{"id":418116,"date":"2025-12-22T08:58:00","date_gmt":"2025-12-22T07:58:00","guid":{"rendered":"https:\/\/climatescience.press\/?p=418116"},"modified":"2025-12-22T08:58:25","modified_gmt":"2025-12-22T07:58:25","slug":"a-spherical-cow-climate-model-that-actually-works","status":"publish","type":"post","link":"https:\/\/climatescience.press\/?p=418116","title":{"rendered":"A Spherical Cow Climate Model That Actually Works"},"content":{"rendered":"\n
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From Watts Up With That?<\/a><\/p>\n\n\n\n

Guest Post by<\/em><\/strong> Willis Eschenbach<\/a> (@weschenbach on X, personal blog at\u00a0Skating Under The Ice<\/a>)<\/em><\/strong><\/p>\n\n\n\n

Here\u2019s a science joke about the dangers of oversimplified models.<\/p>\n\n\n\n

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A dairy farmer with low milk production asks a physicist for help. After some months, the physicist eventually reports back: \u201cI have found out how to solve the problem.\u201d<\/em><\/p>\n\n\n\n

\u201cAbout time!\u201d, said the farmer, \u201cWhat is it?\u201d<\/em><\/p>\n\n\n\n

The physicist replies, \u201cFirst, assume a spherical cow in a vacuum.\u201d<\/em><\/p>\n<\/blockquote>\n\n\n\n

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In 2023 I wrote post entitled Testing a Constructal Climate Model<\/a>, where I took a first cut at making a computer implementation of Adrian Bejan\u2019s ideas about a Constructal model of the climate.<\/p>\n\n\n\n

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\u2019s the JA Jones Distinguished Professor of Mechanical Engineering at Duke University, and his writings have over 100,000 citations.<\/p>\n\n\n\n

The Constructal Law says that flow systems far from equilibrium evolve to maximize access to flow. Rivers don\u2019t meander randomly\u2014they organize to maximize water transport. Animal circulatory systems don\u2019t just happen\u2014they 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.<\/p>\n\n\n\n

Now, that sounded reasonable enough \u2026 but does it actually work? Can you build a working climate model based on that principle?<\/p>\n\n\n\n

Turns out you can. And as my post above showed, it\u2019s a very simple model.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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\u2019s one of them. I introduced myself to him, and he said Oh, you\u2019re Willis Eschenbach. I\u2019ve seen your work. It\u2019s very impressive!<\/p>\n\n\n\n

Zowie, sez I. A win for the reformed cowboy!<\/p>\n\n\n\n

There were many interesting presentations at the Constructal Conference, but that interaction alone was worth the ticket to Miami \u2026 however, I digress.<\/p>\n\n\n\n

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 \u2014 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\u2019s an expanded version of my presentation at the conference.<\/p>\n\n\n\n

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The model I\u2019m about to show you treats the Earth as a smooth sphere\u2014no 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\u2019s it. Two zones. And from those two zones, using the Constructal Law to maximize heat flow between them, the model reproduces the actual Earth\u2019s temperature, circulation patterns, and year-to-year variations with remarkable accuracy.<\/p>\n\n\n\n

Let me show you how it works, and then we\u2019ll look at what it tells us about climate sensitivity. Spoiler alert: the news is good if you\u2019re worried about catastrophic warming.<\/p>\n\n\n\n

The Basic Setup<\/strong><\/p>\n\n\n\n

Figure 1 shows the concept. We divide Earth into a hot zone,extending from the equator to some latitude \u03b8, and two cold zones, poleward of \u03b8. Heat flows from hot to cold, driven by the temperature difference.<\/p>\n\n\n

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Figure 1. The spherical cow \u2026 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.<\/em><\/figcaption><\/figure>\n<\/div>\n\n\n

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

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Figure 2a. The actual Earth\u2019s 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\u2014Sahara, Arabia, Australia, Gobi\u2014are actually in the cold zone despite being hot. They radiate more energy than they absorb.<\/em><\/figcaption><\/figure>\n<\/div>\n\n\n

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.<\/p>\n\n\n

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Figure 2b. Annual average hot and cold zones, 2001 to 2024<\/em><\/figcaption><\/figure>\n<\/div>\n\n\n

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.<\/p>\n\n\n\n

The Energy Balance<\/strong><\/p>\n\n\n\n

Being a simple fellow, I started with Bejan\u2019s 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, \u201calbedo_H\u201d):<\/p>\n\n\n\n

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Energy in = (projected area) \u00d7 (1 \u2212 albedo_H) \u00d7 (solar constant) [Equation 1]<\/p>\n<\/blockquote>\n\n\n\n

The hot zone radiates energy to space, reduced by the greenhouse fraction \u201cgreenhouse_H\u201d. This is the percentage of upwelling thermal radiation from the surface that is absorbed by the clouds, water vapor, and the radiatively active gases.<\/p>\n\n\n\n

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Energy out = (area) \u00d7 (1 \u2212 greenhouse_H) \u00d7 \u03c3 \u00d7 T_H^4  [Equation 2]<\/p>\n<\/blockquote>\n\n\n\n

where \u03c3 is the Stefan-Boltzmann constant and greenhouse_H is the fraction of surface radiation absorbed by the atmosphere.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

The difference between incoming and outgoing power in each zone must equal the heat flow q between the zones. Energy is conserved\u2014it has to go somewhere.<\/p>\n\n\n\n

So far, nothing fancy. Just bookkeeping.<\/p>\n\n\n\n

The Heat Transport<\/strong><\/p>\n\n\n\n

Here\u2019s 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:<\/p>\n\n\n\n

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q = C \u00d7 (T_H \u2212 T_L)(3\/2)<\/sup>  [Equation 3]<\/p>\n<\/blockquote>\n\n\n\n

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.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

Now we have five unknowns (T_H, T_L, q, the fraction x of Earth\u2019s surface in the hot zone, and the conductance factor C), and three equations. How do we close the system?<\/p>\n\n\n\n

Enter the Constructal Law<\/strong><\/p>\n\n\n\n

Here\u2019s where Bejan\u2019s 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\u2019s climate engine is permanently coupled to its \u201cbrake\u201d\u2014all the power it generates gets dissipated immediately through friction in winds and ocean currents.<\/p>\n\n\n\n

For such a system\u2014an engine hard-wired to its brake\u2014maximum power production equals maximum dissipation, which means maximum heat flow. The atmosphere and ocean organize themselves to achieve this.<\/p>\n\n\n\n

So the fourth equation is simply:<\/p>\n\n\n\n

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dq\/dx = 0  [Equation 4]<\/p>\n<\/blockquote>\n\n\n\n

Making It Work<\/strong><\/p>\n\n\n\n

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\u2019s optimization within optimization.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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Figure 3a. Results of my\u00a0previous model<\/a>\u00a0incarnation, which shows an anomalous trend in the modeled temperatures<\/em><\/figcaption><\/figure>\n\n\n\n

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.<\/p>\n\n\n\n

So how many parameters does this model have? Let\u2019s count:<\/p>\n\n\n\n