1. Introduction
The turbulence induced by nonbreaking surface waves was first noted in an analysis by Phillips (1961). The analysis postulated that the production is due to the straining associated with the waves and is balanced by viscous dissipation. Using a model based on rapid distortion theory (RDT), Teixeira and Belcher (2002) further studied the interaction of turbulence with a monochromatic irrotational surface wave. The modeling results indicated that the turbulence distorted by surface waves behaves in a way similar to that of Langmuir turbulence, but strikingly different from the turbulence distorted by mean shear. The dominant turbulence structure is attributed to Stokes drift, which tilts the vertical vorticities into the horizontal and subsequently stretches them into elongated streamwise vortices.
The potential impact of this mechanism in the upper ocean was suggested by Ardhuin and Jenkins (2006) and Babanin (2006), who showed the importance of wave-induced turbulence in the attenuation of swell propagating across the Pacific Ocean.
A number of laboratory experiments have been conducted to show evidence of the turbulence generated by nonbreaking surface waves. The incipience of turbulence was detected on mechanically generated laboratory waves using dye dispersion in Babanin (2006) and using particle image velocimetry in Babanin and Haus (2009). Beyá et al. (2012) reproduced the experimental setup of Babanin and Haus (2009), also using dye to visualize turbulent mixing; their experiments, however, showed no evidence of turbulent mixing. Dai et al. (2010), on the other hand, observed significant enhancement in thermal destratification in the presence of surface waves. Savelyev et al. (2012) recently conducted laboratory experiments using a thermal-marking velocimetry technique to quantify turbulent velocities at the water surface. The thermal image of the water surface clearly revealed an elongated streaky structure, which indicates the formation of streamwise vortices beneath the water surface. The measurements confirmed the production of wave-induced turbulence with a growth rate consistent with the model of Teixeira and Belcher (2002).
Numerical simulations to study the wave–turbulence interaction process were also carried out by Babanin and Chalikov (2012) and Savelyev et al. (2012). The numerical model decomposes the flow into a two-dimensional potential flow of surface waves and a three-dimensional vortical flow of turbulence. The momentum transfer from the potential waves to the turbulence is accounted for through additional terms in the equations for the vortical flow. The backward transfer from the vortical flow to the waves and attenuation of the surface waves due to dissipation are not taken into account in the model. The results confirmed the production of turbulence due to wave motion.
In this study, more direct evidence is presented from a direct numerical simulation of monochromatic surface waves propagating over a turbulent field. The present numerical simulation differs from those of Babanin and Chalikov (2012) and Savelyev et al. (2012) in that the primitive momentum and mass conservation equations subject to the fully nonlinear boundary conditions on the exact water surface z = η(x, y, t) are solved. The numerical model thus resolves the wave–turbulence interaction process of various length scales and avoids the ambiguity in parameterizing the near-surface subgrid turbulence. The focus of the present numerical study is to reveal the characteristics and inherent structure of wave-induced turbulence.
2. Numerical simulation
Details of the numerical model are reported in Tsai and Hung (2007). As in the simulation of Tsai et al. (2013), temperature is treated as a passive tracer; the buoyancy effect due to temperature fluctuations hence does not modify the vertical momentum equation. To set up the ambient turbulence in an otherwise quiescent flow, the evolution of a fluctuating solenoidal velocity field without the mean flow beneath a free-slip surface is first computed, as in Tsai et al. (2005). The spinup simulation is performed until the flow reaches an equilibrium state, in which the inertial and dissipating ranges of the energy spectra converge. The velocity field of a progressive, monochromatic surface wave is then superimposed on the turbulence field to form the initial velocity field for the simulation. The initial turbulent kinetic energy (TKE) is about 10−3 of the total wave energy.
We consider a gravity wave with a wavelength of λ = 7.5 cm and initial steepness of 0.25. The length, width, and depth of the computational domain are 4λ, 2λ, and 0.8λ, respectively, discretized by 512, 256, and 128 grids. The simulation is carried out for 30 wave periods (T0). During this time interval, the accumulated energy loss attributed to numerical discretization is about 0.07% of the initial total energy and remains less than 0.4% of the energy dissipation by viscosity. The numerical errors in energy conservation are tolerable for the purposes of the present study.
The mobile water surface poses the difficulty of evaluating statistical properties in a fixed coordinate system for the region encountering the passage of surface waves. To resolve this problem, the flow variables are transformed into a wave-following curvilinear coordinate system (e.g., Hsu et al. 1981) defined by ζ = (z + H)/(η + H), where H is the water depth. Since the flow is dominated by the motions of surface waves, the wave-correlated component of a flow variable f can be defined as the difference between the phase average over spanwise y and the average over the x–y plane: 
For the results presented in the following sections, the variables are nondimensionalized by choosing k−1 = λ(2π)−1 and c0 = (gk−1)0.5 as the characteristic length and velocity, respectively.
3. Growth of turbulence
Evolution of the surface-integrated and volume-integrated TKE Ek′ and the corresponding streamwise, spanwise, and vertical components, Eu′, Eυ′, and Ew′, where Ek′ = Eu′ + Eυ′ + Ew′, are shown in Figs. 1a and 1b. Immediately after the start of the simulation, the TKE decreases drastically, showing a rapid adjustment of the imposed initial fluctuation to realistic turbulence and the initiation of wave–turbulence interaction. After the short adjustment stage (t ≈ 0 ~ 5T0), the TKE increases monotonically with time.
Evolutions of the (a) surface and (b) total TKE Ek′ and the streamwise, spanwise, and vertical components, Eu′, Eυ′, and Ew′, where Ek′ = Eu′ + Eυ′ + Ew′, and the corresponding nondimensional growth rates, βk′, βu′, βυ′, and βw′.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Evolutions of the (a) surface and (b) total TKE Ek′ and the streamwise, spanwise, and vertical components, Eu′, Eυ′, and Ew′, where Ek′ = Eu′ + Eυ′ + Ew′, and the corresponding nondimensional growth rates, βk′, βu′, βυ′, and βw′.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Evolutions of the (a) surface and (b) total TKE Ek′ and the streamwise, spanwise, and vertical components, Eu′, Eυ′, and Ew′, where Ek′ = Eu′ + Eυ′ + Ew′, and the corresponding nondimensional growth rates, βk′, βu′, βυ′, and βw′.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
To quantify the growth trend, the nondimensional growth rate of TKE is defined as (Teixeira and Belcher 2002) βk′ = (E0k′)−1dEk′/d(t/T0), where the temporal derivative is computed by fitting Ek′ within an interval of 2T0; E0k′ is the TKE at the beginning of the interval. The corresponding growth rates βk′, βu′, βυ′, and βw′ of Ek′, Eu′, Eυ′, and Ew′ for t > 10T0 are also depicted in Fig. 1.
The surface TKE and its components (Fig. 1a) grow exponentially with increasing growth rates up to t ≈ 22T0 and then continue growing at reduced rates. The surface turbulent motion is highly anisotropic with Eυ′ ≫ Eu′ > Ew′. The spanwise component is about 4 times or more that of the streamwise component, which is consistent with the measurements of Savelyev et al. (2012). The growth rates of Eu′ and Eυ′ vary around 0.1, which are also close to the measurements.
The total TKE (Fig. 1b) also grows exponentially with an increasing growth rate in the early stage and then continues growing at a reduced rate. The growth rate ranges from 0.2 to 0.8, which is consistent with the simulation result of Savelyev et al. (2012) (≈0.58 for ak ≈ 0.18) and the modeling result of Teixeira and Belcher (2002) (≈0.35 for ak ≈ 0.2). Similar growth trends are also observed in the streamwise and spanwise turbulent intensities. The vertical component of the total TKE, however, grows with an increasing growth rate and dominates the turbulent flow in the entire period. The components of the total TKE therefore exhibit anisotropy with Ew′ > Eυ′ > Eu′. This is different from the case of turbulence underneath a nondeformable interface and distorted by mean shear (Tsai 1998; Tsai et al. 2005), in which Eu′ > Eυ′ > Ew′. These distinct turbulent structures near the water surface and in the bulk suggest a different turbulence production mechanism from that of shear turbulence (Teixeira and Belcher 2002, 2010).
4. Flow structures
To further elucidate the structures of the underlying turbulent flow, vertical distributions of the root-mean-square (rms) velocities, σu, συ, and σw, and the rms turbulent velocities, σu′, συ′, and σw′, are depicted in Fig. 2. The distribution of σu resembles that of σw; both are dominated by surface waves. These distributions vary insignificantly in time, indicating the slow attenuation of the waves. In contrast, significant temporal variations can be observed in the distributions of rms turbulent velocities. The profile of συ is equivalent to that of συ′ since 〈υ〉 ≈ 0.
Vertical distributions of (a) the rms streamwise velocity σu, (b) the rms spanwise and vertical velocity συ and σw, (c) the rms streamwise turbulent velocity σu′, (d) the rms spanwise turbulent velocity συ′, and (e) the rms vertical turbulent velocity σw′ at various time instances ranging from t = 15T0 to 30T0 increasing by 3T0.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Vertical distributions of (a) the rms streamwise velocity σu, (b) the rms spanwise and vertical velocity συ and σw, (c) the rms streamwise turbulent velocity σu′, (d) the rms spanwise turbulent velocity συ′, and (e) the rms vertical turbulent velocity σw′ at various time instances ranging from t = 15T0 to 30T0 increasing by 3T0.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Vertical distributions of (a) the rms streamwise velocity σu, (b) the rms spanwise and vertical velocity συ and σw, (c) the rms streamwise turbulent velocity σu′, (d) the rms spanwise turbulent velocity συ′, and (e) the rms vertical turbulent velocity σw′ at various time instances ranging from t = 15T0 to 30T0 increasing by 3T0.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
The profiles of the turbulent velocities exhibit vertical variability and strong anisotropy, supporting the characteristic features of TKE evolutions in Fig. 1. Isotropy of the turbulence field can only be observed in a narrow region below ζ ≈ 0.8 (z ≈ −0.16λ). Above this isotropic region and approaching the water surface, σw′ increases more rapidly than σu′ and συ′, that is, σw′ > σu′ > συ′; σw′ reaches its maximum at ζ ≈ 0.94 (z ≈ −0.05λ). Drastic change in the structure of the turbulence field occurs in the surface layer immediately beneath the water surface (0 > z > ≈ −0.05λ); the vertical turbulent velocity decreases rapidly approaching the surface, accompanying a drastic increase of the spanwise component, while the streamwise turbulent velocity increases gradually and then decreases near the water surface. In this surface layer, the anisotropic structure of the turbulence field evolves from σw′ > σu′ > συ′ in the submerged water to συ′ > σu′ > σw′ in the region immediately beneath the water surface. At the water surface, συ′ is about twice σu′, consistent with the numerical and experimental results of Savelyev et al. (2012).
In addition to the anisotropic turbulent structure near the surface, previous experimental and numerical results of Savelyev et al. (2012) and the RDT model of Teixeira and Belcher (2002) also suggested the dependence of turbulence intensity on wave phase. Figure 3 depicts the distributions of the phase-averaged wavy velocities and rms turbulent velocities on the x–z plane at t = 30T0. The velocity field is dominated by the motion of surface waves, as shown in Figs. 3a and 3b. The maximum streamwise turbulent velocity is observed near the crest slightly toward the front face of the wave, consistent with the observation of Savelyev et al. (2012) and the model of Teixeira and Belcher (2002), whereas the spanwise turbulent velocity attains its maximum on the backward surface between crest and trough. Note that the intensified spanwise turbulence is confined within a thin layer immediately beneath the surface between the crests. The vertical turbulent velocity attains its maximum in a submerged depth near the backward surface between the wave crest and trough.
Representative contour distributions of the phase-averaged (a) streamwise wavy velocity, (b) vertical wavy velocity, (c) rms streamwise turbulent velocity, (d) rms spanwise turbulent velocity, and (e) rms vertical turbulent velocity at t = 30T0. The spanwise wavy velocity is not shown as it is much smaller than the other velocity components. The horizontal locations of maximum turbulent velocities are marked with vertical arrows. Note that the horizontal and vertical coordinates have the same scale. The wave propagates from left to right.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Representative contour distributions of the phase-averaged (a) streamwise wavy velocity, (b) vertical wavy velocity, (c) rms streamwise turbulent velocity, (d) rms spanwise turbulent velocity, and (e) rms vertical turbulent velocity at t = 30T0. The spanwise wavy velocity is not shown as it is much smaller than the other velocity components. The horizontal locations of maximum turbulent velocities are marked with vertical arrows. Note that the horizontal and vertical coordinates have the same scale. The wave propagates from left to right.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Representative contour distributions of the phase-averaged (a) streamwise wavy velocity, (b) vertical wavy velocity, (c) rms streamwise turbulent velocity, (d) rms spanwise turbulent velocity, and (e) rms vertical turbulent velocity at t = 30T0. The spanwise wavy velocity is not shown as it is much smaller than the other velocity components. The horizontal locations of maximum turbulent velocities are marked with vertical arrows. Note that the horizontal and vertical coordinates have the same scale. The wave propagates from left to right.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
5. Surface signatures
Savelyev et al. (2012) attributed the anisotropy of near-surface turbulence to coherent streamwise eddies, which form elongated thermal streaks in the infrared image. In the present numerical simulation, surface streaming similar to that observed in the experiment also emerges on the water surface. Instantaneous distributions of the streamwise velocity, streamwise turbulent velocity, spanwise velocity, and temperature on the water surface at t = 30T0 are shown in Fig. 4. The streamwise velocity distribution is dominated by the motions of surface waves (Fig. 4a). Filtering out the wavy component, the resulting distributions of turbulent streamwise velocity and the corresponding spanwise and temperature contours are characterize by elongated streaks carrying cooler and fast-moving fluids. Between these narrow, cool streaks are relatively warmer bands of slow-moving and diverging flows.
Representative contour distributions of the (a) streamwise velocity, (b) streamwise turbulent velocity, (c) spanwise velocity, and (d) temperature on the water surface at t = 30T0. The wave propagates from left to right. Examples of cool streaks and warm bands are marked by circular dots and plus signs, respectively.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Representative contour distributions of the (a) streamwise velocity, (b) streamwise turbulent velocity, (c) spanwise velocity, and (d) temperature on the water surface at t = 30T0. The wave propagates from left to right. Examples of cool streaks and warm bands are marked by circular dots and plus signs, respectively.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Representative contour distributions of the (a) streamwise velocity, (b) streamwise turbulent velocity, (c) spanwise velocity, and (d) temperature on the water surface at t = 30T0. The wave propagates from left to right. Examples of cool streaks and warm bands are marked by circular dots and plus signs, respectively.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
To reveal the underlying flow structure forming the streaky surface signatures, distributions of streamwise-averaged temperature; streamwise, spanwise, and vertical velocities; and streamwise vorticity on the cross-stream y–ζ plane at t = 30T0 are shown in Fig. 5. The distribution of temperature consists of distinct downward cool and upward warm tongues, emerging as cool streaks and warm bands, respectively, at the water surface (Fig. 5a). Beneath the cool streaks there are fast-moving streamwise jets and downwellings (Figs. 5b,d) accompanying convergent flows immediately beneath the water surface and divergent flows in the submerged water. An opposite velocity configuration emerges beneath the warm surface bands. The above two velocity fields form counterrotating pairs of circulatory structure. This flow pattern is also evidenced by the distribution of streamwise vorticity, which clearly reveals the array of counterrotating vortex pairs. The vortical flow structure is identical to that of Langmuir circulations. The averaged spacing between the streaks (≈0.3λ) and the depth of the vortical cells (≈0.15λ) approximate that of Langmuir cells, which arise from the interaction between the wind-driven shear flow and surface waves of a similar wavelength and steepness (Tsai et al. 2013); the characteristic velocity and vorticity of the cells, however, are one order of magnitude less than that of Langmuir cells.
Streamwise-averaged distributions on (y, ζ) plane of various flow properties: (a) temperature, (b) streamwise velocity, (c) spanwise velocity, (d) vertical velocity, and (e) streamwise vorticity at t = 30T0. The positive streamwise velocity points out of the plane, the positive spanwise velocity points to the right, the positive vertical velocity is upward, and the positive vorticity is counterclockwise. The locations of examples of cool downwelling and warm upwelling are marked by downward and upward arrows, respectively. The accompanying convergent/divergent flows are indicated by horizontal arrows.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Streamwise-averaged distributions on (y, ζ) plane of various flow properties: (a) temperature, (b) streamwise velocity, (c) spanwise velocity, (d) vertical velocity, and (e) streamwise vorticity at t = 30T0. The positive streamwise velocity points out of the plane, the positive spanwise velocity points to the right, the positive vertical velocity is upward, and the positive vorticity is counterclockwise. The locations of examples of cool downwelling and warm upwelling are marked by downward and upward arrows, respectively. The accompanying convergent/divergent flows are indicated by horizontal arrows.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Streamwise-averaged distributions on (y, ζ) plane of various flow properties: (a) temperature, (b) streamwise velocity, (c) spanwise velocity, (d) vertical velocity, and (e) streamwise vorticity at t = 30T0. The positive streamwise velocity points out of the plane, the positive spanwise velocity points to the right, the positive vertical velocity is upward, and the positive vorticity is counterclockwise. The locations of examples of cool downwelling and warm upwelling are marked by downward and upward arrows, respectively. The accompanying convergent/divergent flows are indicated by horizontal arrows.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
6. Energetics budget
The vertical variations of the averaged TKE transport terms normalized by the TKE density at t = 20T0 and 25T0 are shown in Fig. 6. The pressure strain term Π′, which is associated with the blocking effect of the water surface, converts streamwise TKE to the spanwise component immediately beneath the surface and vertical TKE to horizontal components beneath the viscous sublayer (Tsai et al. 2005). Production arising through the Stokes shear P′ contributes positively to TKE budget; its impact, however, is not as significant as that in wind-driven Langmuir turbulence (McWilliams et al. 1997) and shear-distorted turbulence. Instead, the advection of turbulence by the velocity straining of the waves A′ dominates TKE transport near the water surface. This is consistent with the analysis of Phillips (1961). Transport from waves to turbulence W′ is, in general, positive except within a small depth near the water surface. This extraction of turbulence energy from surface waves is relatively weak and results in a slow attenuation of the waves.
Vertical distributions of averaged terms in the TKE budget equation normalized by the TKE density at t = 20T0 (solid lines) and 25T0 (dashed lines): (a) transport by pressure perturbation Π′, (b) transport by turbulent eddies T′, (c) mechanical production from the mean flow P′, (d) transport between wave and turbulence W′, (e) advection of turbulence by waves A′, (f) transport by molecular diffusion D′, and (g) viscous dissipation rate ε′.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Vertical distributions of averaged terms in the TKE budget equation normalized by the TKE density at t = 20T0 (solid lines) and 25T0 (dashed lines): (a) transport by pressure perturbation Π′, (b) transport by turbulent eddies T′, (c) mechanical production from the mean flow P′, (d) transport between wave and turbulence W′, (e) advection of turbulence by waves A′, (f) transport by molecular diffusion D′, and (g) viscous dissipation rate ε′.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
Vertical distributions of averaged terms in the TKE budget equation normalized by the TKE density at t = 20T0 (solid lines) and 25T0 (dashed lines): (a) transport by pressure perturbation Π′, (b) transport by turbulent eddies T′, (c) mechanical production from the mean flow P′, (d) transport between wave and turbulence W′, (e) advection of turbulence by waves A′, (f) transport by molecular diffusion D′, and (g) viscous dissipation rate ε′.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0121.1
The transport terms, including transport of energy by pressure perturbation, by turbulent eddies, from the Stokes shear, between turbulence and wave perturbation, and by molecular diffusion, all contribute in comparable amounts to the energetics budget. These productions, however, are less than that attributed to the advection and distortion of turbulence by the surface waves.
7. Concluding remarks
The present numerical simulation of gravity surface waves propagating over a turbulence field reveals a significant growth of the initially weak turbulence and so supports previous theories, observations, and modeling that the nonbreaking surface waves induce turbulence. The numerical results also confirm the formation of an array of counterrotating streamwise vortex pairs beneath the water surface, which induce elongated surface streaks. The characteristic spanwise and vertical scales of these vortices resemble those of Langmuir cells. Such a resemblance in integral length scales suggests the similar dynamics of the two flow systems. Shear production of initial turbulence acts as the source of fluctuating vortices for the subsequent wave–turbulence interaction. Stokes drift then tilts vertical vorticities into the horizontal direction and subsequently stretches and amplifies it into elongated streamwise vortices. It is postulated that the present turbulence generated by surface waves could be considered as the limiting case of Langmuir turbulence with no shear (Teixeira and Belcher 2010).
Babanin (2006) hypothesized that the orbital motions of a monochromatic wave will transit from laminarity to turbulence if the amplitude-based Reynolds number, Rew = a2ων−1, exceeds a threshold of approximately 3000, where a is the wave amplitude, ω is the angular frequency, and ν is the kinematic viscosity. The hypothesis was tested on mechanically generated laboratory waves using dye dispersion. Subsequent experiments by various groups, however, resulted in different or conflicting conclusions. The particle image velocimetry measurements in Babanin and Haus (2009) showed a lower critical Rew of 1300 (or 2300 if the wave amplitude is scaled to the surface). The experiments of Beyá et al. (2012), however, did not detect turbulent dye dispersion for Rew up to 7000. Dai et al. (2010), on the other hand, observed significant enhancement in thermal destratification in the presence of surface waves for Rew as low as 780. In contrast to these experiments, systematic measurements were conducted in Savelyev et al. (2012) by varying the wavelength and steepness. No clear threshold of Rew has been identified from the results. The present numerical simulation also observed wave-induced turbulence at the wave Reynolds number (≈300) one order of magnitude lower than that proposed by Babanin (2006). In fact, the experimental results of Dai et al. (2010) and Savelyev et al. (2012) indicate the intensification of turbulence to be proportional to the steepness ak and the characteristic length k−1. Using the linear dispersion relation for rewriting the wave Reynolds number as Rew = a2ων−1 ≈ (ak)2k−1.5g0.5ν−1, and since 
Acknowledgments
This work was supported by a grant from Taiwan Ministry of Science and Technology under contract NSC 101-2611-M-002-MY3.
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