# LAGRANGIAN FLUID DYNAMICS OF PLANET EARTH - 2024/5

Module code: MAT3048

## Module Overview

Motivated by problems in meteorology and oceanography, this module applies a range of mathematical techniques to the characteristion of fluid flows. Geometry, vector calculus, differential equations, symmetry, dynamical systems theory, and analysis are all combined with fluid motion to produce a deeper understanding of atmosphere and ocean dynamics.

### Module provider

Mathematics & Physics

### Module Leader

BRIDGES Tom (Maths & Phys)

### Number of Credits: 15

### ECTS Credits: 7.5

### Framework: FHEQ Level 6

### Module cap (Maximum number of students): N/A

## Overall student workload

Independent Learning Hours: 64

Lecture Hours: 33

Seminar Hours: 5

Guided Learning: 15

Captured Content: 33

## Module Availability

Semester 1

## Prerequisites / Co-requisites

MAT2011 Linear PDEs

## Module content

**The following topics will be covered in every presentation of MAT3048:** kinematics of fluids: Lagrangian particle path formulation; Eulerian fluid flow; mass conservation and divergence free vector fields, and dynamical systems aspects of particle motion. Kinematics of vorticity; point vortices; vortex dynamics on the surface of a sphere. Changes of reference frame; rotating fluids, and the Biot-Savart law. Conservation laws and the circulation theorem. Quasi-geostrophic theory in meteorology. Shallow water hydrodynamics and water wave dynamics in oceanography.

**A selection from the following will be covered:** integral transport theorems, Lagrangian variational principles for flow, geometry of divergence free vector fields, mixing and chaotic particle paths, vortex lines and sheets, theory of averaging, and models for hurricanes and typhoons.

## Assessment pattern

Assessment type | Unit of assessment | Weighting |
---|---|---|

School-timetabled exam/test | In-Semester Test (50 min) | 20 |

Examination | End-of-Semester Examination (2 hours) | 80 |

## Alternative Assessment

n/a

## Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate:

- Understanding of fundamental concepts in Lagrangian fluid dynamics and ability to develop and apply them to a new context.
- Subject knowledge through recall of key definitions, formulae and derivations.
- Analytical ability through the solution of unseen problems in the test and examination.

Thus, the

__summative assessment__for this module consists of:

- One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 to 4.
- A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 5.

__Formative assessment__

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

__Feedback__

Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal feedback given in lectures. Students also receive verbal feedback in office hours.

## Module aims

- The module aims to cover the range of mathematics required to understand the dynamics of fluid motion in the context of meteorology and oceanographic flows on planet earth.

## Learning outcomes

Attributes Developed | ||

001 | Students will demonstrate understanding of the derivation of the Eulerian and Lagrangian particle path formulation of fluid motion and its kinematics and applications. | K |

002 | Students will understand the role of vorticity, variational principles, moving frames, rotation and conservation laws in fluid dynamics. | KCT |

003 | Students will be able to develop the geometry and analysis required for modelling fluid motion. | KC |

004 | Students will understand the stability of fluid flows, from stability of particle motion to stability of Eulerian velocity fields. | KC |

005 | Students will understand the application of mathematics to the equations of fluid mechanics, with particular attention to equations in meteorology (quasi-geostrophic theory) and oceanography (theory of water waves). | KC |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

## Methods of Teaching / Learning

The __learning and teaching__ strategy is designed to:

Equip students with the knowledge, experience and confidence to apply the techniques of Lagrangian Fluid Dynamics to practical problems.

The __learning and teaching__ methods include:

- Three one-hour lectures per week for eleven weeks, with typeset notes to complement the lectures. Lectures are delivered using blackboards and whiteboards for real-time presentation.
- Five one-hour tutorials are devoted to the discussion of example sheets, giving opportunities for students to ask questions and to practice methods taught.
- There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk

Upon accessing the reading list, please search for the module using the module code: **MAT3048**

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

**Digital Capabilities:*** *The SurreyLearn page for MAT3048 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

**Employability:*** *Students are equipped to analyse fluid motion, a skill valued in various industries including: aerospace; environmental science; and energy.

**Global and Cultural Capabilities: **During tutorials, student engagement in discussions naturally cultivates the sharing of the different cultures from which the students originate.

**Resourcefulness and Resilience**: The problem-solving aspects of MAT3048 fosters resourcefulness. Through the often intricate scenarios of fluid dynamic problems in MAT3048, students develop resilience in tackling dynamic challenges.

**Sustainability: **The tools covered in MAT3048 enable students to explore mathematical models for fluid motion, relevant to ecosystems and water resources. Thus MAT3048 contributes to sustainability by enabling students to analyse fluid behaviour in environmental contexts.

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.