Shallow water equations with temperature gradients and well-balanced f-wave finite volume Riemann solvers, NERC GW4+ DTP, PhD in Mathematics Ref: 3324

About the award


Lead Supervisor

Dr Hamid Alemi Ardakani, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, Penryn Campus, University of Exeter

Additional Supervisors

Dr Bob Beare, Department of Mathematics, College of Engineering Mathematics and Physical Sciences, Streatham Campus, University of Exeter

Professor Andrew Hogg, University of Bristol

Location: University of Exeter, Penryn Campus, Penryn, Cornwall TR10 9FE

Main Information

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 index-linked stipend for 3.5 years (currently £14,777 p.a. for 2018/19);
  • Payment of university tuition fees;
  • A research budget of £11,000 for an international conference, lab, field and research expenses;
  • A training budget of £4,000 for specialist training courses and expenses.

Up to 30 fully-funded studentships will be available across the partnership.

Students from EU countries who do not meet the residency requirements may still be eligible for a fees-only award but no stipend.  Applicants who are classed as International for tuition fee purposes are not eligible for funding.

Project details

The shallow water equations are a set of hyperbolic PDEs which are widely used in geophysical fluid dynamics, 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 (in one and two dimensions) 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 an augmented well-balanced f-wave Riemann finite volume solver for the shallow water equations over variable bottom topography in one dimension with horizontal temperature gradient, by including an extra equation for the flux of the momentum equation. The starting point for this part of the project are the works of George (2008) and Alemi Ardakani et al. (2016). Possible extensions of the theory and numerics to two-layer fluid flows.

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 and fluid sloshing in a rectangular container, 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.


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.

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.

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 and Fortran is desirable.

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.
  • Two References (applicants are recommended to have a third academic referee, if the two academic referees are within the same department/school).

Reference information
You will be asked to name two referees as part of the application process.  It is your responsibility to ensure that your two referees email their references to, as we will not make requests for references directly; you must arrange for them to be submitted by 7 January 2019

References should be submitted 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 midnight on 7 January 2019.  Interviews will be held between 4 and 15 February 2019.

If you have any general enquiries about the application process please email  Project-specific queries should be directed to the 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:7th January 2019
Value:£14,777 per annum for 2018-19
Duration of award:per year
Contact: PGR Enquiries