Spray-Mediated Air-Sea Gas Exchange: The Governing Time Scales

Abstract

It is not known whether sea spray droplets can act as agents that influence air-sea gas exchange. We begin to address that question here by evaluating the time scales that govern spray-mediated air-sea gas transfer. To move between the interior of a spray droplet and the atmospheric gas reservoir, gas molecules must complete three distinct steps: (1) Gas molecules must mix between the interior surface and the deep interior of the aqueous solution droplet; time scale τ_aq estimates the rate of this transfer; (2) Molecules must cross the droplet’s interface; time scale τ_int parameterizes this transfer; and (3) The molecules must transit a “jump” layer between a spray droplet’s exterior surface and the atmospheric gas reservoir; time scale τ_air dictates the rate of this transfer. The same steps, in reverse order, pertain to gas molecules moving from an atmospheric reservoir to a drop’s interior. For the six most plentiful gases, excluding water vapor, in the atmosphere—helium, neon, argon, oxygen, nitrogen, and carbon dioxide—τ_air, τ_int, and τ_aq are shorter than the time scales that quantify the rate at which a newly formed spray droplet’s temperature, radius, and salinity evolve. We therefore conclude that, following the assumptions herein, a model for spray-mediated air-sea gas exchange can assume that the gas concentration in spray droplets is always in instantaneous equilibrium with the local atmospheric gas concentration.

1. Introduction

In the open ocean, when the wind blows strongly enough and long enough, the sea surface reaches a dynamic equilibrium where the work done by the wind on the sea is dissipated in part by breaking waves, which result in the formation of whitecaps (e.g., [1,2]. The bubbles and sea spray associated with breaking waves and their associated whitecaps enhance the air-sea exchange of heat, moisture, and gases [3]. However, not all bubbles upon bursting at the sea surface produce sea spray droplets [4]), and not all sea spray droplets are formed by the bursting of whitecap bubbles (Figure 10 in reference [5].

While many authors have assessed the role of sea spray droplets in the air-sea exchange of heat and moisture [6,7,8,9,10], Andreas and Monahan (2000) [11] gained further insights by looking at the role played by bubbles, and the moisture-laden air they contained, in the air-sea exchange of heat and moisture. Likewise, many research groups have considered the role of bubbles bursting on the sea surface in enhancing the air-sea exchange of a wide range of gases, particularly when the wind approaches speeds above 10 m s⁻¹ [12,13,14,15,16,17] Subsequently, many other researchers took up the study of bubble-mediated gas transfer [18,19,20,21,22,23,24,25,26]. Noting that the role of sea spray in gas transfer had not been assessed, our group was motivated to evaluate sea spray’s potential role in addition to the established role of bubbles in this exchange.

A recent brief publication dealing with the time scales relevant to an evaluation of the role of sea spray droplets in air-sea gas exchange identified the importance of evaluating timescales involved in both droplet physics and gas diffusion [3]. Recognizing the benefits of a full systematic development of the time scales associated with the evolution of these droplets, and of the steps associated with gas moving into, and out of, these airborne droplets, we have undertaken to provide this comprehensive treatment in expanded form herein.

We therefore begin this formal treatment of how and whether sea spray might influence the rate of air-sea gas transfer following the approach that Andreas and colleagues used for assessing how sea spray affects air-sea heat and moisture transfer [9,27,28,29,30,31]. Microphysical modeling of the temperature and radius evolution of individual spray droplets underlies most of this work [32,33,34,35]. In turn, we will begin our study of spray-mediated gas transfer by first studying the microphysical, thermodynamic, and gas transfer properties of individual spray droplets. We quantify these processes with time scales that ultimately show us where the rate-limiting step is for spray-mediated gas transfer and, consequently, inform our decision on how next to proceed.

The difference in partial pressures of the gases in the sea and in the near-surface air (Δp) is dictated by a gas’s Henry’s Law Constant (K_H). In the case of sea spray, the dynamic nature of the evaporating droplet leads to a shifting K_H due to both changes in temperature and salinity. As wind speeds increase, the nature of sea spray also changes, shifting from small film droplets and moderate jet drops at wind speeds between 4 to 12 m s⁻¹ to a second sea spray source represented by the typically larger spume drops tearing off wave crests at higher wind speeds. The spume drops become increasingly important as wind speeds increase (>12 m s⁻¹). A critical analysis of the timescales involved in the gas exchange of sea spray involves the distinction between small droplets that remain in air long enough to reach equilibrium and large droplets that return to the sea before doing so, sometimes virtually unchanged. The former become hypersaline as they evaporate and dissolved gases experience a corresponding decrease in solubility and increase in partial pressure. The cutoff between the small and large droplet pools is greatly dependent on the relative humidity [26,36,37,38]. Ascertaining the fraction of sea spray that reaches equilibrium under a given set of conditions enables one to reasonably gauge the effusion of gas into the atmosphere from small drops. The larger drops will contribute to gas exchange primarily through changes in temperature and modest changes in volume. These interactions require accurate estimates of the timescales controlling gas transfer.

2. Theoretical Background

2.1. Spray Droplet Evolution

Microphysical modeling provides insights into spray droplet behavior [9,32,35,39], as shown in Figure 1 for a droplet whose initial radius (r₀) was 100 μm. Initially, droplet temperature (T) and radius (r) follow exponential relations:

$$\frac{T\left(t\right)-T_{eq}}{T_w-T_{eq}}=\exp \left(-t/{\mathsf{\tau }}_T\right)$$

(1)

$$\frac{r\left(t\right)-r_{eq}}{r_0-r_{eq}}=\exp \left(-t/{\mathsf{\tau }}_r\right)$$

(2)

In Equations (1) and (2), t is the time since the droplet formed, T_w, the sea surface temperature (assumed to be the initial droplet temperature); T_eq and r_eq, the droplet’s equilibrium temperature and radius and τ_T and τ_r, the corresponding e-folding times to reach these equilibrium values.

The microphysical quantities T_eq, r_eq, τ_T, and τ_r all depend on the initial droplet radius, temperature, and salinity and on the environmental conditions like air temperature (T_a), relative humidity (RH), and barometric pressure. In particular, for droplets smaller than 100 μm, the curves in Figure 1 all shift to the left, i.e., to earlier times. For larger droplets, the curves all shift to the right, i.e., to later times. Nevertheless, for all radii, there is always the three-order-of-magnitude separation between τ_T and τ_r that Figure 1 depicts. These microphysical quantities are estimated using fast algorithms by Andreas (2005a).

Figure 2 shows the range of droplet sizes that we expect will be important for spray-mediated gas transfer, with sea spray droplets ranging in radius from 0.5 to 500 µm. This figure presents the spray generation function (F) as a volume flux. The number of droplets of radius r₀ produced per square meter of sea surface, per second, per micrometer increment in droplet radius is commonly denoted dF/dr₀ [5,28,40]. Therefore, the volume flux in Figure 2 is $\left(4\mathsf{\pi }{r}_0^3/3\right)dF/d{r}_0$, for any spray-mediated process. Spume droplets carry most of the water and therefore, as with spray-mediated heat and moisture transfer [9,30].), may be responsible, under high-wind conditions, for most of the spray-mediated air-sea gas exchange. While the flux of smaller spray droplets, those produced by the bursting of whitecap bubbles varies as roughly the cube of the wind speed, it should be acknowledged that the wind dependence of the flux of the larger spume drops is not as well constrained.

2.2. Gases and Spray Droplets

The time scales that govern how gas molecules interact with water droplets are based on work by Seinfeld and Pandis [38], Pruppacher and Klett [43], and Lamb and Verlinde [44]. Consideration must be made, however, for differences wherein cloud droplets are often smaller than sea spray droplets, and are never as saline.

Figure 3 schematically shows how an individual spray droplet may exchange an arbitrary gas with an atmospheric reservoir. While gas exchange with cloud droplets is usually an invasion process, spray-mediated transfer, on the other hand, can involve both invasion due to changes in partial pressures and initial cooling of the droplet and evasion due to subsequent increases in temperature and evaporation. Though we hypothesize evasion will be the primary direction for spray-mediated air-sea gas transfer, both directions are addressed.

Gas transfer across an air-water interface is controlled by K_H where,

$$C_d\left(r_0\right)=K_H p_g\left(r_0\right)$$

(3)

Here, p_g(r₀) is the partial pressure (or fugacity; Pilson 2013 [45], p. 416) of arbitrary gas g at the external surface of a droplet, and C_d(r₀) is the concentration of the gas just inside the droplet. Here, p_g and C_d are in atm and mol L⁻¹, respectively. Hence, K_H has units of mol (L atm)⁻¹. Using the ideal gas law to convert p_g to the concentration of the gas in air (C_a). Equation (3) thus becomes

$$C_d\left(r_0\right)=K_H R T C_a\left(r_0\right)$$

(4)

where R ($8.20574\times {10}^{-2} \mathrm{atm} \mathrm{L} \mathrm{mo}{\mathrm{l}}^{-1} {\mathrm{K}}^{-1}$) is the universal gas constant, and T is in Kelvins. Note, the product K_H R T is dimensionless. Equation (4) explains the apparent discontinuity in concentration in Figure 3 at the surface of the spray droplet.

For the gases of interest—helium, neon, argon, oxygen, nitrogen, and carbon dioxide—K_H generally increases with decreasing temperature and decreasing salinity. Therefore, in contrast to interfacial and bubble-mediated transfer, for which the ocean’s temperature and salinity and thus, K_H can be assumed to be constant during a gas flux measurement by eddy-covariance (i.e., for 30–60 min), the Henry’s law coefficient for an evolving spray droplet is continually changing. This may have important implications for measurements conducted in high-sea-spray conditions.

Figure 4 is an example of how K_H for each of our six gases would change during the droplet evolution that Figure 1 depicts. All K_H values increase slightly as the droplet initially cools; but if the droplet remains in the air long enough to reach radius equilibrium, its increasing salinity causes K_H to decrease to between 53% and 65% of its original value (see Appendix A). In other words, with time, a spray droplet generally becomes a less hospitable reservoir for dissolved gas, augmenting evasion.

2.3. Microphysics

Pruppacher and Klett ([43], p. 761) derive an equation for the rate of change of gas mass (m_g) in a solution droplet under the assumption that the gas in the droplet is well mixed:

$$\frac{d{m}_g}{dt}=4\mathsf{\pi }r(t){D^{\prime }}_g\left[\frac{C_d(t)}{K_H(t)R{T}_d(t)}-C_a(\infty )\right]$$

(5)

Here, t is time; the droplet’s radius r(t), gas concentration C_d(t), and temperature T_d (t) and the Henry’s law coefficient K_H(t) all depend on time. C_a(∞) is the gas concentration in air far from the droplet (Figure 3). ${D^{\prime }}_g$ is the modified diffusivity of the gas in air (cf. Andreas 1989, 2005b, [34,39]):

$${D^{\prime }}_g=\frac{D_g}{\frac{r}{r+{\Delta }_g}+\frac{D_g}{r\mathsf{\alpha }}{\left(\frac{2\mathsf{\pi }{M}_g}{R{T}_a}\right)}^{1/2}}$$

(6)

In this expression, D_g is the molecular diffusivity of the gas in air; M_g, the molecular weight of the gas; and T_a, the air temperature. Equation 6 incorporates two effects that dictate gas transfer to or from a small water droplet. The left term in the denominator accounts for surface curvature. If the droplet is small enough (i.e., r < 10Δ_g) the air and gas may no longer behave as continuous fluids; the Δ_g, the gas “jump” length, which is approximately the mean free path of the gas in air (λ_g; see Figure 3), determines what “small enough” is. Gas molecules cross this jump length as dictated by the kinetic theory of gases (e.g., [46], pp. 188, 234–235). Thus, in the second term in the denominator of (2.6), we recognize ${\left(2\mathsf{\pi }{M}_g/R{T}_a\right)}^{1/2}$ as $4/{\overline{v}}_g$, where ${\overline{v}}_g$ is the average speed of an ideal gas molecule according to the Maxwell-Boltzmann speed distribution ([46], pp. 60–64). Finally, in (2.6), α is the mass accommodation coefficient of the gas: the ratio of the number of gas molecules that stick to a droplet over the number that strike the droplet. Consequently, α must be one or less. There are few estimates of the value of α for our six gases. In his comparable microphysical model for heat and water vapor exchange with a spray droplet, Andreas [32,39]), used α_T = 0.7 for heat exchange and a range of α_v = 0.036 to 1.0 for water vapor [3]. The impact of α for each gas, is assessed through sensitivity calculations for a range of available values.

Recognizing in Equation (5) that $m_g=\left(4\mathsf{\pi }{r}^3/3\right)C_d$, (5) becomes

$$\frac{d{C}_d}{dt}=-\frac{3{D^{\prime }}_g}{r^2}\left(\frac{C_d}{K_{H}R{T}_d}-C_a\right)$$

(7)

Further, if we assume that r is the initial droplet radius r₀ and K_H and T_d are constant, (7) has the solution

$$C_d\left(t\right)=K_{H}R{T}_d{C}_a+\left(C_w-K_{H}R{T}_d{C}_a\right)\exp \left(\frac{-3{D^{\prime }}_{g}t}{r_0^2{K}_{H}R{T}_d}\right)$$

(8)

Here, C_w is the gas concentration in the surface seawater. Note, this assumes that the gas concentration in the droplet, C_d, is uniform. Note, T_d does change as the drop evolves, as does the radius; thus, K_H may change appreciably in saline droplets.

Equation (8) implies that gas concentration in the droplet evolves exponentially over an e-folding time identified with the subscript PK in Equation (9) because it is derived based on Pruppacher and Klett [43].

$${\mathsf{\tau }}_{PK}=\frac{r_0^2{K}_{H}R{T}_d}{3{D^{\prime }}_g}$$

(9)

But following Lamb and Verlinde ([44], p. 507), we recognize that τ_PK has two parts when we insert ${D^{\prime }}_g$ from (6):

$${\mathsf{\tau }}_{PK}=\frac{K_{H}R{T}_d}{3D_g}\left[\frac{r_0^3}{r_0+{\Delta }_g}+\frac{D_g{r}_0}{\mathsf{\alpha }}{\left(\frac{2\mathsf{\pi }{M}_g}{R{T}_a}\right)}^{1/2}\right]$$

(10)

The first term dictates how rapidly gas molecules cross the jump layer and, therefore, is the time scale in air (see Figure 3):

$${\mathsf{\tau }}_{air,PK}=\frac{r_0^3{K}_{H}R{T}_d}{3D_g\left(r_0+{\Delta }_g\right)}$$

(11)

The second term reflects the rate at which a droplet releases (or entrains) gas molecules across its interface and thus is the droplet’s interfacial time scale (again, see Figure 3):

$${\mathsf{\tau }}_{\mathrm{int},PK}=\frac{r_0{K}_{H}R{T}_d}{3\mathsf{\alpha }}{\left(\frac{2\mathsf{\pi }{M}_g}{R{T}_a}\right)}^{1/2}$$

(12)

While Equations (11) and (12) are derived here following Lamb and Verlinde ([44] p. 507), who derive the same estimate as (12) for the interfacial time scale (i.e., τ_int,LV = τ_int,PK), they ignore the jump layer (and, thereby, [cf. (11)]).

$${\mathsf{\tau }}_{air,LV}=\frac{r_0^2{K}_{H}R{T}_d}{3D_g}$$

(13)

By assuming that the gas concentration at the external surface of the spray droplet is equal to that of the bulk air, Seinfeld and Pandis ([38], pp. 538, 549–551, 580–582) estimate the gas time scale in air as

$${\mathsf{\tau }}_{air,SP}=\frac{r_0^2}{4D_g}$$

(14)

Following Kumar [47], Seinfeld and Pandis ([38], pp. 551–554) start with the equation for diffusion of gas within a spherical droplet:

$$\frac{d{C}_d}{dt}=D_{g,sw}\left(\frac{d^2{C}_d}{d{y}^2}+\frac{2}{y}\frac{d{C}_d}{dy}\right)$$

(15)

Notice, in contrast to Pruppacher and Klett’s [43] derivation, C_d is now a function of radial position within the droplet. Here, also, D_g,sw is the molecular diffusivity of a gas in seawater; and y is the radial distance from the center of the droplet, which is at y = 0. An assumed initial condition is, without losing generality, that at time zero, the initial droplet gas concentration C_d,t=0 is C_w.

C(y, t = 0) = C_w for y ≤ r₀

(16)

A second boundary condition, because of the spherical symmetry, is that the droplet is well mixed, and at the center of the droplet (y = 0) and for all time t

$$\frac{d{C}_d}{dy}=0$$

(17)

The key part of this approach is identifying the flux boundary condition at the surface of the droplet. Here, the net flux across the droplet’s interface is (Seinfeld and Pandis [43], pp. 551–552; Lamb and Verlinde [44], pp. 503–504; cf. Bohren and Albrecht [46], pp. 233–234)

$${\mathsf{\varphi }}_{net}=\frac{\mathsf{\alpha }\left[C_a\left(r_0\right)-C_d\left(r_0\right)\right]}{K_H{\left(2\mathsf{\pi }{M}_{g}R{T}_d\right)}^{1/2}}$$

(18)

As Figure 3 implies, in this, C_a(r₀) is the gas concentration at the surface of the droplet just outside of it; C_d(r₀) is the gas concentration just inside the droplet.

Kumar [47] and Seinfeld and Pandis ([38], pp. 552–553) complete this analysis to derive the solution for the gas concentration within the droplet, which is, for y ≤ r₀,

$$C_d\left(y,t\right)=C_a\left(r_0\right)\left[1-{\displaystyle \sum _{n=1}^{\infty }\frac{2r_0^{2}H \sin \left({\mathsf{\beta }}_{n}y/r_0\right)}{y{D}_{g,sw}\left(L+L^2+{\mathsf{\beta }}_n^2\right)\sin \left({\mathsf{\beta }}_n\right)}}\right]\exp \left(-\frac{{\mathsf{\beta }}_n^2{D}_{g,sw}t}{r_0^2}\right)$$

(19)

In this,

$$H=\frac{\mathsf{\alpha }}{K_H{\left(2\mathsf{\pi }{M}_{g}R{T}_d\right)}^{1/2}}$$

(20)

$$L=\frac{r_{0}H}{D_{g,sw}}-1$$

(21)

and the β_n are the “n” positive roots of

$${\mathsf{\beta }}_n\cot \left({\mathsf{\beta }}_n\right)+L=0$$

(22)

We ultimately solve (22) by Newton’s method although Carslaw and Jaeger ([48], p. 492) tabulate β_n for n = 1 to 6. Kumar [47] and Seinfeld and Pandis ([38], p. 553), however, explain that just the first root, β₁, provides an adequate solution in (19).

In (19), the e-folding time is

$${\tau }_{aq,SP}=\frac{r_0^2}{{\beta }_1^2{D}_{g,sw}}$$

(23)

For large values of L, nominally above 100, β₁ from (22) is π (e.g., Carslaw and Jaeger [29], p. 492) and

$$\underset{L\to \infty }{\lim }{\tau }_{aq,SP}=\frac{r_0^2}{{\pi }^2{D}_{g,sw}}$$

(24)

We will see this case often in our subsequent calculations because L is large when r₀ is large, or the gas is not very soluble (i.e., a small K_H)—see (20) and (21). Of our six gases, all except carbon dioxide are weakly soluble.

This final time scale, τ_aq, quantifies mixing inside the aqueous solution droplet. The other two primary sources (Pruppacher and Klett [43], p. 764; Lamb and Verlinde [44], p. 508), likewise, arrive at the same estimate for this time scale, which we will therefore designate

$${\mathsf{\tau }}_{aq}=\frac{r_0^2}{{\mathsf{\pi }}^2{D}_{g,sw}}$$

(25)

Equation (25) “was derived” under the assumption that the fluid within the droplets is motionless; gas molecules thus would move only through molecular diffusion, D_g,sw. Ample evidence exists, however, that the fluid even in small droplets develops circulations when a shear stress occurs at the droplet’s surface (Clift et al. 1978, pp. 36–38, 127–129; Pruppacher and Klett [43], pp. 386–393). Consequently, mixing of the gas within a droplet is surely much faster than τ_aq suggests. We thus consider (25) as the lower limit for mixing of gas molecules within a spray droplet.

2.4. Droplet Residence Time

A droplet’s residence time in air is the boundary condition under which these processes are considered. A spray droplet can exchange heat, water vapor, and gases with air only during its (often brief) lifetime between creation and re-entry into the ocean. As spume droplets, which are formed by the wind tearing drops right off of the wave crests, accomplish most of the spray-mediated heat and moisture exchange, Andreas and colleagues [9,28,30,31,49] based a residence time scale on their behavior where

$${\tau }_f=\frac{h_{1/3}/2}{u_f\left(r_0\right)}$$

(26)

Here, u_f(r₀) is the terminal fall velocity of droplets of radius r₀. Also, h_1/3 is the significant wave height; consequently, h_1/3/2 (≡A_1/3) is the significant wave amplitude, the height above mean sea level where many spray droplets originate. h_1/3 can be taken from measurements; but in the absence of measurements of wave height, we estimate h_1/3 from Andreas and Wang’s [50] algorithm in which h_1/3 goes as the square of the 10 m wind speed. Equation (26) models the residence time in air of a droplet that has reached terminal fall velocity. τ_f represents the minimum time aloft wherein droplets that have not reached terminal velocity would reside longer.

3. Spray Droplet Time Scales

Figure 5a–c compares time scales over a range of radii for one set of environmental conditions and for three gases: helium, oxygen, and carbon dioxide. In each plot, the τ_f, τ_r, and τ_T curves are the same, because these times do not depend on the gas. Helium (M_g = 4 g/mol), oxygen (M_g = 32 g/mol), and carbon dioxide (M_g = 44 g/mol) span the range from smallest to largest in our set of six gases, and thereby demonstrate how the size of the gas molecule influences τ_air, τ_int, and τ_aq. Any spray-mediated transfer, whether it is for sensible heat (τ_T), water vapor (τ_r), or gas molecules (τ_air, τ_int, τ_aq), must occur over a time less than the droplet’s residence time, τ_f. This time scale decreases as the radius increases, because u_f increases with radius. The τ_f curves in Figure 5a–c are for a 10-m elevation wind speed of 12 m s⁻¹; the τ_f curve will rise for increasing wind speed (because h_1/3 increases), allowing more spray-mediated exchange to take place. The τ_f curve, as an example, is above τ_r for radii up to about 30 μm. Droplets of 30 μm and smaller thus can reach radius equilibrium and become more saline before they fall back into the ocean. Droplets larger than 30 μm, where τ_f < τ_r, on the other hand, fall back into the water before losing much water and, consequently, return to the ocean with approximately their original salinity. A droplet’s temperature evolution is much faster. Thus, droplets up to about 200 μm in radius have time to reach an equilibrium temperature of T_eq at this wind speed before they fall back into the sea.

Our several estimates of gas time scales are comparable to and even shorter than τ_T. The mixing of gas molecules within a spray droplet, parameterized as τ_aq is a maximum value that does not recognize any fluid motion in a spray droplet that could certainly enhance gas mixing by at least an order of magnitude. Consequently, we suspect that the true τ_aq will be at least ten times less than depicted.

Mixing within the droplets falls into two categories. (1) Small droplets in the film and jet droplet range for which mixing is primarily molecular that reach radial and temperature equilibrium while the droplets are airborne. The change in radius and consequent increase in salinity significantly alters the dissolved gas fugacity and drives evasion such that, it can be expected, the majority of the dissolved gas at the time of droplet formation is evaded and delivered to the atmosphere; and (2) The internal mixing in larger droplets of r > 50 µm needs careful consideration, as these droplets are more likely to return to the surface ocean before reaching radial equilibrium. There are few studies that address these larger droplet dynamics, and the majority of these are generated as spume droplets resulting from tearing off wave crests. There are, however, several analogous processes that have been evaluated in rain droplet studies (50 < r < 2 000 µm). As spume drops fall within this size range, the dynamics of raindrops may shed light on the internal mixing processes that apply to these large marine-derived droplets.

Drops for which r < 500 µm can be effectively treated as spheres that experience oscillations, which can be reasonably predicted from water surface tension and density. These oscillations introduce dynamic pumping or mixing of the gases in the droplet [51]. At drop radii above 500 µm, surface tension is overcome by weight force, and drop geometry deviates from the purely spherical. At r > 500 µm eddy shedding and sideways drift begins to occur [52] and collisions between large drops add to the oscillations and further enhance these. Thus, these larger drops (50 µm < r) are turbulently mixed.

For the air time scales, τ_air, in Figure 5a–c we show only τ_air,PK, (2.11), because it is near the Lamb and Verlinde [44] scale, (2.13), and because the Seinfeld and Pandis [38] scale, (2.14), is based (unrealistically) only on molecular diffusion of gas molecules to and from a spray droplet. Note that τ_air,PK is orders of magnitude less than τ_T, which we consider to represent very fast exchange. τ_air,PK does, however, increase by roughly three orders of magnitude between helium and carbon dioxide as a consequence of how increasing molecular weight slows molecular velocities in air. The final time scale shown in Figure 5a–c quantifies exchange across a droplet’s interface, and we represent this exchange with τ_int,PK, (2.12), which, although increasing with the increasing molecular weight of the gas molecule, always represents extremely fast gas transfer.

Figure 6 shows the range of τ_air,PK values for our six gases. As with Figure 5a–c, τ_air,PK increases monotonically with the molecular weight of the gas. In effect, the mean molecular speed predicted by the kinetic theory of gases decreases as the molecular weight increases (see (Equation (12)). Still, all our gases of interest can cross the jump layer well within the time granted by a droplet’s flight from creation and back to the sea surface. Note that τ_f in Figure 6 is based on a wind speed of 12 m s⁻¹; at higher wind speeds, when spray concentrations would rapidly increase, τ_f gets longer with the square of the wind speed.

3.1. Implications of the Time Scales

Our comparisons of the time scales τ_air, τ_int, and τ_aq in Figure 5a–c yield significantly different results based on the approach of choice. The time for gas molecules to transit the jump layer that we base on Pruppacher and Klett’s [43] approach (i.e., τ_air,PK) is so short that we could assume the gas concentration right at the exterior surface of a spray droplet is always C_a(∞). Pruppacher and Klett’s approach also yields an interfacial time scale (τ_int,PK) that is even shorter for most droplet sizes. The caveat, though, is that their approach assumes that the gas is well mixed inside the droplets. Because our estimate of this internal mixing time scale, τ_aq, is orders of magnitude longer than τ_int,PK, this assumption seems inappropriate. We have discussed how, because of fluid motion within spray droplets, the actual time scale for internal mixing will be shorter than τ_aq, but we are uncertain whether it will be the 4 orders of magnitude shorter that would be necessary to justify Pruppacher and Klett’s approach.

At this point, it appears essential that we make at least a crude estimate of how fluid motion within spray droplets may affect τ_aq. LeClair et al. ([53]; also [54], pp. 127–129; [43], pp. 386–393) report observations and theoretical predictions for the surface velocity (v_s) inside droplets in a shear flow. That shear flow generally is a consequence of the droplet’s falling at terminal velocity (u_f). The ratio v_s/u_f is predicted to be a function of the droplet Reynolds number, $\mathrm{Re}=2r_0{u}_f/\mathsf{\nu }$, where v is the kinematic viscosity of air.

Table 1. Estimated effects of fluid motion on enhancing mixing within spray droplets. The r₀ v_s is the estimate of the motion-induced diffusion within a spray droplet of the given radius. Compare these values with D_g,sw, which ranges from $1.7\times {10}^{-9}$ to $2.3\times {10}^{-9} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$ for our six gases. In Re, the viscosity of air is calculated at 20 °C and is $1.504\times {10}^{-5} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$.

r0	uf	Re	vs/uf a	vs	r0 vs
(μm)	(m s−1)			(m s−1)	(m2 s−1)
1	1.2 × 10−4	1.6 × 10−5	0.01	1.2 × 10−6	1.2 × 10−12
10	0.123	0.0164	0.01	1.2 × 10−4	1.2 × 10−9
50	0.254	1.69	0.012	3.1 × 10−3	1.9 × 10−7
100	0.725	9.65	0.016	0.012	1.2 × 10−6
200	1.67	44.4	0.024	0.030	1.0 × 10−5
500	4.10	272	0.042	0.16	1.0 × 10−5

^a Estimated from Table 6 and Figure 7 in LeClair et al. (1972).

shows our estimates of v_s for a range of droplet sizes. A reasonable method for estimating the effect of fluid motion within a droplet is to calculate a motion-induced diffusivity as r₀ v_s (e.g., [55], pp. 8–11). Table 1 also shows this estimate, which ranges from $1.2\times {10}^{-12} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$ for 1 μm droplets to $8.0\times {10}^{-5} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$ for our largest droplets (500 μm). For comparison, the molecular diffusivities at 20 °C for our gases of interest range from $1.7\times {10}^{-9} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$ for carbon dioxide to $2.3\times {10}^{-9} {\mathrm{m}}^2 {\mathrm{s}}^{-1}$ for helium. That is, for all but the smallest droplets that we are considering, fluid motion within the droplets could increase the effective gas diffusivity by several orders of magnitude.

As just one sensitivity calculation to demonstrate this effect, we repeat in Figure 7 our time scales for carbon dioxide from Figure 5c, but here increase the molecular diffusivity in seawater, D_g,sw, by a factor of 10. We also show τ_f for a 10-m elevation wind speed of 20 m s⁻¹. Increasing the effective diffusivity of carbon dioxide within spray droplets by a factor of 10 decreases τ_aq by a factor of 10. As a result, for U₁₀ = 12 m s⁻¹, the τ_f crossover with the τ_aq curve increases from a radius of about 150 μm to over 300 μm; more large droplets become involved in the gas transfer. For a wind speed of 20 m s⁻¹, the crossover moves out to r₀ of 400 μm. Even more large droplets participate fully in spray-mediated gas transfer with increasing wind speed. Furthermore, if we interpret Table 1 literally, it suggests that, because r₀ v_s increases with increasing droplet radius, internal mixing (τ_aq) is even more efficient than in Figure 7 with its enhanced internal mixing (10 D_g,sw). In other words, the τ_aq curve would fall farther below the τ_f curve at a larger radius than Figure 7 depicts. Our results also indicate that the choice of a mass accommodation coefficient within reasonable limits (0.01 ≤ α ≤ 1.0) does not alter our conclusions about how sea spray mediates gas transfer, at least for our six gases of interest.

Consequently, a conceptual picture is emerging from this discussion in which gas transfer in each of the three potential rate-limiting steps is so fast that the gas in ambient air can almost always be assumed to be in instantaneous equilibrium with the evolving spray droplet. Only for the largest radii and for weak to moderate winds, say less than 15 m s⁻¹, may droplets fall back into the ocean before establishing equilibrium with the atmospheric gas reservoir.

3.2. Initial Gas Concentration in Spray Droplets

Our implicit assumption throughout has been that spray droplets are created with the same gas concentration as the near-surface ocean, C_w. For spume droplets, which form at the crests of breaking waves where turbulence in the seawater produces effective vertical mixing, this assumption is a good one. It should be noted that gas transfer for spume droplets returning to the sea can be in either direction and is primarily controlled by the temperature differential between air and sea. In cases where air temperature is lower than the sea temperature, net cooling of the spume aqueous phase results, thereby reducing the fugacity of a dissolved gas in spume. This leads to gas retention or, where the partial pressure of the gas in air is sufficiently high, gas invasion from the atmosphere to the droplet. That is, cooled spume drops may act as sinks for atmospheric gases or as attenuators that dampen biogenic gases such as DMS derived in the sea.

For the jet and film droplets that derive from bursting bubbles, however, we need to look more closely at this assumption. In terms of gas transfer at the air-sea interface, all six of our gases are controlled by flow through a diffusive sublayer on the water side of the interface [56,57]. The gas concentration changes through this sublayer from C_w, the concentration in bulk seawater at the bottom of the layer, to C_a(∞), the concentration in the ambient air at the top of the layer. Broecker and Peng [58] estimate the thickness of this sublayer as being between about 100 and 300 μm.

The bubbles that produce jet and film droplets most often have diameters less than 300 μm [42], and would thus burst from within this aqueous sublayer where the gas concentration in the water is somewhere between C_w and C_a(∞). Because the water entrained into jet droplets comes primarily from concentric spherical shells beginning with one corresponding to the interior surface of the parent bubble [59], jet droplets should have gas concentrations nearer to C_w than to C_a(∞). For bubbles larger than 300 μm in diameter, it is reasonable to assume that droplets (primarily spume) carry gas concentrations of C_w.

Film droplets, on the other hand, derive from the bubble cap, which is a thin film over the bubble at the air-sea interface. That film cap can be of order of 1 μm thick and is, therefore, not much of a barrier for gas transfer between the air in the bubble and the ambient air. We thus expect that the gas concentration in the film droplets, which derive from this bubble cap, will be near K_HC_a(∞) or equilibrated to the bubble air. Film droplets may thus play little role in mediating air-sea gas transfer of the gases considered here, though they may be important in the gas exchange of biogenic gases concentrated in the surface films.

3.3. Surface-Active Material

Surface-active material—primarily organic—that is known to collect on bubbles as they rise in the water column and to form as monolayers on the sea surface [60,61,62,63] can be entrained onto spray droplets and, thus, affect their chemical and physical properties. Such contamination could affect several aspects of what we are assuming about spray droplets and the time scales that we have been discussing.

For example, a contaminated surface will retard evaporation from a spray droplet, and thus lengthen τ_r. Surface contamination can also cause a spherical liquid droplet to behave more like a rigid sphere ([64], p. 237; [54], p. 125), thereby lowering the drag coefficient (e.g., Pruppacher and Klett [43], pp. 382, 385–386; [54], pp. 129–131). Alternately, the surface may experience more drag due to irregularities.

Surface contamination alters the drag coefficient of droplets by changing—generally slowing—their internal circulation. Consequently, the internal mixing time scale for a droplet would not be a small fraction of τ_aq, as we had earlier explained, but would, with surface contamination, be pushed back toward τ_aq. In other words, the diffusion of a gas within a spray droplet would slow if the surface were contaminated, but the time scale would still be no longer than τ_aq, (2.25).

A last effect of surface contamination on a spray droplet would be to its surface tension. Surface-active material tends to lower the surface tension of water (Pruppacher and Klett [43], p. 130). Lower surface tension means droplets have less tendency to be spherical; they therefore become more susceptible to breakup. The ultimate result is that the near-surface droplet size distribution may shift to smaller droplets [65].

4. Conclusions

Sea spray gas diffusion follows a series of steps, including (1) diffusion between the deep interior of the droplet and its interior surface, (2) diffusion across the air-droplet interface, and (3) diffusion between the air-side boundary layer and the bulk atmosphere. If a gas undergoes a series of reactions while dissolved in sea spray, such as in the case of CO₂, then (4) the timescales of these kinetics are also important. Of the six gases, 5 can be described by the non-reactive timescales 1 to 3. Only carbon dioxide participates in the additional fourth step, and these timescales will be considered in future work.

This comprehensive analysis confirms that, regardless of the model, the diffusional timescales occur at significantly faster timescales than the rates of droplet temperature and radius change. It is therefore possible to proceed with a sea-spray gas-exchange model that assumes that these latter rate-limiting steps control gas transfer. Although, for “clean” sea spray, the interfacial diffusion was not rate limiting, it is important to consider the effects on diffusion rates for sea spray that contains biogenic surfactants to establish thresholds that may impede diffusion.