The evolution of a stratified turbulent cloud

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We present direct numerical simulations (DNS) of unforced stratified turbulence with the objective of testing the strongly stratified turbulence theory. According to this theory the characteristic vertical scale of the turbulence is given by v ∼ u h /N, where u h is the horizontal velocity scale and N the Brunt-Väisälä frequency. Combined with the hypothesis of the energy dissipation rate scaling as ∼ u 3 h / h , this theory predicts inertial range scalings for the horizontal spectrum of horizontal kinetic energy and of potential energy, according to E(k h ) ∝ k −5/3 h . We begin by presenting a scaling analysis of the horizontal vorticity equation from which we recover the result regarding the vertical scale, v ∼ u h /N, highlighting in the process the important dynamical role of large-scale vertical shear of horizontal velocity. We then present the results from decaying DNS, which show a good agreement with aspects of the theory. In particular, the vertical Froude number is found to reach a constant plateau in time, of the formin all the runs. The derivation of the dissipation scaling ∼ u 3 h / h at low Reynolds number in the context of decaying stratified turbulence highlights that the same scaling holds at high R = ReFr 2 h 1 as well as at low R 1, which is known (see Brethouwer et al., J. Fluid Mech., vol. 585, 2007, pp. 343-368) but not sufficiently emphasized in recent literature. We find evidence in our DNS of the dissipation scaling holding at R = O(1), which we interpret as being in the viscous regime. We also find k ∼ u 3 h / h and p ∼ u 3 h / h (with = k + p ), in our high-resolution run at earlier times corresponding to R = O(10), which is in the transition between the strongly stratified and the viscous regimes. The horizontal spectrum of horizontal kinetic energy collapses in time using the scaling E h (k h ) = C 1 2/3 k k −5/3 h and the horizontal potential energy spectrum is well described byThe presence of an inertial range in the horizontal direction is confirmed by the constancy of the energy flux spectrum over narrow ranges of k h . However, the vertical energy spectrum is found to differ significantly from the expected E h (k v ) ∼ N 2 k −3 v scaling, showing that Fr v is not of order unity on a scale-by-scale basis, thus providing motivation for further investigation of the vertical structure of stratified turbulence.

Section: Visualization Of Flow Field and Emergence Of Strongly Stratimentioning

confidence: 77%

Section: Numerical Methodologymentioning

confidence: 92%

Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number

Maffioli

Davidson

2015

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“…Rotation and stratification co-exist in many geophysical phenomena, and have close analogies (Veronis 1970). A turbulent cloud under the effect of stratification has recently been investigated by Maffioli et al (2014). In this study, we consider rotation alone which is particularly relevant to the equatorial plane in the core of the Earth, where a pronounced radial heat flux is thought to be a localized source of turbulence and hence inertial waves (Olson 2013;Davidson 2014).…”

mentioning

confidence: 99%

Evolution of a turbulent cloud under rotation

Ranjan

Davidson

2014

Localized patches of turbulence frequently occur in geophysics, such as in the atmosphere and oceans. The effect of rotation, Ω, on such a region (a 'turbulent cloud') is governed by inhomogeneous dynamics. In contrast, most investigations of rotating turbulence deal with the homogeneous case, although inhomogeneous turbulence is more common in practice. In this paper, we describe the results of 512 3 direct numerical simulations (DNS) of a turbulent cloud under rotation at three Rossby numbers (Ro), namely 0.1, 0.3 and 0.5. Using a spatial filter, fully developed homogeneous turbulence is vertically confined to the centre of a periodic box before the rotation is turned on. Energy isosurfaces show that columnar structures emerge from the cloud and grow into the adjacent quiescent fluid. Helicity is used as a diagnostic and confirms that these structures are formed by inertial waves. In particular, it is observed that structures growing parallel to the rotation axis (upwards) have negative helicity and those moving antiparallel (downwards) to the axis have positive helicity, a characteristic typical of inertial waves. Two-dimensional energy spectra of horizontal wavenumbers, k ⊥ , versus dimensionless time, 2Ωt, confirm that these columnar structures are wavepackets which travel at the group velocities of inertial waves. The kinetic energy transferred from the turbulent cloud to the waves is estimated using Lagrangian particle tracking to distinguish between turbulent and 'wave-only' regions of space. The amount of energy transferred to waves is 40 % of the initial at Ro = 0.1, while it is 16 % at Ro = 0.5. In both cases the bulk of the energy eventually resides in the waves. It is evident from this observation that inertial waves can carry a significant portion of the energy away from a localized turbulent source and are therefore an efficient mechanism of energy dispersion.

“…(From Thorpe (1982, figure 3). ) (Park, Whitehead & Gnanadeskian 1994) to a set of parallel vertical bars that oscillates continually through a short distance along the tank (Thorpe 1982) or oscillates once at steady speed through part of the length of the tank (Maffioli et al 2014). More complex grids are used, e.g.…”

Section: Turbulence and Layersmentioning

confidence: 99%

“…Table 1 lists comparative values of Fr 0 , Re 0 and of I 0 , a measure of the characteristic turbulent buoyancy Reynolds number or isotropy index, given by I 0 = u 3 0 /bνN 2 or ε/νN 2 , where u 0 is the r.m.s. velocity fluctuations, determined (following Maffioli et al 2014) as u 0 = 0.05U where U is the grid speed (or aω where a is the grid amplitude and ω its oscillation frequency) and ε = u 3 0 /b. The data used to derive values in table 1 are those given in or derived from tables or, when details are otherwise lacking, in the captions of figures in the respective papers.…”

Section: Types Of Laboratory Experiments and Their Characteristics Frmentioning

confidence: 99%

Layers and internal waves in uniformly stratified fluids stirred by vertical grids

Thorpe

2016

Laboratory experiments in which uniformly stratified fluids are stirred by horizontally moving vertical grids, or arrays of vertical rods, are reviewed to examine their consistency and to compare their findings, particularly those relating to the generation of layers. Selected experiments are of three types, those in which (a) turbulence spreads from a horizontally confined region where it is continuously generated by an oscillating grid; (b) grid stirring is maintained throughout a rectangular tank; or (c) a 'cloud' of turbulence decays after a short period of horizontally localized grid mixing. In all the experiments turbulence is generated over the full vertical extent of the tank. In the experiments of types (a) and (c) layers of comparable scale are observed to intrude into the ambient fluid from the turbulent region. In the type (b) experiments, layers form only when the time interval between the passage of the grid through the stratified fluid is sufficiently long, allowing turbulence to decay substantially between grid strokes. Two mechanisms are found to be dominant in the production of layers. In experiments of type (a) and (c) overturning eddies in the turbulent region of scale significantly larger than the Ozmidov length scale collapse and spread, intruding and forming layers in the adjoining laminar region. Internal shear waves propagating ahead of the intrusions have a vertical wavelength that is approximately twice the layer height. In type (b) experiments, layers are formed through a process described by Holford & Linden (Dyn. Atmos. Oceans, vol. 30, 1999a, pp. 173-198): the bending of vortices shed by the grid bars. The height of the layers is approximately half the vertical wavelength of internal shear waves that travel at the speed of the grid. It is proposed that the flow field of the shear waves bends the vortices, resulting in diapycnal mixing that forms the layers. The relationship of layers and internal shear waves in the experiments is therefore as follows: in type (a) and (c) experiments internal waves are generated by layers intruding from the turbulent region into the quiescent stratified region, but in experiments of type (b) internal waves drive and dictate the height of the layers; layers are generated as a consequence of vortex bending by internal waves. There is insufficient evidence to establish whether zigzag instability, either of vortex pairs or of vortex streets shed by a moving grid, accounts for the layers in any of the three types of grid experiments. The Phillips and Posmentier instability may reinforce layers formed by other processes. The generation of pancake vortices or vortical mode motion is left for later review.