Introduction

Under windy conditions, the ocean no longer has a simple interface with the atmosphere. Crashing waves produce bubbles that penetrate to depths exceeding two meters, while spume droplets literally are torn from the wave crests. Both of these effects act to increase the effective surface area of the ocean relative to calm conditions.

The ocean surface acts as an interface for the exchange of energy and mass between the ocean and the atmosphere. Therefore, any quantity crossing the air-sea interface will experience enhanced transport due to the increase in effective ocean surface. Of particular interest in energy budget calculations are the transport of heat and moisture across the air-sea interface. Clearly, when the ocean and atmospheric boundary layer are at different temperatures, sensible heat will be transferred across the air-sea boundary. Furthermore, unless the relative humidity of the marine boundary layer is exceptionally high, sensible heat will also be transported via moisture saturation of entrained air bubbles and evaporation of spray droplets.

The large heat fluxes that seem necessary to sustain hurricanes and other strong cyclonic storms are not accounted for by simple interfacial fluxes of sensible and latent heat. It has been suggested that the necessary heat flux may be accounted for by considering the enhanced heat and moisture flux from bubbles and sea spray.

The problem of calculating the flux enhancement breaks down into two separate functions. We must first determine the flux of bubbles at the sea surface as a function of windspeed, then calculate the heat flux based on temperature and humidity conditions of the atmosphere and ocean. Likewise, we must determine spray production as a function of windspeed and make similar calculations.

Finally, there is considerable interest in determining the flux of sea salt into the marine boundary layer as a consideration of total aerosol load in the atmosphere. Implications include effects on atmospheric optical depth and total available surface area for chemical reactions and particle growth.

Modeling bubble generation

We are interested in knowing the flux of bubbles to the sea surface, from which we can derive the associated heat and moisture fluxes. Breaking waves generate foam that contains bubbles, and, in addition, crashing waves entrain near-surface air, driving bubbles down to several meters below the sea surface. Therefore, bubbles will be present only where the ocean exhibits whitecap and foam coverage. This leads to the conclusion that in order to estimate total bubble flux, we need to know both the flux of bubbles to the sea surface in whitecap covered areas and the average area of whitecap coverage.

Monahan and O'Muircheartaigh (1980) and Monahan (1989) classify whitecap coverage as consisting of two basic types. The first type is that which is directly produced by spilling wave crests, and are denoted as Stage A (W_A) whitecaps. The bubbles associated with W_A are characterized by a broad spectrum in droplet size and an extremely dense concentration of bubbles. The second type of whitecap coverage, denoted as Stage B (W_B) whitecaps, evolve from W_A over time. The bubbles associated with W_B have a much narrower size distribution and tend to have a larger fractional coverage of sea surface. Observations and analysis indicate that the fractional surface coverage of W_A and W_B are nearly proportional to the third power of the ten-meter wind speed, U₁₀. The exact relationships between fractional whitecap coverage and ten-meter wind speed are as follows:

(Monahan) (1)

(Monahan & O'Muircheartaigh) (2)

where W_A and W_B are fractions of the sea surface covered for for wind speeds U₁₀ expressed in meters per second.

The second piece of information we need is the concentration of bubbles under a given area of whitecap coverage. This quantity is expressed as a number of bubbles per cubic meter of sea water per increment in bubble radius. Acoustic measurements by Deane (1997) give a bubble spectrum, which Andreas and Monahan (1992) uses as an upper bound in his estimates of heat and moisture transfer (Figure 1).

Now, to determine the flux of bubbles across the surface of the ocean, we need to estimate the rise velocity of bubbles submerged beneath the surface. For the purposes of determining the total flux of bubbles across the air-sea interface, Andreas assumes that all bubbles reach their terminal rise velocity quickly. According to Clift et al. (1978) and Thorpe (1982), the terminal rise velocity can be roughly divided into two values, depending on bubble size. Considering the entire range of bubbles up to 6 mm in radius , which is the largest radius reported by Dean (1997), the volume flux of air across the air-sea interface under Stage A whitecaps is determined from

(3)

where u(R) is the terminal rise velocity of bubbles of radius R, and is the number concentration of bubbles of radius R per cubic meter of sea water. This calculation yields the resulting volume flux

(4)

This value, with units of cubic meters of air per square meter per second, represents an upper bound on the volume flux of air across the air-sea interface, as we have used the largest spectrum of bubble radii and the terminal rise velocities of the bubbles.

The volume flux of air crossing the air-sea interface under Stage B whitecaps is found in exactly the same manner. However, the bubble concentration spectrum determined by Monahan (1988, 1989) gives an upper bound of 150 µm on bubble radius.

The volume of air V_B crossing the air-sea interface under Stage B whitecaps is thus:

(5)

We now have all the information necessary to estimate the volume flux of bubbles across the air-sea interface as a function of wind speed. The total air volume flux will be the simple product of air volume flux per square meter of whitecap coverage and whitecap coverage as a function of windspeed. Thus, we have

(6)

for the volume of air due to Stage A whitecaps, and

(7)

Clearly, in any further calculations, it is acceptable to neglect the volume flux contribution from Stage B whitecaps, as the V_BW_B is four orders of magnitude less than V_AW_A.

Determining the bubble-mediated transfer of heat and moisture

Studies show that in windy conditions (i.e., conditions likely to produce crashing waves), seawater temperature is uniform to a depth of several meters (Donlon and Robinson, 1997). Furthermore, because the heat capacity of air is much smaller than that of seawater, even the largest bubbles reach thermal equilibrium within one second. The terminal rise velocity for bubbles larger than 1 mm is given by Woolf (1993) as 0.25 m s^-1 in 20›C seawater. Thus, all bubbles submerged below 25 cm will reach temperature equilibrium with the ocean, whether the temperature of the entrained air is greater than or less than the ocean temperature. Andreas and Monahan (2000) assume that since breaking waves plunge many bubbles deeper than 25 cm, and that since many bubbles plunged to a depth less than 25 cm are less than 1mm in diameter, virtually all entrained air comes to thermal equilibrium with the ocean before crossing the interface back into the boundary layer. As a result, the ocean converts entrained air from temperature T_h to temperature T_w. , where T_h is the temperature of air at height h above the ocean surface and T_w is the ocean surface layer temperature. The amount of warming will of course be more dramatic when there is a significant difference in the temperatures of the atmosphere and ocean, as is observed at high latitudes.

The term T_h is somewhat problematic, as the height from which air is entrained can vary from 1 cm to 1 m or more. Andreas and Monahan (2000) instead model T_h in terms of T₁₀, the 10 m temperature

(8)

where t_* is a flux scale relating sensible heat flux to the average temperature profile, ç_h is an empirical profile correction that is a function L, the Obukhov length, and k is the von Kármán constant.

Thus, the sensible heat flux across the air-sea interface due to bubbles is

(9)

where C_H10 and C_D10 are dimensionless bulk transfer coefficients for sensible heat and momentum flux (Andreas and Monahan, 2000).

To determine the amount of latent heat transported by bubbles across the air-sea interface, Andreas and Monahan adopt terms from Pruppacher and Klett (1978) and Andreas (1989) describing the vapor pressure at the interior surface of a bubble submerged in seawater. The discussion of the derivation of the formula for the latent heat flux due to bubbles Q_bL goes beyond the scope of this paper. Andreas and Monahan report the flux as

(10)

where C_E10 is the dimensionless bulk transfer coefficient for latent heat.

Modeling spray generation

Several spray generation functions have been suggested in the literature, though none are entirely satisfactory (Andreas et al., 1995, Andreas 1998). For any given particle radius and ten meter wind speed, estimations on number production vary by several orders of magnitude, as shown in Figure 2. Some confusion comes from the several ways which the spray generation function can be defined. Andreas et al. (2001) and Andreas (2002) clarify notation and review the available functions. For the purposes of this paper, I retain the spray generation function used by Andreas (1992).

All of the generation functions reviewed by Andreas (1992) are based on empirical data sets from experiments such as JASIN (Joint Air-Sea Interaction Experiment) and STREX (Storm Transfer and Response Experiment), and from laboratory experiments using wind tunnels and water tanks. Ultimately, Andreas (1992) decides on a generation function divided into three domains, based on droplet radius

for (8a)

for (8b)

for (8c)

Notice that these fluxes are given in terms of r₈₀, which is the radius of a droplet at a reference humidity of 80%. The equations Andreas (1992) derives to determine the heat and moisture transfer of spray droplets as reported in the next section depend on r₀, which is larger than r₈₀. In order to use the generation function with the equations for determining the heat and moisture transfer Andreas uses the following relation to convert between initial and reference radius

(9)

where

(10)

Thus, by simple differentiation,

(11)

Furthermore, the coefficients C₁, C₂, and C₃, are fitted to the 10 meter wind speed U₁₀, and are reproduced in Table 1.

Determining the heat and moisture transfer of spray droplets

According to Andreas (1992), examination of the effects of sea spray enhancement on air-sea heat and moisture exchange dates back to the 1940s. Andreas (1989, 1990b) develops the characteristic time constants ã_T and ã_r to determine the time for a droplet to reach equilibrium with its environment and the characteristic time constant ã_f to determine the atmospheric residence time for a droplet. Andreas' model shows that

(12)

where T_W is the seawater temperature, at which a droplet presumably starts, and T_eq is the equilibrium temperature of the droplet. Likewise,

(13)

where r₀ is a droplet's initial radius and r_eq is the radius of a droplet that has reached moisture equilibrium with its environment.

Finally, Andreas defines the characteristic fall time ã_f as

(14)

where u_f is the Stokes fall speed modified for large Reynolds numbers as described in Andreas (1989, 1990b), and A_1/3 is the significant wave amplitude. The significant wave amplitude is defined as the mean level above sea level of the highest one third of all ocean waves. Clearly, A_1/3 is a function of windspeed. Andreas (1992) chooses from the literature the following relation between A_1/3 and the ten meter wind speed

(15)

where U₁₀ is measured in meters per second, and A_1/3 is measured in meters.

Using the characteristic fall time ã_f Andreas (1992) calculates the sea spray mediated transfer of sensible and latent heat to the atmosphere, denoted Q_sS and Q_sL respectively. A droplet returning to the surface of the ocean will have a temperature

(16)

Thus, the transfer of sensible heat by all droplets of a given initial radius will be

(17)

where á_W is the density of seawater, c_ps is the specific heat of seawater at constant pressure, and dF/dr is the aforementioned sea spray generation function. Clearly, the term represents the volume flux of droplets with initial radius r₀. According to Figure 3, all except the largest drops have ã_f > ã_T which implies that all available sensible heat will be transferred between droplet and environment before the droplet returns to the surface of the ocean.

Andreas determines the formula for Q_sL similarly. The radius of a spray droplet with initial radius r₀ returning to the ocean surface is given by

(18)

However, according to Figure 3, ã_f is not always greater than ã_r. Therefore, two equations are needed to represent the transfer of latent heat from droplets to the atmosphere. The difference between the two equations lies in the condition that in the first, the droplets have not reached moisture equilibrium with the environment, while in the second, they have. In the first case, the transfer of latent heat by all droplets of a given initial radius will be

for (19)

and in the second case,

for (20)

where L_V is the latent heat of vaporization of water and again á_W is the density of seawater and the quantity is the volume flux of spray droplets with initial radius r₀. Note that in both equations, the evaporative process actually extracts heat from the atmosphere.

Comparing the heat and moisture transfer of sea spray and bubbles to the usual interfacial fluxes of sensible and latent heat

Andreas and Monahan (2000) report the usual interfacial fluxes of sensible and latent heat, H_S and H_L, as

(21)

(22)

where á is the density of seawater, c_p is the specific heat of air at constant pressure, C_H10 is the dimensionless bulk transfer coefficient for sensible heat, U₁₀ is the 10 m wind speed, T_W is the seawater temperature, T₁₀ is the 10 m temperature, L_v is the latent heat of vaporization of seawater, C_E10 is the dimensionless bulk transfer coefficient for latent heat, q_w is the specific humidity of air in saturation with seawater of temperature T_W, and q₁₀ is the 10 m specific humidity. The values for the bulk transfer coefficients C_H10 and C_E10 are evaluated from the COARE bulk flux algorithm by Fairall et al. (1996). The combined interfacial and bubble mediated sensible and latent heat transfer, H_S,T and H_L,T result from combining (9) and (21) and (10) and (22) to give (23) and (24)

and

Andreas and Monahan (2000) calculate modification factors f_S and f_L which describe the bubble-mediated enhancement to the interfacial transfer of heat and moisture across the air-sea boundary. These factors are

(25)

and

(26)

As the graphs in Figure 4 show, for all but the highest windspeeds, f_S and f_L differ from unity by less than 5%. This leads to the conclusion that bubbles do not significantly enhance heat and moisture transfer beyond the usual aerodynamic bulk transfer given by (21) and (22).

Andreas (1992) summarizes the comparison of sea spray-mediated sensible and latent heat fluxes with the usual interfacial fluxes in Table 2, making calculations from eqs. (17) and (19) for sea spray contributions and (21) and (22) for the usual interfacial fluxes. Generally, the contribution of sea spray-mediated transfer of sensible heat falls well short (by an order of magnitude) of the bulk transfer. However, depending on conditions, the sea spray-mediated transfer of latent heat often has the same order of magnitude as the usual bulk transfer of latent heat. The latent heat transfer is sensitive to the atmospheric temperature, as droplets evaporate more efficiently in warm air than in cool air.

Comments and Conclusions

The models used by Andreas show that little sensible heat is transferred across the air-sea interface as a result of air entrainment by crashing waves and its subsequent release back into the atmosphere. Also, the sea spray-mediated transfer of sensible heat is shown to be less than 10% of the usual bulk sensible heat transferred across the air-sea interface. Thus, it is difficult to measure and distinguish as a separate quantity. However, the sea spray-mediated transfer of latent heat is shown to be significant in some circumstances, particularly when the atmospheric temperature is warm (>20°C) and RH is low. These results lead to the conclusion that overall, bubbles and sea spray do not significantly contribute to the transfer of sensible and latent heat to the atmosphere, and are thus unlikely to account for the energy necessary to sustain large cyclonic storms.

The model is not without flaws, however. For instance, the model assumes fixed profiles for such quantities as relative humidity and temperature above the surface of the ocean. This assumption, however, can only cause an overestimation on the transfer of latent heat. Therefore, it does not change the conclusions based on the model's results.

Other researchers in the field take issue with Andreas findings which are in contradiction with experimental findings. Katsaros and de Leeuw (1994) point out that the Humidity Exchange Over the Sea (HEXOS) program determined that the bulk exchange coefficients are not wind speed dependent below 18 m s^-1, though Andreas' model treats them this way.

Finally, Andreas concludes that the energy transfer to the atmosphere from sea spray is large enough to warrant consideration in global climate models. Given that forcings on the order of a few watts per square meter are sufficient to cause measurable climate change, this suggestion is reasonable.

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