This exercise will not be related to statistics. Instread, you get the chance to write your very own, simple “climate model” that calculates the temporal evolution of the mean temperature of Earth for different simplified, idealised Earths.


Preparation for this exercise will only require the close study of the contents of previous lectures. Make sure you are familiar with the concepts of black body radiation, radiative fluxes, Stefan Boltzman Law, heat capacity and dynamic equilibrium.

The essential topics are covered in the Building a Climate III.



Learning goals


  • implementation of a very simple energy balance climate model

  • using loops for real life problems

What to Submit

Use the skeleton script below (or create your own in your preferred language) and add your lines of code where prompted. Submit the same script with your own code. The script should be named [your surname]_e105.[ext], where [ext] is the file extension. In case of Python, this would be .py, for Matlab it is .m, etc. Furthermore, submit the figure(s) of temperature evolution your script calculates.

Creating a Simple Climate Model


The concept of black body radiation is essential in understanding radiative fluxes (the input and output) of Earth. Specifically, it lets us calculate the nature of incoming and outgoing radiation, as well as the mean global temperatures on Earth or any planet if the whole system is in equilibrium.

Study the script below. Certain parameters are already defined for you. Follow the instructions of the script carefully and add your own code where asked. The final script will be able to calculate the temporal evolution of mean global temperatures on different simplified and idealised versions of Earth until an equilibrium is reached.

import numpy
import matplotlib.pyplot as plt
Fixed Parameters

SB_SIGMA - Stefan-Boltzman constant in W/(m**2 K**4)
PI       - delicious old pi
SB_SIGMA = 5.7E-8
PI = 3.141592653589793 

Sun's Parameters
F_SOL - Temperature of the sun (K)
R_SOL - Radius of the sun (m)
T_SOL = 5800
R_SOL = 6.96000E8

Earth's Parameters
R - distance of Earth from sun (m)
R = 1.5E11

C_AT - heat capacity: atmosphere only
C_OS - heat capacity: ocean surface 50m (real ca. 70m) 
C_OU - heat capacity: upper ocean layer 250m (real ca. 360m)  
C_OD - heat capacity: ocean 3900m, 70% of Earth surface   

Note: Heat capacity is given in J/(K*m^2) to simplify calculations.

The specific heat capacity of air (Cp) is ~1000 J/(kg*K) or, given the density 
of air (ro) of 1 kg/m^3, the volumetric heat capacity Cv of ~1000 J/(K*m^3).
[ Cv=Cp*ro=1000 J/(K*m^3) ]. 

In ro*Cp*dT/dt=dQ/dx, Q is heat flow in W/m^2 or J/(s*m^2), which is given by 
f_net. To get dT, we need to divide heat flow by a unit of length. In this exercise,
volumetric heat capacity (c=ro*cp) is already multiplied by 10000 (m), i.e. the 
average thickness of the troposphere, and therefore skip the division of the heat  
flow by length.

C_AT= ro*Cp*H where H is the height of the troposphere (~10km).
C_AT = 1.0E7 
C_OS = 2.0E8 
C_OU = 1.0E9 
C_OD = 1.6E10   

ALBEDO_0  - albedo mean constant
TRANSM_0  - transmissivity mean constant
ALBEDO_0 = 0.3 
TRANSM_0 = 0.64262

Tasks for You!

1) Calculate the sun's output F_SOL and radiative flux PHI_SOL

# your code goes here

2) Calculate F_SW_IN_0, the mean incoming shorwave radiation per area on Earth in W/m^2, i.e. the solar constant.

# your code goes here

3) Calculate the predicted temperature on Earth T_EARTH assuming no atmosphere

# your code goes here

You'll notice that the temperature of task 3 deviates somewhat from the actual temperature on Earth. 
What we need is to consider greenhouse gas related transmissivity.

4) Let temperature evolve over time!
Initialise temperature and let it evolve over time until it reaches equilibrium. 
Make sure to include albedo and transmissivity in your calculations.

HINT: To calculate temperature change for a specific time step based on net incoming radiation,
remember dT (change in temperature) = dQ (change in heat/energy) / C (heat capacity), incoming 
radiation is measured in W/m^2 (J/(s*m^2)) and heat capacity is given in J/(K*m^2) here.

Note: When calculating the radiation emitted by Earth, a multiplication by the factor of 0.95 is
necessary to take into consideration the emissive propertiesof the Earth's surface.

4-a: Initialise temperature at 300K and let it evolve using a heat capacity of the atmosphere only.
4-b: Change the initial temperature, run your code again and see what happens. Does this make sense?
4-c: Pick a different heat capacity and see what happens. Does this make sense?

# your code goes here


Late submissions won’t be accepted!