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 This derivation is based on unpublished work by Nathan Urban – many thanks!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.

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5 thoughts on “A Simple Stochastic Climate Model: Deriving the Backbone

  1. A little nuance on interpreting the equation: S depends on T0, but T0 isn’t fundamentally the initial temperature. It’s the equilibrium temperature. We often treat the two as the same, assuming the pre-industrial climate was approximately in equilibrium. But if you started the model out of equilibrium at some other temperature, S wouldn’t change. (In this model, that is; it can be state-dependent in general.) Also, the direct dependence of S on T0 is lost once you start considering non-Planck feedbacks (or at least, there’s no first-principles way of determining the dependence in a model this simple). I presume you’ll discuss this in the next installment.

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