Scaling relations of supersonic turbulence in star-forming molecular clouds

We present a direct numerical and analytical study of driven supersonic MHD turbulence that is believed to govern the dynamics of star-forming molecular clouds. We describe statistical properties of the turbulence by measuring the velocity difference structure functions up to the fifth order. In particular, the velocity power spectrum in the inertial range is found to be close to E(k) \~ k^{-1.74}, and the velocity difference scales as ~ L^{0.42}. The results agree well with the Kolmogorov--Burgers analytical model suggested for supersonic turbulence in [astro-ph/0108300]. We then generalize the model to more realistic, fractal structure of molecular clouds, and show that depending on the fractal dimension of a given molecular cloud, the theoretical value for the velocity spectrum spans the interval [-1.74 ... -1.89], while the corresponding window for the velocity difference scaling exponent is [0.42 ... 0.78].Comment: 17 pages, 6 figures include

Density-PDFs and Lagrangian Statistics of highly compressible Turbulence

We report on probability-density-functions (PDF) of the mass density in numerical simulations of highly compressible hydrodynamic flows and the corresponding structure formation of Lagrangian particles advected by the flows. Numerical simulations were performed with $512^3$ collocation points and 2 million tracer particles integrated over several dynamical times. We propose a connection between the PDF of the Lagrangian tracer particles and the predicted log-normal distribution of the density fluctuations in isothermal systems

Dominance margins for feedback systems

The paper introduces notions of robustness margins geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and in engineering, a theory of robustness for behaviors away from equilibria is lacking. The proposed framework addresses this need in the framework of p-dominance theory, which aims at generalizing stability theory for the analysis of systems with low-dimensional attractors. Dominance margins are introduced as natural generalisations of stability margins in the context of p-dominance analysis. In analogy with stability margins, dominance margins are shown to admit simple interpretations in terms of familiar frequency domain tools and to provide quantitative measures of robustness for multistable and oscillatory behaviors in Lure systems. The theory is illustrated by means of an elementary mechanical example.The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645

Structure Function Scaling of a 2MASS Extinction Map of Taurus

We compute the structure function scaling of a 2MASS extinction map of the Taurus molecular cloud complex. The scaling exponents of the structure functions of the extinction map follow the Boldyrev's velocity structure function scaling of supersonic turbulence. This confirms our previous result based on a spectral map of 13CO J=1-0 covering the same region and suggests that supersonic turbulence is important in the fragmentation of this star--forming cloud.Comment: submitted to Ap

Structure Function Scaling in Compressible Super-Alfvenic MHD Turbulence

Supersonic turbulent flows of magnetized gas are believed to play an important role in the dynamics of star-forming clouds in galaxies. Understanding statistical properties of such flows is crucial for developing a theory of star formation. In this letter we propose a unified approach for obtaining the velocity scaling in compressible and super--Alfv\'{e}nic turbulence, valid for arbitrary sonic Mach number, \ms. We demonstrate with numerical simulations that the scaling can be described with the She--L\'{e}v\^{e}que formalism, where only one parameter, interpreted as the Hausdorff dimension of the most intense dissipative structures, needs to be varied as a function of \ms. Our results thus provide a method for obtaining the velocity scaling in interstellar clouds once their Mach numbers have been inferred from observations.Comment: published in Physical Review Letter

Mean field variational bayes for elaborate distributions

We develop strategies for mean field variational Bayes approximate inference for Bayesian hierarchical models containing elaborate distributions. We loosely define elaborate distributions to be those having more complicated forms compared with common distributions such as those in the Normal and Gamma families. Examples are Asymmetric Laplace, Skew Normal and Generalized Extreme Value distributions. Such models suffer from the difficulty that the parameter updates do not admit closed form solutions. We circumvent this problem through a combination of (a) specially tailored auxiliary variables, (b) univariate quadrature schemes and (c) finite mixture approximations of troublesome density functions. An accuracy assessment is conducted and the new methodology is illustrated in an application. © 2011 International Society for Bayesian Analysis

The Protostellar Mass Function

The protostellar mass function (PMF) is the Present-Day Mass Function of the protostars in a region of star formation. It is determined by the initial mass function weighted by the accretion time. The PMF thus depends on the accretion history of protostars and in principle provides a powerful tool for observationally distinguishing different protostellar accretion models. We consider three basic models here: the Isothermal Sphere model (Shu 1977), the Turbulent Core model (McKee & Tan 2003), and an approximate representation of the Competitive Accretion model (Bonnell et al. 1997, 2001a). We also consider modified versions of these accretion models, in which the accretion rate tapers off linearly in time. Finally, we allow for an overall acceleration in the rate of star formation. At present, it is not possible to directly determine the PMF since protostellar masses are not currently measurable. We carry out an approximate comparison of predicted PMFs with observation by using the theory to infer the conditions in the ambient medium in several star-forming regions. Tapered and accelerating models generally agree better with observed star-formation times than models without tapering or acceleration, but uncertainties in the accretion models and in the observations do not allow one to rule out any of the proposed models at present. The PMF is essential for the calculation of the Protostellar Luminosity Function, however, and this enables stronger conclusions to be drawn (Offner & McKee 2010).Comment: 16 pages, 8 figures, published in Ap

Simulating Supersonic Turbulence in Magnetized Molecular Clouds

We present results of large-scale three-dimensional simulations of weakly magnetized supersonic turbulence at grid resolutions up to 1024^3 cells. Our numerical experiments are carried out with the Piecewise Parabolic Method on a Local Stencil and assume an isothermal equation of state. The turbulence is driven by a large-scale isotropic solenoidal force in a periodic computational domain and fully develops in a few flow crossing times. We then evolve the flow for a number of flow crossing times and analyze various statistical properties of the saturated turbulent state. We show that the energy transfer rate in the inertial range of scales is surprisingly close to a constant, indicating that Kolmogorov's phenomenology for incompressible turbulence can be extended to magnetized supersonic flows. We also discuss numerical dissipation effects and convergence of different turbulence diagnostics as grid resolution refines from 256^3 to 1024^3 cells.Comment: 10 pages, 3 figures, to appear in the proceedings of the DOE/SciDAC 2009 conferenc

Synthetic Molecular Clouds from Supersonic MHD and Non-LTE Radiative Transfer Calculations

The dynamics of molecular clouds is characterized by supersonic random motions in the presence of a magnetic field. We study this situation using numerical solutions of the three-dimensional compressible magneto-hydrodynamic (MHD) equations in a regime of highly supersonic random motions. The non-LTE radiative transfer calculations are performed through the complex density and velocity fields obtained as solutions of the MHD equations, and more than 5x10^5 synthetic molecular spectra are obtained. We use a numerical flow without gravity or external forcing. The flow is super-Alfvenic and corresponds to model A of Padoan and Nordlund (1997). Synthetic data consist of sets of 90x90 synthetic spectra with 60 velocity channels, in five molecular transitions: J=1-0 and J=2-1 for 12CO and 13CO, and J=1-0 for CS. Though we do not consider the effects of stellar radiation, gravity, or mechanical energy input from discrete sources, our models do contain the basic physics of magneto-fluid dynamics and non-LTE radiation transfer and are therefore more realistic than previous calculations. As a result, these synthetic maps and spectra bear a remarkable resemblance to the corresponding observations of real clouds.Comment: 33 pages, 12 figures included, 5 jpeg figures not included (fig1a, fig1b, fig3, fig4 fig5), submitted to Ap