ALBERT

Description: Following the Kolmogorov technique, an exact relation for a vector third-order moment $mathbi{J}$ is derived for three-dimensional incompressible stably stratified turbulence under the Boussinesq approximation. In the limit of a small Brunt–Väisälä frequency, isotropy may be assumed which allows us to find a generalized $4/ 3$-law. For strong stratification, we make the ansatz that $mathbi{J}$ is directed along axisymmetric surfaces parameterized by a scaling law relating horizontal and vertical coordinates. An integration of the exact relation under this hypothesis leads to a generalized Kolmogorov law which depends on the intensity of anisotropy parameterized by a single coefficient. By using a scaling relation between large horizontal and vertical length scales we fix this coefficient and propose a unique law.

Description: This paper builds upon the investigation of Augier et al. (Phys. Fluids, vol. 26 (4), 2014) in which a strongly stratified turbulent-like flow was forced by 12 generators of vertical columnar dipoles. In experiments, measurements start to provide evidence of the existence of a strongly stratified inertial range that has been predicted for large turbulent buoyancy Reynolds numbers Rt = εK/(νN2), where εK is the mean dissipation rate of kinetic energy, ν the viscosity and N the Brunt-Väisälä frequency. However, because of experimental constraints, the buoyancy Reynolds number could not be increased to sufficiently large values so that the inertial strongly stratified turbulent range is only incipient. In order to extend the experimental results toward higher buoyancy Reynolds number, we have performed numerical simulations of forced stratified flows. To reproduce the experimental vortex generators, columnar dipoles are periodically produced in spatial space using impulsive horizontal body force at the peripheries of the computational domain. For moderate buoyancy Reynolds number, these numerical simulations are able to reproduce the results obtained in the experiments, validating this particular forcing. For higher buoyancy Reynolds number, the simulations show that the flow becomes turbulent as observed in Brethouwer et al. (J. Fluid Mech., vol. 585, 2007, pp. 343-368). However, the statistically stationary flow is horizontally inhomogeneous because the dipoles are destabilized quite rapidly after their generation. In order to produce horizontally homogeneous turbulence, high-resolution simulations at high buoyancy Reynolds number have been carried out with a slightly modified forcing in which dipoles are forced at random locations in the computational domain. The unidimensional horizontal spectra of kinetic and potential energies scale like C1εK2/3kh-5/3 and C2εK2/3kh-5/3(εP/εK), respectively, with C1 = C2 ≃ 0.5 as obtained by Lindborg (J. Fluid Mech., vol. 550, 2006, pp. 207-242). However, there is a depletion in the horizontal kinetic energy spectrum for scales between the integral length scale and the buoyancy length scale and an anomalous energy excess around the buoyancy length scale probably due to direct transfers from large horizontal scale to small scales resulting from the shear and gravitational instabilities. The horizontal buoyancy flux co-spectrum increases abruptly at the buoyancy scale corroborating the presence of overturnings. Remarkably, the vertical kinetic energy spectrum exhibits a transition at the Ozmidov length scale from a steep spectrum scaling like N2kz-3 at large scales to a spectrum scaling like CKεK2/3kz-5/3, with CK = 1, the classical Kolmogorov constant. © 2015 Cambridge University Press.

Description: Efforts to model accretion disks in the laboratory using Taylor-Couette flow apparatus are plagued with problems due to the substantial impact the end plates have on the flow. We explore the possibility of mitigating the influence of these end plates by imposing stable stratification in their vicinity. Numerical computations and experiments confirm the effectiveness of this strategy for restoring the axially homogeneous quasi-Keplerian solution in the unstratified equatorial part of the flow for sufficiently strong stratification and moderate layer thickness. If the rotation ratio is too large, however (e.g. Ωo/Ωi= (ri/ro)3/2, where Ωo/Ωi is the angular velocity at the outer/inner boundary and ri/ro is the inner/outer radius), the presence of stratification can make the quasi-Keplerian flow susceptible to the stratorotational instability. Otherwise (e.g. for Ωo/Ωi= (ri/ro)1/2), our control strategy is successful in reinstating a linearly stable quasi-Keplerian flow away from the end plates. Experiments probing the nonlinear stability of this flow show only decay of initial finite-amplitude disturbances at a Reynolds number Re = O(104). This observation is consistent with most recent computational (Ostilla-Mónico, et al. J. Fluid Mech., vol. 748, 2014, R3) and experimental results (Edlund & Ji, Phys. Rev. E, vol. 89, 2014, 021004) at high Re, and reinforces the growing consensus that turbulence in cold accretion disks must rely on additional physics beyond that of incompressible hydrodynamics. © © 2016 Cambridge University Press.

Description: We investigate the spectral properties of the turbulence generated during the nonlinear evolution of a Lamb-Chaplygin dipole in a stratified fluid for a high Reynolds number Re= 28 000 and a wide range of horizontal Froude number Fhin and buoyancy Reynolds number R= ReFh2. The numerical simulations use a weak hyperviscosity and are therefore almost direct numerical simulations (DNS). After the nonlinear development of the zigzag instability, both shear and gravitational instabilities develop and lead to a transition to small scales. A spectral analysis shows that this transition is dominated by two kinds of transfer: first, the shear instability induces a direct non-local transfer toward horizontal wavelengths of the order of the buoyancy scale Lb = U/N, where U is the characteristic horizontal velocity of the dipole and N the Brunt-Väisälä frequency; second, the destabilization of the Kelvin-Helmholtz billows and the gravitational instability lead to small-scale weakly stratified turbulence. The horizontal spectrum of kinetic energy exhibits a K2/3 kh5/3 power law (where kh is the horizontal wavenumber and K is the dissipation rate of kinetic energy) from kb = 2πLb to the dissipative scales, with an energy deficit between the integral scale and kb and an excess around kb. The vertical spectrum of kinetic energy can be expressed as E(kz)= C N N2kz-3+ CK 2/3kz-5/3} where CN and C are two constants of order unity and kz is the vertical wavenumber. It is therefore very steep near the buoyancy scale with an N2kz-3 shape and approaches the K2/3kz-5/3 spectrum for k z〉 ko, ko being the Ozmidov wavenumber, which is the cross-over between the two scaling laws. A decomposition of the vertical spectra depending on the horizontal wavenumber value shows that the N2kz-3 spectrum is associated with large horizontal scales kh 〈kband the K2/3kz- 5/3 spectrum with the scales khgt;kb. © 2012 Cambridge University Press.

Description: Recently, Deloncle, Billant & Chomaz (J. Fluid Mech., vol. 599, 2008, p. 229) and Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239) have performed numerical simulations of the nonlinear evolution of the zigzag instability of a pair of counter-rotating vertical vortices in a stratified fluid. Both studies report the development of a small-scale secondary instability when the vortices are strongly bent if the Reynolds number Re is sufficiently high. However, the two papers are at variance about the nature of this secondary instability: it is a shear instability according to Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) and a gravitational instability according to Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239). They also profoundly disagree about the condition for the onset of the secondary instability: ReF2h 〉 O(1) according to the former or ReFh 〉 80 according to the latter, where Fh is the horizontal Froude number. In order to understand the origin of these discrepancies, we have carried out direct numerical simulations of the zigzag instability of a Lamb-Chaplygin vortex pair for a wide range of Reynolds and Froude numbers. The threshold for the onset of a secondary instability is found to be ReF2h â 4 for Re â 3000 and ReFh = 80 for Re â 1000 in agreement with both previous studies. We show that the scaling analysis of Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) can be refined to obtain a universal threshold: (Re â̂' Re0)F2h â 4, with Re0 â 400, which works for all Re. Two different regimes for the secondary instabilities are observed: when (Re â̂' Re0)F2h â 4, only the shear instability develops while when (Re â̂' Re0)F2h â 4, both shear and gravitational instabilities appear almost simultaneously in distinct regions of the vortices. However, the shear instability seems to play a dominant role in the breakdown into small scales in the range of parameters investigated. © Cambridge University Press 2011.

Description: The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the 'four-fifths law' of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as (ϵd)1/3, where ϵ is the mean dissipation rate and d the mean distance between the shocks, and that the pth-order displacement and velocity structure functions scale as (ϵd)p/3r/d, where r is the separation. Then we carry out a series of forced simulations with resolutions up to 76802, varying the Froude number, Ff = (ϵLf)1/3/c, where Lf is the forcing length scale and c is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as E ∼ Lfc, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson-Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as d ∼ F1/2fLf. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, Ff → 0, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given Ff, no matter how small, shocks will prevail if the Reynolds number is sufficiently large. © 2019 Cambridge University Press.

Description: A new formulation of the spectral energy budget of kinetic and available potential energies of the atmosphere is derived, with spherical harmonics as base functions. Compared to previous formulations, there are three main improvements: (i) the topography is taken into account, (ii) the exact three-dimensional advection terms are considered, and (iii) the vertical flux is separated from the energy transfer between different spherical harmonics. Using this formulation, results from two different high-resolution GCMs are analyzed: the Atmospheric GCM for the Earth Simulator (AFES) T639L24 and the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecast System (IFS) T1279L91. The spectral fluxes show that the AFES, which reproduces quite realistic horizontal spectra with a k−5/3 inertial range at the mesoscales, simulates a strong downscale energy cascade. In contrast, neither the k−5/3 vertically integrated spectra nor the downscale energy cascade are produced by the ECMWF IFS.