1. Introduction
Estimation of eddy fluxes is not the only motivation for studying the statistics of sea surface winds. Recent years have seen considerable interest in the effect on oceanic variability of fluctuations in atmospheric forcing, for example, in the context of the dynamics of El Niño–Southern Oscillation (e.g., Penland 1996; Kleeman and Moore 1997; Thompson and Battisti 2000), of midlatitude gyre variability (e.g., Moore 1999; Sura et al. 2001), of the stability of the meridional overturning circulation (e.g., Aeberhardt et al. 2000; Monahan 2002a,b), and of variability in open-ocean deep convection (e.g., Kuhlbrodt et al. 2001; Kuhlbrodt and Monahan 2003). In a recent study, Sura (2003) fit observed surface winds from the Southern and Pacific Oceans to a scalar stochastic differential equation, and found evidence that the dynamics is characterized by multiplicative noise [noise with intensity dependent on the state variable; see Penland (2003a,b) for a discussion]. This result was interpreted as reflecting increasing gustiness at stronger wind speeds. We will demonstrate that the results of Sura (2003) can largely be accounted for by boundary layer momentum dynamics subject to fluctuating forcing, in an appropriate limit, and that the multiplicative noise can be understood as arising from turbulent fluctuations in the surface drag coefficient.
A description of the data used in this study is presented in section 2. The relationship between the mean and the skewness fields of both the zonal and meridional winds is characterized in section 3. Section 4 describes the boundary layer momentum equations subject to fluctuating forcing, and provides a minimal theory for the observed relationship between the mean and skewness of wind components. In section 5, the white-noise limit of the model presented in section 4 is compared to the results of Sura (2003). Finally, a discussion and conclusions are presented in section 6.
2. Data
Five datasets were analyzed in this study:
Six-hourly National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis sea level pressure, surface air temperature, and 995-hPa (the lowest sigma level of the reanalysis model) zonal and meridional winds, available on a 2.5° × 2.5° grid from 1 January 1948 to 31 December 2002 (Kalnay et al. 1996). This is the primary dataset considered in this study. [Data available online at the National Oceanic and Atmospheric Administration–Cooperative Institute for Research in Environmental Sciences (NOAA–CIRES) Climate Diagnostics Center, http://www.cdc.noaa.gov/.]
Six-hourly European Centre for Medium-Range Weather Forecasts (ECMWF) 40-yr reanalysis period (ERA-40) 10-m zonal and meridional winds, available on a 2.5° × 2.5° grid from 1 September 1957 to 31 August 2002 (Simmons and Gibson 2000). (Data available online at http://data.ecmwf.int/data/d/era40/.)
Six-hourly 10-m zonal and meridional wind data from a blend of National Aeronautics and Space Administration's (NASA's Quick Scatterometer (QuikSCAT) scatterometer observations with NCEP analyses, available on a 0.5° × 0.5° grid from 19 July 1999 to 31 March 2003 (Chin et al. 1998; Milliff et al. 1999). [Data available online at the NCAR Data Support Section (DSS), http://dss.ucar.edu/datasets/ ds744.4/.]
Variationally gridded six-hourly Special Sensor Microwave Imager (SSM/I) 10-m zonal and meridional wind observations, available on a 1° × 1° grid from 1 July 1987 to 31 December 2001 (Atlas et al. 1996). [Data available online at the NASA Jet Propulsion Laboratory (JPL) Distributed Active Archive Center, http://podaac.jpl.nasa.gov.]
Hourly 4-m zonal and meridional winds from 65 buoys in the Tropical Atmosphere–Ocean (TAO) array (McPhaden et al. 1998). (Data available online at the TAO Project Office at http://www.pmel.noaa.gov/tao/data_deliv/.) Wind data were reported from these buoys intermittently between 1990 and 2001; the number of hourly measurements at individual buoys varied from a minimum of 4000 to more than 80 000.
3. Moments of surface ocean winds
A linear relationship between the mean and skewness fields of both the zonal and meridional sea surface wind components is evident from Figs. 1 and 2: positive (negative) mean wind components are associated with negative (positive) skewness. This spatial anticorrelation is further illustrated in Fig. 3, which presents scatterplots over all ocean grid points of the mean versus the skewness of the zonal and meridional winds for the NCEP– NCAR reanalysis, the blended QuikSCAT–NCEP analysis, and the TAO buoy datasets. Similar plots for the ERA-40 and SSM/I surface winds (not shown) display the same relationship between the mean and skewness fields. The spatial correlation coefficients between the mean and skewness fields of the surface winds for the different datasets are displayed in Table 1; these correlation coefficients display essentially no seasonal variability.
Zonal wind skewness is seen to range between approximately −1 and 1.5; skewness toward the upper and lower end of this range is strong enough to be manifest in the probability density functions. Kernel density estimates (Silverman 1986) of the probability density functions of the zonal wind from the NCEP– NCAR reanalysis at three representative grid points [(50°S, 110°W), (20°N, 160°W), and (50°N, 60°W)] are presented in Fig. 4. The negative and positive skewness characteristic of the zonal winds in the Southern Ocean and the tropical Pacific respectively are evident in this figure, as is the relatively weak skewness in the central North Atlantic. It is interesting to note the relatively weak skewness of the wind components at Ocean Station Papa (50°N, 145°E) evident in Figs. 1 and 2, which is consistent with the observation in Monahan and Denman (2004) that the zonal and meridional winds at this location are nearly Gaussian.
The negative skewness of zonal winds in the Southern Ocean was noted by Sura (2003), who attributed the skewness to the presence of multiplicative noise (that is, noise with state-dependent intensity) in an empirical stochastic equation for the zonal wind. In the next two sections, we will demonstrate that while the dynamics of the winds may involve multiplicative noise (in an appropriate limit, to be defined in section 5), the skewness of the wind components and the relationship between the skewness and the mean do not require multiplicative noise, but can be understood as a consequence of the nonlinear drag law characteristic of turbulent boundary layers.
4. A minimal theory for the skewness of sea surface wind components
The sea surface momentum roughness length z0 is determined by the sea state, which is affected both by remotely generated swell and by surface waves generated by local winds. Consequently, z0 is a function of the local wind speed |u|, and the drag coefficient will depend on |u| even in the neutral (L → ∞) limit in which buoyant effects on turbulence are negligible. The precise form of the dependence of the neutral drag coefficient on swell and local winds is a field of active research, and a number of different formulations have been proposed (e.g., Mahrt et al. 1996; Rieder 1997; Taylor 2000; Grachev and Fairall 2001; Taylor and Yelland 2001; Fairall et al. 2003; Mahrt et al. 2003). These formulations differ in detail, but there is a general consensus that the drag coefficient increases with |u|, except perhaps for very light winds.
Evidently, fluctuations in u and υ can result from fluctuations in
the pressure gradient forces,
the drag coefficient cD,
the wind components U and V above the surface layer, and
the size of the viscosity κ.
Figure 5 displays scatterplots of the mean versus the skewness of the simulated wind components for different parameter values. Clearly, the mean and skewness of the simulated wind components are anticorrelated, with a linear dependence that matches favorably that illustrated in Fig. 3. Note that the skewness is a result of nonlinearities in the drift term, and not of multiplicative noise as suggested by Sura (2003); by assumption, the intensities of fluctuations in
In this analysis, we have neglected the dependence of the neutral drag coefficient on the wind speed. Several numerical integrations of Eq. (31) were carried out in which
An intuitive understanding of the relationship between the mean and skewness of the surface wind components is straightforward. Consider the atmospheric surface layer subject to zonal forcing Πu with zero mean and with fluctuations that are equally as likely to be positive as negative. By symmetry, the mean and skewness of the resulting zonal wind will both be zero. Now suppose that the zonal forcing Πu has a non-zero mean, which we will take (without loss of generality) to be positive, so that the mean u will be positive. Because of the nonlinear surface drag, positive anomalies in u will be subject to stronger friction than negative anomalies. Consequently, a positive perturbation in the forcing will produce a weaker response than a negative perturbation of the same magnitude, and the symmetric fluctuations will produce an asymmetric response. In particular, a tail toward slower zonal winds will develop in the distribution, so the zonal wind will be negatively skewed. As the mean of Πu increases, the asymmetry in drag between positive and negative u anomalies will also increase, and so the skewness will become more strongly negative. By the same token, negative mean Πu will yield negative mean zonal winds that are positively skewed, with the skewness becoming more positive as the mean of u becomes more negative. In this manner, the nonlinear surface drag results in the anticorrelation of the mean and skewness fields of zonal surface winds. A corollary of this argument is that for a given mean zonal forcing, the skewness of u should decrease as the variance of u decreases, because the difference in drag between positive and negative fluctuations of typical (e.g., unit standard deviation) magnitude will decease. In fact, this behavior is observed in Fig. 5. The scatter of points with relatively low skewness for relatively high wind speed correspond to relatively low standard deviations. This effect explains the greater scatter in skewness for easterly zonal winds than for westerly zonal winds evident in Fig. 3. In some parts of the Tropics (e.g., the eastern tropical Pacific), surface zonal winds are subject to mean forcings of about the same magnitude as surface winds in the midlatitudes, but of considerably weaker variability. Consequently, these are regions of relatively weak skewness. The preceding arguments hold of course for meridional winds as well.
It is admittedly naive to assume that
The strong linear relationship between the mean and skewness of the surface wind components has been shown to be a consequence of the nonlinearity of the surface drag, which follows from well-established similarity theories for turbulent boundary layers. A number of previous studies of the momentum balance of monthly averaged surface winds (e.g., Deser 1993; Ward and Hoskins 1996; Chiang and Zebiak 2000) have employed a surface drag law in which the surface stress varies linearly with wind speed to represent the combined effects of surface friction and downward mixing of momentum into the surface layer. While such a linear drag law may be appropriate for the equilibrium balance of spatially and temporally averaged winds, it is clearly inadequate for understanding the character of nonequilibrium fluctuations in the winds on the daily and shorter time scales that play important roles in turbulent transfers between the ocean and the atmosphere.
5. Stochastic dynamics of sea surface winds
A recent effort to characterize the variability of sea surface winds is the study of Sura (2003), in which the drift and diffusion functions of SDEs for the zonal and meridional winds were estimated empirically from 6-hourly blended QuikSCAT–NCEP analysis data. Sura assumed that the zonal and meridional components of the wind can be described individually by one-dimensional stochastic differential equations, that is, that the 6-hourly zonal and meridional winds are individually Markov processes. However, the discussion in the previous section suggests that this assumption is inconsistent with dynamical balances in the surface layer when the forcings Πu and Πυ decorrelate on time scales of the same order as the decorrelation time scale of the winds. Given that marine boundary layers evolve slowly relative to terrestrial boundary layers, it is also to be expected that the decorrelation time scales of Γ and K will not be much less than 6 h. In fact, Rieder (1997) found that changes in the drag coefficient follow changes in surface wind speed with a lag of about 4 h, suggesting that the turbulent marine boundary layer has an adjustment scale on the order of several hours. Furthermore, u and υ are coupled frictionally, and even in the limit of white-noise fluctuations in Πu, Πυ, Γ, and K, neither u nor υ is individually Markov. Nevertheless, it is instructive to consider the limit of Eq. (31) in which these fluctuations become white relative to those of the winds, and to compare the resulting system to the results of the empirical analysis of Sura (2003). For the sake of simplicity, we will neglect the dependence of the drag coefficient on the wind speed.
Sura (2003) attributes the quadratic dependence of Δ(u) on u to burstiness in the turbulent boundary layer, suggesting that as midlatitude wind speeds increase, so too does the variability. In fact, there is no evidence of this behavior in the 6-hourly winds. Figure 7 displays a scatterplot of the absolute value of the 6-hourly zonal wind increment, |u(t + 6 h) − u(t)|, as a function of [u(t + 6 h) + u(t)]/2 for winds at the representative point (57°S, 0°). It is evident that instead of the 6-h wind increments being larger at larger wind values, they are generally smaller. Inspection of similar plots at different grid points indicates that this relationship is independent of latitude and longitude. The increase in wind variability at higher wind speeds may be a feature of the winds at shorter time scales, but there is no evidence of this behavior at the 6-hourly time scales used in Sura (2003) to estimate the SDE parameters. The previous analysis suggests that multiplicative noise in the τγ → 0 limit arises not from burstiness in the winds, but from turbulent fluctuations in the magnitude of the drag coefficient.
6. Conclusions
This study has demonstrated the existence of a strong spatial anticorrelation between the mean and skewness fields of surface wind components over the ocean, and has shown that this relationship can be understood as a consequence of the nonlinear surface drag predicted by Monin–Obukhov similarity theory for a turbulent boundary layer. As this relationship is evident in the NCEP–NCAR and ERA-40 reanalysis products, in blended QuikSCAT scatterometer–NCEP analysis data, in SSM/I satellite data, and in TAO buoy data, it is not simply an artifact of the observing system or the reanalysis algorithm. Furthermore, it has been demonstrated that the surface layer momentum dynamics subject to fluctuation forcing are consistent with the results of the empirical study by Sura (2003), although the interpretation of the results is different. In particular, it is suggested that the presence of multiplicative noise does not arise from increasing gustiness with higher wind speeds, as suggested by Sura, but from fluctuations in the turbulent drag.
A number of approximations were used in this study that cannot be expected to hold in the real world. First, the downward mixing of momentum into the surface layer, which has been ignored in this study, has been demonstrated to be a potentially significant component of the surface layer momentum budget in the Tropics (e.g., Deser 1993; Chiang and Zebiak 2000). Second, advection terms in the momentum budget have been ignored; while this may be justifiable for the equilibrium momentum budget of monthly averaged winds (Deser 1993), it may be important during periods of strong winds on shorter time scales. Third, Monin–Obukhov similarity theory predicts that buoyancy effects in a nonneutral boundary layer introduce Richardson number– dependent correction terms to the drag coefficient. Fourth, it has been assumed that all fluctuating quantities in the momentum equations (Πu, Πυ, Γ, and K) can be represented as independent Gaussian random variables; as is described in section 4, there are compelling reasons why this should not be true. More complex models of boundary layer turbulence could be used to develop a better understanding of the fluctuations in the drag coefficient and the eddy viscosity, and the relationship of these distributions to the surface winds. Such an extension of the present study would be necessary for the detailed quantitative analysis of the relationships between the moments of sea surface wind components. Finally, the statistics of Πu, Πυ, γ, and μ have been assumed to be stationary, when in fact they will display both diurnal and annual variability (Deser 1994). Nonetheless, the model presented in this study is able to reproduce the essential features of the relationship between the mean and skewness of surface winds, and is broadly consistent with the empirical results of Sura (2003). An investigation of the reasons for those differences that do exist between the results of the present study and those of Sura (2003) are an interesting direction of future study. Thus, while the simplifying approximations may limit the model's quantitative accuracy, they do not seem to have significantly degraded its qualitative utility.
Previous studies of the oceanic response to fluctuating atmospheric forcing (e.g., Moore 1999; Sura et al. 2001; Monahan 2002a; Kuhlbrodt and Monahan 2003) have assumed Gaussian fluctuations, while the results presented in the present study demonstrate that in the Tropics and midlatitudes the surface wind components are generally skewed. In particular, a leading theory of El Niño–Southern Oscillation variability posits that the tropical Pacific coupled atmosphere–ocean system acts as a damped linear oscillator subject to Gaussian perturbations (e.g., Penland 1996; Kleeman and Moore 1997; Thompson and Battisti 2000). Such linear models cannot account for the marked asymmetry of the magnitude and structure of sea surface temperature anomalies between El Niño and La Niña events that has been noted in observations (e.g., Monahan 2001; Hannachi et al. 2003; An and Jin 2004; Monahan and Dai 2004). This asymmetry is generally ascribed to nonlinearities in ocean dynamics, but may also reflect the strong positive skewness of the zonal wind in the western equatorial Pacific. An interesting question is the extent to which the strength of El Niño events relative to La Niña events simply reflects the fact that westerly wind bursts in the western equatorial Pacific are not matched by easterly wind bursts.
A natural extension of the present study would be an investigation of the dependence on time scale of the relationship between the mean and skewness of surface wind components (e.g., the relationship between monthly averaged winds and the skewness of submonthly fluctuations), and the incorporation of this information into eddy flux closure schemes. In particular, such a study could provide important constraints on the development of stochastic parameterization schemes, which have been proposed as improvements to the representation of subgrid-scale phenomena in models of the climate system (Palmer 2001; Imkeller and Monahan 2002). Errors in the representation of turbulent fluxes between the ocean and the atmosphere are a potentially significant source of systematic errors in global climate models, and the development of more accurate parameterization schemes is an important aspect of improving our understanding of this complex coupled system.
Acknowledgments
The author acknowledges support from the Natural Sciences and Engineering Research Council of Canada, by the Canadian Foundation for Climate and Atmospheric Sciences, and by the Canadian Institute for Advanced Research Earth System Evolution Program. The author is grateful to Norm McFarlane, Ken Denman, and Alexandra Guerrero-Martinez for helpful discussions. The author would like to thank the three anonymous referees whose thoughtful comments considerably improved this manuscript.
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Mean, std dev, and skewness fields of NCEP–NCAR reanalysis zonal winds
Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<2037:ASMFTS>2.0.CO;2
Mean, std dev, and skewness fields of NCEP–NCAR reanalysis meridional winds
Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<2037:ASMFTS>2.0.CO;2
Grid point by grid point scatterplots of the mean vs the skewness of the zonal and meridional sea surface wind components for the NCEP–NCAR reanalysis data (red dots), the blended scatterometer–NCEP analysis data (black dots), and the TAO buoy data (green circles)
Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<2037:ASMFTS>2.0.CO;2
Kernel density estimates of the probability density functions of the zonal wind, u, at 50°S, 110°W; 20°N, 160°W; and 50°N, 60°W
Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<2037:ASMFTS>2.0.CO;2
Scatterplots of the mean vs the skewness of the zonal and meridional components of the simulated wind components.
Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<2037:ASMFTS>2.0.CO;2
Drift and diffusion functions corresponding to the system Eq. (48) that would be estimated using the methodology of Sura (2003)
Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<2037:ASMFTS>2.0.CO;2
Scatterplot of the absolute 6-h increment of the zonal wind, |u(t + 6 h) − u(t)|, as a function of the zonal wind strength [u(t + 6 h) + u(t)]/2 at 57°S, 0°
Citation: Journal of the Atmospheric Sciences 61, 16; 10.1175/1520-0469(2004)061<2037:ASMFTS>2.0.CO;2
Spatial correlation coefficients between the mean and skewness fields of the zonal and meridional winds from the NCEP– NCAR reanalysis data, the ERA-40 reanalysis, the blended Quik SCAT–NCEP analysis data, the SSM/I passive microwave data, and the TAO buoy data
Stochastic boundary layer model parameter ranges estimated from NCEP–NCAR reanalysis data