Using boundary layer equilibrium to reduce uncertainties in transport models and CO 2 flux inversions

This paper reexamines evidence for systematic errors in atmospheric transport models, in terms of the diagnostics used to infer vertical mixing rates from models and observations. Different diagnostics support different conclusions about transport model errors that could imply either stronger or weaker northern terrestrial carbon sinks. Conventional mixing diagnostics are compared to analyzed vertical mixing rates using data from the US Southern Great Plains Atmospheric Radiation Measurement Climate Research Facility, the CarbonTracker data assimilation system based on Transport Model version 5 (TM5), and atmospheric reanalyses. The results demonstrate that diagnostics based on boundary layer depth and vertical concentration gradients do not always indicate the vertical mixing strength. Vertical mixing rates are anti-correlated with boundary layer depth at some sites, diminishing in summer when the boundary layer is deepest. Boundary layer equilibrium concepts predict an inverse proportionality between CO 2 vertical gradients and vertical mixing strength, such that previously reported discrepancies between observations and models most likely reflect overestimated as opposed to underestimated vertical mixing. However, errors in seasonal concentration gradients can also result from errors in modeled surface fluxes. This study proposes using the timescale for approach to boundary layer equilibrium to diagnose vertical mixing independently of seasonal surface fluxes, with applications to observations and model simulations of CO 2 or other conserved boundary layer tracers with surface sources and sinks. Results indicate that frequently cited discrepancies between observations and inverse estimates do not provide sufficient proof of systematic errors in atmospheric transport models. Some previously Correspondence to: I. N. Williams (inw@uchicago.edu) hypothesized transport model biases, if found and corrected, could cause inverse estimates to further diverge from carbon inventory estimates of terrestrial sinks.


Introduction
Coupled carbon-climate models predict that the fraction of anthropogenic CO 2 emissions absorbed by ecosystems will decrease over the 21st century (Friedlingstein in assessing the net impact of anthropogenic emissions on global climate.
Uncertainty in the global distribution of ecosystem CO 2 sources and sinks has been attributed to errors in the transport models used for CO 2 inversions (Fung et al., 1983;Denning et al., 1995;Gloor et al., 1999;Dargaville et al., 2003;Baker et al., 2006;Yang et al., 2007;Stephens et al., 2007), yet the problem of identifying these errors remains. Net ecosystem exchange varies seasonally over mid-latitudes but corresponding spatial concentration gradients are weaker in summer when CO 2 uptake coincides with stronger atmospheric mixing and transport. This covariance between atmospheric dynamics and ecosystem exchange, known as the atmospheric rectifier effect, produces enhanced annual-mean near-surface concentrations relative to the overlying at- 15 mosphere even over annually-balanced ecosystems (Fung et al., 1983;Denning et al., 1995Denning et al., , 2008. Conventional wisdom is that the depth of boundary layer mixing plays the dominant role in the rectifier effect, since solar forcing drives both boundary layer mixing and ecosystem exchanges. Analyses of field measurements have shown that the dilu-20 tion of concentrations by boundary layer depth variations is of comparable magnitude to net ecosystem exchange over the diurnal cycle (Raupach, 1991;Raupach et al., 1992;Levy et al., 1999;Lloyd et al., 2001;Styles et al., 2002), suggesting that boundary layer depth is a key indicator of mixing strength. Atmospheric transport models are known to underestimate summer boundary layer depths, which could lead to un- 25 derestimation of inferred northern terrestrial CO 2 sinks and further increase the discrepancy between atmospheric inverse and carbon inventory estimates of the global carbon budget (Denning et al., 1995;Chen et al., 2004;Yi et al., 2004;Yang et al., 2007;Denning et al., 2008). However, systematic differences between transport model model errors in terms of the mechanisms producing observed CO 2 gradients. We will demonstrate that the lower-tropospheric CO 2 budget is strongly timescale dependent, so that the transport and mixing derived from any one field study is relevant only in the context of the timescale over which the measurements were taken or analyzed. We use this timescale dependence to develop an equilibrium boundary layer model 10 for vertical CO 2 gradients and to develop a new diagnostic of transport and mixing that can be applied to transport model simulations and observations at seasonal and longer timescales. We will show that the equilibrium boundary layer model serves as a useful test of hypotheses proposed to reconcile atmospheric CO 2 inversions with carbon inventory estimates of global carbon sinks.

Mixed-layer approximation
Atmospheric transport and mixing cannot be observed directly, so it is necessary to consider simplified tracer conservation equations to interpret differences between models and observations. A common simplifying assumption is based on the observation 20 that dry convective mixing is sufficiently fast to maintain well-mixed trace gases (CO 2 ) in the boundary layer (Emanuel, 1994). The boundary layer satisfying this well-mixed assumption is hereafter referred to as the mixed-layer. The mixed-layer tracer conservation equation in the absence of horizontal advection (considered in Sect. 4

.3) and Introduction
where Reynolds averaging is implied; h is the mixed-layer height; c f and c m are the free troposphere and mixed-layer mixing ratios, respectively; ρ is atmospheric molar density; w is the vertical wind velocity evaluated at the mixed layer height; and subscripts 5 f and m denote quantities at the free-troposphere level just above the mixed-layer and averaged within the mixed layer, respectively. The surface flux (F ) in Eq. (1) is the net ecosystem exchange, and balances the sum of CO 2 storage in the mixed-layer and entrainment of free-troposphere CO 2 owing to growth of the mixed-layer into the freetroposphere (sum of the first two left-hand side terms), as well as vertical advective 10 transport averaged over the mixed-layer (third LHS terms). Note that there is no contribution to the vertical advective tendency by vertical motion exceeding the growth rate of the mixed-layer, since net loss of mixed-layer air to the free-troposphere changes only the volume of the mixed-layer and not the mixed-layer concentration. Horizontal advection may be significant in these cases and is considered in later sections. 15 The mixed-layer approximation (Eq. 1) is not strictly valid at night, when turbulence becomes shallow and surface emissions (e.g., ecosystem respiration) accumulate in the shallow nocturnal boundary layer. However the rapid decay of turbulence after sunset leaves a relic mixed-layer, or so-called residual layer, extending from the nocturnal boundary layer to the depth of the afternoon mixed-layer, in which CO 2 concentrations 20 are similar to the afternoon mixed-layer (Yi et al., 2001). Since h is a limit of integration in the derivation of Eq. (1), it can be taken without additional approximation to be the total depth of the residual-layer and mixed-layer system. Equation (1) can be integrated over the diurnal cycle by assuming (1) that depth-integrated advective tendencies (the third term of Eq. 1) are much larger in the residual layer than the nocturnal boundary 25 layer, due to the much larger residual layer depth and wind speed; and (2) that the nighttime vertical concentration gradient between free-troposphere and residual layer is steady and given by the daytime or afternoon concentration gradient. Since CO 2 is Introduction conserved, any CO 2 that accumulates in the shallow nocturnal boundary layer is accounted for by the daytime mixed-layer concentration. These approximations, which we apply here, have been used in various forms to close the CO 2 budget over diurnal cycles (Chou et al., 2002;Bakwin et al., 2004;Helliker et al., 2004;Yi et al., 2004).

5
Further approximations to Eq. (1) have been made by neglecting either entrainment and storage Helliker et al., 2004) or vertical advection (e.g. Yi et al., 2004), referred to here as the equilibrium and non-equilibrium approximations, respectively. The choice of approximation is crucial to the interpretation of observations and transport model errors. To inform the appropriate approximation, we apply 10 dimensional analysis to Eq. (1). The ratio of the sum of CO 2 storage and entrainment to vertical advection is scaled according to the ratio of a mixed-layer relaxation time (τ) to a characteristic timescale (T ) of Eq. (1). The mixed-layer relaxation time is given by the ratio of characteristic mixed-layer depth to vertical velocity (τ = H/W ). The timescale (T ) is determined by averaging Eq. (1) over time. We will refer to the ratio 15 of these timescales as t * (t * = H/W T ). Although both equilibrium and non-equilibrium approximations have been used to interpret seasonal variability in atmospheric CO 2 , equilibrium assumes that the seasonal cycle is long compared to the mixed-layer relaxation time (i.e. t * 1) whereas non-equilibrium assumes that this ratio is large (t * 1). Dimensional analysis therefore predicts that the equilibrium assumption becomes in-20 creasingly valid at longer timescales and for stronger atmospheric circulations. We will test these predictions in the following section. Introduction  (Fischer et al., 2007). Winter wheat grows from November through June over 40% of the region (Fischer et al., 2007), mostly to the southeast (Riley et al., 2009), with the remaining area dominated by pasture (40%) and a mixture 10 of C 3 and C 4 crops (20%) that grow from April through August (Cooley et al., 2005).

Timeseries
A precision gas system (Bakwin et al., 1995) was used to obtain mixed-layer CO 2 concentrations at 15 min intervals on the 60 m tower. Pressurized air samples have been collected approximately weekly on the 60 m tower and in the free-troposphere at SGP 15 for subsequent analysis at NOAA/ESRL. The continuous mixed-layer CO 2 concentrations were compared with the flask-based measurements to account for and correct possible drifts. These flask-based measurements include, but are not limited to, CO 2 , CH 4 , CO, N 2 O, CO, SF 6 , H 2 , and 13 C and 18 O in CO 2 . The sample collection procedures have been described in detail in Conway et al. (1994). Typically, air was pumped 20 into a pair of 2.5 L glass flasks, connected in series, and slightly pressurized above ambient pressure. Analyses to follow required free-troposphere mixing ratios just above the mixed-layer (Fig. 1b), which were obtained from the first available flask sample just above the CT/TM5 mixed-layer depth (typically available at altitudes of 457.2, 609.6, 914, 1219.2, 1524, 1828.8, 2133.6, 2438.4, and 2743.2 m above ground, and higher, 25 with an uncertainty of about 30 m). The timeseries of mixed-layer CO 2 concentrations ( Fig. 1) provides evidence that analyses of short-term field observations cannot be extrapolated to explain the seasonal rectifier effect. As the averaging time increases the growth and decay of mixedlayer concentrations does not correspondingly increase, as seen by comparing the 1 day and 90 day running averages of mixed-layer CO 2 in Fig. 1a. Therefore we con-5 clude that the magnitude of mixed-layer CO 2 storage decreases when the length of the running averages increases. Similar results have been obtained previously (Davis et al., 2003) based on observations at a tall-tower in Wisconsin (hereafter LEF). The magnitude of mixed-layer depth variations also does not increase between daily and seasonal timescales (Fig. 1c), suggesting that the importance of entrainment to the 10 seasonal rectifier effect may also be overemphasized if analyses of the CO 2 budget are extrapolated from daily or synoptic timescales to the seasonal cycle. These conclusions are made more quantitative in the following.

Calculation of budget terms
Entrainment and storage were analyzed according to Eq. (1) from day-to-day changes 15 in afternoon (01:00-04:00 p.m. LT), as described in Sect. 2.1. Mixed-layer depths were obtained from the CarbonTracker data assimilation system (hereafter CT/TM5). CT/TM5 combines CO 2 surface exchange models and a global, two-way nested atmospheric transport model driven by meteorological fields from the European Centre for Medium-Range Weather Forecasts (ECMWF). Mixed-layer depth and the three-20 dimensional distribution of CO 2 mole fractions and surface fluxes were available at 1 • × 1 • spatial resolution every 3 h.
Vertical advection was not archived in the 2008 CT/TM5 dataset, and was recreated here with vertical velocities from the ECMWF interim reanalysis (Fig. 1d), based on the same general circulation model and parameterization schemes in CT/TM5 (Peters 25 et al., 2007). Differences between our recreated advective tendencies and CT/TM5 are expected in part due to differences in the forecast time-steps used to create each dataset, and interpolation of the ECMWF meteorological data from 1. Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | resolution in CT/TM5. We averaged the two CT/TM5 grid-points closest to each of our study sites for use with the ECMWF reanalysis vertical velocities, due to the difference in resolution of these products. Additional spatial averaging had little effect on the results. Resolution sensitivity of the recreated vertical advective tendency was investigated by spatially averaging ECMWF velocities over the 1 through 4 grid-cells nearest the SGP site, and was found to be small. Sensitivity was further tested using vertical velocities from three independent data assimilation systems (RUC, NCEP, and NARR), which were qualitatively similar to ECMWF velocities and did not have a notable effect on our results. 10 Observations from the SGP tower and assimilated data from CT/TM5 were used to quantify the timescale dependence of lower-tropospheric CO 2 . The timeseries for each budget term in Eq. (1) was divided into non-overlapping segments of length ranging from 1 to 90 days, and each budget term was averaged over each of these segments (storage and entrainment were added together to form a single budget term). This 15 procedure resulted in a statistical ensemble of CO 2 budgets, with one ensemble for each of the averaging times ranging from 1 to 90 days. After taking the magnitude of each term, we calculated the ensemble-median of the ratios of terms. The ratios of budget terms ( Fig. 2) confirm that the relative importance of entrainment and storage is a strong function of timescale at the three sites tested, with entrainment 20 and storage becoming an order of magnitude smaller than surface fluxes by monthly and longer timescales. These results are based on surface fluxes from the CT/TM5 data assimilation system and entrainment and storage calculated using mixing ratios from the same dataset. The three test sites are SGP, described in the section above; LEF (45.95 • N, 90.27 • W), characterized by a managed forest of mixed northern hard- 25 wood, aspen, and wetlands; and Harvard Forest (HFM, 42.54 • N, 72.17 • W), a managed, deciduous forest, and were chosen to correspond to the North American aircraft flask sampling locations used in a related study (Stephens et al., 2007) Stephens et al., 2007) were excluded from our analysis because the calculation of vertical advection may be influenced by differences in model topography and resolution between CT/TM5 and the ECMWF interim reanalysis. The CT/TM5 concentrations used here were vertically averaged over the mixed layer to define the mixed-layer concentrations.

5
The results (Fig. 2) indicate a general timescale dependence of the CO 2 budget for continental mixed-layers. The comparison confirms that entrainment and storage are of secondary importance relative to both advective transports and surface fluxes at seasonal timescales. The results hold for budget terms calculated using mixing ratios from CT/TM5 (Fig. 3b) and from tower and aircraft observations (Fig. 3a).

Scaling relationship for mixed-layer tracer budgets
Advection and net surface flux dominate the budget of mixed-layer CO 2 at seasonal timescales, but at a rate which varies quantitatively between sites in North America and between seasons (Fig. 3b). The scaling relationship developed in Sect. 2.1 predicts that mixed-layer budgets depend not only on the averaging time, but on the ratio of the 15 averaging time to the boundary layer relaxation time, which depends on the strength of the local circulation. To test this hypothesis we reproduced the results of Fig. 3a,b in a scatter plot where each point represents the sum of entrainment and storage (y-axis), and advection (x-axis), corresponding to a given averaging time (Fig. 4a,d). The scatter plots were made by binning the results of the ensemble budget (shown in Fig. 3) into 20 10 bins according to averaging time, where the bin spacing is allowed to be uneven so that there are equal numbers of ensemble members in each bin. The same timescale dependence shown in Fig. 3 is recovered in Fig. 4a,d except the scatter plots reveal the individual magnitudes of entrainment and storage, and advection, and their standard deviations (numbered labels in Fig. 4a,d represent mean averaging times for each bin). 25 The differences between mixed-layer budgets at different timescales and continental locations can be understood in terms of the averaging time and the strength of the local circulation. To demonstrate this relationship we scaled the vertical advection 11464 Introduction

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Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | term by the non-dimensional number t * 0 (Sect. 2.2), calculated with mixed layer depths and vertical velocities (H 0 ,W 0 ) averaged across the three sites (SGP, LEF, HFM). The budget terms were binned again using t * 0 as opposed to the averaging time. Typical values of t * 0 (and t * ) range from 1.5 to 0.01 for averaging times of 1 day to 90 days, respectively. The results (Fig. 4b,e) confirm that the ratio of entrainment plus storage 5 to advection scales linearly with t 0 . Differences in the slopes of the linear relations, particularly between HFM and the other two sites, are eliminated when further scaled according to t * , where the contribution of differences in local atmospheric circulation is taken into account (Fig. 4c,f) by calculating mixed layer depths and vertical velocities at each site. 10 We conclude that the mechanisms found to control tracer vertical gradients over the course of a field campaign could vary considerably from the mechanisms operating over longer timescales and at different locations. Storage and entrainment are relatively large components of the atmospheric CO 2 budget over the diurnal cycle, and are proportional to mixed-layer depth variations, so that mixed-layer depth variations 15 explain a large portion of the mixing between the free-troposphere and boundary layer at the daily timescale. However this mixing is controlled by the vertical advective transport at longer timescales, which does not necessarily scale with mixed-layer depth. The following sections develop a new diagnostic of vertical mixing valid at seasonal timescales. 20 4 Boundary layer quasi-equilibrium

Theoretical basis
We have shown that mixed-layer depth is not a reliable indicator of mixing strength when t * is small, typically at seasonal timescales. In this limit the tracer conservation Eq.
(1) yields a balance between vertical advective transport and net surface flux. Here 25 we explore the finite timescale over which equilibrium is approached, and whether this ACPD 11,2011 Boundary layer equilibrium and CO 2 inversions timescale is diagnostic of mixing rates. For example, consider the assumption that freetroposphere mixing ratios vary slowly in time compared to mixed layer mixing ratios. Equation (1) can then be written as where ∆c is the difference in mixing ratio (c m − c f ) between the free troposphere and 5 mixed layer, and vertical velocity has been rewritten in terms of the vertically integrated horizontal divergence (bracketed term in Eq. 2) using the continuity equation (the horizontal wind velocity, u, is written entirely in the x-direction for brevity). If horizontal wind divergence and surface flux are slowly varying in time compared to perturbations in the mixed-layer depth and vertical mixing ratio difference, we obtain a solution Equation (3) predicts that tracer vertical gradients relax toward equilibrium at a rate determined by the rate of mass divergence in the mixed-layer. Horizontal advection and non-linear vertical tracer advection have been neglected, however the slowly varying components of these terms can be considered part of the steady forcing (i.e., can be 15 combined with the surface flux, F ) without changing the predicted exponential decay of perturbations. This solution compares favorably to observational data, shown in the following section. In the long-time limit (t ∂ x u −1 ) we obtain the equilibrium solution given by The weighting of the vertical gradient (∆c) by h reflects the fact 20 that the divergent circulation transports more mixed-layer mass per surface area as the depth of the mixed-layer increases, requiring either larger surface flux or smaller tracer vertical gradients to balance tracer transport. Although fluctuations in h can still produce fluctuations in ∆c at any given time, a statistically steady state is expected when the relaxation time (given by ∂ x u −1 ) is short compared to the averaging time. 25 In this limit we obtain the equation in Bakwin et al. (2004) and Helliker et al. (2004), where the continuity equation has been used to write the average divergence in terms of the average vertical velocity, and over-bars represent time-averaging (e.g., 90 day seasonal averages), and F is understood to be the slowly varying (seasonal) component of surface flux. The solution (Eq. 3) is consistent with our earlier scaling arguments (Sect. 2.2), which predicted a balance between advective tracer transport 5 and surface fluxes for long averaging times relative to the relaxation time (i.e. t * 1). This quasi-equilibrium solution (Eq. 4) is analogous to that of boundary layer entropy or convective-available potential energy in the quasi-equilibrium theory for tropical climate (Raymond, 1997). Equation (4) illustrates why systematic errors in vertical concentration gradients among transport models do not necessarily reflect systematic errors in 10 modeled transport or mixing, since ∆c is also affected by possible systematic errors in the surface flux (F ).

Relaxation times
Vertical CO 2 gradients, when taken alone, are poor indicators of lower-tropospheric mixing and transport because of the presence of strong surface fluxes, as shown by 15 Eq. (4) and the scaling arguments of Sect. 2.2. Since we are interested in evaluating the mixing and transport properties of models, we wish to separate the influence of surface fluxes from that of mixing in a diagnostic model. Here we demonstrate that the relaxation timescale is suitable for this purpose. Differences in relaxation times can be understood through Eq. (3) in terms of the modeled divergent wind, which could re-20 flect errors in parameterized turbulent mass flux, resolved advection, or parameterized moist convection in transport models (as discussed in Sect. 1). The results will also support the equilibrium approximation (Eq. 4) by demonstrating that relaxation times are generally much shorter than the seasonal timescale and shorter than the synoptic timescale. 25 We compared the observed relaxation time to that predicted by Eq. (3) by comparing the autocorrelation of observed fluctuations in the vertical CO 2 gradient to that predicted by Eq. (3). Equation (2)  (2) to yield, where C is the fluctuation given by h∆c − h∆c, and density is assumed constant over 5 the relaxation timescale. The autocorrelation function is the exponential decay given by where |t − t | is the time lag of the autocorrelation (e.g. Chatfield, 2004). The derivation is strictly for autocorrelations of fluctuations in h∆c, but the autocorrelations of ∆c 10 were nearly identical in all the cases examined here. Adding fluctuations (F ) about the mean surface flux to the right hand side of Eq. (5) yields the familiar Langevin equation (see North et al., 1993, for an analogous application to climate sensitivity), which has the same autocorrelation function (Eq. 6) as long as the fluctuations have the statistical properties of Gaussian white noise. Although the latter assumption is not strictly valid 15 for surface CO 2 fluxes, the solution (Eq. 6) proves to be a useful quantitative description of observed concentration autocorrelations. A comparison of observed (SGP) and theoretical autocorrelations is shown in Fig. 5, where winter (December-February) and summer (June-August) are shown separately in panels Fig. 5a and Fig. 5b, respectively. Observed vertical gradients were calcu-20 lated from the difference in mixing ratios between the 60 m tower and aircraft flask samples at SGP. The gray lines in Fig. 5 indicate the theoretical exponential decay calculated from the ECMWF reanalysis divergent wind field (i.e., using Eq. 6, with the full horizontal divergence ∂ x u + ∂ y v ). The results confirm the theoretical prediction that vertical mixing ratio gradients approach a time-mean value approximately exponentially, at a rate determined by the rate of mass divergence in the mixed-layer. Although the quasi-equilibrium theory is based on linear assumptions (i.e. covariation of w and ∆c is neglected), the exponential decay of autocorrelations still holds if the non-linear component of advection is slowly varying relative to the relaxation time. Relaxation 5 times were generally less than 3 days, or shorter than the typical synoptic timescale and much shorter than the seasonal timescale. The above comparison was repeated for the CT/TM5 dataset at SGP, LEF, and HFM. The difference between the summer and winter relaxation times at SGP and HFM is captured by the theoretical solution to the conservation equation (gray lines in Fig. 6), 10 but the summer CT/TM5 concentration gradient at HFM decays faster than theory predicts (Fig. 6c). Note that the theoretical gray lines for SGP in Fig. 5a,b are the same as those in Fig. 6a,d, except for the addition of two years (2001,2002) of data from the CT/TM5 analyses. Removing the additional two years had no effect on the agreement between observations, theory, and CT/TM5, seen by comparing panels Fig. 5a,b, and  Fig 6a,d. Based on this comparison we conclude that there is no evidence for errors in the divergent wind field of the CT/TM5 data assimilation system at SGP. The disagreement between theory and CT/TM5 in summer at HFM (and in winter at LEF) did not improve upon changing the averaging time from 90 to 45 days or 180 days. These results warrant future investigations using data from the measurement towers at LEF 20 and HFM.
Our results show that the rate at which CO 2 vertical gradients relax toward equilibrium can be used to obtain a more reliable mixing diagnostic than using CO 2 vertical gradients alone. One drawback to this method is the requirement that surface fluxes be steady or have the statistical properties of Gaussian white noise on timescales shorter 25 than the relaxation timescale. This assumption also applies to the horizontal advection and cloud mass fluxes included in the more general form of the flux (F ) in Eq. (3). The analysis described here could be further tested in future studies using other conserved tracers whose surface fluxes are less likely to be correlated at these short timescales. Autocorrelations using SF 6 were reported (Denning et al., 1999) and were similar to our results using CO 2 , but were not interpreted in terms of the divergent wind field. Our results provide a theoretical basis for relating the autocorrelation to transport statistics from atmospheric models.

5
The equilibrium approximation was further tested by extending Eq. (4) to include horizontal advection and comparing the equilibrium surface fluxes to surface fluxes from the CT/TM5 data assimilation system. Horizontal winds were interpolated to the SGP site, using the interpolation scheme internal to CT/TM5, and gradients were taken from the CT/TM5 data assimilation system. The full non-linear horizontal advection was calculated (i.e., gradients and winds were multiplied before seasonal averaging). We also tested the linearity assumption made in deriving the equilibrium solution (Eq. 4) by calculating the full non-linear vertical advection and dividing this term into linear and nonlinear components following standard Reynolds decomposition, based on 90 day averages (e.g., w = w + w , ∆c = ∆c + ∆c , where primes denote departure from the 15 90 day average). A 90 day running average was then taken to write the vertical advection term as the sum of a linear and nonlinear component (i.e., w∆c + w ∆c ). The full mixed-layer CO 2 budget at SGP is shown in Fig. 7 and is decomposed into the sum of storage and entrainment, vertical advection, and horizontal advection. The CO 2 budget was also calculated using CT/TM5 data for SGP, LEF and HFM (Fig. 8). 20 The total vertical advection (sum of linear and non-linear terms) and the non-linear term are each shown separately in Fig. 8. Vertical advection is typically the dominant term at SGP. The non-linear advection is also significant, at times accounting for most of the vertical advective transport, and highlights the importance of transport by synopticscale disturbances. Horizontal advection is smaller than vertical advection but is of CT/TM5 analyses, as shown in Fig. 9. We extended the mixed-layer approximation to diagnose vertical concentration gradients, as discussed in the following.

Application to vertical concentration gradients
Previous studies assumed that the mixed-layer acts as a mixing chamber in which concentrations of gases exchanged at the surface are either diluted or concentrated 5 through deep summer or shallow winter mixed layer depths. Studies based on these analogies have emphasized the role of mixed-layer depth in explaining differences between observed and simulated vertical CO 2 gradients between the mixed-layer and the free-troposphere above. Having shown that the analogy to a mixing chamber breaks down at long timescales, we instead hypothesize that vertical concentration gradients 10 come into equilibrium with transport into and out of the mixed-layer and ecosystem exchanges at seasonal timescales. We used the equilibrium approximation to Eq. (1), including horizontal advection, with vertical advection divided into linear and non-linear components (as in Sect. 4.3), and inverted the slowly-varying component of vertical advective transport to obtain the 15 vertical concentration gradient (here given by ∆c = c m − c f ), according to where the last two RHS terms are the zonal and horizontal advection, respectively. Vertical concentration gradients between the free-troposphere and mixed-layer were calculated by applying Eq. (7) to CO 2 concentrations and surface fluxes from CT/TM5. 20 Density differences between the free-troposphere and mixed layer account for less than 10% of the variation in concentration gradients and have been neglected in Eq. (7) to simplify our discussion, but were included in all our computations. The resulting equilibrium CO 2 vertical gradients (Fig. 10a) at all three sites (SGP, LEF, HFM) compared favorably with those obtained directly from CT/TM5 (Fig. 10b), al- 25 though there are differences in the magnitude of the seasonal cycle in some years. We did not compare panels (Fig. 10a,b)  and the equilibrium approximation were within the sensitivity of our results to increasing or decreasing the mixed-layer depth by one vertical model level. Rather we point out that the equilibrium approximation is successful in predicting the seasonal cycle and the differences in annual mean vertical gradient between the three sites. The CO 2 transport considered in this diagnostic is that by the subsiding vertical wind, which 5 is in approximate equilibrium with mixed-layer entrainment of free-tropospheric CO 2 . This diagnostic contrasts with previous diagnostics of vertical concentration gradients, which considered transient changes in day-to-day mixed-layer depth as the dominant mechanism of exchange between the free-troposphere and mixed-layer. The latter diagnostic is based on the common assumption that seasonal variations in mixed-layer depth drive annual-mean vertical CO 2 gradients. This assumption is tested in the following section.

Application to the seasonal rectifier effect
Annual-mean vertical CO 2 gradients result in part from the covariation of atmospheric dynamics with net ecosystem exchanges over the seasonal cycle, known as the ver-15 tical rectifier effect, which is frequently attributed to seasonal variation of mixed-layer depths. Mixed-layer depth variations do not explicitly appear in the equilibrium description of vertical concentration gradients (Eq. 7) because, as discussed previously, their direct effect on the dilution of concentrations is an order of magnitude smaller than transport at seasonal timescales. Therefore, we investigate the possibility that sea-20 sonally varying transport could explain the rectifier effect. Composite seasonal cycles were calculated by further averaging the 90 day running average subsidence velocities (i.e., the solid black line in Fig. 1d) over the seven annual timeseries extending from 1 January-31 December, for the years that CT/TM5 data were available (January 2001(January -2008. 25 The results (solid lines in the lower three panels of Fig. 11) confirm that vertical wind at the mixed-layer depth varies seasonally and therefore contributes to the seasonal rectifier effect. Seasonal variability in the vertical wind at both SGP and LEF is such 11472 Introduction that mixed-layer concentrations would be enhanced relative to the free-troposphere if the surface flux (F in Eq. 7) were seasonally varying but annually balanced, as in idealized modeling studies (Denning et al., 1995). Subsidence is strongest in winter at HFM, which may help to explain why vertical concentrations gradients at HFM are comparable to those at SGP and LEF, even though the net surface flux is significantly 5 more positive there (c.f. panels Figs. 10,9). The difference between vertical velocities at HFM and at the other two sites (SGP, LEF) can be explained by topographic forcing of flow over the Appalachian mountains (e.g. Saltzman and Irsch, 1972), which is strongest in winter due to stronger surface westerlies. According to Eq. (7), the coincidence of larger vertical wind with the strongest respiratory fluxes in winter helps to 10 reduce winter mixed-layer concentrations relative to the free troposphere, bringing the winter vertical gradient at HFM closer to those of other sites. The mechanism by which vertical transport contributes to the seasonal rectifier effect is clarified in height-time cross sections of the vertical wind field (Fig. 11, top three panels) with seasonal variability in mixed-layer depth overlaid (solid black lines in the 15 top three panels of Fig. 11). Recall that only the downward vertical velocity is relevant to the vertical advective tendency of mixed-layer concentrations, since vertical transport out of the mixed-layer by itself has no effect on mixed-layer concentrations. Note that the 90 day average subsidence velocity at the 90 day average mixed layer depth (as indicated by the top three panels of Fig. 11) is not exactly the same as the 90 day 20 average of daily subsidence velocity calculated at daily mixed layer depths (shown in the bottom three panels of Fig. 11 in solid black lines), but the differences are small. The monotonic increase in vertical velocity with height in the lowest 1-2 km is not surprising, since vertical velocity must vanish at the earth's surface in the absence of topography. More importantly, from Fig. 11 the seasonal rectifier effect depends on 25 both the seasonal variation in mixed-layer depth and seasonal variation in subsidence rates.
The effect of deeper summer mixed layers is to mix concentrations exchanged at the earth's surface to a depth above that of the winter mixed layer, where transport by the vertical and horizontal wind is correspondingly stronger. This effect can be seen in the lower three panels of Fig. 11, where the vertical wind at annually-averaged mixed-layer depths (dashed line) is compared to the wind at seasonally varying mixed-layer depths (solid line). This comparison indicates that seasonal variability in mixed-layer depth is essential to seasonal variability in transport at SGP and LEF, however the same com-5 parison yields the opposite result at HFM, where mixed layer depth variations act to reduce seasonal variations in vertical transport. These results underscore the importance of considering seasonal variations in mixed layer depth and transport together when explaining vertical CO 2 gradients, and that mixed layer depth is not always a good indicator of the strength of mixing and transport.

Discussion
Boundary layer equilibrium has important consequences for the interpretation of CO 2 flux inversion errors. Given the surface flux, equilibrium predicts a simple inverse proportionality between CO 2 vertical gradients and the vertical wind (∆c ≈ −F w −1 , recalling that ∆c = c m − c f ), which is consistent with the hypothesis that weaker than 15 observed summer vertical gradients resulted from overestimated vertical mixing in atmospheric models, while stronger than observed winter gradients resulted from underestimated mixing (Stephens et al., 2007). However it is also evident (from Eq. 7) that the inverse proportionality between errors in vertical gradients and mixing will only hold if there are no errors in the surface flux, horizontal advective transport, or non-linear 20 vertical advective transport (i.e., synoptic-scale eddies). For example, systematic biases in vertical gradients could also result from the specification of the first-guess fluxes used in forward transport simulations prior to inversion, known as prior fluxes. As a counter example to the hypothesis that vertical mixing is systematically biased in transport models, consider a hypothetical transport model simulation having per-25 fect vertical mixing and transport but subject to specification of overestimated summer prior surface fluxes (shown schematically in Fig. 12 that subsidence (w < 0) in this simulation always matches that of the real atmosphere. Overestimated summer fluxes (i.e., F less negative than observed) require larger freetropospheric concentrations in the forward simulation (solid circles in Fig. 12a) than in observations (squares in Fig. 12a), and correspondingly weaker vertical gradients, according to ∆c ≈ −F w −1 . Unless the inversion of this transport is tightly constrained 5 to observed concentrations, the post-inversion vertical gradient may remain underestimated, falsely indicating summer overestimated mixing, while free-tropospheric concentrations would remain overestimated, falsely indicating underestimated summer mixing. This example illustrates why either CO 2 concentrations or vertical CO 2 gradients taken alone are not reliable indicators of transport model errors, and how opposite 10 conclusions regarding CO 2 mixing can be drawn when free-troposphere concentrations are considered separately of mixed-layer concentrations. Contradictory hypotheses of both summer overestimated and underestimated mixing could be reconciled if specified prior surface fluxes are overestimated, as discussed above, without having to invoke systematic transport model biases. Possible overesti-15 mation of prior summer fluxes has been suggested as a result of seasonality in fossil fuel emissions (Gurney et al., 2005;Erickson et al., 2008) or underestimated seasonal amplitudes of specified prior biospheric fluxes (Peters et al., 2007;Yang et al., 2007), which have not been considered in past analyses of CO 2 inversion sensitivity to transport. Previous studies using SF 6 , an ideal tracer due to its well known and sea-20 sonally invariant emissions, found no systematic errors in summer modeled transport (Gloor et al., 2007;Patra et al., 2009). Overestimated prior fluxes would enhance predicted boundary-layer CO 2 concentrations relative to the free-troposphere in forward transport model simulations (Fig. 12a), requiring compensating stronger summer biospheric uptake in the inversions when constrained by observed concentrations Gurney 25 et al., 2005). This process could explain why the northern land sink is larger in inverse estimates than carbon inventories.
The possibility of underestimated summer mixing (Yang et al., 2007) should be examined carefully, because erroneously weak vertical mixing, if it exists and is corrected in transport models, could increase the discrepancy between inversion estimates and land carbon inventories. Increased mixing rates (more negative w) would result in stronger modeled vertical gradients relative to observations (∆c more negative in ∆c ≈ −F w −1 ), requiring an even larger land carbon sink in the inversions. The process is shown schematically in Fig. 12c (circles and squares indicate model and observa-5 tions, respectively) and contrasted with that of overestimated summer mixing (Fig. 12b).
In the case of overestimated summer mixing, the hypothetical perfect prior flux would be adjusted to take up more CO 2 in order to minimize differences between the forwardtransport simulation and observations in the inversion procedure. Overestimated summer mixing would therefore be more desirable with regard to explaining why the land carbon sink is stronger in inversion estimates than in land carbon inventories. Underestimated model mixed layer depths Denning et al., 2008) have been cited as a mechanism for underestimated summer mixing, and as an explanation for underestimated seasonal amplitudes of modeled free-tropospheric and column-average CO 2 (Yang et al., 2007), presumably due to dilution and transient en- 15 trainment associated with the growth of deep mixed-layers. However the direct effect of mixed layer depths on the dilution of concentrations is an order of magnitude smaller than transport at seasonal timescales (see Sect. 2.2). We therefore consider the indirect effect of mixed layer depths through the dependence of subsidence rates on depth above the surface. Seasonality in mixed layer depth can oppose seasonality in the ver-20 tical gradient of subsidence, for example at HFM, a site exhibiting the same biases in modeled vertical CO 2 gradients as other sites (Stephens et al., 2007). Seasonality in mixing strength is reversed at HFM relative to other sites even though the seasonality in mixed-layer depth is not. Mixed-layer depth is therefore a poor indicator of mixing rates. It is necessary to consider both the divergent wind field and the mixed-layer 25 depth together when assessing transport model performance.
We have not considered convective cloud mass fluxes, and so may underestimate the extent to which spring and summer flux inversions could be in error due to transport model errors. However, the resolved-scale subsidence velocity in the data assimilation Introduction systems used here is required to reflect the aggregate effects of transport and mixing at the sub-grid scale, due to separate enforcement of mass and energy balance at both sub-grid and global scales in the models underlying these datasets (Lawrence and Salzmann, 2008). This process is reflected in the reasonable agreement between surface fluxes and vertical gradients from data assimilation systems and those 5 reconstructed using the equilibrium approximation and resolved-scale model winds. Improvements to the parameterization of mixing and transport by convective clouds are still needed, especially since these parameterizations were designed to realistically simulate energy balances for climate studies rather than tracer transport. General circulation models have a well-known bias in the diurnal timing of deep convection (e.g. Guichard et al., 2004), with a preference toward local noon, whereas observed continental convection has a preference for the evening and nighttime (e.g. Wallace, 1975). Enhanced deep-convection during peak respiration would tend to reduce near-surface CO 2 concentrations relative to the free troposphere, enhancing the summer depletion of mixed-layer CO 2 relative to the free-troposphere. This process 15 could explain why summer near-surface CO 2 is more depleted in observations than transport models despite apparent lack of similar biases in other trace gases, such as SF 6 (Gloor et al., 2007).

Conclusions
Previous hypotheses for transport model errors assumed that boundary layer depth and 20 vertical CO 2 gradients reflect the strength of vertical mixing in the lower-troposphere. However seasonality in the strength of transport and mixing is anti-correlated with boundary layer depth at some sites, diminishing in summer when the boundary layer is deepest. Vertical CO 2 gradients are also unreliable indicators of mixing because surface fluxes and mixing both contribute to vertical concentration gradients. 25 We developed a new diagnostic model to understand vertical CO 2 gradients and their relation to atmospheric transport and mixing, based on the concept of boundary layer equilibrium, and demonstrated its application to long-term observations and a global data assimilation system. The finite timescale over which observed concentration gradients relax toward equilibrium is diagnostic of the rate at which boundary layer air is exchanged with the free-troposphere. This diagnostic does not depend on model variables such as vertical velocity and is independent of seasonal surface fluxes.

5
Boundary layer equilibrium is an idealized concept that cannot account for the full complexity of atmospheric transport and mixing. Yet even in terms of this simplified description, many observations cited as evidence for systematic biases in atmospheric transport models are insufficient to prove that such biases exist, and in some cases model errors proposed to reconcile carbon inventory and inverse estimates of global 10 carbon sinks could confound these estimates. Atmospheric transport model errors are widely cited as contributing to uncertainty in CO 2 flux inversions, but the problem of identifying or refuting these errors remains. The methods developed here can be used to diagnose the several factors leading to uncertainty in global inversions of surface CO 2 fluxes. 15 observed from a very tall tower, Glob. Change Biol., 9, 1278-1293 Denman, K. L., Brasseur, G., Chidthaisong, A., Ciais, P., Cox, P. M., Dickinson, R. E., Hauglustaine, D., Heinze, C., Holland, E., Jacob, D., Lohmann, U., Ramachandran, S., da Silva Dias, P. L., Wofsy, S. C., and Zhang, X., Couplings between changes in the climate system and biogeochemistry, in:  ACPD 11,2011 Boundary layer equilibrium and CO 2 inversions