Effect of Langmuir cells on bubble dissolution and air sea gas exchange

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi:10.1029/2010jc006203, 2010 Effect of Langmuir cells on bubble dissolution and air sea gas exchange David Chiba 1 and Burkard Baschek 1 Received 15 February 2010; revised 15 June 2010; accepted 12 July 2010; published 22 October 2010. [1] Gases are exchanged between the atmosphere and ocean by diffusion through the sea surface and by dissolution of air bubbles injected by breaking wind waves. Langmuir cells enhance the contribution from bubbles by keeping them under water for longer thus increasing their dissolution. We determine the importance of Langmuir cells by using a bubble model to calculate the amount of gas that dissolves from bubbles as a function of wind speed, gas saturation, and injection depth and a Langmuir cell model to estimate the effect of the associated downwelling currents on bubble dissolution. The calculations are preformed for the eight gases N 2,O 2,CO 2, He, Ne, Ar, Kr, and Xe, and the results are then compared with the total gas exchange determined by common descriptions of air sea gas exchange. The contribution of gas bubbles to air sea gas exchange increases with wind speed and can reach 77% for O 2, 98% for N 2, and 16% for CO 2 at a wind speed of 20 m s 1 and a gas saturation of 95%. The additional effect of Langmuir cells on the total gas exchange at a gas saturation of 95% is 19% for O 2, 35% for N 2, and 0.7% for CO 2. The importance of Langmuir cells for air sea gas exchange generally increases with gas saturation and decreases with the solubility of the gas. Citation: Chiba, D., and B. Baschek (2010), Effect of Langmuir cells on bubble dissolution and air sea gas exchange, J. Geophys. Res., 115,, doi:10.1029/2010jc006203. 1. Introduction [2] At high wind speeds, air bubbles can significantly enhance air sea gas exchange especially for poorly soluble gases [Woolf and Thorpe, 1991; Woolf, 1997]. After the bubbles have been injected into the ocean by breaking waves, they rise back to the sea surface due to their buoyancy while losing some of their gas to the surrounding water. The amount of gas that dissolves from the bubbles depends on their size and injection depth, as well as the solubility and diffusivity of the gas. The dissolution of gas from the bubbles, and hence their contribution to air sea gas exchange, can be further increased by downwelling currents such as in tidal fronts [e.g., Baschek, 2003; Baschek et al., 2006; Baschek and Jenkins, 2009] or in Langmuir cells [Thorpe, 1984b; Woolf and Thorpe, 1991; Thorpe et al., 2003]. Langmuir cells [Langmuir, 1938] often form in the surface layer of the ocean when a wind is present (Figure 1). The threshold wind speed for the initiation of cells varies from 2 ms 1 [Weller and Price, 1988] to 8 ms 1 [Smith, 1992] depending on the duration of the wind. The associated downwelling currents can reach 0.08 0.2 ms 1 [Scott et al., 1969; Leibovich, 1983; Thorpe et al., 1994] and are likely to have a significant effect on air sea gas exchange as they are on the same order of magnitude as the rise 1 Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, California, USA. Copyright 2010 by the American Geophysical Union. 0148 0227/10/2010JC006203 speed of the bubbles [Fan and Tsuchiya, 1990], thus holding the bubbles under water for longer and increasing gas dissolution. [3] However, the contribution of bubbles to air sea gas exchange remains uncertain. Air sea gas exchange is often implemented in global and regional models to predict the oceanic uptake of gases such as CO 2 or O 2. The commonly used descriptions of gas exchange [e.g., Liss and Merlivat, 1986; Wanninkhof, 1992; Nightingale et al., 2000; Monahan, 2002] assume that gas diffusion is directed from the atmosphere to the ocean at oceanic gas saturations of less than 100% and vice versa at saturations larger than 100%. This assumption, however, neglects the fact that bubbles tend to drive the ocean to supersaturated equilibrium conditions due to the increased gas pressure inside the bubbles caused by hydrostatic pressure and surface tension forces. [4] The importance of air bubbles for air sea gas exchange has been investigated in several studies. Woolf and Thorpe [1991] modeled the dissolution of bubbles and calculated the resulting gas flux and supersaturations. Woolf [1993, 1997] determined the transfer velocity for CO 2 due to bubbles and estimated the significance of bubble transfer for CO 2, showing that bubbles are more important for less soluble gases than for highly soluble gases. The affect of Langmuir cells was explored by Thorpe [1984b], Woolf and Thorpe [1991], and Thorpe et al. [2003] who showed that the associated downwelling velocities enhance bubble dissolution by increasing the time bubbles are submerged. [5] While previous estimates of bubble gas exchange assume to be proportional to whitecap coverage [Woolf and 1of13

Figure 1. Sketch of air sea gas exchange processes. A wind blowing over the sea surface creates waves of amplitude a and enhances the diffusive gas exchange through the air sea interface. At higher wind speeds, air bubbles are injected by breaking waves forming an active white cap coverage W. Bubble density and void fraction v decay exponentially with depth (decay scale g). Hydrostatic pressure and surface tension increase the gas pressure inside the bubble and force the gas into dissolution. Streaks of foam and debris form when Langmuir cells are present. The vertical velocities w in the associated vortices of width L x and depth L z influence the bubble mediated air sea gas exchange. Thorpe, 1991; Woolf, 1993; Andreas and Monahan, 2000; Zhang and Yuan, 2004], we provide a novel approach in estimating the amount of bubbles dissolved that depends on the active white cap coverage (the white caps that occur immediately after wave breaking), the duration of wave breaking, and the injected void fraction as function of depth and wave height. We introduce a modified Langmuir cell model to better simulate observations found in the field and have implemented recent observational data and parameterizations that were not available during previous studies, thus presenting a first estimate of the effect of Langmuir cells on air sea gas exchange. [6] In order to estimate the total amount of gas that dissolves in the ocean at a given wind speed due to bubble entrainment by breaking waves, we proceed as follows: first, we calculate from observational data published in literature the total amount of air entrained by bubbles (section 2). Then, we use a bubble model [Thorpe, 1982, 1984a; Baschek et al., 2006] to determine the percentage of gas for N 2,O 2,CO 2, He, Ne, Ar, Kr, and Xe that dissolves while the bubbles rise back to the sea surface (section 3). And finally, we assess the effect of Langmuir cells on bubble dissolution with a simple 2 D Langmuir cell model (section 4). The results are then compared with commonly used descriptions of the total air sea gas exchange by Liss and Merlivat [1986] and Wanninkhof [1992] and we discuss the limitations of the models and the associated errors (section 5). 2. Air Entrainment by Gas Bubbles [7] In steady conditions, the rate V e at which gas is entrained into the ocean in the form of air bubbles is given per unit area by V e ¼ W Z vz ðþdz; ð1þ where W is the active white cap coverage (stage A white caps), t the duration of wave breaking, and v(z) the void fraction as function of depth z. V e is estimated by using values for W, t, and v(z) from the literature, which is difficult as most of the available data sets are sparse and have been partially measured in laboratories, lakes, or nearshore environments which may not be representative of open ocean conditions. No comprehensive data set exists that comprises all relevant variables at the same time and it is therefore unclear how well the different data sets can be merged. The presented estimates provide nevertheless an approximation of the importance of air bubbles and Langmuir cells to air sea gas exchange. We have generally used 2of13

Figure 2. Measurements of white cap coverage for white caps of type A (active white caps) by Bondur and Sharkov [1982] (circles), Mironov and Dulov [2007] (crosses), and Monahan and Woolf [1989] (solid line). parameterizations that will lead to a best, yet conservative, estimate of bubble mediated gas exchange as discussed in the following. 2.1. White Cap Coverage [8] In equation (1) we use the active white cap coverage W, also termed stage A white caps or dynamic foam, which occurs immediately after a wave breaks. It contains the full bubble spectrum below it and is therefore directly applicable to our calculations. Within seconds of the wave ceasing to break the majority of bubbles rise back to the surface and the active white cap is transformed into a stage B white cap of decaying foam and smaller bubbles that are still rising to the surface [Monahan, 1988, Monahan and Woolf, 1989; Monahan, 2001]. The active white cap coverage is about an order of magnitude less than the total white cap coverage since the decaying white caps spread considerably due to the turbulence of the crashing wave environment [Monahan, 2001]. In Figure 2, data for active white cap coverage from Bondur and Sharkov [1982], Monahan and Woolf [1989], and Mironov and Dulov [2007] are shown. The most comprehensive data set was obtained by Monahan and Woolf [1989] giving the lowest values of the three data sets and hence providing a conservative estimate of air entrained by breaking waves. We use the relationship by Monahan [2001] derived from this data W ¼ 3:16 10 7 u 3:2 10 for u 10 > 4:5 ms 1 : ð2þ The importance of different sea state conditions in the context of white cap coverage [Woolf, 2005], is addressed below by describing the bubble entrainment depth as a function of wave height. For a more realistic wavefield, white capping may also be described as a function of the average length of breaking crests per unit area and wave speed increment, as described by Melville and Matusov [2002] and Thomson et al. [2009]. 2.2. Duration of Wave Breaking [9] The amount of time that an active white cap persists is equivalent to the duration of a void fraction event or the duration of active wave breaking t, which has been measured by several authors using acoustical measurements [Snyder et al., 1983; Lamarre and Melville, 1992; Ding and Farmer, 1994]. The values provided by Ding and Farmer [1994] range between 1.23 s and 1.73 s, which is larger than those provided by Lamarre and Melville [1992] (0.1 1.0 s) and Snyder et al. [1983] (0.38 0.94 s). Since a large value for t yields the most conservative estimate for our results, we use relationship by Ding and Farmer [1994] based on comprehensive measurements taken in the open ocean, 600 nm WNW off San Diego ¼ u 0:27 10 0:31: ð3þ 2.3. Void Fraction [10] Numerous measurements show that the void fraction decays exponentially with depth z as v(z) =v 0 e z/g, where g is the bubble entrainment depth [e.g., Medwin and Breitz, 1989; Deane and Stokes, 2002; Johnson and Cooke, 1979; Lamarre and Melville, 1994] and v 0 the surface void fraction. The measured values for g range from 0.25 to 2 times the wave height H. We are using the smallest value of 0.25 [Baschek, 2003] since it yields a conservative estimate and is more suitable for large waves, whereas the larger values have been derived for very small waves in laboratory setups [e.g., Hwang et al., 1990; Chanson and Jaw Fang, 1997]. This has been confirmed by Hoque and Aoki [2005] who found values of g = H/4 for plunging breakers and g = H/3.75 for spilling breakers. 3of13

[11] The wave height H is given as a function of wind speed by Sverdrup and Munk [1947] H ¼ k u2 10 g ; who have collected wave height data from many different sites around the world. k is a constant that varies depending on where the data was taken. While many values for k are given we chose a middle value of k = 0.15 that corresponds to the open ocean data taken in the North Atlantic. More recent results from a study in the German Bight [Emeis and Turk, 2009] closely match this equation. [12] The surface void fraction v 0 under breaking waves was measured by various authors in the laboratory [Hoque and Aoki, 2005; Blenkinsopp and Chaplin, 2007; Lamarre and Melville, 1994], in the surf zone [Deane, 1997; Stanton and Thornton, 2000], or in the open ocean [Lamarre and Melville, 1992; Terrill et al., 2001; Wu, 1988]. Since wave heights, associated entrainment depths, and void fraction in laboratories studies differ greatly from open ocean measurements and since it is unclear if the void fraction generated by plunging breakers in the surf zone are representative for open ocean conditions, we will use the results by Lamarre and Melville [1992] who have carried out the most comprehensive study in the open ocean. [13] During their experiment 100 km off the coast of Delaware, wind speed and significant wave height H ranged from 4.5 m s 1 to 14.9 m s 1 and 1.7 m to 2.8 m, respectively. The void fraction was measured at a depth of 0.2 m with maximal values of 24%. To derive an average value we have assumed that the average void fraction of each event is 1/3 of the maximum void fraction given in the paper. We have fitted it to an exponential function proportional to exp(z/g) and with g = 1/4 H in order to calculate the surface void fraction v 0, yielding an overall average of 1.3%. While this is significantly lower than the values of most studies, it provides a conservative estimate of the contribution of gas bubbles to air sea gas exchange. 3. Bubble Model [14] In order to calculate the rate V d at which gas from the bubbles dissolves in the water, the rate of total gas entrained by bubbles V e is multiplied with the percentage of gas 8 that dissolves in the ocean before the bubbles rise back to the sea surface V nolc d ð4þ ¼ 8 nolc V e and V LC d ¼ 8 LC V e : ð5þ This value is altered by vertical currents such as in Langmuir cells (LC). In order to assess the effect of Langmuir cells on the dissolution rate, and hence gas exchange, we run the bubble model to calculate the dissolution rate for a given bubble size distribution and compare it to a run where the bubble model has been coupled to a Langmuir cell model (section 4). [15] The diffusive gas exchange between bubble and water is driven by the difference between the partial pressure of a gas j in a bubble P j and the gas pressure in the water far from the bubble P j w. The gas exchange rate is calculated for the eight gases N 2,O 2,CO 2, He, Ne, Ar, Kr, and Xe with [Thorpe, 1982, 1984a] dn j dt ¼ 4rD j S j Nu j P j Pj w ; ð6þ where n j is the number of moles. The partial gas pressure in the bubble is given by P j = x j (p atm rgz +2g/r), with the mole fraction in dry air x j, the atmospheric pressure p atm, density r, gravitational acceleration g, surface tension coefficient g, and bubble radius r. The gas pressure in water is P w j = (1 + 0.01s j ) x j p atm, where s j is the percent supersaturation of gas in water. S j is the Bunsen solubility, D j the diffusivity, and Nu j the Nusselt number, which are calculated with formulas from Woolf and Thorpe [1991] for a temperature of 15 C. The time derivate of the bubble radius is given by dr dt ¼ " # 3RT X dn j dp 4r 2 r 3P 2 1 ; ð7þ dt dt r j where R is the universal gas constant, T the water temperature, and P the total gas pressure inside the bubble. The change of hydrostatic pressure dp/dt in the above equation is determined by the sum of the bubble rise speed w b, the vertical current speed in the Langmuir cell w LC, and the turbulent current speed w turb dp=dt ¼ gðw b þ w LC þ w turb Þ: ð8þ The bubble rise speed is given for dirty bubbles by a formula from Fan and Tsuchiya [1990] as function of bubble radius. We consider the bubbles to be dirty since bubbles in the ocean often have a layer of surfactants affecting their rise speed [Thorpe, 1982]. While it has been suggested by Leifer et al. [2000] that bubbles with r > 300 mm have a rise speed equivalent to clean bubbles this would only have a negligible effect on our results (Table 1). [16] The coupled ordinary differential equations (6), (7), and (8), which describe the gas exchange from a single bubble, were solved with an explicit Runge Kutta formula in MATLAB. This one step solver only needs the solution at the immediately preceding time point, so that the vertical current speed can be easily described as a function of time. A time step of 0.1 s is used for a total time interval of 5 min. [17] We have implemented turbulence in equation (8) as a random walk function of normal distribution with zero mean and a standard deviation of 1, multiplied with a turbulent vertical velocity of 0.1 m s 1 and displacement of half the wave height [see also Thorpe, 1984a]. However, since the effect of turbulence only leads to a reduction of <3% for N 2 and O 2 and an increase of <7% for CO 2, we are not using it in the main model runs. [18] The bubble model is used to calculate the amount of gas that dissolves from a single bubble, which varies with bubble radius and injection depth and hence wave height and wind speed. For each model run, we are injecting bubbles of 80 different radii ranging from 3 to 5000 mm at 30 different depths between 0.01 m and 15 m, with a total of 2400 individual bubbles. Both, the radii and injection depths are spaced logarithmically with the highest resolution for 4of13

Table 1. Results From the Sensitivity Study a Changed Parameter 5 m/s 10 m/s 15 m/s 2 white cap coverage 2 2 2 1/2 white cap coverage 0.5 0.5 0.5 2 duration of wave breaking 0.5 0.5 0.5 1/2 duration of wave breaking 2 2 2 2 surface void fraction 2 2 2 1/2 surface void fraction 0.5 0.5 0.5 5 m/s 10 m/s 15 m/s Changed Parameter O 2 N 2 CO 2 O 2 N 2 CO 2 O 2 N 2 CO 2 2 downwelling speed 1.54 1.67 1.08 1.64 1.82 1.12 2.03 2.25 1.22 1/2 downwelling speed 0.76 0.71 0.96 0.78 0.71 0.96 0.82 0.76 0.96 2 wave height/entrainment depth 1.02 1.02 1.02 2.17 2.03 2.03 2.76 2.61 2.34 1/2 wave height/entrainment depth 1 1 1 0.74 0.76 0.75 0.41 0.44 0.46 2 cell depth 0.79 0.72 0.97 0.79 0.72 0.97 0.84 0.78 0.97 1/2 cell depth 1.10 1.12 1.02 1.31 1.38 1.05 1.16 1.21 1.03 2 cell width 1 1 1 1 1 1 1 1 1 1/2 cell width 1 1 1 1 1 1 1 1 1 2 ratio down /upwelling 0.82 0.77 0.97 0.82 0.77 0.97 0.86 0.82 0.97 1/2 ratio down /upwelling 1.23 1.29 1.03 1.22 1.28 1.04 1.19 1.25 1.04 Temperature 10 C 1.00 0.98 1.02 1.00 0.98 1.02 1.00 0.98 1.02 With turbulence 0.90 0.88 1.02 0.95 0.94 1.03 0.96 0.96 1.06 For r > 300 mm clean bubbles 1.001 1.001 1 1.001 1.001 1 1.000 1.000 1 a One variable at a time has been changed relative to the main coupled model run (changed parameter column). The 5, 10, and 15 m/s columns indicate the factor by which this description would change the estimate of the amount of gas dissolved by bubbles and Langmuir cells as calculated with the coupled model. small bubbles and shallow depths where the horizontal and vertical gradient is largest. The amount of gas dissolved from each bubble is then multiplied with a bubble size distribution N(r,z) determined from measurements yielding the total percentage of gas dissolved 8. [19] Numerous measurements show that the shape of the bubble size distribution is given by N(r) r a [e.g., Baldy and Bourguel, 1987; Crawford and Farmer, 1987; Hwang et al., 1990; Deane, 1997; Deane and Stokes, 1999, 2002], while the depth distribution can be described with a decaying exponential function e z/g [e.g., Medwin and Breitz, 1989; Deane and Stokes, 2002; Johnson and Cooke, 1979; Lamarre and Melville, 1994], yielding Nr; ð zþ ¼ Br e z= : ð9þ Several observations [Deane, 1997; Deane and Stokes, 1999, 2002] have shown that the bubble size distribution immediately after bubble injection is divided into two parts with spectral slopes a = 3/2 for r 1000 mm and a = 10/3 for r > 1000 mm. Here a is constant with depth, g = H/4 is the average bubble entrainment depth (see section 2.3), and the scaling factor B is determined by fitting the bubble size distribution to the observed total void fraction. The change of the spectral slope due to dissolution, buoyancy, and the effect of vertical velocities is taken into account by the bubble model. The sensitivity of the model results to water temperature, gas saturation, wave height, and turbulence is discussed below. 4. Langmuir Cell Model [20] The downwelling velocities associated with Langmuir cells reach several cm s 1 and keep the bubbles underwater for longer, hence increasing the amount of gas that dissolves from the bubbles, while upwelling velocities reduce the gas exchange. We are using a simple 2 D model of Langmuir cells to assess their effect on air sea gas exchange (Figure 3). We describe the cells with a stream function S given by Thorpe et al. [2003] that has been modified so that the ratio R of maximal downwelling to upwelling velocities matches the observations [Weller and Price, 1988] and allows for different cell width L x and depth L z S ¼ B L xw max sinðx=l x Þsinðz=L z Þ : ð10þ coshðx=lxþþa The horizontal and vertical velocities are derived from this equation with u = ds/dz and w = ds/dx. Herew max is the maximum downwelling velocity and x and z are the horizontal and vertical space coordinates. The coefficient A is used to change the ratio R of maximum downwelling to upwelling velocity (Figure 3d). For R = 1 and L x =L z, equation (10) reduces to the equation given by Thorpe et al. [2003]. For our calculations we use R = 3 (Figure 3b) as an average ratio matching observations from Weller and Price [1988] showing an upwelling speed of 1 2 cms 1 and a downwelling speed of 3 6 cms 1. Li et al. [2009] found lower downwelling to upwelling ratios of R = 1.2 1.6. However, we use the Weller and Price [1988] values since they provide a conservative bound for the results (section 6). Coefficient B is used to scale equation (10) so that the maximum downwelling velocity w max matches equation (11). 4.1. Downwelling Velocity [21] Weller and Price [1988] compiled data from several papers describing maximum downwelling velocities w max as a function of wind speed u 10 (Figure 4). The linear rela- 5of13

Figure 3. Contour plots showing the streamlines of Langmuir cells with a ratio R of maximum downwelling to upwelling velocities of (a) 1:1 and (b) 3:1 as function of Langmuir cell width L x and depth L z. (c) The corresponding vertical velocities along a horizontal cross section at the center depth of the cells is shown. (d) Values for coefficients A (solid line) and B (dashed line) used to adjust the ratio R of downwelling to upwelling velocities (equation (10)). tionship by Leibovich [1983] represents an average value and is therefore used in this paper wmax b ¼ ð 8:5 u 10 3Þ10 3 : ð11þ 4.2. Langmuir Cell Size [22] The Langmuir cell depth L z was determined by Plueddemann et al. [1996] by using the bubble entrainment depth as indicator during conditions when Langmuir cells were present (Figure 5). Since the entrainment depth in the data set is much larger than the wave height and hence the initial injection depth g, the bubbles must have been carried to depth by the strong vertical currents associated with Langmuir cells [Baschek and Farmer, 2010]. The results are roughly consistent with estimates by Scott et al. [1969], who used the vertical temperature gradient as indicator, a relationship that has been confirmed by Kukulka Figure 4. Relationship between wind speed and maximum downwelling velocity w max in Langmuir cells (redrawn from Weller and Price [1988]). We use the linear relationship from Leibovich [1983] in this study (equation (11)). 6of13

Figure 5. Relationship between wind speed and depth/width of Langmuir cells. Scott et al. [1969] used the vertical temperature gradient to determine cell depth (red crosses). Data from Plueddemann et al. [1996] (orange squares) show the entrainment depth of gas bubbles, while Zedel and Farmer [1991] (gray shaded area; dark color shows higher likelihood of occurrence) and Thorpe et al. [1994] (blue diamonds), and Plueddemann et al. [1996] (green circles) show measurements of mean cell spacing divided by 2. The linear relationships for cell width L x and depth L z used in this paper are shown as dashed and solid lines (equation (12)), respectively. et al. [2009] who investigated the effect of Langmuir cells on upper layer mixing. Langmuir cells may vary greatly in size and have been recorded to have depths of up to 50 m [Smith, 1992; Li et al., 2009] which is significantly larger than typical parameterizations. However, the downwelling speeds for the large cells found by Li et al. [2009] are at 5 m depth approximately the same as the downwelling speed of the smaller cells [Weller and Price, 1988] and may therefore have a similar effect on bubble dissolution. [23] The Langmuir cell width L x was determined from measurements of the horizontal cell spacing by Zedel and Farmer [1991], Thorpe et al. [1994], and Plueddemann et al. [1996] (cell spacing = 2 L x ; Figure 5). We are using the description by Plueddemann et al. [1996], which is consistent with observations by Smith et al. [1987] showing that the size of the cell width is less than the cell depth. However, the choice of parameterization for the cell width is not critical as it does not affect the calculated gas exchange rate as shown in the sensitivity study (Table 1). The solid and dashed lines in Figure 5 are derived from linear fits through the data, providing relationships between wind speed and Langmuir cell depth L z and width L x L z ¼ 0:539 u 10 0:432 and L x ¼ 0:079 u 10 þ 1:76: ð12þ 4.3. Modeling [24] In order to calculate the effect of Langmuir cells on the dissolution of gas bubbles we are injecting gas bubbles at varying horizontal locations within the Langmuir cells in respect to areas of vertical downwelling or upwelling. We use 30 evenly spaced horizontal points throughout the Langmuir cell width while keeping the same bubble radii and injection depths as for the previously described model runs, so that each Langmuir cell run includes 72,000 individual bubbles. The amount of gas dissolved is calculated for each individual bubble with the 1 D bubble model (equations (6) (8)) while the vertical currents in the Langmuir cells w LC (equation (8)), which influence the amount of gas dissolved, are taken into account. 5. Results and Discussion [25] After the total amount of gas entrained by bubbles was calculated with equation (1), the bubble model (equations (6) (8)) was used to determine how much of the injected gas dissolves in the ocean yielding the bubblemediated gas exchange rate. The Langmuir cell model in equation (9) was used to estimate the effect of the vertical velocities in Langmuir cells on the dissolution of bubbles. Due to the choice of parameters used in these calculations, 7of13

Figure 6. Percentage of gas injected by bubbles that dissolves before the bubbles rise back to the sea surface. It is shown for O 2,N 2,andCO 2 as function of gas saturation and wind speed (a c) without the contribution of Langmuir cells and (d f) with the contribution of Langmuir cells. the results provide a conservative estimate of the contribution of gas bubbles to air sea gas exchange. The associated errors and uncertainties are discussed below. [26] Only a small percentage 8 of the gas injected into the ocean in form of gas bubbles dissolves before the bubbles rise back to the sea surface (equation (5)). This percentage is shown in Figure 6 for O 2,N 2, and CO 2 as function of gas saturation and wind speed. The zero line indicates where the average bubble dissolution is zero. Multiplied with the absolute amount of gas injected (equation (5)) this yields the bubble mediated gas exchange rate shown in Figure 7 as the solid lines. If the effect of Langmuir cells is also taken Figure 7. The bubble mediated gas exchange rate is plotted for (a) O 2, (b) N 2, and (c) CO 2 as a function of wind speed, both for conditions when Langmuir cells are (dotted line) and are not (solid line) present. The total gas exchange rate is shown as a dash dotted line [Liss and Merlivat, 1986] and dashed line [Wanninkhof, 1992]. The gas saturation for all gases is 95%. 8of13

Figure 8. Gas entrainment rates due to bubble injection as a function of wind speed. The results of this study (solid line) are compared to descriptions given by [Andreas and Monahan 2000], who used a descriptions for the active white cap coverage by Monahan [1989] and bubble spectra by Deane [1997] with a maximum bubble radius of 3 (dashdotted line) and 6 mm (dashed line). Results for a revised bubble spectrum by Deane [1997] (see discussion by [Andreas and Monahan, 2000]) and a maximum bubble of 3 mm are shown by the dotted line. into account the gas exchange rate is higher for all gases as indicated by the dotted lines. As comparison, the total airsea gas exchange was calculated according to Liss and Merlivat [1986] (dash dotted line) and Wanninkhof [1992] (dashed line). These descriptions contain the diffusion through the sea surface as well as the contribution of gas bubbles, but do not allow for supersaturated conditions as discussed below. [27] The bubble mediated gas exchange is zero for wind speeds below 4.5 m s 1 (equation (2)) and increases with wind 5 speed proportional to u 10 to u 6 10, depending on the gas. This steep increase of the contribution of gas bubbles to airsea gas exchange results from the fact that white cap coverage, bubble injection depth, periodicity of wave breaking, and void fraction all depend on wind speed. The observed dependency of white cap coverage on wind speed is by itself proportional to u 3.2 10 [Monahan, 2001] (equation (2)). The bubble mediated gas exchange increases therefore quicker than the total gas exchange, which is proportional to u 2 10 [e.g., 3 Wanninkhof, 1992] or u 10 [e.g., Monahan, 2002], implying that the contribution of diffusion through the sea surface generally decreases for larger wind speeds. [28] The rate of total gas entrained V e given by equation (1) agrees well with parameterizations by Wu [1988] showing that the number of bubbles in a plume is proportional to u 4.5 10. In contrast, parameterizations by Andreas and Monahan [2000] and estimates by Zhang and Yuan [2004] assume that V e is simply proportional to the active white cap coverage, or u 3.2 10, while the void fraction stays constant with wind speed. However, several studies [Medwin and Breitz, 1989; Deane and Stokes, 2002; Johnson and Cooke, 1979; Lamarre and Melville, 1994] have shown that the bubble entrainment depth, and hence the depth integrated void fraction, increases with wind speed. For u 10 <11ms 1, our description of the total gas entrainment rate (Figure 8, solid line) yields therefore smaller values than the descriptions by Andreas and Monahan [2000], and for u 10 >11ms 1 it agrees reasonably well with the middle value of this study which is based on bubble spectra by Deane [1997] and a maximum bubble radius of 6 mm. [29] The importance of bubble mediated gas exchange relative to the total gas exchange is shown in Figure 9 as solid lines for a gas saturation of 95% for O 2,N 2, and CO 2. This contribution of bubble mediated gas exchange to the aeration of water is further enhanced by the downwelling currents associated with Langmuir cells; the additional effect by Langmuir cells on the total gas exchange is shown as dashed lines. For a wind speed of u 10 =15ms 1 and a gas saturation of 95%, gas bubbles contribute 1%, 19%, and 8% to the total air sea exchange of O 2,N 2, and CO 2 with an additional 3%, 6%, and 0.1% due to the effect of Langmuir cells. For u 10 =20ms 1 and a gas saturation of 95%, the contribution of gas bubbles increases significantly to 77% Figure 9. The contribution of bubble mediated gas exchange (solid lines) and the additional effect of Langmuir cells (dashed lines) to the total air sea gas exchange for (a) O 2, (b) N 2, and (c) CO 2 at a gas saturation of 95%. 9of13

Figure 10. The bubble mediated gas exchange rate for O 2 as a function of wind speed for conditions when Langmuir cells are (dotted line) and are not (solid line) present. The total gas exchange rate is shown as a dash dotted line [Liss and Merlivat, 1986] and dashed line [Wanninkhof, 1992]. The gas saturation is (a) 95%, (b) 98%, and (c) 105%. for O 2, 98% for N 2, and 16% for CO 2, while the additional contribution by Langmuir cells is 19%, 35%, and 0.7%. The bubble mediated gas exchange and the effect of Langmuir cells are therefore very significant for gases with a low solubility and it is small for gases with a high solubility. Due to the strong dependence of the bubble mediated gas exchange and the contribution of Langmuir cells on wind speed even short storm events with high wind speed of u 10 > 15 m s 1 play an important role in the aeration of water. [30] A comparison of our calculated bubble gas exchange and the total gas exchange is shown in Figure 10 for gas saturations of 95%, 98%, and 105%. When comparing the bubble mediated gas exchange and the total gas exchange caution needs to be exhibited at gas saturations close to 100%. While our model calculations include the physics of gas bubbles which tend to drive the water to supersaturation, the descriptions for the total gas exchange used for comparison [Liss and Merlivat, 1986; Wanninkhof, 1992] do not allow for supersaturated equilibrium conditions while the bubble mediated gas exchange is implicitly included. This results in a mismatch at gas saturations close to 100%, when the total and diffusive gas exchange approach zero and are hence smaller than the bubble mediated component (Figure 10). [31] We are therefore comparing the bubble mediated gas exchange for model runs when Langmuir cells are present with model runs when no Langmuir cells are present (Figure 11), which has the additional advantage that all variables used to calculate the absolute value of gas entrained (equation (1)), cancel out and only the entrainment depth, bubble size distribution, and wave height are used, thus reducing the uncertainty of the estimate. The percentage increase of the bubble mediated gas exchange due to the effect of Langmuir cells is shown for the eight gases N 2,O 2,CO 2, He, Ne, Ar, Kr, and Xe as a function of wind speed and gas saturations of 98%, 100%, and 102%. [32] At a gas saturation of 100%, Langmuir cells increase the dissolution of gas bubbles by 47% to 81% for N 2, 32% to 55% for O 2, and 5% to 6% for CO 2. The importance of Langmuir cells increases significantly with gas saturation and is especially pronounced at wind speeds of u 10 <10ms 1, where Langmuir cells increase the bubble mediated gas exchange by a factor of 3, or by 200%, for gases with a low solubility such as N 2 or Ne (Figure 11c). The increase in bubble dissolution is more pronounced at low wind speeds since the absolute contribution of gas bubbles to air sea gas exchange is small (Figure 6) and therefore even a small increase in the bubble dissolution by Langmuir cells has a large relative effect. [33] The small effect of Langmuir cells on the dissolution of CO 2 is due to its distinct chemical properties. Highly soluble gases such as CO 2 reach equilibrium relatively quickly. The increased subduction time of the bubbles due to the downwelling currents in Langmuir cells has therefore only little effect on its dissolution. Furthermore, the bubble gas exchange from CO 2 is largely driven by larger bubbles [Baschek, 2003], which have large rise speeds and are therefore much less likely to be drawn down than small bubbles. 6. Errors and Sensitivity Study [34] The results of this paper depend on the chosen descriptions for the variables in equation (1) and the bubble model (equations (6) (8)). Even for the relatively well studied variables such as wave height and white cap coverage, the results can vary by a factor of 3 between different descriptions of the parameters. For the less studied variables, such as void fraction and periodicity of a breaking wave, the results can differ by more than an order of magnitude. [35] While the uncertainties of our results can be greatly reduced when more data becomes available in the future, we estimate the current uncertainties of the total amount of gas dissolved by bubbles and Langmuir cells in a sensitivity study where only one variable at a time has been changed keeping all other variables the same as in the main coupled model runs. The results of the sensitivity study are shown in Table 1. The first column shows the variables that we have increased or decreased by a factor of two and the right hand columns show the factor by which the resulting sensitivity run differs from the main run. [36] Different parameterizations of the white cap coverage, duration of wave breaking, and surface void fraction 10 of 13

Figure 11. The effect of Langmuir cells on the dissolution of gas bubbles as function of wind speed for the eight gases N 2,O 2,CO 2, He, Ne, Ar, Kr, and Xe. The percentage increase of gas exchange relative to model runs without Langmuir cells for a gas saturation of (a) 98%, (b) 100%, and (c) 102% is shown. translate directly into a change of the amount of gas dissolved according to equation (1), i.e., a change of the white cap coverage or surface void fraction by a factor of two, or a change in the duration of wave breaking by factor of 0.5, all result in twice the amount of gas dissolved. [37] The results of our study are especially sensitive to the parameterizations of wave height and bubble entrainment depth at wind speeds u 10 >10ms 1. For example, a factor of 2 increase of the description of wave height, which is equivalent to a doubling of the bubble entrainment depth, results in an increase of dissolved O 2 and CO 2 by a factor of 2.8 and 2.3 at 15 m s 1. Changes in cell depth (equation (12)) and the ratio of upwelling to downwelling speed within the Langmuir cells (equation (10)) have a much smaller effect on the order of 10 30%. The choice of the cell width (equation (12)) does not affect the model results and a temperature of 10 C instead of 15 C results in a change of the dissolved gas of <2% for O 2 and CO 2. The effect of turbulence implemented as random walk equation (8) leads to a decrease of dissolved N 2 and O 2 of <3% and an increase of dissolved CO 2 of up to 7% at u 10 = 15 m s 1. [38] In our model calculations we have also made approximations by assuming that the saturation of each gas is constant, all the white caps have the same bubble distribution below them, and each white cap has the same life time and regeneration period. These assumptions may be reasonable when conditions are steady, but may no longer be valid if they are changing with time requiring a more complex approach. It is also assumed that Langmuir cells cover the entire surface area of the ocean when they are present and in our model all variables depend directly or indirectly on wind speed. This assumption may not be accurate if the parameters are significantly affected by other factors, such as the duration and variability of the wind, wave steepness, or the type of wave breaking that is occurring. These factors may indeed be partially responsible for the large range of the available descriptions for some parameters. [39] Since our simple analytical model is aimed to provide a general estimate of the contribution of gas bubbles to air sea gas exchange and the effect of Langmuir cells on bubble dissolution, we have chosen parameters that are typical for open ocean conditions. The results may therefore be different for fetch limited conditions or in the shallow water environment of the coastal ocean. 7. Summary [40] We have used a 1 D model to determine the contribution of gas bubbles to air sea gas exchange at different wind speeds and gas saturations. A 2 D Langmuir cell model is used to estimate the additional enhancement of bubble dissolution and hence gas exchange by the downwelling currents associated with Langmuir cells. 11 of 13

[41] The contribution of gas bubbles to air sea gas exchange increases significantly with wind speed and is proportional to approximately u 5 10 to u 6 10. At a wind speed of u 10 =15m s 1 and a gas saturation of 95%, gas bubbles contribute 15%, 19%, and 8% to the total air sea exchange of O 2,N 2, and CO 2. The additional contribution of Langmuir cells is 3%, 6%, and 0.1%. This value increases significantly for a wind speed of u 10 =20ms 1, yielding a contribution of gas bubbles to gas exchange of 77% for O 2, 98% for N 2, and 16% for CO 2, while Langmuir cells contribute 19%, 35%, and 0.7%. The air sea exchange of gases with a low solubility (e.g., N 2 ) is therefore particularly affected by gas bubbles and Langmuir cells. The increase of the bubblemediated exchange of gases with low solubility (e.g., Ne, N 2 ) by Langmuir cells at wind speeds of u 10 >10ms 1 is generally 40% to 70% and can reach 200% at supersaturated conditions and low wind speeds. 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