A mass-weighted atmospheric isentropic coordinate for mapping chemical tracers and computing inventories

Abstract. We introduce a transformed isentropic coordinate Mθe, defined as the dry air mass under a given equivalent potential temperature surface (θe) within a hemisphere. Like θe, the coordinate Mθe follows the synoptic distortions of the atmosphere, but unlike θe, has a nearly fixed relationship with latitude and altitude over the seasonal cycle. Calculation of Mθe is straightforward from meteorological fields. Using observations from the recent HIPPO and Atom airborne campaigns, we map the CO2 seasonal cycle as a function of pressure and Mθe, where Mθe is thereby effectively used as an alternative to latitude. We show that the CO2 cycles are more constant as a function of pressure using Mθe as the horizontal coordinate compared to latitude. Furthermore, short-term variability of CO2 relative to the mean seasonal cycle is also smaller when the data are organized by Mθe and pressure than when organized by latitude and pressure. We also present a method using Mθe to compute mass-weighted averages of CO2 on a hemispheric scale. Using this method with the same airborne data and applying corrections for limited coverage, we resolve the average CO2 seasonal cycle in the Northern Hemisphere (mass weighted tropospheric climatological average for 2009–2018), yielding an amplitude of 7.8 ± 0.14 ppm and a downward zero-crossing at Julian day 173 ± 6.1 (i.e., late June). Mθe may be similarly useful for mapping the distribution and computing inventories of any long-lived chemical tracer.



Introduction
The spatial and temporal distribution of long-lived chemical tracers like CO2, CH4, and O2/N2 typically includes regular seasonal cycles and gradients with latitude (Conway and Tans, 1999;Ehhalt, 1978;Randerson et al., 1997;Rasmussen and Khalil, 1981;Tohjima et al., 2012). These patterns are evident in climatological averages but are potentially distorted on short 25 time scales by synoptic weather disturbances, especially at middle to high latitudes (i.e. poleward of 30° N/S) (Parazoo et al., 2008;Wang et al., 2007). With a temporally-dense dataset such as from satellite remote sensing or tower in-situ measurements, https://doi.org/10.5194/acp-2020-841 Preprint. Discussion started: 19 August 2020 c Author(s) 2020. CC BY 4.0 License. climatological averages can be created by averaging over this variability. For temporally sparse datasets such as from airborne campaigns, it may be necessary to correct for synoptic distortion.
One method for reducing the impact of synoptic variability is to evaluate tracer data on isentropic coordinates, i.e. based on 30 potential temperature θ (Hess, 2005;Miyazaki et al., 2008;Parazoo et al., 2011Parazoo et al., , 2012. As air parcels move with synoptic disturbances, θ and the tracer tend to be similarly displaced so that the θ-tracer relationship is relatively conserved (Keppel-Aleks et al., 2011). Furthermore, vertical mixing tends to be rapid on θ surfaces, so θ and tracer contours are often nearly parallel (Barnes et al., 2016). On the other hand, θ varies greatly with latitude and altitude over seasons due to changes in heating and cooling with solar insolation, which complicates the interpretation of θ-tracer relationships on seasonal time scales. 35 During analysis of airborne data from the HIAPER Pole-to-Pole Observations (HIPPO) (Wofsy, 2011) and the Atmospheric Tomography Mission (ATom) (Prather et al., 2018) airborne campaigns, we have found it useful to transform potential temperature into a mass-based unit, Mθ, which we define as the total mass of dry air under a given isentropic surface in the hemisphere. In contrast to θ, which has large seasonal variation, Mθ has a more stable relationship to latitude and elevation, while varying in parallel with θ on synoptic scales. Also, for a tracer which is well-mixed on θ, a plot of this tracer versus Mθ 40 can be directly integrated to yield the inventory of the tracer, because Mθ directly corresponds to the mass of air. We note that a similar concept to Mθe has been introduced in the stratosphere by Linz et al. (2016).
Several choices need to be made in the definition of Mθ, including defining boundary conditions (e.g. in altitude and latitude) for mass integration and whether to use potential temperature θ or equivalent potential temperature θe. Here, for boundaries, we use the dynamical tropopause (based on potential vorticity unit, PVU) and the Equator, thus integrating the dry air mass of 45 the troposphere in each hemisphere. We also focus on Mθ defined using equivalent potential temperature (θe) to conserve moist static energy in the presence of latent heating during vertical motion, which improves alignment between mass transport and mixing especially within storm tracks in mid-latitudes (Parazoo et al., 2011;Pauluis et al., 2008Pauluis et al., , 2010. We call this tracer Mθe. In this paper we describe the method for calculating Mθe and discuss its variability on synoptic to seasonal scales. We also 50 discuss the time variation of the θe-Mθe relationship within each hemisphere and explore the stability of Mθe and θe-Mθe relationship using different reanalysis products. To illustrate the application of Mθe, we map CO2 data from two recent airborne campaigns (HIPPO and ATom) on Mθe. Further, we show how Mθe can be used to accurately compute the average CO2 concentration over the entire troposphere of the Northern Hemisphere using measurements from the same airborne campaigns.
We examine the accuracy of this method and propose an appropriate way to sample the atmosphere with aircraft to compute 55 the average of a chemical tracer within a large zonal domain.

Equivalent potential temperature (θe) and dry air mass (M) of the atmospheric fields 65
We compute θe (K) using the following expression: from Stull (2012). T (K) is the temperature of air, w (kg water vapor per kg air mass) is the water vapor mixing ratio, R d (287.04, J kg -1 K -1 ) is the gas constant for air, C pd (1005.7 J kg -1 K -1 ) is the specific heat of dry air at constant pressure, P 0 (1013.25, mbar) is the reference pressure at the surface, and L v (T) is the latent heat of evaporation at temperature T. L v (T) is 70 defined as 2406 kJ kg -1 at 40 °C, and 2501 kJ kg -1 at 0 °C and scales linearly with temperature.
We compute water vapor mixing ratio (w) from relative humidity provided by the reanalysis products and the saturation mixing ratio of water vapor following Bolton (1980). We compute the total air mass of each grid cell x at time t, Mx(t), from the product of pressure range and surface area, and divided by a latitude and height dependent gravity constant provided by Arora et al. (2011). The dry air mass is then computed 75 by subtracting the water mass, computed from relative humidity, saturation water vapor mass mixing ratio, and total air mass of the grid cell.
Since this study focuses on tracer distributions in the troposphere, we compute Mθe with an upper boundary at the dynamical tropopause defined as the 2 PVU (potential vorticity units, 10 -6 K kg -1 m 2 s -1 ) surface. As PV is not a standard output product in NCEP2, we linearly interpolate PV from ERA-Interim to NCEP2 on its pressure coordinate. 80 ERA-Interim and NCEP2 include hypothetical levels below the true land/sea surface which we exclude in the calculation of Mθe.

Determination of Mθe
We show a schematic of the conceptual basis for the calculation of Mθe in Figure 1. To compute Mθe, we sort all tropospheric 85 grid cells in the hemisphere by increasing θe, and sum the dry air mass over grid cells following M θ e (θ e , t) = ∑M x (t)| θ e x <θ e (2) where Mx(t) is the dry air mass of each grid cell x at time t, and θ e x is the equivalent potential temperature of the grid cell. The sum is over all grid cells with θ e x less than θe.
This calculation yields a unique value of Mθe for each value of θe. We refer to the relationship between θe and Mθe as the "θe-90 Mθe look-up table", which we generate at daily resolution. We provide this look-up table for each hemisphere computed from ERA-Interim from 1980 to 2018 with daily resolution and 1 K interval (see data availability). Mθe surfaces are always exactly parallel to θe surfaces, which decrease with latitude and generally increase with altitude.

Spatial and temporal distribution of Mθe
Whereas, the θe surfaces vary by up to 20 degrees in latitude over seasons, the meridional displacement of Mθe is much smaller, as expected, because the displacement of atmospheric mass over seasons is small. Particularly in latitudes of the Hadley circulation (Equatorward of 30° N/S) and in summer, Mθe and θe both exhibit strong secondary maxima at the surface, driven 100 by high water vapor. From the contours in Figure 2, this surface branch of high Mθe and θe appears disconnected from the upper tropospheric branch. In fact, we expect these two branches are connected through air columns undergoing deep convection, which are not resolved in the zonal means shown in Figure 2. The existence of these two branches limits some applications of Mθe, as discussed in Section 4. Figure 3 shows the zonal average meridional displacement of θe and Mθe with daily resolution. In summer, Mθe surfaces 105 displace poleward in the lower troposphere but equatorward in the upper troposphere. The displacements in the lower troposphere (925 mbar) are greater in the Northern Hemisphere, where the Mθe = 140 (10 16 kg) surface, for example, displaces poleward by 10 degrees in latitude between winter and summer ( Figure 3b). The seasonal displacement of Mθe surfaces is closely associated with the seasonality of vertical sloping of θe surfaces ( Figure 2). As the mass under each Mθe surface is always constant, the change in tilt must cause the meridional displacement. In the summer, the tilt is steeper (due to increased 110 deep convection) so Mθe surfaces move poleward in the lower troposphere but move equatorward in the upper troposphere.
Beside the seasonal variability, Figure 3 also shows evident synoptic-scale variability.
Since the tilting of θe surfaces has an impact on the seasonal displacement of Mθe surfaces, the contribution of different pressure levels to the mass of a given Mθe bin must also vary with season. In Figure 4, we show these contributions as two daily snapshots on 1 January 2009 and 1 July 2009. Low Mθe bins consist of air masses mostly below 500 mbar near the Pole. As 115 Mθe increases, the contribution from the upper troposphere gradually increases while the contribution from the surface to 800 mbar decreases to its minimum at ~ 100 to 120 (10 16 kg). The contribution from the surface to 800 mbar increases as Mθe increases above 120 (10 16 kg). On the other hand, the contribution of air mass below 800 mbar is always higher in the summer hemisphere at high Mθe bins.

θe-Mθe relationship 120
Figure 5 compares the temporal variation of Mθe of several given θe surfaces (i.e., θe-Mθe look-up table) computed from different reanalysis products for 2009. The deviations are indistinguishable between ERA-Interim and MERRA-2, except near θe = 340 K, where MERRA-2 is systematically lower than ERA-Interim by 1.5 to 6.5 (10 16 kg). NCEP2 shows slightly larger deviations from ERA-Interim, but less than 8.5 (10 16 kg). The products are highly consistent in seasonal variability, and they also show agreement on synoptic time scales. The small difference between products is expected because of different 125 resolutions and methods (Mooney et al., 2011). We expect these differences would be negligible for most applications of Mθe. Figure 5 shows that, in both hemispheres, Mθe reaches its minimum in summer and maximum in winter for a given θe surface, with the largest seasonality at the lowest θe (or Mθe) values. The seasonality decreases as θe increases, following the reduction in the seasonality of shortwave absorption at lower latitudes (Li and Leighton, 1993). The seasonality is smaller in the Southern Hemisphere, consistent with the larger ocean area and hence greater heat capacity and transport (Fasullo and Trenberth, 2008;130 Foltz and McPhaden, 2006). Figure 5 also shows that Mθe has significant synoptic-scale variability but smaller than the seasonal variability. Synoptic variability is typically larger in winter than summer, as discussed below.

Relationship to diabatic heating and mass fluxes
A key step of the application of Mθe for interpreting tracer data is the generation of the look-up table that relates θe and Mθe. In this section, we address a tangential question of what controls the temporal variation of the look-up table, which is not 135 necessary for the application but may be of fundamental meteorological interest.
As shown in Appendix A, the temporal variation of the lookup (J s -1 K -1 ) is the effective diabatic heating, integrated over the full θe surface per unit width in θe, mT(θ e , t) (kg 140 s -1 ) is the net mass flux across the tropopause and mE(θ e , t) (kg s -1 ) is the net mass flux across the Equator, including all air with equivalent potential temperature less than θe. Qdia has contributions from internal heating without ice formation (Q int ′ ), heating from ice formation (Q ice ), sensible heating from the surface (Q sen ), surface evaporation (Q evap ), turbulent diffusion of heat (Q diff ), and turbulent transport of water vapor (Q H 2 O ) following Q dia (θ e , t) = Q int ′ (θ e , t) + Q ice (θ e , t) + Q sen (θ e , t) + Q evap (θ e , t) + Q diff (θ e , t) + Q H 2 O (θ e , t) (4) 145 The terms Q evap and Q H 2 O are expressed as heating rates by multiplying the underlying water fluxes by L v (T)/C pd . In order to quantify the dominant processes contributing to temporal variation of Mθe, the terms in Eqs. 3 and 4 must be linked to diagnostic variables available in the reanalysis or model products. Although there was no perfect match with any of the three reanalysis products, MERRA-2 provides temperature tendencies for individual processes, which can be converted to heating rates per Eq. 4 following 150 where i refers a specific process (Q int ′ , Q ice , etc.), ( There are 5 heating terms provided in the MERRA-2 product, which we can approximately relate to terms in Eq. 4, as shown in Table 1. The first three terms (Q rad , Q dyn , and Q ana ) can be summed to yield Q int ′ , the forth (Q trb ) is equal to the sum of 155 Q diff and Q sen , and the fifth (Q mst ) approximates the sum of Q ice and Q evap . MERRA-2 does not provide terms corresponding to Q H 2 O or Q evap but Q mst represents heating due to moist processes, which includes Q ice plus water vapor evaporation/condensation within the atmosphere. This water vapor evaporation/condensation should be approximately equal to Q evap with small time lag when integrated over a θe surface because mixing is preferentially along the θe surface and water vapor released into a θe surface by surface evaporation will tend to transport and precipitate from the same θe surface within a 160 short time period (Bailey et al., 2019). Thus, the MERRA-2 term for heating by moist processes should approximate Q ice + Q evap . Figure 6a compares the temporal variation of M θ ė computed by integrating dry air mass (i.e., θe-Mθe look-up table) with Mθe computed from the sum of the diabatic heating terms from MERRA-2 (via Eq. 3 to Eq. 5). The comparison focuses on the θe = 300 K surface, which does not intersect with the Equator or tropopause, so that the two mass flux terms (m T , m E ) vanish. 165 These two methods have a high correlation at 0.71. We do not expect perfect agreement because M θ ė computed by the sum of heating neglects turbulent water vapor transport, and only approximates Q evap as discussed above. This relatively good https://doi.org/10.5194/acp-2020-841 Preprint. Discussion started: 19 August 2020 c Author(s) 2020. CC BY 4.0 License. agreement nevertheless demonstrates that the formulation based on MERRA-2 heating terms includes the dominant processes that drive temporal variations in the look-up table. were over the Pacific Ocean, while on ATom, southbound transects were over the Pacific Ocean and northbound transects were over the Atlantic Ocean. The flight tracks are shown in Figure 7a. We aggregate data from each campaign into northbound and southbound transects within each hemisphere, but only use data from the Northern Hemisphere. We only consider tropospheric observations by excluding measurements from the stratosphere, which is defined by observed water vapor less than 50 ppm and either O3 greater than 150 ppb or detrended N2O to the reference year of 2009 less than 319 ppb. Water vapor 190 and O3 were measured by the NOAA UCATS (UAS Chromatograph for Atmospheric Trace Species) instrument and were interpolated to 10-sec resolution. N2O was measured by the Harvard QCLS (Quantum Cascade Laser System) instrument.
Furthermore, we exclude all near-surface observations during take-offs, landings, and missed approaches, which usually show high CO2 variability due to strong local influences. In-situ measurements of CO2 were made by 3 different instruments on both HIPPO and Atom. Of these, we use the CO2 measurements made by the NCAR Airborne Oxygen Instrument (AO2) with a 195 2.5 seconds measurement interval (Stephens et al., submitted to AMTD, 2020), for consistency with planned future applications to APO (atmospheric potential oxygen) computed from AO2. The differences between instruments are small for our application (Santoni et al., 2014). The data used in this study are averaged to 10-sec resolution and we show the detrended CO2 values along each airborne campaign transect for the Northern Hemisphere in Figure 7b, Since we focus on the seasonal https://doi.org/10.5194/acp-2020-841 Preprint. Discussion started: 19 August 2020 c Author(s) 2020. CC BY 4.0 License.
cycle of CO2, all airborne observations are detrended by subtracting an interannual trend fitted to CO2 measured at the Mauna 200 Loa Observatory (MLO) by the Scripps CO2 Program. This trend is computed by a stiff cubic spline function plus 4-harmonic terms with linear gain to the MLO record. Mθe is computed from ERA-Interim in this section.

Mapping Northern Hemisphere CO2
A conventional method to display seasonal variations in CO2 from airborne data is to plot time series of the data at a given location or latitude and different pressure levels (Graven et al., 2013;Sweeney et al., 2015). In Figure 8, we compare this 205 method using HIPPO and ATom airborne data, binning and averaging the data from each airborne campaign transect by pressure and latitude bins, with our new method, binning the data by pressure and Mθe. For each latitude bin, we choose a corresponding Mθe bin which has approximately the same meridional coverage in the lower troposphere. We remind the reader that Mθe decreases poleward, while also generally increasing with altitude ( Figures 2 and 3).
As shown in Figure 8, the transect average of detrended CO2 (shown as points) from both binning methods resolve well-210 defined seasonal cycles (based on 2-harmonic fit) in all bins, with higher amplitudes near the surface (low pressure) and at high latitude/ low Mθe. However, binning by Mθe leads to much smaller variations of the mean seasonal cycle (shown as solid curves) with pressure, as expected, because moist isentropes are preferential surfaces of mixing. Also, within individual pressure bins, the short-term variability relative to the mean cycles based on the distribution of all detrended observations (not shown as points but denoted as 1 σ values in Figure 8) is smaller when binning by Mθe (F-test, p < 0.01), except in the lower 215 troposphere of the highest Mθe bin (90-110 10 16 kg). The smaller short-term variability is expected because Mθe tracks the synoptic variability of the atmosphere. When binning by latitude, the smallest short-term variability is found at the lowest bin (surface-800 mbar) and the largest short-term variability is found in the highest bin (500 mbar-tropopause), except the highest latitude bin (45° N-55° N). When binning by Mθe, in contrast, the short-term variability in the middle pressure bin is always smaller than the higher and lower pressure bins (F-test, p < 0.01), except for the 50 to 70 Mθe bin, where the difference between 220 the lowest and middle pressure bins is not significant (based on 1 σ levels). The lower variability in the mid troposphere may reflect the suppression of variability from synoptic disturbances, leaving a clearer signal of the influence of surface fluxes of CO2 and stratosphere-troposphere exchanges. We compare the variance of detrended airborne observations within each Mθepressure bin with its fitted value. The fitted seasonal cycle of each bin explains 63.2% to 90.5% of the variability for different bins, with higher fractions in the middle troposphere. 225 Figure 8 also shows the CO2 seasonal cycle at MLO, which falls within a single Mθe-pressure bin (90-110 10 16 kg, 500-800 mbar) at all seasons. Although the airborne data in this bin span a wide range of latitudes (~ 10° N-75° N), the seasonal cycle averaged over this bin is very similar to the cycle at MLO (airborne cycle leads by ~10 days with 1.0% lower amplitude).
It is also of interest to examine how CO2 data from surface stations fit into the framework based on Mθe. Figure 9 compares the CO2 seasonal cycle of five NOAA surface stations (Dlugokencky et al., 2019) with the cycle from the airborne observations binned into selected Mθe bins. These surface stations are chosen to be representative of different Mθe ranges. For the comparison, we chose Mθe bins that span the seasonal maximum and minimum Mθe value of the station. These bins are narrower than the bins used in Figure 8, in order to sharply focus on the latitude of the station. To maximize sampling coverage, we bin the airborne data only by Mθe without pressure sub-bins. For mid-and high latitude surface stations (right three panels), the seasonal amplitude of station CO2 and corresponding airborne CO2 are close (within 4-5%), while airborne cycles lag by 2-3 235 weeks. The lag presumably represents the slow mixing from the mid-latitude surface to the high latitude mid-troposphere (Jacob, 1999). In contrast, for low latitude stations (left two panels) which generally sample trade winds, the seasonal cycles differ significantly, indicating that the air sampled at these stations is not rapidly mixed along surfaces of constant Mθe or θe with air aloft. As mentioned above (Section 3.1), surfaces of high Mθe within the Hadley circulation have two branches, one near the surface and one aloft. A timescale of several months for transport from the lower to the upper branch can be estimated 240 from the known overturning flows based on air mass flux streamfunctions (Dima and Wallace, 2003). This delay, plus strong mixing and diabatic effects (Miyazaki et al., 2008), ensures that the lower and upper branches are not well connected on seasonal time scales. Our results nevertheless demonstrate that the Mθe framework combining airborne and surface data could help understand details of atmospheric transport both along and across θe surfaces.

Computing the hemispheric mass-weighted average CO2 mole fraction 245
We next illustrate the use of Mθe for computing the mass-weighted average of a long-lived chemical tracer by performing this exercise for CO2 in the Northern Hemisphere. We calculate the Northern Hemisphere tropospheric mass-weighted average CO2 from each airborne transect using a method that assumes that CO2 is uniformly mixed on θe surfaces throughout the hemisphere. HIPPO-1 Northbound is excluded here due to the lack of data north of 40° N. We use the θe-Mθe lookup table of the corresponding date to assign a value of Mθe to each observation based on its θe. The observations for each transect are then 250 sorted by Mθe. The hemispheric average CO2 is calculated by trapezoidal integration of CO2 as a function of Mθe and divided by the total dry air mass as computed from the corresponding range of Mθe.
To illustrate the Mθe integration method, we choose HIPPO-1 Southbound and show CO2 measurements and ΔCO2 inventory (Pg) as a function of Mθe in Figure 10. The Northern Hemisphere tropospheric average detrended CO2 is computed by integrating the area under the curve (subtracting negative contributions) and dividing by the maximum value of Mθe within the 255 hemisphere (here 195.13×10 16 kg). This yields a mass-weighted average detrended CO2 of 1.13 ppm for the full troposphere of the Northern Hemisphere. The trapezoidal integration has a high accuracy because the data are dense over Mθe. The ΔCO2 inventory is dominated by the domain Mθe < 120, with less than 4.1% contributed by the additional ~38.8% of the air mass outside this domain (Fig. 10b).
We compute a Northern Hemisphere mass-weighted average detrended CO2 for each airborne campaign transect and fit the 260 time series to a 2-harmonic fit to estimate the seasonal cycle ( Figure 11). We find that the cycle has a seasonal amplitude of https://doi.org/10.5194/acp-2020-841 Preprint. Discussion started: 19 August 2020 c Author(s) 2020. CC BY 4.0 License. 7.9 ppm and a downward zero-crossing at Julian day 179, where the latter is defined as the date when the detrended seasonal cycle changes from positive to negative.
To address the error in fitted amplitude and zero crossing, we consider two main sources: (1) irreproducibility in the CO2 measurements and (2) limited coverage in space and time. For the first contribution, we compute the difference between mass-265 weighted average CO2 from AO2 and mean mass-weighted average CO2 from Harvard QCLS, Harvard OMS, and NOAA Picarro for each airborne campaign transect, while masking values that are missing in any of these datasets. We compute the standard deviation of these differences (± 0.15 ppm) for mass-weighted average CO2 of each airborne campaign transect as the 1σ level of uncertainty. We further compute the uncertainties for the seasonal amplitude of ± 0.11 ppm and for the downward zero-crossing of ± 0.83 days, which are calculated from 1000 iterations of the 2-harmonic fit, allowing for random 270 Gaussian uncertainty (σ = ± 0.11 ppm) for each transect.
For the contribution to the error in the amplitude and phase from limited space/time coverage, we use simulated CO2 data from the Jena CO2 Inversion Run ID: s04oc v4.3 (Rödenbeck et al., 2003(Rödenbeck et al., , 2018. This model includes full atmospheric fields from 2009 to 2018, which we detrend using the cubic spline fit to the observed MLO trend. From these detrended fields, we compute the climatological cycle of the Northern Hemisphere average by integrating over all tropospheric grid cells (cutoff at PVU = 275 2) to produce a daily time series of the hemispheric mean, which we take as the model "truth". We fit a 2-harmonic function to this "true" time series to compute a "true" climatological cycle over the 2009-2018 period (Table 3), which is our target for validation. We then subsample the Jena CO2 Inversion along the HIPPO and ATom flight tracks and process the data similarly to the observations, using the Mθe integration method and a 2-harmonic fit. The comparison shows that the Mθe integration method yields an amplitude which is 1% too large and yields a downward zero-crossing date which is 6 days too late. We view 280 these offsets as systematic biases, which we correct from the observed amplitude and phase reported above. The uncertainties in these biases are hard to quantify, but we take ±100 % as a conservative estimate. We thus allow an additional random error of ± 0.08 ppm in amplitude and ± 6.0 days in downward zero crossing for uncertainty in the bias. Combining the random and systematic error contributions leads to a corrected Northern Hemisphere tropospheric average CO2 seasonal cycle amplitude of 7.8 ± 0.14 ppm and downward zero-crossing of 173 ± 6.1 days. This corrected cycle is an estimate of the climatological 285 average from 2008-2019.
The error due to limited space/time coverage can be divided into three components: limited seasonal coverage (17 transects over the climatological year), limited interannual coverage (sampling particular years instead of all years), and limited spatial coverage (under-sampling the full hemisphere). We quantify the combined biases due to both limited seasonal and interannual coverage by comparing the two-harmonic fit of the full "true" daily time series of the hemispheric mean to a two-harmonic fit 290 of that data subsampled on the actual mean sampling dates of the 17 flight tracks. We isolate the bias associated with limited seasonal coverage by repeating this calculation, replacing the "true" daily time series with the daily climatological cycle. The bias associated with limited spatial coverage is quantified as the residual. Combining these results, we estimate that the limited https://doi.org/10.5194/acp-2020-841 Preprint. Discussion started: 19 August 2020 c Author(s) 2020. CC BY 4.0 License. seasonal, interannual, and spatial coverage, account for biases in the downward zero-crossing of 1.1, 1.4, and 3.5 days respectively, all in the same direction (too late). The seasonal amplitude bias due to individual components are all small (< 295 0.5%).
It is of some interest to compare our estimate of the Northern Hemisphere average cycle with the cycle at Mauna Loa, which is also broadly representative of the hemisphere. Our comparison in Figure 11 shows small but significant differences in both amplitude and phase, with the MLO amplitude being ~ 11.5% smaller than the hemispheric average and lagging in phase by ~ 1 month. There are also differences in the shape of the cycle, with the MLO cycle rising more slowly from October to 300 February, but more quickly from February to May. These features at least partly reflects variations in the transport of air masses to the station (Harris et al., 1992;Harris and Kahl, 1990).
In Figure 12, we compare the Mθe integration method with an alternate latitude-pressure weighted average method, with no correction for synoptic variability. For this method, we bin flight track subsampled Jena CO2 Inversion data into sin(latitude)pressure bins with 0.01 and 25 mbar as intervals respectively, while all bins without data are filtered. We further compute a 305 weighted average CO2 for each airborne campaign transect. The root-mean-square errors (RMSE) to the true average of the Mθe integration method are ± 0.32 and ± 0.27 ppm for HIPPO and ATom campaigns, respectively, which are smaller than the RMSE of the simple latitude-pressure weighted average method at ± 0.82 and ± 0.53 ppm.
We also evaluate the biases in the hemispheric average season cycles computed with the simple latitude-pressure weighted average method. As summarized in Table 3, the latitude-pressure weighted average method yields a larger error in seasonal 310 amplitude (Mθe method 1.0 % too large, latitude-pressure method 20.8% too large), while both methods show similar phasing error (6 to 7 days late). The larger error associated with the latitude-pressure weighted average method is consistent with strong influence of synoptic variability. This synoptic variability could potentially be corrected using model simulations of the 3dimensional CO2 fields (Bent, 2014). The Mθe integration method appears advantageous because it accounts for synoptic variability, and easily yields a hemispheric average by directly integrating over Mθe. 315 The relative success of the Mθe integration method in yielding accurate hemispheric averages using HIPPO and ATom data is attributable partly to the extensive data coverage. To explore the coverage requirement for reliably resolving hemispheric averages, we also test the integration method when applied to simulated data with lower coverage. We start with the same coverage as for ATom and HIPPO but select only subsets of the points in four groups: poleward of 30° N, Equator to 30° N, surface to 600 mbar, and 600 mbar to tropopause. We also examine whether we can only utilize observation along the Pacific 320 transect by excluding measurements along the Atlantic transects (ATom northbound). We further explore the impact of reduced sampling density by subsampling the Jena CO2 Inversion based on the spatial coverage of the Medusa sampler, which is an airborne flask sampler that collected 32 cryogenically dried air samples per flight during HIPPO and ATom (Stephens et al., submitted to AMTD, 2020). We further randomly retain 10%, 5%, and 1% of the full flight track subsampled data, repeating each ratio with 1000 iterations. We compute the detrended average CO2 from these nine simulations by the Mθe integration method and then compute the RMSE relative to the detrended true hemispheric average, together with the seasonal magnitude and day of year of the downward zero-crossing, as summarized in Table 3. HIPPO-1 Northbound is excluded in all these simulations. The number of data points of each simulation and number of observations of the original HIPPO and ATom data sets are summarized in Table S1. These results show that limiting sampling to either equatorward or poleward of 30° N yields significant error (24.3% smaller and 24.9% larger seasonal amplitude, respectively). Additionally, there is a ~ 25 days lag in 330 phase if sampling is limited to equatorward of 30° N. However, restricting sampling to be exclusively above or below 600 mbar, or only along the Pacific transect does not lead to significant errors. Randomly reducing the sampling by 10-to 100fold or only keeping Medusa spatial coverage also have minimal impact. This suggests that, to compute the average CO2 of a given region, it may be sufficient to have low sampling density provided that the measurements adequately cover the full range in θe (or Mθe). 335

Discussion and summary
We have presented a transformed isentropic coordinate, Mθe, which is the total dry air mass under a given θe surface in the troposphere of the hemisphere. Mθe can be computed from meteorological fields by integrating dry air mass over θe surfaces, and different reanalysis products show a high consistency. The θe-Mθe relationship varies seasonally due to seasonal heating/cooling of the atmosphere via radiative heating and moist processes. The seasonality in the relationship is greater at 340 low θe compared to high θe, and is greater in the Northern than the Southern Hemisphere. The θe-Mθe relationship also shows synoptic-scale variability, which is mainly driven by dynamic dissipation of energy. Mθe surfaces show much less seasonal displacement with latitude and altitude than surfaces of constant θe, while being parallel and exhibiting essentially identical synoptic scale variability. As a coordinate for mapping tracer distributions, Mθe shares with θe the advantages of following displacements due to synoptic disturbances and aligning with surfaces. Mθe has the additional advantage of being 345 approximately fixed in space seasonally, which allows mapping to be done on seasonal time scales, and having units of mass, which provides a close connection with atmospheric inventories.
As a coordinate, Mθe is probably better viewed as an alternative to latitude, due to its nearly fixed relationship with latitude over season, rather than as an alternative to altitude (or pressure), as typically done for potential temperature (Miyazaki et al., 2008;Miyazaki and Iwasaki, 2005;Parazoo et al., 2011;Tung, 1982;Yang et al., 2016). Even though the contours of constant 350 Mθe extend over a wide range of latitudes (from low latitudes at the Earth surface to high latitudes aloft), a close association with latitude is provided by the point of contact with Earth's surface. Also, Mθe is nearly always monotonic with latitude (increasing equatorward) while it is not necessarily monotonic with altitude in the lower troposphere ( Figure 2).
As a first application, we have illustrated using Mθe to map the seasonal variation of CO2 in the Northern Hemisphere, using data from the HIPPO and ATom airborne campaigns. This application shows that Mθe has several advantages as a coordinate 355 compared to using latitude: (1) variations in CO2 with pressure are smaller at fixed Mθe than at fixed latitude, and (2) the scatter about the mean CO2 seasonal cycle is smaller when sorting data into pressure/Mθe bins than into pressure/latitude bins. We https://doi.org/10.5194/acp-2020-841 Preprint. Discussion started: 19 August 2020 c Author(s) 2020. CC BY 4.0 License.
have also shown that, at middle and high latitudes, the CO2 seasonal cycles that are resolved in the airborne data (binned by Mθe but not pressure) are very similar to the cycles observed at surface stations at the appropriate latitude, with a phase lag of ~ 2 to 3 weeks. At lower latitudes, CO2 cycles in the airborne data (binned similarly by Mθe) are less consistent with surface 360 data, as expected due to slow transport and diabatic processes within the Hadley Circulation. For characterizing the patterns of variability in airborne CO2 data, we expect the advantages of Mθe over latitude will be greatest for sparse datasets, allowing data to be binned more coarsely with pressure or elevation while still resolving features of large-scale variability, such as seasonal cycles or gradients with latitude.
As a second application, we use Mθe to compute the Northern Hemisphere tropospheric average CO2 from the HIPPO and 365 ATom airborne campaigns by integrating CO2 over Mθe surfaces. With a small correction for systematic biases induced by limited hemispheric coverage of the HIPPO and ATom flight tracks, we report a seasonal amplitude of 7.8 ± 0.14 ppm and a downward zero-crossing at Julian day 173 ± 6.1. This hemispheric average cycle may prove valuable as a target for validation of models of surface CO2 exchange.
Our analysis also clarifies that computing hemispheric averages with the Mθe integration method depends on adequate spatial 370 coverage. The coverage provided by the HIPPO and ATom campaigns appears more than adequate for computing the average seasonal cycle of CO2 in the Northern Hemisphere, and the errors for this application remain small if the coverage is limited to either above or below 600 mbar, or reduced to retain only 1% of the measurements. Most critical is maintaining coverage in latitude, or Mθe surfaces. The Mθe integration method of computing hemispheric averages assumes that the tracer is uniformly distributed and instantly mixed on θe (Mθe) surfaces. We have shown that systematic gradients in CO2 are resolved with pressure 375 at fixed Mθe, which reflects the finite rates of dispersion on θe surfaces. Further improvements to the integration method seem possible by integrating separately over different pressure levels, taking account of the different mass fraction in different pressure bins (e.g. Figure 4). The need is especially relevant for high Mθe bins which are less completely mixed, and which tend to intersect the Equator or have separate surface branches. For these Mθe bins, it would be more appropriate to integrate over Mθ in the upper and lower atmosphere separately. This complication is of minor importance for computing the mass-380 weighted average CO2 cycle, because the cycle of CO2 is small in these air masses.
The definition of Mθe requires horizontal and vertical boundaries for the integration of dry air mass. We use the dynamic tropopause (based on PVU) and the Equator as boundaries, which is appropriate for integrating tropospheric inventories in a hemisphere. Other boundaries may be more appropriate for other applications. For example, Mθe could be computed from the lowest θe surface in the Southern Hemisphere with a latitude cutoff at 30° S, to apply to airborne observations only over the 385 Southern Ocean. On the other hand, the boundary choice only influences Mθe surfaces that actually intercept the boundaries, making the choice less important at high latitude in the lower troposphere (lowest Mθe surfaces). Some tropospheric applications may also benefit by integrating over dry potential temperature (θ) rather than θe.
Based on our promising results for CO2, we expect that Mθe may be usefully applied as a coordinate for mapping and computing inventories of many tracers, such as O2/N2, N2O, CH4, and the isotopes of CO2, whose residence time is long compared to the 390 time scale for mixing along isentropes. Mθe may also prove useful in the design phase of airborne campaigns to ensure strategic coverage. Our results show that, to study the seasonal cycle of a tracer on a hemispheric scale, it is critical to have welldistributed sampling in Mθe.

Appendix A: Temporal variation of Mθe
Following Walin's derivation for cross-isothermal volume flow in the ocean (Walin, 1982), we show how M θ ė = ∂ ∂t M θ e (θ e , t) can be related to energy and mass fluxes. We start by deriving the relationship for Mθ (based on potential temperature θ) but 415 later generalize to apply to Mθe. https://doi.org/10.5194/acp-2020-841 Preprint. Discussion started: 19 August 2020 c Author(s) 2020. CC BY 4.0 License.
Substituting Eq. A1 to Eq. A3 into Eq. A5 and differentiating with respect to θ yields C pd θ ∂ ∂t ∂M θ (θ, t) ∂θ = C pd θ ( ∂F T (θ, t) ∂θ + ∂F E (θ, t) ∂θ + ∂F I (θ, t) ∂θ ) + C pd F I (θ, t) + where, Differentiating Eq. A4 with respect to θ, and multiplying C pd ⋅ θ yields Subtracting Eq. A8 from Eq. A6, we obtain Eq. A9 divided by C pd plus Eq. A4 yields Eq. A10 illustrates the temporal variation of Mθ, where Qint includes radiative heating (i.e. sum of shortwave and longwave heating), dynamic dissipation of heat, and latent heat releasing due to evaporation and condensation. 440 To modify Eq. A10 to apply to Mθe rather than Mθ, it is necessary to replace all θ with θe, and additionally account for the following: 1. Condensation and evaporation is conserved on the θe surfaces, but the gaining and losing of water vapor through surface evaporation and water vapor transport contributes to θe. This contribution can be computed as the product of latent heat of evaporation and the extra water vapor content. Thus, the surface contribution (Q S ) needs to include both sensible heating of 445 the atmosphere (Q sen ) and the water vapor flux from the surface into the atmosphere (Q evap ). Similarly, the diffusion term within the atmosphere (Q diff ) needs to include both heat and water vapor (Q H 2 O ).
2. Internal heating (Q int ) needs to exclude latent heat releasing due to evaporation and condensation of liquid water, which cancel in θe, but it still needs to include heating from ice formation, which does not cancel in θe. We subtract this ice component from the rest of the internal heating, yielding two terms Q int ′ and Q ice , with Q int = Q int ′ + Q ice . 450 Therefore, we can write the temporal variation of Mθe as

Competing interests
The authors declare that they have no conflict of interest.

Acknowledgements
The original Mθe concept arose out of discussions during the ORCAS field campaign that included Ralph Keeling, Colm Sweeney, Eric Kort, Matthew Long, and Martin Hoecker-Martinez. We would like to acknowledge the efforts of the full 460 HIPPO and ATom science teams and the pilots and crew of the NCAR/NSF GV and NASA DC-8, the NCAR and NASA project managers, field support staff, and logistics experts. In this work, we have used the HIPPO and ATom 10-sec merge files, supported by the National Center for Atmospheric Research (NCAR). NCAR is sponsored by the National Science        Table 1      https://doi.org/10.5194/acp-2020-841 Preprint. Discussion started: 19 August 2020 c Author(s) 2020. CC BY 4.0 License. Figure 12: Comparison between the Northern Hemisphere average CO2 from full integration of the simulated atmospheric fields from the Jena CO2 Inversion (cutoff at PVU = 2) and from two methods that use the same simulated data subsampled with 540 HIPPO/ATom coverage: (1) the Mθe integration method (blue) and (2) simple integration by sin(latitude)-pressure (red). We divide the comparison into HIPPO (left) and ATom (right) temporal coverage. The lower panel shows the Error for individual tracks using alternate subsampling methods.

565
Variable Definition Unit θ ′ (r, t) Potential temperature at location r and time t. K θ Potential temperature of the chosen isentropic surface. K

R(θ, t)
A region in which θ ′ (r, t) < θ shown as shaded area in Figure A1.