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

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

### Fourth AGM Meeting

Geophysical Fluid Dynamics

The fourth AGM meeting will be devoted to Geometric Methods for Ocean Modelling. The meeting will be articulated along two related themes. The first is the application of geometric methods to the problem of parameterisations of the impact of unresolved eddies and waves on currents. The second theme is the geometry of thermohaline transport in the ocean. The conceptual model of neutral transport, that is, transport restricted to a distribution of planes determined by the salinity and temperature structure, has recently been connected to a series of mathematics topics (Frobenius integrability and the Carathérodory–Chow–Rashevsky theorem, Reeb graphs, hypo-elliptic PDEs). The meeting will offer a first opportunity to bring these connections to the attention of the geometric mechanics community.

Topics expected to be discussed include:

(1) Geometric parameterisations of eddies and waves. Energetically consistent parameterizations. Lagrangian mean theories of wave–mean flow interactions; applications to surface gravity waves and internal waves. Numerical Lagrangian averaging.

(2) The geometry of neutral transport. Neutral transport and sub-Riemannian geometry. Neutral diffusion motion and its large-scale consequences. Construction and topology of approximately neutral surfaces.

Theme: Geophysical Fluid Dynamics

Date: Monday 5 June 2023, 11:00 – 17:40 & Tuesday 6 June 2023, 9:00 – 14:00

Venue: The University of Edinburgh, 50 George Square, Room G.02

Organiser: Jacques Vanneste, University of Edinburgh

Timetable

Monday 5 June 2023

Mini-session on ice-dynamics 11:00–12:25

11:00–11:40 Clara Henry (Tubingen)

Antarctic ice rise dynamics and the consequences for ice sheet evolution

11:45–12:25 James Maddison (Edinburgh)

Uncertainty quantification in glaciological models

12:30–13:30 Lunch

Applied Geometric Mechanics Programme 13:30–17:40

13:30–14:10 Andrew Gilbert (Exeter)

Geometry and vortex motion

14:15–14:55 Han Wang (Edinburgh)

Deep learning to disentangle internal tides and balanced flows

15:00–15:30 Coffee

15:30–16:10 Remi Tailleux (Reading)

Crocco-Vazsonyi and lateral stirring in the oceans

16:15–16:55 Geoff Stanley (Victoria)

The topology of neutral surfaces and their exact geostrophic streamfunction

17:00–17:40 Natalyia Balabanova (Toledo)

Point vortices on non-orientable manifolds

Tuesday 6 June 2023

Applied Geometric Mechanics Programme 9:00–14:00

9:00–9:40 Hamid Alemi Ardakani (Exeter)

A variational principle for the generalized Whitham equations

9:45–10:25 James Maddison (Edinburgh)

The GEOMETRIC parameterization: From paper to GCMs

10:30–11:00 Coffee

11:00–11:40 Hossein Kafiabad (Durham)

A grid-friendly approach for computing generalised Lagrangian mean

11:45–12:25 Théo Lavier (Heriot-Watt)

Optimal transport and the compressible semi-geostrophic equations

12:30–13:10 David Dritschel (St Andrews)

The dynamics of non-hydrostatic rotating shallow-water flows

13:10–14:00 Lunch

Acknowledgements

The fourth meeting of the Applied Geometric Mechanics Network is supported by the London Mathematical Society, the Edinburgh Mathematical Society, and the Engineering and Physical Sciences Research Council under grant number EP/T023139/1.

Titles and Abstracts

Clara Henry (Tubingen)

Antarctic ice rise dynamics and the consequences for ice sheet evolution

Antarctic ice rise dynamics and the consequences for ice sheet evolution Ice rises form in ice shelves in Antarctica where otherwise floating ice is locally grounded due to elevated bed topography. Ice rises regulate the flow of ice towards the ocean and contain various flow regimes (divide flow, vertical shear flow, lateral shear zones and grounding zones) on small spatial scales, meaning they are ideal locations to study ice flow dynamics. The current velocity field and evidence of past flow re-organisation are reflected in the englacial isochronal stratigraphy which can be captured using radar measurements. Comparisons between the observed and modelled isochronal stratigraphy allow the validation of model parameters and boundary conditions. Using the finite element model Elmer/Ice to solve the three-dimensional Stokes equations, we investigate ice rises with isothermal and thermo-mechanically coupled simulations. With an idealised setup, we show that ice rise geometry and the drag imposed on the upstream ice shelf respond with hysteresis to sea level variation. We present a blueprint for the three-dimensional simulation of the isochronal stratigraphy of an ice rise using data integration and investigate the role of the degree of nonlinearity of the strain rate – stress dependence on ice rise flow dynamics.

James Maddison (Edinburgh)

Uncertainty quantification in glaciological models

Variational assimilation has a natural Bayesian interpretation, with mismatch and regularization terms corresponding to likelihood and prior factors in Bayes theorem. The optimal state estimate corresponds to a maximum a posteriori estimate, and the Bayesian interpretation then allows one to add uncertainty measures to this optimal state estimate. In particular one can, in principle, compute posterior variances. This approach is applied to a glaciological model. Satellite observations are combined with a finite element shallow shelf approximation code, obtaining an optimal estimate of the system state, together with uncertainty estimates. The computational techniques used to tackle this high dimensional uncertainty quantification problem are discussed.

Andrew Gilbert (Exeter)

Geometry and vortex motion

A geometrical viewpoint of inviscid incompressible fluid dynamics highlights vorticity as the key field which generates the velocity field and is in turn transported, stretched and rotated, that is Lie-dragged, in the fluid flow. In this setting it is most natural to consider the velocity as a vector field, the momentum as a one-form (or co-vector) field, and the vorticity as a two-form field, making use of the metric and corresponding volume form. Such a view point is not only helpful in the abstract, but also gives practical ways of writing down the equations for vortex motion in a Lagrangian framework, where the coordinate system follows the evolution of a slender vortex. This talk will describe how one can write down the equations for vortex motion using such a coordinate system, which is general is both non-orthogonal and time-dependent. We will apply the framework to recover classic results on the motion of slender vortex rings.

Work with Steve Childress, New York University.

Han Wang (Edinburgh)

Deep learning to disentangle internal tides and balanced flows

Internal tides (ITs) are inertia-gravity waves generated by large-scale oceanic tidal currents flowing over topography, important to oceanographers due to their roles in problems such as deep/upper ocean mixing. Conventionally, for altimetric observations of Sea Surface Height (SSH) data, ITs have been extracted by harmonically fitting over observed time sequences. However, in presence of strong time-dependent phase shifts induced by interactions with mean flows or changes in stratifications, harmonic fits do not work well for data with coarse temporal sampling. Such problem would be exacerbated in the upcoming Surface Water Ocean Topography(SWOT) satellite mission due to the finer spatial scales to be resolved. However, SWOT’s wide swaths will un-precedentedly produce SSH snapshots that are spatially two-dimensional, which allows the community to treat tidal extraction as an operation on two-dimensional images. Here, we regard tidal extraction purely as an image translation problem. We design and train what we call ”Toronto Internal Tide Emulator” (TITE), a conditional Generative Adversarial Network, which, given a snapshot of raw SSH, generates a snapshot of the embedded tidal component. The presentation will introduce a recent work (see (Wang et al., 2021)) where we train and test TITE on a set of idealized numerical eddying simulation . No temporal information or physical knowledge is required for TITE to work in this scenario. We test TITE on data whose dynamical regimes are different from the data provided during training. Despite the diversity and complexity of data, it accurately extracts tidal components in most individual snapshots considered and reproduces physically meaningful statistical properties. Predictably, TITE’s performance decreases with the intensity of the turbulent flow. Ongoing work where we simplify the algorithm will be discussed too.

Remi Tailleux (Reading)

Crocco-Vazsonyi and lateral stirring in the oceans

Lateral stirring in the oceans has been commonly assumed to preferentially take place along ‘neutral surfaces’, along which fluid parcels are supposedly able to exchange position without experiencing restoring buoyancy forces. However, despite their physical justification being seemingly rooted in momentum considerations, the neutral directions have never in fact been explicitly connected to the actual momentum balance equations governing fluid motions. In this work, I show that to establish such a connection, the key is to invoke Crocco-Vazsonyi theorem, which allows one to rewrite the Navier-Stokes equations in their thermodynamic form and to remove the dynamically inert part of the stratification. Such an approach yields a somewhat different definition of neutral surfaces and neutral directions. Similarities and differences with existing approaches, as well as implications for ocean mixing parameterisations, will be discussed.

Geoff Stanley (Victoria)

The topology of neutral surfaces and their exact geostrophic streamfunction

A neutral surface is a 2D manifold such that the ocean's 3D salinity and temperature, when restricted to this surface, have gradients that are aligned and proportional. Alignment of gradients implies alignment of contours of level sets, so salinity and temperature are functionally related on neutral surfaces. This functional relationship is multivalued, with different single-valued branches applying in different 2D subsets of the neutral surface. These 2D regions are determined by the Reeb graph, whose nodes represent extrema and saddle points, and whose edges represent the domains of single-valued branches. This topological theory of neutral surfaces is used to create a new class of approximately neutral surfaces that obeys the alignment of gradients, and also to derive the exact geostrophic streamfunction on neutral surfaces and its approximation on non-neutral but well-defined surfaces.

Natalyia Balabanova (Toledo)

Point vortices on non-orientable manifolds

Hamiltonian approach to the dynamics of point vortices on a surface involves the symplectic form on the surface together with the vorticity of the point vortices. On the other hand, nothing prevents us from considering systems of point vortices on non- orientable manifolds; however, most elements from the classical Hamiltonian approach involve the choice of orientation. In this talk we will discuss the setup for considering point vortices on non- orientable manifolds, with particular emphasis on the case of the Möbius band.

This is joint work with Dr. James Montaldi.

Hamid Alemi Ardakani (Exeter)

A variational principle for the generalized Whitham equations

An averaged Lagrangian for the generalized Green–Naghdi equations for fluid sloshing in rotating coordinates is derived. The Green–Naghdi model has a form of potential vorticity conservation, which can be obtained from the particle-relabeling symmetry property of the Lagrangian. The assumption of zero-potential-vorticity flow is applied to the Green–Naghdi Lagrangian functional to derive a new set of Boussinesq-like evolution equations, which are a generalization of the Whitham equations for fluid sloshing in three dimensions. If time permits, variational principles for wave-body-slosh interactions will be discussed.

James Maddison (Edinburgh)

The GEOMETRIC parameterization: From paper to GCMs

The GEOMETRIC parameterization originates by noting the relationship between eddy momentum fluxes and eddy potential vorticity fluxes in the quasigeostrophic limit. Crucially the eddy momentum fluxes have bounds in terms of the eddy energy, and this allows for the construction of eddy parameterization schemes which are energetically constrained, conserve momentum by construction, and have a natural geometric interpretation. The development of the GEOMETRIC parameterization will be discussed. The scheme originates from fundamental theory, and has since been implemented in models of increasing complexity, most recently being implemented in the NEMO ocean circulation model.

Hossein Kafiabad (Durham)

A grid-friendly approach for computing generalised Lagrangian mean

Lagrangian averages are central to the modern theory of wave-mean-flow interactions and are shown to have many advantages over their Eulerian counterparts. Despite the theoretical advances in defining and analysing this class of averages, their wide-spread application has not been realised mainly due to computational difficulties. To make Lagrangian means more accessible we put forward a novel approach in which the Lagrangian means of various fields (including particle positions) are computed by solving partial differential equations (PDEs). The PDEs can be discretised in a variety of ways, e.g. using the same discretisation as that employed for the governing dynamical equations, and solved on-the-fly to minimise the memory footprint. We illustrate the application of this tool in simulations of geophysical flows.

Théo Lavier (Heriot-Watt)

Optimal transport and the compressible semi-geostrophic equations

The semi-geostrophic equations for a compressible fluid, first analyzed by Cullen and Maroofi [2003], provide a simplified model of the formation and evolution of atmospheric fronts. I will describe the use of semi-discrete optimal transport theory to construct a numerical particle method. This method is structure preserving in the sense that numerical solutions conserve energy. I will then present numerical results and discuss the challenges we faced in implementing this numerical method. Using this approach, we give a constructive proof of the existence of global-in-time weak solutions as the limit of spatially discrete approximations. This work directly extends the work of Bourne et al. [2022] from the incompressible to the compressible setting.

This is joint work with David Bourne, Charlie Egan, and Beatrice Pelloni at Heriot-Watt University.

David Dritschel (St Andrews)

The dynamics of non-hydrostatic rotating shallow-water flows

The shallow-water equations are usually derived by making two approximations: first that the horizontal scales are much larger than vertical ones, and second that the vertical acceleration is negligible (hydrostatics). In fact, the second approximation is not required, and indeed this enables one to model shorter horizontal scales comparable to the fluid depth. Some recent results using this extended model are discussed, including a new extension to include bottom topography.

For any enquiries, please contact Hamid Alemi Ardakani.

The fourth meeting will be held on Monday 5 June 2023 and Tuesday 6 June 2023 at the University of Edinburgh. If you would like to attend the meeting, please register by sending an email to h.alemi-ardakani@exeter.ac.uk