Measurements of small-scale vorticity, turbulence velocity, and dissipation rates of turbulence kinetic energy ɛ were taken in a littoral fetch-limited surface wave boundary layer. Drifters deployed on the surface formed convergence streaks with ∼1-m horizontal spacing within a few minutes. In the interior, however, no organized pattern of velocity, vorticity, or turbulence mixing intensity was found at a similar horizontal spatial scale. The turbulent Langmuir number La was 0.6–1.3, much larger than the 0.3 of the typical open ocean, suggesting comparable importance of wind-driven turbulence and Langmuir circulation. Observed ɛ are explained by the wind-driven shear turbulence. The production rate of turbulence kinetic energy associated with the vortex force is about 10−7 W kg−1, slightly smaller than that generated by the wind-driven turbulence. The rms values of the streakwise component of vorticity σζ|| and the vertical component of vorticity σζz have a similar magnitude of ∼0.02 s−1. Vertical profiles of ɛ, σζ||, and σζz showed a monotonic decrease from the surface. Traditionally, surface convergence streaks are regarded as signatures of Langmuir circulation. Two large-eddy simulations with and without Stokes drift were performed. Both simulations produced surface convergence streaks and vertical profiles of ɛ, vorticity, and velocity consistent with observations. The observations and model results suggest that the presence of surface convergence streaks does not necessarily imply the existence of Langmuir circulation. In a littoral surface boundary layer where surface waves are young, fetch-limited, and weak, and La = O(1), the turbulence mixing in the surface mixed layer is primarily due to the wind-driven shear turbulence, and convergence streaks exist with or without surface waves.
In the oceanic surface mixed layer, primary turbulent processes include wind-driven shear turbulence, convective turbulence, surface wave breaking, and Langmuir circulation (LC). Previous results provide reliable wind-driven shear and convective turbulence scalings (Shay and Gregg 1986; Lombardo and Gregg 1989). Recent studies report progress on the parameterization of turbulence mixing due to surface wave breaking (Terray et al. 1996; Drennan et al. 1996; Anis and Moum 1995). These successful turbulence scalings are obtained from extensive field observations and laboratory experiments.
Langmuir circulation is the least understood of the turbulent processes. Extensive field observations of surface drifters, velocity, and turbulence mixing associated with LC have been made (Farmer and Li 1995; Weller and Price 1988; Thorpe et al. 1994, 2003; Smith 1992). However, the flow vorticity, one of the most vital variables associated with the generation of LC, has never been measured directly. Laboratory experiments cannot realistically simulate high-Reynolds-number vorticity and its interaction with surface waves. Therefore, no proper parameterization scheme for Langmuir turbulence is available.
It is generally believed that LC is generated by the vortex force as a result of the interaction between the Stokes drift velocity uS and the background vorticity (Craik and Leibovich 1976). The interaction between uS and the cross-streak component of vorticity ζ⊥ generates upwelling and downwelling of LC, the Craik–Leibovich type-1 (CL-1) vortex force. The interaction between uS and the vertical component of vorticity ζz generates the convergence and divergence velocity of LC, the Craik–Leibovich type-2 (CL-2) vortex force. It is commonly accepted that the CL-2 vortex force is the primary generation mechanism for LC. Numerical simulations have produced results to resemble LC observations by prescribing vortex forces as the external forcing. However, there have been no direct field observations of vortex forces because of the lack of vorticity measurements.
Here, we present direct oceanic observations of vorticity ζ, turbulence velocity, and dissipation rates of turbulence kinetic energy ɛ in a fetch-limited littoral surface wave boundary layer. Conventionally, surface convergence streaks are considered the signature of LC. We discuss two primary turbulent processes, wind-driven shear instability and LC, and compare theoretical scalings with observations. Simulations with a large-eddy-simulation model result in good agreement with observations, whether including or excluding the surface Stokes drift. Results of our analysis and numerical model simulations propose that in a fetch-limited boundary layer, where La = u*/uS = O(1), the wind-driven shear instability dominates LC and may generate surface convergence streaks. In the following analysis, both LC and wind-driven shear instability are considered potential processes for the observed surface convergence streaks.
The experimental setup and instruments are described in section 2. Observations of the surface waves, the wind stress, the Stokes drift velocity, and the surface mixed layer properties are described in section 3. Time series of basic observations are given (section 4). In section 5, we present the surface signature of convergence streaks, then discuss properties of observed ɛ and compare it with the existing theories of turbulence scaling. Direct observations and spectral properties of vorticity and velocity in the upper ocean are presented in section 6. In section 7, results of large-eddy simulations are compared with observations. Estimates of the vortex force and production rates of turbulence kinetic energy due to the vortex force are given, and a summary of results is stated.
2. Experiment and instruments
A field experiment was conducted to obtain direct observations of turbulence intensity and vorticity in the upper ocean in the surface mixed layer. Data presented here were collected on 31 August 2001 near Shilshole Bay in Puget Sound, Washington (Fig. 1). About 3.8 h of measurements were taken. The experimental site was chosen to avoid complex oceanic processes in the coastal waters other than the expected local processes including LC and the wind-driven shear turbulence, to avoid ship traffic, and to have a sufficiently deep surface mixed layer so that observations in this layer from our towed instrument would be possible. During our experiments the surface mixed layer was mostly shallower than 4 m.
a. Sensors on EMVM
The primary instrument was the electromagnetic vorticity meter (EMVM). Structural details and the principle of operation are described by Sanford et al. (1999). A pair of vorticity meters (VMs), a microstructure shear probe, a pressure sensor, a set of conductivity and temperature sensors, and a suite of altitude sensors are mounted on the 600-lb EMVM tow body. During the experiment, the EMVM was mostly towed at constant depths in the upper 4 m, and was occasionally profiled vertically to obtain the structures of stratification and shear. Time series of the instrument depth are shown in Fig. 3.
The VM sensor measures two components of velocity tangent to the surface of the VM sensor plate, and one component of vorticity normal to the surface of the VM sensor. Two VM sensors were mounted on the EMVM. One of the sensors was mounted vertically on the side of the EMVM and measured the streamwise velocity (along the axis of the EMVM tow body), the vertical velocity, and the spanwise component of vorticity (normal to the axis of the EMVM tow body). The other VM sensor was mounted horizontally on the bottom of the EMVM tow body and measured the streamwise velocity, the spanwise velocity, and the vertical component of vorticity. Measurements of velocity and vorticity by the VM sensors represent averages of the true velocity and vorticity fields over a circular area of ∼0.1-m radius, the distance between which the electrical voltage differential is generated by the conductive fluid, that is, the seawater, flowing through the permanent magnetic field of the VM plates. Velocity and vorticity fluctuations at scales smaller than 0.1 m are attenuated in VM measurements. Sensor response functions of velocity and vorticity measurements taken by the VM sensor are detailed by Sanford et al. (1999).
One shear probe was mounted at the front end of the EMVM tow body. It provided measurements of the dissipation rate of turbulence kinetic energy ɛ. The TC sensor and the pressure sensor provided temperature, salinity, density, and pressure measurements.
b. Surface wave buoy
A 0.9-m-diameter TRIAXYS directional wave buoy was used to measure surface waves. It had three accelerometers, three rotation rate sensors, and a flux-gate compass. Resolution was ∼1 cm for heave and 1° for wave direction. It resolved wave periods of 1.6–30 s. Before taking EMVM measurements, the wave buoy was deployed, drifted freely during the course of the experiment, and was repositioned when it drifted too far from the R/V Henderson. Surface wave data were sampled at 4 Hz. Statistics of surface waves were processed in 10-min segments, transmitted to the R/V Henderson via modem, and recorded internally in the wave buoy. Wave data were available at a 15-min interval.
c. Surface drifters: Popcorn
Popcorn was used as the surface drifter to identify surface convergence streaks associated with LC. During the experiment popcorn was deployed four times and each time convergence streaks were formed within 1–2 min. After deploying popcorn, the R/V Henderson was turned to a reciprocal course that was about perpendicular to popcorn convergence streaks.
d. Onboard sensors
An R. M. Young wind sensor was mounted ∼10 m above the water surface. Measurements of wind speed and direction, relative to the ship, were sampled at 1 Hz. Differential GPS data, sampled at 1 Hz, were used to compute the ship velocity. Measurements of the ship’s heading and velocity, and measurements from the wind sensor, were combined to obtain the absolute wind velocity, that is, relative to the earth.
A camera was mounted in front of the R/V Henderson bridge. It took pictures at 2 Hz of the water surface, centering where the cable of the EMVM entered the water. This camera was intended to capture the surface signatures of the LC, for example, drifters converged on streaks, to identify the periods when the R/V Henderson steamed across convergence streaks.
An infrared radiometer was used to measure surface skin temperature in an area of ∼0.3-m diameter with a precision of ∼0.1°C. It operated at 8–14-μm wavelengths and sampled at 1 Hz. We found a skin temperature anomaly at O(1 m) scale. Unfortunately, it is not possible to differentiate the skin temperature anomaly generated by surface waves from that by other turbulent processes such as LC and wind-driven shear turbulence.
3. Surface forcing and surface mixed layer
a. Surface wind stress
The relation between the surface friction velocity u* and the surface wind speed is often expressed by an empirical drag coefficient or by a similarity form described as
where U10 is the wind speed at 10 m above the sea surface, κ = 0.4 is the von Kármán constant, and z0 is the surface roughness length in meters. Because we had no direct estimate of the net surface heat flux, deviations of (1) due to the density stratification were ignored. Effects of surface waves on the estimate of u* are embedded in (1) via z0 and Cd.
Estimates of u* were computed and compared using five expressions of z0 or Cd. The first expression follows that of Large and Pond (1981), which suggests that Cd is a constant at U10 < 10 m s−1, and increases linearly with the wind speed at U10 > 10 m s−1. The second expression uses Charnock’s relation, z0 = 0.11u2*g−1 (Charnock 1955), which is appropriate for waves in the open ocean. For fetch-limited waves, the wave age Cpu−1* needs to be considered, where Cp = gω−1peak is the phase speed of surface waves at the frequency ωpeak of the surface wave spectral peak. The third expression follows Smith et al. (1992), including effects of the wave age, such that z0 = (0.48u*C−1p)u2*g−1. The fourth expression follows Johnson et al. (1998) assuming z0 = [1.89(u*C−1p)1.59]u2*g−1. The fifth expression follows Drennan et al. (2003) assuming z0 = 13.4(u*C−1p)3.4ση, where ση is the standard deviation of surface elevations. This expression is valid for rough pure wind waves when u*z0ν−1 > 2.3, where ν is the kinematic molecular viscosity of air, but not applicable to swells. During the experiment the surface wind was weaker than 8 m s−1 and the significant wave height Hs < 0.3 m. Therefore, this expression is unlikely to be applicable.
The first four expressions yield similar estimates of the drag coefficient Cd (1−1.4 × 10−3) and differ by less than 10% (Fig. 4), whereas the fifth expression yields a considerably lower value. The average of friction velocity estimates computed from the first four expressions is used for this analysis. The friction velocity u* varies from 0.002 to 0.01 m s−1, the roughness length z0 varies from 2 × 10−5 m to 3 × 10−4 m, and the surface wind stress varies from 0.01 to 0.1 N m−2.
b. Surface waves and Stokes drift
Surface wave data were reprocessed postcruise by Axys Environmental, Ltd. Frequency–directional spectra, frequency spectra, and statistics of surface waves were available at a 15-min interval. Processed wave spectra have a directional resolution of 3° and frequency resolution of 0.005 Hz. The postcruise-processed surface wave data have a Nyquist frequency of 0.64 Hz.
The average of the 16 observed directional surface wave spectra exhibited two primary groups of waves, although the wind was persistently southerly (Fig. 5). Here, the direction is described in degrees clockwise from true north, unless stated otherwise. One group of waves propagated southeast, ∼150°, at a dominant frequency of 0.16 Hz (Fig. 6). This group occurred at 4000–6000 s (Fig. 4) and was generated by passing ships. The second group of waves propagated northeast, ∼35°, with a direction bandwidth of 25° and a dominant frequency of 0.35 Hz. This group included local wind waves and swells propagating from south of the experiment site. The wave direction was 35° clockwise from the local wind, presumably due to the combined effects of an upwind headland, the bathymetry, and the surface current. The significant wave height varied between 0.2 and 0.3 m, and the period of significant waves varied from 2.5 to 3.8 s, excluding waves at 4000–6000 s (Fig. 4).
The Stokes drift velocity uS is a net Lagrangian velocity of water particles in the presence of surface waves. The Stokes drift velocity for a single plane deep-water wave is (Kundu 1990)
Here a is the amplitude of surface waves, and k and ω are wavenumber and wave frequency, respectively. For deep-water waves, k = ω2g−1. The Stokes drift velocity decays two times faster in the vertical than the surface wave.
For deep-water waves of a broad spectrum of frequencies and directions, the Stokes drift velocity can be computed using the direction–frequency wave spectrum Φη(ω, θ). The direction–frequency distribution of the Stokes drift velocity is
The ω3 dependence in (3) emphasizes the importance of high-frequency surface waves in producing Stokes drift velocity. If the spectral slope of Φη(ω, θ) is flatter than ω−4, the frequency integration of uS(ω, θ, z) in (3) will not converge. The observed Φη had an ω−5 spectral slope at high frequencies. Nonetheless, the Nyquist frequency is only about twice the frequency of the spectral peak, so the Stokes drift velocity contributed by high-frequency surface waves is not fully resolved.
The average frequency–direction distribution of the surface Stokes drift velocity uS(ω, θ, z = 0) during the 4-h experiment (Fig. 5) shows that it was dominant at frequencies >0.3 Hz and in the direction of 0°–90°.
The frequency distribution of the surface Stokes drift velocity is obtained by integrating (3) over azimuthal directions at the sea surface (Fig. 6b). The peak of the Stokes drift velocity is barely resolved and high-frequency components of the Stokes drift velocity are missed. We assume an ω−5 spectral shape for Φη, extrapolate the surface wave spectrum to high frequencies, and compute the Stokes drift velocity again to estimate the missing fraction of Stokes drift velocity. The total Stokes drift velocity is about twice the resolved Stokes drift velocity. In the following analysis, we do not compensate for the Stokes drift velocity missed in field measurements. The Stokes drift velocity induced by the previously mentioned low-frequency ship waves is insignificant compared to that induced by surface waves at higher frequencies.
Integrating over all frequencies, the directional distribution of the Stokes drift velocity is expressed as
where ω0 = 0.1 Hz is chosen to avoid low-frequency noise and ωNq is the Nyquist frequency. The east (uS) and the north (υS) components of the Stokes drift velocity are computed as
Vertical profiles of uS, υS, and the speed of the Stokes drift velocity are shown in Fig. 6c; surface values of uS and υS are 0.006 and 0.004 m s−1, respectively; and the speed is 0.0072 m s−1. The vertical e-folding scale is 0.7 and 0.5 m for uS and υS, respectively, and is 0.5 m for the speed. These e-folding scales are shorter than those commonly computed based on the frequency of significant waves, that is, 0.5gω−2peak ≈ 1 m, because the Stokes drift velocity is dominated by high-frequency short surface waves, not by significant waves.
The computed Stokes drift velocity has a magnitude comparable to the friction velocity, ∼0.006 m s−1 (Fig. 6c). However, the vertical shear of the wind-driven current, on the basis of the law-of-wall, has a z−1 scaling and is one order of magnitude greater than the vertical shear of the Stokes drift velocity (Fig. 6d), even if corrected for the missing Stokes drift velocity at high frequencies.
Following McWilliams et al. (1997), the shear production of turbulence kinetic energy (TKE) in the wind-driven surface mixed layer, for example, u′w′∂zU, where U includes the Stokes drift velocity and the wind-driven current, is a result of the interaction between the turbulent Reynolds stress u′w′ with the vertical shear of the wind-driven current and with the vertical shear of the Stokes drift. The ratio of the vertical shear of the Stokes drift velocity to the vertical shear of the wind-driven current, LS = ∂zuS(u*κ−1z−1)−1, is a proper parameter to measure the relative importance of the turbulent energy produced by LC and by the wind-driven current (Fig. 7). The largest value of LS, <0.3, was at 0.5-m depth. Below 1-m depth, where most measurements were taken, LS was <0.1. Below 2-m depth, LS is <0.02. After 14 000 elapsed seconds, LS increased because the wind weakened from 8 to 4 m s−1, whereas surface waves remained strong (Fig. 4). Unfortunately, during this period, EMVM measurements were taken below 0.5-m depth. During the experiment, the production rate of turbulence kinetic energy by LC was expected to be at most 30% of that by the wind-driven shear at 0.5-m depth, and much weaker at greater depths.
c. Surface mixed layer
Four CTD-quality vertical profiles were obtained by profiling the EMVM tow body vertically. These profiles of potential density indicate that even in the upper layer the water was weakly stratified (Fig. 8c). Profiles were taken within 1–2 h and 1–2 km apart, showing strong variability. The first, taken closer to shore, shows a 2-m surface mixed layer overlying a weakly stratified layer at 5–8 m. Vertical profiles 2 and 4 show a weakly stratified layer in the upper 4 m. The third profile, taken close to the center of the channel, shows a weakly stratified layer in the upper 2 m overlying a strong pycnocline at 2–3 m. The buoyancy frequency in the upper 2 m is 0.005 s−1.
4. Summary of observations
During the 4-h experiment, a southerly wind of 1–10 m s−1 persisted at the mean direction of −4.5° relative to true north with a standard deviation of 11.9° (Fig. 3b). For the following analysis, observations are divided into three periods: A, a weak-wind period between 1730 and 7200 s when the wind speed was 4.0 ± 0.7 (standard deviation) m s−1; B, a strong-wind period between 7200 and 13 400 s when the wind speed was 6.5 ± 0.8 m s−1; and C, a decaying-wind period between 13 400 and 15 520 s when the wind speed decreased from 8 to 4 m s−1.
Most measurements were taken when the EMVM was towed at depths of 0.5, 1, 2, 3, and 4 m, besides four vertical profiles (Fig. 3a). Popcorn was deployed four times (red lines in Fig. 3a); they converged and formed streaks within 1–2 min. The R/V Henderson steamed approximately perpendicular to the wind at 1–2 kt. When popcorn streaks were present, the Henderson steamed perpendicular across the streaks (Fig. 3c). The surface Stokes drift velocity was comparable to the friction velocity before 12 000 s and became greater than the friction velocity after 12 000 s (Fig. 3d).
a. Coordinate system
When the Henderson steamed perpendicular to the wind, a right-hand coordinate is used. The || axis is downwind and along convergence streaks, the ⊥ axis is cross-wind and cross-streaks, and the z axis is positive upward (Fig. 8a). The VM sensor mounted on the side of the EMVM measures the along-streak vorticity (ζ||), the cross-streak velocity (u⊥), and the vertical velocity (w). The VM sensor mounted on the bottom measures the vertical vorticity ζz, the cross-streak velocity u⊥, and the along-streak velocity u||. The two VM sensors are separated vertically by 0.5 m. Spatial structures of vorticity and velocity expected by the conventional view of LC are illustrated in Fig. 8.
b. Ship motions
The ship motion due to surface waves was transmitted to the EMVM tow body through the winch cable. A set of bungee cords was attached to the EMVM winch cable to decouple the EMVM from the ship motion. This effort was only partially successful; the vertical velocity spectrum computed from EMVM measurements exhibits a clear spectral peak at the frequency of surface waves, and sometimes at higher harmonics, presumably due to the resonance of bungee cords (Fig. 9).
5. Surface signature of convergence streaks
Convergence streaks of 0.5–2-m horizontal separation (Fig. 8b) formed within a few minutes after popcorn deployment. Another estimate of the streak separation, obtained by replaying pictures of passing popcorn streaks captured by the camera, yielded a similar result. The streak separation was estimated based on the ship speed and the time interval between consecutive passing popcorn streaks. The accuracy of this estimate was ∼0.5 m, set by the 0.5-s sampling interval of pictures and the ∼1 m s−1 ship speed.
Previous observations proposed that the separation of LC convergence streaks is approximately twice the thickness of the surface mixed layer. The observed 0.5–2-m horizontal spacing of convergence streaks is shorter than expected from LC.
Microstructure shear probe measurements were taken at 400 Hz. Dissipation rates of turbulence kinetic energy ɛ were computed by fitting observed shear spectra to the Panchev spectrum (Panchev and Keisch 1969). Previous experiments in a tidal channel proved that the microstructure shear probe mounted on the EMVM provided quality estimates of ɛ (Sanford et al. 1999). To resolve the spatial variation of ɛ within the scale of the streak separation, shear spectra were computed and estimates of ɛ at 10 Hz were obtained to correspond to a 0.1-m spacing at the ship speed of 1 m s−1.
A parameter Q = n/N was defined to measure the quality of estimates of ɛ, where n is the number of spectral estimates that agree with the fitted Panchev spectrum within the 95% confidence interval, and N is the total number of spectral estimates. Estimates of ɛ with Q > 0.5 are shown in Fig. 3e. They vary between 10−8 and 10−5 W kg−1, representing the mixture of the temporal and spatial variations. Some estimates of ɛ are poor (Q < 0.5; when ship wakes were present as the ship changed speed or direction, or when the EMVM was profiling vertically).
During the experiment, 14 time segments of EMVM measurements were taken at fixed depths: one short segment at 0.5-m depth, three segments at ∼1-m depth, five segments at ∼2-m depth, four segments at ∼3-m depth, and one segment at ∼4-m depth (Fig. 3). During some segments, the ship direction or speed changed. These segments were further divided into smaller segments so that the ship speed and direction were nearly constant within each segment.
Vertical profiles of observed ɛ within three periods of different surface forcing are shown in Fig. 10. The averages of the friction velocity in periods A, B, and C, are 0.0043, 0.0075, and 0.0054 m s−1, respectively. The averages of the surface Stokes drift velocity in periods A, B, and C are 0.0053, 0.0093, and 0.0106 m s−1, respectively.
Vertical profiles of ɛ in all three periods show a monotonic decrease with depth. In period C, ɛ at 0.5-m depth is slightly weaker than at 2-m depth. However, the difference is not statistically meaningful because only 29 s of measurements are available at 0.5-m depth. Averaged ɛ decreases from 3 × 10−7 W kg−1 at 1-m depth to 0.6 × 10−7 W kg−1 at 3-m depth in period A, and from 10−6 W kg−1 at 1-m depth to 2 × 10−7 W kg−1 at 3-m depth in period B. The decrease of a factor of 5 from 1- to 3-m depth is slightly greater than that predicted by the turbulence mixing generated by the vertical shear of wind-driven current scaled as u3*(κz)−1, that is, a decrease of a factor of 3 from 1- to 3-m depth (Lombardo and Gregg 1989; Agrawal et al. 1992). Although the vertical decaying scale of observed ɛ is slightly shorter than that predicted by the law-of-wall theory, the overall magnitude of turbulence mixing intensity in 1–3-m depth is consistent with the law-of-wall scaling.
In period C the Langmuir number La was the lowest and the contribution of the turbulence mixing by the LC was expected to be the most significant during the entire experiment. Observed ɛ in 2–3-m depths was about a factor of 2 greater than that predicted by the law-of-wall scaling in period C. Assuming this factor of 2 deviation was due to LC, it should have the turbulence mixing intensity of O(10−7) W kg−1, which, however, would be difficult to differentiate from that generated by the vertical shear of the wind-driven current.
Observations of turbulence mixing are significantly correlated with those predicted by the law-of-wall scaling with a correlation coefficient of 0.9 (Fig. 11). This further supports the dominance of wind-driven shear turbulence relative to LC during our observations.
To further examine the importance of turbulence mixing induced by LC relative to that by the wind-driven shear, the ratios ɛu*/ɛobs were plotted as functions of uS, u*, and La (Fig. 12). The Langmuir number La varies between 0.6 and 1.1. There is no apparent trend between ɛu*/ɛobs and La. Results of numerical simulations by Li et al. (2005) suggest that the wind-driven turbulence mixing dominates that induced by LC at La > 0.7, which is consistent with our observations.
Surface waves breaking could also produce turbulence mixing. Terray et al. (1996) divided the near-surface turbulence regime into three layers: 1) a constant ɛ layer for z < =0.6Hs where surface waves break and dump energy, 2) a transition layer ɛ ∝ z−2 where both wind and wave forcing are important, and 3) a deeper layer where the law-of-wall scaling, that is, ɛ ∝ z−1, is valid.
Drennan et al. (1996) studied similar scaling, normalizing the depth by the peak wavenumber kp. In the transition zone, ɛ scales with the wave energy flux divergence kpF as
where F = u2*c is the wave energy flux. The effective phase speed c is a function of wave age Cp/uair*, where Cp is the phase speed of the peak wave and uair* is the friction velocity on the air side.
During the experiment, Cp was ∼5 m s−1, the average uair* was ∼0.12, 0.22, and 0.16 m s−1 in periods A, B, and C, respectively, and the corresponding inverse wave age C−1puair* was 0.02, 0.05, and 0.03. Terray et al. (1996) report that the effective phase speed c is linearly proportional to u* for the inverse wave age between 0.03 and 0.07. No data were available for the inverse wave age smaller than 0.03. The effective wave speed in periods B and C are 1 and 0.4 m s−1 (no estimate of c for period A is available) and the average wave energy fluxes are 5.6 × 10−5 and 1.1 × 10−5 m3 s−3, respectively. At 1-m depth, ɛ predicted by (7) are 1.4 × 10−5 and 0.27 × 10−5 W kg−1 in periods B and C, respectively. These predictions are at least one decade greater than the observations. Any reasonable adjustment of the effective wave speed c could not explain the discrepancy. Therefore, we conclude that the surface wave energy flux did not penetrate to our measurement depths, which were deeper than the transition zone.
In the absence of surface wave breaking, Langmuir circulation, or convective turbulence, the law-of-wall theory often provides a proper turbulence scaling in the upper ocean. Our observations were taken below the depth of possible surface wave breaking, an order of significant wave height O(0.1 m), and were taken during the daytime in the absence of convective turbulence. Therefore, possible explanations for deviations between observed turbulence mixing and the law-of-wall scaling are LC, other unidentified turbulent mechanisms such as internal wave breaking, errors in ɛ estimates, or errors in wind stress estimates.
6. Direct observations of vorticity and velocity in the upper ocean
This study presents the first set of field observations of vorticity in the upper ocean. Both streakwise and vertical components of vorticity were observed when the R/V Henderson steamed perpendicular to surface convergence streaks (Fig. 8).
a. Velocity observations
Time series of velocity and vorticity measurements are high-pass filtered at 0.1 Hz (Fig. 13). Fluctuations of velocity and vorticity are complicated by the combined vertical and temporal variations. The variance of velocity is mostly dominated by surface waves. Averaged over all observations, the rms values are 0.04, 0.03, and 0.13 m s−1, for cross-streak, along-streak, and vertical components of velocity, respectively. A short segment of 2000-s measurements (Fig. 14) shows the obvious effect of surface waves at a period of ∼3 s on velocity fluctuations. The rms values are 0.05, 0.06, and 0.18 m s−1, for cross-streak, along-streak, and vertical components of velocity, respectively.
Vertical profiles of standard deviations of all three velocity components are shown in Fig. 15. Velocity measurements are high-pass filtered at 1 s to avoid effects of surface waves. All three velocity components show a magnitude of cm s−1, and decrease with depth. The streakwise component of velocity u|| decreases the most rapidly with depth, on average from 0.04 m s−1 at 0.5-m depth to ∼0.01 m s−1 at depths greater than 2 m. Below 2-m depth, the standard deviation of streakwise velocity remains nearly constant in all three periods and the histogram is narrow. At 1-m depth, the streakwise velocity has a standard deviation of 0.025 m s−1 in period B and 0.015 m s−1 in period A. The stronger streakwise velocity in period B is likely due to the stronger wind and wave forcing during this period.
Vertical profiles of the standard deviation of the cross-streak velocity u⊥ also exhibit a decrease of amplitude with depth. The magnitude is about 0.015, 0.015, 0.01, and 0.008 m s−1 at 0.5-, 1-, 2-, and >3-m depths, respectively. It is the strongest in period B, and the weakest in period A. The cross-streak velocity is weaker than the streakwise velocity. Li et al. (2005) also report a stronger streakwise velocity than the cross-streak velocity when La > 0.7 indicating that wind-driven turbulence dominates the Langmuir turbulence. It is consistent with our observations taken in a dynamic regime of La = O(1). Our observed cross-streak velocity is about twice that of friction velocity.
The vertical velocity has the strongest amplitude of the components. In period A it has a nearly constant standard deviation of 0.02 m s−1 at 1–3-m depths. In period B it decreases from ∼0.03 m s−1 at <2-m depth to ∼0.02 m s−1 at >2-m depth. In period C the vertical velocity has a standard deviation of ∼0.04 m s−1 at 0.5-m depth, and similar magnitudes as in period B at 2- and 3-m depths. Li et al. (2005) found a stronger vertical velocity fluctuation in convective turbulence. Our measurements, however, were not taken in a surface convection period. The observed larger vertical velocity is due to the effects of surface waves at higher wave frequency harmonics that escaped the high-pass filtering.
b. Observation of vorticity
Time series of 0.1-Hz high-pass-filtered streakwise vorticity ζ|| and vertical vorticity ζz are shown in Fig. 13, and a short segment of the time series at 1-m depth is shown in Fig. 14. A vorticity contribution directly from surface waves is not expected here because surface waves are irrotational and the observations were taken deeper than the surface wave breaking zone. Vorticity fluctuations associated with wake vortices of the instrument could contaminate measurements. Instrument motion due to surface waves is mostly vertical; therefore, the wake vortices’ effect is most severe for the bottom vorticity meter. Because a significant increase of vorticity variance was observed when the vertical acceleration rate of the instrument was negative (not shown), those measurements were excluded from the following analysis, for example, gaps of vorticity data in Fig. 14.
The amplitude of the streakwise vorticity ζ|| is generally greater than that of the vertical vorticity ζz. Vertical profiles of the standard deviation of the streakwise vorticity σζ|| and the standard deviation of the vertical vorticity σζz show a slight decrease with depth (Fig. 16) and vary between 0.01 and 0.1 s−1. Averaging observations at different depth ranges, σζ|| decreases from 0.05 s−1 at 0.5-m depth to 0.028 s−1 at 4-m depth. It is the strongest in period B and the weakest in period A, in accordance with the surface wind forcing.
Vertical profiles of σζz show a similar structure as those of σζ||, but with an amplitude ∼20% weaker, decreasing from about 0.04 s−1 at 0.5-m depth to about 0.02 s−1 at 4-m depth.
We examined the ratio of the standard deviations of two vorticity components as a function of friction velocity and Stokes drift velocity (Fig. 17). The ratio does not depend on the friction velocity, the Stokes drift velocity, or the Langmuir number. For isotropic turbulence, the expected ratio is one. Our observations are consistent with isotropic turbulence.
c. Spectral properties of velocity and vorticity
Spectra of three velocity components and two vorticity components are shown in Fig. 18. Spectra were computed using measurements taken at fixed depth, as used to compute histograms (Figs. 15 and 16). Frequency spectra have been converted to wavenumber spectra, using the Taylor frozen-field assumption (Lumley and Terray 1983). The streakwise and cross-streak velocity spectra have a similar shape and magnitude. Both show a clear peak at wavenumbers of 2–3 m−1 due to surface waves. The vertical velocity spectrum shows the most distinct peak of surface wave frequency. At wavenumbers smaller than 1 m−1, cross-streak and streakwise velocity spectra show a −2 spectral slope and the vertical velocity spectrum is almost white, characteristic of internal waves. At wavenumbers greater than 5 m−1, several peaks at the higher harmonics of surface waves exist. The streakwise velocity is greater than the cross-streak velocity (as mentioned previously). The excess streakwise velocity variance appears to be at wavenumbers of 3–10 m−1 (wavelength of 0.6–2 m), similar to the scale of our observed separation of surface convergence streaks. Another distinction in the velocity spectra is the additional peak in the cross-streak velocity spectrum at 1 m−1, probably due to ship motion.
Averaged spectra of streakwise vorticity ζ|| and vertical vorticity ζz have a similar magnitude and shape: white at low wavenumbers and roll-off at wavenumbers greater than ∼3 m−1 at a slope slightly slower than −2 (Fig. 18b). Variances of observed vorticity mostly reside near 3 m−1, around the roll-off wavenumber. Sanford et al. (1999) discuss the vorticity spectrum obtained from EMVM measurements, concluding that the white vorticity spectrum is expected from 3D turbulence, and a spectral roll-off at high wavenumbers is due to the sensor attenuation effect. The observed vorticity here represents a lower bound estimate of real vorticity in the ocean. The actual attenuation of vorticity depends on the spectral content of vorticity at wavenumbers beyond the roll-off wavenumber. The vertical vorticity spectrum is 5–10 times greater than the streakwise vorticity spectrum at wavenumbers of 0.05–0.2 m−1.
7. Large-eddy-simulation model
A large-eddy-simulation model was run with background conditions and surface forcings similar to experimental observations. The model addressed the relative importance of the vortex force and the wind-driven shear instability and simulated the observed phenomenon, relatively organized convergence streaks at the sea surface and unorganized flows in the interior.
The large-eddy-simulation model was developed by Moeng (1984) to study the atmospheric planetary boundary layer, and modified by McWilliams et al. (1997) for the oceanic mixed layer. The model domain has a horizontal dimension of 30 m × 30 m and a vertical dimension of 15 m. The grid resolution is 0.6 m in the two horizontal directions and 0.15 m in the vertical direction.
The simulation setup was guided by the experimental measurements. A surface mixed layer of 5-m depth is set. The initial temperature distribution is uniform in the mixed layer followed by a thermocline with a temperature gradient of 0.06°C m−1. The friction velocity is 0.006 m s−1.
Two simulations are performed, with and without the vortex force. The first simulation is forced by the surface wind stress and the vortex force with the Stokes drift velocity of 0.006 m s−1, implying La = 1. The simulation spinup period is u*βt = 50, where β−1 is the e-folding depth of the observed Stokes drift, where the time is scaled by the turnover time of near-surface eddies generated by the vortex force of surface waves. The turbulence statistics are then computed during the quasi-equilibrium phase by averaging over the horizontal domain and over 10 eddy turnover times. The second simulation is forced by the surface wind stress alone assuming a zero vortex force, that is, La = ∞. We compare results of these two simulations to understand whether the observed surface convergence streaks are a unique feature of the vortex force.
Model results of vertical profiles of ɛ, standard deviations of vorticity components, and velocity components are similar to observations averaged in period A, when La is the closest to one (Fig. 19). Magnitudes of model results are mostly within a factor of 2 from observations, except for the vertical velocity because the observed vertical velocity is contaminated by effects of surface waves. Interestingly, model results with and without the surface vortex force show little difference. Both simulations predict convergence streaks at the sea surface of 0.5–5-m horizontal spacing and unorganized flow structures in the interior, supporting our observations. Therefore, the presence of surface convergence streaks does not necessarily indicate the existence of LC. Tsai (2001) performs a similar simulation and demonstrates that the surface convergence streaks of O(10 cm) scale can be formed by mechanisms similar to those generating low-speed streaks within a sublayer next to a no-slip wall.
8. Discussion and summary
a. Vortex force
Although LC is not the dominant process at work during the experiment, direct observations of vertical vorticity and Stokes drift velocity are used to estimate the CL-2 vortex force, which is generally thought responsible for the instability of LC. Our direct estimates of the vortex force should be useful for future studies of LC.
The vortex force varies in time and at depth. Averaged over the entire experiment period, the rms values of vertical vorticity σζz are 0.04, 0.035, and 0.03 s−1 (Fig. 16); and the Stokes drift velocities are 0.003, 0.0013, and 0.0004 m s−1 (Fig. 6) at depths of 0.5, 1, and 2 m, respectively. The corresponding estimate of the vortex force, |ζz| × uStokes, is 12, 4.5, and 1.2 × 10−5 m s−2. It decreases by a factor of 10 between depths of 0.5 and 2 m. This estimate from direct oceanic observations may provide a guide for numerical model simulations, especially in the littoral regime.
b. Energy production rate
The production rate of turbulence kinetic energy due to the CL-2 vortex force is estimated by averaging the product of cross-streak velocity and the CL-2 vortex force within segments of constant depths (Fig. 10). Both positive and negative estimates of production rates are found. Negative values for production rate estimates may reflect the insufficient measurements required to compute their correlation.
Magnitudes of estimated production rates decrease with increasing depths at a rate faster than the law-of-wall scaling and observed ɛ. The estimated magnitude of production rates, 1–4 × 10−7 W kg−1 in the upper 1 m, is smaller than the law-of-wall scaling and observed turbulence by about a factor of 2 at 0.5–1-m depth. This supports the conclusion that wind-driven turbulence is stronger than that generated by LC in an environment of La = O(1).
The goals of this experiment were to measure vorticity and turbulence in the upper ocean, to study dominant turbulent processes in the littoral wave boundary layer, and to provide an estimate of the vortex force, which is believed responsible for the LC but has never been measured. In a littoral wave boundary layer where surface waves are often fetch limited and young, the LC is weak and the wind-driven turbulence dominates the Langmuir turbulence. Our observations and large-eddy simulations conclude that surface convergence streaks may be generated via wind-driven shear turbulence, rather than by LC.
Observed turbulence dissipation rates were in close agreement with those predicted by the wind-driven turbulence. The vertical shear of the wind-driven current was much stronger than the vertical shear of the Stokes drift velocity. The observed Langmuir number indicates that wind-driven turbulence should be the dominant turbulent process. Here, an estimate is provided of vorticity in the upper ocean of the order of 0.01 s−1, and a CL-2 vortex force of 10−4 m s−2. The rate of turbulence kinetic energy produced by the CL-2 vortex force is estimated to be O(10−7) W kg−1. These direct oceanic observations of vorticity and vortex force may provide guidance for numerical model simulations in littoral regimes.
We thank Debra Simecek-Beatty and William Lehr at NOAA/HAZMAT, Nick Bond at NOAA/PMEL, and Skip Albertson at EAP/DOE Washington for providing data and help to determine the experimental site before the cruise. Discussions with Ramsey Harcourt and Eric D’Asaro were encouraging during the course of analysis. Especially, we thank John Dunlap, Arthur Bartlett, and Zoli Szuts for cruise preparation and experiment execution. This work was supported by the National Science Foundation.
Corresponding author address: R.-C. Lien, Applied Physics Laboratory, University of Washington, Seattle, WA 98105. Email: firstname.lastname@example.org