2.1. Experimental campaigns

Three separate experimental campaigns were conducted between 2020 and 2023. Experiment 1 was also reported in Smeltzer et al. (Reference Smeltzer, Rømcke, Hearst and Ellingsen2023), where the focus was on the change in turbulent enstrophy and wave scattering.

The three experiments investigate essentially the same phenomenon, but with fundamental differences in experimental design and the nature of the acquired measurement data. A summary is provided in table 1.

2.1.1. Experiment 1: wave groups

Cases 1.A-1.D in tables 1–3 are from Experiment 1. Wave groups were generated that propagated upstream atop the turbulent flows. The water depth $h$ was $0.4$ m. The velocity field was measured using stereoscopic particle image velocimetry (SPIV), measuring all three velocity components in a plane perpendicular to the mean flow located a distance $8.38$ m downstream from the active grid (i.e. $83.8$ grid units), and $L_{\textit{FOV}}=7.38$ m downstream of the trailing edge of the surface plate. Two 25-megapixel cameras were mounted on either side of the test section, viewing the field of view at $\pm 45^\circ$ to the $x$ -axis as shown in figure 1 b). The field of view was $0.12\times 0.14$ m $^2$ . Particle images recorded by the two cameras were processed using a final pass of $48\times 48$ pixel interrogation window and a 50 % overlap, resulting in a velocity vector spacing of about $0.8$ mm. The free-surface intersection with the SPIV plane was detected from laser-induced fluorescence (LIF) images recorded by a camera viewing the plane at an oblique angle from the air side. A small amount of rhodamine-6G was added to the water generating image contrast between the air and water regions, and the free surface was detected from the image intensity gradient. Further details can be found in Smeltzer et al. (Reference Smeltzer, Rømcke, Hearst and Ellingsen2023).

Table 3.

Wave quantities derived from measured values in table 2. Here, $\lambda _0=2\pi /k_0$ is the carrier wavelength, $c(k_0)=\sqrt {g/k_0}$ is the carrier-wave phase velocity, $u_s(0)=(ak_0)^2c(k_0)$ is the Stokes drift velocity of the carrier wave at the surface, $T_0=\lambda _0/c(k_0)$ is the intrinsic wave period, $\tau _0$ the intrinsic group width from (3.2) and $u_{{rf}}$ is the Eulerian return flow from (4.2).

For each wave group, SPIV/LIF measurements taken during three time intervals were used: well before the group arrived, and at the leading and trailing edges of the group envelope, referred to as intervals 1, 2 and 3, respectively, as shown in figure 2 b). We consider the difference in mean velocity between intervals 1 and 3 here, with interval 2 as a check to verify that the change is indeed due to waves. Values for all three intervals in all cases can be found in the Supplementary Materials available at https://doi.org/10.1017/jfm.2026.11163. The duration of the measurements for each interval, $T_{\textit{PIV}}$ , was $10$ s and PIV and LIF were sampled at $f_{\textit{ac}}=8$ Hz. After each group, residual waves from reflections were allowed to dissipate for approximately five minutes before the next wave group was generated. The above procedure was performed a total of $N_{\textit{ens}}=60$ times to produce ensemble statistics, except for the case 1.C.2 which was performed $20$ times. In case 1.B, only vertically orientated bars of the active grid were actuated. For case 1.A, the grid was stationary with the wings aligned with the flow in the position of least blockage (see also Appendix A). The experimental conditions are listed in table 1.

Figure 2.

(a) Example surface elevation of a single wave group from Experiment 1 as a function of time $t$ , measured by a wave probe at the measurement location. (b) Example from Experiment 1 of an ensemble-average group surface elevation amplitude envelope as a function of time normalised by the measured group temporal width $\tau$ for case 1.D (see (3.1)). The time intervals for SPIV measurement (1–3) are shown with vertical dashed lines. (c) Surface elevation measurements of one ensemble from Experiment 3 (cases 3.A and 3.B), which shows the onset of a regular-wave train. The red box indicates the interval used for analysis.

The wave groups were generated with the wavemaker motion having carrier frequency $f_0 = 1.02$ Hz and a Gaussian amplitude envelope of the form

(2.1) \begin{equation} S(t) = S_0 \exp\left [-\frac {(t-T_{{wm}}/2)^2}{2\tau _{{wm}}^2}\right ]\!, \end{equation}

for $0\leqslant t\leqslant T_{{wm}}$ , where $S_0$ is the peak stroke, $\tau _{{wm}}= 6\,{\textrm{s}}$ is the group width in time as applied to the wavemaker signals and $T_{{wm}} =24\,{s}$ was the duration over which the wavemaker plungers were actuated. The surface elevation for one wave group measured at the SPIV measurement location is shown in figure 2(a).

3.1. Flow and wave characteristics

The measured physical characteristics of mean flow, turbulence and waves are listed in table 2, and some derived quantities we will make use of are quoted in table 3 for convenience.

An assumption of deep water is well satisfied since in all our cases $k_0h\gtrsim 3.6$ hence $1\gt \sqrt {\tanh (k_0h)}\gt 0.999$ . Here, and henceforth, $U_0$ is the mean absolute surface speed in the absence of waves (i.e. the mean surface velocity is $\boldsymbol{U}_0=(-U_0,0,0)$ ) and in all our cases the velocity profile is sufficiently depth uniform that vertical shear affects wave dispersion negligibly (e.g. Ellingsen & Li Reference Ellingsen and Li2017). The wavenumber $k_0$ and the wavemaker carrier frequency $f_0$ are related approximately by $2\pi f_0 = \sqrt {gk_0} {-} U_0k_0$ , or equivalently $f_0 = (c(k_0){-}U_0)/\lambda _0$ with $c(k_0)$ from (1.2) and the wavelength is $\lambda _0=2\pi /k_0$ . We will use the measured (as opposed to derived) value of $\lambda _0$ and the wavenumber $k_0$ accordingly. Values for these quantities are listed in tables 2 and 3, respectively. The time it takes for the waves to propagate from wavemaker to the edge of the surface plate is around $40$ s for Experiment 1, $25$ s for cases 2.A.1 and 2.B.1, $56$ s for cases 2.A.2 and 2.B.2 and $35$ s for Experiment 3.

The root-mean-square (r.m.s.) of the turbulent velocity fluctuation after subtracting the average is defined as $\boldsymbol{u}_{\textit{rms}}=[u_{\textit{rms}},v_{\textit{rms}}, w_{\textit{rms}}]$ representing the streamwise, spanwise, and vertical components, respectively. The turbulent kinetic energy is as $e=({1}/{2}) |\boldsymbol{u}_{\textit{rms}}|^2$ . In cases 2.A–3.B, only $u_{\textit{rms}}$ and $w_{\textit{rms}}$ are available, so we use instead $e=({1}/{2}) (u_{\textit{rms}}^2+2w_{\textit{rms}}^2)$ , an assumption of cross-plane isotropy, as is the standard convention for grid turbulence (e.g. Comte-Bellot & Corrsin Reference Comte-Bellot and Corrsin1966; Lavoie et al. Reference Lavoie, Burattini, Djenidi and Antonia2005) and was observed for our facility by Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021). The assumption is not quite satisfied for our flow as table 2 shows for Experiment 1. However, given the inevitable anisotropy of the flow as the free surface is approached (also with no waves) and consequent depth variation of $e$ closer to the surface than about one integral scale (the blockage layer, see, e.g. Aarnes et al. Reference Aarnes, Babiker, Xuan, Shen and Ellingsen2025), representing turbulent kinetic energy (TKE) by a single number is always a simplified picture. The precise value of $e$ does not significantly affect our conclusions, and given that no simple expression for $e$ in terms of $u_{\textit{rms}}$ and $w_{\textit{rms}}$ can reconcile all cases in Experiment 1, we choose to go with the conventional expression.

The mean velocity $U_0$ was calculated from averaging all streamwise velocities over the lower part of the field of view; averaging was performed over $-12$ cm $\lt z\lt -5$ cm for Experiment 1 and $-20$ cm $\lt z\lt -10$ cm for Experiments 2 and 3. Full profiles of the Eulerian-mean velocity profiles for all experimental cases may be found in the Supplementary Materials.

When comparing turbulent quantities for the various cases one should bear in mind that these are measured at a fixed position in space, whereas the turbulence becomes gradually weaker as it travels downstream because of dissipation. The change in Eulerian-mean current is an integrated effect of wave–turbulence interactions that occurred upstream, where the turbulence intensity is in general a little higher. This is true of all cases, but because of the lower mean velocity, the turbulence that reaches the field of view in cases 3.A and 3.B has decayed for longer. A detailed study of turbulent decay in our lab with similar flow conditions as Experiment 1 was reported by Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021). They found that for all cases, once the turbulence is fully developed TKE decays approximately as $\sim 1/x$ downstream for all cases. Thus, although the absolute values of TKE and Reynolds stresses are different at different positions, their relative magnitudes are expected to be constant. We consider only trends with respect to TKE, and ratios between Reynolds stresses which it seems reasonable to assume are well captured.

In previous experiments by Nepf & Monismith (Reference Nepf and Monismith1991) and Klopman (Reference Klopman1994, Reference Klopman1997), secondary motion in the form of a pair of streamwise rolls were measured when waves were propagated on a current, triggered by the CL2 mechanism. We could measure the cross-plane flow in Experiment 1, and while some weak in-plane mean flow was observed, of the order of $1$ cm s⁻¹ or less, the pattern was not consistent with a coherent pair of rolls. The resulting convection, in magnitude and direction, would be insufficient to convect boundary-layer turbulence into the near-surface regions where turbulence and waves interact to any noticeable degree. During the travel from grid to measurement position, the vortex force near the boundaries has far less time to accelerate a similar amount of water as in Klopman (Reference Klopman1994) and would likely not have time to develop. It is not perhaps so surprising that secondary flows would be different given that geometrical size and aspect ratios are rather different, and the current’s travel time from grid to measurement point is less than half that in either of the mentioned flows even for our slowest flow in Experiment 3.

It was found by Smeltzer et al. (Reference Smeltzer, Rømcke, Hearst and Ellingsen2023) that the turbulent integral length scale, characteristic of the largest turbulent structures, is important for wave–turbulence interactions, especially so for the scatting of waves by turbulence. An advantage of our active-grid turbulence generation is that the integral scale can be varied independently of TKE. The most pertinent integral scale is related to correlation of streamwise turbulent velocity for points separated in the streamwise direction, $L_x^x$ . Due to differences in the way the experimental data were acquired, no single method for estimating $L_x^x$ can be applied to all cases. Estimating the integral length scales from experimental data uniquely and quantitatively is notoriously difficult, and the various methods in common use produce quantitatively different results. Additional challenges pertain to active-grid turbulence due to the slow spatial decay of the autocorrelation function (Puga & LaRue Reference Puga and LaRue2017; Mora et al. Reference Mora, Muñiz Pladellorens, Riera Turró, Lagauzere and Obligado2019). In Experiment 1, the integral scale was estimated with a zero-crossing method as described by Mora & Obligado (Reference Mora and Obligado2020), and we used the same method to calculate the integral scale for Experiment 3, as listed in table 2. Since the data in Experiment 2 have low time resolution the same method cannot be applied there, and using another, spatially based method would not give directly comparable numbers, so we provide no integral scale for Experiment 2. We note that integral scales are significantly shorter in Experiment 3 than in Experiment 1 at similar turbulence levels, which can be explained by the lower mean-flow velocity $U_0$ .

Several practical aspects in evaluation of the wave and flow characteristics reported in table 2 differed for the three experiments, and are described in further detail below.

3.1.1. Wave group measurements (Experiment 1)

The mean flow and turbulence statistics without waves presented in table 2 were evaluated over the first interval, where any influence from waves can be assumed to be negligible. The mean flow in the negative $x$ -direction had approximately constant (absolute) value $U_0$ , found from averaging as described above. A small spanwise and vertical mean velocity was found, everywhere below $1$ cm s⁻¹, mostly much less, which we attribute to the channel flow conditioning not being perfect (achieving a perfectly straight and uniform flow in a channel of our size is very challenging). Notably, unlike the streamwise velocity, the cross-plane velocities did not change after the passage of the wave group, which implies that there is negligible contribution secondary flow due to wave–current interactions near the sidewalls mentioned above.

The characteristic peak wave amplitude $a_{{p}}$ and the observed group temporal width $\tau$ were estimated from the ensemble-averaged amplitude envelope of the wave groups as measured by the probes near the SPIV/LIF laser sheet as seen in figure 2(b). The average envelope was fitted to a Gaussian function of the form

(3.1) \begin{equation} a(t) = a_{{p}} \exp\! \left [-\frac {(t-t_{{p}})^2}{2\tau ^2}\right ]\!, \end{equation}

with $t_{{p}}$ the temporal location of the group peak. The peak wave steepness is $k_0a_{{p}}$ .

We define an intrinsic temporal group width $\tau _0$ , listed in table 3 as

(3.2) \begin{equation} \tau _0 = \tau \frac {c_g(k_0)-U_0}{c_g(k_0)}, \end{equation}

with $c_g$ defined in (1.2). The intrinsic temporal group width is expressed in a reference frame without mean flow and reflects the time scale during which the ambient turbulence interacts with the wave groups.

3.1.2. Measurements with regular waves (Experiments 2 and 3)

The ambient flow statistics were evaluated from PIV measurements acquired without waves. Similarly to Experiment 1, the mean-flow profile varied only slightly across the measurement plane, and a representative absolute value $U_0$ is given in table 2.

The wave steepness $ak_0$ was calculated using the average wave amplitude from the wave probe measurements in the proximity of the PIV measurement location. The amplitude of each individual wave oscillation varied slightly during the experiments, especially in cases with the highest level of turbulence, likely due to wave–turbulence interactions (e.g. Smeltzer et al. Reference Smeltzer, Rømcke, Hearst and Ellingsen2023). The variation is not of central interest to the present study, and thus only the mean steepness value is reported.

The regular waves were present throughout the entire test section during the experiments. The grid-generated and measured turbulence thus interacted with the waves over a length $L_{\textit{FOV}}$ as given in §§ 2.1.2 and 2.1.3, with associated interaction time $T_{\textit{int}}=L_{\textit{FOV}}/U_0$ . The frequency at which turbulence encounters waves, i.e. as seen by the moving flow, equals the intrinsic wave frequency (in Hz)

(3.3) \begin{equation} f_{\textit{enc}} = f_0 + U_0/\lambda _0 = \sqrt {gk_0}/2\pi. \end{equation}

3.2. Measured change in Eulerian-mean velocity

The measured changes in Eulerian-mean velocities presented below are of the order of millimetres per second, yet, while this is the same order of magnitude as our PIV measurement accuracy, one should bear in mind that the variance of an average from hundreds of independent measurements is far lower than that of single measurements. It is crucial that a careful analysis of errors and statistical convergence be performed in order to establish confidence that our results are accurate and reliable. In Appendix B, we report results of these tests, supported by further data in the Supplementary Materials.

3.2.1. Velocity change after the passage of wave groups (Experiment 1)

We now consider the measured Eulerian-mean velocity change for Experiment 1, i.e. cases 1.A–1.D. Vertically sheared flows in our laboratory have been found to be stable and therefore the wave-modified Eulerian-mean velocity profile remains close to unchanged throughout interval 3 (also found to be true for strongly sheared currents, see § 4 of Pizzo et al. Reference Pizzo, Lenain, Rømcke, Ellingsen and Smeltzer2023). In figure 3(a) we show as an example the mean streamwise velocity profiles for case 1.D during the three measurement intervals, denoted $U_I(z)$ for intervals $I = \{1,2,3\}$ (see figure 2 b). As can be seen, the mean-flow speed had increased in the direction opposite to wave propagation after the passage of the wave groups. Similar plots for all cases are provided in the Supplementary Materials. Error bars are omitted from the figure for reasons of visibility, but discussed in Appendix B. The net Eulerian-mean-flow change $\Delta U=U_3 {-} U_1$ is shown in figure 3(b) for flow cases 1.A–1.D as expressed in the legend. For all cases, $\Delta U\lt 0$ near the surface, and decays to small absolute values at depths $|k_0z|\gtrsim 1$ .

Figure 3.

Change in Eulerian-mean current due to the passage of a wave group. The waves travelled in the positive $x$ -direction, against the current. (a) An example of mean streamwise velocity depth profile before the arrival of a wave group, $U_1(z)$ , and after the group has passed, $U_3(z)$ , here for case 1.D; (b) mean streamwise velocity difference $\Delta U = U_3-U_1$ as a function of depth for the flow cases of Experiment 1. Error bars are omitted for visibility – see analysis in Appendix B; (c) the slope of $\Delta U(z)$ relative to the Stokes drift gradient (a prime denotes derivation with respect to $z$ ). Light smoothing (moving average with window size $8$ mm) was applied to the curves in panel (c) for better visibility.

Interestingly, $\Delta U$ clearly depends on the level of ambient turbulence; while the waves have approximately equal properties in cases 1.A, 1.B, 1.C.1 and 1.D, the velocity change is far higher at the highest turbulence level (case 1.D) than at the lowest level (case 1.A), with the intermediate cases 1.B and 1.C.1 in between. Moreover, comparing cases 1.C.1 and 1.C.2 where two wave steepnesses are tested on the same turbulent current indicates a positive correlation between $k_0a_{{p}}$ and current change $\Delta U$ . Put together, the results in figure 3(b) provide a strong indication that the change in current is due to an interaction between waves and turbulence.

We can exclude the possibility that the measured $\Delta U$ is due to the Eulerian return flow under groups of waves, as measured by van den Bremer et al. (Reference van den Bremer, Whittaker, Calvert, Raby and Taylor2019). The return flow follows the group, i.e. it is very weak in intervals 2 and 3 which lie outside the main group itself. It is also near-uniform in depth, whereas the current measured in figure 3(a) is strongly depth dependent.

It is instructive to plot the ratio between ${\rm d}(\Delta U)/{\rm d} z$ and $-{\rm d} u_s/{\rm d} z$ as a function of $z$ , shown in figure 3(c), because it gives some indication of how far the turbulent flow has transitioned towards a new, quasi-equilibrium state. We will later present theory and evidence that at the end of the ‘spin-up’ period, this ratio should, in the final state, be approximately equal to $u_{\textit{rms}}^2/w_{\textit{rms}}^2$ nearest the surface. Since our bulk turbulence is slightly anisotropic (in the cases reported in Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021), $1.2\lesssim u_{\textit{rms}}^2/w_{\textit{rms}}^2\lesssim 1.4$ ), a final current change $|\Delta U|\gtrsim |u_s|$ is expected nearest the surface. We shall later see that in the regular-wave cases where the equilibrium is likely reached, this relation holds well. Since none of the changes in currents in cases 1.A–1.D are close to reaching these values, it seems that the passing of the wave group has not led to a wave–turbulence interaction of sufficient duration for a final state to be reached, and the flow is still relatively early in the ‘spin-up’ stage.

There are at least two striking observations to make in figure 3(c). First, with the exception of the low-turbulence case, 1.A, the ratio between the slopes is close to constant with depth, which illustrates that $\Delta U \sim \exp (2k_0z)$ near the surface in these cases. The scaling is not perfect, particularly at the shallowest depths; this should not be surprising since the depth dependence of wave–turbulence interaction should scale not only with the wavelength, but also the turbulent integral length scale, which delimits the vertical extent of the topmost layer where the kinematic boundary condition at the surface begins to limit the vertical extent of turbulent eddies (the blocking effect, see, e.g. Teixeira & Belcher Reference Teixeira and Belcher2002). Second, while the value of $\Delta U(z)$ after the passing of a group depends strongly on the turbulence level and steepness, there is no such trend for the relative slope ( $-\Delta U^{\prime }(z)/u_s^{\prime }(z)$ ), case 1.A excepted.

In § 4.4 we will develop a RDT model describing the early onset of wave–turbulence interaction, which we can compare with the measurements in figure 3(b), given this evidence that the combined wave/turbulence flow is still far from fully developed in Experiment 1.

In conclusion, the evidence suggests that the change in Eulerian-mean current observed in our experiments is due to wave–turbulence interaction, and increases with increasing turbulence and increasing wave steepness.

3.3. Velocity change under regular waves (Experiments 2 and 3)

Cases 2.A–3.B all consider turbulence interacting with regular (i.e. continuous and periodic) waves. For cases 2.A.1–2.B.2, we evaluate the mean streamwise velocity in the presence of waves, and subtract off the mean velocity profile from the ambient flow case without waves. This velocity difference we define as $\Delta U(z)$ . Although the flow is wavy, the time series is long enough for the Eulerian-mean current, averaged over all $2000$ PIV images, to be well converged in the sense that further measurements would affect it insignificantly. See details in Appendix B.

The turbulent current is affected by the waves during the time it takes it to traverse the test section, a distance $L_{\textit{FOV}}$ as defined in §§ 2.1.2 and 2.1.3. The interaction time is thus $T_{\textit{int}}=L_{\textit{FOV}}/U_0$ , approximately $29$ s for Experiment 2 and $46$ s for Experiment 3, the latter having a slower flow speed. This corresponds to a number of wave cycles, $T_{\textit{int}}/T_0,$ between $36$ (cases 2.A and 3.A) and $83$ listed in table 4. Due to the slower flow speed. Cases 3.A and 3.B interact for considerably longer than 2.A and 2.B, both in terms of absolute time (in seconds) and in terms of number of intrinsic wave periods $T_0$ , i.e. $T_{\textit{int}}/T_0$ is the number of wave cycles that the turbulence has encountered.

Table 4.

Derived wave quantities for use in RDT analysis. For cases 1.A–1.D, $T_{\textit{int}}=\sqrt {\pi }\tau _0$ is used with $\tau _0$ from (3.2), while for cases 2.A–3.B, $T_{\textit{int}}=L_{\textit{FOV}}/U_0$ ; $\beta _{{f}}(0)$ is found from (4.28) and (4.29) for groups and regular waves, respectively. Note that it is related to $T_{\textit{int}}$ via (4.33).

In figure 4 we show $\Delta U(z)$ for cases 2.A–3.B, where the different wave steepness values $ak_0$ are labelled in the legend. Several characteristic behaviours can be observed. First, all graphs have similar dependence on depth, with the highest absolute value nearest to the surface, decreasing down to a level where $k_0z$ is roughly in the range between $-1$ and $-2$ , then turning and slowly becoming more negative again. (The non-monotonicity (turning) would not be visible in figure 3, where measurements were only made for $k_0z\gt -1.2$ .) Indeed, some cases see the induced Eulerian-mean current take positive values over a certain depth range. (One may note that the current profile is similar in shape and magnitude to those of Rashidi et al. Reference Rashidi, Hetsroni and Banerjee1992). Second, there is again a clear tendency that higher steepness leads to a larger vertical variation of $\Delta U(z)$ . Finally, there appears to be a near-constant offset between graphs (an addition to the mean flow), which increases with steepness, and we will find in § 4.1 that it can be explained, at least in part, as the approximately depth-uniform Eulerian return flow $u_{{rf}}$ .

Figure 4.

The wave-induced current $\Delta U$ under regular waves as a function of depth $k_0z$ for cases 2.A.1, 2.A.2, 2.B.1, 2.B.2, 3.A and 3.B in panels (a)–( f), respectively. For each case, different wave steepness values $ak_0$ are shown as indicated in the legend. The dashed lines are the theoretical Stokes drift profiles at the same location for each case, shown as $-u_s(z)$ , that is, with opposite sign to the Stokes drift. The filled circles at $k_0z=-4$ and $0$ indicate the theoretical value of the Eulerian return flow, $u_{{rf}}$ .

The reverse of the theoretical Stokes drift $-u_s(z)$ from (1.1) is plotted in all panels of figure 4 with dashed lines of the same colour for comparison, calculated for each case. For all cases we note that $|\Delta U|\lt u_s$ nearest the surface, but that the depth variation ${\rm d} |\Delta U|/{\rm d} z\sim {\rm d} u_s/ {\rm d} z$ , is higher in some cases, smaller in others. We will return to these points later when comparing the results with theory.

To further illustrate how the Eulerian-mean-current change depends on wave steepness, we plot the value of $\Delta U$ at a set reference depth as a function of steepness $ak_0$ in figure 5. The value at the shallowest depth available for all cases is used, i.e. $k_0z=-0.27$ . For cases 2.A.1 and 2.B.1, $\Delta U$ appears to scale as $(ak_0)^2$ , indicated by the dashed line. Cases 2.A.2 and 2.B.2 do not adhere to the scaling, possibly due to the high wave steepness values involved, so that interactions of order $(ak_0)^3$ and higher may become significant. A further possible explanation is the lack of scale separation: the wavelength is likely only slightly larger than the integral length scale in these two cases (based on comparison with cases 1.A–1.D in table 2, which have similar $U_0$ – unfortunately, a direct comparison of $L_x^x$ is not possible, as explained). The comparatively large integral length scale and high turbulence levels mean that stronger angular scattering of waves on turbulent velocity changes is to be expected (Villas Bôas & Young Reference Villas Bôas and Young2020; Smeltzer et al. Reference Smeltzer, Rømcke, Hearst and Ellingsen2023), a process which does not scale with steepness. Figure 5 should be interpreted only qualitatively, since the value of $\Delta U$ at a constant value of $k_0z$ is not entirely comparable between cases with different turbulence properties. As discussed in connection with figure 3(c) and at length in § 4.4, $\Delta U$ depends not only on $k_0$ but also on the turbulent integral length scale and anisotropy, which varies between cases.

Figure 5.

The wave-induced current under regular waves at the ‘reference’ depth $k_0z=-0.27$ as a function of wave steepness for cases 2.A–3.B as indicated in the legend. The dashed line is proportional to $(ak_0)^2$ .

4.1. Irrotational Eulerian return flow

As is well known, the passage of a deep-water wave group does not entail a net mass flux, because the Stokes drift is cancelled in a depth-integrated sense by an Eulerian current in the opposite direction (Longuet-Higgins & Stewart Reference Longuet-Higgins and Stewart1962) known as the return flow. The current has been observed in laboratory studies, e.g. van den Bremer et al. (Reference van den Bremer, Whittaker, Calvert, Raby and Taylor2019).

The return flow is found beneath a passing wave group, following the group and tending rapidly to zero ahead of and behind the group. For this reason, velocity measurements in cases 1.A–1.D which are taken before and after the passage of the wave group, are expected to see negligible influence from such flow. Cases 2.A–3.B, on the other hand, occur under regular waves – in practice a very long wave group. Since the length of the ‘group’ is far longer than the depth of the channel, the current will be approximately depth-uniform and, in the system following the mean current without waves, satisfying Lagrangian mass conservation, i.e.

(4.2) \begin{equation} u_{\textit{rf}} =-\frac1h \int_{-h}^0 u_s(z;h){\rm d} z \approx -\frac1h \int_{-\infty}^0 u_s(z){\rm d} z = - \frac{k_0a^2}{2h}c(k_0), \end{equation}

where we have assumed infinite water depth (i.e. $k_0h\gg 1$ ) for the Stokes drift profile, as we argued above, and inserted $u_s$ from (1.1). The relation is valid when the depth is greater than about half a wavelength but much smaller than the group length ( $\sim L_{\textit{FOV}}$ in our experiment), both of which are well satisfied in our experiments. We note that (4.2) is only approximate, altered somewhat by the effect of the channel’s bottom and sidewalls (see e.g. van den Bremer & Breivik (Reference van den Bremer and Breivik2017) for a discussion).

4.2. Statistical theory of Eulerian flow generation

Pearson (Reference Pearson2018) argued that, when waves appear in a turbulent field, the combined flow will undergo a transition to a new statistically steady state. We will tailor his argument to our current application and discuss consequences in the context of our experimental observations. Momentum conservation, expressed through the time-averaged Navier–Stokes equations or the Craik–Leibovich equations as appropriate, implies that an Eulerian-mean current will manifest during the ‘spin-up’ period, before a quasi-steady state is reached (in our case slowly decaying due to dissipation).

4.2.1. Turbulent statistics in the transition period

While quiescent-water irrotational waves and turbulence may each be steady in isolation (except for their decay due to dissipation), together they form a system in disequilibrium which will go through a transient change to a new quasi-steady state. This is similar to the model of ‘spin-up from rest’ employed in theory for Langmuir turbulence (e.g. McWilliams et al. Reference McWilliams, Sullivan and Moeng1997). Revisiting and, to some extent, adapting the arguments of Pearson (Reference Pearson2018) sheds light both on the early-stage ‘spin-up’ stage and the final situation. We first assume that a triple decomposition of the turbulent and wavy flow can be performed according to (4.1) (by no means a trivial point, an intricacy we shall return to later) and that all averaged quantities vary slowly as a function of $x$ over a wavelength so that their derivative can be neglected to leading order. Moreover, we assume that any changes in the mean flow develop slowly compared with a wave period. This is reasonable because we have a strong indication that the interaction during passage of the wave groups in Experiment 1, which effectively lasts for approximately $T_{\textit{int}}\sim 7$ wave periods, is far from enough to reach the quasi-equilibrium state: the ratio $|\Delta U'(z)/u_s'(z)|$ only reaches approximately $0.5$ , as figure 3(c) shows, whereas we shall see in the next section (in turn backed by experiments) that in the final quasi-equilibrium state this fraction is $\gtrsim 1$ . The wave field, and consequently the Stokes drift, changes slowly, so $\partial _t\boldsymbol{u}_s$ is negligible. The flow is assumed to be unforced, without, e.g. wind stress, and there is no influence from buoyancy or the Coriolis force. The details of the dissipation are not important to the argument, so we shall not consider them beyond assuming that viscous decay is slow compared with the mean-flow effects of wave–turbulence interactions (which is reasonable in light of the observations by Jooss et al. Reference Jooss, Li, Bracchi and Hearst2021), so that a quasi-steady state can be reached. The final assumption is that $\bar {\boldsymbol{u}}$ and $\boldsymbol{u}_s$ are both oriented along the $x$ -axis and vary slowly except with coordinate $z$ .

The turbulent flow is observed to develop slowly at the relevant scales, and can be assumed to be little affected by a Lagrangian average over a wave period. Wave averaging the equations of motion yields the so-called Craik–Leibovich equation which under the above assumptions may be written in the form (e.g. Suzuki & Fox-Kemper Reference Suzuki and Fox-Kemper2016)

(4.3) \begin{equation} \partial _t \check {\boldsymbol{u}} + (\check {\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{\nabla })\check {\boldsymbol{u}} = \boldsymbol{u}_s\times \check {\boldsymbol{\omega }} -\boldsymbol{\nabla }\left(\check {p} + { \frac 12} u_s^2 + \boldsymbol{u}_s\boldsymbol{\cdot }\check {\boldsymbol{u}} \right) + \text{diffusion,} \end{equation}

where $\check {\boldsymbol{\omega }}=\boldsymbol{\nabla }\times \check {\boldsymbol{u}} =(0,\partial _z\bar {u},0) + \boldsymbol{\nabla }\times \boldsymbol{u}'$ denotes the (Eulerian) wave-averaged vorticity field. Taking the $x$ -component and ignoring diffusion gives

(4.4) \begin{equation} \partial _t \check {u} + \bar {u}\partial _xu'+ (\boldsymbol{u}'\boldsymbol{\cdot }\boldsymbol{\nabla })u'+ w'\partial _z\bar {u} \approx -\partial _x (p' + u_s u' ), \end{equation}

where we have used $\bar {w}=0$ . Performing a Reynolds average over turbulent motions (denoted with an overbar), while noting that $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}'=0$ and that averaged quantities are independent of $x$ , eventually yields

(4.5) \begin{equation} \partial _t \bar {u} \approx - \partial _z \overline {u'w'}. \end{equation}

Equation (4.5) has two important consequences. First, that an inhomogeneous field of turbulence will drive a mean current for as long as a vertical gradient of the shear stress exists (and dominates over viscous forces). Second, when a steady state has been reached, i.e. $\partial _t\bar {u}=0$ , then $\overline {u'w'}$ is approximately constant with respect to depth (see also Pearson Reference Pearson2018, for more discussion), or, more precisely, its vertical gradient is balanced by viscous diffusion. Physically, $\overline {u'w'}\neq 0$ represents a vertical redistribution of streamwise momentum, allowing $\boldsymbol{u}=\boldsymbol{u}(z)$ to transition from its initial value to a presumed final steady-state value, so when the transition period is ended, this shear stress should thus be small compared with TKE. Equation (4.5) paves the way for further analysis of the ‘spin-up’ of the Eulerian-mean current, because the right-hand side can be related to the underlying physical process using a model based on RDT.

4.3. Statistics in the quasi-equilibrium state

Following Harcourt (Reference Harcourt2013) and Pearson (Reference Pearson2018), it can be readily argued that the development of a mean flow via the turbulent shear stress $\overline {u'w'}$ is due to interaction between pre-existing turbulence and Stokes drift.

Multiplying the $x$ -component of (4.3) by $w'$ and the $z$ -component by $u'$ , adding them together and averaging, yields

(4.6) \begin{equation} \frac {\partial \overline {u'w'}}{\partial t}+\overline {w'w'} \frac {{\rm d} \bar {u}}{{\rm d} z} =- \overline {u'u'}\frac {{\rm d} u_s}{{\rm d} z} - \frac {{\rm d} }{{\rm d} z}\overline {u'w'w'} + \textrm{diffusion}, \end{equation}

where we have employed the chain rule and the fact that $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}'=0$ . The diffusion term contains viscous dissipation and turbulent pressure fluctuations, both of which may reasonably be assumed to be small in an oceanographic setting (Harcourt Reference Harcourt2015; Pearson Reference Pearson2018) (unfortunately we cannot directly ascertain how accurate this assumption is in the present context; see also Pearson, Grant & Polton Reference Pearson, Grant and Polton2019). We should bear in mind that this assumption is questionable for our experiment, where waves are shorter and turbulence levels considerably higher than in typical field settings. This is especially true for cases 2.A.2 and 2.B.2. For instance, angular diffusion of the waves can be highly significant when the waves are not long compared with the turbulent integral scale as found by Smeltzer et al. (Reference Smeltzer, Rømcke, Hearst and Ellingsen2023) which would cause a spatially variable Stokes drift. The approximation is more justified the larger the Stokes drift ( $\sim (ak_0)^2c(k_0)$ ) is compared with turbulent velocities ( $\sim u_{\textit{rms}}$ ). In cases 2.A.2 and 2.B.2 the turbulence is strong and the waves short, and turbulent diffusion and pressure correlation terms may not be negligible compared with the Stokes drift contributions even at surface level. Unfortunately, we are not in a position to quantify the diffusive terms in (4.6) from our data.

We now assume that a quasi-steady state has been reached, so that the explicit time derivative of $\overline {u'w'}$ is negligible. The term $\overline {u'w'w'}$ represents net vertical transport, and is expected to become very small in a quasi-steady state. Assuming the diffusion is small, it follows that

(4.7) \begin{equation} \overline {w'w'} \frac {{\rm d} \bar {u}}{{\rm d} z} \approx - \overline {u'u'}\frac {{\rm d} u_s}{{\rm d} z}, \end{equation}

which is to say that an Eulerian-mean current must have been created with the opposite sign compared with Stokes drift.

Note carefully that the relation (4.7) will only hold as long as the right-hand side term is large enough to dominate over the terms in (4.6) we have neglected. Since the Stokes drift decreases exponentially with depth, our assumptions will surely be highly suspect deeper than $k_0z \sim -2$ , possibly above if the turbulence is strong.

In a situation with two horizontal directions (such as in an ocean wave model), the above argument easily generalises to

(4.8) \begin{equation} \overline {w'w'} \frac {{\rm d} \bar {\boldsymbol{u}}_h}{{\rm d} z} \approx - \overline {\bar {\boldsymbol{u}}_h^{\prime}\boldsymbol{\cdot }\bar {\boldsymbol{u}}_h^{\prime}}\frac {{\rm d} \boldsymbol{u}_s}{{\rm d} z} ,\end{equation}

with $\boldsymbol{u}_s$ and $\bar {\boldsymbol{u}}_h$ lying in the horizontal plane. We emphasise that if a constant, depth-uniform current is initially present (as in our experiment), the current $\bar {\boldsymbol{u}}$ in (4.7) and (4.8) is the change in current due to wave–current interaction, or alternatively the current measured in the reference system following the original Eulerian-mean flow.

Beneath a free surface, turbulence is not isotropic, yet $\overline {u'u'}$ and $\overline {w'w'}$ can be expected to be of the same order of magnitude except very near the mean surface level where $\overline {w'w'}$ tends to zero. It is worth noticing that (4.7) is not a perfect cancellation between Eulerian flow and Stokes drift as suggested by the experimental (re)analysis of Monismith et al. (Reference Monismith, Cowen, Nepf and Thais2007), but when turbulence is close to isotropic, the remaining Lagrangian-mean current could be difficult to distinguish from zero in an experiment. On the basis of available evidence it seems a reasonable conjecture that the wave-turbulence-generated anti-Stokes flow explains, at least partly, this surprising cancellation.

A key aspect to notice in (4.7) for modelling purposes is that the degree of cancellation of the mean Lagrangian current does not depend on the overall turbulence level, only on the anisotropy of turbulent fluctuations within the near-surface layer of thickness $\sim 1/k_0$ where Stokes drift is non-negligible. On the other hand, the rate of growth of the Eulerian current is proportional to $\partial _z\overline {u'w'}$ and is higher the more intense the pre-existing turbulence is. In an ocean setting some level of turbulence is nearly always present, so the partial cancellation of Stokes drift according to (4.7) is to be expected.

4.3.1. Comparison with experiment

In order to test (4.7), it is necessary to extract the variances $\overline {u'u'}$ and $\overline {w'w'}$ from the wave-turbulence PIV data. Accomplishing this with the best possible accuracy is paramount in order to estimate turbulent second-order moments, because the wave velocities far exceed those of the turbulence near the surface, so even a small percentage of the wave motion erroneously identified as turbulence could lead to considerable overestimates of $\overline {u'u'}$ and $\overline {u'w'}$ (less so $\overline {u'w'}$ due to the phase difference between $\tilde {u}$ and $\tilde {w}$ ).

In cases 3.A and 3.B one could employ the now-standard technique of PhCA (Umeyama Reference Umeyama2005; Buckley & Veron Reference Buckley and Veron2017) to average out the wave motion, because the wave phase at each point in time and space can be estimated. For cases 2.A and 2.B the task is more difficult because measurements are made with frequency lower than the wave’s, and no phase information is available (Experiment 2, when performed, was not planned with this task in mind). Estimates of $\overline {u'u'}$ and $\overline {w'w'}$ can be made in all cases regardless of acquisition rate using the method of POD which decomposes the measured velocity field into spatial modes ordered according to their ‘energy’ content. A more detailed discussion of the performance of this method and validation may be found in Appendix C. In cases 3.A and 3.B direct comparison with PhCA can be made; for the purpose of comparing with (4.7), the two methods give very similar, but not quite identical results.

Although PhCA is in common use while POD has not been employed for this purpose previously to our knowledge, there are fundamental reasons why a data-driven method such as POD, in general, is more appropriate. In particular, PhCA relies on assumptions which typically, including in our case, are not fully met. A further discussion of this point is found in Appendix C, where we argue that POD is the preferred method of decomposition also in cases 3.A and 3.B.

The two sides of (4.7) are compared in figure 6 for cases 2.A to 3.B, the cases whose measurements were taken in the streamwise-vertical plane so that waves and turbulence can be separated. The POD procedure was employed for triple decomposition to evaluate turbulent second moments. The dashed lines show the vertical derivative of the Stokes drift, ${\rm d} u_s/{\rm d} z=2\sqrt {gk_0}(ak_0)^2\exp (2k_0z)$ , while the solid line is the measured value of $-(\overline {w'w'}/\overline {u'u'}){\rm d}\bar {u}/{\rm d} z$ , which is hypothesised to equal the Stokes drift gradient in the quasi-steady state. For completeness, the Reynolds stresses themselves are plotted in Appendix D.

Figure 6.

Test of (4.7) for cases 2.A to 3.B. The theoretical Stokes drift gradient ${\rm d} u_s/{\rm d} z$ (the derivative of (1.1)) is shown as dashed lines of corresponding colour. Separation of turbulence from waves was performed with POD, discussed in Appendix C.

Panels (e) and ( f) of figure 6 are from Experiment 3 where the separation of waves and turbulence could be performed and validated with confidence, as discussed in Appendix C, but for cases 2.A and 2.B in panels (a)–(d) the uncertainly is difficult to quantify. For this reason, considerable caution should be exercised when interpreting the results in figure 6(a)–(d), particularly nearest the surface where the wave motion is most energetic, so that even a small percentage of wave motion remaining in the calculation of $\overline {u'u'}, \overline {w'w'}$ could cause a significant overestimation. We have decided to include the figures nevertheless, to illustrate the overall similarity of trend.

Given the highly irregular and unsteady nature of our strongly turbulent flow, the agreement is striking in many of the cases. As argued above, the assumptions behind (4.7) are expected to be reasonable when Stokes drift is sufficiently high compared with turbulent velocities, i.e. $k_0z \gtrsim -2$ and sufficiently large $ak_0$ , and subject to corrections if diffusion levels are high. The relatively poor fit for cases 2.A.2 and 2.B.2 is therefore not very surprising for the latter reason. The very low values of $ak_0$ in cases 2.A.1.1 and 2.A.2.1 is a likely reason why (4.7) is unlikely to be reasonable for these cases – again the Stokes drift term in (4.19) may not dominate over terms which are neglected. Indeed, the good performance in case 3.A.1 is more striking, but tallies with the fact that TKE is only half that of case 2.B and less than a quarter of that of case 2.A, with correspondingly less diffusion.

Some further words of caution are warranted. As discussed previously, turbulence was measured only at one position near the downstream end of the channel, while the wave–current interaction occurred upstream. We therefore cannot guarantee that the ratio $\overline {u'u'}/\overline {w'w'}$ has been the same throughout the time of interaction – indeed the theory of Teixeira & Belcher (Reference Teixeira and Belcher2002) indicates this ratio has increased in magnitude near the surface during the spin-up period due to wave–current interaction. We hesitate to offer an explanation for the surprising fact that the experimental curves in figure 6 appear to curve back near the surface, corresponding to a similar behaviour of the fraction $\overline {u'u'}/\overline {w'w'}$ (see also figure 12 i), yet we remind the reader of the difficulty of separating waves and turbulence near the surface for these cases; we cannot rule out that the effect is spurious.

4.4. A model based on RDT

In § 3.2.1, we measured the change in Eulerian current after the passage of a wave group. Unlike for the regular-wave cases, the shear of the resulting current is considerably smaller than the gradient of the Stokes drift, so the two sides of (4.7) are highly dissimilar, which signifies that the influence of the group of waves has not brought the flow near the quasi-steady state. Since evidence suggests that the flow is still ‘spinning up’ when the group has passed, a theory which describes the development soon after the onset of waves is expected to describe the physical process in further detail.

The early onset of the (change in the) Eulerian current is governed approximately by (4.5), and we will use RDT to study how its right-hand side – that is, the Reynolds stress $\overline {u'w'}$ – depends on depth $z$ and time while changes are still moderate. The theoretical predictions will be compared with the observations made for cases 1.A–1.D, shown in figure 3(b). Teixeira & Belcher (Reference Teixeira and Belcher2002) showed using RDT that a shear stress that varies in the vertical is generated by the passage of a progressive monochromatic surface wave over isotropic turbulence. We will show next that something similar occurs for a finite wave group, such as considered in the experiments of the present study. Since the effect we consider is inviscid in nature, we will neglect viscosity in the following.

The RDT is a theory first proposed by Batchelor & Proudman (Reference Batchelor and Proudman1954) where the straining of turbulence due to the distortions from the surrounding flow is assumed to dominate over that due to turbulence acting on itself, so that the term $(\boldsymbol{u}'\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}'$ is negligible in the Navier–Stokes equation, yielding a linearised theory. The leading cause of change for the turbulence is presumed to be the mean Lagrangian motion of turbulence-containing parcels of fluid due to the larger-scale surrounding motion. The RDT approach is formally valid whenever the distortions applied to turbulence are sudden. This amounts to assuming that the distortions (for example, mean-flow gradients) are applied over a time shorter than an eddy turn-over time. The spectral formulation of RDT also requires that there is a spatial scale separation between the turbulence and the mean flow. In the present wave-associated mean flow, this corresponds to $\lambda \gg L$ where $\lambda$ is the wavelength of the waves and $L$ is the integral length scale of the turbulence (cf. Teixeira & Belcher Reference Teixeira and Belcher2002). Both of these criteria are reasonably well satisfied in the Experiments 1.A–1.D where the relevant turbulent length scale, $L_x^x$ is at most half a wavelength. Visual inspection of the velocity field clearly shows that the typical coherent eddies are much smaller than $\lambda _0$ . The values of $L_x^x$ and details of its calculation with the method of Mora & Obligado (Reference Mora and Obligado2020) were reported in Smeltzer et al. (Reference Smeltzer, Rømcke, Hearst and Ellingsen2023), and are quoted in table 2. In practice, RDT is known to provide useful results even when these conditions are not strictly fulfilled (e.g. Hunt & Carruthers Reference Hunt and Carruthers1990; Mann Reference Mann1994; Cambon & Scott Reference Cambon and Scott1999).

In the present case, the effect that is essential in order to explain the generation of a Eulerian-mean current is not related to the individual wave oscillations but rather to the systematic tilting and stretching of the vorticity of the turbulence by the Stokes drift of the wave. Therefore, we adopt a linearised version of the Craik–Leibovich equation, (4.3), whose key term is the ‘vortex force’ $\boldsymbol{u}_s\times \check {\boldsymbol{\omega }}$ . Applying a simplified phase-averaged formalism is a good approximation because individual wave cycles have little effect on the net stretching and tilting of vortices, and it is the integrated effect of the whole wave group which matters; this is illustrated, e.g. in figure 3 of Teixeira & Belcher (Reference Teixeira and Belcher2010) (regular waves are considered there, but the arguments hold equally for the relatively long wave groups we consider here). Only a small uncertainty is thereby introduced which is far smaller than the quantitative effects of the simplifications underlying RDT itself. The corresponding vorticity equation, which will be used in RDT, is

(4.9) \begin{equation} \frac {\partial \boldsymbol{\omega }'}{\partial t} + (\boldsymbol{u}_s\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{\omega }' = (\boldsymbol{\omega }'\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}_s. \end{equation}

In (4.9), $\boldsymbol{\omega }' = \boldsymbol{\nabla }\times \boldsymbol{u}'$ is the turbulent vorticity. As before, $\boldsymbol{u}_s$ is orientated along the $x$ axis and is independent of $x,y$ and $t$ . The mean background current is assumed to be initially depth uniform; hence, it has only trivial effect on the flow system, and we set it to zero by a change of coordinate system. It should be noted that these equations represent the effect of the Stokes drift of an irrotational surface wave; the rotational correction to the wave motion due to the small change in Eulerian current (see Ellingsen Reference Ellingsen2016) is neglected.

Let us first consider the turbulence away from the air–water interface, which to a first approximation can be considered homogeneous and isotropic, being denoted by the superscript $(H)$ . In this section, it is useful to let the subscript $i=1,2,3$ denote the component of a vector or tensor along directions $x,y$ and $z$ , respectively. The turbulent, homogeneous (or bulk) vorticity $\boldsymbol{\omega }'^{(H)}$ is expressed as a three-dimensional Fourier integral, as

(4.10) \begin{equation} \omega '^{(H)}_i(\boldsymbol{x},t)=\iiint \hat {\omega }^{(H)}_i(\boldsymbol{\kappa },t) \, {e}^{{i} \boldsymbol{\kappa } \boldsymbol{\cdot }\boldsymbol{x}} \, {\rm d} \kappa _1 \, {\rm d} \kappa _2 \, {\rm d} \kappa _3, \end{equation}

where the hat denotes a Fourier transformed turbulent perturbation quantity and $\boldsymbol{\kappa }=(\kappa _1,\kappa _2,\kappa _3)$ is the wavenumber vector.

Two equations result from (4.9) together with (4.10)

(4.11) \begin{align} \frac {{\rm d} \hat {\omega }^{(H)}_i}{{\rm d} t}&= \frac {\partial u_{si}}{\partial x_j} \hat {\omega }^{(H)}_j , \end{align}

(4.12) \begin{align} \frac {{\rm d} \kappa _i}{{\rm d} t} &= - \frac {\partial u_{sj}}{\partial x_i}\kappa _j. \end{align}

Since $u_{si} = u_s \delta _{1i}$ , (4.11)–(4.12) reduce to

(4.13a) \begin{align} \frac {{\rm d} \hat {\omega }^{(H)}_1}{{\rm d} t} &= \frac {{\rm d} u_s}{{\rm d} z}\hat {\omega }^{(H)}_3, \quad \frac {{\rm d} \hat {\omega }^{(H)}_2}{{\rm d} t}=0,\quad \frac {{\rm d} \hat {\omega }^{(H)}_3}{{\rm d} t}=0, \end{align}

(4.13b) \begin{align} \frac {{\rm d} \kappa _1}{{\rm d} t}&=0, \quad \frac {{\rm d} \kappa _2}{{\rm d} t}=0, \quad \frac {{\rm d} \kappa _3}{{\rm d} t}=-\frac {{\rm d} u_s}{{\rm d} z}\kappa _1. \end{align}

These equations can be integrated in time to yield

(4.14a) \begin{align} \hat {\omega }_1^{(H)}(t)&=\hat {\omega }_{10}^{(H)}+\hat {\omega }_{30}^{(H)} \int _0^t \frac {{\rm d} u_s}{{\rm d} z}{\rm d} t', \quad \hat {\omega }_2^{(H)}(t)=\hat {\omega }_{20}^{(H)}, \quad \hat {\omega }_3^{(H)}(t)=\hat {\omega }_{30}^{(H)}, \end{align}

(4.14b) \begin{align} \kappa _1(t)&=\kappa _{10}, \quad \kappa _2(t)=\kappa _{20}, \quad \kappa _3(t)=\kappa _{30}-\kappa _{10} \int _0^t \frac {{\rm d} u_s}{{\rm d} z}{\rm d} t', \end{align}

where the subscript ‘ $0$ ’ applied to a variable denotes its value at the initial time, before the turbulence is distorted. It is convenient to define

(4.15) \begin{equation} \beta = \int _0^t \frac {{\rm d} u_s}{{\rm d} z}{\rm d} t' = \frac {{\rm d} \Delta x}{{\rm d} z}, \end{equation}

where $\Delta x(z,t)$ is the total fluid parcel displacement in the $x$ -direction associated with the Stokes drift.

In order to calculate statistics of the turbulent velocity, it is necessary to relate the turbulent velocity fluctuations before and after distortion. Continuity of $\boldsymbol{u}'$ implies

(4.16) \begin{equation} \boldsymbol{\nabla} ^2 \boldsymbol{u}' = - \boldsymbol{\nabla } \times \boldsymbol{\omega }' ,\end{equation}

which in spectral space becomes

(4.17) \begin{equation} \hat {u}_i^{(H)} = \textrm {i} \varepsilon _{\textit{ijl}} \frac {\kappa _j}{\kappa ^2} \hat {\omega }_l^{(H)}, \end{equation}

where $\varepsilon _{\textit{ijl}}$ is the Levi-Civita permutation symbol and $\kappa = |\boldsymbol{\kappa }|$ .

We wish to relate the turbulent velocity after distortion by the Stokes drift to the velocity of the initial undistorted turbulence. The Fourier transform of the velocity of the homogeneous turbulence after distortion by the Stokes drift can be related to the corresponding Fourier transform of the vorticity using (4.17). In turn, the vorticity of the homogeneous turbulence after distortion can be related to the initial turbulent vorticity using (4.14a )–(4.14b ). Finally, the initial turbulent vorticity can be related to the initial turbulent velocity through

(4.18) \begin{equation} \hat {\omega }_{i0}^{(H)} = \textrm {i} \varepsilon _{\textit{ijl}} \kappa _{j0} \hat {u}_{l0}^{(H)}, \end{equation}

which comes from the definition of vorticity. From all of the above one finds that the distorted velocity can finally be expressed as

(4.19a) \begin{align} &\hat {u}_1^{(H)}(t)=\left ( 1 + \frac {\kappa _1 \kappa _3 \beta }{\kappa ^2} \right ) \hat {u}_{10}^{(H)}+ \frac {\kappa _1^2 \beta }{\kappa ^2} \hat {u}_{30}^{(H)}, \end{align}

(4.19b) \begin{align} &\hat {u}_3^{(H)}(t) = \left ( \frac {\kappa _0^2}{\kappa ^2} - \frac {\kappa _1 \kappa _{30} \beta }{\kappa ^2} \right ) \hat {u}_{30}^{(H)} - \frac {\kappa _{12}^2 \beta }{\kappa ^2} \hat {u}_{10}^{(H)}, \end{align}

where $\kappa _{12}=(\kappa _1^2+\kappa _2^2)^{1/2}$ , $\boldsymbol{\kappa }_0=(\kappa _{10},\kappa _{20},\kappa _{30})$ and $\kappa _0=|\boldsymbol{\kappa }_0|$ , and we only focus on $\hat {u}_1$ and $\hat {u}_3$ because these are the velocity components necessary to calculate the shear stress.

The previous calculations only apply to the turbulence far away from the air–water interface, at least a distance $\sim L$ where $L$ is a characteristic integral length scale (but affected by the Stokes drift, because of the scale separation $L \ll \lambda$ ). In order to take into account the blocking effect of the air–water interface, where we assume that the interface affects the turbulence essentially as a frictionless wall for depths of $O(L)$ (because the air–water interface has a large density contrast), this effect can be accounted for by adding an irrotational correction to the homogeneous turbulent velocity field, as done before by Hunt & Graham (Reference Hunt and Graham1978) and Teixeira & Belcher (Reference Teixeira and Belcher2002). Note that if the waves that generate the Stokes drift are irrotational (which is true to a good degree of approximation), then this irrotational correction remains irrotational. Hence, the turbulent velocity components affected both by the Stokes drift and by blocking can be expressed as two-dimensional Fourier integrals along the horizontal directions

(4.20) \begin{equation} u'_i(\boldsymbol{x},t) = \iint \hat {u}_i(\kappa _1,\kappa _2,z) e^{i(\kappa _1 x+ \kappa _2 y)} {\rm d}\kappa _1 \, {\rm d}\kappa _2, \end{equation}

because the inhomogeneity imposed by blocking does not allow Fourier transformation in the vertical direction.

Based on Hunt & Graham (Reference Hunt and Graham1978) and Teixeira & Belcher (Reference Teixeira and Belcher2002), the Fourier transforms of $u'_1$ and $u'_3$ can be written

(4.21a) \begin{align} &\hat {u}_1= \int \left ( \hat {u}_1^{(H)} e^{i \kappa _3 z} - i \frac {\kappa _1}{\kappa _{12}} e^{\kappa _{12} z} \hat {u}_3^{(H)} \right ) {\rm d}\kappa _3, \end{align}

(4.21b) \begin{align} &\hat {u}_3 = \int \hat {u}_3^{(H)} \big ( e^{i \kappa _3 z} - e^{\kappa _{12} z} \big ) {\rm d}\kappa _3. \end{align}

For a flow associated with a surface wave, the blocking condition actually applies perpendicularly to the wavy air–water interface, which when adopting a model accounting for the individual wave cycles requires the use of a curvilinear coordinate system, as in Teixeira & Belcher (Reference Teixeira and Belcher2002). However, since our present model only includes the Stokes drift effect, no explicit wavy deformations of the interface are accounted for, and the surface is assumed to be in its average state, i.e. flat and horizontal at $z=0$ . The solutions (4.21) take this into account.

We wish to evaluate $\overline {u'w'} = \overline {u'_1 u'_3}$ after some time of deformation. This can be done by using (4.14b), (4.19), (4.20), and also noting that, by definition

(4.22) \begin{equation} \overline {\hat {u}_{i0}^{(H)}(\boldsymbol{\kappa }_0)^* \hat {u}_{j0}^{(H)}(\boldsymbol{\kappa }_0')} = \varPhi _{ij}^{(H)}(\boldsymbol{\kappa }_0) \delta (\boldsymbol{\kappa }_0 - \boldsymbol{\kappa }_0'), \end{equation}

where an asterisk denotes the complex conjugate, $\varPhi _{ij}^{(H)} (\boldsymbol{\kappa }_0)$ is the three-dimensional spectrum of the initial homogeneous and isotropic turbulence and $\delta$ is the three-dimensional Dirac delta, to show that $\overline {u'_1 u'_3}$ is given by

(4.23) \begin{equation} \overline {u'_1 u'_3} = \iiint \Big ( M_{13} \varPhi _{13}^{(H)}+M_{11} \varPhi _{11}^{(H)} + M_{33} \varPhi _{33}^{(H)} \Big ) {\rm d}\kappa _1 \, {\rm d}\kappa _2 \, {\rm d}\kappa _{30}, \end{equation}

where

(4.24a) \begin{align} M_{13}=&\left ( 1+ \frac {\kappa _1 \kappa _3 \beta }{\kappa ^2} \right ) \left ( \frac {\kappa _0^2}{\kappa ^2} - \frac {\kappa _1 \kappa _{30} \beta }{\kappa ^2} \right ) - \frac {\kappa _1^2 \kappa _{12}^2 \beta ^2}{\kappa ^4} \notag \\ &-\left [ \left ( 1 + \frac {\kappa _1 \kappa _3 \beta }{\kappa ^2} \right ) \left ( \frac {\kappa _0^2}{\kappa ^2} - \frac {\kappa _1 \kappa _{30} \beta }{\kappa ^2} \right ) - \frac {\kappa _1^2 \kappa _{12}^2 \beta ^2}{\kappa ^4} \right ] e^{\kappa _{12} z} \cos ( \kappa _3 z) \nonumber \\ &+ 2 \frac {\kappa _1 \kappa _{12} \beta }{\kappa ^2} \left ( \frac {\kappa _0^2}{\kappa ^2} - \frac {\kappa _1 \kappa _{30} \beta }{\kappa ^2} \right ) e^{\kappa _{12} z} \sin (\kappa _3 z), \end{align}

(4.24b) \begin{align} M_{11}=&-\frac {\kappa _{12}^2 \beta }{\kappa ^2} \left [ 1 + \frac {\kappa _1 \kappa _3 \beta }{\kappa ^2} - \left ( 1 + \frac {\kappa _1 \kappa _3 \beta }{\kappa ^2} \right ) e^{\kappa _{12} z} \cos (\kappa _3 z) +\frac {\kappa _1 \kappa _{12} \beta } {\kappa ^2} e^{\kappa _{12} z} \sin (\kappa _3 z) \right ], \end{align}

(4.24c) \begin{align} M_{33}=&\left ( \frac {\kappa _0^2}{\kappa ^2} - \frac {\kappa _1 \kappa _{30} \beta }{\kappa ^2} \right )\! \left [ \!\frac {\kappa _1^2 \beta }{\kappa ^2} \left ( 1 - e^{\kappa _{12} z} \cos (\kappa _3 z) \right ) - \frac {\kappa _1}{\kappa _{12}} \left ( \!\frac {\kappa _0^2}{\kappa ^2} - \frac {\kappa _1 \kappa _{30} \beta }{\kappa ^2} \!\right )\! e^{\kappa _{12} z} \sin (\kappa _3 z) \!\right ]\!. \end{align}

To calculate the shear stress using (4.23), an expression for $\varPhi _{ij}^{(H)}$ must be assumed. We employ the model (Batchelor Reference Batchelor1953)

(4.25) \begin{equation} \varPhi _{ij}^{(H)}(\boldsymbol{\kappa }_0) = \left ( \delta _{ij} - \frac {\kappa _{i0} \kappa _{j0}}{\kappa _0^2} \right ) \frac {E(\kappa _0)}{4 \pi \kappa _0^2}, \end{equation}

where $E(\kappa _0)$ is the energy spectrum of the homogeneous and isotropic turbulence, and since the calculations are inviscid, a suitable assumption for the energy spectrum is that first introduced by Von Kármán (Reference Von Kármán1948), expressed as

(4.26) \begin{equation} E(\kappa _0)= \frac {g_2 q^2 L (\kappa _0 L)^4}{\left [ g_1 + (\kappa _0 L)^2 \right ]^{17/6}}, \end{equation}

where $q$ is a characteristic r.m.s. value of a turbulent velocity component and $L$ is the longitudinal integral length scale. The constants $g_1 \approx 0.558$ and $g_2 \approx 1.196$ ensure that the definitions of $q$ and $L$ are consistent.

Note that while a number of simplifying assumptions have been made which mean that full quantitative agreement with observations is not to be expected, the above theory involves no fitting or particular tailoring to our specific system. The turbulence is described by just two scalar parameters pertaining to the homogeneous bulk, $q$ and $L$ , a crude simplification which nevertheless is expected to capture the most essential features. Importantly, these are quantities obtainable from point measurements in experiments as well as in the field.

4.5. Input parameters

Three quantities must be evaluated in order to obtain numerical values for $\overline {u'w'}$ from (4.23): $L, q$ and $\beta$ .

The simplest to determine is $q$ , which according to RDT satisfies

(4.27) \begin{equation} q^2 = {\frac{2}{3}}e, \end{equation}

assuming isotropic turbulence, where the TKE $e$ is found in table 2.

The exact choice of turbulent integral length scale $L$ involves some amount of judgement. We shall take the most immediate available option and use the streamwise integral scale $L_x^x$ for $L$ , reported for cases 1.A–1.D by Smeltzer et al. (Reference Smeltzer, Rømcke, Hearst and Ellingsen2023). One does well to note that estimating the integral scale from experimental data is not trivial and several methods exist, which tend to yield significantly different results. The difference between the experiments means that no single method of calculating $L_x^x$ can be used for all, so direct quantitative comparison is dubious; however, RDT is only expected to describe the ‘spin-up phase’ cases 1.A–1.D from Experiment 1, which are directly comparable.

The displacement $\beta$ defined in (4.15) is determined by the duration and extent of interaction between waves and turbulence in the time between generation by the active grid, and measurement a distance $L_{\textit{FOV}}$ downstream. For a Gaussian wave packet like those in cases 1.A–1.D, van den Bremer et al. (Reference van den Bremer, Whittaker, Calvert, Raby and Taylor2019) showed that the final value of $\beta$ after the passage of the group, $\beta _{{f}}$ (subscript ‘ ${f}$ ’: final), takes the form

(4.28) \begin{equation} \beta _{{f}}(z) = 4 \sqrt {\pi } \sigma _0 k_0 (k_0a_{{p}})^2{e}^{2 k_0 z}, \end{equation}

where $\sigma _0$ is the spatial standard deviation of the Gaussian wave group and $k_0a_{{p}}$ the peak steepness. By definition of wave group, $\sigma _0= c_g(k_0) \tau _0$ (see (1.2) and table 2). The expression presumes that the turbulence moving downstream from the active grid has interacted with the whole wave group before reaching the field of view; this is true in our Experiment 1.

It might be interesting to also calculate the values of $\beta _{{f}}$ for the cases with regular waves, which gives an indication of the extent of interaction between waves and turbulence before the turbulent flow reaches the field of view. The cases 2.A–3.B consider regular waves so that the interaction occurs at a near-constant rate during the travel time $t_{\textit{travel}}$ , hence in this case we may approximate

(4.29) \begin{equation} \beta = t_{\textit{travel}}\frac {{\rm d} u_s}{{\rm d} z} = \frac {2k_0L_{\textit{FOV}}}{U_0}(ak_0)^2c(k_0){e}^{2 k_0 z}. \end{equation}

Values of $\beta _{{f}}(0)$ for all cases are given in table 4, and vary by more than an order of magnitude from the longest waves of lowest steepness to the short and steep waves.

4.6. The RDT results

Figure 7(a) shows the profile of the Reynolds shear stress provided by RDT after passage of the Gaussian wave group, using the input parameters specified above. Figure 7(b) shows the vertical derivative of this shear stress, which is proportional to the forcing of the mean current, according to (4.5). The results are expected to represent a similar situation to our Experiment 1 where wave groups were employed and the flow is still in a state of transition when the wave group has passed. One may also note the similarity in shape with the measured currents also for regular waves presented in figure 4, indicative that the same process has been at play.

Figure 7.

Results from RDT: (a) profile of the normalised shear stress $\overline {u'w'}/q^2$ as a function of depth $k_0 z$ ; (b) profile of the normalised vertical derivative of the shear stress; (c) the normalised shear stress as a function of $\beta$ for $k_0 z=-0.4335$ , the depth where $\overline {u'w'}/q^2$ attains its maximum magnitude. The dashed line is the 1:1 line, illustrating a linear dependence. (d) The RDT estimates of the anti-Stokes velocity profile after the passage of the wave groups, corresponding to figure 3(b).

Next, figure 7(c) shows the dependence of the shear stress at its maximum on $\beta$ (which is proportional to the square of the wave slope $ak_0$ as (4.28) and (4.29) show). The vertical derivative of the shear stress at the surface, to which the current at the surface must be proportional, is clearly proportional to this maximum. This implies that the current at the surface (departing from a quiescent flow) must be proportional to $(ak_0)^2$ , which is consistent with the results of figure 5 for cases 2.A and 2.B.

A shear stress profile such as depicted in figure 7(a) would, according to (4.5), lead to a current intensifying continuously and indefinitely in time. However, the current will generate a shear stress of opposite sign to that associated with the Stokes drift, and when these shear stresses have evolved so that they cancel each other, the current will reach a steady state (see Pearson Reference Pearson2018). Moreover, dissipation, which has been neglected in the present RDT treatment, is always present, and would act in a similar way to limit the current growth.

4.6.1. Effective interaction time

While the primary goal of our RDT model is to elucidate the physical process behind our observations, we have also used (4.23) to obtain a rough estimate of the current resulting from the wave groups in Experiment 1 (cases 1.A–1.D) encountering the incoming turbulence. In order to obtain quantitative values, we use a simplified procedure: noting that $\partial _z \overline {u'w'} \propto (ak_0)^2$ , we estimate the current after the passage of the group as

(4.30) \begin{align} U_{\textit{RDT}}(z) &\approx -\!\int \partial _z \overline {u'w'}(t) {\rm d} t \approx -(\partial _z \overline {u'w'})_{\textit{RDT}}\int _{-\infty }^\infty \frac {\tilde {a}(t)^2}{a^2_{{p}}} {\rm d} t\notag \\ &\approx -(\partial _z \overline {u'w'})_{\textit{RDT}}T_{{\textit{int},\textit{grp}}}, \end{align}

where we have introduced an `effective interaction time' for a wave group,

(4.31) \begin{equation} T_{{\textit{int},\textit{grp}}} = \frac 1{a_{{p}}^{2}}\int _{-\infty }^\infty \tilde {a}(t)^2{\rm d} t, \end{equation}

an estimate for the duration for which the turbulent current has been influenced by the waves before it reaches the point where it is measured, taking into account that the wave amplitude varies throughout the group. As seen by the turbulence, $\tilde {a}(t)$ is a Gaussian with $\tau _0$ from (3.2) replacing $\tau$ in (3.1), inserting which into (4.31) gives

(4.32) \begin{equation} T_{{\textit{int},\textit{grp}}} = \sqrt {\pi }\tau _0. \end{equation}

We next insert into (4.30) the calculated Reynolds stress vertical gradient shown in figure 7(b) (multiplied by $k_0q^2$ to obtain a dimensional value), using values for $k_0, q=2e/3,L=L_x^x$ and $T_{{int,grp}}$ from table 4. The resulting estimated velocity profiles are shown in figure 7(d).

It is worth noting the relation between the effective interaction time and the fluid displacement $\beta$ . For regular waves, the turbulence interacts constantly with waves throughout its passage from grid to field of view, so the interaction time in the presence of regular waves is $T_{{int,reg}} = L_{\textit{FOV}}/U_0$ . Using (4.28) and noting that for deep-water waves, $\sigma _0=c_g\tau _0=({1}/{2}) c\tau _0$ , we find that for regular waves as well as wave groups it holds that

(4.33) \begin{equation} \beta (z) = 2k_0 T_{\textit{int}} u_s(z). \end{equation}