PhD Position F/M Reduced-order modelling of internal tide for altimetry data assimilation

The Inria Rennes - Bretagne Atlantique Centre is one of Inria's eight centres and has more than thirty research teams. The Inria Center is a major and recognized player in the field of digital sciences. It is at the heart of a rich R&D and innovation ecosystem: highly innovative PMEs, large industrial groups, competitiveness clusters, research and higher education players, laboratories of excellence, technological research institute, etc.

The thesis will be directed by Etienne Mémin and supervised by Noé Lahaye and Gilles Tissot, at Inria in Rennes.

The "Odyssey" inter-institute team involves members of Ifremer, Laboratoire d'Océanographie Physique et Spatiale (UMR 6523), IMT Atlantique in Brest and the Inria centre at the University of Rennes. It will provide an extremely rich working environment and collaborations for the candidate. The main aim of this team is to develop innovative, cross-disciplinary research areas (satellite observation / physical modelling / applied mathematics / numerical methods) based on the analysis of observation data and numerical modelling, in order to improve our understanding and knowledge of ocean dynamics.

The candidate will also benefit from the intense research activity around the
stochastic modelling of surface dynamics (ERC STUOD).

Internal tide waves are disturbances in currents and densities that propagate in the ocean, and are generated by the interaction of the astronomical tide and the underwater topography.  They play a major role in ocean dynamics, as they contribute to the transfer and dissipation of energy and to mixing in the ocean, affecting global ocean circulation and its climatic role.

Despite their importance, however, they remain poorly represented in general circulation models, due to the wide range of spatial and temporal scales they cover and their complex and intrinsically non-linear dynamics. In addition, their estimation using observation methods such as satellite altimetry is currently imperfect, mainly because of the temporal variability of the waves and the temporal sampling of the satellites. Among the main dynamic processes involved, the interactions between internal waves and “balanced” turbulence (eddies and jets) are a major source of uncertainty in our understanding of ocean dynamics.

In this thesis, we will seek to develop approaches for modelling the internal tide that take account of these interactions with balanced turbulence and provide a basis for estimating the internal tidal wave field from satellite observations. This thesis is connected with the ‘SWOT’ wide swath oceanographic mission, operational since December 2023, and for which the separation between internal waves and equilibrated flows is crucial.

The methodology used in this thesis will be based on resolvent analysis. This method, derived from fluid mechanics, consists of investigating the spectrum associated with realisations of a non-linear flow by searching for the response/forcing pairs of the linearised system (the resolvent operator), in which the forcing represents the non-linear terms. By constructing a modified formulation of this formalism, we can extract the incoherent part of the wave as being a response to the interaction between the coherent part, assumed to be regular in time, and fluctuations in the balanced dynamics, the latter then appearing in the forcing term. The idea is to identify two bases: one associated with currents and one associated with waves, where each of the modes of one base is dynamically linked to the corresponding mode of the other base.

First, the candidate will implement this formulation in an idealised context, the reference for which will be provided by simulations in the (non-linear) rotating shallow water model. Then, he/she will extend this formulation to a more realistic context, in particular by allowing the multi-frequential character of the coherent part of the internal tide to be taken into account. Another advantage of the proposed formulation, compared with so-called a posteriori strategies, is that it avoids problems of convergence of the basis, which is a fundamental obstacle in the realistic framework.

Secondly, the candidate will formulate a reduced model based on a Galerkin projection of the dynamic equations (linearised shallow water equations) on the basis of resolved modes.  The aim will then be to set up a data assimilation model using this reduced model and a variational formalism (4Dvar) to estimate the internal tide from a set of sea level observations. Again, this assimilation system will first be deployed and tested in an idealised configuration, but it is expected that the thesis will include the application of this methodology to realistic configurations: first on realistic numerical simulations (which give access to a reference solution, or “ground truth”) and then on data from the wide-swath satellite “SWOT”.

The thesis project is at the interface between geophysical fluid dynamics and applied mathematics. Either a background in physics, fluid mechanics or physical oceanography with strong mathematical skills and an interest in numerical simulation, or a background in applied mathematics with a strong interest in fluid mechanics modelling and numerical simulation is sought.

Basic programming skills in Python or Julia and Fortran or C++ will be appreciated.

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
• Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities
• Access to vocational training
• Social security coverage

Salary gross : 2200€