## 1. Introduction

Current hurricane models are unable to produce high wind speeds when using traditional air–sea flux parameterizations. Emanuel (1995) showed that the ratio of the enthalpy exchange coefficient to the drag coefficient controls the maximum wind speed and that traditional parameterizations produce too low of a ratio when extrapolated to high wind speeds. Consequently, some combination of a higher enthalpy exchange coefficient and a lower drag coefficient is necessary to simulate hurricanes in models. Such uncertainty indicates that our knowledge of these coefficients, which is largely based on databases of empirical formulas at lower wind speeds, does not extrapolate to high wind speed conditions in the field. While the drag coefficient seems to depart from previous extrapolated estimates at high wind speeds as seen in both laboratory and field experiments (Donelan et al. 2004; Powell et al. 2003), the behavior of the enthalpy coefficient remains largely unknown. Without a strong reduction of the drag coefficient, even a slightly increasing enthalpy exchange coefficient cannot constitute a sufficiently high coefficient ratio. Sea spray has the potential not only to increase the enthalpy exchange coefficient dramatically but also to decrease the drag coefficient at high winds. Numerous studies have examined the impact of sea spray on the drag and enthalpy fluxes (e.g., Edson and Fairall 1994; Andreas 1998; Pattison and Belcher 1999; Meirink 2002), but no clear consensus exists.

The great uncertainty of the sea spray source function ultimately leads to unresolved spray-mediated momentum, heat, and mass fluxes. Spume droplets, which form when the wind tears water globules from the crest of breaking waves, presumably dominate the sea spray problem at high wind speeds because of their relative size. Yet, very little is known about the production of spume droplets. Most previous studies have assumed a constant form, found empirically, for the sea spray generation function with increased wind forcing. Various theoretical models for droplet distributions in a turbulent carrier fluid (e.g., Hinze 1955), however, predict a reduction of the mean diameter with wind speed. If the droplet distribution does indeed shift to smaller droplets as predicted, the amount of wind energy necessary to produce each unit mass of spray increases. Andreas et al. (1995) argue that the energy required to work against surface tension and form droplets roughly scales as the wind energy flux into the ocean. Thus, the amount of mass produced relative to the wind energy does not necessarily increase.

Herein we present a semiempirical derivation of the spume source function. Section 2 provides a general overview of a plausible mechanism for the generation and breakup of ligaments and spume drops. We use experimental data of single ligament formation events from round water jets along with an estimate of event occurrences at the air–sea interface (given in section 3) to find the spume generation function, which consequently spans a wider range of sizes than most previous functions. With the inclusion of large drops, we also consider the vertical transport and suspension of the produced drops after formation in section 4. Finally, in section 5 we compare the corresponding mass and energy fluxes of our source function with previous studies.

## 2. Spume production process

Instead of assuming that the droplet distribution at a given height is produced instantaneously, we consider a possible spume production process. From wind–wave experiments (Koga 1981) as well as experiments investigating the breakup of a liquid sheet (e.g., Ahmed et al. 2008; Li and Ashgriz 2006) and a round water jet (Lasheras et al. 1998; Marmottant and Villermaux 2004), the following mechanism for spume production is proposed. The shear between the light, fast-moving air and the heavy, slow-moving water produces (Kelvin–Helmholtz)^{1} instabilities. Subsequently, a secondary instability, transverse to the wind, develops. The wind elongates these modulations to form coherent ligaments, which are eventually torn away from the surface, forming globules. These globules, after initial coalescence, are subject to fragmentation by shear forces and by turbulent fluctuations.

While there may be additional coalescence along the way, the equilibrium state ultimately consists of the stable, fully broken-up (atomized) droplets (Pilch and Erdman 1987). Moreover, some of the largest globules fall back to the ocean before being fully atomized. The ultimate formation process is not of critical importance here because we use empirical relationships, but we note here that discrepancies between previous estimates and measurements of the sea spray source function might stem from the fact that the equilibrium distribution is not reached instantaneously. Any empirical sea spray generation function found above the surface (e.g., Wu 1993; Andreas 1992) will most likely be closer to the atomized distribution that is transported vertically, while investigations of the production process (e.g., Anguelova et al. 1999; Koga 1981) will more closely resemble the globule distribution.

### a. Ligament formation

^{2}the average diameter of a ligament

*ξ*and the transverse spacing between ligaments

*λ*

_{⊥}are powers of the interfacial Weber number We

_{δ}=

*ρ*

_{a}u_{a}^{2}

*δ*/

*σ*:where

*ρ*is the density,

*u*is the velocity,

*δ*=

*u*/(

_{a}*du*/

_{a}*dz*)|

_{max}is the vorticity thickness, and

*σ*is the surface tension. The subscripts

*a*and

*w*denote the air and water phases, respectively. The air velocity relative to the surface is roughly

*u*=

_{a}*U*

_{10}−

*U*

_{0}, where

*U*is the mean velocity and the subscripts 10 and 0 denote 10-m and surface heights, respectively. The maximum velocity gradient is taken as

*du*/

_{a}*dz*|

_{max}≈

*u*

_{*}

^{2}/

*ν*, where

_{a}*u*

_{*}is the friction of air and

*ν*is the kinematic viscosity of air. Finally, the diameter of a sphere with the equivalent volume as the average ligament

_{a}*D*

_{0}is found to be a constant multiple of the vorticity thickness

*D*

_{0}= 4.334

*δ*.

### b. Atomization

*D*

*D*

_{0}. They also found that these droplets follow a gamma distribution,where

*x*=

*D*/

*D*

*n*

^{−1}. According to Villermaux et al. (2004),

*n*, between 2 and 4 for their data, increases slightly with wind speed and is a function of the ratio

*D*

*ξ*taken here to beConsequently, the variance of the drop distribution with normalized diameter decreases slightly as the wind forcing increases. Also, to conserve volume, the drop distribution is rescaled aswhere

*N*is the number of drops per ligament.

_{l}*υ*is the slip velocity between the droplet and the airflow. This defines a maximum stable diameter of the orderFirst, consider a falling droplet in a quiescent carrier fluid. The maximum slip velocity is the terminal velocity of the drop. Because of regimes of high Reynolds numbers and shape deformation, the fall velocity is not equivalent to the Stokes’ terminal velocity (accounting for the added mass) for larger droplets. Instead the settling velocity is taken to bewith droplet Reynolds number Re

_{s}_{d}(

*D*) =

*D*

*υ*

_{s}^{2}

*ν*and the correction Stokes’ flow,

_{a}*f*=

*f*′, whereis the high Reynolds number correction for solid particles from Clift and Gauvin (1971) andis the shape deformation correction for very large water droplets. Hinze (1949) found that the critical Weber number for a falling droplet is approximately 22. Now consider a falling droplet in a turbulent flow where the critical shear Weber number can be closer to 10 (Wierzba 1990). In this case, the maximum slip velocity is

_{C}f*υ*=

_{s}*σ*

_{x}^{2}+

*υ*

_{f}^{2}

*σ*is the standard deviation of the horizontal air velocity.

_{x}*ϵ*is the mean turbulent kinetic energy (TKE) dissipation rate and the corresponding maximum stable diameter, defined as the diameter for which 95% by volume is contained in drops with

*D*≤

*D*, is

_{t}*κ*≈ 0.4 is the von Kármán constant,

*B*is the production/destruction of TKE due to buoyancy, and

_{p}*d*Π/

*dz*is the divergence of the wave-induced energy flux (Mueller and Veron 2009a; Hara and Belcher 2004). The air–sea momentum flux is defined as

*τ*=

*ρ*

_{a}u_{*}

^{2}and is modeled according to Mueller and Veron (2009b).

Figure 1 plots the mean volume equivalent diameter *D*_{0} for each ligament as well as the maximum diameter in the resulting spray, defined as the diameter at which 95% by volume is contained within smaller drops *D*_{95%}. The theoretical maximum stable diameters from the critical shear and turbulent Weber numbers are also given. Because the dissipation changes with height above the surface, a constant near-surface height, which is equivalent to the peak radius at *U*_{10} = 15 m s^{−1} and lies outside the viscous sublayer for all wind speeds greater than 15 m s^{−1}, is used for the calculation of the TKE dissipation rate. For reference, the distribution peaks from Fig. 3 of Anguelova et al. (1999) are plotted at their 10-m equivalent wind speed. Immediately after initial coalescence upon separation from the surface but before atomization, these elliptical globules are of a similar size. The wind speed dependence of the maximum droplet diameter characteristically follows the maximum diameter derived from the critical turbulent Weber number *D _{t}* (Hinze 1955). Although drops that are larger than those predicted to be stable by theoretical diameter maxima can exist in the flow for some finite amount of time before breakup, all vertically transported drops appear to be stable. This can be seen from the maximum droplet size

*D*that the turbulence can transport vertically, and it will be discussed further in section 4.

_{f}## 3. Source function

*d*

**c**per unit area to be

*L*(

**c**) = Λ(

**c**)

*d*

**c**, where Λ(

**c**) is the spectral length per unit area. To find

*L*, we use the two-dimensional empirical wave spectrum (and wave age–fetch dependence) from Elfouhaily et al. (1997) coupled with the surface stress model from Mueller and Veron (2009b) with the implicit assumption that the wind and waves are in equilibrium (i.e., the spectral dissipation due to breaking is equivalent to the spectral wind energy input). The corresponding area swept by breaking wave crests of incremental wave speed

*d*

**c**per unit area per unit time is

*A*=

**c**Λ(

**c**)

*d*

**c**. From this area, the active whitecap coverage can be modeled aswhere

*T*is the active breaking time and is proportional to the wave period

_{b}*T*(Rapp and Melville 1990; Phillips et al. 2001; Melville and Matusov 2002). The active breaking time lies somewhere in the range from 0.1

*T*to

*T*. Using these values, the corresponding whitecap A coverage values fall in the conventional range (see Fig. 1 of Anguelova and Webster 2006). The lowest reported white cap coverage data seem to be those from Monahan and Woolf (1989), and when matched to Eq. (14), we find

*T*< 0.1

_{b}*T*, which seems rather low. Thus, we take

*T*≈ 0.2

_{b}*T*hereafter, which produces a whitecap A coverage roughly equivalent to that reported in Monahan (1993).

*dW*, the total length of breaking wave crests per unit area per unit time is

*λ*=

_{b}*T*

_{b}**c**.

*L*

_{sp}provides the length of spume production sites per unit area per unit time. The number of ligaments formed per unit area per unit time can now be found from the length of spume production sites divided by the transverse spacing between ligaments [Eq. (1)],Finally, the spume source function, which is a function of both wind speed and fetch, isand by substitution can be expressed in more familiar variables aswhere

*p*(

*D*/

*D*

Figures 2a and 2c, respectively, plot the spume surface production function for multiple 10-m wind speeds at a laboratory scale fetch (22 m) and a field scale fetch (75 km). The diameters of produced drops span a range on the order of 10 *μ*m to 10 mm. The peaks of our distributions give somewhat smaller droplet diameters than those reported by Anguelova et al. (1999), given in Fig. 1. This discrepancy could be due to their imaging limitations; the resolution of their images was roughly 1 mm. As noted previously, the distribution peaks show more droplets and smaller peak diameters with increasing wind speed. The corresponding volume production functions in Figs. 2b and 2d show that the peaks of the volume distribution only shift upward slightly with wind speed. The effect of transitioning to smaller droplets suggests that the additional energy from increased wind forcing primarily goes to the atomization of the droplets rather than to the increase of volume/mass production.

Figure 3 plots the total production rate of spume drops at both laboratory and field fetches. For direct comparison to the spume components of previous spray generation functions, the lower integration limit for the diameter is 100 *μ*m, which is just above the jet droplet range (6–100 *μ*m; Andreas 1998). Note that the lower limit for the Wu (1993) function is 150 *μ*m. From our model, the total production of spume drops seems to be slightly less at a laboratory fetch. This is a result of less normalized breaking relative to the 75-km fetch.

In the limit that every dominant wave crest breaks, the length of breaking crests per unit area per unit time depends only on the number of crests that pass through a unit area per unit time. As the dominant waves get longer with increasing fetch, the corresponding phase speeds only increase as the square root of the wavelength according to the deep water dispersion relationship. In other words, the number of crests passing through a unit area per second is greater at shorter fetches. This means that there must be more normalized breaking at the 75-km fetch. Otherwise, there would be more spume production at the 22-m fetch.

Nevertheless, the wind speed dependence of our spume production function follows the function from Fairall et al. (1994) at all wind speeds but is about an order of magnitude lower. This discrepancy in magnitude likely stems from the difference between whitecap A and whitecap B coverage. Whereas Fairall et al. (1994) find the magnitude from whitecap B coverage, our model is tuned with whitecap A coverage because spume drops presumably form only at active breaking sites. Furthermore, our generation function does not consider jet and film spray droplets that can form within the more expansive whitecap B coverage.

## 4. Vertical flux

*υ*(

_{f}*D*) = 2

_{f}*σ*, where

_{z}*σ*= 1.25

_{z}*u*

_{*}is the standard deviation of the turbulent vertical velocity. Assuming the distribution of the turbulent vertical velocity is Gaussian, a simple probabilistic relationship between the produced droplets and the vertically transported droplets is the complementary error function (erfc) of the ratio between the settling velocity and the standard deviation of vertical turbulent velocity. As this ratio goes to zero, erfc goes to one. When the settling velocity is greater than two standard deviations of the turbulent velocity, erfc quickly goes to zero. In other words, all of the droplets with settling velocities greater than the most extreme upward turbulent velocities immediately fall back to the ocean, while nearly all of the droplets with very small settling velocities are able to be suspended. The resulting distribution of drops transported upward takes the following form:with corresponding volume flux,

Figures 2a and 2c show the vertically transported distribution of spume for both laboratory and field fetches, and Figs. 2b and 2d plot the corresponding volume flux. For reference, a line with the typical diameter relationship *D*^{−8} (e.g., Andreas 1992; Fairall et al. 1994; Wu 1993) for large drops is also plotted. Compared to the source functions, the peak diameter of the vertically transported spray does not shift nearly as much with wind speed. In fact, the peaks in Fig. 2 show that the diameter of the vertically transported spray increases with wind speed until the wind reaches roughly 30 m s^{−1}; at greater wind speeds, the peak diameter reduces as before. The average value for these peaks of the vertically transported spray is roughly 300 *μ*m, which agrees with the peaks of the spume functions reviewed by Andreas (2002). Nevertheless, Fig. 2 illustrates a possible reason why previous investigations have found or assumed a constant peak diameter even though theoretical studies predict smaller droplets with increased wind forcing.

## 5. Mass and energy fluxes

*μ*m and 10 mm. While the lower limit is a conventional number, the upper limit is roughly the maximum diameter at

*U*

_{10}= 9 m s

^{−1}, where spume production begins as can be seen in Fig. 1. The mass of spume droplets produced at the surface appears to be nearly constant with wind forcing. Again this is because of the balance between decreasing droplet size and increasing number of drops formed with increased wind forcing. This balance will be explored in more detail below.

The mass of spume droplets that can be suspended, however, increases sharply with wind forcing at low to moderate wind forcing. Eventually, the wind forcing becomes sufficiently strong to suspend virtually all of the droplets produced. Therefore, even the mass of suspended drops plateaus at extremely high wind speeds. An upper limit for mass production is somewhat intuitive because there is only so much surface from which to tear water. In fact, the spray function from Fairall et al. (1994) scales the normalized Andreas (1992) spray function at *U*_{10} = 11 m s^{−1} with whitecap B coverage. This leads to the maximum mass flux, that is, where the whitecap B coverage is 100%, at approximately *U*_{10} = 40 m s^{−1} or *u*_{*} = 2 m s^{−1}. The limit of the mass flux for our source function at high wind forcing is about 4 times lower than the limit from Fairall et al. (1994) also plotted in Fig. 4 (gray dashed line), which again roughly corresponds to the difference between whitecap coverages A and B. In any event, our two curves provide natural bounds for previous spume production functions. The spume source function (upper curve) represents the drops that are generated just above the crest of breaking waves, some of which may never be seen as they never separate substantially from the surface. The suspended spume function (lower curve) represents the droplets that can be suspended in the air. The spume distribution that resides in the air for some finite amount of time lies somewhere between these two curves.

*dF*/

*dD*for

*dN*/

*dD*to find the vertically transported energy flux

*E*. The integration limits are the same as those in the mass flux calculation. Figure 5 shows the energy flux for the spume source function and the suspended spume function. For reference, the energy fluxes from Andreas (1998), Fairall et al. (1994), and Wu (1993) are also shown. Again, the limit of our energy flux at high wind speeds is about 3 times lower than the limit from Fairall et al. (1994) because of the difference between whitecap A coverage and whitecap B coverage. This also may explain the slope of our energy flux, as we do not consider the smaller jet and film droplets that result from whitecap B coverage. Moreover, the extrapolated energy flux from Andreas (1998) does not increase at the rate of energy supplied from the wind to the ocean at high values of friction velocity because our modeled friction velocity increases linearly with wind speed at high winds. Even though the mass flux of our source function plateaus, the shift to smaller droplets results in more free energy with increased wind forcing.

The rate at which the wind provides energy to the surface, *E _{t}* ∝

*ατU*

_{0}∝

*βu*

_{*}

^{3}, is also plotted in Fig. 5, where the surface drift velocity is approximated as

*U*

_{0}≈ 15.2(

*τ*/

_{ν}*ρ*)

_{w}^{1/2}(Wu 1983) with

*τ*as the average viscous stress per unit area from Mueller and Veron (2009b). To match

_{ν}*E*with

_{t}*E*and

_{N}*E*at high wind speeds,

*α*and

*β*are

*σ*/150 and

*α*/10, respectively. From such small values, it is evident that much of the energy supplied to the surface does not appear to go toward generating spume droplets. Whitecap formation and other wave breaking processes presumably consume the rest of the energy when the air–sea boundary is close to equilibrium. Furthermore, the formation energy required to produce the source distribution of drops is presumably less significant than the energy required to form the suspended droplets. The reason is that the suspended droplets not only take mechanical energy at formation but also consume heat energy as they evaporate and get even smaller.

## 6. Conclusions

The proposed spume generation process produces drop sizes that span the empirical range. Immediately after separation from the surface, the spume drops have diameters on the order 0.01–10 mm. The vertically transported and completely atomized drops are closer to the more familiar 10–500-*μ*m range. Although the proposed source function implies considerably larger mass and energy fluxes at low to moderate wind speeds, most of this discrepancy is due to its vast size range compared to other source functions. Excluding the drops greater than 1.5 mm in diameter, which are typically ignored because they cannot be transported vertically, our source function closely resembles currently accepted generation functions. The uncertainty in many of the terms stresses the need for further study of the individual processes presented here as well as direct measurements of the production and suspension of droplets.

We thank Ed Andreas for numerous fruitful discussions as well as his helpful suggestions on an early draft. This work was supported by ONR Grant N00014-05-1-0609 to Veron.

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^{1}

To the best of the authors’ knowledge, Phillips (1969) was the first to speculate that spume droplets originate from a Kelvin–Helmholtz instability.

^{2}

The coefficients and power laws presented here may differ slightly from those given in Marmottant and Villermaux (2004) because we use least squares fits to their air–water data.