The Probability Distribution of
Sea Surface Wind Speeds:
Effects of Variable Surface Stratification
and Boundary Layer Thickness
Adam Hugh Monahan([email protected])
School of Earth and Ocean Sciences
University of VictoriaP.O. Box 3065 STN CSC
Victoria, BC, Canada, V8W 3V6
Submitted to Journal of Climate
March 30, 2009
1
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 influ-
ence, 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 con-
sidered 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 variabil-
ity 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.
1
1 Introduction
Air-sea exchanges of momentum, energy, and material substances, of fundamental impor-
tance 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 influ-
enced by, these fluxes. Standard bulk parameterisations express air-sea fluxes as functions
of the surface wind speed averaged over the “main timescales” of atmospheric boundary-
layer 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 non-
linear 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, 1995). Ac-
curate 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 scatterom-
eter on the Quick Scatterometer (QuikSCAT) satellite (Jet Propulsion Laboratory, 2001;
Chelton et al., 2004). These data have provided an unprecedented opportunity to charac-
terise 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
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 charac-
teristic 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
skew(y) = S
(
mean(y)
std(y)
)
, (1)
where
S(u) =Γ(
1 + 1
x
)
− 3Γ(
1 + 1
x
) (
1 + 2
x
)
+ 2Γ3(
1 + 1
x
)
[
Γ(
1 + 2
x
)
− Γ2
(
1 + 1
x
)]3/2, x = u1.086 (2)
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
1, in which 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 Figure 1 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 1 is the relationship between moments as simulated by an idealised
3
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 1.
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 we = 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 we 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 we, 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 we 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
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:
1. The surface drag coefficient cd 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, 1997)
2. Deepening of the boundary layer turbulently mixes momentum from the free atmo-
sphere 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 dis-
tributed across fluid parcels throughout the depth of the boundary layer. As the bound-
ary layer becomes thicker, these interfacial fluxes are therefore diluted and weakened;
5
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) 10-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
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: (1) 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
Expressing the above ideas quantitatively: the boundary-layer momentum budget is given
by
du
dt=
A︷︸︸︷
U s
τs+
B︷︸︸︷ηu
τs−
C︷ ︸︸ ︷
cd(w, T )
hwu +
D︷ ︸︸ ︷we
h(U(h + δ) − u)+
E︷ ︸︸ ︷
σuW1 (3)
dv
dt=
ηv
τs
− cd(w, T )
hwv +
we
h(V (h + δ) − v) + σuW2. (4)
The “non-local” ageostrophic forcing is expressed as the sum of mean (terms A) and fluc-
tuating (term B) components (by definition there is no component of the mean forcing in
the cross-mean wind direction). The quantity τs is a characteristic surface wind adjustment
timescale, specified so that mean(u) ∼ Us. Fluctuations in large-scale forcing are modelled
as are red-noise processes with autocorrelation timescale τη and mean zero:
d
dtηu = − 1
τηηu +
√
2(τη + τs)σenv
τηW3 (5)
d
dtηv = − 1
τη
ηv +√
2(τη + τs)σenv
τη
W4. (6)
The noise terms are scaled so that σ2env 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+ηu)/τs has standard deviation σenv). 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 σu (term E). The random processes Wi are
mutually uncorrelated white noise processes:
mean(Wi(t)Wj(s)) = δ(t − s)δij. (7)
Variability in boundary layer depth is driven by an imbalance in tendencies between
stratifying and mixing processes:
d
dth = − 1
τhh + w∗
e +ξ
τh. (8)
8
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
τh, 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 τξ:
d
dtξ = − 1
τξ
ξ +Σξ
τξ
W5 (9)
(where W5 is a white noise process uncorrelated with Wj , j = 1, ..., 4).To ensure that the
boundary layer height does not become negative, h is not allowed to decrease below a mini-
mum value, hmin. To a good approximation (exact in the absence of the lower limit hmin) h
has the stationary standard deviation
std(h) =Σξ
√
2(τh + τξ)(10)
and autocorrelation function
chh(t) =1
τh − τξ
[
τh exp
(
−|t|τh
)
− τξ exp
(
−|t|τξ
)]
. (11)
The rate at which momentum is mixed from the free atmosphere into the boundary layer
is determined by the entrainment velocity
we = w∗e +
1
τh
max (ξ, 0) . (12)
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 stratifica-
tion, wind speeds, or the state of the free atmosphere. In reality, the turbulent entrainment
9
rate at the top of the marine boundary layer is influenced by a number of factors, includ-
ing 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 me-
chanical 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), (13)
10
while above z = h(t), in the free atmosphere, the horizontal winds relax on a timescale τr to
the “large-scale” environmental profile with shear Λ = (Λu, Λv)
Uenv(z) = (Uenv(z), Venv(z)) = (Us + Λuz, Vs + Λvz), (14)
so
∂tU(z, t) =1
τr(Uenv(z) − U(z, t)) z ≥ h(t). (15)
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. (16)
Unstable stratification (T < 0) enhances turbulence and increases the rate of turbulent
momentum exchange with the underlying surface, increasing cd. Conversely, stable strati-
fication (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 surface
ocean waves by surface winds increases surface roughness and therefore 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
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
11
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 Gar-
diner (1997) 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, 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 variabil-
ity 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
12
computed:
p(w) =∫
p(w|T )p(T ) dT . (17)
For simplicity, the following analysis will assume that the distribution of T is Gaussian with
mean µT and standard deviation σT :
p(T ) =1
√
2πσ2T
exp
(
−(T − µT )2
2σ2T
)
. (18)
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∗eτh (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, τη → 0. In this
limit we obtain the SDE
d
dtu =
U s
τs− cd(w, T )
hwu − w∗
e
hu +
√
2
τsσenvW3 + σuW1 (19)
d
dtv = −cd(w, T )
hwv − w∗
e
hv +
√
2
τs
σenvW4 + σuW2, (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 ) =1
N1
exp
(
2τs
2σ2env + σ2
uτs
{
U s
τsu − w∗
e
2h(u2 + v2) − 1
h
∫√
u2+v2
0
cd(w′, T )w′2 dw′
})
(21)
(as discussed in Monahan, 2006a). Integrating over wind direction, we obtain the marginal
13
pdf of the wind speed (conditioned on T )
p(w|T ) =1
N2
wI0
(
2Usw
2σ2env + σ2
uτs
)
exp
(
− 2τs
2σ2env + σ2
uτs
{w∗
e
2hw2 +
1
h
∫ w
0
cd(w′, T )w′2 dw′
})
.
(22)
The quantities N1 and N2 are normalisation constants. Combining Eqns. (18) and (22)
through (17), we obtain the marginal pdf of the wind speed p(w).
Moments of w computed from the pdf p(w) are contoured in Figure 5 as functions of µT
and σT (over realistic ranges) for various values of U s 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 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 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 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 strat-
ification cannot account for the inability of the idealised model to simulate large positive
14
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 sea-
sonal 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, 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., 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 1. 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, 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 differ-
ential equations were integrated numerically (for 15 years of model time, with output saved
every six hours) using a standard forward-Euler technique (Kloeden and Platen, 1992). 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 (µT = σT = 0 K).
15
The results of these simulations are displayed in Figure 7. The moment relationship from
the model of Monahan (2006a) (displayed in Figure 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 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 particu-
lar, 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 as-
sociated 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 bound-
ary layer momentum budget through episodic downwards mixing of momentum from the free
atmosphere and modulation of the strength of interfacial turbulent momentum fluxes rela-
tive 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 we held fixed at
mean(we) = w∗e + mean(max(ξ/τh, 0)) (23)
16
(Figure 8, left panel) 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 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 tem-
perature 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
17
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) influ-
ences the simulated moments of the surface wind speed pdf, it has an insubstantial
effect on the modelled relationship between surface wind speed moments. In particu-
lar, 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 partic-
ular, 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 down-
wards 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-
18
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, 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 wind speed moments into closer agreement with the observed relation-
ship, 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 consid-
eration 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
19
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 in-
crease 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
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.
21
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, 978–983.
Csanady, G., 2001: 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, 16, 571–591.
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, Cam-
bridge, UK, 316 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, 318 pp.
Jet Propulsion Laboratory, 2001: SeaWinds on QuikSCAT Level 3: Daily, Gridded Ocean
Wind Vectors. Tech. Rep. Tech. Rep. JPL PO.DAAC Product 109, California Institute of
Technology.
22
Kara, A. B., H. E. Hurlburt, and A. J. Wallcraft, 2005: Stability-dependent exchange coef-
ficients for air-sea fluxes. J. Atmos. Ocean. Tech., 22, 1080–1094.
Kloeden, P. E. and E. Platen, 1992: Numerical Solution of Stochastic Differential Equations.
Springer-Verlag, Berlin, 632 pp.
Mahrt, L. and J. Sun, 1995: The subgrid velocity scale in the bulk aerodynamic relationship
for spatially averaged scalar fluxes. Mon. Weath. Rev., 123, 3032–3041.
Medeiros, B., A. Hall, and B. Stevens, 2005: What controls the mean depth of the PBL? J.
Climate, 18, 3157–3172.
Monahan, A. H., 2004: A simple model for the skewness of global sea-surface winds. J.
Atmos. Sci., 61, 2037–2049.
Monahan, A. H., 2006a: The probability distribution of sea surface wind speeds. Part I:
Theory and SeaWinds observations. J. Climate, 19, 497–520.
Monahan, A. H., 2006b: The probability distribution of sea surface wind speeds. Part II:
Dataset intercomparison and seasonal variability. J. Climate, 19, 521–534.
Monahan, A. H., 2007: Empirical models of the probability distribution of sea surface wind
speeds. J. Climate, 20, 5798–5814.
Penland, C., 2003: Noise out of chaos and why it won’t go away. Bull. Am. Met. Soc., 84,
921–925.
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, 1557–1566.
23
Simmons, A. and J. Gibson, 2000: The ERA-40 Project Plan. ERA-40 Project Report Series
No. 1, ECMWF, Reading, RG2 9AX, UK. 63 pp.
Small, R., et al., 2008: Air-sea interaction over ocean fronts and eddies. Dyn. Atmos. Oceans,
45, 274–319.
Spall, M. A., 2007: Midlatitude wind stress - sea surface temperature coupling in the vicinity
of oceanic fronts. J. Climate, 20, 3785–3801.
Stevens, B., 2002: Entrainment in stratocumulus-topped mixed layers. Q. J. R. Meteorol.
Soc., 128, 2663–2690.
Stull, R. B., 1997: 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, 21, 621–637.
Wanninkhof, R., S. C. Doney, T. Takahashi, and W. R. McGillis, 2002: The effect of us-
ing time-averaged winds on regional air-sea CO2 fluxes. Gas Transfer at Water Surfaces,
M. A. Donelan, W. M. Drennan, E. S. Saltzman, and R. Wanninkhof, Eds., American
Geophysical Union, 351–356.
24
Figure Captions
Figure 1: 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 1:1 line. The observed wind speeds are taken from level
3.0 gridded daily QuikScat SeaWinds observations from 1999-2008, 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)).
Figure 2: As in Figure 1, for four different values of the entrainment velocity we = 0m/s
(blue), we = 0.01 m/s (red), we = 0.02 m/s (magenta), we = 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
2.5◦ × 2.5◦ grid from 65◦S to 60◦N from 1 September 1957 to 31 August 2002; downloaded
from http://data.ecmwf.int/data/d/era40/).
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 cd(w, T ).
Figure 5: Contour plots of leading three moments of sea surface winds (mean(w), std(w),
skew(w)) as functions of µT and σT , computed from Eqns. (17), (18),and (22). Upper
panel: (U s, σenv) = (0, 7) ms−1; middle panel: (U s, σenv) = (5, 4) ms−1; bottom panel:
(U s, σenv) = (10, 2) ms−1.
25
Figure 6: As in Figure 1, for the idealised model of the boundary layer momentum budget
accounting for dependence of the drag coefficient on surface stratification. Left panel:
σT = 0 K. Right panel: σT = 3 K. In both panels, µT = -3K (blue dots), µT = -1K (maroon
dots), µT = 1K (green dots), and µT = 3K (yellow dots).
Figure 7: As in Figure 1, 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
we = mean(we) = w∗e + mean(max(ξ/τh, 0)) held constant (red dots). Right panel:
moments of simulated wind with boundary layer depth held constant in momentum budget
(red dots).
26
Table Captions
Table 1: Model coordinates and state variables.
Table 2: Model parameters and standard values.
27
S(mean(w)/std(w))
skew
(w)
−0.5 0 0.5 1
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
0.100 0.316 1.000 3.162 10.000
Figure 1: 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 1:1 line. The observed wind speeds are taken from level
3.0 gridded daily QuikScat SeaWinds observations from 1999-2008, 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)).
28
S(mean(w)/std(w))
skew
(w)
−0.5 0 0.5 1
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
0.100 0.316 1.000 3.162 10.000
Figure 2: As in Figure 1, for four different values of the entrainment velocity we = 0m/s
(blue), we = 0.01 m/s (red), we = 0.02 m/s (magenta), we = 0.05 (green).
29
mean(h)/std(h)
skew
(w)
0 1 2 3 4 5 6 7−1
−0.5
0
0.5
1
0.010 0.025 0.063 0.158 0.398
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
2.5◦ × 2.5◦ grid from 65◦S to 60◦N from 1 September 1957 to 31 August 2002; downloaded
from http://data.ecmwf.int/data/d/era40/).
30
0.25 0.250.250.750.75
1.25
1.251.25
1.251.25
1.75 1.751.75
2.25 2.25 2.25
0.5
0.5
1
11
1.5
1.51.5
1.5
2 2 2 2
2.5 2.5 2.5
T (K)
w (
ms−
1 )
−6 −4 −2 0 2 4 6
5
10
15
20
25
30
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 cd(w, T ).
31
σ T (
K)
mean(w) (ms−1)
−4 −2 0 2 4
1
2
3
6.4
6.5
6.6
std(w) (ms−1)
−4 −2 0 2 4
1
2
3
3.063.083.13.123.143.16
skew(w)
−4 −2 0 2 4
1
2
3
0.32
0.34
0.36
σ T (
K)
−4 −2 0 2 4
1
2
3
5
5.2
5.4
−4 −2 0 2 4
1
2
3
2.322.342.362.382.42.422.44
−4 −2 0 2 4
1
2
3
0.25
0.3
µT (K)
σ T (
K)
−4 −2 0 2 4
1
2
3
6.2
6.4
6.6
6.8
µT (K)
−4 −2 0 2 4
1
2
3
1.7
1.72
µT (K)
−4 −2 0 2 4
1
2
3
−0.22
−0.2
−0.18
Figure 5: Contour plots of leading three moments of sea surface winds (mean(w), std(w),
skew(w)) as functions of µT and σT , computed from Eqns. (17), (18),and (22). Upper panel:
(U s, σenv) = (0, 7) ms−1; middle panel: (U s, σenv) = (5, 4) ms−1; bottom panel: (U s, σenv) =
(10, 2) ms−1.
32
S(mean(w)/std(w))
skew
(w)
σT = 0 K
−0.5 0 0.5 1
−0.5
0
0.5
1
0.100 0.316 1.000 3.162 10.000
S(mean(w)/std(w))
skew
(w)
σT = 3 K
−0.5 0 0.5 1
−0.5
0
0.5
1
0.100 0.316 1.000 3.162 10.000
Figure 6: As in Figure 1, for the idealised model of the boundary layer momentum budget
accounting for dependence of the drag coefficient on surface stratification. Left panel: σT = 0
K. Right panel: σT = 3 K. In both panels, µT = -3K (blue dots), µT = -1K (maroon dots),
µT = 1K (green dots), and µT = 3K (yellow dots).
33
S(mean(w)/std(w))
skew
(w)
−0.5 0 0.5 1
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
0.100 0.316 1.000 3.162 10.000
Figure 7: As in Figure 1, 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
S(mean(w)/std(w))
skew
(w)
we = const.
−0.5 0 0.5 1
−0.5
0
0.5
1
0.100 0.316 1.000 3.162 10.000
S(mean(w)/std(w))
skew
(w)
h = const
−0.5 0 0.5 1
−0.5
0
0.5
1
0.100 0.316 1.000 3.162 10.000
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
we = mean(we) = w∗e +mean(max(ξ/τh, 0)) held constant (red dots). Right panel: moments
of simulated wind with boundary layer depth held constant in momentum budget (red dots).
35
Variable Definition
z vertical coordinate
t time
h(t) boundary-layer thickness
u(t) along-mean horizontal wind component
v(t) cross-mean horizontal wind component
u(t) = (u(t), v(t)) surface horizontal vector wind
w(t) =√
u2 + v2 surface horizontal wind speed
U(z, t) along-mean wind component of“large-scale” wind profile
V (z, t) cross-mean wind component of “large-scale” wind profile
U(z, t) = (U(z, t), V (z, t)) “large-scale” horizontal wind profile
T (t) air-sea temperature difference (SAT-SST)
Table 1: Model coordinates and state variables
36
Parameter Definition Standard value
Λ = (Λu, Λv) environmental wind shear (3,0) ms−1km−1
τr free atmosphere momentum relaxation timescale 6 h
τs adjustment timescale of large-scale wind profile to forcing 18 h
τη relaxation timescale of non-local ageostrophic forcing 12 h
Us = (U s, 0) mean surface wind from non-local forcing variable
σenv standard deviation of non-local forcing variable
w∗e background entrainment velocity 0.009 ms−1
hmin minimum boundary layer depth 10 m
τh boundary layer relaxation timescale 1 d
τξ boundary layer forcing autocorrelation timescale 3 h
Σξ boundary layer forcing strength parameter variable
σu scaling factor for “local” fluctuations in momentum forcing 0.01 ms−1/2
cd(w, T ) surface drag coefficient variable
Table 2: Model parameters and standard values.
37
Top Related