The Probability Distribution of Sea Surface Wind Speeds: Effects of Variable Surface Stratification and Boundary Layer Thickness

Adam Hugh Monahan School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada

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Abstract

Air–sea exchanges of momentum, energy, and material substances of fundamental importance to the variability of the climate system are mediated by the character of the turbulence in the atmospheric and oceanic boundary layers. Sea surface winds influence, and are influenced by, these fluxes. The probability density function (pdf) of sea surface wind speeds p(w) is a mathematical object describing the variability of surface winds that arises from the physics of the turbulent atmospheric planetary boundary layer. Previous mechanistic models of the pdf of sea surface wind speeds have considered the momentum budget of an atmospheric layer of fixed thickness and neutral stratification. The present study extends this analysis, using an idealized model to consider the influence of boundary layer thickness variations and nonneutral surface stratification on p(w). It is found that surface stratification has little direct influence on p(w), while variations in boundary layer thickness bring the predictions of the model into closer agreement with the observations. Boundary layer thickness variability influences the shape of p(w) in two ways: through episodic downward mixing of momentum into the boundary layer from the free atmosphere and through modulation of the importance (relative to other tendencies) of turbulent momentum fluxes at the surface and the boundary layer top. It is shown that the second of these influences dominates over the first.

Corresponding author address: Adam Hugh Monahan, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3065, STN CSC, Victoria, BC V8W 3V6, Canada. Email: monahana@uvic.ca

Abstract

Air–sea exchanges of momentum, energy, and material substances of fundamental importance to the variability of the climate system are mediated by the character of the turbulence in the atmospheric and oceanic boundary layers. Sea surface winds influence, and are influenced by, these fluxes. The probability density function (pdf) of sea surface wind speeds p(w) is a mathematical object describing the variability of surface winds that arises from the physics of the turbulent atmospheric planetary boundary layer. Previous mechanistic models of the pdf of sea surface wind speeds have considered the momentum budget of an atmospheric layer of fixed thickness and neutral stratification. The present study extends this analysis, using an idealized model to consider the influence of boundary layer thickness variations and nonneutral surface stratification on p(w). It is found that surface stratification has little direct influence on p(w), while variations in boundary layer thickness bring the predictions of the model into closer agreement with the observations. Boundary layer thickness variability influences the shape of p(w) in two ways: through episodic downward mixing of momentum into the boundary layer from the free atmosphere and through modulation of the importance (relative to other tendencies) of turbulent momentum fluxes at the surface and the boundary layer top. It is shown that the second of these influences dominates over the first.

Corresponding author address: Adam Hugh Monahan, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3065, STN CSC, Victoria, BC V8W 3V6, Canada. Email: monahana@uvic.ca

1. Introduction

Air–sea exchanges of momentum, energy, and mass, which are of fundamental importance to the variability of the climate system, are mediated by the character of the turbulence in the atmospheric and oceanic boundary layers. Sea surface winds influence, and are influenced by, these fluxes. Standard bulk parameterizations express air–sea fluxes as functions of the surface wind velocity averaged over the “turbulent time scales” of the atmospheric boundary layer (e.g., are “eddy averaged”). For those fluxes depending linearly on the surface wind speed (e.g., heat and moisture, to a first approximation), fluxes averaged over longer time scales or over some spatial domain are given by bulk formulas in terms of the average wind speed. However, for other fluxes (e.g., momentum, gases, aerosols) the nonlinear dependence on the surface wind speed implies that the mean flux is not the flux associated with the mean wind speed (e.g., Wanninkhof et al. 2002). A further complication arises in the computation of fluxes spatially averaged over some domain [e.g., a general circulation model (GCM) grid box]: in general, there is a difference between the area-averaged mean wind speed and the magnitude of the area-averaged mean vector wind (e.g., Mahrt and Sun 1995). Accurate computations of (space or time) average fluxes require the development of models of the probability distribution of sea surface winds.

A new era in the study of sea surface winds was ushered in with the introduction of high-resolution (in both space and time), global-scale observations from the SeaWinds scatterometer on board the Quick Scatterometer (QuikSCAT) satellite (Jet Propulsion Laboratory 2001; Chelton et al. 2004). These data have provided an unprecedented opportunity to characterize the probability density function (pdf) of observed surface winds (both vector winds and wind speed) on a global scale. In particular, these wind observations have been shown to be characterized by distinct relationships between statistical moments (mean, standard deviation, and skewness), corroborating results first obtained using surface wind fields from reanalysis products (Monahan 2004, 2006b). In particular, Monahan (2006a) demonstrated that the skewness of the sea surface wind speed is a decreasing function of the ratio of the mean to the standard deviation. Where this ratio is small, the wind speeds are positively skewed; where this ratio is intermediate in size, the wind speeds are unskewed; and where this ratio is large, speeds are negatively skewed. This relationship between moments is also characteristic of the Weibull distribution, which has been widely used as an empirical model of the pdf of surface wind speeds over both land and water (e.g., Monahan 2006a and references therein). For a Weibull distributed variable y, to a very good approximation,
i1520-0442-23-19-5151-e1
where
i1520-0442-23-19-5151-e2
and where Γ(x) is the gamma function. A plot of the observed relationship between moments for sea surface winds (contours) and for a Weibull variable (thick black line) is presented in Fig. 1. In this plot, the horizontal axis has been scaled so that the Weibull relationship appears as a 1:1 line. The observed relationship between moments clusters around the Weibull line, although with a somewhat steeper slope and pronounced curvature to the lower left. While it is evident from Fig. 1 that the Weibull distribution is a good approximation to the pdf of sea surface winds, it must be emphasized that this distribution is an empirical model without a mechanistic basis.

Also plotted in Fig. 1 is the relationship between moments simulated by an idealized model of the boundary layer momentum budget (Monahan 2004, 2006a). In this model, the horizontal momentum tendency includes contributions from surface turbulent momentum fluxes (quadratic in the surface wind speed), the turbulent exchange of momentum between the boundary layer and the free atmosphere, and “ageostrophic tendencies” with specified mean and fluctuating components (with the latter modeled as Gaussian white noise). The idealized model is able to capture important aspects of the observed relationship between moments, particularly the low-skewness curvature. However, an evident weakness of the model is its inability to capture the high values of skewness in the upper-right part of Fig. 1.

A parameter to which skew(w) is sensitive (where w denotes the surface wind speed) in this idealized model is the rate at which momentum is exchanged between the atmospheric surface layer and the air aloft. In Monahan (2006a), the surface layer was specified to have a constant thickness H within the boundary layer and the mixing was expressed in terms of an “eddy viscosity” K; equivalently, we can express this rate in terms of a momentum entrainment velocity, we = K/H. The values taken by skew(w) (Fig. 2) increase in response to increases in we. For large values of we, skew(w) has a maximum value of about 0.6 and does not take substantially negative values. In fact, for large we, this curve corresponds to the situation in which the joint distribution of the surface vector wind components is bivariate Gaussian. The model of Monahan (2004, 2006a) predicts that this limiting behavior will occur when the downward mixing of momentum becomes a much larger component of the momentum budget than surface drag.

The model considered in Monahan (2004, 2006a) was based upon a number of simplifying approximations; among these were the specification of neutral surface stratification and a fixed boundary layer thickness. In fact, the boundary layer momentum budget is influenced by local surface stratification and boundary layer depth variations in three distinct ways:

  1. The surface drag coefficient cd is a function of surface stratification (equivalently the surface buoyancy flux, assuming downgradient fluxes in the surface layer) such that an unstable stratification enhances surface turbulence (through buoyant generation of turbulence kinetic energy) and thereby increases surface drag (as the increased turbulent mixing allows more efficient momentum exchange with the surface). Conversely, stable stratification suppresses surface turbulence (through buoyant consumption of turbulence kinetic energy) and decreases surface drag. Monin–Obukhov theory parameterizes these effects through a correction to the neutral stability drag coefficient that depends on the Obukhov length L (such that L < 0 for surface heat flux to the atmosphere, and L > 0 for surface heat flux to the ocean; e.g., Stull 1997).

  2. Boundary layer thickness changes influence the turbulent momentum exchange between the free atmosphere and the boundary layer. While this turbulent exchange exists even for a boundary layer of constant thickness (as the thickness tendency associated with boundary layer top turbulent mixing may be balanced or exceeded by restratifying processes such as radiative cooling to space or large-scale subsidence (e.g., Medeiros et al. 2005)), it is generally stronger when the boundary layer is deepening.

  3. In the well-mixed slab boundary layer approximation, momentum tendencies produced by turbulent momentum fluxes at the surface and the boundary layer top are distributed across fluid parcels throughout the depth of the boundary layer. As the boundary layer becomes thicker, these interfacial fluxes are therefore diluted and weakened; conversely, as the boundary layer becomes shallower, these fluxes are concentrated and strengthened (e.g., Samelson et al. 2006).

Other tendencies driven by horizontal gradients in surface heat fluxes or boundary layer depth (e.g., mesoscale thermal circulations) are nonlocal and manifest as one contribution among several to the pressure gradient force.

The separation between the influence of the surface stratification and boundary layer depth variations is somewhat artificial, as variations in the air–sea temperature difference play an important role in driving variability in the marine boundary layer thickness (e.g., Samelson et al. 2006; Small et al. 2008). However, boundary layer thickness variations are also driven by processes other than surface fluxes, such as those associated with clouds at the boundary layer top (e.g., Stevens 2002; Medeiros et al. 2005). The focus of this study will be on the direct influence on the wind speed pdf of the surface buoyancy fluxes (though modification of the drag coefficient) and of boundary layer thickness variability (by whatever process this is generated). As a first approximation, potential interactions between these influences will be neglected.

Evidence of a relationship between the variability in the boundary layer height (denoted h) and the shape of the wind speed pdf is suggested by the negative correlation between December–February (DJF) 10-m ocean skew(w) and the ratio mean(h)/std(h) (Fig. 3), as determined from the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40; Uppala et al. 2006). It is evident from Fig. 3 that in the ERA-40 reanalysis the wind speed skewness tends to be most positive at locations where the boundary layer variability is large (relative to the mean boundary layer depth) and the skewness tends to decrease as variability in h decreases. Of course, this anticorrelation is not perfect, and the representation in the reanalysis data of nonassimilated quantities such as surface winds and (particularly) boundary layer thickness must be treated with caution. Nevertheless, the relationship illustrated in Fig. 3, which is derived from a comprehensive global numerical weather prediction system with a sophisticated boundary layer scheme, suggests that at least one of the missing factors in the idealized boundary layer momentum budget model is an active boundary layer.

The present study generalizes the idealized model of the boundary layer momentum budget developed in Monahan (2004, 2006a) to consider the effects on the pdf of sea surface winds of surface stratification and variations in boundary layer depth. The generalized model is still a highly simplified representation of marine boundary layer physics and is designed to capture the essential qualitative features of the wind speed pdf rather than to provide a quantitatively precise characterization. The generalized model of the boundary layer momentum budget is described in section 2, followed in section 3 by a consideration of the effects of accounting for surface stratification. The influence of variability in boundary layer depth on the pdf of sea surface winds is considered in section 4, and conclusions follow in section 5.

2. Idealized boundary layer momentum budget model

The original idealized boundary layer momentum budget model of Monahan (2004, 2006a) represented surface vector wind tendencies as resulting from an imbalance between four forces: 1) a mean “large scale” ageostrophic tendency, 2) fluctuations in the ageostrophic forcing, 3) surface drag, and 4) downward mixing of momentum from above the (fixed depth) surface layer. Because the layer thickness was fixed, the character of the winds above z = H did not need to be modeled explicitly and their contributions to the momentum tendencies were subsumed into the mean and fluctuating ageostrophic forcing. In the present study, entrainment of momentum from the free atmosphere is a variable and potentially intermittent process and is, thus, modeled explicitly.

Expressing the above ideas quantitatively, the boundary layer momentum budget is given by
i1520-0442-23-19-5151-e3
i1520-0442-23-19-5151-e4
The large-scale ageostrophic forcing is expressed as the sum of mean (term A) and fluctuating (term B) components. The quantity τs is a characteristic surface wind adjustment time scale, specified so that mean(u) ∼ Us. Fluctuations in large-scale forcing are modeled as red-noise processes with an autocorrelation time scale τη and mean zero:
i1520-0442-23-19-5151-e5
i1520-0442-23-19-5151-e6
The noise terms are scaled so that is approximately the contribution to the variance of u (or υ) associated with the large-scale ageostrophic forcing. More precisely, a variable z described by dz/dt = (−z + ηu)/τs has standard deviation σenv.

Surface and boundary layer top eddy momentum fluxes are represented by terms C and D, respectively. The second of these terms represents the entrainment flux as a finite-differenced eddy diffusion, in which boundary layer fluid is mixed with fluid a distance δ above the boundary layer with the entrainment velocity we. The distance δ can be interpreted as being a measure of the thickness of the entrainment layer between the boundary layer and free atmosphere. Along with large-scale forcing of the boundary layer momentum budget, local fluctuating forcing is represented as white-noise forcing with a scaling coefficient, σu (term E).

Variability in boundary layer depth (interpreted as the depth of the active mixing layer rather than the well-mixed layer) is driven by an imbalance in tendencies between stratifying and mixing processes:
i1520-0442-23-19-5151-e7
The first term in Eq. (7) represents the average tendency of restratifying processes (e.g., large-scale subsidence, radiative cooling), which causes boundary layer heights to decrease on a time scale τh, while the second term represents a baseline turbulent entrainment velocity that acts to deepen the mixed layer. The third term in this equation describes the net effect of variability in restratifying and entrainment rates and is described for simplicity as a red-noise process with an autocorrelation time scale, τξ:
i1520-0442-23-19-5151-e8
To ensure that the boundary layer height does not become negative, h is not allowed to decrease below a minimum value, hmin. To a good approximation (exact in the absence of the lower limit hmin), h has the stationary standard deviation,
i1520-0442-23-19-5151-e9
and autocorrelation function,
i1520-0442-23-19-5151-e10
In the above equations, the random processes i(i = 1, 2, 3, 4, 5) are mutually uncorrelated white-noise processes:
i1520-0442-23-19-5151-e11
The rate at which momentum is mixed from the free atmosphere into the boundary layer is determined by the entrainment velocity:
i1520-0442-23-19-5151-e12
The first of these terms is the constant background entrainment rate, while the second is associated with only those fluctuations in mixed layer depth that tend to deepen the mixed layer (as restratifying processes do not unmix the boundary layer).

As modeled, the variability of the boundary layer thickness is not explicitly influenced by surface stratification, wind speeds, or the state of the free atmosphere. In reality, the turbulent entrainment rate at the top of the marine boundary layer is determined by a number of factors, including the strength of the boundary layer top inversion, the generation of turbulence kinetic energy within the boundary layer (influenced by wind shears and surface buoyancy fluxes), and the radiatively driven generation of turbulence within boundary layer top clouds (e.g., Stevens 2002). Surface winds modulate both surface fluxes and the mechanical generation of turbulence, and the strength of the boundary layer top inversion can be expected to depend on h. From the perspective of the main focus of this study, namely, the direct influences on the pdf of the sea surface wind speed of the surface stratification (through changes of the drag coefficient) and boundary layer depth variability, the fact that the boundary layer depth is variable is more important than the precise details of why it is variable. Nevertheless, the neglect of feedbacks between state variables and boundary layer tendencies is a substantial approximation. A more detailed representation of boundary layer tendencies including the influence of the winds would involve a substantial increase in the complexity of the model (e.g., explicitly modeling potential temperature and moisture). The specified boundary layer dynamics represent a compromise between model simplicity and fidelity to nature motivated by the main concerns of the present study.

Over the time scale of the adjustment processes in the lower free atmosphere the boundary layer may deepen, shallow, and deepen again; if this variability is sufficiently rapid, the second deepening will bring the boundary layer top into contact with free-atmospheric air that retains some memory of earlier contact with the boundary layer. It is therefore desirable to incorporate into the model a simplified prognostic representation of the free-atmospheric wind profile: U(z, t) = [U(z, t), V(z, t)]. Within the boundary layer, U(z, t) and u(t) are defined to coincide:
i1520-0442-23-19-5151-e13
Above z = h(t), in the free atmosphere, the horizontal winds relax on a time scale τr to the large-scale environmental profile with constant shear Λ = (Λu, Λυ):
i1520-0442-23-19-5151-e14
where
i1520-0442-23-19-5151-e15
That is,
i1520-0442-23-19-5151-e16
Each level in the free atmosphere relaxes independently toward the large-scale profile; no vertical transport processes in the free atmosphere are modeled explicitly.
The surface stratification influences the momentum budget directly through changes in the character of surface layer turbulence. A natural measure of surface stratification is the difference T between the surface air temperature (SAT) and the sea surface temperature (SST):
i1520-0442-23-19-5151-e17
Unstable stratification (T < 0) enhances the turbulence and increases the rate of turbulent momentum exchange with the underlying surface, increasing cd. Conversely, stable stratification (T > 0) inhibits surface turbulence and reduces cd. The drag coefficient is also a function of the sea surface wind speed w (e.g., Csanady 2001), as the generation of the surface ocean waves by surface winds increases the surface roughness and drag. At moderate to strong wind speeds over a developed sea, cd is an increasing function of w. For very weak winds, cd may also increase as w decreases, in accordance with the characteristics of drag over an aerodynamically smooth surface. The functional dependence of cd on T and w in the observations displays considerable scatter (as a result of varying conditions and the difficulty of measurements), so there is no uniquely agreed upon functional form for this relationship. In this study, we will make use of the drag coefficient cd(w, T) given by a local polynomial approximation (Kara et al. 2005) to the drag coefficient from the Coupled Ocean–Atmosphere Response Experiment (COARE) version 3.0 algorithm [based on a large number of observations over a wide range of surface conditions; Fairall et al. (2003)], as illustrated in Fig. 4.

Together, these equations constitute a vector stochastic differential equation (SDE) for the state variables u, υ, h, U, and V (along with the red-noise variables ηu, ηυ, and ξ) listed in Table 1. The model parameters (and standard values) are listed in Table 2. General introductions to SDEs are presented in Gardiner (1997) and Horsthemke and Lefever (2006); an introduction within the context of atmosphere–ocean modeling is presented in Penland (2003). Corresponding to this (nonlinear) SDE is a linear diffusion equation for the associated pdf known as the Fokker–Planck equation (FPE). In some circumstances, the stationary FPE (for the “statistically equilibrated” time-invariant pdf) admits an analytic solution. More generally, state variable pdf’s must be simulated by numerical integration of the associated SDEs Kloeden and Platen (1992).

3. Effects of surface stratification

Similarly to surface winds, variability in the air–sea temperature difference is driven by a combination of large-scale and local processes. Of particular importance are local surface heat fluxes, which are themselves functions of the surface wind speed; as is discussed in Sura and Newman (2008), much of the variability of T can be understood as a response to variability of w. However, as temperature fluctuations have much longer characteristic time scales than surface wind fluctuations (e.g., Sura and Newman 2008), it is meaningful to consider the probability distribution of the surface wind speed in equilibrium with a fixed temperature difference expressed through the conditional probability density function p(w|T). Given the pdf p(T) of the air–sea temperature difference, the pdf of the surface wind speeds p(w) can be computed:
i1520-0442-23-19-5151-e18
For simplicity, the following analysis will assume that the distribution of T is Gaussian with mean μT and standard deviation σT:
i1520-0442-23-19-5151-e19
In fact, while both SST and SAT display nonzero skewness and kurtosis (Sura and Newman 2008), these non-Gaussian features are sufficiently modest that the specification of Gaussian fluctuations in T is a reasonable first approximation.
To consider the direct effects of air–sea temperature differences on the pdf of surface winds, we will examine the model described in section 2 with constant boundary layer depth h = w*eτh (which is 800 m for the standard parameter values). We will further assume that Vs = V(h + δ) = 0 m s−1 and embed the tendency associated with U(h + δ) into Us. The analysis is also facilitated by considering the white-noise limit of the “nonlocal” ageostrophic forcing, τη → 0. In this limit we obtain the SDE:
i1520-0442-23-19-5151-e20
i1520-0442-23-19-5151-e21
with an associated Fokker–Planck equation for the stationary pdf conditioned on T, p(u, υ|T), which is analytically solvable as
i1520-0442-23-19-5151-e22
(as discussed in Monahan 2006a). Integrating over the wind direction, we obtain the marginal pdf of the wind speed (conditioned on T):
i1520-0442-23-19-5151-e23
The quantities 1 and 2 are normalization constants. Combining Eqs. (19) and (23) through Eq. (18), we obtain the marginal pdf of the wind speed p(w).

Moments of w computed from the pdf p(w) are contoured in Fig. 5 as functions of μT and σT (over realistic ranges) for various values of Us and σenv. In general, the dependence of the moments of sea surface wind speed on the mean and standard deviation of the air–sea temperature difference is weak. Both mean(w) and std(w) tend to increase with μT, while skew(w) decreases. The dependence of the moments on σT is weaker and less systematic than that on μT.

The dependence of individual moments of w on μT and σT does not imply a corresponding dependence of the relationship between wind speed moments. Plots of the relationship between skew(w) and S[mean(w)/std(w)] for various values of μT and σT are illustrated in Fig. 6. It is evident that the relationship between moments of p(w) has a very weak dependence on variability in the surface drag coefficient driven by fluctuations in the air–sea temperature difference. Although the dependence of this relationship on the mean air–sea temperature difference μT is somewhat stronger than that on the standard deviation σT, the curves associated with different values of μT are almost indistinguishable.

It thus appears that the direct influence of surface stratification (through modification of the drag coefficient) has little effect on the shape of the pdf of the sea surface wind speeds, providing a posteriori justification for the assumption of neutral stratification in the idealized boundary layer models in Monahan (2004, 2006a). In particular, nonneutral surface stratification cannot account for the inability of the idealized model to simulate the large positive wind speed skewness in the conditions of light mean winds. He et al. (2010) also find that in terrestrial areas classified as “open water” (lakes and coastal regions) the diurnal and seasonal patterns of evolution of the shape of p(w) are much weaker than over open land or forested regions, suggesting a much weaker influence of surface heat fluxes over water where the momentum and thermal roughness lengths both tend to be small (Garratt 1992). Over land, there is evidence that surface buoyancy fluxes (both mean and variability) have a pronounced influence on the character of the surface wind speed pdf (He et al. 2010). In the following section, we will consider the effects of the variable boundary layer depth on the shape of p(w).

4. Effects of variable boundary layer depth

The correlation between the wind speed skewness and the boundary layer depth variability illustrated in Fig. 3 suggests that the specification of a fixed layer depth in the idealized boundary layer momentum budget of Monahan (2004, 2006a) may contribute to this model’s inability to account for the observed large positive values of skew(w) illustrated in Fig. 1. To test this hypothesis, moments of w were computed from the idealized model of the boundary layer momentum budget described in section 2 over broad ranges of the parameters Us, σenv, and std(h). Because the Fokker–Planck equation associated with this model does not admit analytic solutions (other than in the limit considered in section 3), these stochastic differential equations were integrated numerically (for 15 yr of model time, with output saved every 6 h) using a standard forward Euler technique (Kloeden and Platen 1992). As was demonstrated in section 3, the direct influence of air–sea temperature differences on the momentum budget is small, so these numerical simulations were carried out with constant neutral stratification (μT = σT = 0 K).

The results of these simulations are displayed in Fig. 7. The moment relationship from the model of Monahan (2006a) (displayed in Fig. 1) corresponds to that of the present model for simulations with std(h) = 0 m (with the slight difference that the present model has both red-noise ageostrophic forcing and white-noise local forcing). As the variability in h becomes larger, the values of both S[mean(w)/std(w)] and skew(w) increase. Consistent with the relationship between skew(w) and mean(h)/std(h) in the ERA-40 reanalysis (Fig. 3), the idealized boundary layer model predicts that larger positive wind speed skewness is associated with stronger variability in boundary layer depth. In particular, the model is better able to simulate the larger values of skew(w) characteristic of observed surface wind speeds (the top-right part of Fig. 7). Furthermore, the range of moments predicted by the present model fills out the joint pdf of the observed moments more completely than that of the model of Monahan (2006a). Sampling variability will contribute to the breadth of this pdf in the observations, but the fact that pdf’s of comparable breadth are produced by longer datasets [e.g., in reanalysis winds; cf. Monahan (2006b)] suggests that some fraction of this variability is real. While skew(w) in the idealized model is still biased low relative to the observations, accounting for variability in h brings the model into closer agreement with the observed relationship between moments.

As discussed in the introduction, variability in the boundary layer depth influences the boundary layer momentum budget through episodic downward mixing of momentum from the free atmosphere and modulation of the strength of interfacial turbulent momentum fluxes relative to the bulk body forces. These processes can be suppressed individually in the model to assess the importance of each in producing the large positive values of skew(w) displayed in Fig. 7. Model integrations with h varying but we held fixed at
i1520-0442-23-19-5151-e24
(Fig. 8, left) demonstrate that the simulated wind speed moments are essentially unchanged from those of the full model with variable we. In contrast, integrations with we varying but h held fixed in Eqs. (3) and (4) for the surface vector wind momentum budget (Fig. 8, right) differ from those of the full model. This analysis demonstrates that in this model it is the concentration–dilution of turbulent eddy fluxes due to variability in boundary layer thickness rather than the variability of the downward mixing of momentum from aloft that is responsible for the larger positive values of skew(w) simulated by the full model. These results are consistent with those of Samelson et al. (2006), which emphasized the greater importance of variations in boundary layer depth, relative to that of the vertical exchange of momentum, for the coupling between wind stress and sea surface temperature.

5. Conclusions

This study has considered the influences of surface stratification and variable boundary layer thickness on the shape of the probability density function of sea surface wind speeds. As has been shown in previous studies (e.g., Monahan 2006a, 2007), the pdf of the sea surface wind speed is characterized by a relationship between the shape of the pdf (as measured by skewness) and measures of the “size” of the pdf (as measured by the ratio of the mean to the standard deviation). An earlier mechanistic study of the pdf of the sea surface wind speeds using an idealized model of the boundary layer momentum budget, assuming neutral stratification and constant boundary layer depth, resulted in a reasonable first approximation of this relationship between moments (Monahan 2006a). However, this earlier model was not able to account for the large positive wind speed skewnesses seen in the observations in conditions of light and variable winds. A generalization of this earlier idealized model was used to assess the relative importance of surface stability-driven variations in the drag coefficient, the downward mixing of momentum from aloft in a deepening boundary layer, and the dilution (concentration) of eddy momentum fluxes at the surface and the top of the boundary layer as the boundary layer deepens (shallows). The following conclusions were obtained:

  • While surface stratification (as measured by the air–sea temperature difference) influences the simulated moments of the surface wind speed pdf, it has an insubstantial effect on the modeled relationship between surface wind speed moments. In particular, over a broad (and physically realistic) range of values of the mean and standard deviation of T = SAT − SST, the model was unable to simulate the large positive wind speed skewnesses seen in the observations.

  • Accounting for variability in the boundary layer thickness improves the agreement between the observed and simulated relationships between sea surface wind moments. In particular, larger positive values of skew(w) are simulated in conditions of weak and variable winds [small mean(w)/std(w)]. The improvements in agreement between simulated and observed wind speed moments are due primarily to the dilution–concentration of eddy momentum fluxes at the surface and the boundary layer top (relative to the body forces) associated with variations in boundary layer thickness. Episodic variability in the downward mixing of momentum from the free atmosphere in the model had little effect on the relationship between wind speed moments.

The ERA-40 reanalyses are characterized by a relationship between the sea surface wind speed skewness and the boundary layer variability, such that skew(w) is a decreasing function of the ratio mean(h)/std(h). That is, the reanalysis winds are most positively skewed in regions where the variability in the boundary layer thickness is relatively large compared to its mean value. While reanalysis data are not observations, and extreme caution must be exercised in consideration of a derived field such as boundary layer height, this relationship indicates that a complex model containing a broad range of physical processes displays a correlation between the shape of the wind speed pdf and the (relative) variability of the boundary layer thickness in broad agreement with that predicted by the idealized model of the present study.

While surface stratification does not appear to have a substantial direct influence (through the drag coefficient) on the shape of the wind speed pdf over the ocean, the same is not true over land. He et al. (2010) demonstrate that in open and wooded areas in the North American domain there is a strong diurnal cycle in the relationship between mean(w)/std(w) and mean(w), such that for larger values of the ratio the wind speed skewness values are much smaller during the day than they are at night. Furthermore, the study of He et al. (2010) provided evidence that these changes in the shape of the land surface wind speed pdf are produced by surface buoyancy fluxes. In general, thermal roughness lengths are larger over land than over water (e.g., Garratt 1992), so it is physically reasonable that surface stratification should exercise a stronger direct influence on the drag coefficient over land than over water.

While incorporation of variability in boundary layer thickness brings the model-simulated relationship between the wind speed moments into closer agreement with the observed relationship, significant model–observation differences remain. In particular, for larger values of the ratio mean(w)/std(w), the modeled relationship between moments is closer to that of the Weibull distribution than to that of the observed sea surface winds: values of skew(w) are still systematically underestimated. While the present model is more general than that considered in Monahan (2006a), it remains a highly idealized single-column slab model. It is possible that addressing the model deficiencies would require a more complete consideration of the vertical and horizontal momentum transport in the boundary layer. Surface buoyancy fluxes can generate structure within the planetary boundary layer (such as internal boundary layers), which a model such as the one under consideration cannot represent. In this model, the stratification only affects the wind through changes in the surface drag coefficient. Furthermore, the present study has also made the simplifying assumption that variability in the boundary layer thickness can be decoupled from variability in the surface stratification and from the winds themselves. In fact, surface buoyancy fluxes are one contributor (among others) to the dynamics of the boundary layer, and are particularly important in the vicinity of oceanic fronts and eddies (e.g., Spall 2007; Small et al. 2008) where SST changes are particularly pronounced. A more complete model accounting for the influences of various processes driving the boundary layer top entrainment velocity (including the influence of surface winds) would need to represent the profiles of (moist) thermodynamic and radiative processes within the boundary layer (e.g., Stevens 2002; Medeiros et al. 2005). Such a model would represent a dramatic increase in complexity relative to the model considered in the present study; a more thorough consideration of the influences of these various boundary layer processes on the pdf of the surface wind speeds is a potentially important direction of future research.

The analysis presented in this study provides further insights regarding the physical factors that control the shape of the sea surface wind speed pdf. This developing mechanistic understanding holds the promise of improving the estimation of surface fluxes and surface wind power density from observations, and their simulation in GCMs. Sea surface winds are a geophysical field of fundamental importance to the coupled climate system; improvements in our understanding of this field hold out the promise of an improved understanding of past, present, and future climates.

Acknowledgments

The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada. The author would like to thank Yanping He, Philip Poon, and two anonymous referees for valuable comments on this manuscript.

REFERENCES

  • Chelton, D. B., M. G. Schlax, M. H. Freilich, and R. F. Milliff, 2004: Satellite measurements reveal persistent small-scale features in ocean winds. Science, 303 , 978983.

    • Search Google Scholar
    • Export Citation
  • Csanady, G., 2001: Air–Sea Interaction: Laws and Mechanisms. Cambridge University Press, 248 pp.

  • Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16 , 571591.

    • Search Google Scholar
    • Export Citation
  • Gardiner, C. W., 1997: Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Springer, 442 pp.

  • Garratt, J., 1992: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.

  • He, Y., A. H. Monahan, C. G. Jones, A. Dai, S. Biner, D. Caya, and K. Winger, 2010: Probability distributions of land surface wind speeds over North America. J. Geophys. Res., 115 , D04103. doi:10.1029/2008JD010708.

    • Search Google Scholar
    • Export Citation
  • Horsthemke, W., and R. Lefever, 2006: Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology. Springer-Verlag, 318 pp.

    • Search Google Scholar
    • Export Citation
  • Jet Propulsion Laboratory, 2001: SeaWinds on QuikSCAT level 3: Daily, gridded ocean wind vectors. JPL Tech. Rep. JPL PO.DAAC, Product 109, California Institute of Technology, 39 pp.

    • Search Google Scholar
    • Export Citation
  • Kara, A. B., H. E. Hurlburt, and A. J. Wallcraft, 2005: Stability-dependent exchange coefficients for air–sea fluxes. J. Atmos. Oceanic Technol., 22 , 10801094.

    • Search Google Scholar
    • Export Citation
  • Kloeden, P. E., and E. Platen, 1992: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, 632 pp.

  • Mahrt, L., and J. Sun, 1995: The subgrid velocity scale in the bulk aerodynamic relationship for spatially averaged scalar fluxes. Mon. Wea. Rev., 123 , 30323041.

    • Search Google Scholar
    • Export Citation
  • Medeiros, B., A. Hall, and B. Stevens, 2005: What controls the mean depth of the PBL? J. Climate, 18 , 31573172.

  • Monahan, A. H., 2004: A simple model for the skewness of global sea surface winds. J. Atmos. Sci., 61 , 20372049.

  • Monahan, A. H., 2006a: The probability distribution of sea surface wind speeds. Part I: Theory and SeaWinds observations. J. Climate, 19 , 497520.

    • Search Google Scholar
    • Export Citation
  • Monahan, A. H., 2006b: The probability distribution of sea surface wind speeds. Part II: Dataset intercomparison and seasonal variability. J. Climate, 19 , 521534.

    • Search Google Scholar
    • Export Citation
  • Monahan, A. H., 2007: Empirical models of the probability distribution of sea surface wind speeds. J. Climate, 20 , 57985814.

  • Penland, C., 2003: Noise out of chaos and why it won’t go away. Bull. Amer. Meteor. Soc., 84 , 921925.

  • Samelson, R., E. Skyllingstad, D. Chelton, S. Esbensen, L. O’Neill, and N. Thum, 2006: On the coupling of wind stress and sea surface temperature. J. Climate, 19 , 15571566.

    • Search Google Scholar
    • Export Citation
  • Small, R., and Coauthors, 2008: Air–sea interaction over ocean fronts and eddies. Dyn. Atmos. Oceans, 45 , 274319.

  • Spall, M. A., 2007: Midlatitude wind stress–sea surface temperature coupling in the vicinity of oceanic fronts. J. Climate, 20 , 37853801.

    • Search Google Scholar
    • Export Citation
  • Stevens, B., 2002: Entrainment in stratocumulus-topped mixed layers. Quart. J. Roy. Meteor. Soc., 128 , 26632690.

  • Stull, R. B., 1997: An Introduction to Boundary Layer Meteorology. Kluwer, 670 pp.

  • Sura, P., and M. Newman, 2008: The impact of rapid wind variability upon air–sea thermal coupling. J. Climate, 21 , 621637.

  • Uppala, S. M., and Coauthors, 2006: The ERA-40 Re-Analysis. Quart. J. Roy. Meteor. Soc., 131 , 29613012.

  • Wanninkhof, R., S. C. Doney, T. Takahashi, and W. R. McGillis, 2002: The effect of using time-averaged winds on regional air–sea CO2 fluxes. Gas Transfer at Water Surfaces, M. A. Donelan et al., Eds., Amer. Geophys. Union, 351–356.

    • Search Google Scholar
    • Export Citation

Fig. 1.
Fig. 1.

Relationship between wind speed skewness from observations (contours) and an idealized boundary layer model (red dots). The horizontal axis is scaled by the function S(x) [Eq. (2)] so that the relationship between moments for a Weibull-distributed variable (thick black curve) falls along the 1:1 line. The observed wind speeds are taken from level 3.0 gridded daily QuikScat SeaWinds observations from 1999 to 2008, as described in Monahan (2006a). The model is as described in Section 2 with parameter values τη = 0 and std(h) = 0 [corresponding to the model in Monahan (2006a)].

Citation: Journal of Climate 23, 19; 10.1175/2010JCLI3184.1

Fig. 2.
Fig. 2.

As in Fig. 1, but for four different values of the entrainment velocity: we = 0 (blue), 0.01 (red), 0.02 (magenta), and 0.05 m s−1 (green).

Citation: Journal of Climate 23, 19; 10.1175/2010JCLI3184.1

Fig. 3.
Fig. 3.

A kernel density estimate of the joint pdf of the ratio of the mean to the standard deviation of the boundary layer thickness mean(h)/std(h) and the sea surface wind speed skewness skew(w), as estimated for the DJF season from ERA-40 reanalyses (6-hourly data on a 2.5° × 2.5° grid from 65°S to 60°N from 1 Sep 1957 to 31 Aug 2002; information online http://data.ecmwf.int/data/d/era40/).

Citation: Journal of Climate 23, 19; 10.1175/2010JCLI3184.1

Fig. 4.
Fig. 4.

Dependence of the drag coefficient cd × 103 on the wind speed w and air–sea temperature difference T [based on the local polynomial approximation of Kara et al. (2005)]. The sharp corners of some contours for small values of w are associated with change points in the polynomial approximation of cd(w, T).

Citation: Journal of Climate 23, 19; 10.1175/2010JCLI3184.1

Fig. 5.
Fig. 5.

Contour plots of the leading three moments of the sea surface winds [mean(w), std(w), skew(w)] as functions of μT and σT, computed from Eqs. (18), (19), and (23): (top) (Us, σenv) = (0, 7) m s−1, (middle) (Us, σenv) = (5, 4) m s−1, and (bottom) (Us, σenv) = (10, 2) m s−1.

Citation: Journal of Climate 23, 19; 10.1175/2010JCLI3184.1

Fig. 6.
Fig. 6.

As in Fig. 1, but for the idealized model of the boundary layer momentum budget accounting for the dependence of the drag coefficient on the surface stratification: σT = (left) 0 and (right) 3 K. In both panels, μT = −3 (blue dots), −1 (magenta dots), 1 (green dots), and 3 K (yellow dots).

Citation: Journal of Climate 23, 19; 10.1175/2010JCLI3184.1

Fig. 7.
Fig. 7.

As in Fig. 1, but for the idealized boundary layer model with fluctuating boundary layer depth: std(h) = 0 (blue dots), 200 (magenta dots), and 400 m (green dots).

Citation: Journal of Climate 23, 19; 10.1175/2010JCLI3184.1

Fig. 8.
Fig. 8.

As in Fig. 7, but with the results of the boundary layer model for std(h) = (0, 200, 400) m (blue dots). (left) Moments of the simulated wind with entrainment velocity we = mean(we) = w*e + mean[max(ξ/τh, 0)] held constant (red dots). (right) Moments of the simulated wind with boundary layer depth held constant in the momentum budget (red dots).

Citation: Journal of Climate 23, 19; 10.1175/2010JCLI3184.1

Table 1.

Model coordinates and state variables.

Table 1.
Table 2.

Model parameters and standard values.

Table 2.
Save
  • Chelton, D. B., M. G. Schlax, M. H. Freilich, and R. F. Milliff, 2004: Satellite measurements reveal persistent small-scale features in ocean winds. Science, 303 , 978983.

    • Search Google Scholar
    • Export Citation
  • Csanady, G., 2001: Air–Sea Interaction: Laws and Mechanisms. Cambridge University Press, 248 pp.

  • Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16 , 571591.

    • Search Google Scholar
    • Export Citation
  • Gardiner, C. W., 1997: Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Springer, 442 pp.

  • Garratt, J., 1992: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.

  • He, Y., A. H. Monahan, C. G. Jones, A. Dai, S. Biner, D. Caya, and K. Winger, 2010: Probability distributions of land surface wind speeds over North America. J. Geophys. Res., 115 , D04103. doi:10.1029/2008JD010708.

    • Search Google Scholar
    • Export Citation
  • Horsthemke, W., and R. Lefever, 2006: Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology. Springer-Verlag, 318 pp.

    • Search Google Scholar
    • Export Citation
  • Jet Propulsion Laboratory, 2001: SeaWinds on QuikSCAT level 3: Daily, gridded ocean wind vectors. JPL Tech. Rep. JPL PO.DAAC, Product 109, California Institute of Technology, 39 pp.

    • Search Google Scholar
    • Export Citation
  • Kara, A. B., H. E. Hurlburt, and A. J. Wallcraft, 2005: Stability-dependent exchange coefficients for air–sea fluxes. J. Atmos. Oceanic Technol., 22 , 10801094.

    • Search Google Scholar
    • Export Citation
  • Kloeden, P. E., and E. Platen, 1992: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, 632 pp.

  • Mahrt, L., and J. Sun, 1995: The subgrid velocity scale in the bulk aerodynamic relationship for spatially averaged scalar fluxes. Mon. Wea. Rev., 123 , 30323041.

    • Search Google Scholar
    • Export Citation
  • Medeiros, B., A. Hall, and B. Stevens, 2005: What controls the mean depth of the PBL? J. Climate, 18 , 31573172.

  • Monahan, A. H., 2004: A simple model for the skewness of global sea surface winds. J. Atmos. Sci., 61 , 20372049.

  • Monahan, A. H., 2006a: The probability distribution of sea surface wind speeds. Part I: Theory and SeaWinds observations. J. Climate, 19 , 497520.

    • Search Google Scholar
    • Export Citation
  • Monahan, A. H., 2006b: The probability distribution of sea surface wind speeds. Part II: Dataset intercomparison and seasonal variability. J. Climate, 19 , 521534.

    • Search Google Scholar
    • Export Citation
  • Monahan, A. H., 2007: Empirical models of the probability distribution of sea surface wind speeds. J. Climate, 20 , 57985814.

  • Penland, C., 2003: Noise out of chaos and why it won’t go away. Bull. Amer. Meteor. Soc., 84 , 921925.

  • Samelson, R., E. Skyllingstad, D. Chelton, S. Esbensen, L. O’Neill, and N. Thum, 2006: On the coupling of wind stress and sea surface temperature. J. Climate, 19 , 15571566.

    • Search Google Scholar
    • Export Citation
  • Small, R., and Coauthors, 2008: Air–sea interaction over ocean fronts and eddies. Dyn. Atmos. Oceans, 45 , 274319.

  • Spall, M. A., 2007: Midlatitude wind stress–sea surface temperature coupling in the vicinity of oceanic fronts. J. Climate, 20 , 37853801.

    • Search Google Scholar
    • Export Citation
  • Stevens, B., 2002: Entrainment in stratocumulus-topped mixed layers. Quart. J. Roy. Meteor. Soc., 128 , 26632690.

  • Stull, R. B., 1997: An Introduction to Boundary Layer Meteorology. Kluwer, 670 pp.

  • Sura, P., and M. Newman, 2008: The impact of rapid wind variability upon air–sea thermal coupling. J. Climate, 21 , 621637.

  • Uppala, S. M., and Coauthors, 2006: The ERA-40 Re-Analysis. Quart. J. Roy. Meteor. Soc., 131 , 29613012.

  • Wanninkhof, R., S. C. Doney, T. Takahashi, and W. R. McGillis, 2002: The effect of using time-averaged winds on regional air–sea CO2 fluxes. Gas Transfer at Water Surfaces, M. A. Donelan et al., Eds., Amer. Geophys. Union, 351–356.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Relationship between wind speed skewness from observations (contours) and an idealized boundary layer model (red dots). The horizontal axis is scaled by the function S(x) [Eq. (2)] so that the relationship between moments for a Weibull-distributed variable (thick black curve) falls along the 1:1 line. The observed wind speeds are taken from level 3.0 gridded daily QuikScat SeaWinds observations from 1999 to 2008, as described in Monahan (2006a). The model is as described in Section 2 with parameter values τη = 0 and std(h) = 0 [corresponding to the model in Monahan (2006a)].

  • Fig. 2.

    As in Fig. 1, but for four different values of the entrainment velocity: we = 0 (blue), 0.01 (red), 0.02 (magenta), and 0.05 m s−1 (green).

  • Fig. 3.

    A kernel density estimate of the joint pdf of the ratio of the mean to the standard deviation of the boundary layer thickness mean(h)/std(h) and the sea surface wind speed skewness skew(w), as estimated for the DJF season from ERA-40 reanalyses (6-hourly data on a 2.5° × 2.5° grid from 65°S to 60°N from 1 Sep 1957 to 31 Aug 2002; information online http://data.ecmwf.int/data/d/era40/).

  • Fig. 4.

    Dependence of the drag coefficient cd × 103 on the wind speed w and air–sea temperature difference T [based on the local polynomial approximation of Kara et al. (2005)]. The sharp corners of some contours for small values of w are associated with change points in the polynomial approximation of cd(w, T).

  • Fig. 5.

    Contour plots of the leading three moments of the sea surface winds [mean(w), std(w), skew(w)] as functions of μT and σT, computed from Eqs. (18), (19), and (23): (top) (Us, σenv) = (0, 7) m s−1, (middle) (Us, σenv) = (5, 4) m s−1, and (bottom) (Us, σenv) = (10, 2) m s−1.

  • Fig. 6.

    As in Fig. 1, but for the idealized model of the boundary layer momentum budget accounting for the dependence of the drag coefficient on the surface stratification: σT = (left) 0 and (right) 3 K. In both panels, μT = −3 (blue dots), −1 (magenta dots), 1 (green dots), and 3 K (yellow dots).

  • Fig. 7.

    As in Fig. 1, but for the idealized boundary layer model with fluctuating boundary layer depth: std(h) = 0 (blue dots), 200 (magenta dots), and 400 m (green dots).

  • Fig. 8.

    As in Fig. 7, but with the results of the boundary layer model for std(h) = (0, 200, 400) m (blue dots). (left) Moments of the simulated wind with entrainment velocity we = mean(we) = w*e + mean[max(ξ/τh, 0)] held constant (red dots). (right) Moments of the simulated wind with boundary layer depth held constant in the momentum budget (red dots).

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