# Assignment: Climate change in the zero-dimensional EBM#

This notebook is part of The Climate Laboratory by Brian E. J. Rose, University at Albany.

## Learning goals#

Students completing this assignment will gain the following skills and concepts:

• Familiarity with the Jupyter notebook

• Familiarity with the zero-dimensional Energy Balance Model

• Understanding of the adjustment toward equilibrium temperature

• Introduction to the concept of albedo feedback

• Use of numerical timestepping to find the equilibrium temperature

• Python programming skills: arrays, loops, and simple graphs

## Instructions#

• Complete the required problems below.

• Some assignments have optional bonus problems. These are meant to be interesting and thought-provoking, but are not required. Extra credit will be given for interesting answers to the bonus problems.

• Remember to set your cell types to Markdown for text, and Code for Python code!

• Remember to actually answer the questions. Written answers are required (not just code and figures!)

• Make sure that your notebook runs cleanly without errors:

• From the Kernel menu, select Restart & Run All

• Did the notebook run from start to finish without error and produce the expected output?

• If yes, save again and submit your notebook file

• If no, fix the errors and try again.

## Problem 1: Time-dependent warming in the zero-dimensional Energy Balance Model#

In lecture we defined a zero-dimensional energy balance model for the global mean surface temperature $$T_s$$ as follows

$C \frac{dT_s}{dt} = \text{ASR} - \text{OLR}$
$\text{ASR} = (1-\alpha) Q$
$\text{OLR} = \tau \sigma T_s^4$

where we defined these terms:

• $$C$$ is a heat capacity for the atmosphere-ocean column

• $$\alpha$$ is the global mean planetary albedo

• $$\sigma = 5.67 \times 10^{-8}$$ W m$$^{-2}$$ K$$^{-4}$$ is the Stefan-Boltzmann constant

• $$\tau$$ is our transmissivity parameter for the atmosphere.

• $$Q$$ is the global-mean incoming solar radiation, or insolation.

Refer back to our class notes for parameter values.

1. If the heat penetrated to twice as deep into the ocean, the value of $$C$$ would be twice as large. Would this affect the equilibrium temperature? Why or why not?

2. In class we used numerical timestepping to investigate a hypothetical climate change scenario in which $$\tau$$ decreases to 0.57 and $$\alpha$$ increases to 0.32. We produced a graph of $$T_s(t)$$ over a twenty year period, starting from an initial temperature of 288 K. Here you will repeat this calculate with a larger value of $$C$$ and compare the warming rates. Specifically:

• Repeat our in-class time-stepping calculation with the same parameters we used before (including a heat capacity of $$C = 4\times10^8$$ J m$$^{-2}$$ K$$^{-1}$$), but extend it to 50 years. You should create an array of temperatures with 51 elements, beginning from 288 K.

• Now do it again, but use $$C = 8\times10^8$$ J m$$^{-2}$$ K$$^{-1}$$ (representing 200 meters of water). You should create another 51-element array of temperatures also beginning from 288 K.

• Make a well-labeled graph that compares the two temperatures over the 50-year period.

3. What do your results show about the role of heat capacity on climate change? Give a short written answer.

## Problem 2: Albedo feedback in the Energy Balance Model#

For this exercise, we will introduce a new physical process into our model by letting the planetary albedo depend on temperature. The idea is that a warmer planet has less ice and snow at the surface, and thus a lower planetary albedo.

Represent the ice-albedo feedback through the following formula:

$\begin{split} \alpha(T) = \left\{ \begin{array}{ccc} \alpha_i & & T \le T_i \\ \alpha_o + (\alpha_i-\alpha_o) \frac{(T-T_o)^2}{(T_i-T_o)^2} & & T_i < T < T_o \\ \alpha_o & & T \ge T_o \end{array} \right\}\end{split}$

with the following parameter values:

• $$\alpha_o = 0.289$$ is the albedo of a warm, ice-free planet

• $$\alpha_i = 0.7$$ is the albedo of a very cold, completely ice-covered planet

• $$T_o = 293$$ K is the threshold temperature above which our model assumes the planet is ice-free

• $$T_i = 260$$ K is the threshold temperature below which our model assumes the planet is completely ice covered.

For intermediate temperature, this formula gives a smooth variation in albedo with global mean temperature. It is tuned to reproduce the observed albedo $$\alpha = 0.299$$ for $$T = 288$$ K.

• Define a Python function that implements the above albedo formula. There is definitely more than one way to do it. It doesn’t matter how you do it as long as it works!

• Use your function to calculate albedos for a wide range on planetary temperature (e.g. from $$T=250$$ K to $$T=300$$ K.)

• Present your results (albedo as a function of global mean temperature, or $$\alpha(T)$$) in a nicely labeled graph.

1. Now investigate a climate change scenario with this new model:

• Suppose that the transmissivity decreases from 0.611 to 0.57 (same as before)

• Your task is to calculate the new equilibrium temperature. First, explain very briefly why you can’t just solve for it analytically as we did when albedo was a fixed number.

• Instead, you will use numerical time-stepping to find the equilibrium temperature

• Repeat the procedure from Question 3 (time-step forward for 50 years from an initial temperature of 288 K and make a graph of the results), but this time use the function you defined above to compute the albedo for the current temperature.

• Is the new equilibrium temperature larger or smaller than it was in the model with fixed albedo? Explain why in your own words.

## Bonus problem#

Open-ended investigation for extra credit, not required

Something very different occurs in this model if you introduce a strong negative radiative forcing, either by substantially reducing greenhouse gases (which we would represent as an increase in the transmissivity $$\tau$$), or by decreasing the incoming solar radiation $$Q$$.