Evaluating gas transfer velocity parameterizations using upper ocean radon distributions

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2009jc005805, 2011 Evaluating gas transfer velocity parameterizations using upper ocean radon distributions Michael L. Bender, 1 Saul Kinter, 1 Nicolas Cassar, 1,2 and Rik Wanninkhof 3 Received 30 September 2009; revised 10 September 2010; accepted 27 September 2010; published 9 February [1] Sea air fluxes of gases are commonly calculated from the product of the gas transfer velocity (k) and the departure of the seawater concentration from atmospheric equilibrium. Gas transfer velocities, generally parameterized in terms of wind speed, continue to have considerable uncertainties, partly because of limited field data. Here we evaluate commonly used gas transfer parameterizations using a historical data set of 222 Rn measurements at 105 stations occupied on Eltanin cruises and the Geosecs program. We make this evaluation with wind speed estimates from meteorological reanalysis products (from National Centers for Environmental Prediction and European Centre for Medium Range Weather Forecasting) that were not available when the 22 Rn data were originally published. We calculate gas transfer velocities from the parameterizations by taking into account winds in the period prior to the date that 222 Rn profiles were sampled. Invoking prior wind speed histories leads to much better agreement than simply calculating parameterized gas transfer velocities from wind speeds on the day of sample collection. For individual samples from the Atlantic Ocean, where reanalyzed winds agree best with observations, three similar recent parameterizations give k values for individual stations with an rms difference of 40% from values calculated using 222 Rn data. Agreement of basin averages is much better. For the global data set, the average difference between k constrained by 222 Rn and calculated from the various parameterizations ranges from 0.2 to +0.9 m/d (average, 2.9 m/d). Averaging over large domains, and working with gas data collected in recent years when reanalyzed winds are more accurate, will further decrease the uncertainties in sea air fluxes. Citation: Bender, M. L., S. Kinter, N. Cassar, and R. Wanninkhof (2011), Evaluating gas transfer velocity parameterizations using upper ocean radon distributions, J. Geophys. Res., 116,, doi: /2009jc Introduction [2] Sea air fluxes are fundamental properties of the geochemical cycles of CO 2, O 2, dimethyl sulfide (DMS), methane, and halocarbons among other gases. Local fluxes of these gases are generally determined from measurements of the solubility disequilibrium between the mixed layer and the atmosphere together with an estimate of the gas transfer velocity, according to the equation Sea-air flux ¼ kc ð C sat Þ; ð1þ where k is the gas transfer velocity (units of m/d), C is concentration (units of mol m 3 ), and C sat is the saturation 1 Department of Geosciences, Princeton University, Princeton, New Jersey, USA. 2 Now at Division of Earth and Ocean Sciences, Nicholas School of the Environment, Duke University, Durham, North Carolina, USA. 3 Atlantic Oceanographic and Meteorological Laboratory, National Oceanic and Atmospheric Administration, Miami, Florida, USA. Copyright 2011 by the American Geophysical Union /11/2009JC concentration in surface water (i.e., the concentration at which the partial pressure of the gas in surface water equals that in the overlying atmosphere). The gas transfer velocity is often parameterized in terms of wind speed [e.g., Wanninkhof, 1992], although it is widely recognized that other factors, such as sea surface roughness [Frew et al., 2007], must influence its variability. [3] Four approaches involving data from the field have been used to determine gas transfer velocities and their variations with wind speed. One approach is to assume a certain functionality of gas transfer with wind speed and derive constants in the parameterization that match the seaair disequilibrium to the inventory of bomb 14 C in the oceans. This approach was first used by Wanninkhof [1992], who assumed that gas transfer velocity scales with the square of wind speed. Peacock [2004] subsequently showed that the oceanic inventory used by Wanninkhof [1992] had been overestimated, leading Naegler et al. [2006] and Sweeney et al. [2007] to revise down gas exchange rates as estimated by Wanninkhof by order 20%. A second approach involves measuring the changes in the 3 He/SF 6 ratio at the sea surface after injecting these gases into the surface mixed 1of11

2 layer [Ho et al., 2006; Nightingale et al., 2000a]. The helium concentration decreases in the injection patch as a result of dilution and gas exchange, while the SF 6 concentration decrease is primarily due to dilution. One can thus calculate the gas transfer velocity from the time dependence of the concentration ratio. A third approach, often referred to as the micrometeorological approach, uses the CO 2 covariance or gradient methods in the marine air boundary layer [McGillis et al., 2004], the DMS covariance method [Huebert et al., 2004], or the DMS gradient measurements [Zemmelink et al., 2002]. In the gradient method, one determines the sea air flux by measuring the gas gradient above the sea surface. In the covariance method, the flux is determined from the covariance of concentration with vertical motions of air that transfer the gases to the free atmosphere. In both methods of the micrometeorological approach, one also measures gas concentrations in the ocean mixed layer. The gas transfer velocity is then calculated from these properties using equation (1). [4] The fourth field approach to constraining gas transfer velocities involves measuring the mixed layer concentration of 222 Rn, a radioactive gas continuously produced at a known rate by the decay of 226 Ra. In the absence of gas exchange, 222 Rn achieves a steady state concentration in which it is decaying as fast as it is produced by radioactive decay of 226 Ra (state of secular equilibrium ). Because of gas exchange, however, the observed decay rate (or activity) of 222 Rn is less than its production rate. The deficiency of its observed concentration relative to the steady state value is a quantitative measure of the gas transfer velocity as detailed below [Peng et al., 1979]. The time period over which the gas transfer velocity is accessed by the 222 Rn deficiency depends on the mean lifetime (=half life/ln(2)) of 222 Rn (5.5 days) and the flushing time of the mixed layer with respect to gas exchange, but is of order 2 weeks. An extensive data set of 222 Rn measurements was collected as part of the GEOSECS program and Eltanin cruises ( ). Peng et al. [1979] calculated gas transfer velocities for these data. A key point is that they were unable to fully analyze the wind speed dependence because they did not know wind speeds before the ship arrived on station and collected the samples. [5] Two advances allow us to make a more complete analysis of gas transfer velocities summarized by Peng et al. [1979] and their dependence on wind speed. First, meteorological reanalysis products (e.g., those of the National Center of Environmental Prediction (NCEP) and European Centre for Medium Range Weather Forecasting (ECMWF) now allow us to evaluate gas transfer parameterizations in terms of wind speeds during the 2 week period prior to sample collection. Second, in the past two decades, authors have proposed a number of parameterizations of gas transfer velocity in terms of wind speed. We can now challenge these parameterizations, in light of reanalyzed winds, with the historical 222 Rn data summarized by Peng et al. [1979]. [6] Our goal in this paper is to define and narrow uncertainties associated with gas exchange parameterizations. To this end, we reevaluate gas transfer velocities, measured using the 222 Rn method, on cruises of R/V Eltanin and the GEOSECS program Peng et al. [1979]. In section 2 of this paper, we discuss the basis of the 222 Rn method for constraining gas transfer velocities and its limitations. We examine 222 Rn profiles to illustrate some problems in using this property to compute gas exchange velocities, and we make a rough error estimate. In section 3, we assess errors in the reanalyzed winds that will be used to analyze the 222 Rn data. We then use five different gas exchange parameterizations and two wind speed products to predict gas transfer velocities inferred from 222 Rn data and evaluate the parameterizations based on their skill in reproducing 222 Rn observations; this exercise forms the core of our paper. In section 4, we discuss, based on our results, the accuracy of air sea fluxes, calculated using wind speed parameterizations for gas transfer velocities, at local and larger scales. 2. Background: The 222 Rn Method for Determining Gas Transfer Velocities 2.1. Principles [7] 226 Ra is a divalent cation with a 1599 year half life. It is produced by radioactive decay of 230 Th in sediments, diffuses into bottom water, and is subsequently mixed throughout the oceans. It is cycled like a bioactive tracer and thus has a nutrient like profile in the oceans but is only partly depleted at the surface. 226 Ra decays to 222 Rn, a noble gas with a half life of 3.82 days. In a closed system, the 222 Rn concentration will reach a level such that it is decaying at the same rate that it is produced. In the mixed layer, its level is lower, because it is also lost by gas exchange. The gas transfer velocity is determined from the ratio of the 222 Rn decay rate to the 222 Rn production rate, which is identical to the decay rate of 226 Ra [Peng et al., 1979], k ¼ C E C M h; ð2þ C M p where k is the gas transfer velocity (m d 1 ), C E is the equilibrium concentration (assuming radioactive steady state in the absence of gas exchange), C M is the observed concentration of 222 Rn, p is the atmospheric partial pressure of 222 Rn, a is the solubility of radon in water, l is the decay rate constant of 222 Rn (0.181 d 1 ), and h is the depth of the mixed layer. This equation states that the calculated gas exchange velocity scales with the inverse of the observed 222 Rn deficiency, the 222 Rn decay constant, and the depth of the mixed layer. While there is always some 222 Rn in air, its concentration is very low, <5% of the equilibrium value with surface seawater [Smethie et al., 1985], and C M is much greater than pa. Since the decay rate or activity A is proportional to the concentration, we can substitute A for C in equation (2). Therefore, we can simplify equation (2) to k ¼ A E 1 h: ð3þ A M This approach was used to calculate the gas transfer velocities for cruises of R/V Eltanin and the GEOSECS program [Peng et al., 1974] used in this analysis. The limitations of this approach to determine k are examined below. We denote gas transfer velocities calculated from equation (3) with the term k Rn. The k Rn values that we use to evaluate gas transfer parameterizations are from Table 1 of Peng et al. [1974], who summarized results from 89 radon profiles collected on 2of11

3 Table 1. Basin Averaged Wind Speeds and Differences of Reanalyzed and Observed Winds on Days Samples Were Collected a Ship NCEP ECMWF NCEP Obs. ECMWF Obs. North Atlantic 6.8 ± ± ± ± ± 2.7 South Atlantic 6.9 ± ± ± ± ± 2.9 Atlantic Ocean 6.8 ± ± ± ± ± 2.7 North Pacific 8.8 ± ± ± ± ± 2.2 South Pacific 8.2 ± ± ± ± ± 4.3 Pacific Ocean 8.5 ± ± ± ± ± 2.5 Southern Ocean 9.8 ± ± ± ± ± 4.3 Whole ocean 8.2 ± ± ± ± ± 3.2 a Ship, NCEP, and ECMWF columns: mean and standard deviations of wind speeds (m/s) observed on ships, and inferred by reanalysis, on the days that 222 Rn samples were collected on GEOSECS and Eltanin cruises. NCEP Obs. and ECMWF Obs. columns: mean difference and the root mean square differences between ship and reanalyzed winds for both the NCEP and ECMWF data products. the Atlantic and Pacific GEOSECS programs and 16 collected on Eltanin cruises in the Southern Ocean. Wind speeds at 10 m height were determined on board for the two 24 h periods prior to sample collection. The data coverage is global, but we put more emphasis on the Atlantic where reanalyzed winds best match shipboard observations made during the GEOSECS and Eltanin programs Limitations of the 222 Rn Method [8] While the 222 Rn method is a powerful approach and was used extensively in early studies aimed at obtaining local gas transfer velocities over the ocean, it also has wellarticulated limitations [Kromer and Roether, 1983; Roether, 1983; Smethie et al., 1985]. As applied (equation (3)) [Peng et al., 1979], the method implicitly assumes lateral homogeneity, invariant mixed layer depths, and steady state of the 222 Rn deficit in the mixed layer. As well, most radon studies have been carried out on cruises that occupied stations for short periods of time. In these studies, the local wind speed history prior to radon sampling and hence the wind speed parameterization was largely unconstrained [e.g., Peng et al., 1979]. Such snapshot studies were supplemented by several programs involving longer term occupations: BOMEX [Broecker and Peng, 1971], Station PAPA [Peng et al., 1974], and JASIN and FGGE [Kromer and Roether, 1983]. [9] The mixed layer 222 Rn deficiency reflects gas transfer velocity over the 2 weeks prior to sampling, weighted toward the present. At the time of the GEOSECS and Eltanin measurements, however, there was no way to estimate wind speeds or gas exchange velocities before the ship arrived on station. Here we reassess the results from these expeditions using wind speed products, derived by reanalyses of historical meteorological data utilizing numerical weather models, to estimate the evolution of the 222 Rn in the mixed layer at historical sampling locations. We use time weighted wind speed histories, together with parameterizations of gas transfer velocity in terms of wind speed, to estimate the radon gas transfer velocities at the collection times of historical radon profiles (k wt ). We then compare our calculated gas transfer velocities with values inferred from the observed radon deficits (k Rn, equation (3)) [Peng et al., 1979] and examine the differences between k Rn and k wt for various parameterizations. Our basic approach builds on and modifies that of Smethie et al. [1985]. [10] Our approach to examining gas transfer parameterizations has the advantage that it is based on a large radon data set from many oceanic locations. However, it does not account for the following factors. First, there are important uncertainties in reanalyzed winds from the time of the radon observations ( ). Second, we need to make the assumption that mixed layer depths were constant in the 2 weeks prior to collection of 222 Rn samples as there is no robust way to reconstruct the time history of mixed layers. In fact, the following discussion shows that, in many of the 222 Rn profiles measured by Peng et al. [1979], mixing prior to collection was more complex and that mixed layer depth was variable [Deacon, 1981]. Third, internal waves can cause the measured depth of the mixed layer to differ somewhat from the average depth that determines the 222 Rn concentration. Finally, lateral processes can also affect 222 Rn profiles in a way that would cause calculated piston velocities to differ from correct values Examination of 222 Rn Data [11] When calculating k Rn from equation (3) [Peng et al., 1979], one must assume that the mixed layer depth has been constant and there has been no exchange with waters below this depth. To illustrate the reliability of these assumptions, we have plotted a selection of the radon profiles for all GEOSECS stations, arbitrarily selecting those stations ending with the digit 5 (Figure 1). The profile for station 25 (top left hand corner) illustrates the ideal case [Peng et al., 1979], although there are no data points between 24 and 44 m. In this plot, activities of 222 Rn are plotted versus depth as heavy black squares, and temperature is plotted as the open red circles connected by the dashed line. The mixed layer 226 Ra activity was measured on mixed layer samples returned to the laboratory or determined from the activity of 222 Rn below the mixed layer. Mixed layer 226 Ra activity is plotted as the heavy, horizontally dashed, blue line extending from the sea surface to the base of the mixed layer. The bottom of this line indicates the depth of the mixed layer as inferred from temperature profiles [Peng et al., 1979]. At station 25, the 222 Rn activity is constant in the mixed layer at 3.8 disintegrations per minute (dpm)/ 100 kg (A M ), and the 226 Ra activity is 7.1 dpm/100 kg (A E ). The mixed layer depth is 28 m, and the calculated value of k Rn is 4.5 m/d (equation (3)). [12] Data at stations 55, 85, and 215 all show roughly constant concentrations in the mixed layer and concentrations below that are similar to 226 Ra values. These profiles are thus consistent with the simple model underlying equation (2). The profile at station 15 features only duplicate samples from 3of11

4 Figure Rn profiles from GEOSECS stations. Open back boxes, 222 Rn concentration; red circles, potential temperature; horizontally dashed blue line, mixed layer 226 Ra concentration; lower end of horizontally dashed blue line, base of the mixed layer. a single depth lying within the mixed layer. Profiles at other sites indicate complex mixing processes. For example, at station 115 there is a lens of 222 Rn depleted waters at about 90 m depth that presumably originates from the recent subduction of waters that had previously mixed to the sea surface. The same appears to be the case at station Magnitude of Errors in Gas Transfer Velocities Determined Using 222 Rn Data [13] The measurement uncertainty in the 222 Rn activities is typically 0.5 dpm/100 kg and the standard error for a typical case where five mixed layer samples have been analyzed is 0.2 dpm/100 kg. The uncertainty in the 226 Ra activity is about ±0.4 dpm/100 kg (inferred by comparing 226 Ra concentrations estimated from 222 Rn activity below the mixed layer and directly measured concentrations of 226 Ra) [Peng et al., 1979]. The median difference between A E and A M for the entire Peng data set is 1.8 dpm/100 kg (average difference = 2.0), and the errors sum quadratically to 0.5. Therefore, the analytical uncertainty in k Rn due to uncertainty in Rn deficit is ±28%. Uncertainties in mixed layer depth history add to this number. [14] In the context of a one layer model (no horizontal gradients), it is possible to estimate the magnitude of error introduced by violations of the assumptions one invokes when calculating k Rn. Several studies indicate that the mixed layer depth may vary by about a factor of 2 over intervals much shorter than the half life of 222 Rn [Acreman and Jeffery, 2007; Kara et al., 2003; Ohno et al., 2009]. It turns out that shoaling and (generally) deepening of the mixed layer both cause the calculated value of k Rn to be less than the true value. Consider the case where the mixed layer depth (MLD) changes instantaneously by a factor of 2 and refer to equation (3). Shoaling immediately lowers MLD by a factor of 0.5, while leaving A E /A M unchanged, causing k Rn to be half the true value. Deepening doubles MLD, which would tend to increase k Rn immediately after the event. However, doubling also entrains high Rn water from below the mixed layer, decreasing A E /A M, which has the opposite effect. When A E /A M is <2, as is almost always the case, the overall result is for k Rn to decrease. As an example of the magnitude, consider a boxcar time series for MLD, with an instantaneous halving or doubling of MLD every 7 days. In this case, the average value of k Rn is 75% of the true value, and the standard deviation is ±15%. We regard the magnitude of the error as an upper limit, because MLD does not change instantaneously, and weekly changes appear to frequently be less than a factor of 2 in papers cited above. A more realistic estimate for the error associated with entrainment or detrainment might be about 10%, still a significant number. We thus note that that entrainment and detrainment introduce systematic as well as random errors, but we do not attempt to constrain the magnitude or correct for their effect. By way of comparison, Roether and Kromer [1984] suggest an overall uncertainty (analytical plus arti- 4of11

5 factual) of 20% 35% based on time series observations and modeling, when 222 Rn is measured to high precision. [15] In summary, k Rn values have errors associated with analytical uncertainties of about ±28% and a systematic low bias associated with entrainment and detrainment that we estimate to be of order 10%. There are also errors associated with the assignment of mixed layer depth and lateral processes in the oceans that we do not attempt to quantify. 3. Wind Products and Time Weighted Gas Transfer Velocities 3.1. Wind Products [16] We use three wind products in this study. The first is the wind speed observed on the ship during the day the radon profile was collected (wind 1 in Table 1 of Peng et al. [1979]). The second is the NCEP surface wind speed (NCEP reanalysis 1, ncep.reanalysis.html [Kalnay et al., 1996]). The third is ECMWF ERA 40 surface wind speed ( portal. ecmwf.int/data/d/era40_daily/). [17] To assess the consistency of the meteorological reanalysis products for our purposes, we compare reanalyzed and ship winds for the 24 h period associated with collection of the 222 Rn profile. This comparison is appropriate for the time intervals, sampling times, and sampling locations of this study. [18] In Table 1, we summarize mean wind speeds, based on observations and reanalyses, and their standard deviations, by ocean basin. We also summarize means and standard deviations of differences between reanalyzed and ship winds. Agreement between ship and reanalyzed winds is, on average, best in the North and South Atlantic oceans, presumably because the assimilation models are better constrained in these basins with more observations of winds and sea level pressure. For other basins, mean differences between reanalyzed and ship winds are larger, or standard deviations of differences are larger, or both. In the Atlantic, the mean of the difference between ship and reanalyzed winds is slightly smaller for NCEP than for ECMWF, but the standard deviation of the differences is somewhat smaller for ECMWF. In our discussion, we give equal attention to calculations from both sets of reanalyzed winds. [19] In the Atlantic, the standard deviation of differences between ship and reanalyzed winds is about 40%. If the gas transfer velocity scales with the square of wind speed, then, for a 40% standard deviation in the wind speed product differences, instantaneous gas transfer velocities will have a 1 sigma uncertainty of 64% to +96% and 78% to +174% if a cubic dependency is assumed. However, work presented below shows that calculated gas transfer velocities utilizing the wind speed history agree more closely with k Rn values. Two factors contribute to this better agreement. First, gas transfer velocities determined from radon reflect winds over periods longer than a single day, and some short term errors in reanalyzed winds are averaged out. Second, ships observe locally, while the reanalyses simulate conditions over larger areas, which are more appropriate for gas exchange calculations Calculations of Gas Transfer Velocities Wind Speed Parameterizations [20] Five proposed relationships between gas transfer velocity and wind speed are used in this work: [Wanninkhof, 1992, hereafter W92] k ¼ 0:074u 2 * ðsc=660þ 0:5 ; ð4þ [Wanninkhof and McGillis, 1999, hereafter W99] [Ho et al., 2006, hereafter Ho06] k ¼ 0:0068u 3 * ðsc=660þ 0:5 ; ð5þ k ¼ 0:061u 2 * ðsc=660þ 0:5 ; ð6þ [Nightingale et al., 2000b, hereafter N00] k ¼ 0:053u 2 þ 0:024u * ðsc=660þ 0:5 ; ð7þ [Sweeney et al., 2007, hereafter Sw07] k ¼ 0:065u 2 * ðsc=660þ 0:5 : ð8þ [21] The k s are expressed in units of m/d and winds are in m/s. Sc is the Schmidt number defined as the kinematic viscosity of water divided by molecular diffusion of the gas of interest in water. The term (Sc/660) 0.5 normalizes the k to a common value equivalent to CO 2 gas transfer for seawater at 20 C. The Schmidt number is calculated from observed temperature and salinity using equations in Wanninkhof [1992]. [22] Of note is the close correspondence between the Ho06, N00, and Sw07 parameterizations, suggesting that the results on local scale obtained by dual deliberate tracer methods [N00, Ho06] are in accord with the global constraints based on the updated 14 C inventories [Sw07]. As indicated below, our reanalyses of the historical 222 Rn observations also support these parameterizations Calculation of Gas Transfer Velocities Based on Measured Winds on the Day of Sample Collection, Using Wind Speed Parameterizations [23] We make calculations of the gas transfer velocity using wind speed parameterizations and the following measured properties: wind speed measured on board ship during the collection day of the 222 Rn profile [Peng et al., 1979, Table 1], mixed layer temperature, and mixed layer depth. The mixed layer temperature is used to determine the Sc for 222 Rn. We have done this calculation for each of the five gas exchange parameterizations presented above. In these and subsequent calculations, we normalized the gas transfer velocity to a Schmidt number of 660. Below, we compare these k values to k Rn (660) calculated from equation (3) Calculation of Gas Transfer Velocities Based on Weighted Average Reanalyzed Winds, Using Wind Speed Parameterizations [24] By using reanalyzed winds, we can take into account changes in wind speeds prior to collection [Reuer et al., 2007] and calculate weighted average gas transfer velocities and wind speeds. Our calculations assume that, prior to sample collection, MLD, mixed layer temperature, and 226 Ra activity were constant at the measured values. We 5of11

6 Table 2. Differences Between k Rn and k wt, Normalized to a Schmidt Number of 660, for Four Gas Transfer Velocity Parameterizations a K Rn (660) k wt (660): Means and Rms Deviations Region W92 W99 N00 Sw07 NCEP Winds Global 0.2 ± ± ± ± 3.0 North Atlantic 0.4 ± ± ± ± 1.3 South Atlantic 1.1 ± ± ± ± 1.2 North Pacific 0.9 ± ± ± ± 2.2 South Pacific 0.5 ± ± ± ± 2.4 Southern Ocean 0.6 ± ± ± ± 6.2 ECMWF Winds Global 0.2 ± ± ± ± 2.9 North Atlantic 0.3 ± ± ± ± 1.8 South Atlantic 0.3 ± ± ± ± 1.1 North Pacific 1.6 ± ± ± ± 1.9 South Pacific 0.5 ± ± ± ± 2.2 Southern Ocean 0.8 ± ± ± ± 6.0 n k Rn (660): Basin Means k Rn k wt, rms Value k Rn k wt, rms Value/ n Average Magnitude, k Rn k wt North Atlantic South Atlantic North Pacific South Pacific Southern Ocean a Mean differences and root mean square differences between k Rn (gas transfer velocity determined from Rn profiles using equation (3)) and k wt (weighted gas transfer velocities derived from reanalyzed winds/gas exchange parameterizations) calculated using NCEP and ECMWF winds. Results are given for the global ocean and five regional basins; they are calculated for four parameterizations. For example, in the North Atlantic using the W92 parameterization, the mean difference for all stations between gas transfer velocity calculated from Rn data and the parameterization is 0.4 m/d, and the variance in this number (the root mean square value of the differences) is ±1.6 m/d. In the bottom five rows, we tabulate, for each ocean basin, the number of stations (out of a total of 104), the mean k Rn for all stations in that basin, the magnitude of the difference between k Rn and k wt averaged over the four different calculations (the W92 and Sw07 parameterizations and the ECMWF and NCEP wind products), the average rms difference between k Rn and k wt as calculated using the W92 and Sw07 parameterizations and ECMWF and NCEP winds; and the standard error in the difference (average rms value/ n). All k values are normalized to a Schmidt number of 660. allow for variations in wind speed, gas transfer velocity, and associated changes in mixed layer 222 Rn activities. [25] To derive a weighted average gas transfer velocity for a single station, we first look back 30 days prior to the time of collection and divide each day into 6 h periods for which wind speed can be accessed from reanalysis products. Thirty days is a sufficiently long period to obviate any prior influence. We calculate gas transfer velocities (k s) for each 6 h period for each parameterization. We then use these gas transfer velocities to calculate a weighted average value at the time the sample is collected. This calculation can be made using two numerically equivalent approaches; one yields an analytical solution while the second is more intuitive. Derivations are in Appendix A. In the first approach, we give k for each 6 h period a weight that diminishes with increasing time before sample collection. The weight for a given time period takes into account both the decay of 222 Rn between that time period and sample collection, and the extent to which the mixed layer is flushed by gas exchange between that time period and sample collection. In the second approach, we explicitly calculate the 222 Rn burden as a function of time starting 30 days before the sample is collected. For each 6 h time period, we evaluate the change in the 222 Rn concentration (or activity) due to production by 226 Ra decay, loss by 222 Rn decay, and loss by gas exchange. We then calculate the 226 Ra/ 222 Rn activity ratio at the time of sample collection (A Ra /A Rn ), insert this number into equation (3), and calculate k. The two approaches give values for weighted gas exchange coefficients that are identical to about 1%. In practice, we use the first approach (equation (A3)) to calculate weighted gas transfer velocities for each 222 Rn station according to the parameterizations in equations (4) (8). We use the term k wt to refer to weighted average values of gas transfer velocities. We compare k wt with k Rn to assess the accuracy of the various gas transfer parameterizations Calculation of Time Weighted Wind Speeds From Time Weighted Gas Transfer Velocities [26] The time weighted gas transfer velocity implies a time weighted wind speed (w wt ). We calculate this wind speed by substituting k wt into the equation relating gas transfer velocity to wind speed (i.e., equations (4) (8)). Calculating values of weighted wind speed then allow us to examine gas transfer parameterizations with the traditional plot of gas transfer velocity (k Rn ) versus wind speed (u wt ) Some Characteristics of Time Weighted Gas Transfer Velocities and Wind Speeds [27] It is worthwhile to remark on some features of these calculations. First, the weights given to each time period will be different for each gas exchange parameterization: parameterizations giving higher gas transfer velocities will put more weight on time periods closer to sample collection. Since more gas is lost during these late periods, the importance of preceding intervals is diminished. Second, the weighted wind speed for each 222 Rn observation station is different for each gas exchange parameterization. Again, 6of11

7 Figure 2. The k Rn k wt (difference of gas transfer velocities calculated from 222 Rn measurements and weighted gas transfer velocities based on the W92, W99, N00, and Sw07 parameterization) for different oceanic regions using wind speeds from the NCEP reanalysis and ECMWF reanalysis. Symbols represent means, and horizontal bars show root mean square deviations of differences for the individual samples. All k values are normalized to a Schmidt number of 660. parameterizations giving higher gas transfer velocities put more weight on wind speeds closer in time to sample collection. However, this sensitivity is very small, and weighted wind speeds calculated for the various parameterizations differ by only a few percent, and never more than 10%. Third, weights calculated for 222 Rn will not apply to other gases with different Schmidt numbers. Gases with lower Schmidt numbers have higher k s, and will be weighted more heavily for recent winds. 4. Discussion of the Results [28] Here we compare k Rn and k wt values to assess the accuracy of gas transfer wind speed parameterizations. [29] We have calculated gas transfer velocities using winds measured on board ship during the day of sample collection with equations (4) (8), and compared these k values with k Rn values. The differences are larger, by up to 100%, than differences between k Rn and the values of k wt calculated as described in section 3. As an example, we summarize results for gas transfer velocities calculated using the Sw07 parameterization. The rms difference between k calculated using day of collection (shipboard) winds and k Rn is 4.2 m/d for the global data set and 2.5 m/d for the Atlantic data. Using NCEP winds, the rms difference between k wt and k Rn is ±3.0 m/d for the global data set and ±1.3 m/d for the Atlantic. When calculated using ECMWF winds, the rms difference between k wt and k Rn is 2.5 m/d for the global data set and 1.5 m/d for the Atlantic. These results confirm the earlier finding of Smethie et al. [1985] that calculated gas transfer velocities are more accurate when calculated with weighted wind speeds rather than instantaneous wind speeds measured on board ship during the day of collection. [30] Differences between k Rn and k wt, calculated for different gas exchange parameterizations, are summarized in Table 2. In Table 2, the individual cells in the columns cor- 7of11

8 and Figure 2. We attribute the better performance in the Atlantic to more accurate winds and focus our analysis on this basin. In Figure 3, we compare Atlantic Ocean differences between k Rn and k wt values calculated using different parameterizations. In Figure 4, we plot k Rn versus weighted wind speed (w wt ), for the Atlantic and for other basins, superimposed on curves representing different gas transfer parameterizations. Mean and rms differences between k Rn and k wt are largest in the case of W99 and smallest for N00, Ho06, and Sw07. These trends are also apparent in the plot of k Rn versus wind speed (Figure 4). The cubic parameterization of Wanninkhof and McGillis [1999] underestimates k Rn values at low wind speeds and overestimates k Rn at high wind speeds. The quadratic equation of Wanninkhof [1992] tends to overestimate k Rn values over most of the range of Atlantic samples. Showing the most skill are the more recent Figure 3. k Rn k wt (difference of gas transfer velocities calculated from 222 Rn measurements and weighted gas transfer velocities) for the North and South Atlantic Oceans based on wind speeds from the NCEP reanalysis and ECMWF reanalysis. This plot compares differences for the Atlantic only, where wind speeds are most accurate, for four different gas exchange parameterizations. Heavy points represent means, and horizontal bars show root mean square deviations of differences for the individual samples. All k values are normalized to a Schmidt number of 660. responding to specific parameterizations are the mean difference between k Rn and k wt for the individual basins, and the root mean square deviations of the differences for individual stations. These same terms are plotted in Figure 2. [31] The parameterizations generally do a good job of reproducing basin mean values of k Rn. In the Atlantic, the mean difference between k Rn and k wt ranges from 0% 37% of k Rn, depending on parameterization and wind speed product. Differences can be larger in the Pacific and Southern Oceans, where the reanalyzed winds show poorer agreement with shipboard observations. Still, the global average values for k Rn and k wt agree within 5% 21%. [32] Agreement tends to be somewhat poorer between k Rn and k wt when k wt is calculated from the cubic parameterization of Wanninkhof and McGillis [1999] than when calculated from the other three parameterizations (Table 2 and Figures 2 4). The other three parameterizations do not show a consistent ordering of skill across the ocean basins and wind products. With NCEP winds, for example, W92 tends to do better in the Southern Ocean than N00 and Sw07, but worse in the Atlantic. [33] Two metrics indicate that the quadratic parameterizations do better in the Atlantic than in the Pacific or Southern oceans: the mean difference by basin between k Rn and k wt and the root mean square values of the deviations for individual stations. This difference is apparent in Table 2 Figure 4. Gas transfer velocity inferred from 222 Rn data, normalized to a Schmidt number of 660, versus weighted wind speed (w wt ), calculated as described in the text. The w wt values are calculated using the W92 parameterization for gas transfer velocity as a function of wind speed; values calculated with other parameterizations would be nearly identical. Shown (top) for the Pacific and Southern Ocean and (bottom) for the Atlantic Ocean only. Arrows signify data points lying off the vertical scale. 8of11

9 parameterizations of Nightingale et al. [2000b] and Sweeney et al. [2007], as well as the similar equation of Ho et al. [2006] (not shown). [34] For the Nightingale and Sweeney parameterizations, the mean residual between k Rn and k wt of Atlantic samples ranges from 0 to 0.5 m/d against a mean of m/d, corresponding to a fractional value of the mean difference of 0 13%. Gas transfer rates calculated using N00 and Sw07 thus appear to be quite accurate when averaged over a large scale. The same is likely to be true for all the oceans basins now, given the great improvement in wind products since the mid 1970s, but we are unable to argue this point given data analyzed here. [35] Rms residuals for the North and South Atlantic range from 23% to 51% for the N00 and Sw07 parameterizations. The range is 23% 37% if one restricts the comparison to gas transfer velocities calculated with NCEP winds. These values give an estimate of the accuracy to which one can constrain the sea air flux of any gas at a single time and place if the partial pressure difference across the air water interface is accurately measured. [36] The differences between k Rn and k wt are smaller for basin means than for individual samples. Thus basin scale systematic errors in the parameterizations are smaller than errors associated with either measuring k Rn or applying the parameterizations at individual sites. We make the working assumption that errors are random and Gaussian. In this case, we can reduce errors in regional sea air gas fluxes by making measurements at multiple sites; errors then diminish by the square root of the number of measurements. In practice, this approach only applies when sites are far enough separated in time and space that errors are uncorrelated; we speculate that this might require separations on the scale of synoptic weather systems. A test of this approach is to determine whether, for each basin, the mean difference between k Rn and k wt is within the standard error of zero. We find that it is for the Sw07 and N00 parameterizations except for the Southern Ocean. Here the average difference between k Rn and k wt is 1.9 m/d, whereas the standard error is 1.2 m/d (Table 2). 5. Summary and Conclusions [37] Sea air fluxes may be calculated from gas transfer velocities parameterized in terms of wind speed. Instantaneous fluxes can be accurately calculated from instantaneous wind speeds observed on the ship when seawater is sampled. However, average fluxes will be accurate only if the gas transfer velocity is calculated in a way that weights recent wind speeds according to their influence on mixed layer supersaturation when water is sampled. In this paper, we have shown how gas transfer velocities of 222 Rn (k Rn ) may be predicted, using the formalism of Reuer et al. [2007], in a way that takes into account their recent time dependence. The use of weighted gas transfer velocities (k wt ) improves the agreement with values computed from 222 Rn observations, in agreement with the previous result of Smethie et al. [1985] for measurements during the Transient Tracers in the Oceans campaign in the Equatorial Atlantic. [38] We have also evaluated the fidelity with which we can reproduce gas exchange velocities derived from 222 Rn by invoking various parameterizations in the context of the wind speed history. The parameterizations of Nightingale et al. [2000b], Sweeney et al. [2007], and Wanninkhof [1992] reproduce the observed gas transfer velocities calculated from 222 Rn observations with similar accuracy; a given parameterization may perform better or worse depending on the ocean basin and wind speed product that is being evaluated. The parameterization of Wanninkhof [1992] does better in the Southern Ocean but not as well in the Atlantic. The cubic parameterization of Wanninkhof and McGillis [1999] does less well than the quadratic parameterizations. The parameterization of Ho et al. [2006], not analyzed here, is similar to that of Sweeney et al. [2007] (see equations (6) and (8)). [39] The most useful comparison between k Rn and k wt is in the Atlantic, where the parameterizations of Nightingale et al. [2000b] and Sweeney et al. [2007] give gas transfer velocities different from those calculated from the 222 Rn data by 23% 51%, depending on basin (north versus south), parameterization, and wind product. We can account for most of these differences with analytical uncertainties in 222 Rn and 226 Ra activities, variability in mixed layer depth, and errors in winds. If, however, we attribute the entire error to the parameterization, air sea fluxes can be determined at a single station to an accuracy of about ±40%. When making regional mass balances, this accuracy can be improved by multiple measurements, so long as these measurements are well separated in time or space. There is no evidence of systematic differences from one ocean basin to another in the parameterizations. [40] The comparisons in this paper do not allow one to designate a single parameterization as giving the most accurate values of gas transfer velocities. However, the parameterization of Sweeney et al. [2007] is arguably the parameterization of choice: it reproduces k Rn values as well as, or better than, any other, and by design also accurately reproduces the oceanic 14 C inventory. [41] The weighting formalism of Reuer et al. [2007], together with the modification outlined here to account for radioactive decay in the case of radon, provide an approach that can be used to recalculate more accurate gas transfer velocities of unbuffered gases in the upper ocean from mixed layer mass balances that have been measured over the years, and continue to be measured. Appendix A [42] As described in the text, we use two methods to calculate a weighted average value for the gas transfer velocity over the 30 days (120, 6 h time periods) preceding sample collection. The first method involves giving intervals lower weights going back in time. Weights diminish according to how much 222 Rn decayed and how much gas exchange transpired between a given 6 h interval and the time of sample collection. The second involves explicitly calculating the ratio of 226 Ra/ 222 Rn activity (A E /A M in equation (3)) and calculating k according to that equation. [43] Reuer et al. [2007] derived a numerical solution for the gas transfer velocity of a nonradioactive species, produced at a constant rate, as a function of the wind speed 9of11

10 history. The fraction of the mixed layer ( f i ) that is ventilated during the time period t i prior to sample collection is f i ¼ k i Dt=h: ða1þ [44] k i is the gas transfer velocity for period t i, calculated from wind speed over t i and the particular gas transfer parameterization. h is the depth of the mixed layer, assumed to be constant at the value observed when the 222 Rn profile was collected. Dt i is always 6 h in this work. If the gas transfer velocity is 4 m/d and the mixed layer thickness is 40 m, 2.5% of the mixed layer is ventilated per 6 h interval (f i = 0.025). A similar equation applies for all periods going back in time, but the gas transfer velocity calculated for each 6 h period needs to be weighted to account for subsequent losses. The weighting term (w) for the first period prior to collection = 1 f 1 = 0.975, and the weighting for each preceding interval is! t ¼! t 1 ð1 f t 1 Þ: ða2þ [45] For example, for the gas transfer velocity and mixed layer depth given above, the weight for the second period will be ( ), the weight for the fifth period (1.25 days before collection) will be ( ) 5 = 0.881, and the weight of the sixtieth period (15 days before collection) will be ( ) 60 = Thus, the gas transfer velocity for the sixtieth period only counts in the average 0.219/0.975 = times as much as the first time period. [46] The overall equation [Reuer et al., 2007] is k ¼ P120 t¼1 ð1! 120 k t! t : ða3þ Þ P120! t t¼1 [47] Each term of the numerator is a measure of the fraction of gas produced during interval t that is lost by the time the sample is collected. [48] For a radioactive species, equation (10) needs to be modified to account for decay, which progressively diminishes the weight of prior intervals,! t ¼! t 1 ð1 f t 1 Þe Dt : ða4þ [49] l is the decay constant of 222 Rn, d 1, and t is the duration of each time interval. [50] We apply this approach to calculate gas transfer velocities at 222 Rn sampling sites based on the parameterizations in equations (4) (8). As implemented here equation (A3) is solved with 120 time steps of 0.25 day (6 h) each, extending back to 30 days before sample collection. The term (1 w 120 ) corrects for the 222 Rn predating the start of our analysis period. In practice, this term is very close to 1. The grid size of the reanalyzed winds is approximately 2 2. [51] In the second approach, we simply use the mass balance of 222 Rn prior to sample collection to evaluate the 222 Rn activity at the time the sample is collected dn 222Rn dt ¼ A 226Ra Rn N Rn k h N Rn; ða5þ where N 222Rn is the volumetric concentration of 222 Rn atoms. [52] Then at the time of collection, 222Rn N 222Rn ¼ 222Rn Z 0 t¼120 A 226Ra Rn N Rn k h N Rn dt: ða6þ [53] This term equals A M in equation (3), and A Ra226 equals A E. Equation (A6) is integrated numerically, in 6 h time steps, from 30 days before sample collection up to the time of collection. Preexisting 222 Rn is ignored, an appropriate assumption over approximately 8 half lives. k wt is then calculated using equation (3) and the values for h, l 222Rn, and A E /A M. [54] Acknowledgments. We gratefully acknowledge support from the National Science Foundation, National Atmospheric and Oceanic Administration, and National Aeronautics and Space Administration. 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