**FLUX MODELING AND REMOTE SENSING**

**APPLIED TO AIR-SEA INTERACTIONS**

Mark A. Bourassa

*Center for Ocean-Atmospheric Prediction Studies, Florida State
University, Tallahassee, FL 32306-2840, USA*

**ABSTRACT**

An improved model is developed for the dependency of surface turbulent
stress on wave characteristics. Recent studies have used differences
between satellite and in situ observations to gain insights into the
physical processes that might be related to air-sea interaction. Both
scatterometers and buoys provide very accurate measurements of wind
speed. Differences between these measurements can be explained in terms
of the different mechanisms to which the instruments respond. A
physically-based flux model is developed herein. The model is validated
with recent observations gathered under severe conditions. The model
explains more wave-related variability than previous models.

# INTRODUCTION

Surface stress over water is primarily dependent on wind speed, but it
is also dependent on stratification of the atmosphere (atmospheric
stability) and sea state (i.e., characteristics of the wave field). A
typical assumption in GCM modeling of surface fluxes is that the wind
and the waves are in a state of local equilibrium or a prescribed state
of non-equilibrium. This assumption is equivalent to specifying a sea
state. Recent examinations (Bourassa et al., 1999; Taylor and Yelland,
2001) based on in situ data have shown that this assumption is invalid
for most spatial and temporal scales where there is substantial
variability in sea state. Physical mechanisms have been proposed to
account for the dependency of stress on sea state (Kusaba and Masuda,
1988; Geernaert et al., 1990; Toba et al., 1990; Perrie and Toulany,
1990; Maat et al., 1991; Smith et al., 1992; Yelland et al., 1998;
Bourassa et al., 1999, 2001); however, an empirical formulation based on
a much wider range of wind speeds and non-directional sea states (Taylor
and Yelland, 2001) has been shown to provide substantially better
matches to observations that cover a wide range of conditions.
Differences between satellite and in situ observations have been used to
infer the relative importance of several wave characteristics (Quilfen
et al., 2001). The results of this study are correlations, which do not
explain the physical mechanisms through which these wave characteristics
modify surface fluxes. A physical mechanism that utilizes these
satellite-derived insights is developed, and it is found to be largely
consistent with the results of Taylor and Yelland's (2001) empirical
relation. One advantage of this physically-based mechanism is that it
also considers directional sea state (i.e., the wind direction relative
to the direction of wave propagation). This additional consideration
makes the flux model consistent with observations over a wider range of
conditions. The non-directional impacts on surface turbulent stress are
developed and validated herein.

A stress-dependent model for significant wave height is coupled with a
sea state dependent surface flux model to demonstrate the
interdependence of surface fluxes and wave characteristics. It is shown
that significant wave height can be used, in conjunction with wind
speed, as an indicator for the departure from local wind-wave
equilibrium as well as the calculation of non-equilibrium fluxes. This
result is largely consistent with the empirical formulation of Taylor
and Yelland (2001); however, the physical impacts of sea state are
parameterized through the influences of the surface's orbital motion
induced by waves, rather than the slope of the waves. Such a model has
been successfully applied to capillary-wave related surface stress
(Bourassa et al., 1999); however, such wave dominate stress for ten
meter wind speeds (*U*~10~) \< \~5 ms^-1^. The mechanism was not applied
to gravity wave related surface stresses, which dominate greater wind
speeds due to the lack of appropriate data for validation. A recent
study of differences between satellite and in situ observations of
surface vector winds (Quilfen et al., 2001) has shown that the
differences between observed and model radar backscatter are well
correlated to wave slope characteristics, but better correlated to the
wave-induced surface motion. This finding supports the physical
reasoning applied in the flux model developed herein, for low wind
speeds as well as high wind speeds.

Differences from local-equilibrium values are regionally systematic and
result in large changes in stress (−10% and \>50%) and latent heat flux
(−10s to +100s of W m^-2^).

# Data

The impacts of sea state on stress have previously been validated
(Bourassa et al., 1999) for conditions dominated by capillary waves. The
model reproduced observed changes in stress as a function of directional
sea state. The developments in this study are applicable to conditions
where stress is dominated by gravity waves. There are few observational
data sets that contain both stress and wave observations. A preliminary
version of observations from the Storm Wave Study experiment (SWS-2;
Dobson et al., 1999; Taylor et al., 1999) was kindly provided by Peter
K. Taylor. These observations have the additional advantage of covering
a large range of wind conditions: 4 ms^-1^ to 24 ms^-1^, with the
quality control criteria described below. It is also the data set that
Taylor and Yelland (personal communication, 2000) used to demonstrate
that models using the HEXOS parameterization (e.g., Bourassa et al.,
1999) overestimated stresses for high wind speeds.

The following constraints were applied to the SWS-2 observations.

1\) The sonic anemometer's estimate of the speed of sound is less than
332 ms^-1^,

2\) The standard deviation of the platform's heading is less than 8°,

3\) The calculated orbital velocity of the waves is less than 2.5
ms^-1^, and

4\) The dimensionless Monin−Obukhov scale length (*z/L*) is less than
0.18.

They serve as quality control on the stress and wave observations, as
well as remove conditions that were too far from neutral to be
considered examinations of near-neutral conditions.

# Flux Model

The downward momentum flux (**τ**) can be modeled in terms of the
friction velocity (**u~∗~**):

**τ***  = ρ* **u\*** \| **u\*** \| , (1)

where *ρ* is the density of the air. The upward surface turbulent fluxes
of sensible (*H*) and latent heat (*E*) are

H*  = −ρ C~p~ θ***\*** \| **u\*** \| , and (2)

E*  = −ρ L~v~ q***\*** \| **u\*** \| , (3)

where *θ***\*** and *q***\*** are scaling parameters analogous to
*u*~∗~, *C~p~* is the specific heat of air, and *L~v~* is the latent
heat of vaporization. The goal of this study is to improve the accuracy
of modeled values of *u*~∗~, which will improve the accuracy of these
surface turbulent fluxes.

The direct influence of surface waves on flux and airflow
characteristics (**u~∗~** and *z~o~*) is determined by the relation
between **u~∗~** and roughness length (*z~o~*). Given *z~o~*(**u~∗~**)
and the modified log wind relation **U**(*z*), where *z* is the height
above the local mean surface, it is possible to iteratively solve for
**u~∗~**(**U**) and **τ**(**U**). The modified log−wind relation is

$U(z) - U_{s} = \frac{u_{{}_{}}}{k}\left\lbrack \text{ln}\left( \frac{z}{z_{o}} + 1 \right) + j\left( z,z_{o},L \right) \right\rbrack$,
(4)

where *k* is von Kârmân's constant for momentum, and *L* is the
Monin-Obukhov stability length. The influence of atmospheric
stratification in the boundary-layer is modeled through the
Monin-Obukhov stability length (Liu et al*.,* 1979). The profiles of
potential temperature (*θ*) and specific humidity (*q*) have functional
forms similar to the log-wind profile.

$q(z) - q_{s} = \frac{q_{}}{k_{q}}\left\lbrack \text{ln}\left( \frac{z}{z_{oq}} + 1 \right) + j_{q}\left( z,z_{oq},L \right) \right\rbrack$;
and (5)

$q(z) - q_{s} = \frac{q_{}}{k_{q}}\left\lbrack \text{ln}\left( \frac{z}{z_{\text{oq}}} + 1 \right) + j_{q}\left( z,z_{\text{oq}},L \right) \right\rbrack$.
(6)

The parameterizations of roughness lengths for potential temperature
(*z~oθ~*) and specific humidity (*z~oq~*), and *L* are identical to
those used in the BVW (Bourassa-Vincent-Wood) flux model (Bourassa et
al., 1999).

## Dependence of stress on sea state

The dependence of stress on sea state (wave age, *cp* / *u*~∗~ in
earlier versions of this model) typically enters through the gravity
wave contribution in the roughness length parameterization (e.g., Kusaba
and Masuda, 1988; Geernaert et al., 1990; Toba et al., 1990; Perrie and
Toulany, 1990; Maat et al., 1991; Smith et al., 1992; Bourassa et al.,
1999; Taylor and Yelland, 2001). The BVW roughness length (7) is among
the most complicated: it is anisotropic and considers contributions to
surface roughness from three types of surface features.

![](media/image1.wmf) (8)

where the *β*-terms are binary weights for the roughness lengths
associated with (from left to right) associated with an aerodynamically
smooth surface (Nikuradse 1933; Kondo 1975), capillary waves (Bourassa
et al. 1999), and gravity waves (Smith et al. 1992), where *ν* is the
molecular viscosity of air, *U*~orb~ is the orbital speed of the
dominant waves, *b* is a dimensionless constant, *σ* is surface tension,
*ρ~w~* is water density, *c~p~* is the phase speed of the dominant
waves, and *g* is gravitational acceleration. In this case, sea state
(through orbital velocity) also influences the capillary wave roughness
length.

For *U*10 \> 7 m s-1 the roughness length due to capillary waves is
small compared to that for gravity waves. The BVW parameterization of
capillary wave roughness length is dependent on the velocity frame of
reference, which is often considered to match the surface current.
However, the capillary waves and short gravity waves that interact with
the airflow tend to ride on top of the dominant gravity waves. The
exponential term in the capillary wave roughness length is an adjustment
to the velocity frame of reference of the surface current. This frame of
reference adjustment is the second mechanism through which sea state
influences the roughness length.

## Improvements to model

An improvement is made to the BVW (Bourassa et al. 1999) flux model. For
high wind speeds, the BVW stresses were found to greatly overestimate
the stresses observed in the Storm Wave Study experiment (SWS-2; Dobson
et al., 1999; Taylor et al., 1999) experiment (P. K. Taylor, personal
communication, 1999). Both Bourassa et al. (1999) and Taylor (personal
communication, 1999) suggested that such biases were due to
extrapolating the HEXOS parameterization for surface roughness length
(Smith et al. 1992) to conditions for which it was not valid. The HEXOS
parameterization is not applied in the model developed herein: the
roughness length (9) can be written with no explicit dependence on sea
state, where the gravity wave roughness length is a two-dimensional
version of Charnock's equation (Charnock, 1955). The influence of sea
state enters solely through the modification of vertical shear in wind
speed (10), due to a non-zero lower boundary condition: the wave-induced
surface motion.

![](media/image2.wmf) (11)

![](media/image3.wmf) (12)

The orbital speed of gravity waves (*U*~orb~) is

$U_{\text{orb}} = \frac{{pH}_{s}}{T_{p}}$, (13)

where *H~s~* is the significant wave height, and *T~p~* is the period of
the significant waves. For gravity waves, Uorb can easily be related to
the significant slope used by Taylor and Yelland (2001). The fraction of
the orbital velocity (*f*) that modifies the surface wind was set at 50%
in BVW. There was considerable uncertainty in *f*, with little
difference in the accuracy of fluxes for values \>0.45. Both *f* and *a*
must be re-evaluated for use in (14).

The log-wind equation is in a more typical form, when the orbital
velocity term is moved from the left hand side of (15) to the roughness
length:

![](media/image4.wmf) (16)

This version of the roughness length has an explicit dependency on sea
state through the exponential dependence on the orbital velocity (or the
ratio of *H~s~* to *T~p~*).

# Comparisons to Observations

The model is evaluated with SWS-2 observations, and it is compared to
empirical relation (17) developed by Taylor and Yelland (2001) based on
the observations from SWS-2 and data from a wide range of other
conditions. The functional form of the gravity wave portion of (18) is
far different than the empirical relation (19):

![](media/image5.wmf), (20)

where *λ~p~* is the wavelength associated with the dominant waves.
Equation (21) has been shown (Taylor and Yelland, 2001) to be an
excellent match to mean observations over a wide range of sea states.
For moderate and high wind speeds, this empirical relation appears to be
better than existing physically-based algorithms; therefore, it is an
excellent standard of comparison.

Matching the observed roughness length as a function of friction
velocity is an excellent test for roughness length parameterizations
(Bourassa et al., 1999). The roughness length can be accurately
estimated from the observed friction velocity and wind speeds. The mean
roughness length is binned in 0.05 ms^-1^ intervals, and bins with \>8
observations are considered. The differences between (22) and (23),
without the wind speed modified by the orbital velocity, show (Fig. 1)
that for the SWS-2 conditions the orbital velocity could have a large
impact on the roughness length. The mean roughness lengths based on (24)
are similar to the observed means and means based on (25). Both (26) and
(27) are good representations of the mean roughness lengths.

A fit to the mean is not an ideal test for the quality of the roughness
length parameterizations because the comparison roughness length is
cross-correlated with the observed friction velocity. The substantial,
but unknown, uncertainty in friction velocity leads to biases in the
apparent dependence of observation-base roughness length on friction
velocity. Accounting for this bias is beyond the scope of this study.
This bias complicates the estimation of the constant in Charnock's
parameterization (or the modified from in (28)), which is usually
determined from the y-intercept of Fig. 1. For this preliminary study,
the value of the modified *a* in (29) is approximated as 0.68 (4 times
typical values). A better fit to the mean SWS-2 observations could
probably be achieved; however, it will be shown that this approximation
is a considerable improvement over existing models.

The root mean square (rms) difference between observation-based and
modeled roughness length is better indication of accuracy than a fit to
a mean distribution. The rms difference between the observation-based
roughness length and (30), without the consideration of orbital
velocity, is 0.0027 ms^-1^. When orbital velocity is considered, the rms
difference drops to 0.014. This is a substantially better than the rms
difference (0.0022 ms^-1^) that is achieved with Taylor and Yelland's
(2001) parameterization.

![](media/image6.png){width="4.716666666666667in" height="3.5in"}

> Fig. 1. Roughness length as a function of friction velocity for four
> cases: SWS-2 observations (); Eq. (31) without orbital velocity
> considered (upper +); Eq. (32) with orbital velocity considered (lower
> +), Taylor and Yelland (diamonds).

# CONCLUSIONS

Insights gained from examining the differences between satellite and in
situ observations demonstrated a consistency with theory that had
previously only been tested for low and moderate wind speeds. These
observations and theory indicated that the orbital velocity of waves
modifies wind shear, and consequently modifies oceanic surface turbulent
stress. This concept had been difficult to verify because there were few
appropriate in situ observations. The SWS-2 observations provide an
excellent opportunity to test a model that considers orbital velocity. A
physically-based model that considers orbital velocity is developed.
Comparisons to the SWS-2 data find that the model developed herein has
mean distributions similar to Taylor and Yelland's impressive empirical
formulation. However, the new model's rms difference to roughness length
is \~35% smaller than rms differences achieved by Taylor and Yelland:
the new model better accounts for variability from the mean. Further
work is required to optimize parameters in the new model; however, even
with the estimates used herein, the new model is a substantial
improvement over existing models.

**Acknowledgments***.* I thank Peter Taylor for providing the SWS-2
data. COAPS receives it base funding from ONR's Secretary of Navy Grant
to James J. O'Brien. Current support is from NASA's Ocean Vector Winds
Science Team, OSU's SeaWinds project, and the NSF.

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Email address of M. A. Bourassa Bourassa@coaps.fsu.edu

Manuscript received ; revised ; accepted