A Simple Model for the Skewness of Global Sea Surface Winds

Transcription

1 037 A Simple Model for the Skewness of Global Sea Surface Winds ADAM HUGH MONAHAN School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, and Earth Systems Evolution Program, Canadian Institute for Advanced Research, Toronto, Ontario, Canada (Manuscript received October 003, in final form 9 February 004) ABSTRACT A strong linear relationship between the mean and skewness of global sea surface winds (both zonal and meridional) is shown to exist, such that where the wind component is on average positive, it is negatively skewed (and vice versa). This relationship is observed in reanalysis, satellite, and buoy data. This relationship between the mean and skewness fields of sea surface winds follows from the nonlinear surface drag predicted for a turbulent boundary layer by Monin Obukhov similarity theory since forcing perturbations speeding the wind up are subject to a stronger drag force than perturbations slowing it down. Furthermore, it is demonstrated that the results of an empirical fit of observed surface winds to a stochastic differential equation presented in a recent study by Sura are consistent with the white-noise limit of the momentum equations for a turbulent boundary layer subject to fluctuating forcing, albeit with a somewhat different physical interpretation.. Introduction Interactions between the ocean and the atmosphere are mediated through their respective boundary layers, such that the character of the turbulence in these boundary layers determines the rate of exchange of momentum, of energy, and of material substances such as water vapor, aerosols, and gases. Classical similarity theory predicts that the character of the turbulence in the atmospheric boundary layer can be simply related to the density stratification and the vertical shear of the wind, each averaged over a typical eddy time scale of h or so (Stull 997). This relationship can then be used to develop bulk parameterizations of turbulent air sea fluxes as functions of the eddy-averaged winds at a given altitude (typically 0 m). Denoting the eddy-averaging of the variable x by an overbar with an e, x e, standard bulk parameterizations for the turbulent flux of a quantity across the ocean atmosphere interface take the form e e e w F( U, ), () where U is the wind speed and F is a (typically nonlinear) function which will generally depend on the stability of the boundary layer and on the sea state (Taylor 000); this is discussed in more detail in section 4. It is quite often the case, however, that only observations of boundary layer variables averaged over time Corresponding author address: Adam Hugh Monahan, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3055, STN CSC, Victoria, BC V8P 5C, Canada. monahana@uvic.ca scales longer than the eddy time scale are available for estimating air sea fluxes, in which case a time-averaged version of Eq. () must be used. Denoting averaging of the variable x over the time scale T (assumed to be longer than the eddy-averaging time scale) by x T, the T-averaged flux will not equal the flux corresponding to the T-averaged variables if the function F is nonlinear (which it typically is): in general, e e e e e w T F( U, )T F( U T, T). () The importance of time-averaging on estimates of air sea momentum fluxes has been demonstrated by (for example) Wright and Thompson (983), Mahrt et al. (996), and Ponte and Rosen (004), and for air sea CO fluxes by Bates and Merlivat (00). The T-averaged eddy fluxes can, however, be represented in terms of the T-averaged boundary layer quantities plus correction terms involving higher-order moments of fluctuations on time scales shorter than T. Using a Taylor series expansion, we obtain e e e e F( U, )T F( U T, T) F e e e ( U U T) T! U T 3 F e e 3 e ( U U T) 3 T 3! U T, (3) where the remaining terms represent higher-order moments of U, moments of, and cross moments of e e e e e e U and.if F is a polynomial in U and, 004 American Meteorological Society

2 038 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 6 then an expansion of (3) will be exact with a finite number of terms; otherwise, it will be approximate. If the higherorder moments in the expansion (3) can be related to the T-averaged quantities, then workable closures for the eddy fluxes in terms of the T-averaged variables can be derived. This paper demonstrates the existence of a strong linear relationship between the mean and the skewness (normalized third-order moment) of wind components (both zonal and meridional) in the marine boundary layer, such that the more positive the mean wind component is at a given location, the more negative is the skewness (and vice versa). Furthermore, it will be shown that this relationship follows naturally from the nonlinear surface drag predicted by Monin Obukhov similarity theory for a turbulent boundary layer. 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 996; Kleeman and Moore 997; Thompson and Battisti 000), of midlatitude gyre variability (e.g., Moore 999; Sura et al. 00), of the stability of the meridional overturning circulation (e.g., Aeberhardt et al. 000; Monahan 00a,b), and of variability in open-ocean deep convection (e.g., Kuhlbrodt et al. 00; Kuhlbrodt and Monahan 003). In a recent study, Sura (003) 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 (003a,b) for a discussion]. This result was interpreted as reflecting increasing gustiness at stronger wind speeds. We will demonstrate that the results of Sura (003) 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. 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 (003). Finally, a discussion and conclusions are presented in section 6.. 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.5.5 grid from January 948 to 3 December 00 (Kalnay et al. 996). 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, cdc.noaa.gov/.] ) Six-hourly European Centre for Medium-Range Weather Forecasts (ECMWF) 40-yr reanalysis period (ERA-40) 0-m zonal and meridional winds, available on a.5.5 grid from September 957 to 3 August 00 (Simmons and Gibson 000). (Data available online at int/data/d/era40/.) 3) Six-hourly 0-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 grid from 9 July 999 to 3 March 003 (Chin et al. 998; Milliff et al. 999). [Data available online at the NCAR Data Support Section (DSS), ds744.4/.] 4) Variationally gridded six-hourly Special Sensor Microwave Imager (SSM/I) 0-m zonal and meridional wind observations, available on a grid from July 987 to 3 December 00 (Atlas et al. 996). [Data available online at the NASA Jet Propulsion Laboratory (JPL) Distributed Active Archive Center, 5) Hourly 4-m zonal and meridional winds from 65 buoys in the Tropical Atmosphere Ocean (TAO) array (McPhaden et al. 998). (Data available online at the TAO Project Office at noaa.gov/tao/datadeliv/.) Wind data were reported from these buoys intermittently between 990 and 00; the number of hourly measurements at individual buoys varied from a minimum of 4000 to more than No preprocessing, such as filtering or removing the annual cycle, was carried out on any of these datasets. 3. Moments of surface ocean winds The mean, standard deviation, and skewness fields of the sea surface zonal and meridional wind components from the NCEP NCAR reanalysis are presented in Figs. and, respectively. The skewness of a random variable x is the normalized third-order moment, 3 (x x) skewness(x), (4) 3/ (x x) where angle brackets denote long-term time aver-

3 039 FIG.. Mean, std dev, and skewness fields of NCEP NCAR reanalysis zonal winds. aging. The mean sea surface zonal wind field displays the familiar tropical easterlies and midlatitude westerlies, while the meridional wind field shows strong flow only on the eastern flanks of the subtropical highs. The standard deviation fields of both zonal and meridional winds display variability minima in the Tropics and subtropics, and variability maxima in the storm tracks. The skewness field of the zonal wind is generally positive in the Tropics and negative in the midlatitudes, while that of the meridional wind is generally small except along the eastern flanks of the subtropical highs, where skewness is positive in the Northern Hemisphere and negative in the Southern Hemisphere. A linear relationship between the mean and skewness fields of both the zonal and meridional sea surface wind components is evident from Figs. and : 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 ; these correlation coefficients display essentially no seasonal variability. Zonal wind skewness is seen to range between approximately and.5; skewness toward the upper

4 040 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 6 FIG.. Mean, std dev, and skewness fields of NCEP NCAR reanalysis meridional winds. FIG. 3. 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).

5 04 TABLE. 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. Dataset Zonal Meridional NCEP NCAR ERA-40 QuikSCAT SSM/I TAO and lower end of this range is strong enough to be manifest in the probability density functions. Kernel density estimates (Silverman 986) of the probability density functions of the zonal wind from the NCEP NCAR reanalysis at three representative grid points [(50S, 0W), (0N, 60W), and (50N, 60W)] 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 (50N, 45E) evident in Figs. and, which is consistent with the observation in Monahan and Denman (004) 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 (003), 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 Neglecting advection terms, the horizontal momentum equations averaged over a layer from the surface to the altitude h read u p f [ x(h) x(0)], (5) a cos h p fu [ y(h) y(0)], (6) a h where u and are respectively the depth-averaged zonal and meridional components of the wind, and are respectively the latitude and the longitude, a is the radius of the earth, is the air density, p is the pressure, f is the Coriolis parameter, and x and y are respectively the zonal and meridional components of the stress vector. Monin Obukhov similarity theory for the boundary layer (Haltiner and Williams 980) implies that these equations may be expressed as c D / u p f (u ) u a cos h x(h), (7) h cd / p fu (u ) y(h), (8) a h h where [ ] 0 za z a cd k 0 ln m (9) z L is the drag coefficient. The drag coefficient is a function of the reference altitude z a (typically 0 m), of the momentum roughness length z 0, and of the stability of the boundary layer through the term z/l z m m L z /L [ ( )] d ln, (0) 0 where L is the Obukhov length and m is a dimensionless function characterizing buoyant effects on boundary layer turbulence. The Obukhov length L is a function of the local wind speed u (u ) / and the surface buoyancy flux. The von Karman constant k 0 is a fixed number. Note that it has been assumed that the wind at the reference level z a is approximately equal to the wind averaged from z 0tozh. The sea surface momentum roughness length z 0 is determined by the sea state, which is affected both by remotely generated swell and by surface waves generated by local winds. Consequently, z 0 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. 996; Rieder 997; Taylor 000; Grachev and Fairall 00; Taylor and Yelland 00; Fairall et al. 003; Mahrt et al. 003). These formulations differ in detail, but there is a general consensus that the drag coefficient increases with u, except perhaps for very light winds. The simplest parameterization for ( x, y )atz h assumes an eddy viscosity so that d [ (h), (h)] (u*, *), () dz x y zh where (u*, *) is the (nondepth-averaged) wind vector. Equation () can be approximated using finite differences as

6 04 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 6 FIG. 4. Kernel density estimates of the probability density functions of the zonal wind, u, at 50S, 0W; 0N, 60W; and 50N, 60W. [ x(h), y(h)] (U u, V ), () h where U and V are, respectively, the zonal and meridional winds in the atmospheric layer above z h. Evidently, fluctuations in u and can result from fluctuations in the pressure gradient forces, the drag coefficient c D, the wind components U and V above the surface layer, and the size of the viscosity. Variability in U, V, and the pressure gradients will reflect variations in weather above the surface layer. The drag coefficient will fluctuate with changes in stability of the boundary layer (as measured by the Obukhov length L) and the surface wave field (as measured by the roughness length z 0 ). Similarly, will vary depending on the character and intensity of the turbulence in the boundary layer. For convenience, we define K and (3) h u p f KU, (4) a cos p fu KV, (5) a so that / u u(u ) u Ku, (6) / (u ) K, (7) where c D. (8) h As already noted, the statistics of u and will be determined by the fluctuations in u,,, and K; we will model these fluctuations as stochastic processes. The simplest stochastic representation of (6), (7) is obtained by representing u,,, and K as independent Ornstein Uhlenbeck (red noise) processes: where (t) (t), (9) u u u (t) (t), (0) [ (t)], () K K[ (t)], () k 0 (3) h[ln(z a/z 0)] is the neutral (L ) drag coefficient (which may depend on u through z 0 ) divided by h. The fluctuations are assumed to be governed by the stochastic differential equations (SDEs) u W u u, (4) u u W, (5) W 3, (6) W 4, (7) where ( u,,, ) are relaxation time scales, ( u,,, ) tune the strength of the fluctuations, and (Ẇ, Ẇ, Ẇ 3, Ẇ 4 ) are independent white-noise processes: W (t )W (t ) (t t ), i j ij (8) where ij is the Kronecker delta and (t t ) is the Dirac delta function. A basic introduction to stochastic differential equations is given in Penland (003a); more detailed expositions appear in Penland (003b) and Gardiner (997). Note that the Ornstein Uhlenbeck process z described by the stochastic differential equation ż z W (9) will (after initial transients have died out) have a mean of 0, the stationary autocovariance function

7 043 z(t )z(t ) exp( t t /), (30) and will converge to the white-noise process Ẇ in the limit that 0. Together, Eqs. (6) (7) define a stochastic differential equation for the six-dimensional Markov process X (u,, u,,, ): X A(X) BW, (3) where / u( u )(u ) u K( )u / ( )(u ) K( ) u u A, u (3) / u u B, 0 / / / and (33) Ẇ W Ẇ. (34) Ẇ 3 Ẇ 4 In the terminology of SDEs, the deterministic vector A(X) is known as the drift and the matrix B multiplying the white-noise vector is known as the diffusion. As the matrix B is independent of the state variable X, the noise is said to be additive. To demonstrate that the relationship between the mean and skewness of sea surface wind components described in the previous section follows from Eq. (3), the statistics of u and were estimated from the NCEP NCAR reanalysis data as follows. First, the pressure gradient fields were estimated using centered finitedifference approximations to the derivatives, and surface air density was calculated from sea level pressure and surface air temperature. Next, the u and fields were calculated using Eqs. (4) and (5) and the NCEP NCAR 995-hPa u and fields. At each grid point; the sample mean and lag autocovariance functions of u and were estimated, yielding u, u,, and. TABLE. Stochastic boundary layer model parameter ranges estimated from NCEP NCAR reanalysis data. Parameter Min Max u (m s ) u (m s 3/ ) (m s ) (m s 3/ ) The noise strength parameters u and were then estimated from the standard deviation fields (std) of u and using Eq. (30) with t t : std( ), (35) u u u std( ). (36) Note that it follows from Eqs. (35) and (36) that the units of u and arems 3/. As it was found that the autocorrelation times u and were generally on the order of day, a constant value of u day (37) was used in these and all subsequent calculations. Because finite differences of the NCEP NCAR reanalysis surface pressure field are coarse approximations of the derivatives, estimates of the mean forcing fields u and were noisy and uncertain. Estimates of u and were more robust. Upper and lower limits of the estimates of these parameters are listed in Table. For the sake of simplicity, the effect of downward mixing of momentum was ignored by setting K 0, (38) and a constant drag coefficient was assumed: and (39) 80 m 0, (40) where the layer depth h 80 m corresponds to the average thickness of the lowest level of the NCEP NCAR reanalysis model. The numerical integration of Eq. (3) was carried out using a forward-euler scheme, with parameters u,, u, and sampled evenly through the ranges indicated in Table. The numerical integration of stochastic differential equations is described in Kloeden and Platen (99); in brief, for a timestep t, the forward-euler numerical recursion is X X ta(x ) tb, (4) k k k k where k is a vector of mutually independent zero-mean, unit-variance Gaussian random variables generated independently at each time step: T i jij I, (4) where I is the 4 4 identity matrix. Figure 5 displays scatterplots of the mean versus the skewness of the simulated wind components for dif-

8 044 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 6 FIG. 5. Scatterplots of the mean vs the skewness of the zonal and meridional components of the simulated wind components. ferent 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 (003); by assumption, the intensities of fluctuations in u and in this model are independent of the state of the boundary layer. In this analysis, we have neglected the dependence of the neutral drag coefficient on the wind speed. Several numerical integrations of Eq. (3) were carried out in which was taken to depend on u using various formulations described in Taylor (000) and Fairall et al. (003). In none of these integrations was the relationship between the mean and skewness fields of the surface wind components found to be significantly different from that displayed in Fig. 5 for a drag coefficient independent of wind speed. The surface wind speed dependence of the neutral drag coefficient does not appear to play a leading-order role in the relationship between the mean and skewness fields of u and. 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 u 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 u 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 u 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 u 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 u,,, and K should be independent Ornstein Uhlenbeck processes. First, it cannot be expected that u,,, and K should have Gaussian probability distributions and exponential temporal autocorrelation functions. In particular, u and estimated from the NCEP NCAR reanalyses are themselves non-gaussian. However, the spatial distribution of the skewness of u bears no obvious relation to that of u (and similarly for and ), so the skewness of the wind components is not simply a result of skewed forcing fluctuations. Furthermore, the Obukhov length L and the eddy viscosity K are to some extent determined by the boundary layer wind speed, which itself is affected by pressure gradient

9 045 forces. Consequently, fluctuations in and K should not be independent of those in u and. The choice to represent these fluctuations by independent Ornstein Uhlenbeck processes was not based on any physical motivation, but on the desire to have this representation be as simple as possible. The simplicity of the system (3) and its success in characterizing the relationship between the mean and skewness of surface wind components indicates that the approximations lead to a model that is qualitatively useful. 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 993; Ward and Hoskins 996; Chiang and Zebiak 000) 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 (003), 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 u and 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 and K will not be much less than 6 h. In fact, Rieder (997) 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 are coupled frictionally, and even in the limit of white-noise fluctuations in u,,, and K, neither u nor is individually Markov. Nevertheless, it is instructive to consider the limit of Eq. (3) 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 (003). For the sake of simplicity, we will neglect the dependence of the drag coefficient on the wind speed. In the limit that u,,, 0, u,,, and become white-noise processes, as described in section 4. By the Wong Zakai theorem (Penland 003b), the six-dimensional stochastic differential equation (3) converges to a stochastic differential equation for the two-dimensional process u (u, ): U C(U) D(U) W, (43) where / u (u ) u K u C(U), and (44) / (u ) K / u (u ) u Ku 0 D(U). (45) / 0 (u ) K Note that in this limit the SDE involves products of state variables and white-noise processes; the corresponding noise terms are said to be multiplicative (Gardiner 997). The open circle in Eq. (43) indicates that the multiplicative noise terms are to be interpreted in the Stratonovich sense, as is appropriate for the whitenoise limit of a process with a non-zero autocorrelation time (Gardiner 997; Penland 003b). The convergence of Eq. (3) to (43) is a special case of the general convergence problem considered in Majda et al. (00, 003) for the simple situation in which the dynamics of the fast variables ( u,,, ) do not depend on the slow variables u and (although of course the dynamics of the slow variables depend on the fast variables). In the present simple case, the theorem of Wong and Zakai is sufficient to characterize the white-noise limit of the SDE (3). The equivalent Itô drift I C (U) / u (u ) u Ku (u )u K u / (u ) K (u ) K (46)

10 046 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 6 is obtained through the standard transformation I C C D D, (47) i i kj k ij j,k where k for k and denotes partial differentiation with respect to u and, respectively. The difference between Itô and Stratonovich representations of SDEs arises in the presence of multiplicative noise, for which the state dependence of fluctuations can change the average dynamics of the system. Penland (003a) presents an intuitive description of the origins of this noiseinduced drift; technical details are given in Penland (003b), Gardiner (997), and Kloeden and Platen (99). We need to consider the Itô drift C I (U) because, as we will see, the drift and diffusion functions calculated by the algorithm used in Sura (003) are those of an Itô SDE. In the limit that fluctuations in are significantly smaller than those in u, to a good approximation u itself can be described by a Markov process described by the Itô SDE I u C (u) D W D (u)w D (u)w 3 3, (48) with the associated Fokker Planck equation for the probability density function p(u): I T tp [C u (u)p] u(dd p), (49) where D [D D (u) D 3(u)] and (50) D T DD [D D (u) D 3(u)] D (u) D (u) u (u )u K u. (5) Sura (003) assumes that the zonal wind u is a Markov process governed by an Itô SDE of the form u (u) (u)w, (5) where it has been assumed that only a single whitenoise process enters the SDE, so the diffusion is a scalar function of u. This equation is similar in form to the scalar equation (48) for the evolution of u, which we have shown follows from the momentum equations in a turbulent boundary layer in the limit of white-noise forcing and vanishing variability in. Equations (48) and (5) differ, however, in that three independent white-noise processes are present in (48) so the diffusion isa 3 matrix. The algorithm used in Sura (003) to estimate the parameters of the SDE (5) for the zonal wind takes advantage of the fact that the Itô SDE 3 FIG. 6. Drift and diffusion functions corresponding to the system Eq. (48) that would be estimated using the methodology of Sura (003). ẏ E(y) F(y)W (53) (where y is an n-dimensional vector, Ẇ is an m-dimensional white-noise vector, and F(y) isann m matrix function of y) is associated with the Fokker Planck equation for the joint probability density function p(y): p(y) [E p(y)] [FFT t i i i j p(y)] (54) i and thus that the product FF T can be estimated from the limit T F(y 0)F(y 0) lim [y(t t) y 0] t 0 t T [y(t t) y 0] y(t)y 0. (55) Thus, in Sura s methodology, the diffusion matrix F is not estimated directly, but only through the product FF T. Sura notes the nonuniqueness of F satisfying Eq. (55) for n, and asserts that this nonuniqueness vanishes (up to a sign) for the scalar case, n. This is true if m, that is, if only a single white-noise process enters the SDE for the scalar process, but is certainly not true if m, which the preceding analysis suggests is the case relevant to the stochastic dynamics of sea surface winds. In fact, the estimation algorithm presented in Sura (003) does not provide a methodology to estimate the number m of white-noise processes entering the SDE (53), and thus the matrix F cannot be uniquely determined. Note that an SDE with m and diffusion i,j T / u (u) DD [ (u )u K u ], (56) which is approximately quadratic in u, would produce the same Fokker Planck equation as the SDE (48). Consequently, (u) would be the diffusion estimated by the method of Sura (003) from a dataset produced by the I SDE (48). Figure 6 shows plots of the drift C and effective diffusion as functions of u, calculated using characteristic values u ms, u 0.5 ms 3/, K 0, 3ms, 30 s / (corresponding to a 0% standard deviation for fluctuations of with a decorrelation time scale of 3 h). The drift and effective diffusion functions displayed in Fig. 6 bear a striking resemblance to those estimated in Sura (003). In particular, the effective diffusion varies approximately quadratically with u and the drift function decreases monotonically with u and displays inflection points on either

11 047 side of the zonal wind value u eq which balances the pressure gradient, Coriolis, and frictional forces. The physical interpretation of the drift and diffusion coefficients shown in Fig. 6 is that the deterministic component of the dynamics acts to drive the zonal wind to the balanced value u eq while the stochastic component drives the zonal wind away from this equilibrium value, such that typical fluctuations become larger with increasing u. Interestingly, the diffusion functions estimated in Sura (003) have minima at nonzero values of u; in contrast, the minimum of the effective diffusion term (u) is necessarily at u 0. This discrepancy may be a consequence of the simplifying assumptions we have used to obtain the SDE (48), or it may be an artifact of Sura s estimation algorithm. In particular, the removal of the seasonal cycle from the winds in Sura (003) before estimating the drift and diffusion coefficients could lead to biases in the estimated drift and diffusion functions, as the drag is a function of the total wind speed and not its standardized anomaly. Nevertheless, while the agreement between the drift and diffusion functions estimated by Sura (003) and those arising from boundary layer dynamics is not perfect, it is remarkably good. Sura (003) attributes the quadratic dependence of (u) onu 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 6h)u(t), as a function of [u(t 6h)u(t)]/ for winds at the representative point (57S, 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 (003) to estimate the SDE parameters. The previous analysis suggests that multiplicative noise in the 0 limit arises not from burstiness in the winds, but from turbulent fluctuations in the magnitude of the drag coefficient. FIG. 7. 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 6h) u(t)]/ at 57S, 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 (003), 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 993; Chiang and Zebiak 000). 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 993), 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 ( u,,, 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 u,,, and have been assumed to be stationary, when in fact they will display

12 048 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 6 both diurnal and annual variability (Deser 994). 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 (003). An investigation of the reasons for those differences that do exist between the results of the present study and those of Sura (003) 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 999; Sura et al. 00; Monahan 00a; Kuhlbrodt and Monahan 003) 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 996; Kleeman and Moore 997; Thompson and Battisti 000). 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 00; Hannachi et al. 003; An and Jin 004; Monahan and Dai 004). 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 00; Imkeller and Monahan 00). 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. REFERENCES Aeberhardt, M., M. Blatter, and T. F. Stocker, 000: Variability on the century time scale and regime changes in a stochastically forced zonally averaged ocean atmosphere model. Geophys. Res. Lett., 7, An, S.-I., and F.-F. Jin, 004: Nonlinearity and asymmetry of ENSO. J. Climate, 7, Atlas, R., R. Hoffman, S. Bloom, J. Jusem, and J. Ardizzone, 996: A multiyear global surface wind velocity dataset using SSM/I wind observations. Bull. Amer. Meteor. Soc., 77, Bates, N. R., and L. Merlivat, 00: The influence of short-term wind variability on air sea CO exchange. Geophys. Res. Lett., 8, Chiang, J. C., and S. E. Zebiak, 000: Surface wind over tropical oceans: Diagnosis of the momentum balance, and modeling the linear friction coefficient. J. Climate, 3, Chin, T., R. Milliff, and W. Large, 998: Basin-scale, high-wavenumber sea surface wind fields from a multiresolution analysis of scatterometer data. J. Atmos. Oceanic Technol., 5, Deser, C., 993: Diagnosis of the surface momentum balance over the tropical Pacific Ocean. J. Climate, 6, , 994: Daily surface wind variations over the equatorial Pacific Ocean. J. Geophys. Res., 99 (D), Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 003: Bulk parameterization of air sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 6, Gardiner, C. W., 997: Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Springer, 44 pp. Grachev, A. A., and C. W. Fairall, 00: Upward momentum transfer in the marine boundary layer. J. Phys. Oceanogr., 3, Haltiner, G. J., and R. T. Williams, 980: Numerical Prediction and Dynamic Meteorology. Wiley, 477 pp. Hannachi, A., D. Stephenson, and K. Sperber, 003: Probability-based methods for quantifying nonlinearity in the ENSO. Climate Dyn., 0, Imkeller, P., and A. H. Monahan, 00: Conceptual stochastic climate models. Stochast. Dyn.,, Kalnay, E., and Coauthors, 996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77, Kleeman, R., and A. Moore, 997: A theory for the limitation of ENSO predictability due to stochastic atmospheric transients. J. Atmos. Sci., 54, Kloeden, P. E., and E. Platen, 99: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, 63 pp. Kuhlbrodt, T., and A. H. Monahan, 003: Stochastic stability of openocean deep convection. J. Phys. Oceanogr., 33, , S. Titz, U. Feudel, and S. Rahmstorf, 00: A simple model of seasonal open ocean convection. Part II: Labrador Sea stability and stochastic forcing. Ocean Dyn., 5, Mahrt, L., D. Vickers, J. Howell, J. Højstrup, J. M. Wilczak, J. Edson, and J. Hare, 996: Sea surface drag coefficients in the Risø Air Sea Experiment. J. Geophys. Res., 0, ,, P. Frederickson, K. Davidson, and A.-S. Smedman, 003: Sea-surface aerodynamic roughness. J. Geophys. Res., 08, 37, doi:0.09/00jc Majda, A. J., I. Timofeyev, and E. Vanden-Eijnden, 00: A math-

13 049 ematical framework for stochastic climate models. Commun. Pure Appl. Math., 54, ,, and, 003: Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci., 60, McPhaden, M., and Coauthors, 998: The Tropical Ocean Global Atmosphere (TOGA) observing system: A decade of progress. J. Geophys. Res., 03, Milliff, R., W. Large, J. Morzel, and G. Danabasoglu, 999: Ocean general circulation model sensitivity to forcing from scatterometer winds. J. Geophys. Res., 04C, Monahan, A. H., 00: Nonlinear principal component analysis: Tropical Indo Pacific sea surface temperature and sea level pressure. J. Climate, 4, 9 33., 00a: Lyapunov exponents of a simple stochastic model of the thermally and wind-driven ocean circulation. Dyn. Atmos. Oceans, 35, , 00b: Stabilization of climate regimes by noise in a simple model of the thermohaline circulation. J. Phys. Oceanogr., 3, , and A. Dai, 004: The spatial and temporal structure of ENSO nonlinearity. J. Climate, 7, , and K. L. Denman, 004: Impacts of atmospheric variability on a coupled upper-ocean/ecosystem model of the subarctic Northeast Pacific. Global Biogeochem. Cycles, in press. Moore, A. M., 999: Wind-induced variability of ocean gyres. Dyn. Atmos. Ocean, 9, Palmer, T. N., 00: A nonlinear dynamical perspective on model error: A proposal for nonlocal stochastic dynamic parameterisation in weather and climate prediction models. Quart. J. Roy. Meteor. Soc., 7, Penland, C., 996: A stochastic model of IndoPacific sea surface temperature anomalies. Physica D, 98, , 003a: Noise out of chaos and why it won t go away. Bull. Amer. Meteor. Soc., 84, 9 95., 003b: A stochastic approach to nonlinear dynamics: A review. Bull. Amer. Meteor. Soc., 84, ES43 ES5. Ponte, R. M., and R. D. Rosen, 004: Nonlinear effects of variable winds on ocean stress climatologies. J. Climate, 7, Rieder, K. F., 997: Analysis of sea-surface drag parameterizations in open ocean conditions. Bound.-Layer Meteor., 8, Silverman, B. W., 986: Density Estimation for Statistics and Data Analysis. Chapman and Hall, 75 pp. Simmons, A., and J. Gibson, 000: The ERA-40 Project Plan. ERA- 40 Project Report Series No., ECMWF, Reading, United Kingdom, 63 pp. Stull, R. B., 997: An Introduction to Boundary Layer Meteorology. Kluwer, 670 pp. Sura, P., 003: Stochastic analysis of Southern and Pacific Ocean sea surface winds. J. Atmos. Sci., 60, , K. Fraedrich, and F. Lunkheit, 00: Regime transitions in a stochastically forced double-gyre model. J. Phys. Oceanogr., 3, Taylor, P. K., Ed., 000: Intercomparison and validation of ocean atmosphere energy flux fields. Joint WCRP/SCOR Working Group on Air Sea Fluxes Final Report, WMO Tech. Doc. 036, 306 pp., and M. J. Yelland, 00: The dependence of sea surface roughness on the height and steepness of the waves. J. Phys. Oceanogr., 3, Thompson, C., and D. Battisti, 000: A linear stochastic dynamical model of ENSO. Part I: Development. J. Climate, 3, Ward, M. N., and B. J. Hoskins, 996: Near-surface wind over the global ocean J. Climate, 9, Wright, D. G., and K. R. Thompson, 983: Time-averaged forms of the nonlinear stress law. J. Phys. Oceanogr., 3,