Exercise 1, policy goals under uncertainty#

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 \(\overline{B} = -1.3\) 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 \(\Delta T\) caused by a doubling of \(CO_2\) (e.g. from the pre-industrial value 280 ppm to 560 ppm), at equilibrium.

At equilibrium, the energy balance model equation is:

\(0 = \frac{S(1 - α)}{4} - (A - BT_{eq}) + a \ln\left( \frac{2\;\text{CO}_{2\text{PI}}}{\text{CO}_{2\text{PI}}} \right)\)

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

\(0 = \frac{S(1-α)}{4} - (A - BT_{0}),\)

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

\(\text{ECS} \equiv T_{eq} - T_{0} = -\frac{a\ln(2)}{B}\)

import xarray as xr
import numpy as np
import matplotlib.pyplot as plt

from energy_balance_model import ebm

# create ecs(B, a) function

def double_CO2(t):
    return 280 * 2

f, ax = plt.subplots(1)
EBM = ebm(14, 0, 1., double_CO2)
EBM.run(300)
ax.plot(EBM.t, EBM.T - EBM.T[0], label = "$\Delta T (t) = T(t) - T_0$", color = "red")

ax.axhline(ecs(EBM.B, EBM.a), label = "ECS", color = "darkred", ls = "--")

ax.legend()
ax.grid()
ax.set_title("Transient response to instant doubling of CO$_2$")
ax.set_ylabel("temperature [°C]")
ax.set_xlabel("years after doubling")

The plot above provides an example of an “abrupt 2xCO\(_2\)” 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 for the simulation).

The simulation begins at the preindustrial equilibrium, i.e. a temperature °C is in balance with the pre-industrial CO\(_2\) concentration of 280 ppm until CO\(_2\) is abruptly doubled from 280 ppm to 560 ppm. The climate responds by rapidly warming, and after a few hundred years approaches the equilibrium climate sensitivity value, by definition.

# Create a graph to visualize ECS as a function of B (B should be on the x axis)
# calculate the range from -2 to -0.1 with 0.1 as a step size
# Note use plt.scatter for plotting and

Question:

(1) 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.

Answer:

(2) What happens when \(B\) is greater than or equal to zero?

Answer:

Exercise 1.2 - _Doubling CO#

To compute ECS, we doubled the \(CO_2\) in our atmosphere. This factor 2 is not entirely arbitrary: without substantial effort to reduce \(CO_2\) emissions, we are expected to at least double the \(CO_2\) in our atmosphere by 2100.

Right now, our \(CO_2\) concentration is 415 ppm – 1.482 times the pre-industrial value of 280 ppm from 1850.

The \(CO_2\) 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.

def CO2_RCP85(t):
    return 280 * (1+ ((t-1850)/220)**3 * np.maximum(1., np.exp(((t-1850)-170)/100)))

t = np.arange(1850, 2100)
plt.ylabel("CO$_2$ concentration [ppm]")
plt.plot(t, CO2_RCP85(t));

Question:

In what year are we expected to have doubled the \(CO_2\) concentration, under policy scenario RCP8.5?

Hint: the function

np.where()

might be useful

# Enter your code here

Answer:

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 \(CO_2\) 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 for \(B_{avg}\) above. Plot a histogram.

Hint: use the functions

np.random.normal() # with 50000 samples

and plot with

plt.hist()

sigma = 0.4
b_avg = -1.3

samples = # Enter code here

    # plot here
plt.xlabel("B [W/m²/K]")
plt.ylabel("samples")

Generate a probability distribution for the ECS based on the probability distribution function for \(B\) above.

values =  # your code here

values = np.where((values < -20) | (values > 20) , np.nan, values) # drop outlier
plt.hist(values, bins = 20)
plt.xlim([0, 20])
plt.xlabel("Temperature [°C]")

It looks like the ECS distribution is not normally distributed, even though \(B\) is.

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

# your code here

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

Answer:

Exercise 1.5 - Running the model#

In the lecture notebook we introduced a class ebm (energy balance model), which contains:

  • the parameters of our climate simulation (C, a, A, B, CO2_PI, alpha, 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:

def my_co2function(t):
    # here we imply NO co2 increase
    return 280

my_model = ebm(T=14, t=0, deltat=1., CO2=my_co2function)

my_model

Let’s look into our ebm object

attrs = vars(my_model)
print(', \n'.join("%s: %s" % item for item in attrs.items()))

What function do we have?

help(my_model)

# Run the model

EBM = ebm(14, 1850, 1, my_co2function)
EBM.run(10)

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 = 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.

Question: Run a simulation starting at 1850 with policy scenario RCP8.5, and plot the computed temperature graph. What is the global temperature at 2100?

def CO2_RCP85(t):
    return 280 * (1+ ((t-1850)/220)**3 * np.maximum(1., np.exp(((t-1850)-170)/100)))

## Run the model here

We can change values before running the model.

EBM = ebm(15, 1850, 1, my_co2function)
EBM.B = -2
EBM.run(10)

EBM.T

Exercise 1.6 - Application to policy relevant questions (BONUS)#

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

def CO2_RCP26(t):
    return 280 * (1+ ((t-1850)/220)**3 * np.minimum(1., np.exp(-((t-1850)-170)/100)))
def CO2_RCP85(t):
    return 280 * (1+ ((t-1850)/220)**3 * np.maximum(1., np.exp(((t-1850)-170)/100)))

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?