Air–Sea $\mathrm{CO}_{2}$ Gas Transfer Velocity in a Shallow Estuary

Abstract

The air–sea transfer velocity of $\mathrm{CO}_{2}\, (k_{\mathrm{CO}_{2}})$ was investigated in a shallow estuary in March to July 2012, using eddy-covariance measurements of $\mathrm{CO}_{2}$ fluxes and measured air–sea $\mathrm{CO}_{2}$ partial-pressure differences. A data evaluation method that eliminates data by nine rejection criteria in order to heighten parametrization certainty is proposed. We tested the data evaluation method by comparing two datasets: one derived using quality criteria related solely to the eddy-covariance method, and the other derived using quality criteria based on both eddy-covariance and cospectral peak methods. The best parametrization of transfer velocity normalized to a Schmidt number of 600 $(k_{600})$ was determined to be: $k_{600} = 0.3\,{U_{10}}^{2.5}$ where $U_{10}$ is the wind speed in m $\mathrm{s}^{-1}$ at 10 m; $k_{600}$ is based on $\mathrm{CO}_{2}$ fluxes calculated by the eddy-covariance method and including the cospectral peak method criteria. At low wind speeds, the transfer velocity in the shallow water estuary was lower than in other coastal waters, possibly a symptom of low tidal amplitude leading to low intensity water turbulence. High transfer velocities were recorded above wind speeds of 5 m $\mathrm{s}^{-1}$, believed to be caused by early-breaking waves and the large fetch (6.5 km) of the estuary. These findings indicate that turbulence in both air and water influences the transfer velocity.

1 Introduction

The influence of the global oceans on the carbon cycle is of great importance to modelling the atmospheric accumulation of $\mathrm{CO}_{2}$, and a better understanding of air–sea $\mathrm{CO}_{2}$ exchange is likely to reduce the uncertainty in global climate models (Olsen et al. 2005; McGillis and Wanninkhof 2006; Takahashi et al. 2009).

Coastal waters represent only 7–8 % of the global ocean area but are biogeochemically very active systems and account for 14–30 % of global marine primary production (Alvarez et al. 1999; Borges et al. 2004a). Coastal waters, and especially estuaries, are often omitted from global carbon models. However, Frankignoulle et al. (1998), Borges et al. (2006) and Cai (2011) have shown that the $\mathrm{CO}_{2}$ exchange in estuaries is an important component of the global carbon cycle, counterbalancing $\mathrm{CO}_{2}$ uptake in continental shelves.

Air–sea exchange of $\mathrm{CO}_{2}$ can be parametrized as,

$$\begin{aligned} F_{\mathrm{CO}_2} =k_{\mathrm{CO}_{2}} \; K\left( {\Delta p\mathrm{CO}_2 } \right) \end{aligned}$$

(1)

where, $F_{\mathrm{CO}_2 }$ is the corrected air–sea $\mathrm{CO}_{2}$ flux, $k_{\mathrm{CO}_{2}}$ is the transfer velocity, $K$ is the solubility of $\mathrm{CO}_{2}$, and $\Delta p\mathrm{CO}_{2}$ is the difference between the partial pressure of $\mathrm{CO}_{2}$ in the water and in the atmosphere (Weiss 1974; Wanninkhof et al. 2009).

$\Delta p\mathrm{CO}_{2}$ and $K$ can be determined with relatively high precision, and it is thought that the greatest uncertainty when modelling the air–sea flux is the parametrization of $k_{\mathrm{CO}_{2}}$. For example, the estimated global air–sea flux increases by 70 % when using the parametrization of Wanninkhof and McGillis (1999) as opposed to that proposed by Wanninkhof (1992) (Rutgersson et al. 2008). In coastal areas, the uncertainty in $k_{\mathrm{CO}_{2} }$ is even higher (Borges et al. 2004a; Rutgersson et al. 2008).

The transfer velocity is often parametrized as a function of wind speed, but it is also known to be affected by fetch, water currents, wave state, atmospheric stability, sea spray, surface films and rain (Upstill-Goddard and Frost 1999; Wanninkhof et al. 2009). Although earlier studies have tried to construct functions that include multiple parameters, the influence of the wave field still needs further validation, and this may lead to a more precise parametrization of the transfer velocity (Fangohr and Woolf 2007; Wanninkhof et al. 2009). Other studies indicate that the magnitude of $k_{\mathrm{CO}_{2}}$ is determined by turbulence in both the air and the water, making water-side convection and mixed-layer depth important controlling factors (Rutgersson and Smedman 2010; Rutgersson et al. 2011; Read et al. 2012).

Coastal waters are often more turbulent than open oceans because of increased bottom stress. Thus the exchange coefficient is likely to be larger for these waters than for the open oceans. On the other hand, it is argued that the air–sea exchange of gas in coastal waters is reduced by extra biological surface films and shorter fetch (Marino and Howarth 1993; Raymond et al. 2000; Borges et al. 2004a).

The large discrepancy in the transfer-velocity parametrizations could be explained by uncertainties in transfer-velocity measurements. The eddy-covariance method, hereafter the EC method, is considered to be the most direct method to determine the $\mathrm{CO}_{2}$ flux used for determining transfer velocities. This method is challenging as it requires high frequency, high precision measurement of atmospheric gas concentrations and the correction of measured fluxes for the influences of water vapour and temperature. At sea, measurements of turbulent velocity can be influenced by platform motion, whilst local airflow can be distorted by the set-up platform or vessel (Fairall and Larsen 1986; Wanninkhof et al. 2009; Griessbaum et al. 2010).

In the present work the aim is to obtain a reliable parametrization of the air–sea $\mathrm{CO}_{2}$ gas transfer velocity for a shallow water estuary. The transfer velocities are determined from measurements of $\Delta p\mathrm{CO}_{2}$ and the turbulent flux of $\mathrm{CO}_{2}$, with data filtered in order to increase the quality of the dataset. The transfer velocity is parametrized as a function of wind speed or the closely related friction velocity $u_*$ to account for both turbulence and sea state (Wanninkhof et al. 2009) furthermore it is based on two datasets. One dataset using fluxes determined solely with the EC method, and a second dataset applying quality criteria from the cospectral peak method, hereafter the CSP method (Sørensen and Larsen 2010). The reported transfer velocities are compared to earlier work, and possible explanations are offered to account for the observed differences.

2 Sampling and Analytical Methods

2.1 Location and Measuring Period

$p\mathrm{CO}_{2},\, \mathrm{CO}_{2}$ fluxes and meteorological data were collected from March to July 2012 in the inner part of Roskilde Fjord (Fig. 1) (55$^\circ$41.5N; 12$^\circ$4.92E). Roskilde Fjord is situated in the northern part of Zealand, Denmark, and flows into the Kattegat through the Isefjord. It is a semi-closed water area throughout which seawater mixes with fresh water from rivers and land to yield a salinity of 7–19 $\permille$. Roskilde Fjord can therefore be categorized as an estuary (Conley et al. 2000). The northern and southern parts of the estuary are separated by a low threshold that results in a long residence time of the water in the southern part (Flindt et al. 1997; Fauser et al. 2009). Roskilde Fjord can have markedly heterogeneous waters (Flindt et al. 1997); thus, the collected data only represent the inner part of the estuary known as Roskilde Bredning (Table 1).

Fig. 1

Map of the study site in Roskilde Fjord, Denmark. The black dot marks the location of the pier from which data were collected

Table 1 Physical data for Roskilde Fjord and Roskilde Bredning (Flindt et al. 1997; Clarke et al. 2003; Fauser et al. 2009; DMI 2011)

2.2 Partial Pressure of $\mathrm{CO}_{2}$ in the Air

Air is sampled continuously from the pier sketched in Fig. 2 and the concentration of $\mathrm{CO}_{2}(\mathrm{C})$ is determined using a Licor LI-6262 $\mathrm{CO}_{2}/\mathrm{H}_{2}\mathrm{O}$ closed path infrared analyzer. The concentration and partial pressure of $\mathrm{CO}_{2}$ are calculated using relations from the manufacturer’s instruction guide. The air intake is placed above the pier, 2 m from the edge, and air is sampled for 10 min every half hour. Between each sample of air, nitrogen is sampled for 5 min. It is assumed that the air above the water surface is well mixed, both horizontally and vertically, and this assumption was tested during a cruise in October 2012 measuring air $p\mathrm{CO}_{2}$ with the PP Systems EGM-4 Environmental gas monitor for $\mathrm{CO}_{2}$, finding heterogeneity of $\pm$4 ppm at the entire Roskilde Fjord (data not shown).

Fig. 2

Top-down view of the instrumental set-up at the pier in Roskilde Bredning. The larger red dot marks the water intake for $p\mathrm{CO}_{2}$ sampling. The black line on the flux mast is the supporting boom. The smaller dot on the flux mast marks the location of the sonic anemometer and $\mathrm{CO}_{2}$ sensor, and the blue dot on the pier marks the air intake for the air $p\mathrm{CO}_{2}$

2.3 Partial Pressure of $\mathrm{CO}_{2}$ in the Water

To establish $p\mathrm{CO}_{2}$ in the water in Roskilde Bredning, a buoy with a water intake 0.2 m below the water surface was anchored 10 m from the edge of the pier (Fig. 2). It is assumed that the water is homogenous, thus the water at the buoy represents the surface water in the source area of the measured $\mathrm{CO}_{2}$ flux. The homogeneity was tested in April and July 2012; at both times a difference of $\approx {\pm }$10 $\upmu$atm was recorded between the water area at the pier and a water area 2 km north-west of the pier. If such heterogeneity existed during the entire measuring period, the uncertainty of $\Delta p\mathrm{CO}_{2}$ is estimated to be approximately 2 %. This uncertainty could increase during periods with a larger salinity gradient caused by intruding sea water or a high amount of run-off water from land. Furthermore a heterogeneous benthic community with patches of e.g. eelgrass could create a larger spatial variation of water $p\mathrm{CO}_{2}$ (Polsenaere et al. 2012).

At the pier, water for $p\mathrm{CO}_{2}$ measurements was pumped continuously through an insulated water hose into a shower-head equilibrator at a flow rate of between 70 and 120 l $\mathrm{h}^{-1}$ (see Hilligsøe et al. (2011) and Sejr et al. (2011) for details of the instrument design). Shower-head equilibrators reach equilibrium in a short amount of time and have low uncertainty (Bakker et al. 1997; Bates et al. 1998).

For 10 min once every half-hour, the air from the equilibrator is dried and analyzed in the same infrared analyzer (LI-6262) as was used for the atmospheric $p\mathrm{CO}_{2}$. After every 10 min of sampling equilibrated air, a sample of nitrogen follows to secure a consistent reference level.

2.4 $\mathrm{CO}_{2}$ Fluxes Using Eddy-Covariance and Cospectral Peak Methods

The EC method was employed for measuring air–sea $\mathrm{CO}_{2}$ fluxes. To increase the reliability of the calculated transfer velocity, the CSP method introduced by Sørensen and Larsen (2010) is used to identify and eliminate periods with potentially problematic $\mathrm{CO}_{2}$ fluxes.

The EC method is the most common direct method for measuring gas fluxes (Fairall et al. 2000). Fluxes are determined by correlating the turbulent fluctuations of a scalar concentration with the turbulent fluctuations of the vertical wind velocity. The flux of e.g. $\mathrm{CO}_{2}$ is then derived as

$$\begin{aligned} F_{\mathrm{CO}_2 } =\overline{w^{\prime }{\rho _c}^{\prime }}+\overline{w}\,\overline{\rho _c } \end{aligned}$$

(2)

where $w$ is the vertical wind component, $\rho _{c}$ is the mass density of $\mathrm{CO}_{2}$, the overbar denotes the mean, and the primes represent the deviation from the mean (Sørensen and Larsen 2010).

Flow distortion and platform movement influence the flux when using the EC method. To rectify this shortcoming, additional flux methods have been introduced: the inertial dissipation method (Fairall and Larsen 1986) and the CSP method (Sørensen and Larsen 2010). In the present study, the cospectra are analyzed according to the CSP method since the inertial dissipation method is sensitive to the choice of the Kolmogorov constant (Sørensen and Larsen 2010). The basic assumption of the CSP method is that a universal form for the cospectrum exists, and that the full integral, given the covariance, can be derived from the integral of a limited spectral range (Sørensen and Larsen 2010).

When investigating the flux of a scalar quantity e.g. $\mathrm{CO}_{2}\,(C)$, the flux can be estimated as

$$\begin{aligned} \overline{w^{\prime }C^{\prime }} \approx \frac{1}{0.25}nCo_{{wC}} (n)_\mathrm{peak} \end{aligned}$$

(3)

for 0.01 $\le$ normalized frequency $(f) \le$ 0.2 using the normalized cospectrum $n\,{Co}_{wC} (n)$, with $n$ being the frequency (Hz) (Sørensen and Larsen 2010), where the location of the peak is a function of stability (Norman et al. 2012) and where $f=nz/U$. In the present study, the CSP method is used to validate the $w^{\prime }C^{\prime }$ cospectra according to the universal spectrum presented by Kaimal et al. (1972) and Sørensen and Larsen (2010). The peak location in the frequency interval 0.01–0.2 Hz and peak magnitude of 0.25 in each sample’s cospectrum determine whether or not the sample needs to be eliminated.

On a mast at a pier in Roskilde Bredning, a horizontal boom with a three-dimensional sonic anemometer, Metek USA-1, and a $\mathrm{CO}_{2}$ sensor, the LI-7500 open-path $\mathrm{CO}_{2}/\mathrm{H}_{2}\hbox {O}$ analyzer, is mounted, reaching 1.2 m out across the water side. The sonic anemometer operates at a sampling rate of 20 Hz, and the LI-7500 determines mass density of $\mathrm{CO}_{2}$ and water vapour at the same frequency. The sensors are placed horizontally 0.5 m apart and 5 m above the average water level.

The fluxes used in this study are calculated using EddyPro 4.0.0 (2012) followed by post-processing of binned cospectra, where all spectral samples are averaged in exponentially-spaced frequency bins. The selected processing options and methods are shown in Appendix.

The calculated fluxes are based on half-hour sampling.

2.5 Meteorological Parameters

The Obukhov stability parameter $(z/L)$ is calculated from

$$\begin{aligned} L=\frac{{-T_\mathrm{p} u_*}^{3}}{\kappa g\frac{H_0 }{\rho _\mathrm{a} c_\mathrm{p} }} \end{aligned}$$

(4)

where $z$ denotes the measurement height and $L$ is the Obukhov length, and where $T_\mathrm{p}$ is the potential temperature, $\kappa$ is the von Karman constant $\approx$ 0.41, $g$ is the acceleration due to gravity $(\approx$9.81 m $\mathrm{s}^{-1}), H_{0}$ is the uncorrected ambient sensible heat flux, $\rho _\mathrm{a}$ is moist air density and $c_\mathrm{p}$ is moist air heat capacity.

Here $u_*$ is determined according to

$$\begin{aligned} u_*=\left( {\left( {\overline{u^{\prime }w^{\prime }}} \right) ^{2}+\left( {\overline{v^{\prime }w^{\prime }}} \right) ^{2}} \right) ^{1/4}, \end{aligned}$$

(5)

where $\overline{u^{\prime }w^{\prime }}$ and $\overline{v^{\prime }w^{\prime }}$ are the components of the kinematic momentum flux in the along-wind and in the cross-wind direction, respectively.

Wind speed is measured at 5 m $(U_{5})$, however $k$ is commonly parametrized as a function of the wind speed at 10 m $(U_{10})$, thus $U_{10}$ is calculated using the integrated Businger–Dyer relationships (Stull 1988),

$$\begin{aligned} U_z =\frac{u_*}{\kappa }\left[ {\hbox {ln}\left( {\frac{z}{z_0 }} \right) -\varPsi _M \left( {\frac{z}{L}} \right) } \right] \end{aligned}$$

(6a)

during stable conditions

$$\begin{aligned} \varPsi _M \left( {\frac{z}{L}} \right) =-4.7\left( {\frac{z}{L}} \right) \end{aligned}$$

(6b)

or during unstable conditions

$$\begin{aligned} \varPsi _M \left( {\frac{z}{L}} \right) =2 \ln \left[ {\frac{\left( {1+x} \right) }{2}} \right] +{\ln }\left[ {\frac{\left( {1+x^{2}} \right) }{2}} \right] -2\,\tan ^{-1}\left( x \right) +\left( {\frac{\pi }{2}} \right) \end{aligned}$$

(6c)

where

$$\begin{aligned} x=\left[ {1-\left( {15\frac{z}{L}} \right) } \right] ^{1/4}. \end{aligned}$$

(6d)

Here, $\varPsi _M$ is the integrated form of the dimensionless wind shear, $\phi _{m}, U_{z}$ is wind speed at height $z$ and $z_{0}$ is the aerodynamic roughness length (Businger et al. 1971). This is calculated using the traditional approach based on sonic-measured wind speed at 5 m $(U_{5}), u_*$ and the calculated $z/L$. The measuring height from sea level to the sensors varies with up to 1.5 m depending on tide, waves and amount of water held back by the wind. The maximum change of $U_{10}$ caused by this sea level change is equal to $\pm$0.25 m $\mathrm{s}^{-1}$ and is not taken into account.

2.6 Data Quality Control and Filtering

To ensure that only reliable $\mathrm{CO}_{2}$ fluxes and $\Delta p\mathrm{CO}_{2}$ values are used, the dataset is refined using nine criteria listed in Table 2. Based on criterion 1, samples are eliminated if $w$or $\mathrm{CO}_{2}$ data are missing or if the LI-7500 is clearly affected by high relative humidity or rain. Criterion 2 eliminates samples collected when airflow is from the land sector (wind direction 000$^\circ$–225$^\circ$), or from directions that lead to pier-induced flow distortion (wind direction 225$^\circ$–250$^\circ$ and 290$^\circ$–360$^\circ$). Due to set-up limitations, this criterion eliminates a very large percentage of the samples ($>$70 %), however, tests have shown that if all wind directions from the water sector are included and flow distortion thereby ignored, uncertain samples are eliminated in the subsequent criteria (numbers 3–6). Samples are eliminated by criterion 3 if the calculated sample-wise sonic pitch angle exceeds 5$^\circ$, indicating a change in the second rotation angle caused by the edge of the pier. The specifics of the experimental set-up when measuring EC fluxes determines the degree of flow distortion and sonic pitch angle (Dellwik et al. 2010), and the amount of data eliminated in criteria 2 and 3 could be changed with a change of mast location or height. If samples do not pass a test of steady state and flux-variance similarity and thereby do not apply to the assumptions of eddy covariance, they are flagged according to the 0-1-2 system created by Mauder and Foken (2004), and only samples flagged 0 are retained, as required by criterion 4. The flux-variance similarity test defines how well turbulence is developed and poorly developed turbulence can be a result of set-up-induced additional mechanical turbulence. Non-stationarity can be caused by e.g. changing meteorological fields. Under stable atmospheric conditions the along-wind distance that corresponds to 90 % of the flux becomes very large (in this case modelled according to Kormann and Meixner (2001) as exceeding 5.5 km), making the assumption of homogeneity in the footprint area more unlikely. Therefore, samples are eliminated when $z/L > 0.15$ (criterion 5). Samples are eliminated by criterion 6 if the equilibrator was clogged or if $\Delta p\mathrm{CO}_{2}$ data were lacking. This problem occurs when measuring with a continuously running shower-head equilibrator (Bakker et al. 1996). Furthermore, when $\Delta p\mathrm{CO}_{2}$ is close to zero, the calculation of the transfer velocity becomes more sensitive to error, and the heterogeneity of the footprint area has a higher significance (Borges et al. 2004a). As mentioned in Sect. 2.3, the heterogeneity was $\approx {\pm }$10 $\upmu$atm, and to reduce the effect of this, samples are eliminated if the absolute value of $\Delta p\mathrm{CO}_{2} < 30\, \upmu$atm (criterion 7). A possible heterogeneity in the footprint area could e.g. be caused by a spatial heterogeneity of the benthic community (Fenchel and Glud 2000).

Table 2 The criteria used to eliminate unreliable measurements

Criteria 8 and 9 evaluate the $w^{\prime }C^{\prime }$ cospectra. Samples are eliminated if the peak is not located between 0.01 and 0.5 Hz (criterion 8), as suggested by Sørensen and Larsen (2010) and Norman et al. (2012), or if the maximum of the peak is outside the interval 0.15–0.35 $n{Co}_{wC}(n)/C_*u_*$ (criterion 9). If these two criteria are not met the cospectrum does not agree with the universal cospectrum proposed by Kaimal et al. (1972). An example of an accepted and a rejected cospectrum are shown in Fig. 3, the peak in Fig. 3a is located as described by criteria 8 and 9, while the peak of the rejected cospectrum (Fig. 3b) is located at $f = 0.01$ with $n{Co}_{wC}(n)/C_*u_*= 0.6$. The deviation in the low frequency part of the cospectrum is expected to be caused by flow distortion or advected gravity waves (Stull (1988)), whereas deviations in the higher frequencies are generated by white noise (El-Madany et al. 2013).

Fig. 3

Normalized $w^{\prime }{C^{\prime }}_{2}$ cospectra $n{Co}_{Cw}(n)/C_*u_*$ (black diamonds) plotted on logarithmic x and y axes with the running mean fit using a window width of five (black bold line) and the universal cospectrum suggested by Kaimal et al. (1972) (grey line): a An example of an accepted cospectrum (March 10, 2034 UTC) and b an example of a rejected cospectrum (March 13, 0341 UTC)

Eliminations based on criteria 1 through 7 result in a dataset (DS1) containing 157 samples (3 % of the original samples), and when adding criteria 8 and 9, a second dataset is formed (DS2) containing only 69 valid samples (1 % of the original samples) (Table 2).

To estimate the importance of using the CSP method (criteria 8 and 9) as a quality check, parametrizations of $k_{\mathrm{CO}_{2}}$ are determined based on both DS1, which is not filtered by the CSP method, and on DS2, in which a larger amount of instrumental and set-up induced error is eliminated by taking the CSP method into account.

Outliers in both datasets are ultimately eliminated by Peirce’s criterion (Peirce 1852; Ross 2003). This method uses mean, standard deviation and the ratio of the maximum allowable deviation of a measured value from the mean to the standard deviation ($R$ value) to eliminate one or several doubtful observations.

2.7 Calculation and Parametrization of $\mathrm{CO}_{2}$ Transfer Velocity

When rearranging Eq. 1, the transfer velocity of $\mathrm{CO}_{2}$ can be calculated as (Upstill-Goddard et al. 1990; Jacobs et al. 2002)

$$\begin{aligned} k_{\mathrm{CO}_2 } =\frac{F_{\mathrm{CO}_2 } }{K\Delta p\mathrm{CO}_{2} } \end{aligned}$$

(7)

where $K$ is determined according to Weiss (1974). The salinity is kept at 10 $\permille$, and the water temperature is measured as water enters the equilibrator.

$k_{\mathrm{CO}_2 }$ is normalized to fresh water at a temperature of 20 $^\circ$C, using the Schmidt number (Sc) proposed by Wanninkhof (1992) and Raymond et al. (2000), where $T$ is the water temperature in $^\circ$C (Wanninkhof 1992; Raymond et al. 2000)

$$\begin{aligned}&k_{600} /k_{\mathrm{CO}_2 } =\left( {\frac{600}{{Sc}_{\mathrm{CO}_2 } }} \right) ^{-1/2}, \end{aligned}$$

(8a)

$$\begin{aligned}&Sc_{\mathrm{CO}_{2} } =1911.1-\left( {118.11\,T} \right) +\left( {3.4527\,T^{2}} \right) -\left( {0.04132\,T^{3}} \right) , \end{aligned}$$

(8b)

and $k_{600}$ is subsequently parametrized as a function of $u_*, U_{5}$ and $U_{10}$. As in similar studies, the parametrizations are determined by binning and averaging $k_{600}$ in 0.05 m $\mathrm{s}^{-1}$ intervals for $u_*$, and 1 m $\mathrm{s}^{-1}$ intervals for $U_{5}$ and $U_{10}$.

3 Results

3.1 Air–Sea Transfer Velocity of $\mathrm{CO}_{2} (k_{600})$

$k_{600}$ in Roskilde Bredning range from $-4.89$ to 1.72 mm $\mathrm{s}^{-1}$ with a mean and error of mean of 0.16 $\pm$ 0.5 mm $\mathrm{s}^{-1}$ for DS1 and $-0.79$ to 1.29 mm $\mathrm{s}^{-1}$ with a mean and error of mean of 0.24 $\pm$ 0.04 mm $\mathrm{s}^{-1}$ for DS2 (for comparison with earlier studies e.g. Raymond and Cole (2001) or Borges et al. (2004b) we hereafter use cm $\mathrm{h}^{-1}$ thus $k_{600}$ range from $-1{,}\!761$ to 618 cm $\mathrm{h}^{-1}$ with a mean and error of mean of 59 $\pm$ 19 cm $\mathrm{h}^{-1}$ for DS1 and $-283$ to 465 cm $\mathrm{h}^{-1}$ with a mean and error of mean of 88 $\pm$ 14 cm $\mathrm{h}^{-1}$ for DS2).

Negative $k_{600}$ is not physically possible (Borges et al. 2004b), but the samples cannot be eliminated by any of the objective quality criteria listed in Table 2. Negative $k_{600}$ is found when $\mathrm{H}_{2}\mathrm{O}$ fluxes are high (latent heat flux (LE) $>$ 100 W $\mathrm{m}^{-2})$, indicating that the $\mathrm{CO}_{2}$ fluxes used to calculate the $k_{600}$ in these cases are affected by $\mathrm{H}_{2}\mathrm{O}$ as shown in Fig. 4. Earlier studies of air–sea $\mathrm{CO}_{2}$ fluxes have shown a discrepancy in EC measured $\mathrm{CO}_{2}$ fluxes due to cross correlation between $\mathrm{CO}_{2}$ and $\mathrm{H}_{2}\mathrm{O}$ as a result of hygroscopic particles contamination on the sensor head (Kohsiek 2000; Prytherch et al. 2010). In four samples high $\mathrm{H}_{2}\mathrm{O}$ fluxes are not responsible for negative transfer velocities and these episodes are connected with high sensible heat fluxes $(H >$ 140 W $\mathrm{m}^{-2})$. The tendency is not seen in DS1, and the elimination of samples having $LE > 100~\mathrm{W\,m}^{-2}$ is only introduced in DS2.

Fig. 4

Correlation between corrected $\mathrm{CO}_{2}$ fluxes $(F_{\mathrm{CO}_2 } )$ and latent heat fluxes (LE) in dataset 2 (DS2) with the negative transfer velocities marked

The best fit in the DS1 parametrizations is a power-law regression with $U_{5}$ (Table 3). The tendency shown in Fig. 5 is for $k_{600}$ to increase with increasing wind speed, making a linear or exponential function a reasonable regression type. The power-law fit is preferred here since it explains the greatest amount of variation in the dataset.

Fig. 5

a Parametrization of the transfer velocity of $\mathrm{CO}_{2} (k_{600})$ by measured wind speed at 5 m using dataset 1 $(U_{5}1)$. Error bars and shaded area represent the 95 % confidence level of each wind class mean and power fit, respectively. b Distribution of used data points

Table 3 List of calculated bin-average parametrizations of $k_{600}$ in Roskilde Bredning

All three parametrizations $(U_{5}1, U_{10}1$ and $u_*1)$ are characterized by large scatter in the data, but the scatter is lowest in $U_{5}1$ due to the binning in $U_{5}$ which eliminates, in this case, many of the negative $k_{600}$ values as outliers. To compare $U_{5}1$ with earlier work the $U_{5}1$ parametrization is converted to a $U_{10}$ parametrization, still using the bins and eliminating the outliers from $U_{5}1$. This gives a parametrization based on DS1 with a $R^{2} = 0.51$ and $p$ value $=$ 0.045

$$\begin{aligned} k_{600} ={2.9U_{10}}^{1.3}. \end{aligned}$$

(9)

In DS2 the tendency is for $k_{600}$ to increase with increasing wind speed until 13 m $\mathrm{s}^{-1}$ followed by a decrease in $k_{600}$ from 13 to 17 m $\mathrm{s}^{-1}$ (data not shown), which is in direct conflict with common theory (Liss and Merlivat 1986). The low $k_{600}$ at high wind speeds may be caused by the increased influence from sea spray and flow distortion (Wyngaard 1988; Grelle and Lindroth 1994; Griessbaum et al. 2010). Thus, the decrease of $k_{600}$ in DS2 with increasing wind speed showed that the set-up in Roskilde Bredning can only be used at $U_{10}< 13 \,\mathrm{m}\,\mathrm{s}^{-1}$. With this in mind, the parametrization with the highest level of significance is given by a power function based on $U_{10}$ inferred from $u_*$ and measured $U_{5} (U_{10}2)$ as shown in Fig. 6. The parametrization in Eq. 10 gives a $R^{2} = 0.86$ and $p = 0.003$

$$\begin{aligned} k_{600} ={0.3U_{10}}^{2.5}. \end{aligned}$$

(10)

$U_{10}2$ and $u_*2$ are very similar in $R^{2}$ and the $p$ value, however $U_{10}$ is once again selected instead of $U_{5}$ and $u_*$ because it allows comparison of the results to those reported in earlier work.

Fig. 6

a Parametrization of the transfer velocity of $\mathrm{CO}_{2} (k_{600})$ by calculated wind speed at 10 m using dataset 2 $(U_{10}2)$ after the elimination of the samples with $\mathrm{CO}_{2}$–$\mathrm{H}_{2}\mathrm{O}$ cross-correlation and high wind speeds. Error bars and shaded area represent the 95 % confidence level of the wind-class mean and power fit, respectively. b Distribution of used data points

The deviations in all parametrizations imply that $k_{600}$ is not solely a function of $u_*$ or wind speed and that experimental uncertainties or unfulfilled assumptions still exist (Jacobs et al. 1999; Nightingale et al. 2000; Weiss et al. 2007). The variability with wind speed in $U_{10}2$ is of similar size or smaller than in the studies by Raymond and Cole (2001), Weiss et al. (2007) and Borges et al. (2004b), with an $R^{2}$ of 0.53, 0.81 and 0.95, respectively. The $U_{10}$ parametrization in Eq. 10 is therefore deemed equally solid as the three above-mentioned studies. However, it should be noted that the number of binning intervals in the present study, and thus the size of the datasets, are smaller than those employed by Weiss et al. (2007) and Borges et al. (2004b). This is due to the shorter measuring period, as well as the quality check and filtering of the data samples.

3.2 $\Delta p\mathrm{CO}_{2}$ and $\mathrm{CO}_{2}$ Fluxes

The new parametrization of $k_{600}$ can be used when determining the $\mathrm{CO}_{2}$ exchange and $\mathrm{CO}_{2}$ budget in coastal areas such as Roskilde Fjord. In the current study Eqs. 1 and 10 have been used to calculate the average $\mathrm{CO}_{2}$ flux of Roskilde Bredning in the period for which water-side $\Delta p\mathrm{CO}_{2}$ and wind-speed data were available from March to July 2012. Figure 7 shows time series of measured water $p\mathrm{CO}_{2}, \Delta p\mathrm{CO}_{2}$ and calculated $\mathrm{CO}_{2}$ flux. Roskilde Bredning is mainly a sink of $\mathrm{CO}_{2}$ from March to July with a mean $\mathrm{CO}_{2}$ flux and standard deviation of the mean of $-0.7 \pm 0.03\, \upmu \mathrm{mol} \,\,\mathrm{m}^{-2}\, \mathrm{s}^{-1}$ and $\Delta p\mathrm{CO}_{2}$ of $-91 \pm 2.7\, \upmu$atm. It seems reasonable to conclude that the area is a $\mathrm{CO}_{2}$ sink in the spring and early summer as indicated for similar estuaries (Algesten et al. 2004; Zemmelink et al. 2009). The $\mathrm{CO}_{2}$ flux is equivalent to a primary production rate of approximately 269 g C $\hbox {m}^{-2} \hbox {y}^{-1}$, which makes the primary production in the inner part of Roskilde Fjord at the average level of other Danish estuaries (Conley et al. 2000).

Fig. 7

Measured water $p\mathrm{CO}_{2}, \Delta p\mathrm{CO}_{2}$ (the secondary y axis) and calculated $\mathrm{CO}_{2}$ flux $(F_{\mathrm{CO}_2 } )$ (the primary y axis) in Roskilde Bredning from March to July 2012. Data are lacking in May, thus we add a break of the x axis in this period. $F_{\mathrm{CO}_2 }$ are calculated using Eqs. 1 and 10

The diurnal cycle of the $\Delta p\mathrm{CO}_{2}$ with the lowest values during daytime and the highest during nighttime is the opposite of that expected from temperature variations (Kuss et al. 2006); thus, it is concluded that the variation is caused by biological production and respiration with modification from both mixing and heating/cooling.

4 Discussion

4.1 Quality and Application

$\Delta p\mathrm{CO}_{2}$ in the present study resembles data from the Baltic Sea, but it is significantly smaller than those registered in Randers Fjord, Denmark and the Scheldt off the Belgian coast (Table 4). These differences could be due to physical influences, such as longer residence time of the water, higher input of ground water and non-ventilated water from rivers in the Scheldt and Randers Fjord or different biological dynamics (Marino and Howarth 1993; Borges et al. 2006).

Table 4 List of $\mathrm{CO}_{2}$ data from studies comparable to Roskilde Bredning

The mean $\mathrm{CO}_{2}$ flux of Roskilde Bredning is less than one third of the $\mathrm{CO}_{2}$ flux reported in the Danish Wadden Sea (Table 4), which, with regard to water depth and geographical location, is an area similar to Roskilde Fjord. The difference could be caused by the different $\Delta p\mathrm{CO}_{2}$ indicating that the Danish Wadden Sea has a higher primary production than Roskilde Bredning. However, Roskilde Bredning is a greater sink of $\mathrm{CO}_{2}$ than both Randers Fjord or the Scheldt (Borges et al. 2004a). The discrepancy may also be due to the different experimental approach. The chamber method is known to shield the water surface, thereby preventing wind-created turbulence, resulting in smaller $\mathrm{CO}_{2}$ fluxes (Liss and Merlivat 1986). The modelled $\mathrm{CO}_{2}$ flux of the Baltic Sea by Norman et al. (2013) shows an uptake of $\mathrm{CO}_{2}$, but the average of Roskilde Bredning is more than five times larger than the maximum uptake of the Baltic Sea. The $\mathrm{CO}_{2}$ flux estimated using the EC method in the Baltic Sea in May 2004 (Weiss et al. 2007) is within the same range as Roskilde Bredning; thus, the size of the calculated $\mathrm{CO}_{2}$ flux in the present study does not seem unrealistic for a coastal area.

The parametrizations presented in Eqs. 9 and 10 differ from those of earlier work shown in Fig. 8, however there is a large spread in the already existing parametrizations of $k_{600}$. Applying criteria 8 and 9 has a notable impact on the result, which would not have become apparent if only DS1 had been analyzed.

Fig. 8

Parametrizations of $\mathrm{CO}_{2}$ transfer velocity $(k_{600})$ by wind speed in dataset 1 (Eq. 9) and dataset 2 (Eq. 10) compared to earlier work. The parametrizations are calculated using floating chambers (Marino and Howarth 1993; Raymond and Cole 2001; Borges et al. 2004b), eddy covariance (Weiss et al. 2007) or the deliberate and natural tracer methods (Upstill-Goddard et al. 1990; Clark et al. 1994; Carini et al. 1996; Raymond and Cole 2001; Kuss et al. 2004)

Equation 9 increases rapidly at low wind speed values, and at 4 m $\mathrm{s}^{-1}$, it predicts a greater value of $k_{600}$ than any of the other parametrizations considered. A less strong increase at low wind speeds is seen in Eq. 10; at $U_{10} >$ 5 m $\mathrm{s}^{-1}$, Eq. 10 shows the greatest $k_{600}$ of all of the parametrizations, and Eq. 9 is 50 % lower at $U_{10}$ = 12 m $\mathrm{s}^{-1}$. Equation 10 is similar to that of Raymond and Cole (2001) but with a larger increase at lower wind speeds, resulting in an offset of approximately 1.5 m $\mathrm{s}^{-1}$ between the two parametrizations. Equation 9 from DS1 is similar to Kuss et al. (2004) or to Borges et al. (2004b), but when including the elimination based on the CSP method, it is argued that Eq. 9 overestimates the low wind speed $k_{600}$ and greatly underestimates the high wind speed $k_{600}$.

The difference of Eq. 10 to the other parametrizations can both be a result of methodological or physical variations between the studies or research sites. The parametrizations shown in Fig. 8 are calculated using either floating chambers (Marino and Howarth 1993; Raymond and Cole 2001; Borges et al. 2004b), EC (Weiss et al. 2007) or the deliberate and natural tracer methods (Upstill-Goddard et al. 1990; Clark et al. 1994; Carini et al. 1996; Raymond and Cole 2001; Kuss et al. 2004). The use of floating chambers when measuring $\mathrm{CO}_{2}$ fluxes has been shown to disturb the turbulence regime and presented limited agreement with other flux measuring methods (Broecker and Peng 1984; Belanger and Korzun 1991). Furthermore floating chambers give a snapshot in time and cover a small water-surface area (Raymond and Cole 2001). The mass-balance techniques using tracers have uncertainties caused by variations in the mixed-layer depth, shifts in wind speed over the course of the observations and difficulties accounting for the biological effect on the natural tracers (Wanninkhof et al. 2009). Although the dataset presented here is a small sample size, the $\mathrm{CO}_{2}$ flux and $\Delta p\mathrm{CO}_{2}$ data are of high quality and exclude errors due to non-ideal conditions.

The transfer velocity is not only controlled by wind speed, and some fraction of the differences between the parametrizations may be due to the physical differences between the study sites. The earlier studies were conducted in either open water systems such as the Arkona Sea (Weiss et al. 2007) and the Baltic Sea (Kuss et al. 2004) with water depths of 45 and 27–249 m, respectively, or river systems such as the Hudson River (Marino and Howarth 1993; Clark et al. 1994) with a water depth of 14 m, width of 800 m and large influence from tidal currents. The parametrization of $k_{600}$ in Roskilde Bredning is characterized by low k values at wind speeds $<$2 m $\mathrm{s}^{-1}$ and increasing steeply at wind speeds above 5 m $\mathrm{s}^{-1}$. We speculate that this pattern could be a result of low tidal amplitude, which cause less mixing and turbulence at low wind speeds as observed in other Danish fjords (Borges et al. 2004a). The relatively large fetch combined with shallow depth could cause early wave breaking and high whitecap coverage contributing to the steep increase in $k_{600}$ (Upstill-Goddard and Frost 1999; Nightingale et al. 2000; Borges et al. 2004a). This hypothesis was unfortunately not tested through the study period but a comparison between measured and modelled significant wave heights ($H_\mathrm{s}$) in Roskilde Bredning, October 2012, showed that measured $H_\mathrm{s}$ are a factor of 10 smaller than the modelled $H_\mathrm{s}$ (using the Pierson–Moskowitz value $H_\mathrm{s} = 0.02466 {U_{10}}^{2}$ (Carter 1982)). The low $H_\mathrm{s}$ in Roskilde Bredning indicates high whitecap coverage as a result of depth-limited breaking waves (Battjes and Janssen 1978; Katsardi et al. 2012).

The $\mathrm{CO}_{2}$–$\mathrm{H}_{2}\hbox {O}$ cross correlation, contaminating the measured $\mathrm{CO}_{2}$ fluxes and thereby causing negative $k_{600}$, can be eliminated by application of the PKT correction (Prytherch et al. 2010), although the PKT correction can be problematic at small $\mathrm{H}_{2}\hbox {O}$ fluxes (Else et al. 2011; Huang et al. 2012), as during our measurement period. For these reasons it is estimated that the cross correlation is small and the PKT correction was therefore not applied to the $\mathrm{CO}_{2}$ fluxes.

Equation 10 is a parametrization based on $\mathrm{CO}_{2}$ flux and $\Delta p\mathrm{CO}_{2}$ data where all theoretical assumptions are attempted to be met and methodological uncertainty is minimized, and is thus proposed for application to shallow estuaries. The data screening criteria used are recommended for use in studies requiring the highest quality flux data. Criteria 8 and 9 should change with changing stability as atmospheric stability is known to change the location of the peak in the normalized cospectra (Kaimal et al. 1972; Norman et al. 2012). During stable conditions it is expected that the peak of the normalized cospectrum is located at higher frequencies than in neutral or unstable conditions (Norman et al. 2012). The importance of different stabilities was not possible to examine since the atmosphere was near neutral in the investigated period.

In future studies of the $\mathrm{CO}_{2}$ budget concerning shallow water estuaries, Eq. 10 will give a calculated air–sea $\mathrm{CO}_{2}$ flux resembling the actual $\mathrm{CO}_{2}$ flux to a greater extent than earlier used parametrizations. The difficulty of choosing the correct parametrization in a shallow estuary is discussed in Zemmelink et al. (2009) and Eq. 10 would be a better choice than parametrizations based on data from rivers or the open sea.

4.2 Uncertainties

It is often difficult and expensive to measure air–sea fluxes by the EC method and many set-ups introduce flow or motion distortion (Sørensen and Larsen 2010; Griessbaum et al. 2010). Most of the motion distortion can easily be eliminated in coastal regions if measurements are conducted from a fixed platform, such as a pier, or directly from a mast mounted on the coastline. But depending on the shore line such set-ups can introduce flow distortion, which, in the present study, may result in a large number of uncertainties. The effect of the flow distortion was eliminated to the greatest extent possible, but it did not eliminate the decrease in $k_{600}$ at high wind speed. This can only be avoided by changing the set-up at the pier.

An uncertainty introduced in most coastal areas is the water heterogeneity that can cause the measured water $p\mathrm{CO}_{2}$ to differ from the $p\mathrm{CO}_{2}$ in the flux footprint area. The size and location of the footprint area is dependent on atmospheric stability, wind speed and wind direction, where stable conditions and high wind speeds lead to a larger footprint area. The water heterogeneity was investigated twice during the study period, with results that indicate very homogenous water conditions. However, the homogeneity was only examined by measurements at two points in space. Ideal conditions for determining $k_{600}$ would be a footprint area with the highest contribution to the measured flux located at the water intake for the $p\mathrm{CO}_{2}$ measurements. In the present study the highest contribution to the measured flux is on average located $\approx$130 m from the water intake as modelled according to Kormann and Meixner (2001) or Kljun et al. (2004). A heterogeneous footprint area can furthermore affect the spectra, as is seen with $u$ and $v$ in Murthy et al. (2009), where large eddies in the stable atmosphere have a long memory and have to travel far in homogenous terrain to reach equilibrium with the surface.

In addition to heterogeneous waters, the heterogeneous surrounding terrain at a coastal site might have an effect on the measured $\mathrm{CO}_{2}$ flux as air masses can be advected from land (Leinweber et al. 2009). The study site is mainly surrounded by agricultural land but is located 5 km north of the city Roskilde (50,000 population), 3 km north-east of 6 $\hbox {km}^{2}$ forest and 200 m south-west of the research facility DTU Risø. Advected air masses from Roskilde and DTU Risø are seen in the wind directions 000$^\circ$–065$^\circ$ and 110$^\circ$–180$^\circ$ where the average atmospheric $p\mathrm{CO}_{2}$ values shown in Fig. 9a are above 390 $\upmu$atm. The effect of the agricultural area east of the mast can be seen by low and highly variable atmospheric $p\mathrm{CO}_{2}$ in the wind sector from 065$^\circ$–110$^\circ$ (Fig. 9a, b). The similar pattern in the south-east wind sector is estimated to be advected air from the forest areas. The wind directions used to calculate $k_{600}$ are not affected by the surrounding non-water terrain, since this wind sector has a low average atmospheric $p\mathrm{CO}_{2}$ with a low amount of variation, hence less pronounced diurnal and seasonal variations (Leinweber et al. 2009).

Fig. 9

Atmospheric $p\mathrm{CO}_{2}\, (p\mathrm{CO}_{2\mathrm{Air}})$ and wind direction in 10$^\circ$ wind spans where red represent the wind directions used to find $k_{600}$: a wind-span mean; and b wind-span standard deviation

5 Conclusions

We estimate the air–sea transfer velocity of $\mathrm{CO}_{2}$ in a shallow estuary $(k_{600})$, and present a data screening method that increases the quality of data upon which parametrizations of $k_{600}$ in coastal regions are based. The data screening resulted in two different datasets (DS1 and DS2) from which two different $k_{600}$ parametrizations were derived.

Values of $k_{600}$, based on measurements made between March and July 2012 in an estuary (Roskilde Bredning), were generally higher than estimates reported in earlier work for coastal waters. It is proposed that these higher values of $k_{600}$ are caused by early-breaking waves produced by the shallow water (3 m) and large fetch (6.5 km) at Roskilde Bredning. At low wind speeds, $k_{600}$ was found to be lower than most other parametrizations, which could relate to low intensity water turbulence in the estuary unrelated to wind action (e.g. tidal currents).

The low $k_{600}$ at low wind speeds and the rapid increase with increasing wind speed were only evident in dataset DS2. This supports the conclusion that $\mathrm{CO}_{2}$ fluxes from which $k_{600}$ is calculated must be critically evaluated and, if possible, complimented by a second flux method e.g. the cospectral peak method to achieve a robust parametrization. Furthermore, the sample cospectra quality control based on the cospectral peak method indicates that the set-up at the pier does not generate reliable $k_{600}$ values at wind speeds above 13 m $\mathrm{s}^{-1}$, and suggests that the physically incongruous negative $k_{600}$ values are a result of the $\mathrm{CO}_{2}$–$\mathrm{H}_{2}~\mathrm{O}$ cross-correlation effect on $\mathrm{CO}_{2}$ fluxes determined from eddy covariance.

These results suggest that the choice of parametrization, when calculating a $\mathrm{CO}_{2}$ flux in an estuary, is of even greater importance than was previously thought. When selecting a parametrization, it is therefore necessary to ensure sufficient similarity between study sites with respect to e.g. water depth, fetch, and size of tide. This can only be achieved when parametrizations are dependent not only on wind speed, but also on additional variables.

Appendix

In the Advanced Mode of EddyPro 4.0.0 the settings shown in Table 5 are chosen using the listed methods.

Table 5 The chosen processing options and methods used in the calculation of $\mathrm{CO}_{2}$ fluxes using EddyPro, version 4.0.0