Exercises: planetary energy balance

Exercises: planetary energy balance#

For these exercises you’ll need some pen and paper and your notes from the class. We also recommend strongly 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)

Below we use mostly the notations from Marshall & Plumb, but the differences are small.

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

Constants

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 (S0_sun) at the location of Earth, given \(Q = 3.87 \times 10^{26} W\) and \(r = 150 \times 10^{9} m\). Find out by how much the solar constant (which is the direct solar irradiance at the top of the atmosphere) fluctuates over a year due to Earth’s varying distance from the Sun.

# your answer here, or on pen & paper

Ex 1: A planet in another solar system has a solar constant S = 2000 W m-2 , and the distance between the planet and the star is 300 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 30 million km away 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.

  • Try to understand each step of the derivation of \(T_e\) and \(T_s\) (see e.g. Ch. 2.1 and Ch. 2.3.1 of the Marshall \(\&\) Plumb book) and write the steps down in such a way that you can present the derivation to your fellow students.

  • If you do not understand a step, write it down and we will discuss it during our next meeting.

# 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.

  • What is different in this new assumption? Try again to understand each step of the derivation to find a relation between \(T_s\) and \(T_e\) (see e.g. Ch. 2.3.2 of the Marshall & Plumb book) and write the steps down in such a way that you can present the derivation to your fellow students.

  • What is the theoretical minimum / maximum value of \(\epsilon\) and what does that mean for the relation between atmospheric and ground temperatures in the “leaky” model ?

# 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 opaque to longwave radiation but transparent to shortwave radiation with an initial albedo of 0.3 and solar constant of 1367 W/m 2 ).

Hint: Here you can again assume an epsilon of 1 (you can solve ex. 4 independent of ex. 3)

# your answer here, or on pen & paper

Ex 5: For a very opaque atmosphere with N layers opaque to longwave radiation, the equilibrium surface temperature is:

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

  • Try to understand the reasons for the equation (no full derivation necessary, but an explanation of the physical meaning)

  • Hint: Consider how radiation is absorbed and emitted by each layer and how the number of layers affects the surface temperature.

# your answer here (best to do that with pen & paper)

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