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Mathematical modelling of the shallow-water equations with temperature gradients. PhD in Mathematics (NERC GW4+ DTP)

  • Full or part time
  • Application Deadline
    Monday, January 06, 2020
  • Competition Funded PhD Project (European/UK Students Only)
    Competition Funded PhD Project (European/UK Students Only)

Project Description

Lead Supervisor
Dr Hamid Alemi Ardakani, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, University of Exeter

Additional Supervisors
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 accommodation 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

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.

Funding Notes

NERC GW4+ funded studentship available for September 2020 entry. For eligible students, the studentship will provide funding of fees and a stipend which is currently £15,009 per annum for 2019-20.

References

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.

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