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Register a new userThe Probability Distribution of Density Fluctuations in Supersonic Turbulence
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We study density fluctuations in supersonic turbulence using both theoretical methods and numerical simulations. A theoretical formulation is developed for the probability distribution function (PDF) of the density at steady state, connecting it to the conditional statistics of the velocity divergence. Two sets of numerical simulations are carried out, using either a Riemann solver to evolve the Euler equations or a finite-difference method to evolve the Navier-Stokes (N-S) equations. After confirming the validity of our theoretical formulation with the N-S simulations, we examine the effects of dynamical processes on the PDF, showing that the nonlinear term in the divergence equation amplifies the right tail of the PDF and reduces the left one, the pressure term reduces both the right and left tails, and the viscosity term, counter-intuitively, broadens the right tail of the PDF. Despite the inaccuracy of the velocity divergence from the Riemann runs, as found in our previous work, we show that the density PDF from the Riemann runs is consistent with that from the N-S runs. Taking advantage of their much higher effective resolution, we then use the Riemann runs to study the dependence of the PDF on the Mach number, $mathcal{M}$, up to $mathcal{M}sim30$. The PDF width, $sigma_{s}$, follows the relation $sigma_{s}^2 = ln (1+b^2 {mathcal M}^2)$, with $bapprox0.38$. However, the PDF exhibits a negative skewness that increases with increasing $mathcal{M}$, so much of the growth of $sigma_{s}$ is accounted for by the growth of the left PDF tail, while the growth of the right tail tends to saturate. Thus, the usual prescription that combines a lognormal shape with the standard variance-Mach number relation greatly overestimates the right PDF tail at large $mathcal{M}$, which may have a significant impact on theoretical models of star formation.
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The rich structure that we observe in molecular clouds is due to the interplay between strong magnetic fields and supersonic (turbulent) velocity fluctuations. The velocity fluctuations interact with the magnetic field, causing it too to fluctuate. Using numerical simulations, we explore the nature of such magnetic field fluctuations, $vec{delta B}$, over a wide range of turbulent Mach numbers, $mathcal{M} = 2 - 20$ (i.e., from weak to strong compressibility), and Alfven Mach numbers, $mathcal{M}_{text{A}0} = 0.1 - 100$ (i.e., from strong to weak magnetic mean fields, $B_0$). We derive a compressible quasi-static fluctuation model from the magnetohydrodynamical (MHD) equations and show that velocity gradients parallel to the mean magnetic field give rise to compressible modes in sub-Alfvenic flows, which prevents the flow from becoming two-dimensional, as is the case in incompressible MHD turbulence. We then generalise an analytical model for the magnitude of the magnetic fluctuations to include $mathcal{M}$, and find $|vec{delta B}| = delta B = c_ssqrt{pirho_0}mathcal{M}mathcal{M}_{text{A}0}$, where $c_s$ is the sound speed and $rho_0$ is the mean density of gas. This new relation fits well in the strong $B$-field regime. We go on to study the anisotropy between the perpendicular ($ B_{perp}$) and parallel ($ B_{parallel}$) fluctuations and the mean-normalised fluctuations, which we find follow universal scaling relations, invariant of $mathcal{M}$. We provide a detailed analysis of the morphology for the $delta B_{perp}$ and $delta B_{parallel}$ probability density functions and find that eddies aligned with $B_0$ cause parallel fluctuations that reduce $B_{parallel}$ in the most anisotropic simulations. We discuss broadly the implications of our fluctuation models for magnetised gases in the interstellar medium.
We conduct numerical experiments to determine the density probability distribution function (PDF) produced in supersonic, isothermal, self-gravitating turbulence of the sort that is ubiquitous in star-forming molecular clouds. Our experiments cover a wide range of turbulent Mach number and virial parameter, allowing us for the first time to determine how the PDF responds as these parameters vary, and we introduce a new diagnostic, the dimensionless star formation efficiency versus density ($epsilon_{rm ff}(s)$) curve, which provides a sensitive diagnostic of the PDF shape and dynamics. We show that the PDF follows a universal functional form consisting of a log-normal at low density with two distinct power law tails at higher density; the first of these represents the onset of self-gravitation, and the second reflects the onset of rotational support. Once the star formation efficiency reaches a few percent, the PDF becomes statistically steady, with no evidence for secular time-evolution at star formation efficiencies from about five to 20 percent. We show that both the Mach number and the virial parameter influence the characteristic densities at which the log-normal gives way to the first power-law, and the first to the second, and we extend (for the former) and develop (for the latter) simple theoretical models for the relationship between these density thresholds and the global properties of the turbulent medium.
The probabilistic approach to turbulence is applied to investigate density fluctuations in supersonic turbulence. We derive kinetic equations for the probability distribution function (PDF) of the logarithm of the density field, $s$, in compressible turbulence in two forms: a first-order partial differential equation involving the average divergence conditioned on the flow density, $langle abla cdot {bs u} | srangle$, and a Fokker-Planck equation with the drift and diffusion coefficients equal to $-langle {bs u} cdot abla s | srangle$ and $langle {bs u} cdot abla s | srangle$, respectively. Assuming statistical homogeneity only, the detailed balance at steady state leads to two exact results, $langle abla cdot {bs u} | s rangle =0$, and $langle {bs u} cdot abla s | srangle=0$. The former indicates a balance of the flow divergence over all expanding and contracting regions at each given density. The exact results provide an objective criterion to judge the accuracy of numerical codes with respect to the density statistics in supersonic turbulence. We also present a method to estimate the effective numerical diffusion as a function of the flow density and discuss its effects on the shape of the density PDF.
Simulations generally show that non-self-gravitating clouds have a lognormal column density ($Sigma$) probability distribution function (PDF), while self-gravitating clouds with active star formation develop a distinct power-law tail at high column density. Although the growth of the power law can be attributed to gravitational contraction leading to the formation of condensed cores, it is often debated if an observed lognormal shape is a direct consequence of supersonic turbulence alone, or even if it is really observed in molecular clouds. In this paper we run three-dimensional magnetohydrodynamic simulations including ambipolar diffusion with different initial conditions to see the effect of strong magnetic fields and nonlinear initial velocity perturbations on the evolution of the column density PDFs. Our simulations show that column density PDFs of clouds with supercritical mass-to-flux ratio, with either linear perturbations or nonlinear turbulence, quickly develop a power-law tail such that $dN/d log Sigma propto Sigma^{-alpha}$ with index $alpha simeq 2$. Interestingly, clouds with subcritical mass-to-flux ratio also proceed directly to a power-law PDF, but with a much steeper index $alpha simeq 4$. This is a result of gravitationally-driven ambipolar diffusion. However, for nonlinear perturbations with a turbulent spectrum ($v_{k}^{2} propto k^{-4}$), the column density PDFs of subcritical clouds do retain a lognormal shape for a major part of the cloud evolution, and only develop a distinct power-law tail with index $alpha simeq 2$ at greater column density when supercritical pockets are formed.
We investigate the possibility of generating and studying turbulence in plasma by means of high-energy density laser-driven experiments. Our focus is to create supersonic, self-magnetized turbulence with characteristics that resemble those found in the interstellar medium (ISM). We consider a target made of a spherical core surrounded by a shell made of denser material. The shell is irradiated by a sequence of laser pulses sending inward-propagating shocks that convert the inner core into plasma and create turbulence. In the context of the evolution of the ISM, the shocks play the role of supernova remnant shocks and the core represents the ionized interstellar medium. We consider the effects of both pre-existing and self-generating magnetic fields and study the evolution of the system by means of two-dimensional numerical simulations. We find that the evolution of the turbulent core is generally, subsonic with rms-Mach number $M_tapprox 0.2$. We observe an isotropic, turbulent velocity field with an inertial range power spectra of $P(k)propto k^{-2.3}$. We account for the effects of self-magnetization and find that the resulting magnetic field has characteristic strength $approx 3times 10^{4}$ G. The corresponding plasma beta is $approx 1times 10^{4}$--$1times 10^{5}$, indicating that the magnetic field does not play an important role in the dynamical evolution of the system. The natural extension of this work is to study the system evolution in three-dimensions, with various laser drive configurations, and targets with shells and cores of different masses. The latter modification may help to increase the turbulent intensity and possibly create transonic turbulence. One of the key challenges is to obtain transonic turbulent conditions in a quasi-steady state environment.
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