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

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.

Training:

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.

Useful links:

For information relating to the research project please contact the lead Supervisor via [Email Address Removed]
https://emps.exeter.ac.uk/mathematics/staff/ha397

Prospective applicants:

For information about the application process please contact the Admissions team via [Email Address Removed].
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Please note that applications received via other routes including a standard programme application route will not be considered for the studentship funding.