Image: D.D. Holm, T. Schmah & C. Stoica. Geometric Mechanics and Symmetry:

From Finite to Infinite Dimensions, Oxford University Press (2009).

### Third AGM Meeting

Wave Current Interactions

The third meeting will be held on Thursday 30 March 2023 at Imperial College London. If you would like to attend the meeting, please register by sending an email to h.alemi-ardakani@exeter.ac.uk

The third AGM meeting in 2023 will be devoted to the topic of Wave Current Interactions. This topic lies at the heart of the NASA/ESA mission entitled Surface Water and Ocean Topography (SWOT) (link here).

Wave Current Interactions are also an important component of the Stochastic Transport in the Upper Ocean (STUOD) project. The STUOD project is funded by an European Research Council (ERC) Synergy Grant. It involves three institutions (Imperial College London, INRIA Rennes, and Ifremer, Brest). The STUOD project aims to create a complete pipeline for the analysis of upper ocean dynamics: problem formulation, mathematical modelling, rigorous analysis, parallel computational software and implementation, performance evaluation, data analysis and data assimilation, with improvements achieved by repetitions of this cycle. The STUOD endeavour draws on the team’s combined expertise in mathematics, physics, scientific computing and oceanography to chart pathways to impact ranging from theory to applications. More details of the STUOD project’s aims and its recent advances toward achieving its succession of goals may be found on the STUOD website (link here).

Mathematical advances based on geometric mechanics have been applied in the STUOD project to guide the development of the needed aspects of data-driven modelling of ocean dynamics into a rigorous theoretical framework which partners with computational simulation and data assimilation in support of the physical oceanographic interpretations of the satellite images arising from SWOT and other satellites that are managed and made accessible to oceanographic researchers under the auspices of Ifremer, Brest.

Theme: Wave Current Interactions

Date: Thursday 30 March 2023, 13:00 – 17:00

Venue: Department of Mathematics, Imperial College London, Huxley Building, Room 747, 180 Queens Gate SW7 2AZ

Organiser: Darryl Holm, Imperial College London

Timetable

12:30–13:00 Arrival and Coffee

13:00–13:50 Jacques Vanneste (University of Edinburgh)

Scattering of swell by vortices

13:50–14:40 Tom Bridges (University of Surrey)

Patching together fields of water waves with distinct wave-mean properties

14:40–15:20 Coffee break

15:20–16:10 Baylor Fox-Kemper (Brown University)

Waves Affect and Detect Climate

16:10–17:00 Oliver Street (Imperial College London)

Interactions between mean flows and inertial gravity waves

Titles and Abstracts

Jacques Vanneste (University of Edinburgh)

Scattering of swell by vortices

The refraction of surface gravity waves by currents leads to the spatial modulation of the wave field and, in particular, to spatial inhomogeneities in the significant wave height. This can be captured by an asymptotic theory based on the smallness of the ratio between the typical current velocity and the wave group speed. For swell-like, highly directional wave spectra, the angular spread of the spectrum is another small parameter. I will explain how the relative size of the two parameters controls the structure of the modulations in significant wave height, and I will illustrate the results for simple localised flow structures such as vortices and dipoles. A broad conclusion is that the formation of caustics, which have attracted a great deal of attention, is unlikely for realistic wave spectra. The theoretical predictions are tested against numerical simulation using WAVEWATCH III.

Tom Bridges (University of Surrey)

Patching together fields of water waves with distinct wave-mean properties

This talk is about constructing connecting orbits between distinct fields of waves. The prototype is a travelling front which is asymptotic to distinct Stokes waves (different wavenumber, amplitude, frequency and mean flow). The fields are joined by a heteroclinic connection. Construction involves satisfying jump conditions (the far-fields must be compatible) which use conservation laws. Then, once hyperbolicity is established, dynamical systems theory is used to construct the connection. Two numerical methods have been successfully implemented: shooting and a

boundary value problem formulation. Extension to 2+1 will also be discussed. This work is with Dave Lloyd, Dan Ratliff, Michael Shearer, and Pat Sprenger. It emerged from the Newton Institute programme on nonlinear waves in Autumn 2022.

Baylor Fox-Kemper (Brown University)

Waves Affect and Detect Climate

Ocean surface gravity waves, the ones familiar to surfers and beachgoers, are mostly generated by winds. Because they are the most direct way that winds affect the ocean, they both affect the climate system and are used in the detection of climate change. It is difficult to measure the winds everywhere on earth, but detection of ocean waves from satellites provides a near-global view into their continual change. Over long distances, changes in the waves accumulate the history of the winds that

blow over them, so changes in the waves are both an impact (through coastal inundation) and detector of climate change. Waves also link closely to ocean turbulence, ocean currents, and sea ice breakup. Nobelist Irving Langmuir first described a key consequence of ocean turbulence driven by waves (windrows) where

flotsam such as pollution (oil and plastics), seafoam, and seaweed accumulates. Accounting for this wave-driven turbulence in climate models has improved our ability to project future global warming, and future improvements in modeling waves’ effects on sea ice and currents lie ahead.

Oliver Street (Imperial College London)

Interactions between mean flows and inertial gravity waves

(Joint work with D. Holm and R. Hu)

We derive the generalised Lagrangian mean (GLM) theory as a phase-averaged variational principle expressed as a composition of two smooth invertible maps. Following Gjaja and Holm 1996, we consider 3D inertial gravity waves (IGWs) in the Euler–Boussinesq fluid approximation. We provide closure models for GLM IGWs at leading order in 3D complex vector WKB wave asymptotics.

For any enquiries, please contact Hamid Alemi Ardakani.