Anguelova and Webster (2006) describe in detail the general concept of estimating the whitecap fraction W from measurements of the brightness temperature TB of the ocean surface with satellite-borne microwave radiometers. Salisbury et al. (2013) describe the basic points of the algorithm estimating W (hereafter referred to as the W(TB) algorithm). Briefly, the algorithm obtains W by using measured TB data for the composite emissivity of the ocean surface and modeled TB data for the emissivity of the rough sea surface and areas that are covered with foam (Bettenhausen et al., 2006; Anguelova and Gaiser, 2013). An atmospheric model is necessary to evaluate the contribution from the atmosphere to TB. Minimization of the differences between the measured and modeled TB data in the W(TB) algorithm ensures minimal dependence of the W estimates on model assumptions and input variables.

Wind speed U10 is one of the required inputs to the atmospheric, roughness, and foam models (Anguelova and Webster, 2006; Salisbury et al., 2013). Wind speed data come from the SeaWinds scatterometer on the QuikSCAT platform or from the Global Data Assimilation System (GDAS), based on whichever matches up better with the WindSat data in time and space within 60 min and 25 km; hereafter, we refer to both QuikSCAT or GDAS wind speed values as U10 from QuikSCAT or U10QSCAT. The use of U10QSCAT in the estimates of satellite-based W is anticipated to lead to some intrinsic correlation when/if a relationship between W and U10QSCAT is sought.

The W data used in this study are obtained from TB at 10 and 37 GHz, W10 and W37; data for 37 GHz are shown in Fig. 1a. The W10 and W37 data approximately represent different stages of the whitecaps because of different sensitivity of microwave frequencies to foam thickness (Anguelova and Gaiser, 2011). Data of W10 are an upper limit for predominantly active wave breaking (stage A whitecaps; Monahan and Woolf, 1989) partially mixed with decaying (stage B) whitecaps, while W37 data quantify both active and decaying whitecaps. Because decaying foam covers a much larger area of the ocean surface than active whitecaps (Monahan and Woolf, 1989), W37 data are usually larger than W10 data. Comparisons to historic and contemporary in situ W data in Fig. 1b confirm the approximate representations of stage A whitecaps (cyan squares) and A plus B whitecaps (blue diamonds) by W10 (green) and W37 (magenta), respectively. Anguelova et al. (2009) have quantified the differences between satellite-based and in situ W data using both previously published measurements and time–space matchups of W, and discussed possible reasons for the discrepancies.

The satellite-based W data are gridded into a 0.5∘ × 0.5∘ grid cell together with the variables accompanying each W data point, namely U10QSCAT, T from GDAS, time (average of the times of all samples falling in each grid cell), and statistical data generated during the gridding including the root mean square (rms) error, standard deviation (SD), and count (the number of individual samples in a satellite swath averaged to obtain the daily mean W for a grid cell). In this study, we used daily matchups of W, U10, and T data for each grid cell for the year 2006. To reiterate, this data set – consisting of W10 and W37 accompanied with three environmental variables (U10, Udir and T) – is an early version of the whitecap database; the extended W database used by SAL13 (Sect. 1) contains three additional variables suitable to quantify explicitly the effects of wave field and atmospheric stability on W. Due to large data gaps in both space and time, the daily W data cannot be interpolated to provide better coverage (Fig. 1a). Therefore, only the available data are used without filling the gaps for areas where data are lacking. This global data set was used to assess the globally averaged wind speed dependence of W.

The annual global W distributions show regions with valid data points ranging from 100 to 300 samples per grid cell per year when both ascending and descending satellite passes are considered. Thus, different regions were selected using two criteria, namely (i) regions with a high number of valid data points, and (ii) a selection representative of different conditions in the Northern and Southern hemispheres (NH and SH).

With these criteria, 12 regions of interest were selected (Fig. 2) and W, U10, and T data for each region were extracted from the whitecap database. The coordinates of the selected regions are listed in Table 1, together with the corresponding number of samples (data points) and minimum, maximum, mean, and median values for wind speed and SST for January and July. For 90 % of the regional and monthly data used in the study, the percent difference (PD, defined as the difference between two values divided by the average of the two values) between mean and median values of U10 and T is less than 4 and 9.5 %, respectively. With medians and means approximately the same, the U10 and T data have normal distributions; i.e., outliers, though existing, do not affect the mean values significantly. All analyses presented here use the mean U10 and T values.

Figure 3 shows the seasonal cycles of the mean U10 and T values for 4 of the selected 12 regions (4, 5, 6, and 12) chosen to visualize the full range of regional variations of U10 and T data. With the large number of samples, the mean U10 and T values plotted in Fig. 3 are determined within 95 % confidence interval (CI) of the order of 10-2 (Table 1). That is, any uncertainty due to sampling is removed, and Fig. 3 represents seasonal variations well, which we will use in our analyses. Variability of SST within each region is visualized with error bars (±1 SD) in Fig. 3b. The distinct regional SST variations suggest the effect of SST can be discerned with our data and thus used to parameterize the effect of SST on W. The variability of U10 within regions is higher (wider error bars are not plotted to avoid clutter), which suggests that the use of the global data set to obtain a generalized wind speed dependence of W (Sect. 2.1) is reasonable.

Regions 2–11 are all in the open ocean; region 1 was selected for its landlocked position (Fig. 2). Region 6 in the Pacific doldrums is used as a reference for the lower limit of U10 (Fig. 3a), while region 12 is included to represent the lowest T values (Fig. 3b). Four regions (2, 3, 7, and 8) are at latitudes between 0 and 30∘ S and N (tropics and subtropics) representing the trade winds zone. These are regions with persistent (easterly) winds blowing over approximately the same fetches (except region 8) in oceans with different salinity (Tang et al., 2014) and primary production (Falkowski et al., 1998) (a proxy for surfactant concentrations). Region 4 is in the NH temperate zone representing long-fetched westerly winds. Region 5 covers the latitudes between 40 and 50∘ S known as “The Roaring Forties” for the strong westerly winds there, and is characterized with longer fetch. Differences in the seasonal cycles of U10 and T in regions 4 and 5 (Fig. 3) suggest more uniform conditions and longer fetches in the SH temperate zone. We have chosen regions 8 and 9 to represent different zonal conditions and to gauge the effect of narrower range of SST variations (as compared to the SST range in region 5). Chosen at the same latitude, regions 9–11 have approximately the same SST, salinity, and surfactants but represent different wind fetches, shortest for region 10 and longest for region 9. Overall, the chosen regions cover the full range of global oceanic conditions and, while representative of diverse regional conditions, each one has distinct regional characteristics.

Ideally, when deriving a W(U10) parameterization, the data for W and U10 should come from independent sources. The intrinsic correlation between W and U10 that might have arisen from the use of U10 from QuikSCAT in the estimates of W from TB (Sect. 2.2.1), might affect the relationship between W and U10 developed here. To evaluate the magnitude of such intrinsic correlation, we used U10 from the ECMWF (U10ECMWF), which is considered to be a more independent source. Note though that even the ECMWF data are generated by assimilating observational data sets (e.g., from buoys) in a coupled atmosphere–wave model (Goddijn-Murphy et al., 2011).

To compile this “independent” data set, we made time–space matchups between the W37 data and U10ECMWF from the 3-hourly ECMWF data for 2006. For each W-U10QSCAT pair at a time t from the original W database, there is a corresponding W-U10ECMWF pair of data within an interval t ± 1.5 h. This matching procedure differs from the W-U10QSCAT matching which was done at the WindSat swath resolution, before gridding the variables for the whitecap database (Anguelova et al., 2010). To speed up calculations, and because this already provides a statistically significant amount of data, we used only ascending satellite overpasses. Wind speeds above 35 m s-1 were discarded. Besides ECMWF wind data, for consistency we also extracted ECMWF SST values.

Figure 4a shows all ECMWF wind speed data that have been matched in time and space with the available U10QSCAT data for March 2006. The majority of the data is clustered in the range of 5–10 m s-1 (dark red). To characterize the difference between the two wind speed sources, the correlation between U10 from ECMWF and U10 from QuikSCAT was determined as the best linear fit forced through zero U10ECMWF=0.953U10QSCAT, with a coefficient of determination R2=0.824. For comparison, the unconstrained fit between U10QSCAT and U10ECMWF is also shown in Fig. 4a (dashed line); both fits are very close (they almost overlap) with identical correlation coefficients (R2=0.824 for the unconstrained fit). Similarly, Fig. 4b compares T from ECMWF and GDAS showing almost 1 : 1 correlation. That is, the two data sources provide almost the same values for T.

On average, U10 from ECMWF is about 5 % lower than U10 from QuikSCAT. This U10 difference can be explained to some extent with the effect of atmospheric stability because QuikSCAT provides equivalent neutral wind which accounts for the stability effects on the wind profile (Kara et al., 2008; Paget et al., 2015), while the ECMWF model gives stability-dependent wind speeds (Chelton and Freilich, 2005).

Having the correlation between U10 from the whitecap database and U10 from the ECMWF quantified (as well as for T), one can evaluate differences caused by the use of different data sources. Equation (7) could also be useful when one decides to use ECMWF data because of their availability at 6 or 3 h intervals as compared to the availability of W, U10, and T matchups twice a day (Sect. 2.2.1).

We first analyze the satellite-based W data to derive a general W(U10) expression (i.e., the trend of W with U10). We apply Eq. (6) with coefficients (n, a, b) left as free parameters to global data sets of W10, W37, and both together (W10 and W37). Table 2 shows the results for the regression coefficients determined from the fitting procedure within the 95 % CI. Each set of coefficients (n, a, b) accounts implicitly for U10 and all secondary factors.

To consistently interpret and explicitly quantify regional and seasonal variations of W data, it is necessary to analyze all W data – global, regional, and at different frequencies – with the same mean trend given by the W(U10) expression. Because Table 2 shows different wind speed exponents, we need to establish a general (unifying) n value. With sampling uncertainty removed from the determination of these n values (see the 95 % CIs in Table 2), we now investigate the variations of the wind speed exponents among data sets. The mean of the free-parameter n values in Table 2 is 1.82 with lower and upper limits of the 95 % CI of 0.88 and 2.77. A value of n=2 is within this 95 % CI and is thus a reasonable choice for such general (unifying) wind speed exponent. We further verified such a choice by applying two-sample t test for equal means to the n values in Table 2 and n=2. The t test showed that the mean of the wind speed exponents n determined as free parameters is not statistically different from n=2 (p>0.05). On this ground, we adjust the free-parameter wind speed exponents to n=2, a quadratic wind speed dependence of W.

Quadratic wind speed dependence here is not unprecedented. The first reported W(U10) relationship of Blanchard (1963) was quadratic. With careful statistical considerations, Bondur and Sharkov (1982) derived a quadratic W(U10) relationship for residual W (strip-like structures, in their terminology). Parameterizations of W in waters with different SST have also resulted in wind speed exponents around 2 (see Table 1 in Anguelova and Webster, 2006). Quadratic wind speed dependence is also consistent with the wind speed exponents of SAL13 in Eq. (1).

With the adjustment of the free-parameter n in Eq. (6) to a general (unifying) wind speed exponent n=2, for all subsequent analyses, we use a functional form for W(U10) modified from Eq. (6) to W=aU10+b2. Following Monahan and Lu (1990), we derive an expression W(U10) in the form of Eq. (8a) by plotting W1/2 as a function of U10QSCAT. Applying linear regression, we find an expression: W1/2=mU10+c, which is then rearranged and squared to provide coefficients a=m2 and b=c/m in Eq. (8a) (results in Sect. 3.1.1). All linear fits are done on the W data associated with U10 from 3 to 20 m s-1. The lower limit of 3 m s-1 is chosen as a threshold for observing whitecaps. This restriction is reasonable in light of the SAL13 analysis in which W data with a relative standard deviation (σW/W)>2 were removed: the discarded W data were about 10 % of all W data, mostly in regions with low wind speeds of around 3 m s-1. We exclude the high wind speed regime in order to avoid uncertainty due to (i) fewer data points in this regime; and (ii) anticipated larger uncertainty in the W data from the W(TB) algorithm.

For the intrinsic correlation analysis, the W-U10ECMWF data pairs are used in a similar fashion to make W1/2(U10ECMWF) linear fits and derive from them a relationship between the satellite-based W data and the ECMWF wind speeds. The two global W(U10) parameterizations for the two wind speed sources are then compared to evaluate the magnitude of the intrinsic correlation (results in Sect. 3.1.2).

Because Fig. 4 and Eq. (7) give the possibility to evaluate discrepancies due to the use of different sources for U10 and T, we use U10 and T from the whitecap database in all subsequent analyses and results. In this way, with the intrinsic correlation characterized, we restrict the uncertainty in our analyses by using the close matching up of W, U10, and T data in the whitecap database. This decision is reasonable considering that both data sets can be used in practice for different applications. The collocated data in the whitecap database (involving QuikSCAT) are most suitable for analysis (as done in this study). Meanwhile, W data from the whitecap database combined with forcing data from a global model (such as ECMWF or other) are useful for forecasts and climate simulations.

With n=2 for the general wind speed dependence determined, we then apply Eq. (8b) to the regional monthly subsets of W10 and W37 data. All available data per month were used, ranging from 22 to 31 days of data. Once again, scatter plots of W1/2(U10) were generated and the best linear fits were determined providing coefficients m and c for each region for each month for W10 and W37. The regional and seasonal variations of coefficients a and b are analyzed to inform us how to parameterize them in terms of SST, i.e., obtain a(T) and b(T) (results in Sect. 3.2).

To quantify how a(T) and b(T) are influenced by different wind speed dependences – our empirically determined (adjusted) wind speed exponent n=2 (Eq. 8a) or the physically reasoned cubic wind speed dependence (Eq. 5b) – we also analyzed scatter plots of W1/3(U10) and derived a respective set of coefficients a(T) and b(T).

We quantify differences between new and previously published parameterizations with two metrics (results in Sect. 3.3): (i) the PD between W values obtained with different parameterizations; and (ii) significance tests (Student t test and ANOVA) of the differences between W values obtained with new and previous W parameterizations.

Efforts to include wave parameters in W parameterizations are well justified because, after wind speed, the most important secondary factor that accounts for variability in W is the wave field (SAL13). Lacking wave characteristics, the early version of the whitecap database is not suitable for deriving an explicit expression for the wave field influence on W. However, we have investigated the effect of rising and waning winds on the W(U10) relationship (results in Sect. 3.1.3); increasing–decreasing winds are considered a proxy for undeveloped–developed seas (Stramska and Petelski, 2003; CAL08).

It is not feasible to determine whether winds are rising or waning from satellite-based wind speed data because of their low temporal resolution (twice a day at a given location). As wind speed provided by ECMWF is available every 3 h, U10ECMWF values were used to examine the wind conditions at the satellite overpass time associated with a W data point. Wind speed difference between two 3 h intervals ΔU10 has been used to detect changing winds. Wind speed differences of ΔU10 from 1 to 5 m s-1 in steps of 1 m s-1 were used to examine the sensitivity of the analysis to the choice of ΔU10 in identifying rising or waning winds. Higher ΔU10 values are associated with the passage of stronger atmospheric low-pressure systems, which come with higher wind speeds and thus stronger wind forcing of waves. The U10QSCAT values were correlated with W using Eq. (8). Only data for 37 GHz from the ascending satellite overpass were used.

The main assumptions of M86 for the SSSF based on the discrete whitecap method – constant values for τ and dE/dr (Sect. 1) – are usually questioned (Lewis and Schwartz, 2004; de Leeuw et al., 2011; Savelyev et al., 2014). It is not expected for either of these assumptions to hold for wave breaking at various scales and under different conditions in different locations. The SSSF proposed by Smith et al. (1993) on the basis of measured size-dependent aerosol concentrations is one of the first formulations to demonstrate that the shape factor cannot be constant. Norris et al. (2013a) also demonstrated that the aerosol flux per unit area whitecap varies with the wind and wave conditions.

Recently, Callaghan (2013) showed that the whitecap timescale is another source of often overlooked variability in SSSF parameterizations based on M86. Because W typically includes foam from all stages of whitecap evolution, Callaghan (2013) suggested that the adequate timescale for the aerosol productivity from a discrete whitecap is not just its decay time (as in Eqs. 4 and 4′), but the sum of the whitecap formation and decay timescales τ′. The value of τ′ varies from breaking wave to breaking wave, but an area-weighted mean whitecap lifetime can be calculated for any given observational period to account for this natural variability. Analyzing the lifetimes of 552 oceanic whitecaps from a field experiment, Callaghan (2013) found that the area-weighted mean τ′ varies by a factor of 2.7 (from 2.2 to 5.9 s). We refer the reader to Callaghan (2013) for an SSSF that accounts for SSA flux variability by explicitly incorporating whitecap timescale τ′.

Despite these questionable assumptions, the SSSF based on the discrete whitecap method in the form of M86 has been widely used in many models (Textor et al., 2006). Therefore, to those who have worked with M86 until now, a meaningful way to demonstrate how the new satellite-based W data, and W parameterizations based on them, would affect estimates of SSA flux is to hold everything else constant (e.g., the whitecap timescale and productivity in the shape factor) and clearly show differences caused solely by the use of new W expression(s) as a magnitude factor. On these grounds, the choice of the SSSF based on the M86 whitecap method is a suitable baseline for comparisons.

Though the chosen size range of 1–10 µm for SSA particles is limited, it is well justified for the purposes of this study with the following arguments.

Generally, the division of the SSA particles into sizes of small, medium, and large modes (de Leeuw et al., 2011, their Sect. 8) is well warranted when one considers the climatic effect to be studied (Sect. 1). For example, submicron particles are important for scattering by SSAs (direct effect) and the formation of cloud condensation nuclei (indirect effect), while super-micron particles are important for heat exchange (via sensible and latent heat fluxes) and heterogeneous chemical reactions (which need surface and volume to proceed effectively). However, in this study we do not focus on how the choice of the size distribution will affect the SSA estimates, nor do we aim to present estimates of specific effect on the climate system. Rather, with a fixed size distribution, we explore how parameterizing W data, which carry information for the influences of many factors, would affect estimates of SSA emission (Sect. 1). In this sense, we can choose to use any published size distribution as a shape factor.

The chosen size range is the range of medium (super-micron) mode of SSA particles. The size distribution of M86 is valid within this range (Sect. 2.4). The M86 size distribution, in its original or modified form, is widely used in GCMs and CTMs (Textor et al., 2006, their Table 3). The size range of 1–10 µm is a recurrent part of the various size ranges used in all (or at least most) SSSFs (see Table 2 in Grythe et al., 2014; hereafter G14).

The chemical composition of the SSA particles is another argument favoring the chosen size range. The super-micron particles consist, to a good approximation, solely of sea salt, whereas in biologically active regions, the submicron size range additionally includes organic material, with an increasing contribution as particle size decreases (O'Dowd et al., 2004; Facchini et al., 2008; Partanen et al., 2014). Since the organic mass fraction in submicron SSA particles is still highly uncertain (Albert et al., 2012), we focus on the medium-mode SSA emissions.

We evaluate the discrepancy expected due to neglecting particles below 1 µm using the G14 report of SSA production rate for dry particle diameters Dp=r80 obtained with M86 over two different size ranges: 4.51 × 1012 kg year-1 for the size range of 0.8 µm < r80 < 8 µm and 5.20 × 1012 kg year-1 for size range of 0.1 µm < r80 < 10 µm. The different size ranges bring a difference between the two G14 estimates of about 14 %. Neglecting particles with r80 < 0.1 µm would not significantly change the results presented here because they contribute on the order of 1 % to the overall mass (Facchini et al., 2008).

Because total whitecap fraction, rather than only the active breaking crests, provides bubble-mediated production of SSAs, we use W37 data to estimate the emission of medium-mode SSAs. The calculations use a modeling tool (Albert et al., 2010) in which the W(U10) parameterization of MOM80, as incorporated in Eq. (4), was replaced with the newly derived W(U10,T) parameterization (Eq. 4′). The resulting size-segregated droplet number emission rate was converted to mass emission rate using the approximation r80=2rd≡Dp, where rd and Dp are the particle dry radius and diameter, respectively (e.g., Lewis and Schwartz, 2004; de Leeuw et al., 2011), and a density of dry sea salt of 2.165 kg m-3.

Following the arguments of our approach (Sect. 2.1) and evaluating the wind speed exponents determined as free parameters (Sect. 2.3.1), we found that a quadratic wind speed exponent (n=2) fits reasonably well both W10 and W37 data sets. For the same data shown in Fig. 5, Fig. 6 shows the linear regression of the square root of W vs. U10: W1/2=10.23×10-3U10-10.82×10-310GHzW1/2=10.38×10-3U10+18.57×10-337GHz, with coefficients of determination R2 of 0.996 and 0.951, respectively. From Eq. (9), we obtain the following global average wind speed dependence of W using U10 from QuikSCAT: W10=10.47×10-5U10-1.0582W37=10.77×10-5U10+1.7892, where W is a fraction (not a percentage).

Figure 6c compares W(U10) in Eqs. (10)–(11) to W(U10) of SAL13. The trends are close, implying that having a different wind speed exponent is largely balanced by corresponding changes to the parametric coefficients. Indeed, the PD between our quadratic W(U10) and SAL13 W(U10) at 37 GHz ranges from 0.5 to 10 % over the wind speed range of 3–20 m s-1. ANOVA and Student tests show that these differences are not statistically significant. That is, the global quadratic W(U10) parameterization approaches the predictions of the SAL13 parameterization, which has a more specific wind speed exponent (n=1.59). Note that we do not expect our W(U10) parameterization to be distinctly different from that of SAL13 because both studies use the same data for W and U10 (though from different versions of the whitecap database). Rather, we aim to identify a general W(U10) trend in order to perform consistent regional analysis.

The y intercept for W10 (Eq. 10) is negative and, following the usual interpretation, yields a threshold wind speed of about 1.1 m s-1 for whitecap inception. This is in the range of previously published values from 0.6 (Reising et al., 2002) to 6.33 (Stramska and Petelski, 2003). Meanwhile, the positive y intercept b for W37 (Eq. 11) is meaningless at first glance and intriguing upon some pondering. While stabilized residual foam and/or foam from biological sources are possible (Sect. 3.1), it is not known whether such mechanisms are capable of providing a measurable amount of foam patches which produce bubble-mediated sea spray efficiently.

We propose broader interpretation of b in Eqs. (10)–(11), be it negative or positive. Generally, it is expected that the atmospheric stability (Kara et al., 2008) and fetch (through the wave growth and development) cause inception of the whitecaps at lower or higher wind speed. One can consider the range of values for b mentioned above (0.6 to 6.33) as an expression of such influences. We suppose that b can also incorporate the effect of the seawater properties on the extent of W. The net result of all secondary factors may be either negative or positive b.

Specifically, we promote the hypothesis that a positive y intercept b can be interpreted as a measure of the capacity of seawater with specific characteristics, such as viscosity and surface tension – which are governed by SST, salinity, and surfactant concentration – to affect W. Undoubtedly, none of these secondary factors creates whitecaps per se. Rather, they prolong or shorten the lifetime of the whitecaps via processes governed by the seawater properties. For instance, surfactants and salinity influence the persistence of submerged and surface bubbles. This yields variations of bubble rise velocity that replenish the foam on the surface at different rates. Long-lived decaying foam added to foamy areas created by subsequent breaking events would augment W; conversely, conditions that shorten bubble lifetimes would reduce W (or at least not add to W).

A positive y intercept can be thought of as a mathematical expression of this static forcing (as opposed to dynamic forcing from the wind) that given seawater properties can sustain. That is, at any given location, this static forcing acts as though higher wind speed of magnitude (U10+b) is producing more whitecaps than U10 alone. By parameterizing coefficients a and b in terms of different variables, one can evaluate how much the static forcing affects W in different geographic regions. By developing parameterizations a(T) and b(T) (Sect. 2.1), here we quantify only one static influence.

To quantify the possible intrinsic correlation in the derived W(U10) parameterization (Eqs. 10–11), we derived W(U10) using ECMWF wind speeds instead of the QuikSCAT wind speeds (Sect. 2.3.2). Figure 7a shows a scatter plot of W1/2 vs. U10ECMWF (only data for 37 GHz are shown); dashed and solid lines show unconstrained and zero-forced fits, respectively. The linear regression (given in the figure legend) is used to obtain the global average wind speed dependence using U10 from ECMWF as follows: W37=8.1×10-5U10+3.332. The positive intercept here is interpreted as in Sect. 3.1.1. Using Eq. (12), parameterized W values are plotted as a function of U10ECMWF in Fig. 7b. Increased scatter of the W data is evident when comparing Figs. 7b and 5d. We use different metrics to detect and evaluate possible intrinsic correlation.

The change of the coefficient of determination R2 of the W(U10) relationship when QuikSCAT winds are substituted with the ECMWF winds is one sign for the presence of intrinsic correlation. Physically, we expect a strong correlation between W and U10, and we see this clearly in Fig. 6b which shows R2=0.951 for W1/2 and U10QSCAT. However, the correlation coefficient might not be as high as in Fig. 6 if U10 were from a more independent source. We see this when comparing Figs. 6b and 7a. The W1/2-U10 correlation is still strong in Fig. 7a, but the plot shows more scatter and slightly lower correlation with R2=0.826.

Figure 8 visualizes the change in the spread of the W data with a plot of the residuals (biases) between the W data and the derived W parameterizations (Eqs. 11 and 12) as a function of wind speed; Fig. 8a is for U10QSCAT and Fig. 8b is for U10ECMWF. Larger biases are evident when U10ECMWF is used. The rms deviation between W data and parameterized W values increases from ΔW=0.214 % for the data set using U10QSCAT to ΔW=0.367 % for the data set using U10ECMWF.

The slopes in Figs. 6b and 7a differ by about 14 %. We evaluate how this translates into differences in W37 values as predicted by Eqs. (11) and (12). We found the PD between W37(U10QSCAT) and W37(U10ECMWF) to be less than ±19 % for wind speeds of 4–20 m s-1. Specifically, the W37 values obtained with U10QSCAT and U10ECMWF are approximately equal for wind speed of 8 m s-1. Below 8 m s-1, W37(U10ECMWF) is higher than W37(U10QSCAT) by up to 18.6 %. Above 8 m s-1, W37(U10ECMWF) is smaller than W37(U10QSCAT) by up to 14.8 %. The difference goes up to 27 % for wind speeds of 3 m s-1.

While different metrics suggest that the intrinsic correlation is present and may contribute to these differences, it is not the only reason for the discrepancies. Different matching procedures (Sect. 2.2.3) and the difference of about 5 % between the U10 values from the two different sources (Fig. 4a) also contribute to the W discrepancies from Eqs. (11) and (12). We therefore conclude from the PD values that the effect of the intrinsic correlation alone on W is most likely less than about 10 % for most frequently encountered wind speeds.

Figure 9 shows global W1/2 at 37 GHz as a function of rising and waning U10 from QuikSCAT for March 2006; both rising and waning winds were identified with a wind speed difference ΔU10=1 m s-1 (Sect. 2.3.4). The lines are fits of the data to Eq. (8); solid lines are zero-forced fits. Table 3 shows the slopes m and intercepts c together with R2 of the fits for rising and waning winds speeds for ΔU10 from 1 to 5 m s-1.

Note the difference between W1/2 as a function of rising and waning wind speeds in the range of 10–20 m s-1 (blue colors): more variability is seen at these wind speeds when the wind is waning than in cases when the wind is rising. Larger variability for waning winds lowers their R2 values compared to R2 for rising wind speeds for most ΔU10 (Table 3). For both rising and waning wind speeds, R2 values decrease with ΔU10 increasing. The reason for this is that higher ΔU10 threshold selects W1/2 values associated with more extreme wind conditions (Sect. 2.3.4). Because such conditions are rarer, less W1/2 values are selected yielding an increase in the spread of data points.

Table 3 shows that slopes m for both free and zero-forced fits do not differ substantially for either rising or waning wind speeds for any ΔU10 threshold. The intercepts c of the free fits increase with the wind threshold for both rising and waning winds. The intercepts are larger for waning winds than for rising. These results yield rising-vs.-waning average PD of 10 and 29 % for coefficients a and b, respectively.

The rise–wane wind effect, as detected in this study, is not pronounced compared to findings in previous studies that use in situ wind speed data. Goddijn-Murphy et al. (2011) studied wind history and wave development dependencies on in situ W data using wave model (ECMWF), satellite (QuikSCAT), and in situ data for U10. These authors detected significant effects only with in situ U10. The limited wave field effect in our study might be traced back to the method through which U10 was determined: wind speeds from satellites are spatial averages of scatterometric or radiometric observations that take a snapshot of the surface as it is affected by both wind history and wind local conditions, whereas in situ data for wind speed are single-point values averaged over a short time and hence representative of a relatively small area. The effect of the spatial averaging of the satellite data over a much larger area (i.e., the satellite footprint) might be that information on wind history is lost in the process. Limited results on the effect of the wave field obtained with a proxy analysis of the wind history using data paired with a less-than-optimal matching procedure (Sect. 2.2.3) do not justify further consideration in this study.

Table 4 exemplifies the results from Eq. (8b): listed are the slopes m and the intercepts c for W1/2-U10 relationships at 10 and 37 GHz in March 2006 in all 12 regions together with coefficients of determination R2 and 95 % CIs from the fitting procedure, as well as mean U10 and T values. The results in Table 4 attest that with satellite-based data sets, the sampling uncertainty in determining relationships is removed. The remaining geophysical (i.e., regional and seasonal) variations of coefficients a and b, which are obtained from coefficients m and c, are investigated here. Figure 10 shows examples of the W1/2 vs. U10QSCAT relationships for different regions and seasons. Figure 10a and b show scatter plots for the Gulf of Mexico (region 1) at both frequencies for January 2006. Statistics are presented in the figure's legend and the fit lines are shown in red. Figure 10c and d show the fit lines W1/2(U10) for 10 and 37 GHz in region 5 for all months, while Fig. 10e and f demonstrate variations of the fit lines W1/2(U10) for both frequencies over all regions for March 2006.

Figure 10 shows that the variations of the W1/2(U10) relationships at 10 GHz are smaller than those for 37 GHz. Focusing on the results for 37 GHz, we note that geographic differences from region to region for a fixed time period (Fig. 10f) yield more variability in the W1/2(U10) relationship than seasonal variations at a fixed location (Fig. 10d). Because the 37 GHz data provide more information for secondary forcing than the 10 GHz data, the remainder of the data analysis in this study is illustrated with results for W37 data. Note that all procedures and analyses described for W37 data have also been carried out for the W10 data and final results are reported (Sect. 3.3).

Figure 10 also shows that variations of W1/2 caused by U10 from 3 to 20 m s-1 are much larger than the regional and seasonal variations of W1/2. While this is expected (because U10 is a primary forcing factor), this also points out that we need to evaluate whether these regional and seasonal variations are statistically significant. For this, we grouped the values of a and b in two ways: (1) by month, with the full range of geographical variability (over all 12 regions) for each month; and (2) by region, with the full range of seasonal variability (over all 12 months) for each region. The ANOVA test applied to both groups showed that the seasonal variations are not statistically significant, while the regional variations are.

We illustrate this in Fig. 11 with values for b; similar graphs for a show the same results. Figure 11a shows the seasonal cycle for the regionally averaged b values with error bars (±1 SD) representing the regional variability. It is clear that the seasonal variations of the regionally averaged b values lie within the regional variability. That is, variations of b from month to month are statistically undistinguishable. Figure 11b illustrates why variations of b from region to region are significantly different. The graph shows the annually averaged b values for each region with error bars representing the seasonal variability. It is clear that the geographical variations are not lost in the seasonal variability.

The regional differences in Fig. 11b are the variations that we want to quantify with coefficients a and b in terms of secondary factors. The deviations of the regional regression coefficients a and b from the regression coefficients A=10.77×10-5 and B=1.789 of the general W(U10) dependence (Eq. 11) give a sense for the magnitude of these variations. The PD between the annually averaged 〈a〉 and A is about 5 % (average for all regions); the average PD between 〈b〉 and B is 50 %. These regional differences can be caused by any or all other secondary factors. It is not trivial to separate (deconvolve) the effects of different factors influencing W data. Because our proxy analysis of the wave field effect produced limited results (Sect. 3.1.3), quantification of the regional differences in terms of wave field with the data we use is not practical. Meanwhile Fig. 3b shows that SST is a distinct characteristic for different regions. This suggests that quantifying the variations of coefficients a and b in terms of SST is a viable possibility. We thus proceed with deriving expressions a(T) and b(T) for the regional variations of the W data; such results are useful to evaluate how well SST can account for the regional variations.

We derived a(T) and b(T) for W data at 37 GHz by relating annually averaged a and b values to the annually averaged T for each region (Fig. 12). Figure 12c shows the monthly values of coefficients b for each region and thus demonstrates how the data points in Fig. 12b have been formed; a similar procedure is used for the data points in Fig. 12a. As in Fig. 11b, the error bars (±1 SD) represent the seasonal variability of SST (horizontal bars) and coefficients a and b (vertical bars). A second order polynomial is fitted to the data points in Fig. 12a; a linear fit is applied to the data in Fig. 12b. The coefficients of determination for the derived SST dependences are R2=0.57 for a(T) and R2=0.87 for b(T). Such R2 values are consistent with the expectation that SST, being a static secondary factor, affects W more via the intercept b than via the slope a.

To evaluate the performance of the quadratic vs. cubic wind speed dependence in Eq. (6), we also derived SST-dependent coefficients a(T) and b(T) for n=3 following the same procedure as for the case of n=2. We applied Eq. (5b) with n=3 to W37 data for all months in regions 4, 5, 6, and 12; we verified that differences due to the use of 4 instead of 12 regions are not significant. Coefficients a and b were calculated from the m and c values and graphs similar to those in Fig. 12 were produced. Linear fits for both a and b were applied to these graphs.

The W(U10,T) parameterization for 37 GHz is used here. The W values from SAL13 (37 GHz) and MOM80 are used as references for PD calculations and significance tests (Sect. 2.3.3). All parameterizations are run for wind speeds from 3 to 20 m s-1.

Figure 13a compares W values from the derived W(U10,T) parameterization at three fixed SST values (T=2, 12, and 28 ∘C). Large changes of SST (from 2 to 28 ∘C) bring relatively small variations between the wind speed trends of W at different T values. The PDs between the three curves are no more than 15 %; indeed, significance tests show that the W values at any T remain statistically the same. In addition, W values at any T are not significantly different from the W predictions of the global quadratic W(U10) parameterization.

These results qualitatively illustrate the relative contributions of the implicit and explicit accounts for SST effect in the derived parameterization. Namely, large part of the SST and other influences on W is taken care of implicitly by using the quadratic wind speed exponent. Much smaller variations are explicitly expressed with the temperature-dependent coefficients. Taken together, the set of parametric coefficients – n=2, a(T), and b(T) – accounts for the (i) full SST effect (i.e., influence on both the trend and the spread of the W data); and (ii) globally averaged effects of all other secondary factors (i.e., influences only on the trend of W data).

We verify the validity of this deduction by comparing in Fig. 13b W values obtained with the quadratic and cubic W(U10,T) parameterizations at T=20 ∘C; MOM80 and SAL13 at 37 GHz are shown for reference. The W values from the cubic W(U10,T) parameterization are not statistically different from those obtained with either the quadratic W(U10,T) or SAL13 for low winds (< 10 m s-1). Different trends of the W values at higher wind speeds suggest that accounting explicitly for SST via a(T) and b(T) in the physically expected cubic wind speed dependence is not sufficient to replicate the satellite-based W data. In other words, when the wind speed exponent n is not adjusted to the data but instead follows the physically determined cubic dependence, explicit representation of the SST effect alone via the parametric coefficients a(T) and b(T) cannot account for all observed variations of W. The implication is that when using the cubic wind speed exponent, more secondary factors should be introduced explicitly.

The PD between the trends of the derived W(U10,T) and MOM80 W(U10) is from 5 to 175 % with the largest PDs for wind speeds below 7 m s-1. Figure 13 illustrates this with the different trends of the two parameterizations.

Here, we evaluate how well the derived whitecap fraction parameterizations model the trend and spread of the satellite-based W data. The parameterized W values are calculated using U10 and T from the whitecap database (Sect. 2.2.1).

Figure 14a compares W values predicted with both new parameterizations, W(U10) and W(U10,T), to the same in situ data plotted in Fig. 1b and to independent satellite-based W data for 10 and 37 GHz from 17 March 2007. Comparisons to the in situ W data demonstrate order-of-magnitude consistency of the W values from the new parameterizations. The new global W(U10) parameterizations (black symbols in Fig. 14a) follow reasonably well the wind speed trends of the satellite-based W data. The W values predicted with the new W(U10,T) parameterization (red and cyan symbols in Fig. 14a) are spread as the satellite-based W data. The cluster of W values predicted with W(U10,T) are statistically different from the MOM80 W(U10) parameterization. This is the most important result of this study: we demonstrate that by accounting for at least one secondary factor, we are able to model both the trend and the spread of the W values.

Note in Fig. 14a that the new W(U10,T) parameterization does not predict the spread of the satellite-based W data entirely. This suggests that accounting explicitly for SST in a W parameterization is not enough to replicate all natural variability (spread) of W. This is consistent with our general understanding of the need to explicitly include many secondary factors in W parameterizations, not just SST (Sect. 2.1).

Though SST entails small variations in the trend of W with U10 (Figs. 13a and 14a), an important consequence of the newly derived W(U10,T) parameterization is that it shapes significantly different spatial distribution compared to cubic and higher wind speed dependences like that of the MOM80. Figure 14b shows a difference map between the global annual average W distributions for 2006. MOM80 relationship yields a wider W range with higher values in regions with the highest wind speeds. In particular, this occurs between about 40 and 70∘ in the Southern Ocean and in the North Atlantic. The latitudinal variations from the Equator to the poles are more pronounced when using the MOM80 relationship as compared to Eqs. (13)–(14). The new W(U10,T) parameterization provides a global spatial distribution with similar patterns, but the absolute values are lower at high latitudes and higher at low latitudes. Note that in most studies, as in this study, W(U10) of MOM80 is extrapolated beyond the range of the data from which it was derived (Sect. 1). This could contribute to the large differences between the two parameterizations at higher wind speeds (and especially in cold waters).