# A Simple Stochastic Climate Model: Climate Sensitivity

Last time I derived the following ODE for temperature T at time t:



where S and τ are constants, and F(t) is the net radiative forcing at time t. Eventually I will discuss each of these terms in detail; this post will focus on S.

At equilibrium, when dT/dt = 0, the ODE necessitates T(t) = S F(t). A physical interpretation for S becomes apparent: it measures the equilibrium change in temperature per unit forcing, also known as climate sensitivity.

A great deal of research has been conducted with the aim of quantifying climate sensitivity, through paleoclimate analyses, modelling experiments, and instrumental data. Overall, these assessments show that climate sensitivity is on the order of 3 K per doubling of CO2 (divide by 5.35 ln 2 W/m2 to convert to warming per unit forcing).

The IPCC AR4 report (note that AR5 was not yet published at the time of my calculations) compared many different probability distribution functions (PDFs) of climate sensitivity, shown below. They follow the same general shape of a shifted distribution with a long tail to the right, and average 5-95% confidence intervals of around 1.5 to 7 K per doubling of CO2.

Box 10.2, Figure 1 of the IPCC AR4 WG1: Probability distribution functions of climate sensitivity (a), 5-95% confidence intervals (b).

These PDFs generally consist of discrete data points that are not publicly available. Consequently, sampling from any existing PDF would be difficult. Instead, I chose to create my own PDF of climate sensitivity, modelled as a log-normal distribution (e raised to the power of a normal distribution) with the same shape and bounds as the existing datasets.

The challenge was to find values for μ and σ, the mean and standard deviation of the corresponding normal distribution, such that for any z sampled from the log-normal distribution,





Since erf, the error function, cannot be evaluated analytically, this two-parameter problem must be solved numerically. I built a simple particle swarm optimizer to find the solution, which consistently yielded results of μ = 1.1757, σ = 0.4683.

The upper tail of a log-normal distribution is unbounded, so I truncated the distribution at 10 K, consistent with existing PDFs (see figure above). At the beginning of each simulation, climate sensitivity in my model is sampled from this distribution and held fixed for the entire run. A histogram of 106 sampled points, shown below, has the desired characteristics.

Histogram of 106 points sampled from the log-normal distribution used for climate sensitivity in the model.

Note that in order to be used in the ODE, the sampled points must then be converted to units of Km2/W (warming per unit forcing) by dividing by 5.35 ln 2 W/m2, the forcing from doubled CO2.

# Uncertainty

Part 5 in a series of 5 for NextGen Journal
Read Part 1, Part 2, Part 3, and Part 4

Scientists can never say that something is 100% certain, but they can come pretty close. After a while, a theory becomes so strong that the academic community accepts it and moves on to more interesting problems. Replicating an experiment for the thousandth time just isn’t a good use of scientific resources. For example, conducting a medical trial to confirm that smoking increases one’s risk of cancer is no longer very useful; we covered that decades ago. Instead, a medical trial to test the effectiveness of different strategies to help people quit smoking will lead to much greater scientific and societal benefit.

In the same manner, scientists have known since the 1970s that human emissions of greenhouse gases are exerting a warming force on the climate. More recently, the warming started to show up, in certain patterns that confirm it is caused by our activities. These facts are no longer controversial in the scientific community (the opinion pages of newspapers are another story, though). While they will always have a tiny bit of uncertainty, it’s time to move on to more interesting problems. So where are the real uncertainties? What are the new frontiers of climate science?

First of all, projections of climate change depend on what the world decides to do about climate change – a metric that is more uncertain than any of the physics underlying our understanding of the problem. If we collectively step up and reduce our emissions, both quickly and significantly, the world won’t warm too much. If we ignore the problem and do nothing, it will warm a great deal. At this point, our actions could go either way.

Additionally, even though we know the world is going to warm, we don’t know exactly how much, even given a particular emission scenario. We don’t know exactly how sensitive the climate system is, because it’s quite a complex beast. However, using climate models and historical data, we can get an idea. Here is a probability density function for climate sensitivity: the greater the area under the curve at a specific point on the x-axis, the greater the probability that the climate sensitivity is equal to that value of x (IPCC, 2007):

This curve shows us that climate sensitivity is most likely around 3 degrees Celsius for every doubling of atmospheric carbon dixoide, since that’s where the area peaks. There’s a small chance that it’s less than that, so the world might warm a little less. But there’s a greater chance that climate sensitivity is greater than 3 degrees so the world will warm more. So this graph tells us something kind of scary: if we’re wrong about climate sensitivity being about 3 degrees, we’re probably wrong in the direction we don’t want – that is, the problem being worse than we expect. This metric has a lot to do with positive feedbacks (“vicious cycles” of warming) in the climate system.

Another area of uncertainty is precipitation. Temperature is a lot easier to forecast than precipitation, both regionally and globally. With global warming, the extra thermal energy in the climate system will lead to more water in the air, so there will be more precipitation overall – but the extra energy also drives evaporation of surface water to increase. Some areas will experience flooding, and some will experience drought; many areas will experience some of each, depending on the time of year. In summary, we will have more of each extreme when it comes to precipitation, but the when and where is highly uncertain.

Scientists are also unsure about the rate and extent of future sea level rise. Warming causes the sea to rise for two different reasons:

1. Water expands as it warms, which is easy to model;
2. Glaciers and ice sheets melt and fall into the ocean, which is very difficult to model.

If we cause the Earth to warm indefinitely, all the ice in the world will turn into water, but we won’t get that far (hopefully). So how much ice will melt, and how fast will it go? This depends on feedbacks in the climate system, glacial dynamics, and many other phenomena that are quantitatively poorly understood.

These examples of uncertainty in climate science, just a few of many, don’t give us an excuse to do nothing about the problem. As Brian, a Master’s student from Canada, wrote, “You don’t have to have the seventh decimal place filled in to see that the number isn’t looking good.”. We know that there is a problem, and it might be somewhat better or somewhat worse than scientists are currently predicting, but it won’t go away. As we noted above, in many cases it’s more likely to be worse than it is to be better. Even a shallow understanding of the implications of “worse” should be enough for anyone to see the necessity of action.

# Storms of my Grandchildren

I hope everyone had a fun and relaxing Christmas. Here’s a book I’ve been meaning to review for a while.

The worst part of the recent book by NASA climatologist James Hansen is, undoubtedly, the subtitle. The truth about the coming climate catastrophe and our last chance to save humanity – really? That doesn’t sound like the intrinsic, subdued style of Dr. Hansen. In my opinion, it simply alienates the very audience we’re trying to reach: moderate, concerned non-scientists.

The inside of the book is much better. While he couldn’t resist slipping in a good deal of hard science (and, in my opinion, these were the best parts), the real focus was on climate policy, and the relationship between science and policy. Hansen struggled with the prospect of becoming involved in policy discussions, but soon realized that he didn’t want his grandchildren, years from now, to look back at his work and say, “Opa understood what was happening, but he did not make it clear.”

Hansen is very good at distinguishing between his scientific work and his opinions on policy, and makes no secret of which he would rather spend time on. “I prefer to just do science,” he writes in the introduction. “It’s more pleasant, especially when you are having some success in your investigations. If I must serve as a witness, I intend to testify and then get back to the laboratory, where I am comfortable. That is what I intend to do when this book is finished.”

Hansen’s policy opinions centre on a cap-and-dividend system: a variant of a carbon tax, where revenue is divided evenly among citizens and returned to them. His argument for a carbon tax, rather than cap-and-trade, is compelling, and certainly convinced me. He also advocates the expansion of nuclear power (particularly “fourth-generation” fast nuclear reactors), a moratorium on new coal-generated power plants, and drastically improved efficiency measures.

These recommendations are robust, backed up with lots of empirical data to argue why they would be our best bet to minimize climate change and secure a stable future for generations to come. Hansen is always careful to say when he is speaking as a scientist and when he is speaking as a citizen, and provides a fascinating discussion of the connection between these two roles. As Bill Blakemore from ABC television wrote in correspondence with Hansen, “All communication is biased. What makes the difference between a propagandist on one side and a professional journalist or scientist on the other is not that the journalist or scientist ‘set their biases aside’ but that they are open about them and constantly putting them to the test, ready to change them.”

Despite all this, I love when Hansen puts on his scientist hat. The discussions of climate science in this book, particularly paleoclimate, were gripping. He explains our current knowledge of the climatic circumstances surrounding the Permian-Triassic extinction and the Paleocene-Eocene Thermal Maximum (usually referred to as the PETM). He explains why neither of these events is a suitable analogue for current climate change, as the current rate of introduction of the radiative forcing is faster than anything we can see in the paleoclimatic record.

Be prepared for some pretty terrifying facts about our planet’s “methane hydrate gun”, and how it wasn’t even fully loaded when it went off in the PETM. Also discussed is the dependence of climate sensitivity on forcing: the graph of these two variables is more or less a parabola, as climate sensitivity increases both in Snowball Earth conditions and in Runaway Greenhouse conditions. An extensive discussion of runaway greenhouse is provided, where the forcing occurs so quickly that negative feedbacks don’t have a chance to act before the positive water vapour feedback gets out of control, the oceans boil, and the planet becomes too hot for liquid water to exist. For those who are interested in this scenario, Hansen argues that, if we’re irresponsible about fossil fuels, it is quite possible for current climate change to reach this stage. For those who have less practice separating the scientific part of their brain from the emotional part, I suggest you skip this chapter.

I would recommend this book to everyone interested in climate change. James Hansen is such an important player in climate science, and has arguably contributed more to our knowledge of climate change than just about anyone. Whether it’s for the science, for the policy discussions, or for his try at science fiction in the last chapter, it’s well worth the cover price.

Thoughts from others who have read this book are welcome in the comments, as always.