Guest Post by Willis Eschenbach (@WEschenbach at X, my own blog is here.)
I’d like to take a moment to discuss some implications of my peer-reviewed paper about my implementation of a Constructal climate model. The paper is available here. I’ll steal some images from the paper for this discussion
To understand the model, it’s necessary to understand the Constructal Law. The Constructal Law is the most recently discovered fundamental law of thermodynamics. It applies to flow systems far from equilibrium, which is much of what we see in the world around us.
The Constructal Law was discovered by Adrian Bejan in 1996. His statement of it is as follows:
“For a flow system to persist in time it must evolve in such a way that it provides easier access to its currents.”
In terms of the Earth’s climate as a whole, what this means is that the climate system is always working to maximize the flow of power from the tropics to the poles and from there out to space.
This is the missing link in the current generation of climate models. They view the climate as a passive, linear system where if, say, the albedo increases by some amount X, then some other variable will perforce change by an amount Y. But that’s not the case.
Instead, the climate is dynamic. It responds to changing conditions, not randomly, but always following the Constructal Law by evolving in the direction of increasing the throughput of power from the tropics to the poles.
Now, Bejan and Reis applied the Constructal Law to the Earth’s climate and derived the equations governing the situation.
Fig. 1. Conceptual Constructal model. Earth’s surface is divided into a hot equatorial zone (area AH, temperature TH) and the cold polar zones (total area AL, temperature TL). Heat flow q is transported from hot to cold zones by atmospheric and oceanic circulation.
Clearly, that is an extremely simplified model. There’s no ocean, no mountains, just a simple sphere with some unspecified means of transporting the heat from the tropics to the poles.
The model calculates four different variables—the average temperatures TH and TL of the hot and cold zones, the area AH of the hot zone, and the power flow “q”.
The model calculates the temperature using two observable climate variables—the albedo (how much solar energy is reflected back to space) and the “greenhouse factor” (how much upwelling surface radiation is absorbed by the atmosphere). The short version is that the albedo regulates how quickly the power from the sun enters the system, and the greenhouse factor regulates how quickly that power leaves the system. Obviously, the temperature of the system will depend on the ratio of the two—if more power is entering than leaving the system will warm, and vice versa.
But as I said above, this is not a simple linear relationship. What happens is that the climate system is constantly evolving, subject to the constraints of power in and power out, to maximize the power throughput.
However, what Bejan and Reis didn’t do was to actually implement that model on a computer and test it against the actual Earth’s climate. So I set out to do that, and succeeded far beyond my expectations.
Here’s what the system looks like on the real Earth. Just as in the model, there is a clear hot zone and two cold zones, and the boundary between them is roughly linear and symmetrical north/south.
Fig. 2. Actual Earth hot equatorial zone (area AH, temperature TH) and cold polar zones (total area AL, temperature TL). Note that unlike the oceans that follow the theoretical boundaries shown in Fig. 1, the desert areas are in the cold zone.
And despite the model not having deserts and mountains and oceans and all the rest, here’s how well the model is able to emulate the actual average temperatures of the hot and cold zones.
Fig. 3. Annual mean temperatures from CERES observations (blue/cyan) and Constructal model (red/orange). Top: hot zone temperature TH. Bottom: cold zone temperature TL. Note that the model is NOT tuned to reproduce these temperatures, it is only tuned to reproduce the interzonal temperature difference.
Other than a slight variation around 2010, the computed and actual temperatures are so close that they overlap each other almost exactly.
And here are the annual changes in the anomalies of the actual and modeled power flow “q” from the tropics to the poles.
Fig. 4. Anomalies (about data mean) of the CERES satellite data (blue) and modeled Constructal results (red) for the amount of power flowing from the Equator to the Poles.
This graph is clear evidence that the model is actually capturing how the climate works. It does indeed follow the Constructal Law, maximizing the power flowing from the tropics to the poles just as the Constructal Law predicts.
Finally, here’s how well the model calculates the area of the hot zone “AH”.
Figure 5. As in Figure 1, but overlaid with what the model calculates as the boundaries between the hot and cold zones.
As I said, I was quite surprised by this. I didn’t expect the model to emulate the Earth anywhere near as well as it did.
Finally, I used the model to see what the climate sensitivity would be if the greenhouse factor were increased by, say, a doubling of CO2. I got an answer of an increase of 1.1°C from a doubling of CO2. This is at the lower end of the other previous estimates of this sensitivity, but it is not the lowest.
Figure 6. The climate sensitivity estimated from doubling the CO2 in the Constructal climate model.
However, this does not include any changes in the albedo or any effect of the warming on the prevalence or timing of thunderstorms and cumulus fields, so it is likely an upper bound on the climate sensitivity.
Let me close this section by noting that, as shown in Figure 6 above, despite hundreds of thousands of hours of computer time and human study, the uncertainty and spread of the estimates of the constant called the “climate sensitivity” have increased over time. I view this as evidence that the underlying theory is incorrect. That theory says that the changes in temperature are a simple linear function of the downwelling radiation times the constant climate sensitivity. The Constructal Law says that this is not the case, and the model demonstrates that.
Conclusions and Implications
1) The model shows that the Earth’s climate system is indeed ruled by the Constructal Law, in that it maximizes tropical-polar power flow. Any model that does not include this active evolution of the system is an incorrect representation of reality.
2) Two variables alone, the albedo and the greenhouse factor, are sufficient to explain the changes in the surface temperature, as well as the variations in the equatorial-polar energy flow and the area of the hot zone. So while the model clearly shows that the greenhouse effect is real and important … the model also shows that it’s only half the story.
3) The climate sensitivity, which is how much the temperature will rise from a doubling of CO2, is likely to be quite small. So it looks like Thermageddon™ is cancelled, sorry, no refunds of the billions spent on meaningless climate gestures.
4) While it is at least somewhat possible to provide some direction and bounds on the future evolution of the greenhouse factor, that’s only half of the equation. The other half is the albedo. The variations in albedo are mostly ruled by the clouds, which everyone agrees are the part of the climate that is least understood and hardest to model, measure, or predict. As a result, the state of our current knowledge of clouds makes any long-term projections of future climate states … well … let me call them wildly hubrimistic and leave it at that.
Late night here, the redwood forest is wrapped in fog. The fog channels sound and makes it carry, so I can hear the currently high Pacific Ocean surf grumbling and gnawing on the coast six miles (10 km) to the west. Ah, dear friends, what an amazing, entrancing, awesome world it is our privilege to inhabit.
I can only wish the best of life for all of you,
w.
PS—Yes, you’ve heard it before: when you comment, please QUOTE the exact words you are discussing. It helps greatly to avoid the misunderstandings that are the bane of the intarwebs.
PPS:
hubrimistic
adjective
hu·bri·mis·ticˌˈhyü-brə-ˈmi-stik
: of, relating to, or characterized by feeling or showing unwarranted hope for the future, with just a soupçon of hubris thrown in for good measure
The Bejan papers have used 2 different values for Earth’s radius as follows:
Int. J. Energy Res. (2005); 29:303–316 R = 5000 km
International Journal of Heat and Mass Transfer 49 (2006) 1857–1875 R = 6600 km
Int. J. Global Warming, Vol. 4, Nos. 3/4, (2012) R = 6600 km
Your value R = 6371 km
The balance equations that are solved are given in “Section 3.1 Numerical solution method”
“Section 3.2 Model improvements” mentions some changes to the original equation system.
“Finally, the original Bejan and Reis model (2005, 2006) neglected the power absorbed by the ocean. This led to a constant trend in the modeled high and low temperatures. Accordingly, this absorption by the oceans is added as a tunable parameter.”
The changes to the model equations that this improvement makes are not reflected in the Equations in Section 3.1. The reported results of the calculations cannot be independently verified.
As a point of interest, the three papers by Bejan and coworkers contain several physical quantities: Sun temperature, Earth-Sun distance, deep space temperature, and several others; I have summarized Earth’s radius values above. The Bejan papers make run-time adjustments to the conductance, which you also change, is another example. A Table listing the physical parameters from the Bejan papers and your paper, along with comparisons with calculated values for T_H, T_L, q, and x would be interesting.
A possible numerical solution method verification case might be: What results do your equations give for T_H, T_L, q, and x if the Sun is turned off.
Let me know if I have messed up in reading your paper.
The original model equations show power flowing from the hot and cold zones out to space, and from the hot zone to the cold zone.
The model equations are modified by simply adding a term for power flowing into the ocean from both the hot and cold zones.
And since it all depends on solar input, if the solar input is 0, the temperature goes to 0K.
Regards,
w.
An objective of Verification of the software and solution method is to ensure that the equations are correctly solved. Analytical solutions or thought problems or degenerate cases can be used as a part of the process.
So, while the expected results for the situation of no energy input from the Sun can be determined without actually solving any equations, the requirement is to determine that when the equations are solved, the solution method gives the expected results. The other dependent variables T_L, T_H, X, q, and power output, for example, can also be checked.
Two other situations might also be considered: (1) x->1, and (2) x->0.
A related aspect is to ensure that the software/code agrees with the specification documents. In this regard, even the specification document, Eqs. [13] – [16], is incomplete. The changes discussed following presentation of the equations include both (1) quantities that are not in the equations, and (2) numerical values that also are not presented.
Results presented in the paper cannot be checked and independent replication of the results is impossible. Notably, Verification is always required to precede Validation.
Thanks, Dan. In the set of equations, the variables are scaled as fractions of the total input solar. As a result, it’s not possible to set solar to zero, as it gives meaningless values for the variables.
Here are the relevant equations:
A1 <- ((asin(x) + x * sqrt(1 – x^2)) / (2 * pi * x)) * (1 – rhoh)
A2 <- (((pi / 2) – asin(x) – x * sqrt(1 – x^2)) / (2 * pi * (1 – x))) * (1 – rhoc)
eq1 <- x * A1 – x * (1 – gammah) * th^4 – q – imbalance*hfactor
eq2 <- (1 – x) * A2 – (1 – x) * (1 – gammac) * tl^4 + q- imbalance*(1-hfactor)
eq3 <- cc * (th – tl)^(3/2) – q
where x is the hot fraction, th and tl are hot and cold zone temperatures, q is the power flow from hot to cold, rhoh and rhoc are albedos for the hot and cold zones, gammah and gammac are the greenhouse factors for the hot and cold zones, imbalance is the oceanic absorption, cc is the conductance and the hfactor is how much of the absorption is in the hot zone.
This is merely an implementation of the underlying relationships, which state
• Hot zone in – hot zone out – hot zone ocean absorption = power flow hot->cold
• Cold zone in + power flow – cold zone ocean absorption = cold zone out
• Power flow = (hot temp – cold temp)^3/2 * conductance
You are welcome to implement them as you see fit. I used a double optimization simply because my math-fu is not strong enough to solve them directly.
Best regards,
w.
“The variations in albedo are mostly ruled by the clouds, which everyone agrees are the part of the climate that is least understood and hardest to model, measure, or predict. As a result, the state of our current knowledge of clouds makes any long-term projections of future climate states … well … let me call them wildly hubrimistic and leave it at that.”
An AMO based prediction of changes in European sunshine hours:
Central-European sunshine hours, relationship with the Atlantic Multidecadal Oscillation, and forecast | Scientific Reports
Willis,
You are really on to something here.
The predominant planetary mechanism for meridional energy transport (equator to the poles) is in the Oceans and not in the Atmosphere.
During Glacial times, the Bering Strait is closed by the presence of the Beringia land bridge between Siberia and Alaska.
Therefore, the Arctic Ocean becomes ponded with only one marine energy transport route into it from the Norwegian Sea. This leads to throttling (think back-pressure) and so the energy transport system must work harder during lower sea level glaciations. This leads inevitably to a greater temperature gradient in glacial times.
A nice extension of your model that is worth persuing
Best to you
Philip
Most of the post is very good and I agree with it, but this statement isn’t quite right:
Maximal heat flow from the tropics to the poles (meridional transport) is a sign of a colder Earth overall. Maximal zonal (west to east) flow is a sign of a warmer Earth. This is shown clearly in Chris Scotese’s work. More here:
Here is an intro:
https://andymaypetrophysicist.com/2022/08/16/the-sun-climate-effect-the-winter-gatekeeper-hypothesis-iii-meridional-transport-the-most-fundamental-climate-variable/
The tropics radiate much more total heat to space than the poles because they are warmer, but the poles do radiate more than they receive. This is because much of the heat in the poles is delivered there (advected) by winds and ocean currents. More delivered heat to the poles makes the poles warmer, but not the planet. The planet warms when zonal winds dominate.
See here:
https://andymaypetrophysicist.com/2022/10/24/meridional-transport-the-most-fundamental-climate-variable/
Other than that, the post is excellent, thanks. A statement that says what I think you mean, but is more accurate is:
While the Earth is always trying to maximize the flow of heat out of the tropics and toward the poles, the Earth is actually warmer over all when zonal flow is dominant. When meridional flow dominates, the poles become warmer and the Earth as a whole gets colder. In the geological past, lower equator to pole gradients always correlate with a warmer planet, higher gradients with a colder planet.
If you change it to that, everything else is OK.
Thanks, Andy. You say:
“Maximal heat flow from the tropics to the poles (meridional transport) is a sign of a colder Earth overall. “
But that’s certainly not what is happening today. A warmer earth today is associated with an increased MT. Here’s the situation with seasonal variations removed.
What am I missing here?
Best regards,
w.
This linear regression model appears to be highly uncertain. One of the first things you learn is that with a functional relationship every independent variable value should have one and only one dependent variable value. Anything different must be treated as uncertainty in measurement or as an incorrect model.
If you assume the regression equation accurately depicts the correct model, then the measurement uncertainty would be somewhere around ±0.5 petawatts.