1. Introduction

Air/sea fluxes of mass, momentum, and energy are mediated by the character of turbulence in both the atmospheric and oceanic boundary layers. In particular, the surface turbulent momentum flux is affected by and feeds back on sea surface wind speeds, where in this study the term “sea surface wind” will refer to the eddy-averaged wind at the standard anemometer height of 10 m. Wind shears drive the boundary layer turbulence, which in turn determines the rate at which momentum is exchanged with the upper ocean. Because surface fluxes generally have a nonlinear dependence on the sea surface wind speed, the area- or time-averaged fluxes are not equal to the fluxes diagnosed from the averaged winds. This fact is of potential importance to the calculation of air/sea fluxes in general circulation models (GCMs), in which only gridbox averaged vector winds are available to calculate gridbox-averaged fluxes (e.g., Cakmur et al. 2004). Calculation of the spatially averaged fluxes requires knowledge of wind speed moments of higher order than the mean; in fact, there is value to having the entire probability density function (pdf). Furthermore, the gridbox-averaged wind speed will not generally equal the magnitude of the gridbox-averaged vector wind; it is easy to show that higher-order moments of the vector winds will influence the mean wind speed (e.g., Mahrt and Sun 1995). This motivates the need for developing parameterizations of sea surface wind pdfs that take as input grid-scale-resolved variables, in particular moments of the vector winds. Improved parameterizations of sea surface wind speed pdfs are also relevant to improving observational diagnoses of air/sea fluxes (e.g., Wanninkhof 1992; Wanninkhof and McGillis 1999; Taylor 2000; Wanninkhof et al. 2002). Note that we are assuming spatial stationarity of surface wind statistics on the gridbox scale, so probability distributions in space and in time are interchangeable. That is, we assume that collocated individual observations of the wind taken at different times will follow the same distribution as simultaneous individual observations of the wind taken at different locations (on scales where boundary conditions are homogeneous). This assumption is analogous to a weak version of Taylor’s “frozen turbulence” hypothesis (weak because it is applied to the marginal distributions of the wind field, not full joint distributions).

The beginnings of a systematic theory for the probability distribution of sea surface wind speeds were presented in Monahan (2006a). In this earlier study, it was shown that the moments of wind speed are functionally related: the skewness is a decreasing, concave upward function of the ratio of the mean to the standard deviation, such that the skewness is positive when this ratio is small, near zero where the ratio is intermediate, and negative when the ratio is large. This relationship, which is shared with the Weibull distribution traditionally used to parameterize surface wind speed pdfs, was shown to follow from elementary boundary layer physics. In particular, the fact that the wind speed skewness can become negative was demonstrated to be a consequence of skewness in the vector winds resulting from the nonlinear dependence of surface drag on wind speed (Monahan 2004b). These facts were used to construct empirical parameterizations of sea surface vector wind and wind speed pdfs, using Gram–Charlier expansions of a Gaussian (e.g., Johnson et al. 1994) to produce along-mean wind component distributions with nonzero skewness and kurtosis. While this approach was able to demonstrate the importance of simulating the pdf of sea surface wind speeds of non-Gaussian structure in the vector winds, the Gram–Charlier distribution is problematic in that it is not guaranteed to be positive-definite (e.g., Johnson et al. 1994; Jondeau and Rockinger 2001). This is theoretically unsatisfactory and practically limiting in that the associated random variable is not realizable.

The present study has two primary goals. The first is to further characterize the deviations in observed sea surface wind pdfs from Weibull structure discussed in Monahan (2006a, b) through the use of relative entropy, a natural metric for the difference between probability distributions. Second, and more significantly, the present study further develops empirical parameterizations of sea surface wind speed pdfs in terms of moments of the vector winds. The importance of anisotropic fluctuations in vector winds is first considered, and then the wind speed pdf parameterizations arising from four different non-Gaussian distributions of the vector winds are compared. Section 2 of this study describes the dataset used in the analysis and establishes notation. Deviations of observed sea surface wind speed pdfs from the Weibull distribution are characterized and discussed in section 3. The empirical sea surface wind speed pdf parameterizations are developed and assessed in section 4, and a discussion and conclusions are presented in section 5.

2. Data

The sea surface wind dataset considered in this study consists of level-3.0 gridded daily SeaWinds scatterometer 10-m zonal and meridional wind observations from the National Aeronautics and Space Administration Quick Scatterometer (NASA QuikSCAT) satellite (Jet Propulsion Laboratory 2001), available on a 1/4° × 1/4° grid from 19 July 1999 to the present (31 December 2005 for the present study). These data are available to download from the NASA Jet Propulsion Laboratory (JPL) Distributed Active Archive Center (http://podaac.jpl.nasa.gov). The SeaWinds data have been extensively compared with buoy and ship measurements of surface winds (Ebuchi et al. 2002; Bourassa et al. 2003; Chelton and Freilich 2005); the root-mean-squared errors of the remotely sensed wind speed and direction are both found to depend on wind speed, with average values of ∼1 m s⁻¹ and ∼20°, respectively. Because raindrops are effective scatterers of microwaves in the wavelength band used by the SeaWinds scatterometer, rainfall can lead to errors in estimates of sea surface winds. The SeaWinds level-3.0 dataset flags those data points that are estimated as likely to have been corrupted by rain (Jet Propulsion Laboratory 2001); these data points have been excluded from the present analysis.

No further processing of the data, such as filtering or removing the annual cycle, was carried out on this dataset. There is significant seasonal variability in the probability distributions of both vector winds and wind speeds (Monahan 2006b); the data analyzed in this study were not seasonally stratified because of the relatively short duration of the SeaWinds dataset. The analysis has been repeated with 6-hourly National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis 10-m winds, which span the much longer period from 1 January 1948 to the present. Differences between the statistical structure of SeaWinds and NCEP–NCAR sea surface winds were considered in Monahan (2006b), in which it was shown that while the moment fields of the two datasets agree to leading order, they display some regional differences in detail (particularly over the Southern Ocean and Tropics). The results of the analysis carried out using the NCEP–NCAR data are qualitatively the same as those reported in the following sections. Because the SeaWinds data are a more direct representation of observed sea surface winds than reanalysis products (which are a hybrid of observations and general circulation model output), they were chosen as the dataset on which this study will focus.

In this study, we will consider surface wind vector components defined relative to a local coordinate system. The notation we use is u = wind component along local time-mean wind vector, υ = wind component across local time-mean wind vector (positive to the left), u = (u, υ) wind vector, and w = wind speed.

The use of a local coordinate system does not lead to any computational difficulties in this study, as only single-point statistics are considered.

The present study makes frequent use of the skewness and kurtosis of the sea surface vector winds and wind speeds; for a random variable x, these quantities are defined, respectively, as the normalized third- and fourth-order moments:

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where std is the standard deviation. Both of these quantities are zero for Gaussian random variables.

3. Sea surface wind speed pdfs: Observations

The first row of Fig. 1 displays the fields of mean(w), std(w), and skew(w) estimated from SeaWinds observations. The mean(w) field is characterized by high values in the extratropics, secondary maxima along the equatorward flanks of the subtropical highs, and minima in the equatorial doldrums and subtropical horse latitudes. Maxima in std(w) occur in the midlatitude storm tracks, while values in the subtropics and Tropics are generally small. Finally, skew(w) is positive in the NH midlatitudes and between the bands of westerlies and easterlies in the SH, it is near zero around the Southern Ocean, and it is negative throughout most of the Tropics. A detailed discussion of these fields and their physical origin is given in Monahan (2006a).

Historically, wind speed pdfs have been parameterized using the two-parameter Weibull distribution (e.g., Hennessey 1977; Justus et al. 1978; Conradsen et al. 1984; Pavia and O’Brien 1986; Isemer and Hasse 1991; Deaves and Lines 1997; Pang et al. 2001), although it has been noted that the fit is not exact (e.g., Stewart and Essenwanger 1978; Takle and Brown 1978; Tuller and Brett 1984; Erickson and Taylor 1989; Bauer 1996). Deviations of observed wind speed pdfs from Weibull structure were discussed in Monahan (2006a,b) in terms of differences between observed moments (mean, standard deviation, and skewness) and those of the best-fit Weibull distributions. A more comprehensive measure of the difference between two probability distributions, p₁(x) and p₂(x), is their relative entropy:

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(e.g., Kleeman 2002, 2006; Kleeman and Majda 2005). Although it is nonnegative, the relative entropy is not a Euclidean distance measure [in particular, it is not commutative: ρ(p₁||p₂) ≠ ρ(p₂||p₁)]. Nevertheless, relative entropy is a useful metric of the difference between probability distributions. In particular, the relative entropy between a given pdf and the Gaussian with the same covariance matrix has been used as a measure of non-Gaussianity (e.g., Abramov et al. 2005). In a comparable fashion, we will consider the relative entropy between the observed sea surface wind speed pdf, p_w(w), and the Weibull distribution with the best-fit parameters, p_wbl(w), as a measure of non-Weibull structure; a map of estimated ρ(p_w||p_wbl) is given in Fig. 2. Algorithms for the estimation of relative entropy are an active field of research (e.g., Kleeman and Majda 2005; Haven et al. 2005); the estimates presented in Fig. 2 were obtained by first calculating the maximum entropy distribution constrained by the wind speed mean, standard deviation, skewness, and kurtosis (section e of appendix B), and then numerically integrating Eq. (3). The solid black contour in Fig. 2 is the 95% confidence level for the null hypothesis that the observed relative entropy arises due to sampling fluctuations from an underlying Weibull distribution. This value was calculated to be ρ = 6 × 10⁻³ from a Monte Carlo simulation assuming the wind dataset consisted of 1000 statistical degrees of freedom (5.5 yr of daily data with an approximately 2-day autocorrelation e-folding time; Monahan 2004b).

Statistically significant deviations from Weibull structure are evident throughout the Tropics and in the NH midlatitudes, consistent with the skewness difference field illustrated in Monahan (2006a). The best-fit Weibull pdf at a typical midlatitude point [30°N, 165°W, where ρ(p₁||p₂) = 0.013] is compared to a kernel density estimate (Silverman 1986) of the observed wind speed in Fig. 3. While the two pdfs are not dramatically different, the observed wind speed pdf is narrower with a longer tail and more positive skewness than the corresponding Weibull pdf. Non-Weibull structure in surface wind speed pdfs is modest but robust and widespread, motivating the characterization of sea surface wind speeds pdfs that are more accurate than the Weibull distribution. Furthermore, the Weibull distribution is specified in terms of parameters (the shape and scale parameters; see section a of appendix B) that are not directly related to the grid-scale quantities produced by GCMs. In the next section, we will develop parameterizations of the sea surface wind speed that are of greater fidelity to nature than the Weibull distribution and use as input information moments of the sea surface vector winds.

4. Sea surface wind speed pdfs: Empirical models

The primary goal of this study is the development of parameterizations of wind speed pdfs that take as input moments of the vector winds; a natural place to start is the derivation of the wind speed pdf from the vector wind pdf. Given the joint pdf p_uυ(u, υ) of the vector wind, the wind speed pdf p_w(w) can be obtained through a change to polar coordinates and an integration over wind direction:

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where the factor of w before the integral comes from the Jacobian of the coordinate transformation as required by conservation of probability. This calculation can be simplified by assuming that (i) the vector wind components u and υ are independent so their joint distribution factors as the product of the marginals:

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and (ii) that the distribution of υ is Gaussian:

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[where by definition mean(υ) = 0]. Then

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Different parameterizations of p_w(w) then follow from different models of the pdf of the along-mean wind component u.

It was demonstrated in Monahan (2004b) that the sea surface vector winds are non-Gaussian, such that the mean and skewness fields are anticorrelated for both the zonal and meridional components. This relationship between moments was demonstrated to follow from the nonlinear dependence of surface drag on sea surface wind speed. Maps of the skewness and kurtosis fields estimated from the SeaWinds data for the along- and cross-mean wind components are presented in Fig. 4. It is evident that the distribution of the along-mean wind component displays non-Gaussian structure, with strongly negative skewness in both the Tropics and the Southern Ocean, where mean(u) is strongly positive [note that mean(u) is positive by definition], and strongly positive kurtosis in the Tropics. Deviations of the distribution of υ from Gaussianity are considerably weaker: skew(υ) is nonzero in the subtropical highs and along the ITCZ, but the magnitudes are considerably smaller than those of skew(u). Similarly, kurt(υ) takes positive values in the Tropics, but much less so than kurt(u). Thus, while u is strongly non-Gaussian, the approximation that υ has a Gaussian distribution is reasonable to first order.

The statistical dependence of u and υ was discussed in Monahan (2006a); Fig. 5 displays a plot of the squared correlation coefficient between u and υ (the fraction of variance accounted for by a linear relation between u and υ). Clearly, linear dependence between the along- and cross-wind components is weak, except over the equatorial oceans and monsoon regions where seasonal variability is strong. Of course, a vanishing correlation coefficient does not imply independence unless the joint distribution of u and υ is Gaussian. The stochastic boundary layer model in Monahan (2006a) predicts that u and υ should be uncorrelated but not independent. A more general measure of dependence employs the mutual information,

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defined as the relative entropy between the joint distribution p_uυ(u, υ) and the product of the marginals p_u(u)p_υ(υ); this quantity vanishes only if the joint distribution factors as the product of its marginals, that is, when u and υ are independent [Eq. (5)]. The quantity

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varies between 0 (u, υ independent) and 1 (u, υ perfectly dependent) and is equal to the squared correlation coefficient between u and υ when p_uυ(u, υ) is Gaussian; C² is a non-Gaussian generalization of the squared correlation coefficient. A map of C² (not shown) is not significantly different from the map of the squared correlation coefficient in Fig. 5, indicating that while u displays significant non-Gaussian structure, the covariability between u and υ is not strongly nonlinear. Thus, while the assumption of independence of u and υ is violated in a few localized regions, it is not a bad approximation on a global scale and it considerably simplifies the computation of the wind speed pdf from the pdf of vector winds.

We will now proceed to develop several different wind speed pdf parameterizations p_w(w) following from different models for the pdf of the along-mean wind component, p_u(u). The accuracy of these pdfs will be assessed by comparing the fields of mean(w), std(w), and skew(w) simulated using the fields of the vector wind moments estimated from the SeaWinds data to the observed wind speed moment fields.

a. Gaussian fluctuations in u

While it is evident from Fig. 4 that the distribution of u is non-Gaussian, to assess the importance of this non-Gaussian structure to the pdf of w we first consider the structure of p_w(w) following from Gaussian p_u(u). The case of isotropic Gaussian winds,

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with σ_u = σ_υ = σ, was considered in Cakmur et al. (2004) and Monahan (2006a); the integral in Eq. (7) can be evaluated to yield

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where I₀(z) is the zeroth-order associated Bessel function of the first kind (Gradshteyn and Ryzhik 2000). The second row of Fig. 1 displays maps of the mean(w), std(w) and skew(w) fields following from the pdf (11); the first row of Fig. 6 displays the differences between these modeled fields and those estimated from SeaWinds observations. The isotropic standard deviation σ was calculated from observations as

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As was discussed in Monahan (2006a), the speed pdf following from isotropic Gaussian fluctuations in the vector wind underestimates mean(w) and overestimates std(w) relative to observations (except in the Tropics and the North Pacific) and has a positive bias in skew(w) throughout the Tropics and over much of the Southern Ocean. In particular, the pdf (11) cannot characterize the negative skew(w) in the Tropics and the band of near-zero skew(w) over the Southern Ocean.

The observed pdf of sea surface wind speeds is characterized by a distinct relationship between the moments: skew(w) is a strong function of the ratio mean(w)/std(w), such that the skewness is positive where the ratio is small, near zero where the ratio is intermediate, and negative where the ratio is large. This relationship between moments is also characteristic of the Weibull distribution. Figure 7 displays a Gaussian kernel estimate of the observed joint distribution of mean(w)/std(w) with skew(w) along with a scatterplot of these fields simulated by the model pdf (11). As is discussed in Monahan (2006a), the relationship between wind speed moments following from the assumption of isotropic Gaussian fluctuations in the vector wind falls within the observed relationship for regions of weak mean wind speed or large wind speed variability but is unable to capture this relationship for larger values of the ratio mean(w)/std(w).

Clearly, isotropic Gaussian fluctuations in the vector wind are not consistent with observed distributions of sea surface wind speeds. These biases may arise because of the assumption of isotropic vector wind fluctuations; to test this, the integral (7) was evaluated with p_u(u) given by Eq. (10) with σ_u ≠ σ_υ. The integral for p_w(w) can be evaluated as the infinite series of Bessel functions:

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(see appendix A). The quantity

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arises as a natural anisotropy parameter; as λ → 0 and the fluctuations become isotropic, the pdf (13) reduces to (11). The mean, standard deviation, and skewness of w calculated from Eq. (13) using observed values of mean(u), std(u), and std(υ) are presented in the third row of Fig. 1; the corresponding difference maps with the observed moment fields are presented in the second row of Fig. 6. These fields are essentially the same as those of the pdf of w following from the assumption of isotropic Gaussian fluctuations. Furthermore, there is no substantial improvement on the characterization of the relationship between the ratio mean(w)/std(w) and skew(w) (Fig. 7). Evidently, no improvement in the parameterization of the wind speed pdf follows from accounting for anisotropy in the Gaussian vector wind fluctuations. In fact, as illustrated in Fig. 4, the vector winds are manifestly non-Gaussian. We now proceed to consider the importance for parameterizing the pdf of wind speeds of accounting for non-Gaussian structure in the vector winds.

5. Discussion and conclusions

The distributions considered in section 4 are not on an equal footing: the Gaussian distribution requires specification of only the mean and standard deviation of the along-mean sea surface wind component u; the bi-Gaussian and centered gamma distributions require the specification of mean, standard deviation, and skewness; and the Gram–Charlier and constrained maximum entropy distributions require the specification of mean, standard deviation, skewness, and kurtosis. The superiority of the Gram–Charlier and constrained maximum entropy distributions over the Gaussian distribution in the parameterization of the pdf of sea surface wind speeds, w, is not surprising given the fact that the former take as input more information than the latter. From the perspective of practical implementations of this parameterization (e.g., calculating gridbox-averaged air/sea fluxes in a GCM), knowledge of higher-order moments of u is required. In fact, skew(u) and kurt(u) can be very well predicted by mean(u) and std(u), as boundary layer dynamics places strong constraints on the relationship between these moments [e.g., Monahan (2004a, b, 2006a); Fig. 8]. Thus, all that is needed for highly accurate parameterizations of the pdf of sea surface wind speeds in terms of gridbox-scale vector winds are the mean and standard deviation of u. The former is a standard GCM output field, and parameterizations for the latter are already in use (e.g., Cakmur et al. 2004). As the Gram–Charlier and constrained maximum entropy distributions can therefore be expressed solely in terms of mean(u) and std(u), they can be employed in practical computations with no more information than is required by the much less accurate Gaussian parameterization [as was demonstrated for the Gram–Charlier distribution in Monahan (2006a)].

This study has extended the results of Monahan (2006a) in two primary ways. First, through consideration of the relative entropy between the observed wind speed pdf and the best-fit Weibull distribution, the degree of and spatial structure of non-Weibull behavior in sea surface wind speeds has been characterized. The strongest non-Weibull structure appears in the Tropics and subtropics, consistent with the fact that the observed relationship between the ratio mean(w)/std(w) and skew(w) deviates most from the associated relationship for a Weibull variable where skew(w) is negative (Figs. 2 and 7).

Second, empirical parameterizations of the pdf of w based on different models for the joint pdf of the sea surface vector winds have been compared. It was shown in Monahan (2006a) that incorporation of skew(u) and kurt(u) information through a Gram–Charlier distribution improved the performance of the parameterized pdf of w relative to the model assuming isotropic Gaussian fluctuations considered in Cakmur et al. (2004). In the present study, an analytic expression for the pdf of w was found for anisotropic Gaussian fluctuations in the sea surface vector wind, and it was shown that accounting for anisotropy in the fluctuations did not improve the accuracy of the parameterization of this pdf. This result reinforces the conclusion of Monahan (2006a) that the observed non-Gaussian structure in the pdf of the sea surface vector winds contributes importantly to observed structure in the pdf of w. Four different non-Gaussian models for u were then compared: two, the bi-Gaussian and centered gamma distributions, allow specification of skew(u) independently from mean(u) and std(u); two others, the Gram–Charlier and constrained maximum entropy distributions, allow the independent specification of both skew(u) and kurt(u). To first order, all four distributions provide a significantly improved representation of the pdf of w so long as the values of skew(u) and kurt(u) are related in the manner characteristic of observations. This relationship between moments is satisfied approximately by the bi-Gaussian and centered gamma distributions, but must be explicitly imposed for the Gram–Charlier and constrained maximum entropy distributions. Although to a first approximation the structure of the pdf of w is captured by all four distributions, the representation is the best for the constrained maximum entropy distribution. In an information theoretic sense, the constrained maximum entropy pdf is the least-biased distribution among all pdfs with specified moments (e.g., Jaynes 1957; Majda et al. 2005); it is interesting that this distribution provides the best representation of the pdf of u.

We have demonstrated that highly accurate parameterizations of the pdf of sea surface wind speeds can be developed in terms of the mean and standard deviation of the along-mean wind component, in association with a parameterization of the skewness and kurtosis of u in terms of the lower-order moments. This latter parameterization can be empirical (as in Monahan 2006a) or mechanistic (as in Monahan 2004b). Mechanistic parameterizations are preferable to empirical ones, particularly in the context of simulating a changing climate; however, existing mechanistic models are qualitatively accurate but quantitatively inaccurate (Monahan 2004b, 2006a). A goal of future study is the refinement of these mechanical models of the relationship between the moments of sea surface vector winds and their inclusion in parameterizations of sea surface wind speeds appropriate for use in operational climate models. It is to be anticipated that improvements to diagnosed air/sea fluxes will be most dramatic in situations when the vector winds are most variable, and will be due either to differences between the mean flux and the flux associated with the mean wind speed or to differences between the magnitude of the mean vector wind and the mean wind speed. The latter biases, characteristic of situations with strong variability in vector wind direction such as found in conditions of free convection, are the focus of gustiness parameterizations (e.g., Redelsperger et al. 2000; Williams 2001). The incorporation of buoyancy effects in the mechanistic surface wind speed parameterization presented in Monahan (2006a), in order to better represent free convective situations, is another goal of future study. Exchanges between the atmosphere and the ocean are at the heart of the coupled dynamics of the climate system; the development of improved parameterizations of surface wind pdfs is an important component of improving simulations of past, present, and future climates.