The role of parametric instabilities in turbulence generation and proton heating: Hybrid simulations of parallel propagating Alfv\'en waves

Large amplitude Alfv\'en waves tend to be unstable to parametric instabilities which result in a decay process of the initial wave into different daughter waves depending upon the amplitude of the fluctuations and the plasma beta. The propagation angle with respect to the mean magnetic field of the daughter waves plays an important role in determining the type of decay. In this paper, we revisit this problem by means of multi-dimensional hybrid simulations. In particular, we study the decay and the subsequent nonlinear evolution of large-amplitude Alfv\'en waves by investigating the saturation mechanism of the instability and its final nonlinear state reached for different wave amplitudes and plasma beta conditions. As opposed to one-dimensional simulations where the Decay instability is suppressed for increasing plasma beta values, we find that the decay process in multi-dimensions persists at large values of the plasma beta via the filamentation/magnetosonic decay instabilities. In general, the decay process acts as a trigger both to develop a perpendicular turbulent cascade and to enhance mean field-aligned wave-particle interactions. We find indeed that the saturated state is characterized by a turbulent plasma displaying a field-aligned beam at the Alfv\'en speed and increased temperatures that we ascribe to the Landau resonance and pitch angle scattering in phase space

Testing Conditional Independence of Discrete Distributions

We study the problem of testing \emph{conditional independence} for discrete distributions. Specifically, given samples from a discrete random variable $(X, Y, Z)$ on domain $[\ell_1]\times[\ell_2] \times [n]$, we want to distinguish, with probability at least $2/3$, between the case that $X$ and $Y$ are conditionally independent given $Z$ from the case that $(X, Y, Z)$ is $\epsilon$-far, in $\ell_1$-distance, from every distribution that has this property. Conditional independence is a concept of central importance in probability and statistics with a range of applications in various scientific domains. As such, the statistical task of testing conditional independence has been extensively studied in various forms within the statistics and econometrics communities for nearly a century. Perhaps surprisingly, this problem has not been previously considered in the framework of distribution property testing and in particular no tester with sublinear sample complexity is known, even for the important special case that the domains of $X$ and $Y$ are binary. The main algorithmic result of this work is the first conditional independence tester with {\em sublinear} sample complexity for discrete distributions over $[\ell_1]\times[\ell_2] \times [n]$. To complement our upper bounds, we prove information-theoretic lower bounds establishing that the sample complexity of our algorithm is optimal, up to constant factors, for a number of settings. Specifically, for the prototypical setting when $\ell_1, \ell_2 = O(1)$, we show that the sample complexity of testing conditional independence (upper bound and matching lower bound) is \[ \Theta\left({\max\left(n^{1/2}/\epsilon^2,\min\left(n^{7/8}/\epsilon,n^{6/7}/\epsilon^{8/7}\right)\right)}\right)\,. \

Scale dependence and cross-scale transfer of kinetic energy in compressible hydrodynamic turbulence at moderate Reynolds numbers

We investigate properties of the scale dependence and cross-scale transfer of kinetic energy in compressible three-dimensional hydrodynamic turbulence, by means of two direct numerical simulations of decaying turbulence with initial Mach numbers M = 1/3 and M = 1, and with moderate Reynolds numbers, R_lambda ~ 100. The turbulent dynamics is analyzed using compressible and incompressible versions of the dynamic spectral transfer (ST) and the Karman-Howarth-Monin (KHM) equations. We find that the nonlinear coupling leads to a flux of the kinetic energy to small scales where it is dissipated; at the same time, the reversible pressure-dilatation mechanism causes oscillatory exchanges between the kinetic and internal energies with an average zero net energy transfer. While the incompressible KHM and ST equations are not generally valid in the simulations, their compressible counterparts are well satisfied and describe, in a quantitatively similar way, the decay of the kinetic energy on large scales, the cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. There exists a simple relationship between the KHM and ST results through the inverse proportionality between the wave vector k and the spatial separation length l as k l ~ 3^1/2. For a given time the dissipation and pressure-dilatation terms are strong on large scales in the KHM approach whereas the ST terms become dominant on small scales; this is owing to the complementary cumulative behavior of the two methods. The effect of pressure dilatation is weak when averaged over a period of its oscillations and may lead to a transfer of the kinetic energy from large to small scales without a net exchange between the kinetic and internal energies. Our results suggest that for large-enough systems there exists an inertial range for the kinetic energy cascade ...Comment: 14 pages, 10 figure

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Plasma turbulence and kinetic instabilities at ion scales in the expanding solar wind

The relationship between a decaying strong turbulence and kinetic instabilities in a slowly expanding plasma is investigated using two-dimensional (2D) hybrid expanding box simulations. We impose an initial ambient magnetic field perpendicular to the simulation box, and we start with a spectrum of large-scale, linearly polarized, random-phase Alfvénic fluctuations that have energy equipartition between kinetic and magnetic fluctuations and vanishing correlation between the two fields. A turbulent cascade rapidly develops; magnetic field fluctuations exhibit a power-law spectrum at large scales and a steeper spectrum at ion scales. The turbulent cascade leads to an overall anisotropic proton heating, protons are heated in the perpendicular direction, and, initially, also in the parallel direction. The imposed expansion leads to generation of a large parallel proton temperature anisotropy which is at later stages partly reduced by turbulence. The turbulent heating is not sufficient to overcome the expansion-driven perpendicular cooling and the system eventually drives the oblique firehose instability in a form of localized nonlinear wave packets which efficiently reduce the parallel temperature anisotropy. This work demonstrates that kinetic instabilities may coexist with strong plasma turbulence even in a constrained 2D regime

The oblique firehose instability in a bi-kappa magnetized plasma

In this work, we derive a dispersion equation that describes the excitation of the oblique (or Alfv\'en) firehose instability in a plasma that contains both electron and ion species modelled by bi-kappa velocity distribution functions. The equation is obtained with the assumptions of low-frequency waves and moderate to large values of the parallel (respective to the ambient magnetic field) plasma beta parameter, but it is valid for any direction of propagation and for any value of the particle gyroradius (or Larmor radius). Considering values for the physical parameters typical to those found in the solar wind, some solutions of the dispersion equation, corresponding to the unstable mode, are presented. In order to implement the dispersion solver, several new mathematical properties of the special functions occurring in a kappa plasma are derived and included. The results presented here suggest that the superthermal characteristic of the distribution functions leads to reductions to both the maximum growth rate of the instability and of the spectral range of its occurrence

Nonlinear theory of mirror instability near threshold

An asymptotic model based on a reductive perturbative expansion of the drift kinetic and the Maxwell equations is used to demonstrate that, near the instability threshold, the nonlinear dynamics of mirror modes in a magnetized plasma with anisotropic ion temperatures involves a subcritical bifurcation,leading to the formation of small-scale structures with amplitudes comparable with the ambient magnetic field

Magnetic field turbulence in the solar wind at sub-ion scales: in situ observations and numerical simulations

We investigate the transition of the solar wind turbulent cascade from MHD to sub-ion range by means of a detail comparison between in situ observations and hybrid numerical simulations. In particular we focus on the properties of the magnetic field and its component anisotropy in Cluster measurements and hybrid 2D simulations. First, we address the angular distribution of wave-vectors in the kinetic range between ion and electron scales by studying the variance anisotropy of the magnetic field components. When taking into account the single-direction sampling performed by spacecraft in the solar wind, the main properties of the fluctuations observed in situ are also recovered in our numerical description. This result confirms that solar wind turbulence in the sub-ion range is characterized by a quasi-2D gyrotropic distribution of k-vectors around the mean field. We then consider the magnetic compressibility associated with the turbulent cascade and its evolution from large-MHD to sub-ion scales. The ratio of field-aligned to perpendicular fluctuations, typically low in the MHD inertial range, increases significantly when crossing ion scales and its value in the sub-ion range is a function of the total plasma beta only, as expected from theoretical predictions, with higher magnetic compressibility for higher beta. Moreover, we observe that this increase has a gradual trend from low to high beta values in the in situ data; this behaviour is well captured by the numerical simulations. The level of magnetic field compressibility that is observed in situ and in the simulations is in fairly good agreement with theoretical predictions, especially at high beta, suggesting that in the kinetic range explored the turbulence is supported by low-frequency and highly-oblique fluctuations in pressure balance, like kinetic Alfv\'en waves or other slowly evolving coherent structures.Comment: Manuscript submitted to Frontiers Astronomy and Space Sciences, Research Topic: Improving the Understanding of Kinetic Processes in Solar Wind and Magnetosphere: From CLUSTER to MM