Third-order Els\"asser moments in axisymmetric MHD turbulence

Incompressible MHD turbulence is investigated under the presence of a uniform magnetic field \bb0. Such a situation is described in the correlation space by a divergence relation which expresses the statistical conservation of the Els\"asser energy flux through the inertial range. The ansatz is made that the development of anisotropy, observed when $B_0$ is strong enough, implies a foliation of space correlation. A direct consequence is the possibility to derive a vectorial law for third-order Els\"asser moments which is parametrized by the intensity of anisotropy. We use the so-called critical balance assumption to fix this parameter and find a unique expression.Comment: 10 pages, 2 figures, will appea

Theory for helical turbulence under fast rotation

Recent numerical simulations have shown the strong impact of helicity on \ADD{homogeneous} rotating hydrodynamic turbulence. The main effect can be summarized through the following law, $n+\tilde n = -4$, where $n$ and $\tilde n$ are respectively the power law indices of the one-dimensional energy and helicity spectra. We investigate this rotating turbulence problem in the small Rossby number limit by using the asymptotic weak turbulence theory derived previously. We show that the empirical law is an exact solution of the helicity equation where the power law indices correspond to perpendicular (to the rotation axis) wave number spectra. It is proposed that when the cascade towards small-scales tends to be dominated by the helicity flux the solution tends to $\tilde n = -2$, whereas it is $\tilde n = -3/2$ when the energy flux dominates. The latter solution is compatible with the so-called maximal helicity state previously observed numerically and derived theoretically in the weak turbulence regime when only the energy equation is used, whereas the former solution is constrained by a locality condition.Comment: 4 page

Tournament MAC with Constant Size Congestion Window for WLAN

In the context of radio distributed networks, we present a generalized approach for the Medium Access Control (MAC) with fixed congestion window. Our protocol is quite simple to analyze and can be used in a lot of different situations. We give mathematical evidence showing that our performance is tight, in the sense that no protocol with fixed congestion window can do better. We also place ourselves in the WiFi/WiMAX framework, and show experimental results enlightening collision reduction of 14% to 21% compared to the best known other methods. We show channel capacity improvement, and fairness considerations

Nonlinear diffusion equations for anisotropic MHD turbulence with cross-helicity

Nonlinear diffusion equations of spectral transfer are systematically derived for anisotropic magnetohydrodynamics in the regime of wave turbulence. The background of the analysis is the asymptotic Alfv\'en wave turbulence equations from which a differential limit is taken. The result is a universal diffusion-type equation in ${\bf k}$-space which describes in a simple way and without free parameter the energy transport perpendicular to the external magnetic field ${\bf B_0}$ for transverse and parallel fluctuations. These equations are compatible with both the thermodynamic equilibrium and the finite flux spectra derived by Galtier et al. (2000); it improves therefore the model built heuristically by Litwick \& Goldreich (2003) for which only the second solution was recovered. This new system offers a powerful description of a wide class of astrophysical plasmas with non-zero cross-helicity.Comment: 20 pages, 3 figure

Weak turbulence theory for rotating magnetohydrodynamics and planetary dynamos

A weak turbulence theory is derived for magnetohydrodynamics under rapid rotation and in the presence of a large-scale magnetic field. The angular velocity $\Omega_0$ is assumed to be uniform and parallel to the constant Alfv\'en speed ${\bf b_0}$. Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length $d \equiv b_0/\Omega_0$. In the large-scale limit ($kd \to 0$; $k$ being the wave number), the left- and right-handed waves tend respectively to the inertial and magnetostrophic waves whereas in the small-scale limit ($kd \to + \infty$) pure Alfv\'en waves are recovered. By using a complex helicity decomposition, the asymptotic weak turbulence equations are derived which describe the long-time behavior of weakly dispersive interacting waves {\it via} three-wave interaction processes. It is shown that the nonlinear dynamics is mainly anisotropic with a stronger transfer perpendicular ($\perp$) than parallel ($\parallel$) to the rotating axis. The general theory may converge to pure weak inertial/magnetostrophic or Alfv\'en wave turbulence when the large or small-scales limits are taken respectively. Inertial wave turbulence is asymptotically dominated by the kinetic energy/helicity whereas the magnetostrophic wave turbulence is dominated by the magnetic energy/helicity. For both regimes a family of exact solutions are found for the spectra which do not correspond necessarily to a maximal helicity state. It is shown that the hybrid helicity exhibits a cascade whose direction may vary according to the scale $k_f$ at which the helicity flux is injected with an inverse cascade if $k_fd < 1$ and a direct cascade otherwise. The theory is relevant for the magnetostrophic dynamo whose main applications are the Earth and giant planets for which a small ($\sim 10^{-6}$) Rossby number is expected.Comment: 4 figures, 33 page