Mathematical modelling of the shallow-water equations with temperature gradients. PhD in Mathematics (NERC GW4+ DTP) Ref: 3693
About the award
Dr Hamid Alemi Ardakani, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, University of Exeter
Dr Tristan Pryer, Department of Mathematical Sciences, University of Bath
Dr Bob Beare, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, University of Exeter
Location: University of Exeter, Penryn Campus, Penryn, Cornwall, TR10 9FE
This project is one of a number that are in competition for funding from the NERC GW4+ Doctoral Training Partnership (GW4+ DTP). The GW4+ DTP consists of the GW4 Alliance of research-intensive universities: the University of Bath, University of Bristol, Cardiff University and the University of Exeter plus five unique and prestigious Research Organisation partners: British Antarctic Survey, British Geological Survey, Centre for Ecology & Hydrology, the Natural History Museum and Plymouth Marine Laboratory. The partnership aims to provide a broad training in the Earth, Environmental and Life sciences, designed to train tomorrow’s leaders in scientific research, business, technology and policy-making. For further details about the programme please see http://nercgw4plus.ac.uk/
For eligible successful applicants, the studentships comprises:
- A stipend for 3.5 years (currently £15,009 p.a. for 2019/20) in line with UK Research and Innovation rates
- Payment of university tuition fees;
- A research budget of £11,000 for an international conference, lab, field and research expenses;
- A training budget of £3,250 for specialist training courses and expenses.
- Travel and accommmodation is covered for all compulsory DTP cohort events
- No course fees for courses run by the DTP
We are currently advertising projects for a total of 10 studentships at the University of Exeter
Students who are resident in EU countries are eligible for the full award on the same basis as UK residents. Applicants resident outside of the EU (classed as International for tuition fee purposes) are not eligible for DTP funding. Residency rules are complex and if you have not been resident in the UK or EU for the 3 years prior to the start of the studentship, please apply and we will check eligibility upon shortlisting.
The shallow water equations are a set of hyperbolic PDEs which are widely used in geophysical fluid dynamic, ocean currents, sloshing dynamics, flows in rivers and reservoirs, and ocean engineering. A class of high resolution wave propagation finite volume methods is developed for hyperbolic conservation laws by LeVeque (1997). These methods are based on solving Riemann problems for waves that define both first order updates to cell averages and also second order corrections which can be modified by limiter functions to obtain high resolution numerical solutions. Ripa (1993, 1995) derived a new set of shallow water equations for modelling ocean currents. The governing equations can be derived by vertically integrating the density, horizontal pressure gradient and velocity field in each layer of multi-layered ocean models. Ripa's model includes the horizontal temperature gradients which are of prime importance for modelling ocean currents, and result in the variations in the fluid density within each layer.
Project Aims and Methods
The interest in this project is to develop the background theoretical and numerical schemes for the Ripa system and its variants for flows over variable topography and cross-section using f-wave-propagation finite volume Riemann methods. The three key themes of the research project are:
- Develop the background theory and augmented Riemann finite volume solvers for the shallow water equations over variable bottom topography in one dimension with horizontal temperature gradient. The starting point for this part of the project are the works of George (2008) and Alemi Ardakani et al. (2016).
- Extend the derivation of the 1-D shallow water equations with horizontal temperature gradient over variable topography to include variable cross-section of the domain of integration, and extend the augmented Riemann solvers to incorporate the source terms due to the variable cross-section. The starting point would be the well-balanced augmented Riemann solvers of Alemi Ardakani et al (2016).
- Develop well-balanced and positivity preserving finite volume methods for the two-dimensional form of the Ripa system for modelling ocean currents using the two-dimensional f-wave Riemann solvers of Bale et al. (2002).
The lead supervisor would be happy to adapt or change the project to better match the interests of the student.
You should have or expect to achieve at least a 2:1 Honours degree, or equivalent, in Mathematics, Physics or Engineering. Experience in Fluid Dynamics and programming in MATLAB Python or Fortran is desirable.
The PhD student will be meeting the lead supervisor every week and will be taught the background mathematics and numerical analysis required for the project. Moreover, the student will be regularly meeting the second supervisors to receive necessary advice and trainings. In addition, the student will be encouraged to attend the relevant Magic courses to their PhD topic and also attend summer schools, conferences and workshops to interact with their world leading mathematicians.
References / Background reading list
Alemi Ardakani, H., Bridges, T. J., Turner, M. R. 2016 Shallow-water sloshing in a moving vessel with variable cross-section and wetting-drying using an extension of George's well balanced finite volume solver. J. Comput. Phys. 314, 590-617.
Bale, D., LeVeque, R. J., Mitran, S., Rossmanith, J. A. 2002 A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24, 955-978.
George, D. L. 2008 Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. J. Comput. Phys. 227, 3089-3113.
LeVeque, R. J. 1997 Wave propagation algorithms for multidimensional hyperbolic systems. J. Comput. Phys. 131, 327-353.
Ripa, P. 1993 Conservation laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn. 70, 85-111.
Ripa, P. 1995 On improving a one-layer ocean model with thermodynamics. J. Fluid Mech. 303, 169-201.
Applicants should have obtained, or be about to obtain, a First or Upper Second Class UK Honours degree, or the equivalent qualifications gained outside the UK. Applicants with a Lower Second Class degree will be considered if they also have Master’s degree. Applicants with a minimum of Upper Second Class degree and significant relevant non-academic experience are encouraged to apply.
All applicants would need to meet our English language requirements by the start of the project http://www.exeter.ac.uk/postgraduate/apply/english/.
How to apply
In the application process you will be asked to upload several documents. Please note our preferred format is PDF, each file named with your surname and the name of the document, eg. “Smith – CV.pdf”, “Smith – Cover Letter.pdf”, “Smith – Transcript.pdf”.
- Letter of application outlining your academic interests, prior research experience and reasons for wishing to undertake the project.
- Transcript(s) giving full details of subjects studied and grades/marks obtained. This should be an interim transcript if you are still studying.
- If you are not a national of a majority English-speaking country you will need to submit evidence of your current proficiency in English.
You will be asked to name 2 referees as part of the application process, however we will not expect receipt of references until after the shortlisting stage. Your referees should not be from the prospective supervisory team.
If you are shortlisted for interview, please ensure that your two academic referees email their references to the firstname.lastname@example.org, 7 days prior to the interview dates. Please note that we will not be contacting referees to request references, you must arrange for them to be submitted to us by the deadline.
References should be submitted by your referees to us directly in the form of a letter. Referees must email their references to us from their institutional email accounts. We cannot accept references from personal/private email accounts, unless it is a scanned document on institutional headed paper and signed by the referee.
All application documents must be submitted in English. Certified translated copies of academic qualifications must also be provided.
The closing date for applications is 1600 hours GMT Monday 6 January 2020. Interviews will be held between 10 and 21 February 2020. For more information about the NERC GW4+ DPT please visit https://nercgw4plus.ac.uk
If you have any general enquiries about the application process please email email@example.com. Project-specific queries should be directed to the lead supervisor.
During the application process, the University may need to make certain disclosures of your personal data to third parties to be able to administer your application, carry out interviews and select candidates. These are not limited to, but may include disclosures to:
- the selection panel and/or management board or equivalent of the relevant programme, which is likely to include staff from one or more other HEIs;
- administrative staff at one or more other HEIs participating in the relevant programme.
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|Application deadline:||6th January 2020|
|Value:||£15,009 per annum for 2019-20|
|Duration of award:||per year|
|Contact: PGR Enquiriesfirstname.lastname@example.org|