Handout (95.0 kB)

A number of current models for the air-sea exchange of radiatively important gases, and for the rate of production at the sea surface of marine aerosol particles, are cast in terms of the fraction of the ocean surface instantaneously covered by whitecaps. Since whitecap coverage can be inferred from satellite data, e.g. the enhanced microwave brightness temperature of the sea surface, such expressions are of immediate use to the remote sensing community. For many other modelers, a gas transfer coefficient, or sea surface aerosol flux, expressed in terms of the wind speed, and perhaps other meteorological parameters, would be of great utility.

A recent extensive review of the literature, published and unpublished, by the authors has uncovered 262 equations describing whitecap coverage in terms of wind speed, atmospheric stability, surface water temperature, fetch, duration, wave height, wave age, and other independence environmental parameters. Some of these formulations involve upwards of two dozen such parameters. With the above mentioned applications in mind, the authors have initially focused their attention on the subset of 88 equations where the fraction of the sea surface covered by decaying foam patches, i.e. Stage B whitecaps, is described by a simple power-law in U (Eq. 1), as was first done by D.C. Blanchard in 1963.

W_{B }= C_{o} U^{n} (1)

These 88 W_{B}(U)
expressions were then subject to a culling to remove those where their
originators had, *a priori*, taken the power-law exponent, n, to be a
simple integer (for computational convenience), or a value based on theoretical
considerations (e.g., the n of 3.75 of Wu, 1979). When the remaining list of
77 equations was further edited to remove those W_{B}(U) expressions
derived from observations where the lower atmosphere was explicitly stable, or
unstable (as opposed to near-neutral), and those equations derived from the analysis
of whitecap data obtained when the wind was of markedly limited duration or
fetch, the remaining set of equations was reduced in number to 66. Finally, when
the 10 W_{B}(U) equations based on the analysis of whitecap coverage
derived from the necessarily subjective visual estimates of lighthouse
keepers, and the two such equations derived from a data set where the winds had
been measured aboard one ship and the whitecap observations recorded on
another, were likewise struck from the list, the authors were left with 54 W_{B}(U)
power-law equations upon which to base their subsequent statistical analyses.

Before discussing some of the
findings of these analyses, it should be mentioned that the various authors
responsible for these 54 W_{B}(U) equations had recourse to only 19 W_{B},U-data
sets. The majority of these equations (29) were each derived from the
consideration of one of 9 data sets, taken in whole are in part. The other
equations (25) are based on the interpretation of 19 composite data sets, that
collectively encompass at least portions of 17 different W_{B},U-data
sets, 7 of which have been also been treated individually. The BOXEX+ data
set of Monahan (1971) was used alone or in combination with one or more other
data sets in the derivation of more than 40% (23 of 54), while the East China
Sea data set of Toba and Chaen (1973) was used in the derivation of almost as
many (22 of 54) , of the W_{B}(U) equations. These two data sets,
taken alone or together, represent the sole data base used by the various
authors in deriving 24% of the 54 W_{B}(U) equations.

As an initial demonstration
of the utility of the extensive list of W_{B}(U) power-law equations,
the authors have taken the 47 equations associated with data sets that included
surface sea water temperatures and regressed n, the power-law exponent versus
the average surface water temperature, T_{W}. They did this because
previously, using the five W_{B},U-data sets which were the only ones then
available to them, and applying a variety of statistical approaches, they
arrived at the five n_{avg},T_{W} points displayed on Fig. 10
in Monahan and O'Muircheartaigh (1986). These points in turn yield Eq. 2.

n(T_{W})
= 1.750 + 0.0574 T_{W} (2)

Applying now the same
approach to the 47 n-values in the current list of W_{B}(U) power-law
equations yields Eq. 3.

n(T_{W}) =
2.272 + 0.0365 T_{W} (3)

While Eq. 2 is based on 22
analyses of 5 W_{B},U-data sets, Eq. 3 is based on the 47 analyses of
no less than 15 W_{B},U-data sets, taken individually or in various
combinations.

Again applying the same
approach to the 40 power-law coefficients, C, in the current list of W_{B}(U)
equations (those equations for each of which an explicit value for C was given,
and whose associated data sets contained values for the surface sea water
temperature) yields, based on a simple linear fit in logC,T_{W} space,
the following Eq. 4 for C(T_{W}) .

C(T_{W}) =
4.81 x 10^{-5} x 10^{-0.0303T}_{w}
(4)

Combining Eq. 3 and Eq. 4
results in Eq. 5, the first explicit power-law formulation of W_{B} in
terms of U and T_{W}:

W_{B} = 4.81 x 10^{-5}
x 10^{-0.0303T}_{w} x U^{2.272 + 0.0365T}_{w}
(5)

The current finding, as
summarized in Eq. 3, validates the previous result as to the apparent T_{W}-dependence
of n. As was pointed out in Monahan and O'Muircheartaigh (1986), both typical
wind duration and mean sea water temperature vary latitudinally. As the
latitude increases, the wind duration, and the mean surface water temperature,
tend to decrease. While other factors may well be involved, and are enumerated
in the 1986 paper, the authors posit that it is the typical decrease in wind
duration with increasing latitude that is the primary cause of the finding
summarized in Eq. 3.

In the absence of time series
data documenting the duration of the local wind, Eq. 5 is put forward as a
preferred expression for estimating W_{B }from current U and T_{W}
measurements.