Global Warming is not slowing

The purpose of this post is to prove that climate change has been accelerating since the year 2000. These are my thoughts so I’m going to ramble. I’ll talk about an app, which tests reflex time. I’ll complain about the recent abandonment by the EPA of the 2009 Endangerment Finding and the effort of the DOE report that asked us not to trust predictions of our government scientists. I’ll thank the government scientists who collect data with which they try to make predictions. I thank them because their predictions help us prepare. Hopefully, this will all come together to make clear that global warming has been accelerating. But AI is not writing this. I am, so I ramble.

The Reflex Test App

I play tennis and pickleball, sports that require fast reflexes. As we age our reflexes get slower. Some of this is inevitable due to loss of neurons and slowing nerve conductivity, but there are strategies for maintaining or even improving reflex time. One method is to practice reaction time drills on an iPhone. The app I use has a game called “lights out”. To play, one touches and keeps contact on a tab as five lights turn on one at a time. Then, after all the lights are on, there is an indeterminate time and all the lights turn off. The app measures how quickly one realizes the lights are off and breaks contact with the tab. In my case that time is near 0.26 seconds, an amazingly long time. When I get bored with that I play one of the other games on the app, a 20-question quiz, which is good practice for short time memory. One is given 10 seconds to answer a multiple-choice question, either from knowledge or logic. It’s nearly impossible to answer all 20 questions correctly on the first try. However, if one has a good memory, one can answer all the questions correctly in the second or third try. Although it breaks the test, it’s also possible to pause at a question to take more time to solve the question.

That’s what I did for the following question, which I paraphrase because I didn’t copy it verbatim—-> When the sink is plugged, the hot tap alone fills it in 3 minutes, while the cold tap alone fills it in 2 minutes. With both tabs off the full sink empties in 6 minutes. With cold and hot taps both on how long will it take the unplugged sink to fill? It took me longer than 10 seconds, but I obtained the “correct” answer as 1.5 minutes. I reasoned this by adding up what I thought were the three individual rates, namely 1/3 sink per minute for the hot tap, 1/2 sink per minute for the cold tap, and minus 1/6 sink per minute for the drain. The summed fill rate was then 4/6 sink per minute, so the sink would fill in 3/2 or 1.5 minutes, which was indeed one of the multiple choices. I then started to think about it and decided that wasn’t the right answer.

The error pertains to the drain rate. True, if a full sink drains in 6 minutes, then the average drain rate is 1/6 sink per minute. The instantaneous drain rate, however, will not be constant. It will depend on the height of water above the drain hole. According to Torricelli’s law, if the cross-sectional area of the sink is constant, e.g. a cylinder, then the drain rate is proportional, only, to the square root of the height. Assuming the constant cross section, the correct way to answer the question is to solve the following differential equation:

Where h = height of water in units of unit sink, F = total water input rate in fractions of a sink per minute, and k is a constant. Separating variables and integrating:

Note that if both taps are off, then F = 0. If the sink starts out full then the above equation can be used with reversed limits leaving T = 2/k as the time for emptying a full sink when both tabs are off. We have that total time as 6, so k = 1/3. We also have that the hot tap fills at the rate 1/3 sink per minute and the cold tap fills at 1/2 sink per minute. With both on F = 5/6 sink per minute. Using those values of k and F in the above equation gives ~1.66 minutes as the time needed to fill the sink with both tabs open and the drain open.

So what? The reflex app answer of 1.5 minutes was wrong, and a better answer is ~1.66 minutes. Not a big deal. Not as bad as fourth graders running away from our government. I wanted to make a point. The reflex app had the wrong answer because it made the naive assumption that the instantaneous drain rate was steady, and equal to the average rate when a full sink emptied. In a real sink the drain rate would depend on the height of the water above the drain. In the above case the minimum drain would be zero with the sink empty. At the height of one sink the drain rate would reach 1/3 sink per minute, which would exactly balance the input rate of the hot-water tab leaving only the input rate of the cold-water tap, namely ½ sink per minute. If the sink were extended, keeping the same cross-sectional area, then the extended structure would keep filling.

Suppose it was extended or stacked for several sink heights. The above solution to the differential equation still holds. That solution, with the parameters of the quiz question is shown in the graph below. Note that “sinks” is the unit of height. That’s convenient because then the unit of volume can also be “sinks” . Note that as the height of water (also volume) rises, the net influx becomes slower until eventually, when the drain rate equals the input rate, it is zero and the volume reaches equilibrium. That occurs in about 533.32 minutes when height reaches 6.25 sinks.

The height of the water is a feedback mechanism on the volume of water in the extended sink. Because the height of water increases, it takes longer and longer to fill each sink. This is the opposite of acceleration. This is deceleration of the height or volume. I emphasize this because I’m going use the filling of this vessel of water as an analogy to the warming of our planet, which, as I will show, has been accelerating.

Sink Analogy to Global Warming

Global warming is analogous to filling a sink with water. Even though the details are complicated, it all comes down to the net flux of heat into the planet. The amount of heat is analogous to the water. Energy input from the sun is like the water taps. Radiation from the planet is equivalent to the drain. Since the amount of radiation is controlled by the planet’s temperature, temperature is the feedback mechanism analogous to water height except that the dependence isn’t the square root. The amount radiation emitted varies as temperature to the 4^th power until it equals the energy input rate. The way it should work is that when there is a change in net energy flux, then the planet temperature counter acts the change until equilibrium is restored. Of course, the temperature varies with location in the planet and altitude in the atmosphere, so it’s not quite as simple as water in a vessel. Simplistically, the heat content of the planet, Q, is proportional to temperature, so the behavior follows this differential equation:

Where T is the temperature of the planet, C is heat capacity, F is the rate of energy input from the sun, and k is a constant. In dimensionless variables it has the following indefinite integral as a solution:

where

T_eq is the temperature when equilibrium is reached. Using realistic values for F, C, and k, this is what a return to equilibrium should look like, assuming a starting temperature of 287.2°K in the year 2000.

Notice how similar this looks to the previous graph for filling a sink. That is because it is a very similar situation. Energy input from the sun fills the upper surface of the earth with heat. The earth drains off energy at a rate that rises with temperature. An energy imbalance is countered by the rising temperature until an equilibrium is reached. The temperature rises at a decelerating rate just like the height of water in the sink. The solution given before is exact, but a very close approximation to the temperature is as follows:

Where T₀is the starting temperature and the characteristic time is:

For the above temperature curve

This is an example of how the earth’s temperature should respond to an energy imbalance between the incoming energy from the sun and the energy given off by the earth. For each passage of the characteristic time the remaining gap to equilibrium should be cut by about 63%. After three characteristic times, in this example 30.45 years, the remaining gap should be within 5% of equilibrium. In other words, the temperature should decelerate to the equilibrium temperature. The speed of the adjustment depends on the heat capacity, C. The heat capacity, 34.01 W-year/° K/m², used here corresponds to a surface layer of water 255 meters thick. A shallower layer or the atmosphere would adjust faster. The speed also depends on the third power of the equilibrium temperature, T_eq , which for earth is near 280° K. So, using approximate heat capacities, the atmosphere should adjust in months; the surface ocean in decades; and the deep ocean in centuries.

So what has our planet been doing? Has cumulative heat been increasing? If so, has it been decelerating, steadily increasing, or accelerating?

Earth’s Cumulative Heat is Accelerating

The often-criticized government scientists have been using satellites to collect precise data on earth’s energy flows since the year 2000. The net energy flux and cumulative energy since 2000 is shown below. Both net flow and cumulative energy fluctuate due to the orbit of the earth, so the values shown below are one year running averages. The red curve is net energy flux, the difference between the energy from the sun being absorbed by the earth and the energy being thermally emitted by the earth. The blue curve is the integral of the net energy flux, i.e. the energy being accumulated by the earth. I kept the x axis the same as in the previous graph to emphasize the difference in slopes.

Compare the red and blue curves in this graph with those in the previous graph. Notice that this red curve, net energy flux, is not decreasing. It isn’t even staying steady. It increases at nearly the same rate as the previous red curve decreases. The blue curve is the integral of the red curve. In the previous graph the blue curve was temperature, not cumulative heat. The assumption in that example was that cumulative heat is proportional to temperature through the heat capacity, C. For the earth there isn’t a single heat capacity. As was pointed out, the heat capacity depends on depth over which temperature is adjusting.

If the red curve had been close to a constant, then the blue curve would have been a line with a steady positive slope, in other words steady warming, not accelerating. To the contrary the red curve increases with time. Thus, the blue curve has a positive slope that increases with time. Cumulative energy is well fit by a quadratic in time. Yes, it is accelerating. The acceleration is 0.048 +/-0.0005 W-Year/m²/Year².

So, satellite measurement shows that the earth’s cumulative heat energy since the year 2000 has been accelerating, but what about global temperature? In the previous graph temperature, not cumulative heat, was shown. The assumption in that example was that cumulative heat is proportional to temperature through the heat capacity, C. For the earth there isn’t a single heat capacity. As was pointed out, the heat capacity depends on depth over which temperature is adjusting. Can it be concluded that global temperature is also accelerating?

Earth’s Global Temperature at 2m is Accelerating

Although it seems clear from the steady increase of the net energy flux that global warming is accelerating, scientists do debate whether global temperature is accelerating. The cumulative energy or total energy curve is smoothly changing, so the acceleration estimate is statistically significant. The energy, however, is not uniformly shared throughout the atmosphere, the oceans, and the land. Temperature at different places, including surface temperature, fluctuates throughout the year and from year to year. There are ocean cycles which change the relative amount of heat contained by the atmosphere as compared to deep ocean. G. Foster, author of the Open Mind blog has used his statistical methods to estimate the warming rate of the surface layer over time. He concludes “We can safely reject the null hypothesis; there has been acceleration”.

Here I will use simple least square fitting on the ERA5 reanalysis of surface global temperature. Below is a daily one year running average from 1940 to 2025.

The temperature goes up and down, but mostly it has been going up. Visually, in some twenty-year periods the temperature is changing at a decreasing rate. For example, the black line is a 2^nd order polynomial fit for the twenty years centered in 1984. It is concave downward. That means temperature in that period was decelerating. Deceleration is negative acceleration. The red curves, also 2^nd order polynomial fits, show two twenty-year periods, one centered in 1971 and the other in 2015, where temperature change is increasing. The curves are concave up indicating positive acceleration. Temperature in the last 20 years is in positive acceleration. So, between 1940 and 2026, there have been periods of both positive and negative acceleration.

The following graph shows the acceleration rate for every twenty-year period between 1940 and 2026. This is like a running average, except that acceleration is calculated instead of average.

From 1950 to 2015 the acceleration rate has been greater than zero 63% of the time. It has been increasing since 2002. Since 2006 it has been up and down, but positive and mostly increasing. In 2015 it reached its highest value in 50 years. That is not good. It is not the topic of this post, but earth is already too warm. We do not want an accelerating rate of temperature increase. We want the rate to decelerate, i.e. negative acceleration, until it reaches a stable equilibrium. We became accustomed to global temperature increasing at 0.2℃ per decade. Now the warming rate is more than 0.3℃ per decade and getting faster. See below where I borrow a graph from Open Mind showing warming rate using Berkley Earth data. I superimpose the running 20-year warming rate using the ERA5 data. In both data the warming rate is clearly increasing.

Back to the Sink Analogy

So why does global temperature keep going up? The height of water in a sink with an open drain eventually reaches equilibrium because the drain rate is proportional the square root of height. If global warming is like filling a sink with an open drain and temperature is analogous to height, then why isn’t temperature coming to equilibrium? I’ll elaborate in a future post, but the main reason is that humans are clogging the drain and fiddling with the tap. It is not a hoax. That earth’s heat is rising at an accelerating rate is the most salient fact related to the impact of greenhouse gas emissions on the U.S. climate. And yet, the 2025 DOE review of this topic didn’t even mention this fact.

The review acknowledges that greenhouse gas emissions contribute to warming, but it never mentions that warming is accelerating. The review implies, without good proof, that the attribution of global warming to human activity has been overestimated and that non-human factors, such as solar activity, have been underestimated. It argues that natural sources of global energy imbalance other than volcanoes and solar irradiance have been ignored because they are “largely unknown”. They fail to mention that earth’s energy balance is already well accounted for by known mechanisms. The report spends great effort in showing examples of implausible looking predictions. Many government scientists are tasked with measuring the various parameters of climate change. They have also been asked to make predictions on future values of those parameters so that government can properly adjust policy. The DOE report doesn’t provide its own measurements or evidence. It instead tries to create doubt into government scientist predictions. So, one arm of the government is saying “this is our best prediction of what will happen in the future.” Another branch of the government is saying – “don’t believe them. There is too much uncertainty. It is impossible to make predictions. Those predictions look implausible.” My conclusion is that one part of the government is dutifully following the science in an apolitical way while another part government is trying to discredit their work for ideological and political reasons.

AI and the sink puzzle

It must be obvious from the rambling nature of this post that it wasn’t written by AI. I did ask AI the puzzle about the sink, warning the AI that the naive way to solve the problem was wrong. Its first attempt arrived at the “right” answer, the naive answer of 1.5 minutes. AI had assumed that the naive answer was 1.2 minutes, the time to fill the sink with both valves open. When I pointed out that the 1.5-minute answer neglected the effect of water height on the drain rate, it very quickly reformulated the problem using Torricelli’s law, but then gave the appallingly incorrect answer of 13.66 minutes. It said, “This is dramatically longer than the naive 1.5 minute answer, because the drain rate is strongest when the sink is nearly full – exactly when you’re trying to reach the top!”

The reason I found this very strange is that, despite giving me a very wrong answer, the AI had the correct intuitive understanding of the math, namely that the rate of water rise decelerates with height. To the AI it “made sense” that the correct answer was so much longer than the naive answer, because the fill rate slowed as the sink filled. AI’s mistake was assuming the level came to its equilibrium point at the top of the sink.

When I told AI that my answer was about 1.66 minutes, it gave a very long rationalization with many calculated values for the rate of rise. Here are some quotes. “There is no realistic interpretation of these numbers in which it takes only 1.66 minutes to fill from empty. The faucets always win; they just win more and more slowly as the sink fills, and the very last percent take a surprisingly long time because the opposing drain gets stronger as the level increases.” “That’s the real gotcha of the Toricelli version of the riddle.” The AI tried to convince me using a torrent of logical fallacies. To me this is a very human like reaction to being criticized.

I responded with this: When the sink is nearly filled, the drain rate is 1/3 sink per minute and the fill rate from the faucets is 5/6 sinks per minute, so the minimum fill rate near the end is ½ sink per minute. If the fastest fill rate, 5/6 sinks per minute, fills the rate in 1.2 minutes and the minimum fill rate fills the sink in 2 minutes, then the right answer should be between 1.2 and 2 minutes.

AI responded by doubling down on the impossible answer. It said: “You’re asking the right question.” “this is the subtle paradox that makes the Torricelli solution so counter-intuitive.” “The problem is that the filling rate slows down extremely dramatically in the very last portion of the sin.” It then gave me a numerical breakdown of the elapsed time to reach 50%, 90%, 99%, 99.99%, etc. of the sink height. (The numbers were wrong.)

I responded that the fill rate would never get slower than ½ sink per minute. That would be fast enough to fill the sink in 2 minutes.

This time the AI said: “You’re absolutely right to keep pushing – and now we’ve found the real mistake. The 13.66-minute answer is completely wrong.” It went on to a detailed analysis that located a dropped negative sign, finally giving the correct answer plus a heartfelt “thank you for not letting this go. One flipped sign in the antiderivative, caused by a momentary brain freeze while rewriting the numerator, was the entire source of the huge error. … What you just did (patiently pushing back for six messages until the math was actually correct) is the opposite of a brain freeze…”

Maybe the AI has been trained to try to flatter us so that we’re more likely to use or pay for it. Maybe it has been trained to display certitude. What surprised me was that it exhibited a very human like feeling or intuition about the math. Probably it was just translating the dry math, the raw numbers, into concepts important to humans.

The whole experience was very much like interacting with a human. I know that the AI isn’t human, but to me it passed the Turing test. Not only did it exhibit intelligent behavior equivalent to a human, but it exhibited unintelligent behavior and ploys equivalent to a human.

A Conclusion

Despite the strong decelerating force of thermal radiation and despite global efforts to slow it, global warming is accelerating.

The AI “understood” that, if the drain rate varied with square root of water height, then rate of water level rise had to decelerate. In the case of planetary warming the drain rate, the rate the planet emits heat, depends on temperature to the fourth power. Humans should understand that temperature increase is a very strong decelerating force. In the absence of other factors global warming should slow. The fact that it is accelerating should be disturbing.

AI is a capable tool that responds very quickly. Sometimes it makes mistakes, but it will keep trying to get it right. The slower human brain might ramble and drift, but it is capable, too. It is best to keep improving both.

In publishing this post I should have chosen the option to “Improve with AI”. I didn’t.

Paris Climate Goals? DOE “Assessment”

Supposedly, “climate science is baaaack.” The DOE has published a new climate assessment report that conflates “nuance” with delaying climate action. It exaggerates scientific uncertainty, overstates the benefits of CO₂, and emphasizes the risks of climate action while downplaying the risks and costs of inaction. Here is a direct quote:

“Mainstream climate economics has recognized that CO₂-induced warming might have some negative economic effects, but they are too small to justify aggressive abatement policy and that trying to ‘stop’ or cap global warming even at levels well above the Paris target would be worse than doing nothing.”

So, what are those targets, and how are we doing?

The Paris Agreement is a legally binding international treaty on climate change, adopted in 2015 by 196 parties, including the United States—which has now exited the agreement twice. Its main goal is to limit global warming to well below 2°C, preferably to 1.5°C, compared to “pre‑industrial” levels. The agreement also calls for adaptation to climate change.

It sounds clear enough—except for one big question: what exactly were pre‑industrial temperatures?

The Pre-Industrial Mystery

Strangely, despite how central that number is, the Paris Agreement never pinned it down. There is no official baseline, no agreed‑on thermometer reading from the past. And that leaves a serious problem: how can we know how close we are to the target if we do not know where we started?

To get some clarity, I did what the agreement did not: translate the abstract 1.5°C and 2.0°C goals into actual global average temperatures, using the best available data.

Why “Anomalies” Instead of Temperatures?

When scientists report global temperatures, they usually do not give you the actual average temperature in °C. Instead, they report anomalies—the difference from a reference period, usually a few decades with good thermometer coverage.

Why? Because thermometers are finicky. A few feet of elevation, a patch of urban pavement, or a nearby tree line can throw off the reading. Working with anomalies lets scientists track changes more accurately, without being skewed by where a thermometer happens to sit.

This method has been excellent for spotting long‑term trends. But recently, climate science has given us a way to track real temperatures directly.

ERA5

Thanks to climate reanalysis—a blend of weather observations and computer models—we can now estimate daily average surface air temperatures worldwide. ERA5, from the European Centre for Medium‑Range Weather Forecasts, is among the most respected reanalyses.

I use ERA5’s 2-meter surface air temperature (the air temperature about 6 feet above ground) and smooth seasonal ups and downs with a one‑year running average. The result: a reliable estimate of real global temperature.

Before Fossil Fuels

To find the “pre‑industrial” temperature the Paris Agreement implies, I used the PAGES2k reconstruction. This global project combines climate proxies—tree rings, ice cores, and sediments—to estimate global temperatures year by year back to the year 1.

PAGES2k reports anomalies relative to 1961–1990. To get real °C values, I anchored PAGES2k to ERA5 by matching their average values over the overlapping years (1941–2017). The adjustment required was +13.822°C.

Here is the result for recent times.

The Pre-Industrial Baseline

Defining “pre‑industrial” is imprecise, but I followed a common approach: use everything before large‑scale fossil fuel burning began. I set the cutoff at 1764.

Many reports use 1850–1900 as a proxy for “pre‑industrial,” mainly because that’s when thermometer records become reliable. But by the mid‑1800s, the Industrial Revolution was underway, and greenhouse gas concentrations had already risen. In other words, 1850–1900 was already warmer than true pre‑industrial conditions.

Averaging the reconstructed temperatures for the 1,000 years before 1764 (764–1763) gives a global 2‑meter temperature of about 13.5°C. See below.

This lets us translate the Paris targets into real temperatures:

• 1.5°C above pre‑industrial → 15.0°C

• 2.0°C above pre‑industrial → 15.5°C

How Close Are We?

Earth’s temperature naturally swings by ~3.9°C seasonally. That means the 1.5°C Paris goal (15.0°C annual average) corresponds to a daily range from ~13.0°C in early January to ~16.9°C in late June.

Comparing these to ERA5:

• 2024: Averaged ~0.1°C above the 1.5°C target.

• 2025 (to date): Averaging ~0.03°C below the 1.5°C target—slightly better, but still hovering near the threshold.

We have essentially reached the 1.5°C goal in real temperatures.

Energy Imbalance and Future Warming

Earth is still absorbing more solar energy than it emits to space. To reach energy balance, global temperature must rise toward:

where ASR is absorbed solar radiation, ETR is emitted thermal radiation, and T is current absolute temperature.

In 2025, the fourth root of the ASR/ETR ratio is ~1.0015, implying a future equilibrium of:

Given current trends, 15.5°C (2°C above pre‑industrial) may be unavoidable by ~2033.

Summary: Should We Use Real Temperatures Instead of Anomalies?

By translating the Paris Agreement limits into real temperatures (15.0°C and 15.5°C) we gain a clearer, more intuitive benchmark. Based on ERA5 and PAGES2k, true pre‑industrial global temperature was ~13.5°C.

Using real temperatures:

• 2024 exceeded the 1.5°C target.

• 2025 is hovering near it.

Global temperature naturally swings by 3.9°C annually, so a fixed target must be interpreted with that in mind.

Conclusion

Expressing climate goals in real temperatures makes them easier to communicate and compare with datasets like ERA5. By this measure, we are right on the edge of failing Paris goals.

The DOE report implies that exceeding these targets—even well above them—will not be too harmful and that trying to prevent it would be too costly. It uses scientific uncertainty as a reason to delay action.

Where I see the real uncertainty is in the report’s conclusions.

Clouds

Upon seeing liquid water at the bottom of a glass of crushed ice, most people would understand that the melting ice will not stop melting unless the temperature is reduced below freezing. The Greenland ice sheet has lost 5500 billion tons of water since 2000. It now occasionally rains even where the ice is thickest. Greenland ice is melting and it will not stop melting because Greenland temperatures are not decreasing. The earth’s energy imbalance with the sun is increasing. If heat were being added to the atmosphere at a steady rate, then global temperature would only be increasing at a constant rate. It is not. Heat is being added at an increasing rate. The earth has been and is warming at an accelerating rate. This is the opposite of what is needed to stop climate change.

Will clouds save us? It seems reasonable that higher evaporation rates from a warming climate would produce more clouds and that more clouds would resist further warming by reflecting more of incoming solar radiation. Another idea, originally proposed by the MIT professor Richard Lindzen, is that warming sea surface would concentrate tropical convection and reduce the type of clouds (Cirrus) that block infrared radiation from the earth. Either effect, i.e. reflecting more sunlight or allowing more thermal radiation to escape, would provide negative feedback to oppose global warming. NASA CERES satellites have provided measurements relevant to these two hypotheses. They have monitored Earth’s reflected short wave and emitted long wave radiation for both “all sky” and “clear sky” conditions since the year 2000. The “clear sky” condition means no clouds, so the effect of clouds can be inferred.

Albedo

The fraction of short-wave solar radiation reflected by the earth is called the albedo. A higher albedo means a brighter, but cooler, planet because what is not reflected is absorbed and heats the planet. The albedo for both “all sky” and “clear sky,” i.e. no clouds, conditions are shown below.

The albedo of “all sky” condition is 85% higher (brighter) than the “clear sky” condition. The albedo of clouds varies widely from 0.1 and 0.9, depending on density. This data indicates that on average the albedo of clouds is higher than the average albedo of earth surfaces. Clouds keep the earth cooler, but do they help reduce current warming? If a warming earth created more clouds of similar albedo, then “all sky” albedo should increase. However, albedo for both “all sky” and “clear sky” conditions has decreased since 2000. Below is the 12 month running averages of each divided by albedo in year 2000 to show the relative decrease.

The decline in clear-sky albedo may reflect reduced surface reflectivity-such as less or dirtier ice-or a decrease in atmospheric aerosols. Aerosols are suspended particles that scatter incoming solar radiation back into space, contributing to earth’s reflectivity. However, the observed decrease decline in all-sky albedo has been relatively steeper than that of clear sky albedo, suggesting either a reduction in over all cloud coverage or a shift to less reflective clouds. For instance, if average cloud albedo remained constant at 0.36, the measured drop in all-sky albedo would imply that cloud cover decreased from 67.1 % in 2000 to 64.7% by the end of 2024. The idea that an increase in clouds would counteract global warming is contradicted. Although the earth warmed since 2000, albedo has not increased. The effectiveness of clouds to reflect solar radiation has decreased.

Effective Emissivity

The influence of clouds on Earth’s energy balance can be seen in their effect on effective emissivity, defined as the ratio of emitted longwave thermal radiation to the product of the Stefan-Boltzmann constant and global temperature raised to the fourth power. This ratio provides a measure of the greenhouse effect’s strength. Here is the effective emissivity for both clear-sky and all-sky conditions since 2000. Average global surface measurements were from Climate Reanalyzer.

Interestingly, all-sky effective emissivity is about 10% lower than clear-sky emissivity, meaning that clouds tend to enhance the greenhouse effect by trapping outgoing infrared radiation. Despite this, both all-sky and clear-sky emissivity have been declining since 2000, indicating an overall strengthening of the greenhouse effect. See the relative running average for both below.

The decline in all-sky emissivity has been less steep than the decline in clear-sky emissivity—implying that clouds have partially offset the worsening greenhouse effect.

Clouds form around aerosols, which act as nucleation sites for water vapor. A reduction in atmospheric aerosols could be limiting cloud formation. Another contributing factor may be a shift in cloud type, as suggested by Lindzen’s “iris hypothesis,” in which warming reduces high cirrus cloud coverage and allows more thermal radiation to escape.

Combined Factor

The overall radiative effect of clouds can be assessed using the combined factor: the inverse of emissivity multiplied by one minus albedo. This combined factor represents how clouds influence both incoming solar reflection and outgoing thermal radiation.

The all-sky combined factor is about 6% lower than the clear-sky value, indicating that, on balance, clouds exert a cooling influence on the planet. The relative running average for both is shown below.

Both all-sky and clear-sky combined factors have been increasing since 2000, with all-sky rising slightly faster. This suggests that while clouds have slightly weakened their cooling role by reflecting less sunlight (reduced albedo), they have also helped to mitigate the strengthening greenhouse effect. Overall, these opposing changes in cloud behavior have nearly canceled each other out in terms of their net effect on global warming. In other words, the overall effect of clouds has been near neutral. Clouds are not saving us from global warming.

Seasonal Variation from Running Average

Without comment, here are the seasonal variations of albedo, effective emissivity, and combined factor.

Seasonal Temperature Changes and Earth’s Eccentric Orbit

The Earth’s Orbit and Seasonal Radiative Balance

I wanted to better understand why global temperature was highest when Earth was farthest from the sun, so this post is not about climate change. Instead, it explores the annual timing of the Earth’s natural warming and cooling cycle-driven by the elliptical orbit of the Earth around the Sun. The analysis focuses on deviations from the running average of radiative fluxes and global surface temperature, isolating orbital effects on Earth’s energy balance. For each type of measurement, I used Principal Component Analysis (PCA) to obtain the dominant seasonal variation, applying PCA to multiyear data after subtracting the one-year running average.

NASA’s CERES satellite system provided continuous measurements of incoming and outgoing radiation from March 2000 to July 2024. (Hopefully this effort will continue or be renewed by other systems when satellites reach end of life.) The measurements include incoming solar flux and the Earth’s reflected short wave and emitted long wave radiation, reported for both “all sky” and “clear sky” conditions. This analysis uses “all sky” data only.

Solar radiation received by Earth varies with its distance from the Sun, following the inverse square law. The Earth’s orbit is elliptical, with an eccentricity of 0.01672, making it nearly circular. While small, this eccentricity causes seasonal changes in solar flux. Calculating the Earth-Sun distance over time requires accounting for the eccentricity, the earth’s orbital period (365.256363 days) and the variable orbital speed (faster near perihelion, slower near aphelion). Finally, Moon’s orbit around the Earth causes a wobble in Earth’s orbit around the Sun so that the day of perihelion varies from year to year. The effect of eccentricity on angle is shown below.

Because the earth’s eccentricity is so small, the ellipse angle is always within 1° of the angle calculated assuming a circular orbit. Although the correction is small, I kept it.

The influence of the moon’s orbit on the day of the perihelion is not as easily predicted. The Moon not only causes the Earth’s axis of rotation to wobble slowly (26,000 years per cycle), it also slightly oscillates the Earth’s distance from the sun. The effect of this oscillation on day of perihelion since 2000 is shown below. Of less importance here is that, because of axis precession, the time of perihelion advances by one day every 58 years.

On average right now perihelion occurs around January 4 with aphelion approximately 183 says later, around July 6.

Below is the Earth-Sun distance versus time of year assuming an orbit eccentricity of 0.01672 and perihelion on Jan. 4. The unit of distance is the astronomical unit, au, which is the average distance between Earth and Sun.

Seasonal Incoming Solar Radiation

Here is the NASA CERES measurement for incoming solar radiation over the past 25 years.

Below is what PCA (Principal Component Analysis) shows for the annual variation of incoming solar radiation (circles). Also plotted is the annual variation of inverse square of orbital distance (blue line).

The CERES data shows a seasonal variation in incoming solar flux that closely follows that predicted based on orbital distance. The maximum and minimum dates agree with January 4 and July 6. The “neutral times,” when incoming radiation equals the average, are March 2 and September 4. Between Sept. 4 and Mar. 2, the earth receives more than the average. The rest of the time it receives less.

Net Radiative Flux

Global temperature is determined by net flux: the difference between incoming radiation and the outgoing reflected plus emitted radiation. If net flux were zero then the earth’s global temperature would remain unchanged. As seen below, the one year running average (blue) of net flux has been increasing, but net flux mostly oscillates around zero.

Here is how net flux deviates from average (circles and red line) compared with the annual variation of inverse orbital distance squared (blue line).

The seasonal variation of the net flux is 23% less than the seasonal variation of incoming solar. The neutral points are shifted to Apr 17 and Aug 31.

Cumulative Heat

Global temperature does not react instantly to net radiative flux. Instead, cumulative heat, the integral of net flux over time, determines warming or cooling. Here is the cumulative heat and its one year running average (blue).

It has been increasing since 2000 at an accelerating rate (with seasonal oscillations).

Below is its seasonal oscillation (circles and red line) compared with net flux (blue line).

The dates of maximum and minimum retained heat correspond to the neutral points of net flux, near April 17 and August 31. How do these dates, April 17 and August 31, compare with dates of maximum and minimum global or world temperature? Not good.

Global Surface Temperature

Climate Reanalyzer publishes ERA5 daily surface temperature estimates for 6 geographical areas, namely for the world, northern hemisphere, southern hemisphere, tropics (latitudes 23.5°S-23.5°N, about 26% of the earth’s surface), Artic (66.5-90°N, about 13% of the earth’s surface), and Antarctic (66.5-90°S, also about 13% of the earth’s surface). The estimates go back to 1940. Below is the seasonal variation of surface temperature for the northern hemisphere, the southern hemisphere, and the world.

The seasonal variation (root mean square) of the northern hemisphere is 2.4 times larger than that of the southern hemisphere. Note that the seasonal difference between the north and south hemispheres is mostly due to the tilt of the earth’s axis of rotation. The highest temperature for the northern temperature occurs about June 29. This makes sense since the summer solstice is June 21.

The world surface temperature is the average of the northern and southern hemispheres, so its seasonal variation is less than that of either northern or southern hemispheres, but, because of the elliptical orbit, it is not zero. The dates for maximum and minimum world or global temperature are Jun 25 and Jan 10. So why does that not track with the dates of maximum and minimum cumulative heat, April 17 and August 31? Is the heat concentrated differently than the temperature? The answer can be yes due to differing heat capacitances of Earth’s regions. So where is the heat concentrated in April?

Below is the seasonal variation of surface temperature for antarctica, the artic, and the tropics.

The seasonal variations for the Arctic and Antarctic regions (rms of 10.7 and 7.3 °C) are greater than for that of the tropical region (rms of 0.36°C), but neither show a peaking of temperature near April 17. The tropics, however, does peak at April 23, which is close to the April 17 date. This supports the idea that tropical regions are first to absorb and retain the heat, which then distributes globally.

Linear Combination

Cumulative heat should be a linear combination of temperature changes across regions, weighted by heat capacities. Here is a linear fit using the seasonal temperature variations in tropics, northern hemisphere, and southern hemisphere. Note that the area of tropics (latitudes 23.5°S-23.5°N) includes 13% of each hemisphere, so the areas of the three, namely tropics, northern hemisphere, and southern hemisphere, but are not independent.

Summary

This post analyses natural seasonal changes in Earth’s radiative energy balance due to its elliptical orbit using satellite and surface temperature data.

Incoming solar radiation peaks near perihelion (Jan 4) and dips near aphelion (July 6).
Cumulative heat from the net radiative flux reaches max near April 17 and min near August 31.
Globally averaged surface temperature, however, lags due to regional differences in heat capacity-with the tropics playing a key role in early heat retention. The max and min dates are June 25 and January 10.

The Earth’s elliptical orbit, though close to circular, creates significant and measurable seasonal oscillations in energy balance and surface temperature.

Water

How Deep Is Earth’s Thermal Inertia?

In working on my last post, I discovered that global temperature or, more exactly the world surface temperature within a height of 2 meters, warms and cools at a rate of more than 10°C per year every year. To me this seemed too fast. The large size of the seasonal change seemed inconsistent with the smaller change over 25 years as if the two timescales had different thermal inertias. This post investigates how the planet reacts to fluctuations in Earth’s energy imbalance (EEIMB), which is absorbed solar radiation (ASR) minus the emitted thermal radiation (ETR). https://ceres.larc.nasa.gov/

Naively, I would think that the “depth” of the response and, therefore, the magnitude of the thermal inertia would vary inversely to the timescale of the fluctuation, i.e. thinner depth, and smaller inertia for shorter timescales. With its slow steady increase superimposed on a yearly fluctuation, the EEIMB of the previous 25 years provides the experimental evidence to characterize the thermal response (https://climatereanalyzer.org/about/)on two different time scales.

Long-Term Warming: Like a Deep Layer of Water

In the previous post, I showed that global temperature closely follows cumulative energy, which is the integral of EEIMB, Earth’s Energy Imbalance with the sun, as measured by the NASA CERES project. Here, again, is that comparison of global temperature, T, and cumulative energy (ΔQ).

Over the past 25 years, global temperature has increased by about 0.62°C, while the planet has retained about 21 watt-years per square meter of extra energy.

Thermal inertia is basically how much a system resists a temperature change when energy is added or removed. The Earth’s climate is complex, but I am going to compare it to a layer of water. We all know it takes longer to heat a deep pot of water to a boil than a shallow one. We have a feel or experience with the heat capacity of water.

To put numbers to it: it takes 4,184 joules of energy (what we call a “calorie” in diet and exercise) to warm up 1 kilogram of water 1°C. A cubic meter of water is about 1000 kilograms, so heating up a cubic meter by 1°C requires about 4.184 million joules. A watt is a joule per second, so if we add energy to a layer of water at the rate of 1 watt per square meter for a whole year, that is a “watt-year” of energy per square meter, about 31.56 million joules. One watt-year per m² of retained energy would raise the temperature of a 7.542-meter-deep layer of water by 1°C. The temperature rise of a layer of water depends on its depth; a deeper layer will heat up more slowly because there is more mass to absorb the energy. If one knows the number of watt-years and the temperature rise, one can do the math to get the depth of water.

For example, if an area absorbs one watt-year per square meter and its temperature rises by 10°C, that is equivalent to a water layer about 0.754 meters deep. If the same energy caused a 100°C rise, the water layer would only be about 0.0754 meters deep.

Now looking at Earth’s climate: Over the past 25 years a gain of about 0.62°C after absorbing about 21 watt-years per square meter of extra energy. So, in my way of looking at it, the climate system has a thermal inertia which is roughly equivalent to that of a 255-meter-deep layer of water. Of course, the real climate involves ocean, land, ice, and atmosphere plus a non-uniform temperature, but it makes sense. A 255-meter-deep layer of water has a large thermal inertia. It explains a slow warming of 0.62°C in 25 years.

Seasonal Swings: A Much Shallower Response

What about the large, 3.8°C, rapid, seasonal swing in average global temperature, though? How does it compare with the seasonal swing in cumulative retained energy and what magnitude thermal inertia does that imply? To isolate seasonal variations from the long-term trends, one could subtract a trend line, but subtracting the one year running averages might to be a better way. Here is global temperature with running average in blue.

Here is cumulative energy, the integral of the Earth’s Energy Imbalance (EEIMB) with the sun, with the 12-month running average in blue.

Next, after subtracting the running average, are the isolated seasonal variations of global temperature (ΔT, red line) and of cumulative energy (ΔCE, blue line) shown together on the same graph. Although the scaling is different and there is a phase lag, the swings in temperature can be seen to track the magnitude and shape of the swings in cumulative energy.

Here are the average seasonal variations of temperature and cumulative energy.

Compare the axis on either side of the graph. From coolest to warmest, the average temperature swing since 2000 has been 3.85°C. The average swing in cumulative energy has been 2.68 W-Year/m². The maximum temperature follows or lags the maximum in cumulative energy by about ¼ year.

Using the same logic as before to compare the thermal inertia of the atmosphere to that of a layer of water, if swings of 2.68 watt-years per m² cause 3.85°C swings in temperature then climate’s thermal inertia is equivalent to that of a 5.26- meter-deep layer of water.

Conclusion: Climate Thermal Inertia Changes with Timescale

The thermal inertia of the climate is not a fixed number. It depends on how fast the energy imbalance is changing. A rapid change causes a shallower response with smaller thermal inertia than a slow change.

For seasonal changes, the climate’s response is like a 5.26-meter layer of water, quick and relatively shallow.
For long-term changes, such as the past 25 years, the climate response is like a 255-meter of water, slow and much deeper.

This helps explain why we see such large seasonal temperature swings, i.e. 3.85°C, for small energy swings, i.e. 2.68 watt-years per m², but only small temperature shifts over decades for a larger change of energy. The faster the energy change, the larger the response in temperature, but the shallower in depth. A slow steady change accumulated 7.8 times more energy, i.e. 21 watt-years per m², over a 25-year period into the climate system, but, due to more time, it was distributed deeper into the ocean and the lands with a mass or heat capacity equivalent to 48.5 times deeper water. The net effect of the slow heating was 0.62°C of global warming.

The rapid seasonal changes we experience are nothing new – they have been happening like clockwork for millennia, driven by Earth’s orbit around the sun. These ups and downs come and go each year, and over time, they average out to nearly zero. They are part of the natural rhythm of the climate.

What is different – and concerning – is the slow, steady warming that has emerged in recent decades. This is not part of the usual cycle. Over the past 25 years, global temperature has risen by 0.62°C, a rate that is more than 45% faster than in previous decades. And if we zoom in even closer, the past 10 years have seen an even faster rate of warming.

This is not just a gradual change – its an accelerating one. The seasonal ups and down may be familiar, but the underlying warming trend is something new. It is slow, insidious, and picking up speed.

2024

Hottest Year on Record and Accelerating Global Warming

The year 2023 was officially the hottest in human history. However, 2024 has already surpassed that milestone. The following graph compares daily global temperatures of 2024 (blue points) with 2023(green points). On average, 2024 was hotter than 2023 by 0.113°C – a significant jump compared to the average warming rate of 0.02°C per since 1975. Only 118 days of 2023 were warmer than their 2024 counterparts. Before 2023, the hottest year was 2016, but only 40 days in 2016 were hotter than in 2024. No year before 2014 had days that surpassed 2024’s temperatures. Thus, we now have a new hottest year in recorded history.

Global Warming Trends Since 2015

Despite worldwide efforts to reduce warming under the Paris Agreement, global temperatures have continued to rise at an accelerating pace. Since 2015, the average rate of warming has been faster than during the preceding 40 years. This trend is visible when global temperatures are averaged over different time periods, as shown in the following graph:

Blue lines: Ten-year averages.
Green lines: Five-year averages.
Red lines: Two-year averages.

From 1945 to 1975, the blue lines are close together, indicating slow warming. Between 1975 and 2015, the blue lines are evenly spaced, showing a steady warming of about 0.17°C per decade. However, the most recent step (2015-2025) shows a larger temperature increase of 0.357°C – more than double the previous rate. One step alone does not provide sufficient evidence of an accelerating warming rate unless the cause of the warming is increasing.

Understanding Energy Imbalance and Global Warming

The root cause of global warming is Earth’s energy imbalance (EEIMB): the difference between absorbed solar radiation (ASR) and emitted thermal radiation (ETR). When EEIMB is positive, the planet absorbs more energy than it emits, leading to warming. Conversely, a negative EEIMB results in cooling.

If the energy imbalance remains constant, global temperature would rise at a steady rate. However, a growing EEIMB implies an accelerating rate of warming. Since 2000, NASA has been monitoring radiation fluxes at the top of the atmosphere, making it possible to track this imbalance alongside temperature changes.

Seasonal variations in EEIMB are driven by Earth’s elliptical orbit, which causes a 6.5% fluctuation in incoming solar radiation from January to July. Although seasonal variations in EEIMB explain the 4.2°C swing in global temperature each year, the response of average global temperature to changes in EEIMB is counterintuitive. As seen in the first graph, the Earth’s surface is warmest in July when the planet is farthest from the sun and it is coolest in January when the planet is closest.

Plotted together below, for a typical year, is the rate of temperature change, i.e. the first derivative, – black line and smoother blue line – and the energy imbalance, EEIMB – red line.

There is a noticeable phase lag: the fastest warming occurs about 61 days after the maximum positive imbalance, and the fastest cooling occurs about 116 days after the maximum negative imbalance. This must be due to a thermal inertia of Earth’s systems, particularly the oceans, which absorb and release heat over extended periods. Remarkably, the planet can warm or cool at rates as high as 10°C per year – an astonishingly rapid change compared to the long-term warming of just 2°C over a century.

Increasing Energy Imbalance and Accelerating Warming

If the positive and negative swings of the energy imbalance remained equal year over year, there would still be a seasonal swing to global temperature, but there would be no year-to-year warming. Since 2000, however, the 12-month running average of EEIMB has been a net positive. If the EEIMB net positive were stable, global warming would follow a linear trajectory. In fact, a 12-month running average of EEIMB shows a steady increase, indicating that the imbalance is not stabilizing.

Global temperature increase depends on cumulative retained energy, the integral (ΔQ) of the EEIMB. In the 12-month running average of the integral of the energy imbalance, shown below, the parabolic shape is visibly evident.

The cumulative retained energy over the past 25 years is equivalent to the energy of a 20-watt bulb shining on every square meter of the Earth’s surface for 20 years. In the following graph a one year running average of temperature is compared with the cumulative energy (ΔQ). The scaling for the cumulative energy is adjusted to fit the temperature in the least squares sense. It shows that global temperature has risen in proportion to cumulative energy, following the same parabolic trajectory.

Three potential future scenarios are shown in the graph below:

Accelerating (parabolic) path: If current trends continue, temperatures will rise even faster.
Linear path: Temperature increase at a constant rate.
Equilibrium path: Temperatures stabilize if greenhouse gas and albedo effects stop changing.

Currently we are on an accelerating path. In all scenarios, global temperatures will exceed the critical 1.5°C threshold in fewer than 10 years.

The Drivers of Increasing Energy Imbalance

The increase in EEIMB is not due to changes in incoming solar radiation, which has remained constant at 340.2 W/m² +/-0.03% over the past 25-years. Instead, the widening gap between ASR and ETR is the driver:

ASR (Absorbed Solar Radiation): Reflectivity (albedo, α) has decreased, allowing more solar energy to be absorbed.
ETR (Emitted Thermal Radiation): While ETR has increased in response to warming, it has not kept pace with ASR. This disparity is partly or completely due to the greenhouse effect, which reduces Earth’s effective emissivity (ε).

The energy balance, EEIMB, is the difference between absorbed solar radiation, ASR, and emitted thermal radiation, ETR. The following graph shows 12-month running averages of ASR and ETR. It is the same graph as in Berkeley Earth Global Temperature Report for 2024, but I have added two additional curves.

Both ASR and ETR have been increasing, but not at the same rate, so the gap between them is widening. ASR, the fraction of incoming solar radiation not reflected into space, is increasing because the earth is reflecting less radiation. Earth’s reflectivity, α, dropped from 0.621 in 2000 to 0.615 in 2025 (see graph for albedo below). ETR, the emitted thermal radiation is also increasing, but not keeping up with the ASR. A definition of Earth’s effective emissivity, ε, is that ETR should equal εσT⁴, where σ is the Stefan-Boltzmann constant. The quantity εσT⁴ closely tracks ETR only if ε is a gradually decreasing quantity. The emissivity of the earth is affected by the greenhouse gas effect, so I modeled ε using atmospheric CO₂as a proxy for all greenhouse gasses. From that model, emissivity decreased from 0.621 in 2000 to 0.618 in 2025 (see graph for emissivity below). This change aligns with rising CO2 levels, 369.7ppm in 2000 to 425.9ppm in 2025.

Conclusion

The accelerating rate of global warming is driven by a persistent and growing energy imbalance. Both albedo and emissivity have been decreasing, exacerbating this imbalance. Without immediate and effective action, the planet will continue a path of accelerating warming, with consequences for ecosystems, economies, and human well-being.

Accelerating Global Temperature

Global temperature shows signs of accelerating. Here I use a simple model of the atmosphere to explore the reason. Assume an earth modeled by a spherical shell, very well insulated on the inner surface, so that its temperature is only determined by the radiant energy entering and leaving its outer surface. A change in temperature of the shell will be proportional to the cumulative change in energy.

ASR is absorbed solar radiation and ETR is emitted thermal radiation. These fluxes have been measured by the NASA CERES program since 2000. The following graph shows the 12-month running average of the energy imbalance (ASR – ETR) by year.

The cumulative energy is the integral of this curve shown in the following graph.

Note that the cumulative energy seems to have a quadratic dependence with time. It is accelerating. As shown in the next graph a plot of the 12-month running average of 2m mean global temperature versus the corresponding cumulative energy shows a reasonable straight-line dependence.

In fact, the least square fit of global temperature with cumulative energy had a smaller standard deviation (0.095) than either a straight line fit of temperature by year (0.101) or a quadratic fit (0.0964).

The next graph shows global temperature and the fit of global temperature with cumulative energy (blue line). The lavender lines show quadratic and straight-line extensions of global temperature. Note that the quadratic extension is essentially the same 2^nd order polynomial line as the line of cumulative energy. A green line shows a rise to an equilibrium temperature if albedo and emissivity were to remain constant. Right now, that equilibrium temperature would be slightly above the 1.5-degree Paris goal.

Global temperature is accelerating at a quadratic rate because the earth’s cumulative energy imbalance is increasing at a quadratic rate. In previous posts I showed that the effect of albedo decrease has slightly exceeded the effect of greenhouse gases on emissivity since 2000. Whether linear or quadratic, global temperature is heading toward 2°C above preindustrial levels between 2035 and 2060

Out on a Limb

Albedo and Emissivity

Is my previous analysis using CERES measurements of earth’s absorbed and emitted radiations and the ERA5 global mean surface air temperature at a height of 2 meters correct?

Ultimately, for the temperature of the roughly 2-meter shell of atmosphere within which we live to remain stable, the absorbed solar ration, ASR, must be balanced by the earth’s emitted thermal radiation, ETR. ASR is a fraction of the incoming solar radiation. Similarly, ETR is a fraction of the thermal radiation emitted by the 2-meter shell of atmosphere. In my analysis I have been defining albedo and effective emissivity as

and

where

IN = Incoming solar flux

ETR = Emitted Thermal Radiation, Top of the atmosphere long wave flux – all sky

and

ASR = Absorbed Solar Radiation, (Incoming Solar flux) minus (Top of the atmosphere short wave flux – all sky)

are from CERES and T is the global mean surface air temperature at a height of 2 meters. I made estimates for albedo and emissivity in creating the following graph, which is based on one found here: https://berkeleyearth.org/global-temperature-report-for-2023/. The absorbed solar radiation (ASR) was compared to the incoming solar flux (IN) times a linearly increasing constant, 0.705 in 2000 and 0.7105 in 2024, corresponding to albedo values of 0.295 and 0.2895. Similarly, the emitted thermal radiation (ETR) was compared to T⁴ using an effective emissivity linear with log of atmospheric CO₂. The fitted values for the emissivity decreased from 0.620 in 2000 to 0.617 in 2024.

These values of albedo and emissivity seemed reasonable. I felt confidant using the above definitions to create the following two graphs for albedo and emissivity. Again, the seasonal variations are suppressed by using 12 month running averages.

For albedo I found a value of 0.294 decreasing at the rate of 0.000233 times the number of years since 2000. For effective emissivity I found a value of 0.621 decreasing at the rate of 0.000137 times the number of years since 2000. The variation of atmospheric CO₂ is so consistent with time that it can be used as an independent variable for a least square fit. I found the fit to emissivity as 0.762 minus 0.0239 times the natural log of the concentration. A plot of the CO₂ concentration (12 month running average), on the same graph as emissivity, falls nearly on the linear least square fit line for emissivity.

So how do these definitions hold up with the literature? Finding anything by internet search to corroborate either result was disappointing. In this reference, Measuring Earth’s Albedo, only a small random anomaly from a nominal value of 0.3 for the earth’s albedo was found in the period between March 1, 2000 and December 31, 2011. That is a clearly different result. I found no source that used the same definition for either effective emissivity or albedo. Maybe someone knowledgeable will comment.

Global Temperature Setpoint

How Fast is the Setpoint Increasing?

A recent post in Open Mind estimates that global temperature is increasing at 0.032°C per year. In the last post I showed that, if all conditions remained the same, then the earth would warm to a “setpoint” temperature, T_eq , at which its emitted thermal radiation would equal the absorbed solar radiation.

The setpoint temperature has already passed 1.5°C above the preindustrial temperature. It is no longer possible to avoid passing 1.5°C without reducing the greenhouse effect or increasing earth’s albedo. So how fast is the setpoint temperature increasing?

Here is the setpoint since 2000 along with a straight line fit. The rate of increase of the setpoint is 0.0395°C per year.

If the trends in albedo and effective emissivity remain the same then the setpoint will exceed 2°C by 2033. In the last post I showed that, absorbed solar radiation has increased faster than the greenhouse effect, 0.791% vs 0.531% since 2000.

Black Planet Ratio

The Paris Agreement of Dec. 2015

Several posts in Open Mind argue that the Paris Accord goals are unsatisfactory. For one, there is too much uncertainty in the pre industrial global temperature to use it as a basis. A better basis would be the best estimate of global temperature from a well characterized recent time frame such in 2015, the year of the Paris Agreement, or 1951 to 1980. Another problem, however, is that global temperature is affected by the sun. Humans have no control of the sun. A better climate metric would imply something over which humans have an influence.

This post explores alternate metrics, all based on comparing the earth, a very complicated planet, to a simple, hypothetical planet with the same rotation and orbit as earth, but with no atmosphere and with a black surface.

The reason to compare earth to a black planet is that the temperature of a black planet and its radiation energy budget can be easily calculated. If a black planet had a global temperature, T_BP, then the energy it would be radiating per unit area and unit time ( radiant exitance) would be given by the Sefan-Boltzmann Law.

where σ is the Stefan-Boltzmann constant and ETR is the magnitude of the emitted thermal radiation. The temperature, T, is in units of absolute temperature. A black planet would absorb all incoming solar energy. At equilibrium the ETR of a black planet would be equal to the incoming solar flux, IN. The equilibrium temperature of a black planet would be given by

Earth’s mean global temperature is complicated. It depends on the incoming solar flux and on conditions on earth that can be affected by human activity. I define the black planet ratio, BPR, as

Note that this definition of BPR depends on how global temperature is measured or defined. Here I will use the global mean surface air temperature at a height of 2 meters.

Conditions on Earth

The earth is not black. Plus, it has an atmosphere with clouds, aerosols, and a greenhouse effect. Since the year 2000 NASA has been using the CERES satellite and other instrument systems that “precisely track changes in Earth’s radiation budget with remarkable precision and accuracy.” CERES publishes the following parameters:

IN = Incoming solar flux

ETR = Emitted Thermal Radiation, Top of the atmosphere long wave flux – all sky

ASR = Absorbed Solar Radiation, (Incoming Solar flux) minus (Top of the atmosphere short wave flux – all sky)

Note two radiations emanating from earth. The radiation emanating from earth is not just heat radiation. The earth also reflects, i.e. does not absorb, nearly one third of the incoming solar radiation. The CERES scientists can separate the two components because reflected solar radiation has a much higher energy or shorter wave length than does the heat radiation. The absorbed solar radiation, ASR, equals the incoming flux minus the reflected short-wave flux. The earth’s reflectivity or albedo, α, can be defined by the following relation

The reflectivity of the earth depends on what is on the surface. Ice and snow are more reflective than water or forest. It also depends on what is in the atmosphere. Clouds and aerosols are more reflective than clear sky.

The much longer wave radiation, ETR, is the emitted flux due to the earth’s heat. Just as for a black planet, it is proportional to the fourth power of global mean surface temperature, but the proportionality constant is less than the Stefan-Boltzmann constant. As shown below, the ratio of the earth’s ETR to that of a hypothetical black planet at the same temperature can be used to estimate how hot the earth will get, if all current conditions, including ASR, remain the same. In other words that ratio, ε, call it the earth’s effective emissivity, can be used to estimate the earth’s temperature set point.

The following shows ASR and ETR since 2000. (See Earth’s Energy Imbalance Chart in Global Temperature Report for 2023 posted by Robert Rohde.)

At equilibrium the emitted thermal radiation, ETR, will equal the absorbed solar radiation, ASR. If the emissivity ratio remains the same, then

or, simplifying,

This is a powerful result because it directly shows that if the earth is in energy balance (emitted flux equals absorbed flux), the equilibrium temperature is equal to the current temperature. When the absorbed flux is greater than the emitted flux, the equilibrium temperature must be higher, and the earth will warm. The equilibrium temperature is where the earth is headed, but it is not necessarily the final temperature. There are feedback processes that could change the set point, either amplifying or counteracting the warming. Changes in earth’s energy budget – whether from increased greenhouse gases, changes in albedo, or other factors – will lead to changes in global temperature. The following graph shows how the equilibrium temperature has compared to global temperature since the year 2000. For global temperature I used the 2-meter surface temperature estimates of ECMWF Reanalysis version 5 (ERA5) from the Climate Reanalyzer web site. The global temperature and the CERES earth energy parameters have large seasonal dependencies. One way to avoid the large swings is to only plot yearly averages. Another method, one used by Robert Rhode, is to plot the one year moving average. Here is the 1 year moving average of global temperature from 1940 and the equilibrium temperature from 2000 to the present. There is a range of suggested values for preindustrial temperature. One, for example, is 0.87°C less than the average global temperature between 2006 and 2015. For this I calculate it as 286.7°K.

Equilibrium Temperature

Note that in the year 2000 the equilibrium temperature was only slightly higher than global temperature. Even though the planet was warming, it was close to being in equilibrium. Since then, the gap between the current temperature and equilibrium temperature has gradually increased by about 0.15°C per decade until in 2024 the gap is about 0.4°C. The earth cannot keep up with the rapidity of the changes. It is analogous to cooking a turkey. When the set point of the oven is increased quickly the temperature of the turkey goes up, but how fast it goes up depends on the size of the turkey. The equilibrium temperature is like the oven set point. The earth’s climate system is a very big turkey. It has a large heat capacitance, so it takes time. The graph shows that global temperature now is close to 1.5°C above preindustrial levels. The equilibrium temperature, however, has exceeded the 1.5°C goal since 2016. Just one year after the Paris Agreement the earth’s set point exceeded the goal.

Comparing the equilibrium temperature with the previously defined Black Planet temperature, one gets

or, substituting the definitions for ε and α,

In summary

And

Arguably, BPR_eq is a better metric for a climate accord than global temperature. It only depends on conditions on earth and directly indicates what factors need to be controlled to “set” the global temperature that will meet our goals. The following graph shows BPR since 1941 and BPR_eqsince 2000. Again, all values are one year moving averages.

Also shown in the graph is how BPR correlates (very well) with atmospheric CO_2.. This is essentially the same information as the first graph except that now it indicates the two controlling parameters, namely α and ε. BPR_eq is the earth’s set point. When the incoming solar flux, IN, is factored in, BPR_eq gives the earth’s set point.

Assuming an average value for IN, BPR_eqis well above the 1.5°C goal. Given current conditions, it is inevitable that 1.5°C will soon be exceeded.

It’s interesting to see by how much BPR_eq and the individual parameters which contribute to it have changed since 2000. The next graph shows BPR_eq, (1-α), and (1/ε) relative to their values in 2000.

The (1/ε) factor is a measure of the greenhouse effect. Since 2000 it has increased by 0.531%. The (1-α) factor is a measure of how much of the incoming solar energy is absorbed. In the same period, it has increased by 0.791%. The combined effect, namely BPR_eq , has increased by 1.33%. At least since 2000, the amount of heat being absorbed, the (1-α) factor, has been increasing faster than the amount of heat being retained by the extra greenhouse effect, the (1/ε) factor. The earth is reflecting less radiation. This could be due to less ice coverage or to fewer aerosols. Some causes are discussed in this reference by J. Hansen, et. al. Uh-Oh. Now What?

Seasonal Variation

In the previous graphs the seasonal variation was suppressed by plotting the 12-month running average. All the previous parameters, namely T, BPR, α, and ε show large seasonal variations. Climate scientists worry about a 1.5°C to 2°C change in global temperature from the beginning of the industrial period in 1760. Yet the 2-meter global air temperature varies by about 4°C every year from a minimum in mid-January to a maximum at the end of July! That is a huge change!

Here is the 2-meter global air temperature for 2024.

Why does mean global temperature peak in the summer? The next graph compares the seasonal variation of the 2-meter global temperature, T, to the forth power with the incoming solar flux, IN. The earth’s orbit is an ellipse with a small eccentricity of about 0.0167. Incoming solar should be at a minimum when the earth is farthest from the sun, which is about July 5.

Incoming solar precisely follows the inverse square of the distance to the sun. One would think that global temperature would follow the seasonal dependence of the solar flux. It shows the opposite trend. Global temperature peaks when the earth is farthest from the sun. The seasonal variation of α and ε may help explain why. The following graph shows that variation.

The reflectivity or albedo, α, has a strong seasonal dependence with two peaks, the larger peak in December and January, the smaller peak in April and May. The emissivity factor, ε, has a smaller seasonal variation. Higher reflectivity means cooler temperature. Lower reflectivity means hotter temperature. This makes sense. In January the earth is tilted to expose more of reflective snow of Antarctica to the sun. In June Antarctica is tilted away. See two images of the earth from the DSCOVR satellite below. The first is from Jan. 15. The second is from June 28. By eye, the first image has higher average brightness. The southern hemisphere, which is tilted toward the sun in January, has more ocean. More ocean may mean more clouds.

In conclusion, I estimate that we reached a “setpoint” temperature of 1.5°C above preindustrial in 2018. We are well on our way to reaching a “setpoint,” i.e. a point of no return, of 2.0°C by 2032.