Entanglement of helicity and energy in kinetic Alfven wave/whistler turbulence

The role of magnetic helicity is investigated in kinetic Alfv\'en wave and oblique whistler turbulence in presence of a relatively intense external magnetic field $b_0 {\bf e_\parallel}$. In this situation, turbulence is strongly anisotropic and the fluid equations describing both regimes are the reduced electron magnetohydrodynamics (REMHD) whose derivation, originally made from the gyrokinetic theory, is also obtained here from compressible Hall MHD. We use the asymptotic equations derived by Galtier \& Bhattacharjee (2003) to study the REMHD dynamics in the weak turbulence regime. The analysis is focused on the magnetic helicity equation for which we obtain the exact solutions: they correspond to the entanglement relation, $n+\tilde n = -6$, where $n$ and $\tilde n$ are the power law indices of the perpendicular (to ${\bf b_0}$) wave number magnetic energy and helicity spectra respectively. Therefore, the spectra derived in the past from the energy equation only, namely $n=-2.5$ and $\tilde n = - 3.5$, are not the unique solutions to this problem but rather characterize the direct energy cascade. The solution $\tilde n = -3$ is a limit imposed by the locality condition; it is also the constant helicity flux solution obtained heuristically. The results obtained offer a new paradigm to understand solar wind turbulence at sub-ion scales where it is often observed that $-3 < n < -2.5$.Comment: 26 pages, submitted to the special issue of JPP "Present achievements and new frontiers in space plasmas

Meromorphy and topology of localized solutions in the Thomas–MHD model

The one-dimensional MHD system first introduced by J.H. Thomas [Phys. Fluids 11, 1245 (1968)] as a model of the dynamo effect is thoroughly studied in the limit of large magnetic Prandtl number. The focus is on two types of localized solutions involving shocks (antishocks) and hollow (bump) waves. Numerical simulations suggest phenomenological rules concerning their generation, stability and basin of attraction. Their topology, amplitude and thickness are compared favourably with those of the meromorphic travelling waves, which are obtained exactly, and respectively those of asymptotic descriptions involving rational or degenerate elliptic functions. The meromorphy bars the existence of certain configurations, while others are explained by assuming imaginary residues. These explanations are tested using the numerical amplitude and phase of the Fourier transforms as probes of the analyticity properties. Theoretically, the proof of the partial integrability backs up the role ascribed to meromorphy. Practically, predictions are derived for MHD plasmas

A Universal Law for Solar-Wind Turbulence at Electron Scales

The interplanetary magnetic fluctuation spectrum obeys a Kolmogorovian power law at scales above the proton inertial length and gyroradius which is well regarded as an inertial range. Below these scales a power law index around $-2.5$ is often measured and associated to nonlinear dispersive processes. Recent observations reveal a third region at scales below the electron inertial length. This region is characterized by a steeper spectrum that some refer to it as the dissipation range. We investigate this range of scales in the electron magnetohydrodynamic approximation and derive an exact and universal law for a third-order structure function. This law can predict a magnetic fluctuation spectrum with an index of $-11/3$ which is in agreement with the observed spectrum at the smallest scales. We conclude on the possible existence of a third turbulence regime in the solar wind instead of a dissipation range as recently postulated.Comment: 11 pages, will appear in Astrophys.

Wave turbulence in magnetized plasmas

The paper reviews the recent progress on wave turbulence for magnetized plasmas (MHD, Hall MHD and electron MHD) in the incompressible and compressible cases. The emphasis is made on homogeneous and anisotropic turbulence which usually provides the best theoretical framework to investigate space and laboratory plasmas. The solar wind and the coronal heating problems are presented as two examples of application of anisotropic wave turbulence. The most important results of wave turbulence are reported and discussed in the context of natural and simulated magnetized plasmas. Important issues and possible spurious interpretations are also discussed

Anisotropic fluxes and nonlocal interactions in MHD turbulence

We investigate the locality or nonlocality of the energy transfer and of the spectral interactions involved in the cascade for decaying magnetohydrodynamic (MHD) flows in the presence of a uniform magnetic field $\bf B$ at various intensities. The results are based on a detailed analysis of three-dimensional numerical flows at moderate Reynold numbers. The energy transfer functions, as well as the global and partial fluxes, are examined by means of different geometrical wavenumber shells. On the one hand, the transfer functions of the two conserved Els\"asser energies $E^+$ and $E^-$ are found local in both the directions parallel ($k_\|$-direction) and perpendicular ($k_\perp$-direction) to the magnetic guide-field, whatever the ${\bf B}$-strength. On the other hand, from the flux analysis, the interactions between the two counterpropagating Els\"asser waves become nonlocal. Indeed, as the ${\bf B}$-intensity is increased, local interactions are strongly decreased and the interactions with small $k_\|$ modes dominate the cascade. Most of the energy flux in the $k_\perp$-direction is due to modes in the plane at $k_\|=0$, while the weaker cascade in the $k_\|$-direction is due to the modes with $k_\|=1$. The stronger magnetized flows tends thus to get closer to the weak turbulence limit where the three-wave resonant interactions are dominating. Hence, the transition from the strong to the weak turbulence regime occurs by reducing the number of effective modes in the energy cascade.Comment: Submitted to PR

A weak turbulence theory for incompressible magnetohydrodynamics

We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0ê[parallel R: parallel]. Numerically and analytically, we find energy spectra E± [similar] kn±[bot bottom], such that n+ + n− = −4, where E± are the spectra of the Elsässer variables z± = v ± b in the two-dimensional case (k[parallel R: parallel] = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made