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Mathematical modelling of internal waves in rotating stratified shear flows

About This PhD Project

Project Description

Internal waves are waves supported by density gradients. They are ubiquitous in fluids such as the oceans and the atmosphere. The waves have a strong effect on offshore structures, underwater cables and submersibles. They also contribute to the ocean mixing processes.

The Korteweg - de Vries (KdV) type integrable and near-integrable long wave weakly-nonlinear models became the basic paradigm for interpretation of oceanic observations and modelling results [1]. When background rotation of the planet is taken into account, the KdV equation is replaced by the Ostrovsky equation [2]. Recently, the model has been generalised to include the combined effects of rotation and background shear flow (current) [3, 4]. The KdV model for internal waves has been generalised to include cylindrical divergence [5], which is relevant to the modelling of the waves generated in straits, river-sea interactions zones and by interaction with localised topographic features.

The research will be a systematic study of nonlinear internal waves modelled mathematically by such weakly nonlinear models. It will involve a combination of asymptotic approximations combined with numerical simulations.

The project will require basic knowledge of differential equations, fluid mechanics, and good numerical skills.

Loughborough University

Loughborough University is a top-ten rated university in England for research intensity (REF, 2014) and an outstanding 66% of the work of Loughborough’s academic staff who were eligible to be submitted to the REF was judged as ‘world-leading’ or ‘internationally excellent’, compared to a national average figure of 43%.

In choosing Loughborough for your research, you’ll work alongside academics who are leaders in their field. You will benefit from comprehensive support and guidance from our Doctoral College, including tailored careers advice, to help you succeed in your research and future career.

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How to apply

All applications are made online, please select the school/department name under the programme name section and include the quote reference number: KK/DT-Un1/2020


[1] R. Grimshaw (ed.) Solitary Waves in Fluids. Advances in Fluid Mechanics, Vol. 47, WIT Press, 2007.

[2] L.A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Oceanology 18 (1978) 119-125.

[3] A. Alias, R.H.J. Grimshaw, K.R. Khusnutdinova, On strongly interacting internal waves in a rotating ocean and coupled Ostrovsky equations, Chaos 23 (2013) 023121.

[4] A. Alias, R.H.J. Grimshaw, K.R. Khusnutdinova, Coupled Ostrovsky equations for internal waves in a shear flow, Phys. Fluids 26 (2014) 126603.

[5] K.R. Khusnutdinova and X. Zhang, Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech. 794 (2016) 17-44.

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