Energy transfer in Hall-MHD turbulence: cascades, backscatter, and dynamo action

Scale interactions in Hall MHD are studied using both the mean field theory derivation of transport coefficients, and direct numerical simulations in three space dimensions. In the magnetically dominated regime, the eddy resistivity is found to be negative definite, leading to large scale instabilities. A direct cascade of the total energy is observed, although as the amplitude of the Hall effect is increased, backscatter of magnetic energy to large scales is found, a feature not present in MHD flows. The coupling between the magnetic and velocity fields is different than in the MHD case, and backscatter of energy from small scale magnetic fields to large scale flows is also observed. For the magnetic helicity, a strong quenching of its transfer is found. We also discuss non-helical magnetically forced Hall-MHD simulations where growth of a large scale magnetic field is observed.Comment: 25 pages, 16 figure

Shell to shell energy transfer in MHD, Part II: Kinematic dynamo

We study the transfer of energy between different scales for forced three-dimensional MHD turbulent flows in the kinematic dynamo regime. Two different forces are examined: a non-helical Taylor Green flow with magnetic Prandtl number P_M=0.4, and a helical ABC flow with P_M=1. This analysis allows us to examine which scales of the velocity flow are responsible for dynamo action, and identify which scales of the magnetic field receive energy directly from the velocity field and which scales receive magnetic energy through the cascade of the magnetic field from large to small scales. Our results show that the turbulent velocity fluctuations are responsible for the magnetic field amplification in the small scales (small scale dynamo) while the large scale field is amplified mostly due to the large scale flow. A direct cascade of the magnetic field energy from large to small scales is also present and is a complementary mechanism for the increase of the magnetic field in the small scales. Input of energy from the velocity field in the small magnetic scales dominates over the energy that is cascaded down from the large scales until the large-scale peak of the magnetic energy spectrum is reached. At even smaller scales, most of the magnetic energy input is from the cascading process.Comment: Submitted to PR

On the inverse cascade of magnetic helicity

We study the inverse cascade of magnetic helicity in conducting fluids by investigating the detailed transfer of helicity between different spherical shells in Fourier space in direct numerical simulations of three-dimensional magnetohydrodynamics (MHD). Two different numerical simulations are used, one where the system is forced with an electromotive force in the induction equation, and one in which the system is forced mechanically with an ABC flow and the magnetic field is solely sustained by a dynamo action. The magnetic helicity cascade at the initial stages of both simulations is observed to be inverse and local (in scale space) in the large scales, and direct and local in the small scales. When saturation is approached most of the helicity is concentrated in the large scales and the cascade is non-local. Helicity is transfered directly from the forced scales to the largest scales. At the same time, a smaller in amplitude direct cascade is observed from the largest scale to small scales.Comment: Submitted to PR

Marginally unstable Holmboe modes

Marginally unstable Holmboe modes for smooth density and velocity profiles are studied. For a large family of flows and stratification that exhibit Holmboe instability, we show that the modes with phase velocity equal to the maximum or the minimum velocity of the shear are marginally unstable. This allows us to determine the critical value of the control parameter R (expressing the ratio of the velocity variation length scale to the density variation length scale) that Holmboe instability appears R=2. We then examine systems for which the parameter R is very close to this critical value. For this case we derive an analytical expression for the dispersion relation of the complex phase speed c(k) in the unstable region. The growth rate and the width of the region of unstable wave numbers has a very strong (exponential) dependence on the deviation of R from the critical value. Two specific examples are examined and the implications of the results are discussed.Comment: Submitted to Physics of Fluid

Rigidity of stationary black holes with small angular momentum on the horizon

We prove a black hole rigidity result for slowly rotating stationary solutions of the Einstein vacuum equations. More precisely, we prove that the domain of outer communications of a regular stationary vacuum is isometric to the domain of outer communications of a Kerr solution, provided that the stationary Killing vector-field \T is small on the bifurcation sphere.Comment: Minor corrections, submitted versio

Stratified shear flow instabilities at large Richardson numbers

Numerical simulations of stratified shear flow instabilities are performed in two dimensions in the Boussinesq limit. The density variation length scale is chosen to be four times smaller than the velocity variation length scale so that Holmboe or Kelvin-Helmholtz unstable modes are present depending on the choice of the global Richardson number Ri. Three different values of Ri were examined Ri =0.2, 2, 20. The flows for the three examined values are all unstable due to different modes namely: the Kelvin-Helmholtz mode for Ri=0.2, the first Holmboe mode for Ri=2, and the second Holmboe mode for Ri=20 that has been discovered recently and it is the first time that it is examined in the non-linear stage. It is found that the amplitude of the velocity perturbation of the second Holmboe mode at the non-linear stage is smaller but comparable to first Holmboe mode. The increase of the potential energy however due to the second Holmboe modes is greater than that of the first mode. The Kelvin-Helmholtz mode is larger by two orders of magnitude in kinetic energy than the Holmboe modes and about ten times larger in potential energy than the Holmboe modes. The results in this paper suggest that although mixing is suppressed at large Richardson numbers it is not negligible, and turbulent mixing processes in strongly stratified environments can not be excluded.Comment: Submitted to Physics of Fluid

Shell to shell energy transfer in MHD, Part I: steady state turbulence

We investigate the transfer of energy from large scales to small scales in fully developed forced three-dimensional MHD-turbulence by analyzing the results of direct numerical simulations in the absence of an externally imposed uniform magnetic field. Our results show that the transfer of kinetic energy from the large scales to kinetic energy at smaller scales, and the transfer of magnetic energy from the large scales to magnetic energy at smaller scales, are local, as is also found in the case of neutral fluids, and in a way that is compatible with Kolmogorov (1941) theory of turbulence. However, the transfer of energy from the velocity field to the magnetic field is a highly non-local process in Fourier space. Energy from the velocity field at large scales can be transfered directly into small scale magnetic fields without the participation of intermediate scales. Some implications of our results to MHD turbulence modeling are also discussed.Comment: Submitted to PR