Long nonlinear oceanic surface and internal waves in 3D (Ref: MA/KK-SF1/2025)

About the Project

Solitary waves and undular bores are long nonlinear waves which owe their existence to the balance between nonlinear wave steepening effects and linear wave dispersion. In particular, internal solitary waves commonly occur in the stratified flows of coastal seas, fjords and lakes. Surface and internal waves have a strong effect on offshore and underwater structures. They also contribute to the ocean mixing processes, which is important for climate. The Korteweg - de Vries (KdV) and Ostrovsky type long-wave weakly-nonlinear models became the basic paradigm for interpretation and modelling of long oceanic waves [1].

Recently, the cylindrical KdV-type models were also developed for the ring waves in stratified fluids [2,3,4]. The models include cylindrical divergence and underlying current, which is relevant, in particular, to the waves generated in straits and river-sea interactions zones. The derivation of such reduced models is based on the existence of a suitable far-field modal decomposition [2]. The proposed work will continue the research on 3D waves. It will be a systematic study of surface and internal waves which can be modelled mathematically by Kadomtsev-Petviashvily type equations [5] and their extensions. The research will employ asymptotic approximations combined with numerical simulations both at the level of reduced mathematical models and suitable parent systems.

Specific objectives:

1. To study the waves described by versions of the KP equation associated with different geometries using a combination of asymptotic and numerical methods.
2. To study the mappings between the solutions of these equations.
3. To investigate the effects of variable environment (e.g. slowly-varying bottom topography, currents, etc.) using the KP-type models.

The project will require good knowledge of fluid mechanics, nonlinear wave theory, differential equations, and familiarity with computational packages such as MATLAB, Mathematica or similar.

References:

1. R. Grimshaw (ed.) Solitary Waves in Fluids. Advances in Fluid Mechanics, Vol. 47, WIT Press, 2007.
2. K.R. Khusnutdinova and X. Zhang, Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech. 794 (2016) 17-44.
3. C. Hooper, K. Khusnutdinova, R. Grimshaw, Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current, J. Fluid Mech. 927 (2021) A37.
4. D. Tseluiko, N. Alharthi, R. Barros, K. Khusnutdinova, Internal ring waves in a three-layer fluid on a current with a constant vertical shear, Nonlinearity 36 (2023) 3431-3466.
5. B.P. Kadomtsev and V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (1970) 539-541.

94% of Loughborough’s research impact is rated world-leading or internationally excellent. REF 2021

Supervisors

• Primary supervisor: Karima Khusnutdinova

Entry requirements

A minimum of a 2:1 honours degree (or equivalent international qualification) in mathematics, physics or a related discipline.

English language requirements

Applicants must meet the minimum English language requirements. Further details are available on the International website.

Tuition fees for 2025-26 entry

• UK fee - £5,006 Full-time degree per annum
• International fee - £22,360 Full-time degree per annum

Fees for the 2025-26 academic year apply to projects starting in October 2025, January 2026, April 2026 and July 2026.

How to apply

All applications should be made online. Under programme name, select Mathematical Sciences. Please quote the advertised reference number: MA/KK-SF1/2025 in your application.

To avoid delays in processing your application, please ensure that you submit a CV and the minimum supporting documents.

The following selection criteria will be used by academic schools to help them make a decision on your application. Please note that this criteria is used for both funded and selffunded projects.

Please note, applications for this project are considered on an ongoing basis once submitted and the project may be withdrawn prior to the application deadline, if a suitable candidate is chosen for the project.