Numerical and analytic descriptions of cosmic-ray transport

It is not trivial to solve the equations that describe charged particle transport with the aid of computers, for instabilities, inaccuracies, and subtle artifacts are well known afflictions of numerical analysis. Two specific points are discussed. First to avoid inaccuracies, pitch angle scattering must be treated with great care. In particular, slightly inappropriate numerical formulations give rise to mean free paths that are in error by large factors. Secondly, A previously unrecognized artifact, numerical dispersion, is very similar to the physical phenomenon of dispersion. To avoid misinterpretations arising from this similarity, the spatial increment of the finite difference grid must be a small fraction of the mean free path. These points are illustrated by calculations based upon finite difference approximations to the transport equation

Helicopter-V/STOL dynamic wind and turbulence design methodology

Aircraft and helicopter accidents due to severe dynamic wind and turbulence continue to present challenging design problems. The development of the current set of design analysis tools for a aircraft wind and turbulence design began in the 1940's and 1950's. The areas of helicopter dynamic wind and turbulence modeling and vehicle response to severe dynamic wind inputs (microburst type phenomena) during takeoff and landing remain as major unsolved design problems from a lack of both environmental data and computational methodology. The development of helicopter and V/STOL dynamic wind and turbulence response computation methology is reviewed, the current state of the design art in industry is outlined, and comments on design methodology are made which may serve to improve future flight vehicle design

The diffusive idealization of charged particle transport in random magnetic fields

The transport of charged particles diffusing in a random magnetic field parallel to a relatively large guiding field is presented. The same coefficient of diffusion is obtained by three methods. Two corrections must be added to the expression in which the diffusive flux is proportional to the gradient of the density. Explicit expressions are given for a characteristic time and a characteristic length which describe the corrections. The well known divergence of the coefficient of diffusion, which is implied by the quasilinear analysis of pitch angle scattering, does not occur if the scattering rate is finite at 90 deg pitch angle. This effect is illustrated by formulas which give the coefficient of diffusion when the quasilinear expression is perturbed by a variable amount of isotropic scattering

Modulation of cosmic ray electrons

The origin and variations of the steep spectrum of electrons observed below 20 MeV is explained by a simple model in which the spectrum of interplanetary cosmic rays is decomposed, at low energies, into two independently varying components

The dispersive evolution of charged-particle bunches in random magnetic fields

Shortly after a strongly anisotropic beam of charged particles is injected along a guiding magnetic field on which is superimposed a small random conponent, the particle density can be represented by a Gaussian profile whose center moves with the coherent velocity and whose width increases with time at a rate controlled by the coefficient of dispersion. Both parameters depend upon the mean free path, which characterizes scattering by the random fields, and the focusing length, which characterizes spatial variations of the guiding field. These dependencies are known explicitly for the coherent velocity. Formulae for coefficient of dispersion are available only in the limits of very weak and very strong focusing. A new expression for coefficient of dispersion, which spans this gap, is presented

Complexity models in design

Complexity is a widely used term; it has many formal and informal meanings. Several formal models of complexity can be applied to designs and design processes. The aim of the paper is to examine the relation between complexity and design. This argument runs in two ways. First designing provides insights into how to respond to complex systems – how to manage, plan and control them. Second, the overwhelming complexity of many design projects lead us to examine how better understanding of complexity science can lead to improved designs and processes. This is the focus of this paper. We start with an outline of some observations on where complexity arises in design, followed by a brief discussion of the development of scientific and formal conceptions of complexity. We indicate how these can help in understanding design processes and improving designs