Nonlinear waves and turbulence in space plasmas contributions

Etienne Behar, Fouad Sahraoui

Laboratoire de Physique des Plasmas, CNRS - École polytechnique - Sorbonne Université - Observatoire de Paris, Université Paris-Sud, Université Paris-Saclay, F-91128 Palaiseau, France

We present our current work on the analysis of MMS data carried out in particular with particle velocity distribution functions (VDF). We propose methods that tackle the high time resolution of these four-dimensional data sets, in various reference frames and coordinate systems. In particular, we explore the feasibility of obtaining spatial and time derivatives of the VDF, with the inherent price in terms of time resolution/integration. Together with field measurements, these derivatives enable the quantification of the various terms of the Poisson-Vlasov equations, with the ultimate goal of a direct measurement of the energy exchange taking place between fields and particles, as a function of velocity, following the effort initiated by Howes et al. 2017 and Chen et al. 2019.

Authors: E. Bello-Benítez, G. Sánchez-Arriaga, T. Passot, D. Laveder and E. Siminos. There exist a broad variety of nonlinear-wave phenomena in the solar wind. Different types of stable large-amplitude solitary waves are typically observed in these plasmas. The study of small amplitude waves can be described by well-known equations: Korteweg-de-Vries (KdV), modified KdV, Derivative Nonlinear Schrödinger (DNLS) and triple-degenerate DNLS. However, magnetohydrodynamic (MHD) fluid equations are more suitable for the analysis of large-amplitude structures, which is the approach used in this work [1] —to be precise, MHD equations with Hall effect and Finite Larmor Radius (FLR) corrections to the double adiabatic pressure tensor. Assuming travelling wave solutions, the system of partial differential equations yields a set of 5 ordinary differential equations (ODEs) governing the spatial profile of the velocity and magnetic-field vectors —if double adiabatic equations of state are used for the gyrotropic pressures. The procedure to derive these equations follows Ref. [2], but some discrepancies are shown [1]. The existence of solitary-wave solutions in different parametric regimes is rigorously proved in this system of ODEs using concepts and tools from the theory of dynamical systems. Two key features of the concerning ODEs are: (1) the system is reversible and (2) the existence of an invariant which allows reducing the effective dimension of the system from 5 to 4. These characteristics are guaranteed if equations of state are used for the pressures. Nevertheless, only stable structures have physical interests and are expected to be observed in space. The global stability of the solitary waves is investigated by numerical spectral simulations using two different closures for the pressures: (1) double adiabatic equations and (2) evolution equations including the FLR work terms [3], which guarantee energy conservation and better reproduces the real physics. In case (1), it is found that the solitary waves may have a stable core even if the background is unstable. The background instability seems to disappear when the energy-conserving model (2) is considered. In this case, stable solitary waves are found that survive long time without significant deformation.

References

[1] E. Bello-Benítez, G. Sánchez-Arriaga, T. Passot, D. Laveder and E. Siminos, Phys. Rev. E 99, 023202 (2019).

[2] E. Mjølhus, Nonlin. Proc. Geophys. 16, 251 (2009).

[3] P. L. Sulem and T. Passot, J. Plasma Phys. 81, 325810103 (2015).

F. Califano, G. Arrò Dipartimento di Fisica "E. Fermi", Università di Pisa, Pisa, Italy S.S. Cerri Department of Astrophysical Sciences, Princeton University, Princeton, 08544 USA

It has been observed experimentally the occurrence of a new process, namely electron-only reconnection, where the reconnection dynamics is driven only by electrons (e-rec-only) [1]. Recently, a theoretical study in the context of plasma magnetized turbulence has given evidence about the possibility to drive e-rec-only by fluctuations at scales of the order of the ion scale length [2] (see Faganello abstract, this Conference). By considering two Vlasov simulations of magnetized plasma turbulence where “standard” reconnection or e-rec-only separately occur, we make a compared study of the turbulence statistical properties, in particular of the structure functions in order to separate the contribution of the ions at the so-called dissipative scale. We found, in agreement with experimental [3] and theoretical [4] studies a non-Gaussian statistics in both the fluid and sub-ion range with a transition from an intermittent to a self-similar behavior. Our main finding here is that the transition is observed at a scale length of the order of several de instead that around di independently from the ion dynamics. The transition seems to be driven mainly by the small scale electron dynamics around the reconnection structures where the electron inertial terms become non-negligible.

[1] T.D. Phan et al., Electron magnetic reconnection without ion coupling in Earth’s turbulent magnetosheath, Nature, 2018

[2] F. Califano, S.S. Cerri, M. Faganello, D. Laveder, M. W. Kunz, Electron-only magnetic reconnection in plasma turbulence, Astrophys. Journal, submitted

[3] Kiyani et al., Global Scale-Invariant Dissipation in Collisionless Plasma Turbulence, Phys. Rev. Lett., 2009

[4] E. Leonardis et al., Multifractal scaling and intermittency in hybrid Vlasov-Maxwell simulations of plasma turbulence, Physics of Plasmas, 2016

• This contribution has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 776262 (AIDA, www.aida-space.eu)

Turbulence and kinetic processes in magnetized space plasmas have been extensively investigated over the past decades via theoretical models, in-situ spacecraft measurements, and numerical simulations. In particular, multi-point high-resolution measurements from the Cluster and MMS space missions brought to light an entire new world of kinetic processes, taking place at the plasma microscales, and exposed new challenges for their theoretical interpretations. A long-lasting debate concerns the nature of ion and electron scale fluctuations in solar-wind turbulence and their dissipation via collisionless plasma mechanisms. Alongside observations, numerical simulations have always played a central role in providing a test ground for existing theories and models.

In this talk, the current advances achieved with 3D3V kinetic simulations, as well as the main questions left open (or raised) by these studies will be discussed. This includes assessing the spectral properties and intermittency of turbulent fluctuations in the sub-ion range$[1]$ and the existence of an anisotropic turbulent cascade involving the entire phase space$[2]$ (i.e., a cascade of free energy that is anisotropic with respect to the ambient magnetic field in both real and velocity space). Finally, also preliminary combined results from recent numerical studies will be presented to assess similarities and/or differences in the properties of kinetic-scale plasma turbulence, estimated from these state-of-the-art 3D kinetic simulations$[1,2,3,4]$.

$[1]$ Cerri, Servidio & Califano, ApJL 846, L18 (2017)

$[2]$ Cerri, Kunz & Califano, ApJL 856, L13 (2018)

$[3]$ Franci em et al., ApJ 853, 26 (2018)

$[4]$ Groselj em et al., PRL bf120, 105101 (2018)

Fluid and plasma turbulence is a longstanding problem in physics. Studying its dynamics can help understanding various processes such as mass transport and energy dissipation, in particular in collisionless systems like most of the astrophysical plasmas. The solar wind heating problem, which is manifested by a slower decrease of the ion temperature as function of the heliocentric distance than the prediction from the adiabatic expansion model of the wind, is one example of such problems where turbulence can help give an explanation.

A way to study fluid or plasma turbulence is to estimate the total energy cascade rate, which is the energy transferred from the largest scales into the dissipative scales of the system. This is made possible by the use of exact laws, which link the energy cascade rate to the physical variables of the flow. Significant progress has been made in recent years on deriving various forms of exact laws for different compressible flows: HydroDynamics (HD), MagnetoHydroDynamics (MHD) and Hall-MagnetoHydroDynamics (HMHD). Some of them were used successfully to estimate the energy cascade rate in the solar wind and the magnetosheath, but at the expense of making additional assumptions that made different mathematical terms involved in the laws accessible to in-situ measurements.

Here we present an alternative exact law for compressible Hall-MHD turbulence. This law is more compact and easier to compute in numerical simulations and spacecraft data, thus reducing the memory load and time required to compute the energy cascade rate. We also show the validity of this new law in the limit of compressible HD using high-resolution simulation data of HD turbulence spanning the subsonic and supersonic regimes.

The non-linear nature of the Einstein’s equations of general relativity suggests that space-time can be turbulent. Such a turbulence is expected during the primordial universe (first second) when gravitational waves (GW) have been excited through eg. the merger of primordial black holes. The analytical theory of weak GW turbulence, published in 2017 [1], is built from a diagonal space-time metric reduced to the variables t, x and y [2]. The theory predicts the existence of a dual cascade driven by 4–wave interactions with a direct cascade of energy and an inverse cascade of wave action. In the latter case, the isotropic Kolmogorov-Zakharov spectrum N(k) has the power law index -2/3 involving an explosive phenomenon. In this context, we developed a fourth-order and a second-order nonlinear diffusion models in spectral space to describe GW turbulence in the approximation of strongly local interactions [3]. We showed analytically that the model equations satisfy the conservation of energy and wave action, and reproduce the power law solutions previously derived from the kinetic equations. We show numerically by computing the second-order diffusion model that, in the non-stationary regime, the isotropic wave action spectrum N(k) presents an anomalous scaling which is understood as a self-similar solution of the second kind. The regime of weak GW turbulence is actually limited to a narrow wavenumber window and turbulence is expected to become strong at larger scales. Then the phenomenology of critical balance can be used. The formation of a condensate may happen and its rapid growth can be interpreted as an accelerated expansion of the universe that could be at the origin of the cosmic inflation. We can show with this scenario that the fossil spectrum obtained after inflation is compatible with the latest data obtained with the Planck/ESA satellite [4].

[1] Galtier & Nazarenko, Turbulence of weak gravitational waves in the early universe, Phys. Rev. Lett. 119, 221101 (2017).

[2] Hadad & Zakharov, Transparency of strong gravitational waves, J. Geom. Phys. 80, 37 (2014).

[3] Galtier, Nazarenko, Buchlin & Thalabard, Nonlinear diffusion models for gravitational wave turbulence Physica D 390, 84 (2019).

[4] Galtier, Nazarenko & Laurie, Cosmic inflation driven by space-time turbulence (2019).

Coherent magnetic structures such as magnetic vortex chains have been observed in the solar wind close to the Earth by the Cluster space mission (Perrone et al. (2016, 2017)). Making use of a gyrofluid model, we investigate the existence of analytical solutions of magnetic vortex type and study their stability. The adopted model can provide a nonlinear description of turbulent collisionless magnetized plasmas accounting for ion finite Larmor radius, equilibrium temperature anisotropy and fluctuations of the component of the magnetic field parallel to the direction of a strong and uniform guide field. The model possesses a noncanonical Hamiltonian structure which provides a convenient framework for the use of analytical tools, such as the Energy-Casimir method for determining stability conditions. We carry out investigations for some asymptotic regimes of the model, such as for instance in the limit of a large ion-to-electron perpendicular equilibrium temperature ratio, with negligible electron inertia effects, and compare our results with those found recently in the framework of a reduced magnetohydrodynamics model (Jovanovic et al. 2018).

• D. Perrone, O. Alexandrova, O. W. Roberts, S. Lion, C. Lacombe, A. Walsh, M. Maksimovic and I. Zouganelis. The Astrophysical Journal, 849:49, 2017

• D. Perrone, O. Alexandrova, A. Mangeney, M. Maksimovic, C. Lacombe, V. Rakoto, J. C. Kasper, and D. Jovanović. The Astrophysical Journal, 826:196, 2016

• D. Jovanović, O. Alexandrova, M. Maksimović, M. Belić. J. Plasma Phys., vol. 84, 2018

In space plasmas, turbulence, magnetic reconnection and shock propagation are ubiquitous physical processes that have been traditionally studied using a one-fluid resistive MHD description.

Within the theoretical framework of two-fluid MHD, we retain the effects of the Hall current and electron inertia. Also, this description brings two new spatial scales into play, such as the ion and electron inertial lengths. We perform numerical simulations of the two-fluid equations and study the physical processes arising at sub-ion and even electron scales both three important phenomena in space plasmas: turbulence, magnetic reconnection and perpendicular shocks.

When a stationary turbulent regime is established, our simulations show changes in the slope of the energy power spectrum at the ion and electron inertial lengths, in agreement with the slopes obtained using dimensional analysis. Using non-dissipative two-fluid simulations, we confirm that magnetic reconnection arises only when the effects of electron inertia are retained. In a stationary regime, we obtain that the reconnection rate is proportional to the ion inertial length, as it also emerges from a scaling law derived from dimensional arguments. Finally, using 1D two-fluid simulations, we show the generation of fast-mode perpendicular shocks with a thickness of a few electron inertial lengths.

Cometary induced magnetospheres are archetypes of mass-loaded, partially collisional, partially ionised plasmas, characterised by a wide range of varying plasma parameters, where the interplay between collisionless and collisional processes are essential to give a global picture of the plasma dynamics. While several cometary fly-by missions have enabled to pave the way towards the exploration of cometary environments, the Rosetta mission was the first space mission to escort a comet along its orbit around the Sun. During more than two years (2014-2016), the Rosetta orbiter has monitored comet 67P/CG and its ionised environment, at heliocentric distances ranging from 1.2 to 3.8 AU accounting for a variety of cometary activity, and at distances from the comet nucleus ranging from 1500 km down to the comet nucleus surface itself during the Rosetta Orbiter’s final descent. This was the first extensive, long-term, in situ survey of the expanding ionosphere of a comet which interaction with the solar wind forms an induced magnetosphere. In this context, I will review the results obtained from in situ observations made by the different instruments of the Rosetta Plasma Consortium (RPC), combined to state-of-art numerical modelings of cometary plasma environments, to give an overview of the current understanding of the structure and dynamics of a cometary induced magnetosphere. Among different mechanisms, I will show how plasma waves traces the signature of plasma mixing at different interfaces (e.g., electron temperature discontinuities, strong density gradients) and describe some acceleration mechanisms at play in the inner cometary plasma.

We present a local (in space and time) approach to the study of scale-to-scale energy transfers in magnetohydrodynamic (MHD) turbulence. This approach is based on performing local averages of the physical fields, which amounts to filtering scales smaller than some parameter $\ell$. A key step in this work is the derivation of a local Kármán-Howarth-Monin relation which can be interpreted as a coarse-grained energy balance. This provides a local form of Politano and Pouquet’s 4/3-law without any assumption of homogeneity or isotropy, which is exact, non-random, and connects well to the usual statistical notions of turbulence. After a brief presentation of this approach, we first apply it to turbulent data obtained via a three dimensional direct numerical simulation of the forced, incompressible MHD equations from the John Hopkins turbulent database. The local Kármán-Howarth-Monin relation holds well. The space statistics of local cross-scale transfers is studied, their means and standard deviations being maximum at inertial scales, and their probability density functions (PDFs) displaying very wide tails. Events constituting the tails of the PDFs are shown to form structures of strong transfers, either positive or negative, which can be observed over the whole available range of scales. As $\ell$ is decreased, these structures become more and more localized in space while contributing to an increasing fraction of the mean energy cascade rate. Second, we show how the same approach can be applied to spacecraft data where the main difficulty lies in the fact that measurements are restricted to few points, in one small region of space at a time, and a single scale. We compare our approach to results obtained from Cluster and MMS data using the LET proxy, and highlight its importance to the understanding of solar wind turbulence and solar wind heating.

A puzzling property of fast solar wind magnetic fluctuations is that, despite their large amplitude, they induce little variations in the strength of the magnetic field, thus maintaining a low level of compressibility in the plasma.

At the same time, in addition to the well-known Kolmogorov MHD inertial range spectrum with slope -5/3, larger scales of fast streams are characterised by a shallower slope, close to -1. This 1/f range is considered the energy reservoir feeding the turbulent cascade at smaller scales, although its origin is not well understood yet.

These aspects are usually addressed as separate properties of the plasma, however, we suggest that a link between the two exists and we propose a phenomenological model in which a 1/f spectrum for large scales can be derived as a consequence of the low magnetic compressibility condition. Remarkably this model, although simple, can capture most of the variability observed in situ in the solar wind and explain spectral differences in wind regimes. Moreover, our model provides a prediction for the evolution of the 1/f range close to the Sun that it will be possible to test soon thanks to the forthcoming observations of Parker Solar Probe.

We investigate the transition of the solar wind turbulent cascade from MHD to sub-ion range by means of in situ observations and hybrid numerical simulations. First, we focus on the angular distribution of wave-vectors in the kinetic range, between ion and electron scales, using Cluster magnetic field measurements. Observations suggest the presence of a quasi-2D gyrotropic distribution around the mean field, confirming that turbulence is characterised by fluctuations with $k_\perp>>k_|$ in this range; this is consistent with what is usually found at larger MHD scales, and in good agreement with our hybrid simulations.

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, with higher magnetic compressibility for higher beta. Moreover, we observe that this increase has a gradual trend from low to high beta in the 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 the prediction based on kinetic Alfvén waves (KAW), especially at high beta, suggesting that in the kinetic range explored the turbulence is supported by KAW-like fluctuations.

Rolled-up vortices associated to the Kelvin-Helmholtz instability (KHI) have been detected by in-situ observations around the Earth, Saturn and Mercury magnetospheres due to the interaction with the solar wind. KHI in magnetized plasmas have been widely studied numerically in the framework of a fluid, hybrid, and full kinetic approach, while only very few studies have focused on the physics of electrons because of computational constraints. In this work we present a full kinetic particle in cell study of the KHI spanning a range of scales going from fluid to electron scales. The simulation is initialized with an extended fluid equilibrium including finite ion Larmor radius effects. Our large-scale configuration includes two-possible alignment of the vorticity with the background magnetic field each one corresponding to the interaction of the solar wind with the dawn and dusk side of a planet. We discuss electron heating and acceleration by analyzing temperature anisotropy and particle distribution functions. Two fluid simulations have suggested that KHI instability can lead to the onset of the mirror instability. Our full kinetic approach confirms such hypothesis. We discuss the formation of mirror modes in our simulations.

F. Sahraoui (1), L. Z. Hadid (2), N. Andrés (1,3), F. Galtier (1,4), S. Y. Huang (5), R. Ferrand (1), and S. Banerjee (6)

(1) LPP, CNRS - Ecole Polytechnique – Sorbonne Université - Univ. Paris-Sud - Observatoire de Paris, Université Paris-Saclay, Palaiseau, 91128, France

(2) Swedish Institute of Space Physics, Uppsala, Sweden

(3) Instituto de Astronomia y Fisica del Espacio, UBA-CONICET, CC. 67, suc. 28, 1428, Buenos Aires, Argentina

(4) Institut Universitaire de France

(5) School of Electronic Information, Wuhan University, Wuhan, China

(6) Indian Institute of Technology, IIT, Kanpur, India

Compressible turbulence has been a subject of active research within the space physics community over the past years. It is thought to be essential for understanding the physics of the solar wind (for instance the heating of the fast wind), planetary magnetospheres and the interstellar medium (star formation). Using recently derived exact laws of compressible isothermal MHD and the THEMIS and CLUSTER spacecraft data we investigate the physics of the fast and slow solar winds and the Earth magnetosheath. We emphasize the role of density fluctuations in enhancing both the energy cascade rate and the turbulence spatial anisotropy by analyzing different types of turbulent fluctuations (magnetosonic and Alfvénic-like), and show how kinetic instabilities can regulate the energy cascade rate. This has motivated further investigation of the sub-ion scale cascade using MMS high time resolution data and the exact laws of the Hall-MHD model (see talk by Andrés et al.). Preliminary results on the estimation of the fluid cascade rate at sub-ion and its possible connection to kinetic dissipation will be discussed.

After discussing some open problems concerning Alfvén and kinetic Alfvén wave turbulence in the solar wind, and the transition between these two regimes, we introduce a two-field reduced gyrofluid model which includes ion finite Larmor radius corrections, parallel magnetic fluctuations and electron inertia, and thus covers a spectral range extending from MHD to electron scales [1]. The model reproduces the usual phenomenology of balanced turbulence in the regimes of dispersive, kinetic and inertial Alfvén waves and provides, as suggested by preliminary direct numerical simulations, an efficient tool to address the sub-ion dynamics in the imbalanced regime. Furthermore, starting from the kinetic equations of weak turbulence, a nonlinear diffusion model retaining only strongly local interactions is derived and phenomenologically extended to strong turbulence by a suitable modification of the transfer time which, in the case of balanced turbulence, is consistent with critical balance [2]. The associated scale anisotropy turns out to be affected by the degree of imbalance. In this framework, Landau damping is modeled using the dissipation rate given by the linear kinetic theory, with a modification of the transfer time taking into account the effect of temperature homogenization along the magnetic field lines. Extension of the gyro-fluid model including coupling to slow magnetosonic waves and thus permitting the decay instability will be briefly discussed.

[1] T. Passot, P.L. Sulem and E. Tassi, Gyrofluid modeling and phenomenology of low βe Alfvén wave turbulence, Phys. Plasmas, 25, 042107, 2018.

[2] T. Passot and P.L. Sulem, Imbalanced kinetic Alfvén wave turbulence: from weak turbulence theory to nonlinear diffusion models for the strong regime, J. Plasma Phys., in press.

Multi-dimensional Eulerian simulations of the hybrid Vlasov-Maxwell model[1] have been employed to investigate the role kinetic effects in turbulent plasmas at typical ion scales. Numerical results suggest that kinetic effects manifest through the deformation of the ion distribution function, with patterns of non-Maxwellian features being concentrated near regions of strong magnetic gradients. The velocity-space departure from Maxwellian of the ion velocity distributions has been also recovered in observational data from spacecraft. In a recent paper, Servidio et al.[2] investigated the velocity-space cascade process suggested by the highly structured shape of the ion velocity distribution detected by the NASA Magnetospheric Multiscale mission. Through a tree-dimensional Hermite transform, these authors pointed out a power-law distribution of moments and provided a theoretical prediction for the scaling, based on a Kolmogorov approach. Here, the possibility of a velocity-space cascade is investigated in the strongly magnetized case, in kinetic simulations of turbulence at ion scales. Through the Hermite decomposition of the ion velocity distribution from the simulations, we found that (i) the plasma displays spectral anisotropy in velocity space, due to the presence of the background magnetic field, (ii) the distribution of energy is in agreement with the prediction in Ref. [2] and (iii) the activity in velocity space shows a clear intermittent character in space, being enhanced close to coherent structures, such as the reconnecting current sheets produced by turbulence. Finally, in order to explore the possible role of inter-particle collisions, collisional and collisionless simulations of plasma turbulence have been compared using Eulerian Hybrid Boltzmann-Maxwell simulations, that explicitly model the proton-proton collisions through the nonlinear Dougherty operator.

This talk has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 776262 (AIDA, www.aida-space.eu)

[1] Valentini, F. et al., J. Comput. Phys. 225 2007, 753-770

[2] Servidio, S. et al., Phys. Rev. Lett. 119 2017, 205101

Debye is a proposed and pre-selected mission concept in response to ESA’s F-class call. Debye will consist of a main spacecraft with instrumentation to measure electrons, ions, electric fields, and magnetic fields; and up to three deployable spacecraft that measure magnetic fields only. The deployable spacecraft will fly around the main spacecraft, covering different and varying baselines. In this configuration, Debye will measure electron-scale fluctuations and their effects on the electron distribution function. The key science question for the Debye mission is: How are electrons heated in astrophysical plasmas? In order to answer this top-level science question, Debye's first objective is to identify the nature of electron-scale turbulent fluctuations. Then it will measure the rapid transfer of energy from the fields to the particles through high-cadence and high-resolution electron measurements. Finally, Debye will study the partition of energy between particle species and the dependence of the energy transfer on the plasma conditions.

In this presentation, we discuss the science questions and our proposed pathways to science closure for the Debye mission. Moreover, we discuss the implications of Debye science for the turbulence-research communities in the fields of space, astrophysics, and laboratory plasma physics.