A Spherical Cow Climate Model That Actually Works

Guest Post by Willis Eschenbach (@weschenbach on X, personal blog at Skating Under The Ice)

Here’s a science joke about the dangers of oversimplified models.

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A dairy farmer with low milk production asks a physicist for help. After some months, the physicist eventually reports back: “I have found out how to solve the problem.”

“About time!”, said the farmer, “What is it?”

The physicist replies, “First, assume a spherical cow in a vacuum.”

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In 2023 I wrote post entitled Testing a Constructal Climate Model, where I took a first cut at making a computer implementation of Adrian Bejan’s ideas about a Constructal model of the climate.

For those who missed my earlier post, the Constructal Law is the most recently discovered fundamental law of thermodynamics. It was first described by Adrian Bejan in 1996. He’s the JA Jones Distinguished Professor of Mechanical Engineering at Duke University, and his writings have over 100,000 citations.

The Constructal Law says that flow systems far from equilibrium evolve to maximize access to flow. Rivers don’t meander randomly—they organize to maximize water transport. Animal circulatory systems don’t just happen—they evolve to maximize nutrient flow. And according to Adrian Bejan, the climate should organize itself to maximize heat flow from the hot tropics to the cold poles.

Now, that sounded reasonable enough … but does it actually work? Can you build a working climate model based on that principle?

Turns out you can. And as my post above showed, it’s a very simple model.

In any case, after writing that post, I got invited to present my work in November at the 15th Constructal Law Conference, hosted by the Florida International University College of Engineering and Computing in Miami.

I went in part because I wanted to meet Adrian Bejan, who was slated to be in attendance. I have a number of scientific heroes, and he’s one of them. I introduced myself to him, and he said Oh, you’re Willis Eschenbach. I’ve seen your work. It’s very impressive!

Zowie, sez I. A win for the reformed cowboy!

There were many interesting presentations at the Constructal Conference, but that interaction alone was worth the ticket to Miami … however, I digress.

Originally, my plan for the conference was to just present the work shown in my post linked above. But then I realized that was a lazy copout — that model was two years old. I had to take it further forward. So I did more research and analysis, which is what this post is about. Basically, it’s an expanded version of my presentation at the conference.

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The model I’m about to show you treats the Earth as a smooth sphere—no land, no ocean, no mountains, no ice sheets, nothing but a ball heated by the sun. It divides this ball into two zones: a hot equatorial zone and cold polar zones. That’s it. Two zones. And from those two zones, using the Constructal Law to maximize heat flow between them, the model reproduces the actual Earth’s temperature, circulation patterns, and year-to-year variations with remarkable accuracy.

Let me show you how it works, and then we’ll look at what it tells us about climate sensitivity. Spoiler alert: the news is good if you’re worried about catastrophic warming.

The Basic Setup

Figure 1 shows the concept. We divide Earth into a hot zone,extending from the equator to some latitude θ, and two cold zones, poleward of θ. Heat flows from hot to cold, driven by the temperature difference.

Figure 1. The spherical cow … er, spherical Earth model. There is a hot equatorial zone of area A_H with an average temperature T_H. There are two cold polar zones of total area A_L with an average temperature T_L). Heat flow q moves from hot to cold.

Now, before anyone objects that this is ridiculously oversimplified, let me show you what the real Earth looks like in terms of energy balance:

Figure 2a. The actual Earth’s hot and cold zones, as determined by the gridded top-of-atmosphere radiation balance from CERES satellite data. The tropics export energy (red), and the polar regions import it (blue). Note that the deserts—Sahara, Arabia, Australia, Gobi—are actually in the cold zone despite being hot. They radiate more energy than they absorb.

The similarity is striking. The real Earth organizes itself into these zones naturally. The model just captures this organization mathematically. And the division into these zones is surprisingly stable over time.

Figure 2b. Annual average hot and cold zones, 2001 to 2024

Note how the Sahara, Arabia, and Gobi deserts protrude down into the hot zone. This leads to some offsets in the results, as discussed below.

The Energy Balance

Being a simple fellow, I started with Bejan’s simple energy balance equations. The hot zone receives solar energy based on its projected area, reduced by its albedo (the fraction of incoming solar radiation reflected by the planet, “albedo_H”):

Energy in = (projected area) × (1 − albedo_H) × (solar constant) [Equation 1]

The hot zone radiates energy to space, reduced by the greenhouse fraction “greenhouse_H”. This is the percentage of upwelling thermal radiation from the surface that is absorbed by the clouds, water vapor, and the radiatively active gases.

Energy out = (area) × (1 − greenhouse_H) × σ × T_H^4  [Equation 2]

where σ is the Stefan-Boltzmann constant and greenhouse_H is the fraction of surface radiation absorbed by the atmosphere.

The same things are true for the cold zones. They absorb solar radiation mediated by albedo and emit thermal radiation to space, regulated by the greenhouse fraction.

The difference between incoming and outgoing power in each zone must equal the heat flow q between the zones. Energy is conserved—it has to go somewhere.

So far, nothing fancy. Just bookkeeping.

The Heat Transport

Here’s where it gets interesting. How much heat flows from the hot zone to the cold zone? Bejan and Reis showed that for buoyancy-driven atmospheric circulation, the heat flow should vary as:

q = C × (T_H − T_L)^(3/2)  [Equation 3]

where q is the heat flow and C is a thermal conductance factor representing how easily the heat is conducted from the hot to the cold zones.

According to Bejan, the 3/2 power law comes from the physics of convection. Warmer air rises, cooler air sinks, and the flow rate depends on the temperature difference in a very specific way. The constant C depends on atmospheric properties and circulation patterns.

Now we have five unknowns (T_H, T_L, q, the fraction x of Earth’s surface in the hot zone, and the conductance factor C), and three equations. How do we close the system?

Enter the Constructal Law

Here’s where Bejan’s insight comes in. The system evolves to maximize the heat flow q. Why? Because the climate is a heat engine, with the tropics as the hot reservoir and the poles as the cold reservoir. But unlike a car engine that delivers power to the wheels, Earth’s climate engine is permanently coupled to its “brake”—all the power it generates gets dissipated immediately through friction in winds and ocean currents.

For such a system—an engine hard-wired to its brake—maximum power production equals maximum dissipation, which means maximum heat flow. The atmosphere and ocean organize themselves to achieve this.

So the fourth equation is simply:

dq/dx = 0  [Equation 4]

Making It Work

I implemented this as a nested optimization problem in R. For any given value of x, I first solve for the values of T_H, T_L, and q that satisfy the three energy balance equations. Then I vary x to find the value that maximizes q. Finally, I optimize the value of C, the conductance, to agree with the distance between T_H and T_L. (T_H minus T_L) It’s optimization within optimization.

The original Bejan/Reis model used global average values for albedo and greenhouse effect. I improved this by using separate values for each zone, derived from CERES satellite data. Tropical regions have lower albedo (less ice and snow) and higher greenhouse effect (more water vapor) than polar regions. This matters.

I also added one more refinement. As you can see in my previous model analysis linked to above, the original model had a slight drift in temperatures over time.

Figure 3a. Results of my previous model incarnation, which shows an anomalous trend in the modeled temperatures

After too much investigation, I realized that this is because my model neglected the heat absorbed by the oceans. This flow was also not included in the Bejan model shown in Figure 1 above. I added this as a small tunable parameter for each zone.

So how many parameters does this model have? Let’s count:

• Albedo in hot zone (measured from CERES data)
• Albedo in cold zone (measured from CERES data)
• Greenhouse fraction in hot zone (measured from CERES data)
• Greenhouse fraction in cold zone (measured from CERES data)
• Conductance C^(3/2) (tuned parameter)
• Ocean heat absorption, hot zone (tuned)
• Ocean heat absorption, cold zone (tuned)

That’s four measured parameters and three tuned ones. Compare that to the thousands of parameters in comprehensive climate models. It’s a spherical cow.

The Results: Temperatures

OK, enough theory. Does it work? Figure 3b shows 24 years of data (2001-2024) comparing CERES satellite surface temperature observations to the Constructal model emulation. This is the corrected version of Figure 3a above.

Figure 3b. Annual mean temperatures. Blue/cyan: CERES satellite observations. Red/orange: Constructal model predictions. Top: hot zone temperature T_H. Bottom: cold zone temperature T_L.

The agreement with the actual real-world temperatures of the hot and cold zones is remarkable, given that nowhere is the model tuned to produce them. At this scale, they are so close that you can only see the difference between reality and the model around 2010.The model gets the absolute temperatures right within 0.1°C on average. It also tracks the year-to-year variations. It captures the slight warming trend. And remember—this is a model of a featureless sphere with no continents, no oceans, no mountains, no ice sheets, nothing.

Next, here’s a close-up of the hot zone temperatures shown in Figure 3b above:

Figure 4. Detailed view of hot zone temperatures. CERES data (blue) and model (red). Root mean square error: 0.20°C. Correlation of interannual variations: r = 0.56, p < 0.001.

The main difference between the model and reality is that the swings in the modeled temperature are somewhat larger than the swings in the real world. I suspect this is because the model assumes that the losses are all temperature-controlled, and also, in the model, there’s no atmospheric absorption of the sunlight. Both of these omissions increase the size of the swings.

Next, the cold zone results:

Figure 5. Cold zone temperatures. CERES data (cyan) and model (orange). RMSE: 0.13°C. Correlation: r = 0.71, p < 0.001.

The fit in the cold zone is better than in the hot zone, although again the swings are exaggerated in the model.

This shows that the model isn’t just matching the average temperatures—it’s capturing the dynamic annual changes in temperature as well.

The Hot Zone Boundary

The model predicts that the boundary between hot and cold zones should be at about 36° North and South latitude. The actual boundary, based on where top-of-atmosphere radiation balance crosses zero, is at around 34°. The model is off by about 2°, or about 220 kilometers.

Not bad for a model that doesn’t know about Hadley cells, polar cells, El Nino/La Nina, jet streams, Gulf Streams, or any of the detailed atmospheric and oceanic circulation patterns. Those patterns emerge naturally from maximizing heat transport.

Figure 6 shows how the boundaries compare by year:

Figure 6. Boundaries of Earth’s hot zone. Blue: CERES observations. Red: Model predictions. The model boundary is about 2.4° wider than observed.

The model tracks the variations well. The offset in the boundaries exists because in the real Earth, the major deserts (Sahara, Arabia, Australia, Gobi) are in the cold zone—they radiate more than they absorb. See Figures 2a and 2b above. But in the smooth-sphere model, those latitudes are in the hot zone. This makes the model’s hot zone larger than reality.

Despite this, the model captures the year-to-year changes quite well. Figure 7 shows the anomalies (variations around the mean) of the latitude of the boundary between hot and cold:

Figure 7. Anomalies in hot zone area. Blue: CERES. Red: Model. When you remove the offset, the model tracks the variations quite well.

This is doing a workmanlike job of calculating both the annual variations in the size of the hot zone and the overall trend of hot zone size over the period of record.

And to close the circle on this part of the analysis, here’s Figure 2 with the Constructal model hot zone boundaries overlaid on it.

Figure 8. As in Figure 2, but with the modeled boundaries overlaid on it (yellow/gray dotted lines).

Here you can see why the actual earth’s total hot zone area is slightly smaller than that of the model—the areas in green which are cold zone intruding into the model-simulated hot zone.

Heat Flow

Now for the crucial test. The Constructal Law says the system maximizes heat flow. Does the model get the heat flow right?

Figure 9 shows the total poleward heat transport:

Figure 9. Annual heat flow from tropics to poles, in petawatts (PW = 10^15 watts, left scale) and the equivalent global heat flow (petawatts divided by global surface area, W/m², right scale). Blue: CERES. Red: Model.

The model shows a significantly higher heat flow than observed (14.9 PW modeled vs. 12.3 PW observed). This is for the same reason the area of the hot zone is larger as discussed above—those deserts that are in the cold zone in the real world but in the hot zone in the model. This means the model is absorbing more solar power than the real world in the hot zones and transferring it to the poles. So the total flow is too large, by a globally averaged value of ~ 5 W/m². But look at the heat flow anomalies below:

Figure 10. Heat flow anomalies. Blue: CERES. Red: Model. RMSE: 0.04 PW. The model captures year-to-year variations almost perfectly.

That RMSE of 0.04 PW is tiny compared to the year-to-year variations. The model is faithfully capturing the dynamics of heat transport using only annually updated albedo and greenhouse parameters as input. That validates that we are seeing the Constructal Law at work.

Climate Sensitivity

Finally, what does this model tell us about climate sensitivity—how much warming would we get from doubling atmospheric CO₂?

To determine that, I ran the model with a uniform increase in downwelling radiation of 3.7 W/m², the IPCC standard assumed forcing from doubled CO₂. The model predicts:

• Hot zone warms by 1.09°C
• Cold zone warms by 1.12°C
• Global average warms by 1.10°C

So the equilibrium climate sensitivity is about 1.1°C per doubling of CO₂.

But wait, there’s more. And this is important. This value is a maximum estimate because the model doesn’t include negative feedbacks from a host of emergent climate phenomena like thunderstorm responses, El Nino/La Nina events, tornadoes, and other phenomena which I’ve written about extensively in my thunderstorm thermostat posts. Those feedbacks all oppose surface warming by transferring heat from the surface to the atmosphere. As a result, the real climate sensitivity is likely lower than shown above.

In addition, total known fossil fuel reserves contain about 4,780 gigatonnes of CO₂. This is not total proven reserves, those we know we can get to within budget. This is everything we know about, whether or not it actually is economically recoverable with today’s technology.

At 17.4 Gt CO₂ added per each ppmv increase, that will increase the atmospheric CO₂ by 275 ppmv … and since we have ~ 420 ppmv of CO₂, it’s unlikely that we’ll ever see a doubling of CO₂ from here.

And with a sensitivity of 1.1°C from doubling, this implies a maximum temperature increase on the order of log2((275 + 420) / 420) * 1.1 = 0.8°C … Thermageddon™ cancelled, sorry, no refunds.

Is this sensitivity believable? The IPCC’s range for climate sensitivity is 1.5°C to 4.5°C per doubling. The Constructal model puts it at 1.1°C, below the IPCC’s lower bound. However, this is in the same range of observational estimates from Lewis and Curry and others who’ve analyzed the actual warming we’ve seen.

For comparison, here’s the history of climate sensitivity estimates:

Figure 11. Historical estimates of equilibrium climate sensitivity. The range of the estimates has actually gotten wider over time. That tells you something about how well we understand the climate.

Climate sensitivity is a, perhaps the, biggest unknown in climate science. Note that the uncertainty of the climate sensitivity has increased, despite hundreds of thousands of hours of computer time and human research. That’s not how science is supposed to work. To me, it strongly suggests our current climate models have been barking up the wrong tree.

What This Tells Us

The success of this ultra-simple spherical cow model reveals some profound things about how climate works:

1. Optimization trumps complexity. You don’t need to simulate every cloud, every ocean eddy, every rainstorm to understand the big picture. The climate organizes itself according to simple principles—it maximizes heat transport, subject to radiative constraints.

2. The ocean’s role is simpler than we thought. The model treats Earth as a uniform sphere—no explicit ocean, no currents, no thermohaline circulation. Yet it works, and works very well. This suggests the ocean’s main climate role is facilitating heat transport (captured in that conductance parameter C), not creating fundamentally new dynamics that require detailed representation.

3. Circulation patterns emerge. The model predicts a hot/cold boundary near 34°N/S—where the Hadley cell meets the Ferrel cell in the real atmosphere. The model doesn’t know about these circulation cells. They emerge from the optimization. The atmosphere organizes itself into Hadley and polar cells because that configuration maximizes heat transport.

4. Less is more. This model has seven parameters (four measured, three tuned). Comprehensive climate models have hundreds. Yet this simple model matches observations as well as or better than the complex ones for these fundamental variables of hot and cold zone temperatures, hot zone area, and heat flow. Occam’s razor suggests we should pay attention to that.

5. It demonstrates and validates the importance of the albedo and the poorly-named “greenhouse effect”. Clearly, those two variables alone have an overwhelming importance in the way that the climate organizes itself.

6. It makes intuitive sense. Albedo and the greenhouse fraction respectively control the amount of energy entering and leaving the system. And from basic physics, this makes sense—we’d expect the global temperature to be a function of the energy entering the system minus energy leaving the system. However, it is not a simple function of those two values. Instead, following the Constructal Law, the system constantly rearranges itself to maximize the energy flow given those two values.

7. Current climate models are going down the wrong path. This model clearly shows the underlying Constructal structure and action of the climate, a structure that is not emulated by the current generation of climate models.

Limitations

Now, I’m not claiming this model can do everything. It can’t. It has obvious limitations:

• Only two zones, so no gradients within zones and no longitudinal (east-west) variations
• Annual averages only, so no seasonal or daily cycles
• Simplified transport physics lumped into one parameter
• No ocean, no land. If nothing else, the thermal conductance factor C will be different in those two regimes.
• No explicit clouds, water vapor changes, ice-albedo feedback, or vegetation
• Simple greenhouse treatment

Future work should address these. A multi-zone model with finer resolution. Adding seasonal cycles. Explicitly modeling thunderstorm convection and ocean circulation. Including the desert areas as part of the cold zones. Coupling to detailed radiative transfer models.

But the point is, it works now despite its simplicity. That tells us something important about nature’s organizing principles.

Not Maximum Entropy Production

Some folks confuse the Constructal Law with Maximum Entropy Production (MEP). They’re related but different.

MEP says systems evolve to maximize entropy generation—maximum dissipation. For Earth’s climate, that’s sort of true. But Bejan’s Constructal Law is more general. It says flow systems evolve to maximize flow access, not necessarily dissipation. For an engine that can deliver power externally (like your car), Constructal Law predicts minimum dissipation. For an engine coupled to its brake (like Earth), it predicts maximum dissipation.

Earth just happens to be the second case. The Constructal Law explains both. MEP only explains the Earth. The success of this model validates the more general Constructal principle.

Where We Stand

Let me summarize what we’ve got here:

• A computational climate model based on Constructal optimization of heat flow
• Validated against 24 years of CERES satellite data
• Matches absolute temperatures within 0.1°C average (RMSE 0.13-0.20 K)
• Matches hot zone area within 2° latitude
• Captures interannual temperature variability and trends of the two zones (correlations 0.56-0.74)
• Predicts heat flux variations with RMSE of 0.04 PW
• Uses minimal parameterization—four measured quantities plus three tuned parameters
• Predicts climate sensitivity of 1.1°C per doubling of CO₂ as an emergent property, not a tuned value

This ultra-simple smooth sphere model, with no ocean or land or topography, successfully reproduces observed temperatures, circulation patterns, and power flow with unprecedented accuracy. I don’t know of any comprehensive climate model that achieves this level of skill, regardless of its level of complexity.

And it shows that every once in a while … an über-simplified model of a spherical cow in a vacuum actually does work …

CODA:

The Constructal climate model has two input variables—greenhouse fraction and albedo. It’s worth noting that the albedo and the greenhouse fraction are both functions inter alia of thunderstorms and clouds.

The surface is constantly emitting thermal radiation upwards (“upwelling radiation”). Some of this upwelling radiation is absorbed by the clouds and the “greenhouse gases”. The greenhouse fraction is the percentage of the upwelling radiation that is absorbed in the atmosphere. Figure 12 below shows how that differs around the world.

Figure 12. Average greenhouse fraction, CERES data.

As you can see, the areas of largest absorption enclosed by the 45% greenhouse fraction contour line are the great band of thunderstorms that stretches around the world at the Inter-Tropical Convergence Zone (ITCZ) just north of the equator.

Next, here is a map of the albedo, overlaid with the 45% greenhouse fraction contour line.

Figure 13. Average albedo, CERES data

Note that the areas of high albedo line up with the areas of high greenhouse fraction.

And to verify that this is indeed related to thunderstorms, here is the rainfall from the Tropical Rainfall Measuring Mission satellite observation.

Figure 14. Average monthly rainfall as measured by the TRMM

And that’s all I have for now, dear friends … tomorrow is a new day, always more to learn.

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Here on our Northern California coastal hillside, we had rain yesterday—blessed rain after a dry spell. The weather systems that brought it organized themselves according to the same principles this model uses: maximize heat and moisture transport from where there’s an excess to where there’s a deficit. It’s beautiful to see the same physics working at scales from individual storms to the global circulation.

My warmest regards to all, my thanks to Adrian Bejan for illuminating conversations about Constructal theory, and to Umit Gunes and Pezhman Manipour for organizing the conference.

w.

As Is My Custom: When you comment, please quote the exact words you’re discussing, so we can all be clear on the exact topic under discussion.

References and Data

CERES data are from NASA Langley Research Center (https://ceres.larc.nasa.gov/). Key papers:

• Bejan & Reis (2005), “Thermodynamic optimization of global circulation and climate,” Int. J. Energy Res.
• Reis & Bejan (2006), “Constructal theory of global circulation and climate,” Int. J. Heat Mass Transfer
• Loeb et al. (2018), “Clouds and the Earth’s Radiant Energy System (CERES) EBAF data product,” J. Climate