Wide-field-of-view PIV measurements above and below microscale breaking waves

Since the wide PIV field of view is able to capture multiple waves at a time, we can follow the evolution and propagation of individual waves, and their associated airflow and water dynamics. In Fig. 4, we show a chronological sequence of air-side PIV fields (only the horizontal velocity component is shown) over a period of 0.82 s.

For clarity, we have marked one of the waves with a blue triangle during the entire time interval, and the following wave with a red triangle (starting at time \(t_1+0.27\) s).

The sequence of velocity fields shows that the waves significantly modulate the airflow. In fact, at time \(t_1,\) for example, the streamwise velocity is high above the crest of the blue-marked wave, and drops significantly past the crest, where it even becomes negative. The young, slow, strongly wind-forced wave is acting as an obstacle (such as a hill) and shelters the region past its crest. This phenomenon is known as a separated sheltering event, or airflow separation Jeffreys (1925); Belcher and Hunt (1998). In fact, we observe sheltering events (separated and non-separated) past the crests of nearly all the waves shown in this figure. In the cases where separation takes place, the airflow boundary layer detaches from the surface just past the crest of a wave and generally reattaches on the windward side of the next wave (see, for example, the blue-marked wave at times \(t_{1}\) and \(t_{1}\)+0.09 s). This observation is consistent with previous laboratory and field studies (e.g., Buckley and Veron 2016; Buckley et al. 2020; Buckley 2015). Here, however, thanks to the unprecedented width and resolution of the field of view, we are able to directly observe, instances where the sheltered region extends beyond the crest of the next wave, only to reach the windward face of the following one. This is apparent at \(t_{1}\)+0.18 s, where the blue-marked wave causes the airflow to “skip” the small wave just downwind of it. This is, to the best of our knowledge, the first direct observation of such an event.

This phenomenon is evidently highly intermittent; since an instant later (\(t_{1}+0.27\) s), the size of the sheltered region decreases significantly. At the same time, the red-marked wave, located upwind of the blue-marked wave, creates a large sheltered event, causing a large region of reduced velocities and increased variability. At \(t_{1}+0.36\) s, this sheltered region appears to merge with the sheltered regions above the two next waves, producing a large sheltering event that extends nearly up to the right edge of the field of view. This effect is short-lived, as it subsides 0.09 s later, at \(t_{1}+0.45\) s.

Then, from \(t_{1}\) + 0.54 s to \(t_{1}\) + 0.82 s, the blue-marked wave appears to be intermittently within the sheltered region caused by the red-marked wave. It should be mentioned that since we can follow individual waves over multiple images, the speed of an individual wave can be readily obtained from the provided sequence. Here, the blue-marked wave travels at a speed above 0.6 m/s at \(t_1\) and decelerates rapidly to reach approximately 0.3 m/s 0.82 s later. In fact, the height of the blue wave also decreases until the wave almost disappears at \(t_{1}\) + 0.82 s, yet capillary waves remain clearly visible near its crest. Therefore, the system offers the ability to not only examine the effects of air- and water-side dynamics on the development and breaking of gravity-capillary waves, but also the influence of wave groups and associated modulations of individual waves as they travel through individual wave packets.

Full size image

In order to evaluate the complex coupling mechanisms between wind stress, water-side turbulence, and microscale breaking events, it is important to resolve microscale roughness elements at the water surface. This is especially critical in the near crest region, where parasitic capillary waves are believed to interact with the air and water turbulence. In an effort to connect the observed airflow kinematics over individual waves (shown in Fig. 4) with their microscale surface geometry features, we now examine the ability of the imaging system to resolve surface slope and capillary waves along the dominant waves.

Full size image

As shown in Fig. 3 (Panels d,e), the raw LIF and PIV images resolve well the geometry of the water surface, including capillary waves near the crests of the dominant waves. In Fig. 5, we show an example LIF surface \(\eta (x)\) (panel a), and its associated slope \(\partial {\eta }/\partial {x}\) (panel b). Important slope variations are clearly visible near the crests of the dominant waves, where capillary waves are present. However, the slope signal is a function of both dominant and capillary wave geometry, rendering a systematic identification of capillary waves difficult. In order to systematically identify the presence, or absence, of parasitic (bound) capillary waves in the vicinity of the wave crests above (resp. below) which we measure air-side (resp. water-side) kinematics, an empirical mode decomposition (EMD) of the LIF surfaces is performed, following the methodology used by Xu and Perlin (2023). Thanks to a sifting process, the EMD can decompose the instantaneous surface \(\eta (x)\) into a sum of oscillatory components called intrinsic mode (IMF) functions (noted here \(\eta _{n}(x)\)), and a monotonic function (residue) r(x) (Zeiler et al. 2010):First, the envelopes of the primary signal \(\eta\)(x) are computed. Then, their means are subtracted from the original surface. To be validated as an IMF, the number of local minima and maxima of the resulting signal has to differ at most by one and its mean has to be zero. If these two conditions are not fulfilled, the sifting process continues with the previous signal as input signal. Otherwise, the IMF mode is stored. Finally, if the function resulting from the subtraction of the IMF mode from the input signal is monotonic, it is identified as the residue (r(x) in Eq. 1) and the algorithm stops. Otherwise, the procedure restarts with this function as the input signal (Zeiler et al. 2010). Using this method, we are able to subtract the capillaries (highest frequency signal which corresponds to the first mode of the EMD, see Fig. 5c), from the primary waves (contained in the remaining lower frequency modes, shown in panel d).

Full size image

In Fig. 6, we show the first IMF corresponding to the surface of the first three air PIV snapshots from Fig. 4. At \(t_{1}\), the first and third waves seem to be steep enough to generate a capillary wave train past their crests. These are represented by the high-amplitude oscillations in the first mode of the EMD. The small amplitude variability is assumed to be noise.

At \(t_{1}\) + 0.09 s, airflow separation still occurs past the crest of the first wave (Fig. 4) but no capillaries are detected past the crest (Fig. 6). This is also the case past the last dominant wave at \(t_{1}\) + 0.09 s. This observation points to the complexity of airflow separation dynamics over wind waves. While past studies have suggested that airflow separation can only take place over breaking waves (Banner and Melville 1976; Gent and Taylor 1976), more recent measurements have shown airflow separation events over microscale breaking waves, (Buckley and Veron 2017; Buckley et al. 2020).

As described in the Methods section above, water-side PIV measurements were achieved in the same conditions as the air-side PIV. In Fig. 7, we show a sequence of wide-field-of-view water-side PIV snapshots (horizontal velocity component), taken within a time interval of 0.73 s. We observe that the orbital velocities of the dominant waves dominate the velocity field, with positive horizontal velocities measured below crests, and negative velocities below troughs. Here, we follow the evolution of the highest, steepest wave (labeled by a blue triangular marker). This wave, traveling on average at 0.55 m/s, displays high velocity values (\(\sim 0.5\) m/s) just below its crest. From \(t_{2}\) to \(t_{2}\) + 0.36 s, the wave grows and the horizontal velocity below its crest gradually increases.

At \(t_{2}\) + 0.45s, the wave is skewed to the right and there is a bulge, prior to the formation of the capillaries. This instant is further investigated in Section 3.2.2.

From \(t_{1}\) + 0.54s to \(t_{1}\) + 0.73s, as the wave evolves to the end of the field of view, parasitic capillary waves are generated. Strong positive and negative horizontal velocity patches are visible under the capillaries, an indicator of a strong mixing in this particular region. This observation is in general agreement with past numerical, observational and theoretical studies of steep, microscale breaking waves (Deike et al. 2015; Ebuchi et al. 1987; Longuet-Higgins 1992). We note here that the wave upwind of the blue-marked wave gradually steepens with time, while the wave downwind becomes flatter. This is likely due to the modulation of these deep water wind waves by the wave group, whereby individual waves steepen, (microscale) break, and flatten as they travel through a group. However, as shown above, these changes in wave geometry are usually coupled with rapidly varying, turbulent sheltering events, which may in turn control how momentum is partitioned between turbulent, wave-induced, and viscous contributions, to be ultimately transferred to the waves and surface currents (Buckley et al. 2020).

In Fig. 8, we focus on the instantaneous airflow velocity field, acquired above a train of 3 wind waves. As in the sequence of velocity fields from Fig. 4, the horizontal velocity field shown here (Panel a) displays clear signs of sheltering past the crests of all three waves. In fact, the vertical velocities (Panel b) are largely consistent with sheltering dynamics. Over the first, steeper wave, the airflow broadly moves upward upwind of the crest, and downward downwind, but with increased variability near the surface on the leeward side, i.e., within the sheltered region. Past the second wave and up to the right-hand edge of the snapshot, the vertical velocities show important variability, possibly caused by the pronounced airflow separation event over the first wave. Nonetheless, the following waves also modulate the airflow, as upward motion can be observed just before the crest very near the surface, as well as alternating upward–downward motions within the sheltered regions of the second and third waves.

The observation of airflow separation over the first, steep wave is in general agreement with past measurements with narrower fields of view (Buckley and Veron 2017). Here, however, thanks to the wide field of view, we can demonstrate that the high variability observed within the right-hand (downwind) portion of the field is influenced by the airflow separation event taking place above waves farther upwind. This suggests that the position of a wind wave within the group likely has an impact on the wind-induced momentum and energy fluxes into that wave. The instantaneous spanwise vorticity (Panel c), a Galilean invariant, unequivocally confirms that all three waves observed here are causing airflow separation: a high (negative) vorticity layer is present on the windward side of the waves, which is characteristic of an attached boundary layer. Past the crest of the waves, this high vorticity layer detaches from the surface, then disintegrates, and is finally regenerated on the windward face of the next wave. After the separation point, close to the surface, the vorticity is either considerably reduced or is even slightly positive in the region over which airflow reversal happens. The detached, free high-vorticity layers disintegrate into high-magnitude negative and positive vortices, which are shed away from the top of the wave crests.

Reul (1998) suggested that the vortices caused by airflow separation events can be advected several wavelengths downwind of mechanically generated (spilling) breaking waves that they studied. Here, our wide field of view allows us to directly observe that vorticity structures caused by airflow separation over wind waves can also extend farther downwind than 1 wavelength, and that the created wakes can even overcome the following waves, directly influencing the airflow structure above them. This is particularly visible if one follows the negative vorticity layers ejected away from the surface, past the crest of the first wave.

In order to compute turbulent and wave-coherent velocities, a triple decomposition of instantaneous velocities was performed. The instantaneous horizontal velocity component can be written as \(u=\langle u \rangle +u'=\bar{u}+\tilde{u}+u'\), where \(\langle u \rangle\) is the phase averaged velocity field, \(\bar{u}\) the mean velocity profile, \(\tilde{u}\) the wave-induced component and \(u'\) the turbulent velocity. Phase averages are produced the same way as described by Buckley and Veron (2016) and Tenhaus et al. (2024): First, a Hilbert transform of the surface data, \(\mathcal {H}\{\eta (x)\}\), is computed for each wide-field-of-view frame, withwhere P.V is the Cauchy principal value of the integral. This is done using MATLAB’s hilbert function, which makes use of the fact that the Fourier transform of the Hilbert transform of a signal is equal to two times its Fourier transform (e.g., Melville 1983, their equations 2.1\(-\)2.8). The phase domain (from 0 to \(2\pi\)) is divided into 72 bins. Data corresponding to a specific phase are then averaged together (conditional averaging). The algorithm is run for both surface elevation data and velocities. For this study, averages were performed over 660 2D PIV velocity fields for both air and water cases. Each 2D velocity field consists of over 2.7 million vectors.

The resulting wave-induced velocities (horizontal component, Fig. 8d) reflect the wave-induced modulations of the airflow, with alternating positive and negative patterns. Close to the surface, the velocity contours are tilted downwind and are positive above the windward side of crests and negative above the leeward side. Away from the surface, regions of positive (resp. negative) \(\tilde{u}\) are found above crests (resp. troughs). This pattern is in good agreement with past laboratory measurements (Buckley and Veron 2016, 2019) and numerical simulations (Sullivan et al. 2000).

The instantaneous turbulent flux field \(-u'w'\) (Fig. 8e) carries the signature of the airflow separation taking place past the crests of the three waves. The positive and negative velocity perturbations within the sheltered region are significant in that they exceed the mean, wave-induced velocity reduction that takes place on average downwind of crests.

Intense negative and positive turbulent stresses (Fig. 8e) are visible above the leeward side of the waves: The airflow separation enhances the production of turbulent shear. This result is in general agreement with past observations of airflow kinematics over single wave crests (Tenhaus et al. 2024; Buckley and Veron 2016, 2017). Here, with a spatial snapshot of turbulence over 3 consecutive waves, we can directly observe the substantial spatial variability of the turbulence, from one wave to another. We observe intense turbulence above the lee of the first, steepest wave, followed directly by reduced turbulence over the small, second wave, and again an important increase on the lee of the third wave. These phase-locked features point toward the likely important role of turbulent (sheltering) events for energy transfer between wind and waves (Belcher and Hunt 1998; Buckley et al. 2025).

The surface viscous stress (or viscous momentum flux, Fig. 8f) is given bywhere µ is the air or water dynamic viscosity, and \(u_t\) and \(w_n\) are the velocity components projected in the \(\zeta\) and \(\xi\) directions, which are, respectively, tangential and normal to the instantaneous surface \(\eta (x)\) (Buckley and Veron 2016). Here, only the first measured velocity vectors within the viscous boundary layer were used, and the results were smoothed via a cubic spline method.

We observe important along-wave modulations of the viscous stress. It reaches a maximum slightly upwind of the wave crests and a minimum downwind of the crest. The viscous stress even reaches negative values downwind of the first and third waves, where air flow reversal was observed (Panel a). This means that at those locations of the surface, momentum is transferred from the water to the air. While the observed pattern agrees well, once again, with previous measurements (Tenhaus et al. 2024; Buckley et al. 2020), the wide field of view allows us to begin looking at the influence of neighboring wind-wave kinematics on individual waves. Here, we note that the second (smaller) wave experiences intense viscous stress just windward of its crest, in spite of being less steep and less high than its upwind neighbor, for example. A possible explanation is the intense downward sweep of high velocity airflow just downwind of the airflow separation event past the first wave’s crest (Panel a).

In Fig. 9, we examine in detail the water-side dynamics observed at \(t_2\) + 0.45 s (from Fig. 7). The instantaneous horizontal (Panel a) and vertical components (Panel b) of the velocity are broadly consistent with orbital motions of linear waves (e.g., Kundu and Cohen 2004). Both plots show that the magnitude of these motions is high near the surface but rapidly decreases with increasing depth. The wave-coherent velocity contours (Fig. 9d) further support this agreement, with positive \(\tilde{u}\) below crests, and negative values below troughs.

The relatively steep wave in the middle of the field of view has similar geometrical characteristics as the airflow separating waves observed above. Therefore, we assume that this wave is also causing airflow separation. In fact, the wave exhibits a bulge at its crest and is skewed to the right.

This bulge (or toe) is likely a source of instabilities according to the theoretical work of Longuet-Higgins (1992) and precedes the generation of parasitic capillaries (see also again Fig. 7, at time \(t_2\) + 0.54 s, i.e., 0.09 s later, where capillaries are indeed observed). Underneath the crest, the horizontal velocity magnitude is particularly strong.

The vorticity map (Fig. 9c) is somewhat noisy, especially on the left side of the field of view. On the right side of the image, 2 lines of positive and negative vorticity tilted to the right are assumed to be due to artifacts produced by laser light reflections.

The crest of the steep, middle wave is dominated by negative vorticity, except below the forward face of the toe, where a small region of high positive vorticity is detected. Directly behind this area, strong negative vorticity is being produced and seem to diffuse along the crest to the windward side of the wave. Xu and Perlin (2023), in their study of small, mechanically generated waves, also observed vorticity below capillaries, in particular strong negative vorticity below troughs, and weak positive and negative vorticity below the crest of the parent wave. Here, the observed positive vorticity is comparatively somewhat shifted downstream and much more intense. Negative vorticity is prevalent below the leeward side of the wind wave shown here, even though capillaries have not yet formed. While the vorticity below the capillaries of mechanically generated waves of Xu and Perlin (2023) was advected downstream, here, the vorticity seems to propagate upwind, suggesting that additional effects due to the wind drift layer are probably modifying the underwater dynamics. In addition, thanks to our ability to track vorticity structures over the width of our field of view, we can directly observe that vorticity produced under a wave crest does not extend beyond one dominant wavelength, as was hypothesized by Siddiqui and Loewen (2010). This is in contrast with the air side, where airflow separation can influence the airflow well beyond one wavelength (see again Fig. 8c).

Below the crest of the steep middle wave, we observe regions of intense turbulent stress (Panel e). Just past the bulge, a small region of intense negative turbulent stress is visible, which decays rapidly with depth. Intense positive turbulent shear stress is found along and below the upwind face of the crest, and up to the end of the bulge.

We note that the intense vorticity and turbulence observed under the steep wave’s crest confirm that this wave is microscale breaking. As mentioned above, this was also suggested by the appearance of parasitic capillaries an instant later.

The derived surface viscous stress (Fig. 9f) is consistent with the air-side observations, being high below the crests and low below the troughs. The peak is particularly visible below the toe, likely because of the high shear region present here. The viscous stress notably drops below zero downwind of the bulge, an indicator that airflow separation is likely taking place over this microscale breaking wave.