Turbulent flow. Let us now consider the case that the fluid is moving much faster, such that downstream of an obstruction in the flow turbulence arises in the matter current. In Figure 3.3 we have indicated a typical stream line showing that the flow is now characterized by a finite vorticity in the re- gion behind the obstruction. Let us now repeat the exercise, assuming that fluid elements arrive at the obstruction with aligned magnetization vectors. Following a fluid element when it traverses the region with finite circulation it is immediately obvious that even for a fixed precession rate the non-Abelian charge/magnetization becomes multivalued when it has travelled around the vortex! Henceforth, at long times the magnetization will average away and the spin current actually disappears at the ’sink’ associated with the rotational Abelian flow. This elementary example highlights the essence of the problem dealing with non-Abelian ’hydro- dynamics’: the covariant conservation principle underlying everything is good enough to ensure a local conservation of non-Abelian charge so that one can reliably predict how the spin current evolves over infinitesimal times and distances. However, it fails to impose a global conservation. This is neatly illustrated in this simple hydrodynamical example: at the moment the mass flow becomes topologically non-trivial it is no longer possible to construct globally consistent non-Abelian flow patterns with the consequence that the spin currents just disappear. Although obscured by irrelevant details, the above motive has been recognized in the literature on spin flow in semiconductors where it is known as D’yakonov-Perel spin relaxation. We hope that the analogy with spin-transport in solids is helpful for the community that is trying to find out what is actually going on in the quark-gluon fireballs. Because one has to deal eventually with the absence of hydrodynamics we are pessimistic with regard to the possibility that an elegant description will be found, in a way mirroring the state of spintronics. We will instead continue now with our exposition of the remarkable fact that the rigidity associated with order parameters is not only simplifying the hydrodynamics (as in the Abelian case) but even making it possible for hydrodynamics to exist! Fluid element Non−Abelian charge Direction of laminar flow Electric field
Turbulent flow the Gaussian Puff Model In practice the contaminant will not be immediately well mixed in the air and there will be some transient period of flow dispersion and mixing potentially on a scale similar to the dwelling time of individuals in the area. When bacteria are aerosolised they become subject to local turbulent forces (causing random movement of the air packet within which the aerosol is contained) as well as longer range advective processes (transport of the packet by the dominant airflow). There are a number of models for the atmospheric dispersion of aerosol material. It is not within the scope of PANDHUB to perform a systematic review of such techniques because we are interested in a hub setting specifically. Stockie provides a helpful review of the assumptions made and the derivation of the classical Gaussian plume model and the reader is referred to this article and references therein for detail (Stockie 2011). The aim is to solve the three-dimensional advection-diffusion equation &C + ∇. (uC) = ∇. (K ∇C) + S − yC &t where C(t, x) is the concentration of the infectious agent in time and space. For illustration at this stage we make the further assumptions to I above:
