ALBERT

Description: The dynamics of a counter-rotating pair of columnar vortices aligned parallel to a stable density gradient are investigated. By means of numerical simulation, we extend the linear analyses and laboratory experiments of Billant & Chomaz (J. Fluid Mech. vol. 418, p. 167; vol. 419, pp. 29, 65 (2000)) to the fully nonlinear, large-Reynolds-number regime. A range of stratifications and vertical length scales is considered, with Frh 〈 0.2 and 0.1 〈 Frz 〈 10. Here Frh ≡ U/(NR) and Frz ≡ Ukz/N are the horizontal and vertical Froude numbers, U and R are the horizontal velocity and length scales of the vortices, N is the Brunt-Väisälä frequency, and 2π/kz is the vertical wavelength of a small initial perturbation. At early times with Frz 〈 1, linear predictions for the zigzag instability are reproduced. Short-wavelength perturbations with Frz 〉 1 are found to be unstable as well, with growth rates only slightly less than those of the zigzag instability but with very different structure. At later times, the large-Reynolds-number evolution diverges profoundly from the moderate-Reynolds-number laboratory experiments as the instabilities transition to turbulence. For the zigzag instability, this transition occurs when density perturbations generated by the vortex bending become gravitationally unstable. The resulting turbulence rapidly destroys the vortex pair. We derive the criterion η/R ≈ 0.2/Frz for the onset of gravitational instability, where η is the maximum horizontal displacement of the bent vortices, and refine it to account for a finite twisting disturbance. Our simulations agree for the fastest growing wavelengths 0.3 〈 Frz 〈 0.8. Short perturbations with Frz 〉 1 saturate at low amplitude, preserving the columnar structure of the vortices well after the generation of turbulence. Viscosity is shown to suppress the transition to turbulence for Reynolds number Re ≲ 80/Frh, yielding laminar dynamics and, under certain conditions, pancake vortices like those observed in the laboratory. © 2008 Cambridge University Press.

Description: We present numerical simulations of stably stratified, vortically forced turbulence at a wide range of Froude numbers. Large-scale vortical forcing was chosen to represent geophysical vortices which break down at small scales where Coriolis effects are weak. The resulting vortical energy spectra are much steeper in the horizontal direction and shallower in the vertical than typical observations in the atmosphere and ocean, as noted in previous studies. We interpret these spectra in terms of the vertical decoupling which emerges in the strongly stratified limit. We show that this decoupling breaks down at a vertical scale of U/N, where N is the Brunt-Väisälä frequency and U is a characteristic horizontal velocity, confirming previous scaling arguments. The transfer of vortical energy to wave energy is most efficient at this vertical scale; vertical spectra of wave energy are correspondingly peaked at small scales, as observed in past work. The equilibrium statistical mechanics of the inviscid unforced truncated problem qualitatively predicts the nature of the forced-dissipative solutions, and confirms the lack of an inverse cascade of vortical energy. © 2004 Cambridge University Press.

Description: We present numerical simulations of randomly forced internal gravity waves in a uniformly stratified Boussinesq fluid, and compare the resulting vertical wavenumber energy spectra with the saturation spectrum Ez(kz) = c N2kz -3 (N is the Brunt-Väisälä frequency) observed in the atmosphere and ocean. Overall, we have been unsuccessful at reproducing the observed spectrum in our simulations. Our spectra are shallower than kz-3, although they steepen towards it with increasing stratification as long as wave breaking (in the form of static instability) is resolved. The spectral amplitude increases like N1.1 rather than N2. For a single stratification, our spectrum agrees well with the saturation spectrum with c = 0.1, but only because it is spuriously steepened by insufficient resolution. We show that overturning occurs when the length scale lc = urms/N is larger than the dissipation scale, where urms is the root mean square velocity. This scale must be at least three times larger than the dissipation scale for the energy spectrum to be independent of Reynolds number in our simulations. When this condition is not satisfied, the computed energy spectrum must be interpreted with caution. Finally, we show that for strong stratifications, the presence of vortical energy can have a dramatic effect on the spectrum of wave energy due to the efficiency of interactions between two waves and a vortical mode. Any explanation of the energy spectrum involving resonant interactions must take into account the effect of vortical motion. © 2005 Cambridge University Press.

Description: We present numerical simulations of forced rotating stratified turbulence dominated by vortical motion (i.e. with potential vorticity). Strong stratification and various rotation rates are considered, corresponding to a small Froude number and a wide range of Rossby numbers Ro spanning the regimes of stratified turbulence (Ro = ∞) to quasi-geostrophic turbulence (Ro ≪ 1). We examine how the energy spectra and characteristic vertical scale of the turbulence vary with Rossby number between these two regimes. The separate dependence on N/f, where N is the Brunt-Väisälä frequency and f is the Coriolis parameter, is found to be of secondary importance. As the macroscale Ro decreases below 0.4 and the microscale Ro (at our resolution) decreases below 3, the horizontal wavenumber energy spectrum steepens and the flat range in the vertical wavenumber spectrum increases in amplitude and decreases in length. At large Rossby numbers, the vertical scale H is proportional to the stratified turbulence value U/N, where is the root mean square velocity. At small Ro takes the quasi-geostrophic form (f/N)L, where is the horizontal scale of the flow. Implications of these findings for numerical atmosphere and ocean modelling are discussed. © 2006 Cambridge University Press.

Description: In this paper large-eddy simulations (LES) of forced stratified turbulence using two common subgrid scale (SGS) models, the Kraichnan and Smagorinsky models, are studied. As found in previous studies using regular and hyper-viscosity, vorticity contours show elongated horizontal motions, which are layered in the vertical direction, along with intermittent Kelvin-Helmholtz (KH) instabilities. Increased stratification causes the layer thickness to collapse towards the dissipation scale, ultimately suppressing these instabilities. The vertical energy spectra are relatively flat out to a local maximum, which varies with the buoyancy frequency mathsfbi boldsymbol {mathsf {#1}}}let le =leqslant let leq =leqslant let ge =geqslant let geq =geqslant def Pr {mathit {Pr}}def Fr {mathit {Fr}}def Rey . The horizontal energy spectra depend on the grid spacing Δ if the resolution is fine enough, the horizontal spectrum shows an approximately -5/3 slope along with a bump at the buoyancy wavenumber kb = N/u{rms}, where u{rms} is the root-mean-square (r.m.s.) velocity. Our results show that there is a critical value of the grid spacing Δ, below which dynamics of stratified turbulence are well-captured in LES. This critical Δ depends on the buoyancy scale Lb and varies with different SGS models: the Kraichnan model requires Δ 〈 0.47 Lb, while the Smagorinsky model requires Δ 〈 0.17 Lb. In other words, the Smagorinsky model is significantly more costly than the Kraichnan approach, as it requires three times the resolution to adequately capture stratified turbulence. © 2014 Cambridge University Press.

Description: The parameter regime of strong stable density stratification and weak rotation is an important one in geophysical fluid dynamics. These conditions exist at intermediate length scales in the atmosphere and ocean (mesoscale and sub-mesoscale, respectively), and turbulence here links large-scale quasi-geostrophic motions with small-scale dissipation. While major advances in the theory of stratified turbulence have been made over the last few decades, many open questions remain, particularly about the nature of the energy cascade. Recent numerical experiments and analysis by Augier, Chomaz & Billant (J. Fluid Mech., vol. 713, 2012, pp. 86-108) present a remarkably vivid illustration of the nonlinear interactions that drive such turbulence. They consider a columnar vortex dipole, which naturally three-dimensionalizes under the influence of strong stratification. Kelvin-Helmholtz instabilities subsequently transfer energy directly to small scales, where the flow transitions into three-dimensional turbulence. This direct link between large and small scales is quite distinct from the usual picture of a turbulent cascade, in which nonlinear interactions are local in scale. But how important is this mechanism in the atmosphere and ocean? © 2013 Cambridge University Press.

Description: The dynamic Smagorinsky model for large eddy simulation (LES) of stratified turbulence is studied in this paper. A maximum grid spacing criterion of Δ/L〈inf〉b〈/inf〉 〈 0.24 is found in order to capture several of the key characteristics of stratified turbulence, where Δ is the filter scale and L〈inf〉b〈/inf〉 is the buoyancy scale. These results show that the dynamic Smagorinsky model needs a grid spacing approximately twice as large as the regular Smagorinsky model to reproduce similar results. This improvement on the regular Smagorinsky eddy viscosity approach increases the accuracy of results at small resolved scales while decreasing the computational costs because it allows larger Δ. In addition, the eddy dissipation spectra in LES of stratified turbulence present anisotropic features, taking energy out of large horizontal but small vertical scales. This trend is not seen in the non-stratified cases, where the subgrid-scale energy transfer is isotropic. Statistics of the dynamic Smagorinsky coefficient c〈inf〉s〈/inf〉 are investigated; its distribution is peaked around zero, and its standard deviations decrease slightly with increasing stratification. In line with previous findings for unstratified turbulence, regions of increased shear favour smaller c〈inf〉s〈/inf〉 values; in stratified turbulence, the spatial distribution of the shear, and hence c〈inf〉s〈/inf〉, is dominated by a layerwise pancake structure. These results show that the dynamic Smagorinsky model presents a promising approach for LES when isotropic buoyancy-scale resolving grids are employed. © 2015 Cambridge University Press.

Description: Direct numerical simulations are used to investigate potential enstrophy in stratified turbulence with small Froude numbers, large Reynolds numbers, and buoyancy Reynolds numbers (Reb) both smaller and larger than unity. We investigate the conditions under which the potential enstrophy, which is a quartic quantity in the flow variables, can be approximated by its quadratic terms, as is often done in geophysical fluid dynamics. We show that at large scales, the quadratic fraction of the potential enstrophy is determined by Reb. The quadratic part dominates for small Reb, i.e. in the viscously coupled regime of stratified turbulence, but not when Re b ≳ 1. The breakdown of the quadratic approximation is consistent with the development of Kelvin-Helmholtz instabilities, which are frequently observed to grow on the layerwise structure of stratified turbulence when Reb is not too small.©2013 Cambridge University Press.

Description: There is considerable interest in the fragmentation and loss of natural and semi-natural habitats, but few studies have examined the dynamics and mechanisms of change. A temporal analysis of landscape change on the South Downs in Sussex, UK, provides a clear description of the process of change over the 20 years 1971–1991. Transition probabilities were calculated from digital interpretations of an aerial photography time series of West Sussex. The analysis enabled quantitative comparison of landscape mosaics within different landscape ecoregions and under different management regimes to be made. Past changes in land use have produced a fragmented downland landscape. The key land conversion sequences identified show a substantial transition towards arable production, often at the expense of the internationally and nationally important unimproved grassland systems. A geographical information system facilitated greater understanding of the environmental and topographical characteristics of land converted to arable and other uses, and highlighted areas for protection and potential restoration. The patterns of land-use conversion observed in the study provide a landscape-scale planning tool for assessing the potential impact of agri-environmental policies, plans, and programmes in semi-natural grassland habitats.