the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A massweighted isentropic coordinate for mapping chemical tracers and computing atmospheric inventories
Ralph F. Keeling
Eric J. Morgan
Nicholas C. Parazoo
Britton B. Stephens
We introduce a transformed isentropic coordinate ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, defined as the dry air mass under a given equivalent potential temperature surface (θ_{e}) within a hemisphere. Like θ_{e}, the coordinate ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ follows the synoptic distortions of the atmosphere but, unlike θ_{e}, has a nearly fixed relationship with latitude and altitude over the seasonal cycle. Calculation of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ is straightforward from meteorological fields. Using observations from the recent HIAPER PoletoPole Observations (HIPPO) and Atmospheric Tomography Mission (ATom) airborne campaigns, we map the CO_{2} seasonal cycle as a function of pressure and ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, where ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ is thereby effectively used as an alternative to latitude. We show that the CO_{2} seasonal cycles are more constant as a function of pressure using ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ as the horizontal coordinate compared to latitude. Furthermore, shortterm variability in CO_{2} relative to the mean seasonal cycle is also smaller when the data are organized by ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ and pressure than when organized by latitude and pressure. We also present a method using ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ to compute massweighted averages of CO_{2} on a hemispheric scale. Using this method with the same airborne data and applying corrections for limited coverage, we resolve the average CO_{2} seasonal cycle in the Northern Hemisphere (massweighted tropospheric climatological average for 2009–2018), yielding an amplitude of 7.8 ± 0.14 ppm and a downward zerocrossing on Julian day 173 ± 6.1 (i.e., late June). ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ may be similarly useful for mapping the distribution and computing inventories of any longlived chemical tracer.
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The spatial and temporal distribution of longlived chemical tracers like CO_{2}, CH_{4}, and O_{2} ∕ N_{2} typically includes regular seasonal cycles and gradients with latitude and pressure (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 timescales by synoptic weather disturbances, especially at middle to high latitudes (i.e., poleward of 30^{∘} N and 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, 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.
A common approach to correct synoptic distortion is to use transformed coordinates rather than geographic coordinates (i.e., pressure–latitude), to take into account atmospheric dynamics and transport barriers. Such coordinate transformation has been used, for example, to reduce dynamically induced variability in the stratosphere using equivalent latitude rather than latitude as the horizontal coordinate (Butchart and Remsberg, 1986); to diagnose the tropopause profile using a tropopausebased rather than surfacebased vertical coordinate (Birner et al., 2002); to study the transport regime in the Arctic using a horizontal coordinate based on the polar dome (Bozem et al., 2019); and to study UTLS (upper troposphere–lower stratosphere) tracer data by using tropopausebased, jetbased, and equivalent latitude coordinates (Petropavlovskikh et al., 2019). In the troposphere, a transformed coordinate, the isentropic coordinate (θ), has been widely applied to evaluate the distribution of tracer data (Miyazaki et al., 2008; Parazoo et al., 2011, 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 (KeppelAleks 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). However, 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 timescales.
During analysis of airborne data from the HIAPER PoletoPole 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 massbased unit, M_{θ}, 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, M_{θ} has a more stable relationship to latitude and altitude, while varying in parallel with θ on synoptic scales. Also, for a tracer which is wellmixed on θ, a plot of this tracer versus M_{θ} can be directly integrated to yield the atmospheric inventory of the tracer, because M_{θ} directly corresponds to the mass of air. We note that a similar concept to ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ has been introduced in the stratosphere by Linz et al. (2016), in which M(θ) is defined as the mass above the θ surface, to study the relationship between age of air and diabatic circulation of the stratosphere.
Several choices need to be made in the definition of M_{θ}, including defining boundary conditions (e.g., in altitude and latitude) for mass integration and whether to use potential temperature θ or equivalent potential temperature θ_{e}. Here, for boundaries, we use the dynamical tropopause (based on the potential vorticity unit, PVU) and the Equator, thus integrating the dry air mass of the troposphere in each hemisphere. We also focus on M_{θ} defined using equivalent potential temperature (θ_{e}) 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 midlatitudes (Parazoo et al., 2011; Pauluis et al., 2008, 2010). We call this tracer ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$.
In this paper we describe the method for calculating ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ and discuss its variability on synoptic to seasonal scales. We also discuss the time variation in the θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ relationship within each hemisphere and explore the stability of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ and the θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ relationship using different reanalysis products. To illustrate the application of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, we map CO_{2} data from two recent airborne campaigns (HIPPO and ATom) on ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$. Further, we show how ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ can be used to accurately compute the average CO_{2} 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 the average of a chemical tracer within a large zonal domain.
2.1 Meteorological reanalysis products
The calculation of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ requires the distribution of dry air mass and θ_{e}. For these quantities, we alternately use three reanalysis products: ERAInterim (Dee et al., 2011), NCEP2 (Kanamitsu et al., 2002), and ModernEra Retrospective analysis for Research and Applications Version 2 (MERRA2) (Gelaro et al., 2017). All products have a 2.5^{∘} horizontal resolution. NCEP2 has a daily resolution, and we average 6hourly ERAInterim fields and 3hourly MERRA2 fields to yield daily fields. ERAInterim has 32 vertical levels from 1000 to 1 mbar, with approximately 20 to 27 levels in the troposphere. NCEP2 has 17 vertical levels from 1000 to 10 mbar, with approximately 8 to 12 levels in the troposphere. MERRA2 has 42 vertical levels from 985 to 0.01 mbar, with approximately 21 to 25 levels in the troposphere.
2.2 Equivalent potential temperature (θ_{e}) and dry air mass (M) of the atmospheric fields
We compute θ_{e} (K) using the following expression:
from Stull (2012). T (K) is the temperature of air; w (kg of water vapor per kg of 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 defined as 2406 kJ kg^{−1} at 40 ^{∘}C and 2501 kJ kg^{−1} at 0 ^{∘}C and scales linearly with temperature.
Following Bolton (1980), we compute the water vapor mixing ratio (w) from relative humidity (RH; kg kg^{−1}) provided by the reanalysis products and the formula for the saturation mixing ratio of water vapor (P_{s,v}; mbar) modified by Wexler (1976).
We compute the total air mass of each grid cell x at time t, M_{x}(t), shown in Eq. (4), from the product of the pressure range and surface area and divided by a latitude and heightdependent gravity constant provided by Arora et al. (2011). The surface area is computed by using latitude (Φ), longitude (λ), and the radius of the Earth (R, 6371 km). The total air mass of each grid cell is computed from
where Δ represents the difference between two boundaries of each grid cell.
The gravity constant (g; kg m^{−2}) is computed following Arora et al. (2011) as
where the reference gravity constant (g_{0}) is assumed to be 9.78046 m s^{−2} and the height (h) in units of meters is computed from
where H is the scale height of the atmosphere and assumed to be 8400 m.
The dry air mass is then computed by subtracting the water mass, computed from relative humidity, the saturation water vapor mass mixing ratio, and the total air mass of the grid cell (Eq. 3). Since this study focuses on tracer distributions in the troposphere, we compute ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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.
ERAInterim and NCEP2 include hypothetical levels below the true land or sea surface, for example, the 850 hPa level over the Himalaya, which we exclude in the calculation of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$.
2.3 Determination of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$
We show a schematic of the conceptual basis for the calculation of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ in Fig. 1. To compute ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, we sort all tropospheric grid cells in the hemisphere by increasing θ_{e} and sum the dry air mass over grid cells following
where M_{x}(t) is the dry air mass of each grid cell x at time t and ${\mathit{\theta}}_{{\mathrm{e}}_{x}}$ is the equivalent potential temperature of the grid cell. The sum is over all grid cells with ${\mathit{\theta}}_{{\mathrm{e}}_{x}}$ less than θ_{e}.
This calculation yields a unique value of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ for each value of θ_{e}. We refer to the relationship between θ_{e} and ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ as the “θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ lookup table”, which we generate at daily resolution. We provide this lookup table for each hemisphere computed from ERAInterim from 1980 to 2018 with a daily resolution and from the lowest to the highest θ_{e} surface in the troposphere with 1 K intervals (see data availability).
3.1 Spatial and temporal distribution of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$
Figure 2 shows snapshots of the distribution of zonal average θ_{e} and ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ with latitude and pressure at two arbitrary time slices (1 January 2009, 1 July 2009). ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ is not continuous across the Equator because it is defined separately in each hemisphere. By definition, each ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surface is exactly aligned with a corresponding θ_{e} surface, and ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces have the same characteristics as θ_{e} surfaces, which decrease with latitude and generally increase with altitude. Whereas the zonal average θ_{e} surfaces vary by up to 20^{∘} in latitude over seasons, the meridional displacement of zonal average ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ is much smaller, with less than 5^{∘} in latitude poleward of 30^{∘} N and S, as expected, because the zonal average displacement of atmospheric mass over seasons is small. This small seasonal displacement is closely associated with the seasonality of vertical sloping of θ_{e} surfaces (Fig. 2). As the mass under each ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surface is always constant, the change in tilt must cause the meridional displacement. In the summer, the tilt is steeper (due to increased deep convection), so ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces move poleward in the lower troposphere but move equatorward in the upper troposphere.
${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces at given meridians (Fig. 3) in the Northern Hemisphere show clear zonal asymmetry, with larger and more complex displacements compared to the zonal averages, associated with differential heating by land and ocean and orographic stationary Rossby waves (Hoskins and Karoly, 1981; Wills and Schneider, 2018). For example, over the Northern Hemisphere ocean at 180^{∘} E (Fig. 3a) and from the summer to winter, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces move poleward in the middle to high latitudes (e.g., poleward of 45^{∘} N) but move equatorward in the mid to lowlatitude lower troposphere (e.g., equatorward of 45^{∘} N, 900–700 mbar), with the magnitude smaller than 10^{∘} latitude in both. In comparison, over the Northern Hemisphere land at 100^{∘} E (Fig. 3b) and from the summer to winter, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces move equatorward by up to 30^{∘} latitude, except in the highlatitude middle troposphere (e.g., poleward of 70^{∘} N, ∼ 500 mbar), where the flat ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces lead to slightly poleward displacements. In the Southern Hemisphere, in contrast, the summertowinter displacements of the 180 and 100^{∘} E sections are similar to the zonal average.
At lower latitudes, the zonal averages of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ and θ_{e} both exhibit strong secondary maxima near the surface associated with the Hadley circulation (equatorward of 30^{∘} N and S) and in the summer, driven by high water vapor. From the contours in Fig. 2, this surface branch of high ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ and θ_{e} appears disconnected from the upper tropospheric branch. In fact, these two branches are connected through air columns undergoing deep convection, which are not resolved in the zonal means shown in Fig. 2 but are resolved in some meridians (e.g., Fig. 3a). We also note that, over the land at 100^{∘} E (Fig. 3b), the two disconnected ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ and θ_{e} branches in the Northern Hemisphere summer are displaced poleward compared to the zonal average, consistent with a northward shift of the intertropical convergence zone (ITCZ) over southern Asia. The existence of these two branches may limit some applications of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, as discussed in Sect. 4.
Figure 4 shows the zonal average meridional displacement of θ_{e} and ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ with a daily resolution. In summer, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces 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 ${M}_{{\mathit{\theta}}_{\mathrm{e}}}=\mathrm{140}\times {\mathrm{10}}^{\mathrm{16}}$ kg surface, for example, displaces poleward by 10^{∘} in latitude between winter and summer (Fig. 4b). Beside the seasonal variability, Fig. 4 also shows evident synopticscale variability.
Since the tilting of θ_{e} surfaces has an impact on the seasonal displacement of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces, the contribution of different pressure levels to the mass of a given ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bin must also vary with season. In Fig. 5, we show these contributions as two daily snapshots on 1 January and 1 July 2009. Low ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bins consist of air masses mostly below 500 mbar near the pole. As ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ increases, the contribution from the upper troposphere gradually increases while the contribution from the surface to 800 mbar decreases to its minimum at around 100 × 10^{16} to 120 × 10^{16} kg. The contribution from the surface to 800 mbar increases as ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ increases above 120 × 10^{16} kg. The mass fraction shows only small variations with season, with the lower troposphere (surface to 800 mbar) contributing slightly less in the low${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bands and slightly more in the high${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bands in the summer, which is closely related to the seasonal tilting of corresponding θ_{e} surfaces.
3.2 θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ relationship
Figure 6 compares the temporal variation in ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ of several given θ_{e} surfaces (i.e., θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ lookup table) computed from different reanalysis products for 2009. The deviations are indistinguishable between ERAInterim and MERRA2, except near θ_{e}=340 K, where MERRA2 is systematically lower than ERAInterim by 1.5 × 10^{16} to 6.5 × 10^{16} kg. NCEP2 shows slightly larger deviations from ERAInterim but by less than 8.5 × 10^{16} kg. The products are highly consistent in seasonal variability, and they also show agreement on synoptic timescales. The small difference between products is expected because of different resolutions and methods (Mooney et al., 2011). We expect these differences would be negligible for most applications of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$.
Figure 6 shows that, in both hemispheres, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ reaches its minimum in summer and maximum in winter for a given θ_{e} surface, with the largest seasonality at the lowest θ_{e} (or ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$) values. The seasonality decreases as θ_{e} 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; Foltz and McPhaden, 2006). Figure 6 also shows that ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ has significant synopticscale variability although smaller than the seasonal variability. Synoptic variability is typically larger in winter than summer, as discussed below.
3.3 Relationship to diabatic heating and mass fluxes
A key step of the application of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ for interpreting tracer data is the generation of the lookup table that relates θ_{e} and ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$. In this section, we address a tangential question of what controls the temporal variation in the lookup table, which is not necessary for the application but may be of fundamental meteorological interest.
As shown in Appendix A, the temporal variation in the lookup table, ${\dot{M}}_{{\mathit{\theta}}_{\mathrm{e}}}=\frac{\partial}{\partial t}{M}_{{\mathit{\theta}}_{\mathrm{e}}}\left({\mathit{\theta}}_{\mathrm{e}},t\right)$, can be related to underlying mass and heat fluxes according to
where $\frac{\partial {Q}_{\mathrm{dia}}\left({\mathit{\theta}}_{\mathrm{e}},t\right)}{\partial {\mathit{\theta}}_{\mathrm{e}}}$ (J s^{−1} K^{−1}) is the effective diabatic heating, integrated over the full θ_{e} surface per unit width in θ_{e}; m_{T}(θ_{e}t) (kg s^{−1}) is the net mass flux across the tropopause; and m_{E}(θ_{e}t) (kg s^{−1}) is the net mass flux across the Equator, including all air with equivalent potential temperature of less than θ_{e}. Q_{dia} has contributions from internal heating without ice formation (${Q}_{\mathrm{int}}^{\prime}$), 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 (${Q}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$) following
The terms Q_{evap} and ${Q}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ 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 in ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, the terms in Eqs. (8) and (9) 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, MERRA2 provides temperature tendencies for individual processes, which can be converted to heating rates per Eq. 9 following
where i refers to a specific process (${Q}_{\mathrm{int}}^{\prime}$, Q_{ice}, etc.), ${\left(\frac{\mathrm{d}T}{\mathrm{d}t}\right)}_{x}$ (K s^{−1}) is the temperature tendency of grid cell x, M_{x} (kg) is the mass of grid cell x, and Δθ_{e} is the width of the θ_{e} surface.
There are five heating terms provided in the MERRA2 product, which we can approximately relate to terms in Eq. (9), as shown in Table 1. The first three terms (Q_{rad}, Q_{dyn}, and Q_{ana}) can be summed to yield ${Q}_{\mathrm{int}}^{\prime}$; the fourth (Q_{trb}) is equal to the sum of Q_{diff} and Q_{sen}; and the fifth (Q_{mst}) approximates the sum of Q_{ice} and Q_{evap}. MERRA2 does not provide terms corresponding to ${Q}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ or Q_{evap}, but Q_{mst} represents heating due to moist processes, which includes Q_{ice} plus water vapor evaporation and condensation within the atmosphere. This water vapor evaporation and condensation should be approximately equal to Q_{evap} with a small time lag when integrated over a θ_{e} surface because mixing is preferentially along θ_{e} surfaces and water vapor released into a θ_{e} surface by surface evaporation will tend to transport and precipitate from the same θ_{e} surface within a short time period (Bailey et al., 2019). Thus, the MERRA2 term for heating by moist processes (Q_{mst}) should approximate Q_{ice}+Q_{evap}.
Figure 7a compares the temporal variation in ${\dot{M}}_{{\mathit{\theta}}_{\mathrm{e}}}$ computed by integrating the dry air mass (i.e., θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ lookup table) with ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ computed from the sum of the diabatic heating terms from MERRA2 (via Eqs. 8 to 10). The comparison focuses on the θ_{e}=300 K surface, which does not intersect with the Equator or tropopause, so the two mass flux terms (m_{T}, m_{E}) vanish. These two methods have a high correlation at 0.71. We do not expect perfect agreement because ${\dot{M}}_{{\mathit{\theta}}_{\mathrm{e}}}$ computed by the sum of heating neglects turbulent water vapor transport (${Q}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$), and only approximates Q_{evap} as discussed above. This relatively good agreement nevertheless demonstrates that the formulation based on MERRA2 heating terms includes the dominant processes that drive temporal variations in the lookup table. Figure 7a shows poorer agreement from late August to October, which we also find in other years (Figs. S1 and S2 in the Supplement) and on lower (e.g., θ_{e}=290 K, Fig. S3) but not higher (e.g., θ_{e}=310 K, Fig. S4) surfaces, where the two methods agree better. The poor agreement may reflect a partial breakdown of the assumption that Q_{mst} approximates the sum of Q_{ice} and Q_{evap}, but further analysis is beyond the scope of this study.
Figure 7b further breaks down the sum of the heating terms in Eqs. (8) and (10) from MERRA2 into individual components. Each term clearly displays variability on synoptic to seasonal scales. To quantify the contribution of different terms on the different timescales, we separate each term into a seasonal and synoptic component, where the seasonal component is derived by a twoharmonic fit with a constant offset and the synoptic component is the residual. We estimate the fractional contribution of each heating term on seasonal and synoptic timescales separately in Table 2, using the method in Sect. S1 in the Supplement. On the seasonal timescale, the variance is dominated by radiative heating and cooling of the atmosphere and moist processes (including both ice formation and extra water vapor from surface evaporation) together, with prominent counteraction between them. On the synoptic timescale, dissipation of the kinetic energy of turbulence dominates the variance.
Similar analyses on different θ_{e} surfaces and in different years (Figs. S1 to S4) all show that a combination of radiative heating and moist processes dominates the temporal variation in ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ on the seasonal timescale, while dissipation of the kinetic energy of turbulence dominates on the synoptic timescale.
To illustrate the potential application of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ for interpreting sparse data, we focus on the seasonal cycle of CO_{2} in the Northern Hemisphere as resolved by two series of global airborne campaigns, HIPPO and ATom. HIPPO consisted of five campaigns between 2009 and 2011, and ATom consisted of four campaigns between 2016 and 2018. Each campaign covered from ∼ 150 to ∼ 14 000 m and from nearly pole to pole, along both northbound and southbound transects. On HIPPO, both transects 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 Fig. 8a. 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 of less than 50 ppm and either O_{3} greater than 150 ppb or detrended N_{2}O to the reference year of 2009 of less than 319 ppb. Water vapor and O_{3} were measured by the NOAA UCATS (Unmanned Aerial Systems Chromatograph for Atmospheric Trace Species; Hurst, 2011) instrument and were interpolated to a 10 s resolution. N_{2}O was measured by the Harvard QCLS (Quantum Cascade Laser System; Santoni et al., 2014) instrument. Furthermore, we exclude all nearsurface observations within ∼ 100 s of takeoffs and within ∼ 600 s of landings as well as missed approaches, which usually show high CO_{2} variability due to strong local influences. In situ measurements of CO_{2} were made by three different instruments on both HIPPO and Atom. Of these, we use the CO_{2} measurements made by the NCAR Airborne Oxygen Instrument (AO2) with a 2.5 s measurement interval (Stephens et al., 2020), for consistency with planned future applications of 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 a 10 s resolution, and we show the detrended CO_{2} values along each airborne campaign transect for the Northern Hemisphere in Fig. 8b. Since we focus on the seasonal cycle of CO_{2}, all airborne observations are detrended by subtracting an interannual trend fitted to CO_{2} measured at the Mauna Loa Observatory (MLO) by the Scripps CO_{2} Program. This trend is computed by a stiff cubic spline function plus fourharmonic terms with linear gain to the MLO record. ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ is computed from ERAInterim in this section.
4.1 Mapping Northern Hemisphere CO_{2}
A conventional method to display seasonal variations in CO_{2} 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 Fig. 9, we compare this 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 ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$. For each latitude bin, we choose a corresponding ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bin which has approximately the same meridional coverage in the lower troposphere. We remind the reader that ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ decreases poleward while also generally increasing with altitude (Figs. 2 to 4).
As shown in Fig. 9, the transect averages of detrended CO_{2} (shown as points) from both binning methods resolve welldefined seasonal cycles (based on twoharmonic fit) in all bins, with higher amplitudes near the surface (low pressure) and at high latitudes (low ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$). However, binning by ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ leads to much smaller variations in the mean seasonal cycle (shown as solid curves) with pressure, as expected, because moist isentropes are preferential surfaces for mixing. Also, within individual pressure bins, the shortterm variability relative to the mean cycles based on the distribution of all detrended observations (not shown as points but denoted as 1σ values in Fig. 9) is smaller when binning by ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ (F test, p < 0.01), except in the lower troposphere of the highest ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bin (90 × 10^{16}–110 × 10^{16} kg). The smaller shortterm variability is expected because ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ tracks the synoptic variability in the atmosphere. When binning by latitude, the smallest shortterm variability is found at the lowest bin (surface–800 mbar) and the largest shortterm variability is found in the highest bin (500 mbar tropopause), except the highest latitude bin (45–55^{∘} N). When binning by ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, in contrast, the shortterm variability in the middle pressure bin is always smaller than the higher and lower pressure bins (F test, p < 0.01), except for the 50to70 ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bin, where the difference between the lowest and middle pressure bins is not significant (based on 1σ levels). The lower variability in the middle troposphere may reflect the suppression of variability from synoptic disturbances, leaving a clearer signal of the influence of surface fluxes of CO_{2} and stratosphere–troposphere exchanges. We compare the variance of detrended airborne observations within each ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$–pressure 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.
Figure 9 also shows the CO_{2} seasonal cycle at MLO, which falls within a single ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$–pressure bin (90 × 10^{16}–110 × 10^{16} kg, 500–800 mbar) at all seasons. Although the airborne data in this bin span a wide range of latitudes (∼ 10–75^{∘} N), the seasonal cycle averaged over this bin is very similar to the cycle at MLO (airborne cycle leads by ∼ 10 d with 1.0 % lower amplitude). This small difference is within the 1σ uncertainty in our estimation from airborne observation, and some difference is expected, since we choose a ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$–pressure bin wider than the seasonal variation in ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ and pressure at MLO.
It is also of interest to examine how CO_{2} data from surface stations fit into the framework based on ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$. Figure 10 compares the CO_{2} seasonal cycle of five NOAA surface stations (Cooperative Global Atmospheric Data Integration Project, 2019) with the cycle from the airborne observations binned into selected ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bins. These surface stations are chosen to be representative of different ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ ranges. For the comparison, we chose ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bins that span the seasonal maximum and minimum ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ value of the station. These bins are narrower than the bins used in Fig. 9, in order to sharply focus on the latitude of the station. To maximize sampling coverage, we bin the airborne data only by ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ without pressure subbins. For mid and highlatitude surface stations (right three panels), the seasonal amplitude of station CO_{2} and corresponding airborne CO_{2} are close (within 4 %–5 %), while airborne cycles lag by 2–3 weeks. The lag presumably represents the slow mixing from the midlatitude surface to the highlatitude midtroposphere (Jacob, 1999). In contrast, for lowlatitude 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 ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ or θ_{e} with air aloft. As mentioned above (Sect. 3.1), surfaces of high ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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 from the known overturning flows based on air mass flux stream functions (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 timescales. Our results nevertheless demonstrate that the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ framework combining airborne and surface data could help understanding of details of atmospheric transport both along and across θ_{e} surfaces.
4.2 Computing the hemispheric massweighted average CO_{2} mole fraction
We next illustrate the use of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ for computing the massweighted average of a longlived chemical tracer by performing this exercise for CO_{2} in the Northern Hemisphere. We calculate the Northern Hemisphere tropospheric massweighted average CO_{2} from each airborne transect using a method that assumes that CO_{2} is uniformly mixed on θ_{e} surfaces throughout the hemisphere (Barnes et al., 2016; Parazoo et al., 2011, 2012). We exclude airborne observation from HIPPO1 Northbound due to the lack of data north of 40^{∘} N. We use the θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ lookup table of the corresponding date to assign a value of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ to each observation based on its θ_{e}. The observations for each transect are then sorted by ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$. The hemispheric average CO_{2} is calculated by trapezoidal integration of CO_{2} as a function of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ and divided by the total dry air mass as computed from the corresponding range of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$.
To illustrate the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ integration method, we choose HIPPO1 Southbound and show CO_{2} measurements and ΔCO_{2} atmospheric inventory (Pg) as a function of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ in Fig. 11. The Northern Hemisphere tropospheric average detrended ΔCO_{2} is computed by integrating the area under the curve (subtracting negative contributions) and dividing by the maximum value of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ within the hemisphere (here 195.13 × 10^{16} kg). This yields a massweighted average detrended ΔCO_{2} 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 ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$. The ΔCO_{2} atmospheric inventory is dominated by the domain ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ < 120 ×10^{16} kg (midlatitude to high latitude), which has a large CO_{2} seasonal cycle driven by a temperate and boreal ecosystem, with less than 4.1 % contributed by the additional ∼ 38.8 % of the air mass outside this domain in the low latitudes or upper troposphere (Fig. 11b), where ΔCO_{2} differs less from the subtracted baseline.
We compute Northern Hemisphere massweighted average detrended ΔCO_{2} for each airborne campaign transect and fit the time series to a twoharmonic fit to estimate the seasonal cycle (Fig. 12). We find that the cycle has a seasonal amplitude of 7.9 ppm and a downward zerocrossing on 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 our estimation of the Northern Hemisphere massweighted average CO_{2} seasonal cycle from HIPPO and Atom airborne observation, we consider two main sources: (1) irreproducibility in the CO_{2} measurements and (2) limited coverage in space and time. For the first contribution, we compute the difference between massweighted average CO_{2} from AO2 and mean massweighted average CO_{2} 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 the massweighted average CO_{2} 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 zerocrossing of ±0.83 d, which are calculated from 1000 iterations of the twoharmonic fit, allowing for random Gaussian uncertainty (σ $=\pm $0.15 ppm) for each transect.
^{∗} Each simulation yields 17 data points of different dates over the seasonal cycle from 17 airborne campaign transects. RMSE of each simulation is computed with respect to the true value.
For the contribution to the error in the amplitude and phase from limited special and temporal coverage, we use simulated CO_{2} data from the Jena CO_{2} inversion Run ID s04oc v4.3 (Rödenbeck et al., 2003). 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 = 2) to produce a daily time series of the hemispheric mean, which we take as the model “truth”. We fit a twoharmonic 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 CO_{2} inversion along the HIPPO and ATom flight tracks and process the data similarly to the observations, using the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ integration method and a twoharmonic fit. The comparison shows that the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ integration method yields an amplitude which is 1 % too large and yields a downward zerocrossing date which is 6 d too late. We view 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 d in downward zerocrossing for uncertainty in the bias. Combining the random and systematic error contributions leads to a corrected Northern Hemisphere tropospheric average CO_{2} seasonal cycle amplitude of 7.8 ± 0.14 ppm and downward zerocrossing of 173 ± 6.1 d. This corrected cycle is an estimate of the climatological average from 2009 to 2018.
The error due to limited spatial and temporal 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 (undersampling the full hemisphere). We quantify the combined biases due to both limited seasonal and limited interannual coverage by comparing the twoharmonic fit of the full true daily time series of the hemispheric mean to a twoharmonic fit of those 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 seasonal, interannual, and spatial coverage account for biases in the downward zerocrossing of 1.1, 1.4, and 3.5 d respectively, all in the same direction (too late). The seasonal amplitude biases due to individual components are all small (< 0.5 %).
It is of 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 Fig. 12 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 February but more quickly from February to May. These features at least partly reflect variations in the transport of air masses to the station (Harris et al., 1992; Harris and Kahl, 1990).
In Fig. 13, we compare the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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 CO_{2} 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 weighted average CO_{2} for each airborne campaign transect. The rootmeansquare errors (RMSEs) to the true average of the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ integration method are ±0.32 and ±0.27 ppm for the HIPPO and ATom campaigns respectively, which are smaller than the RMSEs of the simple latitude–pressure weighted average method at ±0.82 and ±0.53 ppm.
We also evaluate the biases in the hemispheric average seasonal 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 amplitude (${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ method 1.0 % too large, latitude–pressure method 20.8 % too large), while both methods show a similar phasing error (6 to 7 d 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 CO_{2} fields (Bent, 2014). The ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ integration method appears advantageous because it accounts for synoptic variability and easily yields a hemispheric average by directly integrating over ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$.
The relative success of the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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 transect by excluding measurements along the Atlantic transects (ATom northbound). We further explore the impact of reduced sampling density by subsampling the Jena CO_{2} 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., 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 CO_{2} from these nine simulations by the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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 zerocrossing, as summarized in Table 3. HIPPO1 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 datasets 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 d lag in 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 has minimal impact. This suggests that, to compute the average CO_{2} of a given region, it may be sufficient to have low sampling density provided that the measurements adequately cover the full range in θ_{e} (or ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$).
We have presented a transformed isentropic coordinate, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, which is the total dry air mass under a given θ_{e} surface in the troposphere of the hemisphere. ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ can be computed from meteorological fields by integrating dry air mass under a specific θ_{e} surface, and different reanalysis products show a high consistency. The θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ relationship varies seasonally due to seasonal heating and cooling of the atmosphere via radiative heating and moist processes. The seasonality in the relationship is greater at low θ_{e} compared to high θ_{e} and is greater in the Northern Hemisphere than in the Southern Hemisphere. The θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ relationship also shows synopticscale variability, which is mainly driven by the dissipation of the kinetic energy of turbulence. ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces show much less seasonal displacement with latitude and altitude than surfaces of constant θ_{e} while being parallel and exhibiting essentially identical synopticscale variability. As a coordinate for mapping tracer distributions, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ shares with θ_{e} the advantages of following displacements due to synoptic disturbances and aligning with surfaces of rapid mixing. ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ has the additional advantages of being approximately fixed in space seasonally, which allows mapping to be done on seasonal timescales, and having units of mass, which provides a close connection with atmospheric inventories.
As a coordinate, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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 ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ extend over a wide range of latitudes (from low latitudes at the Earth's surface to high latitudes aloft), a close association with latitude is provided by the point of contact with the Earth's surface. Also, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ is nearly always monotonic with latitude (increasing equatorward), while it is not necessarily monotonic with altitude in the lower troposphere (Figs. 2 and 3).
As a first application, we have illustrated using ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ the seasonal variation in CO_{2} in the Northern Hemisphere, with data from the HIPPO and ATom airborne campaigns. This application shows that ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ has several advantages as a coordinate compared to using latitude: (1) variations in CO_{2} with pressure are smaller at fixed ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ than at fixed latitude, and (2) the scatter about the mean CO_{2} seasonal cycle is smaller when sorting data into pressure–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bins than into pressure–latitude bins. We have also shown that, at middle and high latitudes, the CO_{2} seasonal cycles that are resolved in the airborne data (binned by ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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, CO_{2} cycles in the airborne data (binned similarly by ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$) are less consistent with surface data, as expected due to slow transport and diabatic processes within the Hadley circulation. For characterizing the patterns of variability in airborne CO_{2} data, we expect the advantages of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ over latitude will be greatest for sparse datasets, allowing data to be binned more coarsely with pressure or elevation while still resolving features of largescale variability, such as seasonal cycles or gradients with latitude.
As a second application, we use ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ to compute the Northern Hemisphere tropospheric average CO_{2} from the HIPPO and ATom airborne campaigns by integrating CO_{2} over ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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 zerocrossing on Julian day 173 ± 6.1. This hemispheric average cycle may prove valuable as a target for validation of models of surface CO_{2} exchange.
Our analysis also clarifies that computing hemispheric averages with the ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ integration method depends on adequate spatial coverage. The coverage provided by the HIPPO and ATom campaigns appears more than adequate for computing the average seasonal cycle of CO_{2} 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 ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces. The ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ integration method of computing hemispheric averages assumes that the tracer is uniformly distributed and instantly mixed on θ_{e} (${M}_{{\mathit{\theta}}_{\mathrm{e}}}$) surfaces. We have shown that systematic gradients in CO_{2} are resolved with pressure at fixed ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$, which reflects the finite rates of dispersion on θ_{e} 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., Fig. 5). The need is especially relevant for high ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bins which are less completely mixed and which tend to intersect the Equator or have separate surface branches. For these ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ bins, it would be more appropriate to integrate over M_{θ} in the upper and lower atmosphere separately. This complication is of minor importance for computing the massweighted average CO_{2} cycle, because the cycle of CO_{2} is small in these air masses.
The definition of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ requires horizontal and vertical boundaries for the integration of dry air masses. We use the dynamic tropopause (based on potential vorticity units) 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, ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ could be computed from the lowest θ_{e} surface in the Southern Hemisphere with a latitude cutoff at 30^{∘} S, to apply to airborne observations only over the Southern Ocean. On the other hand, the boundary choice only influences ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces that actually intercept the boundaries, making the choice less important at high latitudes in the lower troposphere (lowest ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ surfaces). Some tropospheric applications may also benefit by integrating over dry potential temperature (θ) rather than θ_{e}.
Based on our promising results for CO_{2}, we expect that ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ may be usefully applied as a coordinate for mapping and computing atmospheric inventories of many tracers, such as O_{2} ∕ N_{2}, N_{2}O, CH_{4}, and the isotopes of CO_{2}, whose residence time is long compared to the timescale for mixing along isentropes. ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ 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 ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$.
Following Walin's derivation for crossisothermal volume flow in the ocean (Walin, 1982), we show how ${\dot{M}}_{{\mathit{\theta}}_{\mathrm{e}}}=\frac{\partial}{\partial t}{M}_{{\mathit{\theta}}_{\mathrm{e}}}\left({\mathit{\theta}}_{\mathrm{e}},t\right)$ can be related to energy and mass fluxes. We start by deriving the relationship for M_{θ} (based on potential temperature θ) but later generalize to apply to ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$.
All definitions are summarized in Table A1, and Fig. A1 is the schematic diagram of mass and energy flux.
All mass and heat fluxes into region R(θ,t) are counted as positive. The heat fluxes through the tropopause, Equator, and surface of region R(θ,t) can be divided into an advective (F(θ,t)) and a turbulent (D(θ,t)) component. Integrating over the tropopause and equatorial boundary, we have
where C_{pd} is the heat capacity of dry air in units of J kg^{−1} K^{−1}.
Based on the continuity of mass and energy for region R(θ,t), we obtain
Substituting Eqs. (A1) to (A3) into Eq. (A5) and differentiating with respect to θ yields
where
Differentiating Eq. (A4) with respect to θ and multiplying C_{pd}⋅θ yields
Subtracting Eq. (A8) from Eq. (A6), we obtain
Equation (A9) divided by C_{pd} plus Eq. (A4) yields
Equation (A10) illustrates the temporal variation in M_{θ}, where Q_{int} includes radiative heating (i.e., sum of shortwave and longwave heating), dissipation of the kinetic energy of turbulence, and latent heat release due to evaporation and condensation.
To modify Eq. (A10) to apply to ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ rather than to M_{θ}, it is necessary to replace all θ with θ_{e} and additionally account for the following:

Condensation and evaporation is conserved on the θ_{e} surfaces, but the gaining and losing of water vapor through surface evaporation and water vapor transport contributes to θ_{e}. 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 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 (${Q}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$).

Internal heating (Q_{int}) needs to exclude latent heat releasing due to evaporation and condensation of liquid water, which cancel in θ_{e}, but it still needs to include heating from ice formation, which does not cancel in θ_{e}. We subtract this ice component from the rest of the internal heating, yielding two terms ${Q}_{\mathrm{int}}^{\prime}$ and Q_{ice}, with ${Q}_{\mathrm{int}}={Q}_{\mathrm{int}}^{\prime}+{Q}_{\mathrm{ice}}$.
Therefore, we can write the temporal variation in ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ as
We provide R code to generate θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ lookup tables from ERAInterim meteorological fields at https://doi.org/10.5281/zenodo.4420417 (Jin, 2021a).
All HIPPO 10 s merge data are available from https://doi.org/10.3334/CDIAC/HIPPO_010 (Wofsy et al., 2017b). Besides, all HIPPO Medusa merge data are available from https://doi.org/10.3334/CDIAC/HIPPO_014 (Wofsy et al., 2017a). All ATom 10 s and Medusa merge data are available from https://doi.org/10.3334/ORNLDAAC/1581 (Wofsy et al., 2018).
CO_{2} data from Mauna Loa Observatory are available from the Scripps CO_{2} Program at https://scrippsco2.ucsd.edu/assets/data/atmospheric/stations/in_situ_co2/monthly/monthly_in_situ_co2_mlo.csv (last access: 10 July 2020; Keeling et al., 2001.). Other surface station CO_{2} data, including Trinidad Head, Cold Bay, Barrow, Cape Kumukahi, and Sand Island, are provided by the NOAA ESRL GMD flask sampling network (http://www.cmdl.noaa.gov/ccgg/trends, last access: 9 July 2020) and downloaded from Observation Package (ObsPack) at https://doi.org/10.25925/20190812 (Cooperative Global Atmospheric Data Integration Project, 2019).
The Jena CO_{2} inversion data are available at the project website: https://doi.org/10.17871/CarboScopes04oc_v4.3 (Rödenbeck, 2005). Run ID s04oc v4.3 was used in this study.
θ_{e}–${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ lookup tables with daily resolution and 1 K intervals in θ_{e} from 1980 to 2018 computed from ERAInterim are available at https://doi.org/10.5281/zenodo.4420398 (Jin, 2021b).
The supplement related to this article is available online at: https://doi.org/10.5194/acp212172021supplement.
YJ carried out the data analysis and derivations. All sections in the initial draft were prepared by YJ and RFK, with critical revisions from all coauthors. BBS made important contributions to the improvement of the application part. EJM and NCP raised useful suggestions regarding the definition of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$. ER provided valuable suggestions for analyzing the relationship between diabatic heating and mass fluxes.
The authors declare that they have no conflict of interest.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
The original ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ concept arose out of discussions during the ORCAS field campaign that included Ralph F. Keeling, Colm Sweeney, Eric Kort, Matthew Long, and Martin HoeckerMartinez. We would like to acknowledge the efforts of the full HIPPO and ATom science teams and the pilots and crew of the NSF/NCAR GV and NASA DC8 as well as the NCAR and NASA project managers, field support staff, and logistics experts. In this work, we have used the HIPPO and ATom 10 s merge files, supported by the National Center for Atmospheric Research (NCAR). NCAR is sponsored by the National Science Foundation under Cooperative Agreement No. 1852977. We thank the Harvard QCLS, Harvard OMS, NOAA UCATS, and NOAA Picarro teams for sharing measurements. We thank NOAA ESRL GML for providing surface station CO_{2} data measured at Trinidad Head, Cold Bay, Barrow, Cape Kumukahi, and Sand Island. We thank Christian Rödenbeck for sharing the Jena CO_{2} inversion run. We thank the two anonymous reviewers for their valuable comments and efforts.
This research has been supported by the National Science Foundation (grant nos. ATM0628575, ATM0628519, ATM0628388, AGS1547797, and AGS1623748) and the NASA (grant no. NNX15AJ23G).
This paper was edited by Andreas Engel and reviewed by two anonymous referees.
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 Abstract
 Introduction
 Methods
 Characteristics of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$
 Applications of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ as an atmospheric coordinate
 Discussion and summary
 Appendix A: Temporal variation in ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement
 Abstract
 Introduction
 Methods
 Characteristics of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$
 Applications of ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$ as an atmospheric coordinate
 Discussion and summary
 Appendix A: Temporal variation in ${M}_{{\mathit{\theta}}_{\mathrm{e}}}$
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement