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

Transcription

1 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 P.O. Box 3065 STN CSC Victoria, BC, Canada, V8W 3V6 Submitted to Journal of Climate March 30, 2009

2 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 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 which 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 to consider the influence (in the context of an idealised model) of boundary layer thickness variations and non-neutral 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 observations. Boundary layer thickness variability influences the shape of p(w) in two ways: through episodic downwards mixing of momentum into the boundary layer from the free atmosphere, and 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.

3 Introduction 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 turbulence in the atmospheric and oceanic boundary layers. Sea surface winds influence, and are influenced by, these fluxes. Standard bulk parameterisations express air-sea fluxes as functions of the surface wind speed averaged over the main timescales of atmospheric boundarylayer turbulence (e.g. are eddy-averaged ). For those fluxes depending linearly on surface wind speed (e.g. heat and moisture, to a first approximation), fluxes averaged over longer timescales, or over some spatial domain, are given by the bulk formulae in terms of the averaged wind speed. However, for other fluxes (e.g. momentum, gases, aerosols) the nonlinear dependence on 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) gridbox): in general, there is a difference between the area-averaged mean wind speed and the magnitude of the mean vector wind (e.g. Mahrt and Sun, 995). 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 the Quick Scatterometer (QuikSCAT) satellite (Jet Propulsion Laboratory, 200; Chelton et al., 2004). These data have provided an unprecedented opportunity to characterise 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 2

4 be characterised by quite specific 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 pdf of sea surface wind speed is characterised by a distinct relationship between statistical moments, such that the skewness 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, the winds 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 where skew(y) = S ( ) mean(y), () std(y) S(u) = Γ ( ) ( ) ( + x 3Γ + x + 2 x) ( ) + 2Γ 3 + x [ ( ) ( )] 3/2, x = u.086 (2) Γ + 2 x Γ 2 + x and 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 Figure, in which the horizontal axis has been scaled so that the Weibull relationship appears as a : 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 Figure that the Weibull distribution is a good approximation to the pdf of sea surface winds, it is worth emphasising that this distribution is an empirical model without mechanistic basis. Also plotted in Figure is the relationship between moments as simulated by an idealised 3

5 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 surface wind speed), turbulent downwards mixing of momentum from the free atmosphere, and ageostrophic tendencies with specified mean and fluctuating components (with the latter modelled as Gaussian white noise). Evidently, the idealised model is able to capture important aspects of the observed relationship between moments, particularly the low-skewness curvature. An evident weakness of the model is its inability to capture the high values of skewness in the upper right of Figure. A parameter to which skew(w) is sensitive (where w denotes the surface wind speed) in the Monahan (2006a) model is the rate at which momentum is mixed from above down into the atmospheric surface layer. In Monahan (2006a) this layer was of a specified 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 rate w e = K/H (which when H is the depth of the boundary layer corresponds to the entrainment rate between the boundary layer and the free atmosphere). As w e is changed, so too changes the range of values taken by skew(w) (Figure 2). In particular, the maximum values taken by skew(w) increase with increasing w e, asymptoting at the uppermost line for which skew(w) takes a maximum value of about 0.6 and does not take substantially negative values. In fact, this large w e 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 this limit to occur when the downwards mixing of momentum becomes a much more significant contribution to the momentum budget than surface drag. The model considered in Monahan (2004, 2006a) is based upon a number of simplifying approximations; among these were the specification of neutral surface stratification and fixed 4

6 surface layer thickness. In fact, the boundary-layer momentum budget is influenced by local surface stratification and boundary layer depth variations in three distinct ways:. The surface drag coefficient c d is a function of surface stratification (equivalently - assuming downgradient fluxes in the surface layer - surface buoyancy fluxes) such that an unstable stratification enhances surface turbulence (through buoyant generation of turbulent 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 turbulent kinetic energy) and decreases surface drag. Monin-Obukhov theory parametrises these effects through a correction term 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, 997) 2. Deepening of the boundary layer turbulently mixes momentum from the free atmosphere into the boundary layer, inducing a boundary layer momentum tendency. This turbulent mixing exists even for a boundary layer of constant thickness (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)), but is stronger when the boundary layer itself 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; 5

7 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 non-local and manifest through the pressure gradient force. The separation between the influence of surface stratification and boundary layer depth variations is somewhat artificial, as variations in 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 by boundary layer top clouds (e.g. Stevens, 2002; Medeiros et al., 2005). The focus of this study will be on the direct influence of surface buoyancy fluxes (though modification of the drag coefficient) and of boundary layer thickness variability (by whatever processes this is generated). Evidence of a relationship between variability in boundary layer height (denoted h) and the shape of the wind speed pdf is suggested by the negative correlation between December- January-February (DJF) 0-m ocean skew(w) and the ratio mean(h)/std(h) (Figure 3), as determined from the European Centre for Medium Range Weather Forecasts (ECMWF) ERA-40 reanalysis product (Simmons and Gibson, 2000). It is evident from Figure 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 non-assimilated quantities such as surface winds and (particularly) boundary layer thickness must be regarded skeptically. Nevertheless, Figure 3 is derived from a fully complex GCM with a sophisticated boundary layer scheme. As such, the relationship illustrated in Figure 3 is suggestive that at least 6

8 one of the missing factors in the idealised boundary layer momentum budget is an active boundary layer. The present study generalises the idealised 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 generalised 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 characterisation. The generalised model of the boundary layer momentum budget is described in Section 2, followed by a consideration of the effects of accounting for surface stratification in Section 3. 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 Idealised Boundary Layer Momentum Budget Model The original idealised boundary layer momentum budget of Monahan (2004, 2006a) modelled surface vector wind tendencies as resulting from an imbalance between four forces: () a mean non-local ageostrophic tendency, (2) fluctuations in the ageostrophic forcing, (3) surface drag, and (4) downwards mixing of momentum from above the (fixed-depth) layer. Because the layer thickness was fixed, the character of the winds above z = H did not need to be modelled explicitly and the associated 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 modelled explicitly. 7

9 Expressing the above ideas quantitatively: the boundary-layer momentum budget is given by du dt dv dt = = A {}}{ B {}}{ U s η u + τ s C {}}{ c d (w, T) h wu + τ s η v c d(w, T) τ s h D {}}{ E {}}{ w e (U(h + δ) u)+ σ u Ẇ (3) h wv + w e h (V (h + δ) v) + σ uẇ2. (4) The non-local ageostrophic forcing is expressed as the sum of mean (terms A) and fluctuating (term B) components (by definition there is no component of the mean forcing in the cross-mean wind direction). The quantity τ s is a characteristic surface wind adjustment timescale, specified so that mean(u) U s. Fluctuations in large-scale forcing are modelled as are red-noise processes with autocorrelation timescale τ η and mean zero: d dt η u = η u + τ η d dt η v = η v + τ η 2(τ η + τ s ) σ env Ẇ 3 (5) τ η 2(τ η + τ s ) σ env Ẇ 4. (6) τ η The noise terms are scaled so that σ 2 env is approximately the contribution to the variance of u (or v) associated with the non-local ageostrophic forcing (more precisely, a variable z described by dz/dt = ( z+η u )/τ s has standard deviation σ env ). Surface and boundary-layer top eddy momentum fluxes are given by terms C and D, respectively. Along with large-scale forcing of the boundary layer momentum budget, local fluctuating forcing is represented as white-noise forcing with scaling coefficient σ u (term E). The random processes Ẇ i are mutually uncorrelated white noise processes: mean(ẇi(t)ẇj(s)) = δ(t s)δ ij. (7) Variability in boundary layer depth is driven by an imbalance in tendencies between stratifying and mixing processes: d dt h = τ h h + w e + ξ τ h. (8) 8

10 The first term in Eqn. (8) represents the average tendency of restratifying processes (e.g. subsidence, radiation to space) which cause boundary layer heights to decrease on a timescale τ h, while the second term represents a baseline turbulent entrainment velocity which acts to deepen the mixed layer. Finally, the third term describes the net effect of variability in restratifying and entrainment rates and is described for simplicity as a red-noise process with autocorrelation timescale τ ξ : d dt ξ = τ ξ ξ + Σ ξ τ ξ Ẇ 5 (9) (where Ẇ5 is a white noise process uncorrelated with Ẇj, j =,..., 4).To ensure that the boundary layer height does not become negative, h is not allowed to decrease below a minimum value, h min. To a good approximation (exact in the absence of the lower limit h min ) h has the stationary standard deviation std(h) = Σ ξ 2(τh + τ ξ ) (0) and autocorrelation function c hh (t) = τ h τ ξ [ ( τ h exp t ) ( τ ξ exp t )]. () τ h τ ξ The rate at which momentum is mixed from the free atmosphere into the boundary layer is determined by the entrainment velocity w e = w e + τ h max (ξ, 0). (2) The first of these terms is the constant background entrainment rate, while the second is associated with those fluctuations in mixed layer depth which tend to deepen the mixed layer (restratifying processes do not unmix the boundary layer). Note that as modelled boundary layer variability is not influenced by surface stratification, wind speeds, or the state of the free atmosphere. In reality, the turbulent entrainment 9

11 rate at the top of the marine boundary layer is influenced by a number of factors, including the strength of the boundary layer-top inversion, generation of turbulent kinetic energy within the boundary layer (a process which is influenced by surface buoyancy fluxes) and the radiatively-driven generation of turbulence within boundary layer-top clouds (e.g. Stevens, 2002). From the point of view of the main concerns of this study, viz. the direct influences of surface stratification (through the drag coefficient) and boundary layer depth variability on the pdf of sea surface winds, 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 of state variables on boundary layer tendencies (other than the simple relaxation term on h) is a substantial approximation. 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. 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 (involving for example the introduction of new prognostic variables such as boundary layer potential temperature). 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 a short period of time 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 atmosphere wind profile U(z, t) = (U(z, t), V (z, t)). Within the boundary layer, U(z, t) and u(t) coincide: U(z, t) = u(t) 0 < z < h(t), (3) 0

12 while above z = h(t), in the free atmosphere, the horizontal winds relax on a timescale τ r to the large-scale environmental profile with shear Λ = (Λ u, Λ v ) U env (z) = (U env (z), V env (z)) = (U s + Λ u z, V s + Λ v z), (4) so t U(z, t) = τ r (U env (z) U(z, t)) z h(t). (5) The free atmospheric winds that are mixed down into the boundary layer are those at a small height δ above the top of the boundary layer. The surface stratification influences the momentum budget directly through changes in the character of boundary layer turbulence. A natural measure of surface stratification is the difference T between surface air temperature (SAT) and sea surface temperature (SST) T = SAT SST. (6) Unstable stratification (T < 0) enhances turbulence and increases the rate of turbulent momentum exchange with the underlying surface, increasing c d. Conversely, stable stratification (T > 0) inhibits surface turbulence and reduces c d. The drag coefficient is also a function of the sea-surface wind speed w (e.g. Csanady, 200), as the generation of surface ocean waves by surface winds increases surface roughness and therefore drag. At moderate to strong wind speeds over a developed sea, c d is an increasing function of w. For very weak winds, c d may also increase as w decreases in accordance with the characteristics of drag over an aerodynamically smooth surface. The functional dependence of c d on T and w in 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 c d (w, T) given by a local polynomial

13 approximation Kara et al. (2005) to 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 Figure 4. Together, these equations constitute a vector stochastic differential equation (SDE) for the state variables u, v, h, U, and V. General introductions to SDEs are presented in Gardiner (997) and Horsthemke and Lefever (2006); an introduction in the context of climate modelling 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 pdfs must be simulated by numerical integration of the associated SDEs (Kloeden and Platen, 992). 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 timescales than surface wind fluctuations (e.g. Sura and Newman, 2008), it is meaningful to consider the probability distribution of 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 surface wind speeds p(w) can be 2

14 computed: p(w) = p(w T)p(T) dt. (7) For simplicity, the following analysis will assume that the distribution of T is Gaussian with mean µ T and standard deviation σ T : p(t) = ( exp (T µ T) 2 ). (8) 2πσT 2 2σT 2 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-order approximation. To consider the direct effect of air-sea temperature differences on the pdf of surface winds, we will consider the model described in Section 2 with constant boundary layer depth h = w eτ h (which is 800 m for the standard parameter values). The analysis is also facilitated by considering the white-noise limit of the non-local ageostrophic forcing, τ η 0. In this limit we obtain the SDE d dt u = U s c d(w, T) τ s h d dt v = c d(w, T) h wu w e h u + wv w e h v + 2 τ s σ env Ẇ 3 + σ u Ẇ (9) 2 τ s σ env Ẇ 4 + σ u Ẇ 2, (20) with an associated Fokker-Planck equation for the stationary pdf conditioned on T, p(u, v T), which is analytically solvable: p(u, v T) = N exp ( 2τ s 2σ 2 env + σ2 u τ s { Us u w e τ s 2h (u2 + v 2 ) }) u 2 +v 2 c d (w, T)w 2 dw h 0 (as discussed in Monahan, 2006a). Integrating over wind direction, we obtain the marginal (2) 3

15 pdf of the wind speed (conditioned on T) p(w T) = ( ) ( 2U s w 2τ s wi 0 exp N 2 2σenv 2 + σuτ 2 s 2σenv 2 + σuτ 2 s { w e 2h w2 + w } ) c d (w, T)w 2 dw. h 0 The quantities N and N 2 are normalisation constants. Combining Eqns. (8) and (22) through (7), we obtain the marginal pdf of the wind speed p(w). (22) Moments of w computed from the pdf p(w) are contoured in Figure 5 as functions of µ T and σ T (over realistic ranges) for various values of U s and σ env. 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 µ T, while skew(w) decreases. The dependence of the moments on σ T is weaker and less systematic than that on µ T. The dependence of individual moments of w on µ T and σ T 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 µ T and σ T are illustrated in Figure 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 µ T is somewhat stronger than that on the standard deviation σ T, the curves associated with different values of µ T 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 sea surface wind speeds, providing a posteriori justification for the assumption of neutral stratification in the idealised boundary layer models in Monahan (2004, 2006a). In particular, non-neutral surface stratification cannot account for the inability of the idealised model to simulate large positive 4

16 wind speed skewness in conditions of light mean winds. He et al. (2009) also find that in terrestrial areas classified as open water (lakes and coastal regions) the diurnal and seasonal evolution of p(w) is 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, 992). 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., 2009). In the following section, we will consider the effects of variable boundary layer depth on the shape of p(w). 4 Effects of Variable Boundary Layer Depth The correlation between wind speed skewness and boundary layer depth variability illustrated in Figure 3 suggests that the specification of a fixed layer depth in the idealised 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 Figure. To test this hypothesis, moments of w were computed from the idealised model of the boundary layer momentum budget described in Section 2 over broad ranges of the parameters U s, σ env, 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 5 years of model time, with output saved every six hours) using a standard forward-euler technique (Kloeden and Platen, 992). It was demonstrated in Section 3 that 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 (µ T = σ T = 0 K). 5

17 The results of these simulations are displayed in Figure 7. The moment relationship from the model of Monahan (2006a) (displayed in Figure ) 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 non-local 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. In particular, the incorporation of variability in boundary layer thickness allows the model to better characterise the large positive values of skew(w) seen in observations. Consistent with the relationship between skew(w) and mean(h)/std(h) in the ERA-40 reanalysis (Figure 3), the idealised boundary layer model predicts that large positive wind speed skewness should be associated with strong variability in boundary layer depth. Furthermore, the range of moments predicted by the present model fills out the cloud of observed moments more completely than that of the model of Monahan (2006a). Sampling variability will contribute to the breadth of this cloud in observations, but the fact that clouds of comparable breadth are produced by longer datasets (e.g. in reanalysis winds; c.f. Monahan, 2006b) suggests that some fraction of this variability is real. While skew(w) in the idealised model is still biased low relative to 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 boundary layer depth influences the boundary layer momentum budget through episodic downwards 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 Figure 7. Model integrations with h varying but w e held fixed at mean(w e ) = w e + mean(max(ξ/τ h, 0)) (23) 6

18 (Figure 8, left panel) demonstrate that the simulated wind speed moments are essentially unchanged from those of the full model with variable w e. In contrast, integrations with w e varying but h held fixed in Eqns. (3) and (4) for the surface vector wind momentum budget (Figure 8, right panel) demonstrate that in this model the variable downwards mixing of momentum from aloft is not responsible for producing the larger positive values of skew(w) produced by the full model. These results are consistent with those of Samelson et al. (2006), in which the importance to the coupling between wind stress and sea surface temperature of variations in boundary layer depth relative to downwards mixing of momentum was emphasized. 5 Conclusions This study has considered the influence 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 sea surface wind speed is characterised 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 sea surface wind speeds using an idealised model of the boundary layer momentum budget, assuming neutral stratification and constant boundary layer depth, resulted in a reasonable first order approximation to this relationship between moments (Monahan, 2006a). However, this earlier model was not able to account for the large positive wind speed skewnesses seen in observations in conditions of light and variable winds. A generalisation of this earlier idealised model was used to assess the relative importance of surface stability-driven variations in the drag coefficient, the 7

19 downwards 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 modelled 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 observations. Accounting for variability in 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)). These improvements in agreement between simulated and observed surface 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. The episodic downwards 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 characterised by a relationship between sea surface wind speed skewness and 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 variability in boundary layer thickness is relatively large compared to its mean value. While reanalysis data are not observations, and extreme caution must be exercised in con- 8

20 sideration 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 idealised 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. (2009) 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, this study 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, 992), 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 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 modelled relationship between moments is more similar to that of the Weibull distribution than that of the observed sea surface winds: values of skew(w) are still systematically underestimated. Of course, while the present model is more general than that considered in Monahan (2006a), it remains a highly idealised single-column slab model. It is possible that the model deficiencies would be addressed through a more complete consideration of vertical and horizontal momentum transport in the boundary layer. Furthermore, the present model represents as constant in time the profile to which the free atmosphere 9

21 winds relax; in reality, this profile varies with the large-scale weather. The present study has also made the simplifying assumption that variability in boundary layer thickness can be decoupled from variability in 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 influence 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 influence of these various boundary layer processes on the pdf of surface wind speeds is a potentially important direction of future research. The analysis presented in this study provides further insight regarding the physical factors that control the shape of the sea surface wind speed pdf. This developing mechanistic understanding holds the promise of improvements to 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; as we improve our understanding of this field, so our understanding improves of the climate past, present, and future. 20

22 Acknowledgements The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada. The author would like to thank Yanping He for valuable comments on this manuscript. 2

23 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, Csanady, G., 200: Air-Sea Interaction: Laws and Mechanisms. Cambridge University Press, Cambridge, UK, 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, 6, Gardiner, C. W., 997: Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Springer, 442 pp. Garratt, J., 992: The Atmospheric Boundary Layer. Cambridge University Press, Cambridge, UK, 36 pp. He, Y., A. H. Monahan, C. G. Jones, A. Dai, S. Biner, D. Caya, and K. Winger, 2009: Land surface wind speed probability distributions in North America: Observations, theory, and regional climate model simulations. J. Geophys. Res., in review. Horsthemke, W. and R. Lefever, 2006: Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology. Springer-Verlag, Berlin, 38 pp. Jet Propulsion Laboratory, 200: SeaWinds on QuikSCAT Level 3: Daily, Gridded Ocean Wind Vectors. Tech. Rep. Tech. Rep. JPL PO.DAAC Product 09, California Institute of Technology. 22

24 Kara, A. B., H. E. Hurlburt, and A. J. Wallcraft, 2005: Stability-dependent exchange coefficients for air-sea fluxes. J. Atmos. Ocean. Tech., 22, Kloeden, P. E. and E. Platen, 992: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin, 632 pp. Mahrt, L. and J. Sun, 995: The subgrid velocity scale in the bulk aerodynamic relationship for spatially averaged scalar fluxes. Mon. Weath. Rev., 23, Medeiros, B., A. Hall, and B. Stevens, 2005: What controls the mean depth of the PBL? J. Climate, 8, Monahan, A. H., 2004: A simple model for the skewness of global sea-surface winds. J. Atmos. Sci., 6, Monahan, A. H., 2006a: The probability distribution of sea surface wind speeds. Part I: Theory and SeaWinds observations. J. Climate, 9, Monahan, A. H., 2006b: The probability distribution of sea surface wind speeds. Part II: Dataset intercomparison and seasonal variability. J. Climate, 9, Monahan, A. H., 2007: Empirical models of the probability distribution of sea surface wind speeds. J. Climate, 20, Penland, C., 2003: Noise out of chaos and why it won t go away. Bull. Am. Met. Soc., 84, 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, 9,

25 Simmons, A. and J. Gibson, 2000: The ERA-40 Project Plan. ERA-40 Project Report Series No., ECMWF, Reading, RG2 9AX, UK. 63 pp. Small, R., et al., 2008: Air-sea interaction over ocean fronts and eddies. Dyn. Atmos. Oceans, 45, Spall, M. A., 2007: Midlatitude wind stress - sea surface temperature coupling in the vicinity of oceanic fronts. J. Climate, 20, Stevens, B., 2002: Entrainment in stratocumulus-topped mixed layers. Q. J. R. Meteorol. Soc., 28, Stull, R. B., 997: An Introduction to Boundary Layer Meteorology. Kluwer, Dordrecht, 670 pp. Sura, P. and M. Newman, 2008: The impact of rapid wind variability upon air-sea thermal coupling. J. Climate, 2, Wanninkhof, R., S. C. Doney, T. Takahashi, and W. R. McGillis, 2002: The effect of using time-averaged winds on regional air-sea CO 2 fluxes. Gas Transfer at Water Surfaces, M. A. Donelan, W. M. Drennan, E. S. Saltzman, and R. Wanninkhof, Eds., American Geophysical Union,

26 Figure Captions Figure : Relationship between wind speed skewness from observations (contours) and an idealised boundary layer model (red dots). The horizontal axis is scaled by the function S(x) (Eqn 2) so that the relationship between moments for a Weibull distributed variable (thick black curve) falls along the : line. The observed wind speeds are taken from level 3.0 gridded daily QuikScat SeaWinds observations from , as described in Monahan (2006a). The model is as described in Section 2 with parameter values τ η = 0 and std(h) = 0 (corresponding to the model in Monahan (2006a)). Figure 2: As in Figure, for four different values of the entrainment velocity w e = 0m/s (blue), w e = 0.0 m/s (red), w e = 0.02 m/s (magenta), w e = 0.05 (green). Figure 3: Kernel density estimate of joint pdf of the ratio of the mean to the standard deviation of boundary layer thickness mean(h)/std(h) and sea surface wind speed skewness skew(w), as estimated for the DJF season from ERA-40 reanalyses (6-hourly data on a grid from 65 S to 60 N from September 957 to 3 August 2002; downloaded from Figure 4: Dependence of drag coefficient on 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 c d (w, T). Figure 5: Contour plots of leading three moments of sea surface winds (mean(w), std(w), skew(w)) as functions of µ T and σ T, computed from Eqns. (7), (8),and (22). Upper panel: (U s, σ env ) = (0, 7) ms ; middle panel: (U s, σ env ) = (5, 4) ms ; bottom panel: (U s, σ env ) = (0, 2) ms. 25

27 Figure 6: As in Figure, for the idealised model of the boundary layer momentum budget accounting for dependence of the drag coefficient on surface stratification. Left panel: σ T = 0 K. Right panel: σ T = 3 K. In both panels, µ T = -3K (blue dots), µ T = -K (maroon dots), µ T = K (green dots), and µ T = 3K (yellow dots). Figure 7: As in Figure, for the idealised boundary layer model with fluctuating boundary layer depth: std(h) = 0 m (blue dots), std(h) = 200 m (magenta dots), and std(h) = 400 m (green dots). Figure 8: As in Figure 7, with the results of the boundary layer model for std(h) = (0, 200, 400) m (blue dots). Left panel: moments of simulated wind with entrainment velocity w e = mean(w e ) = we + mean(max(ξ/τ h, 0)) held constant (red dots). Right panel: moments of simulated wind with boundary layer depth held constant in momentum budget (red dots). 26

28 Table Captions Table : Model coordinates and state variables. Table 2: Model parameters and standard values. 27

29 skew(w) S(mean(w)/std(w)) Figure : Relationship between wind speed skewness from observations (contours) and an idealised boundary layer model (red dots). The horizontal axis is scaled by the function S(x) (Eqn 2) so that the relationship between moments for a Weibull distributed variable (thick black curve) falls along the : line. The observed wind speeds are taken from level 3.0 gridded daily QuikScat SeaWinds observations from , as described in Monahan (2006a). The model is as described in Section 2 with parameter values τ η = 0 and std(h) = 0 (corresponding to the model in Monahan (2006a)). 28

30 skew(w) S(mean(w)/std(w)) Figure 2: As in Figure, for four different values of the entrainment velocity we = 0m/s (blue), we = 0.0 m/s (red), we = 0.02 m/s (magenta), we = 0.05 (green). 29

31 0.5 skew(w) mean(h)/std(h) Figure 3: Kernel density estimate of joint pdf of the ratio of the mean to the standard deviation of boundary layer thickness mean(h)/std(h) and sea surface wind speed skewness skew(w), as estimated for the DJF season from ERA-40 reanalyses (6-hourly data on a grid from 65 S to 60 N from September 957 to 3 August 2002; downloaded from 30

32 w (ms ) T (K) Figure 4: Dependence of drag coefficient on 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 c d (w, T). 3

33 σ T (K) mean(w) (ms ) std(w) (ms ) skew(w) σ T (K) σ T (K) µ T (K) µ (K) T µ (K) T Figure 5: Contour plots of leading three moments of sea surface winds (mean(w), std(w), skew(w)) as functions of µ T and σ T, computed from Eqns. (7), (8),and (22). Upper panel: (U s, σ env ) = (0, 7) ms ; middle panel: (U s, σ env ) = (5, 4) ms ; bottom panel: (U s, σ env ) = (0, 2) ms. 32

34 σ T = 0 K σ T = 3 K skew(w) skew(w) S(mean(w)/std(w)) S(mean(w)/std(w)) Figure 6: As in Figure, for the idealised model of the boundary layer momentum budget accounting for dependence of the drag coefficient on surface stratification. Left panel: σ T = 0 K. Right panel: σ T = 3 K. In both panels, µ T = -3K (blue dots), µ T = -K (maroon dots), µ T = K (green dots), and µ T = 3K (yellow dots). 33

35 skew(w) S(mean(w)/std(w)) Figure 7: As in Figure, for the idealised boundary layer model with fluctuating boundary layer depth: std(h) = 0 m (blue dots), std(h) = 200 m (magenta dots), and std(h) = 400 m (green dots). 34

36 w e = const. h = const skew(w) skew(w) S(mean(w)/std(w)) S(mean(w)/std(w)) Figure 8: As in Figure 7, with the results of the boundary layer model for std(h) = (0, 200, 400) m (blue dots). Left panel: moments of simulated wind with entrainment velocity w e = mean(w e ) = we +mean(max(ξ/τ h, 0)) held constant (red dots). Right panel: moments of simulated wind with boundary layer depth held constant in momentum budget (red dots). 35

37 Variable z t h(t) u(t) v(t) u(t) = (u(t), v(t)) w(t) = u 2 + v 2 U(z, t) V (z, t) U(z, t) = (U(z, t), V (z, t)) T(t) Definition vertical coordinate time boundary-layer thickness along-mean horizontal wind component cross-mean horizontal wind component surface horizontal vector wind surface horizontal wind speed along-mean wind component of large-scale wind profile cross-mean wind component of large-scale wind profile large-scale horizontal wind profile air-sea temperature difference (SAT-SST) Table : Model coordinates and state variables 36

38 Parameter Definition Standard value Λ = (Λ u, Λ v ) environmental wind shear (3,0) ms km τ r free atmosphere momentum relaxation timescale 6 h τ s adjustment timescale of large-scale wind profile to forcing 8 h τ η relaxation timescale of non-local ageostrophic forcing 2 h U s = (U s, 0) mean surface wind from non-local forcing variable σ env standard deviation of non-local forcing variable we background entrainment velocity ms h min minimum boundary layer depth 0 m τ h boundary layer relaxation timescale d τ ξ boundary layer forcing autocorrelation timescale 3 h Σ ξ boundary layer forcing strength parameter variable σ u scaling factor for local fluctuations in momentum forcing 0.0 ms /2 c d (w, T) surface drag coefficient variable Table 2: Model parameters and standard values. 37