Statistical Downscaling Prediction of Sea Surface Winds over the Global Ocean

Transcription

1 7938 J O U R N A L O F C L I M A T E VOLUME 26 Statitical Downcaling Prediction of Sea Surface Wind over the Global Ocean CANGJIE SUN AND ADAM H. MONAHAN School of Earth and Ocean Science, Univerity of Victoria, Victoria, Britih Columbia, Canada (Manucript received 11 October 2012, in final form 7 March 2013) ABSTRACT The tatitical prediction of local ea urface wind from large-cale, free-tropopheric field i invetigated at a number of location over the global ocean uing a tatitical downcaling model baed on multiple linear regreion. The predictand (the mean and tandard deviation of both vector wind component and wind peed) calculated from ocean buoy obervation on daily, weekly, and monthly cale are regreed on upperlevel predictor field from reanalyi product. It i found that in general the mean vector wind component are more predictable than mean wind peed in the North Pacific and Atlantic, while in the tropical Pacific and Atlantic the difference in predictive kill between mean vector wind component and wind peed i not ubtantial. The predictability of wind peed relative to vector wind component i interpreted by an idealized model of the wind peed probability denity function, which indicate that in the midlatitude the mean wind peed i more enitive to the vector wind tandard deviation (which generally are not well predicted) than to the mean vector wind. In the tropic, the mean wind peed i found to be more enitive to the mean vector wind. While the idealized probability model doe a good job of characterizing month-to-month variation in the mean wind peed in term of the vector wind tatitic, month-to-month variation in the tandard deviation of peed are not well modeled. A erie of Monte Carlo experiment demontrate that the inconitency in the characterization of wind peed tandard deviation i the reult of difference of ampling variability between the vector wind and wind peed tatitic. 1. Introduction Sea urface wind play a central role in influencing the exchange of heat, momentum, and ma between the ocean and the atmophere (e.g., Garratt 1992; Bate et al. 2001; Jone and Toba 2001; Donelan et al. 2002). A well, ea urface wind repreent a potentially ignificant energy reource (e.g., Liu et al. 2008; Capp and Zender 2009) and high ea urface wind repreent hazard to hipping (e.g., Sampe and Xie 2007). While the output of global climate model repreent the bet tool available for tudying large-cale climate variability, it i generally not directly relevant for inference about local climate. The coare reolution and approximate parameterization of ubgrid-cale procee both limit the accuracy of the repreentation of local variability, epecially in the planetary boundary layer. In particular, local urface wind may be influenced by Correponding author addre: Adam Monahan, School of Earth and Ocean Science, Univerity of Victoria, P.O. Box 3065 STN CSC, Victoria BC V8W 3V6, Canada. monahana@uvic.ca local mall-cale procee that are not reolved well in climate model. The proce of downcaling i deigned to relate local, mall-cale variability to variability on large cale. Dynamical downcaling approache thi problem by neting finely reolved, local dynamical model within coarely reolved, large-cale model. In contrat, tatitical downcaling (SD) i a complementary trategy employed to empirically downcale largecale variability through tatitical method. Although dynamical downcaling ha the merit of being phyically baed and not auming a tationary climate, potentially ignificant drawback uch a poible error aociated with imperfect parameterization of key procee (e.g., cloud and boundary layer procee), ytematic biae, coare patial reolution, and extremely high computational demand contrain the ue of pure dynamical downcaling. Although SD ha the weakne that it aume tatitically tationary relationhip between large-cale and local variable (a repreented by hitorical obervation), it i inexpenive and eay to implement. Thi tudy conider the tatitical relationhip between local ocean wind and large-cale free-tropopheric DOI: /JCLI-D Ó 2013 American Meteorological Society

2 15 OCTOBER 2013 S U N A N D M O N AHAN 7939 circulation baed on buoy obervation and reanalyi product. Following earlier work by Monahan (2012a) and Culver and Monahan (2013), SD i ued to invetigate the predictability of ea urface wind (both wind peed and vector wind component) meaured at buoy over the global ocean on daily, weekly (10 day), and monthly time cale. Along with the abolute predictive kill of urface-wind tatitic, the predictability of the tatitic of wind peed relative to thoe of the vector wind component i conidered in thi tudy. A imple multiple linear regreion will be ued for the SD in place of a more ophiticated analyi (e.g., tepwie linear regreion) in order to minimize the chance of overfitting the model. With appropriate crovalidation, multiple linear regreion produce tatitically robut prediction model. In the preent tudy, the number of tatitical degree of freedom available for building the tatitical model can be fairly mall (e.g., 12 year of 3 month for a given eaon at a typical buoy). The number of model parameter increae with the complexity of a tatitical model, requiring larger dataet for their robut etimation and to avoid overfitting the model. The ue of a relatively imple tatitical model reduce the potential rik of kill inflation due to model overfitting. Thi tudy ha four primary goal: 1) Characterize the predictive information that free-tropopheric largecale predictor carry for the tatitic of local ea urface wind acro a range of wind climate. 2) Explore the predictive kill of different urface-wind tatitic (mean and tandard deviation) on different temporal cale (daily, weekly, and monthly). 3) Invetigate the relationhip between the predictability of mean wind peed and that of the vector wind component preented in Culver and Monahan (2013) over a wider range of wind climate uing an idealized probability ditribution model (IPM) of the wind peed probability denity function introduced by Monahan (2012a). 4) Ae the modeling kill of the IPM for the ubmonthly tandard deviation of wind peed acro the global ocean. Monahan (2012a) invetigated the predictability of urface wind in the ubarctic northeat Pacific off of wetern Canada, while Culver and Monahan (2013) tudied the tatitical predictability of hitorical land urface wind over central Canada. Other tudie (e.g., Salameh et al. 2009; van der Kamp et al. 2012) have invetigated the predictability of variou vector wind component in region of complex topography. In contrat to land urface wind, ea urface wind are le influenced by tationary local feature (e.g., topography or fixed urface inhomogeneitie). Therefore, the connection between ea urface wind and upper-level large-cale atmopheric field i expected to be impler than that for urface wind over land. Furthermore, the range of wind climate i much greater over ocean than over land (becaue of the much weaker urface drag over water). Thu, conideration of ea urface wind allow for the analyi of urface wind SD in a relatively idealized etting over a relatively large parameter range. Thi tudy doe not conider the temporal tructure of wind [which i conidered in detail in Monahan (2012b)]. The focu i on the intantaneou prediction of urface wind tatitic on variou averaging time cale from large-cale free tropopheric predictor on the ame time cale. Section 2 decribe the data ued in thi tudy, and ection 3 preent both the methodology to be ued and the prediction reult of urface-wind tatitic. Section 4 introduce the idealized model ued to undertand the relationhip between the predictability of the tatitic of the vector wind component and wind peed. A dicuion and concluion are preented in ection Data Thi tudy aee the cro-validated tatitical predictability of the tatitic of hitorical ea urface wind obervation from a total of 52 moored ocean buoy (the predictand) uing free-tropopheric large-cale circulation data from global reanalyi product (the predictor). a. Buoy data The buoy conidered in thi tudy are ituated in the tropical and North Pacific and Atlantic Ocean, with data duration of between 8 and 28 year (Table 1). The Southern Ocean and Indian Ocean are not conidered in thi tudy a we were not able to find buoy obervation in thoe location of ufficient duration to etablih robut tatitical relationhip with the flow aloft. The wind and direction data from the 52 buoy were obtained from four ource: 1) Prediction and Reearch Moored Array in the Atlantic (PIRATA) project 10-min averaged data (meaured at 3 4 m above mean ea level) from five buoy in the tropical Atlantic (downloaded from 2) Tropical Atmophere Ocean (TAO)/Triangle Tran- Ocean Buoy Network (TRITON) project 10-min averaged wind data (meaured at 3 4 m above mean ea level) from 12 buoy in the tropical Pacific (downloaded from didel/didel-pir.html); 3) National Data Buoy Center (NDBC) hourly report of 8-min averaged data (approximately 5 m above mean ea level) from 31 buoy off of the wet and eat

3 7940 J O U R N A L O F C L I M A T E VOLUME 26 TABLE 1. The location, duration, anemometer height, and data archive of the buoy conidered in thi tudy. Buoy identification (ID) i given where applicable. Buoy ID Latitude (8N) Longitude (8W) Duration Anemometer height (m) Source NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC NDBC PIRATA PIRATA PIRATA PIRATA PIRATA TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON TAO/TRITON JMA JMA JMA JMA coat of North America (downloaded from and 4) Japan Meteorological Agency (JMA) three-hourly report of 10-min averaged data (approximately 5 m above mean ea level) from four buoy in the northwet Pacific; (downloaded from kihou.go.jp/kaiyou/db/veel_ob/data-report/html/ index_e.html).

4 15 OCTOBER 2013 S U N A N D M O N AHAN 7941 Thee data were then ued to calculate mean of vector wind component (projection along 36 direction around the compa) and wind peed on daily, weekly and monthly time cale. Standard deviation of thee quantitie on ubaveraging time cale were alo calculated. Other than removing miing data from the buoy time erie, no other preproceing wa carried out on thee dataet. b. Global reanalyi product Ten-meter and 850-hPa field (zonal wind U, meridional wind V, and temperature T) were obtained from National Center for Environmental Prediction (NCEP)/Department of Energy (DOE) Reanalyi 2 data (downloaded from data/gridded/data.ncep.reanalyi2.html). Wind peed field W were computed from the U and V field. Thee data are available 4 time daily from January 1979 to December 2011 at a reolution of Thedowncaling predictor were calculated from the 850-hPa data. An analyi uing predictor at other preure level demontrated that predictive information i largely independent of predictor preure level throughout the free tropophere (Sun 2012). c. Contruction of urface-wind predictand The tatitic of both wind peed and vector wind component were predicted on three different time cale (daily, weekly, and monthly) in thi tudy. For all prediction, the ame averaging time cale wa ued for both the predictor and predictand. Prediction were carried out eparately in each calendar eaon [December February (DJF), March May (MAM), June Augut (JJA), and September November (SON)] to minimize the influence of nontationaritie in the relationhip between predictor and predictand reulting from the eaonal cycle. An inpection of the eaonal variation in urface wind data (not hown) demontrated that thee calendar eaon characterize the dominant nontationarity in the data. A ummary of the predictand i a follow: 1) mean wind peed on the pecified averaging time cale w, 2) ubaveraging time cale tandard deviation of wind peed w, 3) mean vector wind component in the direction along the bai vector ~e, ~u ~e (Thee component are conidered at 108 increment around the compa. By contruction, projection eparated by 1808 are the ame up to a ign.), and 4) ubaveraging time cale tandard deviation of the vector wind component along ~e, ~u~e. Throughout thi paper, an overbar will denote averaging on daily, weekly, or monthly time cale. If the time cale i not explicitly pecified when dicuing a given reult, it will hold on any time cale. In particular, w 2 i the mean of the quare of the wind peed. Two further tatitic, m and, are alo calculated: p 1) the amplitude of the average vector wind m 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 1 y 2 and p 2) the iotropic vector wind tandard deviation 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/2)( 2 u 1 2 y ), where u and y are mean of two orthogonal vector wind component and u and y are tandard deviation of two orthogonal vector wind component (u and y generally denote arbitrary orthogonal component, unle otherwie explicitly pecified). The quantitie m and arie in the context of the idealized model of the wind peed probability denity function conidered in ection Reult of downcaling prediction a. Spatial correlation map A decribed in the previou ection, 850-hPa U, V, T, and W averaged on daily, weekly, and monthly time cale were ued a the predictor in thi tudy. The trength and patial cale of the tatitical relationhip between urface wind tatitic and upper-level largecale predictor can be aeed through inpection of patial correlation field. Correlation field of each of mean zonal wind u and wind peed w with U, V, W, and T during DJF for one buoy at Atlantic Ocean are diplayed in Fig. 1. Both the monthly and daily time cale are diplayed. It can be een that u i trongly correlated with U on large cale: poitive correlation are found locally while negative correlation field are found to the north. The other predictor field alo how large-cale correlation tructure with u In contrat, the abolute correlation value for w are much maller than thoe for u, particularly on the monthly time cale. The horizontal cale of trong correlation increae with the averaging time cale: on the daily time cale, the patial cale of the correlation field are maller than thoe on monthly time cale (Sun 2012). Thi reult i conitent with the fact that on the ynoptic cale, the influence exerted on urface wind by large-cale circulation i more local while on longer time cale large-cale teleconnection pattern become more important. b. Combined EOF analyi It i evident from Fig. 1 that predictive information for urface wind tatitic i patially ditributed within individual predictor field, and that ome of thi information i common acro thee different field. To efficiently ditribute the predictor variance among the

5 7942 J O U R N A L O F C L I M A T E VOLUME 26 FIG. 1. (left) Correlation map of mean zonal wind at buoy with large-cale predictor at 850 hpa on monthly time cale: (top) U, (econd row) V, (third row) W, and (bottom) T. The poition of the buoy i indicated by the white dot. The white boxe in the panel denote the domain ued for the EOF decompoition of large-cale predictor field. (left center) A in (left), but for monthly-mean wind peed. (right center) A in (left), but on a daily time cale. (right) A in (left center), but on a daily time cale. mallet number of time erie, a combined empirical orthogonal function (EOF) analyi following that in Monahan (2012a) i carried out. Prediction on daily, weekly, and monthly time cale are made uing 26, 16, and 6 combined PC, repectively, a predictor. The number of predictor choen for each time cale were elected a the minimum number needed to explain over 85% of the total variance in the four large-cale predictor field. The prediction reult are not enitive to reaonable change in the number of predictor included in the model, or to reaonable change in the EOF domain. To maintain the conitency acro different time cale, the EOF domain are the ame on daily, weekly, and monthly time cale at a particular ite. A the patial cale of correlation map are maller at horter time cale, more PC predictor are included in the downcaling model on daily time cale than monthly time cale to reproduce the maller tructure of variability. A the number of tatitical degree of freedom i larger on maller averaging time cale, the SD model can accommodate more predictor without overfitting. A leave one year out cro-validation trategy i employed in the multiple linear regreion model to prevent model overfitting. For example, prediction of the firt year were determined from a regreion model built with data from all other year. The prediction of the econd year were then obtained in a imilar way (only the econd year data were withheld when etimating the model parameter). When the prediction for all the year were obtained, the r 2 value (i.e., quare of

6 15 OCTOBER 2013 S U N A N D M O N AHAN 7943 FIG. 2. Monthly-time-cale DJF r 2 prediction kill at three repreentative buoy. Shown are vector wind mean (olid red line) and tandard deviation (red dahed line) in 36 direction, the mean wind peed (blue line), and the wind peed tandard deviation (dahed blue line). The black circle denote a reference prediction kill of r correlation between prediction and obervation) wa computed to meaure the prediction kill. c. Predictability of urface wind tatitic: A cae tudy of three repreentative buoy Statitic of the vector wind component (both mean and tandard deviation) in 36 direction around the compa, along with the mean and tandard deviation of wind peed, were predicted at all buoy on daily, weekly, and monthly time cale. Figure 2 how the DJF predictive kill (r 2 ) of each of the urface-wind tatitic for the monthly time cale at three repreentative buoy. It i evident that the predictive kill of vector wind component are generally aniotropic, a had been previouly noted for land urface wind by van der Kamp et al. (2012) and Culver and Monahan (2013). The peed prediction i iotropic by contruction, a wind peed i a calar quantity. Previou tudie have uggeted that the maximum prediction kill of vector wind component i aligned with topographic feature in mountainou area (van der Kamp et al. 2012), although vector prediction aniotropy i alo oberved in region with little topographic variability (Culver and Monahan 2013). We were unable to determine any dominant factor determining the magnitude or orientation of thi aniotropy. For example, for the buoy conidered in thi tudy, the maximum prediction kill were aligned both along and acro hore. Note alo that at buoy the predictive kill of the bet predicted mean vector wind i much better than that of mean wind peed, while at buoy the mean wind peed i a well predicted a the bet predicted mean vector wind component. Buoy repreent an intermediate cae. d. Wind tatitic predictability ditribution Map of the DJF prediction kill (correlation r 2 value) of the bet predicted mean vector wind component max[r 2 (~u ~e)] (i.e., the vector wind component that ha the highet predictive kill among all the component in 36 direction), the mean wind peed w, the bet predicted tandard deviation of vector wind component max[r 2 ( ~u~e )], and the tandard deviation of wind peed w on monthly time cale at all the 52 buoy are hown in Fig. 3. Several general reult follow from thee prediction map. 1) A wa found in Monahan (2012a) and Culver and Monahan (2013), the prediction kill of the bet predicted mean vector wind component are generally higher than thoe of the mean wind peed acro all the 52 buoy. There i no general relationhip between the predictability of the mean peed and the wort predicted mean vector wind component (not hown). 2) The buoy which have relatively high prediction kill of mean wind peed are generally located in tropical region. Through the midlatitude, the prediction kill of mean wind peed are generally coniderably lower. There i no general relationhip between the predictability of peed and proximity to land. 3) The ubaveraging time cale tandard deviation of both vector wind component and wind peed are generally poorly predicted at all geographic location. Correponding map for the other calendar eaon and averaging time cale produce reult conitent with thee general reult (Sun 2012). Thee reult are alo illutrated by catterplot (acro 52 buoy and 4 eaon) of the relative predictability of the vector wind component and wind peed tatitic (Fig. 4). Each point in thee plot repreent the prediction kill of the pecified urface-wind tatitic in one eaon at one buoy on the pecified time cale. In general, we ee that mean quantitie are generally better

7 7944 J O U R N A L O F C L I M A T E VOLUME 26 maller than thoe of the bet predicted vector wind component. Thi reult wa alo obtained for ea urface wind in the northeat ubarctic Pacific (Monahan, 2012a) and for land urface wind acro Canada (van der Kamp et al. 2012; Culver and Monahan 2013). Monahan (2012a) introduced an idealized probability model of the wind peed probability ditribution to invetigate the reaon for thee difference in predictability. Auming that fluctuation in the vector wind are iotropic, uncorrelated, and Gauian, Monahan (2012a) howed that the mean wind peed can be modeled a a function p of the magnitude of the mean vector wind m 5 u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 1 y 2 and the iotropic tandard deviation 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1/2)( 2 u 1 2 y ) : m w(m, ) 5 F, (1) where (u, y) and ( u, y ) are the mean and tandard deviation of orthogonal vector wind component. The expreion for F i a follow (Rice 1945): rffiffiffi p x 2 F(x) 5 exp 1 1 x I 0 x x2 2 I 1 x 2 4, (2) FIG. 3. Cro-validated DJF r 2 predictive kill on the monthly time cale. (top) Bet predicted vector wind component; (econd row) mean wind peed; (third row) bet predicted tandard deviation of vector wind component; and (bottom) tandard deviation of wind peed. predicted than tandard deviation, particularly on horter averaging time cale. Furthermore, the bet predicted vector wind component i almot alway better predicted than the mean wind peed. To invetigate the relative predictability of the tatitic of vector wind component and wind peed, we now turn to an idealized model of the wind peed probability ditribution. 4. Interpretation of the relative predictability of vector wind and wind peed tatitic a. A Gauian model of the vector wind probability denity function The reult of the previou ection howed that the prediction kill of mean wind peed are generally where I j (x) i the aociated Beel function of the firt kind of order j. It hould be pointed out that no explicit aumption are made about the temporal autocorrelation tructure of the wind component. To ae the performance of thi model, we compared the IPM modeled w and the actual w on monthly time cale uing 10-m urface wind data from NCEP/DOE Reanalyi 2. We have alo aeed the IPM performance with buoy data; the reult are conitent with thoe obtained with the reanalyi data (Sun 2012). For each of the four calendar eaon, the following calculation were carried out: 1) at each grid point and for each month in the record, we computed m and from monthly mean and tandard deviation of the 10-m zonal wind and meridional wind; 2) thee value of m and are ued to compute monthly w uing the IPM; and 3) we calculated the correlation between the modeled monthly w from the IPM and the monthly w computed directly from NCEP/DOE Reanalyi 2 data. The quare of the correlation (r 2 ), which decribe the fraction of variance held in common between the two time erie, provide a linear meaure of the model performance in modeling mean wind peed. It wa found (not hown) that the modeled mean wind peed from the IPM ha a high correlation with the mean

8 15 OCTOBER 2013 S U N A N D M O N AHAN 7945 FIG. 4. (top) The prediction kill (cro-validated r 2 ) of the tandard deviation of wind peed relative to thoe of the mean wind peed, (middle) the bet predicted tandard deviation of vector wind component relative to the bet predicted mean of vector wind component, and (bottom) the mean wind peed relative to the bet predicted mean of vector wind component. (left) The daily time cale prediction, (center) the weekly time cale prediction, and (right) the monthly time cale prediction.

9 7946 J O U R N A L O F C L I M A T E VOLUME 26 FIG. 5. (a) Senitivity of w to m and a function of u and (b) enitivity of w to m and a function of u. wind peed from the reanalyi data on a global cale (r 2 above 0.9 at all grid point). The mean wind peed model derived from the IPM i demontrated to work well acro the global ocean. In later ection, we will demontrate that thi model i le ucceful in modeling month-to-month variation of the ubmonthly wind peed tandard deviation. b. Senitivity of wtom and Having provided evidence that the IPM i able to characterize the variability of w in term of the variability of m and, we can ue thi model to invetigate the enitivity of w to change in thee vector wind tatitic. While w i afunctionofm and, the enitivitie of the mean wind peed to m and are function of the ratio m/ alone: h m m w 5 w F i m 5 m 5 F0 m 5 F 0 m and (3) h m w 5 w F i m 5 5 F 2 m F0 m. (4) For convenience we can define the bounded calar quantity u (Monahan 2012a; Culver and Monahan 2013): u 5 tan 21 m. (5) The enitivitie of w to m and a function of u can be computed numerically and are hown in Fig. 5a. From the enitivity plot, it i clear that in the low u regime, w i more enitive to the variability of than of m. In contrat, in the high u regime w i more enitive to the variability of m. For intermediate value of u, w ha imilar enitivity to both and m. Thi reult indicate that in a high u regime, variability of w i determined by variability of m, irrepective of the variability of. In contrat, in a low u regime, variation in w are determined by thoe of and are inenitive to change of m. Thee different regime can be illutrated by conidering the kill of modeled w by the IPM allowing m to vary from month to month at each point while holding contant at it climatological value. The reult of thi calculation for DJF are diplayed in Fig. 6a. Conitent reult are obtained for other eaon. For comparion, we alo calculated monthly time cale u value at each grid point uing the monthly time cale m and, and then averaged thee acro all month to produce one climatological u field (Fig. 6b). Thee map demontrate the following: 1) The u regime are geographically organized. In general, high u value are found in the tropical Pacific and Atlantic a well a the Indian Ocean. The low u regime i in the ubtropical and ubpolar latitude. Intermediate u value are found predominantly in the midlatitude. 2) The model with fixed repreent month-to-month variation in w well in ome region (e.g., the tropic) and poorly elewhere (e.g., ubpolar and ubtropical region). 3) The two map in Fig. 6a and 6b match cloely. They clearly indicate that where high u dominate, the model with fixed can uccefully repreent monthly-timecale variability of w, while where low u prevail, the

10 15 OCTOBER 2013 S U N A N D M O N AHAN 7947 FIG. 6. (a) Modeling kill of DJF mean wind peed by the IPM [Eq. (1)] with month-to-month variation in m but held contant (at it long-term average value). (b) Climatological DJF u ditribution on monthly time cale, with poition of all 52 buoy. (c) A in (b), but on a weekly time cale. (d) A in (b), but on a daily time cale. Note that the color bar for (a) i between 0 and 1, while for (b) (d) it i between 0 and 1.5. model with fixed cannot accurately characterize month-to-month variation in w. Thi reult i conitent with the enitivity plot in Fig. 5a: for low u regime, w computed from the IPM with fixed climatological i not very accurate becaue variation of w are more enitive to thoe of in thi regime. In contrat, in the high u regime, the performance of the model with fixed remain good a w i primarily dependent on m. Note that the map in Fig. 6a and 6b are imilar but not identical, becaue there i not expected to be an exactly linear relationhip between u and the modeling kill r 2. The reult of thi analyi ugget that the calar quantity u i a good meaure of the dependence of w on m and for oberved ea urface wind. Map of the DJF u ditribution on each of the three averaging time cale (daily, weekly, and monthly) with the location of all the 52 buoy uperimpoed are preented in Fig. 6b d. It can be een that u generally decreae a averaging time cale increae: on daily time cale, the mid-to-high u regime (above 0.9) dominate all the buoy. On weekly time cale, an intermediate u regime ( ) appear on the flank of the urface weterlie. On the monthly time cale, the low u regime appear in ubtropical and ubpolar latitude. A the variability of extratropical ea urface wind i tronget on the ynoptic time cale of everal day, ubdaily variability i maller than the ubweekly or ubmonthly variability. In conequence, i much maller than m on the daily time cale, reulting in a broadly ditributed high u regime. It i noteworthy that in the tropical Pacific and Atlantic, a high u regime generally dominate mot buoy on daily, weekly, and monthly time cale. In the tropic the major form of variability, uch a the Madden Julian ocillation (MJO), have time cale much longer than thoe of midlatitude ynoptic eddie. The relatively teady tropical trade wind reult in m value that are much larger than, reulting in a high u regime. The u field alo diplay eaonal variation (Sun 2012). The ditribution of the buoy conidered provide good coverage of the full u range on weekly and monthly time cale. In conequence, we have a repreentative ample of buoy within each of the three u regime with which to etablih tatitical relationhip. c. Predictability of w relative to m and The reult of the previou ection ugget that the predictability of w relative to that of m and i a function of u. Scatterplot of SD predictive kill (correlation r 2 value) of w relative to thoe of m and are hown in Fig. 7 for daily, weekly, and monthly averaging time cale. Correponding value of u are indicated by color. The SD prediction of each wind tatitic on each of three averaging time cale were done eparately for each of the four calendar eaon and each of the 52 buoy (reulting in a total of 208 point in each plot). The following are oberved: 1) On the daily time cale, for which u i conitently large, the predictive kill of w i trongly correlated with the predictive kill of m acro all tation and eaon. In contrat, the predictive kill of w ha no trong relationhip with that of (except for the mallet value of u). 2) On weekly and monthly time cale, the point in the catterplot of r 2 (m)withr 2 (w) gather around the 1:1 line for high u value, while the point are more broadly cattered for low u value. In contrat, in the catterplot of r 2 ()with r 2 (w), the data point gather around the 1:1 line for low

11 7948 J O U R N A L O F C L I M A T E VOLUME 26 FIG. 7. The correlation-baed predictive kill of w relative to that of (left) m and (right) on (top) daily, (middle) weekly, and (bottom) monthly averaging time cale. The color of the data point denote the value of u. One-to-one line are given in olid blue. u value and catter away from the 1:1 line for high u value. The above reult indicate that u i a good metric for characterizing the tatitical predictability of w relative to that of m and. In the high u regime, good prediction of w require accurate prediction of m. On the other hand, in low u regime, how well w can be predicted depend on the predictability of. In a medium u regime, the ituation i complicated a w ha comparably trong and nonlinear dependence on m and. While thee reult have been demontrated uing a linear tatitical downcaling model, the enitivity reult of the IPM ugget that they hould hold irrepective of how the prediction are made. d. Predictability of w relative to that of the bet predicted vector wind component Having related the predictability of w to that of m and, we are on the way toward undertanding the predictability of w relative to the vector wind component ~u ~e. In the low u regime, the predictive kill of w i determined by that of. Prediction of the iotropic tandard deviation are not better than max[r 2 ( ~u~e )]; it follow from Fig. 4 that will generally not be a well predicted a the max[r 2 (~u ~e)]. It follow that in the low u regime, in which w variation are dominated by thoe of, the predictability of w hould be le than that of the bet predicted mean vector wind component. On the

12 15 OCTOBER 2013 S U N A N D M O N AHAN 7949 other hand, in the high u regime, the predictive kill of w i determined by that of m. To complete the connection between the predictability of peed and of vector wind component, we need to relate the predictability of the amplitude of the mean vector wind m to the predictability of the vector wind component themelve. Culver and Monahan (2013) provided the following theoretical expreion for the linear predictability of m relative to that of the vector wind component aligned along the long-term mean wind: r 2 mean(u) 2 (m) 5 td(u) 2 1 mean(u) 2r2 (u), (6) where r 2 (u) and r 2 (m) are the (correlation-baed) predictabilitie of the wind component u along the longterm mean and of m by a ingle predictor x (aumed to have a Gauian ditribution), repectively. The reult in Eq. (6) i baed on the aumption that variation of u (e.g., from month to month) are iotropic, uncorrelated, and Gauian. The quantitie mean(u) and td(u) are the mean and tandard deviation of u over the entire record, repectively. For convenience, we introduce the quantity g 5 mean(u)/td(u), which give u 1 21 r 2 (m) 5 g r 2 (u). (7) From Eq. (7) it i clear that the predictability of m i bounded above by that of the along-mean wind component. Furthermore, r 2 (u) will itelf be bounded above by the predictability of the bet predicted component (by definition). Figure 8 diplay the predictive kill of m relative to that of u in high u regime (u $ 1) in relation to g on daily, weekly, and monthly time cale. It can be een that in general, when g 1, the predictability of m approache that of u. On the other hand, when g decreae, the predictive kill of m become maller than that of u. A dicued in the previou ection, a majority of the buoy are in a high u regime on daily time cale acro all eaon and location, while on weekly and monthly time cale, many buoy are in a medium or low u regime. A a reult, fewer data point with u. 1 are diplayed for the plot on weekly and monthly time cale. It hould be emphaized that g increae with averaging time cale. In Fig. 8 it can be oberved that the upper limit of g increae from 6 to 10 a the averaging time cale increae from daily to monthly. On longer averaging time cale, a maller fraction of the variance of the wind i retained in variation of u, and a larger fraction i contained in ubaveraging time cale variability (i.e., ). For an increaingly large fraction of FIG. 8. The predictive kill of m relative to that of the along-mean vector wind component u in high u regime (u $ 1) in relationhip to g [Eq. (7); a indicated by the color of the data point]. tation and eaon, a the averaging time cale i increaed the variability in the averaged vector wind u become much maller than the climatological mean wind [i.e., td(u) mean(u)]. In general, horter averaging time cale are aociated with larger value of u and maller value of g, while longer averaging time cale with maller u and larger g. Predictability of w i generally maller than that of the vector wind component on long averaging time cale becaue of lower u value (and the fact that ha weak

13 7950 J O U R N A L O F C L I M A T E VOLUME 26 predictability). In contrat, predictability of w can be limited on hort averaging time cale when m i more poorly predicted than u becaue of the mall value of g. It i poible that there may be an optimal averaging time cale on which the relationhip among u, m, and w i balanced in uch a way a to yield optimal predictability of w relative to u. e. Prediction of ubaveraging time cale tandard deviation of wind peed with the IPM The IPM introduced above ha been hown to be able to uccefully model w in term of m and. An analytic expreion for the tandard deviation of wind peed w can alo be derived from the IPM. By definition, 2 w 5 w2 2 w 2, (8) from which it follow that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h m 2 m w m 2 2 F 5 G. (9) i From Eq. (9), we have an exact expreion for the wind peed tandard deviation w in term of m and. A with the modeled w, the enitivitie m w and w are function of the calar variable u, a illutrated in Fig. 5b. In contrat to modeled w, the modeled w i mot enitive to variation in over the entire u range. In the ame way that we aeed the ability of the IPM to repreent oberved variability of w in term of variability in m and, we now conider a imilar calculation to tet the performance of Eq. (9) in capturing oberved the month-to-month variability in the ubmonthly tandard deviation of wind peed: 1) at each grid point, we calculated monthly m and from monthly mean and tandard deviation of 10-m NCEP/DOE Reanalyi 2 zonal wind and meridional wind; 2) from thee, we computed monthly w uing Eq. (9); and 3) we then calculated the correlation between the modeled month-to-month variation in w from Eq. (9), and thoe directly computed from NCEP/DOE Reanalyi 2 data [the quared value of the correlation (r 2 ) i ued to ae the model performance in modeling the month-to-month change in the ubmonthly tandard deviation of wind peed]. The reult of thi calculation for DJF are preented in Fig. 9a. In contrat to the highly accurate repreentation of w by the IPM on a global cale, the model fail to repreent month-to-month change in w over mot of the ocean. It can be een that in the midlatitude and high-latitude region (of both the Northern and FIG. 9. (a) The modeling kill of DJF ubmonthly time cale w from the IPM [Eq. (9)]. (b) A in (a), but with the non-gauian vector wind model obtained from Eq. (11). Southern Hemiphere), the r 2 prediction kill of the model i generally below 0.5. In the tropical region, the IPM generally perform better although it performance i till poor in a number of place. The failure in reproducing month-to-month variation of the ubmonthly tandard deviation of wind peed from Eq. (9) i perplexing: we hould be able to obtain knowledge of thee tatitic, a long a we have the correct wind peed probability denity function p w (w). The fact that we can imulate w but not w with our model lead u to reexamine the three aumption on which the IPM i baed: that the vector wind fluctuation are Gauian, uncorrelated, and iotropic. While thee approximation are reaonable for modeling firt-order tatitic (mean wind peed), they may not be good approximation for modeling the econd-order tatitic. Monahan (2006) demontrated that the along- and acro-wind component u and y are cloe to being uncorrelated and have nearly iotropic fluctuation on a global cale (with ome exception in monoon and ITCZ region). However, the kewne and kurtoi of the along-mean vector wind component can differ ubtantially from zero (Monahan 2006). Therefore, we will invetigate the influence of the non-gauianity of vector wind component on modeling the tandard deviation of wind peed.

14 15 OCTOBER 2013 S U N A N D M O N AHAN ) WIND SPEED PDF FROM NON-GAUSSIAN VECTOR WINDS We firt decompoe the vector wind into component along and acro the time-mean vector wind. In thi ection, thee will be denoted by u and y repectively. Following Monahan (2006), non-gauian urface wind component are included in the model through a Gram Charlier expanion (Johnon et al. 1994) of the probability denity function (PDF) of the along-mean wind component a follow: " # p u (u) 5 pffiffiffiffiffiffi 1 (u 2 u)2 exp 2 2p 2 2 h n u 2 u 6 He 3 1 k u 2 u i 24 He 4, (10) where He 3 (x) 5 x 3 2 3x, He 4 (x) 5 x 4 2 6x 2 1 3, n 5 kew(u) 5 mean[(u 2 u) 3 ]/td(u) 3 (the monthly along-mean wind kewne), and k 5 kurt(u) 5 fmean[(u 2 u) 4 ]/td(u) 4 g 2 3 (the monthly alongmean wind kurtoi). The cro-mean wind component i modeled a Gauian (a i broadly conitent with obervation; Monahan 2006, 2007). Note that p u (u) defined in thi way i not trictly nonnegative, and therefore i not necearily a proper PDF. Neverthele, the reulting function ha the correct moment and i a ueful model of the PDF of u o long a realization of thi random variable are not required (e.g., Johnon et al. 1994). A more complicated expreion for the wind peed PDF can then be obtained: h p w (w) n u 6 3 He 1 k u wu 24 He 4 I i w w 2 h u u ih wu 22n 1 khe 8 2 I 0 2 h u nhe 2 2 k u wu 3 He 3 I i h u ih 1 I 2 wu 2 i w wu i 1 I k w 4 h wu wu wu io w 3I I I exp wu n 2 k 3I w2 1 u 2 2 2, (11) where I j i the aociated Beel function of the firt kind of order j (Monahan 2006). A with previou analye, we ue thi model to imulate month-to-month variation in w given oberved month-to-month variation in u,, n, and k. The performance in capturing w i lightly improved by including month-to-month change in the kewne and kurtoi of u, but model performance remain much poorer than for w at mot location. (Fig. 9b). Evidently, non-gauianity of the vector wind component i not the primary caue degrading the IPM performance in characterizing variability in w. Why hould it be the cae that the model perform well for modeling monthly-mean wind peed but largely fail when modeling ubmonthly wind peed tandard deviation? A we will now how, a contributing factor i related to difference in ampling variability of thee tatitic. 2) SAMPLING VARIABILITY IN MONTH-TO-MONTH FLUCTUATIONS OF m,, w, AND w Having demontrated that the model aumption of Gauian-ditributed vector wind i not the primary caue of the difficulty in modeling month-to-month variation in w, we now ak the quetion: might the poor imulation of w reult from different ampling variability of m,, and w? At any location, within each month, the wind fluctuation have on the order of tatitical degree of freedom a the urface wind generally have an autocorrelation time cale on the order of one to two day (Monahan 2012b). A a reult, there will be nonnegligible ampling variability in all urface wind tatitic, which may be different from one tatitic to another. To ae the influence of ampling variability, a erie of Monte Carlo experiment were conducted to examine how the potential ampling error in m and influence the Gauian model performance in modeling w and w. By contruction in thee idealized calculation the vector wind were Gauian, uncorrelated, and iotropic o the wind peed population tatitic are exactly related to thoe of the vector wind by Eq. (1) and (9). The trength of thi analyi i that it i a perfect model calculation we know exactly what i true about the underlying relationhip between the tatitic of vector wind and wind peed, and within thi can invetigate the role of ampling variability. In our firt experiment, we let u 5 m 0 (1 1 r m d 1 ) and (12) 5 0 (1 1 r d 2 ), (13) where d 1 and d 2 are random number with a uniform ditribution on (2½, ½). We will interpret m 0 and 0 a

15 7952 J O U R N A L O F C L I M A T E VOLUME 26 FIG. 10. Monte Carlo experiment derived modeling kill r 2 of w and w by the Gauian model [Eq. (1) and (9)] for different value of r and r m, and for (left) M and (right) M the climatological mean and tandard deviation of the vector wind, while the random number d 1 and d 2 decribe month-to-month fluctuation in thee tatitic with trength caled by r m and r, repectively. Note that fluctuation of u and repreent true month-to-month variability in the vector wind tatitic. That i, thee are the ignal that we are intereted in capturing with our model. We then generated N realization of u and each repreenting a eparate month. Within each month, we randomly ampled M day (the number of independent wind realization within each month) of vector wind component (u, y) from the Gauian model with the mean (u, 0) and iotropic tandard deviation for that month. For each month, we computed the ample mean and tandard deviation of wind peed from the ample u and y, a well a the ample m and. The ample m and were then ued to compute the w and w from the Gauian model uing Eq. (1) and (9). We then correlated the oberved and modeled wind peed moment, to characterize the modeling kill of w and w. Thi procedure wa repeated for different value of r m and r, with m ranging from 1 to 10 m 21 and kept fixed at 3 m 21 to ae ampling variability under different u regime. An enemble of 300 etimate of the modeling kill wa computed. We conider both M and M The econd of thee i cloer to the real number of tatitical degree of freedom within any month, while the firt i conidered to illutrate how ampling variability change a ample ize increae. The modeling kill of w and w are plotted a function of u in Fig. 10, from which the following can be oberved: (i) The modeling kill of w i generally high with little enitivity to the value of r m and r. Conitent with the reult preented earlier, the model i able to reproduce month-to-month variability in w for different ize of the true ignal trength. (ii) The modeling kill of w can be quite poor for mall value of r (the true month-to-month variability of ). When r 5 0, the modeling kill of w i

16 15 OCTOBER 2013 S U N A N D M O N AHAN 7953 ubtantially poorer than that with r For intance, for M 5 500, when r m , r 5 0 (Fig. 10a), and the r 2 modeling kill of w i about 0.6. In contrat, when r m 5 0, r (Fig. 10c), and the modeling kill of w i cloe to The value of r m (the true month-to-month variability in m) doe not ubtantially influence the modeling kill of w (not hown). (iii) The modeling kill of w increae with M. Under the ame et of r m and r value, the modeling kill of w i better with M than with M For M 5 20, the modeling kill of w i low even when r i relatively large. When r 5 0, the population vector wind tandard deviation remain the ame from month to month o all fluctuation of are produced by ampling fluctuation alone. In thi cae fluctuation in w modeled by Eq. (9) differ ignificantly from the true fluctuation in the wind peed ubmonthly tandard deviation. A r increae, the real month-to-month fluctuation of (the ignal) increae in ize relative to thoe of the ampling fluctuation (the noie). Thu, the ignal-to-noie ratio (SNR) increae, and the model doe a better job of imulating month-to-month change in w. Similarly it i oberved that the modeling kill of w increae a M increae for pecified r and r m value. By increaing M, while the ignal tay the ame, the noie i reduced o the SNR increae and model performance i better. For mall value of the SNR, the IPM ha difficultie modeling the month-tomonth variability in the wind peed tandard deviation even when the oberved vector wind component are Gauian, uncorrelated, and iotropic without approximation. Thi analyi demontrate that the kill of the IPM in imulating month-to-month variation of w i determined by a SNR that i related to the ize of the true month-to-month fluctuation of (characterized by r ) and the number of independent wind realization M within the month. Sampling fluctuation in w are ditinct from thoe of m and, o variability in w i only well predicted when the ignal of true month-to-month variability i ufficiently large relative to the ampling noie. We will now develop a quantitative meaure of the SNR and ue thi to interpret the modeling reult of Fig. 9a. Thi firt two tep of thi analyi are imilar to thoe of the previou Monte Carlo experiment, ampling a broader range of value of r and M (r range from 0 to 1.5; r m i et to 0.3; and M range from 1 to 1000). We define the ignal-to-noie ratio a td(~) mean(~) (r td(~), M) 2 mean(~) (r 5 0, M) 5 SNR 5 td(~) mean(~) (r 5 0, M) C(~)(r, M) , (14) C(~)(r 5 0, M) where C(~) 5 td(~)/mean(~), mean(~) i the enemble mean value of the ampled vector wind tandard deviation over all 120 month, and td(~) i the correponding tandard deviation. A dicued above, it i expected that the ize of the variability in ~ will depend on the true ignal trength r and the number of degree of freedom M. The SNR defined by Eq. (14) characterize the month-to-month fluctuation of (the ignal) in the dataet relative to the ampling variability (noie) given by [td(~)/mean(~)](r 5 0, M). The SNR can be computed from Monte Carlo imulation and compared with the modeling kill of w. Conitent with the qualitative analyi decribed earlier, the modeling kill of w i determined by the SNR a hown in Fig. 11a: a the SNR increae, w i better modeled. To obtain a model r 2 kill better than 0.9, the SNR ha to achieve a value above 3. The relationhip among M, r, and SNR i illutrated in Fig. 11b. Conitent with the previou analyi, the ignal-tonoie ratio increae with both M and r. When r 5 0.2, the number of independent realization M ha to exceed 1000 to get a ignal-to-noie ratio of 3. When r 5 0.9, a SNR of 3 can be obtained with M below 50. Thee reult indicate that for real ea urface wind data with a typical value of M maller than 30, the month-to-month fluctuation of ha to be relatively large to reult in a ufficiently high SNR, to obtain a good modeling kill of w with the IPM. We will now etimate the SNR from the NCEP/DOE Reanalyi 2 urface wind dataet and compare it with the ditribution of modeling kill of w previouly hown in Fig. 9a. To etimate SNR, we need to have the etimate of ubmonthly ~ and the value of M at each grid point. The number of independent wind realization within a month can be etimated a follow: M 5 N T e, (15) where N i the duration of a month and T e i the autocorrelation time cale. In computing T e, the autocorrelation function of the vector wind component wa modeled a a decaying exponential:

17 7954 J O U R N A L O F C L I M A T E VOLUME 26 FIG. 11. (a) The r 2 modeling kill of w a a function of the ignal-to-noie ratio [Eq. (14)] and (b) the SNR a a function of r and M. 2t af(t) 5 exp, (16) where t i the time lag. The value of T e wa then obtained from the oberved autocorrelation function by linear regreion. In fact, the autocorrelation tructure are not generally exponential and in many location over the ocean the vector wind autocorrelation tructure i aniotropic (Monahan 2012b). For thi calculation, T e wa etimated from the zonal wind component, for which the autocorrelation time cale i generally the larget. Uing the etimated value of M, we calculated value of [td(~)/mean(~)](r 5 0, M) at each grid point from the Monte Carlo imulation. The value of [td(~)/mean(~)](r, M) wa etimated from the oberved month-to-month value in the tandard deviation of the vector wind. From thee, the field of SNR wa computed (Fig. 12). Comparion of the SNR map with that of the IPM modeling kill of w (Fig. 9a) demontrate that the agreement between thee two field i generally good. In the tropical region, SNR i generally high. Correpondingly, in Fig. 9a, the modeling kill of w in thee region i relatively good. In contrat, extratropical region have maller SNR value, which correpond with the poor modeling kill of w found in thee region. Although the two map do not match perfectly, their high degree of correpondence indicate that the Gauian model performance in modeling w i trongly related to ampling variability a meaured by the SNR given by Eq. (14). It follow that while the IPM provide a ueful tool for decribing variability of w in term of that of m and, it will not generally be ueful for doing o with w, and preumably for other wind peed tatitic comparably enitive to ampling fluctuation. T e 5. Summary of reult Thi tudy ha invetigated the predictability of local ea urface wind tatitic from thoe of large-cale freetropopheric flow field. A tatitical downcaling (SD) model baed on multiple linear regreion wa ued to predict the mean and tandard deviation of oberved vector wind component and wind peed at 52 ocean buoy on daily, weekly, and monthly time cale. A ummary of our general reult i a follow: 1) The predictive kill of the bet predicted mean vector wind component i generally higher than that of the mean wind peed. Furthermore, the mean quantitie are generally better predicted than the ubaveraging FIG. 12. Spatial ditribution of the monthly-time cale DJF SNR a defined by Eq. (14).