William I. Newman

Implosive accretion and outbursts of active galactic nuclei

A model and simulation code have been developed for time-dependent axisymmetric disk accretion onto a compact object including for the first time the influence of an ordered magnetic field. The accretion rate and radiative luminosity of the disk are naturally coupled to the rate of outflow of energy and angular momentum in magnetically driven (+/- z) winds. The magnetic field of the wind is treated in a phenomenological way suggested by self-consistent wind solutions. The radial accretion speed u(r, t) of the disk matter is shown to be the sum of the usual viscous contribution and a magnetic contribution proportional to r(exp 3/2)(B(sub p exp 2))/sigma, where B(sub p)(r,t) is the poloidal field threading the disk and sigma(r,t) is the disks surface mass density. An enhancement or variation in B(sub p) at a large radial distance leads to the formation of a soliton-like structure in the disk density, temperature, and B-field which propagates implosively inward. The implosion gives a burst in the power output in winds or jets and a simultaneous burst in the disk radiation. The model is pertinent to the formation of discrete fast-moving components in jets observed by very long baseline interferometry. These components appear to originate at times of optical outbursts of the active galactic nucleus.

An inverse-cascade model for self-organized critical behavior

We introduce an inverse-cascade model to explain self-organized critical behavior. This model is motivated by the forest-fire model. In the forest-fire model trees are randomly planted on a grid, sparks are also dropped on the grid resulting in fires in which trees are lost. In the inverse-cascade model single trees are introduced and these combine to form larger and larger clusters. This is the inverse cascade and gives a power-law (fractal) frequency-size distribution of clusters. Model fires eliminate trees from all cluster sizes but significant numbers of trees are lost only from the largest clusters and this loss terminates the power-law scaling. Finally, our model illustrates important differences between critical and self-organized critical behavior.

An exact renormalization model for earthquakes and material failure: statics and dynamics

Abstract Earthquake events are well-known to posses a variety of empirical scaling laws. Accordingly, renormalization methods offer some hope for understanding why earthquake statistics behave in a similar way over orders of magnitude of energy. We review the progress made in the use of renormalization methods in approaching the earthquake problem. In particular, earthquake events have been modeled by previous investigators as hierarchically organized bundles of fibers with equal load sharing. We consider by computational and analytic means the failure properties of such bundles of fibers, a problem that may be treated exactly by renormalization methods. We show, independent of the specific properties of an individual fiber, that the stress and time thresholds for failure of fiber bundles obey universal, albeit different, scaling laws with respect to the size of the bundles. The application of these results to fracture processes in earthquake events and in engineering materials helps to provide insight into some of the observed patterns and scaling - in particular, the apparent weakening of earthquake faults and composite materials with respect to size, and the apparent emergence of relatively well-defined stresses and times when failure is seemingly assured.

Jet Outbursts from Fast Accretion in a Disk with Zebra-Stripe Magnetic Field

The optical, X-ray, and gamma-ray outbursts, as well as the associated formation of relativistically moving components of parsec-scale jets of some active galactic nuclei (AGNs) are interpreted as dynamical events in a magnetized accretion disk of a massive black hole. Here we discuss the theory and simulation results for a time-dependent, axisymmetric disk accretion model, including the influence of an ordered magnetic field that reverses polarity as a function of radial distance in the disk. The accretion rate of the disk is coupled to the rate of angular momentum and energy outflow in magnetically driven jets originating from the ±z surfaces of the disk. The inward radial accretion speed in the disk (u) is the sum of the familiar viscous term and a magnetic term proportional to r3/2B2z/σ due to the jets, where Bz(r, t) is the field at the midplane threading the disk, and σ(r, t) is the disks surface mass density. We consider conditions where the magnetic term is dominant, and we derive coupled nonlinear equations for the evolution of Bz and σ. For general initial conditions, Bz and σ vary with r. Furthermore, Bz necessarily reverses polarity in order to conserve flux. As a result of the polarity reversals, the evolution of σ and Bz leads to the formation of inward facing shocks, where the radial derivatives of Bz and σ are very large. The shocks separate different annular regions of the disk threaded by positive and negative Bz. The kinetic luminosity in the jets is predominantly from the innermost part of the disk. Consequently, the passage of a shock through the inner edge of the disk gives a strong, narrow spike in the jet kinetic luminosity. We interpret this spike as an outburst of an AGN and the associated creation of a new parsec-scale jet component. Also in this picture, the outburst corresponds to a reversal of polarity of Bz in the inner part of the disk. As a result of the jets propagation and radial expansion, this polarity reversal becomes a polarity reversal of B as z varies across the jet component. Consequently, magnetic field annihilation in the jet component may be important, in particular, for accelerating the leptons to the high Lorentz factors needed to explain the observed synchrotron, synchrotron self-Compton, and inverse Compton radiation.

Fractal Trees with Side Branching

This paper considers fractal trees with self-similar side branching. The Tokunaga classification system for side branching is introduced, along with the Tokunaga self-similarity condition. Area filling (D = 2) and volume filling (D = 3) deterministic fractal tree constructions are introduced both with and without side branching. Applications to diffusion limited aggregation (DLA), actual drainage networks, as well as biology are considered. It is suggested that the Tokunaga taxonomy may have wide applicability in nature.

A Lyapunov functional for the evolution of solutions to the porous medium equation to self‐similarity. I

We employ the transformation properties and the integral invariants of the porous medium equation in m dimensions to construct a Lyapunov functional which characterizes the evolution of the profile to its self‐similar solution.

Quantification of Uncertainty in the Measurement of Magnetic Fields in Clusters of Galaxies

We assess the principal statistical and physical uncertainties associated with the determination of magnetic field strengths in clusters of galaxies from measurements of Faraday rotation (FR) and Compton-synchrotron emissions. In the former case, a basic limitation is noted, that the relative uncertainty in the estimation of the mean-squared FR is generally at least . Even greater uncertainty stems from the crucial dependence of the Faraday-deduced field on the coherence length scale characterizing its random orientation; we further elaborate this dependence and argue that previous estimates of the field are likely to be too high by a factor of a few. Lack of detailed spatial information on the radio emission—and the recently deduced nonthermal X-ray emission in four clusters—has led to an underestimation of the mean value of the field in cluster cores. We conclude therefore that it is premature to draw definite quantitative conclusions from the previously claimed, seemingly discrepant values of the field determined by these two methods.

Failure of hierarchical distributions of fibre bundles. I

We consider by computational and analytic means the failure properties of hierarchically organized bundles of fibres with equal load sharing, a problem that may be treated exactly by renormalization methods. We show, independent of the specific failure properties of an individual fibre, that the threshold for failure of a fibre bundle obeys a universal scaling law with respect to the size of the bundle. Moreover, this scaling law is preserved for any hierarchical organization of fibre bundles.

The long-time behavior of the solution to a non-linear diffusion problem in population genetics and combustion☆

Abstract The Fisher (1937) or Kolmogoroff-Petrovsky-Piscounoff (1937) equation exemplifies wave-like phenomena occurring in population genetics and combustion. In an earlier paper, we proposed an extension of this equation and obtained closed form traveling wave, stationary, and “symmetric” solutions. Employing the transformation properties of the extended equation, two integral invariants for the problem are obtained and two Lyapunov functionals, which characterize the evolution of the profile to a uniformly propagating traveling wave, are constructed. A generalization of this modified Fisher equation is proposed and we obtain its integral invariants, traveling wave solutions and wave speeds, as well as the Lyapunov functionals which describe its asymptotic evolution.

Crack fusion as a model for repetitive seismicity

A renormalization group treatment of a skeletal, hierarchical model of crack fusion and suturing yields the experimental stress versus time-to-fracture law. We construct a model that eliminates the intervening size states in the hierarchy but retains the time delay that represents the time-to-fracture observations. We add a source term to replenish microcracks lost by promotion due to fusion. The system is found to be stable for all values of time delay if the rate of replenishment is steady. If we allow the rate of replenishment of microcracks to be coupled to the rate of appearance of the largest size cracks, which we interpret as large-scale seismicity, then a Hopf bifurcation appears and the system is describable as a limit cycle attractor.