Convergent Calculation of the Asymptotic Dimension of Diffusion Limited Aggregates: Scaling and Renormalization of Small Clusters

Diffusion Limited Aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (2-3 particles) to asymptotically large (n \to \infty) clusters. The computed dimension is D=1.713\pm 0.003

On the Hydrodynamic Interaction of Shock Waves with Interstellar Clouds. II. The Effect of Smooth Cloud Boundaries on Cloud Destruction and Cloud Turbulence

The effect of smooth cloud boundaries on the interaction of steady planar shock waves with interstellar clouds is studied using a high-resolution local AMR technique with a second-order accurate axisymmetric Godunov hydrodynamic scheme. A 3D calculation is also done to confirm the results of the 2D ones. We consider an initially spherical cloud whose density distribution is flat near the cloud center and has a power-law profile in the cloud envelope. When an incident shock is transmitted into a smooth cloud, velocity gradients in the cloud envelope steepen the smooth density profile at the upstream side, resulting in a sharp density jump having an arc-like shape. Such a ``slip surface'' forms immediately when a shock strikes a cloud with a sharp boundary. For smoother boundaries, the formation of slip surface and therefore the onset of hydrodynamic instabilities are delayed. Since the slip surface is subject to the Kelvin-Helmholtz and Rayleigh-Taylor instabilities, the shocked cloud is eventually destroyed in $\sim 3-10$ cloud crushing times. After complete cloud destruction, small blobs formed by fragmentation due to hydrodynamic instabilities have significant velocity dispersions of the order of 0.1 $v_b$, where $v_b$ is the shock velocity in the ambient medium. This suggests that turbulent motions generated by shock-cloud interaction are directly associated with cloud destruction. The interaction of a shock with a cold HI cloud should lead to the production of a spray of small HI shreds, which could be related to the small cold clouds recently observed by Stanimirovic & Heiles (2005). The linewidth-size relation obtained from our 3D simulation is found to be time-dependent. A possibility for gravitational instability triggered by shock compression is also discussed.Comment: 62 pages, 16 figures, submitted to Ap

The survival of interstellar clouds against Kelvin-Helmholtz instabilities

We consider the stability of clouds surrounded by a hotter confining medium with respect to which they are in motion, against Kelvin-Helmholtz instabilities (KHI). In the presence of cooling, sound waves are damped by dissipation. Whenever cooling times are shorter than sound crossing times, as they are in the normal interstellar medium, this implies that the instability generated at the interface of the two media cannot propagate far from the interface itself. To study how this influences the overall stability, first we derive an analytic dispersion relation for cooling media, separated by a shear layer. The inclusion of dissipation does not heal the instability, but it is shown that only a small volume around the interface is affected, the perturbation decaying exponentially with distance from the surface; this is confirmed by numerical simulations. Numerical simulations of spherical clouds moving in a surrounding intercloud medium by which they are pressure confined show that these clouds develop a core/halo structure, with a turbulent halo, and a core in laminar flow nearly unscathed by the KHI. The related and previously reported ``champagne effect'', whereby clouds seem to explode from their top sides, is cured by the inclusion of radiative losses.Comment: 13 pages, AASTEX LATEX, accepted for publication in The Astrophysical Journa

Magnetohydrodynamic Simulations of Shock Interactions with Radiative Clouds

We present results from two-dimensional numerical simulations of the interactions between magnetized shocks and radiative clouds. Our primary goal is to characterize the dynamical evolution of the shocked clouds. We perform runs in both the strong and weak magnetic field limits and consider three different field orientations. For the geometries considered, we generally find that magnetic fields external to, but concentrated near, the surface of the cloud suppress the growth of destructive hydrodynamic instabilities. External fields also increase the compression of the cloud by effectively acting as a confinement mechanism driven by the interstellar flow and local field stretching. This can have a dramatic effect on both the efficiency of radiative cooling, which tends to increase with increasing magnetic field strength, and on the size and distribution of condensed cooled fragments. In contrast, fields acting predominately internally to the cloud tend to resist compression, thereby inhibiting cooling. We observe that, even at modest strengths, internal fields can completely suppress low-temperature cooling.Comment: 21 pages, 9 figures, to appear in The Astrophysical Journa

The Magnetohydrodynamics of Shock-Cloud Interaction in Three Dimensions

The magnetohydrodynamic evolution of a dense spherical cloud as it interacts with a strong planar shock is studied, as a model for shock interactions with density inhomogeneities in the interstellar medium. The cloud is assumed to be small enough that radiative cooling, thermal conduction, and self-gravity can be ignored. A variety of initial orientations (including parallel, perpendicular, and oblique to the incident shock normal) and strengths for the magnetic field are investigated. During the early stages of the interaction (less than twice the time taken for the transmitted shock to cross the interior of the cloud) the structure and dynamics of the shocked cloud is fairly insensitive to the magnetic field strength and orientation. However, at late times strong fields substantially alter the dynamics of the cloud, suppressing fragmentation and mixing by stabilizing the interface at the cloud surface. Even weak magnetic fields can drastically alter the evolution of the cloud compared to the hydrodynamic case. Weak fields of different geometries result in different distributions and amplifications of the magnetic energy density, which may affect the thermal and non-thermal x-ray emission expected from shocked clouds associated with, for example, supernovae remnants.Comment: Accepted for publication in Astrophysical Journal; a higher resolution file can be found at http://www.astro.princeton.edu/~msshin/science/shock_cloud.pdf.g

Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model

A fast method is presented for simulating the dielectric-breakdown model using iterated conformal mappings. Numerical results for the dimension and for corrections to scaling are in good agreement with the recent RG prediction of an upper critical $\eta_c=4$, at which a transition occurs between branching fractal clusters and one-dimensional nonfractal clusters.Comment: 5 pages, 7 figures; corrections to scaling include

Growth in non-Laplacian fields

We develop a formal method for assigning rules to lattice-based walkers which allows the modeling of irreversible growth in systems governed by non-Laplacian partial differential equations. The method is used to study diffusive growth in finite concentration fields. Good agreement with analytic results is obtained. The method is subsequently applied to study electrochemical deposition and investigate the interplay between the electrostatic and diffusion fields. We examine the effect of a local (nonuniform) flow field on deposition on a substrate

Velocity-jump instabilities in Hele-Shaw flow of associating polymer solutions

We study fracturelike flow instabilities that arise when water is injected into a Hele-Shaw cell filled with aqueous solutions of associating polymers. We explore various polymer architectures, molecular weights, and solution concentrations. Simultaneous measurements of the finger tip velocity and of the pressure at the injection point allow us to describe the dynamics of the finger in terms of the finger mobility, which relates the velocity to the pressure gradient. The flow discontinuities, characterized by jumps in the finger tip velocity, which are observed in experiments with some of the polymer solutions, can be modeled by using a nonmonotonic dependence between a characteristic shear stress and the shear rate at the tip of the finger. A simple model, which is based on a viscosity function containing both a Newtonian and a non-Newtonian component, and which predicts nonmonotonic regions when the non-Newtonian component of the viscosity dominates, is shown to agree with the experimental data