Near-Surface Turbulence in the Presence of Breaking Waves

a. Dissipation of turbulent kinetic energy

Our goal is to examine the relation between wave breaking and energy dissipation ε. According to Kolmogorov's hypothesis, there exists an inertial subrange where the three-dimensional wavenumber spectrum of a high Reynolds number flow has a universal form that depends only on the energy dissipation. For isotropic turbulence, the one-dimensional wavenumber spectrum S(k) takes the following form (Hinze 1975):

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where k is the wavenumber and A = 1.5 is a universal constant.

This simple relationship between energy dissipation and wavenumber spectra (1) is the basis for estimating ε. The wavenumber spectrum is calculated for each velocity profile using the Hilbert spectral analysis. The velocity profile represents a broadband signal that is decomposed into several narrowband intrinsic mode functions (Huang et al. 1998, 1999). Application of the Hilbert transform on these intrinsic mode functions yields the local wavenumbers and amplitudes as a function of profiling range. The Hilbert transform of the function g(x) is defined by

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Because of a possible singularity at x = t, the integral has to be taken as a Cauchy principal value P. The original function and its Hilbert transform form the analytical function Z(x) = g(x) + iH(x). The amplitude and phase of this signal are

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respectively. The local wavenumber is

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Amplitudes and wavenumbers are calculated from each intrinsic mode function. The wavenumber spectrum is obtained by integration of the squared amplitude along the entire profile. The profiling range (0.72 m) and the coarsest spatial resolution (35 × 10⁻³ m) determine the wavenumber resolution 9 rad m⁻¹ < k < 180 rad m⁻¹. Utilizing the entire velocity profile requires that turbulent properties are uniform along the complete path. To check the validity of this assumption, we split the velocity profile into near field (0–0.36 m) and far field (0.36–0.72 m) prior to calculating the wavenumber spectra, thereby increasing the low wavenumber cutoff to k > 18 rad m⁻¹ as well as decreasing the degrees of freedom in the spectral analysis. Although computationally more elaborate than the commonly used Fourier spectral analysis, the use of the Hilbert spectral method based on intrinsic mode functions is more appropriate for nonstationary and short data records than the Fourier method, which is strictly limited to linear, stationary data.

The characteristic time scale associated with the inertial subrange τ_t = (ν/ε)^1/2 depends on the kinematic viscosity ν and dissipation (Hinze 1975). For typical dissipation values encountered in the near-surface layer, this time scale ranges from O(10⁻² s) to O(1 s). Therefore individual spectra having a 20-Hz sampling rate are averaged to obtain robust spectra at 1-s intervals (Fig. 2b). These averaged spectra are checked for the existence of a k^−5/3 slope, indicative of an inertial subrange. The inertial subrange concept represents an idealized case of isotropic, steady-state turbulence, which is not always applicable in the ocean surface layer. The intermittency of the turbulence generation, advection of turbulence by the wave orbital motion, the proximity of the surface, and the presence of dense bubble clouds will distort this idealized spectral shape at certain times. For the evaluation of the spectral shape, we use maximum likelihood spectral fitting (Ruddick et al. 2000) and allow for a variable wavenumber range in our calculation of the slope. The instrumental noise spectrum required for the maximum likelihood fitting has been determined from data recorded in calm water off the dock at the Institute of Ocean Sciences. Dissipation values are calculated according to (1) from averaged spectra that are consistent with the inertial subrange. For the full range profiles about 50%–75% of the spectra have properties consistent with an inertial subrange. Using only one-half of the profile, this percentage drops to 45%–70% with no significant difference between the near field and far field. However, between 65% and 90% of the averaged spectra are consistent with an inertial subrange in at least one part of the path. The intermittent occurrence of an inertial subrange appears random (Fig. 3). Data records with the highest percentage of resolved inertial subranges occur mainly at moderate wind speeds. However, we did not find any significant correlation between the percentage of resolved subranges and environmental parameters like wind speed or wave steepness that would hold for all deployments. In the following, dissipation estimates are the average of estimates based on the near field, far field, and entire profile. No dissipation estimates are obtained in cases in which none of the three spectra show an inertial subrange.

A major challenge of oceanic turbulent velocity measurements is the separation of wave-induced motion and turbulence. Orbital velocities of the dominant waves are O(1 m s⁻¹) whereas typical turbulent velocities are O(10⁻² m s⁻¹). The magnitude of the orbital motion is a strong function of distance to the surface. At a distance z below the still waterline, the vertical velocity of a linear wave varies as w(z) = aωe^−k|z|, where a is the amplitude and ω is the frequency of an individual wave component. The largest scale resolved in our velocity profile is k = 9 rad m⁻¹ and the wave orbital motion at this wavelength is O(10⁻⁴) of its surface value, which is negligible. Larger waves, which are associated with significant orbital motion at the profile depth, result in a constant velocity offset, which does not affect the spectral level of the inertial subrange (see the appendix). Therefore, a separation of turbulent and wave-induced velocities is not required for our instrumentation setup.

b. Surface elevation

An estimate of the local surface elevation may be obtained from the horizontal velocity record. The wave orbital velocity is the combination of the downwind u and cross-wind υ velocity component c = u ± υ, where the sign depends on the orientation of the float relative to the wave. The surface elevation is

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where the symbol H represents the Hilbert transform and η_x = −∫ H(−u) dt, η_y = ∫ H(υ) dt. In practice, the velocity records are the mean value of the first three velocity bins and are high-pass filtered with a cutoff at 0.04 Hz prior to applying the Hilbert transform. Generally, the total surface elevation is dominated by the downwind component, η_x ≥ 100η_y, and η_y represents only a small correction. Visual observations confirmed that the float is turning into the waves such that the downwind component is nearly always parallel to the wave propagation. However, the float drifts up to 2 m back and forth from the wave motion. This motion can be tracked on the video recordings, and for specific datasets the error due to the instrument motion can be removed from the time series. If the time series is uncorrected, the recorded velocity is reduced and (5) yields the correct phase of the waves but underestimates the amplitude.

a. Velocity and dissipation measurements

An example of the observed velocity field is given in Fig. 5. As expected, the strongest velocities are observed in the downwind direction, with magnitudes >1 m s⁻¹, dominated by the wave orbital motion. The maximum cross-wind and vertical velocities are more than one order of magnitude smaller than the downwind component. This is expected for the vertical component (appendix), however not in the nominal cross-wind direction in a directional wave field. The tethering of the float allowed it to be advected sideways and also to turn into waves propagating in different directions. Both float motions would tend to greatly reduce cross-wind orbital motions from the recording. Visual observations and the video recording confirmed the combination of this float behavior. Furthermore, the vertical sonar and the sonar oriented in the cross-wind direction are not affected by the wake of the float and are used to estimate dissipation rates. Several bursts of rapid velocity fluctuations lasting up to 20 s are recorded at both sensors. Generally, the vertical and cross-wind sonar show the same pattern of alternating quiescent and energetic periods.

The dissipation varies over four orders of magnitude within tens of seconds, with no simple dependence on wind speed (Fig. 6). The lowest dissipation rates throughout all three deployments are O(5 × 10⁻⁶ m² s⁻³) and peak dissipation reaches 10⁻¹ m² s⁻³. Possibly due to a lack of scatterers (mainly bubbles), no reliable velocity data could be obtained during low wind speed periods; therefore, no estimates of dissipation in these calm conditions are available. Generally, there is a good correlation between high dissipation rates and the occurrence of breaking waves (Fig. 6). As mentioned in section 2, not all velocity spectra obtained from our data are consistent with an inertial subrange; consequently, the dissipation time series are irregularly spaced. Generally, there is good agreement within the dissipation rates based on the cross-wind and the vertical sonars. However, the horizontal sonar shows a slightly higher rate of unresolved dissipation measurements, which may be linked to the different sensor depth. Particularly within dense bubble clouds the turbulent velocity field cannot be monitored, resulting in a higher dropout rate at the shallower, horizontal sensor during periods of higher wind speed. Dissipation estimates in the subsequent analysis are based on the vertical velocity profiles.

Oceanic turbulence is expected to have a lognormal distribution (Oakey 1985) and may be described by its mean and standard deviation. Taking deployment I as representative of our open-ocean measurements we find that, on average, dissipation values are about 60 times ε_wl = u³_∗w/(κ|z|), the value predicted for a constant stress layer (Fig. 7). Here u_w∗ is the friction velocity in the water and κ = 0.4 is the von Kármán constant. However, the mean enhancement ratio, that is, the ratio between observed dissipation rates and the dissipation rate in a flow of the given speed and distance past a solid wall, is not a representative measure of the underlying physics; the observed dissipation values do not have a lognormal distribution. The dissipation enhancement is defined as r_ε = log{ε/[u³_w∗/(κ|z|)]}, where the depth z is the streamline of the wave orbital deformation at 1 m from the instantaneous surface, as discussed below. However, here we take a fixed value of z = −1 m. For extremely steep waves this might cause the depth to be overpredicted (underpredicted) beneath the crests (troughs) by up to a factor of 2. The observed probability distribution of dissipation enhancement r_ε is a combination of two nearly Gaussian distributions: (i) a broad distribution about the predicted dissipation level r_ε = 0 for a law-of-the-wall boundary layer overlapping (ii) a narrow distribution centered at r_ε = 2. The highly enhanced distribution in point ii contributes to about 25% of the total distribution and is directly linked to breaking waves as discussed below.

All previous near-surface dissipation studies reported average dissipation values, where the averaging period ranged from tens of seconds to several minutes. Averaging dissipation data over these time periods greatly modifies their apparent distribution. This can be seen by filtering the dataset used for Fig. 7a, originally obtained at 1-s intervals, with a 60-s filter:

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For this averaged dissipation ε_av a very different, monomodal, distribution is found (Fig. 7b). The peak at high dissipation enhancements disappears and only a single 1-min average satisfies r_ε > 2, implying that the observed high dissipation enhancement represents short events. The reduction of occurrences with r_ε < 0.2 suggests that periods consistent with constant-stress layer scaling or less dissipation are also short-lived events in this wind-driven layer. Of course, the mean of the enhancement does not depend on the averaging period. Even the longer period averages do not follow a lognormal distribution.

b. Individual wave analysis

An example of the turbulence and air entrainment associated with an individual breaking wave is shown in Fig. 8. Corresponding video images of the breaking wave are given in Fig. 9. Note that the instrument is moving in and out of the turbulence patch, yielding a bias toward crest samples (Fig. 10). No other breaking waves are observed in the vicinity of this wave or within 60 s prior to or after the event. Thus, the recorded signal can be attributed to an individual breaking wave. The period of this wave, obtained from the horizontal velocity record, is τ = 5 s (Fig. 8a). This is in good agreement with the wave period τ = 5.4 s, corresponding to the propagation speed c_wc of the whitecap, estimated from the video recording and assuming linear wave dispersion: τ = 2πc_br/g. Here g is the gravitational acceleration and c_br = 1.4c_wc the phase speed of the breaking wave (Lamarre and Melville 1994).

The breaking occurs near the crest of the largest wave of the wave group (Fig. 8a). Note that, due to the float motion, different wave phases are sampled with different spatial resolution, making the wave time series appear asymmetric as discussed subsequently. The spatial transformation for this wave is given in (Fig. 11c). Significant velocity fluctuations occur during passage of the whitecap at t = 15.9 s. The standard deviation of the vertical velocity within each individual profile σ(w) increases rapidly from about 1 × 10⁻² to 6 × 10⁻² m s⁻¹ (Fig. 8b). The duration of the intense fluctuations is 1.2 s or approximately one-quarter of the wave period. Figures 11a,b show a more detailed presentation of the vertical velocity field. The top of the measurement profile, which is 1 m below the instantaneous surface, cuts through near-surface streamlines due to the wave orbital deformation. In Fig. 11b data have been mapped onto a coordinate system fixed at the instantaneous surface, but depth coordinates are stretched so as to represent the instantaneous streamlines. Velocity fluctuations at the onset of breaking occur along the entire velocity profile. The velocity field is divided into the depth-averaged component 〈w〉 and the fluctuating part w′. The coherent signal results mainly from rapid oscillations of the float and a weak signal of the differential orbital velocity (appendix). The incoherent signal is somewhat stronger than the coherent component. The strongest signal occurs at the onset of breaking at the crest of the wave. Fluctuations diminish rapidly when the float drifts out of the most active turbulent patch. The subsequent wave crest advects the float back into the turbulent patch, resulting in increased w′ (i.e., t = 20–21 s).

Prior to the breaking event, the inertial subrange of the velocity spectra spans 20–160 rad m⁻¹ (Fig. 12). At the onset of breaking, energy increases across the spectrum but particularly at high wavenumbers, k > 100 rad m⁻¹. Energy in the high wavenumber range dissipates rapidly. Approximately 1.5 s past the onset of breaking, the peak of the energy spectrum has shifted to k ≈ 20 rad m⁻¹ from an initial peak at k ≈ 150 rad m⁻¹ at the beginning of the breaking event. Roughly within the following one second, the spectrum reassumes a spectral form consistent with an inertial subrange.

The background dissipation prior to the breaking event is O(5 × 10⁻⁶ m² s⁻³) (Fig. 8c). The maximum resolved dissipation is 8 × 10⁻³ m² s⁻³ and occurs when the whitecap passes above the sonar. It is interesting to note that 0.7 s prior to the whitecap arrival the dissipation has already increased to 100 times the background level. Within the first second following the breaking event, no inertial subrange was present in either the full profile or the near or far field, and no dissipation can be estimated. Therefore, the observed dissipation enhancement r_ε = O(5 × 10²) is a lower bound for the turbulence associated with this individual wave. Enhanced turbulence levels persist for at least 10 wave periods. The average dissipation rate in the first 10 wave periods following the breaking event is 5 × 10⁻⁵ m² s⁻³. However, the dissipation fluctuates periodically by two orders of magnitude. High dissipation rates O(10⁻⁴ m² s⁻³) reoccur at intervals similar to the wave period. A probable explanation of the periodicity of turbulence intensity might be the stretching of the turbulence patch due to wave orbital motion, combined with the instrument movement in and out of the turbulence patch (Fig. 10).

Air fraction, which is an integral measurement within the upper 0.2 m of the velocity profile, shows a clear signature of the wave breaking (Fig. 8d). Prior to the event, the air fraction was of the order of 5 × 10⁻⁶. Approximately 1 s after the whitecap reaches the float at the surface, the air fraction at 0.85-m depth increases rapidly and reaches the saturation level of the resonator (10⁻⁴) about 3 s after the crest has passed the event. Air fraction stays at this high level for close to two wave periods and decays slowly afterward. Ten wave periods after the event, the average air fraction is more than twice the value prior to the breaking event.

The peak bubble radius prior to the breaking event is a_pk < 100 μm. Air entrainment associated with the whitecap passage transports larger bubbles down to the depth of the upper resonator, resulting in a peak bubble radius >300 μm. The maximum velocity fluctuations of ∼0.06 m s⁻¹ are sufficient to counteract the rise velocity of bubbles with 200-μm radius (Fig. 8b). The more persistent turbulent velocity field generated by the breaking event may keep bubbles with circa 120-μm radius in suspension. This is consistent with the observed bubble size distribution. Approximately 10 s after the air injection the larger bubbles have disappeared at the sensor depth and the peak bubble radius slowly decreases (Fig. 8e). The majority of the remaining suspended bubbles have radii smaller than 120 μm.

Figure 13 shows more examples of turbulence data related to breaking events. In all cases the whitecap passage is associated with increased turbulence. However, if the breaking event is not discrete but occurs as a succession of smaller breaking events over one or two wave periods, several peaks in the turbulence signals are common (e.g., Figs. 13d,e,h,j).

a. Characteristic turbulence signal of breaking events

The individual breaking wave discussed above in detail shows a distinct turbulence and air-fraction signature common to most strong breaking events. For a further systematic analysis of the wave-induced signal, wave-breaking data for 0930–1330 UTC 10 October are conditionally sampled and averaged. During this 4-h period of nearly steady wind, 31 single breaking events with significant air fraction [γ(0.85 m) > 3 × 10⁻⁵, corresponding to the 95th percentile of the air-fraction record] were detected. The development stage of whitecaps reaching the float covers a wide range, and the onset of increased turbulence occurs within ±1 s of the whitecap passage (Fig. 13). Assuming that a whitecap is generated near the crest, we center the time series of individual breaking waves at the occurrence t_b of the crests associated with the breaking events, not the whitecap passage, and normalize it by the local wave period τ_w:t̃ = (t − t_b)/τ_w. The crest association is based on the “riding wave removal” (RWR) approach developed by Banner et al. (2002).

The RWR method progressively detects and removes riding waves through iterative processing starting with the highest frequency resolved. Occurrence and characteristic dimensions of riding waves are retained in a file and the waves are then removed from the elevation series by replacing it with a cubic polynomial spliced to the underlying longer wave form. Subsequently, the association between breaking events and recorded waves is based on minimizing the relative lag between the time of the breaking event and the time of the nearest local crest, while also satisfying a local steepness threshold ãk > 0.075, where ã and k are the local apparent wave amplitude and wavenumber. The occurrence time t_b and wave period τ_RWR of the breakers are then recorded to file. Generally, there is good agreement between the wave period estimates from the whitecap propagation and the RWR-modified surface elevation.

Figure 14 shows the conditionally sampled average evolution of dissipation and bubble characteristics associated with wave breaking. The time series includes two wave periods prior to the breaking event and the five subsequent waves. Before averaging, each individual time series is normalized by its extreme value within these seven wave periods. In case of the surface elevation, normalization is done with respect to the maximum surface elevation. Dissipation, air fraction, and peak bubble radius cover a wider dynamic range, and it is possible that the maximum value, associated with the breaking event, is not resolved properly. Therefore, these quantities are normalized by their minimum value, which represents background conditions not directly affected by the breaking event and is less sensitive to sensor saturation. It is important to note that the RWR method is used only to determine the properties t_b and τ_RWR of the breaking wave. The conditional sampling is based on the original records and therefore does not involve any filtering of the turbulence signal.

The breaking crest is generally the highest crest within the wave group, roughly 50% higher than the previous or subsequent crests (Fig. 14a). Associated with the breaking crests are the largest dissipation levels. Turbulence signals associated with multiple breaking events are incoherent and the conditional sampling technique tends to average them out. Dissipation associated with the main breaker increases rapidly on the forward face of the wave crest. No reliable dissipation estimates could be obtained in the direct vicinity of the breaking crest. It is possible that the resolved dissipation does not include the maximum value and the observed 3000-fold increase represents a lower bound for the dissipation beneath breaking waves. Nevertheless, the data clearly demonstrate high turbulence levels generated by the breaking wave. Comparison with the bubble observations shows that the increase in turbulence occurs up to a quarter wave period prior to the bubble cloud reaching the sensor depth (Figs. 14c,d). This delay between TKE dissipation and bubble injection is very prominent, even in this averaged signal, and is a general feature of the observed turbulence induced by breaking waves. Thus, wave-enhanced turbulence is caused not only by air entrainment and the associated conversion of potential energy to turbulent kinetic energy but by dynamical processes related to the steep wave itself. Indeed, enhanced turbulence might be a prerequisite for the breakup of the surface and bubble generation. A probable source of turbulent kinetic energy is discussed below.

Following the turbulence generation by the breaking wave, the observed dissipation fluctuates with local maxima coinciding with the forward face of the subsequent waves. These fluctuations are not primarily related to the evolution of the turbulent patch but rather to the motion of the float with respect to the turbulence combined with a possible wave-induced stretching of the turbulence patch. The wave orbital motion prior to passage of the crest advects the float into the center of the turbulence patch, whereas on the rear face of the wave the float is pushed farther to the edge of the patch. It is unlikely that the observed dissipation fluctuations are linked to the persistent near-surface vortex generated by the breaking event (Melville et al., 2002) since the vortex setup takes several wave periods.

The evolution of the maximum dissipation within waves following the breaking event (circles in Fig. 14b) indicates the decay of the wave-induced turbulence. Again, the dissipation in the direct vicinity of the breaking crest is not resolved, and we can only give bounds for the decay rate ε ∝ tⁿ, based on a least squares fit of the marked maxima in Fig. 14b. Assuming that the resolved dissipation includes the maximum value, we find the lower bound n = −2.9. If, however, as is much more likely, the largest dissipation is not measured and occurs closer to the crest, the highest observed value (at t̃ = −0.25) has to be excluded from the least squares fit. This case, where the decay rate is determined from observations two to five wave periods after passage of the crest, defines the upper bound of the decay rate n = −4.3. Extrapolating this decay backward yields a maximum enhancement rate at the breaking crest log(ε_max/ε_min) ≈ 4.6, which serves as an upper bound for the average dissipation beneath breaking waves. The upper bound of the decay rate n = −4.3 is in good agreement with the theoretical dissipation decay rate for isotropic turbulence, n = −17/4 (Hinze 1975). This supports our assumption of isotropic wave-induced turbulence in the near-surface region. We should not expect isotropic turbulence to occur very close to the boundary, and we note here that a much faster decay rate (n = −7.6) was inferred in the upper 0.2 m of the water column where dense bubble clouds lead to stratification (Gemmrich 2000).

Air fraction beneath individual breaking waves fluctuates rapidly and over a wide range that exceeds the resonator saturation. Nevertheless, the average air fraction signal (Fig. 14c) highlights the rapid increase in air fraction beneath the breaking crest. The brief large air fractions beneath the rear face of the breaking wave (Gemmrich and Farmer 1999b; Deane and Stokes 2002) cannot be resolved with the present instrumentation. However, the presence of air bubbles can be tracked for at least five wave periods following the breaking event. These increased air fractions are associated with a rapid shift in the bubble size distribution toward larger radii (Fig. 14d) following passage of the breaking event.

b. Wave–turbulence interaction

Laboratory and field studies have shown that surface waves may not be considered entirely irrotational. [For a short review of these experiments see, e.g., Anis and Moum (1995)]. In a rotational wave field, interactions between the mean, wave, and turbulence components of the flow provide additional TKE sources not present in a wall layer flow. The possible wave–turbulence interactions are best identified in the respective kinetic energy equations. For that purpose the velocity and pressure records are separated into a mean, wave-related, and turbulence contribution: υ_i = V_i + υ̃_i + υ^′_i, i = 1– 3, p = P + p̃ + p′, where the wave–turbulence separation is achieved by the phase-averaging method (e.g., Thais and Magnaudet 1996). Assuming a mean flow in the x direction, the kinetic energy of the mean flow (MKE), the wave orbital flow (WKE), and turbulence (TKE) are

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respectively. The resulting equations are for MKE:

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for WKE:

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and for TKE:

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Averaging periods much longer or comparable to the wave period are indicated by overbars and tildes, respectively.

Beneath a breaking crest, the approximate balance between dissipation [(12), term f] and turbulent diffusion of TKE [(12), term d] results in the observed large dissipation values. However, prior to the onset of breaking the diffusion term has to be much smaller and the increased dissipation may be balanced by TKE production based on wave–turbulence interaction. From (10)– (12) it can be seen that there exist two possible pathways to convert WKE into TKE, one indirect and one direct way.

In the indirect pathway, wave energy is transferred to the mean flow and then in turn into turbulence as follows. The wave-induced shear stress in a mean current shear [(11), term a] transfers energy from the waves to the mean current [(10)]. The MKE may then be redistributed into TKE through the classical production term [(10), term b]. Assuming a steady-state balance between dissipation and shear production yields (Anis and Moum 1995)

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where ϕ is the deviation from quadrature of u and w, and a and k are wave amplitude and wavenumber, respectively. Taking representative values a = 1.5 m, k = 0.01 m⁻¹, and the observed mean current shear ∂U/∂z ≈ 0.01 s⁻¹ suggests that a small phase shift ϕ = 6° is sufficient to generate enhanced dissipation levels ε = O(10⁻³ m² s⁻³), observed at the onset of breaking. Phase differences of this magnitude are fully consistent with those previously reported in the literature (see Anis and Moum 1995). A rough estimate of the u–w phase shift can be obtained from the cross-spectral analysis between the surface elevation time series derived from the horizontal velocities and the observed vertical velocity field, as follows. The float follows the surface motion and therefore the observed vertical velocity is reduced to a residual wave orbital motion (appendix):

w_obszwzwz_F(14)

The direct mechanism for wave–turbulence interaction occurs through term b in (11) and (12), representing the Reynolds stresses working against the shear of the rotational component of the wave orbital motion (Thais and Magnaudet 1996). This mechanism requires that the turbulent time scale T_t is much shorter than the wave period τ, T_t ≪ τ. On dimensional grounds this leads to following conditions for the turbulent velocity scale u_t ≪ (Tε)^1/2 and the turbulent length scale l_t ≪ τu_t (Anis and Moum 1995). Here we get u_t ≪ 0.1 m s⁻¹ and l_t ≪ 1 m, and conditions are favorable for this mechanism to be relevant. Thais and Magnaudet (1996) conclude that microbreaking and capillary ripples are the most obvious candidates for the required vorticity generation. These small-scale features may then also trigger larger-scale breaking. Hence, the direct wave–turbulence interaction would be strongest on the forward face of a wave crest and therefore would occur prior to the whitecap passage, consistent with video recordings of the analyzed breaking event. Indeed, recent laboratory studies of microbreakers using infrared imager and particle image velocimetry revealed strong near-surface vorticity in the crest region of microbreakers (Siddiqui et al. 2001).

Here we have shown possible mechanisms for the generation of near-surface turbulence. Recently, Teixeira and Belcher (2002) studied the interaction of weak near-surface turbulence with a monochromatic irrotational wave. They report that the distortion of the turbulence by the Stokes drift leads to the generation of Langmuir turbulence and also produces additional shear stresses. These stresses work against the straining of the wave orbital motion and thus, over many wave periods, further convert WKE into TKE.

c. Depth dependence

In the classical wall-layer flow dissipation decreases with distance from the boundary as ε(z) ∝ |z|⁻¹. Field observations suggest that in a wind-driven sea the surface layer may be divided into three regimes (Terray et al. 1996). Wave breaking directly injects turbulent kinetic energy into the top layer to depth z_b. In this injection layer dissipation is highest and depth independent. Below that depth the wave-induced turbulence diffuses downward and dissipates. In this region of wave-induced turbulence the turbulence decay rate with respect to depth is steeper than for wall-layer dependence. At a depth z_t, sufficiently far from the air–sea interface, the contribution of wave-induced turbulence becomes small compared to local shear production and turbulence properties are well described by the constant-stress-layer scaling. The problem is complicated by the strong temporal and spatial variability of the turbulence as exemplified by our observations. There is no conclusive observational evidence for the vertical extension of the different regimes (z_b, z_t) nor the exact depth dependence of the wave-induced turbulence. For the latter, the scaling ε ∝ |z|ⁿ, n = −2 to −4 (Terray et al. 1996; Drennan et al. 1991), has been suggested, as well as exponential decay ε ∝ e^−|z| (Anis and Moum 1995). Also, there are no observations verifying the existence of a constant dissipation layer at the surface. In their pioneering study, Stewart and Grant (1962) argue that more than one-half of the energy is dissipated above the mean waterline. Therefore, dissipation levels at a fixed distance beneath a crest and beneath a trough should differ, contradicting the assumption of a constant dissipation layer. To a certain extent, our surface-following measurements allow us to evaluate the vertical structure of the near-surface dissipation field, particularly with respect to the wave phase.

We calculate the average dissipation beneath a wave crest, defined as the period between zero-up crossing and zero-down crossing of the RWR-filtered surface elevation and the average dissipation of the subsequent trough. To avoid potential biases due to the unknown intermittency of near-field and far-field dissipation estimates, the following analysis is based on dissipation estimates obtained from entire velocity profiles only. The distribution of the ratio ε_crest/ε_trough for each individual wave (including breaking and nonbreaking) is given in Fig. 15. Cases with crest dissipation values larger and smaller than the dissipation beneath the trough region are found. However, there is a clear bias toward crest dissipation being larger than the trough dissipation. The average ratio is &ap; 1.6.

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Our observations are made 1 m below the free surface. Only for waves with amplitude a ≫ 1 m do the crest dissipation measurements represent conditions above the mean waterline. We conclude that our observations are consistent with Stewart and Grant's (1962) reasoning.

Depending on the individual wave amplitude a_i and the phase of the wave, our observations profile the water column over the range z_s − a_i ≤ z ≤ z_s + a_i, where depth z is referenced to the mean surface elevation averaged over the wave period (positive upward) and z_s = −1 m is the sensor depth. Normalizing the depth by the wave amplitude and sorting the results in depth bins gives information on the dissipation profile beneath waves. Each profile is then normalized by its value ε₀ = ε(z = 0) and all profiles are bin averaged (Fig. 16). Below the mean waterline dissipation stays nearly constant. However, in the crest region dissipation increases rapidly as ε ∝ z^2.3. Extrapolating the dissipation to the crest (z/a = 1), we find that the total dissipation (depth integrated per unit area) in the region above the mean waterline is about 2.5 times the total dissipation between the trough line and the mean waterline. Our observations reach to a maximum depth 1 m below the actual wave troughs; therefore, we do not have reliable depth dependence below approximately z/a < −1.5. It should be noted that the dissipation structure suggested by Terray et al. (1996) is based on a fixed coordinate system and mainly acquired from tower observations, which excluded the region above the trough line. Since the region above the trough line is the most active part of the water column, it is more appropriate to use a wave-following coordinate system for the analysis of near-surface turbulence measurements.

Models of near-surface turbulence rely on a flat surface, commonly positioned at the mean water level. Burchard (2001) stated that the virtual model origin z_org = z(0) should be “located at the base of the unresolved surface layer” at z_org = −z₀, where z₀ is the surface mixing length. However, the conversion from a wavy surface to a flat surface inherent in the model assumes a linear dissipation profile above the trough line. Based on our finding that the dissipation ratio in the trough to crest region is 2:5, we argue that the model origin should correspond to the location of the mean dissipation, which yields z_org = 0.24H_s, or approximately halfway between the mean waterline and the significant wave crests.

Figure 17a shows a comparison of two recent models with our observations. The models predict a mean dissipation profile that is a function of the friction velocity only. However, Greenan et al. (2001) found that the vertical profile of the mean dissipation is sensitive to the directional distribution of the wave energy. Only for a clear separation between swell and windwaves was the modeled dissipation profile consistent with their observations, which started approximately at 2-m depth and included measurements beyond the shallow mixed layer down to about 18 m. To minimize the effects of processes not included in the model, we limit the model– observation comparison to a case of limited swell effects and with u* approximately constant. Deployment II took place after several days of nearly steady wind, generating a unimodal wave field propagating from 320° with peak frequency f_p = 0.11 Hz, without lower frequency swell. The friction velocity during the deployment stayed nearly constant, allowing a comparison of the mean observed dissipation and the model output. Results are presented in nondimensional form |z|/H_s,εH_s/F, where F = cu²_w∗ is the input of turbulent kinetic energy at the model surface and c is the effective phase speed of waves acquiring energy from the wind (Gemmrich et al. 1994). The commonly applied Craig–Banner (1994) model in its modified version (Terray et al. 1999) and the recent generic length-scale model (Umlauf and Burchard 2003) are in reasonable agreement and consistent with the observed mean dissipation rate. The parameters for both models are the input of turbulent kinetic energy F = cu²_w∗, c = 1 m s⁻¹, and the surface mixing length z₀ = 0.2 m (Gemmrich and Farmer 1999a). Placing the model origin at the mean waterline results in modeled dissipation rates that, at the sensor depth, are 1/30–1/40 of the observed value. This supports our assumption of z_org ≈ H_s/4. A critical parameter of these models is the surface mixing length z₀. Here we have set z₀ = 0.2 m, based on the observed scale of temperature fine structures (Gemmrich and Farmer 1999a), which is significantly smaller than z₀ = 0.6H_s found by Soloviev and Lukas (2003) and z₀ ≈ H_s suggested by Terray et al. (1996). However for z_org = 0 the models do not reproduce the observed dissipation for any choice of z₀ (Fig. 17b) and neither for z_org = −z₀. In light of the strong depth dependence of dissipation it becomes clear that the assumption of a flat surface at the mean waterline cannot represent wave-induced turbulence. This holds in the near-surface area, at least to a depth much larger than the significant wave height.

Based on the modeled dissipation profile (Fig. 17) we estimate the total dissipation in the surface layer, D = ∫zm0 ε(z) dz, with mixed layer depth zm. Depending on the model used we find D = 6.5 × 10−4 m3 s−3 (Craig– Banner) and D = 8.8 × 10−4 m3 s−3 (Umlauf–Burchard). Very recently, Melville and Matusov (2002) estimated the energy dissipation in breaking waves from the speed c and distribution of whitecaps, D = ∫ c5Λ(c) dc, where Λ(c)dc is the average length of breaking crests per unit area. Analyzing aerial imaging of breaking waves they found that the measurements of Λ(c) collapse onto a single exponential curve if normalized by the cube of the wind speed. Using this result we find that the average total dissipation during deployment II is D = 9 × 10−4 m3 s−3. This is in good agreement with the model results and supports the assumption that near-surface energy dissipation is strongly linked to wave breaking.