Anisotropic fluxes and nonlocal interactions in MHD turbulence

We investigate the locality or nonlocality of the energy transfer and of the spectral interactions involved in the cascade for decaying magnetohydrodynamic (MHD) flows in the presence of a uniform magnetic field $\bf B$ at various intensities. The results are based on a detailed analysis of three-dimensional numerical flows at moderate Reynold numbers. The energy transfer functions, as well as the global and partial fluxes, are examined by means of different geometrical wavenumber shells. On the one hand, the transfer functions of the two conserved Els\"asser energies $E^+$ and $E^-$ are found local in both the directions parallel ($k_\|$-direction) and perpendicular ($k_\perp$-direction) to the magnetic guide-field, whatever the ${\bf B}$-strength. On the other hand, from the flux analysis, the interactions between the two counterpropagating Els\"asser waves become nonlocal. Indeed, as the ${\bf B}$-intensity is increased, local interactions are strongly decreased and the interactions with small $k_\|$ modes dominate the cascade. Most of the energy flux in the $k_\perp$-direction is due to modes in the plane at $k_\|=0$, while the weaker cascade in the $k_\|$-direction is due to the modes with $k_\|=1$. The stronger magnetized flows tends thus to get closer to the weak turbulence limit where the three-wave resonant interactions are dominating. Hence, the transition from the strong to the weak turbulence regime occurs by reducing the number of effective modes in the energy cascade.Comment: Submitted to PR

Commercial fire-retarded PET formulations - relationship between thermal degradation behaviour and fire-retardant action

Many types of fire-retardants are used in poly(ethylene terephthalate), PET, formulations, and two commercial fire retardants, Ukanol(TM) and Phosgard(TM), have been shown to improve significantly PET flame-retardancy when used as comonomers. Phosgard incorporates a phosphorus atom within the main chain whereas Ukanol incorporates a phosphorus atom as a pendent substituent. Despite their acknowledged effectiveness, the mode of action of these fire retardants remains unclear, and in this paper we present a comparison of the overall thermal degradation behaviour of PET and Ukanol and Phosgard fire retarded formulations. DSC and particularly TGA data show that both Ukanol and Phosgard have some stabilising influence on PET degradation, especially under oxidative conditions. TGA and pyrolysis experiments both clearly indicate that neither additive acts as a char promoter. Only the Phosgard formulation shows any release of volatile phosphorus species which could act in the gas phase. On the other hand, the most striking feature of the pyrolysis experiments is the macroscopic structure of the chars produced by the fire-retarded formulations, which hints at their fire-retardancy action - an open-cell charred foam was obtained upon charring at 400°C or 600°C. This foaming layer between the degrading melt and the flame would lower the amount of fuel available for combustion, and would also limit the feedback of heat to the condensed phase

The Discrete Fundamental Group of the Associahedron, and the Exchange Module

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank $\binom{n+2}{4}$. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type $A_n$ cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank $\binom{n+2}{3}$.Comment: 16 pages, 4 figure

Discrete homology theory for metric spaces

We define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an n n -dimensional cube to a fixed metric space. We prove that the resulting homology theory satisfies a discrete analogue of the Eilenberg–Steenrod axioms, and prove a discrete analogue of the Mayer–Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a ‘fine enough’ rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space