Tag Archives: mathematics

A Simple Stochastic Climate Model: Climate Sensitivity

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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.

A Simple Stochastic Climate Model: Deriving the Backbone

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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.

A New Kind of Science

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Cross-posted from NextGen Journal

Ask most people to picture a scientist at work, and they’ll probably imagine someone in a lab coat and safety goggles, surrounded by test tubes and Bunsen burners. If they’re fans of The Big Bang Theory, maybe they’ll picture complicated equations being scribbled on whiteboards. Others might think of the Large Hadron Collider, or people wading through a swamp taking water samples.

All of these images are pretty accurate – real scientists, in one field or another, do these things as part of their job. But a large and growing approach to science, which is present in nearly every field, replaces the lab bench or swamp with a computer. Mathematical modelling, which essentially means programming the complicated equations from the whiteboard into a computer and solving them many times, is the science of today.

Computer models are used for all sorts of research questions. Epidemiologists build models of an avian flu outbreak, to see how the virus might spread through the population. Paleontologists build biomechanical models of different dinosaurs, to figure out how fast they could run or how high they could stretch their necks. I’m a research student in climate science, where we build models of the entire planet, to study the possible effects of global warming.

All of these models simulate systems which aren’t available in the real world. Avian flu hasn’t taken hold yet, and no sane scientist would deliberately start an outbreak just so they could study it! Dinosaurs are extinct, and playing around with their fossilized bones to see how they might move would be heavy and expensive. Finally, there’s only one Earth, and it’s currently in use. So models don’t replace lab and field work – rather, they add to it. Mathematical models let us perform controlled experiments that would otherwise be impossible.

If you’re interested in scientific modelling, spend your college years learning a lot of math, particularly calculus, differential equations, and numerical methods. The actual application of the modelling, like paleontology or climatology, is less important for now – you can pick that up later, or read about it on your own time. It might seem counter-intuitive to neglect the very system you’re planning to spend your life studying, but it’s far easier this way. A few weeks ago I was writing some computer code for our lab’s climate model, and I needed to calculate a double integral of baroclinic velocity in the Atlantic Ocean. I didn’t know what baroclinic velocity was, but it only took a few minutes to dig up a paper that defined it. My work would have been a lot harder if, instead, I hadn’t known what a double integral was.

It’s also important to become comfortable with computer programming. You might think it’s just the domain of software developers at Google or Apple, but it’s also the main tool of scientists all over the world. Two or three courses in computer science, where you’ll learn a multi-purpose language like C or Java, are all you need. Any other languages you need in the future will take you days, rather than months, to master. If you own a Mac or run Linux on a PC, spend a few hours learning some basic UNIX commands – it’ll save you a lot of time down the road. (Also, if the science plan falls through, computer science is one of the only majors which will almost definitely get you a high-paying job straight out of college.)

Computer models might seem mysterious, or even untrustworthy, when the news anchor mentions them in passing. In fact, they’re no less scientific than the equations that Sheldon Cooper scrawls on his whiteboard. They’re just packaged together in a different form.

How do climate models work?

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Also published at Skeptical Science

This is a climate model:

T = [(1-α)S/(4εσ)]1/4

(T is temperature, α is the albedo, S is the incoming solar radiation, ε is the emissivity, and σ is the Stefan-Boltzmann constant)

An extremely simplified climate model, that is. It’s one line long, and is at the heart of every computer model of global warming. Using basic thermodynamics, it calculates the temperature of the Earth based on incoming sunlight and the reflectivity of the surface. The model is zero-dimensional, treating the Earth as a point mass at a fixed time. It doesn’t consider the greenhouse effect, ocean currents, nutrient cycles, volcanoes, or pollution.

If you fix these deficiencies, the model becomes more and more complex. You have to derive many variables from physical laws, and use empirical data to approximate certain values. You have to repeat the calculations over and over for different parts of the Earth. Eventually the model is too complex to solve using pencil, paper and a pocket calculator. It’s necessary to program the equations into a computer, and that’s what climate scientists have been doing ever since computers were invented.

A pixellated Earth

Today’s most sophisticated climate models are called GCMs, which stands for General Circulation Model or Global Climate Model, depending on who you talk to. On average, they are about 500 000 lines of computer code long, and mainly written in Fortran, a scientific programming language. Despite the huge jump in complexity, GCMs have much in common with the one-line climate model above: they’re just a lot of basic physics equations put together.

Computers are great for doing a lot of calculations very quickly, but they have a disadvantage: computers are discrete, while the real world is continuous. To understand the term “discrete”, think about a digital photo. It’s composed of a finite number of pixels, which you can see if you zoom in far enough. The existence of these indivisible pixels, with clear boundaries between them, makes digital photos discrete. But the real world doesn’t work this way. If you look at the subject of your photo with your own eyes, it’s not pixellated, no matter how close you get – even if you look at it through a microscope. The real world is continuous (unless you’re working at the quantum level!)

Similarly, the surface of the world isn’t actually split up into three-dimensional cells (you can think of them as cubes, even though they’re usually wedge-shaped) where every climate variable – temperature, pressure, precipitation, clouds – is exactly the same everywhere in that cell. Unfortunately, that’s how scientists have to represent the world in climate models, because that’s the only way computers work. The same strategy is used for the fourth dimension, time, with discrete “timesteps” in the model, indicating how often calculations are repeated.

It would be fine if the cells could be really tiny – like a high-resolution digital photo that looks continuous even though it’s discrete – but doing calculations on cells that small would take so much computer power that the model would run slower than real time. As it is, the cubes are on the order of 100 km wide in most GCMs, and timesteps are on the order of hours to minutes, depending on the calculation. That might seem huge, but it’s about as good as you can get on today’s supercomputers. Remember that doubling the resolution of the model won’t just double the running time – instead, the running time will increase by a factor of sixteen (one doubling for each dimension).

Despite the seemingly enormous computer power available to us today, GCMs have always been limited by it. In fact, early computers were developed, in large part, to facilitate atmospheric models for weather and climate prediction.

Cracking the code

A climate model is actually a collection of models – typically an atmosphere model, an ocean model, a land model, and a sea ice model. Some GCMs split up the sub-models (let’s call them components) a bit differently, but that’s the most common arrangement.

Each component represents a staggering amount of complex, specialized processes. Here are just a few examples from the Community Earth System Model, developed at the National Center for Atmospheric Research in Boulder, Colorado:

  • Atmosphere: sea salt suspended in the air, three-dimensional wind velocity, the wavelengths of incoming sunlight
  • Ocean: phytoplankton, the iron cycle, the movement of tides
  • Land: soil hydrology, forest fires, air conditioning in cities
  • Sea Ice: pollution trapped within the ice, melt ponds, the age of different parts of the ice

Each component is developed independently, and as a result, they are highly encapsulated (bundled separately in the source code). However, the real world is not encapsulated – the land and ocean and air are very interconnected. Some central code is necessary to tie everything together. This piece of code is called the coupler, and it has two main purposes:

  1. Pass data between the components. This can get complicated if the components don’t all use the same grid (system of splitting the Earth up into cells).
  2. Control the main loop, or “time stepping loop”, which tells the components to perform their calculations in a certain order, once per time step.

For example, take a look at the IPSL (Institut Pierre Simon Laplace) climate model architecture. In the diagram below, each bubble represents an encapsulated piece of code, and the number of lines in this code is roughly proportional to the bubble’s area. Arrows represent data transfer, and the colour of each arrow shows where the data originated:

We can see that IPSL’s major components are atmosphere, land, and ocean (which also contains sea ice). The atmosphere is the most complex model, and land is the least. While both the atmosphere and the ocean use the coupler for data transfer, the land model does not – it’s simpler just to connect it directly to the atmosphere, since it uses the same grid, and doesn’t have to share much data with any other component. Land-ocean interactions are limited to surface runoff and coastal erosion, which are passed through the atmosphere in this model.

You can see diagrams like this for seven different GCMs, as well as a comparison of their different approaches to software architecture, in this summary of my research.

Show time

When it’s time to run the model, you might expect that scientists initialize the components with data collected from the real world. Actually, it’s more convenient to “spin up” the model: start with a dark, stationary Earth, turn the Sun on, start the Earth spinning, and wait until the atmosphere and ocean settle down into equilibrium. The resulting data fits perfectly into the cells, and matches up really nicely with observations. It fits within the bounds of the real climate, and could easily pass for real weather.

Scientists feed input files into the model, which contain the values of certain parameters, particularly agents that can cause climate change. These include the concentration of greenhouse gases, the intensity of sunlight, the amount of deforestation, and volcanoes that should erupt during the simulation. It’s also possible to give the model a different map to change the arrangement of continents. Through these input files, it’s possible to recreate the climate from just about any period of the Earth’s lifespan: the Jurassic Period, the last Ice Age, the present day…and even what the future might look like, depending on what we do (or don’t do) about global warming.

The highest resolution GCMs, on the fastest supercomputers, can simulate about 1 year for every day of real time. If you’re willing to sacrifice some complexity and go down to a lower resolution, you can speed things up considerably, and simulate millennia of climate change in a reasonable amount of time. For this reason, it’s useful to have a hierarchy of climate models with varying degrees of complexity.

As the model runs, every cell outputs the values of different variables (such as atmospheric pressure, ocean salinity, or forest cover) into a file, once per time step. The model can average these variables based on space and time, and calculate changes in the data. When the model is finished running, visualization software converts the rows and columns of numbers into more digestible maps and graphs. For example, this model output shows temperature change over the next century, depending on how many greenhouse gases we emit:

Predicting the past

So how do we know the models are working? Should we trust the predictions they make for the future? It’s not reasonable to wait for a hundred years to see if the predictions come true, so scientists have come up with a different test: tell the models to predict the past. For example, give the model the observed conditions of the year 1900, run it forward to 2000, and see if the climate it recreates matches up with observations from the real world.

This 20th-century run is one of many standard tests to verify that a GCM can accurately mimic the real world. It’s also common to recreate the last ice age, and compare the output to data from ice cores. While GCMs can travel even further back in time – for example, to recreate the climate that dinosaurs experienced – proxy data is so sparse and uncertain that you can’t really test these simulations. In fact, much of the scientific knowledge about pre-Ice Age climates actually comes from models!

Climate models aren’t perfect, but they are doing remarkably well. They pass the tests of predicting the past, and go even further. For example, scientists don’t know what causes El Niño, a phenomenon in the Pacific Ocean that affects weather worldwide. There are some hypotheses on what oceanic conditions can lead to an El Niño event, but nobody knows what the actual trigger is. Consequently, there’s no way to program El Niños into a GCM. But they show up anyway – the models spontaneously generate their own El Niños, somehow using the basic principles of fluid dynamics to simulate a phenomenon that remains fundamentally mysterious to us.

In some areas, the models are having trouble. Certain wind currents are notoriously difficult to simulate, and calculating regional climates requires an unaffordably high resolution. Phenomena that scientists can’t yet quantify, like the processes by which glaciers melt, or the self-reinforcing cycles of thawing permafrost, are also poorly represented. However, not knowing everything about the climate doesn’t mean scientists know nothing. Incomplete knowledge does not imply nonexistent knowledge – you don’t need to understand calculus to be able to say with confidence that 9 x 3 = 27.

Also, history has shown us that when climate models make mistakes, they tend to be too stable, and underestimate the potential for abrupt changes. Take the Arctic sea ice: just a few years ago, GCMs were predicting it would completely melt around 2100. Now, the estimate has been revised to 2030, as the ice melts faster than anyone anticipated:

Answering the big questions

At the end of the day, GCMs are the best prediction tools we have. If they all agree on an outcome, it would be silly to bet against them. However, the big questions, like “Is human activity warming the planet?”, don’t even require a model. The only things you need to answer those questions are a few fundamental physics and chemistry equations that we’ve known for over a century.

You could take climate models right out of the picture, and the answer wouldn’t change. Scientists would still be telling us that the Earth is warming, humans are causing it, and the consequences will likely be severe – unless we take action to stop it.

A Vast Machine

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I read Paul Edward’s A Vast Machine this summer while working with Steve Easterbrook. It was highly relevant to my research, but I would recommend it to anyone interested in climate change or mathematical modelling. Think The Discovery of Global Warming, but more specialized.

Much of the public seems to perceive observational data as superior to scientific models. The U.S. government has even attempted to mandate that research institutions focus on data above models, as if it is somehow more trustworthy. This is not the case. Data can have just as many problems as models, and when the two disagree, either could be wrong. For example, in a high school physics lab, I once calculated the acceleration due to gravity to be about 30 m/s2. There was nothing wrong with Newton’s Laws of Motion – our instrumentation was just faulty.

Additionally, data and models are inextricably linked. In meteorology, GCMs produce forecasts from observational data, but that same data from surface stations was fed through a series of algorithms – a model for interpolation – to make it cover an entire region. “Without models, there are no data,” Edwards proclaims, and he makes a convincing case.

The majority of the book discussed the history of climate modelling, from the 1800s until today. There was Arrhenius, followed by Angstrom who seemed to discredit the entire greenhouse theory, which was not revived until Callendar came along in the 1930s with a better spectroscope. There was the question of the ice ages, and the mistaken perception that forcing from CO2 and forcing from orbital changes (the Milankovitch model) were mutually exclusive.

For decades, those who studied the atmosphere were split into three groups, with three different strategies. Forecasters needed speed in their predictions, so they used intuition and historical analogues rather than numerical methods. Theoretical meteorologists wanted to understand weather using physics, but numerical methods for solving differential equations didn’t exist yet, so nothing was actually calculated. Empiricists thought the system was too complex for any kind of theory, so they just described climate using statistics, and didn’t worry about large-scale explanations.

The three groups began to merge as the computer age dawned and large amounts of calculations became feasible. Punch-cards came first, speeding up numerical forecasting considerably, but not enough to make it practical. ENIAC, the first model on a digital computer, allowed simulations to run as fast as real time (today the model can run on a phone, and 24 hours are simulated in less than a second).

Before long, theoretical meteorologists “inherited” the field of climatology. Large research institutions, such as NCAR, formed in an attempt to pool computing resources. With incredibly simplistic models and primitive computers (2-3 KB storage), the physicists were able to generate simulations that looked somewhat like the real world: Hadley cells, trade winds, and so on.

There were three main fronts for progress in atmospheric modelling: better numerical methods, which decreased errors from approximation; higher resolution models with more gridpoints; and higher complexity, including more physical processes. As well as forecast GCMs, which are initialized with observations and run at maximum resolution for about a week of simulated time, scientists developed climate GCMs. These didn’t use any observational data at all; instead, the “spin-up” process fed known forcings into a static Earth, started the planet spinning, and waited until it settled down into a complex climate and circulation that looked a lot like the real world. There was still tension between empiricism and theory in models, as some factors were parameterized rather than being included in the spin-up.

The Cold War, despite what it did to international relations, brought tremendous benefits to atmospheric science. Much of our understanding of the atmosphere and the observation infrastructure traces back to this period, when governments were monitoring nuclear fallout, spying on enemy countries with satellites, and considering small-scale geoengineering as warfare.

I appreciated how up-to-date this book was, as it discussed AR4, the MSU “satellites show cooling!” controversy, Watt’s Up With That, and the Republican anti-science movement. In particular, Edwards emphasized the distinction between skepticism for scientific purposes and skepticism for political purposes. “Does this mean we should pay no attention to alternative explanations or stop checking the data?” he writes. “As a matter of science, no…As a matter of policy, yes.”

Another passage beautifully sums up the entire narrative: “Like data about the climate’s past, model predictions of its future shimmer. Climate knowledge is probabilistic. You will never get a single definitive picture, either of exactly how much the climate has already changed or of how much it will change in the future. What you will get, instead, is a range. What the range tells you is that “no change at all” is simply not in the cards, and that something closer to the high end of the range – a climate catastrophe – looks all the more likely as time goes on.”