Covering more than 70% of the Earth's surface, the oceans play key roles on the Earth climate regulation as well as for human societies. Yet, from wave breaking events to the movement of weather systems, the predictive capabilities of models notoriously quickly diminish with increasing lead times, even with the assistance of the world's largest supercomputers. Despite ever-increasing developments to simulate and observe the coupled ocean-atmosphere system, our ability to understand, reconstruct and forecast the ocean dynamics remains fairly limited for numerous applications.

Our motivations are to help break this apparent logjam, and more specifically to bridge model driven and observation-driven paradigms to develop and learn novel stochastic representations of the coupled ocean-atmosphere dynamics. To address these challenges, Odyssey gathers a unique transdisciplinary expertise in Numerical Methods, Applied Statistics, Data Science, Satellite and Physical Oceanography. Methodological developments are primarily implemented and demonstrated through three main objectives: (i) the analysis of mesoscale/submesoscale processes and internal waves, (ii) the monitoring of extremes ocean-atmosphere events and routes to rapid intensifications; (iii) the derivation of forefront deep-learning stochastic data assimilation techniques. The name Odyssey is a short-cut that stands for “Ocean DYnamicS obSErvation analYsis' – the keyword “Analysis” has to be understood in terms of physical understanding, mathematical analysis and data analysis.

The objectives and research actions of the team can be separated in four methodological axes:

A primary focus of the team intends to better characterize poorly known mechanisms of energy redistribution operating at different scales, through the interactions of different physical mechanisms such as hydrodynamical instabilities, internal or wind waves, turbulence and ocean atmosphere feedback exchanges. Our first credo is that an improved physical understanding cannot be achieved uniquely on the basis of sparse-in-time observations alone or from intrinsically imperfect models: data without models are uninformative and models built without data are useless, as models are generally too far from real-world situations of interest. Today, data and models shall thus be combined to tackle uncertainty quantification and probabilistic ensemble forecasting issues, as advanced data-driven representation of ocean dynamics requires; to that end we need to drift from a purely deterministic physics toward stochastic representations. This is the second credo. Many aspects of the models or of the data-model coupling functional still need to be specified or parameterized through dynamically-adapted basis functions, evolving parameters or covariance matrices. Our third credo is that the improved physical understanding of the multi-scale interactions encoded in such parametrizations can be learned or estimated from data.

The research objectives of our group naturally distribute in several challenges, exploring multimodal (differing space-time resolutions, differing passive and active microwave instruments, ...) observations, air-sea exchanges and upper ocean dynamics, bottom boundary turbulent processes, stochastic flow representations, data assimilation and machine learning procedures. All these challenges take place or rely on principles and/or tools of the four methodological contexts introduced above.

Global Earth Observation (GEO) systems, in situ and satellite platforms, have significantly improved our understanding and capability to manage the Earth's environment. Key products today include, among others, merged global ocean surface topography using the different available altimeter missions, global and daily high-resolution sea surface temperature and ocean colour using multi-sensor and platform measurements. One may also cite the mapping of high sea winds from combined radiometer/scatterometer, including very-high resolution synthetic aperture radar observations, and more recently, the fusion of sea state data (largely improved with the recently launched CFOSAT mission, combined with Copernicus Sentinel-1 and 2 measurements). Pushing to higher spatial resolution (about 10 m to 1 km), signatures of tracer variations from imaging instruments can further provide quantitative information, especially for characterizing internal and surface waves in interactions with the ambient underlying upper ocean flow. Note, modern satellite sensor capabilities, sustained under the Copernicus programme, will soon include the new wide-swath Surface Water & Ocean Topography (SWOT) altimeter, to more precisely characterize ocean sea surface height variability. An essential goal is thus to incorporate and combine these high resolution global observations of air-sea exchanges and upper ocean dynamics into our applications of new methods of mathematical analysis, numerical simulations, stochastic analysis and machine learning to create new capabilities. We aim to combine multi-sensor data algorithm developments with advances in mining and learning from multi-modal observations, i.e. satellite and in-situ measurements, including numerical outputs. The scientific targets of this axis are to fully unveil (1) upper ocean mesoscale variability and its associated lateral exchange processes, known as “eddy fluxes”, (2) sub-mesoscale variability and associated upper-ocean vertical exchange processes, known as “vertical exchange”, and finally (3) internal gravity wave variability (induced by winds, tides, and interactions of low-frequency currents with topography). Another central scientific objective is to explore and develop data-model-driven techniques in the context of extreme marine-atmosphere events, to provide new insights for air-sea exchanges processes and adapted parameterization under extreme conditions.

The core of this theme of research addresses modelling and analysis issues in geophysical fluid dynamics.
Within this context, we mainly focus on the study of the dynamics of the upper oceanic circulation.
One overall objective is to devise random models representing the effects of the computationally unresolvable scales of fluid motion on the resolved scales.
Such models are used for ensemble forecasting, uncertainty quantification and data assimilation.
The representation of the fine-scale effects on the coarser scales of motion depends on the level of geophysical fluid approximation pertinent to the data resolution and to the scale of the other physical processes involved.
An important research effort of the team in this context is to pursue the development of a recently established class of models of stochastic transport in fluid dynamics at the most fundamental level.
This class of models, referred to as model under Location Uncertainty (LU), has the advantage to be derived from physical conservation laws expressed through the stochastic transport of fluid parcels.
As such, they are easily extendable to classical approximations of geophysical dynamics. and the stochastic partial differential equations have nearly the same shape as the corresponding deterministic ones.
As for the ocean models, a known hierarchy of approximate stochastic models can be built from the Navier-Stokes equations almost exactly in the same way as in the deterministic setting.
One of their strong assets is to lead to proper energy conservation and provide new approaches to subgrid parameterization, expressed both in terms of fluctuation distributions, and spatial/temporal correlations.

Research activities in the ODYSSEY team on this subject are manifold. First, the mathematical properties of the involved stochastic partial differential equations are poorly known and need to be explored. The overall objective of the challenge is to explore to what extent the known properties of deterministic flow dynamics models are conserved in the stochastic framework. This concerns for instance local well-posedness of the Navier- Stokes equation or of its oceanic representatives. Another issue concerns the physical analysis of such systems. Do the stochastic systems with general noise models still admit some wave solutions (Rossby wave, Gravity waves, internal waves, etc.)? The characterization of the statistical moments associated with those wave solutions are of primal interest from a physical perspective but also to define proper shape functions for the random terms involved. All these issues are currently being studied within the STUOD project. Finally, the ODYSSEY team also addresses the development and validation of new numerical scheme for both deterministic and stochastic models of geophysical flows. In the stochastic case, the numerical approximation of the SPDEs requires the discretization of both the space and time domains. For the spatial discretization classical schemes can be used, however special care must be taken for the temporal schemes. The consistency of several splitting schemes is studied and numerically implemented.

A first research effort in this theme is dedicated to the development of ensemble data assimilation techniques for geophysical problems (in this context, models and observations from e.g. satellites), addressing the issue of linearity and gaussianity hypotheses, which are major limitations of these approaches. Following recent results on the application of particle filters to address these issues on high-dimension problems, we further develop new schemes relying on multiscale dynamical paradigms. Particle filters comprise a class of numerical methods that produce asymptotically consistent approximations of posterior distributions of partially observed systems. We study hierarchical ensemble data assimilation filters, able to handle multiscale interaction in a nested hierarchy of models (from coarse to fine scale). This multiscale capability (not available today even in a simple coarse form) is expected to provide an important analysis tool to study ocean/atmosphere interactions at different scales. The hierarchy of ocean dynamics models rely on the nested capability provided by the stochastic derivation framework described in the second methodological context.

A second axis of work is more dedicated more directly to the development, the implementation and the validation of simplified models of th ocean dynamics, with the main target to couple these models to the observation via data assimilation techniques. These models aims at covering a wide range of motions in the ocean. The mesoscale eddying dynamics (with typical horizontal scales greater than 100 km), such as multi-layer QG models with the inclusion of active temperature tracer (Thermal QG or coupled Surface QG / QG models) and/or surface mixed layer, allowing to couple the dynamics to sea surface temperature data. Higher frequency motions, such as internal waves and internal tides, are addressed using a hierarchy of models based on the rotating shallow water equations (possibly with some linearization). The development of these models mirrors the evolving nature and growing quantity of data available, with recent and new missions such as SWOT or CFOSAT.

This research axis is focused on the exploration and development of data-driven and learning-based schemes and their interactions with model-based approaches, which constitute the state-of-the-art in ocean and atmosphere science. The general goal is to improve the understanding, modeling, forecasting and reconstruction of air-sea exchanges and upper ocean dynamics, as well as bottom turbulent processes, from the in-depth exploration of the existing observation and simulation data. We jointly explore three main complementary data-driven frameworks, including their possible couplings: analog schemes, kernel approaches, especially RKHS (Reproducing kernel Hilbert space), and deep neural network (NN) representations. RKHS and NN naturally arise as they may directly link to model-driven representations (e.g., NN regarded as discrete numerical solvers for ODE/PDE). Analog methods provide simple yet efficient sampling schemes for complex dynamics. Our recent contributions emphasize the relevance of these data-driven frameworks for the modelling, forecasting and assimilation of upper ocean dynamics on toy models. Ongoing studies aim at extending such methodologies for the learning of subgrid processes in full models. Besides, our recent developments illustrated on simplified systems, including for instance the identification of Neural ODE representations for partially-observed systems as well as the identification of stochastic latent dynamics, provide the methodological and numerical basis for the considered challenges.

This research axis specifically investigate the following issues: (i) embedding explicit or implicit physics-informed priors (e.g., stability, conservation laws, stochasticity, chaos...) into data-driven and hybrid representations, (ii) learning latent representations for oceanic flows and air-sea exchanges accounting for flow stochasticity, including extremes (iii) learning schemes when dealing with partially-observed, irregularly-sampled and noisy dynamics, (iv) the joint learning of data-driven representation and associated data assimilation schemes, possibly directly from observation data.

The application domain is mainly geophysical environmental flows, related to ocean dynamics. By designing new approaches for observation analysis, data-model coupling and stochastic representation of fluid flows, the Odyssey group contributes to several application domains of great interest for the community and in which the analysis of complex turbulent flow is key.

Ocean circulations play a major role in the climate and in the biodiversity of ecosystems. These aspects are crucial for the sustainability of the resources of human societies. Understanding and providing tools to predict ocean dynamics is a brick to apprehend our environment and to help making decisions.

The team has nothing special to report. We had quite a few new project funded, and pursued our efforts in the context of previously-launched projects. The new-generation wide-swath altimeter mission SWOT has been succesfully launched in December 2022, and provided a first round data that over-perform the expectations, opening very exciting and promising perspectives for the team.

Recalling that our current paradigm is that process understanding derived from measurements shall foster improved models (theoretical, numerical) for improved both short-term predictions and long-term projections, important efforts have been dedicated on targeting marine-atmosphere extreme events. Indeed, NWP re-analysis (e.g. ERA-5) generally poorly resolve extreme marine-atmosphere events and their surrounding environment. Such spatio-temporal inconsistencies and unreliability of global historical re-analyses can thus hamper more accurate simulation and the projection of future changes in the main characteristics (size, intensity, locations, translation speed) of extreme events. In particular for intense vortex systems (tropical cyclones, polar lows), near-core surface wind structural properties are today still not precisely recorded and re-analyzed. Present-day available model-data cubes must thus be more systematically combined with direct observations (satellite, in situ). In particular, some theoretical and observational evidences have been accumulated and tested to monitor the integrated kinetic energy. Two characteristic scales have been identified and uniquely estimated using high-resolution ocean surface winds from all-weather spaceborne synthetic aperture radar: the radius of significant upward motions in the inflow layer, controlled by the surface wind decay, and the radius of vanishing azimuthal velocity in the outflow layer, associated with the maximum surface winds. By juxtaposing the high-resolution measurements with best-track intensity and size time derivative estimates, the instantaneous knowledge of the two characteristic scales has then been shown to inform on the steadiness of integrated kinetic energy. The resulting criterion of steadiness depends on a multiplicative constant characterizing the system's thermodynamics. Part of this investigation is in the context of Arthur Avenas PhD work.

Within the Marine-Atmosphere eXtreme Sensor Synergy (MAXSS) project, the team builds an advanced and unique workbench to more precisely study these ocean-atmosphere extreme events, from their generation to their impacts. Specifically, efforts have been dedicated to generate new 10-year-long databases:

We present multiscale, non-linear and directional statistical characterizations of images based on high order statistics and information theory. These characterizations allow us to characterize the multiscale properties directionally and to explore their anisotropy. We use this framework to study different turbulent flows from homogeneous and isotropic to inhomogeneous and anisotropic flows.

We first address several challenges that are expected to arise when analyzing future SWOT data: the separation of wave and eddy dynamics, and spatio-temporal sampling issues. In particular, we aim to quantify the contribution of complementary data sources (drifting buoys, satellite temperature or optical imagery) to resolving these various challenges. To this end, we have so far concentrated our efforts on the analysis of idealized, high-resolution global numerical simulations (LLC4320, eNATL60). This year, we have pursued our analysis and valorization efforts following Zoé Caspar-Cohen's PhD thesis. A final article is in preparation (Caspar-Cohen et al. "Combining surface drifters and high resolution global simulations enables the mapping of internal tide surface energy" ). More recently, analyses of altimetry and in situ observations (drifting buoys) were carried out as part of Margot Demol's thesis. This work is the subject of a first manuscript. We actively contributed to the success of the CSWOT experimental campaign (April 2023). Analysis of the corresponding SWOT data is in progress. New activities have been initiated around SWOT data analysis (physics of short internal tidal wave measurements).

We introduce a stochastic approach to model the ocean surface Ekman boundary layer. This model incorporates wind, surface waves, and turbulent mixing effects. A steady version as well as a time dependent version of this generalized Ekman model has been developed. They both consider the vertical mixing effect of Stokes drift in addition to traditional Ekman-Stokes terms. The stochastic approach aligns with traditional parameterizations through random parameter definitions. Numerical simulations are used to assess uncertainties in the Ekman layer, focusing on statistical moment responses and sensitivity analyses of random parameters.

To infer eigenvalues of the infinite-dimensional Koopman operator, we study the leading eigenvalues of the autocovariance matrix associated with a given observable of a dynamical system. For any observable for which all the time-delayed autocovariance exist, we construct a related Hilbert space and a Koopman-like operator that acts on it. We prove that the leading eigenvalues of the autocovariance matrix has one-to-one correspondence with the energy of that observable; the associated eigenvectors correspond to the eigenvectors of the Koopman operator. The proof is associated to several representation theorems of isometric operators on a Hilbert space, and the weak-mixing property of the observables represented by the continuous spectrum.

In the context of mesoscale/submesoscale variability of the surface and shallow subsurface ocean, two geographical sites have been more particularly studied: the region north of Brazil and the Straits of Gibraltar. Both have been sampled experimentally, but have also been the loci of process studies. 1) In the former (article by Subirade et al), we have characterized the number, structure, trajectory, and lifetime of NBC rings using satellite altimetry and the in-situ measurements of the EUREC4A experiment. We have shown that altimetry in the pre-SWOT era yields weaker currents than sampled in situ. Also, it appeared that 4 to 5 NBC rings were spawn each year. These rings interact with the Amazon plume, and form fronts, with a seasonal variability, as shown by Marin Menard's M.Sc. work. Below the NBC rings, submesoscale vortices are generated in the Demerara Bay and later interact vertically with the NBC rings. The vertical interaction of eddies has been studied more theoretically in the paper by Reinaud and Carton. We have also conducted a study on the definition of the boundaries of an oceanic eddy. Using both theory and in situ data, we have shown that it is characterized by a maximum of horizontal to vertical components of Ertel PV (paper by Barabinot et al., 2024). Several criteria have been derived for the value of this ratio, in particular the limitation of existence of a stable boundary by symmetric instability, or the observation of an inflexion point of the isopycnals at this place. 2) In the latter, at very fine scale (article by Roustan et al.), we have shown that the barotropic tide coupled with the Atlantic inflow/Med outflow exchange, leads to hydraulic jumps on Camarinal Sill and to the formation of internal bores. These bores degenerate into internal waves and particularly into solitary waves (ISW), which propagate eastward and to a lesser degree, westward, southward and northward (by reflection on the Moroccan shelf). Bore and wave breaking lead to an intense diapycnal mixing which is well characterized at the interface between the inflow and the outflow. Vertical recirculation and strong turbulent mixing is observed in the bottom (frictional) layer. The dynamics of ISWs and the quantification of mixing are the subject of forthcoming papers (PhD thesis by Jean Baptiste Roustan). Finally, as a whole, the PhD thesis by Ashwita Chouksey has covered the whole Atlantic Ocean with a focus on Mediterranean water eddies.. She has statistically characterized subsurface, submesoscale eddies (including meddies). She has discovered new deep eddies and she has described their interactions. Finally, she has shown that bottom eddies west of the Mid Atlantic Ridge were shielded and achieved very slow motion and interactions.

Current climate models are known to systematically overestimate the rate of deep water formation at high latitudes in response to too deep and too frequent deep convection events. We propose in this study to investigate a misrepresentation of deep convection in Hydrostatic Primitive Equation (HPE) ocean and climate models due to the lack of constraints on vertical dynamics. We discuss the potential of the Location Uncertainty (LU) stochastic representation of geophysical flow dynamics to help in the process of re-introducing some degree of non-hydrostatic physics in HPE models through a pressure correction method. We then test our ideas with idealized Large Eddy Simulations (LES) of buoyancy driven free convection with the CROCO modeling platform. Preliminary results are encouraging, and support future efforts in the direction of enriching coarse resolution, hydrostatic ocean and climate models with a stochastic representation of non-hydrostatic physics.

We have focused on understanding the role of climate variability in observation of climate changes. This was first with Antoine Hochet (Hochet et al., 2023a) done is estimating if the length of satellite records (30 years) is long enough to detect the effect of anthropogenically forced climate change on wave height trends? Using a statistical model to derive Hs from sea level pressure field and exploiting ERA-5 reanalysis data as well as 80 members of the Community Earth System Model v2 large ensemble, we show that, over the North Atlantic (NA), altimetry-based trends are mostly caused by internal variability. This suggests that changes computed over the satellite era are not yet controlled by anthropogenic climate change. Starting from 1993, the date of emergence, defined as the date when the forced signal becomes dominant over the internal variability, is later than 2050 for H s in the NA Then we have focused with Antoine Hochet (Hochet et al., 2023b) on understanding the mechanisms of regional steric sea level variability in the context of regional sea level variability. We have developed a novel method based on steric sea level variance budget that allows to detect the sources and sinks of the variability. Using ECCO state estimate, we show that interannual steric sea level variability is mainly sustained by interannual fluctuating winds via Ekman transport almost everywhere. The damping of the variability is made by both the interannual fluctuating net heat flux from the atmosphere, that largely dominates the atmospheric freshwater fluxes, and the parametrized effect of eddies. It is also found that the parametrized effect of diffusion on the variability is weak in most regions and that, although globally weak, the fluctuations of atmospheric freshwater fluxes are a source of variance close to the Equator in the Pacific Ocean.

Using outputs from the realistic high-resolution (

In parallel, we focus on the surface signature of the internal tide and the incoherence (lack of regularity in time). We show that the typical time of decorrelation varies between 1 month and 1 day, with shorter time associated with regions of strong mesoscale activity and internal tide with the shortest horizontal scale (i.e., high vertical mode number). A paper is about to be submitted to GRL.

In the context of the M2 internship of Olivier Larroque (co-supervized by Vladimir Zeitlin, LMD, IPSL), we investigated the properties of exact dipolar solution that can propagate along the equator while carrying a temperature (density) anomaly. These structures are exact solutions of an asymptotic, weakly non-divergent, system of equation describing the equatorial dynamics, and where tested in a parent model of greater complexity, the thermal rotating shallow water model on the equatorial beta-plane, by means of numerical simulations. A paper is currently under revision in the Journal of Fluid Mechanics.

We address the dynamics of the near-surface. A collaboration with Hereon has launched us on the analysis of a dataset from two campaigns in the Agulhas Current region, where the Diurnal Warm Layer signal is predominant, and in which microstructure measurements have been made. Analysis of near-surface mixing processes is currently underway and will be the subject of an article and a chapter in Mariana Lage's thesis.

We have developed new methods – Geometry-preserving lie group integrators –, for numerical integration of differential equations, that rely on covariance matrices, [Drumetz et al 2023, International Conference on Geometric Science of Information]. In addition, in the context of C. Bonnet PhD thesis (UBS), we have constructed efficient methods for probability distributions based on covariant matrices and optimal transport (published in International Conference on Machine Learning). Finally, in the context of G. Morel postdoc, we improved generative AI models using optimal transport and variational penalization of Euler equations in large dimension (published in Transactions on Machine Learning Research).

In a first paper ("Continuous boundary condition at the interface for two coupled fluids"), we consider two laminar incompressible flows coupled by the continuous law at a fixed interface

In a possible forthcoming paper, we consider the same framework as in the first paper, i.e two incompressible fluids, but this time coupled with a non-linear interface law and driving conditions at the top and at the bottom. Moreover a transport term has been added. We show the existence of weak solutions and the convergence of numerical simulations obtained using the same kind of Scharz algorithms. The evolution case is being processed from a theoretical and a numerical point of view as well

Over the past year we have continued to analyse our numerical solutions GIGATL Gula et al. 2021, which are simulations of the Atlantic Ocean using the CROCO model at meso- and submesoscale resolutions (6 km, 3 km and 1 km) with realistic topography, high-frequency surface forcing and tidal forcing. An example animation showing the surface dynamics (eddies and waves) and the richness of the deep circulation, in particular the coherent eddies, is shown here.

One of the recently published results using these simulations concerns the kinetic energy cascade at the ocean surface and the ability to estimate it from observations [Schubert et al., 2023]. We have shown that the total geostrophic inverse scale kinetic energy flux is linearly related to quantities that can be computed from along-track altimetry, and have presented for the first time its regional distribution and seasonal cycle on scales of 40 to 150 km for large parts of the global ocean based on observations.

Finally, we have also proposed a new form of potential vorticity (PV), rescaled using the rearranged Lorenz density profile, the novelty being that we consider its time evolution. We argue that this rescaled PV is more representative of the dynamics, in particular for assessing the respective effects of mixing and friction on the generation of the geostrophic circulation. The effect of mixing at the global scale, which only modifies the global stratification at rest, is taken into account in the evolution equation of this "objective" definition of PV, in the sense that it scales the PV changes with respect to their effect on the circulation [Morel et al, 2023].

In this research axis we have devised a principle to derive representation of flow dynamics under location uncertainty. Such an uncertainty is formalized through the introduction of a random term that enables taking into account large-scale approximations or truncation effects performed within the dynamics analytical constitution steps. Rigorously derived from a stochastic version of the Reynolds transport theorem, this framework, referred to as modeling under location uncertainty (LU), encompasses several meaningful mechanisms for turbulence modeling. It indeed introduces without any supplementary assumption the following pertinent mechanisms: (i) a dissipative operator related to the mixing effect of the large-scale components by the small-scale velocity;(ii) a multiplicative noise representing small-scale energy backscattering; and (iii) a modified advection term related to the so-called turbophoresis phenomena, attached to the migration of inertial particles in regions of lower turbulent diffusivity. In a succession of works we have shown how the LU modelling can be applied to provide stochastic representations of a variety of classical geophysical flows dynamics. Numerical simulations and uncertainty quantification have been performed on Quasi Geostrophic approximation (QG) of oceanic models. It has been shown that LU leads to remarkable estimation of the unresolved errors opposite to classical eddy viscosity based models. The noise brings also an additional degree of freedom in the modeling step and pertinent diagnostic relations and variations of the model can be obtained with different scaling assumptions of the turbulent kinetic energy (i.e. of the noise amplitude). For a wind forced QG model in a square box, which is an idealized model of north-Atlantic circulation, we have shown that for different versions of the noise the QG LU model leads to improve long-terms statistics when compared to classical large-eddies simulation strategies. For a QG model we have demonstrated that the LU model allows conserving the global energy. We have also shown numerically that Rossby waves were conserved and that inhomogeneity of the random component triggers secondary circulations. This feature enabled us to draw a formal bridge between a classical system describing the interactions between the mean current and the surface waves and the LU model in which the turbophoresis advection term plays the role of the classical Stokes drift. A study of a stochastic version of the primitive equations model is currently investigated within the PhD of Francesco Tucciarone. Preliminary results have been published in the STUOD proceedings.

In another study we explored the calibration of the noise term through dynamic mode decomposition (DMD). This technique is performed on high-resolution data to learn a basis of the unresolved velocity field, on which the stochastic transport velocity is expressed. Time-harmonic property of DMD modes allowed us to perform a clean separation between time-differentiable and time-decorrelated components. Such random scheme is assessed on a quasi-geostrophic (QG) model and has been published in the STUOD proceedings.

In this study we analyze the theoretical properties of a stochastic representation of the incompressible Navier-Stokes equations defined in the framework of the modeling under location uncertainty (LU). We demonstrate, through classical arguments, the existence of martingale solutions for the stochastic Navier-Stokes equations in LU form. We show they are pathwise and unique for 2D flows. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension 3, to a solution of the deterministic Navier-Stokes equation. similarly to the grid convergence property of well-established large-eddies simulation strategies, this result brings some guarantee on the interpretation of the LU Navier-Stokes equations as a consistent large-scale model of the deterministic Navier-Stokes equation.

We investigate how weakening the classical hydrostatic balance hypothesis impacts theoretical properties of the LU primitive equation, such as its well-posedness. The models we consider are intermediate between the incompressible 3D LU Navier-Stokes equations and the LU primitive equations with standard hydrostatic balance. Also, they are expected to be numerically tractable, while accounting well for non-hydrostatic phenomena. Our main result is the well-posedness of a certain stochastic interpretation of the LU primitive equations: we proposed a weak filtered hydrostatic hypothesis, meaning the system we consider accounts for the influence of the transport noise of the vertical velocity component, of which higher frequencies are cut off. This well-posedness result holds with rigid-lid type boundary conditions, and when the horizontal component of noise is independent of depth. However, the vertical component of the noise can remain general. In fact, this assumption can be related to the physical validity domain of the primitive equations. Moreover, we present and study two non-filtered models, in which the transport noise of the vertical component is regularised using eddy-(hyper)viscosity terms.

The rotating shallow water model is a simplification of oceanic and atmospheric general circulation models that are used in many applications such as surge prediction, tsunami tracking and ocean modelling. In this paper we introduce a class of rotating shallow water models which are stochastically perturbed in order to incorporate model uncertainty into the underlying system. The stochasticity is chosen in a judicious way, by following the principles of location uncertainty. We prove that the resulting equation is part of a class of stochastic partial differential equations that have unique maximal strong solutions. The methodology is based on the construction of an approximating sequence of models taking value in an appropriately chosen finite-dimensional Littlewood-Paley space. Finally, we show that a distinguished element of this class of stochastic partial differential equations has a global weak solution. This work in collaboration with Dan Crisan and Oana Lang has been published in the journal of Mathematical Fluid Mechanics.

In this work we investigated the wave solutions of a stochastic rotating shallow water model. This approximate model provides an interesting simple description of the interplay between waves and random forcing ensuing either from the wind or coming as the feedback of the ocean on the atmosphere and leading in a very fast way to the selection of some wavelength. This interwoven, yet simple, mechanism explains the emergence of typical wavelength associated to near inertial waves. Waves that are not in phase with the random forcing are damped at a rate that depends on the random forcing variance. Geostrophic adjustment is also interpreted as a statistical homogeneization process in which, in order to conserve potential vorticity, the small-scale component tends to align to the velocity fields to form a statistically homogeneous random field. We are pursuing this study to devise a stochastic model of wave-current interaction.

We work on simple parameterization for coarse-resolution oceanic models to replace computationally expensive high-resolution ocean models. We focus on the eddy-permitting scale (grid step Rossby radius) and computationally cheap parameterization. We are currently investigating the modification of the diffusion (friction) operator to reproduce the mean velocity observed via measurements or a high-resolution reference solution. To test this new parameterization on a double-gyre quasi-geostrophic model, we are implementing a fast and portable python implementation of the multilayer quasi-geostrophic model. This study has been published in the STUOD proceedings. In another study we have explored a new discretization of the multi-layer quasi-geostrophic (QG) model that models implicitly the sub-grid-scale effects. This new discrete scheme is based on several numerical choices that first ensure an exact material conservation of the potential vorticity. The advection is performed with a weighted essentially non-oscillatory interpolation whose implicit dissipation replaces the usual explicit (bi-)harmonic dissipation. We finally proposed a new method for solving the elliptic equation that warrants reversibility which on a staggered discretization. The method has the advantage to not requiring the tuning of any additional parameter, e.g. additional hyper-viscosity. This work has been recently submitted and we released a very short, concise, and efficient PyTorch implementation of our method to facilitate future data assimilation or machine-learning developments upon this new discretization.

We also have developed a unified QG and the RSW models, exploring the effect of higher order numerics, exploring the potential of WENO interpolation, developing alternative way of describing interfacial stress (bottom, surface, lateral), diagnosing the numerical implicit dissipation, using discrete differential geometry as a framework for discretization, testing Pytorch and Julia as alternatives languages to Fortran, inventing new code architectures. This work is under review in Geoscientific Model Development (EGU journals).

In this paper we characterize numerically through two criteria the Lévy area related to unresolved fluctuation velocities associated to a stochas tic coarse-scale representation of geophysical fluid flow dynamics. We study in particular whether or not the process associated to the random unresolved velocity components exhibits a Lévy area corresponding to a Wiener process, and if the law of this process can reasonably be approached by a centered Dirac measure. This exploration enables us to answer positively to a conjecture made for the constitution of high-order discrete time evolution schemes for stochastic representation defined from stochastic transport.

In this work we focus on the derivation of efficient discrete schemes, for a stochastic version of the shallow water model derived and analyzed in previous works of the team. We in particular pay attention to the devise of ”second order” like methods. This scheme that takes the form of an iterated double advection of the noise allow us to have an implicit, implementation of the noise associated diffusion, bringing a natural equilibrium, at the discrete level between the energy dissipation by the noise and its energy intake. The corresponding scheme corresponding to an extension of entropy conserving schemes proposed in the discrete setting by the Ange Inria team is fully justified in this study.

The aim of this study is to provide a stochastic version under location uncertainty of the compressible Navier-Stokes equations. The modelling under location uncertainty setting is used here to derive a physically consistent stochastic dynamical system for compressible flows. It relies on an extended stochastic version of the Reynolds transport theorem involving stochastic source terms. In a similar way as in the deterministic case, this conservation theorem is applied to density, momentum and total energy in order to obtain a transport equation of the primitive variables, i.e. density, velocity and temperature. For the modelling of ocean dynamics, the transport of mass fraction of species, such as salinity, is also considered. We show that performing low Mach and Boussinesq approximations to this more general set of equations allows us to recover previous versions of incompressible stochastic Navier-Stokes equations and the stochastic Boussinesq equations, respectively. Finally, we provide some research directions on the use of this general set of equations in the perspective of relaxing the Boussinesq and hydrostatic approximation for ocean modelling.

Stochastic linear modeling proposed in Tissot, Mémin, and Cavalieri [J. Fluid Mech. 912, A51 (2021)] is based on classical conservation laws subject to a stochastic transport. Once linearized around the mean flow and expressed in the Fourier domain, the model has proven its efficiency to predict the structure of the streaks of streamwise velocity in turbulent channel flows. It has been in particular demonstrated that the stochastic transport by unresolved incoherent turbulence allows us to better reproduce the streaks through lift-up mechanism. In the present work, we focus on the study of streamwise-elongated structures, energetic in the buffer and logarithmic layers. In the buffer layer, elongated streamwise vortices, named rolls, are seen to result from coherent wave-wave nonlinear interactions, which have been neglected in the stochastic linear framework. We propose a way to account for the effect of these interactions in the stochastic model by introducing a stochastic forcing, which replace the missing non- linear terms. In addition, we propose an iterative strategy in order to ensure that the stochastic noise is decorrelated from the solution, as prescribed by the modelling hypotheses. This study has been published in the journal Physical Review Fluids. This work is in collaboration with André Cavalieri (Instituto Tecnologico de Aeronautica, SP, Brésil).

Not only for extreme events, ocean surface waves have been demonstrated to be an important component of coupled earth system models. They affect atmosphere-ocean momentum transfer, break ice floes, alter CO2 fluxes, and impact mixed-layer depth through Langmuir turbulence. In contrast to the goals of third-generation spectral models, the wave information needed for mixing, air-sea, and wave-ice-coupling is much less than a full directional wave spectrum. All present parameterizations – for wave-induced mixing, surface drag, floe fracture, or sea spray – use primarily the wave spectrum's dominant frequency, direction, and energy or quantities that can be estimated from these such as Stokes drift and bending moments. Modest errors in sea state do not strongly affect the impacts of these parameterizations. This minimal data and accuracy need starkly contrasts with the computational costs of spectral wave models as a component of next-generation Earth System Models (ESM). In that context, an alternative, cost-efficient wave modeling framework for air-sea interaction to enable the routine use of sea state-dependent air-sea flux parameterization in ESMs. In contrast to spectral models, the Particle-in-Cell for Efficient Swell Wave Model (PiCLES) is under developments targeting coupled atmosphere-ocean-sea ice modeling. Combining Lagrangian wave growth solutions with the Particle-In-Cell method leads to a periodically meshing wave model on an arbitrary grid that scales in an embarrassingly parallel manner. The set of equations solves for the growth and propagation of a parametric wave spectrum's peak wavenumber and total wave energy, which reduces the state vector size by a factor of 50-200 compared to spectral models. Ideally, PiCLES will only require a fraction of the cost of established wave models with sufficient accuracy for ESMs–rivaling that of spectral models in the open ocean. We will evaluate PiCLES against WaveWatchIII in efficiency and accuracy and discuss planned extensions of its capability. This work is in collaboration with M. Hell, B. Fox-Kemper and T. Protin (PhD)

In collaboration with Pierre Ailliot [Tandeo et al, 2023] and within a data-driven context, we have demonstrated our ability to obtain accurate and reliable predictions of a partially observed Lorenz-63 system, where only the second and third components are observed and access to the equations is not allowed. This was done by a combination of machine learning and data assimilation techniques. The key aspects were the following: the introduction of latent variables, a linear approximation of the dynamics and a database that is updated iteratively, maximizing the likelihood. Interestingly, we found that the latent variables inferred by the procedure are related to the successive derivatives of the observed components of the dynamical system. The method is also able to reconstruct accurately the local dynamics of the partially observed system. Overall, the proposed methodology is simple, is easy to code and gives promising results, even in the case of small numbers of observations.

The state of the atmosphere, or of the ocean, cannot be exhaustively observed. Crucial parts might remain out of reach of proper monitoring. Also, defining the exact set of equations driving the atmosphere and ocean is virtually impossible because of their complexity. The goal of this study is to obtain predictions of a partially observed dynamical system without knowing the model equations. In this data-driven context, we focus on the Lorenz-63 system, where only the second and third components are observed and access to the equations is not allowed. To account for those strong constraints, a combination of machine learning and data assimilation techniques is proposed. The key aspects are the following: the introduction of latent variables, a linear approximation of the dynamics and a database that is updated iteratively, maximizing the likelihood. We find that the latent variables inferred by the procedure are related to the successive derivatives of the observed components of the dynamical system. The method is also able to reconstruct accurately the local dynamics of the partially observed system. Overall, the proposed methodology is simple, is easy to code and gives promising results, even in the case of small numbers of observations.

We investigated the application of a physically relevant stochastic dynamical model in ensemble Kalman filter methods. Ensemble Kalman filters are very popular in data assimilation because of their ability to handle the filtering of high-dimensional systems with reasonably small ensembles (especially when they are accompanied with so called localization techniques). The stochastic framework used in this study relies on Location Uncertainty (LU) principles which model the effects of the model errors on the large-scale flow components. The experiments carried out on the Surface Quasi Geostrophic (SQG) model with the localized square root filter demonstrate two significant improvements compared to the deterministic framework. Firstly, as the uncertainty is a priori built into the model through the stochastic parametrization, there is no need for ad-hoc variance inflation or perturbation of the initial condition. Secondly, it yields better MSE results than the deterministic ones. This work has been published in QJRMS.

In another study, we investigated the calibration of the stochastic noise in order to guide the realizations towards the observational data used for the assimilation. This is done in the context of the stochastic parametrization under Location Uncertainty (LU) and data assimilation. The new methodology is rigorously justified by the use of the Girsanov theorem, and yields significant improvements in the experiments carried out on the Surface Quasi Geostrophic (SQG) model, when applied to Ensemble Kalman filters. The particular test case studied here shows improvements of the peak MSE from 85% to 93%.

Using an idealized configuration in a 1-layer rotating shallow water model, we study the evolution of an inertia-gravity wave interacting with a turbulent mesoscale jet. The resulting incoherent inertia-gravity wave field is then analyzed using several methods: spectral POD, extended spectral POD and resolvent analysis. The goal of this study is twofold: 1) better understand and characterize the loss-of coherence of the inertia-gravity wave when interacting with the turbulent background flow and 2) extract the relevant modes of variability to formulate a reduced order model that is able to capture and predict the inertia-gravity wave, given some knowledge of the mesoscale flow contribution. The latter is oriented towards the devise of data-assimilation models for incoherent internal wave fields in the ocean. A paper summarizing the results as been published in the STUOD proceedings (2022), and another one is in preparation.

We define generative neural network architectures to model stochastic 1-dimensional fields with turbulent velocity statistics. The main ideas are 1) to use architectures mimicking the structure of classical stochastic models such as random wavelet cascades and 2) to introduce Kolmogorov and Obukhov laws in both the training and validation of the models. Two approaches are used: an unsupervised one which does not require turbulent data and only needs the desired statistics to be imposed, and one based on GANs which requires turbulent data.

We have studied how fine-scale ocean dynamics affect carbon export and its fate in the water column. The mesoscale and submesoscale currents play an important role, not only creating very strong heterogeneity in particle production at the surface, but also driving horizontal and vertical velocities that affect the exchange between the surface layer and the interior of the ocean. We have used numerical Lagrangian experiments and a realistic high-resolution ocean model to train a neural network to predict the surface origin of particles trapped in a deep sediment trap, with success and suggesting an application to satellite data. The results have been submitted to the EGU Journal Ocean Science.

A methodological framework for ensemble-based estimation and simulation of high dimensional dynamical systems such as the oceanic or atmospheric flows is proposed. To that end, the dynamical system is embedded in a family of reproducing kernel Hilbert spaces (RKHS) with kernel functions driven by the dynamics. In the RKHS family, the Koopman and Perron–Frobenius operators are unitary and uniformly continuous. This property warrants they can be expressed in exponential series of diagonalizable bounded evolution operators defined from their infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. The RKHS family enables us the devise of strikingly simple ensemble data assimilation methods for trajectory reconstructions in terms of constant-in-time linear combinations of trajectory samples. Such an embarrassingly simple strategy is made possible through a fully justified superposition principle ensuing from several fundamental theorems. This study has been published in the journal Physica D: nonlinear phenomena. Duting the PhD of Mael Jaouen we extend the numerical experimentation to a wind-forced three-layers QG model.

We focused our efforts on learning closure terms for the representation of subgrid-scale processes, and more broadly on learning corrections to a reference model in simulations of geophysical flows.We have applied an “a-posteriori” learning method introduced in (Frezat et al., 2022) to non-differentiable direct simulation codes. Our contributions explore both emulator-based methods (Frezat et al., 2023) and Euler-type approximations for computing the gradient of the a posteriori learning cost (Ouala et al., 2023).

We developed several data-driven variational data assimilation methods, addressing various methodological challenges tackled, namely:

This work applies to several topics, and in particular the short- and mid-term prediction of sea level anomaly from real data (in the Gulf Stream area – H. Goeogenthum PhD thesis) and the prediction of multispectral image reflectance dynamics (A.Frion PhD thesis). We also proposed a CNN model with "attention mechanism" for the prediction of chlorophyll concentration from atmospheric and oceanic physical drivers, for the long term reconstruction of past chlorophyll time series at global scale (J. Rousillon PhD thesis). Finally, in the context of P. Aimé PhD work and in collaboration with S. Sharma (postdoc), we studied different evaluation metrics as well as various AI methods for merging multi-spectral and panchromatic data.

We are also developing original end-to-end approaches for learning neural data assimilation methods based on both variational formulations (Fablet et al., 2021) and Kalman filtering methods (Ouala et al., 2022). Our contributions over the past year concern in particular the quantification of uncertainties in the 4DVarNet scheme, for example by exploiting Bayesian variational inference-type formulations (Lafon et al., 2023) and stochastic PDE-type representations of the underlying dynamics. We are developing various simulated and real case studies to demonstrate these approaches (e.g. surface currents, turbidity, sea surface height) (Febvre et al., 2023; Fablet et al., 2023). This line of work is in the context of the PhD thesis (at IMT) of Quentin Febvre, Hugo Georgenthum and Simon Bennaïchouche, and in collaboration with Said Ouala (postdoc IMT) and Maxime Beauchamp (postdoc IMT).

Simulation and analysis of trajectometric data are specific issues for ocean observation (e.g., ocean surface drift, ship trajectories, marine animal movements...). We are exploring learning methods for the simulation and analysis of these different types of trajectory data. This includes both new GAN methods for the simulation of bird trajectories [Roy et al., 2022], conditional simulation of drift trajectories [Botvinko et al., 2022], short-term prediction of ship trajectories [Nguyen et al., 2024] or the exploitation of ship trajectory data for the estimation of marine currents [Benaichouche et al., 2022].

The CLIMARCTIC project aims at improving our understanding of climate change in the arctic, both at regional and global scales. Pierre Tandeo is co-PI and R. Fablet and L. Drumetz participate to WP1.

The MEDIATION project aims at improving and developping better numerical code of the ocean dynamics. E. Mémin is co-PI of WP2 “parametrisation stochastique et quantification d'incertitude” and participate to WP3 “Modèles sous maille”. P. Tandeo and C. Granero Belinchon participate to WP4 “IA pour les codes océaniques”.

“From regional to global impacts of climate change in the Arctic : an interdisciplinary perspective” (procjet “Océan 2030”, PI: C. Lique, LOPS Ifremer). F. Sévellec is in charge of WP1.

“ Physics-Informed AI for Observation- driven Ocean AnalytiX” (PI: R. Fablet)

“Bridging geophysics and MachinE Learning for the modeling, simulation and reconstruction of Ocean DYnamics”. (PI: R. Fablet)

"Modelling the Internal Tide in the Ocean" project aims at developing a data assimilation model for the ocean internal tide field, in the context of the SWOT mission. (PI: N. Lahaye)

“ Statistical ChAracterization of multi-scaLE complex Systems with information theory ” (PI: C. Granero Belinchon)

“Impacts of DEep submEsoscale Processes on the ocEan ciRculation” (PI: J. Gula), 2020 – 2025. The goals of the DEEPER project are to quantify the impacts of deep submesoscale processes and internal waves on mixing and water mass transformations. In addition, the DEEPER project will explore ways of parameterizing these impacts using the latest advances in machine learning.

“Assessing the Role of forced and internal Variability for the Ocean and climate Response in a changing climate” (PI: F. Sévellec), 2022–2024.

“Ocean state Analog in-SItu analyses System (PI: N. Kolodziejczyk – CNAP, LOPS), 2022–2024

ALESE is a MITI CNRS project

PhD thesis of Merveille Talla, on the development of diffusion generative models applied to turbulent flows. Collaboration with Dominique Heitz and Valentin Resseguier (ACTA Inrae Rennes team).

This project (“Koopman operator modelling of non-linear dynamical systems for ensemble methods”) consists in expressing the Koopman operator associated with a high-dimensional geophysical dynamical system in a family of reproducing kernel Hilbert spaces. The interest is to learn the non-linear dynamics, locally in the phase space, in order to solve efficiently ensemble data assimilation problems. Multi-layer quasi-geostrophic models representative of the Gulf stream area is considered in this work.

The Britany ARED project " Analyse Mathématique de Modèles Stochastiques réalistes de la Dynamique Océanique" in collaboration with Arnaud Debussche (ENS/MINGUS) funds 50 percent of the PhD thesis of Antoine Moneyron.

“Analog Methods to Identify Global Atmospheric Simulations” (PI: P. Tandeo).