The Probability Distribution of Sea Surface Wind Speeds ...web.uvic.ca/~monahana/wind_speed_pdf_I.pdf · The probability distribution of sea surface wind speeds, w, is considered

The Probability Distribution of Sea Surface Wind Speeds. Part I: Theory andSeaWinds Observations

ADAM HUGH MONAHAN

School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, and Earth System Evolution Program,Canadian Institute for Advanced Research, Toronto, Ontario, Canada

(Manuscript received 2 December 2004, in final form 25 May 2005)

ABSTRACT

The probability distribution of sea surface wind speeds, w, is considered. Daily SeaWinds scatterometerobservations are used for the characterization of the moments of sea surface winds on a global scale. Theseobservations confirm the results of earlier studies, which found that the two-parameter Weibull distributionprovides a good (but not perfect) approximation to the probability density function of w. In particular, theobserved and Weibull probability distributions share the feature that the skewness of w is a concave upwardfunction of the ratio of the mean of w to its standard deviation. The skewness of w is positive where the ratiois relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero wherethe ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio isrelatively large (such as the equatorward flank of the subtropical highs). An analytic expression for theprobability density function of w, derived from a simple stochastic model of the atmospheric boundarylayer, is shown to be in good qualitative agreement with the observed relationships between the momentsof w. Empirical expressions for the probability distribution of w in terms of the mean and standard deviationof the vector wind are derived using Gram–Charlier expansions of the joint distribution of the sea surfacewind vector components. The significance of these distributions for improvements to calculations of aver-aged air–sea fluxes in diagnostic and modeling studies is discussed.

1. Introduction

Models of the probability distribution of sea surfacewind speeds play a central role in a number of problemsin meteorology, oceanography, and climate; these in-clude wind power meteorology (Petersen et al.1998a,b), remote sensing of sea surface winds (e.g.,Wentz et al. 1984; Meissner et al. 2001), and estimatesof air–sea exchanges of heat, momentum, moisture, andgases (e.g., Wright and Thompson 1983; Thompson etal. 1983; Isemer and Hasse 1991; Wanninkhof 1992;Wanninkhof and McGillis 1999; Taylor 2000; Wan-ninkhof et al. 2002). In particular, turbulent air–seafluxes depend on eddy-averaged quantities such as thefriction or piston velocities, which for many applica-tions are parameterized in terms of the sea surface windspeed. These bulk parameterizations of air–sea fluxes

are typically nonlinear in the sea surface wind speed, sothe space or time average flux is not generally equal tothe flux that would be diagnosed from the averagedwind. In fact, the average flux will generally depend onhigher-order moments of the sea surface wind speed,such as the standard deviation and skewness. Fromboth diagnostic and modeling perspectives, there is aneed for parameterizations of the probability distribu-tion of sea surface wind speeds.

A number of empirical studies have demonstratedthat surface wind speed distributions over both landand sea can be well represented by the two-parameterWeibull distribution (Hennessey 1977; Justus et al.1978; Conradsen et al. 1984; Isemer and Hasse 1991;Deaves and Lines 1997; Pang et al. 2001), although ithas been noted that the fit is not exact (e.g., Stewart andEssenwanger 1978; Takle and Brown 1978; Tuller andBrett 1984; Erickson and Taylor 1989; Bauer 1996).Most previous efforts to characterize wind speed prob-ability distributions in both coastal (e.g., Dixon andSwift 1984; Tuller and Brett 1984; Pryor and Barthelmie2002) and open-ocean (e.g., Pavia and O’Brien 1986;

Corresponding author address: Dr. Adam Hugh Monahan,School of Earth and Ocean Sciences, University of Victoria, P.O.Box 3055 STN CSC, Victoria, BC V8P 5C2, Canada.E-mail: [email protected]

VOLUME 19 J O U R N A L O F C L I M A T E 15 FEBRUARY 2006

© 2006 American Meteorological Society 497

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Isemer and Hasse 1991) locations have relied on datafrom in situ observations and have thus been con-strained by limited sampling in both space and time. Inparticular, the study of seasonal and latitudinal varia-tion of Weibull parameters by Pavia and O’Brien(1986) used ship-based wind speed observations from asingle year (1983), with very poor spatial resolutionover the tropical and Southern Hemisphere oceans. Incontrast, the study of Isemer and Hasse (1991) usedover 30 years’ worth of ship-based wind speed estimatesobtained from visual inspection of the sea state, but thisanalysis was limited to the North Atlantic.

A new era in the study of sea surface wind speedsbegan with the advent of satellite anemometry, usingboth active and passive remote sensing devices (e.g.,Atlas et al. 1996; Bentamy et al. 1999; Kelly 2004). Sat-ellite-borne instruments have provided global observa-tions of sea surface winds with unprecedentedly highresolution in space and time, allowing statistically sig-nificant characterization of the probability density func-tions (PDFs) of surface wind speeds in previouslypoorly sampled oceanic regions. In general, remotelysensed sea surface wind speeds agree reasonably wellwith in situ observations and surface analysis fields, al-though some biases (dependent on region and windspeed) are found (e.g., Bentamy et al. 1999; Ebuchi1999; Meissner et al. 2001; Ebuchi et al. 2002; Bourassaet al. 2003; Curry et al. 2004; Yuan 2004; Chelton andFreilich 2005), some of which are associated with strongsea surface currents (e.g., Kelly et al. 2001, 2005; Chel-ton et al. 2004). In Bauer (1996), the PDFs of remotelysensed winds for November 1992 were considered, butthe short duration of the observations necessitated acoarse spatial resolution of three zonal bins corre-sponding to the Tropics and to the Northern and South-ern Hemisphere extratropics.

The first part of the present study characterizes thespatial structure of both the moments and the best-fitWeibull parameters of 10-m sea surface wind speedsusing 6 yr of level 3.0 gridded observations from theSeaWinds scatterometer on the National Aeronauticsand Space Administration (NASA) Quick Scatterom-eter (QuikSCAT) satellite (Jet Propulsion Laboratory2001). The high temporal and spatial resolution (ap-proximately daily and 1/4° � 1/4°, respectively) of thedataset, along with its relatively long duration, allows aglobal characterization of the annual PDFs of sea sur-face wind speeds. The objective of this part of thepresent study is not to critique or ground-truth theSeaWinds observations of sea surface winds; suchanalyses appear in other studies (e.g., Ebuchi et al.2002; Bourassa et al. 2003; Chelton and Freilich 2005)in which it is demonstrated that SeaWinds observations

of sea surface winds generally agree favorably with insitu buoy and ship observations. Rather, these observa-tions are considered in this study as a source of state-of-the-art data for the characterization of sea surfacewind PDFs. Furthermore, we will investigate the extentto which the SeaWinds data are accurately representedby the Weibull distribution. It will be demonstratedthat while the data are approximately Weibull, thereare systematic deviations from Weibull behavior thatcannot be accounted for by sampling variability. Acomparison of the PDFs obtained in this study withthose derived from other sea surface wind datasets,along with an analysis of the seasonal evolution of thewind speed PDFs, is presented in a companion paper(Monahan 2006, hereafter Part II).

As other authors have noted (e.g., Kestens and Teu-gels 2002), a limitation of most previous studies of thePDF of sea surface wind speeds is that they have beenprimarily empirical and have not provided a theoreticalexplanation of the observed wind speed probability dis-tribution. Several studies (e.g., Tuller and Brett 1984;Wentz et al. 1984; Meissner et al. 2001; Cakmur et al.2004, hereafter CMT) have noted that a Rayleigh dis-tribution (a special case of the Weibull distribution)arises for the wind speed if the vector wind componentsare assumed to be individually Gaussian with zeromean and independent, isotropic fluctuations. Underthese assumptions, the joint PDF of the zonal and me-ridional wind components, respectively denoted as Uand V, is

p�U, V� �1

2��2 exp��U2 � V2

2�2 �, �1�

where � is the standard deviation of both U and V.Transforming from the orthogonal coordinates U, V tothe polar coordinates w, � (respectively the wind speedand compass direction),

U � w sin�, �2�

V � w cos�, �3�

noting that probability must be conserved under thecoordinate transformation:

p�U, V� dU dV � p�U, V�w dw d� � p�w, �� dw d�,

�4�

and integrating over the angle � yields the marginalprobability density function for the wind speed alone:

498 J O U R N A L O F C L I M A T E VOLUME 19

p�w� � �0

2�

p�w, �� d� �w

�2 exp��w2

2�2�. �5�

While the assumptions leading to the PDF [Eq. (5)]may be a good approximation globally (although seesection 5 below), they certainly are not true in generallocally. By the same token, while sea surface windspeeds may be approximately Rayleigh on a globalscale (e.g., Meissner et al. 2001; Yuan 2004), this isgenerally not the case locally or regionally (e.g., Tullerand Brett 1984; Pavia and O’Brien 1986; Bauer 1996;Wanninkhof et al. 2002). A second goal of this study isthe development of physical and empirical models ofthe observed PDF of sea surface wind speed, and thedevelopment of analytic expressions for this PDF. Inparticular, we will demonstrate that the structure of thePDF of w cannot be accounted for without taking intoaccount the non-Gaussian structure of the vector windPDF. It was shown in Monahan (2004b) that the zonaland meridional components of the sea surface vectorwind are each systematically skewed, such that a com-ponent is positively (negatively) skewed when the meanof the component is negative (positive). This spatialanticorrelation of the mean and skewness fields wasshown to be a natural consequence of the nonlineardrag law characteristic of the surface boundary layer.We will show that the skewness, as well as the kurtosis,of the surface vector wind components play an impor-tant role in determining the PDF of sea surface windspeeds.

The sea surface wind dataset used in this study isdescribed in detail in section 2. A review of the prop-erties of the Weibull distribution is presented in section3, and a discussion of the observed statistics of the seasurface wind speeds is presented in section 4. Section 5describes an analysis of the accuracy of the character-ization of sea surface wind speeds on a global scale bythe Rayleigh distribution. A physically based model ofthe wind speed PDFs is presented in section 6. Thistheoretical model is shown to capture essential featuresof the PDF of sea surface wind speeds, but it is notquantitatively accurate. A class of more quantitativelyaccurate empirical models is described in section 7. Asummary and conclusions are given in section 8.

2. Data

The sea surface wind dataset considered in this studyconsists of level 3.0 gridded daily SeaWinds scatter-ometer 10-m zonal and meridional wind observationsfrom the NASA QuikSCAT satellite (Jet PropulsionLaboratory 2001), available on a 1/4° � 1/4° grid from

19 July 1999 to the present (15 March 2005 for thepresent study). These data are available for downloadfrom the NASA Jet Propulsion Laboratory (JPL) Dis-tributed Active Archive Center (see http://podaac.jpl.nasa.gov). The SeaWinds data have been extensivelycompared with buoy and ship measurements of surfacewinds (Ebuchi et al. 2002; Bourassa et al. 2003; Cheltonand Freilich 2005); the root-mean-square errors of theremotely sensed wind speed and direction are bothfound to depend on wind speed, with average values of1 m s�1 and 20°, respectively. Because raindrops areeffective scatterers of microwaves in the wavelengthband used by the SeaWinds scatterometer, rainfall canlead to errors in estimates of sea surface winds. TheSeaWinds level 3.0 dataset flags those data points thatare estimated as likely to have been corrupted by rain(Jet Propulsion Laboratory 2001); these data pointshave been excluded from the present analysis. No fur-ther processing of the data, such as filtering or remov-ing the annual cycle, was carried out on this dataset.

A second dataset used in this study is the number ofobservations per month and 1° � 1° grid box of seasurface wind speed in the Comprehensive Ocean–Atmosphere Data Set (COADS). These data are avail-able for download from the International Research In-stitute for Climate Prediction/Lamont-Doherty EarthObservatory (IRI/LDEO) Climate Data Library (seehttp://ingrid.ldeo.columbia.edu).

3. A brief review of the Weibull distribution

We begin with a brief review of the properties of theWeibull distribution; a more complete discussion is pre-sented in Johnson et al. (1994). A random variable xcharacterized by a two-parameter Weibull distributionhas the PDF

p�x� �b

a �x

a�b�1

exp��x

a�b�. �6�

The parameters a and b denote, respectively, the scaleand shape parameters of the distribution. Averages ofpowers of x are given simply by

mean�xk� � ak��1 �k

b�, �7�

where is the gamma function. In particular, the mean,standard deviation, skewness, and kurtosis of X aregiven by

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mean�x� � a��1 �1b�, �8� std�x� � a��1 �

2b� � �2�1 �

1b��1�2

, �9�

skew�x� �

��1 �3b� � 3��1 �

1b��1 �

2b� � 2�3�1 �

1b�

��1 �2b� � �2�1 �

1b��3�2 , and �10�

kurt�x� �

��1 �4b� � 4��1 �

3b��1 �

1b� � 6��1 �

2b��2�1 �

1b� � 3�4�1 �

1b�

��1 �2b� � �2�1 �

1b��2 � 3; �11�

where the skewness and kurtosis are, respectively, thenormalized third- and fourth-order moments:

skew�x� �mean��x � mean�x� 3�

std3�x�, and �12�

kurt�x� �mean��x � mean�x� 4�

std4�x�� 3. �13�

Skewness is a measure of the asymmetry of a PDF: apositive (negative) skewness of a variable x indicatesthat the PDF is characterized by an elongated tail in thedirection of positive (negative) fluctuations away fromthe mean, so the mean of x is larger (smaller) than themost likely value of x. A variable x has positive (nega-tive) kurtosis if its PDF is more sharply peaked(broadly peaked) and has longer (shorter) tails than aGaussian distribution with the same mean and standarddeviation. Both the skewness and kurtosis are zero fora Gaussian distribution.

The dependence of mean(x), std(x), skew(x), andkurt(x) on the Weibull parameters a and b is illustratedin Fig. 1; note in particular that the skewness and kur-tosis depend only on the parameter b. For b � 3.6, thedistribution of x is positively skewed; for b � 3.6, theskewness is negative. The kurtosis of x is close to zerofor b � 2 but increases rapidly as b drops below thisvalue. Note that the Weibull distribution for b � 3.6 isa close approximation to a Gaussian distribution.

It can be shown that

mean�ln x� � ln a ��

b, and �14�

var�ln x� ��2

6b2 , �15�

where � � 0.57721 is Euler’s constant [Conradsen et al.1984; note the sign error in their Eq. (9)]. Furthermore,Eqs. (8) and (9) can be approximately inverted to yield

b � �mean�x�

std�x� �1.086

, and �16�

a �mean�x�

��1 � 1�b��17�

(e.g., Justus et al. 1978).A number of estimators of Weibull parameters a and

b exist (Conradsen et al. 1984; Pang et al. 2001); theseinclude

1) estimates obtained from Eqs. (14)–(15), using thesample estimates of mean(ln x) and std(ln x);

2) maximum likelihood estimates (Conradsen et al.1984), and

3) estimates from the approximate Eqs. (16)–(17), us-ing sample estimates of mean(x) and std(x).

Monte Carlo experiments using simulated Weibull data(not shown) indicate that each of these estimators isunbiased, although estimator 1 has a relatively largevariance. The performances of estimators 2 and 3 areessentially equivalent, so there is no reason in principleto prefer one over the other. Being based on the samplemean and standard deviation, estimator 3 is easiest tocompute; this estimator will be used for the remainderof this study.

4. Statistical features of the observed sea surfacewind speeds

The mean, standard deviation, skewness, and kurto-sis fields of w estimated from the SeaWinds data aredisplayed in Fig. 2. Large values of mean(w) occur inthe westerly belts of the Northern and Southern Hemi-spheres; secondary maxima in the easterly belt occur onthe equatorward flanks of the subtropical highs.Minima of mean(w) occur in the equatorial doldrumsand subtropical horse latitudes. The standard deviationof w is largest in the midlatitude extratropics (in the


storm tracks) and generally decreases toward the equa-tor, but with a local maximum along the intertropicalconvergence zone (ITCZ). In general, w is positivelyskewed in the extratropics and negatively skewed in theTropics, as has been noted by Bauer (1996). Major ex-ceptions to this pattern are the band of positivelyskewed wind speeds over the tropical Indian Ocean andwestern Pacific, where mean wind speeds are small, andover the Southern Ocean, where the skewness of w isgenerally close to zero. The kurtosis field is muchnoisier than those of mean(w), std(w), or skew(w); thisis not surprising, considering the limited duration of thewind speed dataset.

The upper and lower panels in Fig. 3 display, respec-tively, the Weibull scale and shape parameter fieldsestimated from the SeaWinds data. The structure of thescale parameter a field is essentially identical (up to ascaling factor) to that of mean(w), which is to be ex-pected in light of Eq. (8) and Fig. 1. The Weibull shapeparameter b is generally close to 2 throughout the ex-tratropics, rising to between 3 and 4 over the SouthernOcean. Values of b are generally larger in the Tropics

than in the extratropics, with local extrema (where thevalue of b can exceed 5) on the equatorward flanks ofthe subtropical high pressure cells where the meanwinds are strong and the variability is weak [cf. Eq. (16)and Fig. 1]. The association of high values of theWeibull shape parameter with regions of strong andsteady winds was noted previously in Pavia andO’Brien (1986, hereafter PO) and Isemer and Hasse(1991).

The distribution of the Weibull b parameter over theNorth Atlantic illustrated in Fig. 3 agrees well with theresults presented in Isemer and Hasse (1991). Differ-ences between the estimates of the Weibull parametersin this study with those presented in PO are consider-ably greater. In the northern extratropics, the results ofthe present study agree reasonably well with those ofPO. In the Tropics and Southern Hemisphere extra-tropics, the differences are considerably larger. Esti-mates in PO of the shape parameter b in the Tropics aremuch smaller than those in the present study; in fact,PO find that b within a given longitude band is gener-ally a minimum, rather than a maximum, in the Tropics.

FIG. 1. Dependence of the mean, std dev, skewness, and kurtosis of the Weibull-distributed variable x on theWeibull parameters a and b [Eqs. (8)–(11)].


Furthermore, both Weibull a and b parameters are gen-erally underestimated over the Southern Ocean in POas compared to the present study. While the precisedataset used in PO was not available for the presentstudy, the sampling density of their dataset is presum-ably comparable to that of COADS. The upper panelof Fig. 4 illustrates the number of wind speed observa-tions (per 1° � 1° grid box) for the year 1983 in theCOADS database, on a logarithmic scale; this map re-sembles a similar map of sampling density presented in

PO. It is evident that while the North Atlantic andsubpolar North Pacific Oceans are well sampled, theTropics (particularly the Pacific) and the SouthernOcean are very poorly sampled. In contrast, the num-ber of observations (per 1/4° � 1/4° box) is much moreuniform in the SeaWinds dataset, as is illustrated in thelower panel of Fig. 4. There are relatively few data inregions of high precipitation where a relatively largenumber of rain-flagged measurements were rejected[particularly along the ITCZ and South Pacific conver-

FIG. 2. Mean, std dev, skewness, and kurtosis fields of sea surface wind speed estimated fromdaily SeaWinds observations. The thick black line in the plot of skew(w) is the zero contour.


Fig 2 live 4/C

gence zone (SPCZ)], and in meridional bands around90°E and 90°W resulting from the satellite samplingprotocol, but away from the Antarctic ice margin innone of these pixels are there fewer than 1500 obser-

vations. Throughout the World Ocean, therefore, thepresent SeaWinds dataset has enough observations tocharacterize the statistical structure of the sea surfacewind speeds (on less than interannual time scales) with

FIG. 3. Same as in Fig. 2, but for Weibull a and b parameters.

FIG. 4. Number of wind speed observations (top) (per 1° � 1° grid box) in the COADSdatabase for 1983 and (bottom) (per 1/4° � 1/4° grid box) in the SeaWinds database for1999–2005. Note the logarithmic scales of the two maps.


Fig 3 4 live 4/C

confidence. In the COADS dataset (which we presumeis similar to that used in PO), these regions of highstatistical confidence are restricted to the North Atlan-tic and the subarctic North Pacific. Furthermore, as isclear from Eq. (16) and Fig. 1, small values of b in theTropics are inconsistent with the negative skewness ofw in these regions noted in Bauer (1996) and thepresent study. We conclude that differences betweenthe results of the present study and those of PO arisebecause of the limited spatial coverage of the datasetused in the earlier study.

The skewness and the shape parameter b of aWeibull-distributed variable are uniquely related, as isillustrated in Fig. 1. The fact that the map of the shapeparameter b (Fig. 3) reflects the map of the skewnessfield (Fig. 2) in a manner consistent with this relation-ship provides evidence that sea surface wind speeds areWeibull to a good approximation; note that onlymean(w) and std(w), but not skew(w), were used toestimate b. The relationship between skew(w) and theratio mean(w)/std(w) [which by Eq. (16) is closely re-lated to b] is clearly illustrated by the plot of their jointPDF presented in the upper left panel of Fig. 5. The

thick black line represents the theoretical curve for aWeibull variable, Eq. (10). Evidently, in the observa-tions as for a Weibull variable, skew(w) is a concaveupward function of the ratio mean(w)/std(w), such thatthe function is positive for small values of this ratio andnegative for large values. Similarly, the relationship be-tween skew(w) and kurt(w) in the observations is simi-lar to that for a Weibull variable (Fig. 5, lower leftpanel). From Eqs. (10) and (11) both skew(w) andkurt(w) for a Weibull variable are determined uniquelyby the shape parameter b, so a plot of skew(w) againstkurt(w) traces out a 1D curve, around which the jointPDF of the observations falls. The agreement betweenthe moment relationships in the SeaWinds data andthose for a Weibull variable reinforces the conclusionthat these data are Weibull to a good approximation.

Inspection of Fig. 5, however, indicates distinct non-Weibull structure in the PDF of sea surface windspeeds. The upper left panel of Fig. 5 demonstrates thatthe observed joint PDF of mean(w)/std(w) andskew(w) lies above the Weibull curve for low values ofthe ratio and lies below it for large values. Furthermore,the slope of the relationship between skew(w) and the

FIG. 5. (top) Kernel density estimates of joint PDFs of mean(w)/std(w) and skew(w) for (left) SeaWinds data and(right) synthetic Weibull data. The contour intervals are logarithmically spaced. The thick black line is the theo-retical curve for a Weibull variable. (bottom) Same as in top, but for the joint PDF of skew(w) and kurt(w).


ratio mean(w)/std(w) is steeper than the Weibull curvefor low values of the ratio and shallower for highervalues. A map of the skewness field for the Weibullvariable with estimated mean(w) and std(w) fromSeaWinds is presented in the upper panel of Fig. 6.Comparing this field with the SeaWinds skew(w) fieldin Fig. 2, it is clear that the negative skewness in theTropics is less negative for the Weibull variable than forthe SeaWinds data, and that the positive skewness inthe NH extratropics is not as positive for the Weibullvariable as for the SeaWinds data. These differencesare further emphasized in the lower panel of Fig. 6,which displays the difference in skewness fields be-tween the Weibull and SeaWinds variables. The differ-ence field between kurt(w) and the Weibull kurtosisfield obtained using a and b (not shown) is generallypositive but is very noisy.

It is of course possible that the wind speeds areWeibull and the apparent non-Weibull structure in theSeaWinds skew(w) field arises simply because of sam-pling variability. To assess the probability of the differ-ences having arisen by chance, a Monte Carlo approachwas taken. Values of the Weibull parameters a and bwere sampled on a regular grid spanning the range ofthese parameters observed in the SeaWinds data; foreach pair, 50 realizations of Weibull time series wererandomly generated. Each of these time series con-

sisted of 1000 independent realizations of the randomvariable; given that at most locations the SeaWindsdataset consists of between 2500 and 3500 daily obser-vations with an autocorrelation e-folding time of 1–2days, this number of degrees of freedom is consistentwith those of the observations. The right-hand panelsof Fig. 5 display estimates of the joint PDFs ofmean(w)/std(w) with skew(w) (upper panel), and ofskew(w) with kurt(w) (lower panel), as estimated fromthis synthetic Weibull dataset. When compared to theSeaWinds PDFs, the synthetic Weibull PDFs clustermore tightly and are more symmetrically distributed,around the theoretical Weibull curves. This fact sug-gests that the deviations from Weibull behavior of theSeaWinds data illustrated in Figs. 5 and 6 are not simplyartifacts of sampling variability, but rather reflect actualnon-Weibull structure in the SeaWinds characterizationof sea surface wind speeds. The possibility remains thatthe non-Weibull structure is an artifact of the SeaWindsobservations. However, the facts that SeaWinds windobservations have been found to be generally in closeagreement with buoy and ship observations (e.g., Ebu-chi et al. 2002; Bourassa et al. 2003) and that non-Weibull behavior is evident in other sea surface winddatasets (Part II) suggest that this structure is not spu-rious. Finally, it is also possible that the wind speeds areWeibull with seasonally varying parameters and that

FIG. 6. (top) Map of skew(w) for Weibull-distributed variable with mean(w) and std(w)from SeaWinds observations [cf. Eqs. (10) and (16)]. (bottom) Difference in skewness fieldsbetween SeaWinds observations and the equivalent Weibull variable.


Fig 6 live 4/C

the apparent non-Weibull structure arises because ofthis nonstationarity. A seasonally stratified analysis ofsea surface wind speeds presented in Part II of thisstudy suggests that the observed non-Weibull structureis not an artifact of nonstationarity.

Erickson and Taylor (1989) used a modified Kolmog-orov–Smirnov test to demonstrate statistically signifi-cant non-Weibull behavior of surface winds in thecoarse resolution National Center for Atmospheric Re-search (NCAR) Community Climate Model (CCM1).The preceding analysis extends the results of Ericksonand Taylor (1989) by considering the structure of ob-served rather than modeled winds, and by quantifyingthe differences between the observed wind speed dis-tributions and Weibull distributions, rather than justnoting the existence of these differences.

We have thus demonstrated that while the observedsea surface wind speeds deviate systematically fromWeibull behavior, the Weibull approximation providesa reasonable first-order characterization of mean(w),std(w), skew(w), and kurt(w). This does not guaranteethat extreme events are accurately represented by theWeibull distribution; however, averages of polynomialfunctions of w are relevant to a broad range of climateand wind power applications (Petersen et al. 1998a,b;Taylor 2000; Jones and Toba 2001; Donelan et al. 2002),for which the correct representation of these momentsis adequate. After a brief discussion of the accuracy ofthe Rayleigh distribution in characterizing global seasurface wind speeds, in section 6 we will proceed todevelop a theoretical explanation of the observed rela-tionships between the wind speed field moments.

5. How well does the Rayleigh distributionrepresent global sea surface wind speeds?

The global distribution of sea surface wind speeds isoften modeled as a Rayleigh distribution (e.g., Wan-

ninkhof 1992; Wanninkhof and McGillis 1999; Meissneret al. 2001; Yuan 2004), the special case of the Weibulldistribution for which b � 2. As was noted in section 1,this distribution is appropriate if the vector wind com-ponents are individually Gaussian with mean zero andindependent, isotropic fluctuations. We proceed to in-vestigate the accuracy of the Rayleigh distribution indescribing the global distribution of SeaWinds windspeeds and of the validity of the assumptions fromwhich the Rayleigh model rigorously follows.

The joint PDF p(U, V) of the global zonal and me-ridional wind components (respectively denoted U andV), obtained from all sea surface grid points, is dis-played in the left-hand panel of Fig. 7; note that thestructure of p(U, V) reflects both local and regionalvariability in the vector wind. Plots of the marginal dis-tributions of U and V are given in the center panel ofFig. 7. The marginal distributions of a joint distributionp(U, V) are the distributions of U and V individually,obtained by integrating the joint distribution, respec-tively, over V and U. The anisotropic local minimum inp(U, V) near (U, V) � (0, 0) m s�1 is presumably anartifact of the SeaWinds data; errors in wind directionare known to be greatest at very low wind speeds (e.g.,Ebuchi et al. 2002). It is evident that the vector windcomponent fluctuations are neither independent norGaussian: the variance of V is greater for U � 0 than itis for U � 0, and the marginal PDF of U is positivelyskewed and bimodal with peaks at about �5 m s�1 [al-though this bimodality may be an artifact resulting fromof the minima in p(U, V) around (0, 0) m s�1]. In fact,the joint PDF of U and V is suggestive of two statisticalpopulations: a first characterized by relatively weakfluctuations, with U � 0 on average, and a second withstronger fluctuations and U � 0 on average. Thesepopulations correspond to the easterly and westerlywind belts, respectively; the larger variance of V for

FIG. 7. (left) Joint PDF of observed global zonal and meridional wind components. (middle) Marginal PDFs of observed global zonal(solid line) and meridional (dashed line) wind components. (right) PDF of observed global wind speed (solid line) and best-fit Rayleighdistribution (dashed line).


U � 0 than for U � 0 evident in Fig. 7 simply reflectsthe fact that the storm tracks occur in the westerlies.

The right-hand panel of Fig. 7 displays both the ob-served PDF of w and the Weibull distribution withb � 2 (so the distribution is Rayleigh) and a �mean(w)/(3/2) {an essentially identical PDF is ob-tained using Eq. (5) with �2 � [std2(U) � std2(V)]/2}.The observed PDF of w is somewhat narrower than theRayleigh PDF, with a somewhat larger most likely windspeed (7 m s�1 in contrast to 6.5 m s�1 for the Rayleighdistribution). Despite these small differences, the Ray-leigh PDF is a reasonable approximation to the globalPDF of w.

Non-Rayleigh structure in the global PDF of w canarise due to violations of any of the assumptions, de-scribed above, from which the Rayleigh distributionfollows rigorously. First, it is evident that the marginaldistributions of U and V are non-Gaussian. Second, al-though the vector winds are uncorrelated (the correla-tion coefficient is �0.02), they are not independent, aseasterly winds are associated with smaller variability in� than are westerlies. Note that the independence oftwo variables always implies a vanishing correlation co-efficient, but not vice versa; the correlation coefficientis only a linear measure of dependence. For example, anormal random variable x with zero mean and unit vari-ance is uncorrelated with the variable y � x2, but thetwo are clearly not independent. Statistical indepen-dence of two variables x and y requires that their jointPDF p(x, y) factor as the product p(x)p(y), where p(x)and p(y) are the marginal distributions of x and y, andp(U, V) does not factor as p(U)p(V). Direct calculationyields mean(U) � 0.21 m s�1 and mean(V) � 0.13m s�1, as well as std(U) � 6.7 m s�1 and std(V) � 5.6m s�1. The means of the vector wind components arean order of magnitude smaller than their standard de-viations, so the assumption of a zero mean vector windis reasonable. However, as typical fluctuations in thezonal wind are 20% larger than typical fluctuations inthe meridional wind, the assumption of isotropy is vio-lated. Non-Rayleigh behavior in the global PDF of wmay arise, then, from non-Gaussian structure in thePDFS of the zonal and meridional winds, from statisti-cal dependence between U and V, or from nonisotropicfluctuations in the vector wind.

6. Wind speed PDFs: Stochastic boundary layermodel

To obtain a physical understanding of the observedPDFs of sea surface wind speeds, we will consider thesimple stochastic model for boundary layer winds in-troduced in Monahan (2004b). This model has been

demonstrated to be useful in understanding observedrelationships between moments of the sea surface vec-tor winds (Monahan 2004a,b) and is in good qualitativeagreement with empirical stochastic models of sea sur-face winds (Sura 2003; Sura and Sardeshmukh 2004,manuscript submitted to J. Atmos. Sci., hereafter SS04).At any given location over the sea surface, we will de-fine the surface wind vector components relative to alocal coordinate system:

u � wind component along local mean wind vector,

� � wind component across local mean windvector �positive to the left�.

Denoting the vector wind by u � (u, �), the eddy-aver-aged horizontal momentum equation can be written

�u�t

� u · �u � �1�p � f k � u �

1

��u�u3�

�z,

�18�

where p is the pressure, � is the air density, f is theCoriolis parameter, and u3 is the vertical velocity com-ponent. An analytically tractable model can be ob-tained as follows. First, Eq. (18) is integrated from thesurface z � 0 to an altitude z � h in the mixed layer.Second, horizontal advection of momentum is ne-glected; that is, a “single-column model” approximationis made. Third, the surface eddy momentum flux is rep-resented in terms of a standard Monin–Obukhov bulkparameterization with drag coefficient cd. Finally, theeddy momentum flux from above z � h is expressed interms of a “finite-differenced” eddy flux:

u�u3 � �K

h�U � u�, �19�

where K is a kinematic eddy viscosity and U representsthe wind vector above z � h. The resulting differentialequation can be expressed as

dudt

� � �cd

hwu �

K

h2 u, �20�

where we have defined the quantity

� � �1�p � f k � u �

K

h2 U. �21�

For the sake of convenience, we will assume that �does not depend on u; in particular, we assume that theageostrophic residual between the pressure gradientforce and the Coriolis force does not depend on thewind vector u. Away from the equator, this approxima-tion is similar to a small Rossby number approximation.Finally, we will assume that the forcing � is fluctuatingaround some mean value:


�u�t� � ��u� � �W1�t�, �22�

��t� � �W2�t�, �23�

where the fluctuations are taken to be isotropic andwhite in time:

�Wi�t1�Wj�t2�� ij �t1 � t2� �24�

(where angle brackets denote ensemble averaging),with a strength that is tuned by the parameter �. Notethat as by definition the average cross-mean wind iszero, so the average of �� must also be zero. The re-sulting equations for u and � read

u � ��u� �cd

hwu �

K

h2 u � �W1, �25�

� � �cd

hw� �

K

h2 � � �W2. �26�

Equations (25)–(26) are a stochastic differential equa-tion (SDE) for the surface wind vector; an introduc-tion to SDEs can be found in Penland (2003a,b). Thesurface drag force depends on the wind speed w ��u2 � �2, the depth h of the atmospheric layer con-sidered [taken as in Monahan (2004b) to be 80 m],and the drag coefficient cd. In general, nonneutralstratification of the boundary layer and modificationof the local sea state by surface winds both result ina dependence of cd on w (through the Obukhovlength in the first instance and the roughness lengthin the second). Furthermore, other factors such assurface surfactants and remotely generated swell in-troduce variations in the drag coefficient that areunrelated to the local winds. For simplicity, we willneglect the effects of stratification and swell and con-sider the parameterization of the neutral drag coef-ficient for fully developed seas introduced by Taylorand Yelland (2001), as modified in Fairall et al.(2003) to include a correction for flow over an aero-dynamically smooth surface during conditions oflight winds. The drag coefficient is determined by thesurface roughness length z0 through

cd ��2

�ln�10m�z0� 2 , �27�

where � � 0.4 is von Kármán’s constant and

z0 � �4.11 � 10�6s2m�1�w2 �0.11�

�cdw, �28�

with � the kinematic viscosity of water. In the Taylorand Yelland (2001) parameterization, the roughnesslength depends on the significant wave height and dom-inant wave period, both of which are taken to dependon local wind speed; these dependences are combined

to yield the first term on the right-hand side of Eq. (28).Note that as the roughness length depends explicitly onthe drag coefficient, Eqs. (27) and (28) must be solvediteratively. The dependence of the neutral drag coeffi-cient on the surface wind speed remains a subject ofactive research (e.g., Jones and Toba 2001; Fairall et al.2003), and questions remain as to its precise formula-tion. Calculations using different expressions for thewind speed dependence of the drag coefficient (notshown) demonstrate that the following results are notqualitatively sensitive to which of the various param-eterizations of cd suggested in the literature are used.

One of Einstein’s major insights in his annus mirabi-lis (Einstein 1956) was that the PDF of the solutions ofan SDE satisfies a diffusion equation that has come tobe referred to as a Fokker–Planck equation (e.g., Gar-diner 1997; Penland 2003a,b). In particular, the station-ary joint PDF of u and �, p(u, �), associated with theSDEs (25)–(26) satisfies the Fokker–Planck equation:

0 ��

�u ��u� �cd

hwu �

K

h2 u�p

��

�� cd

hw� �

K

h2 ��p ��2

2 ��2p

�u2 ��2p

��2�,

�29�

which has the solution

p�u, �� N 1 exp� 2

�2��u�u �K

2h2 �u2 � �2�

�1h �0

�u2��2

cd�w�w2 dw��, �30�

where N 1 is a normalization constant. Note that thePDF Eq. (30) is symmetric in �, so

��

� ��

�

u�p�u, �� du d� � 0. �31�

Fluctuations in u and � are therefore uncorrelated, al-though they are not independent [i.e., p(u, �) does notfactor as the product of the marginal distributions of uand �]. Independence of u and � holds only in the un-physical case of linear surface drag, cd � k/w. An ana-lytic expression for the PDF of w can be obtained fromEq. (30) as follows: moving to polar coordinates

u � w cos�, and �32�

� � w sin�; �33�

such that the average value of � is zero (by construc-tion), conservation of probability under a coordinatechange requires that the joint PDF p(w, �) must satisfy


p�u, �� du d� � p�w cos�, w sin�� w dw d�

� p�w, �� dw d�. �34�

Thus,

p�w, �� N 1w exp� 2

�2��u�w cos� �K

2h2 w2

�1h �0

w

cd�w�w2 dw��. �35�

The marginal distribution p(w) for the wind speed w isobtained by integrating p(w, �) over the angle �. Ananalytic expression for this integral follows from thefact that (for n an integer)

In�z� �1

2� �0

2�

d� exp�z cos�� cosn�, �36�

where In is the modified Bessel function of order n(Abramowitz and Stegun 1972). Performing the inte-gral, we obtain the closed-form expression:

p�w� � N 1wI0�2��u�w

�2 � exp��2

�2 � K

2h2 w2

�1h �0

w

cd�w�w2 dw��. �37�

The mean, standard deviation, and skewness of windspeed w from Eq. (37) are contoured as functions of��u� and � in Fig. 8; the ranges of these parameterswere chosen so that the ranges of the simulated mo-ments were quantitatively similar to those observed

(Fig. 2). A typical boundary layer value of K � 1 m2 s�1

was used. The mean wind speed is an increasing func-tion of both ��u� and �. The standard deviation of wis determined primarily by �, displaying only a weakdependence on ��u�. Finally, skew(w) depends onboth ��u� and �; in particular, the skewness of w isnegative when the forcing has a large mean but rela-tively small fluctuations; as the magnitude of the fluc-tuations increases, the skewness eventually becomespositive.

Inspection of Fig. 2 indicates that in general the sur-face wind speed distribution is negatively skewed in theeastern equatorial Pacific and along the equatorwardflanks of the subtropical highs, regions characterized byrelatively high mean wind speeds and relatively lowvariability. The regions of strongest positive skewnessare the Northern Hemisphere midlatitudes, character-ized by intermediate mean wind speeds and strong vari-ability. Finally, the Southern Ocean is characterized byhigh mean(w), intermediate std(w), and skew(w) closeto zero. Qualitatively, the relationship between the spa-tial structures of the mean(w), std(w), and skew(w)fields is as predicted by the PDF Eq. (37).

The agreement of the relationships between mo-ments from the observations and from the PDF Eq.(37) is most obvious in a plot of skew(w) as a functionof mean(w)/std(w) (Fig. 9). Values of mean(w), std(w),and skew(w) for the PDF Eq. (37) were sampled on aregular grid of the parameters ��u� and � over therange of values displayed in Fig. 8; although skew(w) isnot exactly a single-valued function of the ratiomean(w)/std(w), its values cluster tightly around a one-dimensional curve. Like the associated curve for a

FIG. 8. Predicted mean, std dev, and skewness of wind speed w from the PDF Eq. (37), contoured as functions of the mean forcing��u� and fluctuation strength �.


Weibull distribution, this curve is concave upward andruns through the middle of the observed joint PDF ofmean(w)/std(w) and skew(w), taking positive valueswhen the ratio mean(w)/std(w) is small and negativevalues when this ratio is large. In fact, for values of thisratio of approximately 2 and greater, the curve pre-dicted by the stochastic boundary layer model is at leastas good a representation of the observed joint PDF asis the Weibull curve. For lower values of the ratio, how-ever, the performance of the model is not so good: thepredicted skewness does not take values greater thanapproximately 0.5, considerably below the maximumobserved skewness. It is evident that while both theWeibull distribution and the PDF Eq. (37) capture as-pects of the relationships between moments observedin sea surface wind speeds, neither is entirely accurate.It should be emphasized, however, that the PDF Eq.(37) arises from physical arguments with a clear seriesof approximations, while the Weibull characterizationof sea surface wind speeds is entirely empirical. Thesuccess of the Weibull distribution as a useful approxi-mation to the distribution of w evidently arises becauseit imposes the constraints on the relationship betweenmean(w), std(w), and skew(w) that are required by thephysics of the atmospheric boundary layer.

An intuitive understanding of the dependence ofmean(w), std(w), and skew(w) on ��u� and � is straight-

forward. An increase in ��u� will lead to an increase inmean(u), and consequently to an increase in the meanwind speed. The joint PDF of u and � becomes broaderas � increases; this shift of probability mass away fromthe origin increases both the mean amplitude w ofthe vector wind and its variability. Finally, the skewnessof w is determined by the width of the PDF of w rela-tive to its mean value, that is, the ratio mean(w)/std(w).For smaller values of this ratio, the distribution p(u, �)is concentrated around the origin, and the distributionof the magnitude w has a tail toward larger values,so skew(w) is positive. Conversely, for larger values ofthis ratio, the joint PDF p(u, �) is centered away fromthe origin. Because of the anticorrelation of mean(u)and skew(u) discussed in Monahan (2004a, b), p(u, �)will be characterized by a tail extending toward theorigin. As the mass of p(u, �) is concentrated away fromthe origin, this tail will also be present in the margi-nal PDF of w, so skew(w) will be negative. The factthat the PDF Eq. (37) can become negatively skewedis a consequence of the anticorrelation between themean and skewness of the vector wind components,which can be understood to arise because of the non-linearity of the surface drag law. For a linear drag law(cd � k/w), for which the vector winds are Gaussian, theskewness of w from PDF Eq. (37) (not shown) is alwayspositive.

FIG. 9. Plots of the relationship between mean(w)/std(w) and skew(w) for a Weibull dis-tribution [Eqs. (8)–(10), gray line] and for the theoretical PDF Eq. (37) (black line), super-imposed on the kernel density estimate of the joint PDF of mean(w)/std(w) and skew(w) fromSeaWinds observations (contoured on a logarithmic scale).


7. Wind speed PDFs: Empirical parameterization

While the stochastic boundary layer model [Eqs.(25)–(26)] provides physical insight into the PDF of seasurface wind speeds, it is too simple to be quantitativelyaccurate. In particular, the range of values of skew(w)from the PDF Eq. (37) is considerably smaller than theobserved range (Fig. 9). Furthermore, the quantities��u� and � are difficult to obtain from observations.To obtain a more useful and more quantitatively accu-rate expression for p(w), we will consider another ap-proach, extending an analysis presented in CMT. Thisearlier study assumed that u and � were independentand Gaussian with isotropic fluctuations, but in contrastto the argument leading to the PDF Eq. (5) the meanvector wind was not assumed to be zero. The distribu-tions of u and � are thus

p�u� �1

�2��2exp��

�u � u�2

2�2 �, �38�

p�� 1

�2��2exp��

�2

2�2�, �39�

where u is the magnitude of the average vector windand � is the standard deviation of both u and �. Theassumption of independence implies that the joint PDFof u and � is the product of their marginal distributions:

p�u, �� 1

2��2 exp��u � u�2 � �2

2�2 �, �40�

so

p�w, �� w

2��2 exp��1

2�2 �w2 � u2�� exp�wu

�2 cos��41�

and, integrating over �,

p�w� � I0�wu

�2 � w

�2 exp��1

2�2 �w2 � u2��. �42�

We will refer to the PDF Eq. (42) as the CMT distri-bution (CMT); note that the PDF of the scalar windspeed w has been expressed in terms of the average andvariability of the vector wind. As discussed in CMT,p(w) reduces to the Rayleigh distribution Eq. (5) in thelimit that u → 0, as would be expected. Furthermore, itis worth noting that the CMT distribution arises fromthe boundary layer model [Eqs. (25)–(26)] in the case ofa linear surface drag (cd � k/w), with

u � ��u��K

h2 �k

h��1

, �43�

� � �2K

h2 �2k

h ��1�2

�. �44�

To determine the accuracy of the CMT distribution incharacterizing the probability distribution of sea sur-face wind speeds, u and �2 were estimated from theSeaWinds surface wind data. Because fluctuations inthe along- and cross-mean wind directions are not ex-actly isotropic, � was estimated as

� � �12

�std2�u� � std2�� 1�2

. �45�

The upper panels of Fig. 10 illustrate the mean, stan-dard deviation, and skewness fields of the surface windspeed calculated from the CMT distribution using theobserved u and � fields; these should be compared withthe moment fields from observations presented in Fig.2. Maps of the differences between the observed mo-ment fields and those predicted by the CMT distribu-tion are presented in Fig. 11. The CMT distributiongenerally underestimates mean(w) and overestimatesstd(w); differences are particularly large over theSouthern Ocean, where predicted values of mean(w)can be greater than 0.5 m s�1 too low and of std(w) canbe over 1 m s�1 too large. Furthermore, in both thesubtropics and the Southern Hemisphere middle lati-tudes, the CMT distribution overestimates skew(w); inparticular, the CMT distribution is unable to reproducethe negative skewnesses of w observed in the subtropicsand the band of near-zero skewness over the SouthernOcean. It was argued in the previous section that thesefeatures of the wind speed skewness field can be un-derstood to arise from the strong negative skewness ofthe along-mean wind component of the vector wind inthese regions. As the skewness of the vector wind com-ponents is assumed to be zero in the derivation of theCMT distribution, it is not surprising that this distribu-tion is unable to reproduce these features of the ob-served sea surface wind speed skewness field.

The derivation of the CMT distribution presentedabove can be extended to include non-Gaussian along-mean wind surface wind components as follows. As waspointed out in Thompson et al. (1983), the distribution

p�u� �1

�2��2�1 �

�

6He3�u � u

� �� exp��

�u � u�2

2�2 � �46�


FIG. 10. Mean, std dev, and skewness of w from CMT, D(�eff, 0) D(�eff, �eff), and D (�, �) distributions.

FIG. 11. Same as in Fig. 10, but for predicted moments of w minus the observed moments.


Fig 10 11 live 4/C

will have mean u, standard deviation �, and skewness �,where

He3�x� � x3 � 3x �47�

is the Hermite polynomial of order 3 [following thenotation of Gradshteyn and Ryzhik (2000, p. xxxvii)].The distribution Eq. (46) is a low-order Gram–Charlierexpansion (Johnson et al. 1994) of the Gaussian PDFEq. (38). A drawback with the Gram–Charlier expan-sion of a PDF is that the resulting distribution is notguaranteed to be positive definite: negative values ofthe distribution can occur, so that it is no longer a PDF

in strict terms. Algorithms exist to impose the con-straint of nonnegativity on Gram–Charlier distributions(e.g., Jondeau and Rockinger 2001) at the expense ofan analytic expression for the density. In practice, wefind that the Gram–Charlier densities only ever becomeslightly negative for the parameter values used in thepresent analysis.

Proceeding as with the derivation of the CMT distri-bution, assuming independent, isotropic fluctuations inu and �, but using the skewed Gram–Charlier distribu-tion of u Eq. (46) instead of the Gaussian distributionEq. (38), after integrating over the angle � we arrive atthe marginal distribution for the wind speed w:

p�w� �w

�2 exp��w2 � u2

2�2 ��1 ��

6He3�u

��I0�wu

�2 � ��

2 �w

��He2�u

��I1�wu

�2 ��

�

4 �w

��2�u

��I0�wu

�2 � � I2�wu

�2 ��

24 �w

��3�3I1�wu

�2 � � I3�wu

�2 ��, �48�

where

He2�x� � x2 � 1 �49�

is the second-order Hermite polynomial. The PDF Eq.(48) will be denoted the D(�, 0) distribution.

The spatial anticorrelation between mean(u) andskew(u) discussed in Monahan (2004b) suggests that aparameterization can be developed in which the higher-order moment is obtained from the lower-order mo-ment. Such a parameterization will allow the D(�, 0)distribution to be evaluated with no more input datathan is required by the CMT distribution. In fact, asdiscussed in Monahan (2004a) and Monahan (2004b),skew(u) depends (nonlinearly) on std(u) as well. Be-cause we are interested in developing a parameteriza-tion of skew(u) for use in Eq. (48), we will look for amodel of the form

skew�u� � �eff�u, �� , �50�

where � is some residual. Ideally, such a parameteriza-tion would be derived from a mechanistic model such asEqs. (25)–(26); this model, however, is too simple to besufficiently quantitatively accurate for this purpose. Wewill instead estimate �eff statistically from the observedmoments of u. Because �eff is expected to be a signifi-cantly nonlinear function of u and �, it is estimatedusing a standard feed-forward neural network (e.g.,Hsieh and Tang 1998) with five “hidden neurons.” Neu-ral networks are a flexible tool for nonparametric, non-

linear function estimation; for the present application,the statistical models fit to the neural network werefound to outperform polynomial nonlinear regressionmodels with a similar number of parameters (notshown). The left panels of Fig. 12 display maps of theskewness fields of u from observations and from thebest-fit model Eq. (50). While the neural networkmodel generally underestimates the extremes of the ob-served skewness field, it is generally successful in cap-turing the spatial structure of the observed field.

Maps of mean(w), std(w), and skew(w) predicted bythe D(�eff, 0) distribution obtained using the empiricalskewness field �eff are presented in the second row ofFig. 10; the associated difference maps are presented inFig. 11. Similarly to the CMT distribution, the D(�eff, 0)distribution underestimates mean(u) and overestimatesstd(u) in the midlatitudes. However, the absolute errorsare generally considerably smaller (by 40% on average)for the D(�eff, 0) distribution than for the CMT distri-bution. Furthermore, unlike the CMT distribution, theD(�eff, 0) distribution is able to capture the negativeskewness of w in the subtropics and the band of near-zero skewness around the Southern Ocean. However,the subtropical negative skewness of w is overestimatedby D(�eff, 0), leading to a strong negative bias inskew(w) in the subtropics of both hemispheres, and aslight positive bias in skew(w) is evident in the midlatitudes of the Southern Hemisphere. While the D(�eff,0) distribution improves considerably over the CMTdistribution in its characterization of the leading mo-ments of w, considerable biases remain.


The characterization of the probability distributionof sea surface wind speed can be further improved byincluding the kurtosis of the along-mean wind surfacevector wind component in the parameterization. Aswith the skewness, the kurtosis of u is also generallynonzero; the spatial distribution of kurt(u) is presentedin Fig. 12. The kurtosis of u is between �1 and 1throughout most of the midlatitudes, increasing up to avalue of 2 in the Atlantic–Indian Ocean sector of theSouthern Ocean in which mean(u) is largest. Through-out the Tropics, the kurtosis is generally positive, takingvalues in excess of 4 in the central tropical Pacific (al-though the statistical significance of these large valuesis questionable). If the kurtosis of u is known, it can beincluded in an extended Gram–Charlier expansion ofthe Gaussian PDF Eq. (38): the distribution

p�u� �1

�2��2�1 �

�

6He3�u � u

� ��

�

24He4�u � u

� �� exp��u � u�2

2�2 �,

�51�

where

He4�x� � x4 � 6x2 � 3 �52�

is the fourth-order Hermite polynomial, will have speci-fied mean u, standard deviation �, skewness �, and kur-tosis �. Again assuming isotropic and independent fluc-tuations in u and �, we obtain the following expressionfor the probability distribution of w:

p�w� � ��1 ��

6He3�u

��

24He4�u

��I0�wu

�2 � �12 �w

��He2�u

��

3He3�u

��I1�wu

�2 ��

18 �w

��2��2��u

�� He2�u

��I0�wu

�2 � � I2�wu

�2 ��1

24 �w

��3�� u

��3I1�wu

�2 �� I3�wu

�2 ��

192 �w

��4�3I0�wu

�2 � � 4I2�wu

�2 � � I4�wu

�2 ��w

�2 exp��w2 � u2

2�2 �. �53�

This PDF will be denoted as D(�, �).As was the case with skew(u), observations and the

boundary layer model [Eqs. (25)–(26)] indicate that

kurt(u) is related to both mean(u) and std(u). JointPDFs of mean(u), std(u), and skew(u) with kurt(u)from SeaWinds observations are presented in Fig. 13.

FIG. 12. Skewness and kurtosis of along-mean wind component u (top) from observations and (bottom) assimulated by the effective neural-network parameterizations �eff [Eq. (50)] and �eff [Eq. (55)].


Fig 12 live 4/C

Evidently, kurt(u) is generally negative for lowmean(u) and increases as mean(u) increases; kurt(u)generally takes larger values for smaller values of std(u)and decreases as std(w) increases; and kurt(u) increases

with decreasing skew(u). Contour plots of mean(u),std(u), skew(u), and kurt(u) calculated from the mar-ginal PDF for u,

p�u� � ��

�

p�u, �� d� � N 1 exp�2��u�u

�2 ��

�

exp��2

�2h�

0

�u2��2

cd�w�w2 dw� d�, �54�

are presented in Fig. 14. The first three of these plotsagree well with similar plots for a simpler version of themodel presented in Monahan (2004a). Although theboundary layer model considerably underestimates thevalues of kurt(w) compared to observations, the rela-tionships between the observed moments evident inFig. 13 are qualitatively consistent with those presentedin Fig. 14: in general kurt(u) is an increasing function ofmean(u) and a decreasing function of both std(u) andskew(u).

In analogy with Eq. (50), we construct an empiricalmodel

kurt�u� � �eff�u, �� 55�

using a feed-forward neural network. A map of �eff ispresented in Fig. 12; it is broadly consistent with theobserved kurtosis field, although it underestimateskurt(u) over the central Pacific and subtropical NorthAtlantic and Pacific Oceans.

Maps of mean(w), std(w), and skew(w) calculatedfrom the D(�eff, �eff) distribution are presented in thethird row of Fig. 10; the corresponding maps of thedifference with the observed moments appear in thethird row of Fig. 11. Inclusion of information about thekurtosis of u does not significantly improve the repre-sentation of mean(w) or std(w) relative to the D(�eff, 0)distribution, but the representation of skew(w) is con-siderably improved. The bands of negative bias in (w)characteristic of the D(�eff, 0) distribution are largelyeliminated in the D(�eff, �eff) distribution. Including in-

formation about the third- and fourth-order momentsof the vector wind, parameterized in terms of the firstand second moments, results in a representation of thePDF of sea surface wind speed that is a considerableimprovement over the PDF arising from the assump-tion of Gaussian vector wind fluctuations. Note that theD(�eff, �eff) distribution does not make use of any moreinput information than does the CMT distribution.

Finally, differences are relatively small between thefields of mean(w), std(w), and skew(w) predicted bythe D(�eff, �eff) distribution and those predicted by theD(�, �) distribution obtained using the observedskew(u) and kurt(u) fields in Eq. (53) (fourth row ofFigs. 10 and 11). The D(�, �) distribution represents themoments of w marginally better than the D(�eff, �eff)distribution but requires four pieces of information ateach grid point (u, �, �, �) rather than the two (u, �)required by D (�eff, �eff).

It is noteworthy that particularly large biases inD(�, �) occur where the assumptions of isotropic, inde-pendent fluctuations of the vector winds used to obtainEq. (53) are most strongly violated. A map of the cor-relation coefficient between u and � is presented in theupper panel of Fig. 15, while the lower panel displays amap of the ratio of the standard deviations of u and � (ameasure of anisotropy in fluctuations of the vectorwind). In general, fluctuations in u and � are anisotropicand uncorrelated; notable exceptions occur in monsoonregions [the northern Indian Ocean and the easternflanks of the subtropical anticyclones, e.g., Rodwell and

FIG. 13. Kernel density estimates (contoured on logarithmic scales) of joint PDFs of kurt(u) with mean(u), std(u), and skew(u).


Hoskins (2001)] and the ITCZ and SPCZ. As well, fluc-tuations in u are slightly (20%) larger than those of �in the midlatitude storm tracks. Errors in mean(w) andstd(w) for the distribution D (�, �) are greatest in theseregions where fluctuations in the vector wind are mostcorrelated or anisotropic. A seasonally stratified analy-

sis of the correlation structure of the vector wind com-ponents (not shown) indicates that many of these cor-related features (particularly in the Indian Ocean andalong the equator) arise as a consequence of seasonalvariability in the sea surface wind field.

Because of the nonlinear dependence on sea surfacewind speed of bulk formulas for air–sea fluxes, the spa-tially or temporally averaged fluxes will not generallyequal the fluxes associated with the average surfacewind speed. In particular, grid-scale averages of thesefluxes in general circulation models (GCMs) will notequal the fluxes associated with grid-scale winds. Fur-thermore, the average wind speed is not generally equalto the magnitude of the average vector wind: highlyvariable but isotropic fluctuations in the vector windwill be associated with a large average speed but yielda mean vector with small amplitude (e.g., Mahrt andSun 1995; Levy and Vickers 1999). Improvements tocalculations of the fluxes can be obtained through theuse of parameterizations of the PDF of sea surface windspeed, which take as input grid-scale variables. The sur-face wind fields produced by GCMs represent grid-scale-averaged vector winds, but grid-scale wind speeddistributions are required to diagnose air–sea fluxes.The distributions considered in this section provide aspecification of the PDF of sea surface wind speedsfrom the mean and variance of the vector winds. Thefirst of these quantities is a standard model field,while parameterizations of the second exist (e.g.,CMT). The incorporation of these distributions intoGCM calculations of air–sea fluxes is therefore straight-forward in principle; an important and interesting ex-tension of the present study would be an estimate of theeffects on simulations of the climate system of includingthese distributions in the computation of air–sea fluxes.

8. Summary and conclusions

This study has considered the probability distributionof sea surface wind speeds (w) observationally, using6–yr worth of daily sea surface winds on a 1/4° � 1/4°grid from the SeaWinds scatterometer mounted on theNASA QuikSCAT satellite; theoretically, using a sto-chastic model derived using a clear sequence of ap-proximations from the equations governing boundarylayer physics; and empirically, using a class of modelsbased on simplifying assumptions about the PDF of seasurface vector winds. The following results were ob-tained.

• Global fields of mean(w), std(w), and skew(w) havebeen characterized, along with those of the Weibullscale and shape parameters a and b. When these

FIG. 14. Contour plots of mean(u), std(u), skew(u), and kurt(u)as functions of ��u� and � from the marginal PDF Eq. (54) de-rived from the stochastic boundary layer model [Eqs. (25)–(26)].


fields were compared to those characterized in pre-vious studies (e.g., Pavia and O’Brien 1986; Isemerand Hasse 1991; Bauer 1996), it was found that thecharacterizations were in general agreement wherethe data used in these earlier studies were abundant,but substantial differences emerged in regions whereprevious studies suffered from a paucity of data.These differences were concluded to arise as a con-sequence of the limited sea surface wind datasetsavailable to the earlier studies.

• Consistent with the results of earlier studies (e.g.,Hennessey 1977; Justus et al. 1978; Conradsen et al.1984; Isemer and Hasse 1991; Deaves and Lines 1997;Pang et al. 2001), it has been demonstrated that theWeibull distribution is a reasonably accurate approxi-mation of the probability distribution of sea surfacewind speeds on both a global and a local scale. Theobserved relationships between the estimated Weibullscale parameter a and mean(w), and between the es-timated Weibull shape parameter b and skew(w) aresimilar to those for a Weibull distributed variable. Inparticular, in those regions of the Tropics whereskew(w) is negative, the ratio mean(w)/std(w) takeslarge values; over the Southern Ocean, where skew(w)is close to zero, mean(w)/std(w) takes values close tothe 3.6 expected for an almost-Gaussian distribution,and in the NH extratropics, mean(w)/std(w) is smalland skew(w) is positive.

• Despite the reasonable accuracy of the Weibull ap-proximation, it is not an exact characterization of thePDF of sea surface wind speeds . Observed skew(w)is more strongly negative in the Tropics, and morestrongly positive in the NH midlatitudes, than wouldbe expected for a Weibull variable. A Monte Carlosimulation using synthetic Weibull datasets withnumbers of degrees of freedom comparable to thoseof the SeaWinds data indicates that it is highly un-likely that this apparent non-Weibull behavior arisesas a result of sampling fluctuations of a Weibull ran-dom variable.

• An analytic expression for the PDF of w, p(w), wasobtained from the simple stochastic boundary layermodel introduced in Monahan (2004a,b). This PDFwas demonstrated to be in good qualitative agree-ment with the observed relationships between themean, standard deviation, and skewness of w. In par-ticular, the dependence of the sign of skew(w) on theratio of mean(w) to std(w) was demonstrated to fol-low from the anticorrelation of the mean and skew-ness fields of the vector wind components discussedin Monahan (2004a,b). This anticorrelation is a con-sequence of the nonlinear surface drag predicted byboundary layer theory; if the surface drag is modeledas linear, the vector winds are Gaussian and the windspeed skewness is never negative. It was concludedthat the Weibull distribution arises as a useful ap-

FIG. 15. (top) Map of the correlation coefficient between the along- and cross-mean windsurface wind components, u and �. (bottom) Map of the ratio of the std devs of u and �.


Fig 15 live 4/C

proximation to the distribution of w because it im-poses the constraints on the relationship betweenmean(w), std(w), and skew(w) that are required bythe physics of the boundary layer.

• While helpful for developing a qualitative under-standing of the relationships between the moments ofw, the stochastic boundary layer model consideredwas too simple to be quantitatively useful. As well, itrequired as input parameters the mean and variabil-ity of a forcing function that is difficult to estimatefrom data. To construct more quantitatively usefulexpressions for the PDF of w, the PDF described inCMT, which follows from the assumption that thejoint distribution of the vector wind components isbivariate Gaussian, was generalized to allow for non-zero skewness and kurtosis of the along-mean windcomponent u of the vector surface wind. Neglect ofthe skewness and kurtosis of u was shown to producea PDF that tended to underestimate mean(w) andoverestimate std(w) and was unable to reproduce thenegative skewness of w observed over much of thetropical oceans. By incorporating the skewness andkurtosis of the along-mean wind component, param-eterized to depend on the mean and variability of thevector wind, the errors in the representation ofmean(w), std(w), and skew(w) were significantly re-duced. This analysis demonstrates two importantpoints: first, that the moments of the scalar sea sur-face wind speed can be quite accurately predictedgiven a knowledge of the moments of the vectorwind, and second, that the higher-order moments ofthe vector wind can accurately be parameterized interms of the lower moments, as was suggested inMonahan (2004b). The grid-scale-averaged vectorsurface wind is a standard atmospheric general circu-lation model field, and parameterizations exist of thegrid-scale variability of the vector wind (CMT); thesequantities can be used as input to the parameteriza-tion of the PDF of w presented in this study to im-prove representations of grid-scale-averaged fluxes(which are in general not equal to the fluxes associ-ated with grid-scale average winds; e.g., Mahrt andSun 1995; Levy and Vickers 1999).

Only a single global sea surface wind dataset was con-sidered in this study. Furthermore, the analysis was ofdata throughout the entire year, without considerationof seasonal variability. In Part II of this study, the quan-titative details of the PDF of w are compared betweenseasons and with other global surface wind datasets. Ingeneral, it is found that despite quantitative differences,the qualitative features of the relationships betweenmoments of wind speeds described in the annual

SeaWinds data are invariant between different datasetsand seasons.

Given the qualitative utility of the simple stochasticboundary layer model [Eqs. (25)–(26)], an interestingextension of the present study would be the generaliza-tion of this model to include neglected processes. Inparticular, improved quantitative agreement betweenthis model and observations should follow from takinginto account fluctuations in the drag coefficient result-ing from the effects of nonneutral stability and remoteswell (Sura 2003; SS04; Monahan 2004b) and nonwhitenoise structure in the forcing variability. Such an analy-sis would have the further benefit of contributing to-ward the development of physically based parameter-izations of the relationships between the moments ofsea surface vector wind components. Such parameter-izations could then be used in empirically derived PDFssuch as D(�, 0) and D(�, �) instead of the statisticalrelationships used in this study.

Recent studies have emphasized the point that scat-terometer observations do not represent the wind vec-tor relative to a fixed coordinate system, but rather theair motion relative to the underlying sea surface. Inregions of swift surface currents, the sea surface windvector observed by the scatterometer will differ fromthe actual wind vector by the surface current vector(e.g., Kelly et al. 2001, 2005; Chelton et al. 2004). Thisfact has not been taken into account in the presentstudy, in which the sea surface has been treated as a(rough) rigid surface. Differences between the actualsea surface wind and the SeaWinds observations mayaccount for some of the deficiencies of the empiricaland theoretical models presented in this study. An in-teresting direction of future study would be the contri-bution of surface currents to the probability distribu-tion of sea surface winds.

A primary motivation for the study of the probabilitydistribution of sea surface wind speeds from the per-spective of climate studies is the role these distributionsplay in the computation of spatially and/or temporallyaveraged air–sea fluxes of momentum, energy, freshwa-ter, and chemical constituents (e.g., Taylor 2000; Jonesand Toba 2001; Donelan et al. 2002). The Weibull dis-tribution provides a good empirical approximation tothe PDF of w but requires knowledge of the mean andstandard deviation of the wind speed. This paper hasdiscussed both theoretical and empirical parameteriza-tions of p(w), which depend on vector averaged quan-tities of the kind naturally produced by GCMs. Theincorporation of these PDFs into parameterizations ofair–sea fluxes in GCMs, as discussed in CMT, presentsthe possibility of improving the representation of the


surface fluxes that are at the heart of the coupled physi-cal–biogeochemical dynamics of the climate system.

Acknowledgments. The author acknowledges sup-port from the Natural Sciences and Engineering Re-search Council of Canada, from the Canadian Founda-tion for Climate and Atmospheric Sciences, and fromthe Canadian Institute for Advanced Research EarthSystem Evolution Program. The author is grateful toJim Christian and Konstatin Zahariev for helpful dis-cussions. The author would also like to thank PhilipSura and two anonymous referees whose comments sig-nificantly improved this manuscript.

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The Probability Distribution of Sea Surface Wind Speeds ...web.uvic.ca/~monahana/wind_speed_pdf_I.pdf · The probability distribution of sea surface wind speeds, w, is considered