Parametric instability and wave turbulence driven by tidal excitation of internal waves

We investigate the stability of stratified fluid layers undergoing homogeneous and periodic tidal deformation. We first introduce a local model which allows to study velocity and buoyancy fluctuations in a Lagrangian domain periodically stretched and sheared by the tidal base flow. While keeping the key physical ingredients only, such a model is efficient to simulate planetary regimes where tidal amplitudes and dissipation are small. With this model, we prove that tidal flows are able to drive parametric subharmonic resonances of internal waves, in a way reminiscent of the elliptical instability in rotating fluids. The growth rates computed via Direct Numerical Simulations (DNS) are in very good agreement with WKB analysis and Floquet theory. We also investigate the turbulence driven by this instability mechanism. With spatio-temporal analysis, we show that it is a weak internal wave turbulence occurring at small Froude and buoyancy Reynolds numbers. When the gap between the excitation and the Brunt-V\"ais\"al\"a frequencies is increased, the frequency spectrum of this wave turbulence displays a -2 power law reminiscent of the high-frequency branch of the Garett and Munk spectrum (Garrett & Munk 1979) which has been measured in the oceans. In addition, we find that the mixing efficiency is altered compared to what is computed in the context of DNS of stratified turbulence excited at small Froude and large buoyancy Reynolds numbers and is consistent with a superposition of waves.Comment: Accepted for publication in Journal of Fluid Mechanics, 27 pages, 21 figure

Spontaneous generation of inertial waves from boundary turbulence in a librating sphere

In this work, we report the excitation of inertial waves in a librating sphere even for libration frequencies where these waves are not directly forced. This spontaneous generation comes from the localized turbulence induced by the centrifugal instabilities in the Ekman boundary layer near the equator and does not depend on the libration frequency. We characterize the key features of these inertial waves in analogy with previous studies of the generation of internal waves in stratified flows from localized turbulent patterns. In particular, the temporal spectrum exhibits preferred values of excited frequency. This first-order phenomenon is generic to any rotating flow in the presence of localized turbulence and is fully relevant for planetary applications

Thermo-elliptical instability in a rotating cylindrical shell

Journal of Fluid Mechanics (in press, 2006)The linear stability of a rotating flow in an elliptically deformed cylindrical shell with an imposed radial temperature contrast is studied using local and global approaches. We demonstrate that (i) a stabilising temperature profile can either increase or decrease the growth rate of the elliptical instability depending on the selected mode and on the strength of the radial buoyancy force; (ii) when the temperature profile is destabilising, the elliptical instability coexists with 2D convective instabilities at relatively small values of the Rayleigh number, the fastest growing mode depending on the relative values of the Rayleigh number and of the eccentricity; (iii) the elliptical instability totally disappears for larger values of the Rayleigh number. We argue that thermal effects have to be taken into account when looking for the occurrence and influence of inertial instabilities in geophysical and astrophysical systems, especially in planetary cores

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

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

Order Out of Chaos: Slowly Reversing Mean Flows Emerge from Turbulently Generated Internal Waves

We demonstrate via direct numerical simulations that a periodic, oscillating mean flow spontaneously develops from turbulently generated internal waves. We consider a minimal physical model where the fluid self-organizes in a convective layer adjacent to a stably stratified one. Internal waves are excited by turbulent convective motions, then nonlinearly interact to produce a mean flow reversing on timescales much longer than the waves' period. Our results demonstrate for the first time that the three-scale dynamics due to convection, waves, and mean flow is generic and hence can occur in many astrophysical and geophysical fluids. We discuss efforts to reproduce the mean flow in reduced models, where the turbulence is bypassed. We demonstrate that wave intermittency, resulting from the chaotic nature of convection, plays a key role in the mean-flow dynamics, which thus cannot be captured using only second-order statistics of the turbulent motions

Solidification of a binary alloy: finite-element, single-domain simulation and new benchmark solutions

A finite-element simulation of binary alloy solidification based on a single-domain formulation is presented and tested. Resolution of phase change is first checked by comparison with the analytical results of Worster (1986) for purely diffusive solidification. Fluid dynamical processes without phase change are then tested by comparison with previous numerical studies of thermal convection in a pure fluid (de Vahl Davis 1983, Mayne et al. 2000, Wan et al. 2001), in a porous medium with a constant porosity (Lauriat & Prasad 1989, Ni et al. 1997) and in a mixed liquid-porous medium with a spatially variable porosity (Ni et al. 1997, Zabaras & Samanta 2004). Finally, new benchmark solutions for simultaneous flow through both fluid and porous domains and for convective solidification processes are presented, based on the similarity solutions in corner-flow geometries recently obtained by Le Bars & Worster (2006). Good agreement is found for all tests, hence validating our physical and numerical methods. More generally, the computations presented here could now be considered as standard and reliable analytical benchmarks for numerical simulations, specifically and independently testing the different processes underlying binary alloy solidification

Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification

The single-domain, Darcy-Brinkman model is applied to some analytically tractable flows through adjacent porous and pure-fluid domains and is compared systematically with the multiple-domain, Stokes-Darcy model. In particular, we focus on the interaction between flow and solidification within the mushy layer during binary alloy solidification in a corner flow and on the effects of the chosen mathematical description on the resulting macrosegregation patterns. Large-scale results provided by the multiple-domain formulation depend strongly on the microscopic interfacial conditions. No satisfactory agreement between the single- and multiple-domain approaches is obtained when using previously suggested conditions written directly at the interface between the liquid and the porous medium. Rather, we define a viscous transition zone inside the porous domain, where Stokes equation still applies, and we impose continuity of pressure and velocities across it. This new condition provides good agreement between the two formulations of solidification problems when there is a continuous variation of porosity across the interface between a partially solidified region (mushy zone) and the melt

Evidence of the Zakharov-Kolmogorov spectrum in numerical simulations of inertial wave turbulence

Rotating turbulence is commonly known for being dominated by geostrophic vortices that are invariant along the rotation axis and undergo inverse cascade. Yet, it has recently been shown to sustain fully three-dimensional states with a downscale energy cascade. In this letter, we investigate the statistical properties of three-dimensional rotating turbulence by the means of direct numerical simulations in a triply periodic box where geostrophic vortices are specifically damped. The resulting turbulent flow is an inertial wave turbulence that verifies the Zakharov-Kolmogorov spectrum derived analytically by Galtier (Phys. Rev. E, 68, 2003), thus offering numerical proof of the relevance of wave turbulence theory for three-dimensional, anisotropic waves. Lastly, we show that the same forcing leads to either geostrophic or wave turbulence depending on the initial condition. Our results thus bring further evidence for bi-stability in rotating turbulent flows.Comment: 7 pages, 4 figure