MATH35062 Mathematics for a Finite Planet
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MATH35062 Mathematics for a Finite Planet
First assessment: The Greenhouse Gas Effect and the role of carbon dioxide
This project is worth 20% of the total course marks. Read the instructions carefully as the marking scheme is related to the different parts of the description below. You should expect to spend around 5 hours on the assignment.
Assignment: The Greenhouse Gas Effect and the role of carbon dioxide The assignment should be in two parts: Part A: The Greenhouse Gas Effect, and Part B: Energy balance models: solar radiation and carbon dioxide. Part A should be answered as a mini-essay, typeset and in- cluding references; Part B should contain a more standard mathematical solution to the questions (a)-(e) posed below, including the explanation of the answers. The total length must be less than 1000 words (not including the bibliography and diagrams), and each part must have less than 700 words. An equation counts as one word. The word count should be given at the end of each part.
Relation to ILOs
Part A assesses ILO1 (use local, national and international reports to assess and describe the potential environmental and human impact of a problem) and Part B ILO2 (manipulate mathe- matical and/or statistical models to describe aspects of problems from a finite planet).
Instructions
Part A: The Greenhouse Gas Effect ([45 marks])
The first part of the essay should describe what happens to solar radiation incident on the Earth (10 marks) and how greenhouse gases can lead to warming (10 marks), and one of the potential effects of these changes on the human population (10 marks). Additional marks will be given for clarity of writing (10 marks), and good, clear referencing (5 marks). You are particularly encouraged to use the IPCC Assessment Report 6 amongst the references.
Part B: Energy balance models: solar radiation and carbon dioxide. ([55 marks])
In the second part, you should analyse the role of wavelength on energy balance equations. This is designed to describe the role of carbon dioxide in solar radiation capture (a cooling mechanism) and Earth infra-red capture (a warming effect). Bear in mind that these models are very over-simplified and so conclusions cannot be accepted without much more detailed scientific input into the problem. Answer all five questions.
(a) [5 marks] Explain how a standard energy balance model involving only insolation I0 per unit area of the Earth (presented as a disc to the radiation of the sun) and albedo, e, leads to the equation
for the Earth’s temperature Te . Give the standard values of I0 and e and explain carefully
the factor of 4.
(b) [15 marks] The black body radiation spectrum for the power per square metre of the surface of the radiating body at wavelength λ and temperature T is
where h is Planck’s constant, c is the speed of light and k is Boltzmann’s constant. Thus the power per unit area emitted from the surface with wavelengths in an interval [a, b] is
a(b) E(λ, T)dλ, and if λ is in some small range δλ about λc the total emission is approxi- mately E(λc , T)δλ .
Show that the ratio of the power from the Earth to the power from the Sun in a waveband about λc in the Earth’s atmosphere is approximately
where R0 is the radius of the Sun, re is the radius of the Earth’s orbit, T0 is the temperature of the Sun and Te is the temperature of the Earth.
(c) [5 marks] Evaluate Q defined in (b) at the wavelength of highest carbon dioxide absorption which you may round up so that it corresponds to wavenumber 2500 cm−2 . [Be careful of units, the wavenumber is one over the wavelength]. You may assume that
(d) [20 marks] Suppose that the amount of energy absorbed by carbon dioxide is a fraction of the total energy emitted. Thus the insolation effect is (1 尸 e 尸 ε)I0 where ε is the proportion of the insolation absorbed by carbon dioxide in the atmosphere, and similarly the proportion for the Earth’s energy output is η . Explain why the modified energy balance equation from (a) which includes this effect is
Show that the parameters ε and η can be chosen so that the conclusions of (c) hold but there is still a net warming effect on the Earth provided (ignoring quadratic terms in the small parameters η and ε)
Remember that I0 has been defined to be the insolation per unit area of the Earth seen as a disc from the Sun.
(e) [10 marks] The answer to (c) might suggest that the Greenhouse gas effect is less impor- tant than the scattering effect of incident energy from the Sun. [Continued overleaf.]
(i) Explain why (d) shows that this does not necessarily follow even if the analysis of (b) and (c) is valid (which is by not established by the over-simplified model used).
(ii) What does Wien’s Law imply about the relative size of the proportions (no detail is required, which is larger and why?).
(iii) Describe briefly how the modelling of (b) and (d) could be improved to obtain a more convincing scientific description of the role of Greenhouse gases as followed, for example, by the IPCC.
2023-03-06