arXiv subjects

Wolfram Schmidt

Publications and source records attributed to Wolfram Schmidt.

31 records · Page 2Linked to original sources

arXiv2013

The small-scale dynamo: Breaking universality at high Mach numbers

(Abridged) The small-scale dynamo may play a substantial role in magnetizing the Universe under a large range of conditions, including subsonic turbulence at low Mach numbers, highly supersonic turbulence at high Mach numbers and a large range of magnetic Prandtl numbers Pm, i.e. the ratio of kinetic viscosity to magnetic resistivity. Low Mach numbers may in particular lead to the well-known, incompressible Kolmogorov turbulence, while for high Mach numbers, we are in the highly compressible regime, thus close to Burgers turbulence. In this study, we explore whether in this large range of conditions, a universal behavior can be expected. Our starting point are previous investigations in the kinematic regime. Here, analytic studies based on the Kazantsev model have shown that the behavior of the dynamo depends significantly on Pm and the type of turbulence, and numerical simulations indicate a strong dependence of the growth rate on the Mach number of the flow. Once the magnetic field saturates on the current amplification scale, backreactions occur and the growth is shifted to the next-larger scale. We employ a Fokker-Planck model to calculate the magnetic field amplification during the non-linear regime, and find a resulting power-law growth that depends on the type of turbulence invoked. For Kolmogorov turbulence, we confirm previous results suggesting a linear growth of magnetic energy. For more general turbulent spectra, where the turbulent velocity v_t scales with the characteristic length scale as u_\ell\propto \ell^{\vartheta}, we find that the magnetic energy grows as (t/T_{ed})^{2\vartheta/(1-\vartheta)}, with t the time-coordinate and T_{ed} the eddy-turnover time on the forcing scale of turbulence. For Burgers turbulence, \vartheta=1/2, a quadratic rather than linear growth may thus be expected, and a larger timescale until saturation is reached.

astro-ph.CO↗

arXiv2012

Magnetic fields during high redshift structure formation

We explore the amplification of magnetic fields in the high-redshift Universe. For this purpose, we perform high-resolution cosmological simulations following the formation of primordial halos with \sim10^7 M_solar, revealing the presence of turbulent structures and complex morphologies at resolutions of at least 32 cells per Jeans length. Employing a turbulence subgrid-scale model, we quantify the amount of unresolved turbulence and show that the resulting turbulent viscosity has a significant impact on the gas morphology, suppressing the formation of low-mass clumps. We further demonstrate that such turbulence implies the efficient amplification of magnetic fields via the small-scale dynamo. We discuss the properties of the dynamo in the kinematic and non-linear regime, and explore the resulting magnetic field amplification during primordial star formation. We show that field strengths of \sim10^{-5} G can be expected at number densities of \sim5 cm^{-3}.

astro-ph.CO↗

arXiv2011

Statistical properties of supersonic turbulence in the Lagrangian and Eulerian frameworks

We present a systematic study of the influence of different forcing types on the statistical properties of supersonic, isothermal turbulence in both the Lagrangian and Eulerian frameworks. We analyse a series of high-resolution, hydrodynamical grid simulations with Lagrangian tracer particles and examine the effects of solenoidal (divergence-free) and compressive (curl-free) forcing on structure functions, their scaling exponents, and the probability density functions of the gas density and velocity increments. Compressively driven simulations show a significantly larger density contrast, a more intermittent behaviour, and larger fractal dimension of the most dissipative structures at the same root mean square Mach number. We show that the absolute values of Lagrangian and Eulerian structure functions of all orders in the integral range are only a function of the root mean square Mach number, but independent of the forcing. With the assumption of a Gaussian distribution for the probability density function of the velocity increments on large scales, we derive a model that describes this behaviour.

physics.flu-dyn↗

arXiv2011

A semi-analytic model of the turbulent multi-phase interstellar medium

We present a semi-analytic model for the interstellar medium that considers local processes and structures of turbulent star-forming gas. A volume element of the interstellar medium is described as a multiphase system, comprising a cold and a warm gas phase in effective (thermal plus turbulent) pressure equilibrium, and a stellar component. Since turbulence has a critical impact on the shape of the gaseous phases, on the production of molecular hydrogen and on the formation of stars, the consistent treatment of turbulence energy -- the kinetic energy of unresolved motions -- is an important new feature of our model. Besides turbulence production by supernovae and by the cooling instability, we also take into account the forcing by large scale motions. We formulate a set of ordinary differential equations, which statistically describes star formation and the exchange between the different budgets of mass and energy in a region of the interstellar medium with given mean density, size, metallicity and external turbulence forcing. By exploring the behaviour of the solutions, we find equilibrium states, in which the star formation efficiencies are consistent with observations. Kennicutt-Schmidt-like relations naturally arise from the equilibrium solutions, while conventional star formation models in numerical simulations impose such relations with observed efficiency parameters as phenomenological calibrations. Beyond the semi-analytic approach, a potential application is a complete subgrid scale model of the unresolved multi-phase structure, star formation and turbulence in simulations of galaxies or in cosmological simulations. The formulation presented in this article combines various models focusing on particular processes and yet can be adopted to specific applications, depending on the range of resolved length scales.

astro-ph.CO↗

arXiv2010

Numerical and semi-analytic core mass distributions in supersonic isothermal turbulence

We investigate the influence of the turbulence forcing on the mass distributions of gravitationally unstable cores by postprocessing data from simulations of non-selfgravitating isothermal supersonic turbulence with varying resolution. In one set of simulations solenoidal forcing is applied, while the second set uses purely compressive forcing to excite turbulent motions. From the resulting density field, we compute the mass distribution of gravitationally unstable cores by means of a clump-finding algorithm. Using the time-averaged probability density functions of the mass density, semi-analytic mass distributions are calculated from analytical theories. We apply stability criteria that are based on the Bonnor-Ebert mass resulting from the thermal pressure and from the sum of thermal and turbulent pressure. Although there are uncertainties in the application of the clump-finding algorithm, we find systematic differences in the mass distributions obtained from solenoidal and compressive forcing. Compressive forcing produces a shallower slope in the high-mass power-law regime compared to solenoidal forcing. The mass distributions also depend on the Jeans length resulting from the choice of the mass in the computational box, which is freely scalable for non-selfgravitating isothermal turbulence. Provided that all cores are numerically resolved and most cores are small compared to the length scale of the forcing, the normalised core mass distributions are found to be close to the semi-analytic models. Especially for the high-mass tails, the Hennebelle-Chabrier theory implies that the additional support due to turbulent pressure is important.

astro-ph.SR↗

arXiv2009

Large Eddy Simulations of Supersonic Turbulence

Based on ideas by Woodward et al., a subgrid scale model that is applicable to highly compressible turbulence is presented. Applying the subgrid scale model in large eddy simulations of forced supersonic turbulence, the bottleneck effect is largely reduced and, thereby, approximate scaling laws can be obtained at relatively low numerical resolution. In agreement with previous results from PPM simulations without explicit subgrid scale model, it is found that the energy spectrum function for the velocity field with fractional density-weighing, rho^(1/3)u, varies substantially with the forcing, at least for the decade of wavenumbers next to the energy-containing range. Consequently, if universal scaling of compressible turbulence exists, it can be found on length scales much smaller than the forcing scale only.

astro-ph.GA↗

arXiv2009

Two-dimensional AMR simulations of colliding flows

Colliding flows are a commonly used scenario for the formation of molecular clouds in numerical simulations. Due to the thermal instability of the warm neutral medium, turbulence is produced by cooling. We carry out a two-dimensional numerical study of such colliding flows in order to test whether statistical properties inferred from adaptive mesh refinement (AMR) simulations are robust with respect to the applied refinement criteria. We compare probability density functions of various quantities as well as the clump statistics and fractal dimension of the density fields in AMR simulations to a static-grid simulation. The static grid with 2048^2 cells matches the resolution of the most refined subgrids in the AMR simulations. The density statistics is reproduced fairly well by AMR. Refinement criteria based on the cooling time or the turbulence intensity appear to be superior to the standard technique of refinement by overdensity. Nevertheless, substantial differences in the flow structure become apparent. In general, it is difficult to separate numerical effects from genuine physical processes in AMR simulations.

astro-ph.IM↗

arXiv2009

The Fractal Density Structure in Supersonic Isothermal Turbulence: Solenoidal versus Compressive Energy Injection

In a systematic study, we compare the density statistics in high resolution numerical experiments of supersonic isothermal turbulence, driven by the usually adopted solenoidal (divergence-free) forcing and by compressive (curl-free) forcing. We find that for the same rms Mach number, compressive forcing produces much stronger density enhancements and larger voids compared to solenoidal forcing. Consequently, the Fourier spectra of density fluctuations are significantly steeper. This result is confirmed using the Delta-variance analysis, which yields power-law exponents beta~3.4 for compressive forcing and beta~2.8 for solenoidal forcing. We obtain fractal dimension estimates from the density spectra and Delta-variance scaling, and by using the box counting, mass size and perimeter area methods applied to the volumetric data, projections and slices of our turbulent density fields. Our results suggest that compressive forcing yields fractal dimensions significantly smaller compared to solenoidal forcing. However, the actual values depend sensitively on the adopted method, with the most reliable estimates based on the Delta-variance, or equivalently, on Fourier spectra. Using these methods, we obtain D~2.3 for compressive and D~2.6 for solenoidal forcing, which is within the range of fractal dimension estimates inferred from observations (D~2.0-2.7). The velocity dispersion to size relations for both solenoidal and compressive forcing obtained from velocity spectra follow a power law with exponents in the range 0.4-0.5, in good agreement with previous studies.

astro-ph↗

arXiv2008

The Density Probability Distribution in Compressible Isothermal Turbulence: Solenoidal versus Compressive Forcing

The probability density function (PDF) of the gas density in turbulent supersonic flows is investigated with high-resolution numerical simulations. In a systematic study, we compare the density statistics of compressible turbulence driven by the usually adopted solenoidal forcing (divergence-free) and by compressive forcing (curl-free). Our results are in agreement with studies using solenoidal forcing. However, compressive forcing yields a significantly broader density distribution with standard deviation ~3 times larger at the same rms Mach number. The standard deviation-Mach number relation used in analytical models of star formation is reviewed and a modification of the existing expression is proposed, which takes into account the ratio of solenoidal and compressive modes of the turbulence forcing.

astro-ph↗

arXiv2008

Is the Scaling of Supersonic Turbulence Universal?

The statistical properties of turbulence are considered to be universal at sufficiently small length scales, i. e., independent of boundary conditions and large-scale forces acting on the fluid. Analyzing data from numerical simulations of supersonic turbulent flow driven by external forcing, we demonstrate that this is not generally true for the two-point velocity statistics of compressible turbulence. However, a reformulation of the refined similarity hypothesis in terms of the mass-weighted velocity rho^(1/3)v yields scaling laws that are almost insensitive to the forcing. The results imply that the most intermittent dissipative structures are shocks closely following the scaling of Burgers turbulence.

astro-ph↗

arXiv2008

Numerical simulations of compressively driven interstellar turbulence: I. Isothermal gas

We performed numerical simulations of supersonic isothermal turbulence driven by mostly compressive large-scale forcing, using both a static grid and adaptive mesh refinement with an effective resolution N=768^3. After a transient phase dominated by shocks, turbulence evolves into a steady state with an RMS Mach number about 2.5, in which cloud-like structures of over-dense gas are surrounded by highly rarefied gas. The index of the turbulence energy spectrum function beta = 2.0 in the shock-dominated phase. As the flow approaches statistical equilibrium, the spectrum flattens, with beta = 1.9. For the scaling exponent of the root mean square velocity fluctuation, we obtain gamma = 0.43 from the velocity structure functions of second order. These results are well within the range of observed scaling properties for the velocity dispersion in molecular clouds. Calculating structure functions of order p=1,...,5, we find for all scaling exponents significant deviations from the Kolmogorov-Burgers model proposed by Boldyrev. Our results are very well described by a general log-Poisson model with a higher degree of intermittency, which implies an influence of the forcing on the scaling properties. Contrary to previous numerical results for isothermal turbulence, we obtain a skewed probability density function of the mass density fluctuations that is not consistent with log-normal statistics and entails a substantially higher fraction of mass in the density peaks than implied by the Padoan-Nordlund relation between the variance of the density fluctuations and the Mach number. In conclusion, it seems necessary to account for the production mechanism of turbulence in the ISM.

astro-ph↗

arXiv2007

Numerical comparison of Riemann solvers for astrophysical hydrodynamics

The idea of this work is to compare a new positive and entropy stable approximate Riemann solver by Francois Bouchut with a state-of the-art algorithm for astrophysical fluid dynamics. We implemented the new Riemann solver into an astrophysical PPM-code, the Prometheus code, and also made a version with a different, more theoretically grounded higher order algorithm than PPM. We present shock tube tests, two-dimensional instability tests and forced turbulence simulations in three dimensions. We find subtle differences between the codes in the shock tube tests, and in the statistics of the turbulence simulations. The new Riemann solver increases the computational speed without significant loss of accuracy.

astro-ph↗

arXiv2002

Celestial mechanics in Kerr spacetime

The dynamical parameters conventionally used to specify the orbit of a test particle in Kerr spacetime are the energy $E$, the axial component of the angular momentum, $L_{z}$, and Carter's constant $Q$. These parameters are obtained by solving the Hamilton-Jacobi equation for the dynamical problem of geodesic motion. Employing the action-angle variable formalism, on the other hand, yields a different set of constants of motion, namely, the fundamental frequencies $ω_{r}$, $ω_θ$ and $ω_ϕ$ associated with the radial, polar and azimuthal components of orbital motion. These frequencies, naturally, determine the time scales of orbital motion and, furthermore, the instantaneous gravitational wave spectrum in the adiabatic approximation. In this article, it is shown that the fundamental frequencies are geometric invariants and explicit formulas in terms of quadratures are derived. The numerical evaluation of these formulas in the case of a rapidly rotating black hole illustrates the behaviour of the fundamental frequencies as orbital parameters such as the semi-latus rectum $p$, the eccentricity $e$ or the inclination parameter $θ_{-}$ are varied. The limiting cases of circular, equatorial and Keplerian motion are investigated as well and it is shown that known results are recovered from the general formulas.

gr-qc↗