Abstract

Vertical velocities in the ocean boundary layer were measured for two weeks at an open ocean, wintertime site using neutrally buoyant floats. Simultaneous measurements of the surface meteorology and surface waves showed a large variability in both wind and wave properties and only weak correlations between them. Buoyancy forcing was weak. The mean square vertical velocity in the boundary layer σ2w measured from the vertical motion of the floats was proportional to the squared friction velocity u2 estimated from shipboard meteorological measurements using bulk formulas. Thus σ2w = Au2 (the rate of momentum transport from the atmosphere to the ocean is ρu2, where ρ is the density of the water). The deviations from this relation can be attributed entirely to statistical variation and measurement error. The measured values of σ2w were corrected for measurement biases and the nonturbulent contributions of internal waves. The value of the turbulent part of A is 1.75–2 times that measured in laboratory and oceanic solid-wall turbulent boundary layers driven by shear alone. Although surface waves undoubtedly play a large role in the physics of the oceanic boundary layer, their effects on vertical velocity variance are remarkably well parameterized by wind stress in this data.

1. Introduction

Measurements of turbulence properties in the ocean boundary layer are limited. Accordingly, present models of the ocean boundary layer are constructed primarily by analogy with atmospheric boundary layers over land. Over land, the fluid velocity must be zero at the surface but is finite at altitude. Turbulence is generated by instabilities of this sheared flow and carries a vertical flux of horizontal momentum, a stress. In the oceanic analog, the wind stress τ acts to accelerate the ocean surface downwind, thereby setting up an unstable sheared flow in the upper ocean. Instabilities in this flow produce turbulent velocity fluctuations whose average amplitude is proportional to the friction velocity u∗ = τ/ρ, and thus depend on the stress τ. If this is the only source of turbulence, the vertical velocity variance σ2w at depth z is given by

 
σ2wAzu2
(1)

and the vertical velocity variance in the mixed layer is given by

 
formula

where the overbar denotes the vertical averaging operation.

Surface gravity waves make the ocean surface different from the land surface. Gravity waves contain most of the energy in the boundary layer and thus play a large role in the physics of the boundary layer turbulence (Melville 1994). The waves may break, thereby transferring some of their energy and momentum to turbulence and injecting clouds of buoyant bubbles into the ocean. Waves may also be refracted by the nonwave velocities, thereby transferring momentum from the waves to the turbulence. This process, the Craik–Leibovich (CL: Leibovich 1983) interaction, is thought to generate downwind-aligned, counterrotating vortices known as Langmuir cells. The Stokes drift (Phillips 1977) is often used as the key wave parameter in this interaction. Finally, the periodic straining of the turbulence by the surface waves may increase turbulent dissipation (Thais and Magnaudet 1996). It therefore seems sensible that the properties of the oceanic boundary layer should be sensitive to the properties of the surface gravity wave field and thus exhibit a more complex dependence than hypothesized by (1) and (2). We test this sensitivity here.

2. Measurements

Turbulent velocity fluctuations in the oceanic boundary layer are typically a few centimeters per second, much smaller than the one meter per second typical velocities of surface waves. This presents a formidable measurement challenge. This can be overcome by measuring the motion of neutrally buoyant “Lagrangian floats” (D'Asaro et al. 1996). These are designed to follow the three-dimensional motion of water parcels within the boundary layer by maintaining a neutral buoyancy and by having a large drag. The depth of the float is measured using an onboard pressure sensor; its vertical velocity is measured from the rate of change of pressure.

There are two major sources of error in the float's measurement of vertical velocity. First, the floats are slightly buoyant (about 0.5–3 g), a feature imposed to ensure that they do not sink out of the mixed layer. A circular metal drogue with a frontal area of about 1 m2, acted upon by this buoyancy, results in an upward motion of 3–5 mm s−1 of the floats relative to the water. The floats therefore are not exactly Lagrangian and do not necessarily sample the mixed layer uniformly. The magnitude of the resulting error is discussed later in this paper. Second, the approximately 1-m size of the float reduces its response to velocity fluctuations smaller than itself (Lien et al. 1998). This has only a minor effect on the measured velocity variance since most of the velocity variance is at scales larger than the float size.

Surface waves are not an important source of noise in these measurements for several reasons. Pressure fluctuations are zero along particle paths for linear surface waves, thus greatly attenuating the surface wave pressure fluctations measured by the float. The float pressure sensor is mounted at the top of the float and therefore does not exactly follow a particle path. D'Asaro et al. (1996), using float and surface wave measurements, show that the float pressure signal at surface wave frequencies is explained by a model in which the center of the float follows a Lagrangian path and the waves are linear. The pressure fluctuations measured at the float are much less than the surface wave pressure fluctuations in the same depth range. Surface waves are not exactly linear. However, the lowest-order contribution of surface wave nonlinearity to the surface pressure is due to the wave dynamic pressure ρU2/2 where U is the wave velocity (Phillips 1977). For U = 1 m s−1, this is equivalent to only δNL = 0.05 m of depth variation. The wave amplitude δNL will vary with a period of roughly 100 seconds as wave groups propogate past the float. This contributes only about 0.003 m s−1 to the measured float velocity, which is negligible. The small remaining velocity fluctuations at surface wave frequencies are sampled at 1 Hz and reduced to a negligible level by using a 50-s running mean sampled at a 25-s period. Lien et al. (1998) and D'Asaro and Lien (2000) show that Lagrangian frequency spectra of vertical velocity have the same spectral shape in turbulent flows with and without surface waves.1 The vertical velocity variance is concentrated at frequencies far lower than those of surface waves. If surface waves contributed significantly to the velocity spectra, one would expect spectra taken from environments with surface waves to differ from those taken in wave-free environments. None of this is meant to imply that surface waves are unimportant in the dynamics of the mixed layer, only that they do not affect the measurement of float depth from pressure.

Three Lagrangian floats were deployed from the Research Vessel Wecoma for two weeks in deep water off the coast of Vancouver Island in January 1995. The experiment followed the drifting floats 70 km northward from 48.0°N, 127.9°W in 2400–2500 m of water. Their depth–time trajectories are shown in Fig. 1b. Each float was deployed for approximately 24 hours, recovered, and redeployed with its density adjusted to compensate for the slowly changing density of the mixed layer. Frequent vertical profiles of temperature and salinity were made using a SeaBird 9/11 CTD with redundant sensors.2 The CTD profiles were used to define the depth of the mixed layer3 shown by the heavy orange line in Fig. 1b. Individual floats (Fig. 2) moved vertically between the sea surface and approximately the mixed layer depth. Floats placed below the mixed layer (not shown here) are constrained vertically by the strong stratification and undergo much smaller vertical excursions. This illustrates the rapid vertical exchange characteristic of the turbulent boundary layer.

Fig. 1.

(a) Meteorological forcing. Wind stress τ (red) is represented by u2 = τ/ρ and buoyancy flux Jb (blue) is represented by w2 = (JbH)2/3; w2 is scaled relative to u2 in approximate proportion to their ability to force vertical motions in the mixed layer. (The mean square vertical velocity in free convection is about 0.3w2 (Stull 1988); here the mean square vertical velocity forced by wind stress is 1.35u2; (a) therefore scales w2 by 0.22 = 0.3/1.35.) Positive values of w2 imply cooling of the ocean; negative values of w2 imply heating. Direction from which the wind is blowing is shown in green. (b) Depth–time trajectories of floats during experiment. Mixed layer depth H is shown in orange

Fig. 1.

(a) Meteorological forcing. Wind stress τ (red) is represented by u2 = τ/ρ and buoyancy flux Jb (blue) is represented by w2 = (JbH)2/3; w2 is scaled relative to u2 in approximate proportion to their ability to force vertical motions in the mixed layer. (The mean square vertical velocity in free convection is about 0.3w2 (Stull 1988); here the mean square vertical velocity forced by wind stress is 1.35u2; (a) therefore scales w2 by 0.22 = 0.3/1.35.) Positive values of w2 imply cooling of the ocean; negative values of w2 imply heating. Direction from which the wind is blowing is shown in green. (b) Depth–time trajectories of floats during experiment. Mixed layer depth H is shown in orange

Fig. 2.

Depth–time trajectories of floats during one typical day of data. Portions of four float deployments occur during this day as indicated by the red, black, and blue lines. The mixed layer depth, computed from density profiles, is shown in orange. Times of individual density profiles are shown by pulses

Fig. 2.

Depth–time trajectories of floats during one typical day of data. Portions of four float deployments occur during this day as indicated by the red, black, and blue lines. The mixed layer depth, computed from density profiles, is shown in orange. Times of individual density profiles are shown by pulses

Fluxes of buoyancy and momentum were computed from the meteorological measurements on the Wecoma using bulk formulas and are shown in Fig. 1a. Downwelling shortwave and longwave radiation were measured with redundant Eppley radiometers adjusted following Dickey et al. (1994). Temperature, humidity, and wind speed and direction were measured using redundant sensors. Temperature data were screened for occasional coolings due to spray. Wind data from the upwind side of the ship were used. Fluxes were calculated following Fairall et al. (1996) as implemented in the MATLAB Air–Sea toolbox version 2.0 (http://sea-mat.whoi.edu/). Buoyancy flux is the sum of that due to shortwave and longwave radiation and that due to sensible and latent heat fluxes. Average values of these quantities are 41, −22, 4, and −27 W m−2, respectively.

Large variations in wind stress occurred as a series of small storms swept past the site. The depth of the mixed layer tended to increase during periods of strong wind, as expected, since increased wind leads to increased vertical mixing. The depth of the mixed layer tended to decrease during periods of weak wind. This is probably due to the relaxation of horizontal density gradients into the vertical (Rudnick and Ferrari 1999).

3. Analysis and results

The vertical velocity variance σ2w was estimated from the 3.6-h average of all float vertical velocities. Since each float traverses the mixed layer several times during the averaging period, σ2w corresponds approximately to the mixed layer average of w2. Comparison of the time series of u2 and σ2w (Fig. 3) shows a close correspondence. Further averaging of σ2w to reduce the sampling error shows that almost all the variation in σ2w is due to u2 (Fig. 4). Equation (2) is remarkably accurate for these data.

Fig. 3.

Mean square vertical velocity σ2w and squared friction velocity u2. Gray lines are 95% confidence limits of a χ2 distribution. (Degrees of freedom for each average over Tav = 3.6 h and nfloats floats is (Tav/Tc)(〈1/τ〉/τ)nfloats, where τ = H/u∗ is proportional to the time needed for a float to cross the mixed layer and 〈1/τ〉 is the time average of 1/τ. The correlation time Tc = 750 s was estimated from the overall ratio of the mean to the standard deviation assuming a χ2 distribution.)

Fig. 3.

Mean square vertical velocity σ2w and squared friction velocity u2. Gray lines are 95% confidence limits of a χ2 distribution. (Degrees of freedom for each average over Tav = 3.6 h and nfloats floats is (Tav/Tc)(〈1/τ〉/τ)nfloats, where τ = H/u∗ is proportional to the time needed for a float to cross the mixed layer and 〈1/τ〉 is the time average of 1/τ. The correlation time Tc = 750 s was estimated from the overall ratio of the mean to the standard deviation assuming a χ2 distribution.)

Fig. 4.

Root-mean-square vertical velocity σw as a function of friction velocity u∗. Line is σw = 1.35 u∗. Error bars are 95% confidence limits of a χ2 distribution. [Degrees of freedom are the sum of the degrees of freedom ni for for each data segment i of duration T, where ni = T/(τ0u∗/H).]

Fig. 4.

Root-mean-square vertical velocity σw as a function of friction velocity u∗. Line is σw = 1.35 u∗. Error bars are 95% confidence limits of a χ2 distribution. [Degrees of freedom are the sum of the degrees of freedom ni for for each data segment i of duration T, where ni = T/(τ0u∗/H).]

Although buoyancy flux is often the major source of boundary layer turbulence in the oceanic boundary layer at other times and places (Lien et al. 1998; Steffen and D'Asaro 2001) it is of minor significance in these data. No relationships were found between the buoyancy forcing and σ2w. Time series of w2 are uncorrelated with u2, σ2w and that part of σ2w that is uncorrelated with u2.

No relationships were found between surface wave properties and σ2w except those that could be explained by correlations between wave properties and u∗. Scalar surface wave spectra were measured using a single-beam upward-looking sonar mounted on a stable subsurface platform; directional surface wave spectra were measured from a horizontally scanning sonar mounted on a similar platform (Trevorrow 1995). The directional spectra often exhibited complex multipeaked forms reflecting the rapidly changing and spatially variable storm structure. Measured spectra were used to compute various surface wave properties. Figure 5 shows the time series of measured wave height squared σ2ζ and a prediction of σ2ζ dependent only on the local wind and wind stress.4 The measured wave height increases when the wind increases but can remain high for several days after the wind decreases. The wave energy per unit area Ewave and the wave phase speed at the peak of the wave displacement spectrum Cpeak are only weakly correlated with u∗. The wave slope at the peak of the spectrum Slpeak is correlated with u∗, with steeper waves occurring at higher winds. The correlation is stronger for peaks naively chosen from the scalar wave spectra than for the actual windsea peak chosen from the directional wave spectra using the Banner et al. (2000) definition of “significant spectral peak steepness.”5 In contrast, the energy at frequencies well above the spectral peak is described accurately by the Phillips (1985) spectrum and varies closely with u∗. Equivalently, the spectral level of the wave slope spectrum at high frequency Φslope is proportional to u∗. The surface value of the Stokes drift velocity S0 was computed from both the directional wave spectra and, less accurately, from the scalar wave spectra assuming a unidirectional spectrum. These values are only weakly correlated with u∗. The turbulent Langmuir number La = u∗/S0 was more likely to be small at low wind speeds, because S0 is less variable than u∗. Inverse wave “age” u∗/Cpeak is correlated with u∗, with younger waves occurring at higher wind speeds. When any correlation with u∗ was removed, neither S0, La,Ewave, Cpeak, u∗/Cpeak, Φslope, nor Slpeak showed any significant relationship with σ2w.

Fig. 5.

Mean square surface wave displacement calculated from scalar wave spectra (open circles), from directional wave spectra (pluses) and from a model wave spectrum in equilibrium with the wind forcing

Fig. 5.

Mean square surface wave displacement calculated from scalar wave spectra (open circles), from directional wave spectra (pluses) and from a model wave spectrum in equilibrium with the wind forcing

Figure 6 shows A(z). Depth is scaled with H in order to minimize the effect of the large variations in H. Float data with H < 20 m were excluded in order to avoid large vertical stretching of very thin mixed layers. Float data with u∗ < 0.008 m s−1 were also excluded from the averages because, as indicated in Fig. 4, there is a departure from a linear relationship for low values of u∗. The floats are not uniformly distributed in depth as they should be if they followed water parcels perfectly. The distribution (Fig. 6) is surface intensified, presumably owing to the buoyancy of the floats, and becomes more uniform with increasing wind stress. The floats therefore oversample the more energetic upper part of the mixed layer and bias the value of σ2w/u2 upward from 1.27 for a mixed-layer depth average to 1.45 for a time average.

Fig. 6.

Scaled vertical kinetic energy A(z) = σ2w/u2 as a function of scaled depth z/H (curve). Error bars are 95% confidence limits of a χ2 distribution. The degrees of freedom are estimated from the ratio of mean to standard deviation at each depth. The gray profile has been corrected for biases near Z/H = −1. Circles represent σ2w/u2 for a smooth-wall zero-pressure-gradient turbulent boundary layer; stars represent the same for a rough-wall boundary layer (Hinze 1975). Bars indicate probability distribution of float data on an arbitrary scale

Fig. 6.

Scaled vertical kinetic energy A(z) = σ2w/u2 as a function of scaled depth z/H (curve). Error bars are 95% confidence limits of a χ2 distribution. The degrees of freedom are estimated from the ratio of mean to standard deviation at each depth. The gray profile has been corrected for biases near Z/H = −1. Circles represent σ2w/u2 for a smooth-wall zero-pressure-gradient turbulent boundary layer; stars represent the same for a rough-wall boundary layer (Hinze 1975). Bars indicate probability distribution of float data on an arbitrary scale

The profile of A(z) does not go to zero at the mixed layer base, as would be expected for mixed layer turbulence. There are several possible reasons. First, water is exchanged between the mixed layer and the underlying stratification in this region. The float data only include particle trajectories that enter and exit the bottom part of the mixed layer from above and do not sample trajectories that start from or remain in the stratification. The measured trajectories are more energetic than the omitted ones, biasing the average energy high. Harcourt et al. (2001) simulates Lagrangian floats in the Labrador Sea and finds (his Fig. 6) the vertical kinetic energy measured by simulated floats to be high by 29% of the peak value.6 This bias decays upward with a scale of about 0.2H in the simulation. Second, internal waves in the stratified interior will cause vertical velocities throughout the mixed layer. The vertical velocity due to hydrostatic internal waves will decay linearly from its value at the mixed layer base to very close to zero at the surface. Nonhydrostatic internal waves will decay faster (D'Asaro 1978), but these have only a small fraction of the total energy. The average σ2w from five floats deployed in the mixed layer base near the start of the measurements was (6 mm s−1)2 or about 0.18u2. Third, the upward buoyancy of the floats causes them to oversample the more energetic downward-going plumes leaving the surface (D'Asaro et al. 2001) and thus bias the velocity high. Harcourt et al. (2001) models this effect for floats with a bias velocity of 7 mm s−1, which is somewhat larger than is appropriate here. The bias extends across the mixed layer with a value of 25% of the peak σ2w.

The profile of A(z) was corrected for the first two factors (gray line in Fig. 6). The corrections reduce σ2w at the mixed layer base to slightly below zero, indicating that they alone are more than sufficient to explain the measured nonzero value. The effect of the third factor is probably small. The correct profile of turbulent σ2w probably lies between the gray and black curves and is probably closer to the gray curve than the black curve.

For both measured and corrected profiles A(z) reaches a maximum value between 1.9 and 2 in the upper mixed layer. Turbulent boundary layers driven by shear alone (stars and circles in Fig. 6) show maximum values of σ2w/u2 of approximately 1 for both smooth- and rough-walled boundary layers (McPhee and Smith 1976; Hinze 1975). The oceanic profile of turbulent vertical kinetic energy is between 1.75 and 2 times larger than that in these shear-driven boundary layers.

4. Discussion

Surface waves are likely responsible for the enhanced turbulent vertical kinetic energy found here, either through the action of wave breaking or Langmuir circulation. Terray et al. (1996) measured energy dissipation rates near the surface in the ocean boundary layer and found them far above those in wall-bounded boundary layers. They attribute this to surface wave breaking and find the excess energy dissipation equal to the flux of energy Fww from the wind to the waves. Using a large collection of historical surface wave spectra, they find Fww = ρu2C, where C is an effective wave phase speed. For a wide range of measured wave spectra, C/Cpeak is found to be a function of u∗/Cpeak (see their Fig. 6). When waves are “old,” C ≈ 10u∗ and turbulence production due to wave breaking is proportional to u3, as in solid-wall boundary layers, but with a much larger coefficient. The wave spectra measured here indicate that the waves are “old” because u∗/Cpeak < 2.2. Thus Terray et al.'s scaling predicts that the turbulence generated by the waves will scale with u∗, as it does in the absence of waves, but with a much higher turbulence level. A turbulent closure model of this process (Craig and Banner 1994) suggests that the turbulence due to wave breaking should decay rapidly away from the surface and not extend across the entire boundary layer. Physically, this is because the turbulent eddies created by wave breaking are much smaller than the boundary layer thickness. Thus, wave breaking can explain both the high level of turbulence found here and its scaling with u∗, but probably cannot explain why the high level of turbulence is found across the entire boundary layer.

Large-eddy simulation models (Skyllingstad and Denbo 1995; McWilliams et al. 1997) predict vertical kinetic energy profiles for shear-driven, solid-wall boundary layers similar to those shown by the symbols in Fig. 6. When the same models are applied to the ocean, driven by wind stress and surface waves through the CL interaction and using appropriate values of La (D'Asaro and Dairiki 1997), they predict turbulent vertical kinetic energy profiles similar to those shown by the gray line in Fig. 6. The dependence of these results on S0 has not been explored for these models. Thus, it is unclear whether Langmuir circulation dynamics can explain the observed scaling of σ2w on u2.

The wind stress used here is computed using a drag coefficient CD so that τbulk = ρairCDU215, where U15 is the wind speed at 15.4 m. The actual stress imparted by the atmosphere on the ocean τ can deviate by up to 50% from τbulk on the short timescales considered here, owing to time variations in the wind speed and direction and the presence of surface waves that are not aligned with the wind (Rieder and Smith 1998; Dennan et al. 1999). However, these effects are not sufficiently well understood to enable them to be quantified. Large and rapid variations in wind speed are apparent in Fig. 1a; the measured surface wave spectra commonly showed waves propagating from two or more different directions. All of the deviation of σ2w from (1) seen in Fig. 4 could easily be due to deviations of τ from τbulk.

These observations stand in contrast to those of Smith (1998), who found that the variance in one component of the near-surface horizontal velocity was correlated with S20 rather than with u2 for an approximately 40-h period chosen to have strong Langmuir circulations. Other similar measurements yield conflicting results (Pleuddemann et al. 1996). The properties of the near-surface region may indeed scale differently from the bulk of the mixed layer. The situation may be similar to that for rough- or smooth-wall, shear-driven boundary layers. The flow in the region near the wall depends in detail on the nature of the surface; the flow is quite different for a smooth wall than for a rough wall. However, the net effect of the near-wall region on the bulk of the boundary layer is parameterized by the stress alone.

These results do not necessarily imply that the total turbulent kinetic energy in the ocean boundary layer is higher than in similar shear-driven boundary layers. The simulations of McWilliams et al. (1997) suggest that an effect of the CL interaction is to move kinetic energy from the downwind velocity component to the crosswind and vertical components with little change in the total kinetic energy. Estimates of downwind and crosswind kinetic energies during these measurements will be reported in a future paper.

5. Summary

Vertical velocities in the ocean boundary layer were measured for two weeks at an open ocean, wintertime site using neutrally buoyant floats. Simultaneous measurements of the surface meteorology and surface waves showed a large variability in both wind and wave properties and only weak correlations between them. Buoyancy forcing was weak.

The mean square vertical velocity in the boundary layer σ2w measured from the vertical motion of the floats was proportional to the squared friction velocity u2 estimated from shipboard meteorological measurements. Thus, σ2w = Au2 with A = 1.35 ± 0.07. The deviations from this relation can be attributed entirely to statistical variation and measurement error. The slight buoyancy of the floats causes them to oversample the more energetic upper part of the mixed layer and to bias the measurement of σ2w high near the mixed layer base. Internal wave velocities also contribute nonturbulent vertical velocities. Removing these effects reduces the mixed layer average A for turbulent velocities to about 1.0, which is from 1.75 to 2 times that found in solid-wall turbulent boundary layers driven by shear alone.

It is surprising that the measured variations in the boundary layer vertical kinetic energy are not well correlated with the measured variations in surface wave properties. Perhaps this is because the relevant properties of the surface waves are in equilibrium with the wind in these data. The wave measurements, however, show that although the wave breaking rates may be in equilibrium with the wind, their energy, their momentum, and their Stokes drift are not. Other wave properties or other wave measurement systems might yield different results. Similar measurements under short fetch conditions (D'Asaro and Dairiki 1997) produce a higher value of A than found here. Perhaps the relationship between waves and boundary layer turbulence is stronger for these younger waves. These issues need to be understood before surface waves can be safely ignored in boundary layer models.

Acknowledgments

This work was supported by NSF Grant OCE-9711650. Cooperation with David Farmer and his group was crucial to the success of the measurements. Special thanks to Geoff Dairiki who played a key role in the development of the Lagrangian floats and in the analysis of their data. Two anonymous reviewers gave valuable insight and advice.

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Footnotes

Corresponding author address: Eric A. D'Asaro, Applied Physics Laboratory, University of Washington, 1013 NE 40th St., Seattle, WA 98105. Email: dasaro@apl.washington.edu

1

Although Lien et al. (1998) include data from the measurements described in this paper, they include only data from depths deeper than 15 m. Spectra from depths shallower than 15 m have very similar shapes as long as only data from mixed layers deeper than 20 m and u∗ greater than 0.008 m s−1 are considered.

2

Occasional bottle samples were taken and used to verify that the measured salinity was correct to 0.005 psu. Short-term precision of temperature and salinity is better than 0.001°C and 0.001 psu, respectively.

3

The mixed layer depth was defined as the shallowest depth at which the density deviation from the shallowest good value was greater than 0.003 kg m3. The mixed layer depths computed from individual profiles were linearly interpolated onto a fine grid and subjected to a 3.8-h running mean (very close to the averaging time for σw). Note that this mixed layer depth is shallower than the top of the main thermocline Hmt as defined by a deviation of 0.1 kg m3 from the surface density. The value of Hmt slowly increased from about 72 m to about 80 m during the 2 weeks of measurement.

4

The model spectrum for the scalar surface wave spectrum is the Phillips (1985) spectrum, which depends only on u∗, modified by the Pierson and Moskowitz (1964) peak function, which depends only on U20, the wind speed at 20 m height.

5

Banner et. al. compute the wave height associated with the peak by integrating the spectrum from 0.7 to 1.3 times the peak frequency. They do not specify the directional range of integration; a value of ±π/8 around the peak direction is used here.

6

The peak value of σ2w in the Labrador Sea data is very similar to that in Fig. 6, although the mixed layer depth is about 10 times larger.