Joël Sommeria

Why, how, and when, MHD turbulence becomes two-dimensional

A description of MHD turbulence at low magnetic Reynolds number and large interaction parameter is proposed, in which attention is focussed on the role of insulating walls perpendicular to a uniform applied magnetic field. The flow is divided in two regions: the thin Hartmann layers near the walls, and the bulk of the flow. In the latter region, a kind of electromagnetic diffusion along the magnetic field lines (a degenerate form of Alfv6n waves) is displayed, which elongates the turbulent eddies in the field direction, but is not sufficient to generate a two-dimensional dynamics. However the normal derivative of velocity must be zero (to leading order) at the boundaries of the bulk region (as at a free surface), so that when the length scale 1, perpendicular to the magnetic field is large enough, the corresponding eddies are necessarily two-dimensional. Furthermore, if I, is not larger than a second limit, the Hartmann braking effect is negligible and the dynamics of these eddies is described by the ordinary Navier-Stokes equations without electromagnetic forces. MHD then appears to offer a means of achieving experiments on two-dimensional turbulence, and of deducing velocity and vorticity from measurements of electric field.

Emergence of intense jets and Jupiter's Great Red Spot as maximum-entropy structures

We explain the emergence and robustness of intense jets in highly turbulent planetary atmospheres, like that on Jupiter, by a general statistical mechanics approach to potential vorticity patches. The idea is that potential vorticity mixing leads to the formation of a steady organized coarse-grained flow, corresponding to the statistical equilibrium state. Our starting point is the quasi-geostrophic 1-1/2 layer model, and we consider the relevant limit of a small Rossby radius of deformation. Then narrow jets are obtained, in the sense that they scale like the radius of deformation. These jets can be either zonal, or closed into a ring bounding a vortex. Taking into account the beta-eect and a sublayer deep shear flow, we predict organization of the turbulent atmospheric layer into an oval-shaped vortex within a background shear. Such an isolated vortex is centred over an extremum of the equivalent topography, combining the interfacial geostrophic tilt due to the deep shear flow and the planetary beta-eect (the resulting eective beta-eect is locally quadratic). This prediction is in agreement with an analysis of wind data in major Jovian vortices (Great Red Spot and Oval BC).

Characteristics of summer airflow over the Antarctic Peninsula in response to recent strengthening of westerly circumpolar winds

Abstract Summer near-surface temperatures over the northeast coast of the Antarctic Peninsula have increased by more than 2°C over the past 40 years, a temperature increase 3 times greater than that on the northwest coast. Recent analysis has shown a strong correlation between this striking warming trend and significant change in the summer Southern Hemisphere annular mode (SAM), which has resulted in greatly increased summer westerlies across the northern peninsula. It has been proposed that the strengthening westerlies have resulted in increased vertical deflection of relatively warm maritime air over the northern peninsula, contributing significantly to the observed warming and the recent collapse of northern sections of the Larsen Ice Shelf. In this study, laboratory and numerical modeling of airflow incident to the peninsula are employed to further understand this mechanism. It is shown that the effect of the strengthening westerlies has led to a distinct transition from a “blocked” regime to a “flow...

Dynamics of Convectively Driven Banded Jets in the Laboratory

The banded organization of clouds and zonal winds in the atmospheres of the outer planets has long fascinated observers. Several recent studies in the theory and idealized modeling of geostrophic turbulence have suggested possible explanations for the emergence of such organized patterns, typically involving highly anisotropic exchanges of kinetic energy and vorticity within the dissipationless inertial ranges of turbulent flows dominated (at least at large scales) by ensembles of propagating Rossby waves. The results from an attempt to reproduce such conditions in the laboratory are presented here. Achievement of a distinct inertial range turns out to require an experiment on the largest feasible scale. Deep, rotating convection on small horizontal scales was induced by gently and continuously spraying dense, salty water onto the free surface of the 13-m-diameter cylindrical tank on the Coriolis platform in Grenoble, France. A “planetary vorticity gradient” or “ effect” was obtained by use of a conically sloping bottom and the whole tank rotated at angular speeds up to 0.15 rad s 1 . Over a period of several hours, a highly barotropic, zonally banded large-scale flow pattern was seen to emerge with up to 5–6 narrow, alternating, zonally aligned jets across the tank, indicating the development of an anisotropic field of geostrophic turbulence. Using particle image velocimetry (PIV) techniques, zonal jets are shown to have arisen from nonlinear interactions between barotropic eddies on a scale comparable to either a Rhines or “frictional” wavelength, which scales roughly as (/Urms) 1/2 . This resulted in an anisotropic kinetic energy spectrum with a significantly steeper slope with wavenumber k for the zonal flow than for the nonzonal eddies, which largely follows the classical Kolmogorov k 5/3 inertial range. Potential vorticity fields show evidence of Rossby wave breaking and the presence of a “hyperstaircase” with radius, indicating instantaneous flows that are supercritical with respect to the Rayleigh–Kuo instability criterion and in a state of “barotropic adjustment.” The implications of these results are discussed in light of zonal jets observed in planetary atmospheres and, most recently, in the terrestrial oceans.

Decaying grid turbulence in a strongly stratified fluid

Grid turbulence experiments have been carried out in a stably stratified fluid at moderately large Reynolds numbers (160 based on the Taylor microscale). A scanning particle image velocimetry technique is used to provide time-resolved velocity fields in a relatively large volume. For late times, in the low-Froude-number regime, the flow consists of quasi-horizontal motion in a sea of weak internal gravity waves. In this regime the dynamics of the flow is found to be independent of the ambient stratification. Fundamental differences with two-dimensional turbulence, due to the strong vertical shearing of horizontal velocity, are observed. In this regime, a self-similar scaling law for the energy decay and the length-scale evolution are observed. This behaviour reflects a process of adjustment of the eddy aspect ratio based on a balance between the horizontal advective motion which tends to vertically decorrelate the flow and the dissipation due to the strong vertical shear. The characteristic vertical size of the eddies grows according to a diffusion law and is found to be independent of the turbulence generation. The organization of the flow into horizontal layers of eddies separated by intense shear leads to a strong anisotropy of the dissipation: this has been checked by direct measurement of the different tensorial components of the viscous dissipation.

Subgrid-Scale Eddy Parameterization by Statistical Mechanics in a Barotropic Ocean Model

Abstract The feasibility of using a subgrid-scale eddy parameterization, based on statistical mechanics of potential vorticity, is investigated. A specific implementation is derived for the somewhat classic barotropic vorticity equation in the case of a fully eddy-active, wind-driven, midlatitude ocean on the β plane. The subgrid-scale eddy fluxes are determined by a principle of maximum entropy production so that these fluxes always efficiently drive the system toward statistical equilibrium. In the absence of forcing and friction, the system then reaches this equilibrium, while conserving all the constants of motion of the inviscid barotropic equations. It is shown that this equilibrium is close to a Fofonoff flow, like that obtained with truncated spectral models, although the statistical approach is different. The subgrid-scale model is then validated in a more realistic case, with wind forcing and friction. The results of this model at a coarse resolution are compared with reference simulations at a ...

Excitation and breaking of internal gravity waves by parametric instability

We study the dynamics of internal gravity waves excited by parametric instability in a stably stratified medium, either at the interface between a water and a kerosene layer, or in brine with a uniform gradient of salinity. The tank has a rectangular section, and is narrow to favour standing waves with motion in the vertical plane. The fluid container undergoes vertical oscillations, and the resulting modulation of the apparent gravity excites the internal waves by parametric instability. Each internal wave mode is amplified for an excitation frequency close to twice its natural frequency, when the excitation amplitude is sucient to overcome viscous damping (these conditions define an ‘instability tongue’ in the parameter space frequency-amplitude). In the interfacial case, each mode is well separated from the others in frequency, and behaves like a simple pendulum. The case of a continuous stratification is more complex as dierent modes have overlapping instability tongues. In both cases, the growth rates and saturation amplitudes behave as predicted by the theory of parametric instability for an oscillator. However, complex friction eects are observed, probably owing to the development of boundary-layer instabilities. In the uniformly stratified case, the excited standing wave is unstable via a secondary parametric instability: a wave packet with small wavelength and half the primary wave frequency develops in the vertical plane. This energy transfer toward a smaller scale increases the maximum slope of the iso-density surfaces, leading to local turning and rapid growth of three-dimensional instabilities and wave breaking. These results illustrate earlier stability analyses and numerical studies. The combined eect of the primary excitation mechanism and wave breaking leads to a remarkable intermittent behaviour, with successive phases of growth and decay for the primary wave over long timescales.

Electrically driven vortices in a strong magnetic field

A steady isolated vortex is produced in a horizontal layer of mercury (of thickness a ), subjected to a uniform vertical magnetic field. The vortex is forced by an electric current going from an electrode in the lower plane to the circular outer frame. The flow is investigated by streak photographs of small particles following the free upper surface, and by electric potential measurements. The agreement with the asymptotic theory for high values of the Hartmann number M is excellent for moderate electric currents. In particular all the current stays in the thin Hartmann layer of thickness a/M , except in the vortex core of horizontal extension a / M ½ . For higher currents, the size of the core becomes larger and depends only on the local interaction parameters. When the current is switched off, we measure the decay due to the Hartmann friction. A similar study is carried out for a vortex created by an initial electric pulse, and for a pair of vortices of opposite sign. For all these examples, the dynamics can be described by the two-dimensional Navier-Stokes equations with Hartmann friction, except in the vortex cores. Finally a vortex is produced near a lateral wall and a detachment of the boundary layer parallel to the magnetic field occurs, by which a second vortex of opposite sign is generated.

Turbulent vortex dipoles in a shallow water layer

This paper describes an experimental study on turbulent dipolar vortices in a shallow water layer. Dipoles are generated by an impulsive horizontal jet, by which a localized three-dimensional turbulent flow region is created. Dipole emergence is only controlled by the confinement number C=Q/H2tinj whereas the jet Reynolds number Re=Q/v has no influence in the studied range 50 000 2, the flow becomes quasi-two-dimensional and a single vortex dipole emerges in most cases. By qualitative observations and application of particle image velocimetry, the main dipole features have been determined. The shallow water dipoles are characterized by the simultaneous presence of several scales of turbulence: A quasi-two-dimensional main flow at large scale and three-dimensional turbulent motions at small scale. A vertical circulation takes place in the dipole front. A theoretical model i...

Jupiter's and Saturn's convectively driven banded jets in the laboratory

The banded patterns of cloud and wind are among the most striking features of the atmospheres of Jupiter and Saturn, but their dynamical origin remains poorly understood. Most approaches towards understanding zonation so far (also in the terrestrial oceans) have used highly idealized models to show that it might originate from dynamical anisotropy in a shallow turbulent fluid layer due to the planetary β-effect. Here we report the results of laboratory experiments, conducted on a 14-m diameter turntable, which quantitatively confirm that multiple zonal jets may indeed be generated and maintained by this mechanism in the presence of deep convection and a topographic β-effect. At the very small values of Ekman number (≤2 × 10−5) and large local Reynolds numbers (≥2000, based on jet scales) achieved, the kinetic energy spectra suggest the presence of both energy-cascading and enstrophy-cascading inertial ranges in addition to the zonation near twice the Rhines wave number.