Professor Stewart's Hoard of Mathematical Treasures (32 page)
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Authors: Ian Stewart
Tags: #Mathematics, #General
BOOK: Professor Stewart's Hoard of Mathematical Treasures
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Temperature always changes first (schematic picture for illustrative purposes).
Climate science relies heavily on mathematical models of the physical processes that influence the climate, so this is a mathematical issue as well as a scientific one. The best data available to date indicate that this effect is real, with the CO
2
peaks and troughs appearing about 100 years after those of temperature. So does this relationship prove that rising temperatures cause rising CO
2
, rather than the other way round? And what, if anything, does all this have to do with global warming?
2
peaks and troughs appearing about 100 years after those of temperature. So does this relationship prove that rising temperatures cause rising CO
2
, rather than the other way round? And what, if anything, does all this have to do with global warming?
Let’s put our brains in gear first. The graphs are well known to climatologists, and indeed are a big part of the evidence that human production of CO
2
is causing temperatures to rise. If those graphs really do prove that CO
2
is not responsible for rising temperatures, the climatologists are likely to have noticed. Yes, it could all be a big conspiracy, but governments worldwide would be much happier if climate change turned out to be a delusion, and they’re the ones paying for the research. If there’s a
conspiracy, it’s far more likely to be one that tries to suppress evidence of climate change. So it seems likely that the climatologists have worked out why this delay occurs, and have concluded that it does not show that CO
2
plays no significant role in climate change. And indeed they have: it takes 30 seconds on the internet to find the explanation.
2
is causing temperatures to rise. If those graphs really do prove that CO
2
is not responsible for rising temperatures, the climatologists are likely to have noticed. Yes, it could all be a big conspiracy, but governments worldwide would be much happier if climate change turned out to be a delusion, and they’re the ones paying for the research. If there’s a
conspiracy, it’s far more likely to be one that tries to suppress evidence of climate change. So it seems likely that the climatologists have worked out why this delay occurs, and have concluded that it does not show that CO
2
plays no significant role in climate change. And indeed they have: it takes 30 seconds on the internet to find the explanation.
What happens at times A, B, C, D and E?
So why does that 100-year delay happen? The full story is complicated, but the broad outlines aren’t hard to grasp if we think about the schematic picture, where the issues are easier to follow. The key facts are these:
• There is a natural cycle of changes in temperature caused by systematic changes in the Earth’s orbit, the tilt of its axis, and the direction in which the axis points.
• Rises in temperature do indeed cause CO
2
levels to rise, with a time delay of tens or hundreds of years for nature to respond to the temperature change.
2
levels to rise, with a time delay of tens or hundreds of years for nature to respond to the temperature change.
First, observe that most of the time, temperature and CO
2
levels rise together (between times B and C), or fall together (between times D and E). This shows temperature and CO
2
are linked, but it doesn’t tell us which is cause and which is effect. In fact, each causes the other.
2
levels rise together (between times B and C), or fall together (between times D and E). This shows temperature and CO
2
are linked, but it doesn’t tell us which is cause and which is effect. In fact, each causes the other.
What is actually going on here, according to the vast majority of climatologists, is roughly this. At time A the natural cycle causes temperatures to start rising, though not by much. By time B, a century or so later, the effect on CO
2
becomes apparent. This rise feeds back into the temperature, which responds far more
quickly to CO
2
levels than CO
2
levels do to temperature. So the temperature rises. Now temperature and CO
2
reinforce each other though positive feedback, and both climb together (times B to C). At time C, the external temperature cycle and other factors cause temperatures to begin to fall. The CO
2
levels don’t show much effect until time D, but as soon as they do, the fall in CO
2
reinforces the fall in temperature, and both drop together. This continues until time E, when the whole process repeats itself.
2
becomes apparent. This rise feeds back into the temperature, which responds far more
quickly to CO
2
levels than CO
2
levels do to temperature. So the temperature rises. Now temperature and CO
2
reinforce each other though positive feedback, and both climb together (times B to C). At time C, the external temperature cycle and other factors cause temperatures to begin to fall. The CO
2
levels don’t show much effect until time D, but as soon as they do, the fall in CO
2
reinforces the fall in temperature, and both drop together. This continues until time E, when the whole process repeats itself.
Next question: what does this have to do with global warming?
Not a lot.
What we’ve been discussing is a natural free-running cycle, without human intervention. The terms ‘global warming’ and ‘climate change’ do not refer to increasing temperatures or changes in climate as such. They refer, very specifically, to deviations from the natural cycle.
The term ‘global warming’ was used first by scientists who understood that point and also understood that what was being discussed was medium-term average global temperatures, not short-term local ones. It was widely misunderstood, because some parts of the globe may cool for a time, while others get warmer. So the term ‘climate change’ started to be used in the hope of avoiding confusion. But that phrase doesn’t just mean ‘the climate is changing’: that happens during the natural cycle. It means ‘the climate is changing in a way that the natural cycle does not explain’.
In the natural cycle, as we have seen, temperature influences CO
2
and CO
2
influences temperature. When the atmosphere is ‘forced’ by a changing cycle of solar radiation, both quantities respond. The issue of ‘global warming’ is: what do we expect to happen to that cycle if humans cause large quantities of CO
2
to enter the atmosphere? Mathematically, this amounts to giving a big kick to CO
2
, and seeing what the system does. And the answer is: the temperature promptly goes up too, because it responds fairly rapidly to changes in CO
2
.
2
and CO
2
influences temperature. When the atmosphere is ‘forced’ by a changing cycle of solar radiation, both quantities respond. The issue of ‘global warming’ is: what do we expect to happen to that cycle if humans cause large quantities of CO
2
to enter the atmosphere? Mathematically, this amounts to giving a big kick to CO
2
, and seeing what the system does. And the answer is: the temperature promptly goes up too, because it responds fairly rapidly to changes in CO
2
.
So the graphs, with that puzzling time delay, show what a
free-running atmospheric system does when it is forced by variations in incoming radiation. ‘Global warming’ isn’t about that at all. It’s about what this free-running system will do when you give it a kick. We know that human activity has raised CO
2
levels significantly over the past 50 years or so; in fact, they are now distinctly higher than anything found earlier in the ice core record. Look at the right-hand end of the graph of CO
2
on page 164. The proportions of various isotopes of carbon (different forms of the carbon atom with different atomic weight) show that this rise is mainly the result of human activity - and the unprecedented level of CO
2
in modern times confirms that.
free-running atmospheric system does when it is forced by variations in incoming radiation. ‘Global warming’ isn’t about that at all. It’s about what this free-running system will do when you give it a kick. We know that human activity has raised CO
2
levels significantly over the past 50 years or so; in fact, they are now distinctly higher than anything found earlier in the ice core record. Look at the right-hand end of the graph of CO
2
on page 164. The proportions of various isotopes of carbon (different forms of the carbon atom with different atomic weight) show that this rise is mainly the result of human activity - and the unprecedented level of CO
2
in modern times confirms that.
To test the hypothesis that this rise in CO
2
has led to global warming, the mathematical kick that we give to any model of the atmosphere also has to be a rise in CO
2
. So we are asking what effect this rise in CO
2
causes - in that context.
2
has led to global warming, the mathematical kick that we give to any model of the atmosphere also has to be a rise in CO
2
. So we are asking what effect this rise in CO
2
causes - in that context.
To check what happens, and to make it clear that this really is mathematics, I set up a simple system of model equations for how temperature T and carbon dioxide levels C change over time. It’s not ‘realistic’, but it has the basic features we are discussing, and illustrates the key point. It looks like this:
Here temperature is forced periodically (the sin
t
term) which models the changing heat coming from the Sun. Moreover, any change in C produces a proportionate change in T (the 0.25C term), and any change in T produces a proportionate change in C (the 0.1T term). So my model is set up so that higher temperatures cause more CO
2
, and more CO
2
causes higher temperatures, just like the real world. Since 0.25 is bigger than 0.1, temperature responds faster to changes in CO
2
than CO
2
does to changes in temperature. Finally, I subtract 0.01
T
2
and 0.01
C
2
to mimic the cut-off effects known to occur.
t
term) which models the changing heat coming from the Sun. Moreover, any change in C produces a proportionate change in T (the 0.25C term), and any change in T produces a proportionate change in C (the 0.1T term). So my model is set up so that higher temperatures cause more CO
2
, and more CO
2
causes higher temperatures, just like the real world. Since 0.25 is bigger than 0.1, temperature responds faster to changes in CO
2
than CO
2
does to changes in temperature. Finally, I subtract 0.01
T
2
and 0.01
C
2
to mimic the cut-off effects known to occur.
I now solve these equations on my computer, and see what I get. Here are three pictures of how T (black curve) and C (grey curve) change over time. I have plotted 4
y
- 60 rather than y to move the two curves close enough together to see the relationship.
y
- 60 rather than y to move the two curves close enough together to see the relationship.
• When the system is free-running, both T and C fluctuate periodically, and C lags behind T. This is the paradoxical time delay, which according to the TV programme means that rising CO
2
does not cause rising temperatures. But, in our model, rising CO
2
does cause rising temperatures, thanks to the 0.25C term in the first equation, yet we still see that time delay. The time delay is a consequence of non-linear effects in the model, not delays in what affects what.
2
does not cause rising temperatures. But, in our model, rising CO
2
does cause rising temperatures, thanks to the 0.25C term in the first equation, yet we still see that time delay. The time delay is a consequence of non-linear effects in the model, not delays in what affects what.
How temperature (black line) and CO
2
(grey line) vary over time. Note that CO
2
lags behind temperature.
2
(grey line) vary over time. Note that CO
2
lags behind temperature.
• When I give C a sudden brief increase at time 25, both T and C react. However, C still appears to lag behind T, and T doesn’t seem to change much.
The effect of a sudden increase in CO
2
(grey line).
2
(grey line).
• However, if I graph the changes in T and C between the two runs of the equations, then I see that T starts to increase as soon as C does. So a change in C does cause an immediate change in T. What’s interesting here is how temperature continues to increase while the spike in CO
2
is dying down. Non-linear dynamics can be counter-intuitive, which is why we have to use mathematics rather than naive verbal arguments.
2
is dying down. Non-linear dynamics can be counter-intuitive, which is why we have to use mathematics rather than naive verbal arguments.
Differences in CO
2
and temperature between the two runs show that temperature rises immediately CO
2
does.
2
and temperature between the two runs show that temperature rises immediately CO
2
does.
So the issue of ‘global warming’ or ‘climate change’ is not what causes what in the free-running system, where both rising temperature and rising CO
2
cause each other. Climate scientists
don’t dispute that and have known about it for a long time. The issue is: what happens when we know that one of these quantities has suddenly been changed by human activity? That much-trumpeted time delay is irrelevant to this question - in fact, it is misleading. The resulting temperature change begins immediately, and rises.
2
cause each other. Climate scientists
don’t dispute that and have known about it for a long time. The issue is: what happens when we know that one of these quantities has suddenly been changed by human activity? That much-trumpeted time delay is irrelevant to this question - in fact, it is misleading. The resulting temperature change begins immediately, and rises.
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