Nils Erland L. Haugen

Evolving turbulence and magnetic fields in galaxy clusters

We discuss, using simple analytical models and magnetohydrodynamic (MHD) simulations, the origin and parameters of turbulence and magnetic fields in galaxy clusters. Any pre-existing tangled magnetic field must decay in a few hundred million years by generating gas motions even if the electric conductivity of the intracluster gas is high. We argue that turbulent motions can be maintained in the intracluster gas and its dynamo action can prevent such a decay and amplify a random seed magnetic field by a net factor of typically 104 in 5 Gyr. Three physically distinct regimes can be identified in the evolution of turbulence and magnetic field in galaxy clusters. First, the fluctuation dynamo will produce microgauss (μG)-strong, random magnetic fields during the epoch of cluster formation and major mergers. At this stage pervasive turbulent flows with rms velocity of about 300 km s -1 can be maintained at scales of 100-200 kpc. The magnetic field is intermittent, has a smaller scale of 20-30 kpc and average strength of 2 μG. Secondly, turbulence will decay after the end of the major merger epoch; we discuss the dynamics of the decaying turbulence and the behaviour of magnetic field in it. Magnetic field and turbulent speed undergo a power-law decay, decreasing by a factor of 2 during this stage, whereas their scales increase by about the same factor. Thirdly, smaller-mass subclusters and cluster galaxies will produce turbulent wakes where magnetic fields will be generated as well. Although the wakes plausibly occupy only a small fraction of the cluster volume, we show that their area-covering factor can be close to unity, and thus they can produce some of the signatures of turbulence along virtually all lines of sight. The latter could potentially allow one to reconcile the possibility of turbulence with ordered filamentary gas structures, as in the Perseus cluster. The turbulent speeds and magnetic fields in the wakes are estimated to be of the order of 300 km s -1 and 2 μG, respectively, whereas the turbulent scales are of the order of 200 kpc for wakes behind subclusters of a mass 3 x 10 13 M ⊙ and about 10 kpc in the galactic wakes. Magnetic field in the wakes is intermittent and has the scale of about 30 and 1 kpc in the subcluster and galactic wakes, respectively. Random Faraday rotation measure is estimated to be typically 100-200 rad m -2 , in agreement with observations. We predict detectable polarization of synchrotron emission from cluster radio haloes at wavelengths 3-6 cm, if observed at sufficiently high resolution.

The Onset of a Small-Scale Turbulent Dynamo at Low Magnetic Prandtl Numbers

We study numerically the dependence of the critical magnetic Reynolds number Rmc for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number Pm = Rm/Re < 1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rmc → const as Re → ∞ (a small-scale dynamo exists at low Pm) or Rmc/Re = Pmc → const as Re → ∞ (no small-scale dynamo exists at low Pm). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, Rmc grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rmc ~ 500 for Re ~ 3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that Rmc is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve Rmc(Re).

Particle impaction on a cylinder in a crossflow as function of Stokes and Reynolds numbers

A high-order direct numerical simulation code (The Pencil Code) has been used together with the immersed boundary method on a Cartesian grid to simulate particle impaction on a cylinder in a crossflow. The direct numerical scheme concerns only the fluid flow, into which the particles are subsequently coupled through a one-way drag-coefficient law. The immersed boundary method is extended to work with high-order discretization, and the particle impaction efficiency has been measured for Stokes numbers ranging from 0.001 to 40 for a range of different Reynolds numbers. Three modes of impaction on the front side of the cylinder are identified, where, for the large-Stokes-number mode (St > 0.3), an alternative to the traditional Stokes number is presented that provides better scaling. The intermediate impaction mode has a very steep decrease in impaction efficiency as the Stokes number is decreased, and this is identified as the range of Stokes numbers where the viscous boundary layer starts to take effect. The third mode of front-side impaction is for the very small particles with St < 0.1 exactly following the flow but impacting on the cylinder due to their finite radii. There will not be any capture on the front side of the cylinder for impact angles larger than ~56° for this mode. Finally, it is found that the particle impaction on the back side of the cylinder is strongly dependent on the flow Reynolds number, where large Reynolds numbers lead to larger impaction efficiencies. The upper limiting Stokes number of back-side impaction is around 0.13, apparently irrespective of the Reynolds number.

A high-order public domain code for direct numerical simulations of turbulent combustion

A high-order scheme for direct numerical simulations of turbulent combustion is discussed. Its implementation in the massively parallel and publicly available Pencil Code is validated with the focus on hydrogen combustion. This is the first open source DNS code with detailed chemistry available. An attempt has been made to present, for the first time, the full set of evolution and auxiliary equations required for a complete description of single phase non-isothermal fluid dynamics with detailed chemical reactions. Ignition delay times (0D) and laminar flame velocities (1D) are calculated and compared with results from the commercially available Chemkin code. The scheme is verified to be fifth order in space. Upon doubling the resolution, a 32-fold increase in the accuracy of the flame front is demonstrated. Finally, also turbulent and spherical flame front velocities are calculated and the implementation of the non-reflecting so-called Navier-Stokes Characteristic Boundary Condition is validated in all three directions.

Hydrodynamic and hydromagnetic energy spectra from large eddy simulations

Direct and large eddy simulations of hydrodynamic and hydromagnetic turbulence have been performed in an attempt to isolate artifacts from real and possibly asymptotic features in the energy spectra. It is shown that in a hydrodynamic turbulence simulation with a Smagorinsky subgrid scale model using 5123 mesh points, two important features of the 40963 simulation on the Earth simulator [Y. Kaneda et al., Phys. Fluids 15, L21 (2003)] are reproduced: a k−0.1 correction to the inertial range with a k−5∕3 Kolmogorov slope and the form of the bottleneck just before the dissipative subrange. Furthermore, it is shown that, while a Smagorinsky-type model for the induction equation causes an artificial and unacceptable reduction in the dynamo efficiency, hyper-resistivity yields good agreement with direct simulations. In the large-scale part of the inertial range, an excess of the spectral magnetic energy over the spectral kinetic energy is confirmed. However, a trend toward spectral equipartition at smaller scal...

The problem of small and large scale fields in the solar dynamo

Three closely related stumbling blocks of solar mean field dynamo theory are discussed: how dominant are the small scale fields, how is the alpha effect quenched, and whether magnetic and current helicity fluxes alleviate the quenching? It is shown that even at the largest currently available resolution there is no clear evidence of power law scaling of the magnetic and kinetic energy spectra in turbulence. However, using subgrid scalemodeling, some indications of asymptotic equipartition can be found. The frequently used first order smoothing approach to calculate the alpha effect and other transport coefficients is contrasted with the superior minimal tau approximation. The possibility of catastrophic alpha quenching is discussed as a result of magnetic helicity conservation. Magnetic and current helicity fluxes are shown to alleviate catastrophic quenching in the presence of shear. Evidence for strong large scale dynamo action, even in the absence of helicity in the forcing, is presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

High-Resolution Simulations of Nonhelical MHD Turbulence

According to the kinematic theory of nonhelical dynamo action, the magnetic energy spectrum increases with wavenumber and peaks at the resistive cutoff wavenumber. It has previously been argued that even in the dynamical case, the magnetic energy peaks at the resistive scale. Using high resolution simulations (up to 10243 meshpoints) with no large-scale imposed field, we show that the magnetic energy peaks at a wavenumber that is independent of the magnetic Reynolds number and about five times larger than the forcing wavenumber. Throughout the inertial range, the spectral magnetic energy exceeds the kinetic energy by a factor of two to three. Both spectra are approximately parallel. The total energy spectrum seems to be close to k−3/2, but there is a strong bottleneck effect and we suggest that the asymptotic spectrum is instead k−5/3. This is supported by the value of the second-order structure function exponent that is found to be ζ2 = 0.70, suggesting a k−1.70 spectrum. The third-order structure function scaling exponent is very close to unity,—in agreement with Goldreich–Sridhar theory.Adding an imposed field tends to suppress the small-scale magnetic field. We find that at large scales the magnetic energy spectrum then follows a k−1 slope. When the strength of the imposed field is of the same order as the dynamo generated field, we find almost equipartition between the magnetic and kinetic energy spectra.

Detection of turbulent thermal diffusion of particles in numerical simulations

The phenomenon of turbulent thermal diffusion in temperature-stratified turbulence causing a non-diffusive turbulent flux of inertial and non-inertial particles in the direction of the turbulent heat flux is found using direct numerical simulations (DNS). In simulations with and without gravity, this phenomenon is found to cause a peak in the particle number density around the minimum of the mean fluid temperature for Stokes numbers less than 1, where the Stokes number is the ratio of particle Stokes time to turbulent Kolmogorov time at the viscous scale. Turbulent thermal diffusion causes the formation of large-scale inhomogeneities in the spatial distribution of inertial particles. The strength of this effect is maximum for Stokes numbers around unity, and decreases again for larger values. The dynamics of inertial particles is studied using Lagrangian modelling in forced temperature-stratified turbulence, whereas non-inertial particles and the fluid are described using DNS in an Eulerian framework.

Turbophoresis in forced inhomogeneous turbulence

Abstract.We show, by direct numerical simulations, that heavy inertial particles (characterized by Stokes number St) in inhomogeneously forced statistically stationary isothermal turbulent flows cluster at the minima of mean-square turbulent velocity. Two turbulent transport processes, turbophoresis and turbulent diffusion together determine the spatial distribution of the particles. If the turbulent diffusivity is assumed to scale with turbulent root-mean-square velocity, as is the case for homogeneous turbulence, the turbophoretic coefficient can be calculated. Indeed, for the above assumption, the non-dimensional product of the turbophoretic coefficient and the rms velocity is shown to increase with St for small St, reach a maxima for

The effect of turbulent clustering on particle reactivity

{\rm St} \approx 10