Oceanic Whitecaps and the 10m-Elevation Wind Speed: Toward Improved Power-Law Descriptions for Use in Climate Modeling

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A recent extensive review of the literature, published and unpublished, by the authors has uncovered 262 equations describing whitecap coverage in terms of wind speed, atmospheric stability, surface water temperature, fetch,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 Uⁿ              (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.