Utilizing an eigenfunction decomposition, we study the growth and spectra of energy in the vortical and wave modes of a 3D rotating stratified fluid as a function of $ε= f/N$. Working in regimes characterized by moderate Burger numbers, i.e. $Bu = 1/ε^2 < 1$ or $Bu \ge 1$, our results indicate profound change in the character of vortical and wave mode interactions with respect to $Bu = 1$. As with the reference state of $ε=1$, for $ε< 1$ the wave mode energy saturates quite quickly and the ensuing forward cascade continues to act as an efficient means of dissipating ageostrophic energy. Further, these saturated spectra steepen as $ε$ decreases: we see a shift from $k^{-1}$ to $k^{-5/3}$ scaling for $k_f < k < k_d$ (where $k_f$ and $k_d$ are the forcing and dissipation scales, respectively). On the other hand, when $ε> 1$ the wave mode energy never saturates and comes to dominate the total energy in the system. In fact, in a sense the wave modes behave in an asymmetric manner about $ε= 1$. With regard to the vortical modes, for $ε\le 1$, the signatures of 3D quasigeostrophy are clearly evident. Specifically, we see a $k^{-3}$ scaling for $k_f < k < k_d$ and, in accord with an inverse transfer of energy, the vortical mode energy never saturates but rather increases for all $k < k_f$. In contrast, for $ε> 1$ and increasing, the vortical modes contain a progressively smaller fraction of the total energy indicating that the 3D quasigeostrophic subsystem plays an energetically smaller role in the overall dynamics.
We study the morphology of Eulerian vortical structures and their interaction with density interfaces in increasingly turbulent stably-stratified shear layers. We analyse the three-dimensional, simultaneous velocity and density fields obtained in the stratified inclined duct laboratory experiment. We track, across 15 datasets, the evolution of coherent structures from pre-turbulent Holmboe waves, through intermittent turbulence, to full turbulence and mixing. We use the Rortex--Shear decomposition of the vorticity field into a pure rotational part (the rortex vector), and a non-rotational part…
We investigate the dynamics of inertia-gravity wave modes in 3D rotating stratified fluids. We start by deriving a reduced PDE, the GGG model, consisting of only wave-mode interactions. In principle, comparing this model to the full rotating Boussinesq system allows us to gauge the importance of wave-vortical-wave vs. wave-wave-wave interactions in determining the transfer and distribution of wave-mode energy. As in many atmosphere-ocean phenomena we work in a skewed aspect ratio domain (H/L) with Fr = Ro < 1 such that Bu = 1. Our focus is on the equilibration of wave-mode energy and its spect…
Systematic Evaluation of Stencil Configuration, Forcing Scheme, and Resolution Effects in the Stratified Taylor--Green Vortex: A Lattice Boltzmann Study
School of Atmospheric Sciences, Sun Yat-sen University（中山大学大气科学学院）
We construct an explicit steady stratified purely azimuthal flow for the governing equations of geophysical fluid dynamics. These equations are considered in a setting that applies to the Antarctic Circumpolar Current, accounting for eddy viscosity and forcing terms.
A numerical study of decaying stably-stratified flows is performed. Relatively high stratification and moderate Reynolds numbers are considered, and a particular emphasis is placed on the role of helicity (velocity-vorticity correlations). The problem is tackled by integrating the Boussinesq equations in a periodic cubical domain using different initial conditions: a non-helical Taylor-Green (TG) flow, a fully helical Beltrami (ABC) flow, and random flows with a tunable helicity. We show that for stratified ABC flows helicity undergoes a substantially slower decay than for unstratified ABC flo…
Stratified flows forced by internal waves similar to those obtained in the Coriolis platform (LEGI, Grenoble, France) \cite{Savaro2020} are studied by pseudospectral triply-periodic simulations. The experimental forcing mechanism consisting in large oscillating vertical panels is mimicked by a penalization method. The analysis of temporal and spatiotemporal spectra reveals that the flow for the strongest forcing in the experiments is composed of two superposed large and quasi-steady horizontal vortices, of internal waves in box modes and of much weaker waves outside the modes. Spatial spectra …
In this paper we characterize the scaling of energy spectra, and the interscale transfer of energy and enstrophy, for strongly, moderately and weakly stably stratified two-dimensional (2D) turbulence under large-scale random forcing. In the strongly stratified case, a large-scale vertically sheared horizontal flow (VSHF) co-exists with small scale turbulence. The VSHF consists of internal gravity waves and the turbulent flow has a kinetic energy (KE) spectrum that follows an approximate $k^{-3}$ scaling with zero KE flux and a robust positive enstrophy flux. The spectrum of the turbulent poten…
In this paper, we investigate the enstrophy dynamics in relation to objective Eulerian coherent structures (OECSs) and their impact on the enstrophy and scalar transport near the turbulent/non-turbulent interface (TNTI) in flows with and without stable stratification. We confirm that vortex-stretching produces enstrophy inside the boundaries of the OECSs, while viscous diffusion transfers the enstrophy across the boundaries of the structures. Although often overlooked in the literature, viscous dissipation of enstrophy within the boundaries of vortical structures is significant. Conversely, fo…
We use resolvent analysis to determine an unsteady active control setup to attenuate pressure fluctuations in turbulent supersonic flow over a rectangular cavity with a length-to-depth ratio of 6 at a Mach number of 1.4 and a Reynolds number based on cavity depth of 10,000. Large-eddy simulations (LES) and dynamic modal decomposition (DMD) of the supersonic cavity flow reveal the dominance of two-dimensional Rossiter modes II and IV. These predominantly two-dimensional vortical structures generate high-amplitude unsteadiness over the cavity through trailing-edge impingement and create oblique …
The interplay between waves and eddies in stably stratified rotating flows is investigated by means of world-class direct numerical simulations using up to $3072^3$ grid points. Strikingly, we find that the shift from vortex to wave dominated dynamics occurs at a wavenumber $k_R$ which does not depend on Reynolds number, suggesting that partition of energy between wave and vortical modes is not sensitive to the development of turbulence at the smaller scales. We also show that $k_R$ is comparable to the wavenumber at which exchanges between kinetic and potential modes stabilize at close to equ…
We perform two high resolution direct numerical simulations of stratified turbulence for Reynolds number equal to Re~25000 and Froude number respectively of Fr~0.1 and Fr~0.03. The flows are forced at large scale and discretized on an isotropic grid of 2048^3 points. Stratification makes the flow anisotropic and introduces two extra characteristic scales with respect to homogeneous isotropic turbulence: the buoyancy scale, L_B, and the Ozmidov scale, l_{oz}. The former is related to the number of layers that the flow develops in the direction of gravity, the latter is regarded as the scale at …
A key issue in stratified turbulence theory concerns the nature of the link between D(APE), the dissipation rate of available potential energy APE, and W_{r,turbulent}, the turbulent rate of change of background gravitational potential energy GPE_r, which are both controlled by molecular diffusion. For Boussinesq fluids with a linear equation of state, this link is simply W_{r,turbulent}=D(APE), widely interpreted as implying that GPE_r increases at the expense of APE, in contrast with the laminar case where GPE_r increases at the expense of internal energy (IE). This idea is revisited here by…
It is necessary to introduce an external forcing to induce turbulence in a stably stratified fluid. The Heisenberg eddy viscosity technique should in this case suffice to calculate a space-time averaged quantity like the global anisotropy parameter as a function of the Richardson number. We find analytically that the anisotropy increases linearly with the Richardson number, with a small quadratic correction. A numerical simulation of the complete equations shows the linear behaviour.
This paper is the first part of a two-fold study of mixing, i.e. the formation of layers and upwelling of buoyancy, in axially stratified Taylor--Couette flow, with fixed outer cylinder. Using linear analysis and direct numerical simulation, we show the critical role played by non-axisymmetric instability modes, despite the fact that the flow is centrifugally unstable in the sense of Rayleigh's criterion. Interactions between helical modes of opposite handedness leads to the formation of nonlinear coherent structures: (mixed)-ribbons and (mixed)-cross-spirals. These give birth to complex densi…
We address the dynamical and statistical description of stably stratified turbulent boundary layers with the important example of the atmospheric boundary layer with a stable temperature stratification in mind. Traditional approaches to this problem, based on the profiles of mean quantities, velocity second-order correlations, and dimensional estimates of the turbulent thermal flux run into a well known difficulty, predicting the suppression of turbulence at a small critical value of the Richardson number, in contradiction with observations. Phenomenological attempts to overcome this problem s…
An extended immersed boundary method utilizing a semi-implicit direct forcing approach for the simulation of confined incompressible viscous thermal flow problems is presented. The method utilizes a Schur complement approach to enforce the kinematic constraints of no-slip and the corresponding thermal boundary conditions for immersed surfaces. The developed methodology can be straightforwardly adapted to any existing incompressible time marching solver based on a segregated pressure-velocity coupling. The method accurately meets the thermal and the no-slip boundary conditions on the surfaces o…
We study the experimental properties of exchange flows in a stratified inclined duct (SID), which are simultaneously turbulent, strongly stratified by a mean vertical density gradient, driven by a mean vertical shear, and continuously forced by gravity. We focus on the core shear layer away from the duct walls, where these flows are excellent experimentally-realisable approximations of canonical hyperbolic-tangent stratified shear layers, whose forcing allows mean and turbulent properties to reach quasi steady states. We analyse state-of-the-art data sets of the time-resolved density and veloc…
We study the effect of different forcing functions and of the local gradient Richardson number Rig on the vertical mixing of Lagrangian tracers in stably stratified turbulence under the Boussinesq approximation, and present a wave and continuous-time random walk model for single- and two- particle vertical dispersion. The model consists of a random superposition of linear waves with their amplitude based on the observed Lagrangian spectrum of vertical velocity, and a random walk process to capture overturning that depends on the statistics of Rig among other Eulerian quantities. The model is i…
The present paper reports on our effort to characterize vortical interactions in complex fluid flows through the use of network analysis. In particular, we examine the vortex interactions in two-dimensional decaying isotropic turbulence and find that the vortical interaction network can be characterized by a weighted scale-free network. It is found that the turbulent flow network retains its scale-free behavior until the characteristic value of circulation reaches a critical value. Furthermore, we show that the two-dimensional turbulence network is resilient against random perturbations but ca…
We present a numerical investigation of three-dimensional, short-wavelength linear instabilities in Kelvin-Helmholtz (KH) vortices in homogeneous and stratified environments. The base flow, generated using two-dimensional numerical simulations, is characterized by the Reynolds number and the Richardson number defined based on the initial one-dimensional velocity and buoyancy profiles. The local stability equations are then solved on closed streamlines in the vortical base flow, which is assumed quasi-steady. For the unstratified case, the elliptic instability at the vortex core dominates at ea…
A fluid dynamics video of the rotating, weakly stratified Boussinesq equations is presented that illustrates the spontaneous formation of columnar vortices in the presence of stochastic, white noise forcing.
We investigate the stability of katabatic slope flows over an infinitely wide and uniformly cooled planar surface subject to an additional forcing due to a uniform downslope wind field aloft. We adopt an extension of Prandtl's original model for slope flows (Lykosov & Gutman 1972) to derive the base flow, which constitutes an interesting basic state in stability analysis because it cannot be reduced to a single universal form independent of external parameters. We apply a linear modal analysis to this basic state to demonstrate that for a fixed Prandtl number and slope angle, two independent d…
We consider turbulence in a stratified 'Kolmogorov' flow, driven by horizontal shear in the form of sinusoidal body forcing in the presence of an imposed background linear stable stratification in the third direction. This flow configuration allows the controlled investigation of the formation of coherent structures, which here organise the flow into horizontal layers by inclining the background shear as the strength of the stratifica- tion is increased. By numerically converging exact steady states from direct numerical simulations of chaotic flow, we show, for the first time, a robust connec…
Sidewall effects on the onset of interfacial Holmboe waves in stratified exchange flows at high Schmidt number
Univ Toulouse, Toulouse INP, CNRS, IMFT（图卢兹大学，图卢兹国立理工学院，法国国家科学研究中心，流体力学与传热研究所）；Eindhoven University of Technology（埃因霍温理工大学）；Imperial College London（伦敦帝国学院）
Identification and extraction of vortical structures and of waves in a disorganised flow is a mayor challenge in the study of turbulence. We present a study of the spatio-temporal behavior of turbulent flows in the presence of different restitutive forces. We show how to compute and analyse the spatio-temporal spectrum from data stemming from numerical simulations and from laboratory experiments. Four cases are considered: homogeneous and isotropic turbulence, rotating turbulence, stratified turbulence, and water wave turbulence. For homogeneous and isotropic turbulence, the spectrum allows id…
Stationary longitudinal vortical rolls emerge in katabatic and anabatic Prandtl slope flows due to the dominance of the normal component of the buoyancy force over flow shear. Here, we further identify self pairing of these longitudinal rolls as a unique flow structure. The topology of the counter-rotating vortex pair bears a striking resemblance to speaker-wires and their interaction with each other is a precursor to further destabilization and breakdown of the flow field into smaller structures. On its own, a speaker-wire vortex retains its unique topology without any vortex reconnection or …
Recent developments of the weak turbulence theory applied to internal waves exhibit a power-law solution of the kinetic energy equation close to the oceanic Garrett \& Munk spectrum, confirming weakly nonlinear wave interactions as a likely explanation of the observed oceanic spectra. However, finite-size effects can hinder wave interactions in bounded domains, and observations often differ from theoretical predictions. This article studies the dynamical regimes experimentally developing in a stratified fluid forced by internal gravity waves in a pentagonal domain. We find that by changing the…
We present results from direct numerical simulations of the Boussinesq equations in the presence of rotation and/or stratification, both in the vertical direction. The runs are forced isotropically and randomly at small scales and have spatial resolutions of up to $1024^3$ grid points and Reynolds numbers of $\approx 1000$. We first show that solutions with negative energy flux and inverse cascades develop in rotating turbulence, whether or not stratification is present. However, the purely stratified case is characterized instead by an early-time, highly anisotropic transfer to large scales w…
The large-scale structures in the ocean and the atmosphere are in geostrophic balance, and a conduit must be found to channel the energy to the small scales where it can be dissipated. In turbulence this takes the form of an energy cascade, whereas one possible mechanism in a balanced flow at large scales is through the formation of fronts, a common occurrence in geophysical dynamics. We show in this paper that an iconic configuration in laboratory and numerical experiments for the study of turbulence, that of the Taylor-Green or von Kármán swirling flow, can be suitably adapted to the case of…
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…
We present an investigation of rapidly rotating (small Rossby number $Ro\ll 1$) and stratified turbulence where the stratification strength is varied from weak (large Froude number $Fr\gg1$) to strong ($Fr\ll1$). The investigation is set in the context of a reduced model derived from the Boussinesq equations that efficiently retains anisotropic inertia-gravity waves with order-one frequencies and highlights a regime of wave-eddy interactions. Numerical simulations of the reduced model are performed where energy is injected by a stochastic forcing of vertical velocity, which forces wave modes o…
We study experimentally and theoretically transition phenomena caused by the external forcing from Rayleigh-Benard convection with the large-scale circulation (LSC) to the limiting regime of unstably stratified turbulent flow without LSC whereby the temperature field behaves like a passive scalar. In the experiments we use the Rayleigh-Bénard apparatus with an additional source of turbulence produced by two oscillating grids located nearby the side walls of the chamber. When the frequency of the grid oscillations is larger than 2 Hz, the large-scale circulation (LSC) in turbulent convection is…
We derive and test a new heuristic theory for third-order structure functions that resolve the forcing scale in the scenario of simultaneous spectral energy transfer to both small and large scales, which can occur naturally in rotating stratified turbulence or magnetohydrodynamical~(MHD) turbulence, for example. The theory has three parameters, namely the upscale/downscale energy transfer rates and the forcing scale, and it includes the classic inertial range theories as local limits. When applied to measured data, our global-in-scale theory can deduce the energy transfer rates using the full …
We employ a coarse-graining approach to analyze nonlinear cascades in Boussinesq flows using high-resolution simulation data. We derive budgets which resolve the evolution of energy and potential enstrophy simultaneously in space and in scale. We then use numerical simulations of Boussinesq flows, with forcing in the large-scales, and fixed rotation and stable stratification along the vertical axis, to study the inter-scale flux of energy and potential enstrophy in three different regimes of stratification and rotation: (i) strong rotation and moderate stratification, (ii) moderate rotation an…
The paper analyzes the phenomenon of turbulent thermal diffusion in the Earth atmosphere, its relation to the turbulent diffusion and its potential impact on aerosol distribution. This phenomenon was predicted theoretically more than 10 years ago and detected recently in the laboratory experiments. This effect causes a non-diffusive flux of aerosols in the direction of the heat flux and results in formation of long-living aerosol layers in the vicinity of temperature inversions. We demonstrated that the theory of turbulent thermal diffusion explains the GOMOS aerosol observations near the trop…
A formulation is developed to assimilate ocean-wave data into the Numerical Flow Analysis (NFA) code. NFA is a Cartesian-based implicit Large-Eddy Simulation (LES) code with Volume of Fluid (VOF) interface capturing. The sequential assimilation of data into NFA permits detailed analysis of ocean-wave physics with higher bandwidths than is possible using either other formulations, such as High-Order Spectral (HOS) methods, or field measurements. A framework is provided for assimilating the wavy and vortical portions of the flow. Nudging is used to assimilate wave data at low wavenumbers, and th…
We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_B^2$ imposed in the third, vertical direction in a fluid with kinematic viscosity $ν$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this sy…
We present a laboratory study on the instability of internal wave attractors in a trapezoidal fluid domain filled with uniformly stratified fluid. Energy is injected into the system via standing-wave-type motion of a vertical wall. Attractors are found to be destroyed by parametric subharmonic instability (PSI) via a triadic resonance which is shown to provide a very efficient energy pathway from long to short length scales. This study provides an explanation why attractors may be difficult or impossible to observe in natural systems subject to large amplitude forcing.
Mixing and sharpening at the interface of a two-layer fluid forced by random jets
École Normale Supérieure PSL（巴黎高等师范学院）；Institute of Science and Technology Austria（奥地利科学技术研究所）；École Normale Supérieure de Lyon（里昂高等师范学院）；Université de Lorraine（洛林大学）
We report results on rotating stratified turbulence in the absence of forcing, with large-scale isotropic initial conditions, using direct numerical simulations computed on grids of up to 4096^3 points. The Reynolds and Froude numbers are respectively equal to Re=5.4 x 10^4 and Fr=0.0242. The ratio of the Brunt-Väisälä to the inertial wave frequency, N/f, is taken to be equal to 4.95, a choice appropriate to model the dynamics of the southern abyssal ocean at mid latitudes. This gives a global buoyancy Reynolds number R_B=ReFr^2=32, a value sufficient for some isotropy to be recovered in the s…
Low order Carleman simulation of steady state flows at mild Reynolds numbers
Fondazione Istituto Italiano di Tecnologia（意大利理工大学基金会）；Center for Life Nano-Neuroscience at la Sapienza（罗马一大生命纳米神经科学中心）；Istituto per le Applicazioni del Calcolo “Mauro Picone（毛罗·皮科内计算应用研究所）；Italian National Research Council（意大利国家研究委员会）
Horizontal currents in the Bay of Bengal were measured on eight cruises covering a total of 8600 $km$ using a 300 $kHz$ Acoustic Doppler Current Profiler (ADCP). The cruises are distributed over multiple seasons and regions of the Bay. Horizontal wavenumber spectra of these currents over depths of 12--54 m and wavelengths from 2--400 km were decomposed into vortical and divergent components assuming isotropy. An average of across and along track spectra over all cruises shows that the spectral slope of horizontal kinetic energy for wavelengths of 10--80 $km$ scales with an exponent of $-1.7 \p…
