THE CUSP: FROM FINITE TIME SINGULARITIES TO BREAKING WAVES

International audienceWe all have in mind the sound of the ping-pong ball bouncing repeatedly faster and faster off the ground before it comes to a halt. This phenomenon of divergence in a finite duration of a physical quantity-here the frequency of rebounds-carries in physics the name of "singularity in finite time". Another example of a singularity in finite time comes from the wave breaking phenomenon. In the vicinity of this subtle instant preceding breaking, the temporal derivative of the free surface of water diverges as the inverse of the square root of time. A geometric representation of this singular behavior is the cusp, the simplest catastrophe depending on two parameters as classified in his Catastrophes Theory by René Thom. The drawing of the cusp shows that for certain time and space positions, the surface is a single valued function, whereas outside this domain, it takes three possible values. Therefore, the cusp gives a good representation of the tilting of the wave before breaking. The deliminating curves between these two behavior are called caustics and merge at the singularity point. These lines share in fact the exact mathematical description of the pattern drawn by focusing light at the bottom of a cup after reflecting off the cup wall. My wish during my SCIENTIFIC DELIRIUM MADNESS residency was to create an art piece with the intention to share the contemplation of this subtle and fragile instant where the singularity arises, where the curvature of forms changes sign, where waves tilt over. To extend this infinitely small duration, the singularity will be unfolded-as mathematicians say. The result of my creative process is an array of wooden rods whose arrangement makes straight rays to cooperate and interfere in an apparent continuous and smooth motion, giving rise to a regular but cusped surface that sifts and transforms our own perception of the world around it. Like a filter, the CUSP will let wind, sound, light and thoughts to go through its beams. Fig. 1. The CUSP under the full moon in the Djerassi land. (© Patrice Le Gal. Photo: Kristen Stipanov.

The Universal Aspect Ratio of Vortices in Rotating Stratified Flows: Theory and Simulation

We derive a relationship for the vortex aspect ratio $\alpha$ (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt-Vaisala frequencies within the vortex $N_c$ and in the background fluid outside the vortex $\bar{N}$, the Coriolis parameter $f$, and the Rossby number $Ro$ of the vortex: $\alpha^2 = Ro(1+Ro) f^2/(N_c^2-\bar{N}^2)$. This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the background density gradient need not be uniform. Our relation for $\alpha$ has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have $N_c^2 > \bar{N}^2$; weak anticyclones (with $|Ro| \bar{N}^2$. We verify our relation for $\alpha$ with numerical simulations of the three-dimensional Boussinesq equations for a wide variety of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally mixed density; cyclones created by fluid suction from a small localised region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically-computed vortices validate our relationship for $\alpha$, and generally they differ significantly from the values obtained from the much-cited conjecture that $\alpha = f/\bar{N}$ in quasi-geostrophic vortices.Comment: Submitted to the Journal of Fluid Mechanics. Also see the companion paper by Aubert et al. "The Universal Aspect Ratio of Vortices in Rotating Stratified Flows: Experiments and Observations" 201

The Universal Aspect Ratio of Vortices in Rotating Stratifi?ed Flows: Experiments and Observations

We validate a new law for the aspect ratio $\alpha = H/L$ of vortices in a rotating, stratified flow, where $H$ and $L$ are the vertical half-height and horizontal length scale of the vortices. The aspect ratio depends not only on the Coriolis parameter f and buoyancy (or Brunt-Vaisala) frequency $\bar{N}$ of the background flow, but also on the buoyancy frequency $N_c$ within the vortex and on the Rossby number $Ro$ of the vortex such that $\alpha = f \sqrt{[Ro (1 + Ro)/(N_c^2- \bar{N}^2)]}$. This law for $\alpha$ is obeyed precisely by the exact equilibrium solution of the inviscid Boussinesq equations that we show to be a useful model of our laboratory vortices. The law is valid for both cyclones and anticyclones. Our anticyclones are generated by injecting fluid into a rotating tank filled with linearly-stratified salt water. The vortices are far from the top and bottom boundaries of the tank, so there is no Ekman circulation. In one set of experiments, the vortices viscously decay, but as they do, they continue to obey our law for $\alpha$, which decreases over time. In a second set of experiments, the vortices are sustained by a slow continuous injection after they form, so they evolve more slowly and have larger |Ro|, but they also obey our law for $\alpha$. The law for $\alpha$ is not only validated by our experiments, but is also shown to be consistent with observations of the aspect ratios of Atlantic meddies and Jupiter's Great Red Spot and Oval BA. The relationship for $\alpha$ is derived and examined numerically in a companion paper by Hassanzadeh et al. (2012).Comment: Submitted to the Journal of Fluid Mechanics. Also see the companion paper by Hassanzadeh et al. "The Universal Aspect Ratio of Vortices in Rotating Stratifi?ed Flows: Theory and Simulation" 201

Resonances of a rotor/stator cavity in the vicinity of the critical point of SF6

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The transition to turbulence of the torsionnal Couette flow

This work is devoted to the experimental study of the transition to turbulence of a flow confined in a narrow gap between a rotating and a stationary disk. When the fluid layer thickness is of the same order of magnitude as the boundary layer depth, the azimuthal velocity axial gradient is nearly constant and this rotating disk flow is a torsional Couette flow. As in the plane Couette flow or the cylindrical Couette flow, transition to turbulence occurs via the appearance of turbulent domains inside a laminar background. Nevertheless, we show that in the rotating disk case, the nucleation of turbulent spirals is connected to the birth of structural defects in a periodic underlying spiral roll pattern

The linear instability of the stratified plane Couette flow

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonally to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the stream-wise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number Re and the Froude number Fr, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. Fr $\sim$ 1) and above a moderate value of the Reynolds number Re$\gtrsim$700. The instability results from a resonance mechanism already known in the context of channel flows, for instance the unstratified plane Couette flow in the shallow water approximation. The result is confirmed by fully non linear direct numerical simulations and to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of F r and Re indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement

L'instabilité elliptique en géophysique

L'étude de l'instabilité elliptique a été motivée par des problèmes rencontrés en aérodynamique (instabiltés secondaires diverses, vortex de bout d'ailes, turbulence), mais un autre intérêt suscité par cette instabilité elliptique relève de la géophysique. En effet, lorsqu'une planète tourne autour d'un soleil, et (ou) qu'une lune tourne autour de la planète, le noyau liquide de celle-ci subit une déformation elliptique (une marée) causée par le champ de gravitation. La rotation de la planète sur son axe pourrait alors faire résonner des ondes de Kelvin dans le noyau liquide ce qui résulterait en l'apparition de l'instabilité elliptique. Nos résultats expérimentaux et théoriques montrent qu'en effet, le mode dit de "spin-over" apparaît au seuil de l' instabilité dans la géométrie sphérique et qu'une dynamique intermittente existe à plus haut nombre de Reynolds. Finalement, une expérience sous champ magnétique et utilisant un métal liquide, met en évidence la génération d'un champ magnétique directement induit par l'instabilité

Elliptical instability in hot Jupiter systems

Several studies have already considered the influence of tides on the evolution of systems composed of a star and a close-in companion to tentatively explain different observations such as the spin-up of some stars with hot Jupiters, the radius anomaly of short orbital period planets and the synchronization or quasi-synchronization of the stellar spin in some extreme cases. However, the nature of the mechanism responsible for the tidal dissipation in such systems remains uncertain. In this paper, we claim that the so-called elliptical instability may play a major role in these systems, explaining some systematic features present in the observations. This hydrodynamic instability, arising in rotating flows with elliptical streamlines, is suspected to be present in both planet and star of such systems, which are elliptically deformed by tides. The presence and the influence of the elliptical instability in gaseous bodies, such as stars or hot Jupiters, are most of the time neglected. In this paper, using numerical simulations and theoretical arguments, we consider several features associated to the elliptical instability in hot-Jupiter systems. In particular, the use of ad hoc boundary conditions makes it possible to estimate the amplitude of the elliptical instability in gaseous bodies. We also consider the influence of compressibility on the elliptical instability, and compare the results to the incompressible case. We demonstrate the ability for the elliptical instability to grow in the presence of differential rotation, with a possible synchronized latitude, provided that the tidal deformation and/or the rotation rate of the fluid are large enough. Moreover, the amplitude of the instability for a centrally-condensed mass of fluid is of the same order of magnitude as for an incompressible fluid for a given distance to the threshold of the instability. Finally, we show that the assumption of the elliptical instability being the main tidal dissipation process in eccentric inflated hot Jupiters and misaligned stars is consistent with current data.Comment: Icarus (2013) http://dx.doi.org/10.1016/j.icarus.2012.12.01

Flow-induced vibrations of high mass ratio flexible filaments freely hanging in a flow

8 p.The behavior of high mass ratio flexible filaments freely hanging in steady horizontal uniform flows is experimentally and theoretically investigated. When the flow velocity is small, static equilibrium states, where the filaments are inclined to the flow, are observed. Then, above a critical value of the wind velocity, the filaments exhibit periodic oscillations in the vertical plane. The problem is theoretically addressed considering the beam theory equations for the filament dynamics where the action of the flowing fluid is modeled using semi-empirical expressions. These equations are first solved for the stationary equilibrium states. Then, the stability of these steady solutions relatively to small perturbations is analyzed. A good agreement between experimental and theoretical results is found

Experimental study of internal wave generation by convection in water

We experimentally investigate the dynamics of water cooled from below at 0^oC and heated from above. Taking advantage of the unusual property that water's density maximum is at about 4^oC, this set-up allows us to simulate in the laboratory a turbulent convective layer adjacent to a stably stratified layer, which is representative of atmospheric and stellar conditions. High precision temperature and velocity measurements are described, with a special focus on the convectively excited internal waves propagating in the stratified zone. Most of the convective energy is at low frequency, and corresponding waves are localized to the vicinity of the interface. However, we show that some energy radiates far from the interface, carried by shorter horizontal wavelength, higher frequency waves. Our data suggest that the internal wave field is passively excited by the convective fluctuations, and the wave propagation is correctly described by the dissipative linear wave theory