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With the aim of assessing internal wave-driven mixing in the ocean, we develop a new technique for direct numerical simulations of stratified turbulence. Since the spatial scale of oceanic internal gravity waves is typically much larger than that of turbulence, fully incorporating both in a model would require a high computational cost, and is therefore out of our scope. Alternatively, we cut out a small domain periodically distorted by an unresolved large-scale internal wave and locally simulate the energy cascade to the smallest scales. In this model, even though the Froude number of the outer wave, $Fr$, is small such that density overturn or shear instability does not occur, a striped pattern of disturbance is exponentially amplified through a parametric subharmonic instability. When the disturbance amplitude grows sufficiently large, secondary instabilities arise and produce much smaller-scale fluctuations. Passing through these two stages, wave energy is transferred into turbulence energy and will be eventually dissipated. Different from the conventional scenarios of vertical shear-induced instabilities, a large part of turbulent potential energy is supplied from the outer wave and directly used for mixing. The mixing coefficient $\Gamma=\epsilon_P/\epsilon$, where $\epsilon$ is the dissipation rate of kinetic energy and $\epsilon_P$ is that of available potential energy, is always greater than 0.5 and tends to increase with $Fr$. Although our results are mostly consistent with the recently proposed scaling relationship between $\Gamma$ and the turbulent Froude number, $Fr_t$, the values of $\Gamma$ obtained here are larger by a factor of about two than previously reported.

Environmental fluid mechanics underlies a wealth of natural, industrial and, by extension, societal challenges. In the coming decades, as we strive towards a more sustainable planet, there are a wide range of grand challenge problems that need to be tackled, ranging from fundamental advances in understanding and modeling of stratified turbulence and consequent mixing, to applied studies of pollution transport in the ocean, atmosphere and urban environments. A workshop was organized in the Les Houches School of Physics in France in January 2019 with the objective of gathering leading figures in the field to produce a road map for the scientific community. Five subject areas were addressed: multiphase flow, stratified flow, ocean transport, atmospheric and urban transport, and weather and climate prediction. This article summarizes the discussions and outcomes of the meeting, with the intent of providing a resource for the community going forward.

As a necessary preliminary step toward geophysically significant extrapolations, we study the scale effects in internal wave attractors in the linear and nonlinear regimes. We use two geometrically similar experimental set-ups, scaled to factor 3, and numerical simulations (a spectral element method, based on the Nek5000 open solver) for a range of parameters that is typically accessible in laboratory. In the linear regime, we recover the classical viscous scaling for the beam width, which is not affected by variations of the amplitude of the input perturbation. In the nonlinear regime, we show that the scaling of the width-to-length ratio of the attractor branches is intimately related with the energy cascade from large-scale energy input to dissipation. We present results for the wavelength, amplitude and width of the beam as a function of time and as a function of the amplitude of the forcing.

We study experimentally the propagation of internal waves in two different three-dimensional (3D) geometries, with a special emphasis on the refractive focusing due to the 3D reflection of obliquely incident internal waves on a slope. Both studies are initiated by ray tracing calculations to determine the appropriate experimental parameters. First, we consider a 3D geometry, the classical set-up to get simple, 2D parallelogram-shaped attractors in which waves are forced in a direction perpendicular to a sloping bottom. Here, however, the forcing is of reduced extent in along-slope, transverse direction. We show how the refractive focusing mechanism explains the formation of attractors over the whole width of the tank, even away from the forcing region. Direct numerical simulations confirm the dynamics,emphasize the role of boundary conditions and reveal the phase shifting in the transversal direction. Second, we consider a long and narrow tank having an inclined bottom, to simply reproduce a canal. In this case, the energy is injected in a direction parallel to the slope. Interestingly, the wave energy ends up forming 2D internal wave attractors in planes that are transverse to the initial propagation direction. This focusing mechanism prevents indefinite transmission of most of the internal wave energy along the canal.

We present a novel characterization of mixing events associated with the propagation and overturning of internal waves, studied thanks to the simultaneous use of Particle Image Velocimetry (PIV) and Planar Laser Induced Fluorescence (PLIF) techniques. This combination of techniques had been developed earlier to provide an access to simultaneous velocity and density fields in two-layer stratified flows with interfacial gravity waves. Here, for the first time, we show how it is possible to implement it quantitatively in the case of a continuously stratified fluid where internal waves propagate in the bulk. We explain in details how the calibration of the PLIF data is performed by an iterative procedure, and we describe the precise spatial and temporal synchronizations of the PIV and PLIF measurements. We then validate the whole procedure by characterizing the Triadic Resonance Instability (TRI) of an internal wave mode. Very interestingly, the combined technique is then applied to a {precise} measurement of the turbulent diffusivity K_t associated with mixing events induced by an internal wave mode. Values up to K_t=15 mm^2\cdot s^{-1} are reached when TRI is present (well above the noise of our measurement, typically 1 mm^2\cdot s^{-1}), unambiguously confirming that TRI is a potential pathway to turbulent mixing in stratified flows. This work therefore provides a step on the path to new measurements for internal waves.

This paper revisits the problem of tidal conversion at a ridge in a uniformly stratified fluid of limited depth using measurements of complex-valued added mass. When the height of a sub-marine ridge is non negligible with respect to the depth of the water, the tidal conversion can be enhanced in the supercritical regime or reduced in the subcritical regime with respect to the large depth situation. Tidal conversion can even be null for some specific cases. Here, we study experimentally the influence of finite depth on the added mass coefficients for three diffierent ridge shapes. We first show that at low forcing frequency the tidal conversion is weakly enhanced by shallow depth for a semi-circular ridge. In addition, added mass coefficients measured for a vertical ridge show strong similarities with the ones obtained for the semi-circular ridge. Nevertheless, the enhancement of the tidal conversion at low forcing frequency for the vertical ridge has not been observed, in contrast with its supercritical shape. Finally, we provide the experimental evidence of a lack of tidal conversion due to the specific shape of a ridge for certain depth and frequency tuning.

We study the dynamics of Fermi-Pasta-Ulam chains with both harmonic and anharmonic power-law long-range interactions. We show that the dynamics is described in the continuum limit by a generalized fractional Boussinesq differential equation, whose derivation is performed in full detail. We also discuss a version of the model where couplings are alternating in sign., Comment: Corrected coefficients of the nonlinear terms of the generalized fractional Boussinesq equations, minor typos amended

We present the results of a laboratory experimental study of an internal wave field generated by harmonic, spatially-periodic boundary forcing from above of a density stratification comprising a strongly-stratified, thin upper layer sitting atop a weakly-stratified, deep lower layer. In linear regimes, the energy flux associated with relatively high frequency internal waves excited in the upper layer is prevented from entering the lower layer by virtue of evanescent decay of the wave field. In the experiments, however, we find that the development of parametric subharmonic instability (PSI) in the upper layer transfers energy from the forced primary wave into a pair of subharmonic daughter waves, each capable of penetrating the weakly-stratified lower layer. We find that around $10\%$ of the primary wave energy flux penetrates into the lower layer via this nonlinear wave-wave interaction for the regime we study., Comment: 6 pages, 5 figures

Understanding the relaxation of a system towards equilibrium is a longstanding problem in statistical mechanics. Here we address the role of long-range interactions in this process by considering a class of two-dimensional or geophysical flows where the interaction between fluid particles varies with the distance as $\sim$ r^($\alpha$--2) with $\alpha$ \textgreater{} 0. Previous studies in the Euler case $\alpha$ = 2 had shown convergence towards a variety of quasi-stationary states by changing the initial state. Unexpectedly, all those regimes are recovered by changing $\alpha$ with a prescribed initial state. For small $\alpha$, a coarsening process leads to the formation of a sharp interface between two regions of homogenized $\alpha$-vorticity; for large $\alpha$, the flow is attracted to a stable dipolar structure through a filamentation process.

The two-body potential of systems with long-range interactions decays at large distances as $V(r)\sim 1/r^\alpha$, with $\alpha\leq d$, where $d$ is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties is at the origin of ensemble inequivalence, which implies that specific heat can be negative in the microcanonical ensemble and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity implies that ergodicity may be generically broken. We present here a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Relaxation towards thermodynamic equilibrium can be extremely slow and quasi-stationary states may be present. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation., Comment: 118 pages, review paper, added references, slight change of content

We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in $N$-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated $N ~$ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite $N$, dynamics. We then propose and verify numerically a scenario for the relaxation process, relying on the Vlasov equation. When starting from a non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via non stationary states: we characterize numerically this dynamical instability in the finite $N$ system by introducing appropriate indicators. This first step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite $N$ system is initialized in a Vlasov stable homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law $N^{1.7}$. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the $q$-exponential distributions derived from Tsallis statistics., Comment: To appear in Physica A

We present detailed molecular dynamics results for the displacement autocorrelation spectra of the Peyrard-Bishop model of thermal DNA denaturation. As the phase transition is approached, the spectra depend on whether the wavelength is smaller than, or exceeds the correlation length. In the first case, the spectra are dominated by a single peak, whose frequency approaches the bare acoustic frequency of the harmonic chain, and whose linewidth approaches zero as $T_c-T$. In the second case, a central peak (CP) feature is dominant, accounting for most of the weight; the linewidth of the CP appears to be temperature-independent. We also present force autocorrelation spectra which may be relevant for analyzing the statistical properties of localized modes., Comment: Proceedings of the conference on Localization and Energy Transfer in Nonlinear Systems, June 17-21, 2002, San Lorenzo de El Escorial, Madrid, Spain; to be published by World Scientific. 10 pages, 5 figures

We study energy relaxation in a phenomenological model for polymer built from rheological considerations: a one dimensional nonlinear lattice with dissipative couplings. These couplings are well known in polymer's community to be possibly responsible of beta-relaxation (as in Burger's model). After thermalisation of this system, the extremities of the chain are put in contact with a zero-temperature reservoir, showing the existence of surprising quasi-stationary states with non zero energy when the dissipative coupling is high. This strange behavior, due to long-lived nonlinear localized modes, induces stretched exponential laws. Furthermore, we observe a strong dependence on the waiting time tw after the quench of the two-time intermediate correlation function C(tw+t,tw). This function can be scaled onto a master curve, similar to the case of spin or Lennard-Jones glasses., Comment: 8 pages, 10 figures

Using a matched asymptotic expansion we analyze the two-dimensional, near- critical reflection of a weakly nonlinear, internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the amplitude of the incident wave, the dissipation, and the departure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience heals the singularity which is presented by the classical solution in the precisely critical case. In the inviscid critical case, an explicit solution of the initial value problem shows that the buoyancy perturbation and the along-slope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as $1/t$. During the course of this scale reduction, the stratification is `overturned' and the Miles-Howard condition for stratified shear flow stability is violated. However, for all slope angles, the `overturning' occurs before the Miles-Howard stability condition is violated and so we argue that the first instability is convective. Solutions of the simplified dynamics resemble certain experimental visualizations of the reflection process., Comment: To appear in Journal of Fluid Mechanics, latex2e File + 14 pictures

The diffusion of a two-dimensional array of particles driven by a constant force in the presence of a periodic external potential exhibits a hierarchy of dynamical phase transitions when the driving force is varied. This behavior can be explained by a simple phenomenological approach which reduces the system of strongly interacting particles to weakly interacting quasi-particles (kinks). The richness of the strongly coupled system is however not lost because, contrary to a single-Brownian particle, the array shows an hysteretic behavior even at non-zero temperature. The present investigation can be viewed as a first step toward understanding nanotribology., Comment: 4 pages, 3 pictures, revtex to appear in Phys Rev. Lett

We discuss the process by which energy, initially evenly distributed in a nonlinear lattice, can localize itself into large amplitude excitations. We show that, the standard modulational instability mechanism, which can initiate the process by the formation of small amplitude breathers, is completed efficiently, in the presence of discreteness, by energy exchange mechanisms between the nonlinear excitations which favor systematically the growth of the larger excitations. The process is however self regulated because the large amplitude excitations are finally trapped by the Peierls-Nabarro potential., Comment: 15 pages in REVTEX, figures available from the authors.(email: tdauxois@physique.ens-lyon.fr)