Exercise 2, melting the snowball earth

Start after finishing excercise 1

In the last lectures we implemented a energy balance model. The energy balance equation is give by

\begin{gather} \color{brown}{C \frac{dT}{dt}} ; \color{black}{=} ; \color{orange}{\frac{(1 - α)S}{4}} ; \color{black}{-} ; \color{blue}{(A - BT)} ; \color{black}{+} ; \color{grey}{a \ln \left( \frac{[\text{CO}₂]}{[\text{CO}₂]_{\text{PI}}} \right)}, \end{gather}

Recall that in the last lecture we implemented a simplied ice albedo feedback by allowing the albedo to depend on temperature:

\[\begin{split}\alpha(T) = \begin{cases} \alpha_{i} & \text{if }\;\; T \leq -10\text{°C} &\text{(completely frozen)}\\ \alpha_{i} + (\alpha_{0}-\alpha_{i})\frac{T + 10}{20} & \text{if }\;\; -10\text{°C} \leq T \leq 10\text{°C} &\text{(partially frozen)}\\ \alpha_{0} &\text{if }\;\; T \geq 10\text{°C} &\text{(no ice)} \end{cases}\end{split}\]

One thing that we did not adress in the last lecture was the impact of CO\(_2\) increase. We simply set:

\(\ln \left( \frac{ [\text{CO}₂]_{\text{PI}} }{[\text{CO}₂]_{\text{PI}}} \right) = \ln(1) = 0\)

We then evaluated how an increasing solar constant changes the equalibirum temperature on earth. In this excercise you keep \(S\) at \(1368 W/m^2\) and instead increase the \(CO_2\) concentration.

Replot the bifurcations diagramm for \(CO_2\) increase instead of solar radiation.

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

from ipywidgets import interact, interactive, fixed, interact_manual
import ipywidgets as widgets

from IPython.display import HTML
from IPython.display import display

from energy_balance_model import ebm

def CO2_change(t): 
    return 280 + co2

Note that co2 is a global variable here. In general all variables that are assigned in a function call are private (only accessible within the function). However, if this variable is not defined within the function, python looks for a global variable that is called co2 (in your complete code). Therefore, remember to increase co2 for the following task.

Start by running the model for with different C02 concentrations and plot the equlibrium temperature.

Hint: You can copy nearly all code from the Lecture Snowball earth

plt.figure(figsize = (8,6))
plt.plot(co2vec[0:len(co2vec)//2], tvec[0:len(co2vec)//2], color = "blue", label = "cool branch", alpha = 0.3)
plt.plot(co2vec[len(co2vec)//2:], tvec[len(co2vec)//2:], color = "red", label = "warm branch", alpha = 0.3)
plt.axvline(1368, color = "yellow", lw = 5, alpha = 0.2, label = "Pre-industiral / present insolation")

plt.plot(420, 14, marker="o", label="Our preindustrial climate", color="orange", markersize=8)
plt.plot(420, -38, marker="o", label="Alternate preindustrial climate", color="lightblue", markersize=8)
plt.plot(280, -48, marker="o", label="neoproterozoic (700 Mya)", color="lightgrey", markersize=8)

plt.xlabel("CO$_2$ concentration [ppm]")
plt.ylabel("Global temperature T [°C]")

plt.legend()
plt.grid()