A Simple Stochastic Climate Model: Climate Sensitivity

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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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.