Jongsoo Kim

A multidimensional code for isothermal magnetohydrodynamic flows in astrophysics

We present a multidimensional numerical code to solve isothermal magnetohydrodynamic (IMHD) equations for use in modeling astrophysical flows. First we have built a one-dimensional code which is based on an explicit finite-difference method on an Eulerian grid, called the total variation diminishing (TVD) scheme. The TVD scheme is a second-order-accurate extension of the Roe-type upwind scheme. Recipes for building the one-dimensional IMHD code, including the normalized right and left eigenvectors of the IMHD Jacobian matrix, are presented. Then we have extended the one-dimensional code to a multidimensional IMHD code through a Strang-type dimensional splitting. In the multidimensional code, an explicit cleaning step has been included to eliminate nonzero ∇B at every time step. To test the code, IMHD shock tube problems, which encompass all the physical IMHD structures, have been constructed. One-dimensional and two-dimensional shock tube tests have shown that the code captures all the structures correctly without producing noticeable oscillations. Strong shocks are resolved sharply, but weaker shocks spread more. Numerical dissipation (viscosity and resistivity) has been estimated through the decay test of a two-dimensional Alfven wave. It has been found to be slightly smaller than that of the adiabatic magnetohydrodynamic code based on the same scheme. As an example of astrophysical applications, we have simulated the nonlinear evolution of the two-dimensional Parker instability under a uniform gravity.

Three-dimensional Evolution of the Parker Instability under a Uniform Gravity

Using an isothermal MHD code, we have performed three-dimensional, high-resolution simulations of the Parker instability. The initial equilibrium system is composed of exponentially decreasing isothermal gas and a magnetic field (along the azimuthal direction) under a uniform gravity. The evolution of the instability can be divided into three phases: linear, nonlinear, and relaxed. During the linear phase, the perturbations grow exponentially with a preferred scale along the azimuthal direction but with the smallest possible scale along the radial direction, as predicted from linear analyses. During the nonlinear phase, the growth of the instability is saturated and flow motion becomes chaotic. Magnetic reconnection occurs, which allows the gas to cross field lines. This, in turn, results in the redistribution of the gas and the magnetic field. The system approaches a new equilibrium in the relaxed phase, which is different from the one seen in two-dimensional works. The structures formed during the evolution are sheetlike or filamentary, whose shortest dimension is radial. Their maximum density enhancement factor relative to the initial value is less than 2. Since the radial dimension is too small and the density enhancement is too low, it is difficult to regard the Parker instability alone as a viable mechanism for the formation of giant molecular clouds.

AN EXPLICIT SCHEME FOR INCORPORATING AMBIPOLAR DIFFUSION IN A MAGNETOHYDRODYNAMICS CODE

We describe a method for incorporating ambipolar diffusion in the strong coupling approximation into a multidimensional magnetohydrodynamic (MHD) code based on the total variation diminishing scheme. Contributions from ambipolar diffusion terms are included by explicit finite-difference operators in a fully unsplit way, maintaining second-order accuracy. The divergence-free condition of magnetic fields is exactly ensured at all times by a flux-interpolated constrained transport scheme. The super time-stepping method is used to accelerate the time step in high-resolution calculations and/or in strong ambipolar diffusion. We perform two test problems, the steady-state oblique C-type shocks and the decay of Alfven waves, confirming the accuracy and robustness of our numerical approach. Results from the simulations of the compressible MHD turbulence with ambipolar diffusion show the flexibility of our method as well as its ability to follow complex MHD flows in the presence of ambipolar diffusion. These simulations show that the dissipation rate of the MHD turbulence is strongly affected by the strength of ambipolar diffusion.

Enhanced core formation rate in a turbulent cloud by self-gravity

We performed a numerical experiment designed for core formation in a self-gravitating, magnetically supercritical, supersonically turbulent, isothermal cloud. A density probability distribution function (PDF) averaged over a converged turbulent state before turning self-gravity on is well fitted with a lognormal distribution. However, after turning self-gravity on, the volume fractions of density PDFs at a high density tail, compared with the lognormal distribution, increase as time goes on. In order to see the effect of self-gravity on core formation rates, we compared the core formation rate per free-fall time (CFRff) from the theory based on the lognormal distribution and the one from our numerical experiment. For our fiducial value of a critical density, 100, normalized with an initial value, the latter CFRff is about 30 times larger the former one. Therefore, self-gravity plays an important role in significantly increasing CFRff. This result implies that core (star) formation rates or core (stellar) mass functions predicted from theories based on the lognormal density PDF need some modifications. Our result of the increased volume fraction of density PDFs after turning self-gravity on is consistent with power law like tails commonly observed at higher ends of visual extinction PDFs of active star-forming clouds.

A Comparative Study of the Parker Instability in Three Models of the Galactic Gravity

To examine how the nonuniform nature of the Galactic gravity might affect the length and timescales of the Parker instability, we took three models of gravity, including the usual uniform one. In a linear model we let the acceleration perpendicular to the Galactic plane increase linearly with vertical distance z from the midplane. As a more realistic choice, we let a hyperbolic tangent function of z describe the observationally known variation of the vertical acceleration. To make comparisons of the three gravity models on a common basis, we first fixed the ratio of magnetic pressure to gas pressure at α = 0.25, that of cosmic-ray to gas pressure at β = 0.4, and the rms velocity of interstellar clouds at as = 6.4 km s-1; then we adjusted the parameters of the gravity models in such a way that the resulting density scale heights for the three models all have the same value of 160 pc. In the initial equilibrium state, the vertical density structure is given by an exponential, a Gaussian, and a power of the hyperbolic cosine functions of z for the uniform, linear, and realistic gravity models, respectively. Performing linear stability analyses on these equilibria with the same interstellar medium conditions specified by the above α-, β-, and as-values, we calculate the maximum growth rate and corresponding length scale for each of the gravity models. Under the uniform gravity condition the Parker instability has a growth time of 1.2 × 108 yr and a length scale of 1.6 kpc for the symmetric mode. Under the realistic gravity condition it grows in the span of 1.8 × 107 yr for both symmetric and antisymmetric modes and develops density condensations at intervals of 400 pc for the symmetric mode and 200 pc for the antisymmetric one. A simple change of the gravity model has thus reduced the growth time by almost an order of magnitude and its length scale by factors of 4-8. These results suggest that an onset of the Parker instability in the interstellar medium may not necessarily be confined to the regions of high α and β.

The Parker Instability under a Linear Gravity

A linear stability analysis has been done to a magnetized disk under a linear gravity. We have reduced the linearized perturbation equations to a second-order differential equation that resembles the Schrodinger equation with the potential of a harmonic oscillator. Depending on the signs of energy and potential terms, eigensolutions can be classified into continuum and discrete families. When the magnetic field is ignored, the continuum family is identified as the convective mode, while the discrete family is identified as acoustic-gravity waves. If the effective adiabatic index γ is less than unity, the former develops into the convective instability. When a magnetic field is included, the continuum and discrete families further branch into several solutions with different characters. The continuum family is divided into two modes: one is the original Parker mode, which is a slow MHD mode modulated by the gravity, and the other is a stable Alfven mode. The Parker modes can be either stable or unstable depending on γ. When γ is smaller than a critical value γcr, the Parker mode becomes unstable. The discrete family is divided into three modes: a stable fast MHD mode modulated by the gravity, a stable slow MHD mode modulated by the gravity, and an unstable mode that is also attributed to a slow MHD mode. The unstable discrete mode does not always exist. Even though the unstable discrete mode exists, the Parker mode dominates it if the Parker mode is unstable. However, if γ ≥ γcr, then the discrete mode could be the only unstable one. When γ is equal γcr, the minimum growth time of the unstable discrete mode is 1.3 × 108 yr, with a corresponding length scale of 2.4 kpc. It is suggestive that the corrugated features seen in the Galaxy and external galaxies are related to the unstable discrete mode.

ESTIMATION OF MAGNETIC FIELD STRENGTH IN THE TURBULENT WARM IONIZED MEDIUM

We studied Faraday rotation measure (RM) in turbulent media with the rms Mach number of unity, using isothermal, magnetohydrodynamic turbulence simulations. Four cases with different values of initial plasma beta were considered. Our main findings are as follows. (1) There is no strong correlation between the fluctuations of magnetic field strength and gas density. So the magnetic field strength estimated with RM/DM (DM is the dispersion measure) correctly represents the true mean strength of the magnetic field along the line of sight. (2) The frequency distribution of RMs is well fitted to the Gaussian. In addition, there is a good relation between the width of the distribution of RM/RM-bar (RM-bar is the average value of RMs) and the strength of the regular field along the line of sight; the width is narrower, if the field strength is stronger. We discussed the implications of our findings in the warm ionized medium where the Mach number of turbulent motions is around unity.