a. Nonlinear wave codes and their numerical accuracy

We used the fully nonlinear, two-dimensional, periodic domain inviscid model for free surface gravity water waves described in detail in DP. In this model, the following scalings were used: gravitational acceleration g = 1, water density ρ_w = 1, and a carrier wavelength of 2π. The accuracy of the DP code has been checked by several researchers. Among these, the recent study by Skyner (1996) compared numerical predictions from the DP code with subsurface velocity measurements made during the breaking onset process using particle image velocimetry techniques. After a small shift of the numerical data to match the surface profiles, the predicted and measured kinematics were in close (within 2%) agreement. The predicted and measured surface shapes were also in very close agreement. From such studies, the essential features of two-dimensional wave breaking appear to be captured by the DP code well into the overturning regime and conclusions drawn from studying the model behavior are likely to be relevant to the actual fluid dynamics underlying the onset of wave breaking in self-modulating wave groups. Most of the computations reported in this paper are based on the DP code.

For the chirped wave packet (case III) calculations, in which the leading waves have a higher frequency than trailing waves, we used a fully nonlinear numerical model developed by Y. Agnon and N. Drimer (1998, personal communication) with refinements by A. Segre (1998, personal communication). This code (henceforth referred to as the DAS code) solves the fully nonlinear, irrotational free surface boundary value problem using the boundary element method within a numerical wave channel, featuring a piston wave maker at one end of the domain and an energy absorbing “beach” at the other end. This code is capable of representing wave surfaces past the point of overturning, that is, beyond a vertical forward face configuration, without numerical smoothing. Drimer and Agnon (1998, personal communication) document a number of error validation tests. Their computed surface profiles were in close agreement with previous results of Jansen (1986) and New et al. (1985) for steep sinusoidal waves. The wave height versus water depth curve obtained by Drimer and Agnon agrees well with the experimental results of Hansen and Svendsen (1979) for wave shoaling up to breaking. We also checked the accuracy of their model in our present computations by verifying global conservation of water volume and total energy after the piston motion ceases. In all cases that we checked, errors both in wave energy and volume remained smaller than 2% during the subsequent wave propagation.

b. Model runs

A series of runs using the DP code was performed in this study. To eliminate the effects of numerical instabilities and obtain a stable solution in our calculations, following BT, we chose in the DP model a small precision control parameter ε = 0.001, fifth-order backward differencing, the number of points per wave NP = 16 with no numerical smoothing. We took the still water depth h = 50 and used a time step of 0.1T, where T = 2π for all calculations with the DP code. Two classes of initial wave group structures, cases I and II defined below, were used in our DP code runs.

A number of maximally recurrent and breaking wave realizations for case III wave packets were also studied using the DAS code. These runs all started from a still water initial state. A time step of 0.05T was used for all calculations with the DAS code, where T is the reference input period of the piston movement. The number of points along the boundary was chosen to provide the desired channel geometry and resolution of the overturning wave profile. We tested various values of NP on the boundary, especially on the free surface, and found that taking NP = 20 was sufficiently large to ensure a stable result. Zero artificial damping coefficient was used for all the DAS computations. The piston was stopped after the wave group had built up, allowing a chirped wave packet to propagate in the positive x direction.

c. Initial wave group structures

Three classes of initial wave group structures were investigated in this study. The first class (henceforth referred to as case I) had a fundamental carrier wave with two small symmetric sidebands, using as parameters the initial carrier wave amplitude a₀ (or steepness s₀ = a₀k₀, where k₀ = 1) and the number of waves in one modulation length, N. The initial wave group had the structure of a uniform, steady, finite amplitude deep water wave train whose linear approximation is η = a₀ cos(k₀x), where k₀ = 1. On this primary wave are superimposed perturbations having the initial form

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Here, N is the integer number of waves in the group, where 3 ≤ N ≤ 10. This is the range used by DP and provides sideband perturbation wavenumbers in the range [0.667, 1.333]. Also, following DP, the phase angle θ was taken as π/4 as it provides the most rapid initial growth of the sideband modes.

The second class of wave groups (henceforth referred to as case II) had an initial bimodal spectrum of the form

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where a₀ and N are as defined as above. The small phase shift, retained from BT, is inconsequential.

A third class of wave group structures (henceforth referred to as case III) had a more rapidly deforming geometry and was characterized as a “chirped” wave packet. This structure, commonly implemented in wave tank experiments (e.g., RM), comprises carrier waves that coalesce rapidly due to their different phase velocities. These wave packets were produced in the DAS code by driving the piston wave maker with the motion

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where t is time and N is a integer that controls the number of waves produced by the piston, A_p is proportional to the piston amplitude and

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is its angular frequency, λ is the wavelength, and h is the still water depth. To simulate deep water, we took the still water depth near the piston as h = 4 with λ = 2.

a. Evolution to recurrence or breaking

The transition between evolution with recurrence or evolution to breaking occupies a position of central importance in the discussion of breaking criteria. In this context, both model studies and laboratory investigations have typically investigated initial value problems and have adopted a measure based on the initial carrier wave steepness to parameterize their results. This choice of initial steepness parameter warrants discussion, especially as various investigators have reported appreciably different initial steepness thresholds that separate recurrence from breaking cases.

In their study of weak sideband modulating case I wave groups, DP used the initial amplitude a and wavenumber k of the center Fourier mode to characterize the initial steepness (ak)₀ of their wave groups and found that this initial steepness threshold for breaking onset depends on the number of carrier waves N in the group, ranging from (ak)₀ = 0.150 for N = 3 to (ak)₀ = 0.088 for N = 10.

The DP definition of initial steepness requires modification in the context of a full wave modulation (our case II) that results from the superposition of two modes of equal, moderate steepness with a prescribed wavenumber separation. There are two straightforward options for specifying a characteristic initial steepness measure. The first uses the root mean square (rms) steepness

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for a free surface specified by the Fourier series η(x, t) = ΣN0i=1 ai cos(kix − ωit − ϕi). In (4), ai, ki, ωi, and ϕi are, respectively, the amplitude, the wavenumber, the frequency, and the phase of the ith component, and N0 is the number of wave components.

The second initial steepness measure, introduced by RM in their laboratory study of chirped wave packets, defines a spectrally based initial wave steepness ak_c as

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where a_n and k_n are the Fourier amplitudes (1 ≤ n ≤ N_c) of their initially uniform spectral amplitude wave packet, assuming a linear dispersion relation between temporal and spatial variations. Rapp and Melville showed that the energy loss associated with breaking events was well correlated with this parameter.

For cases I and II, the differences in using these two definitions are shown in Fig. 3a. The chirped wave (case III) data from RM shows a threshold level of ak_c ∼ 0.25, yet the rms steepness threshold is typically only 0.055. Thus the critical value of ak_c for breaking not only depends on the number of carrier waves, N, in the group but also on the initial group structure, suggesting its unsuitability as a threshold parameter for characterizing the onset of wave breaking. We conclude that neither of these initial steepness measures provides a consistent, robust indicator for breaking onset.

Finally, we point out that the perturbations imposed on the initial wave group geometry were all in the initially unstable regime. It is of interest to note how the recurrence/breaking margin for different N relate to the theoretical loci of the most unstable initial sideband perturbations (e.g., see Tulin and Waseda 1999, Fig. 1). This comparison is shown on a plot of initial center steepness versus relative bandwidth for case I wave groups in Fig. 3b. This suggests a nonsimple relationship between the initial sideband instability growth rate and the ultimate onset of recurrence or breaking.

b. Evolution characteristics

The complex evolution characteristics for the wave elevation and local energy density shown in Figs. 1–3 and related issues such as breaking onset, timing, and strength are among many fascinating aspects of this problem that motivated our present investigation into whether these processes might be determined by an intrinsic “universal” threshold parameter. In an earlier paper by the second author, BT, it was found that, if the local mean wave energy density or wave action density is used to characterize the evolution of these nonlinear wave groups, the implicit averaging over even a single carrier wavelength loses the spatial localization of the energy maximum. To address this basic issue, BT adopted wave energy and momentum densities averaged locally over one-half of the carrier wavelength as the primary variables, as these variables and their rates of change appeared to have reasonably well-localized maxima. A fundamental result reported by BT was that the envelope maxima of these densities fluctuate on a “fast” timescale, about twice the carrier wave period. This is linked to the well-known crest–trough asymmetry of steep gravity waves and a close analog of this behavior is shown in Fig. 5 below in the context of a more detailed discussion on its origin. Banner and Tian also reported that the corresponding relative growth rates following their envelope maxima have a substantial dynamic range. For a range of case I wave groups containing different numbers of waves, they reported finding a common threshold behavior for the maximum value of either of these local relative growth rates that determined whether breaking subsequently occurred.

c. BT revisited

The breaking onset criterion proposed by BT was based on the behavior on the fast timescale described above. This led to the conclusion that breaking onset could be detected only within less than one-half of the mean carrier wave period. We decided to revisit this important issue, as well as related concerns such as possible sensitivity of the results to other initial wave group geometries, to certain numerical approximations that were made and to the temporal resolution of the BT analysis. To address these goals, we developed an intrinsically more accurate numerical methodology based on a “wave group following” (henceforth WGF) computational approach that allowed us to calculate, with minimal filtering and numerical approximation, the entire “slow” evolution of the energetics of the wave group, as well as the “fast” evolution.

This method underpinned our efforts to understand at what point during the evolution the wave group is destined to proceed irreversibly to break and its connection with the mean energy convergence rate toward the envelope maximum highlighted in Figs. 1 and 2. It also allowed a detailed assessment of the BT methodology and conclusions, which is reported in appendix A. From this assessment, we concluded that

1. while the half-wavelength average adopted by BT had good localization properties for the envelope maximum and its rate of change, our WGF analysis revealed that this averaging masked the difference between breaking and recurrent cases through a subtle smearing of the behavior of the envelope maximum;

2. the envelope maximum migrates between the steepest crest to the steepest trough in a complex fashion, implying that the BT assumption of a constant value of the velocity c_E of the energy envelope maximum is not justified. Making this assumption can lead to significant errors in calculating energy growth rates following the envelope maximum using the Eulerian expression ∂/∂t + c_E∂/∂x. These difficulties are avoided using our WGF approach.

These results convinced us that the complex details are smoothed to such an extent even with half-wavelength averaging that only a local quantity is able to provide a clear resolution of the differences between breaking and recurrence cases. Therefore in the present paper, we focused on the behavior of the depth-integrated local energy density at its maximum along the wave group, as defined in the following section. Rather than working in terms of the dimensional wave energy, we investigated the behavior of a dimensionless measure of the mean maximum local energy density and its dimensionless growth rate, as defined in the following section. This growth rate was then calculated systematically using a WGF technique that allowed us to investigate the entire growth cycle with no additional approximations and negligible filtering.

Aside from the intrinsically improved accuracy, the present calculations extend significantly the scope of those described by BT to the wider class of initial wave group geometries, as described above. In addition, Banner and Song (2002, hereafter Part II) reports the results of our investigation of the influence of wind forcing and a typical background vertical shear current on the evolution to breaking for many typical cases.

As described below, the behavior of our proposed dimensionless growth rate parameter reveals a robust common threshold for all routes to breaking for all the deep water wave cases we have examined, providing an earlier and more decisive indicator for the onset of breaking than previous criteria. It also appears to reflect the relative strength of breaking events.

d. Diagnostic growth rate parameter

Based on the preceding discussion on the WGF technique, we chose a local, nonhorizontally averaged measure of the wave energy at its maximum along the wave group as the basis of a diagnostic parameter that could distinguish between breaking and recurrence evolution. We used the depth-integrated local wave energy density E(x; t), given by the sum of kinetic and potential energy contributions of the fluid particles along the vertical line at x at time t:

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where u and υ are the x and y velocity components, g is gravitational acceleration, ρ_w is the density of water, h is the still water depth, and y = η(x, t) is the free surface elevation.

Following BT, for these irrotational flows the interior velocity field (u, υ) was calculated from the boundary values using Green's theorem and then used to calculate the local energy E(x; t) in (6) for cases I and II. For case III, E could be obtained directly from the code output. The behavior of E at specific times during the evolution for each case was shown in Figs. 1 and 2.

In order to provide “global” criteria, we needed to introduce suitable nondimensional parameters for E and its growth rate following the energy maximum of the wave group. We explored the possibility of using the dimensionless relative local energy growth rate

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but found that this parameter did not provide a robust indicator for resolving the onset of breaking or recurrence for the ensemble of cases we studied.

We investigated two other nondimensional parameters based on E. One was E normalized by the mean energy 〈E_T〉 of the wave group. For cases I and II, 〈E_T〉 is conserved for unforced calculations or is easily calculated at different stages of evolution when wind input is added. The chirped wave packet (case III) presents the only complication for this approach. While the total packet energy is conserved during unforced evolution, the spatial spreading is unbounded. This presents conceptual difficulties in relation to providing a mean energy level for comparison with oceanic cases. Excluding this case, 〈E_T〉 can provide a useful normalization.

Other suitable diagnostic parameters based only on local wave train properties were the parameters s = E/(ρ_wg)k or s² = [E/(ρ_wg)]k². For these, the local wavenumber k was calculated systematically from the x derivative of the unfolded phase function computed from the Hilbert transform of the free-surface profile. Following BT (section 2.3.4 and Fig. 6), suitable low-pass filtering was applied using a bidirectional Butterworth filter determined by the requirement for a smooth distribution of the local wavenumber along the wave profile. Extensive testing indicated that even for extreme waves just prior to breaking, the calculated wavenumber after filtering was within 5% of the mean of the physical wavenumbers determined from the wave profile by direct measurement of successive crests, troughs, and zero-crossings.

The diagnostic parameters s and s² provide measures of local wave steepness without the complication of dealing with issues such as waveform asymmetry. At the instantaneous location x_max of the maximum of s² along the wave group, the corresponding value s²_max reflects the square of the maximum local steepness of the carrier waves. The choice of s²_max rather than s_max circumvented the need to investigate locally large values of D s_max/Dt, which occurred even for modest values of s_max due to intrinsic local unsteadiness in the evolution of the extrema of s_max and are clearly not related to the onset of breaking or recurrence, for which the values of s_max need to be large as well. Use a diagnostic growth rate based on D s²_max/Dt weights D s_max/Dt by the magnitude of s_max, thereby linking maxima of D s_max/Dt with the larger values of s_max appropriate to breaking or recurrence. Accordingly, we adopted μ = s²_max as our preferred diagnostic parameter and investigated its evolution. From this, we calculated the evolution of its local average value 〈μ〉 and the corresponding dimensionless growth rate

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for the various cases, where ω_c is the initial mean carrier wave frequency. The detailed methodology for computing 〈μ〉 and δ is described in appendix B.

As was already seen in Figs. 1 and 2, nonlinear deep water wave group modulation has an associated mean convergence of energy toward the energy maximum. These figures indicate that most of the change in μ is associated with wave energy density changes, with a much smaller contribution arising from changes in k². On this basis, 〈μ〉 and δ could be regarded as surrogate parameters reflecting the resultant of this complex energy flux process. The results of this approach are described in detail in the following section. We conclude by noting that, for periodic boundary condition cases, the alternative nondimensional parameter based on energy E/〈E_T〉 suggested above showed very similar properties for forecasting breaking onset and might be more straightforward to measure in open ocean applications.

a. Growth rate computations

We investigated the behavior of μ(t), 〈μ(t)〉, and δ(t) at each time step as the wave group evolved. For any given time t during the evolution, from the local energy E(x; t) and wavenumber k(x; t), we calculated μ = s²(x_max; t) at the location x_max of the maximum of s² and hence determined its local average (see section 4b below) and nondimensional local average growth rate δ(t). Further, we verified that, if we chose instead to track this quantity either at the maximum of the wave envelope (calculated from the Hilbert transform of the free surface) or at the maximum surface displacement |η|_max, the results for δ(t) were virtually identical.

Typical evolution curves of μ(t), 〈μ(t)〉, and δ(t) are shown in Fig. 4 for case I wave groups just below and above their recurrence threshold. It is seen that the recurrence cases attain their greatest value of δ(t) well before μ(t) reaches its recurrence maximum, where δ(t) = 0.

Also, in the evolution of μ(t), a fast oscillation with period 2T is seen to be superimposed on the longer term evolution growth toward the recurrence maximum or breaking onset. This is due to the strong crest/trough asymmetry of steep nonlinear gravity waveforms shown in the fast evolution sequence of Fig. 5. This figure shows the strong contrast in E for the two times separated by T when the crest and trough occupy the envelope maximum of μ(x; t).

b. The mean growth rate

Based on the clear indication of mean convergence of wave-coherent energy at the energy maximum of the group seen in Figs. 1, 2, and 4, we propose that

1. it is the average growth rate δ(t) that represents the mean convergence rate of wave energy at the energy maximum within the group;

2. this growth rate is a fundamental determinant of recurrence or breaking;

3. the fast O(2T) oscillatory component of μ(t) associated with the kinematics of the asymmetric wave profile is not the primary determinant of recurrence/breaking.

To filter out the rapid fluctuations superimposed on the longer-term trend, we used a method based on smoothed spline fitting that was superior to conventional filtering techniques in regard to managing the end effects at the onset of breaking. Details describing the technique and its accuracy are given in appendix B, but we note that the same “tolerance” parameter in the smoothed spline fitting was used to calculate δ(t) for all wave group cases (I, II, and III) and that the results were insensitive to the choice of this parameter ranging over two orders of magnitude.

For completeness, examples of case II and case III evolution are shown in Figs. 6 and 7. The greater initial modulation depth of case II is seen to provide a more rapid evolution to recurrence or breaking near the recurrence threshold than the comparable case I results shown in Fig. 4. This appears to be associated with its stronger initial local energy distribution contrast within the wave group. The case III evolution is seen to be faster than cases I or II, underlying its popularity in laboratory experiments (e.g., RM).

Systematic computations of μ(t), 〈μ(t)〉, and δ(t) were carried out for a range of N and initial steepness levels for each of the three cases I, II, and III. Our aim was to investigate

1. whether recurrence and breaking could be robustly linked to a threshold behavior of δ(t) and, if so, at what stage of the evolution was the onset of breaking determined;

2. whether the strength of breaking events could be related to the magnitude of δ(t) just preceding breaking onset. This inference was based on the conceptual link between δ(t) and the mean convergence rate (or accumulation rate) of energy at the energy maximum, and hence to the rate at which the breaking crest needed to dissipate the accumulated energy in excess of the maximum accumulation rate corresponding to the recurrence limit.

a. Existence of a common threshold for breaking

Figs. 4, 6, and 7 show typical long-term evolution curves of μ(t), 〈μ(t)〉, and δ(t) for each of the three different initial group cases. They reveal distinctly different generic evolution curve behavior for the recurrence cases and similarly for the breaking cases.

For breaking cases, the diagnostic growth rate δ(t) increases continually until breaking occurs, while for recurrence cases, δ(t) initially increases, then ceases growing at the peak of μ(t), and then begins decreasing. Our results show that this behavior does not depend sensitively on the number of waves N in the group. Table 1 summarizes the key results of our long-term evolution calculations for different wave group lengths N for the three cases I, II, and III. These results include the time t_th when δ(t) attains the upper bound of the breaking onset threshold range δ_th, as discussed in detail below; the time of breaking onset t_br and the corresponding growth rate δ(t_br), or the time of the recurrence peak (t_max) and the corresponding maximum growth rate δ(t_gr,max).

It is interesting to note that the evolution time to breaking differs markedly, both within each of the cases I, II, and III for different N and between the various cases for the same N. The evolution time correlates with the initial degree of departure of the wave group structure from spatial uniformity and on the number of waves in the group. Also, the growth rate evolution curves for δ(t) show varying degrees of smoothness, with case II evolution curves typically showing more intrinsic variability.

On the fundamental question of predicting the onset of breaking or recurrence, reference to Table 1 shows that the maximum growth rate δ^max can clearly distinguish breaking from recurrence for any particular case. For example, δ^max = 0.83 × 10⁻³ for the maximally recurrent case I wave group with N = 5 and initial steepness (ak)₀ = 0.111 while for the comparable marginal breaking case with initial steepness (ak)₀ = 0.112, δ^max = 1.93 × 10⁻³. However, from this table it is also seen that δ^max shows considerable variation among each of the different cases investigated and hence does not provide a unique breaking threshold. Despite considerable effort, we were unable to find an alternative parameter that could provide a unique threshold.

Nevertheless, our results in Table 1 show that the maximum value δ^max of the growth rate δ(t) can be used as the basis of a common threshold parameter that clearly distinguishes recurrence cases from breaking cases and does not depend on the group structure or number of waves N in the group or the type of initial wave group structure. Based on the ensemble of results for the three different classes of initial wave group geometry (cases I, II, and III) in Table 1, we propose that a common growth threshold δ_th for breaking can be taken within the range 1.30 × 10⁻³ to 1.50 × 10⁻³. Breaking is seen to occur whenever δ(t) exceeds δ_th, while the maximum values of δ(t) for all of the recurrence cases remains below δ_th. This key aspect of our findings is summarized graphically in Fig. 8.

b. Time between exceeding threshold and breaking onset

Foreshadowing the results of Part II, it is found that the same proposed threshold range for δ_th is valid in the presence of wind forcing and a background shear current. Further, supplementary results on the influence of intermediate water depth on breaking onset, to be reported elsewhere, indicate that choosing δ_th at the upper boundary 1.50 × 10⁻³ also embraces unforced intermediate water depth propagation in mean water depths of up to at least one-third of the mean wavelength. On this basis, we decided to use δ_th = 1.50 × 10⁻³ as the reference level for assessing other properties of interest, such as the lead time between exceeding the threshold and breaking onset, noting that this choice provides the most conservative estimate of the lead time t_lead.

It is evident from Table 1 that each case has a different time interval t_lead between when the nominal threshold δ_th = 1.50 × 10⁻³ is exceeded and the actual onset of breaking. Some cases break within a short time, comparable to half the carrier wave period, while other cases break only after more than four carrier wave periods have elapsed. To our knowledge, this is the first systematic assessment of the lead time to breaking.

c. Breaking strength

It is observed that individual waves break with varying intensities, often characterized from “spilling” to “plunging,” yet there is presently little quantitative understanding of what controls this observed variation in breaking strength. Rapp and Melville investigated this question observationally in their extensive laboratory study of chirped wave packets, quantifying the relative or energy loss as the change in wave energy density (energy per unit width of the flow for 2D waves) before and after breaking, normalized by the energy density prior to breaking. They explored how this parameter varied as a function of their initial wave steepness parameter ak_c for different wave packet geometries, such as relative bandwidth and distance to breaking. They found that their breaking intensity, for conditions ranging from incipient to plunging breaking waves, correlated strongly with the initial wave steepness parameter ak_c. The trend of their results is shown in Fig. 9.

For comparison, Fig. 10 shows our computed evolution curves for 〈μ(t)〉 and the derived dependence on ak_c of δ^max for case III wave groups with N = 5, for initial steepness levels for marginal recurrence and a number of breaking cases with progressively larger ak_c. While this approach is unable in principle to provide absolute levels of energy loss fraction, we see its value as potentially supporting our viewpoint that it is the mean energy convergence rate to the group maximum that underlies the onset and strength of breaking. Indeed, the favorable correspondence with the observational results in Fig. 9 both with respect to the general shape and threshold initial steepness provide encouraging initial support for our proposed association of breaking strength with δ^max preceding breaking onset.

For comparison, Figs. 11 and 12 show the trend of the calculated results similar to Fig. 10, but for case I and II wave groups with N = 5. The overall trends are not dissimilar, but the marginal breaking steepness ak_c thresholds are significantly reduced while the levels of δ^max are very similar. These differences warrant further observational investigation, especially as it has not been established whether chirped wave studies are representative of ocean wave breaking.

Table 2 summarizes the maximum growth rate estimates and associated parameters of interest for cases I, II, and III for a range of N values. In all cases, δ^max is found to increase with initial wave steepness while the corresponding breaking time t_br decreases rapidly. In addition, from the data for case I, the trend of δ^max with ak_c is shown in Fig. 13 and suggests an appreciable dependence of predicted breaking strength on the number of waves N in the group. Results for case II seen in Fig. 14 show a similar dependence, but for a more limited range of N. This potentially interesting influence of group size on predicted breaking strength could be investigated in future observational studies.