1. Introduction

It is well known that wind-driven ocean waves have a strong group structure. This is evident in wave height records and even to the casual observer. An important feature of these evolving wave groups is the intermittent occurrence of very steep individual waves that can present hazardous conditions for shipping and offshore structures, especially if these waves break. Donelan et al. (1972) and Holthuijsen and Herbers (1986) provide compelling observational accounts that link wave breaking and the group structure of wind waves. Among the many fundamental scientific issues that arise in this context are the relative importance of wind forcing of the waves and also the influence of the concomitant vertical shear in the surface layer region through which the waves propagate.

Part I of this study (Song and Banner 2002) addresses the issue of wave breaking onset in nonlinear wave groups in the absence of wind forcing and background shear. It focuses on identifying and quantifying the role of the nonlinear hydrodynamics in the idealized case of two-dimensional irrotational wave group evolution. It provides an extensive literature review of the topic and proposes new methodologies for quantifying the complex nonlinear behavior of three classes of initial wave group structures. Of particular relevance to the discussion of wind-forced waves and the statistics and structure of extreme waves in such wave groups are the papers of Longuet-Higgins (1984), Boccotti et al. (1993), and Phillips et al. (1993). These papers, however, do not address the underlying issue of determining the onset of wave breaking. The main body of published work on wave breaking onset has been concerned with hydrodynamical aspects and this aspect is addressed in Part I.

Historically, there have been relatively few papers that have focused on wind influence on wave breaking. One notable example was the model of Banner and Phillips (1974), who proposed that the wind drift current at the air–water interface could destabilize water waves by accelerating the onset of the kinematic condition where the horizontal water particle velocity at the wave crest attained the phase speed of the wave form. In that model, we note that the wind is regarded as only generating the wind drift current. This hydrodynamic source is then responsible for the onset of breaking, with the direct aerodynamic forcing regarded as of secondary importance. The motivation for this model had its origin in seeking to explain the remarkably strong attenuation of small-scale wind waves by longer waves in laboratory wind wave tanks, as investigated by Phillips and Banner (1974). A number of authors, notably Wright (1976), Donelan (1987), and Chen and Belcher (2000) have subsequently questioned the primary role of the wind drift current in this process and have proposed alternative mechanisms. Observations of the influence on short waves due to the transient passage of long waves (e.g., Chu et al. 1992) reveal a very fast interaction time with strong breaking of the short waves, indicating a hydrodynamic rather than aerodynamic origin for the short wave attenuation. As this aspect remains to be fully understood, this important phenomenon cannot yet be regarded as fully resolved.

Such studies have focused on elucidating specific mechanisms in established wind wave fields. A different approach addressing the initial spectral development of sideband instabilities was investigated in two recent observational papers that contribute authoritative theoretical and observational insight into the unforced and wind-forced evolution of nonlinear deep water wave trains, focusing particularly on the initial growth rates of sideband instabilities. In their first paper, Tulin and Waseda (1999) investigate the initial evolution and related aspects of the initial instability of deep water wave trains in the absence of wind forcing, while their companion paper, Waseda and Tulin (1999) examines the additional influence of wind forcing. Contrary to earlier results reported by Bliven et al. (1986) and Li et al. (1987), their results for wind forcing showed that wind did not inhibit the growth of sidebands in the case of either two-dimensional or three-dimensional instabilities. Overall, Waseda and Tulin (1999) reported two independent effects of the wind:

1. a modification of the inviscid growth rates for a given modulational frequency, as shown by comparison with “seeded” experiments where initial disturbances are introduced by a prescribed wave paddle motion in the absence of wind;

2. a change in the natural modulational frequency appearing in the presence of wind that is a function of the wave age, as observed in “unseeded” experiments where only wind-generated waves were involved.

Our present investigation is concerned with the long-term evolution to recurrence or breaking onset and is therefore complementary to the earlier studies (e.g., Tulin and Waseda 1999; Waseda and Tulin 1999) that were concerned primarily with initial instabilities. In Part I of our investigation, we report the results of our numerical investigation of unforced long-term evolution of nonlinear deep water wave groups, with a major focus on (i) determinants of breaking event onset, (ii) how far in advance can wave breaking events be predicted, and (iii) what controls their strength. Part I also contains a detailed presentation of our computational approach and the reader is referred to that paper for full details of the methodology used.

Before addressing the issue of wind forcing, we recall that the complementary issue of surface layer shear is another potentially important background process that has been associated with the destabilization of wave trains. The results of Teles da Silva and Peregrine (1988) and Millinazzo and Saffman (1990), among others, suggest that the presence of linear background vertical shear has a potentially strong influence on the structure of a wave train. This was confirmed in Banner and Tian (1998, henceforth BT), for the unforced case, where the wave steepness at breaking was reduced by up to O(20%) in the presence of a strong linear vertical shear current. Our improved computational methodology and revised interpretation of the underlying energetics associated with breaking onset motivated an extension of our investigation to include an assessment of the influence of a linear surface layer shear more typical of well-developed open ocean wave conditions.

In Part II, therefore, we investigate how wind forcing and surface layer shear modify the unforced results reported in Part I, both as separate influences and when both operate concurrently. This allows an assessment of the relative importance of wind forcing and surface layer shear in relation to the fundamental nonlinear fluxes operative in the absence of wind forcing and vorticity.

2. Methodology

To avoid unnecessary duplication, the reader is referred to Part I for a detailed description of our approach and methodology, including the definitions of diagnostic variables, computational techniques, and accuracy. The additional details describing the modeling of wind forcing and surface layer shear in the calculations are given in the appropriate sections below.

3. Influence of wind forcing

Of greater significance and interest is the evolution of the diagnostic energy-related nondimensional parameters μ(t), its local average 〈μ(t)〉 and the corresponding growth rate δ(t) that were defined in section 3d of Part I and subsequently investigated in detail for unforced, irrotational wave groups. Figure 3 shows the influence of surface forcing on 〈μ(t)〉 and δ(t) for the three cases α_∗ = 0.0002, 0.0003, and 0.0016. It is seen that the addition of surface forcing of strength α_∗ = 0.0002 marginally increases the maximum value of 〈μ(t)〉 at the peak of the recurrence cycle, but creates negligible change in the corresponding growth rate δ(t). Only for stronger wind forcing α_∗ = 0.0003 does δ(t) exceed the breaking threshold δ_th in the range (1.30–1.50) × 10⁻³ proposed in Part I on the basis of our results for unforced evolution. Stronger wind forcing results in a faster evolution of 〈μ(t)〉, earlier exceeding of the breaking threshold δ_th and a higher value of the growth rate δ^max just prior to breaking.

We also examined an unforced case (case I, N = 5 and s₀ = 0.12) just beyond the recurrence limit so that breaking occurs for the unforced case. For this case, the influence of surface forcing accelerates the onset of wave breaking, with the time to breaking decreasing from t/T = 60.5 for α_∗ = 0 to t/T = 58.7 and 58.4 for α_∗ = 0.0004 and 0.0016, respectively. The results for 〈μ(t)〉 and δ(t) comparing zero surface forcing (α_∗ = 0) and strong surface forcing (α_∗ = 0.0016) are included subsequently in Fig. 5, which also shows the comparative influence of shear. The steepness at breaking (ak)_br increases from 0.3437 for α_∗ = 0 to 0.3583 for α_∗ = 0.0016. Table 1 contains a summary of the salient results.

While the intuitive expectation is that surface forcing destabilizes the motion of the wave group and accelerates the onset of breaking, we found cases where the opposite can occur. In some cases, an unforced breaking case can become a recurrence case when surface forcing operates while breaking onset can be delayed by similar surface forcing levels for some unforced breaking cases. An example where wind forcing stabilizes the motion of the wave group is case I with N = 10 and s₀ = 0.089. In this case, breaking occurs at t/T = 126.4 without wind forcing, yet breaking does not occur for this same initial wave group geometry when wind forcing of strength α_∗ = 0.0016 is applied.

Another example of wind forcing delaying the breaking is case I with N = 7. Very strong surface forcing α_∗ = 0.0016 does not change the initial steepness s₀ = 0.099 for marginal recurrence and only slightly accelerates the onset of breaking for the marginal breaking case of s₀ = 0.10, decreasing the time to breaking slightly from t/T = 95 for α_∗ = 0 to t/T = 94.9. However, for a slightly larger initial steepness s₀ = 0.11 and the same surface forcing strength, the opposite behavior occurs, with the time to breaking increasing from t/T = 70.8 for α_∗ = 0 to t/T = 72.7 and the steepness at breaking, (ak)_br, increasing from 0.3500 for α_∗ = 0 to 0.3558 for α_∗ = 0.0016. This suggests that the influence of the wind forcing in accelerating breaking onset may depend on the nonlinearity of the windsea.

Nevertheless, our results for wind forcing, summarized in Table 1, reinforce one of the central results of this study concerning the existence of a common breaking threshold δ_th. They confirm the critical aspect that even for extreme surface forcing, δ^max < 1.30 × 10⁻³ continues to apply for each recurrence case, while δ^max exceeds 1.50 × 10⁻³ for all breaking cases that we investigated. Thus for surface forcing levels typical of open ocean conditions, the same critical threshold range of (1.30–1.50) × 10⁻³ for δ_th found for zero surface forcing appears to be applicable when wave-coherent surface forcing is operative.

4. Influence of a uniform surface shear

The actual depth distribution of the oceanic surface-layer shear current is a complex issue (e.g., Craig and Banner 1994), but following BT, we assumed a surface layer current O(0.03U₁₀) that decreased linearly over a depth of one significant wave height H_s. Here, U₁₀ is the mean wind speed at the reference height of 10 m above the mean sea level and the H_s is related to the mean energy 〈E〉 of the waves by H_s = 4〈E〉. Thus we assumed a shear profile of the form U(y) = Ωy, with y = 0 corresponding to mean sea level. The mean shear rate Ω was estimated using H_s calculated from the fully developed Pierson and Moskowitz (1964) windsea model for which

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For U₁₀ ∼ 10 m s⁻¹, this yields Ω ∼ 0.12 s⁻¹. Previously, BT investigated the effect of a strong background shearing current with Ω ∼ 0.2 s⁻¹. In this study, representative shearing current strengths up to Ω = 0.1 s⁻¹ were investigated. We were unable to define a robust reference velocity gradient for scaling Ω. However, when compared with a representative vertical gradient of the orbital motion of the initial carrier waves with typical initial steepness (ak)₀ of O(0.1), it is easily shown that Ω = 0.1 sec⁻¹ corresponds to a nondimensional shear level O(1). In any event, given the continually evolving nonlinear wave field with its fairly broad range of orbital speeds, we decided to retain the prescription of the dimensional shear level Ω (units: s¹). The addition of a background uniform shear is available as an option in the DP model code. After suitable scaling, this background shear was configured within the DP model and the corresponding local energy density E given by Eq. (6) of Part I was calculated. The details of this calculation are given in the appendix.

Unlike wind forcing, the presence of the shearing current considered here always destabilizes the wave group from recurrence to breaking and accelerates its onset. In some cases, an unforced case with recurrence can develop into a breaking case in the presence of a uniform surface shear layer. For example, for the marginally stable unforced case I with N = 5 and s₀ = 0.111, breaking occurs at t/T = 82.6 when a weak shear Ω = 0.02 s⁻¹ is present. Our results confirmed the BT findings that the surface shear current accelerates the onset of breaking. For example, for above case (i.e., case I with N = 5 and s₀ = 0.111), the breaking time decreases to t/T = 69.2 when a background linear vertical shear current with Ω = 0.1 s⁻¹ is applied. The corresponding evolution curves of μ(t), 〈μ(t)〉, and δ(t) for this marginal recurrence, unforced case in the presence of shear Ω = 0.02 and Ω = 0.1 s⁻¹, respectively, are shown in Fig. 4.

In a typical case where wind forcing delays the onset of breaking (i.e., case I with N = 7), the unforced marginal recurrence case with s₀ = 0.099 breaks at t/T = 87 when a background shear of strength Ω = 0.1 s⁻¹ is present. For s₀ = 0.11, the breaking time is t/T = 70.8 for Ω = 0. The breaking onset time decreases to t/T = 70.1 for a weaker background shear with Ω = 0.02 s⁻¹ and to t/T = 67 for a stronger shear of strength Ω = 0.1 s⁻¹. Also, we found that shear reduces the critical initial steepness for marginal breaking. This means the critical initial steepnesses s^Rc₀ and s^Bc₀ become smaller when shear is present. For example, s^Rc₀ and s^Bc₀ reduce to 0.107 and 0.108, respectively, from 0.111 and 0.112 for the unforced cases of case I with N = 5 with a shear of strength Ω = 0.1 s⁻¹. Corresponding changes in δ^max are documented in Table 1 and reveal that the presence of shear significantly changes the behavior of wave group evolution and breaking process. For an unforced recurrence case, the presence of shear can make it break. For a breaking case, the presence of shear accelerates its onset and marginally modifies the value of δ^max. However, the onset of breaking is still determined by the common threshold δ_th value in the range (1.30–1.50) × 10⁻³.

5. Combined wind forcing and shear

It is appropriate to consider the effect of simultaneous shear and surface forcing as they usually act in unison. One example of the influence of a background shear for a case where surface forcing accelerates the onset of breaking is shown in Fig. 5. This is case I with N = 5, s₀ = 0.12, α_∗ = 0.0016, and Ω = 0.1 s⁻¹, for which breaking occurs at t/T = 51.4. In comparison, for this same case but with a shear of strength Ω = 0.1 s⁻¹ and zero surface forcing, breaking occurs at t/T = 53.5, while for the very strong surface forcing level α_∗ = 0.0016 in the absence of shear, the breaking time increases to t/T = 58.4.

Another typical example of the influence of a background shear for a case where surface forcing delays the onset of breaking is shown in Fig. 6 for a weak shear with very strong wind forcing. In this example, N = 7, s₀ = 0.11, and a weaker background linear vertical shearing current of magnitude Ω = 0.02 s⁻¹ was used, as stronger shear masked the influence of the surface forcing. For strong surface forcing α_∗ = 0.0016 in the presence of this background shear, breaking occurred at t/T = 70.4. Correspondingly, for very strong surface forcing α_∗ = 0.0016 in the absence of shear, breaking occurred at t/T = 72.7. With the same weak background shear Ω = 0.02 s⁻¹ and zero surface forcing, the time to breaking decreased to t/T = 70.1. Thus the presence of a shearing current accelerates the onset of breaking, even if the shear is weak. This parallels the influence of shear on unforced cases discussed in BT. In passing, we noticed that surface layer shear tends to modify the surface profile and the evolution process of the nonlinear wave group more strongly than the wind forcing, although the modifications arising from either influence should be considered secondary in importance to the nonlinear wave group hydrodynamics. This is apparent from the properties of the representative case I examples summarized in Table 1.

Although a surface layer shearing current more strongly modifies the surface shape and the evolution process than the wind forcing, the wave steepness at breaking was only modified (increased or reduced) slightly (i.e., by ≤5%) in the presence of a linear shear of strength Ω = 0.02 or 0.1 s⁻¹. For example, (ak)_br increased from 0.3069 to 0.3213 when the shear was reduced from Ω = 0.1 to Ω = 0.02 s⁻¹ for case I with N = 5 and s₀ = 0.111. For case I with N = 7 and s₀ = 0.11, (ak)_br changed to 0.3440 and 0.3513, respectively, from the unforced case where (ak)_br = 0.3500 when shears of strength Ω = 0.02 and 0.1 s⁻¹ were present. These modifications to the breaking steepness are considerably smaller than the findings reported by BT for a stronger shear of Ω = 0.2 s⁻¹.

A summary of our results for δ^max and other parameters of interest for a number of representative shear and surface forcing cases is presented in Table 1. Of central importance is that the results validate our proposed critical average growth rate threshold range δ_th = (1.30–1.50) × 10⁻³ for the diagnostic growth rate δ(t). Based on results in Table 1, Fig. 7 summarizes graphically the relationship between the growth rate δ^max and our proposed breaking threshold for various case I wave group configurations for cases of wind forcing of strength α_∗ = 0.0016 in the absence of shear and shear of strength Ω = 0.1 s⁻¹ in the absence of wind forcing.

In section 5c of Part I, it was proposed that the strength of breaking events was proportional to the growth rate δ^max. Figure 8 shows comparative results for the influence of wind forcing and shear on the predicted breaking strength. This figure shows separately the marginal effect on the trend of our proposed breaking strength indicator δ^max due to wind forcing of strength α_∗ = 0.0016 in the absence of shear and also for shear of strength Ω = 0.1 s⁻¹ in the absence of wind forcing. Results are shown for case I wave groups with N = 5 and N = 7 and suggest that the addition of wind forcing or shear does not produce significant variations from the trend of δ^max with increasing initial steepness ak_c for the unforced, irrotational case I wave groups that we investigated. Table 2 summarizes the corresponding maximum growth rates δ^max, t_br, t_th, and t_lead.

6. Conclusions

This study extends the scope of the results obtained for the unforced, irrotational cases of initial wave groups studied in Part I to investigate representative effects of wind forcing, shear, and the combination of these two additional upper ocean influences. Our findings were:

1. The presence of wind forcing and vertical shear typical of upper ocean conditions results in a similar evolution to recurrence or breaking and reinforces our conclusions of Part I. The associated nondimensional diagnostic parameter μ(t) evolves in a complex fashion, with a “fast” oscillation superimposed on a longer-term mean trend. As discussed in Part I, the trend of the local average of this parameter, 〈μ(t)〉, encapsulates the observed systematic mean energy convergence toward (or away from) the maximum energy region within the wave group and this determines the ultimate breaking or recurrence behavior of the wave group. The fast oscillation, associated with the strong crest/trough asymmetry of the carrier waves, is believed to be primarily a kinematic effect.

2. Our results indicate that for a wide range of wave forcing and current shear, for representative examples of the three cases of initial wave group geometry we investigated, breaking or recurrence of deep water wave trains is still determined by the common threshold value δ_th in the range (1.30–1.50) ×10⁻³. This threshold, proposed in Part I for the nondimensional growth rate δ(t) of 〈μ(t)〉 following the wave group maximum for unforced, irrotational wave groups, has been found to be applicable in the presence of these additional influences for the three classes of initial wave group structures we investigated.

3. In Part I, section 5c, it was suggested that the strength of breaking events may be controlled by the mean rate of convergence of energy at the group maximum immediately preceding breaking onset and reflected parametrically by the corresponding maximum value δ^max of δ(t) at the time of breaking onset. The addition of wind forcing and/or shear did not result in significant variations from the unforced trend of δ^max with increasing initial steepness for the set of case I realizations we investigated. In one case (N = 5), the addition of shear and wind forcing each tended to increase the maximum growth rates by a small margin, while in another case (N = 7), the addition of the same wind forcing or shear levels resulted in a comparable reduction. It remains for careful observations to confirm the correspondence between this proposed measure of breaking strength and actual energy losses.