Collisionless galaxy simulations

Three-dimensional fully self-consistent computer models were used to determine the evolution of galaxies consisting of 100 000 simulation stars. Comparison of two-dimensional simulations with three-dimensional simulations showed only a very slight stabilizing effect due to the additional degree of freedom. The addition of a fully self-consistent, nonrotating, exponential core/halo component resulted in considerable stabilization. A second series of computer experiments was performed to determine the collapse and relaxation of initially spherical, uniform density and uniform velocity dispersion stellar systems. The evolution of the system was followed for various amounts of angular momentum in solid body rotation. For initally low values of the angular momentum satisfying the Ostriker-Peebles stability criterion, the systems quickly relax to an axisymmetric shape and resemble elliptical galaxies in appearance. For larger values of the initial angular momentum bars develop and the systems undergo a much more drastic evolution

Numerical simulation of solar coronal magnetic fields

Many aspects of solar activity are believed to be due to the stressing of the coronal magnetic field by footpoint motions at the photosphere. The results are presented of a fully spectral numerical simulation which is the first 3-D time dependent simulation of footpoint stressing in a geometry appropriate for the corona. An arcade is considered that is initially current-free and impose a smooth footpoint motion that produces a twist in the field of approx 2 pi. The footprints were fixed and the evolution was followed until the field relaxes to another current-free state. No evidence was seen for any instability, either ideal or resistive and no evidence for current sheet formation. The most striking feature of the evolution is that in response to photospheric motions, the field expands rapidly upward to minimize the stress. The expansion has two important effects. First, it suppresses the development of dips in the field that could support dense, cool material. For the motions assumed, the magnetic field does not develop a geometry suitable for prominence formation. Second, the expansion inhibits ideal instabilities such as kinking. The results indicate that simple stearing of a single arcade is unlikely to lead to solar activity such as flares or prominences. Effects are discussed that might possibly lead to such activity

Viscous, resistive MHD stability computed by spectral techniques

Expansions in Chebyshev polynomials are used to study the linear stability of one dimensional magnetohydrodynamic (MHD) quasi-equilibria, in the presence of finite resistivity and viscosity. The method is modeled on the one used by Orszag in accurate computation of solutions of the Orr-Sommerfeld equation. Two Reynolds like numbers involving Alfven speeds, length scales, kinematic viscosity, and magnetic diffusivity govern the stability boundaries, which are determined by the geometric mean of the two Reynolds like numbers. Marginal stability curves, growth rates versus Reynolds like numbers, and growth rates versus parallel wave numbers are exhibited. A numerical result which appears general is that instability was found to be associated with inflection points in the current profile, though no general analytical proof has emerged. It is possible that nonlinear subcritical three dimensional instabilities may exist, similar to those in Poiseuille and Couette flow

Advanced PID Controller Synthesis using Multiscale Control Scheme

For many decades, PID controller has been widely applied in industries despite the advancement in many advanced control techniques. Process models such as the First-Order plus Deadtime (FOPDT) has often been used to design or tune PID controller. Numerous PID tuning formulas have been established since the well-known Ziegler-Nichols formula introduced in the 1940s. In this paper, we present the multi-scale control approach to constructing a PID tuning formula based on the FOPDT model. The effectiveness of the proposed PID tuning formula is compared with some of the existing formulas reported in the literature