frequency de®ned by cially given the highly nonlinear nature of breaking Eq. (14). Their results show that, if the nondimensional waves. growth rates exceed 0.2, then the growth rate and the We end this section with a discussion of some relevant envelope become phase locked for a suf®cient time for points from the theory of regular (Stokes) waves. the associated surface wave to acquire enough energy and momentum to break. For nondimensional growth rates that do not exceed 0.2 the growth rate oscillates a. Irregular waves on a faster timescale than the envelope and thus no phase locking occurs and there is continual growth and decay The problem of de®ning a fully local wave speed for in energy and momentum of the associated surface an irregular wave was tackled by Melville (1983). In wave, which therefore does not break. The study sug- his experiments Melville (1983) started with the gen- gests that a growth rate value exceeding the threshold eration of a 2-Hz sinusoidal wave train at the wave of 0.2 would predict the onset of breaking universally. generator. At a ®xed point, x1, the surface elevation of As with Melville (1983), when calculating the local these waves, ␩(x1, t), was sampled at a rate up to 200 wavelength and local angular frequency Banner and Hz. The wave train evolved toward an irregular breaking Tian (1998) do not distinguish between the partial Hil- wave and the Hilbert transform (with respect to time) bert transform with respect to x or t of the surface pro®le of ␩(x1, t) was used to calculate the phase function ␾t ␩(x, t). If they use the partial Hilbert transform with (x1, t) by Eq. (12). The instantaneous angular frequency respect to x then their calculation of k(x, t) will be correct at position x1, ␻(x1, t), was then calculated from Eq. but that of ␻(x, t) may be incorrect. Their calculations (11). To obtain a local wavenumber Melville (1983) of ␻(x, t) may include errors from two sources: that due took the Hilbert transforms (with respect to time) of two to the dif®culty in calculating the Hilbert transforms closely spaced measurements, ␩(x Ϫ⌬x/2, t) and ␩(x 1 1 after time t␷ when ␩(x, t) becomes triple valued and that ϩ⌬x/2, t), to obtain the phase functions ␾t(x1 Ϫ⌬x/ due to the restriction imposed on the time domain in- 2, t) and ␾t(x1 ϩ⌬x/2, t) from Eq. (12). These were tegration, required for the calculation of the partial Hil- then used to de®ne an instantaneous wavenumber at x1 bert transform with respect to time, by the boundary integral method, which breaks down a short time later ␾t(x1 ϩ⌬x/2, t) Ϫ ␾t(x1 Ϫ⌬x/2, t) k(x1, t) ϭ when the tip of the overturning crest touches the forward ⌬x face of the wave. It is interesting to note that in the ®nal implementation of their de®nition, instead of the in- -stantaneous angular frequency ␻(x, t), they use the con ץ ഠ ␾t(x, t) . (16) .x x x stant value given by ␻ for the equivalent linear waveץ []ϭ 1 This is not the same as the de®nition of k(x, t) given in Eq. (7), so the local phase speed de®ned by Melville 1) KINEMATIC BREAKING CRITERION VIA NUMERICAL SIMULATIONS (1983) as the quotient ␻(x1, t)/k(x1, t) is not the same as that given by Eq. (4). It is the present authors' opinion Boundary integral simulations of breaking waves that a general de®nition of local wavenumber based on from which one can compare the ¯uid velocities with

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equivalent linear wave phase speeds have been per- cases the waves had already overturned, t Ͼ t␷ , making formed by Vinje and Brevig (1981) and New et al. redundant the concept of using the ratio u/cp as a break- (1985). Vinje and Brevig made comparisons, for deep ing criterion. Examining Kjeldsen's (1989) Fig. 13, one and shallow water cases, of u[␩(x, t), x, t] within crests sees that the ¯uid particle velocity near the vertical part of plunging breakers and phase velocities, cp, calculated of the front face of the breaking wave is very low, at from an equivalent linear wave. From the two plots in about 0.3 of the linear phase speed, whereas near the their Fig. 4, which show shallow and deep water plung- highest point of the crest it is very much higher, at about ing breakers at time t␷ , one can see that u[z ϭ ␩(x, t␷ ), 1.36 of the linear phase speed. Doubt is cast on Kjeld- x, t␷ ] Ͻ cp for all x. Only from plots in their Fig. 6, sen's experimental results when comparing them with which show the deep water plunging breaker at times the numerical simulations of Vinje and Brevig (1981) t Ͼ t␷ do we see u[z ϭ ␩(x, t), x, t] Ͼ cp for some x and Longuet-Higgins and Cokelet (1976) or the exper- and t. imental and numerical results of Skyner (1996), each of New et al. (1985) performed simulations of plunging whom show a more gradual change in ¯uid particle breakers in water of ®nite depth and achieved results velocity as one moves up the vertical front face to the for shallow water that compare favorably with those of high point or tip of the plunging breaker. Vinje and Brevig (1981). New et al.'s (1985) Figs. 7b Perlin et al. (1996) used particle image velocimetry and 8b, that illustrate their Wave 3, show that (PIV) and particle tracking velocimetry (PTV) to mea- u[␩(x, t), x, t] in the crest of the breaking wave at about sure the ¯uid particle velocities in the crests of plunging the time of the ®rst occurrence of a vertical tangent to breakers. These velocities were compared to wave crest the surface pro®le is almost equal to the phase speed, phase speeds calculated from the equivalent linear wave, 1 cp, of the equivalent linear wave. Subsequently, as the and also the crest high-point de®nition [Eq. (15)]. The ejected jet grows, u[␩(x, t), x, t] in the crest certainly greatest value of u(z, x, t) was measured by PTV in the increases to a value greater than cp. ejecting jet of the overturning wave. The value was Thus, it seems that neither the simulations of Vinje 130% of the equivalent linear wave phase speed, cp. and Brevig (1981) nor those of New et al. (1985) con- However, the greatest value of u(z, x, t) measured before ®rm the kinematic breaking criterion, as de®ned by Eq. the wave overturned was measured by PIV in a crest (1), when using an equivalent linear wave to de®ne the ``with the front face nearly vertical.'' The value was phase speed. 74% of the phase speed. Chang and Liu (1998) used a phase velocity calcu- 2) KINEMATIC BREAKING CRITERION VIA WAVE lated from linear theory and u(z, x, t) measurements FLUME EXPERIMENTS obtained from PIV. They ®nd that, when the wave is Melville and Rapp (1988), Kjeldsen (1989), Perlin et ``close to breaking'' (before the front face is vertical), al. (1996), and Chang and Liu (1998) have all performed the maximum velocity is 0.86 times the linear phase wave ¯ume experiments to measure the ¯uid particle speed, cp. After the wave has overturned they measure velocities in the crests of breaking waves. the maximum velocity in the ejecting jet as 1.07 times Melville and Rapp used laser anemometry to measure cp. the ¯uid particle velocities on the surface of breaking Thus, in none of the experimental studies performed waves by seeding the surface with neutrally buoyant by Melville and Rapp (1988), Kjeldsen (1989), Perlin tracer particles. In their Fig. 3, which shows the surface et al. (1996), or Chang and Liu (1998) are values u(z, x, pro®le and u[␩(x, t), x, t], we see that the maximum t) measured that exceed the phase speed of the equiv- value of u/c measured in the breaking wave is about alent linear wave before the inception of breaking (that 0.575, where c ϭ g/␻ is their de®nition of linear wave is to say, for t Ͻ t␷ ). Thus, because none of these studies speed. measure u Ͼ cp for t Ͻ t␷ , the kinematic breaking cri- Kjeldsen (1989) used two different methods to mea- terion, as de®ned by Eq. (1), is not supported by any sure the ¯uid velocities in the crests of plunging breakers of these studies. after they had overturned but before their tips touched Hwang et al. (1989) performed wave ¯ume experi- the forward face. For the ®rst method, a current meter ments that measured only the surface wave pro®le, ␩(t), was moved along with the wave crest; for the second, at a single point in space. The ¯uid particle velocities high-speed ®lms of the plunging breaker were taken were not measured directly. Instead they obtained values from which the ¯uid velocity normal to the surface of for the x component of the ¯uid particle velocity at the the overturning crest was calculated. In both cases ¯uid surface from, in our notation, ץ Kjeldsen measured u(z, x, t) in the crest that exceeded u ϭ Ht[␩(t)], (17) tץ the equivalent linear wave phase speed, cp, but in both

1 The phase speed of the equivalent linear wave is calculated by which only gives a correct value for u in the linear wave theory approximation. They do, however, de®ne a local Eqs. (2) and (3) as cp ഠ 0.4 since ␭ ϭ g ϭ 1 in New et al.'s (1985) simulations. phase velocity similar, but not identical, to that de®ned

Unauthenticated | Downloaded 09/27/21 02:22 PM UTC 1274 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32 by Eq. (14). The difference is their need to de®ne local such as its 120Њ crest angle [see, e.g., p. 189 of Cokelet wavelength by (1977)]. Equivalence of phase speed and ¯uid particle speed in the crest is a logical conjecture for a periodic ץ k(x, t) ϭ ␾t(x, t) wave that has a sharply pointed crest. A limiting peri- x odic wave has a sharp corner ¯ow (with zero radius ofץ instead of by Eq. (7). Their de®nition is only valid when curvature) at the tip of the crest and surface of the ¯uid

␾t(x, t) ϭ ␾x(x, t), which again only occurs in the linear is a streamline. Viewing the wave from a reference approximation. Although Hwang et al. (1989) do mea- frame in which the surface pro®le is stationary (that is sure u Ͼ cp, it is not clear whether their measurements to say, a frame moving at the phase speed of the wave) were made in the ejecting jet, that is to say, at times the ¯ow at the tip of the crest must be a stagnation point greater than t␷ . Also, they used linear theory to predict in order to avoid in®nite ¯uid particle accelerations. the ¯uid velocities by Eq. (17), and it is well known Hence, the velocity of the ¯uid particle at the tip of the that linear theory overestimates the magnitude of the sharp crest must be equal to the speed of the wave horizontal ¯uid velocities above the mean water level pro®le. (Gudmestad et al. 1988; Skjelbreia et al. 1991; Gud- The work of Longuet-Higgins and others (cited mestad 1993). above) does not contradict or falsify the kinematic breaking criterion. At ®rst sight it may appear to do so since it shows that a becomes unstable b. Regular waves before it reaches its limiting form where the equivalence The discussion so far has been concerned exclusively of phase speed and ¯uid particle speed in the crest is with irregular waves. For regular irrotational waves certain. But the perturbations applied to the almost-high- (Stokes waves) the situation is different. Instabilities est wave by Longuet-Higgins and Cleaver (1994) are that lead regular waves to break have been studied in not stationary in the frame of reference traveling with some detail during the last three decades. Longuet-Hig- the phase speed of the underlying Stokes wave. Thus, gins (1978a,b) has shown by a linear perturbation anal- the perturbations will not only affect the velocity ®eld ysis that Stokes waves are unstable to certain super- of the surface of the wave, but, more importantly to our harmonic and subharmonic normal-mode amplitude per- discussion, the previously global phase speed of the turbations. Longuet-Higgins and Cokelet (1978) inves- underlying Stokes wave will be replaced by the local tigated the growth of these instabilities numerically and phase speed of the perturbed Stokes wave. The rela- showed that they do indeed lead to breaking. By con- tionship between the local phase speed and the ¯uid sidering just a single isolated crest of a steep (``almost particle velocities on the surface of the unstable Stokes highest'') Stokes wave Longuet-Higgins and Cleaver wave is not clear. (1994) have shown, again by a linear perturbation anal- ysis, that the crest is unstable, having a normal mode 4. Wave ¯ume experiments with nonlinear waves perturbation with an exponential growth rate. Longuet- Higgins and Dommermuth (1997) found a second nor- a. Experimental apparatus and measurements mal mode of crest instability and traced the development Wave experiments were carried out in a wave ¯ume of the fastest growing instability of the almost-highest facility belonging to the Fluids Group at the University wave numerically, showing that these instabilities cause of Edinburgh, Scotland. The ¯ume had a horizontal bot- the crest to overturn. tom and was 9.77 m long by 0.4 m wide with a still The existence of crest instabilities is important be- water depth of 0.75 m. The x coordinate was measured cause it establishes that there is an instability that is along the ¯ume from the front of the vertical wave essentially a local property of the wave crest and hence paddle, and the z coordinate upward from the still water may occur in other types of waves such as irregular level. Waves were produced at one end of the ¯ume by waves. These crest instabilities, however, are not likely a computer-controlled hinged wave paddle. The paddle to account for all types of wave breaking. Longuet- control incorporated a force feedback signal that enabled Higgins and Tanaka (1997) say in conclusion that their it to partially absorb any waves re¯ected from the op- results can only apply to wind waves in a limiting sense posite end of the ¯ume. At the opposite end of the ¯ume as a typical ocean wave spectrum displays strong wave was a passive wave absorber constructed from open cell grouping and a wide range of length scales and both of foam. [See Skyner (1992) for more technical details these features will be associated with other types of about the ¯ume.] wave instability. The motion of the wave paddle is calculated by linear The purpose of the present study is to examine the wave theory, according to which we can specify an kinematic wave breaking criterion that is said to apply arbitrary linear surface elevation with one spatial to wave crests universally. Indeed, it was from the con- dimension, ␩(x, t), as the sum of N Airy waves each jectured equivalence of horizontal particle velocity and with an amplitude a , wavenumber k , angular fre- phase speed applied to a limiting regular wave that n n quency ␻ , and initial phase ⑀ , by the equation Stokes derived other aspects of the limiting regular wave n n

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TABLE 1. Initialization parameters for the set of waves examined function of time from the time of the initial triggering in the wave ¯ume. signal to the wave paddle. For this purpose a series of Spatial Temporal capacitance wave gauges, with electrical resistances giv- focal focal ing a linear response to immersion depth (Morrison Amplitude, A point, xf point, tf 1996), were placed at intervals of 0.1 m along the wave Wave Wave type (m) (m) (s) ¯ume from 0.4 to 7.5 m from the wave paddle. The 1 Plunging 0.143 4.2 15 surface elevation was recorded from the wave gauges 2 Spilling 0.110 3.6 15 at intervals of 1/16 s for a duration of 32 s from the 3 Spilling 0.109 3.4 15 4 Nonbreaking 0.107 4.0 15 initial triggering of the wave paddle. 5 Nonbreaking 0.106 4.0 15 In addition to the surface elevation measurements, 6 Nonbreaking 0.105 4.0 15 ¯uid particle velocity measurements were made in the wave crests using the PIV technique (Skyner 1992; Quinn 1995). These PIV measurements were made at N the positions and times of the start of breaking, (x␷ , t␷ ). ␩(x, t) ϭ a cos(kxϪ ␻ t ϩ ⑀ ), (18) ͸ nn nn PIV is a reliable measurement technique that can yield nϭ1 accurate measurements of the velocity ®eld under break- along with the linear dispersion relationship ing and nonbreaking waves. Skyner (1996) has made direct comparisons between PIV measurements of ¯uid 2 ␻nnϭ gk , particle velocities and predictions of a numerical model where g is the acceleration due to gravity. based on the boundary integral method. He found the Experiments were carried out on the six different predicted and measured kinematics to be in good agree- waves described in Table 1. The waves used in this study ment. will be referred to by their numbers in the ®rst column The PIV measurement system used in this study is of the table. The third column gives the maximum am- now described. A 15-W argon-ion continuous-wave la- ser produced a beam directed onto a rotating octagonal plitude, A ϭ⌺an, of the wave calculated by linear wave theory and the fourth and ®fth columns give the focal mirror and then onto a parabolic mirror. The rate of rotation of the octagonal mirror was adjusted by a ro- points (position of maximum amplitude) in space, xf , 2 tation speed controller unit to give a time between scans and in time, tf , chosen so that ⑀n ϭ ␻ntf Ϫ knxf . These three parameters, together with the shape of the ampli- of 1.70 ms. From the parabolic mirror the laser beam tude spectrum, are de®ning inputs to the program that was re¯ected upward through the bottom of the wave controlled the wave paddle. The amplitude spectrum for ¯ume to give a pulsating light sheet. The water was each of the waves in Table 1 is given by seeded with neutrally buoyant pollen grains that were illuminated by the pulsating light sheet. PIV images Amfϩ s, n ϭ 2,...,N Ϫ 1 were taken by a Kodak ES1 CCD camera ®tted with a a ϭ n n N (mf ϩ s)/2, n ϭ 1, N, Nikon 55-mm lens. The resolution of the CCD camera Ά n was 1008 ϫ 1018 pixels but that of the frame grabber where m ϭϪ0.005 476, s ϭ 0.008 629, and card, which was an Oculus-F/64, was 1024 ϫ 1024 f ϭ f ϩ (n Ϫ 1)␦ f, n ϭ 1,...,N, pixels so all ®nal images had the latter resolution but n 1 with narrow blank strips along the left and bottom edges. where f 1 ϭ 0.4375 and ␦ f ϭ 0.03125. Opening of the camera shutter was triggered by a signal Knowing the form of the spectrum at a point near the from the wave paddle controller and the exposure time paddle in the ¯ume is not considered directly relevant was suf®ciently long to include 3±5 pulses of the light to this study of the kinematic breaking criterion. Indeed, sheet. This yielded individual PIV images of 3±5 ex- wave spectra in the frequency domain are global mea- posures of each pollen particle in the ®eld of view as sures over time whereas wave breaking is local in time it moved across the image with the ¯ow of the ¯uid. with effects distributed across frequencies. For the ex- When analyzed this is equivalent to time averaging each perimental waves, the spectrum produced by, or mea- PIV velocity measurement over 5.1±8.5 ms. sured near, the wave paddle will be different from that Figure 2 shows a PIV image of Wave 3. It is typical measured farther down the wave ¯ume. For this reason, of the PIV images of the spilling breakers. The parasitic detailed descriptions of the spectra of the experimental capillaries beginning to form on the front face, the sharp waves at various positions along the ¯ume are not re- crest shape, and the streak lines of the seeding particles ported. are all clearly visible. Measurements were taken of the surface elevation as Each separate PIV image was then analyzed to cal- a function of distance from the wave paddle and as a culate the ¯uid particle velocities in the ®eld of view. The analysis was carried out with the aid of a PIV anal- 2 These amplitudes predicted by linear theory are signi®cantly dif- ysis program called EdPIV (Dewhirst 1998, appendix ferent from those measured in the experiments. This is to be expected C). EdPIV performs local autocorrelations over a series for highly nonlinear waves. of cells into which the image is divided. The interro-

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Adrian (1990), Huang et al. (1997), Westerweel et al. (1997), and Westerweel (1997). Generally one would not expect the total velocity measurement error to exceed 2%±3% in well-prepared PIV experiments. Speci®cally, we estimate the rms error from Fig. 1 of Westerweel et al. (1997). Typical pixel (px) displace- ments in our PIV images are about 20 px. Seeding par- ticle image diameters are about 3 px. From Fig. 1 of Westerweel et al. (1997) we see that the expected value of the rms displacement error will be no more than about 0.2 px. This gives an upper bound on the rms error in the velocity measurements of about 1%. Considering now the velocity biasing from velocity gradients. According to Keane and Adrian (1990, sec- tion 5) the dimensionless parameter that represents the performance of PIV with regard to velocity biasing is

M | ⌬u | ⌬t/dI, where M is the magni®cation in units of px mϪ1,|⌬u | is the maximum variation of the velocity Ϫ1 u in m s across the interrogation region of size dI px, and ⌬t is the time in seconds between light pulses. The FIG. 2. PIV image of Wave 3 at t␷ ϭ 242/16 s. The image shows condition given by Keane and Adrian (1990, section 5) the parasitic capillaries beginning to form on the front face, the sharp- for optimum performance of PIV with regard to velocity er crest shape, and the streak lines of the seeding particles. biasing is M|⌬u|⌬t Ͻ 0.05. gation algorithm employed by EdPIV uses a ®xed search dI window with autocorrelation by FFT and Gaussian peak detection. A velocity vector is obtained for each cell. Transposing to make | ⌬u | the subject of the equation The choice for the size of the interrogation regions, or we obtain an upper bound on | ⌬u | given by cells, is motivated by two factors: that they are large 0.05d enough to contain the multiple exposures of individual |⌬u| Ͻ I . seeding particles as they track across the image and that M⌬t they a small enough for the velocity vector produced to Calculating the value of the right-hand side for our ex- represent a local measurement. All images in these ex- periments we have, for Waves 2±6, dI ϭ 64 px, M ϭ periments had dimensions of 1024 ϫ 1024 pixels and 6489 px mϪ1, and ⌬t ϭ 0.001 70 s. This sets the upper were analyzed with interrogation regions of 64 ϫ 64 bound of | ⌬u | Ͻ 0.290 m sϪ1. The maximum recorded pixels that overlapped by 32 pixels in both x and z | ⌬u | from any of the Waves 2±6 is about 0.15 m sϪ1, directions, where x is the horizontal distance from the well within the optimum limit. wave paddle and z is the vertical height above mean Similarly for Wave 1 we have dI ϭ 64 px, M ϭ 2473 water level. This gave a 31 ϫ 31 array of velocity vec- px mϪ1, and ⌬t ϭ 0.006 98 s leading to an upper bound tors for each of the PIV images. Many of these velocity of | ⌬u | Ͻ 0.185 m sϪ1. The maximum recorded | ⌬u | vectors, however, were discarded because they occurred from Wave 1 is about 0.175 m sϪ1, which is within the above the ¯uid surface or because the associated inter- optimum limit. rogation region crossed the ¯uid surface or they gave spurious results for some other known reason. Removal of spurious vectors from each image was done by hand, c. Velocity extrapolation to the free surface based on the principle of continuity of ¯uid ¯ow. The EdPIV PIV analysis program, as explained in section 4a, yields ¯uid particle velocities that are av- b. Errors in the velocity measurements erages over interrogation cells that lie wholly within ¯uid regions. Thus, the velocities are measured close There are two sources of error that may be of concern to, but not at, the ¯uid surface. To obtain the latter, the in our PIV measurements of ¯uid velocity in the break- measured velocities, uPIV(z, x, t), were extrapolated to ing wave crests. The ®rst is rms error associated with the surfaces, z ϭ ␩PIV(x, t), which were also measured the accuracy of estimation of the location of the dis- from the PIV images, using the following ®tting func- placement correlation peak. The second is downward tion biasing of the velocity measurement due to gradients in the velocity ®eld. Appropriate references are Keane and ufitted(z, x, t) ϭ ␣ 1(x, t) cosh(␣ 2(x, t) ϩ ␣ 3(x, t)z). (19)

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same reference frame as the ¯uid velocities. The frame of reference in which the experiments are performed, be it one in which the average Eulerian velocity of the ¯uid underneath the wave is zero (corresponding to Stokes ®rst de®nition of wave speed), one in which the total volume ¯ow rate underneath the wave is zero (cor- responding to Stokes second de®nition of wave speed), or any other, is irrelevant to the kinematic breaking cri- terion. Any relative motion due to a transformation be- tween inertial reference frames would simply add a con- stant value to both the measured phase speed and the measured ¯uid particle speed. If the kinematic criterion held in one inertial reference frame, then it would hold in all inertial reference frames. A simple method, called Wheeler stretching (Wheeler

FIG. 3. Graphs of the PIV measured x velocities, uPIV, with their 1970), was used to obtain approximate predictions of standard deviations, along with the ®tted x velocities, u®tted, extrap- ¯uid particle velocities on the surface of the waves from olated to the positions of the ¯uid surface, ␩ , for three values of PIV the wave gauge data. We stress that only the PIV mea- x along Wave 3 at t␷ ϭ 242/16 s. Note that the error bars are aligned horizontally in the key but vertically on the plot. sured velocities are used in the subsequent analyses; the Wheeler approximated values of ¯uid velocities at the

surface, denoted by uw(␩,x,t), are only used in some The ®tting parameters, ␣1, ␣ 2, and ␣3, were calculated subsequent ®gures to aid the reader in visualizing the using the Levenberg±Marquardt method as discussed by form of the wave outside the small region in which PIV Press et al. (1992). The form of the function in Eq. (19) yielded accurate measurements of the velocity. is the same as that obtained from linear wave theory.3 For Wave 1, standard deviations on the extrapolated The values of uPIV(z, x, t), with z Ͻ ␩PIV(x, t), used horizontal surface ¯uid particle velocities and the ve- by the ®tting routines were the averages over three PIV locities themselves are plotted, along with the surface images, each taken of exactly the same region of the elevations, in Fig. 4. The standard deviations in the wave at precisely the same time in its evolution but measurements of the surface elevations are extremely from three different realizations of the wave. Represen- small and are not plotted in the ®gure. tative standard deviations, ␴(uPIV), one for each x co- Some of the extrapolated velocity measurements ordinate, were obtained using all values of uPIV (taken could be checked by hand, directly from the PIV images, over the range of z coordinate and three PIV images) by measurements of seeding particles moving very close having the same x and t coordinates. The surface ele- to the ¯uid surface. They agreed to within 2% of the vations, ␩PIV(x, t), were averages over the three PIV extrapolated predictions. For example, measurements images of each wave. The ®tting was performed on each made by hand from the PIV photos for Wave 1 at x ϭ set of averaged uPIV with constant x and t but differing 4.141 m and t ϭ 234/16 s and averaged over ®ve laser z coordinate. The resulting ®tting parameters and their pulses show that the maximum horizontal ¯uid particle covariances were used in conjunction with Eq. (19) to velocity at the position of maximum curvature of the obtain values for the extrapolated horizontal ¯uid par- tip of the crest is as large as 1.38 m sϪ1, which is almost ticle velocities, u®tted(z ϭ ␩PIV(x, t), x, t), along with their exactly the result given by the extrapolation. Thus, the standard deviations, ␴(u®tted). Figure 3 shows examples authors are satis®ed that the method used for extrapo- of results obtained by using this ®tting procedure on lation of the ¯uid velocities gives accurate values at the Wave 3 and is representative of the other waves also. water surface. It can be seen that the velocities are extrapolated by, at It is important that the surface elevation data mea- most, about 1 cm into the crest. sured by the wave gauges is consistent with that ob- Since the ¯uid velocities crucial to our analysis were tained from the PIV measurements and Fig. 4 shows the measured directly, and did not depend on predictions of high level of consistency for Wave 1, given that the a wave theory, any distinction between the use of wave gauges were placed 0.1 m apart. Stokes' ®rst or second de®nition of wave speed (see, e.g., Fenton 1985) and the associated frame of reference is unnecessary. The phase speeds were measured in the d. Calculation of phase velocities for nonlinear waves

3 From linear theory the x component of the ¯uid particle velocity Values for each of the three de®nitions of phase speed under a wave with surface elevation given by ␩(x, t) ϭ a cos(kx Ϫ given in section 2 were calculated for each wave from ␻t)is the discrete space and time surface elevation samples coshk(h ϩ z) measured by the wave gauges. A value for the ®rst u(z, x, t) ϭ a␻ cos(kx Ϫ ␻t). sinhkh de®nition, given by Eq. (2), was calculated from the

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FIG. 4. Extrapolated horizontal surface ¯uid particle velocities, u®tted(z ϭ ␩PIV, x, t), obtained from the PIV images

for the plunging breaker labeled Wave 1. Also shown are comparisons of the surface elevations, ␩PIV(x, t), obtained

from the PIV measurements and those obtained from the wave gauge measurements, ␩gauge(x, t). Note that the error bars are aligned horizontally in the key but vertically on the plot. zero-downcrossing of the wave pro®le. The results are noise. The values chosen as the frequency cutoffs for low- given in Table 2. pass ®ltering were 6.250 Hz and 4.123 mϪ1. Values for the third de®nition of local phase speed, de®ned by Eq. (15), were found by plotting the positions 1) PLUNGING BREAKER:WAVE 1 of the surface elevation maximum, xc(t), against time over the time interval of 14±15.5 s. In this range the For Wave 1 three sets of PIV velocity data were ob- 2 data were ®tted to the equation xc(t) ϭ a ϩ bt ϩ ct tained for each of the three times t ϭ 234/16, 233/16, and the phase speed calculated as c (t) ϭ b ϩ 2ct. The p and 232/16 s. We take t␷ ϭ 234/16 s, as, with our ex- results of these calculations for cp(t) for all waves of perimental apparatus, this was the closest we could get Table 1 are shown in Table 3. (Ͻ1/32 s) to the time of the start of breaking at t␷ . The Values for the second de®nition, cp(x, t), given by Eq. results of the phase speed calculations for Wave 1 are (14), were calculated from the discrete partial Hilbert trans- shown in Figs. 5a±c. forms of the surface elevation and their spatial and tem- For plunging breaker of Wave 1 the Hilbert transform poral derivatives. In all the ®gures showing cp(x, t) the cannot be calculated if the surface elevations are not raw un®ltered calculation results are shown along with single valued, which presents a problem. For this reason results calculated after ®rst low-pass ®ltering ␩(x, t). The the analysis was applied to the wave just before (Ͻ1/32 frequency cutoffs for the low-pass ®ltering were chosen s) the front face became vertical. But, of course, shortly by examination of the corresponding amplitude spectra after this time the surface elevation becomes triple val- plotted on a log scale. On a log scale each spectrum ued as the wave overturns and this will affect the wave showed a high frequency tail that had a constant amplitude, gauge readings. The capacitance gauge data readings implying that the tail in this region consisted mainly of will give the height of the surface as the sum of the ¯uid volume in a column. Thus, as the triple valued surface of the plunging breaker passes around a gauge, TABLE 2. Wavelengths (to nearest 0.1 m), periods (to nearest 1/16 s), and phase speeds of equivalent linear wave for the waves the gauge will measure the surface elevation as the sum in Table 1. of the two ¯uid regions of the wave and not include the air region in contact with the gauge. The triple valued Wave- Phase surface function of the wave is effectively collapsed to Position Time length Period speed, cp Wave (m) (s) (m) (s) (m sϪ1) a single valued function. Another point where some ambiguity in wave gauge measurements may occur is 1 4.1 234/16 3.6 1.6250 2.22 2 4.0 240/16 3.5 1.5625 2.24 in the whitecap region that occurs after breaking. 3 4.0 242/16 3.3 1.5625 2.11 Note that, contrary to the case for the extrapolated 4 4.0 237/16 3.3 1.5625 2.11 ¯uid particle velocities at the surface, simple hand cal- 5 4.0 237/16 3.3 1.5625 2.11 culations of cp(x, t) cannot be made from the PIV im- 6 4.0 237/16 3.3 1.5625 2.11 ages. Nevertheless, a value for the nonlocal phase speed

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TABLE 3. Fitting parameters and phase speeds de®ned by Eq. (15) for the waves in Table 1. Fitting parameters Phase speed,

Position Time cp(t) Wave (m) (s) ab c(m sϪ1) 1 3.8 232/16 Ϫ53.926 5.9203 Ϫ0.13392 2.03 1 3.9 233/16 Ϫ53.926 5.9203 Ϫ0.13392 2.02 1 4.1 234/16 Ϫ53.926 5.9203 Ϫ0.13392 2.00 2 4.0 240/16 Ϫ80.073 9.5968 Ϫ0.26604 1.61 3 4.0 242/16 Ϫ65.459 7.6127 Ϫ0.19968 1.57 4 4.0 237/16 Ϫ75.858 9.0797 Ϫ0.24914 1.70 5 4.0 237/16 Ϫ77.821 9.3357 Ϫ0.25884 1.69 6 4.0 237/16 Ϫ77.596 9.3350 Ϫ0.25851 1.68 of the crest can be obtained from the PIV images for Eq. (2) and de®ned to be equal to that of the equivalent Wave 1 from the motion of the (near) vertical front linear wave, the truly local de®nition based on the partial surface. This is not the same as any of the three de®- Hilbert transforms of the surface elevation and given nitions of phase speed introduced in section 2, but acts by Eq. (14), and ®nally the de®nition based on the ob- as an approximate cross check to the values for phase served speed of the surface elevation maximum at the speed calculated from the wave gauge data according crest and given by Eq. (15). Statements made in the to the de®nitions of section 2. literature, for example, those highlighted in the intro- duction, suggest that experiments should con®rm the hypothesis that the horizontal ¯uid velocity at the sur- 2) SPILLING BREAKERS:WAVES 2 AND 3 face of the breaking waves, Waves 1, 2, and 3 of Table The situation for the spilling breakers, Waves 2 and 1, would exceed the phase speed for at least one of the 3 of Table 1, was slightly different from that of the de®nitions used. The contrary was found to be true for plunging breaker of Wave 1 in that the surface elevations each of the waves in Table 1 and for each of the three were never double valued, and therefore we expect the de®nitions of phase speed used in this study. The results surface elevation data from the wave gauges to be more are summarized in Table 4. One would expect the ratio accurate leading to more precise calculations of the par- uPIV/cp(x, t) in the last column to be at least unity if the tial Hilbert transforms and hence local phase speed, kinematic breaking criterion was valid for these waves. cp(x, t). The only point where some ambiguity in wave The ®rst wave data analyzed were from the plunging gauge measurements may have occurred is in the white- breaker Wave 1. The large ¯uctuations in the un®ltered cap region, but this is expected to be slight for the results for cp(x, t) are not noise. They occur at positions limited whitecap produced in the wake of the spilling of negative maxima and positive minima in the surface breakers. pro®le (Melville 1983, section 4.3). It was therefore The results of the phase speed calculations for Waves deemed preferable to retain all the measured frequencies

2 and 3 at their start times for breaking, t␷ ϭ 240/16 and in the analysis even though low-pass ®ltered results for 242/16 s, respectively, are shown in Figs. 5d and 5e. cp(x, t) are also shown in the ®gures. The high fre- quencies are certainly necessary to correctly represent the sharp crests associated with the breaking waves.4 3) NONBREAKING WAVES:WAVES 4, 5, AND 6 From Table 4 it is evident that, for Wave 1, the local For the nonbreaking Waves 4, 5, and 6 of Table 1 the phase speed is signi®cantly greater than the horizontal surface elevations were never double valued, neither surface particle velocity at, and shortly before, the point were there any whitecaps so we expect the surface el- of breaking. This is the case for each of the de®nitions evation data from the wave gauges to be the most ac- of phase speed. At the breaking point, (x␷ , t␷ ), the lowest curate of the three wave types leading to the most precise value of local phase speed is seen to be given by the calculations of the partial Hilbert transforms and hence un®ltered local phase speed as de®ned by Eq. (14). This Ϫ1 local phase speed, cp(x, t). lowest value is 1.67 m s . The local phase speed cal- The results of the phase speed calculations for Waves culated from the linear approximation is about 33% 4, 5, and 6 at their times for maximum amplitudes, t ϭ greater, and that calculated from the rate of change in 237/16 s, are shown, respectively, in Figs. 6a±c.

4 5. Discussion It should be noted that examination of the Fourier amplitude spectra of the surface elevations showed that the Fourier contributions The relationship between the phase speed and the at the Nyquist frequencies (8 Hz and 5 mϪ1) for the sampling rates used were very small, being in both cases about 10Ϫ4 times the Fourier horizontal ¯uid velocity at the surface of breaking waves component of maximum amplitude. Thus, these sampling rates are has been investigated. Three different de®nitions of short enough to resolve most of the frequencies in the wave evolution phase speed were used: the nonlocal de®nition given by process and there is very little aliasing in any of the spectra.

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FIG. 5. Comparisons of the three de®nitions of phase speed with horizontal ¯uid particle

velocities at the surface, uPIV(z ϭ ␩PIV, x, t), obtained from PIV measurements. Wave 1 at times

(a) t ϭ 232/16 s, (b) t ϭ 233/16 s, and (c) t␷ ϭ 234/16 s; Wave 2 at (d) t␷ ϭ 240/16 s and

Wave 3 at (e) t␷ ϭ 242/16 s. Also shown are the Wheeler stretched predictions for the horizontal

¯uid particle velocities at the surface uw(z ϭ ␩gauge, x, t). position of the maximum of ␩(x, t) is about 20% greater. sense; that is to say, it is not single valued as the wave For Wave 1 the greatest ratio obtained between the hor- overturns. The wave gauges, however, can never yield izontal ¯uid particle velocity in the crest and the local anything other than a single valued function and this is phase speed is about 0.81 at (x␷ , t␷ ). This is signi®cantly used to calculate the partial Hilbert transforms of the less than the proposed breaking criterion value of unity. wave gauge data and, hence, cp(x, t). As explained in section 4d(1), there will be some error The second wave type analyzed were the spilling in calculating the partial Hilbert transforms of the sur- breakers, Waves 2 and 3. For these waves the problem face elevation for a plunging breaker since the surface of the surface elevation becoming triple valued is elim- elevation is not a mathematical function in the strict inated, but, since a foam whitecap zone exists just after

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FIG. 6. As in Fig. 5 but (a), (b), and (c) show Waves 4, 5, and 6, each at time t ϭ 237/16 s. breaking, there may be some small error introduced into respectively. For both of the spilling breakers the max- the partial Hilbert transform calculations and the cal- imum horizontal ¯uid particle velocity at the tip of the culations of their derivatives due to slight ambiguity in crest is about 6% less than the local phase speeds cp(x, t) the position of the ¯uid surface within the whitecap. or cp(t). Figures 5d and 5e show that at the point of breaking For the nonbreaking waves, Waves 3, 4, and 5, a result for the spilling breakers the local phase speed based on similar to that for the spilling breakers is obtained in the partial Hilbert transforms (un®ltered) and the de®- that the two local de®nitions of phase speed give similar nition using the rate of change in position of the max- values at the crest, although the un®ltered calculation imum of ␩(x, t) both give approximately the same value based on Eq. (14) is consistently lower, by about 5%, for the local phase speed. The values obtained from the than that based on Eq. (15). Also, as with all the previous equivalent linear wave approximations are signi®cantly waves, the value of phase speed calculated from the greater, by about 42% and 30% for Waves 2 and 3, equivalent linear wave de®nition gives a poor approx-

TABLE 4. Values of the phase speed calculated from the various de®nitions given in section 2 compared with the values of the horizontal ¯uid particle velocities extrapolated from PIV measurements to the surface of the crest.

Phase speeds (m sϪ1) Breaking

Time Maximum extrapolated uPIV Linear Crest Min. Hilbert criterion Ϫ1 Wave (s) (m s ) cp cp(t) cp(x, t) ratio uPIV/cp(x, t) 1 232/16 1.05 at 3.973 m 2.22 2.03 at 3.8 m 1.45 at 4.1 m 0.72 1 233/16 1.25 at 4.050 m 2.22 2.02 at 3.9 m 1.63 at 4.2 m 0.77 1 234/16 1.35 at 4.141 m 2.22 2.00 at 4.1 m 1.67 at 4.2 m 0.81 2 240/16 1.50 at 3.999 m 2.24 1.61 at 4.0 m 1.58 at 4.0 m 0.95 3 242/16 1.51 at 4.004 m 2.11 1.57 at 4.0 m 1.62 at 4.0 m 0.93 4 237/16 1.11 at 3.994 m 2.11 1.70 at 4.0 m 1.60 at 4.1 m 0.70 5 237/16 1.13 at 3.999 m 2.11 1.69 at 4.0 m 1.60 at 4.1 m 0.71 6 237/16 1.10 at 3.999 m 2.11 1.68 at 4.0 m 1.61 at 4.1 m 0.68

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imation to the value calculated locally at the crest. For t␷ ), x␷ , t␷ ] is exactly equal to the phase velocity of the the nonbreaking waves the discrepancy between phase wave at that point (but it does not exceed it). speed calculated from the equivalent linear wave de®- It is interesting to note that in all waves analyzed in nition and that obtained from the two local de®nitions the wave ¯ume the local phase speed has its lowest is about 32%. minimum in the vicinity of the focal point of the wave, Comparing the results for the plunging and spilling which is where one would expect the surface velocities breakers, we see that the kinematic breaking criterion to be greatest. Also, note that the variation of cp(x, t) is closer to being satis®ed in the crest of the spilling shows that the rear surface is traveling faster than the breaker. This may be explained by recalling the point, front surface in the ®gures for Wave 1, implying that made in section 3b, that the ratio of horizontal ¯uid this wave is still ``bunching up'' while evolving to a particle velocity to local phase speed must be unity at plunging breaker, whereas for the spilling breakers any point on the wave surface where the radius of cur- cp(x, t) is almost symmetrical about the focal point, vature is zero. The radius of curvatures at the tips of implying that the wave is at its (spilling) breaking point. the crests of the spilling breakers are less than those of Thus, for all the waves in this study the maximum the plunging breaker at the time, t␷ , of incipient break- horizontal ¯uid particle velocities on the surfaces of the ing. crests are consistently less than the associated phase Applying a basic hydrodynamic principle, namely the speeds. The expected errors of 2%±3% in the PIV ve- kinematic free surface boundary condition, it can be locity measurements are not enough to account for the shown that, on the surface of the vertical front face of measured discrepancies between u(␩,x,t) and any of the phase speed de®nitions if the kinematic breaking a breaking wave at the point (x␷ , t␷ ), the x velocity criterion were true. An error of about 20% would be component of the ¯uid particles, u[␩(x␷ , t␷ ), x␷ , t␷ ], is exactly equal to x velocity component of that point on required to satisfy the kinematic breaking criterion of the plunging breaker, Wave 1. For the spilling breakers, the surface given by ␩(x␷ , t␷ ). But this statement is of no use as a breaking criterion. For a kinematic breaking Waves 2 and 3, satisfying the kinematic breaking cri- criterion to have any predictive value it would require terion was missed by about 6%. Our experimental re- that the u(␩(x, t), x, t) was equal to or greater than the sults, therefore, are found to be consistent with all but ¯uid surface speed before the occurrence of a vertical one of the studies reviewed previously in section 3. The front face, that is to say, Eq. (1) would hold for some one exception being Chang and Liu (1998), who use linear theory to obtain u[␩(t), t] from single point mea- de®nition of phase velocity cp(x, t). We have not found this criterion to be satis®ed for any of our de®nitions surements of ␩(t). cp, cp(x, t), or cp(t) given by Eqs. (2), (4), and (15). For breaking to occur it is certainly not a logical 6. Conclusions requirement for Eq. (1) to hold. Consider, by way of analogy, a length of taut elastic material. Let the lon- In this study we have attempted to assess the suf®- gitudinal axis be the x direction and the transverse axis ciency of each of three de®nitions of wave phase speed of interest be the z direction. Consider a transverse wave with regard the kinematic breaking criterion, which propagating in the positive x direction along the length states that the breaking of water waves implies the hor- of elastic. It is quite possible for the x component of izontal velocities of the ¯uid particles in the breaking the velocity of the material of elastic to be identically crest exceed the phase speed of the breaking crest at, zero for all time and all positions along the elastic. The or before, the point of breaking [see Eq. (1)]. transverse wave motion is produced entirely by the z For each of the waves studied we found that the phase component of the velocity of the elastic material, yet speeds calculated from the equivalent linear waves give the wave itself has a phase speed in the x direction that poor approximations to the values obtained by using is not zero. In a similar way the z-velocity component any of the local phase speed de®nitions. Moreover, we of the ¯uid in a water wave can increase or decrease found that, with each of these three de®nitions of phase the x component of the water wave's phase velocity. In speed, the kinematic breaking criterion is not satis®ed the water wave the ¯uid particles preceding the front and is therefore not universal. For the plunging breaker face of a wave are ascending and as, before breaking, the criterion is far from satis®ed. For the spilling breaker the front face must be at an angle less that 90Њ to the the criterion is closer to being satis®ed, but is not, even horizontal, this upward motion of ¯uid particles con- within the limits of experimental error. It is concluded tributes to the forward motion of the angled surface of that the truth of the condition u(␩,x,t)/cp(x, t) Ͼ 1in the wave. Thus, increasing the value of the phase ve- a wave crest, at or before the ®rst occurrence of a vertical locity in the direction of travel of the surface wave tangent to the forward face, may be a suf®cient con- regardless of the ¯uid particle velocity in the x direction. dition for breaking but it is not a necessary condition. Only when the front face is vertical does any z com- On the evidence of the results presented here, we suggest ponent of the ¯uid particle velocity have no effect on that, unless and until a local de®nition of phase speed the phase speed of surface wave. In this case u[␩(x␷ , is devised for which no instances can be found that

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