Quantifying the uncertainties of advection and boundary layer dynamics on the diurnal carbon dioxide budget

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: ATMOSPHERES, VOL. 118, , doi:1.12/jgrd.5677, 213 Quantifying the uncertainties of advection and boundary layer dynamics on the diurnal carbon dioxide budget D. Pino, 1 J.-P. Kaikkonen, 2 and J. Vilà-Guerau de Arellano 3 Received 18 September 212; revised 2 June 213; accepted 16 July 213; published 22 August 213. [1] We investigate the uncertainties in the carbon dioxide (CO 2 ) mixing ratio and inferred surface flux associated with boundary layer processes and advection by using mixed-layer theory. By extending the previous analysis presented by Pino et al. (212), new analytical expressions are derived to quantify the uncertainty of CO 2 mixing ratio or surface flux associated to, among others, boundary layer depth, early morning CO 2 mixing ratio at the mixed layer or at the free atmosphere; or CO 2 advection. We identify and calculate two sorts of uncertainties associated to the CO 2 mixing ratio and surface flux: instantaneous and past (due to advection). The numerical experiments are guided and constrained by meteorological and CO 2 observations taken at the Cabauw 213 m tower. We select 2 days (25 September 23 and 12 March 24) with a well-defined convective boundary layer but different CO 2 advection contributions. Our sensitivity analysis shows that uncertainty of the CO 2 advection in the boundary layer due to instantaneous uncertainties represents at 16 LT on 12 March 24 a contribution of 2 ppm and.72 mg m 2 s 1 in the uncertainty of the CO 2 mixing ratio and inferred surface flux, respectively. Taking into account that the monthly averaged minimum CO 2 surface flux for March 24 was.55 mg m 2 s 1,theerroronthesurfacefluxisonthe order of 1%. By including CO 2 advection in the analytical expressions, we demonstrate that the uncertainty of the CO 2 mixing ratio or surface flux also depends on the past uncertainties of the boundary layer depth. Citation: Pino, D., J.-P. Kaikkonen, and J. Vilà-Guerau de Arellano (213), Quantifying the uncertainties of advection and boundary layer dynamics on the diurnal carbon dioxide budget, J. Geophys. Res. Atmos., 118, , doi:1.12/jgrd Introduction [2] Carbon dioxide (CO 2 ) concentrations in the atmosphere have risen significantly after the industrial revolution [Keeling and Whorf, 1994; Conway et al., 1994]. Elevated CO 2 concentrations have drawn a lot of attention among the scientists and policy makers since CO 2 is a greenhouse gas and consequently, is connected to the global climate change [Houghton, 1993; Cox et al., 2; Rörsch et al., 25; Le Treut et al., 27]. Atmospheric CO 2 content is increased by anthropogenic and natural sources of CO 2 and decreased by biospheric and oceanic sinks of CO 2. The trend of growing atmospheric CO 2 therefore suggests an imbalance between the sources and sinks of CO 2, 1 Department of Applied Physics, Universitat Politècnica de Catalunya BarcelonaTech, and Institute of Space Studies of Catalonia (IEEC-UPC), Barcelona, Spain. 2 Department of Applied Physics, Aalto University School of Science, Espoo, Finland. 3 Meteorology and Air Quality Section, Wageningen University, Wageningen, Netherlands. Corresponding author: D. Pino, Department of Applied Physics, Universitat Politècnica de Catalunya BarcelonaTech, Esteve Terrades 5, 886 Castelldefels, Spain. (david.pino@upc.edu) 213. American Geophysical Union. All Rights Reserved X/13/1.12/jgrd causing that during the last half century, around 5% of the CO 2 emissions every year remain in the atmosphere [Le Quéré et al., 29]. Atmospheric CO 2 concentration also exhibits large interannual variability due to variations in terrestrial biosphere [Keeling et al., 1996], and its growth rate is increasing faster than total anthropogenic CO 2 emissions [Raupach et al., 28]. Moreover, by changing the radiative balance of the atmosphere, the increase in the CO 2 concentration and other greenhouse gases also modifies the water cycle and oceanic circulations. To identify different sources and sinks of CO 2 and their contribution to the global carbon cycle, a great deal of effort has been devoted to estimating the CO 2 exchange between the biosphere and the atmosphere on different time and length scales [Peters et al., 27]. [3] It is already known that convective boundary layer dynamics influence the exchange of CO 2 between the mixed layer and the surface [Culf et al., 1997; Jacobs and de Bruin, 1997; Baldocchi et al., 21], between the mixed layer and the FA [Yi et al., 21, 24; Vilà-Guerau de Arellano et al., 24; Lloyd et al., 27; Casso-Torralba et al., 28; McGrath-Spangler and Denning, 21; Kretschmer et al., 212] and also affect the vertical and horizontal distribution of CO 2 in the atmosphere [Wofsy et al., 1988; Denning et al., 1995; Gibert et al., 27]. Our main aim is to analyze how uncertainties in the convective boundary layer dynamics or in the initial conditions lead to uncertainties in the CO 2

2 mixing ratio or surface flux. Accurate convective boundary layer modeling is therefore crucial for forward simulations of CO 2 mixing ratios based on the measured CO 2 surface flux, and for inferring CO 2 surface flux from the observed CO 2 mixing ratios even at local scales. It is thus important to quantify the role of the dynamics of the boundary layer on the CO 2 mixing ratio, and on the inferred CO 2 surface flux [Culf et al., 1997]. [4] The importance of CO 2 horizontal advection in the CO 2 budget has been pointed out by previous research studies based on observational tower measurements. Yi et al. [2] estimated the different terms of the CO 2 budget on a daily scale by using measurements from a very tall tower. They found that CO 2 advection between 3 and 122 m above ground level can be as much as 27% of the diurnally integrated sum of storage and turbulent CO 2 fluxes during July Yi et al. [28] analyze the contribution of advection to Net Ecosystem Exchange during stable conditions in a subalpine forest site. They explicitly estimate CO 2 advection flux by using four different towers and calculating the horizontal gradients of CO 2 mixing ratio and mean winds. In very stable conditions with very low friction velocity, CO 2 advection flux cause the 6 year cumulative total CO 2 fluxtobe82% lower. Sun et al. [27] found that observed CO 2 advection produced by upslope-downslope flows on a hilly terrain can be significant not only during the night but also on daytime. [5] Moreover, several studies have shown, by using mesoscale model results, how uncertainties associated to some boundary layer processes (surface energy fluxes, horizontal or vertical transport), influence the simulation of the CO 2 mixing ratio at the mixed layer or the retrieval of the CO 2 surface flux. It is important to note that the differences between the boundary layer depth simulated by the different models (WRF, RAMS, Meso-NH), or even the different planetary boundary layer schemes for the WRF model [Skamarock et al., 28] can be as large as 6% [Sarrat et al., 27; Kretschmer et al., 212]. Lin and Gerbig, [25], by using ECMWF atmospheric data and a Lagrangian transport model, and Tolk et al. [28], by using RAMS model [Pielke et al., 1992], estimated the CO 2 mixing ratio uncertainties associated to horizontal transport and mesoscale circulations. On diurnal timescales, they found values between 2.5 and 6 ppm. Regarding the uncertainties of CO 2 mixing ratio related to boundary layer depth, Gerbig et al. [28] and Kretschmer et al. [212], by using a Lagrangian transport and WRF model, respectively, found that a 4% uncertainty in the determination of the mixing height produces up to 3 ppm uncertainty in the CO 2 mixing ratio. Moreover, Tolk et al. [29] found that uncertainties in energy fluxes driving the boundary layer evolution lead to 1.9 ppm uncertainty in the mixed-layer CO 2 mixing ratio. Finally, it is important to note that small uncertainties in the CO 2 mixing ratio determination lead to large uncertainties (.1 PgC/yr per 1 6 km 2 ) of the inferred CO 2 surface flux [Houweling et al., 21]. [6] Pino et al. [212] studied the influence of boundary layer processes in CO 2 mixing ratio evolution in convective conditions but excluded horizontal and vertical advection of CO 2 from the analysis. By using observations from a tall tower located on a flat terrain covered with grass, and mixed-layer theory [Lilly, 1968; Betts, 1973; Carson, 1973; 9377 Tennekes, 1973; Tennekes and Driedonks, 1981; Culf et al., 1997; Vilà-Guerau de Arellano et al., 24], Pino et al. [212] obtained a set of analytical expressions for the CO 2 mixing ratio in the mixed layer. With these expressions, they were able to quantify the uncertainties in the CO 2 mixing ratio or surface flux due to uncertainties in different boundary layer variables. [7] Here CO 2 advection is included in theoretical framework, and the subsequent analysis of the diurnal evolution of CO 2 budget presented by Pino et al. [212] is extended and discussed by including CO 2 advection. Moreover, the analysis is completed by analyzing the conditions observed at Cabauw (Netherlands, Beljaars and Bosveld [1997]; Werner et al. [26]; Vermeulen et al. [211]) during 2 days with different CO 2 advection rates. Unlike the results shown by mesoscale model studies cited above, whose complexity makes it difficult to identify the individual boundary layer variables affecting the CO 2 budget, the mixed-layer formulation presented here allows us to clearly differentiate all the processes driving the CO 2 mixing ratio on convective conditions and to study the evolution of the uncertainties during the day. [8] It is already known that the different processes influencing the CO 2 budget change their importance depending on the time scales considered. For long time scales (months), Williams et al. [211] suggest that the storage and entrainment terms are much less important than advection and surface uptake. In turn, on shorter time scales, advection terms are often neglected [Yi et al., 24]. Consequently, the transition from daily to larger time scales should be carefully investigated. The work presented here attempts to bridge between short and long time scales influencing the CO 2 budget and it enables us to analyze and quantify the error made by neglecting the horizontal advection terms when the CO 2 budget is analyzed over daily time scales. [9] In the next section, by taking into account CO 2 advection, we enlarge the theoretical mixed-layer framework presented by Pino et al. [212]. In the third section, by selecting observations during two convective days with different CO 2 advection rates, we analyze the ability of a mixed-layer model (MLM) to reproduce the evolution of the boundary layer variables. In sections 4 and 5, the mixed-layer formulation presented in section 2 is applied to several boundary layers based on the observations but with different initial inversion strengths and the potential temperature FA lapse rates. The paper ends with some conclusions. 2. Mixed-Layer Equations of the CO 2 Budget [1] In convective conditions, the diurnal evolution of well-mixed CO 2 mixing ratio in the mixed layer is driven by surface and entrainment fluxes and advection. Despite its simplicity, mixed-layer theory, that assumes an instantaneous mixing of scalars in the mixed layer, is a very appropriate modeling tool to understand the diurnal variations of the budget of CO 2 or other atmospheric compounds [Vilà-Guerau de Arellano et al., 24; Vinuesa and Vilà-Guerau de Arellano, 25; Ouwersloot et al., 212]. [11] The following derivation of the sensitivity of CO 2 mixing ratio to uncertainties in the mixed-layer variables follows closely the derivation by Pino et al. [212]. If only CO 2 advection is included in the analysis, the time rate of change

3 of the vertically integrated, from the surface to the boundary layer depth, CO 2 mixing ratio in the mixed-layer times the boundary layer depth reads (Ch) =w c s + C h ) + A c h, where the overbar denotes Reynolds average, prime values indicate deviations from the mean quantities, and the subscript always refers to the initial (morning) conditions (at time ). C is the mixed-layer value of the CO 2 mixing ratio, assumed constant with height in the frame of mixed-layer theory; h and w c s denote the boundary layer depth and the Reynolds averaged CO 2 turbulent surface flux at time t, respectively; A c (= U/@x + V/@y, whereu and V are the two components of the horizontal wind velocity in the mixed layer) is the horizontal CO 2 advection rate in the mixed layer. Finally, the CO 2 mixing ratio in the FA, C FA, just above the inversion, is given in the MLM by C FA = C FA + c (h h )+ Z t A FA c dt, (2) where Ac FA is the horizontal advection rate of CO 2 in the FA, and c is the vertical gradient of CO 2 mixing ratio in the FA; both considered constant in time during the day if not otherwise stated. [12] From left to right, each term of equation (1) physically represents the bulk variation of the total amount of CO 2 in the mixed layer, due to the surface, entrainment, and advection fluxes. It is important to note that the second term at the right-hand side, associated with entrainment flux, only contributes to the evolution of C when h is not constant. [13] Following Kowalski and Serrano-Ortiz [27], CO 2 variables are described by using mixing ratio because it is the only scalar intensity variable conserved through compression/expansion processes and by water vapor diffusion. Consequently, the CO 2 surface flux is given in ppm m s 1. Mass fraction CO 2 surface flux units (mg m 2 s 1 ) are only included in some plots to facilitate the interpretation of the results to the readers not used to mixing ratio units. The conversion factor is 1 ppm m s mg m 2 s Sensitivities of the CO 2 Mixing Ratio [14] Substituting (2) into (1) and integrating on time from to t leads to Ch C h = Z t w c s dt + c 2 (h h ) A c Ac FA Z t h dt + + C FA (h h )+Ac FA (t )h. where constant CO 2 advection rates are assumed. Therefore, the CO 2 mixing ratio is expressed as h C = C h + c 2h (h h ) 2 + t h hw c s i + + A c Ac FA Z t h dt + C FA 1 h (4) + Ac FA (t ), h h R where hw c t s i = w c s dt /(t ) is the time averaged CO 2 surface flux. As can be concluded from the analysis of (3) the fourth term on the right-hand side of equation (4), CO 2 mixing ratio depends on the history of the boundary layer depth evolution if advection rates in the mixed layer and in the FA are unequal. [15] Assuming the independence of the variables, the sensitivities of the CO 2 mixing ratio to uncertainties in the boundary layer variables are obtained by taking partial derivatives of C with respect to the corresponding variables. This yields to the following sensitivity expressions: = h h, (5) FA =1 h h, (6) = (h h ) 2, c 2 c s i = t, (8) h = c + 1 h i c h + C C FA, = c h c = 1 h Z FA =(t ) 1 Z t h h dt, (1) h (C FA C ) c h 2 2 (t )hw c s i (A c A FA c ) h dt, (11) h dt. Z t (12) [16] The first five sensitivities (5) (9) are independent of CO 2 advection and were analyzed by Pino et al. [212]. The last three sensitivities, due to the integral term, depend on the history of the boundary layer depth, and the last one is affected by the CO 2 advection. Moreover, only the last sensitivity depends on the value of CO 2 surface flux. Consequently, the analysis of the other sensitivities is valid for any value of w c s. It is also important to note that these sensitivities depend on the initial conditions (C, C FA, h ), consequently, to know the sensitivities or the corresponding uncertainties at some time, these initial conditions have to be known. Equations (5) (12) are used in section 4 to estimate the effect of uncertainties in different boundary layer variables on the uncertainties in the CO 2 mixing ratio. [17] As can be noticed, vertical advection is not considered in the analysis. Regarding this point, and taking into account that MLM assumes well-mixed variables in the mixed layer, /@z = and vertical advection cannot be explicitly included in A c. Despite this fact, the effect of subsidence in C evolution can still be analyzed by using mixedlayer theory. If subsidence (w s ) is included, the entrainment 9378 velocity reads w e w s. Consequently, subsidence changes the entrainment CO 2 flux by modifying entrainment velocity, and hence, it also changes the evolution of the CO 2 mixing ratio and the inferred CO 2 surface flux. However, as Ouwersloot and Vilà-Guerau de Arellano [213] show, if subsidence is considered, an analytical formula for the CO 2 mixing ratio evolution cannot be derived by separation of

4 variables. Therefore, a similar expression to equations (5) (12), which estimates how subsidence influences the CO 2 mixing ratio, cannot be obtained by the methods used here. [18] Regarding the assumption of time and height independence of A c and Ac FA for the sensitivity analysis, in reality, having constant advection during the whole day is rather an exception than a rule. In fact, this would require that the product of large-scale horizontal wind and horizontal gradient of C would remain constant during the day. For this reason, MLM can prescribe time varying CO 2 advection rates, and, in fact, it is assumed to correctly reproduce the observations recorded for one of the analyzed days (12 March 24). Time and height independent CO 2 advection is however used for simplicity in the derivation of expression (3). If the CO 2 advection rate contribution in the mixed layer varies with time, more terms with history dependence will appear at the analysis. [19] Height and time independence of c is also an approximation, which is not generally true. In the absence of advection in the FA the time independence of c can be justified on diurnal time scales since changes in the FA CO 2 mixing ratio through other processes than advection are usually much slower than the changes occurring in the mixed-layer concentration. However, even if the FA concentration profile does not change with time, it does not have to be exactly linear with height and hence c could also depend on height. Constant c has been successfully used in MLM numerical experiments [Vilà-Guerau de Arellano et al., 24] and often it is reasonably satisfactory and justified approximation on a diurnal time scale Uncertainties of the CO 2 Mixing Ratio [2] The set of equations (5) (12) allow us to quantify the contribution of the instantaneous uncertainty of each variable to the uncertainty of the CO 2 mixing ratio. Assuming that the uncertainties are small and independent (for instance, h estimation does not depend on the uncertainty on h ), the uncertainties in the CO 2 mixing ratio due to instantaneous uncertainties of the studied values, u totins (C), reads [Taylor, 1997]: v u u totins (C) = t X ˇˇˇˇ uins() ˇ, (13) where u ins () is the uncertainty at the time of measurement of each individual variable influencing the CO 2 mixing ratio: C, C FA, c, hw c s i, h, h, A c,orac FA. [21] However, due to the integral term appearing in equation (12), past uncertainties in h need to be taken into account in the estimation of CO 2 mixing ratio uncertainty. We, therefore, need a different expression to quantify this error. This uncertainty based on past errors, u past, cannot be calculated from equation (13). Notice also that no general estimation about it can be given without knowledge or assumptions about the evolution of the uncertainty in the boundary layer depth. If the evolution of this uncertainty is known, by using calculus of variations [Gelfand and Fomin, 2], the CO 2 mixing ratio uncertainty due to past uncertainties in h is estimated as ˇ ˇAc A FA ˇ Z t c u past (C) = u his (h) dt, (14) h where u his (h) is the time dependent uncertainty in the boundary layer depth; that is, the historical uncertainty on h before time t Sensitivities of the Averaged Inferred CO 2 Surface Flux [22] The sensitivity analysis presented in the previous section for the CO 2 mixing ratio is also applied for the inferred CO 2 surface flux. The CO 2 surface flux obtained from equation (4) reads hw c s i = 1 h Ch C h C FA t t (h h ) c 2 (h h ) 2 + Z t + Ac FA A c hdt A FA c h, (15) which is equal to the time averaged prescribed surface flux if the values of C and h obtained with the MLM are provided. [23] By taking the partial derivatives of this equation with respect to different boundary layer variables, the following sensitivity equations are c s i = h t, c s i FA = h h t, c s c = (h h ) 2 2(t ), c s i = h t, c s i = 1 h i C FA C + c (h h t c s i = 1 Z t h c t c c s FA c = 1 Z t h dt h, (22) t = 1 h i C C FA c (h h ) A FA c. (23) t Note that the time dependent CO 2 mixing ratio and the CO 2 advection rate, at FA, only contribute to the last sensitivity. The other sensitivities depend on h, t, and the initial conditions. Consequently, the analysis of these sensitivities is valid for any value of C or CO 2 advection rate. Moreover, to assure the independence of the variables for calculating the uncertainties, in this formulation, it is assumed that C is measured independently. [24] In a similar manner, uncertainty in the inferred CO 2 surface flux is also affected by the history of the uncertainties in the boundary layer depth. If the time evolution of h-uncertainty, u his (h), is known, by using variational calculus, this uncertainty reads u past hw c s i ˇ = ˇA FA c t A cˇˇ Z t u his (h) dt. (24) [25] The uncertainty of the CO 2 surface flux, u totins hw c s i, due to instantaneous uncertainties in the different 9379

5 boundary layer variables considered is calculated by using the sensitivities shown at equations (16) (23) as v u totins hw c s i u = t c s i 2 ˇ uins(), (25) where u ins () is the instantaneous uncertainty of each considered variable influencing the inferred CO 2 surface flux. 3. Observations and Mixed-Layer Numerical Experiments [26] Heat fluxes, potential temperature, CO 2 mixing ratio, and flux for both studied days (25 September 23 and 12 March 24) were measured at different heights on the 213 m tower at the Cabauw site (Netherlands), where prevailing winds come from SW [Beljaars and Bosveld, 1997; Hurley and Luhar, 29]. Additionally, a radar wind profiler determined the boundary layer depth. For the characteristics of the site and the calibration of the CO 2 observations, see Hensen et al. [1997]; Beljaars and Bosveld [1997]; Bosveld et al. [24]; Werner et al. [26]; and Vermeulen et al. [211]. [27] The two convective days selected, 25 September 23 and 12 March 24 present z/l.5 and 1 at 5 m level, respectively, where L is the Monin-Obukhov length, but different CO 2 advection rates. Then, the MLM sensitivity expressions presented before are used to analyze how uncertainties in the boundary layer processes and advection influence the temporal evolution of the CO 2 mixing ratio and the inferred surface flux. [28] The two selected days are at the initial and last stages of the growing season; consequently, the minimum observed CO 2 surface flux during the year is larger than that analyzed here. Gerritsen [212] studied CO 2 mixing ratio and flux over observations taken at the Cabauw tower from July 27 to June 28, and found that during this period, the minimum mean 5 m height vertical turbulent flux has values between.4 and.7 mg m 2 s 1. For one of the studied days, the minimum observed value of the CO 2 surface flux lays in this interval. Anyway, as was mentioned in the previous section, it is important to note that only one of the sensitivities depends on the CO 2 surface flux. Therefore, most of the presented analysis is valid for any value of the CO 2 surface flux, that is, for any season, provided that the atmospheric boundary layer is characterized by convective conditions September 23 [29] The first analyzed day, already studied by Casso Torralba et al. [28]; Pino and Vilà-Guerau de Arellano [21];and Pino et al. [212], was a well-characterized convective day with clear skies, except some scattered clouds at the afternoon and approximately constant light winds. Surface fluxes of heat and CO 2 are shown in Figure 2 by Casso-Torralba et al. [28]. MLM numerical experiment and the sensitivity analysis have been previously done for this day by Pino et al. [212]. However, advection was not considered in their study since its contribution to the CO 2 budget was small on this day [Casso-Torralba et al., 28]. To complete the analysis of this day, the mixed- Table 1. Based on the Observations Taken at Cabauw on 25 September 23, Initial and Prescribed Values Used for MLM to Calculate the Temporal Evolution of the Boundary Layer Depth, Mixed-Layer Value of the Scalars (, q, and C), and Their Corresponding Jump at the Inversion (, q, and C) a Initial (Subscrip) and Prescribed Values h (m) 12 w s (m s 1 ) Potential temperature w s (93 17 LT) (K m s 1 ).8 sin ˇ.3 (K) up (K) 3.5 (K m 1 ) h < 95 m h > 95 m (t 54) 27 Specific humidity w q s (8 2 LT) (g kg 1 ms 1 ).87 sin t 432 q (g kg 1 ) 4.3 q (g kg 1 ).8 q (g kg 1 m 1 ) Carbon dioxide w c s (1 173 LT) (ppm m s 1 ).1 sin (t 72) 27 C (ppm) 415 C (ppm) 4 c (ppm m 1 ) 31 3 A c (ppm s 1 ) Ac FA (ppm s 1 ) a i is the FA lapse rate of each variable i, ˇ is the entrainment to surface flux ratio and w s is the subsidence velocity. Time integration ranges from zero to 432 s. layer experiments conducted by Pino et al. [212] are here reproduced, also with CO 2 advection included in the MLM in order to further analyze its contribution to CO 2 mixing ratio evolution. [3] Initial values and temporal evolution of the surface fluxes prescribed for the MLM numerical experiment are presented in Table 1. These values are prescribed following Casso-Torralba et al. [28] and Pino et al. [212], who based the choice of the values on the observations taken at the Cabauw measurement site a8 Local Time (LT, UTC+2). The MLM numerical experiments run from 8 to 2 LT, which was the period of positive latent heat fluxes (sunrise/sunset occurred a73/1932 LT). The diurnal evolution of h, andco 2 mixing ratio obtained from the MLM numerical experiment are presented in Figure 1 together with the observations taken by the radar wind profiler and along the meteorological tower. Note that CO 2 uptake is imposed in the model based on the observations, that is, no coupled land surface model [Monteith and Unsworth, 27] was considered. Consequently, the CO 2 advection has no effect on the other boundary layer variables except the CO 2 mixing ratio and therefore the evolution of h is exactly the same as obtained by Pino et al. [212], approximately fitting the observations (see Figure 1a). During the morning development of the convective boundary layer, the MLM prescribe a lower than observed mixed-layer growth. Later in the afternoon (around 14 and 18 LT), the peak 938

6 a) b) Figure 1. Time evolution of the (a) boundary layer depth and (b) CO 2 mixing ratio observed (symbols) and simulated by MLM (lines) during 25 September 23. Observations of the boundary layer depth were taken by a radar wind profiler. CO 2 mixing ratio was measured at different heights of the Cabauw tower. Initial and prescribed values of the MLM numerical experiment are shown in Table 1. day. Constant wind direction might explain that air with high CO 2 mixing ratio was advected from the Ruhr area (Germany), which is a highly industrialized region. Back trajectories of the air arriving to Cabauw during the afternoon on 12 March 24 calculated by using MM5 mesoscale simulations corroborate this fact. This second day was chosen to be analyzed because it actually presents significant CO 2 advection rate unlike the first analyzed day. It is also used to compare the sensitivities between different days and to show the applicability of the framework on different days. [33] Casso-Torralba et al. [28] concluded that the previous night was weakly stratified due to the constant and high velocity winds. The day was clear, but occasional clouds were observed. Satellite pictures show that the region south of Cabauw was largely covered with clouds, but the Cabauw region was relatively clear. [34] Diurnal evolution of boundary layer variables were again simulated with the MLM from 63 until 1845 LT (UTC+1), when positive latent heat flux was observed (sunrise/sunset occurred a7/1838 LT). Initial and prescribed values used in the MLM numerical experiment are based on the observations taken at Cabauw and they are presented in Table 2. The prescribed advection is based on the CO 2 budget analysis made by [Casso-Torralba et al., 28]. By using the observations taken at the tower they calculated Table 2. Same as Table 1 for the Observations Taken at Cabauw (Netherlands) on 12 March 24 a63 LT a reflectivity observed by the wind profiler is less well defined when more vigorous convection and clouds create many strong humidity gradients [Cohn and Angevine, 2]. [31] To study how advection modifies the modeled CO 2 mixing ratio evolution, an additional numerical experiment with constant CO 2 advection of A c =51 4 ppm s 1 is prescribed from 8 to 2 LT. This advection is deliberately set to be larger than the actual average advection on this day [Casso-Torralba et al., 28] so that the contribution of advection can be seen more clearly in the sensitivity analysis presented in sections 4 and 5. The CO 2 mixing ratio evolution including advection is shown with a solid line in Figure 1b. It deviates from the observations and from the nonadvective MLM numerical experiment because of the prescribed CO 2 advection in the mixed layer. Comparison of surface uptake, entrainment (w e C, wherew e is the entrainment velocity) and advection (A ch ) fluxes reveals that entrainment flux is four times larger than the other contributions from 113 to 13 LT, when the mixed layer grows faster (not shown). Due to the constant value prescribed for A c, advective flux is proportional to h. Consequently, during the afternoon, when surface and entrainment fluxes are low and h is large, advection flux drives the evolution of the CO 2 mixing ratio March 24 [32] The second analyzed day was characterized by low heat and large CO 2 advection rates [Casso-Torralba et al., 28]. Moisture ranged from 3 to 4 g kg 1 and southeasterly winds of about 8 1 m s 1 were measured during the 9381 Initial (Subscrip) and Prescribed Values h (m) 2 w s (m s 1 ) Potential temperature w s, ( LT) (K m s 1 ).9 sin (t 81) 234 ˇ.25 (K) (K) 2. (K m 1 ) 81 3 Specific humidity w q s, ( LT) (g kg 1 ms 1 ).28 sin t 441 q (g kg 1 ) 3.5 q (g kg 1 ).9 q (g kg 1 m 1 ) Carbon dioxide w c s, (83 17 LT) (ppm m s 1 ).25 sin C (ppm) C (ppm) 15.5 c (ppm m 1 ) A c (ppm s 1 ) t < 36 s (t 72) t 36 s t < 9 s s t < 144 s 4 (t-9) s t < 18 s s t < 234 s 4 (t 234) t 234 s 51 4 A FA c (ppm s 1 ) a A c,anda FA c are estimated accordingly to Casso-Torralba et al. [28]. Time integration ranges from zero to 441 s.

7 a) b) c) advection is mainly horizontal between 113 and 153 LT, but during other times, vertical advection might have played an important role. Additionally, by using MM5 mesoscale model [Grell et al., 1994], Kaikkonen [212] analyzed the wind pattern during that day. Simulation results indicate that a front with significant humidity difference reached the Cabauw region around the time when the CO 2 mixing ratio jump was observed. A front with notable CO 2 difference entering the Cabauw region would be a likely explanation to the sudden jump in the CO 2 mixing ratio. This would also cause a nonconstant horizontal CO 2 gradient, which could explain the poor correlation of advection with the horizontal wind speed at the end of the day Figure 2. Time evolution of the (a) boundary layer depth, (b) potential temperature, and (c) CO 2 mixing ratio observed (symbols) and simulated by MLM (lines) during 12 March 24. Observations of the boundary layer depth were taken by a radar wind profiler. Potential temperature and CO 2 mixing ratio were measured at different heights of the Cabauw tower. Initial and prescribed values of the MLM numerical experiment are shown in Table 2. In Figure 2c, only the results for the advective case are shown. a) storage and flux divergence terms of the CO 2 budget. CO 2 advection is obtained as a residual term. Diurnal evolution of h, potential temperature and CO 2 mixing ratio obtained from the MLM numerical experiment are shown in Figure 2 together with the observations taken at Cabauw. During the morning, due to fluctuations on q-variance, wind profiler measurements might have problems to differentiate between several peaks of the reflective index [Cohn and Angevine, 2], and this may explain the observed increase between 9 and 13 LT of the boundary layer depth, not affecting the mixed-layer potential temperature (well mixed) and the CO 2 mixing ratio, and not captured by the MLM. During the afternoon, the observations show a decrease in the boundary layer depth, due to the decrease of surface heat fluxes and to large-scale subsidence. As a consequence, an increase in the CO 2 mixing ratio occurred because less air with low CO 2 mixing ratio is entering from the FA into the mixed layer, and the surface CO 2 uptake is less effective (see the third term of equation (4)). Taking into account that the prescribed subsidence in the MLM for this case is zero, MLM is not able to reproduce the decrease in the boundary layer depth during the afternoon and, as a consequence, MLM underestimates the CO 2 mixing ratio from 14 LT. [35] Another possible explanation for the sudden increase of CO 2 mixing ratio during the afternoon was pointed out by Kaikkonen [212]. By comparing the evolution of the total advection and the height integral of the horizontal wind speed [Yi et al., 2] during that day, he found that Figure 3. Time evolution of the sensitivity of CO 2 mixing ratio to uncertainties in CO 2 advection in five different cases based on the observations taken on (a) 25 September 23 and (b) 12 March 24. In the figure legend, if the potential temperature lapse rate or inversion strength is not shown, the value for the control case for each analyzed day (see text) applies. In Figure 3a, the vertical lines separate the early morning period (h is almost constant), the morning transition (rapid growth of h), and the afternoon period (lower growth rate). b) 9382

8 Figure 4. c based on the observations taken on 25 September 23 averaged between 14 and 16 LT as a function of initial inversion strength and FA potential temperature lapse rate.the control case is shown with a star. The rest of the symbols indicate the studied extreme cases: open circle, =.2, and 5 K with = and black circles, =1 3, and 1 2 Km 1 with =3.5K. 4. Quantification of the Uncertainties Influencing the CO 2 Mixing Ratio [36] Sensitivity of CO 2 mixing ratio to instantaneous uncertainties associated with boundary layer dynamics or CO 2 characteristics is now quantified by using equations (5) (12). Following Pino et al. [212], thechange in the sensitivities with different atmospheric conditions is studied by performing a set of MLM numerical experiments where the initial potential temperature jump at the inversion,, and FA potential temperature lapse rate,, are systematically changed. In all these numerical experiments, FA potential temperature lapse rate, mixed-layer CO 2 advection rate, and CO 2 surface flux are kept constant with height and time. The other atmospheric variables considered correspond to the conditions observed on 25 September 23 and 12 March 24 shown respectively in Tables 1 and 2. Constant CO 2 advection rate is only justified by the observations taken on 25 September 23 but not for the other analyzed day. As mentioned before, if not constant CO 2 advection rate in the mixed layer was considered, new terms with history dependence would appear in equation (4), but in our opinion, it would not significantly modify the main physical interpretation of our uncertainty analysis. [37] To analyze this sensitivity, expressed by equation (1), it is important to first study the evolution of the boundary layer depth. Pino et al. [212] showed (see their Figure 3) that during the morning for the cases based on the observations taken on 25 September 23, h growth is almost suppressed except for the cases with small inversion strength ( <.5K). From 12 LT, has decreased for all the cases below 1.5 K, and the mixed layer can grow faster except if the FA potential temperature lapse rate is large. Differences of more than 15 m in the boundary layer depth are found at the end of the simulation between the cases with the largest and the smallest. This fact affects the evolution of C due to advection (see the last term of equation (1)), the effect being larger during the afternoon when the largest values of the boundary layer depth occurs. The evolution of h for the other analyzed day is similar for the five analyzed cases. [38] To facilitate the interpretation of the results, the influence of and on the CO 2 mixing ratio sensitivity to CO 2 advection in the mixed layer, /@A c is first considered in five different cases for each analyzed day. The control case is the advective case presented in Figure 1 (2) but with =.36 (.8) Km 1, A c =51 4 (.14) ppm s 1, and w c s =.1 (.25) ppm m s 1 constant in time for 25 September 23 (12 March 24). In the other four cases, the prescribed value for or in the control case is changedto.2or5kand1 3 or 1 2 Km 1, respectively. 9383

9 1 x E 3 8E 3 8E 3 7E 3 7E 3 7E 3 6E 3 6E 3 6E 3 6E E 3 5E 3 5E 3 5E E 3 4E 3 4E 3 4E E 3 3E 3 3E 3 3E E 3 2E Figure 5. Same as Figure 4 for /@h.theco 2 advective (A c =51 4 ppm s 1, colored contours) and nonadvective (A c =ppm s 1, black line contours) cases are shown. Note that this sensitivity is negative, so the largest sensitivities in absolute value are shown in blue colors. [39] Time evolution of /@A c is shown in Figure 3 for the control case and for the four extreme cases based on the observations of the two analyzed days: (Figure 3a) 25 September 23 and (Figure 3b) 12 March 24. During the early morning and at the late afternoon when the boundary layer depth is almost not growing, this sensitivity increases as a function of t for most of the analyzed cases because more CO 2 is advected when time passes if the boundary layer depth is almost constant. The decreasing period shown in Figure 3, occurring during the morning transition, can be explained by considering the dimensionless time derivative of /@A c (see equation (1)) d c =1 1 h 2 dh dt Z t h dt. (26) [4] The sensitivity decreases when this expression is negative and this is the case when the time integral of the boundary layer depth and the instantaneous mixed-layer growth rate are large but the boundary layer depth itself is small. The decrease occurs at the morning transition (see Figures 1a, 2a, and Pino et al. [212, Figure 3a]) when the boundary layer depth is still small but its growth rate increases substantially. Before the morning transition, the mixed layer hardly grows and so the value of the time integral term is roughly the elapsed time times the boundary layer depth. Therefore, the decrease in the sensitivity is amplified with large and small since a large delays the morning transition and thus increases the integral 9384 term, while a small enhances the growth rate of the mixed layer. If small and large values are considered the decrease in the sensitivity is small or even absent. Regarding the differences observed between the 2 days, they are based on different mixed-layer growth rate. In the cases based on the observations on 12 March 24, a decrease of /@A c is observed during the morning transition if is not too small (see Figure 3b). [41] Figure 4 shows the sensitivity of CO 2 mixing ratio to uncertainties in A c,(/@a c, see equation (1)), time averaged between 14 and 16 LT for the cases based on the observations taken on 25 September 23 as a function of and. Taking into account that this sensitivity does not depend on the CO 2 advection rates or CO 2 surface flux considered, the Figure 4 is valid for the whole range of dynamics conditions and the CO 2 initial characteristics (see Table 1). The largest sensitivities are obtained with low values of and large values of. This is in agreement with the conclusion made above about the evolution of this sensitivity. Consequently, the behavior seen in this figure can be understood with the considerations given above. Similar results are found for the cases based on the other selected day. Summarizing, the sensitivity of CO 2 mixing ratio to CO 2 mixed-layer advection is enhanced when the boundary layer depth increases continuously but its growth rate is moderate. [42] If A c Ac FA, advection modifies the sensitivity of CO 2 mixing ratio to uncertainties in the time-dependent boundary layer depth with respect to the nonadvective case

10 a) b) Figure 6. Time evolution of the uncertainty of the CO 2 mixing ratio due to the past uncertainties in the boundary layer depth (u past (C), see equation (14)) for the control case based on the observations taken on (a) 25 September 23 (A c = ppm s 1 ) and (b) 12 March 24 (A c =.14 ppm s 1 ). Solid lines show the uncertainty when the past boundary layer depth uncertainty is calculated as the difference between the observed and simulated values (see Figure 1a). Dashed (dotted) lines represent the CO 2 uncertainty in the case of a constant uncertainty of 1 m (1%) in the boundary layer depth during the whole day. (see equation (12)). In general, CO 2 mixing ratio sensitivity to uncertainties in the boundary layer depth is large during the morning, before 12 LT, and significant differences in the value of the sensitivity can be found for the different studied cases. When the mixed layer grows, the sensitivity decreases in absolute value, and at the end of the afternoon, this sensitivity is approximately c /2 for all the studied cases. It is important to note that the values obtained for this sensitivity, due to entrainment, depends on the CO 2 advection rate differences between mixed layer and FA [Laubach and Fritsch, 22]. In our opinion, it is not clear how large/small this advection difference usually is or how large it can be. If the same advection rates occur in the mixed layer and in the FA, the advection analysis of this sensitivity is similar to the nonadvective case analyzed by Pino et al. [212]. [43] Figure 5 shows the sensitivity of CO 2 mixing ratio to boundary layer depth (/@h, equation (12)) for the cases based on the observations taken on 25 September 23 time averaged between 14 and 16 LT as a function of potential temperature lapse rate and initial inversion strength. Advective case is shown with colored contours while the corresponding sensitivity in the nonadvective case is plotted with black line contours. Notice that this sensitivity is always negative for the studied cases and the largest sensitivities for the advective case are plotted in blue in the figure. In both cases the highest sensitivities occur when and are large, which is the case with the shallowest mixed layer. This leads to larger CO 2 mixing ratios and to larger sensitivities. Sensitivities are generally more negative in the advective case due to the integral term of equation (12). Additionally, the sensitivity is also nearly independent of when is small. For small potential temperature lapse rates, the boundary layer depth during the afternoon can have large values independently of the value of. Consequently, the term divided by h 2 in equation (12) is small, and the sensitivity can be approximated by c /2. When larger lapse rates are considered, h is smaller, and this term cannot be neglected, appearing differences in the sensitivities depending on the value of considered. [44] Before studying how instantaneous uncertainties on the different variables influence CO 2 mixing ratio uncertainty by using equation (13), the uncertainty associated with past uncertainties on the evolution of the boundary layer depth shown in equation (14) will be analyzed. Figure 6 shows the time evolution of this uncertainty, u past (C), for the control case based on both analyzed days. For the studied days, the uncertainty evolution, u his (h), is assumed to be the absolute value of the difference between the boundary layer depth observed and obtained by the MLM numerical experiment shown in Figures 1a and 2a. Initial time,, used in equation (14) was 11 or 71 LT, which was the time of the first observation of the boundary layer depth on 25 September 23 and 12 March 24, respectively. Figure 6a (solid line) shows that, for the case based on the observations taken on 25 September 23, the maximum uncertainty is larger than 1 ppm (1 ppm of atmospheric CO 2 is equivalent to 7.81 Gt). There is a maximum before 12 LT, that is, before the rapid growth of the mixed layer occurring at the morning transition (see Figure 1a). This is due because before this time, the mixed layer hardly Table 3. Estimated Magnitudes of Uncertainties in Different Variables, u ins (). The Uncertainty u ins (hw c s i) Is Only Used in the Calculation of CO 2 Mixing Ratio Uncertainty While u ins (C) Is Employed in the Calculation of the Inferred CO 2 Surface Flux Uncertainty Property Uncertainty, u ins () Value C (ppm) 1 C FA (ppm) 1 C (ppm) 1 c (ppm m 1 ) 1 4 h (m) 1 h (m) 1 hw c s i (ppm m s 1 /mgm 2 s 1 ).5/.9 A c (ppm s 1 ) 1 4 Ac FA (ppm s 1 )

11 a) b) Figure 7. Time evolution of the contributing uncertainties to the CO 2 mixing ratio due to uncertainties in different boundary layer variables ıc = j/@j u ins () for the case based on the observations on (a) 25 September 23 and (b) 12 March 24. Uncertainties used in the calculation are shown in Table 3. grows but there is a clear deviation between the observed and the MLM boundary layer depth which increases the result of the integral appearing in expression (14). Between approximately 12 and 14 LT, the boundary layer depth grows faster decreasing u past (C). During the late afternoon, boundary layer depth is again almost constant and u past (C) increase because of the cumulative effects of the integral included in expression (14) and because, specially around 14 LT, and at the end of the numerical experiment, MLM results deviate from the observed boundary layer depth (see Figure 1a). [45] For comparison, the dashed and dotted lines in Figure 6 represent the CO 2 mixing ratio uncertainty due to a constant past uncertainty of 1 m in the boundary layer depth (u his (h) = 1 m, the range resolution of wind profiler located at the Cabauw site) or a past 1% uncertainty of the boundary layer depth (u his (h) =.1h) during the whole numerical experiment, respectively. By including these two values of u his (h) in the integral of equation (14), and comparing the result with expressions (8), and (1), respectively, it can be concluded that the dashed and dotted lines represent 1 ˇˇAc Ac FA ˇ /@hw c s i and.1 ˇˇAc Ac FA ˇ /@Ac. The CO 2 mixing ratio uncertainty remains around 1 ppm also for the constant uncertainty of 1 m, but is lower for an uncertainty of 1%, because, as can be concluded from Figure 1, the uncertainties of this last case are smaller because h remains below 1 m. [46] Figure 6b shows the time evolution of this uncertainty for the case based on the other analyzed day. For 12 March 24, the uncertainties are much larger than on 25 September 23 (see Figure 6a). This is because advection was larger and the mixed layer shallower on 12 March 24 (see expression (14)). Besides the period between 13 and 113 LT, the uncertainties shown in Figure 6b are overestimated since a maximum value of CO 2 advection was used during the whole day. The maximum, occurring around 13 LT, is however not overestimated. This shows that the past uncertainties in the boundary layer depth can indeed cause significant uncertainties on the CO 2 mixing ratio even on a diurnal time scales especially when the mixed layer is shallow and the difference in CO 2 advection between the mixed-layer and the FA is large. [47] To estimate the CO 2 mixing ratio uncertainty, a plausible magnitude of uncertainty has to be assumed for all the variables influencing CO 2 mixing ratio, u ins (). Thesame uncertainties in the different variables are considered for both analyzed days. The instrumental uncertainty in the initial CO 2 mixing ratio, u ins (C ), is assumed to be of the order of 1 ppm. In general, CO 2 mixing ratios can be measured more accurately but the average mixed-layer value of initial CO 2 mixing ratio can have larger uncertainties, for example if the measurements are taken only at few heights before the morning transition. The initial CO 2 mixing ratio at the FA is also assumed to have a similar uncertainty of 1 ppm, even though it is generally more challenging to measure mixing ratios in the FA than in the mixed layer. CO 2 mixing ratio lapse rate in the FA is usually very small or even zero (see Table 2 and Vilà-Guerau de Arellano et al. [24]). We assume its uncertainty to be one order of magnitude less than the value prescribed for the case based on 25 September 23, u ins ( c )=1 4 ppm m 1. The uncertainty in the initial and instantaneous boundary layer depth is estimated to be of the order of 1 m throughout the day. Notice that this uncertainty depends on the method used to determine boundary layer depth [Seibertetal., 2; Seidel et al., 21]. For example Driedonks [1982] calculated that the uncertainty made by acquiring the boundary layer depth from an individual radiosounding can be as much as 1 m. Moreover, this is the range resolution of the wind profiler operating at the Cabauw site. In general, the uncertainty can also be time dependent, but for simplicity, a constant uncertainty is considered here. Error in the CO 2 surface flux is estimated as an average deviation of the observed values from the prescribed value used in the MLM numerical experiment. For 25 September 23, this gives an uncertainty of.5 ppm ms 1 (.9 mg m 2 s 1 ), which is half of the diurnal maximum. Advection in the mixed layer and in the FA are often neglected when they are relatively small so the uncertainty in these variables is estimated to be 1 4 ppm s 1, which would correspond to a relatively low advection rate. The estimated magnitudes of uncertainties are summarized in Table