Introduction Room: Amphithéâtre Hermite

A primer on ocean mesoscale eddy parameterizations for numerical models

• Stephen Griffies (Princeton University)

Room: Amphithéâtre Hermite Abstract: This talk summarizes some ongoing math/physics/numerics questions related to tracer mixing and stirring as realized in ocean models, moving from the small-scale turbulence to the large-scale mesoscale eddy stirring. The presentation will be pedagogical in style, and it touches only a few topics with an aim to engage both students and researchers.

Coexisting fluxes in rotating turbulence

• Anna Frishman

Room: Amphithéâtre Hermite Co-author: Sebastien Gome Abstract: Turbulence is characterized by energy fluxes, whose direction is determined by conservation laws. In 3D rotating turbulence, however, energy is observed to flow simultaneously toward large-scale two-dimensional structures and toward small-scale three-dimensional waves. Using a mean–wave kinetic theory, we derive analytical expressions for these competing bi-directional transfers in the presence of a spontaneously emergent 2D mean flow. We show the direction of the energy transfer is determined by the type of allowed 2D-3D interactions: the mean flow is fed by a sector of modes for which only same-sign-helicity interactions are allowed, while modes which have helicity-mixing interactions extract energy from it. The balance between the two sectors changes as the Rossby- and Reynolds-numbers are varied. We obtain the 2D-3D energy partition as a function of Rossby and Reynolds analytically, in agreement with fully nonlinear simulations, presenting a unified picture across rotation rates.

Stable Implosions

• Vlad Vicol (New York University)

Room: Amphithéâtre Hermite Co-authors: J. Chen (U Chicago) and S. Shkoller (UC Davis). Abstract: We exhibit a new class of self-similar implosion solutions for the full compressible Euler equations. For any value of the adiabatic exponent, we construct a sequence of implosion profiles that are smooth before collapse and have an explicit similarity exponent. The first profile in this sequence (the ''ground state'') possesses remarkable stability properties, even outside of spherical symmetry.

Horizontal/vertical convective stability

• Stefan Llewellyn Smith (University of California, San Diego)

Room: Amphithéâtre Hermite Co-authors: Michael Le Bars, Clement Audefroy Abstract: Motivated by planetary science, we examine the stability of flows with both buoyancy gradients on horizontal boundaries and vertical buoyancy fluxes entering the domain, hence combining aspects of horizontal convection (HC) and Rayleigh–Bénard (RB) convection. Exact steady states exist in the form of shear flows. Unlike the case of RB and classical shear flow stability, the principles of exchange of stability and Squire's theorem no longer hold, so that the marginal modes are no longer two-dimensional with zero frequency. We explore the stability boundary numerically in the horizontal/vertical Rayleigh number/Prandlt number parameter space. Using scaling arguments, we identify different families of modes: RB modes, central shear modes for small Prandtl numbers, boundary-trapped shear modes and 3D baroclinic modes.

Nonuniqueness in fluid models

• Julien Guillod

Room: Amphithéâtre Hermite Co-authors: Dallas Albritton, Mikhail Korobkov, Xiao Ren, and Vladimír Šverák Abstract: It is well known in physics literature, despite almost no mathematical results, that the steady states of fluid model equations are not unique and appear through bifurcations when the Reynolds number increases. After presenting this, the same methodology will be used for time-dependent problems to obtain the non-uniqueness of solutions to Cauchy problems. Numerical non-uniqueness results will be presented for the Navier-Stokes equations in both two and three dimensions. The physical implications will be discussed in particular.

poster presentation 2min each Room: Amphithéâtre Hermite

Poster session Room: Amphithéâtre Hermite

Ideal Magnetic Reconnection

• Peter Constantin (Princeton University)

Room: Amphithéâtre Hermite Co-author: Zhongtian Hu Abstract: Magnetic fields do not change topology during smooth dynamics of ideal MHD. But topology change does occur in nature. Magnetic resistivity and near singularities have been suggested as a possible explanation. In this talk I will focus on a different explanation, also suggested in the physics literature: reconnection due to magneto-hydrodynamic inertia. I will describe 2D models that exhibit rigorous ideal topology change. The models have global smooth solutions and the reconnection is obtained from merger of a pair of active scalars transported by incompressible velocities they create. Finite time merger without singularities and without resistivity or viscosity is proved rigorously.

Boundary layers and inviscid limits

• Jincheng Yang (Johns Hopkins University)

Room: Amphithéâtre Hermite Co-author: Alexis Vasseur Abstract: I will discuss several recent results on vanishing-viscosity limits for incompressible Navier-Stokes flows near boundaries. I will begin with boundary vorticity estimates and their application to weak inviscid limits near plug flow in a periodic tunnel, giving short-time control of deviations from the shear profile. I will then present unconditional $L^2$ bounds on boundary layer separation between Leray-Hopf Navier-Stokes solutions and smooth Euler flows in bounded domains. Finally, I will discuss joint work on non-characteristic boundaries, where one can quantify energy dissipation and enstrophy production near outflow in terms of the boundary mismatch between Navier-Stokes and Euler flows.

Arnold's Geometry on Finite Meshes

• Peter Korn (Max Planck Institute for Meteorology)

Room: Amphithéâtre Hermite Abstract: Arnold identified ideal fluid motion with geodesics on the group of volume-preserving diffeomorphisms, whose curvature controls hydrodynamic stability. We develop this programme on finite meshes, establishing an approximate finite-dimensional Lie algebra supported by a discrete de Rham complex. This yields well-posed discrete Euler equations that converge to the continuum Euler equations. We derive Arnold's curvature formula and Lorenz's predictability barrier. Finite-dimensional Hopf–Rinow restores geodesic completeness, resolving Shnirelman's obstruction. We connect this to Brenier's relaxed least-action principle, and give a finite-dimensional realisation of the De Lellis–Székélyhidi convex integration scheme, making the construction of wild weak solutions explicit on the mesh. A phase transition emerges as $h\to 0$: local geometry converges while completeness and uniqueness change qualitatively, with curvature providing the geometric link between instability and non-uniqueness.

Bifurcations of shear flows

• Emmanuel Grenier (Beijing Institute of Technology)

Room: Amphithéâtre Hermite Abstract: Generic shear flows are unstable for the incompressible Navier-Stokes equation as the viscosity goes to 0 between the so-called lower and upper marginal stability curves. The aim of this talk is to discuss recent results on the bifurcation which occurs near the upper marginal stability curve.

Diffusive instabilities in seawater

• Remi Tailleux (University of Reading, Department of Meteorology)

Room: Amphithéâtre Hermite Abstract: In two-component seawater, the thermobaric and cabbeling nonlinearities of the equation of state, combined with the disparate molecular diffusivities of salt and heat, give rise to a diverse range of diffusive instabilities. Beyond standard salt finger and diffusive convection instabilities, these include instabilities associated with densification upon mixing. Unlike conventional turbulent diapycnal mixing, which dissipates available potential energy (APE) into background potential energy (BPE), diffusive instabilities can extract energy from the BPE to energise the fluid and provide a source of turbulent kinetic energy. This suggests that in regimes where diffusive instabilities occur, the BPE contains a "latent" form of APE that remains poorly understood. This work reviews the fundamental nature of these processes and identifies the remaining challenges in developing a comprehensive framework for their understanding, drawing on a variety of test cases and examples.

Rotating fingering convection

• Celine Guervilly (Newcastle University)

Room: Amphithéâtre Hermite Co-authors: Martin Gray, Graeme Sarson Abstract: We study double-diffusive convection in the case of an unstable compositional gradient in the presence of a stabilising temperature gradient. Experimental and analytical studies (Hage and Tilgner 2010; Schmitt 2011) have shown that narrow salt fingers (usually encountered in thermohaline convection in “bottom-heavy” layers) can be preferred over large-scale convection in the regime where the stable temperature gradient is much smaller than the destabilising compositional gradient (the “top-heavy” regime). Here, we extend this problem to the context of planetary cores using hydrodynamical numerical simulations in a rotating spherical shell at low Prandtl number. We show that fingering convection can also be preferred over overturning compositional convection for both weak and strong rotation across large regions of parameter space and the transition between the two regimes of convection is gradual.

Asymptotics of inviscid stratified flows

• Lucas Ertzbischoff (Université Paris Dauphine-PSL)

Room: Amphithéâtre Hermite Abstract: I will talk about recent mathematical progress on asymptotic regimes for inviscid stratified fluid models, focusing on two topics that remain only partially understood: (i) the long-time dynamics and (ii) the hydrostatic limit. As a guiding example I will use the classical 2d Euler-Boussinesq system. I will try to connect these two questions for this model, highlighting the role of (in)stability issues and important mechanism such as mixing or dispersion.

Western intensified turbulence

• Antoine Venaille

Room: Amphithéâtre Hermite Co-authors: Lennard Miller, Bruno Deremble Abstract: We first investigate numerically the vanishing-viscosity limit of a two-dimensional wind-driven ocean model. Instead of forming a large-scale condensate, the flow remains strongly out of equilibrium, organizing into a highly energetic turbulent vortex gas coexisting with western-intensified gyres. When stratification is introduced, coherent Gulf Stream–like jets emerge and can dominate the large-scale flow. We map the phase diagram governing their existence within a two-layer quasi-geostrophic model. Guided by this framework, we present high-resolution simulations with a more comprehensive ocean model, suggesting that increased upper-ocean stratification, an inevitable consequence of global warming, can destabilize the Gulf Stream Extension.

Unstable internal waves

• Roberta Bianchini (Consiglio Nazionale delle Ricerche)

Room: Amphithéâtre Hermite Abstract: Stably stratified fluids (e.g., oceans and atmosphere) support internal waves that are fundamental to oceanic circulation and atmospheric dynamics. We present the first rigorous proof of instability for small-amplitude internal waves, establishing the existence of an unstable spectrum for the Boussinesq equations linearised about a traveling wave. The analysis combines a Floquet–Bloch decomposition with a variant of Kato’s similarity transformation, exploiting the wave’s structure. In a specific regime, the resulting growth rates agree with previous theoretical predictions for Triadic Resonant Instability of internal waves.

Cascade transition in geophysical flows

• Alexandros Alexakis (ENS)

Room: Amphithéâtre Hermite Abstract: I will discuss how a cascade can transition from a forward to an inverse cascade in geophysical flows as a parameter is varied. I will review some of the past results and present some of the most resent results in rotating flows and stratified flows.

MHD turbulence and weak solutions

• Laszlo Szekelyhidi (Max Planck Institute for Mathematics in the Sciences)

Room: Amphithéâtre Hermite Co-author: Matteo Giardi Abstract: The ideal magnetohydrodynamic system in three space dimensions consists of the incompressible Euler equations coupled to the Faraday system via Ohm’s law. This system has a wealth of interesting structure, including three conserved quantities : the total energy, cross-helicity and magnetic helicity. Whilst the former two are analogous to the total kinetic energy for the Euler system, magnetic helicity is known to be more robust and of a different nature. In particular, when studying weak solutions, Onsager-type conditions for all three quantities are known, and are basically on the same level of 1/3-differentiability as the kinetic energy in the ideal hydrodynamic case for the former two. In contrast, magnetic helicity does not require any differentiability, only L^3 integrability. In the talk we present and compare some recent constructions of weak solutions and along the way highlight some of the hidden structures in the ideal magnetohydrodynamic system.

Hamiltonian Dysthe equation for hydroelastic waves in a compressed ice sheet

• Catherine Sulem (University of Toronto)

Room: Amphithéâtre Hermite Co-authors: Philippe Guyenne, Adilbek Kairzhan Abstract: This study concerns the motion of nonlinear hydroelastic waves along a com- pressed ice sheet lying on top of a two-dimensional fluid of infinite depth. Applying tech- niques of Hamiltonian perturbation theory, a Hamiltonian Dysthe equation is derived for the slowly varying envelope of modulated wavetrains. The derivation is further complicated by the presence of cubic resonances. A Birkhoff normal form transformation is introduced to eliminate non-resonant triads while accommodating resonant ones. Numerical solutions constructed from the Dysthe equation are compared to direct simulations of the full Euler system, and very good agreement is observed.

Rythmic patterns: dunes, ripples & more

• Bruno Andreotti (LPENS)

Room: Amphithéâtre Hermite Abstract: Periodic patterns spontaneously emerge due to sublimation, erosion/deposition and sediment transport, dissolution, or—when it comes to waves—mechanical deformation of an interface. Starting with sand ripples and dunes, I will thoroughly discuss the various aspects of how these patterns form: linear instability vs pattern coarsening; laminar vs turbulent flow; mixing vs normal stress vs shear stress instability.

Bore wave solutions to Navier-Stokes

• Ian Tice (Carnegie Mellon University)

Room: Amphithéâtre Hermite Abstract: In this talk we will discuss the construction of two-dimensional traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations for a single finite depth layer of constant density fluid. Our construction is based on a rigorous justification of the formal shallow water limit, which postulates that in a certain scaling regime the full free boundary traveling Navier-Stokes system of PDEs reduces to a governing system of ODEs. We find heteroclinic orbits solving these ODEs and, through a delicate fixed point argument employing the Stokes problem in thin domains and a nonautonomous orbital perturbation theory, use these ODE solutions as the germs from which we build bore PDE solutions for sufficiently shallow layers. This is joint work with Noah Stevenson.

Gibbon flow and the d'Alembert's paradox

• Miguel Bustamante (University College Dublin)

Room: Amphithéâtre Hermite Co-authors: Yinshen Xu (UCD), Tiziana Comito (UCD), Johan Hoffman (KTH), John D. Gibbon (Imperial) Abstract: Consider incompressible inviscid flow past an object. D'Alembert (1752) proved that for potential flow, the object experiences no drag force. However, experimental observations find significant drag at high Reynolds numbers, leading to the famous d'Alembert's paradox in fluid mechanics. Prandtl (1904) proposed a milestone solution to this paradox through his boundary layer theory, which attributes drag primarily to the viscous boundary layer. Recently, Hoffman and Johnson (2010) revisited the paradox, bypassing the use of viscosity. In a computational "weak" solution of the 3D Euler equations of flow past a cylinder with slip boundary conditions, they found turbulence to be the primary source of drag. We prove this analytically, showing that Gibbon et al.'s (1999) stagnation-point-like solution of the 3D Euler equations, with appropriate inflow conditions, holds at the rear separation zone. Via a linear instability, nonlinear (stable) streamwise helical vortices form, causing drag.

Exact Energy Transfers in Turbulence

• Mahendra Verma (IIT Kanpur, India)

Room: Amphithéâtre Hermite We developed a mathematical framework called “mode-to-mode energy transfer” to compute energy transfers in fluid flows, in particular turbulence. In this talk, I will describe this general framework and illustrate its application to incompressible and compressible turbulence, turbulent convection, magnetohydrodynamics, dynamo, and quantum turbulence. This is a general framework that enables flux and shell-to-shell energy transfers. Reference: Energy Transfers in Fluid Flows, Cambridge University Press, 2019

Rayleigh-Benard on a logarithmic lattice

• Keaton Burns (Massachusetts Institute of Technology)

Room: Amphithéâtre Hermite Co-authors: Steven Tobias (Univ. Edinburgh), Curtis Saxton (Univ. Leeds), Richard Kerswell (Univ. Cambridge) Abstract: Our ability to numerically study turbulent convection is limited by the high cost of direct numerical simulations (DNS) in the regimes relevant to geophysical and astrophysical flows. This motivates the development of alternatives to DNS which enable faster computation by using reduced models of the full dynamics. Here we explore the use of logarithmic Fourier lattices (LFLs) combined with sparse Chebyshev methods to capture extreme dynamic ranges of spatial scales in Rayleigh-Benard and rotating convection. LFL schemes use a Fourier series with logarithmically rather than linearly distributed wavenumbers. We will discuss ongoing work testing different forms of LFL discretizations by examining their ability to reproduce spectra and transport scalings at extreme parameters. This includes formulations with different lattice spacings, triad weightings, and new modifications for the inclusion of coherent structures.

Solutions to conservation laws unique?

• Sam Krupa (École normale supérieure)

Room: Amphithéâtre Hermite Co-authors: László Székelyhidi, Jr. Abstract: For hyperbolic systems of conservation laws in 1-D, fundamental questions about uniqueness and blow up of weak solutions still remain even for the apparently “simple” systems of two conserved quantities such as isentropic Euler and the p-system. Similarly, in the multi-dimensional case, a longstanding open question has been the uniqueness of weak solutions with initial data corresponding to the compressible vortex sheet. We address all of these questions by using the lens of convex integration, a general method of constructing highly irregular and non-unique solutions to PDEs. Our proofs involve computer-assistance.

Unstable Manifolds of Euler Equations

• Chongchun Zeng (Georgia Institute of Technology)

Room: Amphithéâtre Hermite Co-authors: Zhiwu Lin and Yanbo Wang Abstract: Consider a spectrally unstable steady state $(\rho_0(x), v_0(x))$ of the incompressible stratified Euler equation in certain $d$-dim domain $\Omega$. Assuming the linearized equation satisfies a linear exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of $v_0(x)$, we construct a local unstable manifold of $(\rho_0, v_0)$. The proof is based on the Lyapunov-Perron integral equation method after the Euler equation is reformulated as an ODE on the infinite dimensional manifold of volume-preserving Lagrangian maps where the density is treated as a parameter. Applications to steady states in two space dimensions are also discussed.

Reduced and rescaled equations for RRRBC

• Edgar Knobloch (University of California-Berkeley, Department of Physics)

Room: Amphithéâtre Hermite Co-authors: K. Julien, A. van Kan, B. Miquel, G. Vasil Abstract: Geophysical flows are characterized by parameter values that are far outside those that can be studied in the laboratory or via state of the art numerical simulations. I will describe a formal multiscale asymptotic procedure for rapidly rotating convection that leads to a reduced system of equations valid in the limit of vanishing Ekman number. These equations describe four regimes as the Rayleigh number Ra increases: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger Ra, followed for yet larger Ra by a breakdown of the Taylor columns into disordered plumes, and finally by geostrophic turbulence. When scaled using the asymptotic scales, the full equations can be integrated at Ekman numbers six orders of magnitude smaller than the current state of the art, approaching geophysically realistic values for the very first time. The stationary state results converge to the predictions of the asymptotically reduced equations.

Congestion phenomena in fluids

• charlotte perrin (Institut de Mathématiques de Marseille)

Room: Amphithéâtre Hermite Abstract: This talk addresses recent developments on fluid models with a maximal density constraint. Such constraints arise in the modeling of congestion effects, with applications to geophysical flows such as dense granular avalanches and sea ice dynamics. I will focus on the main theoretical and numerical difficulties induced by this framework, including strong nonlinear effects, transitions between compressible and incompressible regimes, and the resulting challenges for stable and accurate simulations.

On the Instability of Small Stokes Waves

• Miguel Rodrigues (Univ Rennes)

Room: Amphithéâtre Hermite Co-authors: Ziang Jiao, Zhao Yang (Beijing, China), Changzhen Sun (Besançon, France) Abstract: We report on a recent proof that all irrotational planar periodic travelling waves of sufficiently small-amplitude are spectrally unstable as solutions to three-dimensional inviscid finite-depth gravity water-waves equations. The associated temporal growth scales sharply with respect to the amplitude of the wave.

Instabilities around mesoscale eddies

• Michael Le Bars (CNRS, IRPHE UMR 7342)

Room: Amphithéâtre Hermite Co-authors: Antoine Chauchat (IRPHE), Patrice Meunier (IRPHE), Keaton Burns (MIT) Abstract: Our current understanding of ocean mixing remains insufficient to balance the global ocean energy budget, pointing to overlooked local mechanisms. At the edges of mesoscale eddies, horizontal density layering is observed, suggesting enhanced vertical mixing. To investigate its origin, we examine the underlying instabilities. We model this configuration using a solid ellipsoid undergoing differential rotation within a rotating stratified fluid. Combining analytical and experimental approaches, we characterize instabilities across a large range of Rossby, Froude, and Reynolds numbers. Experiments are conducted in a 1 m rotating tank using Particle Image Velocimetry and Schlieren imaging. The base flow is first compared to an exact analytical solution for arbitrary aspect ratios. Observed modes are then compared with a linear stability analysis using the Dedalus solver. We identify in particular viscodiffusive and centrifugal instabilities and assess their contributions to mixing.

The Dynamical Landscape of the AMOC

• Louis-Philippe Nadeau (Université du Québec à Rimouski)

Room: Amphithéâtre Hermite Abstract: While AMOC stability is traditionally viewed through simple box models, these models exhibit a diverse range of behaviors dictated by the background climate. These can be classified into two regimes: abrupt tipping points (saddle-node bifurcations) and millennial-scale oscillations (Hopf bifurcations). This presentation reviews the evolution of conceptual models of the AMOC, identifying how mechanisms like advection, diffusion, sea ice, and stratification drive these distinct behaviors. We introduce a minimal, physically based framework that maps the system's full bifurcation space against global temperature and freshwater forcing. This approach offers a unified perspective on AMOC dynamics, demonstrating that warm, present-day climates are prone to saddle-node collapses, whereas cold, glacial-type climates naturally favor limit-cycle oscillations.

Transition in Stratified Shear Flows

• Colm-cille Caulfield (DAMTP, University of Cambridge)

Room: Amphithéâtre Hermite Abstract: (Vertically) stratified shear flows, where both the background horizontal velocity and buoyancy distribution vary in the vertical (i.e. the direction parallel to gravity) are ubiquitous in geophysical fluid dynamics. A key question is how such flows undergo the transition to turbulence and hence irreversibly mix vigorously. Intuitively, if the buoyancy increases upwards (anti-parallel to gravity), i.e. the fluid is `statically stable’ relative to convection, there should be a competition between the apparently stabilising effect of the buoyancy force and the destabilising effect of shear, quantified classically in terms of a Richardson number, a coupling parameter between the buoyancy and velocity fields. However, transition in stratified shear flows has proved to be significantly more subtle. Behaviour depends on the flow’s Reynolds number and Prandtl number, and indeed the turbulence near transition can be qualitatively different in stratified flows and unstratified flows.

Ekman-inertial instability

• Nicolas Grisouard (University of Toronto)

Room: Amphithéâtre Hermite Abstract: Ekman-inertial instability (EII) occurs in Ro=O(1) jets when the magnitude of the anticyclonic vertical vorticity exceeds that of the Coriolis parameter, and when surface stress differs from interior viscous stress of the thermal wind shear immediately under the surface. EII is to Ekman spirals what symmetric instability is to internal waves. It can grow explosively fast at first due to its non-normal nature and eventually stabilizes to a growth rate equal to that of classical inertial instability. Because of this fast onset, it can outcompete normal-mode instabilities immediately below the surface. We outline the 1D theory and then show, using constant-density, low-noise, 2D initial value problems, that EII outcompetes inertial instability. In baroclinic, low-noise, 2D initial value problems, EII outcompetes symmetric instability, resulting in distinct patterns in energy extraction from the balanced jet.

Long time dynamics 2D Navier-Stokes

• Nader Masmoudi (nyuad)

Room: Amphithéâtre Hermite Abstract : In this talk, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane with Navier-slip boundary conditions. We construct the profile that describes the leading order term when time goes to infinity.