1. Introduction

Global coupled air–wave–ocean models for weather prediction and climate simulations require understanding of the momentum transfer between air and water when waves are present. Importantly, wave conditions effect fluxes of chemicals and the structure of humidity near the ocean surface. As reviewed by Fisher et al. (2015), the impact of the wave field on the characteristics of the lower atmosphere is principally mediated through the wave-coherent boundary layer. When the sea state is a wind sea with little swell present, the turbulence statistics are consistent with Monin–Obukhov similarity theory (MOST), but when the sea state is dominated by swell, MOST fails (Drennan et al. 1999, 2003; Smedman et al. 1999; Högström et al. 2013, 2015, 2018; among others). When the waves are mainly swell and the large-scale atmospheric pressure driven winds low, the swell generates a wind in the lower atmosphere as suggested by Harris (1966) and shown in numerical simulations (Sullivan et al. 2008; Jiang et al. 2016) where the characteristics of the wind profile becomes a function of wave conditions. Analysis of satellite data and reanalysis wave hindcasts (Chen et al. 2002; Hanley et al. 2010; Semedo et al. 2011) indicate that swell is dominant at least 70% of the time in the mid and high latitudes and 95% dominant in the tropics. This in turn suggests that use of MOST to adjust profiles and estimate stress may be inappropriate most of the time in most locations, but since no alternative exists it is used.

The literature on the drag coefficient over water is exceptionally large. Foreman and Emeis (2010, hereinafter FE10) summarize over 20 datasets used to estimate the drag coefficient dividing them by different waterbody types. Most of these studies have no information on the characteristics of the sea or swell conditions. Given the wave statistics from Chen et al. (2002), Hanley et al. (2010), and Semedo et al. (2011) it is likely that there is appreciable swell content and hence the application of MOST to produce estimates of drag or roughness may not be valid. Focus is placed on the approach of Andreas et al. (2012, hereinafter A12), who developed a relationship between the friction velocity u_* and wind speed U that is based primarily on aircraft observations. Following the approach of FE10 they also found that using a u_*–U relationship reduced the scatter relative to a drag coefficient formulation—i.e., a (u_*/U)²-versus-U relationship. Their dataset spanned from light to strong winds (U < 25 m s⁻¹) and contained more cases at lower wind speeds than that of FE10. An important conclusion of A12 is that a significant break in the linear slope of the U–u_* relationship occurred at about U of 8–9 m s⁻¹, which they attributed to a change from smooth to fully rough airflow as the wind speed increased. Forman and Emeis focused on winds largely greater than 8 m s⁻¹ and emphasized rough flow conditions. A significant value of the A12 dataset is that the observations are made systematically from low-flying aircraft in a series of experiments in different ocean/coastal settings. The data were projected to a 10-m height using MOST. Their data had no concurrent wave information, and therefore the possible effect of swell is undetermined, but they noted that data from other experiments containing swell fell within the scatter of their data.

The effect of waves on the drag coefficient has a rich literature (e.g., Harris 1966; Donelan et al. 1997; Rieder and Smith 1998; Smedman et al. 1999, 2003, 2009; Grachev and Fairall 2001; García-Nava et al. 2009, c; Pan et al. 2005; Sullivan et al. 2008; Sahlée et al. 2012; Högström et al. 2013, 2015, 2018; Potter 2015; Jiang et al. 2016; Collins et al. 2018; among others). The consensus is that swell can create significant variation in the drag at lower wind speeds and have moderate effect at higher wind speeds. Recently, Potter (2015) showed swell reduced the drag coefficient at all wind speeds greater than 10 m s⁻¹ as a result of decreased turbulent flux around the swell frequency. Pan et al. (2005) proposed that the effect of swell could be characterized by a combination of the steepness and inverse wave age. Sullivan et al. (2008) and Jiang et al. (2016) use an LES simulation approach for estimating swell impact on the boundary layer. For hurricane wind strength, recent consensus is that the drag coefficient either levels off (Donelan et al. 2004) with increasing wind speed or decreases with increasing wind speed (Powell et al. 2003; Holthuijsen et al. 2012). Data from Holthuijsen et al. (2012) indicate that the drag coefficient is sensitive to the hurricane quadrant, which suggests a dependence on the character of sea/swell dominance in different parts of the storm. However, there is no widely accepted theory for incorporating the effect of swell in any wind speed regime.

Here we examine a set of in situ, near-surface wind, wind stress, and wave spectra observations made in the Gulf of Mexico in 1999 (GoM99). With these data, we can evaluate the A12 parameterization, investigate the impact of swell, and examine the interaction of waves and wind stress at wind speeds below the tropical cyclone levels. We will take a simple approach to focus on the relative energy level of the sea and swell. Although previous research has shown that the impact of swell may be related to the relative direction between the wind and the swell, we will not analyze the direction aspects of the problem because the A12 datasets have no comparable measurements. The paper is organized as follows. In section 2 the Gulf of Mexico dataset is described. Section 3 describes the partition of wave energy into sea and swell and development of relevant descriptive parameters. Section 4 compares the data from GOM99 with A12. In section 5, observations of friction velocity with and without swell are explored. Section 6 provides a discussion of the implications of the analysis to the prediction of stress.

2. Gulf of Mexico data (GOM99)

The GOM99 experiment from 27 April to 20 May 1999 was part of an Office of Naval Research initiative for “Measurements of Atmospheric and Oceanic Parameters Affecting Brightness Temperature in Passive Microwave Radiometry.” GOM99 was conducted in the Gulf of Mexico centered near NDBC Buoy 42036 at 28.5°N 84.517°W approximately 200 km northwest of Saint Petersburg, Florida. The site is near the edge of the continental shelf in a depth of 50–70 m. Three Air–Sea Interaction Spar (ASIS) buoys (Graber et al. 2000) were clustered around the NDBC buoy. The data presented here were analyzed from ASIS buoy Bravo located at 28.4178°N 84.5078°W in a depth of 55 m where 1604 observations of waves, wind and stress were made. An overview of the experiment is provided by Collins (2012), who reanalyzed the data. We removed winds less than 2 m s⁻¹ (183 cases) and removed any incomplete observations (62), leaving 1359 observations. This reduced the wind speed range to 2–16 m s⁻¹. The 2 m s⁻¹ limit was selected because below 2 m s⁻¹ the wind-sea spectrum had too few frequency bands to allow accurate specification of the wind sea.

The ASIS buoy (Graber et al. 2000) is of a spar design tethered to a floating surface buoy that is then anchored to the seafloor to isolate it from mooring effects. Wind and stress measurements used an asymmetric Gill Systems Solent sonic anemometer located at the top of the spar 5.7 m above the water surface. Wind stress was estimated via an eddy correlation method with the winds sampled at 15 Hz. Wave measurements are made by an array of capacitance wave gauges combined with a motion package tracking the movements of the platform. The array consisted of five wires arranged in a pentagon and three wires in an isosceles triangle at the center of the spar. The wires on the pentagon allow sensing directional properties of waves more than 1.8 m long (frequencies greater than 0.87 Hz), while the smaller triangular array senses waves from 0.1 to 1.8 m long (frequencies from 0.87 to 15.6 Hz). The motion package measured six degrees of freedom of the spar movement, and this allowed motion compensation for all sensors. Graber et al. (2000) and Pettersson et al. (2003) provide a validation of the wind and wave measurements. For this study, the wires on the faces of the pentagon were used. A Blackman–Harris window was applied to each 20-min block of sea surface elevation, and these were used to calculate spectra via fast Fourier transform. Fifteen adjacent spectral bins were averaged into larger bins, and spectra from each wire were averaged together.

Collins (2012) compared the wind and wave measurements among the four buoys and found excellent agreement. It is important to note that atmospheric stability was almost always neutral or unstable; stable conditions were present 12% of the time. Given that the buoy is in a 55-m water depth, waves with periods of 6 s or greater are affected by the depth. Some effect on the spectrum for longer waves should be expected; however, the longest peak period in the study is about 9 s. The shoaling coefficient for waves of this period is only about 5%, so we will consider this to be effectively deep water.

3. Sea and swell separation

For simplicity only the one-dimensional frequency spectrum is considered. Each spectrum is divided into sea and swell components based on wind speed where wind-sea sea surface variance ${\sigma }_{\mathrm{sea}}^2$ and swell variance ${\sigma }_{\mathrm{swell}}^2$ are defined by a frequency divide f_s such that the speed of the waves C_w is equal to the wind speed

$C_{\mathit{w}}=g/\left(2\hspace{0.17em}\pi \hspace{0.17em}{f}_s\right)=U$(1)

or C_w/U = 1. The parameters ${\sigma }_{\mathrm{sea}}^2$ and ${\sigma }_{\mathrm{swell}}^2$ are estimated from integration of the one-dimensional (frequency) energy spectrum above f > f_s for the sea and below f < f_s for the swell. The total variance of the wave field ${\sigma }_{\mathrm{total}}^2$ is given by ${\sigma }_{\mathrm{sea}}^2\mathrm{+}{\sigma }_{\mathrm{swell}}^2$. Traditionally the energy in the wave field is described by its significant wave height H_s = 4σ_total (Thompson and Vincent 1985). The sea and swell heights are similarly defined as

$H_{\mathrm{sea}}=4\hspace{0.17em}{\sigma }_{\mathrm{sea}}\text{and}$(2)

$H_{\mathrm{swell}}=4\hspace{0.17em}{\sigma }_{\mathrm{swell}\hspace{0.17em}}.$(3)

Another choice for sea–swell separation is the Pierson–Moskowitz fully developed spectrum (Pierson and Moskowitz 1964) peak frequency f_PM, which yields a C_w/U ~ 1.22. In this case the energy between f_s and f_PM is energy transferred from the actively wind-generated waves via nonlinear wave processes (Hasselmann 1962), or swell, or both but the energy is not directly contributed by the wind. We chose the sea–swell boundary at C_w/U = 1 since waves with C_w/U < 1 can receive direct atmospheric input while waves with C_w/U > 1 do not receive direct wind input. We do not believe the results to be sensitive of this choice. To check this, we recomputed all the principle parameters used in the study for comparison (Table 1) and found in general only small shifts in mean values. The shift in the mean transition frequency and the mean sea and swell frequencies, for example, corresponded to shifts in the associated wave periods of less than 0.6 s, and the shift in mean sea and swell heights was less than 8 cm. The average swell fraction R [see Eq. (4) below] shifted from 0.69 down to 0.59 but the dataset still remained swell dominant; that is, ${\sigma }_{\mathrm{swell}}^2\mathrm{>}{\sigma }_{\mathrm{sea}}^2$.

Table 1.

Comparison of average parameters For different sea–swell split definitions.

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Figure 1 is a plot of wind-sea and swell heights against wind speed. The wind speed ranged from near calm to 18 m s⁻¹. The wind-sea and swell heights each vary up to about 2 m and the significant wave height derived from the full spectrum reached nearly 2.5 m. In Fig. 1, the wind-sea height–versus–wind speed trend is well behaved and increasing, with a few exceptions, in a quadratic manner with wind speed. The H_sea follows the fully developed Pierson–Moskowitz wave height as formulated by Resio et al. (1999) up to 10 m s⁻¹. Above 10 m s⁻¹ the observations are close to Resio et al. (1999) but were limited by duration or fetch and did not reached full development. The few observations that fall substantially off the curve at the higher wind speeds were cases where the wind was observed to spike rapidly and then suddenly drop, typical of a squall line. These spectra are likely not in equilibrium with the wind speed, so they are dropped in the analysis. The swell heights, plotted in red in Fig. 1, appear to be randomly distributed across all but the highest wind speeds. This particular distribution of sea versus swell undoubtedly reflects the short duration of the study, not a full climatology.

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Wave height for the swell (red circles) and sea (blue circles) components of the wave spectrum as a function of wind speed (m s⁻¹), with the division between sea and swell defined by Eq. (1). The estimate of fully developed wave height (green line) for a given wind speed is from Resio et al. (1999).

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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For theoretical purposes, the concepts of a pure wind sea (active wind waves, no swell at all) and pure swell (no wind waves receiving wind input) can be considered. It then becomes important to understand how much swell is sufficient to negate the pure wind-sea concept and likewise how much wind-sea energy is sufficient to negate the pure swell concept. One simple choice is the ratio of swell variance to total variance

$R={\sigma }_{\mathrm{swell}}^2/\left({\sigma }_{\mathrm{swell}}^2+{\sigma }_{\mathrm{sea}}^2\right),$(4)

which we will normally express as a percentage. Ratio R has the value of being intuitive: low R—little swell impact; large R—large impact. Figure 2 provides a plot of R against wind speed U. Very dominant swell (R > 90%) occurs for winds less than 7 m s⁻¹ and represents about 30% of the 1359 observations. Low-swell wind seas have R < 10%, occurred for winds of 6–16 m s⁻¹ but only represent 9% of the observations. Swell was dominant (R > 50%) more than 70% of the time. So, although the Gulf of Mexico is not large compared to an ocean basin, it is sufficiently large to allow swell fields to develop and dominate with nearly pure swell (R > 90%) accounting for 30% of all cases. The wind speed in the experiment ranged up to 18 m s⁻¹, close to the maximum value in the A12 dataset, but the bulk winds were below 12 m s⁻¹. If observations with R < 10% are considered pure wind-sea spectra, Fig. 2 indicates that these conditions only existed at wind speeds greater than 6 m s⁻¹. Hence under the original definition there are no pure wind-sea spectra at low wind speeds, and an alternate definition that includes data from lower wind speeds conditions is desirable.

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The percent of the wave energy in the swell band (R, expressed as percent) as a function of wind speed.

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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Swell steepness ε_swell provides a different perspective on the influence of swell while maintaining many similarities to R. Swell with a low steepness typically have a small wave height and thus a low impact on the wind sea. Wind seas of the same height are steeper because the wave lengths are shorter. Swell with high steepness are very likely to modify the wind field because the swell height is large and breaking may be induced. Analysis of the GOM99 data shows that a swell steepness ε_swell < 0.005 actually encapsulates all of the spectra with R < 10% and includes data at wind speeds down to 2 m s⁻¹, rather than just down to 7 m s⁻¹ for the R < 10% criterion. For this reason, we use ε_swell < 0.005 to define pure wind sea; however, R and ε_swell have value, and we will continue to examine both.

4. Relationship between wind speed and stress ignoring swell

In Fig. 3, we plot the drag coefficient against wind speed with each observation colored by the swell fraction R. The graph exhibits the scatter historically found in such plots (such as Fig. 8 in A12). In Fig. 4, friction velocity is plotted against the wind speed as in A12 (e.g., their Fig. 7) with swell fraction indicated as in Fig. 3. Similar to A12 the scatter in Fig. 3 is greatly reduced, and it is clear that below 6–7 m s⁻¹ swell was dominant (R > 50%).

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Coefficient of drag as a function of wind speed U classified by R: 0%–10% (blue plus signs), 10%–30% (cyan plus signs), 30%–50% (green times signs), 50%–70% (black times signs), 70%–90% (magenta open circles), and 90%–100% (red open circles).

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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Friction velocity u_* vs wind speed U classified by R: 0%–10% (blue plus signs), 10%–30% (cyan plus signs), 30%–50% (green times signs), 50%–70% (black times signs), 70%–90% (magenta open circles), and 90%–100% (red open circles). The linear relationship for U > 8 m s⁻¹ is the black line and for U < 8 m s⁻¹ is the cyan line. A quadratic fit to the full dataset is the green line.

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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Linear and quadratic relationships equivalent to A12 are calculated ignoring wave conditions as

$\begin{array}{cc}{u}_{*}=0.062U-0.24& \text{for}\mathit{U}>{8\hspace{0.17em}\text{m s}}^{\mathrm{-}1}\end{array}.$(7)

The 95% confidence intervals for the coefficient are (0.0562, 0.0668) and for the intercept are (−0.2838, −0.1780), with r² = 0.728 and the F-test statistic p level at 0. For higher winds

$\begin{array}{cc}{u}_{*}=0.034\mathit{U}-0.0039& \text{for}\mathit{U}<{8\hspace{0.17em}\text{m s}}^{\mathrm{-}1}\end{array}.$(8)

The 95% confidence intervals for the coefficient are (0.0330, 0.0351) and for the intercept are (−0.0093, 0.0014), with r² = 0.728 and the F-test statistic p level at 0. Thus, both regressions have nonzero slopes, and the coefficients for each fall outside the 95% confidence bands of the other (as do the intercepts) and therefore statistically have distinctly different variations with U. Our winds were measured at 5.7 m. A12 adjusted their winds to 10 m using MOST and adjusting for air–sea temperature differences. The overall shape of the U–u_* relationships very much resembles that in A12; there are two linear portions with an inflection at a wind speed around 7–8 m s⁻¹. Equations (7) and (8) are plotted over our data in Fig. 4. As in A12 the data points tightly cluster about the bilinear trend. The A12 equations equivalent to Eqs. (7) and (8) for winds corrected to U₁₀ are

$\begin{array}{cc}{u}_{*}=0.0581U_{10}-0.24& \text{for}{\mathit{U}}_{10}>8{.7\hspace{0.17em}\text{m s}}^{\mathrm{-}1}\end{array}\text{and}$(9)

$\begin{array}{cc}{u}_{*}=0.0283U_{10}-0.0053& \text{for}{\mathit{U}}_{10}<8{.7\hspace{0.17em}\text{m s}}^{\mathrm{-}1}\end{array}.$(10)

On the basis of a 1/7th power-law adjustment, this should indicate that our observed winds are about 8% less than those of A12, which would yield coefficients similar to those of A12. A quadratic fit to our data is

$u_{*}=0.0019U^2+0.0154U+0.0375.$(11)

Thus, our equations are very similar to those of A12 (see Fig. 5). We find the consistency between the datasets reassuring because it implies that our data are comparable with A12’s larger, more comprehensive dataset drawn from multiple locations. Our data support FE10’s and A12’s recommendation to forego the traditional drag coefficient approach. These comparisons are done while ignoring information about the wave field.

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Comparison of A12 curves with the pure wind-sea and swell-dominant curves from GOM99.

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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5. Impact of swell in GOM99

Without wave measurements, A12 could not determine the potential impact of swell. Given the global statistics of swell occurrence (Chen et al. 2002; Hanley et al. 2010; Semedo et al. 2011), it is likely that swell is present in open ocean and coastal locations. In our dataset, observed swell and wind-sea information provide an opportunity to consider the impact of swell. We categorized each observation by the percentage of wave energy in the swell part of the wave field R (Fig. 4). The wave conditions in the lower velocity regime (U < 8 m s⁻¹) are swell dominated, with the swell percentages generally 50% or greater. This was also true when we use the alternate definition of the sea–swell split. In the higher velocity regime, the swell percentage is typically less than 30%. This trend with R is particularly evident in stress $u_{\mathrm{*}}^2$ (Fig. 6). At low wind speeds the swell heights are mainly greater than the sea wave heights (Fig. 1), with the maximum swell heights of 1.5–1.7 m only exceeded by the wind-sea heights when U > 10 m s⁻¹.

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Plot of stress parameterized by $u_{*}^2$ (m² s⁻²), vs wind speed (m s⁻¹) from GOM99 classed by R: 0%–10% (blue plus signs), 10%–30% (cyan plus signs), 30%–50% (green times signs), 50%–70% (black times signs), 70%–90% (magenta open circles), and 90%–100% (red open circles). The dark-blue curve is an estimate of $u_{*}^2$ from Eq. (11).

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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a. Friction velocity in absence of swell

Observations lacking swell do not exist in our observations so the goal was to arrive at an approximation for minimal swell. The first definition tried is R < 10%. This yielded many cases but none with wind speeds less than 6 m s⁻¹. We then applied a swell-steepness criterion to isolate those cases in which swell height was low and the wavelength was relatively long, minimizing the disturbance of the air. Jiang et al. (2016) simulations indicate that swell-related friction velocity increases with swell steepness for low winds. We develop a definition for pure wind seas using a ε_swell < 0.005 criterion yielding a U–u_*sea relationship, where u_*sea signifies a u_* with almost no swell present. This definition had the advantages that it included all the points identified as R < 10%, included wind speeds as low as 2 m s⁻¹ and had total wave heights H_s less than 0.25 m for U < 6 m s⁻¹. Although there may be as much swell energy as wind sea, the cases represent the lowest wave cases. The friction velocities were significantly below cases with higher swell steepness at the lowest wind speeds.

Drennan et al. (1999) studied the pure wind-sea problem off the coast of North Carolina and defined a drag coefficient C_d0 for pure sea

$C_{d0}=0.001\left(0.07U+0.6\right)$(12)

that was based on winds greater than 6 m s⁻¹.

Figure 7 is a plot of u_*sea against U using the ε_swell < 0.005 criterion. A linear fit to the data (116 observations) yields

$u_{*\mathrm{sea}}=0.047\hspace{0.17em}U–0.08$(13)

for U > 2 m s⁻¹. The 95% confidence intervals for the coefficient are (0.0437, 0.0497) and for the intercept are (−0.1088, −0.0597), with r² = 0.893 and the F-test statistic p level at 0. If Drennan’s C_do is converted into u_*sea the result plots well through our dataset (Fig. 7).

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Plot of U–u_*0 for pure wind seas as defined by the swell steepness ε_swell < 0.005 criteria. The best-fit line is in red. The black dashed line is Eq. (12) from Drennan et al. (1999) extrapolated from C_d to u_*0.

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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For conditions with minimal swell, Eq. (13) establishes that the relationship between U–u_*sea is linear over the range 2–16 m s⁻¹. It does not have any significant inflection point as seen in the A12 data. However, plots with ε_swell > 0.02 have some inflection. If the A12 explanation for the inflection based on flow transition is accurate, we would expect our pure wind-sea curve to show a significant inflection near 7–8 m s⁻¹. Based on the minimalization of swell in our data we believe MOST is applicable. So our conjecture is that the inflection in the A12 dataset may be due to the presence of swell in their data.

b. Friction velocity in presence of swell

Our data (Figs. 4 and 6) indicate that most of the observations below 7–8 m s⁻¹ have larger R: swell is dominant. We define the set of swell-dominant cases for U < 8 m s⁻¹ by setting ε_swell > 0.005 and derive a best fit (Fig. 8):

$u_{*}=0.034U-\mathrm{0.0021.}$(14)

The 95% confidence intervals for the coefficient are (0.0328, 0.0353) and for the intercept are (−0.0084, −0.0040), with r² = 0.768 and the F-test statistic p level at 0. The data points are generally higher than the pure wind sea for U < 8 m s⁻¹. We note that the swell steepness in our dataset is generally less than 0.04 and leave open the possibility that this curve may not represent a situation with steeper waves. Considering the different degree in scatter shown in Figs. 7 and 8, we examined whether Eqs. (13) and (14) are statistically different. Comparisons indicate that the coefficient for Eq. (14) lies outside the 95% confidence interval for Eq. (13) and vice versa; the same is true for the intercepts. We can conclude then that the wind speed friction velocity relationship when swell is dominant is statistically different from the pure wind-sea situation based on the regression analysis confidence intervals. At the lowest wind speeds the difference in u_* can be a factor of 7.

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Plot of U–u_* for spectra with swell steepness ε_swell > 0.005. The best-fit line is in red. The black dashed line is the pure wind-sea curve.

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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Equations (13) and (14) were computed from datasets with no filtering for different atmospheric stability conditions. Since our data included swell and we cannot use MOST to readjust, we reanalyzed the data excluding observations where the air–sea temperature difference was more than a degree above or below neutral. The resulting regression equations did not test as different than the original unfiltered equations because their estimated coefficient and intercept fell within the respective confidence intervals for Eqs. (13) and (14). So we concluded that atmospheric stability was not a factor for this limited sample. This is likely because the atmosphere was neutral or unstable about 85% of the time and when stable the air–sea temperature difference was generally less than 2°. We do not imply that stability is not in general a problem that must be considered and indeed have other datasets where it is clearly important.

For winds less than 8 m s⁻¹, when we compare our swell-dominated curve [Eq. (14)] with our wave-independent curve [Eq. (8)] and with A12 [Eq. (10)], it is clear they are close (Fig. 9): Eqs. (8) and (14) essentially overlay each other. In our cases elevated values of u_* above u_*sea have been shown to be related to swell-dominant cases. If the points with ε_swell < 0.005 are ignored, the combination of Eqs. (8) and (13) would correspond to A12’s bilinear relationship (Fig. 4) but the controlling process would be the change in swell dominance in our case.

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Plot of U vs u_* for U < 8 m s⁻¹ for data with swell steepness greater than 0.005 (black dots). The best fit [Eq. (14)] is given by the dashed red line. The best fit of all data with U < 8 ignoring swell content [Eq. (8)] essentially overlays Eq. (14) and is shown by magenta dots. The A12 fit for this regime [Eq. (10)] is a green line. Equation (13) (blue line) is plotted to demonstrate the relative location of the pure wind-sea curve.

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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We can evaluate whether swell has an impact in the higher winds (>8 m s⁻¹) by comparing the regression results for our subset that contains all wave conditions [Eq. (7)] to the pure wind-sea case [Eq. (13)]. In this case neither regression coefficient falls within the confidence bands of the other with the same true for the intercepts. This indicates that in the higher wind regime, the relationship between friction velocity and wind speed differs from the pure wind case versus when swell is present. The data points in Figs. 4 and 6 indicate that the swell content is generally less than 30% for U > 8 m s⁻¹. Consequently, swell appears to move the relationship away from the pure wind-sea curve even when the swell is not dominant (R > 50%).

Since we have noted earlier that MOST is not applicable when swell is present, we cannot use MOST to replicate A12’s approach to estimate the flow transition. Our data demonstrate that the elevated stresses at low wind speeds above the pure wind-sea friction velocity appear correlated to the presence of swell. Our pure wind-sea data had no evidence of an inflection such as that observed by A12. If the flow transition hypothesis were correct, our pure wind-sea curve should have an inflection. Addressing the relationship between the boundary layer parameters and roughness fundamentally depends upon the use of MOST. Trying to extend the concept of flow transition which is rooted in the flow systems with fixed roughness elements to a system where the interface is in a random semiperiodic motion may not be possible. We can only suggest that there is enough wave motion due to both sea and swell in our low wind data to make the airflow rough though we suspect that the use of the smooth-rough transition concept may not be applicable for a wiggling boundary. We note that the results of Wu (1994) also suggests that the airflow is rough at these low wind speeds.

The datasets in A12 were carefully collected and it seems reasonable that the flights would have been flown when swell conditions were not extreme. A complicating factor is that swell can also propagate in the same direction of the wind making it very difficult to separate out without measurements of the wave field. This raises the question of how much swell is needed to produce a U–u_* relationship that significantly deviates from the pure wind-sea U–u_*sea one. In our data, the swell impact occurs at winds less than 8 m s⁻¹. During these conditions, swell wave height ranges from 0.25 to 1.7 m with an average of 0.50 m (see Fig. 1); the wind-sea heights range from less than 0.10 m to a maximum of about 0.8 m. The transition also happens about where R ~ 50%. Values of H_swell > 0.3 m are not rare in our data, so it does not appear to take much swell energy at low winds to create the elevated values of friction velocity in the swell-dominant curve. This is because the fully developed wind seas have very small wave heights at low wind speeds.

The swell fraction R is useful as a descriptor of the sea-state but we saw no evidence in the study that it could be used as a quantitative predictor. Figure 2 shows that R has considerable variability in the midrange of velocities (5–12 m s⁻¹) so we computed the average R and plotted it against wind speed. Figure 10 includes two estimates for the average R: one based on our preferred estimate of the sea–swell split (C_w/U = 1) and a second based on C_w/U = 1.22. The first curve (in blue) shows that the average R in the 7–8 m s⁻¹ wind speed range where the A12 inflection occurs is 35%–50%. Our analysis based on the scattering of individual observations suggested that when the wave state became swell dominant, the friction velocities began to exceed the pure wind-sea values. The average R values suggest that this deviation might begin somewhat before strict swell dominance. The green line in Fig. 10 represents the average R values based on the alternate definition C_w/U = 1.22. This definition basically shifts R to lower swell values (R about 10% less than the blue curve). The net result is that the average R value for the 7–8 m s⁻¹ transition point is lower: 25%–40%. So using a C_w/U = 1.22 split for sea and swell would indicate that the inflection occurs with slightly less swell (as defined in this manner). However, the main conclusion follows whichever definition for f_s used: in the lower wind regime as the swell fraction R approaches 50% the observed friction velocity deviates above the pure wind-sea trend.

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Plots of the average R and Pierson–Moskowitz sea height. The average swell fraction is plotted against wind speed for the two sea–swell split definitions C/U = 1 (blue) and C/U = 1.22 (green) for GOM99. The average R for Ocean Station Papa is given in black for C/U = 1. The Resio et al. (1999) estimate of the Pierson–Moskowitz sea height is in red. Down arrows indicate for two GOM99 average R curves the wind speeds and sea heights where swell becomes dominant: average R is 50%. Equivalent locations for Ocean Station Papa are 15 m s⁻¹ and 5 m.

Citation: Journal of the Atmospheric Sciences 77, 11; 10.1175/JAS-D-19-0338.1

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The question becomes whether this result can be generalized to other sites. The upper bound on sea heights in Fig. 1 follow the fully developed wave height curves (Resio et al. 1999) based on the Pierson–Moskowitz spectrum, which are considered the maximum possible actively generated wind-sea heights for a given wind speed. Cases with shorter fetches or durations will have lower H_sea values than the fully developed case. So, at low wind speeds the wind sea is low and not much swell is needed for the swell to be dominant. The 23 studies in FE10 have been categorized by type of water body/fetch and rescaled the data to reflect initiation of rough flow. Their equation coefficients have a spread of about 50% and show similar amount of variability within each water body type. Their focus is on wind speeds higher than the A12 inflection point. We offer the hypothesis that it is the distribution of sea and swell conditions in a dataset that are important. The underlying commonality is the pure wind-sea relationship. For any observation the deviation from that is due to the specifics of the sea–swell characteristics at that time. When we collect a set of data at a site and disregard the sea–swell conditions the scatter seen about the derived drag curve may reflect the peculiarities of what sea–swell conditions were sampled. We expect that the specifics of the A12 curve or those analyzed by FE10 simply reflect the variation of sea and swell in each dataset.

6. Discussion

The GOM99 U–u_* swell agnostic relationships [Eqs. (7) and (8)] closely follow the curves proposed by A12 (considering the difference wind speed measurement heights). Given that our measurements are in situ near the ocean surface, our data support the A12 parameterization with the caveat of mild swell conditions.

We show that there is a significant swell effect on u_*. We define a pure wind-sea U–u_*sea relationship equivalent to Drennan et al.’s (1999) pure wind-sea drag coefficient formulation. We suggest that this is the fundamental equation because it can apply anywhere. The effects of swell are then a perturbation about this curve. Below 8 m s⁻¹ the elevation in u_* above u_*sea appears related to the presence of swell because R > 50%. Our analysis of u_* for winds more than 8 m s⁻¹ suggests that the presence of swell with R in the 20%–30% range is sufficient to push the slope of the U–u_* from the pure wind sea toward the A12 equation.

The swell environment in GOM99 is benign. We do not know how our results would differ in a more extreme swell environment, but the data from Ocean Station Papa (Thomson et al. 2013) illustrate how much the swell environment can vary. In 33 000 observations from Papa, the average swell height was 2.77 m with a lowest swell height of 0.30 m and a maximum swell height of 11 m (as compared with 0.50, 0.25 and 1.8 m, respectively, in GOM99). The average swell steepness in GOM99 is 0.018 as compared with 0.037 at Papa, where swell steepness reached 0.080. The LES simulations of Jiang et al. (2016) demonstrate the significant modifications to the wind profile due to different levels of swell steepness. We would suggest that the occurrence of 6–8-m swell at Ocean Station Papa as did happen in the 2–5 m s⁻¹ wind speed range would produce a significantly different boundary layer than the pure wind-sea case and would yield larger friction velocities. If we hypothesize, based on GOM99 data, that swell-caused deviation in u_* when R > 50% is a general result, would a “swell dominant” curve at Papa equivalent to Eq. (14) be similar to GOM99 or would it be higher ? Would it have an inflection/intersection close to 15 m s⁻¹ where the average R is 50%? Unfortunately, no stress data were measured at Papa, and so it remains an open question. Given the ubiquitous nature of swell and its impact on the wind profile shown in many observational and LES studies, the need is for an alternate to MOST for mixed sea and swell environments. It is particularly important to obtain results for higher wind and swell environments where it is difficult to make detailed observations. We know that we can observe or compute u_* via eddy correlation; however, its interpretation and use when there is a low-level jet associated with swell (e.g., Harris 1966; Sullivan et al. 2008; Jiang et al. 2016) is unclear.

Last, we consider whether the differences that we have shown are important. Our data suggest that either parameterization—this study or A12—is a good estimate of the mean conditions in low-swell environments. We provide an estimate for the pure wind-sea/no swell situation that is applicable on most water bodies. Future work is needed to gather more data with high swell steepness to define u_* sensitivity to swell steepness which appears as a critical parameter (Pan et al. 2005; Sullivan et al. 2008; Jiang et al. 2016). At lower wind speeds, the environment is benign, and it is perhaps less critical to have good estimates of stress even because the impact on the upper atmosphere may not be large. When swell is present, the application of MOST to estimate fluxes within the lower atmosphere may be in error. Since u_* can be more than 5 times u_*sea at the lowest wind speeds, the error may be significant. Prediction of boundary layer characteristics used for climate studies and operational predictions of electromagnetic wave propagation may be degraded. Wave climate data indicate that swell is normally dominant at least 70% of the time so that the impact on annual averages should be evaluated for climate estimates; see, for example, Young (1999), Alves (2006), and Semedo et al. (2011).

Investigators have developed drag coefficient approaches categorized in FE10 in terms of water body type (i.e., open ocean, coastal, lake, etc.). Given that the important factor is the actual state of the sea surface, it may be better to develop empirical curves based on process characterization such as differences between wind and swell directions, swell steepness, ocean current speed or shallow water. However, a more useful approach would be development of algorithms such as Pan et al. (2005) that would be process dependent: functions of the wind and waves. Routine global spectral wave forecasts by major national and international weather centers in the United States, Canada and Europe can now provide the wave information required that was not widely available in previous decades.

7. Summary

An in situ dataset of winds, wind stress, and waves from the GOM99 experiment in the Gulf of Mexico is analyzed for the possible influence of swell on u_*. The data support FE10’s and A12’s assertion that the U–u_* formulation is better than the U–C_d approach because of the reduced scatter. Furthermore, our wave data support the aircraft databased U–u_* parameterization of A12 when swell steepness is low. However, the application of any of these curves for high or steep swell conditions is questioned.

Consideration of sea state suggests a complex picture. We defined a pure wind-sea wind versus stress relationship (U–u_*sea) which is linear over the range 2–16 m s⁻¹ showing no inflection point near 7–8 m s⁻¹, where we would expect an inflection if the A12 hypothesis of flow transition was valid. Above 8 m s⁻¹, the U–u_* equation is closer to the equivalent curve of A12 but even in this regime the regression equations suggest that the presence of a modest amount of swell moves the equation to have a higher slope coefficient than the pure wind-sea case. Below 8 m s⁻¹, when more than a minimal amount of swell (H_swell > 0.30–0.50 m) is present, u_* deviates from the pure wind-sea case and aligns with the lower velocity limb of the A12 parameterization. This yields an apparent inflection point, but our data suggest that this change is related to the dominance of swell because the wind-sea heights are low.