From Andy May Petrophysicist

By Andy May and Philip Mulholland

IGRA2 radiosonde 1991–2025 means (10 hPa bins, 10° latitude slices) show systematic differences between observed mid-tropospheric temperatures, and the CMIP 5 & 6 (Coupled Model Intercomparison Project, see IPCC, 2021) ensemble means as shown by McKitrick & Christy (2020) and Po-Chedley et al. (2022). The difference between the CMIP models and the observations is most noticeable in the tropics as discussed here. While multiple reasons, such as too much weight on the warming effect of additional CO₂, anomalous cloud feedback, or bad weather balloon data, have been proposed (IPCC, 2021, p. 443), no one really knows why they are so poor at reproducing tropospheric temperature profiles.

In previous posts (see here and here) we focused on modeling the troposphere using models based on moist adiabatic theory, a methodology similar to the one used in CMIP6. Our method had three free variables, surface temperature, surface dew point, and overall lapse rate. These models did a good job of reproducing observed tropical air temperatures and dewpoint up to around 250 hPa (~10.6 km). The models did not require any other input, such as greenhouse gas levels or measurements of radiation in and out at the top of the atmosphere. But above ~250 hPa they performed poorly. This opened the possibility that the atmospheric temperature profile, in the tropics below 250 hPa was only dependent upon surface conditions; but above this altitude, a different model, with different forcings was required.

Encouraged by these results, we worked on extending the model to other latitudes and developing a new model to use higher in the troposphere. This post presents a composite model for the global troposphere, it is similar in concept, but not in detail to the composite moist adiabatic/RAE (RAE: radiative advective equilibrium) model of (Cronin & Jansen, 2016). The anchor for our model is the molar density intersection (or simply the “intersection”) discovered by Michael and Ronan Connolly (Connolly & Connolly, 2014a) & (Connolly M. , 2025). It turns out to be the easiest and most general way to divide the troposphere into a lower part that can be modeled using moist adiabatic theory and an upper part that is much more complicated. We call the upper troposphere model the frost-point model or “FPM.”

Tropospheric structure

The intersection (May, 2025) occurs below the traditional tropopause as defined by the WMO (WMO & Ashford, 1957, p. 137). The distance below the tropopause varies from 50 to over 200 hPa (hectopascals of pressure = one millibar) due to the occasional complexity of the lapse rate change from cooling with altitude in the troposphere to warming with altitude in the stratosphere. The tropopause is generally thought to be the altitude where the air temperature stops declining and begins to increase, that is, a lapse rate of zero. While the intersection is easily reproduced and precise, the WMO definition (WMO & Ashford, 1957) of the tropopause is vague and complicated (two full paragraphs) and can result in multiple tropopauses in the mid-latitudes as shown in figure 1. In the tropics the sharper “cold-point” form of the tropopause is seen (see figure 3 in May 2025 or see figures 1 & 2 here).

The upper left plot shows how the intersection is determined. It is the intersection of two best fit lines, an upper one above 125 hPa and a lower from 300 to 750 hPa. The fact that molar density forms a line is not surprising and follows from the ideal gas law, since molar density equals pressure divided by temperature times the gas constant, but the change in slope at the intersection is surprising. This change implies a change in state or gas composition, or both. For more details see here or Connolly & Connolly 2014a.

The Molar Density Intersection

The molar density intersection changes altitude continuously by latitude (see the lower left plot in figure 1), from month to month, and the peak altitude in the tropics moves with the Intertropical Convergence Zone (ITCZ) (May, 2025). The 1991-2025 mean IGRA2 radiosonde characteristics of the air at the intersection in February between 40 and 50°N are listed in figure 1. The abbreviations are: “rh” means relative humidity and “q” means specific humidity. Notably the air is quite dry, and the mean temperature is -51°C.

As noted above, the lower portion of the troposphere, below the intersection, can be accurately modeled using only three free variables, the surface temperature, overall lapse rate, and dew-point depression (temperature – dewpoint). We call this model the DPAH model (Dewpoint Anchor Hypothesis), and it is based on moist-adiabatic theory (May, 2025). Below the intersection, the troposphere behaves like a moist‑adiabatic system because water vapor is abundant enough to control buoyancy, lapse rate, and infrared opacity. Above the intersection, temperatures fall so low that any water vapor left is deposited on ice crystals suspended in the air, breaking the assumptions of moist‑adiabatic theory.

Even though liquid droplets can survive to −38 °C in the absence of ice nuclei, the intersection zone is far colder, around −56 °C on average. At these temperatures, the saturation mixing ratio over ice is extremely small (often below 0.05 g/kg). Any remaining water vapor is rapidly deposited onto existing ice crystals. The air is simply too cold to sustain meaningful water vapor concentrations. The mean specific humidity at the intersection is 0.03 (±0.02) g/kg and the saturation mixing ratio with respect to ice lies between 0.03 and 0.06 g/kg (Stull 2026, saturation-vapor-pressure formulae). The radiosonde mean specific humidity is 0.03±0.02g kg^-1, which coincides with the lower bound of Stull’s range, thus the actual humidity is at or below the minimum possible saturation value.

Key Measurements at the Molar Density Intersection (1991-2025 means)
Latitude Slice (- is south)	Mean Temperature °C	Mean height (m)	Mean Relative Humidity (%)	Mean Specific Humidity (g/kg)	Mean Pressure (hPa)	Mean Molar Density (mol/m3)
-70	-59.9	8,624.3	32.0	0.02	288.2	16.5
-60	-57.0	8,731.7	32.5	0.03	290.5	16.5
-50	-51.7	9,404.6	31.9	0.03	278.5	15.5
-40	-52.2	10,359.2	23.9	0.02	251.9	14.1
-30	-53.9	11,650.2	14.4	0.01	213.7	12.4
-20	-59.6	13,219.3	15.7	0.01	173.4	10.5
-10	-66.2	14,113.0	28.8	0.01	153.5	9.5
0	-67.3	14,281.1	36.1	0.01	150.1	9.4
10	-64.4	13,892.7	23.6	0.01	157.7	9.7
20	-61.3	13,453.6	23.6	0.01	168.2	10.2
30	-55.5	12,036.4	23.6	0.02	204.1	11.9
40	-51.7	10,702.0	25.5	0.03	241.4	13.6
50	-51.4	9,517.0	26.6	0.03	275.7	15.2
60	-48.9	8,753.1	37.2	0.06	306.9	16.7
70	-50.8	8,384.7	28.9	0.03	317.2	17.3
80	-51.2	8,139.2	38.3	0.05	324.3	17.7
Mean	-56.4	10,953.9	27.7	0.03	237.2	13.5
Standard Deviation	5.8	2,198.6	6.8	0.02	60.8	2.9

Table 1. Key radiosonde 1991-2025 mean measurements at the molar density intersection by latitude slice.

Table 1 suggests that at the intersection, ice deposition has removed nearly all the water vapor and the air is not just cold but is water vapor exhausted. It appears that the intersection marks the altitude where supersaturation collapses because ice deposition has caught up.

Since the mean molar density in table 1 is below 15 mol/m³, it is reasonable to assume that radiative cooling dominates in the region. The collapse of water vapor concentration coincides with a shift from moist‑adiabatic to radiatively dominated behavior as discussed by (Cronin & Jansen, 2016). Here, outgoing longwave radiation to space from non-condensing greenhouse gases dominates.

At the intersection, the moist‑adiabat framework loses validity because its key ingredient—water vapor—is no longer present in meaningful quantities. We hypothesize that this transition from water vapor to ice has something to do with the molar density intersection. At the intersection and above it, we need a different model, the FPM model.

The Connolly’s hypothesized that the formation of oxygen and nitrogen multimers (Connolly & Connolly, 2014b) began to occur at the molar density intersection and that they accounted for the change in the molar density versus pressure slope by altering the molar density. For the Connolly multimerization mechanism to explain the molar density-pressure phase change, the implied multimer concentrations would necessarily produce detectable, altitude‑localized spectroscopic and radiative anomalies. These are not observed, despite extensive modern measurements of the radiation emissions of multimers and dimers. In contrast, the conventional explanation—transition from moist convective to radiatively controlled, dry stratified conditions—accounts for the change in molar density-pressure slope without requiring an unobserved layer of multimers.

Dimer (O₂-O₂ and O₂-N₂) collision complexes (or CIAs, collision induced absorption complexes) occur at all pressures and not at a specific altitude. This phenomenon is well-studied (Adkins et al., 2023) & (Finkenzeller & Volkamer, 2022) because the CIAs affect radiation transfer and spectroscopy within the atmosphere. The dimers do occur in the upper atmosphere, but they have not been observed as a condensed phase or a heavy bulk gas that would change the macroscopic molar density slope.

The Connolly multimer hypothesis is certainly not disproven, just not observed in nature. Our idea (again unproven) is that the loss of free water vapor content is the cause of the intersection and acknowledge that more work is needed to determine the root cause. In any case the observed, and easily calculated, molar density intersection provides a convenient upper limit to the applicability of the moist adiabatic model or DPAH.

The modeling results

The DPAH moist-adiabatic model works very well almost everywhere up to the intersection. Above that point, our FPM model produces mixed results depending upon the latitude slice chosen. The tropics model well, as shown in figure 2.

The middle latitudes also model reasonably well, although the frequent double tropopauses in this region can cause problems for the FPM portion.

The only real blow up is in the North Polar Region where modeling is very difficult.

The critical variables are temperature, dewpoint, and lapse rate. We splice the FPM and DPAH model output at the molar density intersection and then compute R² for the combined curves. The R² values are excellent for temperature and the dewpoint in nearly every slice but mixed for the lapse rate as shown in table 2.

R2 for the total column (1010 to 10 hPa)
Latitude slice	R2_T	R2_Td	R2_lapse
-70	98.9%	99.0%	90.5%
-60	99.2%	40.5%	65.7%
-50	98.4%	99.7%	55.6%
-40	98.4%	99.8%	68.1%
-30	99.7%	99.6%	78.4%
-20	99.4%	99.8%	79.1%
-10	96.9%	99.0%	66.0%
0	99.1%	99.8%	53.5%
10	97.5%	99.1%	72.6%
20	98.9%	96.9%	77.4%
30	98.3%	95.8%	89.3%
40	96.7%	94.0%	86.4%
50	98.5%	99.8%	82.1%
60	98.7%	99.2%	83.8%
70	98.3%	98.9%	84.3%
80	96.8%	98.6%	71.8%

Table 2. R² values for the full column (the models are spliced at the molar density intersection) for temperature, dewpoint, and lapse rate.

Methods

To do this work, the IGRA2 radiosonde data were quality controlled and then binned into 10 hPa bins for each 10° latitude slice. Table 3 lists each model parameter, for both models, and provides a short physical description, and ranks the parameters from most to least important based on the variance of the optimized parameter values across all runs. The table also includes the correlations (R²) between each parameter and the column-integrated diagnostics (T, Td, lapse rate), but these R² values do not affect the ranking.

The variance shown in table 3 is not the traditional variance, it is the variance in the optimized parameter values. Table 3 ranks the ten optimized parameters across the 16 latitude slices. The largest variance is 1.09×10⁸ (strato_amp); the smallest is 0.0 (latent_heat_rel). Figure 5 displays the corresponding Z-scores.¹ Z-scores greater than 2 occur at isolated latitudes (e.g., strato_amp at 20°N, td_coeff at 30°N, radiation_out at 50°N). These statistics describe the latitude dependence of the fitted coefficients; they do not establish which processes are “fundamental” or “compensatory.” The DPAH segment uses only three free parameters (T0, dpd, ELR); the FPM segment uses seven. The higher parameter count of FPM is from empirical “curve fitting;” it is not proof that these seven independent physical mechanisms operate or are the only mechanisms at work.

Table 3. The model variables, ranked by optimized parameter variance across latitude bands. The highest variance is “strato_amp” or the strength of stratospheric warming versus latitude. The weakest, or the most consistent across latitudes, is latent heat release.

As shown in table 3, the FPM and DPAH models use different variables and have no variables in common. This suggests that atmospheric cooling above and below the intersection work via completely different processes. Below, water vapor does the work; and above radiation and wind transport do most of the work. Indeed, our FPM model suggests that above the intersection heat is mostly carried away by radiation (radiation_out) and wind (env_loss).

Somewhat counterintuitively, the variables in table 3 that have low scores are important everywhere, those with high scores are only important in some regions (high variance). The special high variance variables required to get a reasonable fit are a bit of a mystery and there are too many of them. The “strato_amp” variable is there mostly to help get the shape of the tropopause; it may have little physical meaning. The “td_coeff” variable controls the shape of the dewpoint as it relates to relative humidity. The “env_loss” variable is always important and is a factor meant to compensate for the very high wind speeds in some portions of the upper atmosphere (May, 2025, fig. 12,13). While the DPAH model is grounded in only three physically meaningful variables, the FPM model is more of a curve fitting exercise. However, it is still very helpful, since it helps us visualize how the upper troposphere might work, and emphasizes how different it is from the lower troposphere. Figure 5 compares the variables listed in table 3 by their Z-score.^[1]

The striking thing about figure 5 is the variation in extremes by latitude. The important variable at 20°S is latent heat release and at 20°N it is the stratospheric warming trend. At 30°N, the main desert-band latitude is the dewpoint coefficient. At 50°N it is lower than normal precipitation and extreme radiation out. Thus, in the upper troposphere FPM model region, the factors controlling heat loss vary tremendously by latitude.

In the desert belt (~30°N), “td_coeff” has a large negative Z-score, meaning the optimizer reduces dewpoint sensitivity to RH relative to the global mean. Here, the lower troposphere is already extremely dry, and the dewpoint depression is large, so the model does not need to amplify dewpoint–RH coupling. The dryness is “baked in” to the column, and “td_coeff” is correspondingly suppressed.

At 50°N, “radiation_out” has a very large positive Z‑score, indicating unusually strong radiative cooling in the upper troposphere at this latitude. This aligns with the high land/ocean ratio and relatively low humidity there. The “env_loss” variable has negative Z‑scores at 40–60°N, meaning the optimizer relies less on the wind/mixing term in these mid‑latitudes. The “precip” variable is also low at 50°N suggesting lower than normal precipitation in that latitude band. Next, we examine the models individually.

The Models

To understand how the DPAH and FPM model parameters behave across latitudes, we examined two complementary diagnostics. First, we compute the variance of each optimized parameter across latitudes, which indicates how strongly the model relies on that parameter to adjust to regional conditions. Parameters with high variance are structurally important locally because the optimization varies them widely.

Second, we compute correlations between each parameter and the R² metrics (temperature, dewpoint, lapse rate) to assess how strongly each parameter influences model skill. Separately, we compute a normalized importance measure for each parameter at each latitude, defined as the absolute deviation from its mean divided by its standard deviation. This normalized importance highlights where each parameter becomes unusually influential across latitude. Figures 6 and 7 plot this normalized importance for each model.

The importance statistics and Z scores reveal which parameters are globally stable and physically fundamental, and which ones vary strongly with latitude because they are responding to real latitude-specific atmospheric differences. The variance, as in table 3, tells us how much a specific variable changes globally, the Z-score tells where it changes and by how much, and the importance shows how much each variable influences the solution at a specific latitude slice.

In figure 6 we see that the optimized variables in the DPAH model travel together once normalized. All are maximal in the tropics and at the poles. They are less significant in the middle latitudes where zonal winds and storms are very important. This is the sort of pattern expected when the troposphere below the molar density intersection is very constrained by intersection height, the observed lapse rate (surface to intersection), the dewpoint depression, and moist adiabatic theory. The DPAH model is anchored by real physical constraints, so the importance factor reflects real atmospheric structure.

Surface temperature (T0) is extremely important at the poles and in the tropics. The lapse rate (ELR) is important over most latitudes, but extremely important at the poles where the intersection is very low. The dewpoint depression (dpd) peaks (positively or negatively) in regions where the humidity structure and the intersection height change quickly. It is significant that it has a peak at 20-30°S and a valley at 20-30°N. The latitudes from 20°S-30°S are dominated by oceans and those from 20-30°N by deserts.

Figure 7 tells us that the troposphere above the intersection is very complex. “strato_amp” or the rapidity of the temperature increase at the base of the stratosphere is very important at 20°N, which is where the height of the intersection drops most rapidly (see figure 1). At 20°S “latent_heat_release” is the dominant variable since this is about the latitude with maximal ocean surface. The “latent_heat_release” variable is also important at 20°N where the intersection height drops most quickly.

The “td_coeff” variable, or the dewpoint to relative humidity factor, is maximal at 30°N, near the aforementioned desert belt where the Sahara, SW USA/Northern Mexico, Gobi, etc. exist. In desert regions entrainment (or the mixing of air parcels with the surrounding air) dilutes rising plumes of moisture, ice formation aloft is reduced, and the dewpoint depression increases throughout the column, as seen in figure 7. The spike in “td_coeff” is physically significant and important.

The “p-trans” variable is the distance of the troposphere to the stratospheric transition above the intersection, it is maximal in the Antarctic and the high northern latitudes where the lower troposphere is very thin. The “env_loss” variable is also high in the polar regions; it estimates the loss of heat to the environment. This can be due to strong winds or melting ice. The “radiation_out” variable also increases slightly.

The noticeable “strato_amp” peak at 20°N, when it is a weak variable at other latitudes is a bit surprising. However, the cold point at 20°S is lower than at 20°N, and to compensate for this the optimizer may have boosted this variable.

Conclusions

DPAH is a reasonable model, based on moist adiabatic theory, that adequately describes heat transfer (or cooling) in the troposphere below the intersection. It reproduces the observed mean temperature and dew-point profiles from the surface to the molar-density intersection with the values listed in Table 2. No explicit CO₂ term appears in the DPAH equations. That absence is a property of the chosen functional form and of the 1991–2025 radiosonde means; it does not constitute a measurement of the radiative contribution of CO₂, if any.

Above the intersection the seven-parameter FPM fit is required to reach a comparable correlation. The increase in the number of free parameters is an empirical result of the optimization; it indicates that a low-parameter description of the observed upper-tropospheric means was not obtained with the functional forms tested. The radiosonde data themselves supply no direct measurement of the relative magnitudes of radiative, advective, or latent-heat fluxes at those altitudes. The result suggests that the cooling mechanism above the intersection is more complex than below it.

This is consistent with the failure of the CMIP climate models in the 300 to 100 hPa region, the region that contains the intersection. It has been proposed that this failure is due to an overemphasis on the greenhouse effect (McKitrick & Christy, 2020), (Po-Chedley et al., 2022), and (IPCC, 2021, p. 443).

Above the intersection there is almost no water vapor, so the driving force below the intersection is gone. Above the intersection CO₂ and other well-mixed greenhouse gases probably play a significant role, below the intersection these models suggest they do not.

The FPM model that we use has too many variables. This suggests that the upper troposphere may be inherently multi-process and has no single overriding physical root across all latitudes.

With so many processes at play in the upper troposphere, the FPM model became more an exercise in curve fitting than a physical model. In the early days of this project, we experimented with a more general, and less empirical Markov model for the upper troposphere. It failed, supporting the idea that a low-parameter physical model is not possible.

The polar regions are dominated by freezing, thawing, and radiation emissions to space. The Northern and Southern Hemispheres act differently because ocean surfaces dominate in the Southern Hemisphere and land surfaces in the Northern. We found it very interesting that the desert regions around 30°N were so visible in the model results and “radiation_out” was so prominent at 50°N where the land/ocean ratio is maximal.

As expected, “p_trans,” or the distance from the molar density intersection to the beginning of the stratosphere, is lowest in the tropics and highest in the middle latitudes where double WMO tropopauses are common. At the scale considered in this post, the DPAH model is fairly solid and shows that in the lower troposphere heat loss is dominated by water vapor processes. Above the molar density intersection, where water vapor is essentially zero, the FPM model has too many degrees of freedom and too many variables to be a proper model, but it is illuminating. It matches observations reasonably well by changing the variable emphasized from latitude to latitude in a reasonable way. Thus, while not a physically meaningful model on its own, it is a good learning tool.

The cause of the sharp molar density intersection is still a mystery. It is unlikely to be multimer formation since multimers occur at all altitudes and are very short lived (Welsh, 1962) & (Hartmann et al., 2011). The dehydration of the air by freezing is also a gradual process, but it could be the cause of the intersection or part of the cause.

This post benefited from conversations with Dr. Ronan Connolly; we are grateful for his help. He disagrees with our assessment of the multimerization theory and argues that our critiques have already been addressed in Connolly & Connolly (2014) We disagree, but interested readers are welcome to read his paper and form their own opinion.

The bibliography can be downloaded here.

1. Parameter value minus global parameter mean, divided by the global standard deviation. The scaled Z-scores in figure 5 can be used to compare the importance of each parameter across latitudes. A Z‑score measures how many standard deviations a parameter at a given latitude lies above or below its global mean. Thus, Z = +2 indicates the parameter is two standard deviations higher than average at that latitude, while Z = −1 indicates it is one standard deviation lower. Because Z‑scores are dimensionless, they allow direct comparison across parameters with very different physical units and magnitudes. Large positive or negative Z‑scores identify latitudes where a parameter plays an unusually strong or unusually weak role in the model solution. ↑

The Tropospheric Model R code can be downloaded here:
https://andymaypetrophysicist.com/wp-content/uploads/2026/08/Tropospheric_model_R_code.zip

You will also need the R code from May 2025 to prepare the IGRA2 data for the model. All the links you need are here:
https://andymaypetrophysicist.com/2025/12/19/the-story-behind-my-paper-on-the-itcz-and-the-hadley-circulation/

This post was just to present the results.