Wavelength limits on isobaricity of perturbations in a thermally unstable radiatively cooling medium

Nonlinear evolution of one-dimensional planar perturbations in an optically thin radiatively cooling medium in the long-wavelength limit is studied numerically. The accepted cooling function generates in thermal equilibrium a bistable equation of state $P(\rho)$. The unperturbed state is taken close to the upper (low-density) unstable state with infinite compressibility ($dP/d\rho= 0$). The evolution is shown to proceed in three different stages. At first stage, pressure and density set in the equilibrium equation of state, and velocity profile steepens gradually as in case of pressure-free flows. At second stage, those regions of the flow where anomalous pressure (i.e. with negative compressibility) holds, create velocity profile more sharp than in pressure-free case, which in turn results in formation of a very narrow (short-wavelength) region where gas separates the equilibrium equation of state and pressure equilibrium sets in rapidly. On this stage, variation in pressure between narrow dense region and extended environment does not exceed more than 0.01 of the unperturbed value. On third stage, gas in the short-wavelength region reaches the second (high-density) stable state, and pressure balance establishes through the flow with pressure equal to the one in the unperturbed state. In external (long-wavelength) regions, gas forms slow isobaric inflow toward the short-wavelength layer. The duration of these stages decreases when the ratio of the acoustic time to the radiative cooling time increases. Limits in which nonlinear evolution of thermally unstable long-wavelength perturbations develops in isobaric regime are obtained.Comment: 21 pages with 7 figures, Revtex, accepted in Physics of Plasma

Logarithmically Slow Expansion of Hot Bubbles in Gases

We report logarithmically slow expansion of hot bubbles in gases in the process of cooling. A model problem first solved, when the temperature has compact support. Then temperature profile decaying exponentially at large distances is considered. The periphery of the bubble is shown to remain essentially static ("glassy") in the process of cooling until it is taken over by a logarithmically slowly expanding "core". An analytical solution to the problem is obtained by matched asymptotic expansion. This problem gives an example of how logarithmic corrections enter dynamic scaling.Comment: 4 pages, 1 figur

Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

We employ Navier-Stokes granular hydrodynamics to investigate the long-time behavior of clustering instability in a freely cooling dilute granular gas in two dimensions. We find that, in circular containers, the homogeneous cooling state (HCS) of the gas loses its stability via a sub-critical pitchfork bifurcation. There are no time-independent solutions for the gas density in the supercritical region, and we present analytical and numerical evidence that the gas develops thermal collapse unarrested by heat diffusion. To get more insight, we switch to a simpler geometry of a narrow-sector-shaped container. Here the HCS loses its stability via a transcritical bifurcation. For some initial conditions a time-independent inhomogeneous density profile sets in, qualitatively similar to that previously found in a narrow-channel geometry. For other initial conditions, however, the dilute gas develops thermal collapse unarrested by heat diffusion. We determine the dynamic scalings of the flow close to collapse analytically and verify them in hydrodynamic simulations. The results of this work imply that, in dimension higher than one, Navier-Stokes hydrodynamics of a dilute granular gas is prone to finite-time density blowups. This provides a natural explanation to the formation of densely packed clusters of particles in a variety of initially dilute granular flows.Comment: 18 pages, 19 figure

Velocity fluctuations of population fronts propagating into metastable states

The position of propagating population fronts fluctuates because of the discreteness of the individuals and stochastic character of processes of birth, death and migration. Here we consider a Markov model of a population front propagating into a metastable state, and focus on the weak noise limit. For typical, small fluctuations the front motion is diffusive, and we calculate the front diffusion coefficient. We also determine the probability distribution of rare, large fluctuations of the front position and, for a given average front velocity, find the most likely population density profile of the front. Implications of the theory for population extinction risk are briefly considered.Comment: 8 pages, 3 figure

Far-from-equilibrium Ostwald ripening in electrostatically driven granular powders

We report the first experimental study of cluster size distributions in electrostatically driven granular submonolayers. The cluster size distribution in this far-from-equilibrium process exhibits dynamic scaling behavior characteristic of the (nearly equilibrium) Ostwald ripening, controlled by the attachment and detachment of the "gas" particles. The scaled size distribution, however, is different from the classical Wagner distribution obtained in the limit of a vanishingly small area fraction of the clusters. A much better agreement is found with the theory of Conti et al. [Phys. Rev. E 65, 046117 (2002)] which accounts for the cluster merger.Comment: 5 pages, to appear in PR

Extinction rates of established spatial populations

This paper deals with extinction of an isolated population caused by intrinsic noise. We model the population dynamics in a "refuge" as a Markov process which involves births and deaths on discrete lattice sites and random migrations between neighboring sites. In extinction scenario I the zero population size is a repelling fixed point of the on-site deterministic dynamics. In extinction scenario II the zero population size is an attracting fixed point, corresponding to what is known in ecology as Allee effect. Assuming a large population size, we develop WKB (Wentzel-Kramers-Brillouin) approximation to the master equation. The resulting Hamilton's equations encode the most probable path of the population toward extinction and the mean time to extinction. In the fast-migration limit these equations coincide, up to a canonical transformation, with those obtained, in a different way, by Elgart and Kamenev (2004). We classify possible regimes of population extinction with and without an Allee effect and for different types of refuge and solve several examples analytically and numerically. For a very strong Allee effect the extinction problem can be mapped into the over-damped limit of theory of homogeneous nucleation due to Langer (1969). In this regime, and for very long systems, we predict an optimal refuge size that maximizes the mean time to extinction.Comment: 26 pages including 3 appendices, 16 figure

Symmetry-breaking instability in a prototypical driven granular gas

Symmetry-breaking instability of a laterally uniform granular cluster (strip state) in a prototypical driven granular gas is investigated. The system consists of smooth hard disks in a two-dimensional box, colliding inelastically with each other and driven, at zero gravity, by a "thermal" wall. The limit of nearly elastic particle collisions is considered, and granular hydrodynamics with the Jenkins-Richman constitutive relations is employed. The hydrodynamic problem is completely described by two scaled parameters and the aspect ratio of the box. Marginal stability analysis predicts a spontaneous symmetry breaking instability of the strip state, similar to that predicted recently for a different set of constitutive relations. If the system is big enough, the marginal stability curve becomes independent of the details of the boundary condition at the driving wall. In this regime, the density perturbation is exponentially localized at the elastic wall opposite to the thermal wall. The short- and long-wavelength asymptotics of the marginal stability curves are obtained analytically in the dilute limit. The physics of the symmetry-breaking instability is discussed.Comment: 11 pages, 14 figure

Thermal Instability-Induced Interstellar Turbulence

We study the dynamics of phase transitions in the interstellar medium by means of three-dimensional hydrodynamic numerical simulations. We use a realistic cooling function and generic nonequilibrium initial conditions to follow the formation history of a multiphase medium in detail in the absence of gravity. We outline a number of qualitatively distinct stages of this process, including a linear isobaric evolution, transition to an isochoric regime, formation of filaments and voids (also known as "thermal" pancakes), the development and decay of supersonic turbulence, an approach to pressure equilibrium, and final relaxation of the multiphase medium. We find that 1%-2% of the initial thermal energy is converted into gas motions in one cooling time. The velocity field then randomizes into turbulence that decays on a dynamical timescale E_k ~ t^-n, 1 < n < 2. While not all initial conditions yield a stable two-phase medium, we examine such a case in detail. We find that the two phases are well mixed with the cold clouds possessing a fine-grained structure near our numerical resolution limit. The amount of gas in the intermediate unstable phase roughly tracks the rms turbulent Mach number, peaking at 25% when M_rms ~ 8, decreasing to 11% when M_rms ~ 0.4.Comment: To appear in the ApJ Letters, April 2002; 5 pages, 3 color figures, mpeg animations available at http://akpc.ucsd.edu/ThermalLetter/thermal.htm