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Shallow water equations with temperature gradients and well-balanced f-wave finite volume Riemann solvers

Project Description

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

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.

Funding Notes

The studentship will provide funding for a stipend which is currently £14,777 per annum for 2018-2019, research costs and UK/EU tuition fees at Research Council UK rates for 48 months (4 years) for full-time students, pro rata for part-time students.


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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