Weather, Macroweather, and the Climate is an excellent book on climate science. Lovejoy engages both climate scientists and climate skeptics and attempts to persuade both. It tries to be accessible to a general audience, but I don’t think it quite makes it. I think that it is accessible to almost any scientist or engineer.

This book is more about climate than about climate change. Lovejoy does not mention climate change until the 6th chapter, out of 7 chapters total. Instead, his main goal is for you to understand the patterns of motion in the atmosphere.

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Prerequisites: Reading graphs that contain a lot of information. You shouldn’t need my explanation of Fractals or my explanation of Fields (not yet available) or a familiarity with power law statistics, but they would be helpful.

Originally written: March 2020.

Confidence Level: While I am not a climate scientist, I am an expert in a neighboring field.

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Lovejoy answers questions that should be foundational but are surprisingly absent in most discussions about climate, like:

• What is the difference between weather and climate? (They have different scaling laws – and there are actually 3 clearly distinct regimes.)
• What is the probability that the warming observed in the industrial era is natural? (about $1/5,000$ – not $1/2$ or $1/3,000,000$)

Atmospheric dynamics is scaling, and best described by anisotropic multifractals. Let’s break down that sentence to see what it means.

Fractals are shapes that look similar when you rescale them – zoom in or zoom out on the shape. The patterns that appear in the atmosphere look similar if you look at large scales or small scales. These patterns are most visible in the shapes of clouds, so the previous sentence says that big clouds look similar to little clouds.

Temperature is not a shape, so temperature itself cannot be a fractal. Instead, temperature is a field – a quantity that can be measured at any location and can be different at different locations. Temperature patterns still look similar if you rescale them. If you look at all of the locations in the atmosphere where the temperature is $30^\circ$, that shape will be a fractal. If you look at all of the locations in the atmosphere where the temperature is $31^\circ$, that shape will be a different fractal. The same is true for any particular temperature in the atmosphere. This is what is meant when we say that the temperature (or pressure or humidity or wind velocity) is a multifractal. One possible definition of turbulence is ‘a physical process that produces multifractal fields’. ^[1]Assuming that it is possible to get multiple scientists to agree on a definition for turbulence, which seems unlikely.

The atmosphere is much wider than it is tall and gets much less dense as you move up, so vertical and horizontal directions are very different. The appropriate way to make the comparison between scales is different if you are moving vertically or horizontally, so the multifractal is anisotropic. (Isotropic means the same in all directions.) If you zoom in or out in time, you also get another, different scaling law.

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Most statistics are Gaussian. Professional statisticians can go their entire career using only Gaussian statistics. The statistics of temperature (or pressure or humidity or wind velocity) are not Gaussian. They have power law tails. The ‘tail’ of a distribution is the low probability of extreme events. Saying that the atmosphere has a power law tail means that it is much more likely to have extreme changes than other statistical systems. If the temperature typically changes by about 10 degrees from day to day, Gaussian statistics suggests than a 40 degree change from day to day would happen maybe twice in your lifetime. In the atmosphere, these sorts of large changes are uncommon, but not nearly that uncommon.

Power law statistics can be used to create the simplest possible models of the Earth’s macroweather and climate. Simple models are rarely found in climate science because the ones based on Gaussian statistics give completely wrong results. Instead, climate scientists typically use GCMs (which stands for both General Circulation Models and Global Climate Models), which are attempts to simulate the entire motion of the atmosphere for decades on an extremely large, fast computer. Simple models using the right statistics can give equally good results with millions of times less computing power.

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Lovejoy is a bit of a contrarian in the climate science community (as am I), so it’s worth pointing out his positions on several of the ongoing debates.

Paleoclimate data are various ways to estimate the temperature of the Earth before we began measuring it directly. There are many proxies to look at that should be related to temperature, including tree rings, layers in ice sheets, and sediment layers below annual algal blooms. I have not looked into this closely enough to have an informed opinion, but am skeptical that it is possible to untangle temperature from all of the other contributing factors and to get enough data to appropriately average over the globe. Lovejoy trusts paleoclimate data more than I do. He thinks that the estimates made since 2003 that involve comparing multiple proxies are trustworthy.

As I have already hinted, Lovejoy does not trust GCMs (General Circulation Models/Global Climate Models). I also distrust GCMs, and for similar reasons, even before I read this book. GCMs attempt to take an extremely complicated physical system (the atmosphere + oceans + ice sheets + … ) and solve it using brute force on a giant computer. The computer keeps track of the average temperature, pressure, wind velocity, and humidity for each square of a $100 km \times 100 km$ grid covering the earth’s surface (and about $50-100$ layers in the vertical direction). Anything that is smaller than $100 km$ across cannot be directly captured in the GCM. Instead, there are phenomenological approximations for these sub-grid processes. The distinction between phenomena larger than your grid which you resolve and phenomena smaller than your grid which you don’t resolve is not physically realistic because the atmosphere is scaling. We should instead be using the scaling laws to relate large and small phenomena.

We both definitely believe that the nonlinear revolution (chaos, fractals, turbulent cascades, etc.) has been underutilized in climate science. Lovejoy works to do just that:

Inasmuch as the randomness is a result of the small-scale sensitivity – the butterfly effect – the stochastic forecast effectively harnesses the butterflies to predict the future skillfully.

p. 284

I will focus this review on a few of the best figures.

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Figure 2.4

This figure shows how much the temperature typically fluctuates on different time scales.

Take the average temperature over the course of a minute. How different is that from the average temperature over the course of the next minute?

Now take the average temperature over the course of a day. How different is that from the average temperature over the course of the next day?

That’s what this plot shows. The horizontal axis is how long you average over – from less than a second to a billion years. The vertical axis is how much the temperature tends to change over that time scale.

The temperature tends to fluctuate by about $0.1^\circ$ to $0.2^\circ C$ ($0.2^\circ$ to $0.4^\circ F$) each second. Each second is typically a few tenths of a degree different from the previous second. The average temperature over 10 days fluctuates by about $10^\circ C$ ($18^\circ F$). Each week is typically 10-20 degrees different from the previous week. The average temperature over 1000 years fluctuate by about $1^\circ C$ ($2^\circ F$). Each millennium is typically about one degree different from the previous millennium. The average temperature over 100,000 years fluctuates by about $8^\circ C$ ($15^\circ F$). Each age is typically about 10 degrees different from the previous age.

The large daily and annual temperature fluctuations are removed from the plot so we can see the rest of the data better. This simplest way to do this is to compare the average temperature for this April to last April instead of to the average temperature for May. Or compare the hour starting at 8 am on Monday to the hour starting at 8 am on Tuesday, instead of the hour starting at 9 am on Monday.

Obviously, we cannot use one data set for this entire plot. The techniques we use to measure temperature over a second are different from the techniques we use to measure temperature over millions of years.

The data for shorter time scales (on the left) involve thermometers in Lovejoy’s home city of Montreal. The data for longer time scales (on the right) involve ice and sediment deposits.

Although this graph is specifically for a single location, plotting the data from other locations results in a similar picture. Colder and drier climates have larger fluctuations than warmer and wetter climates, but the transitions from weather to macroweather to climate are still there. If you average over the entire globe, the temperature fluctuations are smaller, especially for the shorter time scales. This is useful because we can typically only estimate prehistoric data at a few locations, and we have to assume that the fluctuations at these few locations represent the entire world.

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There are three obvious regimes, corresponding to the up-down-up shape of the graph. Lovejoy calls these weather, macroweather, and climate.

For time scales less than 10 days, the slope is increasing. When you average over a longer time, the temperature fluctuations get larger. This means that the dynamics of weather are unstable (it has a positive Lyapunov exponent – although Lovejoy uses the closely related Haar exponent). Weather is inherently unpredictable.

10 days is approximately the time that it takes for a weather system to go around the earth. It is the boundary between weather and macroweather.

During the macroweather regime, the slope is decreasing. When you average over a longer time, the temperature fluctuations start to cancel. The dynamics of macroweather are stable (negative Haar exponent). Macroweather is in principle predictable.

Somewhere between 100 years and 1,000 years, there is another transition. This is more uncertain, in part because it seems to be different in the industrial era than before. This is the boundary between macroweather and climate.

For longer time scales, the slope is increasing again. The dynamics of climate are also unstable. Climate is inherently unpredictable.

If you really trust paleoclimate data, you can see two additional regimes: macroclimate and megaclimate.^[2]You probably shouldn’t trust this data because the lines on the right side of the graph are far apart – different ways of estimating the same quantity give different results. Macroclimate is more predictable. There is an oscillation between ice ages and relatively ice free periods. These oscillations are associated with periodic shifts in Earth’s orbit – we don’t follow a perfect ellipse around the sun because of the influence of other planets, especially Jupiter. Megaclimate, which is probably dominated by continental drift, is also inherently unpredictable.

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The distinction between macroweather and climate is extremely important for people who want to model climate. If you insist that your GCM be stable, then you are inherently going to be modeling macroweather, not climate. This is the standard protocol. Large simulations of unstable dynamics are prohibitively difficult – both conceptually and in terms of computing power. Many climate simulations compare one ‘equilibrium’ climate to another (e.g. natural carbon dioxide levels to current carbon dioxide levels), even though these are really macroweather equilibria, not climate equilibria.

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Figure 6.4 & Company

The purpose of these figures is to construct the simplest possible model for climate change.

This figure has two fairly self-explanatory graphs.

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The first graph is how much carbon dioxide is in the air. In 1900, carbon dioxide accounted for about 300 parts per million, or 0.03%, of the molecules in the atmosphere. In 2000, carbon dioxide concentration was at 365 parts per million, and it continues to rise.

One of the standard questions that climate modelers try to answer is: If we double the amount of $CO_2$ in the atmosphere, relative to its pre-industrial concentration (277 ppm, in 1750), how much will the global temperature change? The fraction of a doubling that has occurred so far is also shown on the vertical axis.

These data were measured on a mountaintop on the island of Hawaii, far away from anything that might influence the carbon dioxide concentration locally. The concentration was also averaged over the course of a year to make the graph smoother.

More recent data can be found here.

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The second graph is the temperature, averaged over the entire world, since 1880.

The vertical axis is not the temperature itself, but the temperature change from the pre-industrial average. The temperature has increased by about $1^\circ C$ ($1.8^\circ F$) since 1900.

This is a combination of data from different sources, averaged together to form a single temperature index.

The ocean data are from satellites. They look at how much infrared light is being emitted from the surface of the ocean and use that to determine the temperature of the surface of the ocean.

This process doesn’t work for land because there are all sorts of different materials over the surface of land. These different materials may emit different amounts of infrared light, even if they are at the same temperature. By contrast, the ocean’s surface is entirely water, a material whose properties we are extremely familiar with.

The land data is from weather stations all around the globe. Each weather station has an accurate thermometer and reports the measured temperature regularly. Weather stations in rural areas are preferred and weather stations in urban areas are often excluded. Cities are hotter than the surrounding countryside because concrete, blacktop, and shingles absorb more sunlight than natural plants. Since all cities have grown over the last 100 years, this could bias the temperature measurements. It is better to only use the rural weather stations.

More recent data can be found here.

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Since we have a physical explanation for why increased carbon dioxide might cause increased temperature (the greenhouse effect), let’s plot temperature vs carbon dioxide concentration.

The shapes of the two previous graphs are the same, so we expect the two to be correlated.^[3]Correlation doesn’t imply causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing ‘look over there’. – xkcd 552, mouseover text.

The vertical axis is how much the temperature has changed from the pre-industrial average.

The horizontal axis is what fraction of a $CO_2$ doubling has occurred.

Since counting the number of doublings is exponential growth (doubling three times means multiplying the original amount by 8), we need to include a log to get rid of the exponential.

This is measured with respect to the pre-industrial average (277 ppm in 1750).

We also have theoretical reasons (which I will not describe here) to believe that the temperature change should be proportional to the log of the carbon dioxide concentration.

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These are definitely correlated. This can be seen even without any statistical tests.^[4]Rutherford, who had strong opinions on what was and was not proper science, once said: “If your experiment requires statistics, you should design a better experiment.” The data should be … Continue reading The fluctuations are much smaller than the trend line.

Notice that the data look more compressed on the left side of the chart (before the 1970s) than on the right side of the chart. This is because the change in carbon dioxide concentration and temperature have been occurring more quickly since then.

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There are two features highlighted on the right side of the graph.

The farthest right is the ‘pause’ much championed by climate change skeptics. Between 1998 and 2014, there was very little warming. Skeptics argued that this means that climate change had stopped on its own and had nothing to do with human activities.

Lovejoy points out that this pause is really just fluctuations about the the trend line shown here. The rapid warming immediately before the pause is actually more anomalous than the pause itself. Neither of them is larger than the deviations from the trend line that had occurred previously.

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This trend line indicates that doubling the carbon dioxide in the atmosphere causes the global temperature to increase by $2.3^\circ C$ ($4.2^\circ F$).

Since the data do nicely follow a straight line, our estimate of the slope of the line wouldn’t change if we only consider data before 1990 and use that to predict the rest of the data.

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The last graph showed how temperature depends on carbon dioxide concentration. We also have measured how the carbon dioxide concentration changes in time. If you put these two things together, we can get a prediction for how temperature changes in time.

The jagged line is the measured temperature vs time. We saw this line above.

To make the smooth line, take the measured carbon dioxide at each time and calculate the corresponding temperature, using the relationship in the previous plot.

Mathematically, we are comparing the empirical temperature as a function of time $T(t)$ with the composition of the functions: temperature vs carbon dioxide concentration and carbon dioxide concentration vs time $T(CO_2(t))$.

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The two graphs remain quite close to each other. This means that we can accurately predict the temperature using the carbon dioxide concentration alone.

This is really a postdiction instead of a prediction because we’re only using data that we already had. But we know we can’t have overfit this data because our model only has one parameter: the slope of the temperature vs carbon dioxide concentration graph.

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This graph shows the difference between the two lines on the previous graph: the actual measured temperature minus the temperature change that we can predict using the carbon dioxide concentration. It shows the temperature changes that cannot be explained simply using the increase in carbon dioxide in the atmosphere.

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These look like random variations. There is no discernible trend line once we’ve removed the effect of carbon dioxide.

We see fluctuations of magnitude of $\pm 0.2^\circ C$ ($\pm 0.4^\circ F$) that last for a few decades. We also see fluctuations of magnitude of $\pm 0.1^\circ C$ ($\pm 0.2^\circ F$) between individual years. We do see any fluctuations over shorter time scales because the data presented are yearly averages.

These fluctuations are smaller than what we saw in Figure 2.4 because we are looking at global averages, not at an individual location. Montreal tends to have annual fluctuations of about $2^\circ C$ ($4^\circ F$) each year and about $0.8^\circ C$ (about $1.5^\circ F$) fluctuations each decade or century. The difference between local fluctuations and global fluctuations shrinks as we look at longer time scales.

These are macroweather fluctuations. Some of them have names, like the El Niño-Southern Oscillation in the Pacific Ocean and the Atlantic Multidecadal Oscillation. Lovejoy prefers not to think of them as distinct phenomena. Instead, they are large examples of atmosphere fluctuations which are similar on many scales.

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Similar macroweather fluctuations can also be seen in paleoclimate data.

The top graph of this image shows globally averaged temperature vs time since 1880 (again). If you subtract out the temperature change caused by carbon dioxide, you get the dashed line. The dashed line is the macroweather fluctuations we saw in the last graph.

The lower graphs show similar data for previous centuries. We can see that the data from 1880-2013 look clearly different from the data for previous centuries. The historical fluctuations look like modern macroweather fluctuations after we remove the trend we predicted using the carbon dioxide concentration.

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Now that we have a decent understanding of the Earth’s temperature in the past, we should look at the future.

Predicting the future climate obviously depends on how much carbon dioxide we add to the atmosphere. If we consider various possibilities, we can predict how much the temperature will change in each of these scenarios. The International Panel on Climate Change (IPCC) focuses on three scenarios: no reduction in carbon dioxide production, some reduction, and a lot of reduction. I will focus on the intermediate case (some reduction of carbon dioxide production). Lovejoy does his analysis for all three cases.

This is another graph of temperature change vs time.

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There are four data sets presented on this graph: orange (and jagged), black, red, and blue. Each data set has error bars, the shaded region around the central line. This model is 90% confident that the actual temperature is within the shaded region.

The jagged orange data is the measured globally averaged temperature for the years 1880-2016.

The black data are the predictions / postdictions from GCMs, compiled by the Coupled Model Intercomparison Project Phase 5 (CMIP5) and reported in the IPCC’s Fifth Assessment Report in 2014. The error bars are determined by running different GCMs with slightly different assumptions and comparing how the climate responds in each run. Since these are massive simulations with thousands of parameters, I am worried about overfitting the historical data – note how much smaller the error bars are for the years 1960-2000.

The red data are from Lovejoy’s simplest possible model that I described above. Take the current amount of carbon dioxide in the atmosphere (in this scenario) and use the relationship between temperature and carbon dioxide to calculate the temperature. The width of the error bars is the typical size of macroweather fluctuations: $\pm 0.2^\circ C$ ($\pm 0.4^\circ F$).

The blue data are another model that Lovejoy presents in the book. The red model assumes that the temperature depends on the carbon dioxide concentration now. The blue model assumes that, when the carbon dioxide concentration changes, it takes some time before the temperature changes as well. The temperature should be calculated using last year’s carbon dioxide instead of this year’s.^[5]It’s actually a weighted average of multiple previous years. As long as carbon dioxide use continues to increase, temperature will as well, so this hasn’t affected our predictions so far. But when we significantly reduce the amount of carbon dioxide we use, the temperature will still increase for a while longer. That is why the blue curve flattens out later and at a higher temperature than the red curve. Since we don’t know how long the delay between carbon dioxide concentration and temperature is, the blue curve also has larger error bars than the red curve. SCRF (scaling climate response function) refers to the blue model.

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The measured data is within the error bars for all three models: GCMs, Lovejoy’s simple model, and Lovejoy’s model with time delay. Lovejoy’s models have much smaller error bars, so they are performing better.

In particular, look at the zoomed in data from 1990-2015, which includes the infamous pause. The measured data grew more slowly than the average prediction from GCMs and got dangerously close to the lower boundary of the error bars. Lovejoy’s models do not have this problem and continue to go right through the center of the measured data.

One of the questions that the IPCC likes to consider is: By what year will the Earth have warmed by $1.5^\circ C$ ($2.7^\circ F$)? GCMs predict that this will occur sometime between 2016 and 2050. Lovejoy’s blue model predicts that this will occur sometime between 2030 and 2040. This is a much more precise prediction and so provides a much better test for the model.

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Conclusion

The atmosphere is a large, turbulent flow. The relevant size of the turbulence ranges from little whirls that chase dry leaves to the size of the planet itself. Trying to describe the atmosphere as a collection of different phenomena, each of which has a particular size, is hopelessly complicated.

Luckily, we don’t have to do that. The atmosphere looks similar to itself if we rescale in horizontal distance, vertical height, and time in the right way. It can be described by an anisotropic multifractal.

The important thing to look at is the scaling law – the rules we use make something look similar to itself when we change its size. The most intuitive way to see this is to plot the magnitude of the fluctuations vs the size we are averaging over. If the data follow a single scaling law, the plot will be a straight line.

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When we look at temperature fluctuations vs time scale (Figure 2.4), we see multiple different scaling laws.

For times less than about 10 days, there is one scaling law with a positive slope. This is the weather. The positive slope means that it is unstable. As you average over longer and longer times, the changes get larger.

For times between 10 days and about 100 years, there is a different scaling law with a negative slope. This is the macroweather. The negative slope means that it is stable. As you average over longer and longer times, the changes get smaller.

For times longer than about 100 years, there is yet another scaling law with a positive slope. This is the climate. The positive slope means that it is unstable. As you average over longer and longer times, the changes get larger.

If you really believe in paleoclimate data, you can see more different scaling laws for time scales longer than 100,000 years.

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The main techniques we use to predict the climate are General Circulation Models / Global Climate Models (GCMs). These are massive computer simulations which attempt to model everything happening in the climate — at least down to a $100 km$ grid. Anything that happens on a scale smaller than this has to be approximated empirically (or if we don’t have relevant empirical measurements, by guessing parameters and seeing if the resulting climate makes sense).

Since they do not take into account the multifractal nature of atmospheric motion and since we do not fully understand all of the sub-grid processes, GCMs do not provide accurate predictions of the climate. Instead, we run many different versions and average the results. This gives us predictions with huge error bars, and still has problems with overfitting the data.

Since we cannot get a fully accurate model of the entire atmosphere, we should try to make much simpler models. A good model is not one that includes the most things. A good model includes the least, while still giving accurate predictions.

The basic explanation for climate change is: Rising $CO_2$ levels (caused by burning fossil fuels) cause the Earth’s temperature to increase. Lovejoy’s model for climate change is that simple. He takes the empirically determined relationship between carbon dioxide concentration and temperature and the measured or projected carbon dioxide concentration as a function of time and uses them to calculate the temperature as a function of time.

Any further complications can be added if and when they are needed. One potentially important complication is to include time delay. The temperature depends on what the carbon dioxide concentration was a year or two ago, not what it is now. As long as the carbon dioxide concentration continues to increase, this gives the same predictions. But once we significantly reduce our carbon dioxide emissions, this would become important.

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The biggest question in the climate change debate is the probability that the warming that we’ve observed is a natural fluctuation. Typically, the globally averaged temperature changes by about $0.2^\circ C$ ($0.4^\circ F$) over the course of a century. During the last hundred years, the globally average temperature has increased by about $1^\circ C$ ($1.8^\circ F$). If you don’t use any statistics, you might think that this is not very surprising and set equal probability for climate change being natural and man-made. If you used Gaussian statistics, you would think that the probability that the temperature increased this much randomly is $1/3,000,000$. But atmospheric fluctuations do not have Gaussian statistics. Using the appropriate power law statistics for the atmosphere, we can conclude that the likelihood of this large of a temperature fluctuation occurring naturally is $1/5,000$. This calculation can be done using (relatively) simple statistical techniques, instead of running simulations of the motion of the atmosphere for decades.