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MATH35062 Mathematics for a Finite Planet
创建日期：2024-08-06 16:13:47
标签： 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.

1

(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∫ b

a

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] EvaluateQ 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

h = 6.63× 10?34 Js, c = 3× 108 ms?1, k = 1.38× 10?23 JK?1

T0 = 6000K, Te = 290K, R0 = 7× 108m, re = 15× 1010m.

(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 e 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

(1? e? ε)1

4

I0 = (1? η)σT 4e .

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)

(1? e)

4

η < ε < (1? e)η.

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.]

2

(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.

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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