MTH2005 — Modelling: Theory & Practice

1 Summary

You will work in a small group to complete a short project applying numerical methods and/or optimisation to real-world problems, complementing the taught material in this module. Each group are required to submit:

.  via ELE, a paper in the style of those published in the SIAM Undergraduate Research Online journal (see the technical speciications below).

.  via email to your project tutor, a zipile containing the code required to replicate your indings.

The  deadline for  submission  of the paper  and zipfle  is  noon  on Thursday 28th March 2024 . Late submissions could be awarded a mark of zero unless mitigation is applied for and granted.

2 How to submit your choices

Submit three choices of project from the list below via the project preferences task on the module ELE page by noon Monday 12th February 2024.

3 Assessment criteria

This project will contribute 30% to your inal module mark.  Marks for your submissions will be awarded as follows:

Code (15 marks): You must submit the complete code you have used to produce the results in your paper. Your code should be modular to allow for testing of each component, efficient, readable, well commented, and you should ensure that variables are speciied as inputs rather than hard-coded.

Your project tutor will provide further guidance as to what they expect to see in your code based on the speciic requirements of your project.

Paper (15 marks): The paper will be assessed according to the following criteria:

. Progress

– Higher:  Substantial evidence of independent investigation of all the threshold ele- ments, and at least one of the higher elements.

– Threshold:  All the threshold elements have been attempted and for the most part have been explored competently.

– Fail:  There are signiicant missing elements from the list of threshold tasks for the investigation and/or engagement is minimal with little independent investigation of

the key ideas.

. Analysis

– Higher:  Substantial progress has been made in presenting and discussing compara- tive analysis of diferent methods for solving the problem.

– Threshold:  Some progress has been made in analysing diferent methods for solving the problem, but this is primarily descriptive or the conclusions are poorly supported by the data.

– Fail:  Little progress has been made to implement and compare diferent methods for solving the problem. The conclusions cannot be justiied by the data selected.

. Mathematical accuracy

– Higher:  All statements made are correct, containing complete hypotheses and proofs where appropriate. Notation is correct and uniform. Assumptions made are clearly identiied.

– Threshold:  Some general results identiied and largely accurate, but occasionally containing some inaccuracies, imprecisions or omissions. Language and/or notation used is occasionally imprecise.

– Fail:  Signiicant and frequent inaccuracies in the statement of results, presentation of examples and use of notation.  Language used is frequently imprecise.

. Clarity and referencing

– Higher:  No spelling, grammatical or editing errors.  A coherent and logical narrative runs through the presentation, binding the material together.  All statements are fully and correctly referenced.

– Threshold:  Occasional spelling, grammatical and/or editing errors.  Occasional loss of low when watching the presentation.  Some material is not clearly referenced, or there are minor errors in the referencing.

– Fail:  Signiicant and frequent spelling, grammatical and/or editing errors.  Little or no logical low or narrative to the presentation.  Little or no attempt to reference material appropriately, or signiicant and persistent errors in the referencing.

. Quality

– Higher:  The paper looks polished and professional — it is obvious that a great deal of care has been taken in the preparation and editing of the inished product.

– Threshold:  Some small errors, but not so signiicant as to seriously detract from what is otherwise a good paper.

– Fail:  Signiicant errors that render the paper di伍cult to read or does not meet the technical speciications.

The marker willirst determine the level of the paper against each category and by means of a best-it principle will determine the overall proile of the paper (higher, threshold, fail).  At the marker’s discretion, a mark will then be awarded within the corresponding level where:  a mark of 0–5 corresponds to a fail; a mark of 6–10 corresponds to threshold level; and a mark of 11–15 corresponds to higher level.

4 Notes on the use of AI

Please note that whilst the use of AI (e.g.  ChatGPT) as a tool to help you work on your project is not forbidden, it is strongly discouraged.  ChatGPT gets some things in numerical methods correct, but it certainly gets other things very wrong.  Please make sure that whatever you hand is in entirely your group’s own work.  Copying work from AI sources would constitute academic misconduct, as detailed in the University’s policy (here).  If the marker suspects misconduct then you may be required to attend a viva to demonstrate your understanding.  Coursework is designed to help solidify your understanding of the material, and you cannot do that if you just copy what an AI produces.

5    Assessing individual contributions to the  projects

By default, all members of the group will get the same project mark.  However, we recognise that sometimes group members either contribute signiicantly more or less than their peers.  To help diagnose this, we will be assessing individual contributions in two ways:

.  We would like each project to have a inal section describing‘contributions to the project’, where you are asked to detail who contributed what to do the project. We ask you to do this using the following categories:

{ Formal Analysis:  Application of statistical, mathematical, computational, or other formal techniques to analyze or synthesize study data.

{ Investigation: Conducting a research and investigation process, speciically perform- ing the experiments, or data/evidence collection.

{ Methodology:  Development or design of methodology; creation of models.

{ Software:  Programming, software development;  designing computer programs; im- plementation of the computer code and supporting algorithms; testing of existing code components.

{ Visualization:   Preparation,  creation  and/or  presentation  of  the  published  work, speciically visualization/data presentation.

{ Writing – Original Draft Preparation:  Creation and/or presentation of the published work, speciically writing the initial draft (including substantive translation).

{ Writing – Review   Editing:  Preparation, creation and/or presentation of the pub- lished work by those from the original research group, speciically critical review, commentary or revision – including pre- or post-publication stages.

These categories are taken from the‘CRediT’taxonomy for deining roles when contribut- ing to a research paper.  An example of the type of statement we are expecting is given below:

Student-1: writing - review and editing (equal), writing - original  draft of question  2 (lead)`. Student-2:  writing – original draft of question 3 (lead); formal analysis  (lead); writing – review and editing (equal).  Student-3:  Software for question 4  (lead); writing  -review and editing (equal) . In addition to the contribution statement, we will also be conducting a‘peer assessment’ as part of the project.  This means that we will be asking the members of each group  to anonymously score their fellow group members on their contributions to the projects.  If (and only if) there is a clear indication from these scores that some members have  contributed substantially more or less to the project, then the project marks of those individuals will be adjusted up or down.

6 Feedback

There will be plenty of opportunity for regular feedback throughout your project.  In particular, you should seek advice from your project tutor each week in the speciied hour slot.

7    Technical speciications

Please ensure that all your submitted outputs conform to the technical speciications detailed below — this is a critically important skill.

Paper:The technical speciications for the paper are adapted from those speciied for submis- sions to the SIAM Undergraduate Research Online journal http://www.siam.org/students/ siuro/authors.php and its sister publications e.g. http://www.siam.org/journals/siopt/ authors.php (URLs correct on February 2, 2024). I have deliberately used the original wording for consistency where appropriate.

A large duplication of another author’s or one’s own work is a sign of poor scholarship. There is also a copyright issue if the source is not cited.   Your  manuscript should provide proper citations, use quotation marks or indentation (for quotations of ive or more lines) to indicate borrowed wording, and minimize duplication.

Papers must be typeset using LATEX and must be submitted in electronic form.   Hard- copy submissions will not be considered.  Each group should submit a .pdf copy of the cor- rectly  compiled  paper.    We  recommend  creating  a  shared  LATEX  document  using  Overleaf www.overleaf.com, for which the University has a license.

Papers may not exceed the equivalent of 20 ordinary journal pages (minimum 11pt font, 2 cm margins) and 3 megabytes and the zipile must not exceed 10 megabytes.  Figures and tables should be labelled consecutively throughout the paper.

The paper should contain each of the following parts:

Title: Titles should be brief and should speciically describe the content of the paper.

Authors:  The name of the group and the candidate numbers of each member of the group should appear here.

Abstract:  An abstract not exceeding 250 words that summarizes the principal techniques and conclusions of the manuscript in relation to known results must accompany the manuscript. Mathematical formulas and bibliographic references in the abstract should be avoided entirely.

Introduction:  The paper must have a clearly written introduction in which the authors outline their new results, describe the motivation for the study, and explain why their work is of interest.The introduction should help the reader to decide whether to read the details in the paper.

Methodology:  A description of the numerical modelling approach and underlying theory applied in the project.

Results/fndings:   Presentation and description of the indings of your study along with igures where appropriate. Main bulk of the report.

Discussion and conclusions:  Summary of the major indings and discussion in the context of existing studies (these can be combined or separate sections).

References:  References should be listed in either alphabetical order or order of citation at the end of the manuscript.  The following reference styles should be used:

.  Journal articles; when possible, titles of journals should be abbreviated in accordance with Mathematical Reviews;  abbreviations are available at http://www.ams.org/msnhtml/ serials.pdf (URL correct on February 2, 2024):

[7] R. T. ROCKAFELLAR, Lagrange multipliers and optimality, SIAM Rev., 35(1993), pp.  183-238.

.  Books, pamphlets, research reports:

[2] B. MANDELBROT, Fractal:  Form, Chance and Dimension, W. H. Freeman, San Francisco, CA, 1977.

.  Paper in a bound collection:

[4] A. NAGURNEY, Parallel computation of economic equilibria,  in  Applications on Advanced Architecture Computers, G. Astfalk, ed., SIAM, Philadelphia, PA, 1996, pp. 265-276.

Acceptable variants on SIAM’s references style are:

[R] R. T. ROCKAFELLAR, Lagrange multipliers and optimality, SIAM Rev., 35 (1993), pp.  183-238.

or

R. T. ROCKAFELLAR (1993), Lagrange multipliers and optimality, SIAM Rev., 35, pp. 183-238.

.  Citations within the text:  A  consistent style should be used, and the style of in-text citations should conform to the reference style chosen. To refer to a speciic page or item in an article or book the following formats maybe used:  [2, p.  51]; [M,p.  51]; Mandelbrot [2, p. 51]; or Mandelbrot (1977, p. 51).

Any queries regarding this coursework brief should be raised with us at your earliest possible convenience. We hope you enjoy your project!

1 Introduction

Hurricanes are low-pressure weather systems that develop mostly over the warm tropical oceans.  They can cause catastrophic loss of life, as shown in 2005 with Huricane Katrina. The physics of the the full hurricane involves a complex mix of vortex dynamics and cloud physics.  Despite this, some of the broad features of a hurricane can be understood with much simpler theory based on the fundamental balances in the vortex.  The aim of this investigation will be to explore such a balanced vortex theory.  This sheet summarises the theory and suggests directions for investigation.

2 Theory

The theory outlined here assumes an axi-symmetric vortex with tangential velocity, v, pressure, φ, and cylindrical polar coordinates (r ,z) where r is the radius and z height.

2.1 Thermal wind: compressible assumption

This section outlines the relationship between the buoyancy and the vortical wind  (the so called thermal wind balance).  The  pressure gradient in the horizontal balances the sum of the centrifugal and Coriolis terms.   In the vertical hydrostatic balance applies,

 ,  = (C, b)                                        (1)

where:

C =  + fv                                           (2)

f is the Coriolis parameter,g gravitational acceleration. Eliminating pressure from (1) gives the thermal wind balance:

 =  (3)

3 Initial investigation

An initial investigation could be to solve for φ, given a prescribed wind proile. An example vortex would be:

v(r, z) = Vmax exp (-  sin  r < 2R

= 0    otherwise                          (4)

and then  solve  for  φ  and  b.   Hv   =  30km,  Vmax    =  40ms-1 ,  R  =  40km, Hρ = 9km, ρs   =  1kgm-3 .  Also, use f = 0.5     10-4   as a typical tropical value.

4 Further investigations

Here are some suggestions for further investigations:

.  Solve for buoyancy (and temperature) in addition to pressure.

.  Calculate the inverse problem:  prescribe buoyancy  and  calculate the wind.

.  Use more realistic windields for the hurricane vortex. The vortex given by (4) is symmetric about its maximum, but more realistic proiles have stronger winds towards the cyclone centre.

.  How do the solutions change for an anti-cyclonic vortex (v < 0).

.  Explore the relative contribution of the centrifugal () and geostrophic (fv) terms in C. Explore how changing the size and magnitude of the vortex, and the Coriolis parameter alter the balance of these terms. What is the relevant non-dimensional parameter for this?

.  This is an approximate steady state vortex.  Explore applying a time evolution of the density, for example a heating from the bottom bound- ary.

Time-dependent and stochastic dynamical systems

Frank Kwasniok

Many applied problems involve dynamical systems described by ordinary di↵erential equa- tions (ODEs). Here you will study two extensions of the usually considered setting which greatly enhance the richness of the modelled behaviour: ODEs with time-dependent param- eters and stochastic di↵erential equations (SDEs).

Dynamical systems with time-dependent parameters may undergo abrupt qualitative, possibly irreversible changes called bifurcations or tipping points.  Equilibria may change stability or disappear completely; equilibria may turn into limit cycles or vice versa.

Stochastic noise terms enable the trajectory to explore regions of state space which are not accessible to the deterministic dynamics. We may, for example, observe random transitions between di↵erent metastable states.  SDEs require special numerical integration schemes which are di↵erent from those for ODEs. A SDE is linked to a Fokker-Planck equation, a partial di↵erential equation (PDE) governing the time evolution of the probability density function.

To receive a pass mark (threshold) for this investigation you must complete the following tasks:

Threshold

.  Numerically generate trajectories of a bistable SDE.

.  Study numerically simple examples of time-dependent ODEs and SDEs.

.  Solve numerically the Fokker-Planck equation for an Ornstein-Uhlenbeck process and a bistable SDE and compare with the analytical solution.

To receive a first class mark (higher) for this investigation you must additionally complete one of the following extension tasks:

Higher

.  Apply the Crank-Nicolson scheme to solve numerically the Fokker-Planck equation of a time-dependent SDE and compare the results with direct simulations of the SDE.

.  Study the notions of weak and strong convergence of integration schemes for SDEs and explore them numerically on examples.

Numerically solving cloud droplet growth by condensation

Dan Partridge

In this assignment you will create your own simple model of the growth of a cloud droplet by condensation of water vapour in a vertically moving adiabatic parcel of air. Such models are traditionally known as adiabatic cloud parcel models in the atmospheric science literature.

They form the basis for the calculation of the number of cloud droplets in global climate models.

You will focus on to creating a simple model in MATLAB that describes how the

supersaturation and subsequently droplet size evolves in time within a cloud as air ascends from cloud base. This process can be modelled using a set of four first order ODEs, which   will be provided to you with the necessary supporting literature.

You should apply numerical methods to solve the given set of ODEs and provide a

comparison of the accuracy and efficiency of different methods. With your final model you

will investigate how varying the initial conditions for different parameters affects the maximum supersaturation and droplet size attained within your simulated cloud.

To receive a pass mark (threshold) for this investigation you must complete the following tasks:

Threshold

•    Question 1

a.  Write a MATLAB code that will model the growth of a single droplet of initial size    1 μm radius assuming constant supersaturation, s (0.30 %) and temperature (282 K)  over a period of 45 minutes. Use a forward Euler time stepping scheme. Repeat

using a 4th order Runge Kutta time stepping scheme. Plot how the droplet radius varies with time for both schemes.

b.   Demonstrate that you have checked the correctness of your numerical solution.

c.   Repeat for a range of initial temperatures and droplet sizes and discuss the

implications of your findings from this simple model of droplet diffusion growth on how the size of a distribution of droplets within a cloud growing by condensation  would evolve over time.

d.   Calculate how long it would take to form precipitation size drops from droplet growth via condensation alone and discuss your findings.

e.   Consider an isolated cloud droplet, with a radius of 7 μm in a cloud at time t is

suddenly moved from its saturated cloudy environment to an unsaturated one

outside the cloud where the relative humidity is 75%. For a range of temperatures

calculate the time it takes for the droplet to evaporate completely due to water vapour diffusion. Discuss your findings with respect to simple observations of cloud edges.

•    Question 2

a.  Write a MATLAB code that will model and plot the evolution of supersaturation and   droplet size with height from cloud base to cloud top for the case in which pressure and temperature are assumed constant. Use a forward Euler time stepping scheme.

b.   Replace your forward Euler time stepping scheme with a higher order time stepping scheme (e.g. 4th order Runge Kutta). Compare the accuracy of the methods in calculating cloud maximum supersaturation for different model timesteps.

To receive a first-class mark (higher) for this investigation you must additionally complete a minimum of: extension task 1 or 2, plus extension task 3.

Higher extension tasks

1.   Extend the model you have created in question 2b to include the ODEs describing change of temperature and pressure with time to solve the full set of four ODEs provided numerically.

Calculate the difference in simulated cloud top droplet size between your new model  and your model from question 2b for two different cloud types: stratiform cloud (cloud depth: 250 m); convective cloud (cloud depth: 1000 m). Briefly discuss your findings.  For your new model which considers the full set of ODEs, for a marine stratocumulus cloud (cloud depth: 250 m, cloud droplet number concentration: 100 cm-3 ) analyse how the cloud top droplet size and cloud maximum supersaturation depend on the input parameter values chosen for: initial droplet radius, droplet number concentration and vertical velocity. Show your results and discuss your findings.

2.   Extend the model you have created in question 2b to consider the growth of a

population of cloud droplets having different initial sizes, i.e. a droplet spectrum , and plot the output.

Analyse how for a marine stratocumulus cloud (cloud depth: 250 m, cloud droplet   number concentration: 100 cm-3 ), the average cloud top droplet size and maximum supersaturation depend on the input parameter values chosen for droplet number   concentration and vertical velocity. Show your results and discuss your findings.

3.   Using your model from higher extension task 1 or 2, demonstrate the first aerosol

indirect effect (also known as the Twomey effect) , by calculating the change in cloud top albedo between two model simulations.

You should design the two simulations such that the parameters controlling the meteorology remain constant for each cloud , but the environmental aerosol concentration (described by cloud droplet number concentration in your model) changes. Show your results and discuss your findings.