The correlation between magnetic pressure and density in compressible MHD turbulence

We study, both analytically and numerically, the behavior of magnetic pressure and density fluctuations in turbulent isothermal magnetohydrodynamic (MHD) flows in a slab geometry. We first consider simple MHD waves, which are the nonlinear analogue of the slow, fast and Alfven linear waves, and show that the dependence of magnetic field strength B on density rho in a simple wave depends on the mode which is considered: for the slow mode, B-2 similar or equal to c(1) - c(2)rho, while for the fast mode, B-2 similar or equal to rho(2). We also perform a perturbative analysis about a circularly-polarized plane Alfven wave to investigate Alfven wave pressure, recovering the results of McKee and Zweibel that B-2 similar or equal to rho(gammae), with gamma(e) similar or equal to 2 at large M-a, gamma(e) similar or equal to 3/2 at moderate M-a and long wavelengths, and gamma(e) similar or equal to 1/2 at low M-a. This wide variety of behaviors implies that a single polytropic description of magnetic pressure is not possible in general, but instead depends on which mode dominates the density fluctuation production. This in turn depends on the angle theta between the magnetic field and the direction of wave propagation and on the Alfvenic Mach number M-a. Typically, at small M-a, the slow mode dominates, and B is anticorrelated with rho. At large M-a, both modes contribute to density fluctuation production, and the magnetic pressure decorrelates from density, exhibiting a large scatter, which however decreases towards higher densities. In this case, magnetic pressure does not act as a restoring force, but rather as a random forcing. These results have implications for the probability density function (PDF) of mass density. The non-systematic behavior of the magnetic pressure causes the PDF to maintain the log-normal shape corresponding to non-magnetic isothermal turbulence, except in cases where the slow mode dominates, in which the PDF develops an excess at low densities because the magnetic random forcing becomes density-dependent. Our results are consistent with the low values and apparent lack of correlation between the magnetic field strength and density in surveys of the lower-density molecular gas, and also with the recorrelation apparently seen at higher densities, if the Alfvenic Mach number is relatively large there.