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## Explaining the Seasons on ‘Game of Thrones’

If you haven’t yet watched the television series Game of Thrones or read George R. R. Martin’s A Song of Ice and Fire books on which the show is based, I would urge you to get started (unless you are a small child, in which case I would urge you to wait a few years). The show and the books are both absolute masterpieces (although, as I alluded, definitely not for kids). I’m not usually a big fan of high fantasy, but the character and plot development of this series really pulled me in.

One of the most interesting parts of the series – maybe just for me – is the way the seasons work in Westeros and Essos, the continents explored in Game of Thrones. Winter and summer occur randomly, and can last anywhere from a couple of years to more than a decade. (Here a “year” is presumably defined by a complete rotation of the planet around the Sun, which can be discerned by the stars, rather than by one full cycle of the seasons.)

So what causes these random, multiyear seasons? Many people, George R. R. Martin included, brush off the causes as magical rather than scientific. To those people I say: you have no sense of fun.

After several lunchtime conversations with my friends from UNSW and U of T (few things are more fun than letting a group of climate scientists loose on a question like this), I think I’ve found a mechanism to explain the seasons. My hypothesis is simple, has been known to work on Earth, and satisfies all the criteria I can remember (I only read the books once and I didn’t take notes). I think that “winters” in Westeros are actually miniature ice ages, caused by the same orbital mechanisms which govern ice ages on Earth.

## Glacial Cycles on Earth

First let’s look at how ice ages – the cold phases of glacial cycles – work on Earth. At their most basic level, glacial cycles are caused by gravity: the gravity of other planets in the solar system, which influence Earth’s orbit around the Sun. Three main orbital cycles, known as Milankovitch cycles, result:

1. A 100,000 year cycle in eccentricity: how elliptical (as opposed to circular) Earth’s path around the Sun is.
2. A 41,000 year cycle in obliquity: the degree of Earth’s axial tilt.
3. A 26,000 year cycle in precession: what time of year the North Pole is pointing towards the Sun.

These three cycles combine to impact the timing and severity of the seasons in each hemisphere. The way they combine is not simple: the superposition of three sinusoidal functions with different periods is generally a mess, and often one cycle will cancel out the effects of another. However, sometimes the three cycles combine to make the Northern Hemisphere winter relatively warm, and the Northern Hemisphere summer relatively cool.

These conditions are ideal for glacier growth in the Northern Hemisphere. A warmer winter, as long as it’s still below freezing, will often actually cause more snow to fall. A cool summer will prevent that snow from entirely melting. And as soon as you’ve got snow that sticks around for the entire year, a glacier can begin to form.

Then the ice-albedo feedback kicks in. Snow and ice reflect more sunlight than bare ground, meaning less solar radiation is absorbed by the surface. This makes the Earth’s average temperature go down, so even less of the glacier will melt each summer. Now the glacier is larger and can reflect even more sunlight. This positive feedback loop, or “vicious cycle”, is incredibly powerful. Combined with carbon cycle feedbacks, it caused glaciers several kilometres thick to spread over most of North America and Eurasia during the last ice age.

The conditions are reversed in the Southern Hemisphere: relatively cold winters and hot summers, which cause glaciers to recede. However, at this stage in Earth’s history, most of the continents are concentrated in the Northern Hemisphere. The south is mostly ocean, where there are no glaciers to recede. For this reason, the Northern Hemisphere is the one which controls Earth’s glacial cycles.

These ice ages don’t last forever, because sooner or later the Milankovitch cycles will combine in the opposite way: the Northern Hemisphere will have cold winters and hot summers, and its glaciers will start to recede. The ice-albedo feedback will be reversed: less snow and ice means more sunlight is absorbed, which makes the planet warmer, which means there is less snow and ice, and so on.

## Glacial Cycles in Westeros?

I propose that Westeros (or rather, the unnamed planet which contains Westeros and Essos and any other undiscovered continents in Game of Thrones; let’s call it Westeros-world) experiences glacial cycles just like Earth, but the periods of the underlying Milankovitch cycles are much shorter – on the order of years to decades. This might imply the presence of very large planets close by, or a high number of planets in the solar system, or even multiple other solar systems which are close enough to exert significant gravitational attraction. As far as I know, all of these ideas are plausible, but I encourage any astronomers in the audience to chime in.

Given the climates of various regions in Game of Thrones, it’s clear that they all exist in the Northern Hemisphere: the further north you go, the colder it gets. The southernmost boundary of the known world is probably somewhere around the equator, because it never starts getting cold again as you travel south. Beyond that, the planet is unexplored, and it’s plausible that the Southern Hemisphere is mainly ocean. The concentration of continents in one hemisphere would allow Milankovitch cycles to induce glacial cycles in Westeros-world.

The glacial periods (“winter”) and interglacials (“summer”) would vary in length – again, on the scale of years to decades – and would appear random: the superposition of three different sine functions has an erratic pattern of peaks and troughs when you zoom in. Of course, the pattern of season lengths would eventually repeat itself, with a period equal to the least common multiple of the three Milankovitch cycle periods. But this least common multiple could be so large – centuries or even millennia – that the seasons would appear random on a human timescale. It’s not hard to believe that the people of Westeros, even the highly educated maesters, would fail to recognize a pattern which took hundreds or thousands of years to repeat.

Of course, within each glacial cycle there would be multiple smaller seasons as the planet revolved around the Sun – the way that regular seasons work on Earth. However, if the axial tilt of Westeros-world was sufficiently small, these regular seasons could be overwhelmed by the glacial cycles to the point where nobody would notice them.

There could be other hypotheses involving fluctuations in solar intensity, frequent volcanoes shooting sulfate aerosols into the stratosphere, or rapid carbon cycle feedbacks. But I think this one is the most plausible, because it’s known to happen on Earth (albeit on a much longer timescale). Can you find any holes? Please go nuts in the comments.

## Two Great TED Talks

Both are about climate modelling, and both are definitely worth 10-20 minutes of your time.

The first is from Gavin Schmidt, NASA climate modeller and RealClimate author extraordinaire:

The second is from Steve Easterbrook, my current supervisor at the University of Toronto (this one is actually TEDxUofT, which is independent from TED):

## What I am doing with my life

After a long hiatus – much longer than I like to think about or admit to – I am finally back. I just finished the last semester of my undergraduate degree, which was by far the busiest few months I’ve ever experienced.

This was largely due to my honours thesis, on which I spent probably three times more effort than was warranted. I built a (not very good, but still interesting) model of ocean circulation and implemented it in Python. It turns out that (surprise, surprise) it’s really hard to get a numerical solution to the Navier-Stokes equations to converge. I now have an enormous amount of respect for ocean models like MOM, POP, and NEMO, which are extremely realistic as well as extremely stable. I also feel like I know the physics governing ocean circulation inside out, which will definitely be useful going forward.

Convocation is not until early June, so I am spending the month of May back in Toronto working with Steve Easterbrook. We are finally finishing up our project on the software architecture of climate models, and writing it up into a paper which we hope to submit early this summer. It’s great to be back in Toronto, and to have a chance to revisit all of the interesting places I found the first time around.

In August I will be returning to Australia to begin a PhD in Climate Science at the University of New South Wales, with Katrin Meissner and Matthew England as my supervisors. I am so, so excited about this. It was a big decision to make but ultimately I’m confident it was the right one, and I can’t wait to see what adventures Australia will bring.

## The Arctic Has Barfed

I was scanning my blog stats the other day – partly to see if people were reading my new post on the Blue Mountains bushfires, partly because I just like graphs – when I noticed that an article I wrote nearly two years ago was suddenly getting more views than ever before:

The article in question highlights the scientific inaccuracies of the 2004 film The Day After Tomorrow, in which global warming leads to a new ice age. Now that I’ve taken more courses in thermodynamics I could definitely expand on the original post if I had the time and inclination to watch the film again…

I did a bit more digging in my stats and discovered that most viewers are reaching this article through Google searches such as “is the day after tomorrow true”, “is the day after tomorrow likely to happen”, and “movie review of a day after tomorrow if it is possible or impossible.” The answers are no, no, and impossible, respectively.

But why the sudden surge in interest? I think it is probably related to the record cold temperatures across much of the United States, an event which media outlets have dubbed the “polar vortex”. I prefer “Arctic barf”.

Part of the extremely cold air mass which covers the Arctic has essentially detached and spilled southward over North America. In other words, the Arctic has barfed on the USA. Less sexy terminology than “polar vortex”, perhaps, but I would argue it is more enlightening.

Greg Laden also has a good explanation:

The Polar Vortex, a huge system of swirling air that normally contains the polar cold air has shifted so it is not sitting right on the pole as it usually does. We are not seeing an expansion of cold, an ice age, or an anti-global warming phenomenon. We are seeing the usual cold polar air taking an excursion.

Note that other regions such as Alaska and much of Europe are currently experiencing unusually warm winter weather. On balance, the planet isn’t any colder than normal. The cold patches are just moving around in an unusual way.

Having grown up in the Canadian Prairies, where we experience daily lows below -30°C for at least a few days each year (and for nearly a month straight so far this winter), I can’t say I have a lot of sympathy. Or maybe I’m just bitter because I never got a day off school due to the cold? But seriously, nothing has to shut down if you plug in the cars at night and bundle up like an astronaut. We’ve been doing it for years.

## The Blue Mountains, Then and Now

During our time in Australia, my partner and I decided on a whim to spend a weekend in the Blue Mountains. This national park, a two-hour train ride west of Sydney, forms part of the Great Dividing Range: a chain of mountains which stretches from north to south across the entire country, separating the vast outback to the west from the narrow strip of coastal rainforest to the east.

For a region so close to Sydney, the Blue Mountains feel surprisingly remote. You can stand at any number of clifftops, gaze out over a seemingly endless stretch of land, and see no sign of civilization whatsoever. Or you can walk down into the valleys between the mountains and explore the rainforest, a vast expanse of ancient gumtrees that’s managed to hide koalas previously thought to have vanished, and possibly even an escaped panther.

Four months later, when we were safely back in Canada, the Blue Mountains bushfires began. It was October, barely even spring in the Southern Hemisphere. To have fires starting so early in the season was virtually unheard of.

The triggers for the fires were decidedly human-caused: arson, a botched army exercise, and sparking power lines. However, unusually hot, dry, and windy conditions allowed the fires to spread far more quickly than they would have in a more normal October.

To get from the clifftops of Echo Point to the walking trails in the valley below, we took the Giant Stairway, which is exactly what it sounds like. Imagine the steepest and narrowest stairway you can manage, cut into the stone cliff and reinforced with metal, and a handrail which you cling to for dear life. Make it 902 stairs long (by my count, so let’s say plus or minus 5) and wind it back and forth around the cliff. After a few minutes walking down the stairway your knees start to buckle, and you require more and longer breaks, but you still can’t see the bottom.

The exhaustion is worth it simply due to the view.

Sometimes we would see swarms of sulfur-crested cockatoos flying over the treetops hundreds of metres below. They looked like tiny white specks at such a distance, but we could still hear them squawking to one another.

The bushfires of 2013 didn’t affect any of the areas we visited in the Blue Mountains – in fact, none of the main tourism regions were damaged. The main losses occurred in residential areas in and around the Blue Mountains. As of October 19th, 208 houses and 40 non-residential buildings had been destroyed.

Despite the huge amount of property loss, there were only two fatalities from the bushfires. This relatively successful outcome was due to mass evacuations organized by the government of New South Wales. At one point a state of emergency was declared, which authorized police to force residents to leave their houses.

As the fires continued to burn out of control, westerly winds blew the smoke and ash right over Sydney. During sunsets the sky over Sydney Harbour turned a bright orange, giving the illusion of a city built on the surface of Mars.

I had heard about lyre birds, widely considered to be among the best mimics of the animal kingdom, many times before. In an elaborate courtship display, the male lyre bird perfectly imitates the songs of nearly every other bird in the forest, one after another like some kind of avian pop-music mashup. Lyre birds blow mockingbirds right out of the water.

Footage from the BBC of a lyre bird imitating camera shutters and chainsaws seemed too good to be true, but its authenticity was bolstered by a similar story from my friend at the climate lab in Sydney. Her neighbours had been doing renovations, and when they were finished the construction equipment went away but the sounds kept going. That’s when they discovered the lyre bird living in the garden.

We saw three or four lyre birds while hiking in the valley that weekend, but for the most part they just wandered around the forest floor, combing through the leaf litter with an outstretched foot and keeping their beaks firmly shut. It was winter in Australia, after all – not courtship season for most birds. On the last day of hiking, we sat by the side of the trail for a rest and a drink of water, while my partner quizzed me on the local bird calls.

“What kind of bird is making that song?”

“An eastern whip-bird, I think.

“Hang on, it just changed into a kookaburra.

“And now it’s a currawong?”

A few minutes later, a male lyre bird strolled out onto the path ahead of us, showing off his fantastic tail feathers and looking extremely pleased with himself.

It is well known among scientists that human-caused climate change increases the risk of severe bushfires. Spells of hot weather will obviously become more common as the planet warms, but so will prolonged droughts, especially in subtropical regions like Australia. Add an initial trigger, like a lightning strike or an abandoned campfire, and you have the perfect recipe for a bushfire.

The current Australian government, which has a history of questionable statements on climate change, really doesn’t want to believe this. Prime Minister Tony Abbott asserted that “these fires are certainly not a function of climate change, they’re a function of life in Australia”, while Environment Minister Greg Hunt cited Wikipedia during a similar statement. I was actually heartened by these events: the ensuing public outcry convinced me that Australians, by and large, do not buy into their government’s indifference on this issue.

It came as a surprise to nobody in the climate science community, and probably nobody in Australia, that 2013 was Australia’s warmest year on record. The previous record, set in 2005, was exceeded by a fairly significant 0.17°C. Even more remarkable was the fact that 2013 was an ENSO-neutral year. For Australia to shatter this temperature record without the help of El Niño indicates that something else (*cough cough climate change*) is at work.

Would the Blue Mountains bushfires have been so devastating without the help of human-caused climate change? In a cooler and wetter October, closer to the historical average, would the initial fire triggers have developed into anything significant? We’ll never know for sure. What we can say, though, is that bushfires like these will only become more common as climate change continues. This is what the future will look like.

## 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.

## 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.