Introduction Room: Amphithéâtre Hermite

Clay Lectures -- Incompressible fluids with inflow/outflow

• Anna Mazzucato (Penn State University)

Room: Amphithéâtre Hermite This lecture is organised in partnership with the Clay Mathematics Institute. I will discuss recent results concerning the well-posedness of the Euler equations with inflow/outflow (injection/suction) at permeable boundary and the vanishing viscosity limit for both Navier-Stokes and the Boussinesq system.

Maths-vives Chair -- Superrotation in Planetary Atmospheres

• Geoffrey Vallis (University of Exeter)

Room: Amphithéâtre Hermite Jets and Superrotation are ubiquitous features in some, but not all, planetary atmospheres. In particular superrotation occurs in slowly rotating terrestrial atmospheres (like Venus), in rapidly rotating gas giants (like Jupiter) and in some exoplanets. In this talk I'll discuss the various mechanisms giving rise to superrotation and their connection to potential vorticity.

2D flows with random initial vorticity

• Milton da Costa Lopes Filho (Universidade Federal do Rio de Janeiro)

Room: Amphithéâtre Hermite We consider a random initial vorticity $\omega_0(x) = \sum_{n\in \mathbb{Z}^2} a_n \phi(x-n)$, where $\phi$ is bounded and compactly supported and $\{a_n\}$ are independent, uniformly bounded, mean $0$, variance $1$ random variables (in other words, $\omega_0$ is an array of randomly weighted vortex blobs). We prove global well-posedness of weak solutions to the Euler equations in $\mathbb{R}^2$ for almost every such initial vorticity.

Kelvin waves along a free-surface vortex

• Jason Barckicke (Université Paris Cité)

Room: Amphithéâtre Hermite Many flows feature a strong vorticity: tornadoes, hurricanes, aircraft wakes ... Vortices also play a key role in the energy cascade of classical and quantum turbulence. Kelvin waves are elementary perturbations of such vortices - helical waves propagating along the vortex line. Many theoretical works have predicted their behaviour but no experimental confirmation has been reported so far. By designing an original experimental set up, consisting in a bathtub vortex without rotation of the tank, we measured the Kelvin wave dispersion relation, with good resolution, for the first time. Their spectral signature is very rich and includes several wave modes. They all match the theoretical predictions. We then turn to investigating Kelvin wave turbulence - a phenomenon carrying energy towards small scales through nonlinear wave interactions. It paves the way to the experimental study of solitons and the dynamics of an array of vortices, whose collective motion is known as 'Tkachenko waves'.

poster presentation 2min each Room: Amphithéâtre Hermite

Poster session Room: Amphithéâtre Hermite

Triple-deck theory for tropical cyclones

• Rupert Klein (Freie Universität Berlin)

Room: Amphithéâtre Hermite Multiscale interactions including boundary layer effects and moist convection are generally thought to be essential for the acceleration of tropical cyclones. Here we present an asymptotic analysis involving a triple-deck matched asymptotic expansion in the vertical and meanfield arguments for multiscale convection in an axisymmetric vortex. The resulting theory suggests transparent explanations of how various contributing processes can constructively interact to produce effective storm intensification.

Quantitative stability of the Rossby—Haurwitz waves of degree two for the Euler equations on the sphere

• Luca Melzi (Imperial College London)

Room: Amphithéâtre Hermite The motion of an incompressible, ideal fluid is described by the Euler equations. Choosing the unit sphere $\mathbb{S}^2$ embedded in $\mathbb{R}^3$ as the domain of interest, the Euler equations represent a suitable model for stratospheric flows. Among such flows, of particular importance in atmospheric dynamics are the Rossby—Haurwitz waves, that are observed in the stratosphere of the Earth and other planets, such as Jupiter, Saturn, Uranus, and Neptune. In this joint work with Matias G. Delgadino (https://arxiv.org/abs/2509.16156), we show that the degree-2 Rossby—Haurwitz travelling waves on the Euler equations on $\mathbb{S}^2$ are orbitally stable. Our proof is short, quantitative, and conceptually easy to follow.

Eye of the cyclone:an experimental study

• Brichet Lyse (Irphé)

Room: Amphithéâtre Hermite One of the most characteristic flow in the atmosphere is the eye of tropical cyclones. Yet, its dynamic formation process is still poorly understood. This study focuses on a small number of parameters to improve our fundamental understanding of these dynamics. We have set up an experimental model in air to analyze the conditions for the formation of the eye of a cyclone in dry rotating convection. The setup consists of a rotating cylindrical tank, heated from below and maintained at a constant temperature on the side. The flow is measured using PIV in a vertical plane near the rotation axis. Within a well-defined parameter space, we observe a persistent descending flow along the rotation axis. Owing to the more extreme control parameters explored here compared with previous studies, the resulting eye is turbulent and non-axisymmetric. Despite this complexity, the eye formation is a robust feature of this idealised dry cyclone, providing that the Rossby number lies ! within a finite range

Quantitative classification of potential Navier-Stokes singularities beyond the blow-up time

• Tobias Barker

Room: Amphithéâtre Hermite It remains an open problem whether or not solutions to the 3D Navier-Stokes equations with smooth data remain smooth for all time. All previously known regularity criteria are formulated in times of a blow-up time (where the solution loses smoothness), which make it practically impossible to use such necessary conditions to test the viability of certain numerically computed candidates. Motivated by these issues, we give the first quantitative classification of potentially singular solutions at any given time in the region of potential blow-up times. The quantitative lower bounds prior to any potential blow-up time (and in the open vicinity of it) are in principle amenable to numerical testing.

Vorticity generation in water waves

• Alan Riquier (École Normale Supérieure - PSL)

Room: Amphithéâtre Hermite We propose to discuss the relevance of the irrotationality assumption commonly made to obtain reduced water waves models (Shallow Water/Saint-Venant, Korteweg-de Vries, Green-Naghdi, etc.). To do so, we investigate the asymptotic behaviour of two boundary layers associated with oceanic flows: one appearing in the vicinity of the free surface and the other lying at the bottom boundary. This is achieved numerically approximating the solution to the Navier-Stokes equations using the finite-element method on a moving mesh.

Physics of Tropical Cyclones

• Kerry Emanuel (MIT)

Room: Amphithéâtre Hermite Outside the immediate subdiscipline of tropical cyclone physics, it is widely held that these vortices are a mode of organization of deep, moist convection. In this lecture, I will dispel that idea and show that TC-like vortices arise from thermodynamic disequilibrium between a fluid and a solid or liquid surface, a disequilibrium that arises from the discontinuity in radiative emissivities between gases and solids or liquids. They can only arise in full turbulent flows in which the lower boundary is thermodynamically “rough”, so that heat flow is rate limited by turbulence and not by molecular diffusion. Phase changes are not necessary. I will focus on the energetics and dynamics of tropical cyclone-like vortices.

Filamentation near monotone zonal caps

• Emeric Roulley (SISSA)

Room: Amphithéâtre Hermite We study the Euler equations on the rotating unit sphere, focusing on the dynamics of vortex caps, i.e. piecewise constant absolute vorticity. Leveraging the area stability of monotone, longitude-independent profiles, we demonstrate that certain ill-prepared initial data within the vortex cap class exhibit an instability characterized by the growth of the interface perimeter. These configurations are nearly equivalent in area to a zonal vortex cap but are perturbed by a localized latitudinal bump. By comparing the longitudinal flows at points along the zonal interface and within the bump region, we track the induced stretching and capture the underlying instability mechanism.

The Fluid Dynamics of Tornadoes

• Richard Rotunno (NSF National Center for Atmospheric Research)

Room: Amphithéâtre Hermite Observations, laboratory experiments and numerical simulations provide the basis for the theoretical fluid dynamics of the tornado. Tornado fluid dynamics is best understood in terms of the dynamics of several subproblems: the two-cell vortex, the boundary-layer beneath a potential vortex, the formation of an end-wall vortex and its vortex breakdown. Vortex breakdown involves further understanding of super- and sub-critical columnar vortices and their connection to each other through the concept of conjugate states. The bulk of this presentation follows Rotunno (2013 Ann Rev Fluid Mech). Recent progress in tornado theory comes in the form of present-day, high-resolution numerical simulations of tornadic supercell thunderstorms which now have the resolution to include surface friction. Many of the characteristics of the above-described tornado fluid dynamics are being reported in the present generation of tornadic supercell simulations (Rotunno and Bluestein 2024 ! Rep Prog Phys).

Coupled Atmosphere-Ocean Models

• Matthias Hieber (MFO Oberwolfach, TU Darmstadt, Allemagne)

Room: Amphithéâtre Hermite In this talk we discuss coupled atmosphere-ocean models described by compressible/incompressible primitive equations subject to either deterministic nonlinear winddriven boundary conditions or stochastic boundary conditions on the interface.

QG dynamics of the LANS-alpha model

• Beth Wingate (University of Exeter)

Room: Amphithéâtre Hermite The Lagrangian-Averaged Navier-Stokes-$\alpha$ (LANS-$\alpha$) model, a fluid dynamics model based on energy-conserving modifications to nonlinear advection, can produce more energetic simulations than standard models, leading to improved fidelity (e.g., in ocean models). However, comprehensive understanding of the mechanism driving this energetic enhancement has proven elusive. To address this, we derive the fast quasi-geostrophic limit of the three-dimensional, stably-stratified LANS-$\alpha$ equations. This provides both the slow, balanced flow and the leading-order fast wave dynamics. Analysis of these wave dynamics suggests that an explanation for the energetic enhancement lies in the dual role of the alpha parameter: increasing $\alpha$ regularizes the dynamics and simultaneously generates a robust landscape of wave-wave resonant interactions where $\alpha$ modifies the role of Burger number. We discuss this using simulations and connect our results to numeri! cal stability issues.

Vortex dynamics for the lake equations

• Lars Eric Hientzsch (Karlsruhe Institute for Technology (KIT))

Room: Amphithéâtre Hermite The lake equations arise as a 2D geophysical model describing the vertically averaged evolution of an incompressible inviscid 3D fluid in a domain with spatially varying topography. We rigorously derive the asymptotic dynamics of point vortices. Specifically, we show that initially sharply concentrated vorticity remains concentrated in a suitable sense. The vortices' trajectories are proven to follow the level lines of the depth function. The result holds for generic concentrated initial data.

Surface Waves and Vortex Interaction

• Delia Ionescu-Kruse ("Simion Stoilow" Institute of Mathematics of the Romanian Academy (IMAR))

Room: Amphithéâtre Hermite The interaction between free-surface waves and localized vorticity structures is a fundamental problem in fluid dynamics, with relevance to geophysical flows. We study this problem by using the general framework for 2D water waves with arbitrary vorticity developed by Ionescu-Kruse and Ivanov (JDE, 2023). In the small-amplitude long-wave Boussinesq and KdV regimes, we derive coupled evolution equations for the free surface and the vortex dynamics. Our analysis shows that the interaction with the vortex does not destroy the surface solitary waves and, for a significant range of the vortex strength, the solitary waves remain practically unaffected. This observation leads to a further simplification of the model, in which the vortex motion beneath propagating solitons is described by a decoupled system of ODEs, capturing the qualitative features of the interaction. Analytical results are complemented by numerical simulations (see Ionescu-Kruse, Ivanov, Todorov, J. N! onlinear Sci, 2026).

Mixing with random cellular flows

• Victor Navarro-Fernandez (Imperial College London)

Room: Amphithéâtre Hermite We study a passive scalar equation on the two-dimensional torus, where the advecting velocity field is given by a cellular flow with a randomly moving center. We prove that the passive scalar undergoes mixing at a deterministic exponential rate, independent of any underlying diffusivity. Furthermore, we show that the velocity field enhances dissipation and we establish sharp decay rates that, for large times, are deterministic and remain uniform in the diffusivity constant. Our approach is purely Eulerian and relies on a suitable modification of Villani's hypocoercivity method, which incorporates a larger set of Hörmander commutators than Villani's original method.

Boundary layers of Navier-Stokes system

• Dongfen Bian (Beijing Institute of Techonology)

Room: Amphithéâtre Hermite The question of the stability of the boundary layers which appear as the viscosity of the fluid goes to zero is a classical question in Fluid Mechanics. In this talk I will discuss recent mathematical results on this question and in particular show that any shear layer is linearly and nonlinearly unstable provided the viscosity is small enough, and that the classical Prandtl boundary layers are always unstable.

Vorticity dynamics of coastal fronts

• Edward Johnson (University College London)

Room: Amphithéâtre Hermite This talk examines how free, long waves propagate along a potential vorticity front beside a vertical coast, using a 1.5-layer quasi-geostrophic model with piecewise-constant potential vorticity. The coastal boundary drives flow via image vorticity and a Kelvin wave, which can either reinforce or oppose Rossby wave dynamics at the front. Front behavior depends on the relative strengths of these three mechanisms, explicitly represented in the model. The richest dynamics—featuring kink solitons (under-compressive shocks) and compound waves—occur when vortical effects dominate. Front evolution follows a fully nonlinear, finite-amplitude equation with first-order dispersion, related to the modified Korteweg–de Vries equation. These behaviours are described analyticaaly via El’s dispersive shock-fitting method. Contour-dynamic simulations confirm that the dispersive long-wave theory accurately captures the full quasi-geostrophic dynamics.

The vanishing limit of a rigid body

• Jiao He (LMO)

Room: Amphithéâtre Hermite We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space R3 . When the small rigid body shrinks to a point in the sense that its density is constant, we prove that the solution of the fluid-rigid body system converges to a solution of the Navier–Stokes equations in the full space.Based on some $L^p$-$L^q$ estimates of the fluid–structure semigroup and a fixed point argument, we obtain a uniform estimate of velocity of the rigid body. This allows us to construct admissible test functions which plays a key role in the procedure of passing to the limit.

GFD on a fast rotating surface

• Bin Cheng (University of Surrey, UK)

Room: Amphithéâtre Hermite We consider the rotating shallow water equations with small (planetary) Rossby and Froude numbers on a surface of revolution with variable Coriolis parameter having opposite signs at the poles. The large variation of the linear operator in the PDE is a possible mechanism of short-time instability as the small parameters tend to zero. However, we prove that such instability does not happen in this case: classical solutions satisfy uniform estimates on a time interval independent of the small parameters. The most novel part of our approach is to find the explicit formula of a modified Laplacian which commutes with the large linear operator of the system. Further, upon a unitary transformation, the solution converges strongly to a limit for which the governing system is identified.

Desingularisation of Vortex Sheets

• David Meyer (ICMAT)

Room: Amphithéâtre Hermite We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff--Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities~$\omega^0_\epsilon\in C^\infty_c(\mathbb{R}^2)$ converging to $\omega^0_{\mathrm{sing}}$ distributionally as $\epsilon\to 0^+$, and show that the corresponding solutions $\omega_\epsilon(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff--Rott system with initial datum $\omega^0_{\mathrm{sing}}$. The regularization relies on a layer construction designed to exploit the key observation that the Kelvin--Helmholtz instability has a strongly anisotropic effect: while vorticities must be analytic in the "tangential" direction, the way layers can be arranged in the "normal" direction is essentially arbitrary.

A Lagrangian Vor-Div SWE Solver

• Anthony Chen (University of Michigan)

Room: Amphithéâtre Hermite In this work, I present a semi-Lagrangian solver for the spherical Shallow Water Equations, written in vorticity-divergence form. The solver makes use of a Biot-Savart law to compute the velocity, and is discretized on a tensor product Chebyshev grid on the cubed sphere. I also discuss some advantages and difficulties associated with using a semi-Lagrangian discretization, as well as the vorticity-divergence formulation of the SWE.

Leapfrogging of Vortex Rings

• Martin Donati (CNRS, LMA Poitiers)

Room: Amphithéâtre Hermite In this talk, we discuss recent developments concerning the motion of concentrated vortex rings. In particular, we outline the proof that, in the appropriate asymptotic regime, two concentrated coaxial vortex rings separated by a small distance exhibit the so-called leapfrogging motion. This highly singular regime lies beyond the scope of the standard tools of vorticity confinement. We introduce a new method to establish the persistence of mass concentration, and then implement a refined iterative scheme, combined with sharper estimates, to control the growth of the vorticity support. The choice of the asymptotic regime will be discussed, along with the technical limitations of our result.

On the stabilizing effect of rotation for 3D fluids

• Haram Ko (Brown University)

Room: Amphithéâtre Hermite It is well known to geophysicists that the rotation of the background can bring about different phenomena not observed in steady fluids. In this talk, I will explain from a mathematical point of view what the rotation introduces in 3D Euler/Navier-Stokes equation, and talk about recent results of how the rotation can enhance the stability (a) in incompressible fluid with high Reynolds number and (b) in compressible Euler equation.

Instability of dusty rotating vortices

• Henrik Latter (DAMTP, University of Cambridge)

Room: Amphithéâtre Hermite We explore the various instabilities that potentially assail vortex cores in the presence of rotation and/or radially drifting solids. We first revisit the classical elliptical instability of rotating purely hydrodynamic vortices, showing that, in certain natural limits, the problem is governed by the ‘inverted Matthieu equation’, which provides a novel and remarkably simple way to conceptualise the parametric instability of the non-modal inertial waves. Second, we construct idealised vortex solutions involving a drifting dust fluid and a two-way drag force, using a multiple scales approach and the presence of a conserved ‘dusty potential vorticity’. These solutions are subject to new small-scale instabilities that can be categorised as ‘resonant drag instabilities’, involving a coupling between the dust advective mode and inertial waves, though as both are non-modal, the classical theory needs significant reworking.

Cumulus Cloud Tilt and Ellipticity

• Dario Falcone (University of California, Davis)

Room: Amphithéâtre Hermite Developing a tractable understanding of the interaction between cumulus clouds’ ellipticity and tilt from shear flows is crucial to expanding theories associated with squall line development and tradewind cumuli climatological feedbacks. Here, we focus on the dynamic interplay between cloud-scale flows and shear flows. To perform this investigation, we implement a Kinematic Representation of Neutrally-buoyant Updraft Tori (KRoNUT) model for cloud-scale motions that allows for tilt and ellipticity. Using a moment closure technique, we then solve for the Dynamics of Neutrally-buoyant Updraft Tori (DoNUT) equations, a coupled non-linear system of ordinary differential equations which govern the temporal evolution of the parameters describing the intensity and geometry of a cloud-scale flow. Using this technique, we analytically and numerically investigate the behavior of the DoNUT equations to better appreciate how this additional geometry influences the life cycle of a cumulus cloud.

Dynamics of aerosol-filled vortices

• Kasturi Shah

Room: Amphithéâtre Hermite Long-lived, self-lofting, diabatically-driven anticyclones co-located with aerosols and trace gases have been detected in the stratosphere following strong wildfire events. An important unresolved dynamical question is the apparent single-signed vorticity anomaly. The expected vorticity structure from localised heating is an anticyclone-above-cyclone dipole and recent idealised dynamical studies which incorporate an evolving heating tracer have indeed revealed this structure. Here, we demonstrate that a potential mechanism for the observed single-signed vortices is the action of background vertical shear. As expected from simple theory, when the vorticity anomalies are weak (small Rossby number) the vertical shear acts similarly on both anticyclone and cyclone. When the vorticity anomalies are strong (O(1) Rossby number), as is relevant to the observed cases, there is a range of shear magnitudes in which the cyclone is torn apart by the shear but the anticyclone survives.

Geometry and vortex motion

• Andrew Gilbert (University of Exeter)

Room: Amphithéâtre Hermite 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.

A point vortex in a half-plane

• Thierry Gallay (Institut Fourier, Université Grenoble Alpes)

Room: Amphithéâtre Hermite As a model for vortex-wall interactions, we consider the two-dimensional incompressible Navier-Stokes equations in a half-plane with no-slip boundary condition and point vortices as initial data. We concentrate on the paradigmatic example of a single vortex in an otherwise stagnant fluid, which is already quite challenging from the mathematical point of view. As a warm-up, we prove that this system has a unique global solution for all values of the Reynolds number, which can be defined in this context as the ratio of the circulation of the vortex to the kinematic viscosity of the fluid. The solution we construct has finite energy for all positive times, and converges to zero in energy norm as time goes to infinity. Our ultimate goal is to understand the motion of the vortex center in the vanishing viscosity limit, but this question is very difficult due to the vortex-induced boundary layer separation, a phenomenon that is well documented in the physical literature.