The energy cascade in a strongly stratified fluid

Lindborg, Erik

doi:10.1017/s0022112005008128

Cited by 411 publications

(610 citation statements)

References 43 publications

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“…Local shear is induced by random low-energy perturbations to the horizontal velocity components at small vertical wave numbers. The prescribed spectrum for the forcing is obtained by repeating the simulations of Lindborg (2006) using his forcing method and so the current flows are very similar in structure to those in that paper.…”

Section: Direct Numerical Simulationsmentioning

confidence: 82%

“…Billant & Chomaz 2001;Lindborg 2006;Brethouwer et al 2007;Riley & Lindborg 2008). Although this asymptotic regime is sometimes referred to as '(strongly) stratified turbulence' in the fluid dynamical literature, this nomenclature can lead to confusion as 'stratified turbulence' is typically used in a much broader sense in the geophysical literature.…”

Section: Introductionmentioning

confidence: 99%

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Robust identification of dynamically distinct regions in stratified turbulence

et al. 2016

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We present a new robust method for identifying three dynamically distinct regions in a stratified turbulent flow, which we characterise as quiescent flow, intermittent layers, and turbulent patches. The method uses the cumulative filtered distribution function of the local density gradient to identify each region. We apply it to data from direct numerical simulations of homogeneous stratified turbulence, with unity Prandtl number, resolved on up to 8192×8192×4096 grid points. In addition to classifying regions consistently with contour plots of potential enstrophy, our method identifies quiescent regions as regions where /νN 2 ∼ O(1), layers as regions where /νN 2 ∼ O(10), and patches as regions where /νN 2 ∼ O(100). Here is the dissipation rate of turbulence kinetic energy, ν is the kinematic viscosity, and N is the (overall) buoyancy frequency. By far the highest local dissipation and mixing rates, and the majority of dissipation and mixing, occur in patch regions, even when patch regions occupy only 5% of the flow volume. We conjecture that treating stratified turbulence as an instantaneous assemblage of these different regions in varying proportions may explain some of the apparently highly scattered flow dynamics and statistics previously reported in the literature.

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Section: Direct Numerical Simulationsmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Robust identification of dynamically distinct regions in stratified turbulence

et al. 2016

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“…Explanations in the literature for the mesoscale spectrum fall into three general categories as follows: (i) an inverse cascade of small-scale energy, produced perhaps by convection (4 -6); (ii) production of gravity waves by divergent f lows (7-11); or (iii) a direct cascade of energy from the large 0scales (12)(13)(14)(15). More recent observations and analysis present new facts that must be accommodated by theory.…”

Section: Previous Explanationsmentioning

confidence: 99%

A theory for the atmospheric energy spectrum: Depth-limited temperature anomalies at the tropopause

Tulloch

Smith

2006

Proc. Natl. Acad. Sci. U.S.A.

111

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The horizontal spectra of atmospheric wind and temperature at the tropopause have a steep ؊3 slope at synoptic scales, but transition to ؊5͞3 at wavelengths of the order of 500 -1,000 km [Nastrom, G. D. & Gage, K. S. (1985) J. Atmos. Sci. 42, 950 -960]. Here we demonstrate that a model that assumes zero potential vorticity and constant stratification N over a finite-depth H in the troposphere exhibits the same type of spectra. In this model, temperature perturbations generated at the planetary scale excite a direct cascade of energy with a slope of ؊3 at large scales, ؊5͞3 at small scales, and a transition near horizontal wavenumber k t ‫؍‬ f͞NH, where f is the Coriolis parameter. Ballpark atmospheric estimates for N, f, and H give a transition wavenumber near that observed, and numerical simulations of the previously undescribed model verify the expected behavior. Despite its simplicity, the model is consistent with a number of perplexing features in the observations and demonstrates that a complete theory for mesoscale dynamics must take temperature advection at boundaries into account.geophysical turbulence ͉ meteorology ͉ atmospheric dynamics I n the 1970s, the National Aeronautics and Space Administration (NASA) instrumented commercial Boeing 747 airliners to collect atmospheric data during their regular flights (1) in an endeavor called the Global Atmospheric Sampling Program (GASP). The resulting data set consists of thousands of flight tracks, a few hundred of which are Ͼ10,000 km long, collected over a 4-year period. Most flights occurred in the midlatitudes and tropics but span the full range of seasons. Because airliners travel at altitudes between 9 and 14 km, the data largely reflect the upper troposphere and lower stratosphere, near the tropopause. Atmospheric wavenumber spectra of horizontal wind and temperature computed from the GASP data set by Nastrom and Gage (ref. 1; hereafter NG85) show a distinct transition from a steep spectral slope of Ϫ3 at synoptic scales (Ϸ1,000-3,000 km) to a shallower slope of Ϫ5͞3 at mesoscales (Ϸ10-500 km), with a fairly distinct transition centered at a horizontal wavelength of Ϸ600 km. Understanding the source and structure of this spectrum has posed a puzzle in atmospheric science for the past 20 years.The spectrum is intriguing because it agrees so well at large scales with Charney's (3) theory of geostrophic turbulence but deviates from that prediction where it shallows. Moreover, the fact that the small-scale slope is Ϫ5͞3 invites multiple explanations, because that is the theoretical slope both for the forward cascade of energy in isotropic, three-dimensional (3D) turbulence, and for the inverse cascade of two-dimensional (2D) turbulence, as well as other systems. At the large scale, Charney argued (3), rotation and stratification conspire to make the atmosphere quasi-2D. Stirring by baroclinic instability (or any planetary mechanism) will induce a forward cascade of potential enstrophy, reflected in a Ϫ3 kinetic energy spectrum below the stirring scale. Mor...

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“…Interestingly, such coherent motions may mask the inverse cascade inducing a positive energy-flux [19], which could explain observations [12]. Other proposed mechanisms (not discussed here) consider 3d or quasi-2d scenarios accounting for stratification and other effects [21][22][23].In this Letter we focus on 2d turbulence forced at two, well separated, scales with the aim of understanding the interplay of oppositely directed cascades in the same range of scales. We thus consider the 2d incompressible Navier-Stokes equation, which for the vorticity ω reads…”

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confidence: 96%

Nonlinear Superposition of Direct and Inverse Cascades in Two-Dimensional Turbulence Forced at Large and Small Scales

2011

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We inquire about the properties of 2d Navier-Stokes turbulence simultaneously forced at small and large scales. The background motivation comes by observational results on atmospheric turbulence. We show that the velocity field is amenable to the sum of two auxiliary velocity fields forced at large and small scale and exhibiting a direct-enstrophy and an inverse-energy cascade, respectively. Remarkably, the two auxiliary fields reconcile universal properties of fluxes with positive statistical correlation in the inertial range.PACS numbers: 47.27.ETurbulence represents a tantalizing nonequilibrium system characterized by cascade processes which, as typical in statistical physics, strongly depends on space dimensionality. In d= 3, kinetic energy, injected at large scales, goes toward smaller ones with positive and constant flux (≈ energy-injection rate ǫ), until is dissipated by molecular diffusion [1]. Between the injection and dissipative scales, the energy spectrum behaves as a powerlaw E(k) ≈ ǫ 2/3 k −5/3 . In d = 2, ideal (inviscid and unforced) fluids preserve both energy ≺ v 2 ≻/2 and square vorticity (enstrophy) ≺ ω 2 ≻ /2 (ω=∇×v). On this basis, Kraichnan [2] predicted that sustaining the flow at a single scale ℓ f (∼ k −1 f ), with energy (enstrophy) injection rate ǫ (η = ǫk 2 f ), generates a double cascade of enstrophy downscale (< ℓ f ) and of energy upscale (> ℓ f ). He also predicted two power laws for the energy spectrum: E(k) ≈ η 2/3 k −3 (but for log-corrections [3]) in the direct enstrophy cascade range; E(k) ≈ ǫ 2/3 k −5/3 for the inverse energy cascade. The direct cascade, with a positive enstrophy flux (≈ η), ends at the dissipative scale. Whilst, in an unbounded domain, the inverse cascade proceeds undisturbed, with a negative energy flux (≈−ǫ), unless large-scale friction stops it at a scale≫ ℓ f [4]. For a recent numerical study of the dual cascade see [5].In 3d-layers, as the atmosphere, both 3d and 2d-phenomenology can be relevant depending on the aspect ratio, the injection and observation scales [6][7][8]. Aircraft measurements [9, 10] of atmospheric-winds revealed that horizontal energy spectra at the troposphere end (at ≈ 10Km altitude) display two power-laws: E(k) ∝ k −5/3 at wave-numbers in the mesoscales (≈10−500Km); E(k) ∝ k −3 at synoptic scales (≈500 − 3, 000Km). Though, 2d phenomenology should dominate at scales larger than the troposphere thickness [11], measured spectra display the steeper and shallower power-laws in reverse order with respect to Kraichnan's scenario. To complicate the picture, the energy flux seems to be positive at 10−100Km [12], suggesting a 3d-like direct energy cascade, though the involved scales may be too large.In the 2d-framework, on which we focus here, several explanations for the observed spectra have been proposed. Interpreting the synoptic −3 spectrum as an enstrophy cascade, forced by instabilities of the horizontal motion at the planetary scale (∼10 4 Km) [11], the −5/3 mesoscales spectrum may result: from a 2d-inverse energy cascade forced b...

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The energy cascade in a strongly stratified fluid

Cited by 411 publications

References 43 publications

Robust identification of dynamically distinct regions in stratified turbulence

Robust identification of dynamically distinct regions in stratified turbulence

A theory for the atmospheric energy spectrum: Depth-limited temperature anomalies at the tropopause

Nonlinear Superposition of Direct and Inverse Cascades in Two-Dimensional Turbulence Forced at Large and Small Scales

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