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, “Thermodynamic optimization of global circulation and climate“, ” Constructal theory of global circulation and climate“, and “Climate change, in the framework of the constructal law“, 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:
“In 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.”
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 “Area High” and “Area Low” 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 “q”, 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 “q”, is transported to the cold zone. There, the flow “q” 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 “q”.
Next, here’s the math of the model, from the third of the papers linked above. Recall from Figure 1 above that “x” is the area fraction, the fraction of the globe occupied by the hot zone.
Daunting … so let me translate for those who like math. For those who don’t, no worries—just skip down to where it says “THEIR MODEL RESULTS“.
And for the three folks still reading this section, ignore equation (26) for now. Next, in the above set of equations, rho (ρ) is the albedo, and gamma (γ) is the “greenhouse factor”, 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 zero—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 “q”, is transported to the cold zone. So “q” is equal to hot zone solar heat input minus hot zone radiation to space. In short, it’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 “q” from the hot zone, and radiates it all to space. So “q” 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 “q” is equal to some constant “C” times the 3/2 power of the difference in temperature between the hot and cold zones.
The final equation (27) specifies that “q” is maximized.
There are four unknowns in the equations—temperatures of the hot and cold zones “TH” and “TL“, the heat flow “q”, and the area fraction “x”. 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 didn’t include the solution. Grrrr.
However, I’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 “q”, 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 “x” 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 “x” puts the limits of the hot zone at about 57°N/S. But that’s not the case in the real world. Here’s the real-world data regarding the heat flow “q”.
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 “x” is about 0.55 of the total surface. This corresponds with a hot zone extending to about 34°N/S. So that was the first problem—the hot zone extends to about 34°N/S, not 57°N/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 -14.75°C. But in the real world, the cold zone poleward of 57°N/S actually has an average temperature of about – 3°C, far from the minus 14°C 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 “C” in Equation (25). This is the “conductance”, 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 “C” was far too small, which meant it required a much greater temperature difference to get the same flow, resulting in a hot zone that’s too hot and a cold zone that’s too cold.
Once the factor “C” 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 “x” (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 “q”, 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 “x” 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 (ρ), which is the albedo, and gamma (γ) which is the “greenhouse factor”.
The model says that if we know the albedo ρ, the greenhouse factor γ, 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 “q” 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. I’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 world’s hot and cold zones
I found this result to be most encouraging. Those model temperatures are calculated based solely on maximizing the heat “q” 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 … 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, there’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 world’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 it’s doing all that using nothing more than the hot zone area “x”, the albedo “rho”, and the greenhouse factor “gamma” to calculate the temperatures that maximize “q”.
That’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. That’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 “q”. Bear in mind that the hot zone fraction could vary from ~0.1 to ~0.9. And there’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 we’d expect the hot zone fraction “x” to be pretty stable as well. In any case, here’s the actual record of the CERES data for “x”, the hot area fraction, along with the constructal model output of the same variable.
Figure 5. The “x” fraction, the amount of the earth’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, it’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 “q” from the hot zone to the cold zone.
TEMPERATURES
For temperatures, I’ve used the CERES surface upwelling longwave data converted using the Stefan-Boltzmann constant and monthly gridded emissivity values. I’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 I’d explain why I think their value for “C”, the “conductance”, 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, they’ve modeled the heat transport via the atmosphere. And they’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 circulation—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 authors’ analysis assumes. Here’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 5°-10° 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 doesn’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 “q” 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 ~1.1°C. (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 earth’s hot zone area and the albedo and greenhouse fractions in the earth’s hot and cold zones, is able to calculate the absolute temperatures of the earth’s hot and cold zones to within a degree or so … a result that I found quite surprising.
Anyhow, that’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 don’t know what will happen yesterday …
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)$par)
par[3]
}
Next, the outer optimization program that calls the inner program.
(bestx=optim(par2,maxq,
control=list(fnscale = -1,reltol=1e-10),
method=”Brent”,
lower=.001,upper=.999)$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])))
Interesting.
But I’m not convinced that the Constructal Law applies to rivers meandering and the formation of ox-bow lakes. Maybe they should have consulted geographers.
Water will always find, due to its liquidity and to gravity, somewhere lower to be. That’s why, in their lower reaches where the land tends to be flatter, they meander, following the lowest level they can find. The erosion and deposition of silt on the outsides and insides (respectively) of bends which leads ultimately to stretches becoming isolated is pure mechanics, nothing to do with mathematical laws, I would suggest.
That was my immediate reaction too . Any meander in a river in fairly flat erodible surface will work to minimize its gradient by maximizing its path .
I Hate maths Willis. But I love reading you, so I have taken penil and paper, and have decided to give your maths a go. I may even end up liking maths.
Thank You.
Gracias, bonita.
w.
Now have another look at the work of myself and Philip Mulholland.
We propose a heat engine built from convective overturning which produces the correct numbers, not only for the Earth but also for Titan and Venus.
I suggest that we have already gone beyond the content of the head post and found the correct solution.
https://pdfs.semanticscholar.org/0387/f80f76699b976ef6ee2f0481b779bad2f2b0.pdf
It is likely that the same principles apply to every situation where there is convective overturning from the gases of stars to the mobile solid of the Earth’s mantle and possibly to galaxies and the universe itself.
The problem with your analysis is that you have the radiation window wrong. It is 66 w/m2 not 40 as proposed by Trenberth who has admitted he was wrong.See here https://cementafriend.wordpress.com/2015/09/25/radiation-window/
I see a claim in your post that it is satellite-measured at 66 W/m2, and a claim that Trenberth knows that and refuses to use it.
What I don’t see is any link to the slightest scrap of evidence to back up any of those claims. They may be right, but you haven’t shown that.
w.
Willis, I was not commenting on your article but on the article linked by Stephen Wilde where there is the number 40 w/m2.
If you looked at the note on my blog you will find it came from a peer reviewed article by (the late) Dr(Ing) Noor Van Andel (who was alive at the time I wrote my post). Van Andel was a highly respected Chemical Engineer in Holland and gave several talks to KMI. He invented an heat exchanger and worked with CO2 climate enhancing in (actual) greenhouses for plant growth. As a Chemical Engineer (also myself) he understood dimensional analysis and the use in heat transfer equations (eg the Prandtl and Nusselt numbers). I corresponded with Van Andel when he was alive. I have no doubt about his knowledge and his correspondence with Trenberth.
As an aside to yourself and to Philip below I find nothing wrong with the analysis by Dr Niklov about dimensional analysis determination of surface temperature on rocky planets with an atmosphere. Engineers have been using dimensional analysis since about 1900 but it seems few calling themselves scientists understand the mathematics.
Thanks, CAF. If you’ve fallen for Nikolov’s rocky planet scam I fear there’s little I can do to help you. It certainly doesn’t do your reputation any good …
Before you do anything else foolish, you might want to read my posts “The Mystery Of Equation 8” and “A Matter Of Some Gravity“, as well as the Washington Post article entitled “Scientists published climate research under fake names. Then they were caught“, and Dr. Roy Spencer’s article “Giving Credit to Willis Eschenbach for setting the Nikolov-Zeller silliness straight“.
Regards,
w.
Haha Willis, unfortunately Dr. Spencer doesn’t understand the difference between “energy” and “power” any better than you do. Claiming him as an authority isn’t going to do your reputation any good, what little there is of it 🙂
If by “Nikolov-Zeller silliness” you are referring to understanding the difference between radiation, energy, heat, entropy, and power, then you do not have anywhere near the foundational physics background necessary to chime in on this topic. Back to Physics 101 for you, Mr. Fisherman-Cowboy! (And for Dr. Spencer too, it seems, more shame to him, since you would think as a climate “science” Dr. he should at least know better)
For a more concrete question, let’s try this: you wrote “(3.7 W/m2 is the amount of radiation increase predicted to occur from a doubling of CO2.)” Where can I measure for myself that 3.7 W/m^2 emitted by CO2, Willis? Whether doubled or not? And what is the target temperature for the measurement?
Willis, The fifth postulate of thermodynamics is “The macroscopic properties of homogeneous PVT systems can be expressed as a function of temperature, pressure and composition only” I have been below 1000m underground where it is warmer than the surface due to the atmospheric pressure. I have been at a height around 4400m in Tibet where it is colder than the sea level surface due to the reduced atmospheric pressure. There are a number of publications on the subject of surface temperature and atmospheric pressure. In a quick look at my files I found beside Niklov & Zeller, Volokin &ReLlez (SpringerPlus 2014), Robert Ian Holmes (Earth Sciences 2018), and I know there is a lengthy article by a Dr Robertson/ Robinson (which I recall was published before the N&Z paper).
I have read your articles and also that from Dr Spencer. I respect Roy S and have had an email from him. However, he is not an engineer and appears not to understand thermodynamics.
Willis, Sorry to take up space but can not find another way to get back to you. In a quick search on the net I found two articles by Dr T D Robinson of NASA 1/ T D Robinson & D C Cattling “An analytic Radiative -Convective Model for Planetary Atmospheres” Astrophysical Journal 2012 2/ T D Robinson & D C Catling “Common 0.1 bar tropopause in the atmospheres set by pressure -dependent infrared transparency ” Nature Geoscience Letter Dec 2013. You really should read the latter from NASA scientists. N & Z just expand on this but do not have the article in their list of 132 references. N& Z paper is “New Insights on Physical Nature of Atmospheric Greenhouse effect deduced from an Empirical Model” Emvion Pollut Climate Change 2017. You may have missed the section “Methods and data -Dimensional analysis (DA) background” Their article is about DA and has many references about that. The answers they get are little different to Robinson & Catling of NASA. (Do you think D & C of NASA are wrong in comparing their model to actual NASA data?)
Happy to correspond with you by email. Wish you well.
CAF, I have no idea what you think the D&C article has to do with this discussion. As to N&Z, I’ve already pointed out that their analysis you list above is a joke. If you can find errors in my analysis, QUOTE MY EXACT WORDS and show why they’re wrong.
w.
CAF, you are hilarious. You say:
You are so woefully uninformed on the subject that you don’t even know that “Volokin & ReLlez” is just nikolov and Zeller spelled backwards, and the journal retracted their article because they lied about their identity …
Next, you say “I have read your articles and also that from Dr. Spencer”. If so, you either failed to understand them or couldn’t find any errors in them.
I say this because rather than point out even one single error in either my articles or that of Dr. Spencer, instead you want to rant about his qualifications …
… and when you start to throw mud like that, it’s a clear and undeniable sign that you have no scientific ammunition.
w.
Willis wrote: “it’s a clear and undeniable sign that you have no scientific ammunition.”
That’s pretty funny, Willis the fisherman. What is your “scientific ammunition”? The last time you took a single physics course was 50 years ago, and it’s not clear that you passed it, either. It is clear that you have forgotten everything they taught you, though. Every statement you have made on this web site (or anywhere else) about thermodynamics that I have seen has been false.
What is a Watt, Willis? How about a Joule? Or a photon?
In response to me pointing out that personal attacks show you’re out of scientific ammunition, stevekj replies with … wait for it … another personal attack.
Steve, if you wish to show I’m wrong, you need to a) quote what I said, and b) demonstrate, not claim but show, that I’m wrong.
Attacking my education just marks you as a gadfly.
Here’s some advice on how you might finally get some traction here. Read it and get back to us. You’re down at the bottom of the pyramid. Move up.
w.
Great job, Willis. Now keep this pyramid in mind for the following two questions:
You wrote (in this article): ““(3.7 W/m2 is the amount of radiation increase predicted to occur from a doubling of CO2.)” Where can I measure those 3.7 W/m^2, Willis?
And in a previous recent article: “Radiant energy (measured in W/m^2)” But is energy normally measured in Watts? Do you know what a Watt is?
Thanks, Steve. To answer your questions. First:
The poorly named “greenhouse effect” was first measured by Ramanathan. He proposed the measurement method used today. This is to take the surface upwelling longwave, and subtract from that the TOA upwelling longwave. The remainder is what is absorbed by the atmosphere and radiated downwards.
When the amount of GHGs of any kind increases, it increases downwelling LW from the atmosphere.
Your second/third questions:
Let me take the second question first. A watt is a joule/second.
First question. We’re talking, not about say potential energy, but about radiant energy flux. The “radiant” part means it’s flowing. It is radiant energy flux, aka “radiant power”, and yes, it’s measured in watts, or watts/m2.
A search of Google Scholar finds about three thousand instances of studies referencing “radiant energy flux”. I’ll leave you to tell them all that they’re idiots.
Radiant energy flux is the amount of radiant energy emitted, reflected, transmitted, or received per unit time. It’s measured in units of Watts (W).
Best regards,
w.
Thanks for sticking to the physics, Willis. Much appreciated. That’s a significant improvement from the last time I asked you this question, when you said “I refuse to engage in pig wrestling.”
You didn’t actually answer my first question (which was “where can I measure these Watts”), but I can guess from what you wrote that if you had explicitly answered it, your answer would have been “at the surface of the Earth”. Feel free to provide a different answer if you don’t like the one I am implying for you.
The flaw in that claim, of course, is that no one has ever actually measured downwelling Watts at the surface of the Earth (from the atmosphere, i.e. at night, and of course at ambient temperature). Does that change your answer?
(Your definition of a Watt is correct, so that’s something at least. We’ll get into Joules soon.)
For my second question, you sound like you are trying to say that “radiant energy” == “radiant flux” == “radiant power”, and yes, that latter concept is indeed measured in Watts. However, if we remove the word “radiant” from all three phrases, would you then conclude that, in general, “energy” == “power”? Or do you only apply this false equivalence when dealing with radiation in particular? Simply because photons are moving?
@cementafriend
The purpose of our analyses is to stress test the current radiative paradigm. To do this we must start with the standard numbers and show by sensitivity testing what the limits of the model are and at what point the logic fails.
By doing this we have shown that the runaway greenhouse whereby the Earth turns into Venus is impossible.
That Back Radiation is the process of adiabatic convection in disguise.
That atmospheric thermal radiant opacity charges the atmospheric reservoir with energy and thereby powers the process of adiabatic convection.
That Earth’s climate is controlled by variations in the Bond albedo and by variation in the Atmospheric Window.
That the role of thermal radiant opacity in climate is completely overstated.
Dazzle with brilliance or baffle them with bullshit. Willis always gets the benefit of the doubt.
You mention Hadley cells…better would be to parameterize Coriolis transport of weather fronts that easily cross the imaginary 30 Degree hadley zone…
EPIC DISCVR a couple of hours ago….
BTW, you have done wonderful extension of Bejan’s too-simple model.
“Everything must be made as simple as possible, but no simpler.” attributed to you-know-who…
Willis,
Also your ocean currents comment is right on…surface currents are a massive transport of heat from equatorial to polar regions, where the water cooled by radiation to outer space finally sinks and the planet spins it back to the equator to resurface, on average, a few hundred years down the road….
Climate models don’t work because the climate system is too complex and includes too many things that are unknown to be properly modeled.
Even if the result is similar to the real world it does not mean that the model is correct. It may achieve that result differently from what happens in the real world. After all, the result is what is being sought by any means available to the model.
Heat transport experiments changes that go against the gradient difference that is supposed to drive it. That’s why the maximum entropy production hypothesis of heat transport, or your “constructal law” model are wrong.
Here is the data on ocean heat transport from the North Atlantic to the Arctic, taken from:
Tsubouchi, T., et al., 2021. Nat. Clim. Change, 11 (1), pp.21–26. doi.org/10.1038/s41558-020-00941-3.
It shows how heat transport to the Arctic by the ocean increased by over 10%, about 30 TW, between 1998 and 2002 when Arctic temperatures were increasing and triggering an accelerated sea ice melting. Obviously it happened as a consequence of the increase in heat transport.
Can your model do that? Can it decrease the temperature difference between the Equator and the pole and at the same time increase heat transport? I seriously doubt it. Nature can.
Never trust climate models, but particularly never trust your own climate models. They have an easier time fooling you.
“All models are wrong; some are useful.” So the real question is: Is this model useful, or at least more useful than the others?
This model stuff is way over my head but what you just said makes sense!
Atmospheric heat transport exceeds that of oceanic heat transport by about 5x. While oceanic transport is governed by meridional advection of sensible heat, atmospheric transport is governed by latent flux.
I agree, but atmospheric heat transport to the Arctic also increased simultaneously with the increase in oceanic transport. The data is in my forthcoming book.
Most oceanic heat transport is driven by the wind. The decrease in the AMOC that so much worries Stephan Rahmstorf has been overcompensated by an increase in heat transport by the gyres, which is coupled to atmospheric transport. The result is more transport, not less, and Rahmstorf can sleep at ease.
Thank you for the response. There is an interesting phenomenon of heat transport compensation between ocean and atmosphere. However, the mechanism seems only to apply to conservation of sensible heat advection between atmospheric and oceanic reservoirs. https://www.nature.com/articles/srep16661
Latent flux, the dominant heat transport reservoir, acts independently.
Can it decrease the temperature difference between the Equator and the pole and at the same time increase heat transport?
yes if you increase the mass trasnport, you can convey more heat at a lower temperature difference, the model does not take in accout mass transport so it can happen.rgds
Is that pole to equator diff really a big change though. Widen the scale and it would appear flat. Check this out.
Arctic is heating faster than the tropics. (misnamed polar amplification).
That graph is very interesting. I noted about 10y ago that most of the “runaway melting” of the Arctic happened between 1997 and 2007. This week I have been looking arctic ice extent and the sept minimum was exactly that same this year as it was in 2007 when Al Bore and the whole clown show kicked off about imminent ice free summers.
Both those trends seem to match the heat flow graph, so I’ll look into that paper , thanks.
The map, where thunderstorms live, ticked a box.
I have always lived between the 33rd and 48th Parallels. {Sad, right?}
Thus, I have never seen a thunderstorm with cloud tops over about 35,000 feet. The map shows areas with cloud tops near 59,000 feet. So, while I have seen impressive storms, I have never seen ITCZ storms.
My thought is that you, W. E., are more inclined to think of the turbocharged Hadley circulation as a driver of south-to-north heat than us poor schmucks that have not seen this.
But what is the point?
Maths and science are two very different things.A lot of scientists seem confused on this.
Validation.
(1) A model without validation, or with only crude validation is no good. I will define “crude validation” as pretend validation. For example: when you have a model which models something in detail – such as the climate – it CANNOT be validated by taking a totalized average and comparing that against another totalized average – such a global average temperature. That is not validation – that is a trick played by modellers on us.
(2) A good validation would attempt to validate the model’s detailed projections against the real world instead. For example – predicted Pacific Southern Ocean Oscillation (La Nina / El Nino) against what is actually experienced. When a model can be validated in this way I will accept that as legitimate validation.
If this post (mine) seems irrelevant to what Willis’ post above then I’ll ask him: (1) what does he intend to show us about climate with this elaborate maths? (2) How does he intend to convince model skeptics – such as myself – that his model means something real? AKA – that it can be validated in some useful way?
Jack Eddyfier October 10, 2023 11:01 am
1) I intended to show that the constructal view of how flow systems evolve is perfectly applicable to climate, and that the climate can be well-emulated with a constructal model that maximizes heat flow. I believe I’ve done that.
2) I’ll never be able to convince skeptics such as yourself of one single fact. Won’t happen. Far too many “skeptics” seem allergic to reality. So I don’t even try. Believe what want to believe, but don’t pretend that I care about what that is or that I’m trying to convince skeptics of something. My goal isn’t to convince people. It’s to give as clear and clean an explanation of what I’m discussing as I can muster.
Don’t like it?
Don’t care …
w.
JE, I want to challenge some of your statements. My summa degree was technically econometrics (economic modeling) but in reality was all about modeling. Any kind, any way. Calculus, linear algebra (Leontief I/O, Markov chains…
There are two ways to meet George Box’ aphorism that ‘all models are wrong, but some are useful.’ Finding the useful ones:
1) I don’t have a problem with individuals such as yourself finding models useful. That’s OK by me, But some modellers clearly think their models are more real than data; and that they need to shout to the world about how their models predict the end of the world. Can you see my point here? To put it bluntly, there’s a bad attitude among certain climate types who say models cannot be questioned because they are just “simple physics”. They literally say that I’m denying simple physics when I question their model assumptions and methods of “validating” their models. That’s not the case. When the model assumptions are wrong – 1.1) “simple physics” is being misused. 1.2) the models are hiding assumptions. A good example of this misuse is Tom Shula’s expose of the greenhouse gas effect fraud.
The function served here it to promote scientism – to corrupt science. So if you have models – by all means use them among yourselves – but stop pretending they predict anything at all other than your own bias and pseudoscience tendencies.
2) This point is irrelevant. Climate models must be complex because the climate is complex.I was criticizing complex climate models, and obscustration of assumptions.
3) When you say “model validation of complex systems is not possible” – there’s an issue there right away. a) you split models into only 2 types: simple and complex. b) some people pretend that the simple ones don’t need to be validated or are validated,
There’s an appalling culture among modellers. They seem to thing we should all care about their models. No, we should tell them to keep those models to themselves until they can show us why we should care.
BTW: Some models I like. If anyone’s modelling solar cycles – I can listen all day to you. I’m all for useful models.
“obscustration” !!
– LOL. I meant obfuscation; but I think I was inventing my own private variation on obscuration.
I think that most of us understood and don’t get or knickers in a knot over a typo. You could have just blamed the lack of a working edit function.
NB : I am overly fond of quotations.
While it is unlikely than anyone who can ask that question will “get where I’m coming from”, I’ll try to answer it anyway.
At the most basic level I believe that Willis “suffers” from a similar affliction that I do, though he has (much !) higher levels of both mathematical and programming ability than I do to actually scratch this specific “itch”.
“The cure for boredom is curiosity. There is no cure for curiosity.” — Dorothy Parker
When it comes to science (and engineering) the following may (?) also help explain “motivations”.
“Science is fun. Science is curiosity. We all have natural curiosity. Science is a process of investigating. It’s posing questions and coming up with a method. It’s delving in.” — Sally Ride
“Research is what I’m doing when I don’t know what I’m doing.” — Wernher von Braun
No doubt this is right up your alley Willis. Thanks interesting. The math is to much for me ; but the concept I get. But what does it all mean? In terms of the “big question” of course.
If we assume a more “ settle science” perspective on green house gas sensitivity ( specifically the magic molecule not water) And we assume the constuctural model concept valid for climate; does this mean the planet will exchange out enough heat to space long run to mitigate any ohhh scary run run runaway greenhouse age of boiling effects?
It’s easy enough to determine the climate sensitivity. We note that the 11 year solar luminosity variation is equal to about 0.3 W per square meter. Then we take a look at the cyclic behavior of the energy imbalance; this will show us how much of an affect 0.3 W. per square meter has. The answer appears to be so close to zero as to be indistinguishable from random noise. Sort of interesting, don’t you think?
Thanks, Brock. The problem is indeed the noise. The maximum monthly change in total solar irradiance (TSI), not decadal but monthly, is 0.25 W/m2. The maximum annual change is 0.34 W/m2
Looking for the effect of a gradual 11-year 0.3W/m2 change resulting from the sunspot cycle is gonna be very hard … and that doesn’t even include the effect on the solar signal from varying cloud albedo, surface albedo, and atmospheric absorption.
So, sadly, it’s not “easy enough to determine the climate sensitivity”. The fact that we can’t detect a small signal in a very noisy environment means little.
w.
There are also various different total solar irradiance datasets.
Hi Willis, nice replication study. I like this chart..sorta reflects status of the heat engine in my mind. Likely fairly slow moving macro view too ..as in climate change not weather.
While it’s true that the EEI data is noisy, I believe running an FFT on the data drives the noise low enough so that you should be able to see even a small signal such as the luminosity variation; without the FFT, you can’t, as you pointed out. What is interesting is there is no periodic behavior in the EEI, even though there is a periodic behavior in downwelling radiation. And, if the IPCC is correct (lol), the luminosity variation will be amplified by a factor of roughly 3. If the luminosity signal were there, we would be able to pull it out of the data. I couldn’t find it when I tried with a 32-point FFT. Hmm, says I.
You neglect the issue of thermal mass. The Earth has a lot of it.
I wouldn’t expect a signal as short as 11 years to have much impact.
The earth changes temperature by as much as a quarter of a degree per month. If a 1-month signal has such an effect, there’s nothing stopping an 11-year signal from making a similar effect.
w.
WE, outstanding. Finally, a conceptually simple model that ‘works’.
As for sensitivity, your 1.1C is most interesting. The no feedbacks CO2 sensitivity is variously given as 1.1C (old Judith Curry post calculation from 2011), or 1.2C (Lindzen Bode plot 2011). I used Monckton’s later ‘irreducible’ equation paper and his input constants to derive exactly 1.16C. All very close to your new model result.
This is likely low in total because there is a much longer time frame slow acting water vapor positive feedback arguably missing from the simple heat transport model. It won’t be picked up much by Ceres because of the ‘short’ Ceres observational timeframe. Callendar’s 1935 curve and the various EBM observational ECS estimates are all about 1.65-1.7C. Callendar is exactly 1.68C. And the 70 year EBM TCR is about 1.35C, again a bit above the ‘no feedbacks’ heat engine model 1.1C derived just over Ceres 22 years.
Very nice. Tune one parameter, add some observations and model several different other observations with surprising accuracy. That’s proper modelling, that is.
Couldn’t that also mean the flow is minimised or paused?
Not entirely sure what that means physically but am thinking it would lead to increased polar ice, increased albedo and flipping into an ice age mode.
The climate does have two modes.
Thanks, M. First, in theory you’re correct. The equation just specifies a peak or a valley in a continuous function.
But in what we’re measuring, “q”, it’s an upside-down “u” shaped plot with respect to “x”, the area fraction. So that equation will always find the maximum.
Second, this is just a mathematical model of the real system. Per the Constructal Law, real flow systems always evolve to maximize flow, not to minimize it.
Best to you,
w.
Thank you for replying.
If the Constructual Law never runs in reverse, it must be driven by entropy. But that still seems to be unproven. Energy flows can (locally) overwhelm the entropy effects.
Maybe I’m too used to chemistry and psephology but in my experience, if something is permitted it will happen sometimes.
“If” is doing a lot of work there, I know, but minimisation of flow does seem to be permitted.
Instead of seeking the ‘most volume of river reaching the sea’ it could pool up and make an inland sea.
MCourtney
Slow or pause, no if there is no wall between zones.
In Bejan’s textbook Convection Heat Transfer, there is a section that deals with something I feel is akin to a flat earth atmosphere. Consider two plates a few feet square spaced 2 inches apart. The space is filled with liquid. Heat is applied to the bottom and the heat creates a convection towards the top cooler plate.
At a certain temperature the fluid breaks into a grid of square “thunderstorms” with a central rising column. This happens as a result of the properties of everything involved. Seeing this happen is strange because a grid of square “storms” looks weird.
Now, increase the temperature of the bottom plate and as the heat flux increases, the pattern changes into perfectly nested hexagons. These two patterns are the most effective heat transfer patterns, depending on the heat flow rate involved. There is no reason to expect that the atmosphere would behave is a less efficient manner.
Willis: is it possible to find a spacial distribution of thunderstorms – more than one – and relating the patterns to the surface temperature? It should work over the ocean. The patterns should vary predictably over the seasons.
Crispin, the second kind of circulation you describe is called “Rayleigh-Benard” circulation. And you can see it in the fields of tropical cumulus, where each cumulus cloud marks the top of one of the ascending columns of air.
Thermally-driven thunderstorms, on the other hand, are curious birds. They need to move in order to survive. They move preferentially to warmer spots in the local area. As a result, there’s no obvious pattern.
If the temperatures continue to climb, a pattern does emerge. The thunderstorms join together into “squall lines”.
w.
Willis: The Model matches the observed temperature well. How well does the Model match the observed specific humidity, relative humidity and cloud cover change?
The model makes no statements about any of those. It’s a model of energy flow equator-poles, not a model of e.g. cloud cover.
w.
Willis:
On a very much lower level, how did the authors come to use the term “constructal”? I could not find a formal definition, and a web search using the quotes came up empty.
Not sure why the name, karlo … hang on …
OK, here’s what ChatGPT says. I asked:
ChatGPT sez:
Works for me …
w.
Danke, I was wondering if it was a term Bejan coined.
Not Willis, but have good google Fu. google ‘Bejan constructal law’ and you will get many good basic explanations. Fundamentally, the structural evolution of transport systems (blood vessels, climate heat engines) over time to optimize transport.That is, how their physical construction predictably evolves toward an optimum. Hence constructal law. (Predictable Constructural evolution == constructal law)
Thanks, Rud.
I agree, the word makes my brain squirm. There was already a perfectly good English word “constructional”
I just had one of comments just disappear. Nothing mean or nasty in it what so ever ,very much on topic. All I asked was what the implications of the model ( and concept) were for the “ big question” of atmospheric sensitivity in terms of exchanging heat out of the atmosphere . In other words if we took a mainstream view on climate would this have the potential to show that the “ heat engine will exchange out heat regardless of the magic molecule in the age of boiling.
Maybe sarc tags are necessary?
Once in many comments, I forget to hit the “Reply” box and then blame the missing comment on WordPress or what is being used. So, maybe rethink the disappear possibility. Of course, I may be the only person that has brain farts! 🙂
On my machine under your comment it says “Awaiting approval”. The site is moderated, the moderators are all volunteers, so it sometimes takes time.
w.
Ok thank you, I figured it was something innocent . Glad there are some real humans involved
It’s refreshing to see some independent scientific thinking and work. All the more so when all we get is nonsensical alarmism
Today’s scare du jour: beer
“”Climate crisis will make Europe’s beer cost more and taste worse, say scientists”” – Grauniad
A good beer crisis might get people’s attention.
Most beer doesn’t have much taste, so this might be an improvement.
A lot of beer is served far too cold which kills the taste.
Some beer has to be served cold in order to kill the awful taste.
So my take is that you are saying that this model pretty much confirms what you have already pointed out. That is that the earth does a lot to regulate the temperature that we experience. That would go a long way to discrediting the notion that CO2 acts as THE control knob for earth’s temperature.
“the earth does a lot to regulate the temperature”
Pretty amazing when you think about it. A big rock spinning around a nuclear reactor- and its surface conditions stay remarkably unchanged for millions of years. Seems almost impossible. Short term warming or cooling aren’t really big changes when you think about it.
The game of energy balance climatology begins and ends with the albedo and greenhouse factor.
In this view, the net forcing and associated temperature change is roughly baked-in when such parameters are prescribed.
However, one can use the analysis to argue for a maximum entropy production hypothesis by presenting if the q is computed to be increasing over time, in spite of the paradoxical decrease of temperature difference between tropic and pole.
One peculiarity of the Earth system is that maximum atmospheric emission to space tends to occur around 60 degrees latitude, not at the pole.
It is around 60 degrees that it perfectly warm enough and yet atmosphere is sufficiently thin enough where maximum emission to space occurs.
At the pole it is optically thin but very cold, and emission is reduced. In the tropical zone, it is hot but the atmospheric is optically thick, and emission reduced.
Optimality occurs at 60 N/S. This is the tipping point where proceeding increasingly poleward results in less emission. This is quite close to the original authors proposition of 57 degrees N/S.
thanks
I should add that the mechanistic interpretation of the apparent paradox is that sustained atmospheric heat transport is governed by gradients in latent flux, not of temperature.
JCM, you say:
First, could you define “maximum atmospheric emission to space”?
Second, do you have a link to a scientific source for your claim? I ask because here’s what CERES says. First, a global view.
Next, a zonal (latitudinal) averages.
Not seeing the peak at 57N/S.
w.
thank you for the response.
yes, the distribution of atmospheric emission is distinct from total radiative emission at TOA. It is the atmospheric emission which is related to the atmospheric flows q, and is an expression of the freedom in heat dissipation.
In the graphic the Ra is atmospheric radiative flux “into atmosphere” More negative values are suggesting of more emission “out” of atmosphere. Anomalous “peaks” are visible around 60 (depicted as troughs in negative space).
I may have other more clear depictions available given time. I hope this helps.
The unannotated figure is from Xhang and Rossow Fig 3 https://journals.ametsoc.org/view/journals/clim/10/9/1520-0442_1997_010_2358_emetbt_2.0.co_2.xml
Similar finding in Fajber 2020
https://dam-oclc.bac-lac.gc.ca/download?is_thesis=1&oclc_number=1334506469&id=bcac6748-c6a5-4a0e-b94f-b5edfc147eb4&fileName=Fajber_Robert_202011_PhD_thesis.pdf
More obvious is Miskolczi and Mlynczak 2004 Fig. 11 for clear sky.
Glad you brought up Miskolczi. I can’t access his most recent work, but vaguely recall that he referenced the ‘constructal law’ therein. However, could be an incorrect recollection on my part.
A useful way to approach energy balance climatology is consider it primarily through 2nd law. From this perspective, energy must be conserved while entropy must increase. Alternatively, energy must be more spread out. It opposes second law to have energy becoming more compacted, on average, in physical space and time.
The rate of entropy production is suspected to the “maximized” within the physical constraints of the system, such that heat is moved around in such a way as to “minimize” temperature. The system achieves this primarily through atmospheric heat transport Q.
Temperature and entropy production are strictly related. The change in entropy (delta S) is equal to the heat transfer (delta Q) divided by the temperature (T). If entropy is to be maximized, temperature is to be minimized, (a greater numerator Q and/or a smaller denominator T). The system has the greatest freedom to meddle with Q, rendering T less sensitive.
The atmosphere has the most freedom to minimize temperature, or to maximize entropy S, by converting energy to-and-from latent heat reservoirs Q, and so this mechanism is exploited. Latent heat reservoirs provide a highly effective means to transport heat vertically and poleward.
The second most important feature is the existence of an atmospheric radiative “window”, such that energy is delivered optimally to where the windows are widest, such that radiative cooling is maximized (temperature is minimized). The width of the windows and the physical constraints on dynamic heat transport dominate the rate of radiative cooling. Windows are wider poleward and at altitude. Without the so-called radiative windows, the heat dynamics of Earth would be dramatically different.
Energy balance is primarily controlled by the dynamic motions of atmosphere which act to minimize temperature differences while maintaining sufficient instability (gradients) to sustain the flows. Too much spreading out of heat and the rate of radiative cooling is not maximized. Likewise, too little spreading out of heat and the same. The “maximum” hypothesis suggests there is physically enforced optimal rate. Picture an inverted U of entropy budget of the y axis and heat transport on the x axis. The maximum rate of sustained entropy production is at the climax.
You seem to be looking at heat fluxes “into the atmosphere” not out of the atmosphere to space. This seems to be looking at surface cooling not planetary cooling.
The slight bump you identifying as max cooling is only present in SH, you claimed it was 60 N/S.
For radiative enthusiasts, 60 N/S represents a “tipping point” where the surface relation to atmosphere inverses.
From 60 towards equator, increasing surface temperature is associated with increasing surface net radiation (the apparent greenhouse effect becomes increasingly more intense with increasing temperature).
From Pole towards 60, increasing surface temperature is associated with decreasing surface net radiation (the apparent greenhouse becomes increasingly less intense with increasing temperature).
The two zones have opposite greenhouse effect relations with surface. The one with positive relation can be said to be a “hot” zone. The one with inverse relation can be said to be a “cold zone”.
In classic theory, atmospheric radiative cooling is said to occur out the bottom and out the top. The approximate relation is that surface net radiation = OLR / 2. Or, surface upward flux = 3/2 OLR.
That is not what tipping point means either. I give up trying to make sense of what you write, I was wasting my time.
cheer up
It has occurred to me there is a more simple description, such that it is no coincidence that the original authors find the critical latitude at ~60, or in their case ~57.
Such a latitude represents the isotherm right at about freezing point of water. In vertical space, the critical height of the mixing layer is also represented by a temperature about the freezing point, or 2km in height. It is no coincidence, and should not be ignored.
Freezing point represents maximum temperature at which atmosphere remains relatively transparent. An optimum and physically meaningful value. Warmer and colder places have a higher apparent greenhouse factor.
The system shall strive to maximize heat transport across this freezing point threshold before the temperature is allowed to increase. Condensation /atmospheric dehumidification, latent heat release, and radiative transmittance is optimal at 2km height or at 60 latitude in climatological mean.
The quality of the discussion here is lacking. It should be understood that that maximized properties of thermal transport Q in the Earth system are not governed by TOA net radiation (solar – LW);
the choice to set the hot/cold threshold at 34°N/S is arbitrary. The maximized properties relate to thermodynamic pressure in non equilibrium steady state. It’s a matter of heat, not radiation. Otherwise, it’s an interesting conceptual framework.
In radiative terms for radiative enthusiasts, the average cold end of the steady state non equilibrium thermodynamic system is the average outgoing emission temperature T (~255-260K). The average warm end temperature is the average surface emission temperature T (290-295K).
There is a spatial separation between between these ends, which is the thermodynamic fluid (S)ystem in question.
This model is very interesting. I’m of the mind, “All models are wrong but some are useful”, and R.J. Hamming’s maxim, “The purpose of computing is insight, not numbers.” But we can still learn something.
I’d like to speak to the mathematics. Equations 23 and 24 are a set of coupled, nonlinear ordinary differential equations. Taken together they form a second-order dynamic system. There are two states, essentially two degrees of freedom. That implies a cyclic solution with two poles that depends on the terms on the right-hand side. These terms all have to do with the solar input and the radiative transfer (albedo and greenhouse).
I get it that the sun is the source of the earth’s energy and the strongest influence of the climate. Thought, we would expect the earth’s climate to contain many more degrees of freedom, which would speak to Javier Vinos’ comment. Two degrees of freedom is not enough to see the nuances of the climate.
But it seems like a nice start. I think we need to find more terms for the right-hand side of equations 23 and 24. At least it’s speaking to climate, unlike the glorified weather models of the IPCC.
Willis, try lowering the albedo a tiny amount to see if you obtain a better fit.
JF
Problem is that albedo varies with clouds, ice, algal blooms and vegetation.
Which do not appear to be independent variables.
That’s one of the most difficult things about trying to model the climate. Everything affects everything, and depending on the time scales used, those effects can change.
Increased rain will change the types and amount of vegetation.
Different vegetation can change both transpiration and albedo, which in turn can impact the amount of rain an area receives.
Snowfall will obviously impact albedo, however the type of plants that grow in the region will also affect how strongly snowfall will impact albedo. Say the region is covered with ever greens. Once the snow melts off the branches of the trees, the snow on the ground is largely hidden from the sun. On the other hand if the landscape is covered with deciduous trees, the sun can more readily hit the snow and hence be bounced back to space.
Changes in flora, will impact the amount and type of fauna, which in turn will impact the amount and types of fauna.
Trying to model just the atmosphere and oceans, is just half the battle. You still need to figure out all the other interactions, and find a way to squeeze them into your model.
Remember the 5 spheres.
Atmosphere
Hydrosphere
CryoSphere
Lithosphere
Biosphere
You have to understand all of them and all of their interactions, before you can ever truly understand climate.
Fascinating read. While most of the maths cause my eyes to glaze over, the narrative explanation of what is going on and how the model works makes a lot of sense. But on the math, if I understood anything, Willis used “H” where the original source used “U” for the temperature areas, where I suppose “U” is upper, for Willis’ “H” for high.
Basil
Climate is chaotic, no matter what “clever” mathematical methods might be applied, they won’t give a reasonable answer over an extended period, say beyond 3 weeks at most. Certainly by the end of the 21st century the real climate and the calculated climate will have diverged dramatically to the point that the the calculated climate will have proven useless except for generating fear.
Hello: there is a thing Ididn´t get, if “q” goes from left to right, means that the colder section is gaining energy, but as we know it is maintained cold, so it has to radiate enough energy to space as to stay so, as the Stefan Botzman law indicates. you need higher temps to radiate more heat or use more area , so we can suppose that cold area is much bigger tha warm one, or conductance of colder atmosphere is much greater than warmer´s.
I am writing a book about climate changes and its political implications so would be very greatefull to receive your comments.
yours truly
Mario Marquinez Otalora
chem engnieer (retird)
Good question, Mario. The poles get little heat from the sun, so even with the importation of “q” it’s still not much energy in total … which leaves the poles cold.
Regards,
w.
I had a problem with method=”Brent”. There is a method Brent with an older version of the stats package, I found BrentMethod as part of the rSTAR package (which is a really long download), but BrentMethod requires more arguments. So if anyone can get WE code to run, please inform.
Not sure what the problem is. First, are you using raw temperatures or scaled temperatures? The linked study and my analysis use a unitless scaled temperature by dividing all temperatures (in K of course) by 392.8. See the study, equations 17 to 22.
I use a function:
It’s the inverse of the “tunscale” function above. Try that first.
Next, the “Brent” is specifying the method for the enclosing function “optim”. It’s not something separate.
I’m running R 4.2.1 GUI 1.79 Big Sur ARM build on my MacBook Pro, with the ‘stats’ package version 4.2.1.
More questions?
Askem …
w.
Willis, it’s good. Your work is always good. From what you say, I have two observations. First: while you prove the model is good at replicating history, you do not mention anything about how well you think it can predict the future. Yes, I know: V&V (validation & verification) is important for estimating forecasting accuracy. The weather forecasting community uses V&V extensively to figure out how well their models are doing. Curious to know what you think of its forecasting potential. Second: thanks for reminding me how much I have forgotten about diffy-Q. Have a nice day!
Willis
To say the least, interesting. I can do math if I have to. But it does not sing to me, so I tend to avoid it except when it is simple and obvious — which this isn’t. Nonetheless I worked through enough of your excellent presentation to possibly understand what you are doing. I agree, all things considered, your results look pretty good
My question. Will this give plausible results for warm periods like the Cretaceous with cool temperate climates at the poles (at least according to the Paleomap project (www.sctese.com))? Or the recent glacial maximum 20,000 years ago with permanent ice at (current) sea level at 41 degrees N (NYC), How about the (probable) “Snowball Earth” of 700B years ago?
An excellent summary – I can see it having implications in geology especially in those areas where there is fluid movement at elevated temperatures that have led to the formation of ore deposits. Here heat transfer is a vital part of the equation in maintaining the environment consistent with allowing mineral transfer in fluid form to its ultimate destination.
So the real challenge is to take the Constructal Law and apply it 18,000 years BP when the ice age was at its greatest extent with sea levels at their lowest level.
Allied with this challenge is the role of the Constructal Law at the start of the ice age about 30,000 years BP and its conclusion some 6,000 years BP later.
Stripping the role of CO2 out of the equation (its changes lag by some 500 years of the changed temperature) means we are left with how the ice formed to the extent it had. Conversely, what caused the ice to disappear?
And on the final aside, there is a water volume discrepancy between the sea levels reached and the extent of the ‘temporary’ glaciers/sea ice area (the permanent glaciers play no role in sea level changes as the annual ice accumulation records indicate in the core logs of the deep ice cores in Greenland and Antarctica attest.
The term “constructural law” is new to me, but I have heard the same concept (I think it is the same concept) referred to as “the principle of maximum entropy.”
So it gives a way to put some expectations on the results that any climate model should produce. Nice. Should make it harder for the climate alarmists get away with their phony scientific claims.
Hi Willis, it’s interesting that such a simple model work so well. It’s almost suspiciously good.
It seems from fig4 that CERES shows about 0.5 degC warming over the last 20y, (cf 0.8 degC in the model) that’s a mean rate of 2.5deg C / century. That seems a lot. Is that right?
UAH seems to show about half that. Is that comparable?
Though you are not using rho and gamma as freely tunable, is the fact that you use CERES to derive them effectively constraining the model to match CERES output ?
cheers.
climategrog October 10, 2023 6:54 pm
I was surprised myself.
That’s what CERES says. Whether its right is a different question.
Unknown. They measure different things.
Yes. But that’s a detail, not the issue. The issue is establishing that yes, the climate system is a gigantic solar-powered heat engine that constantly adapts and evolves to maximize the heat flow from the the tropics polewards.
Back at’cha,
w.
OK thanks for the reply Willis.
I thought CERES was all about radiation flux measurement. Looking at the url from your graph, I don’t see any mention of temperature data https://ceres.larc.nasa.gov/data/ , is that something you have derived. If not do they explain how it is derived ?
IIRC, there is something like a 5W/m^2 inconsistency in their net energy budget, which would lead to a non credible amount of accumulating energy in the system. I believe this is due to the addition of calibration uncertainly in the multple reading they make, so they can only be used to study change, not the net energy budget in absolute terms. Could that be the cause of the apparently exaggerated warming?
I described the derivation of the temperature data in the head post.
w.
Willis:
Indeed. I missed that when checking back, apologies. However, this raises the question of where do the emissivity values come from. They are presumably done by radiation measurements and a temperature, so there’s a chicken and an egg floating around here somewhere.
As I pointed out above CERES derived anomalies (0.5degC) seem to be about twice that shown by UAH (0.25degC) HadCRUFT4 and pretend-to-be-BEST seem to create some more underlying warming (0.35degC). That’s still leaves you about 35% above “ground truth”.
Since the absolute calibration of CERES is recognised to have uncertainty limitations this could be part of it.
Can you say how the emissivity grid you are using is calculated?
Thanks.
The emissivity data is from the MODIS satellite data, as discussed in An Observationally Based Global Band-by-Band Surface Emissivity Dataset for Climate and Weather Simulations.
w.
Many thanks for the explanation, Willis.
At least that should be reasonably good over water and that’s 71% of the story. It seems to avoid the chickens/eggs.
You could probably benefit from some more C tweaking but the demonstration of the heat pump model is interesting. CERES is definitely running hot , probably due to lack of accuracy in absolute calibration. They know the energy balance is not realistic and warn against it IIRC.
I was prepared to take BEST seriously as an independent land record when they started, but since Muller sold out on open collaboration and then they merged with HadSST3 to be “global” it degenerated into a HadCRUFT clone. I don’t see any point in having both and I don’t trust either team.
Averaging land SAT and SST is physically meaningless and artificially boost the average because land warms twice a fast. Probably why the practice is so popular.
https://climategrog.wordpress.com/land-sea-ddt/
Interesting article , thanks for the effort.
Your unsmoothed hot zone temperature in fig 4 does seem to match the bumps and hills in uah lower tropo data, in form, and relative magnitude, so it’s just a question of scaling.
Willis:
You significantly improved the Authors’ model of how heat is distributed around the Earth, but knowing this does nothing to explain why the heating is occurring as it is, which is far more important.
Or am I missing something?.
I have the same thought. From the end of the head article, it says:
“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 ~1.1°C. (3.7 W/m2 is the amount of radiation increase predicted to occur from a doubling of CO2.)”
So the authenticity of the theoretical 3.7 W/m2 from the overall greenhouse effect (involving either a decrease in albedo and/or an increase in radiative trapping or recirculation of power), *that* always seems to be assumed, somehow!
If the constructal model is really all about balancing things to a kind of maximal efficiency, then how do we know that a CO2 enhanced air column wouldn’t release just about as much extra heat into outer space at the top, as it manages to trap or recirculate at the bottom of the air column? In such a situation, the extra power reaching the ground might be zero, or if not zero, it might be much less than the 3.7 watt number, for all we really know! Unless someone has actually doubled the global CO2 fraction lately (just to do the experiment), it all seems quite hypothetical.
We are maybe here assuming a large number of bits of guesswork, just so that conventional climate reasoning can then still be basically true, somehow?
Hi Willis,
Did you calculate the carnot efficiency of the engine?
The construction law makes sense.
For years (centuries) how nature always takes the path of least resistance has baffled science. Almost as though nature knows the future.
“ the climate system can be modeled very accurately by considering the climate as a giant heat engine …”
Surely we can be more clear. No need to be coy. — It is a heat engine. Period. Nothing else exists to drive it and this is the reason for its accuracy.
The trouble with this Constructal law is the same issue with the principle of maximum entropy production, also an Adrian Bejan hypothesis, is that there isn’t any physical principle that demands it. When put to the test it does not always produce correct answers.
Thanks, Kevin. First, I’ve never heard that Bejan was behind the principle of maximum entropy production. Link?
Next, you say there “isn’t any physical principle that demands it.” Please show us the “physical principle” requiring that heat flows from hot to cold … that’s the nature of laws. They are the physical principle.
Finally, you say “when put to the test it does not always produce correct answers.” I suppose it’s possible to be more vague and uninformative than that. Links?
Regards,
w.
Willis,
I forgot entirely about this comment, and it occured to me last night that I’d never checked back on it. I’m stunned that it is from nearly three weeks ago. I hope you still monitor this thread.
My connection with Bejan goes back four and one-half decades to his Ph.D. thesis from MIT. I applied what he had to say there about coupled irreversible flows in a paper in the European journal Geophysics in 1984. That paper is moderately or maybe even widely cited, apparently, as it was reprinted in some special issue in 2006. The point I am making is that I thought and think pretty highly of this work. My skepticism about this constructal law is not because I have distain for the folks involved.
Somewhere along the way Edward Lorenz suggested that natural systems are organized to maximize energy flows. The date of this conjecture is not at present in my mind and I am not sure if Lorenz is even the origin of the conjecture. At any rate my point is the idea was percolating around MIT in the 1970s. Since heat transfer (and by analogy all transport phenomena) is (are) entropy generating, then it must also stand that maximizing energy flows implies maximizing entropy generation. So Bejan must be part of that crowd. A number of people decided to try their hand at applying the idea to several geophysical problems. I don’t recall anything substantial coming from this, and a paper in 2007 by R. Goody, you can find at the Journal of the American Meteorological society showed that for a few specific problems it arrived at wrong results. There are other examples.
The best observation suggesting to me there is no underlying principal here concerns chemical kinetics. There are examples galore of reactions that by thermodynamics should operate like gangbusters, but which in fact barely move. The reason is that there is activation energy to overcome. It’s hard for me to square a principal like maximum energy flow rates with an impediment like activation energy — nature is not organized to maximize flows if it commonly has spontaneous barriers against it. The meadering of rivers is another.
Bejan is also a proponent of using entropy generation minimization to design optimal engineered systems, but this idea has a solid theoretical underpinning in that entropy generated times dead-state temperature is the work potential wasted in a process. So, minimizing entropy generation maximizes available work.
And as long we are speaking of well established principles, the physical principle behind heat transport is that this leads to more probable configurations of molecular speeds; that collisions are more likely to move internal energy from a hot place to a cold one than vice versa. This is a result from statistical mechanics and is pursued in detail in graduate courses in thermal physics. It might be also in an undergraduate course in statistical physics — i don’t know because the physics curriculum has changed some since I took it.
I hope this gives you some food for thought.
Kevin
Willis, could you please define ALL the terms and their physical units used in these equations. Until you have done so, I do not understand how I (or anybody) can comment meaningfully on this article.
In Equation 25, how can “q”, a quantity of energy flux presumably measured in Watts) can be derived from a constant C and temperature (Th and Tl), given that temperature is not a measure of the quantity of energy?
delta Q = c delta T
Willis,
It took me a while to see it, but what you have here is an overfitted model, which has no physical content, it only spits out the information that you put into it.
The model has three independent unknowns: Th, Tl, and x, the fractional area of the hot zone. Of course there are other quantities like C (an input to the model) and q (which is calculated from the independent unknowns).
Meanwhile there are three constraints: conservation of energy for the hot zone, conservation of energy for the cold zone, and the fact that your are optimizing over C to best match the real temperatures.
3 constraints and 3 unknowns means an overfitted model.
To put it another way, the only work the model does is to find x, after which it just uses CoE to find Th and Tl. But by optimizing over C, you are implicitly telling it what x should be. So the model isn’t doing anything.
The reason for some discrepancy between the result and real life is that the actual “cold zone” doesn’t have a uniform temperature, so depending on how you average, the mean will be a bit different.
Nepal2, if you had taken the trouble to read the study, or even my excerpted part of the study in the head post, you’d have seen that there are FOUR unknowns, not three. Here’s what the authors say:
As a result, your analysis is wrong.
In addition, I’m unclear why you think that a model such as this one is meaningless. It clearly demonstrates that Bejan is correct about the system acting to maximize Q, which is very valuable support for his ideas about how the system works.
But hey, if you think you have a better model, break it out …
w.
There is another variable q, but that is controlled by the given heat equation.
If this were a sensible model of conductivity, then it would output a heat flow at each position (or at least each latitude if continuing to assume azimuthal symmetry). Then choosing x to maximize heat flow would just be drawing an imaginary line on the globe where dq/dtheta = 0 , which is exactly what you did in your analysis of CERES data. Any heat flow has such a maximum, and identifying x doesn’t have any physical effect.
But Bejan’s model does not have a sensible, continuous heat flow. Instead it says there is some special latitude where nature averages the temperature of the entire hemisphere below that line, and the entire hemisphere above it, and somehow picks the heat flow across the line based on the global averages. In this model choosing the imaginary dividing line x has an effect, because that determines where the globe cuts off the average.
It’s all very strange and not reminiscent of any physical process I know. Yes, the model “works” in that it finds the line x, and also gets the heat flow at this latitude nearly right.
But outputting two numbers nearly right is very easy, and not good evidence for a model’s predictive power. Particularly when you are optimizing over one fit parameter, and also putting in a huge amount of measured data (albedo and greenhouse effect read off from CERES).
Just my two cents.
Looking at a plot of meriodonal heat transfer, like Fig 1 here, https://hwpi.harvard.edu/files/carlwunsch/files/wunschjclim2005.pdf , makes it easy to identify the line of maximum q (dq/dx = 0) at around +-37 degrees. So we have found the x which maximizes q. But identifying this line doesn’t change anything.
Stick to the original work. The headpost here gets it all wrong and the author should realize this by now.
The first giveaway is that Clausse et al infer a polar temperature of 270K, using a T low of 258K.
The second giveaway is that Clausse et a give a predicted change of T high and T low for both the equator and the pole.
These represent the emission temperatures of surface and atmosphere respectively. I believe they selected an atmospheric emission temperature average about 5km, and a surface (potential) temperature at 0 height.
latitude >35 simply yields the point at which surface begins receiving net heat from atmosphere. Below 35 surface is giving heat to atmosphere net.
It’s a great read on maximization principles and geometric constraints on the Earth system.
https://www.academia.edu/4509041/Climate_change_in_the_framework_of_the_constructal_law
An addendum might best be added in the headpost to warn future readers of the misunderstandings.