Decay laws for three-dimensional magnetohydrodynamic turbulence

Decay laws for three-dimensional magnetohydrodynamic turbulence are obtained from high-resolution numerical simulations using up to 512^3 modes...

Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

Small-scale dynamo action during the formation of the first stars and galaxies. I. The ideal MHD limit

We explore the amplification of magnetic seed fields during the formation of the first stars and galaxies. During gravitational collapse, turbulence is created from accretion shocks, which may act to amplify weak magnetic fields in the protostellar cloud. Numerical simulations showed that such turbulence is sub-sonic in the first star-forming minihalos, and highly supersonic in the first galaxies with virial temperatures larger than 10^4 K. We investigate the magnetic field amplification during the collapse both for Kolmogorov and Burgers-type turbulence with a semi-analytic model that incorporates the effects of gravitational compression and small-scale dynamo amplification. We find that the magnetic field may be substantially amplified before the formation of a disk. On scales of 1/10 of the Jeans length, saturation occurs after ~10^8 yr. Although the saturation behaviour of the small-scale dynamo is still somewhat uncertain, we expect a saturation field strength of the order ~10^{-7} n^{0.5} G in the first star-forming halos, with n the number density in cgs units. In the first galaxies with higher turbulent velocities, the magnetic field strength may be increased by an order of magnitude, and saturation may occur after 10^6 to 10^7 yr. In the Kolmogorov case, the magnetic field strength on the integral scale (i.e. the scale with most magnetic power) is higher due to the characteristic power-law indices, but the difference is less than a factor of 2 in the saturated phase. Our results thus indicate that the precise scaling of the turbulent velocity with length scale is of minor importance. They further imply that magnetic fields will be significantly enhanced before the formation of a protostellar disk, where they may change the fragmentation properties of the gas and the accretion rate.Comment: 11 pages, 9 figures, accepted at A&

Kinetic Energy Decay Rates of Supersonic and Super-Alfvenic Turbulence in Star-Forming Clouds

We present numerical studies of compressible, decaying turbulence, with and without magnetic fields, with initial rms Alfven and Mach numbers ranging up to five, and apply the results to the question of the support of star-forming interstellar clouds of molecular gas. We find that, in 1D, magnetized turbulence actually decays faster than unmagnetized turbulence. In all the regimes that we have studied 3D turbulence-super-Alfvenic, supersonic, sub-Alfvenic, and subsonic-the kinetic energy decays as (t-t0)^(-x), with 0.85 < x < 1.2. We compared results from two entirely different algorithms in the unmagnetized case, and have performed extensive resolution studies in all cases, reaching resolutions of 256^3 zones or 350,000 particles. We conclude that the observed long lifetimes and supersonic motions in molecular clouds must be due to external driving, as undriven turbulence decays far too fast to explain the observations.Comment: Submitted to Phys. Rev. Letters, 29 Nov. 1997. 10 pages, 2 figures, also available from http://www.mpia-hd.mpg.de/theory/preprints.html#maclo