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Award details

University of Exeter funding: NERC GW4+ DTP PhD studentship

Mathematical modelling of the shallow-water equations with temperature gradients. Mathematics PhD studentship (NERC GW4+ DTP funded) Ref: 3997

About the award


Lead Supervisor

Dr Hamid Alemi Ardakani, University of Exeter, Penryn Campus, Mathematics

Additional Supervisors

Prof. Bob Beare, University of Exeter.

Prof. Tristan Pryer, University of Bath.

Location: Penryn Campus, University of Exeter, Falmouth, Cornwall.

This project is one of a number that are in competition for funding from the NERC Great Western Four+ Doctoral Training Partnership (GW4+ DTP).  The GW4+ DTP consists of the Great Western Four alliance of the University of Bath, University of Bristol, Cardiff University and the University of Exeter plus five Research Organisation partners:  British Antarctic Survey, British Geological Survey, Centre for Ecology and Hydrology,  the Natural History Museum and Plymouth Marine Laboratory.  The partnership aims to provide a broad training in earth and environmental sciences, designed to train tomorrow’s leaders in earth and environmental science. For further details about the programme please see

For eligible successful applicants, the studentships comprises:

  • An stipend for 3.5 years (currently £15,285 p.a. for 2020-21) 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.
  • Up to £750 for travel and accomodation for compulsory cohort events.

Project details

Project Background:

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:

1- 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).

2- 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).

3- 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.

Candidate requirements:

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.

Background reading and references:

• 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.

Useful links:

For information relating to the research project please contact the lead Supervisor via

Prospective applicants:

For information about the application process please contact the Admissions team via

Each research studentship project advertisement has an ‘Apply Now’ button linking to an application portal. Please note that applications received via other routes including a standard programme application route will not be considered for the studentship funding.


NERC GW4+ DTP studentships are open to UK and Irish nationals who, if successful in their applications, will receive a full studentship including payment of university tuition fees at the home fees rate.

A limited number of full studentships are also available to international students which are defined as EU (excluding Irish nationals), EEA, Swiss and all other non-UK nationals.

Studentships for international students will only cover fees at the UK home fees rate. However, university tuition fees for international students are higher than the UK home fees rate therefore the difference will need to be funded from a separate source which the student or project supervisor may have to find. Unfortunately, the NERC GW4+ DTP cannot fund this difference from out studentship funding Further guidance on how this will work will be issued in November.

The conditions for eligibility of home fees status are complex and you will need to seek advice if you have moved to or from the UK (or Republic of Ireland) within the past 3 years or have applied for settled status under the EU Settlement Scheme.


Entry requirements

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


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”.

  • CV
  • 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.

Reference information
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, 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 Friday 8 January 2021 2359 GMT .  Interviews will be held between 8th and 19th February 2021.  For more information about the NERC GW4+ DPT please visit

If you have any general enquiries about the application process please email  Project-specific queries should be directed to the lead supervisor.

Data Sharing
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.

Such disclosures will always be kept to the minimum amount of personal data required for the specific purpose. Your sensitive personal data (relating to disability and race/ethnicity) will not be disclosed without your explicit consent.


Application deadline:8th January 2021
Value:£15,285 per annum for 2020-21
Duration of award:per year
Contact: PGR Enquiries