Slicing the earth to study Cloud Fraction and VPD.

According to many sources, the earthโ€™s Cloud Fraction, CF, is the major source of climate change uncertainty. Cloud Fraction varies a lot from northern hemisphere to southern hemisphere and in-between. Global measurement of CF has a narrow range (64% to 60%). With accuracy challenging climate change models. This essay will slice the earth by latitude and analyze the larger cloud fraction range (82% to 52%) and vapor pressure deficit, VPD.

A global map of cloud cover, showing various patterns of clouds across different regions of the Earth in shades of blue and white.

From Watts Up With That?

By: Charles Blaisdell PhD ChE

Abstract

According to many sources, the earthโ€™s Cloud Fraction, CF, is the major source of climate change uncertainty.  Cloud Fraction varies a lot from northern hemisphere to southern hemisphere and in-between.  Global measurement of CF has a narrow range (64% to 60%).  With accuracy challenging climate change models. This essay will slice the earth by latitude and analyze the larger cloud fraction range (82% to 52%) and vapor pressure deficit, VPD.  The results show the logical correlation of sun angle to cloud fraction, VPD, and Enthalpy, En, and an unexpected correlation to land fraction.  The land fraction correlation is most likely related to the lower water evaporation (per unit of global surface) from land vs ocean.  On a global basis land fraction is constant implying that any change in cloud fraction or VPD is related to a change in landโ€™s ET.

Resent research by H. Wu et al (2023), (2), and H Liu et al (2022), (1) have uncovered the effects of El Nino and La Nina on cloud fraction.  This essay will show these perturbations in the cloud fraction do not affect the albedo, VPD, or En.

The H. Liu et al paper (1) also shows land moisture is related to climate change.  Water evaporation change from land is the basis of the Cloud Reduction Global Warming, CRGW, theory (and model), (3).  Global VPD and En are the main variables in the CRGW, theory of climate change.

Introduction

Climate change occurs when the radiation to the earthโ€™s surface does not match the radiation back to space.  Cloud reflectivity and surface reflectivity (albedo) are the main measurements of the total radiation to the surface.  Cloud reflectivity being larger than surface reflectivity.  Total cloud fraction has many sub reflectivity components (all contributing to the uncertainty): Type, height, density, degree of partly cloudy, and probably some others (this author is not a cloud expert, please excuse oversimplification of a complex subject).

Clouds are a small mass of the earthโ€™s total ET mass, but very controlling part of the water cycle.  Water evaporated from the oceans and land by the sunโ€™s radiation is a major part of the water cycle.  Oceans have the largest part of the cycle.

The basic mechanism of cloud formation is the cooling of moist air as it rises (or meets cooler air or lower pressure air or is forced up) and condense and makes clouds and possibly rain.  Where clouds form depends on the global air circulation and where the evaporation of water occurred.   Therefore, the main variables of cloud formation are temperature, pressure, and specific humidity.  Dewpoint is one measure of probability of cloud formation.  Vapor Pressure Deficit, VPD, is another way of looking at dewpoint.   (Particulates or aerosols in the air can promote cloud formation.  Particulate free air can be super saturated and not form a cloud.)  VPD is defined as the difference between the saturated vapor pressure and the actual partial pressure of water.

The equations used in this essay for calculating VPD and En at 1000mb are:

Water saturation pressure, Pws, is from Vaisala Oyj (2013), (4):

Pws = 6.116441*10^((Temp * 7.591386/(240.7263+Temp)))    (in hPa)    Eq 1

(Note: the above is not an Arrhenius equation but give similar results.)

Water vapor, Pw, pressure is from Vaisala Oyj (2013) (4):

Pw  = SH *1000/(621.9907+SH(in hPa)    Eq 2
VPD = Pws โ€“ Pw                      in hPa)   Eq 3

Enthalpy, En, Vaisala Oyj (2013)  (4):

En = Temp * (1.006+0.00189*SH)+2.501*SH(in  kJ/kg (da) )   Eq 4

These equations are not in Clausiusโ€“Clapeyron format but simplified for more convenient use with water

The source of water for clouds is evaporation from land and oceans.  The dominate evaporation variable is the sun angle (followed by cloud fraction): the more orthogonal and the larger the area the more evaporation.     From land the evaporation is from lakes, rivers, groundwater, and vegetation, and all together is called evapotranspiration, ET.  For simplicity this essay will use the term ET for all evaporation of water land or ocean.  The ET (per unit area of earth) from oceans is 6.45 (1000km^3/% of earth) and from land 2.79 (1000km^3/% of earth), 2.3 times smaller (from data in K. Trenberth et al (2011), (10)).  Therefore, it is suspected that the size of land in a slice of earth may influence cloud formation, VPD, and En. 

Global Specific Humidity, SH, is proportional to Global ET and the slice data (Google it).  This essay will use ET and SH interchangeably since for the most part all the graphs deal with multipoint averages and not specific points.  The SH and temperature data is for 1000mb from Physical Science Laboratory, PSL (6).  Higher altitude PSL data all have very good correlation to PSL 1000mb data (not shown too many graphs already)

The preferred indicator of the probability of cloud formation is the Vapor Pressure Deficit, VPD.  VPD can be measured at a single point or altitude or A representative average of many points VPDs can measure the VPD of a slice of the earth, hemisphere, or the whole earth. 

The earthโ€™s Enthalpy, En, has been shown, (9) to be related to the outgoing, LW, radiation, OLR.  VPD and Enthalpy are strongly correlated.  For atmospheric En, changes in En are changes in OLR radiation.

For incoming short wave radiation, ISR,  the albedo (fraction of ISR reflected) from Loeb (2021), (9), and Dubal et al (2022), (7),  will be used as the bests measure of ISR for comparison to cloud fraction, VPD, and En.

H. Wu et al (2023), (1), using reanalyzed data set, ERA5,  has already done similar earth slicing studies to reveal cloud fraction, CF, changes with ocean vs land and is influenced by EL Nino and La Nina.

H Liu et al (2022), (2), continued with ERA5 data to show a decrease in specific humidity, (SH), correlation to CF as well as H. Wu observation of land vs ocean differences and El Nino and La Nina influence.

Cloud data has been re-calculated by many researchers, this essay will use Climate Explorer, CE, (5) data for itโ€™s graphs.  Climate Explorer and ERA5 comparisons are shown in Figure 1.  The shift in both CE and ERA5 data in about 2000 is clearly seen.

Line graph showing the relationship between cloud fraction and year based on Climate Explorer and ERA5 data, with annotations indicating an El Nino La Nina shift.
Figure 1.ย  Comparison of Climate Explorer and ERA5 data

El Nino and La Nina (unpredictable weather patterns, see Figure 2) in the southern Pacific Ocean were correlated to perturbations in cloud coverย (1)ย (2).ย ย The contiguous large shifts from El Nino to La Nina were suspected to be related to cloud fraction shifts.ย  Encouraging the search for better predictors of radiation to the earthโ€™s surface.ย  (For further information on El Nino and La Nina please consult the internet.)

Graph displaying the ONI Index related to El Nino and La Nina events, showing fluctuations suspected to influence cloud fraction perturbations from 1950 to 2015.
Figure 2.ย  Internet graph of El Nino and El Nina perturbations

Methods

Temperature and specific humidity, SH, data from Physical Science Laboratory (6), PSL, for each chosen slice of the earth, VPD, and En were calculated.  Cloud fraction data for each slice of the earth was obtained from Climate Explorer and ERA5 data in H. Liu (2023) et al (2).

The slice land fraction was obtained from an internet graph.  Slice fractions were estimated for each slice, see Figure 3.

Graph illustrating the global land fraction versus latitude, showing the estimated fraction of land in various latitude zones.
Figure 3.ย  ย Slice of the earth vs fraction that is land (graph from internet red lines from this essay)

Results 

The expected horseshoe plots of temperature, cloud present, VPD, and Enthalpy are shown in Figure 4 (A,B,C,D).  Two time periods (start of global warming and current) also plotted in Figure 4 showing where and when the most change occurred.  Note that for temperature, VPD, and En the most change occurred in the middle to northern latitudes, while little change in cloud fraction is indicated there but the cloud change is indicated toward the poles.  The gap in the curves for temperature, VPD, and En is wider in the northern hemisphere.  The northern hemisphere gets more sun and has more land. 

Graph showing the relationship between latitude slice and slice temperature, highlighting data from 1970 to 2024.
Figure 4 (A,B,C,D).ย  Composite graph of Temperature, A, Cloud Percent, B, VPD, C, Enthalpy, D, from earth slicing.

Plotting sun angle instead of latitude range gives two points for each angle which should be the same if sun angle was the only variable related to cloud fraction.ย  In Figure 5, VPD should relate to Cloud Fraction (increasing VPD decrease Cloud Fraction) and Enthalpy (increasing En increases LW radiation out).ย  The Enthalpy correlation is Figure 5 are good for VPD vs En (they should be โ€“ they use the same data in different formulas).ย  The correlation of VPD vs cloud percent is not as strong as expected.ย  One point gives a clue to a possible why.ย  The red circled data point is for a slice that has little to no land in the slice.

Graph depicting the relationship between sun angle and cloud percentage, including a highlighted slice with very little land. The graph shows two distinct trends for cloud percent relative to sun angle and graphs enthalpy measurements.
Figure 5.ย ย  VPD vs CF and En.

To investigate the land correlation further a two variable linear regression table was created from the data in Figure 5, see Table T1.

Table displaying variable data related to land fraction, sun angle, Vapor Pressure Deficit (VPD), and other output values.
Table 1ย  Data from Figure 5 for Regression analysis

The results (see attachment here) of the regression analysis show that the R^2 of the expected sun angle vs cloud percent is only 0.30, when land fraction of the slice is added the R^ jumps to 0.83; therefore, land fraction is playing a role (statically) in the cloud fraction of the slice.  Sun angle and land fraction corelate to VPD better than to cloud fraction with an R^2 of 0.90.  Enthalpy had the best correlation to sun angle and land with an R^2 of 0.95.    Since Enthalpy has been correlated to outgoing LW, OLR, radiation implies that in addition to the expected slice correlation of OLR to sun angle there is also a correlation to land fraction in the slice.   That land fraction in the slice has two components related to cloud fraction: physical area and ET.  The sliced profiles shown in Figure 4 (A, B, C, D) show the shift in slice data with time.  Since the land fraction area is not changing during the time change, this implies that the ET must have changed (ET decreased on land).  On a global basis, CRGW theory (3) mathematically shows how this land decrease in ET can cause a global increase in ET and global warming.

ย The 20 years of CERES data presented by Loeb et al (2021)ย (9)ย and Dubal et al (2021)ย (7)ย show the albedo (the reflectivity of the earth) of the earth decreasing, R^2 = 0.73.ย  This decrease in albedo should correlate to CE cloud fraction decrease.ย  In 20 years of CE data (that matches Dubal 20 years) this correlation is very poor (R^2 = 0.1) (not shown). For about 40 years of CE data the R^2 improves to 0.61, see Figure 6.

A graph comparing CERES data for global albedo and global cloud percent over time, showing trends from 1980 to 2025, with orange and blue lines representing Climate Explorer data and CERES albedo data, respectively.
Figure 6.ย  CERES albedo data (Dubal) vs Cloud Percent (CE)

VPD which has been shown to correlate to cloud fraction (poorly) correlates well to CERES albedo, see Figure 7.  Global Enthalpy (related to OLR) correlates to CERES albedo even better, see Figure 7.  The perturbations of El Nino and La Nina seem to be affecting cloud fraction data but not albedo, VPD, or Enthalpy data.

Scatter plots depicting the relationship between CERES Global Albedo and Global Vapor Pressure Deficit (VPD) on the left, and Global Albedo versus Global Enthalpy on the right, showcasing trends and correlations in climate data.
Figure 7.ย ย  VPD and En vs CERES albedo

Discussion

Slicing the earthโ€™s data has revealed expected correlation of sun angle to cloud fraction, VPD, and Enthalpy and interesting correlation to land fraction.  The land fraction seems to be related to a change in the ET of the land. 

El Nino and La Nina are causing perturbations in the cloud fraction data that are not visible in the albedo, VPD, or Enthalpy data.  This could be related to cloud type change when the circulation patterns switch rotation from El Nino to La Nina (see internet for more information).

Strong correlation of Enthalpy (related to OLR) to albedo suggests one of earthโ€™s albedos components is related to global warming.  Since incoming radiation is relatively constant and the earthโ€™s cloud free albedo is small (about 15% of total albedo) and not changing very much that leaves cloud fraction.  VPD is poorly correlated to cloud fraction on a slice and global basis, but very well correlated to CERES albedo making VPD a candidate to replace cloud fraction in climate change models until a better measurement of cloud fraction is found.  Enthalpy is strongly related to VPD.  The CRGW theory (3) ties all this together.

CRGW theory: a change (-/+) in landโ€™s ET starts a natural chain of events that changes the earthโ€™s energy balance (+/-).  In more words: less ET on land leads to higher land VPD.  Higher VPD on land leads to a plume of hotter air than mixes with upper atmospheric air to reduce clouds over land and ocean.  The reduced clouds let in more sunlight increasing the En of the earth to warm the earth and evaporate more water (the ratio of ocean ET to land ET increases). The final observation is that a lower land ET results in a higher global ET, higher temperature, and lower relative humidity.  The CRGW theory is reversable.

CO2 is innocent but clouds are guilty.

Bibliography

  1. โ€œOpposing trends of cloud coverage over land and ocean under global warmingโ€ (2023) by Huan Liu, Ilan Koren, Orit Altaratz, and Mickaรซl D. Chekr web linkย acp-23-6559-2023.pdf
  1. Atmospheric Moisture Transports from Ocean to Land and Global Energy Flows in Reanalyses (2011) by Kevin E. Trenberth, John T. Fasullo, and Jessica Mackaro web linkย Atmospheric Moisture Transports from Ocean to Land and Global Energy Flows in Reanalyses in: Journal of Climate Volume 24 Issue 18 (2011),

More essays by Author

  1. Where have all the Clouds gone and why care? โ€“ Watts Up With That?
  2. CO2 is Innocent but Clouds are Guilty.ย  New Science has Created a โ€œBlack Swan Eventโ€** โ€“ Watts Up With That?
  3. More on Cloud Reduction.ย  CO2 is innocent but Clouds are guilty (2023). โ€“ Watts Up With That?
  4. An Unexplored Source of Climate Change: Land Evapotranspiration Changes Over Time. โ€“ Watts Up With That?
  5. VPD, Vapor Pressure Deficit a Correlation to Global Cloud Fraction? โ€“ Watts Up With That?
  6. Soundings, Weather Balloons, and Vapor Pressure Deficit โ€“ Watts Up With That?
  7. Not that ET!ย  The Terrestrial ET: EvapoTranspiration, the Unexplored Source of Climate Change โ€“ Watts Up With That?
  8. CRGW 101.ย  A Competitive Theory to CO2 Related Global Warming โ€“ Watts Up With That?
  9. ย More Evidence on Vapor Pressure Deficit, Cloud Reduction, and Climate Change โ€“ Watts Up With That?

Not All That Sensitive

The sensitivity of the surface to changes in absorbed radiation is a central, critical question in climate science. The claim is that the change in global average temperature is equal to the change in absorbed radiation times the โ€œequilibrium climate sensitivityโ€, abbreviated as ECS. The ECS is assumed to be a constant.

ChatGPT

From Watts Up With That?

Guest Post byย Willis Eschenbach

For no particular reason, I got to thinking about the non-global nature of the incoming energy. And this led me to consider the sensitivity of the surface to changes in the radiation absorbed by the surface.

The sensitivity of the surface to changes in absorbed radiation is a central, critical question in climate science. The claim is that the change in global average temperature is equal to the change in absorbed radiation times the โ€œequilibrium climate sensitivityโ€, abbreviated as ECS. The ECS is assumed to be a constant.

However, despite hundreds of thousands of dollars, work-hours, and computer-hours dedicated to determining the value of the ECS, the uncertainty about its value has only continued to increase.

Figure 1. This figure shows a variety of ways that the Equilibrium Climate Sensitivity (ECS) has been estimated. The dotted lines show the uncertainty of the earliest estimates of the ECS, which was said to be between 1.5 and 4.5 degrees C per doubling of CO2 (2xCO2).

Initially, back in the 1970s, the equilibrium climate sensitivity was claimed to be between 1.5 and 4.5 ยฐC of warming from an increase of 3.7 W/m2 in downwelling radiation. This 3.7 W/m2 is the assumed value of the increase in downwelling radiation from a doubling of CO2 (2xCO2).

This can also be expressed as the change of temperature from a one watt per square meter increase in downwelling radiation. The original central estimate of 3 W/m2 per 2xCO2 is the same as 3/3.7 โ‰ˆ 0.8 ยฐC for each radiation increase of 1 W/m2.

However, as you can see in the graphic above, over time, the estimates of the ECS have grown wider and wider. Now, they range from 0.37 degrees C warming resulting from a doubling of CO2 (2xCO2), up to 8.1 degrees C per 2xCO2. This is the same as a range of 0.1 to 2.2 W/m2 for each additional 1 W/m2 of radiation.

I know of no other field of science where this is true, where endless studies of a central constant in the field only show wider and wider uncertainty. To me, this can only mean that the underlying theory is incorrect โ€ฆ but I digress.

Now, the ECS is said to be the climate sensitivity after years of adjusting to the change in radiation. This is because the claim is that the warming is much larger and slower than direct calculations indicate. Mainstream scientists say that various feedbacks increase the warming due to increased radiation, with cloud feedbacks and water vapor feedbacks being the main culprits.

This is discussed in the IPCC AR6 Working Group 1 Chapter 7. They say that the central estimate for the net cloud feedback is 0.42 W/m2 for every 1ยฐC of warming, and that the central estimate for water vapor feedback is another 1.8 W/m2.

In other words, the IPCC says that for every 1 W/m2 increase in radiation, the feedbacks increase that by another 2.22 W/m2. This means that the warming should be far larger than youโ€™d expect just from the underlying increase in radiation.

Keep this in mindโ€”the standard theory says that the warming should be greater than no-feedback warming.

So to start with โ€ฆ if there were no feedbacks, what warming would we expect to occur?

To investigate this question, I started by taking a look at the total radiation being absorbed by the surface. This is the sum of the absorbed sunlight, which is the total sunlight hitting the surface less reflections from the surface, along with the downwelling thermal radiation from the atmosphere. Hereโ€™s a map of the global distribution of the total surface absorbed radiation.

Figure 2. Average Mar 2000 โ€“ Feb 2024 of the total absorbed surface radiation, which is the sum of longwave radiation (โ€œLWโ€, thermal radiation from the atmosphere) and shortwave radiation (โ€œSWโ€, solar radiation).

As a 24/7 average, the surface gets about half a kilowatt per square meter. In Figure 2, you can see the reflective nature of the desert and the poles, which lead to less radiation being absorbed.

Next, I looked at the total change in absorbed surface radiation over the 24-year period of the CERES dataset. Below you can compare the changes in absorbed radiation (top) with the changes in temperature.

Figure 3. Changes in total absorbed surface radiation (top) and changes in surface temperature (bottom) over the period March 2000 to February 2024.

Note that in a number of areas, the changes in absorbed radiation correlate with the surface temperature changes. Also note that, as expected, increasing radiation leads to increasing temperature.

How much would we expect this increase in radiation to increase the temperature? This can be determined from the Stefan-Boltzmann equation, which states that radiation is proportional to the fourth power of the temperature. Using the CERES data and the S-B equation allows us to calculate just what heating weโ€™d expect to occur from that increase in absorbed radiation. Below is the result.

Figure 4. Theoretical change in temperatures from a uniform 1 W/m2 increase in absorbed surface radiation.

This shows that the expected warming is 0.19ยฐC for each 1 W/m2 increase in absorbed radiation (or 0.7 W/m2 per 2xCO2). However, because a given increase in radiation warms cold areas more than warm areas, the warming ranges from 0.15ยฐC per 1 W/m2 (0.6ยฐC per 2xCO2) in the warmest ocean areas up to 0.44ยฐC per 1 W/m2 (1.6 W/m2 per 2xCO2) in Antarctica.

So thatโ€™s our theoretical warming. And the prevailing theory says we should see much greater warming than the theoretical warming due to the feedbacks described above.

To gain further understanding, I looked at how much the observed temperature actually changed in response to the changes the absorbed radiation. Figure 5 below shows the change in temperature that corresponds to a one W/m2 increase in radiation.

Figure 5. Change in the surface temperature for each 1 W/m2 increase in absorbed surface radiation. CERES data from March 2000 โ€“ February 2024.

Now, this reveals a surprise. The climatastrophists keep saying that as the radiation absorbed by the surface increases, the temperature perforce must increase.

And for most of the world this is true, although at greatly different rates as we can see in Figure 5.

But in the area of the โ€œPacific Warm Poolโ€ north of Australia, and the area of the Inter-Tropical Convergence Zone (ITCZ) just above the Equator, the opposite is trueโ€”as the amount of radiation absorbed by the surface increases, the temperature goes down, not up.

I say this is a result of the combined effects of the temperature-threshold driven emergence of both tropical cumulus cloud fields and tropical thermally driven thunderstorms. These cool the surface in a host of ways, and exert a huge negative feedback on temperature increases.

But for the purposes of this post, the mixture of phenomena driving this important oddity are not the focus. Iโ€™m simply looking at the actual response of the surface temperature to changes in absorbed surface radiation. Most places it increases โ€ฆ but not everywhere.

Next, I looked at the trend data in a different way. I looked at a scatterplot of the trend shown in Figure 5 versus the average surface temperature. I also included the theoretically expected temperature change versus the average surface temperature.

Figure 4. Scatterplot, sensitivity of surface temperature to surface absorbed radiation. Actual average, actual land, and actual sea are calculated based on the CERES data. Theoretical results and average are based on the use of the Stefan-Boltzmann equation on the CERES data.

Now, this is quite interesting for a number of reasons.

First, in every case, the results are below the theoretical calculations. This is a clear indication that far from the globe having a strong positive net feedback to any warming, the net feedback is negative.

Next, this data covers 24 years, nearly a quarter century. So this is not the instantaneous sensitivity of the surface to absorbed radiation.

However, itโ€™s likely that the longer-term warming would be greater than shown in Figure 4. In other words, if the amount of absorbed surface radiation were to suddenly stop increasing, the world would continue to warm for a bit.

So โ€ฆ how much will it warm? Well, in my post Lags and Leads, I showed that we can calculate the warming based on the length of time by which the surface warming lags the absorption of the radiation at the surface. As you might expect, this is greater for the ocean than for the land, due to the greater thermal mass of the ocean.

Per the CERES data, the lag in the land warming is about ten days, and the lag in the ocean warming is much larger, about four times that. Using the formula from โ€œLags and Leadsโ€ linked to above, this would increase the land warming by about 2%, and it would increase the ocean warming by about 30%.

So here is the data shown in Figure 4 above, but with the values adjusted as discussed above to account for delayed warming. Iโ€™ve also included the lowest four ECS values from Figure 1 at the head of the post. (The values in that figure have been divided by 3.7 to give them in units of ยฐC per W/m2 rather than ยฐC per 2xCO2.)

Figure 5. As in Figure 4, but with the values adjusted to give the eventual change in the temperature. Purple dots show the lowest four of the ECS estimates shown in Figure 1 at the head of the post (values converted from ยฐC/2xCO2 to ยฐC per W/m2).

Even with these adjustments, the sensitivity of the temperature to absorbed radiation is still well below the theoretical global average. In addition, the four lowest values from Figure 1 are also below the theoretical blackbody values.

And because it is based on observations, this graph includes all possible feedbacks, including cloud feedbacks, cloud aerosol feedbacks, water vapor feedbacks, and all the rest. It is based on how much the surface temperature actually has changed with an increase of 1 W/m2 in absorbed surface radiation.

So Iโ€™ll go out on a limb and say that this value of 0.11ยฐC per W/m2 (0.4ยฐC per doubling of CO2) is a reasonable estimate for the Equilibrium Climate Sensitivity. It includes all feedbacks known and unknown, is adjusted for the different thermal lags in the ocean and the land, it is in the range of the four smallest ECS estimates from mainstream climate studies, and it is quite close to three of those ECS estimates.

As Figure 5 shows, like the four smallest ECS estimates in Figure 1, the net feedback to the warming is negative. We know this because everywhere on the Earth, the warming is less than weโ€™d experience with a blackbody planet.

This negative feedback is as weโ€™d expect, given the remarkable stability of the global surface temperature, which only increased by about 0.2% over the entire 20th Century. Twenty-five years ago, that surprising stability was what first drew me to start studying the climate. If the feedback were positive, there would be much larger variations in the Earthโ€™s temperature.

Finally, it could be argued that this is NOT the equilibrium climate sensitivity (ECS), because Iโ€™ve used the change in radiation absorbed at the surface, while the ECS is based on the change in top-of-atmosphere (TOA) downwelling โ€œgreenhouse effectโ€ radiation (GHE).

As Ramanathan first noted, the TOA GHE radiation can be calculated as the amount of surface upwelling thermal radiation minus the amount of TOA upwelling thermal radiation. And this GHE radiation is closely related to the downwelling radiation absorbed by the surface. Figure 7 shows that relationship.

Figure 7. Absorbed downwelling longwave radiation at the surface, and top of atmosphere (TOA) โ€œgreenhouse effectโ€ downwelling longwave radiation

Because they are so similar, and because the downwelling longwave radiation at the surface is only part of the total absorbed radiation, this will make little difference to the ECS I calculated.

 The CERES data shows that for every 1 W/m2 increase in TOA downwelling greenhouse radiation, as weโ€™d expect, the surface downwelling radiation increases by on the order of one W/m2 (1.1 ยฑ .14 W/m2 per W/m2). So any change in my calculated ECS will be small.

And with that, further affiant sayeth naught.


Fog this morning, which is good news because itโ€™s been hot and I still have another acre to mow. Protip: When your age has two digits and starts with a 7 or above, if you slow down โ€ฆ

โ€ฆ you stop.

And wonder of wonders, a bobcat just walked into the back acreage, what a lovely being!

My very best to all,

w.

Youโ€™ve Heard It Before: Quote the exact words you are commenting on. I can defend my words. I canโ€™t defend your interpretation of my words.

Nikolov & Zeller: Misrepresentation of Critical Satellite Data byย IPCC

The 6th Assessment Report of the Intergovernmental Panel on Climate Change (IPCC AR6) concluded โ€œIt is very likely that well-mixed GHGs [greenhouse gases] were the main driver of tropospheric warming since 1979โ€ (IPCC, 2021; p.5).

From Tallbloke’s Talkshop

Byย tallblokeย 

1. Introduction

The 6th Assessment Report of the Intergovernmental Panel on Climate Change (IPCC AR6) concluded โ€œIt is very likely that well-mixed GHGs [greenhouse gases] were the main driver of tropospheric warming since 1979โ€ (IPCC, 2021; p.5). This statement implies that all known climate forcings have properly been evaluated using the available data, and GHGs have been found to exert a disproportionally large radiative effect on the Global Surface Air Temperature (GSAT) over the past 45 years. However, a close examination of Chapter 7 of the Working Group I (WG1) Contribution to the IPCC AR6 (Forster et al. 2021), which discusses the Earthโ€™s energy budget, climate feedbacks and climate sensitivity, reveals that the observed decrease of Earthโ€™s albedo and the corresponding increase of absorbed shortwave radiation by the Planet for the past 20 years have not been taken into account as contributors to the recent warming. Section 7.2.2 of Chapter 7 entitled โ€œChanges in Earthโ€™s Energy Budgetโ€ acknowledges that there have been multidecadal periods of significant decreasing and increasing trends in surface solar radiation (SSR) called โ€œglobal dimmingโ€ (i.e. from 1950s to 1980s) and โ€œglobal brighteningโ€ (after 1980s), respectively. The report states: โ€œThere is high confidence that these [SSR] trends are widespread, and not localized phenomena or measurement artefacts.โ€ Indeed, the existence of such dimming and brightening multi-decadal periods has been acknowledged by science for more than 10 years (Stanhill et al. 2014; Yuan et al. 2021), but the IPCC AR6 provides no global estimate of the observed positive trend in SSR since 1980s and its impact on GSAT. Instead, the Report simply states โ€œThe origin of these trends is not fully understoodโ€.

With respect to the top-of-the-atmosphere (TOA) solar fluxes, Section 7.2.2 of the IPCC AR WG1 offers no analysis of the substantial decrease in Earthโ€™s shortwave reflectance since 2000 observed by the NASAโ€™s Clouds and the Earthโ€™s Radiant Energy System (CERES) project (Loeb et al. 2018, 2019, 2020 and 2021) and also reported by other research teams (e.g. Dรผbal & Vahrenholt 2021; Stephens et al. 2022). The Report does not discuss the observed 2.0 W m-2 increase in solar-energy uptake by the Planet from 2000 to 2020 nor its contribution to the recent warming. What is even more puzzling, Subsection 7.2.2.1 of the IPCC AR6 WG1 Contribution features graphs in their Fig. 7.3 (on p. 936) showing an increasing reflected solar flux and decreasing outgoing thermal flux since 2000 that are supposedly based on CERES data. However, these trends are opposite of what CERES has actually measured and directly contradict results reported by prior studies. This article presents findings from our investigation of Fig. 7.3 in the IPCC AR6.

2. Figure 7.3 in the IPCC AR6 WG1 Contribution

Page 936 of the IPCC Climate Change 2021: The Physical Science Basis (IPCC, 2021) contains the following figure:

Panel (a) of Fig. 7.3 shows anomalies of the reflected solar flux, panel (b) depicts anomalies of the emitted (outgoing) thermal (longwave) flux, and panel (c) displays net-flux anomalies calculated as a difference between absorbed solar shortwave (SW) and outgoing longwave (LW) anomalies. Since reflection is opposite of absorption, the absorbed SW anomalies are obtained by simply multiplying the reflected solar anomalies by -1. Figure 7.3 also portrays simulation results from 7 climate models that were forced by observed sea-surface temperatures (SST) and sea-ice boundary conditions. Note that the CERES data are depicted as thick black lines while the multi-model means are shown as thick red lines.

The problem with IPCCโ€™s Fig. 7.3 is that the plots of reflected solar and outgoing thermal radiation show opposite temporal trends compared to those found in the actual CERES observations. Figures 1 and 2 below illustrate this fact. At the same time, the net flux (a.k.a. the Earthโ€™s Energy Imbalance or EEI) in the IPCC Figure has a correct trend (Fig. 3).

Figure 1.ย Trends of Earthโ€™s reflected solar radiation in theย CERES datasetย (upper panel) and the IPCC AR6 WG1 Fig. 7.3(a) (lower panel). Note that the 13- vs. 12-month running means do not affect the flux trends.

Figure 2. Trends of Earthโ€™s outgoing thermal radiation in the CERES dataset (upper panel) and the IPCC AR6 WG1 Fig. 7.3(b) (lower panel). Note that the 13- vs. 12-month running means do not affect the flux trends.

Figure 3. Trends of Earthโ€™s net radiative flux (energy imbalance) in the CERES dataset (upper panel) and the IPCC AR6 WG1 Fig. 7.3(c) (lower panel). Note that the net-flux trend has not been altered in the IPCCโ€™s Fig. 7.3.

We discovered this data-trend issue while working on a paper evaluating the contribution of Earthโ€™s albedo decrease since 2000 to recent warming using the CERES dataset.

Upon a close examination, it became clear to us that the trend inversion of reflected solar and emitted thermal flux anomalies appeared to have been achieved by multiplying the original data series by -1. The text of the IPCC AR6 referring to Fig. 7.3 does not mention anything about altering the trends of key global energy-budget parameters, nor does it provide a clear explanation of the reason behind it. However, the caption of IPCCโ€™s Fig. 7.3 contains the following peculiar sentence โ€œAll flux anomalies are defined as positive downwards, consistent with the sign convention used throughout this chapter.โ€ This statement is perplexing, because flux anomalies are independent of the flux direction. Anomalies are simply departures of timeseries data from an arbitrary chosen reference value. In the case of the CERES dataset, flux anomalies are calculated with respect to the mean value of the deseasonalized timeseries over the entire period of observation. Hence, it is the choice of a reference value that solely determines, whether an anomaly is positive or negative irrespective of the flux direction. In other words, the sign of an anomaly is not subject to the definition of a positive flux direction as suggested by the caption of Fig. 7.3. Furthermore, calculating the anomalies of an environmental parameter (such as global temperature, ocean heat content, planetary radiative flux etc.) does not change the temporal trend of the original data set. This is basic knowledge in the field of climate science. In this regard, the statement in the caption of Fig. 7.3 that relates anomalies to a flux direction is physically meaningless and confusing.

Another problem with the IPCCโ€™s Fig. 7.3 is that the net-flux anomalies shown in panel (c), whose trend has not been altered, cannot be calculated from the timeseries shown in panels (a) and (b) as expected. Since the net-flux anomaly is a difference between anomalies of the absorbed solar and emitted thermal fluxes, the graphs in panel (c) are physically and numerically incompatible with the those plotted in panels (a) and (b).

3. Correspondence with IPCC Authors about Fig. 7.3.

In an effort to clarify the above situation, we reached out to the Coordinating Lead Authors of Chapter 7 in the IPCC AR6 WG1 Contribution, Prof. Piers Forster at the University of Leeds (UK) and Prof. Trude Storelvmo at the University of Oslo (Norway). We asked them, who the authors of Section 7.2.2 in Chapter 7 were, where Fig. 7.3 appears. Prof. Forster promptly replied informing us that Section 7.2.2 was written and edited by Dr. Matthew Parmer at the University of Bristoll and UKโ€™s Met Office (UK) and Dr. Chris Smith at the University of Leeds (UK). Dr. Palmer was the lead author of that Section. Prof. Forster also provided us with a link to the IPCC AR6 WG1 repository at GitHub.com, where all data files and Python processing scripts reside used to generate the figures appearing in the WG1 Report.

We searched the GitHub repository and found three text files containing the source data for Fig. 7.3 as well as a Python plotting script used to generate the actual figure panels. There was a TXT file for every panel of Fig. 7.3 providing both CERES observations and climate-model projections covering the period July 2000 โ€“ June 2017. Upon inspecting the data files and the Python script, we discovered that, indeed, the deseasonalized monthly anomalies of reflected solar and emitted thermal fluxes have been multiplied by -1 resulting in inversion of the original trends, and that this data manipulation occurred in the Python script generating the panels of Fig. 7.3. The text files contained unaltered data that exhibited correct trends. Figure 4 depicts the anomalies of reflected solar and outgoing thermal fluxes from CERES observations and climate-model simulations as found in the text data files on GitHub. Figure 5 shows a portion of the Python script, where the signs of shortwave (SW) and longwave (LW) anomalies (code parameters SW_dict and LW_dict) are inverted during the plotting process by placing a minus sign in front of the corresponding data arrays. The Python script inverted the trends of both the CERES data and climate-model projections.       

Figure 4.ย Plots of TOA reflected solar and outgoing thermal flux anomalies found inย text data filesย at the IPCC GitHub repository used to generate Fig. 7.3 in the IPCC AR6 WG1. These data show correct trends.

Figure 5. Portion of the Python script used by the IPCC authors of Section 7.2.2 to plot the TOA anomalies of reflected shortwave (SW), emitted longwave (LW) and net fluxes from both CERES observations and climate model projections. Note the minus sign in front of the arrays containing SW and LW flux data.  

It is worth noting that, although the climate models correctly reproduced the overall pattern of observed changes in planetary SW and LW fluxes, the measured trends of these variables are much steeper compared to the modeled ones. Most importantly, the climate models significantly underestimated the rate of decrease of Earthโ€™s albedo and the associated increase of absorbed solar radiation by the Planet since 2000. The rate of planetary cooling (i.e. emitted thermal flux) was also significantly lower in the model simulations compared to observations. An important detail in this comparison is that all models have been forced with observed sea-surface temperatures (SST) and sea-ice boundary conditions. Had the models been only guided by their internal physical processes without periodic nudging to observations, the above model-data comparison would likely have been much worse.

On July 8, 2024, we sent an email message to Dr. Palmer and Dr. Smith informing them about the findings from our search of the GitHub data repository and asking them to explain the reason for the trend inversion of the SW and LW flux anomalies in Fig. 7.3. We also sought their advice about whether we should use the timeseries shown in Fig. 7.3 or the data found in the source text files, if we decide to create customized graphs of TOA fluxes for a review paper weโ€™ve been working on.

We received a reply from Dr. Palmer on July 10, 2024, where he acknowledged that the reflected solar and outgoing thermal flux anomalies had intentionally been multiplied by -1. However, his explanation for this data manipulation was simply an expansion of the justification stated in the caption of Fig. 7.3 that invokes flux direction. Specifically, Dr. Palmer wrote:

โ€œโ€ฆ reflected SW and outgoing LW are both defined as positive in the upward/outward direction. Therefore, for those timeseries we multiply by -1 so that they are expressed in a way that is consistent with the rest of the chapter. This means, for example, that a decrease in reflected SW means a relative GAIN of energy in the Earth system. Similarly, an increase in outgoing LW means a relative LOSS of energy in the Earth system. Note that in the figure we label these as โ€œglobal solar flux anomalyโ€ and โ€œglobal thermal flux anomalyโ€ rather than โ€œreflected SW fluxโ€ and โ€œoutgoing LW fluxโ€.โ€

As discussed above, this explanation makes no physical sense, because anomalies are always defined with respect to a chosen reference value and, therefore, have nothing to do with flux direction. Also, expressing a timeseries in terms of anomalies is not supposed to change the temporal trend of the original data. Dr. Palmer correctly pointed out that multiplying anomalies of the reflected solar flux by -1 produces a timeseries of a relative energy gain by the system. This new timeseries is called absorbed solar flux, because reflection is opposite of (and complementary to) absorption. Hence, panel (a) of the IPCC Fig. 7.3 essentially shows anomalies of the absorbed solar flux by Earth. The problem is that the caption of Fig. 7.3 labels this panel as โ€œreflected solarโ€, which is misleading. By mislabeling the solar flux in the figure caption while recognizing that Fig. 7.3 (a) depicts a relative gain of solar energy by the Earth system, Dr. Palmer effectively obscured a key natural driver of climate related to the Sun.

On the other hand, multiplying the outgoing thermal flux anomalies by -1 produces nothing meaningful, because (unlike the solar flux) Earthโ€™s LW radiation is always directed outward and does not have a complementary flux directed inward. By showing a decreasing thermal emission from Earth over time as done in Fig. 7.3 (b), the authors of Section 7.2.2 (Dr. Palmer and Dr. Smith) de facto suggest the presence of โ€œheat trappingโ€ in the climate system by increasing concentrations of atmospheric greenhouse gases. However, the 2nd Law of Thermodynamics makes it impossible for an open system with a rising surface temperature such as Earth to have a decreasing outgoing thermal emission. In other words, by inverting the trend of the TOA outgoing LW flux, the IPCC authors misrepresented the physical reality!  

Interestingly, Dr. Palmer advised us to use the data in the text files found in the GitHub repository in case we wanted to create customized plots of CERES and modeled radiative fluxes. We interpreted this as an acknowledgement that the correct data were contained in the text files rather than Fig. 7.3.

In our reply to Dr. Palmer, we listed a series of specific concerns that the trend inversion of reflected solar and outgoing thermal fluxes made in Fig. 7.3 was methodologically inappropriate, because it fundamentally changes the observed behavior of the climate system over the past 20 years and creates a false impression about climate drivers in the minds of researchers and politicians reading the IPCC Report. Dr. Palmer did not address our concerns and instead directed us toward an official IPCC webpage, where we could raise our issues. Since Dr. Palmer did not recognize the misrepresentation of satellite data in Fig. 7.3, it is possible that he was genuinely confused about flux anomalies and how they are calculated judging by the following odd statement he made in one of his replies: โ€œI donโ€™t think there is any fundamental issue here โ€“ just different choices about the sign convention usedโ€.

The truth is that the flux-trend inversions in Fig. 7.3 have enormous implications for both IPCCโ€™s main conclusions and the climate theory; hence, they must be exposed and explained to the Public.

4. Implications of the Data Manipulation in the IPCC Fig. 7.3

Figure 7.3 in the IPCC AR6 WG1 essentially shows an increasing planetary albedo (panel a) and a decreasing infrared cooling to Space (panel b) for the past 2 decades, which is diametrically opposite of satellite observations. While the text of the IPCC WG1 Chapter 7 does not discuss any long-term trends of the TOA reflected solar and emitted thermal fluxes in the 21st-Century, Fig. 7.3 subconsciously suggests that the solar forcing played no role in recent warming and the rising concentrations of atmospheric greenhouse gases due to human industrial activity had increased the retention of heat in the climate system by impeding the outgoing LW radiation. These implications of Fig. 7.3 resulting from the data manipulation align perfectly well with the radiative greenhouse theory of climate change, but contradict directly the physical reality as measured by the CERES Project.

By inverting the trend of the reflected solar flux, the IPCC authors effectively eliminated the need to analyze the cloud-controlled solar forcing and its contribution to recent tropospheric warming while reaffirming at the same time the a-priori assumed pivotal role of greenhouse gases in driving the global surface temperature since 2000. The trend inversion of the outgoing thermal flux further solidifies the false impression that the Earth had warmed in response to โ€œheat trappingโ€ by increasing atmospheric trace gases.

Had the IPCC AR6 WG1 acknowledged the ~2.0 W m-2 increase of solar radiation absorption by the Planet between 2000 and 2020 as measured along the 13-month running mean curve of CERES anomalies (Fig. 6), the following conclusions/claims found in the Reportโ€™s โ€œSummary for Policy Makersโ€ would have been impossible to defend โ€œIt is unequivocal that human influence has warmed the atmosphere, ocean and landโ€ and โ€œIt is very likely that well-mixed GHGs were the main driver of tropospheric warming since 1979โ€ (IPCC, 2021; p.4-5). This is because a solar forcing of 2.0 W m-2 is more than sufficient to explain the entire observed warming for the past 2 decades, thus eliminating the need to invoke any model-generated (i.e. theoretical) radiative forcing by greenhouse gases.

Figure 6. Monthly radiative anomalies of the Earthโ€™s absorbed solar flux estimated from the CERES EBAF 4.2 dataset by multiplying the reported reflected all-sky shortwave anomalies by -1 (in accordance with the fact that absorption is diametrically opposite of reflection).

For example, using the lowest value of IPCCโ€™s transient climate sensitivity estimated from data on total anthropogenic forcing and temperature increase presented in Sections 7.3.5.2 and 7.3.5.3 of the IPCC AR6 WG1 Chapter 7 (Forster et al. 2021), i.e. 0.47 K/(W m-2), the warming caused by solar forcing alone for the period 2000 โ€“ 2020 should have been 2.0*0.47 = 0.94 K. The observed global surface warming over this time interval (calculated as an average temperature rise from 6 datasets) is 0.62 K along the curve of 13-month running means (Fig. 7). Similar results are obtained, if one uses the linear trends depicted in Figures 6 and 7 instead. Assuming the above IPCC climate sensitivity, the mean decadal increase of Earthโ€™s sunlight absorption measured by CERES, i.e. 0.797 (W m-2)/decade (Fig. 6), corresponds to a global surface warming of 0.797*0.47 = 0.37 K/decade, while the actual observed warming is only 0.23 K/decade (Fig. 7). Thus, the observed solar forcing, which is mostly induced by natural cloud-albedo variations (Loeb et al. 2019), leaves no room for any action by greenhouse-gas radiative forcing or amplifying (positive) feedbacks claimed by the IPCC AR6. In addition, according to our analysis, the IPCCโ€™s lowest climate-sensitivity estimate of 0.47 K/(W m-2) is still an overestimation of the Earthโ€™s actual climate sensitivity, 0.29 K/(W m-2) inferred from observed data and a new physical model discussed elsewhere (Nikolov & Zeller, manuscript under review).

Figure 7. Monthly anomalies of the global surface air temperature calculated by averaging data from 6 datasets (HadCRUT5, GISTEMP4, NOAA GlobalTemp, BEST, RSS and NOAA STAR) and a 13-month running mean used to smooth the interannual variability.

In other words, if the measured increase of Earthโ€™s sunlight absorption by CERES in the 21st Century was acknowledged in the IPCC AR6, it would have invalidated the Reportโ€™s central assertion that human carbon emissions were the main driver of climate in recent decades. It is conceivable then that radiative flux anomalies in Fig. 7.3 were manipulated and discussions about long-term CERES trends in Section 7.2.2 were intentionally omitted, precisely because the actual CERES data present a significant empirical challenge (obstacle) to the UNโ€™s political Agenda (established by Resolution A/RES/43/53 of the UN General Assembly in 1988) to promote Anthropogenic Climate Change.

5. Conclusion

Considering the above facts and the enormous global socioeconomic impact of the IPCCโ€™s conclusions and recommendations, we believe that it will be in the Worldโ€™s best interest to launch an independent, critical reevaluation of fundamental premises in the climate theory from the standpoint of modern observations, and establish a new, objective peer-review system that ensures a complete and unbiased representation of all available data in the IPCC Reports. These efforts should be accompanied by a decisive depoliticization of climate science through the passing of appropriate legislation in the form of International Law that restores the financial independence of academic research, criminalizes political interference in scientific inquiry, and incentivizes the adoption of novel approaches to solving climate physics problems.

Testing A Constructal Climate Model

The past climate is being rewritten so fast that we literally donโ€™t know what will happen yesterday โ€ฆ

From Watts Up With That?

Guest Post by Willis Eschenbach

ABSTRACT

A simple constructal model of the operation of the climate system was created by Dr. Adrian Bejan and several others. It posits that the climate system can be modeled very accurately by considering the climate as a giant heat engine turning solar power into mechanical motion. Further, it says that following the constructal law, the heat engine constantly evolves to maximize the heat flow from the tropics to the poles. In this analysis, I examine the inner workings of the model, implement a couple of improvements, and test it against the CERES satellite dataset. Sorry, no spoilers.

CONSTRUCTAL LAW

The Constructal Law, formulated by Professor Adrian Bejan in 1996, is a fundamental principle in physics and engineering that describes the natural tendency of all flow systems, whether inanimate or animate, to evolve and organize in a way that maximizes the flow of matter, energy, or information. This law recognizes that patterns and structures in nature, such as river networks, tree branches, and biological organisms, emerge and evolve to enhance their efficiency in the movement of resources. The constructal law explains things like the endlessly meandering nature of rivers seen in the image above. My previous posts on the Constructal Law are here.

In essence, the Constructal Law states that the design and development of flow systems, whether the branching of blood vessels in the human body, the structure of transportation networks, or even the layout of technology and information networks, are governed by the imperative to reduce flow resistance and facilitate the transfer of resources.

The Constructal Law, as applied to climate, says that natural climate systems, such as atmospheric and oceanic circulation patterns, evolve and organize in a way that maximizes the efficiency of heat and energy flow on Earth. This principle emphasizes that climate systems, like other flow systems, tend to develop structures and patterns that reduce flow resistance and promote the transfer of heat and energy.

In a series of three papers, โ€œThermodynamic optimization of global circulation and climateโ€œ, โ€ Constructal theory of global circulation and climateโ€œ, and โ€œClimate change, in the framework of the constructal lawโ€œ, Adrian Bejan and his co-authors show that the climate can be modeled as a heat engine. Following the Constructal Law, this climate heat engine evolves to maximize its mechanical power output. The authors say:

โ€œIn conclusion, the maximization of the mechanical power output is equivalent to the maximization of the heat current from the hot region to the cold region.โ€

I got to re-reading the final of those three papers the other day, and I realized that I could set up their model on my computer. Let me start with an overview of their model.

Figure 1. The conceptual model.

The top part shows the warm (tropical) and cold (poleward) areas of the global climate heat engine. These areas are marked AH and AL. for โ€œArea Highโ€ and โ€œArea Lowโ€ temperatures. They each have a corresponding temperature TH (temperature high) and TL (temperature low).

The lower part of the diagram shows the various heat currents. The far left downward pointing arrow is heat from the sun to the hot zone. The next arrow, pointing up, is heat radiated from the hot zone to space.

Then we have the horizontal arrow โ€œqโ€, the heat current from the hot zone to the cold zone.

Finally, in the cold zone on the right, we have a downward-pointing solar arrow showing heat from the sun to the cold zone, and an upward-pointing radiation arrow showing heat radiated to space.

In short, the hot zone gets heat from the sun. Some is radiated back to space. The rest, the flow โ€œqโ€, is transported to the cold zone. There, the flow โ€œqโ€ gets radiated back to space along with the heat that the cold zone gets from the sun.

And most important, the Constructal Law says that the system will constantly reorganize itself to maximize the heat flow โ€œqโ€.

Next, hereโ€™s the math of the model, from the third of the papers linked above. Recall from Figure 1 above that โ€œxโ€ is the area fraction, the fraction of the globe occupied by the hot zone.

Daunting โ€ฆ so let me translate for those who like math. For those who donโ€™t, no worriesโ€”just skip down to where it says โ€œTHEIR MODEL RESULTSโ€œ.

And for the three folks still reading this section, ignore equation (26) for now. Next, in the above set of equations, rho (ฯ) is the albedo, and gamma (ฮณ) is the โ€œgreenhouse factorโ€, the fraction of upwelling surface longwave radiation that is absorbed by the atmosphere. And at steady-state, the left-hand side of equations 24 and 25 is zeroโ€”there is no change of temperature with time.

With those as prologue, the first equation (23) describes the hot zone. It says that the hot zone gets heat from the sun. Some is radiated back to space. The rest, the flow โ€œqโ€, is transported to the cold zone. So โ€œqโ€ is equal to hot zone solar heat input minus hot zone radiation to space. In short, itโ€™s just a mathematical description of the bottom left part of Figure 1 above. Simple

The second equation (24) describes the cold zone. It says the cold zone gets heat from the sun, plus the flow โ€œqโ€ from the hot zone, and radiates it all to space. So โ€œqโ€ is equal to the cold zone output to space minus the cold zone solar input. This equation is a mathematical description of the bottom right-hand part of Figure 1 above.

The third equation (25) says that the flow โ€œqโ€ is equal to some constant โ€œCโ€ times the 3/2 power of the difference in temperature between the hot and cold zones.

The final equation (27) specifies that โ€œqโ€ is maximized.

There are four unknowns in the equationsโ€”temperatures of the hot and cold zones โ€œTHโ€ and โ€œTLโ€œ, the heat flow โ€œqโ€, and the area fraction โ€œxโ€. Now, my math-fu is not strong enough to solve those four equations to determine the four unknowns. And unfortunately, the authors of the paper didnโ€™t include the solution. Grrrr.

However, Iโ€™m a determined fellow. After some reflection, I realized that I could use a double optimization process to get the answers.

I wanted to determine the value of x (the size of the hot zone) which gives the largest value for โ€œqโ€, the heat flow from the hot zone to the cold zone. But I only had three equations with four unknowns.

So I divided the problem up by assuming that I knew what โ€œxโ€ was. Using that, I could then use an optimization program to give me the values of TH, TL, and q for any given value of x.

And with that, I could use a second optimization program to give me the value of x that maximized q, the heat flow from the hot zone to the cold zone. See the Appendix below for the R code.

THEIR MODEL RESULTS

Here is their report of the first of their calculations. Using their same numbers, I get the same results that they show below.

Using their values, I was able to reproduce their results very accurately.

PROBLEMS WITH THEIR MODEL

However, there are a couple of issues with their values. First, as they note, their value for โ€œxโ€ puts the limits of the hot zone at about 57ยฐN/S. But thatโ€™s not the case in the real world. Hereโ€™s the real-world data regarding the heat flow โ€œqโ€.

Figure 2. How much heat is moved from the tropics to the poles (positive values), and how much heat is absorbed in the polar regions (negative values). The hot zone is the red to yellow part bordered by the black/white lines. The cold zone is shown in green to blue, outside the black/white lines.

You can see the similarity of this graphic with the model shown in Figure 1 above. However, in the real world, the hot zone fraction โ€œxโ€ is about 0.55 of the total surface. This corresponds with a hot zone extending to about 34ยฐN/S. So that was the first problemโ€”the hot zone extends to about 34ยฐN/S, not 57ยฐN/S.

The second problem is that their equation gives far too cold a result for the cold zone. They say it averages 258.4K, which is -14.75ยฐC. But in the real world, the cold zone poleward of 57ยฐN/S actually has an average temperature of about โ€“ 3ยฐC, far from the minus 14ยฐC they claim.

IMPROVING THEIR MODEL

So of course, being the eternal tinkerer, I had to see whether I could improve their model. The first thing I noticed was that they are using the same albedo and the same greenhouse factor for both the cold and hot zones. But in the real world, both the albedo and the greenhouse factor are very different for the two areas. As a result, their model is giving inaccurate results

Using individual albedo and greenhouse factors for the two areas made the model far more accurate. But there was still a problem. The hot temperatures it calculated were too hot and the cold temperatures were too cold to match the real world. Looking at the equations, I realized that this inter-temperature distance is controlled by the constant โ€œCโ€ in Equation (25). This is the โ€œconductanceโ€, a measure of how much heat flow is generated by a given temperature difference between the hot and cold zones. The value they were using for โ€œCโ€ was far too small, which meant it required a much greater temperature difference to get the same flow, resulting in a hot zone thatโ€™s too hot and a cold zone thatโ€™s too cold.

Once the factor โ€œCโ€ was increased, the results looked very good.

GROUND-TRUTHING THE MODEL

With that model up and running on my computer, I figured that I could test whether in fact, the climate system actually does operate as a gigantic heat engine that is continually evolving to maximize the tropical-polar heat flow. Here was my plan.

The constructal model says that given the albedo and greenhouse factors, for each value of โ€œxโ€ (the area of the hot zone) there will be a preferred temperature for the hot and cold zones. Further, the model says that the average final temperatures will be the ones that maximize โ€œqโ€, the heat flow from the hot zone to the cold zone. I realized we could test those claims using the CERES data.

For each year, the average top of atmosphere net radiation CERES data gives us the observed value of โ€œxโ€ in the constructal model. As mentioned above, x is the fraction of the globe that is exporting heat on average. The CERES data also gives us the information needed to calculate rho (ฯ), which is the albedo, and gamma (ฮณ) which is the โ€œgreenhouse factorโ€.

The model says that if we know the albedo ฯ, the greenhouse factor ฮณ, and the hot zone area x, given those physical constraints the resulting hot and cold temperatures will be the ones that maximize the heat flow โ€œqโ€ from the hot zone to the cold zone.

Here is the performance of the constructal model. Recall that it has only one tuned parameter, C, that regulates how easily the heat flows from the hot zone to the cold zone. Iโ€™ll get back in a bit to why I think their value for C (.181) is far too low. In the meantime, these are the actual (blue/cyan) and modeled (red/orange) temperatures for the hot and cold zones of the planet.

Figure 3. Modeled and actual temperatures of the worldโ€™s hot and cold zones

I found this result to be most encouraging. Those model temperatures are calculated based solely on maximizing the heat โ€œqโ€ flowing from the hot zone to the cold zone, subject to the physical constraints of the albedo and the greenhouse factor. And although the conductance C is tuned, all that tunes is the temperature difference between the hot and cold zones. It does not tune the temperatures themselves. There was no guarantee that tuning the conductance would match the absolute temperatures of the hot and cold zones โ€ฆ but in the event, the match is excellent. I would say that that is very convincing evidence that the constructal model accurately portrays how the climate flow system actually works.

A SECOND TEST

But wait, as they say on TV, thereโ€™s more. Here are closeups of the actual and modeled variations in the yearly average temperatures of the hot and cold zones.

Figure 4. Modeled and actual annual average temperatures of the worldโ€™s hot and cold zones.

Not perfect, but not bad either. So not only does the constructal model give good long-term average temperatures. It also does a decent job of replicating the year-by-year variations in temperature.

And itโ€™s doing all that using nothing more than the hot zone area โ€œxโ€, the albedo โ€œrhoโ€, and the greenhouse factor โ€œgammaโ€ to calculate the temperatures that maximize โ€œqโ€.

Thatโ€™s very clear evidence that in the real world, various physical processes constantly evolve and act to increase the flow of heat from the tropics to the polar regions.

A FINAL TEST

Further evidence that the model is an accurate representation of how the climate heat engine really works is visible in both the size and the stability of the area of the hot zone. The model calculates the average of x, the hot zone fraction of the surface, as being 0.564. The actual CERES 22-year average value for x is 0.556. Thatโ€™s less than a hundredth difference. Once again, the model is accurate.

Regarding stability, remember that x, the hot zone area fraction, is calculated by the model as the hot zone area that maximizes the heat flow โ€œqโ€. Bear in mind that the hot zone fraction could vary from ~0.1 to ~0.9. And thereโ€™s no reason to assume ex-ante that it would remain stable over time.

However, under the constructal model, since the underlying constraints (annual average albedo and greenhouse fraction) are relatively stable weโ€™d expect the hot zone fraction โ€œxโ€ to be pretty stable as well. In any case, hereโ€™s the actual record of the CERES data for โ€œxโ€, the hot area fraction, along with the constructal model output of the same variable.

Figure 5. The โ€œxโ€ fraction, the amount of the earthโ€™s surface that makes up the hot zone.

Clearly, the model is doing an excellent job of representing the real world.

In Figure 5, as in Figs. 3 and 4 above, itโ€™s important to remember that the output (e.g. the modeled x fraction in Fig. 5 above) is not calculated directly from the input. In Figure 5, for example, the x fraction shown in red is not directly calculated from the albedo and greenhouse fraction figures.

Instead, it is the result of a maximization procedure. The x fraction shown in red in Figure 5 is the value of x that, given the physical constraints of albedo and greenhouse fraction, gives the greatest flow โ€œqโ€ from the hot zone to the cold zone.

TEMPERATURES

For temperatures, Iโ€™ve used the CERES surface upwelling longwave data converted using the Stefan-Boltzmann constant and monthly gridded emissivity values. Iโ€™ve checked the results and they are extremely similar to both the Berkeley Earth and the HadCRUT datasets. I use it because it is energy-balanced with the rest of the CERES energy flows.

CONDUCTANCE

I mentioned above that Iโ€™d explain why I think their value for โ€œCโ€, the โ€œconductanceโ€, is too low. This conductance is a measure of how much heat flows between the two zones for some given temperature difference between the zones. In their model, theyโ€™ve modeled the heat transport via the atmosphere. And theyโ€™ve modeled the atmospheric heat transport as being driven by the buoyancy of the warmer, lighter tropical air.

And that is good as far as it goes. But it leaves out a couple of things. One is a main power source driving the Hadley cell circulationโ€”the perennial line of thunderstorms along the inter-tropical convergence zone (ITCZ). These drive air vertically from the surface up to the upper troposphere, and occasionally even into the stratosphere. These thunderstorms turbocharge the Hadley cell circulation, allowing it to move much more heat polewards than if it were driven solely by the general tropical-extratropical temperature differences as the authorsโ€™ analysis assumes. Hereโ€™s a map of where the thunderstorms live.

Figure 6. The altitude of the cloud tops, day/night. High altitude cloud tops are the sign of the tropical thunderstorms driving deep tropical convection. The Inter-Tropical Convergence Zone (ITCZ), where the two atmospheric hemispheres converge, is marked by the band of thunderstorms around the world at 5ยฐ-10ยฐ north of the equator.

The second reason that I think their conductance value is too small is that a large amount of heat is physically moved polewards by the ocean currents. The Agulhas Current in the Indian Ocean and the Gulf Stream in the Atlantic Ocean are constantly transporting warm tropical waters polewards.

In the Pacific, the El Nino/La Nina pumping action periodically strips off the warm top layer of vast areas of the tropical Pacific Ocean and moves that warm water first eastwards and then towards both poles.

Because their model doesnโ€™t include either thunderstorms or ocean currents, their estimate of the conductance is an order of magnitude too small.

CLIMATE SENSITIVITY

This constructal model points out some interesting things about climate sensitivity.

First, sensitivity is a function of changes in rho (albedo) and gamma (greenhouse fraction). But not a direct function. It is the result of physical processes that maximize โ€œqโ€ given the constraints of rho and gamma.

Next, the sensitivity is slightly different depending on whether the changes in albedo and greenhouse fraction are occurring in the hot zone, the cold zone, or both.

Finally, assuming that there is a uniform pole-to-pole increase of 3.7 W/m2 in downwelling radiation from changes in either albedo or greenhouse fraction, the constructal model shows a temperature increase of ~1.1ยฐC. (3.7 W/m2 is the amount of radiation increase predicted to occur from a doubling of CO2.)

CONCLUSIONS

The CERES data shows that the constructal model of the climate system is very consistent with real-world observations. This model views the climate system as a heat engine that, following the constructal law, constantly acts and evolves to maximize the flow of heat from the warm zone of the planet to the cold zone.

This simple three-equation constructal climate model, given only information about the earthโ€™s hot zone area and the albedo and greenhouse fractions in the earthโ€™s hot and cold zones, is able to calculate the absolute temperatures of the earthโ€™s hot and cold zones to within a degree or so โ€ฆ a result that I found quite surprising.

Anyhow, thatโ€™s what I did with my weekend. And meanwhile, back in the real world, the past climate is being rewritten so fast that we literally donโ€™t know what will happen yesterday โ€ฆ

Best to all,

w.

APPENDIX

Here is the R code for the optimization programs. Read the linked paper for the full description of their method.

First, the inner optimization program that calculates TH, TL, and q when given x.

maxq=function(par2){

            theansmax=function(par){

                        th=par[1]

                        tl=par[2]

                        q=par[3]

                        (v1=x*((asin(x)+x*sqrt(1-x^2))/(2*pi*x))*(1-rhoh)-

                                                x*(1-gammah)*th^4-q)

                        (v2=(1-x)*((pi/2-asin(x)-x*sqrt(1-x^2))/(2*pi*(1-x)))*(1-rhoc)-

                                                (1-x)*(1-gammac)*tl^4+q)

                        (v3=1.8*(th-tl)^(3/2)-q)

                        sum(v1^2+v2^2+v3^2)

            }

            par=c(.7,.6,.1)

            x=par2

            (par=optim(par,theansmax)$par)

            par[3]

}

Next, the outer optimization program that calls the inner program.

(bestx=optim(par2,maxq,

control=list(fnscale = -1,reltol=1e-10),

method=โ€Brentโ€,

lower=.001,upper=.999)$par)

Next, some support functions:

surfaream = 5.100656e+14 #earth surface area in sq. m.

qtoq= function(q) q*((5.67e-8)*392.8^4*surfaream)

tunscale=function(tscale) tscale*392.8

xtolat=function(x) degrees(asin(x))

And to get the final output:

(x=bestx)

(nupar=optim(par,theansmax)$par)

q=nupar[3]

qtoq(nupar[3])

xtolat(x)

(th=ktoc(tunscale(nupar[1])))

(tl=ktoc(tunscale(nupar[2])))