The flux density j of a trace gas across a free, smooth or wavy unbroken air–sea interface is governed by the difference in concentration of the gas in air and water (ca and cw) and the gas transfer velocity ks across the water surface: 1j=ksΔc=ks(cw-αca). Because of the discontinuity at the air–water boundary, the solubility α (here, α is equal to the dimensionless Henry solubility Hcc; ) has to be taken into account. The gas transfer velocity ks of a sparingly soluble tracer through a free, smooth or wavy, unbroken surface can be described by 2ks=1βu∗,wSc-n, with the water-side friction velocity u∗,w, a measure for momentum input into the water by the wind, the Schmidt number Sc=ν/D of a tracer, given by the ratio of the kinematic viscosity of water ν and the tracer's diffusion coefficient in water D. The dimensionless parameter β and the Schmidt number exponent n depend on the boundary conditions, with n=2/3 for a smooth water surface and n=1/2 for a rough and wavy surface.

From Eq. (), it is apparent that the transfer velocities of two sparingly soluble gases (A and B) can be converted by Schmidt number scaling: 3ks,Aks,B=ScAScB-n. Commonly, a reference Schmidt number of Sc=600 is chosen, which corresponds to carbon dioxide at 20 ∘C in fresh water.

For gases that have a medium to high solubility, the transfer resistance in the air side has to be taken into account. As first shown by , the total transfer velocity kt can then be expressed by 41kt=α1ka+1ks, with ka being the air-side transfer velocity. For gases with a low solubility, the second term dominates and the first term in Eq. () can be neglected, such that for those gases kt=ks. Inverse transfer velocities can be seen as transfer resistances, such that Eq. () can be written as 5Rt=αRa+Rs. All transfer velocities used in this paper are related to a water-side observer; i.e., they describe how fast a gas is transferred into or out of the water. Air-side observed gas transfer velocities differ by a factor of α.

Bubbles contribute to the gas transfer in two ways. First, they provide an additional surface through which gases can pass. Second, during their generation, by rising through the water-side mass boundary layer of the water surface and by bursting at the water surface, they increase the near-surface turbulence. already considered whitecaps as “low impedance vents”, which “shortcut” the water-side transfer resistance. These bubble effects which intensify near-surface turbulence increase the transfer velocity across the free surface and do not depend on tracer solubility.

The transfer through a closed bubble surface is different from transfer across the free water surface. First, bubbles have a limited lifetime, as they either burst at the water surface or, if they are small enough, completely dissolve. Second, bubbles have a limited volume to take up or release gas. Once a bubble is filled to the equilibrium concentration cbeq given by Henry's law, cbeq=α-1cw, it is inactive for the remainder of its lifetime. For gases with higher solubilities and for small bubbles, this equilibrium is reached faster. And third, bubbles experience an overpressure due to hydrostatic pressure and surface tension. Therefore, small bubbles can completely dissolve and the equilibrium concentration shifts to slightly higher concentrations. Because the measurements reported here are taken far from equilibrium, dissolving bubbles are not important. The transfer velocities themselves are not affected. A detailed analysis of the timescales involved and how they depend on the bubble radius is given in .

Because bubbles form an additional exchange surface, the total water-side transfer velocity kw can be split up into two parts : 6kw=ks+kc, with transfer through the free water surface ks and through the closed surface of submerged bubbles kc. It is important to note here that the bubble-induced gas transfer velocity kc does not include the bubble formation process and the bursting of bubbles when they rise through the surface again. Concerning the gas transfer velocity, these effects cannot be distinguished from other processes generating turbulence close to the water surface, because they do not depend on tracer solubility but only on the Schmidt number. Therefore, bubble-induced gas exchange refers here only to the stages in the lifetime of a bubble with a closed surface and therefore a limited trapped air volume.

Figure  shows a schematic view of the resistances for bubble-mediated gas transfer Rc=kc-1 in relation to the air- and water-sided resistances for transfer through the unbroken water surface.

Many approaches have been made to quantify the bubble-mediated gas transfer kc: gas transfer by single bubbles , transfer in bubble clouds and breaking waves , as well as theoretical models based on bubble dynamics . Bubble-mediated gas transfer also depends on bubble surface conditions. It has been shown that surface active material reduces the gas transfer of single bubbles , while it also decreases the bubbles' rise velocity . During the lifetime of a bubble, these surface conditions can change, as bubbles accumulate surface active material while moving through the water.

Empirical or semi-empirical parameterizations are the state of the art of calculating the bubble-mediated gas transfer kb on the open ocean. Most of these parameterizations link kb to the tracer's solubility and Schmidt number (or diffusion coefficient) as well as the whitecap coverage of the water surface, which in turn depends on the sea state that is usually expressed as a function of the wind speed at 10 m height u10 or the friction velocity u∗,a.

Physically based models distinguish two limiting cases: one for very weakly soluble gases and one for more highly soluble ones. For very weakly soluble tracers, the bubbles act as a very big reservoir. In that case, the bubbles simply provide an additional surface for gas transfer that actively participates in gas transfer for the whole lifetime of a bubble. In this limit, the gas transfer is proportional to the integrated bubble surface area Ab,δr per radius interval δr normalized to the water surface area As and the Schmidt number with the bubble Schmidt number exponent nb and does not depend on solubility: 7kc,lowα=∫Ab,δr(r)kb,600(r)drAs600Scnb=kc,600600Scnb. The transfer velocity kc is the effective bubble-induced transfer velocity related to the free water surface, while kb(r) is the real transfer velocity across the bubble surface of a given radius.

In the limit of high solubility, the bubbles constitute a very small reservoir for the trace gas, so that the higher-solubility tracers will reach concentration equilibrium cbeq=α-1cw very fast. Then, the bubble-mediated gas transfer can only depend on the tracer's solubility and the total bubble volume flux Qb,δr per radius interval δr: 8kc,highα=1α∫Qb,δr(r)drAs=1αQbAs=1αkr. The velocity kr has an intuitive meaning. It is the effective velocity (volume flux per water surface area) averaged over all bubble sizes, with which the air volume is being submerged by breaking waves and rises towards the surface again. call this quantity the air entrainment velocity Va.

The transition solubility, at which the constant bubble-mediated transfer velocity for low solubility kc,lowα changes into the transfer velocity decreasing with increasing solubility can be computed by setting both values equal: 9αt=krkc,600Sc600nb. Based on detailed measurements in a bubble-tank with multiple volatile tracers, showed that the transition between the two regimes can be well described by a simple exponential term (Fig. ). The transfer velocity for bubble-mediated gas exchange results in 10kc=krα1-exp⁡-ααt=kc,600600Scnbα≪αtkr/αα≫αt. For fresh water, Table 9.7 found, under the conditions shown in Fig. , a transition solubility αt=0.23 for fresh water and 0.06 for seawater.

In summary, the total gas transfer velocity ktot for water-side controlled tracers including bubble-mediated gas transfer can be parameterized as 11ktot=ks+kc=ks,600600Sc0.5+krα1-exp⁡-αkc,600kr600Sc0.5, with the three parameters ks,600, kc,600 and kr. Because the measurements were performed with clean water, the Schmidt number exponents for both the transfer across the free water surface and the bubbles' surfaces are set to 1/2. In the limits of low and high solubility, the model equation (Eq. ) simplifies to 12ktot=ks,600+kc,600600Sc0.5α≪αtks,600600Sc0.5+kr/αα≫αt. In the limit of low solubilities, the gas transfer velocity no longer depends on solubility, and simple Schmidt number scaling can be applied. The ratio of gas transfer across bubble surfaces and the free surface is then simply given by the ratio of kc,600 to ks,600.

Whereas, in Eq. (), a parametric approach is given for the whole solubility range, Eq. 12 provide only the high-solubility limit in a semi-empirical approach using field dimethyl sulfide (DMS) and CO2 gas exchange measurements: 13ktot=ANBu∗,a600Sc0.5+Ãbαu∗,a2cp, where in the second term u∗,a5/3(gHs)2/3 is replaced by the simpler term u∗,a2cp using the relation for fetch-limited waves. The term Ãbu∗,a2cp is the air entrainment velocity Va=kr, which, according to , is increasing with the phase speed cp of the peak wave or its significant weight height Hs, respectively. Surprisingly, found the air entrainment velocity decreasing with the wave age cp/u∗,a by estimating the air entrainment velocity directly based on a model for a single breaking wave and the statistics of breaking waves as measured in the field.

The processes mirroring bubbles in the water, which are spray droplets in the air, are less well studied, with the exception of the transfer of water vapor and heat . Only recently, evaluated the timescales governing spray-mediated gas transfer for gases other than water vapor in a similar fashion as did for bubble-mediated transfer more than three decades earlier.

In contrast to bubbles, tracer solubility plays no role for spray droplets as long as the transfer process is controlled by water-sided processes. This is the case for all tracers used. Therefore, spray droplets just constitute an additional exchange surface. The question is only whether the lifetime for gas exchange is longer than the lifetime of the droplets. If this is the case, gas exchange occurs during their whole lifetime. In this upper limit, the gas transfer velocity kd induced by spray droplet is given in analogy to Eq. () by 14kd,upper=∫Ad,δr(r)kd,600(r)drAs600Scnd=kd,600600Scnd. The transfer velocity kd is the effective droplet-induced transfer velocity related to the free water surface, while kd(r) is the real transfer velocity across the droplet surface of a given radius.

If the concentration inside the spray droplet equilibrates faster with the surrounding air than the spray droplet takes to fall back into the water, the spray-induced gas transfer velocity kd depends on the total volume flux Qd of spray generated . This lower limit is given by 15kd,lower=Qd/As. At the highest wind speeds, water is lost from wind-wave tanks because the wind tears offs the wave crests and part of the resulting spray droplets leave the facility with the air flow (Sect. ). Therefore, the volume lost V˙w (see Sect. ) is actually a lower limit for Qd. Because solubility plays no role for the tracers used in the experiment here, spray-induced gas transfer cannot be distinguished from gas transfer across the free water surface.

Another effect may happen however. In the limit of a long droplet lifetime compared to the lifetime for gas exchange (Eq. ), the spray-droplet-induced gas exchange does not depend on the Schmidt number. According to the estimates of , this is the case, except for largest droplet radii and for weak winds less than 15 m s-1. Then, gases with a high diffusivity will no longer show correspondingly higher transfer velocities if gas transfer through the spray droplet surface is significant.

At very high wind speeds, breaking waves disrupt the water surface. It has been shown that the drag coefficient, 16Cd=u∗,a2u10-2, gets saturated or even decreases at wind speeds higher than around 30–35 m s-1 . A two-phase layer forms, consisting of bubble-filled water transitioning to spray-filled air. The turbulence characteristics of this two-phase layer are thought to be controlling the transfer of momentum, which leads to the saturation of the drag coefficient . However, this does not mean that the friction velocity and thus the momentum input from the wind into the water also decrease; they just increase less steeply.

Measurements were performed in two wind-wave tanks, the High-Speed Wind-Wave Tank of Kyoto University, Kyoto, Japan, in October of 2015 and the SUrge STructure Atmosphere INteraction Facility (SUSTAIN), University of Miami, Miami, USA, in May and June of 2017. Table  gives an overview of the technical data of the facilities.

A mass balance method is used to measure the gas transfer velocities in evasion-type experiments. In this approach, all gases are dissolved in the water and the water is mixed well by pumps before the wind is turned on. When the wind is turned on, the main flux is from the water volume to the air, and thus the concentration of the tracer in the water cw decreases exponentially: 17cw(t)=cw(0)exp⁡-k⋅AVwt, with the water volume Vw, the water surface A and the concentration at the start of the experiment cw(0). In the Kyoto high-speed tank, the water lost due to spray was replaced with fresh water, which changes Eq. () to 18cw(t)=cw(0)exp⁡[-k⋅AVw+V˙wVw⋅t], with the water loss rate V˙w/Vw. Thus, knowing the water volume, water surface area and the loss rate and measuring a concentration time series allows us to determine the gas transfer velocity k. A more thorough derivation of Eq. () can be found in .

In both experimental campaigns, a dual membrane inlet mass spectrometer (HPR-40 MIMS, Hiden Analytical, Warrington, UK) was used to measure the tracers' concentrations in water. The water extracted from the wind-wave tank was pumped along one of the inlet membranes where dissolved species diffuse through the membrane directly into the vacuum of the spectrometer where they are ionized and subsequently analyzed with respect to their concentrations. For some tracers, two mass-to-charge ratios were monitored with the MIMS, either because there are sufficiently high concentrations of different isotopes (e.g., Xe) or the tracer molecule is destroyed in the ionization process and forms multiple ions with different masses (e.g., DMS, DFB, HFB).

As mentioned in the previous section, before the start of the evasion experiment, all available gases were dissolved in the water and mixed well. For dissolving gases into the water, membrane contactors (SUSTAIN: Liqui-Cel 8×20 PVC, Kyoto: Liqui-Cel 4×13, Membrana 3M, Charlotte, NC, USA) were used. In Miami, the gases were dissolved directly into the water of the wind-wave tank, while in Kyoto the gases were first dissolved into a holding tank of approximately 7 m3. This water was then mixed into the main water volume of the wind-wave tank using pumps before the start of an experiment. Care was taken that the tracers were mixed into the water as homogeneously as possible. To achieve this, the pumps were kept on even after gas loading was finished, and the concentration was monitored. Only when the concentration was sufficiently stable the pumps were turned off and the experiment was started.

The tracers were chosen in this study to cover a wide range of solubilities and Schmidt numbers. Table  gives an overview of the tracers used sorted by their solubility. Due to technical and logistical reasons, not all tracers could be used in both facilities. Figure  shows the tracers in a Schmidt number–solubility diagram for all conditions encountered. The temperature dependency of the solubility and Schmidt number is apparent. Also shown is for which solubility–Schmidt number combination the air-side resistance equals the water-side resistance (αRa=Rs; see also Eq. ). To calculate the resistances, a rough and wavy water surface was assumed (i.e., n=1/2). Below this, αRa=Rs line, the water-side resistance dominates; therefore, the tracers are called water-side controlled tracers. All tracers used in this study can be classified as water-side controlled tracers, with the exception of methyl acetate (MA), which is partially air-side controlled due to its relatively high solubility.

Bubble size distributions were measured in Kyoto using an optical bright field imaging technique . A Nikon D800 digital SLR camera with a 200 mm f/4 AF-D macro lens looks perpendicular to the wind direction through the water into a purpose-built LED light source. Bubbles entrained in the water scatter the light such that the light no longer reaches the camera sensor, and the bubble appears as a dark circle. Out-of-focus bubbles have a blurry edge, which is used to estimate the 3-D volume the bubbles are in the 2-D images taken by the camera by depth from focus . Two bubble imaging systems were operated during the measurements in Kyoto: one at a fetch of 3 m and the second one at 8 m fetch, approximately 30 cm below the water surface. Calibration and data evaluation is described in detail in .

The relationship between the water-sided friction velocities u∗,w and the wind speeds in 10 m height u10 at which the gas transfer velocities were measured in this study is shown in Fig. a. A clear change in the steepness of the relationship can be seen at a wind speed of approximately 33 m s-1, as indicated by the gray line. The wind speed of 33 m s-1 corresponds to a friction velocity of about 6 cm s-1. Also, the drag coefficient (CD; Fig. b) has a maximum at this wind speed before it levels off.

Up to now, bubble measurements were evaluated only for the Kyoto facility. From the bubble concentration measurements, the total bubble surface was computed and plotted in Fig.  as a dimensionless area in relation to the flat free water surface area, Ab/As. The uncertainties are quite high, because bubbles where measured only at one depth (Sect. ). But still a few important findings can be stated.

The bubble surface area strongly increases with the friction velocity (∝u∗,w3 to ∝u∗,w4) in all facilities and for fresh water and seawater.

Figure  shows the measured gas transfer velocities kmeas in dependency of the water-sided friction velocity u∗,w for three different measurement conditions: (a) fresh water in Kyoto, (b) seawater model in Kyoto and (c) seawater in Miami. Gas transfer increases strongly with the wind speed for all tracers. Beyond a friction velocity of approximately 6 cm s-1, the increase is significantly steeper.

The tracer with the highest solubility, methyl acetate (MA), shows a transfer velocity significantly lower than for the other tracers for all wind speeds and all water types. This reduction in the gas transfer velocity confirms the existence of an additional air-side resistance for this tracer; see Sect. . Due to this additional air-side resistance, MA will be excluded in the following discussion. From Fig. , it is also evident that He has a much higher transfer velocity than the other tracers for all wind speeds. This is caused by its significantly higher diffusion coefficient corresponding to a low Schmidt number, while all other tracers vary in the Schmidt number by at most a factor of 2; see Table .

Figure  also shows dependencies of the form kmeas∝u∗,wx with a variable exponent x. Clearly, the functional dependency between kmeas and u∗,w dramatically changes at a friction velocity of around 6 cm s-1, indicating a new regime starting at this friction velocity. This finding is in good agreement with the earlier measurements of and , who also found a transition to a much steeper increase at 33–35 m s-1 (u10). Also, this wind speed coincides with the change in the u∗,w(u10) relationship discussed in Sect. .

A closer look at the fresh water transfer velocities (Fig. a) reveals an unexpected result. Even for high wind speeds, all tracers (except He and MA) have transfer velocities within a very narrow band, even though their solubility differs by several orders of magnitude. This is a clear indication that transfer through closed bubble surfaces is much slower than the transfer through the water surface for fresh water even at the highest wind speeds.

In seawater and in simulated seawater (Fig. b and c), a clear spacing between the transfer velocities of tracers with different solubilities at high wind speeds can be seen. This means that bubble-induced gas transfer does play a role for seawater.

Once bubbles influence air–sea gas transfer, a separation of the different contributing mechanisms is required. Because of the additional influence of solubility, it is not possible to simply apply Schmidt number scaling. This is why the model combining gas transfer across the free water surface and bubble surface was developed in Sect. ; see Eq. (). Because all measurements were made at high wind speeds with clean water, the Schmidt number exponent was fixed to 0.5. Then, three unknown parameters remain for each measuring condition:

The first parameter is the transfer velocity across the free water surface at a Schmidt number of 600, ks,600.

Because of the multi-tracer approach with more than three tracers covering a wide range of solubilities, it is possible to retrieve all three parameters of the model (Eq. ) for each measuring condition separately and thus to separate the gas transfer across the free water surface from the bubble-induced gas transfer. In addition, the transition solubility αt can be computed according to Eq. (). The model Eq. () was fitted to the data using a least squares algorithm with the free parameters ks,600, kr and kc,600. More information on the algorithm to retrieve those parameters can be found in the Supplement. MA was excluded from the fit due to its additional air-side resistance. Also, He had to be excluded. Including He led to unrealistically low transition solubility of below 0.001. One possible reason for this is the high diffusion coefficient of He and the resulting fast gas transfer from water into the bubbles, which might deplete the He concentration in the water between the bubbles inside bubble clouds. This effect has been observed and described before by . Another explanation is spray-induced gas transfer, which could limit the gas transfer velocity of tracers with high diffusivity as discussed in Sect. .

Also, as a further quality criterion, the fit was required to obey 19kc,600≤kmeas,TSct6000.5-ks,600, for the tracers T = [SF6,CF4]. This was achieved by iteratively narrowing the allowed parameter space of the parameter kc,600. Equation () describes the highest physically reasonable kc,600 (see also Eq. ). At each measuring condition, the regression with three free parameters was performed with 5–14 measured transfer velocities. For some tracers, the concentrations of two different ions of the same tracer were analyzed with the MIMS, which allowed the measurement of two transfer velocities per tracer for a single wind speed condition; see Sect. .

The mean (median) deviation between the measured and the modeled transfer velocity is 7.4 % (6.3 %) of the measured transfer velocity. Out of a total of 242 pairs of measured and modeled values, only 22 deviate by more than 15 %. The maximum deviation found was 31.2 % (acetylene in seawater at u10=62.9 m s-1). This indicates that the regression model is in good agreement with the measured data. Detailed plots comparing measured and modeled values can be found in the Supplement.

Figure  shows the resulting fitted parameters (ks,600, kr and kc,600) and the calculated transition solubility (αt; see Eq. ). The bubble-related parameters (kr and kc,600) are found to be zero for friction velocities below 5.8 cm s-1 for fresh water and seawater. No experiments below 7 cm s-1 were performed for the seawater model. The separation of the gas transfer velocity into its different components gives a detailed insight into the mechanisms of air–sea gas transfer at high wind speeds with unexpected results:

The gas transfer velocity across the free water surface ks,600 normalized to a Schmidt number of 600 (Fig. a) clearly shows a transition to a much steeper increase of the transfer velocity with the friction velocity from ∝u∗,w1 to ∝u∗,w3.0 beyond a friction velocity of about 5.8 cm s-1. It is a substantial effect, resulting in an about 10-fold gas transfer velocity if the water-side friction velocity is increased by a factor of 2 from 6 to 12 cm s-1. This substantial increase of the gas transfer velocity is not related to bubble-induced gas transfer at all and thus valid for all water-side controlled gas tracers independent of the solubility. It is not unexpected that there is no significant difference between seawater and fresh water, because the hydrodynamic conditions do not depend on the salt content of the water, and the normalization of the transfer velocity to a Schmidt number of 600 already takes the small change of the kinematic viscosity between seawater and fresh water into account. It is more surprising that there is no significant difference between the Kyoto and SUSTAIN facilities although they differ significantly in lengths (15.7 m versus 24 m) and width (0.8 m versus 6 m).