On the existence of two-dimensional Euler flows satisfying energy-Casimir stability criteria

The energy-Casimir stability method, also known as the Arnold stability method, has been widely used in fluid dynamical applications to derive sufficient conditions for nonlinear stability. The most commonly studied system is two-dimensional Euler flow. It is shown that the set of two-dimensional Euler flows satisfying the energy-Casimir stability criteria is empty for two important cases: (i) domains having the topology of the sphere, and (ii) simply-connected bounded domains with zero net vorticity. The results apply to both the first and the second of Arnold’s stability theorems. In the spirit of Andrews’ theorem, this puts a further limitation on the applicability of the method. © 2000 American Institute of Physics.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

Crooks' fluctuation theorem for a process on a two dimensional fluid field

We investigate the behavior of a two-dimensional inviscid and incompressible flow when pushed out of dynamical equilibrium. We use the two-dimensional vorticity equation with spectral truncation on a rectangular domain. For a sufficiently large number of degrees of freedom, the equilibrium statistics of the flow can be described through a canonical ensemble with two conserved quantities, energy and enstrophy. To perturb the system out of equilibrium, we change the shape of the domain according to a protocol, which changes the kinetic energy but leaves the enstrophy constant. We interpret this as doing work to the system. Evolving along a forward and its corresponding backward process, we find numerical evidence that the distributions of the work performed satisfy the Crooks relation. We confirm our results by proving the Crooks relation for this system rigorously.

On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow

Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews' theorem is also established for this problem.

Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations

Exact, finite-amplitude, local wave-activity conservation laws are derived for disturbances to steady flows in the context of the two-dimensional anelastic equations. The conservation laws are expressed entirely in terms of Eulerian quantities, and have the property that, in the limit of a small-amplitude, slowly varying, monochromatic wave train, the wave-activity density A and flux F, when averaged over phase, satisfy F = cgA where cg is the group velocity of the waves. For nonparallel steady flows, the only conserved wave activity is a form of disturbance pseudoenergy; when the steady flow is parallel, there is in addition a conservation law for the disturbance pseudomomentum. The above results are obtained not only for isentropic background states (which give the so-called “deep form” of the anelastic equations), but also for arbitrary background potential-temperature profiles θ0(z) so long as the variation in θ0(z) over the depth of the fluid is small compared with θ0 itself. The Hamiltonian structure of the equations is established in both cases, and its symmetry properties discussed. An expression for available potential energy is also derived that, for the case of a stably stratified background state (i.e., dθ0/dz > 0), is locally positive definite; the expression is valid for fully three-dimensional flow. The counterparts to results for the two-dimensional Boussinesq equations are also noted.

Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2[surd radical]2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

Inhomogeneous two-dimensional turbulence in the atmosphere

Faced with the strongly nonlinear and apparently random behaviour of the energy-containing scales in the atmosphere, geophysical fluid dynamicists have attempted to understand the synoptic-scale atmospheric flow within the context of two-dimensional homogeneous turbulence theory (e.g. FJØRTOFT [1]; LEITH [2]). However atmospheric observations (BOER and SHEPHERD [3] and Fig.1) show that the synoptic-scale transient flow evolves in the presence of a planetary-scale, quasi-stationary background flow which is approximately zonal (east-west). Classical homogeneous 2-D turbulence theory is therefore not strictly applicable to the transient flow. One is led instead to study 2-D turbulence in the presence of a large-scale (barotropically stable) zonal jet inhomogeneity.

Physiological, biochemical and transcriptional analysis of onion bulbs during storage

Abstract: During the transition from endo-dormancy to eco-dormancy and subsequent growth, the onion bulb undergoes the transition from sink organ to source, to sustain cell division in the meristematic tissue. The mechanisms controlling these processes are not fully understood. Here, a detailed analysis of whole onion bulb physiological, biochemical and transcriptional changes in response to sprouting is reported, enabling a better knowledge of the mechanisms regulating post-harvest onion sprout development. Biochemical and physiological analyses were conducted on different cultivars ('Wellington', 'Sherpa' and 'Red Baron') grown at different sites over 3 years, cured at different temperatures (20, 24 and 28 degrees C) and stored under different regimes (1, 3, 6 and 6 1 degrees C). In addition, the first onion oligonucleotide microarray was developed to determine differential gene expression in onion during curing and storage, so that transcriptional changes could support biochemical and physiological analyses. There were greater transcriptional differences between samples at harvest and before sprouting than between the samples taken before and after sprouting, with some significant changes occurring during the relatively short curing period. These changes are likely to represent the transition from endo-dormancy to sprout suppression, and suggest that endo-dormancy is a relatively short period ending just after curing. Principal component analysis of biochemical and physiological data identified the ratio of monosaccharides (fructose and glucose) to disaccharide (sucrose), along with the concentration of zeatin riboside, as important factors in discriminating between sprouting and pre-sprouting bulbs. These detailed analyses provide novel insights into key regulatory triggers for sprout dormancy release in onion bulbs and provide the potential for the development of biochemical or transcriptional markers for sprout initiation. Evidence presented herein also suggests there is no detrimental effect on bulb storage life and quality caused by curing at 20 degrees C, producing a considerable saving in energy and costs.

Diversity of condensed tannin structures affects rumen in vitro methane production in sainfoin (Onobrychis viciifolia) accessions

Sainfoin is a non-bloating temperate forage legume with a moderate-to-high condensed tannin (CT) content. This study investigated whether the diversity of sainfoin accessions in terms of CT structures and contents could be related to rumen in vitro gas and methane (CH4) production and fermentation characteristics. The aim was to identify promising accessions for future investigations. Accessions differed (P < 0·0001) in terms of total gas and CH4 productions. Fermentation kinetics (i.e. parameters describing the shape of the gas production curve and half-time gas production) for CH4 production were influenced by accession (P ≤ 0·038), but not by PEG. Accession, PEG and time affected (P < 0·001) CH4 production, but accession and PEG interaction showed only a tendency (P = 0·08). Increase in CH4 due to PEG addition was not related to CT content. Further analysis of the relationships among multiple traits (nutritional composition, CT structure and CH4 production) using principal component analysis (PCA) based on optimally weighted variables revealed differences among accessions. The first two principal component axes, PC1 (57·6%) and PC2 (18·4%), explained 76·0% of the total variation among accessions. Loading of biplots derived from both PCAs made it possible to establish a relationship between the ratio of prodelphinidin:procyanidin (PD:PC) tannins and CH4 production in some accessions. The PD:PC ratio seems to be an important source of variation that is negatively related to CH4 production. These results suggested that sainfoin accessions collected from across the world exhibited substantial variation in terms of their effects on rumen in vitro CH4 production, revealing some promising accessions for future investigations.

Analysis of functional networks involved in motor execution and motor imagery using combined hierarchical clustering analysis and independent component analysis

Cognitive experiments involving motor execution (ME) and motor imagery (MI) have been intensively studied using functional magnetic resonance imaging (fMRI). However, the functional networks of a multitask paradigm which include ME and MI were not widely explored. In this article, we aimed to investigate the functional networks involved in MI and ME using a method combining the hierarchical clustering analysis (HCA) and the independent component analysis (ICA). Ten right-handed subjects were recruited to participate a multitask experiment with conditions such as visual cue, MI, ME and rest. The results showed that four activation clusters were found including parts of the visual network, ME network, the MI network and parts of the resting state network. Furthermore, the integration among these functional networks was also revealed. The findings further demonstrated that the combined HCA with ICA approach was an effective method to analyze the fMRI data of multitasks.

A robust two-dimensional hydrogen-bonded network for the predictable assembly of ternary co-crystals of furosemide

Consideration of the geometrical features of the functional groups present in furosemide has enabled synthesis of a series of ternary co-crystals with predictable structural features, containing a robust asymmetric two-dimensional network.

The cleft ion fountain: a two-dimensional kinetic model

The transport of ionospheric ions from a source in the polar cleft ionosphere through the polar magnetosphere is investigated using a two-dimensional, kinetic, trajectory-based code. The transport model includes the effects of gravitation, longitudinal magnetic gradient force, convection electric fields, and parallel electric fields. Individual ion trajectories as well as distribution functions and resulting bulk parameters of density, parallel average energy, and parallel flux for a presumed cleft ionosphere source distribution are presented for various conditions to illustrate parametrically the dependences on source energies, convection electric field strengths, ion masses, and parallel electric field strengths. The essential features of the model are consistent with the concept of a cleft-based ion fountain supplying ionospheric ions to the polar magnetosphere, and the resulting plasma distributions and parameters are in general agreement with recent low-energy ion measurements from the DE 1 satellite.