homework 9, version 0

Submission by: Jazzy Doe (jazz@mit.edu)

Homework 9: Climate modeling I

18.S191, fall 2020

Let's create a package environment:

Before working on the homework, make sure that you have watched the first lecture on climate modeling 👆. We have included the important functions from this lecture notebook in the next cell. Feel free to have a look!

Exercise 1 - policy goals under uncertainty

A recent ground-breaking review paper produced the most comprehensive and up-to-date estimate of the climate feedback parameter, which they find to be

$$B\approx \mathcal{N}(-1.3,0.4),$$

i.e. our knowledge of the real value is normally distributed with a mean value $\bar{B}=-1.3$ W/m²/K and a standard deviation $\sigma =0.4$ W/m²/K. These values are not very intuitive, so let us convert them into more policy-relevant numbers.

Definition: Equilibrium climate sensitivity (ECS) is defined as the amount of warming $\mathrm{\Delta }T$ caused by a doubling of CO₂ (e.g. from the pre-industrial value 280 ppm to 560 ppm), at equilibrium.

At equilibrium, the energy balance model equation is:

$$0=\frac{S(1-\alpha )}{4}-(A-B{T}_{eq})+a\ln \left(\frac{2\text{CO}{₂}_{\text{PI}}}{\text{CO}{₂}_{\text{PI}}}\right)$$

From this, we subtract the preindustrial energy balance, which is given by:

$$0=\frac{S(1-\alpha )}{4}-(A-B{T}_0),$$

The result of this subtraction, after rearranging, is our definition of $\text{ECS}$:

$$\text{ECS}\equiv T_{eq}-T_0=-\frac{a\ln (2)}{B}$$

The plot below provides an example of an "abrupt 2xCO₂" experiment, a classic experimental treatment method in climate modelling which is used in practice to estimate ECS for a particular model. (Note: in complicated climate models the values of the parameters $a$ and $B$ are not specified a priori, but emerge as outputs of the simulation.)

The simulation begins at the preindustrial equilibrium, i.e. a temperature $T_0=14$°C is in balance with the pre-industrial CO₂ concentration of 280 ppm until CO₂ is abruptly doubled from 280 ppm to 560 ppm. The climate responds by warming rapidly, and after a few hundred years approaches the equilibrium climate sensitivity value, by definition.

$B=$

-1.301

Exercise 1.1 - Develop understanding for feedbacks and climate sensitivity

👉 Change the value of $B$ using the slider above. What does it mean for a climate system to have a more negative value of $B$? Explain why we call $B$ the climate feedback parameter.

👉 What happens when $B$ is greater than or equal to zero?

Reveal answer:

👉 Create a graph to visualize ECS as a function of B.

Exercise 1.2 - Doubling CO₂

To compute ECS, we doubled the CO₂ in our atmosphere. This factor 2 is not entirely arbitrary: without substantial effort to reduce CO₂ emissions, we are expected to at least double the CO₂ in our atmosphere by 2100.

Right now, our CO₂ concentration is 415 ppm – 1.482 times the pre-industrial value of 280 ppm from 1850.

The CO₂ concentrations in the future depend on human action. There are several models for future concentrations, which are formed by assuming different policy scenarios. A baseline model is RCP8.5 - a "worst-case" high-emissions scenario. In our notebook, this model is given as a function of $t$.

👉 In what year are we expected to have doubled the CO₂ concentration, under policy scenario RCP8.5?

Exercise 1.3 - Uncertainty in B

The climate feedback parameter $B$ is not something that we can control– it is an emergent property of the global climate system. Unfortunately, $B$ is also difficult to quantify empirically (the relevant processes are difficult or impossible to observe directly), so there remains uncertainty as to its exact value.

A value of $B$ close to zero means that an increase in CO₂ concentrations will have a larger impact on global warming, and that more action is needed to stay below a maximum temperature. In answering such policy-related question, we need to take the uncertainty in $B$ into account. In this exercise, we will do so using a Monte Carlo simulation: we generate a sample of values for $B$, and use these values in our analysis.

👉 Generate a probability distribution for the ECS based on the probability distribution function for $B$ above. Plot a histogram.

Answer:

It looks like the ECS distribution is not normally distributed, even though $B$ is.

👉 How does $\overline{\text{ECS}(B)}$ compare to $\text{ECS}(\bar{B})$? What is the probability that $\text{ECS}(B)$ lies above $\text{ECS}(\bar{B})$?

👉 Does accounting for uncertainty in feedbacks make our expectation of global warming better (less implied warming) or worse (more implied warming)?

Exercise 1.5 - Running the model

In the lecture notebook we introduced a mutable struct EBM (energy balance model), which contains:

• the parameters of our climate simulation (C, a, A, B, CO2_PI, α, S, see details below)

• a function CO2, which maps a time t to the concentrations at that year. For example, we use the function t -> 280 to simulate a model with concentrations fixed at 280 ppm.

EBM also contains the simulation results, in two arrays:

• T is the array of tempartures (°C, Float64).

• t is the array of timestamps (years, Float64), of the same size as T.

You can set up an instance of EBM like so:

Have look inside this object. We see that T and t are initialized to a 1-element array.

Let's run our model:

Again, look inside simulated_model and notice that T and t have accumulated the simulation results.

In this simulation, we used T0 = 14 and CO2 = t -> 280, which is why T is constant during our simulation. These parameters are the default, pre-industrial values, and our model is based on this equilibrium.

👉 Run a simulation with policy scenario RCP8.5, and plot the computed temperature graph. What is the global temperature at 2100?

Additional parameters can be set using keyword arguments. For example:

Model.EBM(14, 1850, 1, t -> 280.0; B=-2.0)

Creates the same model as before, but with B = -2.0.

👉 Write a function temperature_response that takes a function CO2 and an optional value B as parameters, and returns the temperature at 2100 according to our model.

Exercise 1.6 - Application to policy relevant questions

We talked about two emissions scenarios: RCP2.6 (strong mitigation - controlled CO2 concentrations) and RCP8.5 (no mitigation - high CO2 concentrations). These are given by the following functions:

We are interested in how the uncertainty in our input $B$ (the climate feedback paramter) propagates through our model to determine the uncertainty in our output $T(t)$, for a given emissions scenario. The goal of this exercise is to answer the following by using Monte Carlo Simulation for uncertainty propagation:

👉 What is the probability that we see more than 2°C of warming by 2100 under the low-emissions scenario RCP2.6? What about under the high-emissions scenario RCP8.5?

Exercise 2 - How did Snowball Earth melt?

In lecture 21 (see below), we discovered that increases in the brightness of the Sun are not sufficient to explain how Snowball Earth eventually melted.

We talked about a second theory – a large increase in CO₂ (by volcanoes) could have caused a strong enough greenhouse effect to melt the Snowball. If we imagine that the CO₂ then decreased (e.g. by getting sequestered by the now liquid ocean), we might be able to explain how we transitioned from a hostile Snowball Earth to today's habitable "Waterball" Earth.

In this exercise, you will estimate how much CO₂ would be needed to melt the Snowball and visualize a possible trajectory for Earth's climate over the past 700 million years by making an interactive bifurcation diagram.

Exercise 2.1

In the lecture notebook (video above), we had a bifurcation diagram of $S$ (solar insolation) vs $T$ (temperature). We increased $S$, watched our point move right in the diagram until we found the tipping point. This time we will do the same, but we vary the CO₂ concentration, and keep $S$ fixed at its default (present day) value.

Below we have an empty diagram, which is already set up with a CO₂ vs $T$ diagram, with a logirthmic horizontal axis. Now it's your turn! We have written some pointers below to help you, but feel free to do it your own way.

We used two helper functions:

👉 Create a slider for CO2 between CO2min and CO2max. Just like the horizontal axis of our plot, we want the slider to be logarithmic.

👉 Write a function step_model! that takes an existing ebm and new_CO2, which performs a step of our interactive process:

• Reset the model by setting the ebm.t and ebm.T arrays to a single element. Which value?

• Assign a new function to ebm.CO2. What function?

• Run the model.

👉 Inside the plot cell, call the function step_model!.

Parameters

The albedo feedback is implemented by the methods below:

If you like, make the visualization more informative! Like in the lecture notebook, you could add a trail behind the black dot, or you could plot the stable and unstable branches. It's up to you!

Exercise 2.2

👉 Find the lowest CO₂ concentration necessary to melt the Snowball, programatically.

Exercise XX: Lecture transcript

(MIT students only)

Please see the link for hw 9 transcript document on Canvas. We want each of you to correct about 500 lines, but don’t spend more than 20 minutes on it. See the the beginning of the document for more instructions. :point_right: Please mention the name of the video(s) and the line ranges you edited:

Function library

Just some helper functions used in the notebook.