The decay of Saffman and batchelor turbulence subject to rotation, stratification or an imposed magnetic field

We consider unforced, statistically-axisymmetric turbulence evolving in the presence of a background rotation, an imposed stratification, or a uniform magnetic field. We focus on two canonical cases: Saffman turbulence, in which E(κ → 0) ∼ κ 2, and Batchelor turbulence, in which E(κ → 0) ∼ κ 4. It has recently been shown that, provided the large scales evolve in a self-similar manner, then u ⊥ 2ℓ ⊥ 2ℓ // = constant in Saffman turbulence and u ⊥ 2ℓ ⊥ 4ℓ // = constant in Batchelor turbulence (Davidson, 2009, 2010). Here the subscripts ⊥ and // indicate directions perpendicular and parallel to the axis of symmetry, and ℓ ⊥, ℓ //, and u ⊥ are suitably defined integral scales. These constraints on the integral scales allow us to make simple, testable predictions for the temporal evolution of ℓ ⊥, ℓ //, and u ⊥ in rotating, stratified and MHD turbulence.

The non-local nature of structure functions

Kolmogorov's two-thirds, ((Δv) 2) ∼ e 2/ 3r 2/ 3, and five-thirds, E ∼ e 2/ 3k -5/ 3, laws are formally equivalent in the limit of vanishing viscosity, v → 0. However, for most Reynolds numbers encountered in laboratory scale experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. By creating artificial fields of isotropic turbulence composed of a random sea of Gaussian eddies whose size and energy distribution can be controlled, we show why this is the case. The energy of eddies of scale, s, is shown to vary as s 2/ 3, in accordance with Kolmogorov's 1941 law, and we vary the range of scales, γ = s max/s min, in any one realisation from γ = 25 to γ = 800. This is equivalent to varying the Reynolds number in an experiment from R λ = 60 to R λ = 600. While there is some evidence of a five-thirds law for g > 50 (R λ > 100), the two-thirds law only starts to become apparent when g approaches 200 (R λ ∼ 240). The reason for this discrepancy is that the second-order structure function is a poor filter, mixing information about energy and enstrophy, and from scales larger and smaller than r. In particular, in the inertial range, ((Δv) 2) takes the form of a mixed power-law, a 1+a 2r 2+a 3r 2/ 3, where a 2r 2 tracks the variation in enstrophy and a 3r 2/ 3 the variation in energy. These findings are shown to be consistent with experimental data where the polution of the r 2/ 3 law by the enstrophy contribution, a 2r 2, is clearly evident. We show that higherorder structure functions (of even order) suffer from a similar deficiency.

The application of STEM and in situ controlled dehydration to bacterial systems using ESEM.

Transmission imaging with an environmental scanning electron microscope (ESEM) (Wet STEM) is a recent development in the field of electron microscopy, combining the simple preparation inherent to ESEM work with an alternate form of contrast available through a STEM detector. Because the technique is relatively new, there is little information available on how best to apply this technique and which samples it is best suited for. This work is a description of the sample preparation and microscopy employed by the authors for imaging bacteria with Wet STEM (scanning transmission electron microscopy). Three different bacterial samples will be presented in this study: first, used as a model system, is Escherichia coli for which the contrast mechanisms of STEM are demonstrated along with the visual effects of a dehydration-induced collapse. This collapse, although clearly in some sense artifactual, is thought to lead to structurally meaningful morphological information. Second, Wet STEM is applied to two distinct bacterial systems to demonstrate the novel types of information accessible by this approach: the plastic-producing Cupriavidus necator along with wild-type and ΔmreC knockout mutants of Salmonella enterica serovar Typhimurium. Cupriavidus necator is shown to exhibit clear internal differences between bacteria with and without plastic granules, while the ΔmreC mutant of S. Typhimurium has an internal morphology distinct from that of the wild type.

The application of STEM and in situ controlled dehydration to bacterial systems using ESEM

Transmission imaging with an environmental scanning electron microscope (ESEM) (Wet STEM) is a recent development in the field of electron microscopy, combining the simple preparation inherent to ESEM work with an alternate form of contrast available through a STEM detector. Because the technique is relatively new, there is little information available on how best to apply this technique and which samples it is best suited for. This work is a description of the sample preparation and microscopy employed by the authors for imaging bacteria with Wet STEM (scanning transmission electron microscopy). Three different bacterial samples will be presented in this study: first, used as a model system, is Escherichia coli for which the contrast mechanisms of STEM are demonstrated along with the visual effects of a dehydration-induced collapse. This collapse, although clearly in some sense artifactual, is thought to lead to structurally meaningful morphological information. Second, Wet STEM is applied to two distinct bacterial systems to demonstrate the novel types of information accessible by this approach: the plastic-producing Cupriavidus necator along with wild-type and δmreC knockout mutants of Salmonella enterica serovar Typhimurium. Cupriavidus necator is shown to exhibit clear internal differences between bacteria with and without plastic granules, while the δmreC mutant of S. Typhimurium has an internal morphology distinct from that of the wild type. © 2012 Wiley Periodicals, Inc.

Geometry and interaction of structures in homogeneous isotropic turbulence

A strategy to extract turbulence structures from direct numerical simulation (DNS) data is described along with a systematic analysis of geometry and spatial distribution of the educed structures. A DNS dataset of decaying homogeneous isotropic turbulence at Reynolds number Reλ = 141 is considered. A bandpass filtering procedure is shown to be effective in extracting enstrophy and dissipation structures with their smallest scales matching the filter width, L. The geometry of these educed structures is characterized and classified through the use of two non-dimensional quantities, planarity' and filamentarity', obtained using the Minkowski functionals. The planarity increases gradually by a small amount as L is decreased, and its narrow variation suggests a nearly circular cross-section for the educed structures. The filamentarity increases significantly as L decreases demonstrating that the educed structures become progressively more tubular. An analysis of the preferential alignment between the filtered strain and vorticity fields reveals that vortical structures of a given scale L are most likely to align with the largest extensional strain at a scale 3-5 times larger than L. This is consistent with the classical energy cascade picture, in which vortices of a given scale are stretched by and absorb energy from structures of a somewhat larger scale. The spatial distribution of the educed structures shows that the enstrophy structures at the 5η scale (where η is the Kolmogorov scale) are more concentrated near the ones that are 3-5 times larger, which gives further support to the classical picture. Finally, it is shown by analysing the volume fraction of the educed enstrophy structures that there is a tendency for them to cluster around a larger structure or clusters of larger structures. Copyright © 2012 Cambridge University Press.

Scaling laws for planetary dynamos

We propose new scaling laws for the properties of planetary dynamos. In particular, the Rossby number, the magnetic Reynolds number, the ratio of magnetic to kinetic energy, the Ohmic dissipation timescale and the characteristic aspect ratio of the columnar convection cells are all predicted to be power-law functions of two observable quantities: the magnetic dipole moment and the planetary rotation rate. The resulting scaling laws constitute a somewhat modified version of the scalings proposed by Christensen and Aubert. The main difference is that, in view of the small value of the Rossby number in planetary cores, we insist that the non-linear inertial term, uu, is negligible. This changes the exponents in the power-laws which relate the various properties of the fluid dynamo to the planetary dipole moment and rotation rate. Our scaling laws are consistent with the available numerical evidence. © The Authors 2013 Published by Oxford University Press on behalf of The Royal Astronomical Society.

Ten Chapters in Turbulence

Leading experts summarize our current understanding of the fundamental nature of turbulence, covering a wide range of topics.