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  • View in gallery

    (a) Instrumentation setup from R/P FLIP. (b) Map of the deployment area; R/P FLIP was moored off the coast of southern California in August 2003.

  • View in gallery

    (a) Wind speed (black) and direction (gray) measured with the sonic anemometer for the duration of the experiment. (b) Spectragram of the surface elevation Sηη, measured with the laser altimeter. Note the two peaks at the frequencies of the swell and the wind waves. (c) Significant wave height measured with the altimeter and compared with that obtained from the RDI waves ADCP (ṡ).

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    Infrared images of the surface temperature and surface turbulent velocity field calculated from the passive temperature pattern displacements between each temperature image and the one taken 33 ms later. Shown are the temperature and surface velocity for a wind speed of (a) U10 = 0.5 m s−1 with large upwelling/convection and (b) U10 = 3.8 m s−1 where cold streaks (Langmuir circulations) are clearly visible.

  • View in gallery

    Two-dimensional wavenumber spectra of the (a) surface divergence, (b) vorticity, and (c) surface temperature for a wind speed of 8 m s−1. The spectra were calculated from an average of 120 individual spectra taken over 1 min of data starting at 0503 UTC 23 Aug 2003. (d) Shown are the directions for the wind, from 305°; the wind waves, to 117°; the swell, 97°; the vorticity spectral peak, 52°; the divergence spectral peak, 122°; and the temperature spectral peak, 43°.

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    (a) Directions of the wind, wind waves, swell, and current. (b) Directions of the streaks (i.e., the direction perpendicular to that of the peak spectral densities and the maximum variance) for the vorticity, divergence, shear, and normal deformation.

  • View in gallery

    Two-dimensional wavenumber spectra of the (a) surface divergence, (b) vorticity, and (c) surface temperature for a wind speed of 6.4 m s−1. The spectra were calculated from an average of 120 individual spectra taken over 1 min of data starting at 1118 UTC 26 Aug 2003. (d) Shown are the directions for the wind, from 294°; the wind waves, to 114°; the vorticity spectral peak, 149°; the divergence spectral peak, 66°; and the temperature spectral peak, 33°. The swell direction could not be measured accurately.

  • View in gallery

    Time series of surface displacement and the spatial variance of the along wave surface turbulence over a 120-s record starting at 0500 UTC 24 Aug 2003.

  • View in gallery

    Spectragrams of the (a) surface displacement η and (b) along wave turbulent surface velocity u1 at night. (c),(d) Shown are the squared coherence and phase between the surface displacement and the turbulent surface velocity, respectively.

  • View in gallery

    Variance of the wave-coherent, along wave, surface velocity turbulence with the (a) wind speed and (b) rms surface wave slope. The large symbols show the bin-averaged data, and the solid lines are the fits to the bin-averaged data.

  • View in gallery

    Fraction of to as a function of the wave slope. The large symbols show the bin-averaged data, and the solid gray line shows the rapid distortion theory prediction of Eq. (17).

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Measurements of Ocean Surface Turbulence and Wave–Turbulence Interactions

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  • 1 College of Marine and Earth Studies, University of Delaware, Newark, Delaware
  • 2 Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
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Abstract

The uppermost layers of the ocean, along with the lower atmospheric boundary layer, play a crucial role in the air–sea fluxes of momentum, heat, and mass, thereby providing important boundary conditions for both the atmosphere and the ocean that control the evolution of weather and climate. In particular, the fluxes of heat and gas rely on exchange processes through the molecular layers, which are usually located within the viscous layer, which is in turn modulated by the waves and the turbulence at the free surface. The understanding of the multiple interactions between molecular layers, viscous layers, waves, and turbulence is, therefore, paramount for an adequate parameterization of these fluxes. In this paper, the authors present evidence of a clear coupling between the surface waves and the surface turbulence. When averaged over time scales longer than the wave period, this coupling yields a spatial relationship between surface temperature, divergence, and vorticity fields that is consistent with spatial patterns of Langmuir turbulence. The resulting surface velocity field is hyperbolic, suggesting that significant stretching takes place in the surface layers. On time scales for which the surface wave field is resolved, the authors show that the surface turbulence is modulated by the waves in a manner that is qualitatively consistent with the rapid distortion theory.

Corresponding author address: Fabrice Veron, College of Marine and Earth Studies, University of Delaware, 112C Robinson Hall, Newark, DE 19716. Email: fveron@udel.edu

Abstract

The uppermost layers of the ocean, along with the lower atmospheric boundary layer, play a crucial role in the air–sea fluxes of momentum, heat, and mass, thereby providing important boundary conditions for both the atmosphere and the ocean that control the evolution of weather and climate. In particular, the fluxes of heat and gas rely on exchange processes through the molecular layers, which are usually located within the viscous layer, which is in turn modulated by the waves and the turbulence at the free surface. The understanding of the multiple interactions between molecular layers, viscous layers, waves, and turbulence is, therefore, paramount for an adequate parameterization of these fluxes. In this paper, the authors present evidence of a clear coupling between the surface waves and the surface turbulence. When averaged over time scales longer than the wave period, this coupling yields a spatial relationship between surface temperature, divergence, and vorticity fields that is consistent with spatial patterns of Langmuir turbulence. The resulting surface velocity field is hyperbolic, suggesting that significant stretching takes place in the surface layers. On time scales for which the surface wave field is resolved, the authors show that the surface turbulence is modulated by the waves in a manner that is qualitatively consistent with the rapid distortion theory.

Corresponding author address: Fabrice Veron, College of Marine and Earth Studies, University of Delaware, 112C Robinson Hall, Newark, DE 19716. Email: fveron@udel.edu

1. Introduction

The uppermost layers of the ocean, along with the lower atmospheric boundary layer, play a crucial role in the air–sea fluxes of momentum, heat, and mass, thereby providing important boundary conditions for both the atmosphere and the ocean that control the evolution of weather and climate.

The first models of boundary layers on both sides of the air–sea interface were developed from our understanding of the turbulent flow over rigid flat surfaces and extended to the field after the landmark Kansas experiment on the terrestrial boundary layer (Businger et al. 1971). Consequently, models of neutrally stratified flows are based on the well-known “law of the wall,” which depends on the assumptions of a constant stress layer and horizontal homogeneity. When the flow is stratified, the “law of the wall” is further modified to include the so-called Monin–Obukhov similarity theory. The coupled air–sea boundary layers, however, are dynamically quite different from a solid flat surface. For example, the velocity field is not required to vanish at the interface. The ocean surface responds with drift currents, surface waves, and turbulent eddies over a broad range of scales. There are other phenomena—such as bubble injection, spray ejection, rainfall, foam, and surfactants—which further affect the dynamics and complicate the problem. Consequently, one would expect the dynamics of such an interfacial layer to be significantly different from that over a solid flat surface under similar forcing conditions. Indeed, although there is evidence that, in general and on average, the Monin–Obukov similarity theory holds over the ocean (Edson and Fairall 1998; Edson et al. 2004), there are also some notable differences. Over the last decade or so, it has also become apparent that surface wave processes can play an important role in the kinematics and dynamics of the boundary layers (e.g., Janssen 1989, 1999; Komen et al. 1994; Belcher and Hunt 1998; Hristov et al. 1998; Edson and Fairall 1998; Uz et al. 2002; Sullivan et al. 2004, 2007), and recent measurements and models of the drag of the sea surface on the atmosphere at moderate-to-high wind speeds in fact suggest that much of the momentum transfer at the surface is supported by the form drag of the waves. There is also evidence that the air–sea heat flux can be modulated by the wavy surface (Sullivan and McWilliams 2002; Veron et al. 2008a). In addition, breaking waves generate turbulence (Rapp and Melville 1990; Agrawal et al. 1992; Thorpe 1993; Melville 1994; Anis and Moum 1995; Melville 1996; Terray et al. 1996; Veron and Melville 1999a), which along with small-scale Langmuir circulations and coherent structures (Melville et al. 1998; Veron and Melville 1999b; McWilliams et al. 1997; Veron and Melville 2001; Sullivan et al. 2004, 2007), may lead to enhanced dissipation and mixing with significant departures from the “law of the wall” and may also result in increased heat and gas transfer (Jähne et al. 1987; Hasse 1990; Jähne and Haußecker 1998; Zappa et al. 2001; Garbe et al. 2004; Schimpf et al. 2004; Turney et al. 2005).

With such richness in the phenomena and dynamics at the surface, one can also expect significant interactions between currents, surface waves, and turbulence. A well-known example is Langmuir circulation, which results from the interactions between vorticity and the Stokes drift generated by the surface waves (Leibovich 1983; Thorpe 2004). On smaller time scales, there is also evidence that the turbulence can be significantly coupled with the surface waves (Lumley and Terray 1983; Cheung and Street 1988a,b; Thais and Magnaudet 1995; Teixeira and Belcher 2002).

Because the air–sea transfers of heat and gas partly rely on exchanges through the diffusive surface layers and because these layers are typically smaller than the viscous sublayer, our ability to adequately quantify these fluxes depends on our understanding of the small-scale turbulence that controls the dynamics of the molecular layers.1 In turn, our understanding of the small-scale turbulence depends on our understanding of the multiple interactions between the turbulence, currents, and surface waves.

In this paper, we present results from field experiments that show evidence of coupling between the surface temperature, the surface turbulent velocity fields, and the surface waves. We also show evidence of coupling between the surface waves and the turbulent fields. In section 2, we summarize the experimental setup. In section 3, we present measurements of surface waves and the velocity fields that provide the basis for the results presented here. Section 4 shows the coupling of the waves with the surface kinematic fields and surface temperature over long time scales, when the wave effects are averaged in a manner similar to that which leads to the Craik–Leibovich mechanism (CL II) for the formation of Langmuir circulation (Leibovich 1983). Finally, section 5 shows evidence of distortion of the surface turbulence by the waves over short times scales when the wave field is resolved.

2. Experiments

The measurements described here were obtained from a field experiment conducted from R/P FLIP, moored approximately 150 miles off the coast of southern California, west of Tanner Bank (32°40.20′N, 119°19.46′W; 300-m depth), during 20–26 August 2003.

The main instruments are composed of an integrated active and passive infrared imaging and altimetry system (Veron et al. 2008b), and an eddy covariance atmospheric flux package. Both systems are described in more detail below.

a. Infrared imaging and altimetry system

The active and passive infrared imaging and altimetry system includes an infrared camera (Amber Gallileo), a 60°W air-cooled CO2 laser (Synrad Firestar T60) equipped with an industrial marking head (Synrad FH index) with two computer-controlled galvanometers, a laser altimeter (Riegl LD90-3100EHS), a video camera (Pulnix TM-9701), a 6-degree-of-freedom motion package (Watson Gyro E604), and a single-board computer (PC Pentium 4). All instruments were enclosed in weatherproof, air-conditioned aluminum housing. All instruments and computers were synchronized to within 2 ms and to GPS time. The infrared camera was set to record temperature images (256 × 256 pixels) at 60 Hz, with a 2-ms integration time, yielding better than 15-mK resolution. The footprint of the infrared camera was approximately 2 m × 2 m. The video camera (768 × 484 pixels) was synchronized to the infrared camera and acquired full frames at 30 Hz. The footprint of the infrared camera was contained within the footprint of the video camera. The infrared CO2 laser and accompanying marking head were used to actively lay down patterns of thermal markers on the ocean surface to study the rate of decay of an imposed surface temperature perturbation while tracking the Lagrangian velocity, shear, and vorticity at the surface. Finally, the laser altimeter measured the distance to the water surface at 12 kHz (averaged down to 50 Hz) with a footprint of 5-cm diameter, contained within both the infrared and video images. The system, among other things, yields the velocity at the surface by tracking active thermal markers laid down with the CO2 laser [thermal marking velocimetry (TMV)] or by performing cross-correlation analysis on the passive surface temperature fields [particle image velocimetry (PIV)]. The detailed performance of the passive and active IR measurement system for ocean-surface kinematics is described in Veron et al. 2008b.

b. Eddy covariance system

In addition to the optical infrared system, we used an eddy covariance system to acquire supporting meteorological and atmospheric boundary layer flux data. The eddy covariance system included a three-axis anemometer–thermometer (Campbell CSAT 3), an open path infrared hygrometer/CO2 sensor (LI-7500), a relative humidity/temperature sensor (Vaisala HMP45), and a net radiometer (CNR1). Turbulent fluxes of momentum, heat, and moisture were calculated using the covariance method over 30-min averages. The sonic temperature was corrected for humidity and pressure, rotation angles for correcting the orientation of the anemometer were obtained from the 30-min averages of the velocity components, and the latent heat flux was corrected for density variations (Webb et al. 1980). For the purposes of this paper, the good agreement between the flux covariance data and estimates using the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) 3.0 algorithm (Fairall et al. 1996; Fairall et al. 2003) supports the use of the Monin–Obukhov similarity theory for these open-ocean conditions (Edson et al. 2004), and it gave us confidence that our covariance measurements were adequate measurements of the total flux above the diffusive and wave boundary layers (see Veron et al. 2008b).

The instruments were deployed at the end of the port boom of R/P FLIP, approximately 18 m from the hull at an elevation of 13 m above mean sea level (MSL).

The infrared optical system was set up with the view ports clear of the end of the boom. The meteorological package was placed 15.5 m from the hull on the port boom with all instruments facing upwind with the exception of the net radiometer, which was deployed with its axis downwind (Fig. 1).

We also deployed two fast response, high-resolution, subsurface thermistors [RBR Ltd. (1040; 95 ms) 1-Hz sampling rate] placed at 1.2 and 2 m from the mean sea level and fixed to the hull of R/P FLIP. An upward-looking waves-enabled ADCP [RD Instruments (RDI) Workhorse 600 kHz] was also rigidly mounted to FLIP at 15-m depth and yielded directional wave spectra and significant wave height for the duration of the experiment. Finally, GPS position and R/P FLIP heading were sampled at 50 Hz and used to correct the ADCP data for FLIP motion and to align all other directions to true north.

The experiment described here took place from 20 to 26 August 2003. During this time, a 13.5-s swell was propagating toward 87° ± 17°. For the duration of the experiment, the wind direction was relatively constant from 300° ± 12°. The wind speed, however, picked up from approximately 1 m s−1 at the beginning of the week to 10 m s−1 a few days later as a storm passed (see Fig. 2a). Accordingly, a wind sea developed with waves of periods ranging from 1.7 to 6.3 s, propagating toward 124° ± 12°. Finally, the surface currents were propagating toward 116° ± 21° until 25 August 2003, at which point the current turned significantly. The keel of FLIP was facing 134° ± 4°, providing ideal conditions with wind, swell, surface currents, and wind waves roughly aligned and with the study area at and below the port boom, clear of R/P FLIP’s turbulent wake in both the air and the water.

3. Surface waves and velocity measurements

To examine the coupling between the surface temperature, the surface turbulent velocity fields, and the surface waves, the surface wave elevation and direction of propagation must be measured. This was achieved here by single-point laser altimetry measurements of the surface displacement combined with the direct measurements of the two-dimensional surface velocity over an area encompassing the single-point altimetry data. Details are presented next.

a. Surface altimetry

The optical infrared system was equipped with a 12-kHz single-point laser altimeter with a footprint on the surface of about 5-cm diameter at this operating height. It yielded instantaneous surface height η at 50 Hz after averaging the raw 12-kHz data. The altimeter relies on the travel time measurement of a light pulse, which reflects from the water surface. The altimeter performs significantly better when small-scale surface roughness is present at higher wind speeds. For example, at 0.8 m s−1 (8.4 m s−1), 10% (67%) of the raw returns led to a valid height measurement. When necessary, dropouts were corrected by interpolating the height measurements using piecewise cubic Hermite polynomials. The footprint of the altimeter was situated 22 cm from the center of the infrared image. The altimeter data were used to measure wave parameters (Fig. 2), and the instantaneous distance from the imaging system to the ocean surface was used to calculate an instantaneous image resolution for the evaluation of the surface kinematics.

Figure 2 shows the wind speed and direction along with the spectragram of the surface elevation, and the significant wave height, for the duration of the experiment. The surface elevation spectragram (Fig. 2b) shows the swell peak at 15–20-s period and the wind wave peak that appears as the wind rises from 2 to 8 m s−1 and exhibits a classical downshift in frequency as the wind wave field grows. Figure 2c also shows good agreement between the significant wave height measured with the altimeter and that measured with the waves ADCP.

b. Surface velocity

We have shown that it is possible to calculate the two-dimensional surface velocity u and other kinematic variables from infrared surface temperature images (Veron et al. 2008b). The velocity at the surface can further be projected onto the directions parallel and perpendicular to the wind or wave directions. Here, we employ an algorithm typically employed for PIV and calculate the surface velocity by performing a running normalized cross correlation on subwindows of surface temperature images separated by 33 ms. Each cross correlation yields a local average displacement (over the subwindow), which is then used in conjunction with the instantaneous image resolution and time interval between the successive images to estimate an average surface horizontal velocity. In most cases, we used 8 × 8 pixel subimages with 50% linear overlap and image pairs separated by 166 ms, yielding six velocity maps per second, each having a spatial resolution of approximately 3 cm. Details on the performance of this technique can be found in Veron et al. (2008b). The surface velocity field u obtained is the sum of velocities associated with the mean currents uc, the orbital velocity of the surface waves uw, and the turbulence u′. Figure 3 shows examples of the turbulent surface velocity field ui (i = 1, 2), obtained from the deviation from the image mean velocity (which contains both surface currents and orbital motion) and calculated from an infrared surface temperature image taken under various wind speeds. The velocity fields are shown with the arrows and are overlaid on the first temperature image from which they were calculated. To avoid sky reflectance and other effects, only nighttime temperature time series were used. We show here the turbulent velocity rather than the total velocity for easier visual comparison with the underlying temperature structures. The velocity field shows a variety of phenomena with vortices and regions of surface divergence and convergence that appear, in some cases, to be correlated with the surface temperature structure. For example, the region of warmer surface temperature in Fig. 3a appears to be associated with enhanced surface divergence. We note also that during the experiments presented here, at nighttime, the net air–sea heat flux was positive upward (see Veron et al. 2008b for details).

The benefit of this PIV technique is the ability to simultaneously study processes involved in the evolution of the surface temperature and surface velocity fields, including mean currents, wave orbital motions, and turbulence. In particular, direct surface velocity measurements, along with single-point altimetry, allow us to measure surface wave directional spectra and the spectra of the surface kinematics, as illustrated below.

c. Surface wave directional spectra

For the scope of the work presented here, the single-point altimetry from the laser altimeter and the surface velocity from the passive temperature images were combined to obtain the directional spectra of the surface wave field. Velocities at each corner of the infrared image were obtained by averaging all PIV estimates contained in local 30 cm × 30 cm image areas. These four-corner velocities were then combined with the one-point surface elevation measurements from the laser altimeter. Directional spectra were then computed using the Wave Analysis for Fatigue and Oceanography (WAFO, Lund University) software package. The directional wave frequency spectrum can be associated with the cross spectrum between any two of the considered measured quantities, that is, velocities and elevation (Young 1994). To obtain estimates of the full directional spectrum, one must choose analysis methods capable of inverting the relation between those quantities; we considered two techniques: 1) the iterative maximum likelihood method (IMLM; Pawka 1983), which corrects the directional spreading caused by the conventional MLM technique (Isobe et al. 1984; Capon 1969), and 2) the maximum entropy method (MEM; Lygre and Krogstad 1986). These two techniques showed some differences in the computed spectra, particularly in the spreading of the energy in the vicinity of the wave frequency peaks, but no significant directional differences at the wave energy peaks for swell and wind wave components. Wave peak directions and frequencies also showed a good match with those obtained from a hull-mounted RDI Wave ADCP. Only the peak frequencies and directions were used in the subsequent analysis.

4. Surface kinematics

In addition to measuring the velocity field at O(1)-cm spatial resolution, we can also infer the velocity gradient tensor ∂ui/∂xj (Raffel et al. 1998). This gives the velocity gradient tensor with the same resolution and coverage as the velocity field. From there, one can infer all the two-dimensional first derivative kinematic fields, including divergence D, vorticity ω, shear S, and normal deformation N:
i1520-0485-39-9-2310-e1
or
i1520-0485-39-9-2310-e2
i1520-0485-39-9-2310-e3
i1520-0485-39-9-2310-e4
i1520-0485-39-9-2310-e5
Here, we examine the spatial distribution of the kinematic fields. Figure 4 shows the two-dimensional wavenumber spectra for the divergence and vorticity, which were obtained from the PIV processing of the temperature images. Figure 4 also shows the two-dimensional wavenumber spectrum for the surface temperature. The spectra were obtained by averaging 120 individual spectra of a single velocity or temperature image, taken randomly during one minute of data. The data shown here were collected under wind speeds of 8 m s−1 at the beginning of the wind event at 0503 UTC 23 August 2003 during the development of the wind sea. First, we note that the two-dimensional spectra for the surface divergence, vorticity, and temperature exhibit some anisotropy, indicating that there is some degree of directionality in the structures of these spatial fields. Second, we note that the spectrum of the vorticity (Fig. 4b) is approximately aligned with that of the temperature (Fig. 4c). This is a consequence of the coupling between surface temperature and velocity. We know from previous work (Veron and Melville 2001) that small-scale Langmuir circulations and coherent turbulence (streaks) are present at these scales over this range of wind speeds. Even if instantaneous realizations of the surface temperature and velocity do not clearly exhibit such structures, time averages can reveal their presence. Here, the two-dimensional spectra are averaged over one minute of data. Colder streaks in the temperature images correspond to water parcels that have been organized by the underlying Langmuir cells and have collected in windrows at the surface. These water parcels have been in contact with the air the longest and therefore have had time to exchange significant heat, leading to colder surface temperatures. In addition, these parcels of water also have been exposed to the wind stress the longest and therefore, the “cold” surface temperature streaks also correspond to surface “jets” in which the surface velocity is larger than that of the surrounding fluid. A schematic of these surface jets can be found in Leibovich (1983). As a consequence, there are two horizontal shear layers on each side of the temperature streaks where vertical vorticity is enhanced. Thus, when Langmuir circulations are present, the surface exhibits streaks of enhanced velocity, or surface jets, along with streaks of colder temperature and streaks of vorticity, all in the same direction and aligned with the wind. This is precisely what is shown in Fig. 4. The reader is reminded that the two-dimensional spectrum will show the largest signal in the direction of the largest variance, that is, the direction perpendicular to the streaks. For example, in the data shown in Fig. 4, the vorticity (temperature) spectrum peaks at 52° (43°C), indicating that the streaks in the vorticity fields are oriented along the direction 322°–142° (313°–133°) and roughly aligned with the wind.

Finally, we note that the spectrum of the divergence is perpendicular to that of the vorticity. This is also consistent with the presence of Langmuir circulations. Indeed, Langmuir circulations are generated from the positive feedback between the surface jets, vertical vorticity, and the shearing and stretching of the vertical vortex lines by the Stokes drift (Craik 1977; Leibovich 1977). This means that the surface waves are propagating in the downwind direction, with their wavenumber vector approximately aligned in the direction of the surface streaks. Accordingly, regions of divergence and convergence associated with the nonlinear surface wave field and even perhaps breaking waves will be aligned parallel to the wave fronts, perpendicular to the jets of colder surface temperature and enhanced vertical vorticity. Indeed, for the data shown in Fig. 4, the spectral peak for the divergence is approximately in the direction of the wind and wind waves, indicating that the structure in the surface divergence field is perpendicular to the predominant wind and wind wave directions, and also perpendicular to the streaks in the surface temperature and surface vorticity. Hence, when Langmuir circulations (or Langmuir turbulence) are present, the two-dimensional wavenumber spectra for the vorticity and divergence are perpendicular to one another with the spectrum for the vorticity parallel to that of the surface temperature. The presence of developing Langmuir circulations during the early stage of this wind event was also confirmed by visual inspection of the temperature images and visual observation of the surface during the experiment.

Before proceeding, it is useful to consider conditions and constraints on the surface velocity field that lead to orthogonality of the two-dimensional spectra of the divergence and vorticity, as is observed in Fig. 4. We start with the horizontal components of the mean surface velocity field defined by expanding the velocity in space about an arbitrary origin as follows:
i1520-0485-39-9-2310-e6
where the ui are approximated by quadratic functions of xi, with cubic and higher terms neglected. We can then estimate the divergence and vorticity as
i1520-0485-39-9-2310-e7
i1520-0485-39-9-2310-e8
respectively. When the two-dimensional spectra of the vorticity and divergence are orthogonal, the structures (streaks in the case of Langmuir circulations) in the divergence and vorticity fields are also orthogonal. Consequently, the gradient on the velocity and divergence fields are orthogonal as well. In other words,
i1520-0485-39-9-2310-e9
which leads to the following condition:
i1520-0485-39-9-2310-e10
Providing that χi ≠ 0, this leads to
i1520-0485-39-9-2310-e11
which in turns leads to a surface velocity field of the following hyperbolic form:
i1520-0485-39-9-2310-e12
We deduce here that when the surface velocity field is locally hyperbolic, which leads to significant stretching, then the two-dimensional wavenumber spectra for the vorticity and divergence are also orthogonal. Hyperbolicity is a sufficient condition for the spectra to be orthogonal, but it is not necessary. It is unclear whether or not the presence of Langmuir circulations leads to hyperbolic surface velocity fields and, to our knowledge, this question has not been examined. However, it is well established that Langmuir circulation will lead to significant stretching and, in particular, to the stretching of vortex lines, which incidentally leads to the positive feedback and reinforcement of the circulations. These data raise interesting questions about the nature of the surface velocity fields during periods of growing wind seas and developing Langmuir circulations.

In addition, the hyperbolic surface velocity field in Eq. (12) immediately leads to (∂S/∂xi)(∂N/∂xi) = 0; that is, the two-dimensional wavenumber spectra for the shear and normal deformation are also orthogonal. Although a hyperbolic velocity field is not a necessary condition for this orthogonality, it is easy to demonstrate that (∂D/∂xi)(∂ω/∂xi) = 0 implies (∂S/∂xi)(∂N/∂xi) = 0, and vice versa, consistent with the orthogonality of the shear and normal deformation fields shown here (Fig. 5).

As the wind speed and wind-wave field stabilize during 23 and 24 August 2003, we observe that the relationships between the two-dimensional spectra of the vorticity, divergence, shear, and normal deformation show some additional order. Not only are the two-dimensional spectra perpendicular in pairs—that is, the spectrum of vorticity (shear) is orthogonal to that of the divergence (normal deformation)—but the spectrum for the vorticity (divergence) is orthogonal to that of the normal deformation (shear). This is clearly demonstrated in Fig. 5, which shows the direction of the wind, waves and currents (Fig. 5a) along with the directions of the streaks (i.e., perpendicular to the direction of the peak spectral density) for the kinematic variables (Fig. 5b). This further apparent coupling and ordering in the surface kinematics is presumably a consequence of the stabilization of the surface wind wave field and the wind drift. Using the earlier formulation for the velocity fields, this additional orthogonality yields a particular solution (though not the only one possible) with χi = ±(−1)iγi, which indeed leads to a simpler, more constrained form of the local surface velocity fields
i1520-0485-39-9-2310-e13

We note here that typical hyperbolic surface velocity fields described by Eq. (13) show some two-dimensional stretching and regions where the streamlines converge to form surface streaks in a fashion that is reminiscent of the surface velocity field that one would expect in the presence of Langmuir circulations.

Later still during the week, on 26 August 2003 and beyond, the geometric relationship between all kinematic fields is lost and no obvious coupling can be found (Fig. 5b). Unfortunately, the surface currents turned significantly during these two days (Fig. 5a). Consequently, it is difficult to unequivocally tie the apparent “loss of order” and the breakdown of the orthogonality between kinematics fields to either the drop in wind speed and the relaxation in the surface stress, which presumably destroyed the Langmuir circulations, or to the change in the direction of the surface currents, which at times oppose the wave direction and Stokes drift.

Finally, Fig. 6 shows the two-dimensional wavenumber spectra for the divergence, vorticity, and surface temperature collected under wind speeds of 6.4 m s−1 at the end of the wind event at 1118 UTC 26 August 2003, after the breakdown of the orthogonality between the kinematics fields. Figure 6 shows a somewhat larger spreading in the spectra for both the divergence and vorticity. Still, two-dimensional wavenumber spectra for the vorticity and divergence appear to remain orthogonal. However, only the temperature structures remain aligned with the wind direction.

Together, the data presented earlier illustrate that there are clear couplings that occur on time scales of a minute and longer, which clearly appear out of long-term spectral averages. The time average used here is similar to that used in the development of the CL II mechanism and, perhaps not surprisingly, the data show that the surface kinematic fields are organized in a manner consistent with the presence of Langmuir circulations. These data highlight the need for further study of the relationship between the surface kinematics, surface waves, currents, and even perhaps the surface temperature and gas concentration (i.e., the surface molecular layer).

5. Modulation of turbulence by the waves

We have seen in the preceding section that the surface kinematics is coupled with the developing wind seas, on time scales longer than approximately a minute. This raises the question of the coupling between the surface turbulence, waves, and currents on shorter time scales and, in particular, the coupling between turbulence and surface waves.

As described earlier, the footprint of the laser altimeter was located within the infrared image and therefore within the measured surface velocity and surface kinematic fields. This allows us to examine the modulation of the surface velocity and kinematic fields by the waves. From the infrared images and the resulting velocity images, along with the wave directional spectra obtained from the processing described in section 3c, we have decomposed the surface velocity field into the components u1 and u2, which are aligned with and orthogonal to the direction of propagation of the wind waves, respectively. To avoid sky reflectance and other effects in the infrared images, only nighttime data were used. Also, as for the data shown in Fig. 3, we have subtracted image-mean velocity from every velocity vector of that particular PIV estimate. This operation subtracts both currents uc and wave orbital velocity uw, and yields the surface turbulent velocity field u′. From these we can generate time series of the statistics of the turbulent surface kinematics variables averaged over the footprint of the infrared imager.

For example, Fig. 7 shows time series of the variance of the turbulent surface velocity in the wave direction and the surface displacement η over a 120-s record starting at 0500 UTC 24 August 2003. There is a clear visual correlation between the two variables, with significant bursts of turbulence at the surface that correlate with the passage of surface waves. Also, high turbulent levels are located at or slightly behind the wave crests. Spectral analysis makes this a little clearer.

Figure 8a shows frequency spectragram of the surface displacement η for the duration of the FLIP experiment in August 2003. The quality of the near-IR laser altimeter data is dependent on the roughness of the sea surface with drop outs increasing for lower wind speeds. The data shown here were good for frequencies up to 1 Hz and are low-pass filtered at that frequency. The surface displacement spectra, denoted by Sηη, is typical, showing both wind wave and swell peaks. The wind wave peak appears with the wind event (Fig. 2), showing the classical downshift in wind wave peak frequency with time. The spectra also show a f−4 slope above the wind wave peak. Figure 8b shows that the frequency spectra for the turbulent velocity fluctuations in the direction of the wave propagation u1 exhibit peaks at the local maxima of the wind wave spectra. Figures 8c and 8d show the squared coherence and phase , between u1 and η, respectively. The figures show a peak in the squared coherence of approximately 0.2–0.3, rapidly going to zero at frequencies above approximately 1 Hz. The phase between the turbulence and surface displacement near the peak of the coherence is slightly negative, indicating that the maximum in turbulence intensity lags the maximum surface displacement. This is presumably a consequence of the enhanced wind stress and turbulence on the windward side of the wave. This is corroborated by identical results for the other kinematic variables (not shown). Incidentally, this is consistent with observations of warmer skin temperature also on the windward side of the waves that results from enhanced wind stress and the resulting turbulence, which leads to the disruption of the aqueous viscous layer and the thermal molecular layer, that is, the cool skin (Simpson and Paulson 1980; Miller and Street 1978; Veron et al. 2008a).

Increased turbulence and vorticity near the wave peak could also be a result of the shearing effects on the turbulence and the local stretching of vortex lines by the passing surface waves (Teixeira and Belcher 2002). Next, we examine the effect of the waves on the turbulence over short, wave-resolved time scales.

We wish to extract the wave-coherent turbulent quantities, following the standard Reynolds decomposition u = u + u′, where the overbar represents spatial averages over the PIV estimates (which are distributed over the footprint of the infrared imager). The primes indicate fluctuating quantities taken as deviations from the mean, which can be further decomposed into wave coherent ũ and turbulent components u″. Here, we use a technique similar to that presented in Veron et al. (2008a) to extract the averaged wave coherent quantities. We use the coherence and phase of each variable with the surface displacement η due to the waves. Denoting the spectrum of u1 by , we find that the spectrum of u1 coherent with the waves (i.e., the spectrum of ũ1 is given by .

It follows that the variance of the phase-coherent velocity in the direction of the wave propagation (ũ1) is given by
i1520-0485-39-9-2310-e14

Figure 9 shows the variance of the wave-coherent, along wave, surface velocity turbulence as a function of the wind speed (Fig. 9a) and the surface wave slope2 (Fig. 9b). Although noisy, the data show that there is some fraction of the surface turbulence that is coherent with the surface waves. The data show that the variance of the wave coherent surface turbulence increases with wind speed and wave slope, with perhaps a tighter fit with the wave slope. Figure 9 also shows data taken in shallow water and at lower wind speeds from Scripps Pier during the winter of 2003 (see Veron et al. 2008b for details). The two datasets merge continuously, and the large symbols are the bin-averaged data. The solid lines show the linear fit through the data (forcing a zero intercept for Fig. 9b).

Of further interest is the total fraction of this wave coherent turbulence to the total turbulent intensities. Following the work of Townsend (1976, 71–77) and Teixeira and Belcher (2002), we use the framework of rapid distortion to estimate this ratio. If we assume initially isotropic turbulence and plane straining by a unidirectional, monochromatic linear wave field of amplitude a and wavenumber k, then
i1520-0485-39-9-2310-e15
where β denotes the strain ratio, that is, the ratio of the velocity gradient tensor (or vorticity components) with and without distortion. In this case, β varies as
i1520-0485-39-9-2310-e16
Neglecting terms higher than quadratic in the wave slope ak, we find that
i1520-0485-39-9-2310-e17
which indicates that the normalized wave-modulated turbulence in the direction of the wave propagation is of O(ak). It should be noted that this theoretical estimate relies on assumptions of isotropy and homogeneity in the turbulence. These can be violated near the surface (Belcher et al. 1994) and where the turbulence is injected by localized breaking events; although, as noted by Teixeira and Belcher (2002), even in the latter case, the integral length scale of the turbulence is small compared to the wavelength of the surface breaking wave, thus satisfying the scale separation necessary for the rapid distortion theory to be applicable. On that note, we do not expect the swell to significantly affect the surface turbulence, at least in a rapid distortion sense because the strain rate from the orbital motion of the swell is generally too weak compared to that of the turbulence on itself. In fact, Fig. 8c shows no coherence between the waves and the surface turbulence at the swell frequency. Finally, the estimate above is developed for monochromatic linear waves and not for a full wave spectrum, thus we expect differences between this idealized theoretical prediction and the observed field data. Nevertheless, we feel that it is valuable and interesting to compare the data with the available theoretical estimates. Figure 10 shows the ratio of the variance of the wave coherent turbulence to the total variance in the turbulence as a function of the rms wave slope. The variance ratio in the direction of the waves increases with the wave slope. We should note here that we expect this ratio to be relatively noisy, especially at low wind speeds and wave slopes as becomes smaller. In Fig. 10, we also show the theoretical estimate from Eq. (17). Although there is a large scatter in the data, and an expected flattening due to noisy data for low surface wave slopes, the theoretical estimate agrees reasonably well with the data at the larger slopes. These data appear to be the first field observations qualitatively supporting the use of the rapid distortion theory to predict the modulation of surface turbulence by surface waves.

6. Conclusions

We have shown that there are periods of strong coupling between the surface wave field and wind forcing, and both the surface kinematic and temperature fields. During the early phase of the wind event described here, the two-dimensional wavenumber spectra of the surface vorticity and surface temperature, when averaged over time scales of one minute and longer show that these mean fields are organized in longitudinal structures aligned with the wind direction. In addition, we also show that the vorticity and divergence structures at the surface are orthogonal to each other. This coupling, which is revealed when the surface fields are averaged over many wave periods as is done with the CL II approach, is consistent with the presence of Langmuir circulations (or Langmuir turbulence). As the wind stabilizes, additional order in the surface kinematic fields is found, and the structures in the normal deformation and shear are aligned with that in the fields of surface divergence and vorticity, respectively. This orthogonality can be explained by a surface velocity field of hyperbolic form. In turn, hyperbolicity leads to significant stretching and convergence in the mean surface velocity field, once again consistent with the effects of Langmuir circulations.

Over short time scales where the waves are resolved, we observe the modulation of the surface turbulence by the surface waves. We show that there is a maximum wave-coherent turbulent intensity near the crests of the surface waves. These results are consistent with the modulation of the skin temperature by the surface waves where warmer temperatures, resulting presumably from the destruction of the molecular thermal layer by the turbulence, occur near the crest of the waves. Our data show that the relative wave-coherent turbulent intensities vary with both wind speed and surface wave slope. The data are qualitatively consistent with the rapid distortion theory but more work is needed for more complete quantitative comparisons.

Acknowledgments

We thank Peter Matusov for his help in the design, construction, testing, and deployment of the original instrumentation. We thank the staff of the SIO Marine Development Shop for the fabrication of custom equipment. We thank Bill Gaines, Tom Golfinos and the crew of the R/P FLIP for all their help in the field experiments, as well as John Hildebrand and Gerald D’Spain for sharing space and facilities onboard R/P FLIP. Finally, we thank A. D. Kirwan for his suggestions regarding the surface kinematic fields. This research was supported by NSF grant OCE 01–18449 to WKM and FV, and an ONR (Physical Oceanography) grant to WKM.

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Fig. 1.
Fig. 1.

(a) Instrumentation setup from R/P FLIP. (b) Map of the deployment area; R/P FLIP was moored off the coast of southern California in August 2003.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 2.
Fig. 2.

(a) Wind speed (black) and direction (gray) measured with the sonic anemometer for the duration of the experiment. (b) Spectragram of the surface elevation Sηη, measured with the laser altimeter. Note the two peaks at the frequencies of the swell and the wind waves. (c) Significant wave height measured with the altimeter and compared with that obtained from the RDI waves ADCP (ṡ).

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 3.
Fig. 3.

Infrared images of the surface temperature and surface turbulent velocity field calculated from the passive temperature pattern displacements between each temperature image and the one taken 33 ms later. Shown are the temperature and surface velocity for a wind speed of (a) U10 = 0.5 m s−1 with large upwelling/convection and (b) U10 = 3.8 m s−1 where cold streaks (Langmuir circulations) are clearly visible.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 4.
Fig. 4.

Two-dimensional wavenumber spectra of the (a) surface divergence, (b) vorticity, and (c) surface temperature for a wind speed of 8 m s−1. The spectra were calculated from an average of 120 individual spectra taken over 1 min of data starting at 0503 UTC 23 Aug 2003. (d) Shown are the directions for the wind, from 305°; the wind waves, to 117°; the swell, 97°; the vorticity spectral peak, 52°; the divergence spectral peak, 122°; and the temperature spectral peak, 43°.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 5.
Fig. 5.

(a) Directions of the wind, wind waves, swell, and current. (b) Directions of the streaks (i.e., the direction perpendicular to that of the peak spectral densities and the maximum variance) for the vorticity, divergence, shear, and normal deformation.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 6.
Fig. 6.

Two-dimensional wavenumber spectra of the (a) surface divergence, (b) vorticity, and (c) surface temperature for a wind speed of 6.4 m s−1. The spectra were calculated from an average of 120 individual spectra taken over 1 min of data starting at 1118 UTC 26 Aug 2003. (d) Shown are the directions for the wind, from 294°; the wind waves, to 114°; the vorticity spectral peak, 149°; the divergence spectral peak, 66°; and the temperature spectral peak, 33°. The swell direction could not be measured accurately.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 7.
Fig. 7.

Time series of surface displacement and the spatial variance of the along wave surface turbulence over a 120-s record starting at 0500 UTC 24 Aug 2003.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 8.
Fig. 8.

Spectragrams of the (a) surface displacement η and (b) along wave turbulent surface velocity u1 at night. (c),(d) Shown are the squared coherence and phase between the surface displacement and the turbulent surface velocity, respectively.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 9.
Fig. 9.

Variance of the wave-coherent, along wave, surface velocity turbulence with the (a) wind speed and (b) rms surface wave slope. The large symbols show the bin-averaged data, and the solid lines are the fits to the bin-averaged data.

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

Fig. 10.
Fig. 10.

Fraction of to as a function of the wave slope. The large symbols show the bin-averaged data, and the solid gray line shows the rapid distortion theory prediction of Eq. (17).

Citation: Journal of Physical Oceanography 39, 9; 10.1175/2009JPO4019.1

1

In the case of gas transfer, these molecular layers can also be at the surface of entrained bubbles.

2

Because the surface slope (ηx, ηy) was not directly measured in the experiments, the root-mean-square wave slope is instead approximated from the wave elevation frequency spectrum by , where fp is the peak frequency and k is the wavenumber given by the dispersion relationship. Finally, we estimate ak = 2S. This estimate is valid for the gravity wave range only.

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