Stellar feedback drives the circulation of matter from the disk to the halo of galaxies. We perform three-dimensional magnetohydrodynamic simulations of a vertical column of the interstellar medium with initial conditions typical of the solar circle in which supernovae drive turbulence and determine the vertical stratification of the medium. The simulations were run using a stable, positivity-preserving scheme for ideal MHD implemented in the FLASH code. We find that the majority (≈ 90%) of the mass is contained in thermally-stable temperature regimes of cold molecular and atomic gas at T < 200 K or warm atomic and ionized gas at 5000 K < T < 10 4.2 K, with strong peaks in probability distribution functions of temperature in both the cold and warm regimes. The 200 − 10 4.2 K gas fills 50−60% of the volume near the plane, with hotter gas associated with supernova remnants (30−40%) and cold clouds (< 10%) embedded within. At |z| ∼ 1 − 2 kpc, transition-temperature (10 5 K) gas accounts for most of the mass and volume, while hot gas dominates at |z| > 3 kpc. The magnetic field in our models has no significant impact on the scale heights of gas in each temperature regime; the magnetic tension force is approximately equal to and opposite the magnetic pressure, so the addition of the field does not significantly affect the vertical support of the gas. The addition of a magnetic field does reduce the fraction of gas in the cold (< 200 K) regime with a corresponding increase in the fraction of warm (∼ 10 4 K) gas. However, our models lack rotational shear and thus have no largescale dynamo, which reduces the role of the field in the models compared to reality. The supernovae drive oscillations in the vertical distribution of halo gas, with the period of the oscillations ranging from ≈ 30 Myr in the T < 200 K gas to ∼ 100 Myr in the 10 6 K gas, in line with predictions by Walters & Cox.

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a b s t r a c tThe ideal MHD equations are a central model in astrophysics, and their solution relies upon stable numerical schemes. We present an implementation of a new method, which possesses excellent stability properties. Numerical tests demonstrate that the theoretical stability properties are valid in practice with negligible compromises to accuracy. The result is a highly robust scheme with state-of-the-art efficiency. The scheme's robustness is due to entropy stability, positivity and properly discretised Powell terms. The implementation takes the form of a modification of the MHD module in the FLASH code, an adaptive mesh refinement code. We compare the new scheme with the standard FLASH implementation for MHD. Results show comparable accuracy to standard FLASH with the Roe solver, but highly improved efficiency and stability, particularly for high Mach number flows and low plasma b. The tests include 1D shock tubes, 2D instabilities and highly supersonic, 3D turbulence. We consider turbulent flows with RMS sonic Mach numbers up to 10, typical of gas flows in the interstellar medium. We investigate both strong initial magnetic fields and magnetic field amplification by the turbulent dynamo from extremely high plasma b. The energy spectra show a reasonable decrease in dissipation with grid refinement, and at a resolution of 512 3 grid cells we identify a narrow inertial range with the expected power law scaling. The turbulent dynamo exhibits exponential growth of magnetic pressure, with the growth rate higher from solenoidal forcing than from compressive forcing. Two versions of the new scheme are presented, using relaxation-based 3-wave and 5-wave approximate Riemann solvers, respectively. The 5-wave solver is more accurate in some cases, and its computational cost is close to the 3-wave solver.

Abstract. In the first part of this work ([5]), we introduced an approximate Riemann solver for one-dimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation, and derive explicit wave speed estimates satisfying the stability conditions of [5]. We present a 3-wave solver that well resolves fast waves and material contacts, and a 5-wave solver that accurately resolves the cases when two eigenvalues coincide. A full 7-wave solver, which is highly accurate on all types of waves, will be described in a follow-up paper. We test the solvers on one-dimensional shock tube data and smooth shear waves.