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Register a new userThe Density Variance--Mach Number Relation in Supersonic Turbulence: I. Isothermal, magnetised gas
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It is widely accepted that supersonic, magnetised turbulence plays a fundamental role for star formation in molecular clouds. It produces the initial dense gas seeds out of which new stars can form. However, the exact relation between gas compression, turbulent Mach number, and magnetic field strength is still poorly understood. Here, we introduce and test an analytical prediction for the relation between the density variance and the root-mean-square Mach number in supersonic, isothermal, magnetised turbulent flows. We approximate the density and velocity structure of the interstellar medium as a superposition of shock waves. We obtain the density contrast considering the momentum equation for a single magnetised shock and extrapolate this result to the entire cloud. Depending on the field geometry, we then make three different assumptions based on observational and theoretical constraints: B independent of density, B proportional to the root square of the density and B proportional to the density. We test the analytically derived density variance--Mach number relation with numerical simulations, and find that for B proportional to the root square of the density, the variance in the logarithmic density contrast, $sigma_{ln rho/rho_0}^2=ln[1+b^2mathscr{M}^2beta_0/(beta_0+1)]$, fits very well to simulated data with turbulent forcing parameter b=0.4, when the gas is super-Alfvenic. However, this result breaks down when the turbulence becomes trans-Alfvenic or sub-Alfvenic, because in this regime the turbulence becomes highly anisotropic. Our density variance--Mach number relations simplify to the purely hydrodynamic relation as the ratio of thermal to magnetic pressure $beta_0$ approaches infinite.
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Supersonic turbulence in molecular clouds is a key agent in generating density enhancements that may subsequently go on to form stars. The stronger the turbulence - the higher the Mach number - the more extreme the density fluctuations are expected to be. Numerical models predict an increase in density variance with rms Mach number of the form: sigma^{2}_{rho/rho_{0}} = b^{2}M^{2}, where b is a numerically-estimated parameter, and this prediction forms the basis of a large number of analytic models of star formation. We provide an estimate of the parameter b from 13CO J=1-0 spectral line imaging observations and extinction mapping of the Taurus molecular cloud, using a recently developed technique that needs information contained solely in the projected column density field to calculate sigma^{2}_{rho/rho_{0}}. We find b ~ 0.48, which is consistent with typical numerical estimates, and is characteristic of turbulent driving that includes a mixture of solenoidal and compressive modes. More conservatively, we constrain b to lie in the range 0.3-0.8, depending on the influence of sub-resolution structure and the role of diffuse atomic material in the column density budget. We also report a break in the Taurus column density power spectrum at a scale of ~1pc, and find that the break is associated with anisotropy in the power spectrum. The break is observed in both 13CO and dust extinction power spectra, which, remarkably, are effectively identical despite detailed spatial differences between the 13CO and dust extinction maps. [ abridged ]
Shocks form the basis of our understanding for the density and velocity statistics of supersonic turbulent flows, such as those found in the cool interstellar medium (ISM). The variance of the density field, $sigma^2_{rho/rho_0}$, is of particular interest for molecular clouds (MCs), the birthplaces of stars in the Universe. The density variance may be used to infer underlying physical processes in an MC, and parameterises the star formation (SF) rate of a cloud. However, models for $sigma^2_{rho/rho_0}$ all share a common feature -- the variance is assumed to be isotropic. This assumption does not hold when a trans/sub-Alfvenic mean magnetic field, $vec{B}_0$, is present in the cloud, which observations suggest is relevant for some MCs. We develop an anisotropic model for $sigma_{rho/rho_0}^2$, using contributions from hydrodynamical and fast magnetosonic shocks that propagate orthogonal to each other. Our model predicts an upper bound for $sigma_{rho/rho_0}^2$ in the high Mach number $(mathcal{M})$ limit as small-scale density fluctuations become suppressed by the strong $vec{B}_0$. The model reduces to the isotropic $sigma_{rho/rho_0}^2-mathcal{M}$ relation in the hydrodynamical limit. To validate our model, we calculate $sigma_{rho/rho_0}^2$ from 12~high-resolution, three-dimensional, supersonic, sub-Alfvenic magnetohydrodynamical (MHD) turbulence simulations and find good agreement with our theory. We discuss how the two MHD shocks may be the bimodally oriented over-densities observed in some MCs and the implications for SF theory in the presence of a sub-Alfvenic $vec{B}_0$. By creating an anisotropic, supersonic density fluctuation model, this study paves the way for SF theory in the highly anisotropic regime of interstellar turbulence.
The properties of supersonic isothermal turbulence influence a variety of astrophysical phenomena, including the structure and evolution of star forming clouds. This work presents a simple model for the structure of dense regions in turbulence in which the density distribution behind isothermal shocks originates from rough hydrostatic balance between the pressure gradient behind the shock and its deceleration from ram pressure applied by the background fluid. Using simulations of supersonic isothermal turbulence and idealized waves moving through a background medium, we show that the structural properties of dense, shocked regions broadly agree with our analytical model. Our work provides a new conceptual picture for describing the dense regions, which complements theoretical efforts to understand the bulk statistical properties of turbulence and attempts to model the more complex features of star forming clouds like magnetic fields, self-gravity, or radiative properties.
The interstellar medium of the Milky Way is multi-phase, magnetized and turbulent. Turbulence in the interstellar medium produces a global cascade of random gas motions, spanning scales ranging from 100 parsecs to 1000 kilometres. Fundamental parameters of interstellar turbulence such as the sonic Mach number (the speed of sound) have been difficult to determine because observations have lacked the sensitivity and resolution to directly image the small-scale structure associated with turbulent motion. Observations of linear polarization and Faraday rotation in radio emission from the Milky Way have identified unusual polarized structures that often have no counterparts in the total radiation intensity or at other wavelengths, and whose physical significance has been unclear. Here we report that the gradient of the Stokes vector (Q,U), where Q and U are parameters describing the polarization state of radiation, provides an image of magnetized turbulence in diffuse ionized gas, manifested as a complex filamentary web of discontinuities in gas density and magnetic field. Through comparison with simulations, we demonstrate that turbulence in the warm ionized medium has a relatively low sonic Mach number, M_s <~ 2. The development of statistical tools for the analysis of polarization gradients will allow accurate determinations of the Mach number, Reynolds number and magnetic field strength in interstellar turbulence over a wide range of conditions.
Supersonic isothermal turbulence establishes a network of transient dense shocks that sweep up material and have a density profile described by balance between ram pressure of the background fluid versus the magnetic and gas pressure gradient behind the shock. These rare, densest regions of a turbulent environment can become Jeans unstable and collapse to form pre-stellar cores. Using numerical simulations of magneto-gravo-turbulence, we describe the structural properties of dense shocks, which are the seeds of gravitational collapse, as a function of magnetic field strength. In the regime of a weak magnetic field, the collapse is isotropic. Strong magnetic field strengths lead to significant anisotropy in the shocked distribution and collapse occurs preferentially parallel to the field lines. Our work provides insight into analyzing the magnetic field topology and density structures of young protostellar collapse, which the theory presented here predicts are associated with large-scale strong shocks that persist for at least a free-fall time.
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