The relationship between ocean surface turbulence and air-sea gas transfer velocity: An in-situ evaluation

Although the air-sea gas transfer velocity k is usually parameterized with wind speed, the so-called small-eddy model suggests a relationship between k and ocean surface dissipation of turbulent kinetic energy ϵ. Laboratory and field measurements of k and ϵ have shown that this model holds in various ecosystems. Here, field observations are presented supporting the theoretical model in the open ocean. These observations are based on measurements from the Air-Sea Interaction Profiler and eddy covariance CO2 and DMS air-sea flux data collected during the Knorr11 cruise. We show that the model results can be improved when applying a variable Schmidt number exponent compared to a commonly used constant value of 1/2. Scaling ϵ to the viscous sublayer allows us to investigate the model at different depths and to expand its applicability for more extensive data sets.


Introduction
Gas fluxes across the air-sea interface are computed according to: where ∆P is the partial pressure difference of the particular gas between the ocean and the atmosphere, s is the solubility of the gas, and K is the total gas transfer velocity [1,2]. The technology for field measurements of ∆P , of both DMS and CO 2 , has matured to the point where they are readily accessible, and their values can be determined with high precision. Therefore, the main challenge remaining is to estimate K to accurately model air-sea fluxes, as K is one of the dominant sources of uncertainty in the calculation of these fluxes (e.g., [3]). The water side air-sea gas transfer velocity k is usually parameterized as a function of wind speed (k = f (u10)) [4], as this is a readily available parameter from in-situ and satellite observations. However, wind speed is not the only factor influencing k. As the physical processes that influence k are mostly controlled by the near-surface dissipation rate of turbulent kinetic energy [5], attempts have been made to directly relate k to , originally described by surface renewal models [6,7]. These models describe the air-sea gas exchange by linking the diffusive exchange at the interface with the continuous disruption of the diffusive boundary layer due to turbulent motions. Higher turbulence means more frequent renewal of the water side boundary layer and thus a higher gas exchange rate. Lamont and Scott [8] suggest to model this relation using the small-eddy model: where A is a proportionality constant, 0 the dissipation rate of TKE at the air-sea interface, Sc is the Schmidt number, which is defined as the ratio of the kinematic viscosity of water ν to mass diffusivity of water D so, Sc = ν/D, and n is the Schmidt number exponent. For the surface renewal models, Lamont and Scott [8] predicted a Schmidt number exponent of n = 1/2. This Schmidt number exponent of n = 1/2 is not applicable for interfaces covered with surfactants. For film-covered surfaces, a Schmidt number exponent of n = 2/3 is found [9]. In reality, sea surfaces might be partially covered with surfactants and n varies between 1/2 for a wavy surface and 2/3 for a flat surface depending on the sea surface conditions [10,11]. Laboratory experiments [12,13] and field measurements [14,15,16,17] of k and have shown that this theoretical relation agrees with observations, and that is a good predictor of gas transfer in various ecosystems. These studies cover various types of environmental systems and range from tank experiments to lakes, rivers, estuaries, and coastal oceans as well as the modelworld Biosphere 2 [14,15,16,17]. In addition, these studies cover various types of environmental forcing like wind, tides, and rain and were measured using a variety of instrumentation. This variety of observational setups and environments demonstrates the universality of the small-eddy model described in equation 2. All these studies show a high correlation between k and , though significant variation is found in the empirical proportionality coefficient A (equation 2), usually calculated through linear regression. As the turbulence measurements were carried out at various depths below the surface, the differences in A are most likely related to this, as strongly depends on depth in the surface region of the ocean [18]. Another source of variability in A is the sea surface condition as well as surfactants at the sea surface (e.g., [12]), but Tokoro et al. [15] argue that this should affect and not A. Wang et al. [17] evaluated the small-eddy model under nonbreaking wave conditions based on turbulence measurements directly below the sea surface. Their results indicate that A scales linearly with log( ) and they show that the relation holds for other published studies focusing on similar environments (i.e., lakes with wind and waveinduced surface turbulences), and in laboratory experiments. However, this relation contradicts the measurements of Zappa et al. [14], which were taken in different environmental systems with different forcing.
To the best of our knowledge, no investigations have been performed in the open ocean relating k to . Measurements of eddy covariance CO 2 and DMS air-sea fluxes and of from the Air-Sea Interaction Profiler (ASIP), taken during the Knorr11 cruise, provide an opportunity to investigate the relation between k and in the open waters of the North Atlantic Ocean.

Methods
Measurements were taken during a field campaign in the North Atlantic, aboard the R/V Knorr from late June to mid-July 2011 leaving and returning to Woods Hole, USA [19,20,21,22,23].

Flux measurements
The 3D wind speed was measured at 10 Hz with two Csat3 sonic anemometers mounted at the bow mast of the R/V Knorr. The measurements were corrected for ship motion as described in Miller et al. [24] and Landwehr et al. [25]. The time series of the motion corrected wind speeds were used to calculate the air-side friction velocity u * a and the trace gas fluxes. The CO 2 fluctuations were measured at 10 Hz with a non-dispersive infra-red gas analyzer of the brand LICOR. A diffusion dryer was used to remove ambient water vapor fluctuations [26]. It was shown in Landwehr et al. [27] that this significantly improves the quality of the direct CO 2 flux measurements. The DMS flux measurements and data analysis are described in Bell et al. [28]. The total gas transfer velocities K of CO 2 and DMS were calculated using equation 1 based on 10-min averaged measured data. These K-values combine the effects of processes at the air as well as the water side of the air-sea interface. The water side gas transfer velocity k was calculated and normalized to a Schmidt number of 660 (CO 2 at 25 • C): where Sc x refers to the Schmidt number at the in situ seawater state for either DMS or CO 2 .

ASIP
Presented here are data from four deployments (a total of 283 profiles) of ASIP, an autonomous upwardly rising microstructure profiler. ASIP, which is extensively described in Ward et al. [29], is equipped with a variety of sensors and is calculated from two shear probes. To validate the small-eddy model only the surface values of are of interest. As the vertical resolution of the measured is approximately 0.5 m, all data points within the uppermost 0.5 m of the ocean are declared as surface values ( 0.5 ). These surface values of are compared to the measured k 660 based on the DMS and CO 2 flux measurements (in the following called k DM S and k CO 2 ).

Scaling
It is not straightforward to measure at the air-sea interface and there is little open-ocean data of near-surface available. Therefore, it is helpful to scale close to the air-sea interface, once a universal relation between and k is determined. Lorke and Peeters [5] took the approach to consider the oceanic boundary layer as a shear-driven, flat surface, and scaled with the law of the wall (LOW) [30]. The underlying assumption of this approach is that the total stress in the boundary layer is constant, which leads to a log-linear velocity profile: where κ ≈ 0.41 is the von Kármán constant, z is the distance down from the ocean surface, and u * w is the measured water side friction velocity. The applicability of the assumption that the sea surface is purely shear driven is questioned [31,32,33,34]. Measurements show that higher dissipation rates can be found than those predicted by the LOW. These higher rates are usually ascribed to breaking waves [31,32,33]. The results of this study show that the scaled are nevertheless well correlated to the measured from ASIP. By comparing the scaled and measured surface an offset between them was determined. The profiles of were extrapolated to the viscous sublayer thickness, which has been shown to be a function of wind speed [35,36]. The relationship from Wu [37] was used to determine this thickness, given by z ν = 11ν/u * w , and an offset was subtracted to yield the dissipation rate at the interface ( 0 ). These interface values are on average two orders of magnitude higher than the actual measured 0.5 in the uppermost 0.5 m of the ocean.

Schmidt number exponent
All mentioned field studies on the small-eddy model assumed a constant Schmidt number exponent of n = 1/2 for the full range of their measurements [14,15,16]. However, n depends on the sea surface conditions and is known to vary between 1/2 for a wavy sea surface to 2/3 for a smooth surface [10,11]. Krall [38] carried out experiments investigating the dependence of n on wind speed for different surfactant coverage in the Aeolotron at the university in Heidelberg.
During these experiments, Krall [38] distinguished between three surface conditions: clean surface, 0.052 mmol/l Triton and 0.26 mmol/l Triton, and determined different n(u 10 ) relations for each of them ( figure 1). The higher the surfactants coverage, the higher is the wind speed at which n reaches 2/3, and the steeper is the wind speed dependence. Neither the surfactant coverage nor the mean square slope were measured during Knorr11, so n was parameterized based on wind speed at 10 m height (u 10 ) only. It is assumed that a medium surfactant coverage prevailed during Knorr11, as there were several instances of high chlorophyll values.
The best fit of n(u10) was determined for k DM S and for k CO 2 based on equation 2 with the assumption that the proportionality would be unity (A = 1). The best fit for n(u 10 ) based on k DM S is n = −0.259log(u 10 ) + 0.83 and n = −0.162log(u 10 ) + 0.642 based on k CO 2 . For both gases a logarithmic fit best described the relationship over a linear fit. Overall, the relation based on k DM S yielded better results (RMSD ±4.3 cm hr −1 ) than the one based on k CO 2 (RMSD ±21.5 cm hr −1 ) due to less variability in k DM S measurements. According to figure 1, n(u10) reaches the upper limit at a similar wind speed to the one for medium surfactants [38] and follows its trend whilst being less steep.

Results and discussion
To improve the statistics, the ASIP data are averaged over 90 minutes. Figure 2a and 2b show the results of applying n = f (u10) instead of a constant n = 1/2 value in equation 2. For n = f (u10), 89% of the variation in k DM S can be explained by the small-eddy model, compared to 53% for a constant n. The k CO 2 data are more variable than k DM S (figure 2c and 2d). For n = 1/2, R 2 is found to be 0.16, whereas for n = f (u10), R 2 = 0.2. For both gases, DMS and CO 2 , k is highly variable at low wind speeds, and thus higher n.
The proportionality coefficient A, which is found through data regression, has to be adapted when applying n(u10) instead of a constant n = 1/2. Figure 2b shows that A = 1.0 ± 0.11 for k DM S , which is reasonable as the parametrization of n(u10) is based on the assumption that A = 1 in equation 2. For the commonly used n = 1/2 a proportionality of A = 0.42 ± 0.05 is observed for k DM S . This value is nearly identical to the theoretical value of 0.4 [8] and close to values found in field experiments of Zappa et al. [14] (A = 0.419 ± 0.130), whose depth measurements for ranged from a few cm to 3 m below the sea surface. Vachon et al. [16] estimated A = 0.44 ± 0.01 for large lakes and A = 0.39 ± 0.02 for smaller lakes measuring at 10 cm depth by using the floating chamber method. Thus, when applying the same assumptions as in these studies in lakes, rivers, and estuaries, similar results are obtained in the open ocean.
The proportionality constant A for n = 1/2 (k DM S ) is, however, approximately double that of Tokoro et al. [15], who determined values of A = 0.13 to 0.2 at depths scaled to 86 cm and  also double that of Wang et al. [17], who report values of A = 0.25 measured at 0.25 cm depth and A = 0.31 at 10 cm depth. To further investigate this variability in A, the measured are extrapolated closer to the air-sea interface (the viscous sublayer). These scaled and extrapolated values of feature 6% less variability than those originally measured. This reduction in variability might explain the higher coefficient of determination R 2 at the interface compared to the total uppermost 0.5 m ( figure 2 compared to figure 3). For n = f (u10), the small-eddy model explains 91% of the variability in k DM S at the interface ( 0 ) and 69% for n = 1/2. Again, the relation between k CO 2 and the right-hand side of the small-eddy model is lower than for k DM S , and the small-eddy model explains only 18% of the variability for n = 1/2 and 19% for n = f (u10). For k DM S as well as for k CO 2 , the determined A have lower values when scaled to the air-sea interface compared to the measurements at 0.5 m (0.18 ± 0.03 for n = 1/2 and 0.39 ± 0.06 for n(u10) based on k DM S , and 0.31 ± 0.09 for n = 1/2 and 0.47 ± 0.15 based on k CO 2 ). The value of A for k DM S with n = 1/2 at the interface is significantly closer to those reported by Wang et al. [17] measured at 0.25 cm depth than the A found for the same setting within the whole surface layer. This supports the idea that A depends on the depth of the turbulence measurements, where A has to follow the (z) dependency to balance the small-eddy model.
Once a parametrization for n(u10) and a corresponding A are determined, the small-eddy model is applicable to more extensive gas transfer datasets. This approach can be used to predict k by figure of being scaled based on LOW and thus based on u * w instead of real measurements. Figure 4 shows the prediction of the small-eddy model for the complete Knorr11  data [20]. Bin averaging k reduces variability and results in a linear relationship between k CO 2 and the right hand side of equation 2, with a R 2 value of 0.92. For k DM S a lower correlation is found, as there is a deviation in the transfer velocity at high wind speeds, which has been explained by Bell et al. [20]. If these data points are ignored (u10 > 12 m/s), R 2 reaches 0.78, so the small-eddy model explains a large part of the variability in k DM S for low and medium wind speeds.
No measurements of exist at high wind speeds during Knorr11 (indicated in figure 4). Therefore, a reduction in during high wind speeds could explain the discrepancy between the predicted k DM S and the measured one (figure 4). A reduction in with a simultaneous increase in bubbles could also explain the different behavior in k DM S and k CO 2 [20].

Conclusions
Various studies on gas exchange across the air-sea interface, including a variety of methods and techniques ranging from theoretical approaches to laboratory studies to field campaigns in lakes, coastal areas and estuaries, have verified to be a good predictor for k using the small-eddy model [5,12,13,14,15,16]. Our measurements of k and verify that this theoretical relation holds well in the open ocean.  Figure 4. Bin-averaged measured k against the u * w -based prediction (LOW in equation 4) of the small-eddy model for the full Knorr11 data. Values of 0 are predicted by the LOW for the viscous sublayer depth (z ν = 11ν/u * ) with a determined offset between measurements and predictions being subtracted. For the small-eddy model a varying Schmidt number exponent n = f (u10) is applied. Only the circled data points indicate bins in which more than three actual turbulence measurements exist. For wind speeds higher than 12 m/s, we do not have any direct measurements of . The colour code represents the wind speed, the errors are given by the standard error of k (std/bincounts 1/2 ) and the dashed line represents the 1:1 relation.
When applying the same assumption that n = 1/2, as in former studies, the small-eddy model explains 53% of the variability in k DM S and only 16% of the variability in k CO 2 . Our field data show that this relation can be significantly improved when applying a varying Schmidt number exponent n. This n is know to vary with the sea surface conditions. A parametrization for n based on wind speed improves the predictability of the small-eddy model to 89% for k DM S . This n = f (u10) changes the right-hand side of equation 2 similar to the approach of Wang et al. [17], who scale the proportionality coefficient with A ∝ log( ). In both cases, the expression is shifted towards lower values for low and vice versa.
Wang et al. [17] report different logarithmic parameterizations of A for different depths to retain a linear relation for the small-eddy model. The parametrization of n = f (u10) presented here, however, is universal for different depth levels. In this case, the constant A has to only be adapted for each depth level to avoid any offset in the linear relation between both sides of the small-eddy model. Thus our results support the idea that A depends on depth, and increases towards the air-sea interface as A has to balance the increase in towards the interface. Measurements of reported in earlier studies on this topic were taken at various depths, which makes comparisons of A difficult.
Surface conditions and the function for n (either a constant or n = f (u 10 )) impact values of A. Once an appropriate function for n and a corresponding A for the depth of interest is found, a universal relation between k and can be determined.
In this work the LOW is used to scale from depth measurements to the air-sea interface, but future work will involve the implementation of different scaling approaches to predict [21].