Revisiting Bolgiano–Obukhov scaling for moderately stably stratified turbulence (original) (raw)
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A randomly stirred model for Bolgiano–Obukhov scaling in turbulence in a stably stratified fluid
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2022
A randomly stirred model, akin to the one used by DeDominicis and Martin for homogeneous isotropic turbulence, is introduced to study Bolgiano–Obukhov scaling in fully developed turbulence in a stably stratified fluid. The energy spectrum E ( k ), where k is a wavevector in the inertial range, is expected to show the Bolgiano–Obukhov scaling at a large Richardson number Ri (a measure of the stratification). We find that the energy spectrum is anisotropic. Averaging over the directions of the wavevector, we find E ( k ) = K 0 ε θ 2 / 5 k − 11 / 5 , where ε θ is the constant energy transfer rate across wavenumbers with very little contribution coming from the kinetic energy flux. The constant K 0 is estimated to be of O(0.1) as opposed to the Kolmogorov constant, which is O(1). Further for a pure Bolgiano–Obukhov scaling, the model requires that the large distance ‘stirring’ effects dominate in the heat diffusion and be small in the velocity dynamics. These could be reasons why the Bo...
Energy spectrum of buoyancy-driven turbulence
Physical Review E, 2014
Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux Πu, we demonstrate that for stably stratified flows, the kinetic energy spectrum Eu(k) ∼ k −11/5 , the entropy spectrum E θ (k) ∼ k −7/5 , and Πu(k) ∼ k −4/5 , consistent with the Bolgiano-Obukhov scaling. This scaling arises due to the conversion of kinetic energy to the potential energy by buoyancy. For weaker buoyancy, this conversion is weak, hence Eu(k) follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh Bénard convection, we show that the energy supply rate by buoyancy is positive, which leads to an increasing Πu(k) with k, thus ruling out Bolgiano-Obukhov scaling for the convective turbulence. Our numerical results show that convective turbulence for unit Prandt number exhibits a constant Πu(k) and Eu(k) ∼ k −5/3 for a narrow band of wavenumbers.
Physics of Fluids, 2015
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 = 5.4×10 4 and F r = 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 RB = ReF r 2 = 32, a value sufficient for some isotropy to be recovered in the small scales beyond the Ozmidov scale, but still moderate enough that the intermediate scales where waves are prevalent are well resolved. We concentrate on the largescale dynamics, for which we find a spectrum compatible with the Bolgiano-Obukhov scaling, and confirm that the Froude number based on a typical vertical length scale is of order unity, with strong gradients in the vertical. Two characteristic scales emerge from this computation, and are identified from sharp variations in the spectral distribution of either total energy or helicity. A spectral break is also observed at a scale at which the partition of energy between the kinetic and potential modes changes abruptly, and beyond which a Kolmogorov-like spectrum recovers. Large slanted layers are ubiquitous in the flow in the velocity and temperature fields, with local overturning events indicated by small Richardson numbers, and a small large-scale enhancement of energy directly attributable to the effect of rotation is also observed.
Stably stratified turbulence in the presence of large-scale forcing
Physical Review E, 2015
We perform two high resolution direct numerical simulations of stratified turbulence for Reynolds number equal to Re ≈ 25000 and Froude number respectively of F r ≈ 0.1 and F r ≈ 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, LB, and the Ozmidov scale, ℓoz. 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 which isotropy is recovered. The values of LB and ℓoz depend on the Froude number and their absolute and relative size affect the repartition of energy among Fourier modes in non easily predictable ways. By contrasting the behavior of the two simulated flows we identify some surprising similarities: after an initial transient the two flows evolve towards comparable values of the kinetic and potential enstrophy, and energy dissipation rate. This is the result of the Reynolds number being large enough in both flows for the Ozmidov scale to be resolved. When properly dimensionalized, the energy dissipation rate is compatible with atmospheric observations. Further similarities emerge at large scales: the same ratio between potential and total energy (≈ 0.1) is spontaneously selected by the flows, and slow modes grow monotonically in both regimes causing a slow increase of the total energy in time. The axisymmetric total energy spectrum shows a wide variety of spectral slopes as a function of the angle between the imposed stratification and the wave vector. One-dimensional energy spectra computed in the direction parallel to gravity are flat from the forcing up to buoyancy scale. At intermediate scales a ∼ k −3 parallel spectrum develops for the F r ≈ 0.03 run, whereas for weaker stratification, the saturation spectrum does not have enough scales to develop and instead one observes a power law compatible with Kolmogorov scaling. Finally, the spectrum of helicity is flat until LB, as observed in the nocturnal planetary boundary layer.
Energy spectra of stably stratified turbulence
Journal of Fluid Mechanics, 2012
We investigate homogeneous incompressible turbulence subjected to a range of degrees of stratification. Our basic method is pseudospectral direct numerical simulations at a resolution of 10243102{4}^{3} 10243. Such resolution is sufficient to reveal inertial power-law ranges for suitably comprised horizontal and vertical spectra, which are designated as the wave and vortex mode (the Craya–Herring representation). We study mainly turbulence that is produced from randomly large-scale forcing via an Ornstein–Uhlenbeck process applied isotropically to the horizontal velocity field. In general, both the wave and vortex spectra are consistent with a Kolmogorov-like kensuremath−5/3{k}^{\ensuremath{-} 5/ 3} kensuremath−5/3 range at sufficiently large kkk. At large scales, and for sufficiently strong stratification, the wave spectrum is a steeper kperpensuremath−2{ k}_{\perp }^{\ensuremath{-} 2} kperpensuremath−2, while that for the vortex component is consistent with kperpensuremath−3{ k}_{\perp }^{\ensuremath{-} 3} kperpensuremath−3. Here kperp{k}_{\perp } kperp is the horizontally gathered wave...
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012
The buoyancy subrange of stably stratified turbulence is defined as an intermediate range of scales larger than those in the inertial subrange. This subrange encompasses the crossover from internal gravity waves (IGWs) to small-scale turbulence. The energy exchange between the waves and small-scale turbulence is communicated across this subrange. At the same time, it features progressive anisotropization of flow characteristics on increasing spatial scales. Despite many observational and computational studies of the buoyancy subrange, its theoretical understanding has been lagging. This article presents an investigation of the buoyancy subrange using the quasi-normal scale elimination (QNSE) theory of turbulence. This spectral theory uses a recursive procedure of small-scale modes elimination based upon a quasi-normal mapping of the velocity and temperature fields using the Langevin equations. In the limit of weak stable stratification, the theory becomes completely analytical and y...
Phenomenology of buoyancy-driven turbulence: recent results
2016
In this paper, we review the recent developments in the field of buoyancy-driven turbulence. Scaling and numerical arguments show that the stably-stratified turbulence with moderate stratification has kinetic energy spectrum E_u(k) ∼ k^-11/5 and the kinetic energy flux Π_u(k) ∼ k^-4/5, which is called Bolgiano-Obukhov scaling. The energy flux for the Rayleigh-Bénard convection (RBC) however is approximately constant in the inertial range that results in Kolmorogorv's spectrum (E_u(k) ∼ k^-5/3) for the kinetic energy. The phenomenology of RBC should apply to other flows where the buoyancy feeds the kinetic energy, e.g. bubbly turbulence and fully-developed Rayleigh Taylor instability. This paper also covers several models that predict the Reynolds and Nusselt numbers of RBC. Recent works show that the viscous dissipation rate of RBC scales as ∼Ra^1.3, where Ra is the Rayleigh number.
Buoyancy- to Inertial-Range Transition in Forced Stratified Turbulence
The buoyancy range, which represents a transition from large-scale wave-dominated motions to small-scale turbulence in the oceans and the atmosphere, is investigated through large-eddy simulations. The model presented here uses a continual forcing based on large-scale standing internal waves and has a spectral truncation in the isotropic inertial range. Evidence is presented for a break in the energy spectra from the anisotropic k −3 buoyancy range to the small-scale k −5/3 isotropic inertial range. Density structures that form during wave breaking and periods of high strain rate are analysed. Elongated vertical structures produced during periods of strong straining motion are found to collapse in the subsequent vertically compressional phase of the strain resulting in a zone or patch of mixed fluid. 1. Introduction Much of the large-scale variability in the atmosphere and oceans can be described as internal wave activity, while isotropic turbulence dominates at small scales. Between these extremes, the dynamics is a competition between waves and turbulence. The nature of this intermediate range, called the buoyancy or the saturation range, is highly controversial. A direct numerical simulation which could faithfully span the full range of the scales involved would be a great benefit; however, such simulations remain impractical because of the large range of scales that would need to be represented. On the other hand, as we shall argue below, techniques of large-eddy simulation (LES) should afford us the possibility of at least simulating flow in the buoyancy range and capturing the transition to the inertial range. The goal of this paper is to present some results that might confirm this hope and also give us some insight into the kinds of structures one should be able to observe in the density field of the buoyancy range. To be concrete about spatial scales, we will concentrate on the oceanic application, although much of the basic ideas that follow should hold for the atmospheric problem as well. The spectra of density and velocity fluctuations in the ocean have several distinguishable ranges. As a guide to these ranges, we follow the description in Holloway (1981) and use a similar schematic diagram (figure 1). Here φ represents either the spectrum of the vertical shear or the vertical gradient of temperature as a function of the vertical wavenumber k z. The axis of the vertical wavenumber is
Small-Scale Structure of Strongly Stratified Turbulence
Journal of Physical Oceanography, 2005
The small-scale structure of turbulence subjected to strong stratification is analyzed with rapid distortion theory to evaluate the performance of formulas for predicting dissipation of turbulent kinetic energy and dissipation of scalar variance. The approach is restricted to weak turbulence in strong stratification, like that in the thermocline or the abyssal ocean. Flows with and without mean shear are considered. For unsheared turbulence, the small scales are axisymmetric about the vertical axis, as others have previously assumed. The calculations here complement and extend previous work because they can be used to compute errors in dissipation estimates, develop simpler formulas, and examine the effects of shear and other parameters. For example, effects of the initial conditions can be significant. For sheared turbulence, the small-scale velocity and buoyancy fields are neither isotropic nor axisymmetric about the vertical axis. Although dissipation formulas based on isotropy w...