Assignment 01: planetary energy balance#

For these exercises you’ll need some pen and paper and your notes from the class. I also recommend reading one of (or both) of these book chapters (they explain exactly the same thing but in different styles, you can choose yours):

  • John Marshall & R. Alan Plumb: Atmosphere, Ocean, and Climate Dynamics (Chapter 2)

  • Andrew Dessler: Introduction to Modern Climate Change (Chapters 3 & 4)

In the exercises below I use mostly the notations from Marshall & Plumb.

# These are the modules you will need
import numpy as np
import matplotlib.pyplot as plt


sigma = 5.67e-8  #  W / (m2 K4)
Q_Sun = 3.87e26  #  W
dist_sun_earth = 150e9  #  m
albedo_earth = 0.3

Ex 0: Compute the solar constant at the location of Earth, given \(Q = 3.87 \times 10^{26} W\) and \(r = 150 \times 10^{9} m\)

# your answer here, or on pen & paper

Ex 1: A planet in another solar system has a solar constant S = 2,000 W/m 2 , and the distance between the planet and the star is 100 million km.

  • a) What is the total power output of the star? (Give your answer in watts.)

  • b) What is the solar constant of a planet located 75 million km from the same star? (Give your answer in watts per square meter.)

# your answer here, or on pen & paper

Ex 2: Compute \(T_e\) and \(T_s\) of the Earth using the constants defined and computed above, assuming a one-layer planet opaque to longwave radiation but transparent to shortwave radiation.

# your answer here, or on pen & paper

Ex 3: Using the “leaky” atmosphere model, determine \(\epsilon\) so that \(T_S\) is equal to the observed surface temperature on Earth, about 15°C.

# your answer here, or on pen & paper

Ex 4: As we will discover later, one way to address global warming is to increase the reflectivity of the planet. To reduce the Earth’s temperature by 1 K, how much would we have to change the Earth albedo? (assume a one-layer planet with an initial albedo of 0.3 and solar constant of 1367 W/m 2 ).

# your answer here, or on pen & paper

Ex 5: Either on your own of with the help of one of the books (Exercise 5 in Marshall & Plumb, or Textbook in Dessler), determine that when a very opaque atmosphere as N layers opaque to longwave radiation, the equilibrium surface temperature is:

\[T_s = (N + 1)^{1/4} T_e\]

Now plot the surface temperature of earth as a function of the number of opaque layers in the atmosphere, with N in [1, 100].

# your answer here