EC determines the turbulent flux (F) by calculating the covariance between the vertical wind velocity (w) and concentration of the gas (c): F=w′c′‾, where the macron denotes the time average and the primes denote fluctuations from the mean (x′=xt-x‾, x‾= mean). By convention, a positive flux is upward (emission) and a negative flux is downward (deposition).

With an assumption that turbulent transport is analogous to molecular diffusion (Baldocchi et al., 1988), the flux–gradient theory is theoretically described as follows: F=-KczdC/dz, where Kc is the eddy diffusivity for the gas, and dC/dz is the vertical concentration gradient of the gas. Two of the more popular methods for calculating Kc are the aerodynamic gradient method and the modified Bowen ratio approach.

The AGM method assumes that heat and mass are transported in a similar way within a well-developed surface layer (Erisman and Draaijers, 1995). Kc is related to the interstitial aerodynamic resistance (Ra) (Baldocchi, 1988) as Raz1:z2=∫z2z1dz/Kcz, where z1 and z2 indicate the heights of adjacent levels above canopy (z1>z2).

Using Eqs. (2) and (3), the deposition flux (F) is determined as F=-ΔCRaz1:z2=-C1-C2Raz1:z2, where C1 and C2 indicate the gas concentrations at z1 and z2, respectively.

Ra is calculated as Raz1:z2=(κu∗)-1ln⁡z1-dz2-d+ψhz1-dL-ψhz2-dL, where κ is the von Karman's constant (0.4), u∗ the friction velocity u∗≡-u′w′‾1/2 measured at the reference height, d the zero-plane displacement height, L the Obukhov length, and Ψh the integrated stability correction function for heat using those proposed by Businger et al. (1971) and modified by Högström (1988).

The MBR method is also based on the flux–gradient theory (Eq. 2), but the eddy diffusivity (Kc) is derived from flux and gradient measurements of another scalar (e.g., sensible heat, CO2, H2O) and assumes it is equal to Kc of the gas of interest. In this study, the flux and gradient measurements of CO2 are available at the same heights as O3, so Kc of O3 was calculated from the CO2 measurements as follows: Kc=Kco2=-Fco2Δz/ΔCCO2, where Kco2 is the eddy diffusivity of CO2, Fco2 is the eddy-covariance flux of CO2, ΔCCO2 is the concentration gradient of CO2 over the same height interval as ΔCO3, and Δz is the height interval of the concentration measurements.

Using Eqs. (2) and (6), the O3 flux (F) is calculated as F=Fco2ΔCO3/ΔCCO2.

The newly proposed MGM method is also based on the flux–gradient theory (Eq. 2). It is noted that the flux–gradient theory has long been questioned within the plant canopy due to infrequent but predominant large eddies within the canopy (Wilson, 1989; Raupach, 1989). For example, Bache (1986) suggested that the flux–gradient theory was a reasonable assumption for estimating wind profiles in the upper portion of the canopy but failed to reproduce the secondary wind maximum that was often observed within the trunk space of forests. It should also be noted that most O3 uptake occurs in the upper layers of the canopy where most canopy leaves grow. Within these upper layers the vertical length scales of turbulence are probably smaller than the distance associated with changes in concentration and wind speed gradients (Baldocchi, 1988). Thus, the flux–gradient theory is likely applicable for estimating the vertical flux distribution of air pollutants within a plant canopy as has been used in previous studies (e.g., Baldocchi, 1988; Bash et al., 2010; Wolfe and Thornton, 2011).

Applying the flux–gradient theory within the canopy, a height-dependent flux (F(z)) can then be calculated as F(z)=-Kc(z)dCdz, where z≤h, and Kc(z) is the vertical eddy diffusivity. Based on Eq. (8), the O3 flux at canopy top (F(h)) is defined as Fh=-Ch-C3Ra(h:z3), where Ch and C3 are the concentrations at canopy top (h) and the height of z3 (z3<h), respectively. Ra(h:z3) is related to Kc as Rah:z3=∫z3hdz/Kcz.

According to the aerodynamic gradient method (Eq. 4), the O3 flux above the canopy can be calculated from the concentration gradient between the reference height z1 and the canopy top h (z1 > h) as follows: F=-C1-ChRaz1:h. Based on the assumption of a constant flux layer above the canopy, the O3 flux above the canopy calculated in Eq. (11) should be equal to the O3 flux at the canopy top derived from Eq. (9). Using Eqs. (9) and (11), we can derive that F=-C1-C3Raz1:h+Rah:z3.

Raz1:h is calculated using Eq. (5). Rah:z3 is integrated vertically between the two heights within the canopy using Eq. (10).

Kc(z) is assumed to equal 0.8Km(z), which is the within-canopy eddy diffusivity for momentum transfer (Halldin and Lindroth, 1986). As described in Baldocchi (1988), Km(z) is determined as Kmz=∫0zCmzazuz2dzduz/dz, where a(z) is the leaf area density at height z, and u(z) is the horizontal wind speed within the canopy. Similar to Baldocchi (1988), Km(z) is assumed to be constant below crown closure (about 0.7h) and equal to Km at 0.7h. Thus we suggest here that the level of concentration measurement below canopy (z3) should not be lower that the crown closure of canopy.

The effective drag coefficient (Cm(z)) is assumed to be constant with height (see Thom, 1975) following Baldocchi (1988): Cmz=CamLAIum/uz12, where LAI is the canopy leaf area index, um the mean wind speed within canopy, and u(z1) the wind speed at the reference height z1. The bulk canopy drag coefficient (Cam) is computed as Cam=u∗2/u(z1)2. The mean within-canopy wind speed (um) is calculated as um=1/h∫0huzdz. The within-canopy wind speed profile (u(z)) follows Cionco (1972): uz=uhe-α1-z/h, where uh is wind speed at the canopy top, and α is wind speed attenuation coefficient. The above-canopy logarithmic wind profile is used to scale the wind speed measured at the reference height z1 to the canopy height h: uh=uz1ln⁡h-d-ln⁡z0+ψmh-d/L-ψmz0/Lln⁡z1-d-ln⁡z0+ψmz1-d/L-ψmz0/L, where z0 is the roughness length for momentum, and Ψm is the integrated stability correction function for momentum as proposed by Businger et al. (1971) and modified by Högström (1988).

Assuming a zero concentration on the absorbing surface, the dry deposition velocity (Vd) of O3 can be determined as Vd=-F/C(z1), where C(z1) is the O3 concentration measured at the reference height z1.

The HFEMS (42.54∘ N, 72.18∘ W) is located in central Massachusetts at an elevation of 340 m above sea level. The forest is 80 years old on average and consists of red maple (Acer rubrum) and red oak (Quercus rubra) with scattered stands of Eastern hemlock (Tsuga canadensis), red pine (Pinus resinosa), and white pine (Pinus strobus). The canopy height near the observation tower is up to 23 m with a peak LAI of ∼ 5.0 m2 m-2 during summer. The nearest sources of significant pollution are a secondary road about 2 km to the west of the site and a main highway about 5 km to the north.

A permanent 30 m Rohn 25G tower has been utilized at HFEMS to measure eddy-covariance fluxes of sensible heat, H2O, momentum, CO2, and O3 along with vertical profiles of CO2 and O3 since 1990 (Fig. 1). Eddy-covariance fluxes were measured at a height of 29 m above the ground. For the profile measurements air was continuously sampled from heights of 29, 24.1, 18.3, 12.7, 7.5, 4.5, 0.8, and 0.3 m a.g.l. to determine the concentrations of CO2 and O3. In this study, the upper three levels were used to derive the gradients. Details on the site and the instrumental methods can be found in Munger et al. (1996). Data used in this study are available online at http://atmos.seas.harvard.edu/lab/data/nigec-data.html.

Zhao et al. (2011) retrieved the vertical profile of leaf area density at Harvard Forest from a ground-based lidar scanning. Two tree species groups (i.e., hardwood and conifer) were chosen. According to the species composition around the measurement tower, the average leaf area density used in this study was calculated as 75 % of that of hardwood and 25 % of that of conifer from Zhao et al. (2011), as shown in Fig. 1.

The monthly averaged LAI at HFEMS was derived from the ground-based measurements for most years between 1998 and 2013 using the LICOR LAI-2000 system at 30–40 plots around the tower (Urbanski et al., 2007). As the measurements during January and February were not available, these values were obtained based on extrapolation (Fig. 2). The roughness length (z0) and displacement height (d) were calculated as a function of canopy height (h) and LAI, following Meyers et al. (1998) (see Fig. 2): z0=h(0.215-LAI0.25/10), d=h0.1+LAI0.2/2.

A total of 10 252 hourly measuring points, recorded at HFEMS during 1993–2000, were screened to eliminate the influence of periods associated with instrumental and measurement problems and violation of the use of the flux–gradient theory.

In order to reduce random measurement error in the concentration gradient, O3 concentrations below 1 ppbv were rejected, resulting in approximately 0.1 % of the data being omitted. In addition, periods with [O3] < [NOy] (1.9 %) were excluded to avoid periods when O3 chemical reactions may have exceeded O3 deposition (Munger et al., 1996). Wind speed below 1.0 m s-1 (1.2 %) and drag coefficient below 0.02 (6.6 %) were removed because of probable invalid flux–gradient relationships (Feliciano et al., 2001). Outliers in the data (2.9 %) were removed, omitting any deposition velocity exceeding the maximum achievable deposition velocity Vd,max⁡ (Vd,max⁡=1/Ra+Rb) by more than a factor of 1.5 (Matsuda et al., 2006). Periods with counter-gradient profiles (69.8 %) which represent a downward flux (from EC measurements) while with a negative gradient (upper level minus lower level) or vice versa were rejected (Park et al., 2014). The counter-gradient transport should be mainly due to the non-local nature of turbulent transport within canopies. Large sweep-ejection air motions, associated with coherent structures that can deeply penetrate into the canopy, are believed to be largely responsible for the exchange of momentum, heat, and mass between air above- and within-canopy (e.g, Shaw et al., 1983; Thomas and Foken, 2007). A total of 74.0 % of the data was omitted in the following analysis. This percentage value is slightly smaller than the sum of those from all the criteria due to the overlap of some data points between the criteria.

Figure 3 shows the mean diurnal cycles of O3 concentration at different heights derived from the original data set and from the data after selection. The O3 concentration typically increased during the early morning to reach a daily maximum of over 40 ppbv in the early afternoon and then decreased to ∼ 30 ppbv at night. As shown in Fig. 3a, the gradient between the two heights above the canopy (i.e., 29 and 24.1 m) was only about 0.4 ppbv on average, smaller than that between the levels above the canopy (24.1 m) and inside the canopy (18.3 m) (∼ 0.8 ppbv). The gradients were relatively small during the morning (e.g., 0.1 ppbv at 11:00 LST) compared to the other periods of the day. In the morning, the most effective turbulent exchange between the air above- and within-canopy would substantially reduce the gradients (Sörgel et al., 2011). It is worth mentioning that many earlier studies suggested that the effects of chemistry on O3 flux divergence in the near surface were generally small, likely because the chemical reactions for O3 have larger timescales than the turbulent transport (e.g., Gao et al., 1991; De Arellano and Duynkerke, 1992; Duyzer et al., 1997; Padro et al., 1998; Stella et al., 2012). After screening the data with the criteria, the gradients among these three levels were significantly larger, reaching up to 1.0 and 1.6 ppbv, respectively (see Fig. 3b).

O3 dry deposition velocity (Vd(O3)) measured by the eddy-covariance technique at Harvard Forest typically ranged from 0.14 to 0.53 cm s-1 with a median value of 0.30 cm s-1 during the study period (Table 1). Since the screened deposition velocities still include certain outlying data, the mean value was calculated using data between the 10th and 90th percentiles in order to reduce the influence of the outlying data. Following this approach, the mean Vd(O3) by the EC technique was 0.34 cm s-1, which was significantly smaller than the means calculated by the gradient methods (Table 1). The ratios of the mean Vd(O3) calculated by the modified gradient, modified Bowen ratio, and aerodynamic gradient methods to the mean of the EC technique were 1.18, 1.45, and 2.28, respectively. Previous studies on the inter-comparisons of these methods for O3 are few and the results varied. Muller et al. (2009) found that the mean Vd(O3) by the AGM method was 1.60–3.47 times that of the EC technique at a grassland in southern Scotland. Loubet et al. (2013) showed that the AGM method gave 40 % larger Vd(O3) than the EC technique over a mature maize field in Paris. Keronen et al. (2003) found that Vd(O3) by the AGM and EC methods generally agreed well at a Nordic pine forest, as did Stella et al. (2012) over bare soil in Paris. Droppo (1985) found close Vd(O3) values with the MBR and EC methods at a northeastern US grassland site.

Figure 4 shows the diurnal cycles of Vd(O3) by the EC and gradient methods. Although the trends were similar, the MBR and AGM Vd(O3) were consistently larger than the EC Vd(O3). The EC Vd(O3) was about 0.2 cm s-1 on average during the night and reached a daily maximum of 0.5 cm s-1 around noon. The Vd(O3) by the MBR and AGM methods reached around 0.8 and 1.3 cm s-1 during the daytime, respectively, and remained about 0.4 cm s-1 during the night. The MGM Vd(O3) agreed well with the EC Vd(O3) during the daytime but was slightly larger at night. This discrepancy has been identified in previous studies (Keronen et al., 2003; Stella et al., 2012) and could be due to the fact that nocturnal conditions affect both EC and gradient measurements. The EC technique has been found to underestimate flux during calm nighttime periods at Harvard Forest (Goulden et al., 1996). The stability correction functions used in the gradient methods (AGM and MGM) are subject to large uncertainties under stable conditions (Högström, 1988).

The very large differences in Vd(O3) between the AGM and EC methods are likely caused by a combination of various factors. As can be seen from Eq. (4), any underestimation in the calculation of aerodynamic resistance (Ra) would directly result in the overestimation of Vd. Uncertainties in Ra from using different formulas are generally on the order of 30 % over a whole canopy (Zhang et al., 2003). In the case of Eq. (4), uncertainties can be larger than 30 % if other uncertainties from the related parameters are larger. The potential underestimation of Ra (Eq. 4) also explains the small overestimation in Vd from the MGM method, in which the same Ra formula is used, although it plays a secondary role. Measurement uncertainties in the concentration gradients could also cause large discrepancies between the AGM and EC methods, especially under small gradient conditions. This is supported by the finding that the MBR method also overestimated Vd when compared to the EC measurements.

As shown in Fig. 5, the EC Vd(O3) exhibited a significant seasonal pattern with peak values in summer (∼ 0.5 cm s-1) and small values in winter (0.15–0.28 cm s-1). Both the MGM and MBR methods captured this seasonal cycle, but the MGM method produced a higher Vd(O3) than the EC technique during winter (December–February) and the MBR method produced a significant overestimation in summer (June–September). The monthly AGM Vd(O3) was consistently larger than the EC Vd(O3) and exhibited a less clear seasonal pattern with alternating increases and decreases in the Vd(O3).

As shown in Sect. 4.1, the MGM method performed better than the MBR and AGM methods. This improvement is mainly attributable to smaller errors in the O3 concentration gradients. However, the MGM method increased the complexity of the algorithm and added more model parameters, which may in turn increase the uncertainty of the estimated Vd(O3).

To test the sensitivity of the MGM Vd(O3) values to the key parameters/formulas, calculations were conducted by changing the parameters/formulas within a reasonable range. For some single-value parameters (i.e., roughness length, displacement height, wind speed attenuation coefficient, and leaf area index), sensitivity tests were conducted by increasing or decreasing the value by 10 %.

As shown in Fig. 6 and Table 2, the MGM Vd(O3) was highly sensitive to the changes in wind speed attenuation coefficient and displacement height. Higher wind speed attenuation coefficients could result in lower within-canopy wind speed (Eq. 17) and thus a lower eddy exchange coefficient and Vd(O3) (Table 2). Based on a least-square fitting of within-canopy wind profiles measured at Harvard Forest for noon periods in summer, the attenuation coefficient was estimated to be ∼ 10.6. Cionco (1972) suggested that the attenuation coefficient varies with leaf area. Therefore, the application of this value throughout the whole year could produce an uncertainty in the estimated Vd(O3).

The MGM Vd(O3) increased when the displacement height increased or vice versa (Fig. 6, Table 2). Sakai et al. (2001) calculated the displacement height at Harvard Forest using noon-period measurements and indicated the ratio of displacement height to canopy height was 0.77 in summer with foliated canopy and 0.6 in winter with leafless canopy. In this study, we estimated a similar value in summer (0.79) and a slightly higher value in winter (0.66) using the method proposed by Meyers et al. (1998) (Fig. 2). The overestimation of the displacement height could partly explain the overestimation of Vd(O3) by the MGM method during December to February (Fig. 5).

Figure 6 shows that the MGM Vd(O3) was less sensitive to the changes in roughness length and leaf area index. The relative differences in the estimated Vd(O3) were less than 2 % when the roughness length and leaf area index varied by 10 % (Table 2).

Meyers et al. (1998) provided three typical types of leaf area density profiles, which are significantly different in shape from the profile in Harvard Forest used in this study (see Fig. 7). We conducted sensitivity experiments by replacing the Harvard Forest profile with those of Meyers et al. (1998) to assess the impact of the vertical profile of leaf area density on the determination of Vd(O3). As shown in Fig. 6 and Table 2, the vertical profile of leaf area density impacted the estimated Vd(O3) greatly, with a relative difference in Vd(O3) of greater than 50 %. The profile with higher leaf density in the upper canopy (profile 3) resulted in a higher Vd(O3) while the profile with abundant understory plants (profile 1) led to a lower Vd(O3).

In this study, the stability correction functions proposed by Businger et al. (1971) and modified by Högström (1988) were used, but several others exist, such as those by Dyer (1974), Paulson (1970), and Webb (1970). Figure 6 indicated that uncertainties in the stability correction functions for heat (Ψh) and momentum (Ψm) had little impact on the MGM Vd(O3) values. The relative difference of Vd was less than 4 % for different Ψh and less than 1 % for different Ψm. Stella et al. (2012) found that the variation of Vd(O3) on different Ψh was roughly 10 % on average when using the AGM method. Ψh influences the estimation of Vd due to its impact on the calculation of turbulent transfer above the canopy. Since the MGM method considered both the above- and within-canopy turbulent transfer, the MGM Vd(O3) values were thus less sensitive to the choice of Ψh.