Energy spectra stemming from interactions of Alfven waves and turbulent eddies

We present a numerical analysis of an incompressible decaying magnetohydrodynamic turbulence run on a grid of 1536^3 points. The Taylor Reynolds number at the maximum of dissipation is ~1100, and the initial condition is a superposition of large scale ABC flows and random noise at small scales, with no uniform magnetic field. The initial kinetic and magnetic energies are equal, with negligible correlation. The resulting energy spectrum is a combination of two components, each moderately resolved. Isotropy obtains in the large scales, with a spectral law compatible with the Iroshnikov-Kraichnan theory stemming from the weakening of nonlinear interactions due to Alfven waves; scaling of structure functions confirms the non-Kolmogorovian nature of the flow in this range. At small scales, weak turbulence emerges with a k_{\perp}^{-2} spectrum, the perpendicular direction referring to the local quasi-uniform magnetic field.Comment: 4 pages, 4 figure

The Anisotropy of MHD Alfv\'{e}nic Turbulence

We perform direct 3-dimensional numerical simulations for magnetohydrodynamic (MHD) turbulence in a periodic box of size $2\pi$ threaded by strong uniform magnetic fields. We use a pseudo-spectral code with hyperviscosity and hyperdiffusivity to solve the incompressible MHD equations. We analyze the structure of the eddies as a function of scale. A straightforward calculation of anisotropy in wavevector space shows that the anisotropy is scale-{\it independent}. We discuss why this is {\it not} the true scaling law and how the curvature of large-scale magnetic fields affects the power spectrum and leads to the wrong conclusion. When we correct for this effect, we find that the anisotropy of eddies depends on their size: smaller eddies are more elongated than larger ones along {\it local} magnetic field lines. The results are consistent with the scaling law $\tilde{k}_{\parallel} \sim \tilde{k}_{\perp}^{2/3}$ proposed by Goldreich and Sridhar (1995, 1997). Here $\tilde{k}_{\|}$ (and $\tilde{k}_{\perp}$) are wavenumbers measured relative to the local magnetic field direction. However, we see some systematic deviations which may be a sign of limitations to the model, or our inability to fully resolve the inertial range of turbulence in our simulations.Comment: 13 pages (11 NEW figures), ApJ, in press (Aug 10, 2000?

MHD Turbulence Revisited

Kraichnan (1965) proposed that MHD turbulence occurs as a result of collisions between oppositely directed Alfv\'en wave packets. Recent work has generated some controversy over the nature of non linear couplings between colliding Alfv\'en waves. We find that the resolution to much of the confusion lies in the existence of a new type of turbulence, intermediate turbulence, in which the cascade of energy in the inertial range exhibits properties intermediate between those of weak and strong turbulent cascades. Some properties of intermediate MHD turbulence are: (i) in common with weak turbulent cascades, wave packets belonging to the inertial range are long lived; (ii) however, components of the strain tensor are so large that, similar to the situation in strong turbulence, perturbation theory is not applicable; (iii) the breakdown of perturbation theory results from the divergence of neighboring field lines due to wave packets whose perturbations in velocity and magnetic fields are localized, but whose perturbations in displacement are not; (iv) 3--wave interactions dominate individual collisions between wave packets, but interactions of all orders $n\geq 3$ make comparable contributions to the intermediate turbulent energy cascade; (v) successive collisions are correlated since wave packets are distorted as they follow diverging field lines; (vi) in common with the weak MHD cascade, there is no parallel cascade of energy, and the cascade to small perpendicular scales strengthens as it reaches higher wave numbers; (vii) For an appropriate weak excitation, there is a natural progression from a weak, through an intermediate, to a strong cascade.Comment: 25 pages, to appear in The Astrophysical Journa

Strong Imbalanced Turbulence

We consider stationary, forced, imbalanced, or cross-helical MHD Alfvenic turbulence where the waves traveling in one direction have higher amplitudes than the opposite waves. This paper is dedicated to so-called strong turbulence, which cannot be treated perturbatively. Our main result is that the anisotropy of the weak waves is stronger than the anisotropy of a strong waves. We propose that critical balance, which was originally conceived as a causality argument, has to be amended by what we call a propagation argument. This revised formulation of critical balance is able to handle the imbalanced case and reduces to old formulation in the balanced case. We also provide phenomenological model of energy cascading and discuss possibility of self-similar solutions in a realistic setup of driven turbulence.Comment: this is shorter, 5 page version of what is to appear in ApJ 682, Aug. 1, 200

Loading rates in California inferred from aftershocks

International audienceWe estimate the loading rate in southern California and the change in stress induced by a transient slip event across the San Andreas fault (SAF) system in central California, using a model of static fatigue. We analyze temporal properties of aftershocks in order to determine the time delay before the onset of the power law aftershock decay rate. In creep-slip and stick-slip zones, we show that the rate of change of this delay is related to seismic and aseismic deformation across the SAF system. Furthermore, we show that this rate of change is proportional to the deficit of slip rate along the SAF. This new relationship between geodetic and seismological data is in good agreement with predictions from a Limited Power Law model in which the evolution of the duration of a linear aftershock decay rate over short time results from variations in the load of the brittle upper crust

Role of cross helicity in magnetohydrodynamic turbulence

Strong incompressible three-dimensional magnetohydrodynamic turbulence is investigated by means of high resolution direct numerical simulations. The simulations show that the configuration space is characterized by regions of positive and negative cross-helicity, corresponding to highly aligned or anti-aligned velocity and magnetic field fluctuations, even when the average cross-helicity is zero. To elucidate the role of cross-helicity, the spectra and structure of turbulence are obtained in imbalanced regions where cross-helicity is non-zero. When averaged over regions of positive and negative cross-helicity, the result is consistent with the simulations of balanced turbulence. An analytical explanation for the obtained results is proposed.Comment: 4 pages, 4 figure

Radio-wave propagation through a medium containing electron-density fluctuations described by an anisotropic Goldreich-Sridhar spectrum

We study the propagation of radio waves through a medium possessing density fluctuations that are elongated along the ambient magnetic field and described by an anisotropic Goldreich-Sridhar power spectrum. We derive general formulas for the wave phase structure function, visibility, angular broadening, diffraction-pattern length scales, and scintillation time scale for arbitrary distributions of turbulence along the line of sight, and specialize these formulas to idealized cases.Comment: 25 pages, 3 figures, submitted to Ap

Predicting Failure using Conditioning on Damage History: Demonstration on Percolation and Hierarchical Fiber Bundles

We formulate the problem of probabilistic predictions of global failure in the simplest possible model based on site percolation and on one of the simplest model of time-dependent rupture, a hierarchical fiber bundle model. We show that conditioning the predictions on the knowledge of the current degree of damage (occupancy density $p$ or number and size of cracks) and on some information on the largest cluster improves significantly the prediction accuracy, in particular by allowing to identify those realizations which have anomalously low or large clusters (cracks). We quantify the prediction gains using two measures, the relative specific information gain (which is the variation of entropy obtained by adding new information) and the root-mean-square of the prediction errors over a large ensemble of realizations. The bulk of our simulations have been obtained with the two-dimensional site percolation model on a lattice of size $L \times L=20 \times 20$ and hold true for other lattice sizes. For the hierarchical fiber bundle model, conditioning the measures of damage on the information of the location and size of the largest crack extends significantly the critical region and the prediction skills. These examples illustrate how on-going damage can be used as a revelation of both the realization-dependent pre-existing heterogeneity and the damage scenario undertaken by each specific sample.Comment: 7 pages + 11 figure

Diffusion and dispersion of passive tracers: Navier-Stokes versus MHD turbulence

A comparison of turbulent diffusion and pair-dispersion in homogeneous, macroscopically isotropic Navier-Stokes (NS) and nonhelical magnetohydrodynamic (MHD) turbulence based on high-resolution direct numerical simulations is presented. Significant differences between MHD and NS systems are observed in the pair-dispersion properties, in particular a strong reduction of the separation velocity in MHD turbulence as compared to the NS case. It is shown that in MHD turbulence the average pair-dispersion is slowed down for $\tau_\mathrm{d}\lesssim t\lesssim 10 \tau_\mathrm{d}$, $\tau_\mathrm{d}$ being the Kolmogorov time, due to the alignment of the relative Lagrangian tracer velocity with the local magnetic field. Significant differences in turbulent single-particle diffusion in NS and MHD turbulence are not detected. The fluid particle trajectories in the vicinity of the smallest dissipative structures are found to be characterisically different although these comparably rare events have a negligible influence on the statistics investigated in this work.Comment: Europhysics Letters, in prin

Imbalanced Strong MHD Turbulence

We present a phenomenological model of imbalanced MHD turbulence in an incompressible magnetofluid. The steady-state cascades, of waves traveling in opposite directions along the mean magnetic field, carry unequal energy fluxes to small length scales, where they decay due to viscous and resistive dissipation. The inertial-range scalings are well-understood when both cascades are weak. We study the case when both cascades are, in a sense, strong. The inertial-range of this imbalanced cascade has the following properties: (i) the ratio of the r.m.s. Elsasser amplitudes is independent of scale, and is equal to the ratio of the corresponding energy fluxes; (ii) in common with the balanced strong cascade, the energy spectra of both Elsasser waves are of the anisotropic Kolmogorov form, with their parallel correlation lengths equal to each other on all scales, and proportional to the two-thirds power of the transverse correlation length; (iii) the equality of cascade time and waveperiod (critical balance) that characterizes the strong balanced cascade does not apply to the Elsasser field with the larger amplitude. Instead, the more general criterion that always applies to both Elsasser fields is that the cascade time is equal to the correlation time of the straining imposed by oppositely-directed waves. Our results are particularly relevant for turbulence in the solar wind. Spacecraft measurements have established that, in the inertial range of solar wind turbulence, waves travelling away from the sun have higher amplitudes than those travelling towards it. Result (i) allows us to infer the turbulent flux ratios from the amplitude ratios, thus providing insight into the origin of the turbulence