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## Cover Your Ears and Sing Loudly

At public hearings on the environmental impacts of proposed oil pipelines, Canadians are no longer allowed to discuss climate change: any testimonials concerning how the oil was produced (“upstream effects”) and what will happen when it is burned (“downstream effects”) are considered inadmissible. This new policy was part of a 2012 omnibus bill by the federal government.

So if we refuse to consider the risks, they don’t exist? Or does this government just not care? I’m not sure I want to know the answer.

See the very thoughtful article by Andy Skuce, a geologist who formerly worked in the Alberta oil sands.

## 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.

## Explorers

On Monday evening, a Canadian research helicopter in northwest Nunavut crashed into the Arctic Ocean. Three men from the CGCS Amundsen research vessel were on board, examining the sea ice from above to determine the best route for the ship to take. All three were killed in the crash: climate scientist Klaus Hochheim, commanding officer Marc Thibault, and pilot Daniel Dubé.

The Amundsen recovered the bodies, which will be entrusted to the RCMP as soon as the ship reaches land. The helicopter remains at the bottom of the Arctic Ocean (~400 m deep); until it can be retrieved, the cause of the crash will remain unknown.

Klaus Hochheim

During my first two years of university, I worked on and off in the same lab as Klaus. He was often in the field, and I was often rushing off to class, so we only spoke a few times. He was very friendly and energetic, and I regret not getting to know him better. My thoughts are with the families, friends, and close colleagues of these three men, who have far more to mourn than I do.

Perhaps some solace can be found in the thought that they died doing what they loved best. All of the Arctic scientists I know are incredibly passionate about their field work: bring them down south for too long, and they start itching to get back on the ship. In the modern day, field scientists are perhaps the closest thing we have to explorers. Such a demanding job comes with immense personal and societal rewards, but also with risks.

These events remind me of another team of explorers that died while pursuing their calling, at the opposite pole and over a hundred years ago: the Antarctic expedition of 1912 led by Robert Falcon Scott. While I was travelling in New Zealand, I visited the Scott Memorial in the Queenstown public gardens. Carved into a stone tablet and set into the side of a boulder is an excerpt from Scott’s last diary entry. I thought the words were relevant to Monday night’s tragedy, so I have reproduced them below.

We arrived within eleven miles of our old One Ton camp with fuel for one hot meal and food for two days. For four days we have been unable to leave the tent, the gale is howling about us. We are weak, writing is difficult, but, for my own sake, I do not regret this journey, which has shown that Englishmen can endure hardships, help one another, and meet death with as great a fortitude as ever in the past.

We took risks; we knew we took them. Things have come out against us, and therefore we have no cause for complaint, but bow to the will of providence, determined still to do our best to the last.

Had we lived I should have had a tale to tell of the hardihood, endurance, and courage of my companions which would have stirred the heart of every Englishman.

These rough notes and our dead bodies must tell the tale.

## Bits and Pieces

Now that the academic summer is over, I have left Australia and returned home to Canada. It is great to be with my friends and family again, but I really miss the ocean and the giant monster bats. Not to mention the lab: after four months as a proper scientist, it’s very hard to be an undergrad again.

While I continue to settle in, move to a new apartment, and recover from jet lag (which is way worse in this direction!), here are a few pieces of reading to tide you over:

Scott Johnson from Ars Technica wrote a fabulous piece about climate modelling, and the process by which scientists build and test new components. The article is accurate and compelling, and features interviews with two of my former supervisors (Steve Easterbrook and Andrew Weaver) and lots of other great communicators (Gavin Schmidt and Richard Alley, to name a few).

I have just started reading A Short History of Nearly Everything by Bill Bryson. So far, it is one of the best pieces of science writing I have ever read. As well as being funny and easy to understand, it makes me excited about areas of science I haven’t studied since high school.

Finally, my third and final paper from last summer in Victoria was published in the August edition of Journal of Climate. The full text (subscription required) is available here. It is a companion paper to our recent Climate of the Past study, and compares the projections of EMICs (Earth System Models of Intermediate Complexity) when forced with different RCP scenarios. In a nutshell, we found that even after anthropogenic emissions fall to zero, it takes a very long time for CO2 concentrations to recover, even longer for global temperatures to start falling, and longer still for sea level rise (caused by thermal expansion alone, i.e. neglecting the melting of ice sheets) to stabilize, let alone reverse.

## The Mueller Glacier

Recently I was lucky enough to pay a visit to the South Island of New Zealand. I am actually a Kiwi by birth (that’s why it’s so easy for me to work in Australia) but grew up in Canada. This was my first visit back since I left as a baby – in fact, we were in my hometown exactly 20 years to the day after I left. We didn’t plan this, but it was a neat coincidence to discover.

Among the many places we visited was Aoraki / Mt. Cook National Park in the Southern Alps. It was my first experience of an alpine environment and I absolutely loved it. It was also my first glacier sighting – a momentous day in the life of any climate scientist.

There are 72 named glaciers in the park, of which we saw two: the Hooker Glacier and the Mueller Glacier. The latter is pictured below as seen from the valley floor – the thick, blue-tinged ice near the bottom of the visible portion of the mountain. As it flows downward it becomes coated in dirt and is much less pretty.

Along with most of the world’s glaciers, the Mueller is retreating (see these satellite images by NASA). At the base of the mountain on which it flows, there is a large terminal lake, coloured bright blue and green from the presence of “glacial flour” (rock ground up by the ice). According to the signs at the park, this lake has only existed since 1974.

In the photo above, you can see a large black “sill” behind the lake, which is the glacial moraine showing the previous extent of the ice. Here’s a photo of the moraines on the other side, looking down the valley:

It’s hard to capture the scale of the melt, even in photos. But when you stand beside it, the now-empty glacial valley is unbelievably huge. The fact that it was full of ice just 50 years ago boggles my mind. Changes like that don’t happen for no reason.

## A Simple Stochastic Climate Model: Deriving the Backbone

Last time I introduced the concept of a simple climate model which uses stochastic techniques to simulate uncertainty in our knowledge of the climate system. Here I will derive the backbone of this model, an ODE describing the response of global temperature to net radiative forcing. This derivation is based on unpublished work by Nathan Urban – many thanks!

In reality, the climate system should be modelled not as a single ODE, but as a coupled system of hundreds of PDEs in four dimensions. Such a task is about as arduous as numerical science can get, but dozens of research groups around the world have built GCMs (General Circulation Models, or Global Climate Models, depending on who you talk to) which come quite close to this ideal.

Each GCM has taken hundreds of person-years to develop, and I only had eight weeks. So for the purposes of this project, I treat the Earth as a spatially uniform body with a single temperature. This is clearly a huge simplification but I decided it was necessary.

Let’s start by defining T1(t) to be the absolute temperature of this spatially uniform Earth at time t, and let its heat capacity be C. Therefore,

$C \: T_1(t) = E$

where E is the change in energy required to warm the Earth from 0 K to temperature T1. Taking the time derivative of both sides,

$C \: \frac{dT_1}{dt} = \frac{dE}{dt}$

Now, divide through by A, the surface area of the Earth:

$c \: \frac{dT_1}{dt} = \frac{1}{A} \frac{dE}{dt}$

where c = C/A is the heat capacity per unit area. Note that the right side of the equation, a change in energy per unit time per unit area, has units of W/m2. We can express this as the difference of incoming and outgoing radiative fluxes, I(t) and O(t) respectively:

$c \: \frac{dT_1}{dt} = I(t)- O(t)$

By the Stefan-Boltzmann Law,

$c \: \frac{dT_1}{dt} = I(t) - \epsilon \sigma T_1(t)^4$

where ϵ is the emissivity of the Earth and σ is the Stefan-Boltzmann constant.

To consider the effect of a change in temperature, suppose that T1(t) = T0 + T(t), where T0 is an initial equilibrium temperature and T(t) is a temperature anomaly. Substituting into the equation,

$c \: \frac{d(T_0 + T(t))}{dt} = I(t) - \epsilon \sigma (T_0 + T(t))^4$

Noting that T0 is a constant, and also factoring the right side,

$c \: \frac{dT}{dt} = I(t) - \epsilon \sigma T_0^4 (1 + \tfrac{T(t)}{T_0})^4$

Since the absolute temperature of the Earth is around 280 K, and we are interested in perturbations of around 5 K, we can assume that T(t)/T0 ≪ 1. So we can linearize (1 + T(t)/T0)4 using a Taylor expansion about T(t) = 0:

$c \: \frac{dT}{dt} = I(t) - \epsilon \sigma T_0^4 (1 + 4 \tfrac{T(t)}{T_0} + O[(\tfrac{T(t)}{T_0})^2])$

$\approx I(t) - \epsilon \sigma T_0^4 (1 + 4 \tfrac{T(t)}{T_0})$

$= I(t) - \epsilon \sigma T_0^4 - 4 \epsilon \sigma T_0^3 T(t)$

Next, let O0 = ϵσT04 be the initial outgoing flux. So,

$c \: \frac{dT}{dt} = I(t) - O_0 - 4 \epsilon \sigma T_0^3 T(t)$

Let F(t) = I(t) – O0 be the radiative forcing at time t. Making this substitution as well as dividing by c, we have

$\frac{dT}{dt} = \frac{F(t) - 4 \epsilon \sigma T_0^3 T(t)}{c}$

Dividing each term by 4ϵσT03 and rearranging the numerator,

$\frac{dT}{dt} = - \frac{T(t) - \tfrac{1}{4 \epsilon \sigma T_0^3} F(t)}{\tfrac{c}{4 \epsilon \sigma T_0^3}}$

Finally, let S = 1/(4ϵσT03) and τ = cS. Our final equation is

$\frac{dT}{dt} = - \frac{T(t) - S F(t)}{\tau}$

While S depends on the initial temperature T0, all of the model runs for this project begin in the preindustrial period when global temperature is approximately constant. Therefore, we can treat S as a parameter independent of initial conditions. As I will show in the next post, the uncertainty in S based on climate system dynamics far overwhelms any error we might introduce by disregarding T0.

## A Simple Stochastic Climate Model: Introduction

This winter I took a course in computational physics, which has probably been my favourite undergraduate course to date. Essentially it was an advanced numerical methods course, but from a very practical point of view. We got a lot of practice using numerical techniques to solve realistic problems, rather than just analysing error estimates and proving conditions of convergence. As a math student I found this refreshing, and incredibly useful for my research career.

We all had to complete a term project of our choice, and I decided to build a small climate model. I was particularly interested in the stochastic techniques taught in the course, and given that modern GCMs and EMICs are almost entirely deterministic, it was possible that I could contribute something original to the field.

The basic premise of my model is this: All anthropogenic forcings are deterministic, and chosen by the user. Everything else is determined stochastically: parameters such as climate sensitivity are sampled from probability distributions, whereas natural forcings are randomly generated but follow the same general pattern that exists in observations. The idea is to run this model with the same anthropogenic input hundreds of times and build up a probability distribution of future temperature trajectories. The spread in possible scenarios is entirely due to uncertainty in the natural processes involved.

This approach mimics the real world, because the only part of the climate system we have full control over is our own actions. Other influences on climate are out of our control, sometimes poorly understood, and often unpredictable. It is just begging to be modelled as a stochastic system. (Not that it is actually stochastic, of course; in fact, I understand that nothing is truly stochastic, even random number generators – unless you can find a counterexample using quantum mechanics? But that’s a discussion for another time.)

A word of caution: I built this model in about eight weeks. As such, it is highly simplified and leaves out a lot of processes. You should never ever use it for real climate projections. This project is purely an exercise in numerical methods, and an exploration of the possible role of stochastic techniques in climate modelling.

Over the coming weeks, I will write a series of posts that explains each component of my simple stochastic climate model in detail. I will show the results from some sample simulations, and discuss how one might apply these stochastic techniques to existing GCMs. I also plan to make the code available to anyone who’s interested – it’s written in Matlab, although I might translate it to a free language like Python, partly because I need an excuse to finally learn Python.

I am very excited to finally share this project with you all! Check back soon for the next installment.