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Posts Tagged ‘education’

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.

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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):

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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.

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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.

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I haven’t forgotten about this project! Read the introduction and ODE derivation if you haven’t already.

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.

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.

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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.

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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.

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