Journal of Geophysical Research: Oceans

New sea spray generation function for spume droplets

Authors


Abstract

[1] With increasing wind forcing and development of wind waves, more and more sea spray droplets are produced on the sea surface. On the basis of now available observational data from field and laboratories with 10-m wind speeds ranging from 8 to 41 m/s, it is shown that the traditional approach of using wind speed only fails to describe the increase of spume droplet production, owing to the neglect of effect of wave state. Instead, a nondimensional parameter RB = u*2pν called the windsea Reynolds number (Toba et al., 2006) is very good for characterizing the observational data from laboratories and field, where u* is the friction velocity of air, ωp the angular frequency spectral peak of wind waves, and ν is the kinematic viscosity of air. The windsea Reynolds number RB represents the coupling effect of wind forcing and wind wave state, and can be regarded as a measure of fluid dynamical conditions at the air-sea boundary layer. A new sea spray generation function for spume droplets is proposed as a function of RB. We conclude that spume droplets begin to be produced as RB exceeds 103. The effects of sea spray droplets on air-sea transfers are also estimated with the new model. It is found that the heat and momentum fluxes induced by sea spray droplets become comparable to the interfacial fluxes by bulk formulas when RB is greater than 105 and 106, respectively.

1. Introduction

[2] With increasing wind speed, wind waves at the sea surface develop and break. The breaking waves entrain air and thus produce many bubbles in the sea. When a bubble rises to the sea surface, it bursts and produces from a few to a few hundred film droplets. After the bubble bursts, a water jet is formed upon collapse of the bubble cavity. The shooting jet in the air soon breaks up into a few jet droplets depending on the size of the parent bubble [e.g., Toba, 1961; Andreas, 2002]. The typical radii of bubble-mediated film and jet droplets are typically less than 5 and 20 μm, respectively. Another kind of droplet is called a spume droplet, which results from the mechanical tearing of a wave crest by the wind. Spume droplets torn from the waves are many and large, with minimum radius generally about 20 μm, and no definite maximum radius [Andreas, 2002]. In this article, sea spray droplets refer to film, jet and spume droplets.

[3] Sea spray droplets are usually regarded as having the potential to alter the momentum, heat and moisture exchange processes through the air-sea interface, and they might also have significant influence on gas transfer as indicated by recent studies on the rainfall effect on turbulent mixing [Komori et al., 2004]. The question of whether sea spray can enhance the air-sea heat and moisture fluxes has appeared in the literature off and on for at least 50 years. Answers to this question given by recent studies seem to be contradictory. For example, on the basis of his observational data in the field, Ling [1993] concluded that the sea spray droplets produced by breaking waves in the ocean were a major source of atmospheric moisture and latent heat. Iida and Toba [1999] also suggested that sea spray droplets greatly enhance evaporation from the sea surface several dozen times when wind speed is greater than 20 m/s. On the other hand, Andreas [1992, 1998] and Pattison and Belcher [1999] suggested that droplets were likely to make only a small contribution to the overall heat fluxes, except under storm conditions, when they could be a major source of water vapor. DeCosmo et al. [1996] reported their HEXOS observational results and suggested that sea spray does not significantly affect the surface layer fluxes for the wind speed up to 18 m/s. Wu [1998] also reported that the total moisture flux due to droplets escaping from the sea contributed only 3% of the total evaporation from the sea surface even at wind speed of 30 m/s. Andreas [2004] concluded that spray supports about 10% of the surface stress for a wind of 30 m/s, and supports all of the surface stress for a wind of about 60 m/s.

[4] The main reason for this contradiction is the difference in sea spray generation function (SSGF) dF/dr0, which quantifies how many spray droplets with initial radius r0 are produced per square meter of the surface per second per micrometer increment in droplet radius. The SSGF is usually given as a function of the 10-m wind speed U10. Andreas [2002] reviewed 13 versions of the SSGF for r0 values between 1 and 500 μm, and found that at any given wind speed and r0 these functions range over six orders of magnitude. In Figure 1, we give some examples of the SSGF at wind speed 15 m/s. The uncertainties of the SSGF result from the difficulty in making the required measurements, which hinders us from giving the proper production rate dependences on environmental factors.

Figure 1.

Various sea spray generation functions versus droplet radius for a 10-m wind speed U10 of 15 m/s.

[5] Although the number concentration of smaller droplets is much greater than that of larger droplets, their volumes are much smaller. It appears that spray droplets with initial radii between 10 and 300 μm contribute most to the momentum and heat fluxes [Andreas, 1992, 2004]. In this paper, we will concentrate on the larger droplets that are relevant to air-sea exchange processes: droplets with radii at formation between 30 and 500 μm. Some jet droplets may be in this radius range, but spume droplets will be the vast majority of these larger droplets. Thus, for clarity, “spume droplets” will be used to refer to these size range of droplets in our discussion.

[6] Although wind speed is usually applied to describe the air-sea exchange processes, it is obvious that spume droplet production is directly related not only to the wind forcing but also to the development of wind waves. Thus, instead of wind speed, a parameter that can represent the coupling effect in the air-sea boundary layer is expected. Toba et al. [2006] have shown that a nondimensional parameter,

equation image

called the windsea Reynolds number is the fundamental controlling parameter for the windsea boundary layer and the behavior of air-sea transfers, where u* is the air friction velocity, ωp the angular frequency at the windsea spectral peak, and ν is the air kinematic viscosity.

[7] Wind waves are very special phenomena, which are generated at a shear (frictional) interface between the air and water. In the case of windsea, independent variables are time and space, or duration and fetch if the wind field is homogeneous. Dependent variables are wave property variables such as wave height Hs, wave period Ts, or ωp, and u* as the only external condition, at least under neutral stratification and in the absence of dominant swells. The existence of the similarity in the case of windsea implies that it is sufficient to select only one of these wave-property variables, together with u*, in order to completely describe the dynamical system. So we can introduce a parameter RB shown in equation (1) specifying the coupling effect of wind forcing and wind waves [Toba and Koga, 1986; Zhao and Toba, 2001]. RB may be interpreted as a Reynolds number, since it contains u* as a representative scale of speed, and Ls = u*Ts (=2πu*p) as a representative length scale of the phenomenon under consideration. The Ls is interpreted as the length representing the distance over which a water particle at the surface is driven by the speed u* (representing the wind stress) during the representative wave period Ts. For the same value of a Reynolds number Re (=UL/ν with U and L as the representative scales of speed and length), the equation of motion will have the same solution, even if L, U and/or ν have different values. This is known as the Reynolds' similarity law, which provides the basis for model experiments. Since the kinematic viscosity of water νw has a specific ratio with ν, we need consider only ν. We do not need to consider the surface tension, since it is only related to very high frequency waves, and that part of the windsea spectra is actually controlled by u* [Toba et al., 2006].

[8] After a brief review on spume SSGF presented in section 2, it will be shown in section 3 that the windsea Reynolds number RB, not the wind speed U10 traditionally applied, should be used to describe sea spray production, and a new SSGF as a function of RB for spume droplets will be proposed based on available observational data from field and laboratories. The effects of sea spray droplets on the air-sea momentum and latent heat fluxes will be discussed in section 4. Conclusions will be presented in section 5.

2. Brief Review of the SSGF for Spume Droplets

[9] Many environmental factors affect spume droplet production, and the SSGF is very complicated. The SSGF is assumed to be a function of droplet radius r0, wind speed U10 and wind-wave states such as ωp. So the SSGF may be written as

equation image

It is assumed that if the shape of the droplet size spectrum is independent of wind speed and wind-wave state, the SSGF can be divided into two independent parts: the size spectrum and dependence on environmental parameters.

equation image

The SSGF has usually been integrated over droplet radius as the total production rate on the studies about the effects of droplets on air-sea transfers. Then the shape of the size spectrum f2 (r0) has little effect on the estimation, and the great uncertainty in SSGF is ascribed to f1 (U10, ωp). Traditionally, the dependence on wave state of the SSGF is neglected, and only the wind speed is chosen as the environmental factor to describe the sea spray production. Therefore equation (3) is simplified as

equation image

[10] Many SSGFs as a function of wind speed for spume droplets have been proposed in previous studies, but these various functions are significantly different in magnitude. Ling et al. [1980] first proposed the wind dependence of spume droplet production, which is proportional to the square of wind speed

equation image

[11] On the basis of the laboratory observational data of Wu [1973] and Lai and Shemdin [1974], Monahan et al. [1986] derived the wind speed dependence for spume droplets as

equation image

With a wind speed dependence of exp (2.08U10), Andreas [1992, 1998, 2002] and Wu [1993] regarded that the model proposed by Monahan et al. [1986] provides an unrealistic overestimate of the number of spume droplets.

[12] Wu [1993] indicated that Monahan et al. [1986] incorrectly interpreted his results [Wu, 1973] by directly applying the free stream wind speed in a flume as the wind speed at the standard anemometer height of 10 m above the sea surface. After modification, Wu [1993] gave the wind speed dependence of spume droplet production as

equation image

[13] Andreas [1992] proposed an SSGF that adopted the droplet size spectrum of spume droplets proposed by Monahan et al. [1986] and the wind speed dependence given by Miller [1987]. Although this wind dependence mainly fit bubble-mediated droplets of radius 2 < r0 < 30 μm, it was directly extended into the spume domain by Andreas [1992].

[14] Smith et al. [1993] collected their data on a 10-m tower, which was built on a sloping beach, with instruments about 14 m above mean sea level. Only at high tide, the waterline could reach to the foot of the tower, but at other times, it was roughly 300 m away. So the measured droplets had to travel a long distance to reach the observational tower. Although a large range of wind speeds, from essentially zero to values in excess of 30 m/s, were encountered during this study, the equipment deployed was only able to measure droplets with radii from 2 to 50 μm, mainly in the radius range of film and jet droplets. Smith et al. [1993] suggested that their data can be best represented by

equation image

where the constants α1, α2, and r01, r02 have the values 3.1, 3.3, 2.1 and 9.2 μm, respectively, and r80 = 0.5r0 is the droplet radius at a reference relative humidity of 80% [Andreas, 1992]. The coefficients, A1 and A2 vary with wind speed according the formulae below:

equation image

where U14 is the wind speed at 14 m above sea level. Except for the bubble mediated droplet domain, it is noted that this wind dependence is also strongly influenced by the surf zone and other coastal effects due to the measurements being made on shore. With caution, Smith and Harrison [1998] did not directly extend the wind dependence of equation (9) in their earlier work into the spume domain. Instead, they suggested that the wind dependence for spume droplet production should be replaced by

equation image

[15] On the other hand, Andreas [1998] and Wu [1998] directly adopted equation (9) as their wind dependence for spume droplets to modify their former works [e.g., Andreas, 1992; Wu, 1993], respectively. As Wu [1993] said himself, this kind of extension lacks of physical grounds, especially if the coastal effects involved in this wind speed dependence are considered.

[16] The bubble-mediated film and jet droplet production is directly related to the rate at which air is entrained into the oceanic surface layer. Many models for the bubble-mediated SSGF have been developed in which their production rate is explicitly proportional to whitecap coverage [Monahan et al., 1986; Fairall et al., 1994; Andreas et al., 1995]. Using the size spectrum of Andreas [1992], Fairall et al. [1994] proposed their SSGF for spume droplets that is proportional to the fractional whitecap coverage suggested by Monahan and Ó Muircheartaigh [1980], so their wind dependence was given as

equation image

Therefore they also extended wind dependence for bubble-mediated droplets into the spume domain, as was done by Andreas [1998] and Wu [1998].

[17] From the above analysis, it can be seen that the wind dependence of spume droplet production has been very ambiguous. In addition to the lack of measurements, the main reason is that the wind speed cannot represent the coupling effect at the air-sea boundary layer, where the effect of wind waves must be taken into account. From his and his collaborators' series of works [Toba and Koga, 1986; Iida et al., 1992; Zhao and Toba, 2001; Zhao et al., 2003], Toba et al. [2006] clearly showed that the windsea Reynolds number RB, defined by equation (1), is the fundamental controlling parameter for the windsea boundary layer, and consequently for the behavior of air-sea transfers.

[18] With the combination of field and laboratory data available, we will show that RB is a better parameter than wind speed to describe spume droplet production, and a new SSGF for spume droplets as a function of RB will be proposed in the next section.

3. Dependence of Spume Production on Windsea Reynolds Number

[19] As stated above, RB is a parameter specifying the coupling effect of wind forcing and wind waves. To obtain RB, we have to choose observational data that include wind and wave information in our study. Our observational data set is composed of field measurements by Chaen [1973], and laboratory measurements by Toba [1961], Koga and Toba [1981], and Sugioka and Komori [2005]. The wind speed in our data set ranges from about 8 to 41 m/s in U10.

[20] Systematic observations of sea spray droplets in the lowest atmospheric layer over the oceans were carried out on board of the Hakuho Maru at sea in 1969 and 1970 [Toba and Chaen, 1969; Toba et al., 1970]. The observational data were reported by Chaen [1973] and Toba and Chaen [1973]. During their KH-70-3 cruise on the way to the East China Sea, Typhoon 7008 (Oruga) hit Japan on 5 July 1970, and the ship was anchored off Osaka Bay for 24 hours; however, the typhoon came to the Osaka bay, and the observation was carried out until the wind speed reached 16.6 m/s without rain. The sea spray droplets including large spume droplets were measured at a height of 6 m and 3 m above sea level. These observational data for droplet radii ranging from 28 to 87 μm are used in this analysis.

[21] Experiments on air entrainment in wind waves and droplet production were carried out in a wind flume (21 m long) by Toba [1961]. He measured vertical distribution of droplets up to 350 μm in radius at eight wind speeds. His observational data with wind speeds U10 of 14.9, 16.5, 18.7, 21.4 and 22.8 m/s are used in this analysis. These wind speeds at the 10-m level were obtained by extrapolation using the logarithmic law from wind profile measurements in the flume. Koga and Toba [1981] measured droplet production at three wind speeds, 25.0, 28.2 and 34.2 m/s (in U10) in a wind flume (20 m long). The largest droplets they observed were 873 μm in radius. Their data are also used in the present analyses. Recently, Sugioka and Komori [2005] have measured droplet production in a wind flume (6.5 m long) with a new phase Doppler technique, Phase Doppler Particle Analyzer (PDPA), which is based on the principles of light scattering interferometry. This technique allows for the sizing of spherical particles (typically liquid sprays, but also some bubbles and solid spheres) together with the velocity of the droplets. They measured droplets ranging from 16 to 320 μm in radius at wind speeds of 34.3, 37.7 and 41.3 m/s (in U10). Their data are also integrated in this analysis.

[22] Multiplied by the terminal fall speed wf(r0) of the droplets, the number concentration of droplets measured in experiments can be transformed into SSGF corresponding to the respective reference height [Toba, 1965]. It is impossible to directly compare these SSGFs as they represent the droplet fluxes at different heights above the sea surface. A dry deposition model for downward droplet fluxes developed by Slinn and Slinn [1980] was generalized by Fairall and Larsen [1984] to the case of sea spray droplets characterized by the gravitational settling velocity of a droplet, which is a function of the droplet's terminal fall speed, Stokes number, wind speed and drag coefficient at the reference height. This model has been applied to transfer our SSGFs at various reference heights into values at the sea surface [Andreas, 2002].

[23] Different authors have measured the droplets in different radius ranges. For comparison, the droplet production rates in roughly the same radius ranges, as indicated in the figures, for each observational data, have been selected and calculated. The resulting droplet production rates F as a function of wind speed U10, U103 and windsea Reynolds number RB are plotted in Figures 2a and 2b and Figure 3, respectively. Although the wind speeds in laboratory measurements are much greater than those in field measurements, the droplet production rate for the former is generally smaller than that of the latter. It is clear that the data are scattered so much in Figure 2 that they are impossible to express as a simple analytical function of wind speed. As stated above, the main reason is that the sea spray production is determined by both wind forcing and the development of wind waves, neither of which can be neglected.

Figure 2a.

Droplet production rate versus 10-m wind speed U10.

Figure 2b.

Droplet production rate versus 10-m wind speed U103.

Figure 3.

Same observational data as in Figure 2, except for the production rate versus windsea Reynolds number RB.

[24] When the same data are specified by the windsea Reynolds number RB (Figure 3), it can be seen that field and laboratory data are in agreement, and can be expressed as

equation image

This good agreement is not surprising because RB is a parameter representing the coupling effect of wind and wind waves at the air-sea boundary layer. In fact, RB can also be written as

equation image

where CD is the drag coefficient, g the acceleration of gravity, and β is the wave age of wind waves, which is widely used to describe the degree of windsea development relative to the local wind. With development of wind waves, the wave age increases with fetch ranging from 0 to 1.4. The wind wave in laboratories is usually very young with much smaller wave age due to the limited fetch. On the other hand, the wind wave in the open sea is usually well developed with greater wave age. For example, the wave age in the study of Sugioka and Komori [2005] is on the order of 0.01, and the wave age in the field is usually greater than 0.4 in the open sea. Almost all of the wave ages in the 40 cases observed by Toba and Chaen [1973] are greater than 0.38 with the largest value of 0.89, except for a value of 0.31 in only one case. Thus greater U10 for laboratory data cannot guarantee greater RB owing to their much smaller wave age, and smaller U10 for field data does not guarantee smaller RB owing to their greater wave age. In a word, it is RB, not U10, that specifies the condition of the windsea boundary layer, and RB should be used to describe the sea spray production.

[25] According to various observations, Andreas et al. [1995] suggested that the wind speed threshold for spume production is in the range of 7–11 m/s in the open ocean. However, a definite wind speed threshold cannot be given because spume production depends not only on wind speed, but also on other environmental factors, mainly on the wind-wave state. At this wind speed threshold, it is impossible to produce spume droplets in the laboratory. Therefore wind speed is not a useful quantity to describe the initiation of spume production. By using RB in equation (13) and Figure 3, we suggest that the threshold for spume production is RB > 103, which is valid for either laboratory or field. The photographic in study of direct production of spume droplets by Koga [1981] supports this idea. This threshold corresponds to whitecapping at the air-sea interface, and wind stress as well as gas transfer increases drastically, as pointed out by Toba et al. [2006].

[26] In order to confirm the influence of different droplet radius range on the RB dependence shown in equation (12), we consider all of the data for droplets radius greater than 30 μm that have been measured. Droplet radii up to 350 μm obtained by Toba [1961], 873 μm obtained by Koga and Toba [1981], and 320 μm obtained by Sugioka and Komori [2005] have been used to obtain the production rates. The results versus wind speed U10, U103 and windsea Reynolds number RB are presented in Figures 4a and 4b and Figure 5, respectively. The spume production rate as a function of RB can well be expressed as (Figure 5)

equation image

Except for a slightly greater overall production rate due to the greater radius range included, the inclination of RB in equation (14) is almost the same as that in equation (12). It is shown that RB is a robust parameter to describe the spume droplet production, and the power law shown in equation (12) is almost independent of spume droplet size. It is also noted that the maximum droplet radius measured by Koga and Toba [1981] is up to 873 μm, exceeding the 500 μm range that we are usually interested in. Therefore equation (12) will be used in the rest of the present analyses.

Figure 4a.

Droplet production rate versus 10-m wind speed U10. The production rates are calculated using all available data up to maximum droplet radii from laboratory data of Toba [1961], Koga and Toba [1981], and Sugioka and Komori [2005].

Figure 4b.

Droplet production rate versus 10-m wind speed U103. The same data in Figure 4a are used here.

Figure 5.

Same observational data as in Figure 4, except for the production rate versus windsea Reynolds number RB.

[27] Another important parameter specifying the sea spray intensity is the mass production rate, G, which is defined as

equation image

where ρw is the density of seawater, G is in the unit of kg m−2s−1. G represents how much seawater will be thrown into the air per square meter of the surface per second, regardless the size of droplets. By using the data shown in Figure 2, G is calculated (Figure 6), and the results can be expressed as

equation image

Except for the proportional parameter, equation (16) is very similar to equation (12) with almost the same power of RB. It demonstrates that F and G have the same dynamical dependence on air-sea interaction, and RB is a proper parameter to be selected.

Figure 6.

Same observational data as in Figure 2, except for the mass production rate G (kg m−2 s−1) versus windsea Reynolds number RB.

[28] According to the above analysis, it is clear that RB, not the wind speed traditionally applied, should be used to construct the SSGF for spume droplets. We try to distribute the production rate, (equation (12)), over the droplet size distribution. In fact, we know little about the droplet size spectrum. Fortunately, the SSGF is usually integrated when the effect of sea spray droplets on air-sea transfers is considered, and the concrete form of droplet size spectrum affects the results little. The size spectrum proposed by Monahan et al. [1986] has been widely accepted and applied by other authors, such as Andreas [1992, 1998] and Wu [1993, 1998]. Therefore a similar droplet size spectrum to that of Monahan et al. [1986] is first adopted in this study. In that case, we can write our SSGF for spume droplets as follows:

equation image

where dF/dr0 (expressed in m−2 s−1 μm−1) is the number of spume droplets produced per unit sea surface area per second per unit radius band.

[29] In order to compare our SSGF specified by equation (17) with the traditional models, we express equation (17) in Figure 7 for two values of wave age, 0.2 and 1.2, which represent the young and well-developed wind waves, respectively, for the same wind speed of 20 m/s. On the basis of analysis of observational data of Toba [1961] and Koga and Toba [1981], Iida et al. [1992] proposed a droplet size spectrum in tabulated form (Table 1), which is very different from that in equation (17). Their absolute values of exponents of r0 are much greater than those in equation (17), and almost do not change with the increase of droplet size. The SSGF corresponding to their size distribution is plotted in Figure 7 for comparison. Andreas [2002] conducted a thorough review of 13 versions of SSGF available in the literature, and identified them by magnitude and wind speed dependence in the spume region. He recommended that the most reliable SSGFs are the models proposed by Fairall et al. [1994] and Andreas [1998]. These two models are also plotted in Figure 7. Integrated in the radius range from 30 to 500 μm, the resulting production rates of Fairall et al. [1994], Andreas [1998] and our model of equation (17) are presented in Figure 8.

Figure 7.

Comparison of SSGFs by equation (17) and size spectrum proposed by Iida et al. [1992] with those of Andreas [1998] and Fairall et al. [1994] at U10 = 20 m/s, wave age β = 0.2 and 1.2, which correspond to RB = 2.24 × 104 and 1.34 × 105, respectively.

Figure 8.

Comparison of production rates in the radius range from 30 to 500 μm of our model equations (14) and (17) at β of 0.2 and 1.2 with those of Andreas [1998] and Fairall et al. [1994].

Table 1. Size Spectrum Proposed by Iida et al. [1992]
r0 (μm)Exponent of r0 (RB < 104)Exponent of r0 (RB > 104)
28 ∼ 40−17.22−12.25
40 ∼ 60−17.27−12.33
60 ∼ 88−16.81−12.11
88 ∼ 129−16.45−12.17
129 ∼ 189−16.43−12.16
189 ∼ 276−16.57−12.45
276 ∼ 405 −12.48
405 ∼ 595 −12.79
595 ∼ 873 −13.01

[30] We can see that the wave state strongly affects the production rate and SSGF, which increases with the development degree of wind waves (e.g., with the wave age). The differences in magnitude of production rates and SSGFs for wave ages of 0.2 and 1.2 are one order of magnitude or greater. It is shown in Figures 7 and 8 that our model of equation (17) with wave ages of 0.2 and 1.2 can be regarded as the lower and upper limits of sea spray production, respectively. At low wind speeds, our model of equation (17) is comparable to that of Fairall et al. [1994] and Andreas [1998]. At high wind speeds, spume droplet production is greatly underestimated by Fairall et al. [1994] and Andreas [1998] in the open sea. On the basis of the SSGF of equation (17), we will discuss the effect of spume droplets on momentum and heat fluxes through the air-sea interface in the next section.

4. Effect of Spume Droplets on Air-Sea Fluxes

4.1. Spray Stress

[31] After a droplet is thrown into the air, its motion is affected by turbulent flow and gravitational force. The turbulent flow will suspend the droplets in air, and gravity will pull the droplets down to the sea surface. In general, smaller droplets stay in the air for a longer time than larger droplets, before they fall to the sea surface. During the airborne period, a droplet will be accelerated by the local wind, and transfer its water vapor and sensible heat to the air. Therefore the motion of a droplet in the atmospheric boundary layer is very complicated, and few studies have been done.

[32] Andreas [2004] proposed a model and suggested that for wind speed greater than 10 m/s, all droplets with radii up to 500 μm can reach the local wind speed in less than a second, far before they fall down to the sea, and the droplets take shorter time to reach local wind speed with stronger wind forcing. The accelerated droplets extract momentum from the near-surface wind and slow it. When these droplets fall into the sea, they transfer their momentum to the sea surface. Here we follow the model of Andreas [2004]. Assuming that the spray droplets accelerate to horizontal speed usp (r0) before falling back into the sea, the stress induced by spray droplets can be expressed as

equation image

where ρw is the density of seawater, and rlo and rhi are lower and upper radius limits of the droplets that are important in this process. It is noted that the spray stress is proportional to the spray volume flux, (4πr03/3)dF/dr0, so that a large droplet will play a more important role than a small droplet in this process. Andreas [2004] suggested that all droplets of interest essentially travel at the local wind speed before they fall back into the sea. So usp (r0) is independent of r0, but depends on height, and can be estimated by

equation image

where z0 is the roughness length, κ (=0.4) the von Kármán constant, and usp(zs) is the wind speed at height zs above the sea surface. Here zs is the effective height at which the spume droplets are produced. Andreas [1992] and Iida et al. [1992] suggested that zs values are 0.5Hs and 0.635Hs, respectively. The latter is based on the ideas suggested by Chaen [1973] and Koga and Toba [1981] by means of a statistical consideration, in which the effective sea surface is in the upper half of the probability density distribution of sea surface height along wind waves. We adopt the result of Iida et al. [1992] in our calculation, that is

equation image

where Hs is the significant height of wind waves, and can be estimated by the empirical relationships given by Toba [1972] and Hanson and Phillips [1999],

equation image

where β is the wave age mentioned above. For the roughness length, we adopt the Charnock [1955] formula with the value of coefficient proposed by Wu [1980],

equation image

[33] The results of spray stress calculated by equation (18)(22) are shown in Figures 9 and 10 as a function of wind speed and windsea Reynolds number, respectively. For comparison, the usual interfacial wind stress τ = ρaCDU102 denoted as total stress is also plotted in Figure 9, where ρa is the density of air, and CD is the drag coefficient given by Wu [1980]. When the spray stress is expressed by wind speed (Figure 9), the wave age must be used as another parameter to describe the wave state dependence of spray stress, which is so significant that the difference in magnitude between wave ages of 0.2 and 1.2 is as large as 1 order or more. It is clear that we cannot determine the contribution of spray stress to the interfacial stress only by wind speed as in traditional practice. When the spray stress is expressed by windsea Reynolds number (Figure 10), the wave state effect on the effective spray droplet production height (see equations (20) and (21)) is not appreciable since RB already includes the information on wind waves. From Figure 10, we suggest that the spray stress becomes comparable to the interfacial flux when RB exceeds 106.

Figure 9.

Spray stresses versus wind speed. The thin solid and dashed lines represent the results for β of 1.2 and 0.2, respectively. The thick solid line represents the interfacial stress τ estimated by using the conventional bulk formula by Wu [1980].

Figure 10.

Same spray stress as in Figure 9 versus windsea Reynolds number RB.

[34] Figure 11 shows the ratio of stress induced by spray droplets τsp (equation (18)) to interfacial wind stress τ against RB. The ratio increases with RB a little depending on the wave age β, and is given by

equation image

It can be seen that this ratio is not a constant, but increases with RB and wave age.

Figure 11.

Ratio of τsp/τ as a function of windsea Reynolds number RB.

4.2. Heat Fluxes Induced by Sea Spray Droplets

[35] We are also interested in the effect of heat fluxes induced by droplets on the air-sea transfers. This process is clearly very complicated and little known. Here we adopt the method of Andreas [1992] to estimate the spray heat fluxes. Andreas [1992] provided a model for estimating the sensible and latent heat fluxes induced by sea spray droplets. For the spray latent heat flux, the model gives

equation image

where ρw is the density of seawater, Lv the latent heat of vaporization of water, τf the atmospheric residence time of a droplet, and rf) is the radius when a droplet falls into the sea surface. The residence time that the droplets remain airborne can be estimated by

equation image

where wf is the terminal fall speed that can be calculated by the equation given by Toba [1961] or Andreas [1990]. Substituting equation (20) and equation (21) into equation (25), the residence time of a droplet can be given by

equation image

Andreas [1992] suggested that the residence time for most of the droplets we are interested in is greater than 1 s under strong coupling between wind and wind waves.

[36] An initial droplet radius r0 decreases to rf) owing to evaporation when it returns to the sea surface [Andreas, 1992],

equation image

where req is the droplet radius when it reaches moisture equilibrium with its environment, and τr is the corresponding time. These parameters can be calculated using the equations from Andreas [1989, 1990].

[37] The sea spray droplets also exchange their sensible heat with their environmental air. Andreas [1992] indicated that this process is much faster than the evaporation, and the droplets up to 500 μm can completely transfer their available sensible heat to the ambient air before they fall into the sea. In that case, the sensible heat induced by droplets can be simply written as

equation image

where cps is the specific heat of seawater at constant pressure, and Teq is the temperature of the droplet when it reaches thermal equilibrium with its environment [Andreas, 1989, 1992].

[38] On the other hand, the air-sea latent and sensible heat fluxes can be calculated from the following bulk formulas

equation image
equation image

where ρa is the density of air, cp the specific heat of air at constant pressure, and qs and q10 are the values of relative humidity at the sea surface and 10 m above the mean sea surface. Ta and Tw are the air and water temperatures. CE and CH are the bulk transfer coefficients for latent and sensible heat. We assume that the relative humidity has an average value of about 80% within the constant flux layer. The coefficients CE = 1.2 × 10−3, CH = 1.0 × 10−3 proposed by Smith [1988], the atmospheric temperature Ta = 20°C, and the seawater temperature Tw = 0°C are used in our estimation.

[39] The latent and sensible heat fluxes are plotted in Figure 12 as a function of windsea Reynolds number RB. It is found again that the effect of wave state can automatically be included when RB is used. It is shown that the heat fluxes induced by droplets become comparable to the interfacial fluxes given by equations (29) and (30) when RB exceeds 105. Therefore the effect of droplets on heat flux is more important than that on momentum flux.

Figure 12.

Sensible and latent heat fluxes induced by droplets versus windsea Reynolds number RB.

5. Concluding Remarks

[40] With combination of the available field and laboratory data, it is demonstrated that the sea spray production rate for spume droplets is correlated much better with windsea Reynolds number RB than with wind speed U10. RB can be regarded as the fundamental parameter that controls the windsea boundary layer, and can reconcile the different dynamical conditions with various wind forcing and wave states occurring in laboratory and field. With RB, we can give a definite threshold for the transition of air-sea boundary processes. It is concluded that spume droplets begin to emerge when RB exceeds 103. The heat and momentum fluxes induced by spume droplets become comparable to the interfacial fluxes estimated by bulk formulas when RB exceeds 105 and 106, respectively. Therefore sea spray droplets play an important role in air-sea exchange processes under hurricane and typhoon conditions.

[41] Although these conclusions have been made above on the basis of our now available knowledge and acceptable models of sea spray production, we must admit that there are many uncertainties about sea spray production and its effects on air-sea transfers, especially in storm conditions. For example, the behavior of spray droplets in the atmospheric boundary layer should further be studied in order to estimate their contribution to air-sea exchange more precisely. With increasing computer capability, it is becoming possible to investigate the atmospheric boundary layer by direct numerical simulation and large eddy simulation.

Acknowledgments

[42] The authors would like to acknowledge the financial support of a Grant-in-Aid for Scientific Research (S) (14102016) from the Japan Society for the Promotion of Science, and support by the 21st Century Center of Excellence Program for Research and Education on Complex Functional Mechanical Systems. D. Z. is also supported by a Great Project of The National Natural Science Foundation of China (40490263), the National Basic Research Program of China (2005CB422301), and the Natural Science Foundation of Shandong Province (Z2002E01).

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