We introduce a transformed isentropic coordinate

The spatial and temporal distribution of long-lived chemical tracers like
CO

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 tropopause-based
rather than surface-based 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
tropopause-based, jet-based, and equivalent latitude coordinates
(Petropavlovskikh et al., 2019). In the troposphere, a transformed
coordinate, the isentropic coordinate (

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,

Several choices need to be made in the definition of

In this paper we describe the method for calculating

The calculation of

We compute

Following Bolton (1980), we compute the water vapor mixing
ratio (

The gravity constant (

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

ERA-Interim 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

We show a schematic of the conceptual basis for the calculation of

Schematic of the conceptual basis to calculate

This calculation yields a unique value of

Figure 2 shows snapshots of the distribution of zonal average

Snapshot of the distribution of

At lower latitudes, the zonal averages of

Time series of meridional displacement of selected zonal average

Figure 4 shows the zonal average meridional displacement of

Since the tilting of

Snapshots (1 January 2009 and 1 July 2009) of the mass
distribution of different

Figure 6 compares the temporal variation in

Variability in

Figure 6 shows that, in both hemispheres,

A key step of the application of

As shown in Appendix A, the temporal variation in the look-up table,

Correspondence of heating variables between our derivation (Eq. 9) and MERRA-2.

There are five heating terms provided in the MERRA-2 product, which we can
approximately relate to terms in Eq. (9), as shown in Table 1. The first three
terms (

Figure 7a compares the temporal variation in

Figure 7b further breaks down the sum of the heating terms in Eqs. (8) and (10) from MERRA-2 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 two-harmonic 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.

Fractional contribution of the individual heating terms in Fig. 7b to their sum for

Similar analyses on different

To illustrate the potential application of

A conventional method to display seasonal variations in CO

Seasonal cycles of airborne Northern Hemisphere CO

As shown in Fig. 9, the transect averages of detrended CO

Figure 9 also shows the CO

CO

It is also of interest to examine how CO

We next illustrate the use of

To illustrate the

We compute Northern Hemisphere mass-weighted average detrended

Comparison between the CO

To address the error in our estimation of the Northern Hemisphere mass-weighted
average CO

RMSE, seasonal amplitude, and day of year of the downward
zero-crossing of each simulation based on the Jena CO

For the contribution to the error in the amplitude and phase from limited
special and temporal coverage, we use simulated CO

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 (under-sampling
the full hemisphere). We quantify the combined biases due to both limited
seasonal and limited interannual coverage by comparing the two-harmonic fit of the
full true daily time series of the hemispheric mean to a two-harmonic
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 zero-crossing 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 (

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

In Fig. 13, we compare the

Comparison between the Northern Hemisphere average CO

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 (

The relative success of the

We have presented a transformed isentropic coordinate,

As a coordinate,

As a first application, we have illustrated using

As a second application, we use

Our analysis also clarifies that computing hemispheric averages with the

The definition of

Based on our promising results for CO

Following Walin's derivation for cross-isothermal volume flow in the ocean
(Walin, 1982), we show how

Definition of variables.

All definitions are summarized in Table A1, and Fig. A1 is the schematic diagram of mass and energy flux.

Illustration of terms defined in Table A1. Shaded area denotes
the region

All mass and heat fluxes into region

Based on the continuity of mass and energy for region

To modify Eq. (A10) to apply to

Condensation and evaporation is conserved on the

Internal heating (

Therefore, we can write the temporal variation in

We provide R code to generate

All HIPPO 10 s merge data are available from

CO

The Jena CO

The supplement related to this article is available online at:

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 co-authors. BBS made important contributions to the improvement of the application part. EJM and NCP raised useful suggestions regarding the definition of

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

This research has been supported by the National Science Foundation (grant nos. ATM-0628575, ATM-0628519, ATM-0628388, AGS-1547797, and AGS-1623748) and the NASA (grant no. NNX15AJ23G).

This paper was edited by Andreas Engel and reviewed by two anonymous referees.