MATH35062 Mathematics for a Finite Planet
Final assessment: Glacial Ice Melt
This assessment is worth 80% of the total course marks. Read the instructions carefully as
the marking scheme is related to the different parts of the description below
Assignment: Glacial Ice Melt The assignment should be in IMIP style with a total of not
more than 2000 words. Section 1 (Introduction and Impact) should have at most 300 words,
Section 2 (Mathematical analysis and interpretation) at most 1000 words and Section 3 (Policy
and ethics) should contain at most 1000 words. An equation counts as one word. Any section
which goes over the word limit by more than 10% will be marked out of 90%. The word count
should be given at the end of each part.
Relation to ILOs Section 1 assesses ILO1 (use local, national and international reports to
assess and describe the potential environmental and human impact of a problem) and Section 2
ILO2 (manipulate mathematical and/or statistical models to describe aspects of problems from a
finite planet) and ILO3 (interpret the mathematics in the context of the original problem, and to
reflect critically on the limitations and/or accuracy of the modelling process.). Section 3 assesses
ILO4 (develop policy suggestions based on the mathematical descriptions used, and to assess
the potential impact and identify any ethical issues raised by the policy and to suggest remedies;
evaluate critically the policy proposals and their effects) and all sections reflect on ILO5 (choose
content so as to communicate effectively to a technical audience (for the mathematics) and a
more general audience (for the other areas); write documents aimed at different communities in
a style and with a content that is appropriate to that readership.).
Instructions The assessment is marked out of 100 marks.
Section 1: Introduction and Impact, 20% of total marks
The first section should introduce the ideas used and give a brief description of potential
impacts of glacial ice melt [ 15 marks ]. You should include a ‘hook’ [ 5 marks ] and this
should be different from the hook in the Week 9 IMIP on Sea Level Rise .
Section 2: Mathematical analysis and interpretation, 35% of marks
In this section you should discuss the following simple models of the effects of ice melt and
the impact on albedo. They are all based on the standard averaged energy balance equations (see
lectures) and Archimedes’ Principle (e.g. ). The section should be written as a report, but
it should include the following details. (Note that the parameters given here are indicative and
should be used in the calculations – given the inaccuracy you may prefer to think of this as a
description of Planet X and work on that basis, but do not attempt to replace the given quantities
by parameters obtained from other sources.)
(a) [ 10 marks ] Assume that area of the Antarctic ice sheet (ice lying on the ground and
which is not submerged) is approximately 14 ? 106 km2 and the ice sheet is on average
2150m thick. If all of this was to melt determine the net increase in the sea level of the
You may assume that approximately 70% of the Earth’s surface is oceanic, and ignore
details such as thermal expansion of the liquid water and any increase in the surface area
of the oceans. Use the following values of the Earths radius, Re, the density of ice, ?i and
the density of water ?w:
Re = 6400 km, ?i = 0.91 g cm
3, ?w = 1 g cm3.
(b) [ 5 marks ] Assume that the Arctic sea ice and the Antarctic sea ice (ice which is floating
on the sea) occupy a combined average area of approximately 34? 106 km2 at an average
thickness of 3m. If all of this was to melt determine the net increase in the sea level of the
Earth. (Justify your answer briefly.)
(c) [ 5 marks ] Assume that the mountain glaciers of the world (not including Antarctica and
Greenland) have total volume 1.7 million km3. If all of this was to melt determine the net
increase in the sea level of the Earth. (Justify your answer briefly.)
(d) [ 5 marks ] Assuming that the average height of the mountains on which the mountain
glaciers sit is approximately 6 km above mean sea level (actually the average height of the
Himalayas), would you expect the rate of melt to be faster or slower based purely on the
associated change in pressure? (Justify your answer briefly and comment on other effects
that ought to be taken into account.)
(e) [ 5 marks ]Assuming that the polar ice caps (i.e. Greenland, the Arctic and Antarctica) are
responsible for 20% of the Earth’s albedo e, and the mountain glaciers outside Antarctica
and Greenland have a negligible effect (the remaining 80% being principally atmospheric
and land mass albedo) calculate the corresponding change in the Earths albedo in the form
enew = Keold if all the ice were to melt as described in (a)-(c), whereK is to be determined.
You may assume that the current value of the Earth’s albedo is 0.3, the albedo of rocks
beneath ice sheets and glaciers is er, the ocean has albedo eo, and the albedo for ice and
snow is ei which are given by
er = 0.2, eo = 0.06, ei = 0.7.
The information in (a)-(e) is sufficient to answer this question without using other sources
for areas of regions. The total albedo of a body is the sum of the albedo of a type of surface
multiplied by the proportion of the surface with that property.
(f) [ 5 marks ] Assuming the simple equilibrium relationship I0(1 e) = 4T 4, where I0
is the relevant solar insolation, e is the total Earth’s albedo, is the Stefan-Boltzmann
constant and T is the temperature of the Earth in Kelvins, find the percentage change in
the Earth’s temperature due to the effects in (e). If this percentage is broadly accurate, what
change in temperature does this imply if we assume that the current average temperature
of the Earth is 293 K (i.e. do not take the value of the Earth’s temperature as predicted by
simple energy balance).
Section 3: Policy and ethics, 45% of marks This section should consider the human con-
sequences of mountain glacier melt based on current observations and modelling. Marks will
be assigned for discussion of (a) expected changes and their environmental consequences [ 15
marks ]; (b) adaptation and mitigation strategies [ 10 marks ], and (c) ecological and human eth-
ical issues that arise [ 10 marks ]. You should mention the global effect of sea level rise briefly,
but concentrate on consequences and predictions for regions that are impacted specifically by the
loss of mountain ice glaciers outside the polar regions.
The remaining marks are allocated equally between the style of this section [ 5 marks ] and
the accuracy and content of the references [ 5 marks ].
Underpinning references: The week 9 references on Blackboard will be useful, particularly
the two SROCC reports from 2019 [2, 3] and the Sea Level Rise IMIP . The more recent
reports (AR6) from the IPCC Working Groups can also help [4, 5]. These are all available in the
Assessment Supporting Material Folder on Blackboard.
 P. Glendinning (2021) Sea Level Rise (IMIP Report),Mathematics for a Finite Planet, week
9, Blackboard, University of Manchester.
 IPCC (2019) Summary for Policy Makers, in IPCC Special Report on the Ocean and
Cryosphere in a Changing Climate.
 IPCC (2019) Technical Summary, in IPCC Special Report on the Ocean and Cryosphere in
a Changing Climate.
 IPCC (2021) Summary for Policymakers. In: Climate Change 2021: The Physical Science
Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovern-
mental Panel on Climate Change. Cambridge University Press.
 IPCC (2022) Summary for Policymakers. In: Climate Change 2022: Impacts, Adaptation
and Vulnerability. Contribution of Working Group II to the Sixth Assessment Report of the
Intergovernmental Panel on Climate Change. Cambridge University Press.
 Wikipedia, Archimedes’ Principle, https://en.wikipedia.org/wiki/
Archimedes_principle, last accessed 25 April 2023.
MATH35062 Mathematics for a Finite Planet