## Periodic Reporting for period 1 - SoWHat (Solar Wind Heating and Turbulence)

Reporting period: 2015-11-01 to 2017-10-31

What makes plasma turbulence a particularly fascinating subject is, apart from its ubiquity in the laboratory and in space, its kinetic nature. In order for the free energy

injected into the fluctuations of the electromagnetic field, flows and particle distribution in a plasma to be thermalised, this energy must be transferred from large to small scales in, generally speaking, a 6-dimensional (velocity and position) phase space. In magnetised plasmas, which, at spatial scales larger than the Larmor radii of the particles and time scales longer that those particles’ Larmor periods, can be described by the drift-kinetic or “Kinetic Magnetohydrodynamics” approximations, this scale refinement in phase space can be conceptualised as a superposition of two fundamental processes: nonlinear spatial mixing of both the electromagnetic fields and of the distribution function by the turbulent by the ExB flows across the magnetic field and linear phase mixing of the distribution function due to particle streaming along magnetic-field lines, made vigorous by the absence (or the relatively infrequent occurrence) of Coulomb collisions. The former is turbulence in the usual “fluid” sense. The latter effect, in the linear plasma theory, is known as the Landau damping it involves removal of free energy from the the “fluid” quantities such as density, velocity, magnetic field, etc., and its transfer to higher velocity moments of the perturbed distribution function (fine-scale structure in velocity space).

How the “turbulent cascade” works in the presence of these two types of mixing is a fundamentally interesting question, which, with the advent of high-resolution measurements of turbulence in the solar wind and of kinetic simulations of this turbulence has increasingly preoccupied both theoreticians and modellers (indeed, not only in application to space plasmas, but also, for an even longer time, to fusion ones). In practical terms, is it, one might ask, reasonable to assume that the effect of phase mixing is to damp the spatial part of the turbulent cascade of the “fluid” quantities at a scale-dependent rate equal to the linear Landau damping rate in the system? (This approach is also essentially the one adopted in the so-called Landau fluid closures, albeit at the level of higher-order moments of the distribution function).

In the context of solar-wind turbulence (and, more generally, of collisionless plasma turbulence, of which the solar wind is a particularly well-diagnosed instance), this question arises prominently if one attempts to explain the measured spectra of “compressive” (density and magnetic-field-strength) perturbations. In the fluid theory, these perturbations, which correspond to the slow-wave and entropy modes in low-frequency MHD, are expected to be passively advected by the Alfvénic perturbations, provided the latter are assumed strongly turbulent, anisotropic and critically balanced. As passive tracers, these perturbations should have a spectrum that follows the power-law spectrum of the Alfvénic turbulence and indeed solar-wind measurements show that they do. However, the solar-wind plasma at 1 AU, where these measurements are done, is near-collisionless and in such a plasma, the compressive perturbations, while indeed passive with respect to the Alfvénic ones must in fact be subject to Landau damping (known in this context as Barnes damping) at the rate comparable at each scale to the turbulent cascade rate. Thus, it would appear that there is no conservation of “compressive energy” (variance of density and field-stregth fluctuations) in the inertial range and one might expect that its spectrum should decay steeply, probably in a non-universal way (because at each scale, energy is removed into phase space at a rate at least similar to the rate at which it is passed to the next smaller scale).

injected into the fluctuations of the electromagnetic field, flows and particle distribution in a plasma to be thermalised, this energy must be transferred from large to small scales in, generally speaking, a 6-dimensional (velocity and position) phase space. In magnetised plasmas, which, at spatial scales larger than the Larmor radii of the particles and time scales longer that those particles’ Larmor periods, can be described by the drift-kinetic or “Kinetic Magnetohydrodynamics” approximations, this scale refinement in phase space can be conceptualised as a superposition of two fundamental processes: nonlinear spatial mixing of both the electromagnetic fields and of the distribution function by the turbulent by the ExB flows across the magnetic field and linear phase mixing of the distribution function due to particle streaming along magnetic-field lines, made vigorous by the absence (or the relatively infrequent occurrence) of Coulomb collisions. The former is turbulence in the usual “fluid” sense. The latter effect, in the linear plasma theory, is known as the Landau damping it involves removal of free energy from the the “fluid” quantities such as density, velocity, magnetic field, etc., and its transfer to higher velocity moments of the perturbed distribution function (fine-scale structure in velocity space).

How the “turbulent cascade” works in the presence of these two types of mixing is a fundamentally interesting question, which, with the advent of high-resolution measurements of turbulence in the solar wind and of kinetic simulations of this turbulence has increasingly preoccupied both theoreticians and modellers (indeed, not only in application to space plasmas, but also, for an even longer time, to fusion ones). In practical terms, is it, one might ask, reasonable to assume that the effect of phase mixing is to damp the spatial part of the turbulent cascade of the “fluid” quantities at a scale-dependent rate equal to the linear Landau damping rate in the system? (This approach is also essentially the one adopted in the so-called Landau fluid closures, albeit at the level of higher-order moments of the distribution function).

In the context of solar-wind turbulence (and, more generally, of collisionless plasma turbulence, of which the solar wind is a particularly well-diagnosed instance), this question arises prominently if one attempts to explain the measured spectra of “compressive” (density and magnetic-field-strength) perturbations. In the fluid theory, these perturbations, which correspond to the slow-wave and entropy modes in low-frequency MHD, are expected to be passively advected by the Alfvénic perturbations, provided the latter are assumed strongly turbulent, anisotropic and critically balanced. As passive tracers, these perturbations should have a spectrum that follows the power-law spectrum of the Alfvénic turbulence and indeed solar-wind measurements show that they do. However, the solar-wind plasma at 1 AU, where these measurements are done, is near-collisionless and in such a plasma, the compressive perturbations, while indeed passive with respect to the Alfvénic ones must in fact be subject to Landau damping (known in this context as Barnes damping) at the rate comparable at each scale to the turbulent cascade rate. Thus, it would appear that there is no conservation of “compressive energy” (variance of density and field-stregth fluctuations) in the inertial range and one might expect that its spectrum should decay steeply, probably in a non-universal way (because at each scale, energy is removed into phase space at a rate at least similar to the rate at which it is passed to the next smaller scale).

We have discovered that what in fact happens is rather subtle: while the compressive fluctuations do have a parallel cascade and do phase-mix vigorously, much of their energy flux into phase space due to phase mixing is on average canceled by the return flux from phase space, due to the stochastic version of the plasma-echo effect. The turbulent cascade of compressive energy through a broad range of spatial scales can thus be maintained effectively as if Landau damping were not present at all.

Besides resolving the long-standing (if perhaps not always fully appreciated) puzzle of the density cascade in the solar wind, these results suggest a fundamental conceptual shift for understanding kinetic plasma turbulence generally: rather than being a system where Landau damping plays the role of dissipation, a collisionless plasma is essentially dissipation-less except at very small scales. The universality of “fluid” turbulence physics is thus reaffirmed even for a kinetic, collisionless system.

Besides resolving the long-standing (if perhaps not always fully appreciated) puzzle of the density cascade in the solar wind, these results suggest a fundamental conceptual shift for understanding kinetic plasma turbulence generally: rather than being a system where Landau damping plays the role of dissipation, a collisionless plasma is essentially dissipation-less except at very small scales. The universality of “fluid” turbulence physics is thus reaffirmed even for a kinetic, collisionless system.

Two textbook physical processes compete to thermalise turbulent fluctuations in a collisionless plasma: Kolmogorov’s nonlinear “cascade” to small spatial scales, where dissipation occurs,

and Landau’s damping, which transfers energy to small scales in velocity space via “phase mixing”, also leading to dissipation and entropy production. We show that, in a magnetised plasma,

another textbook process, the plasma echo, will bring back energy from phase space and on average cancel the effect of phase mixing. The energy cascades essentially as it would in a fluid system and thus Kolmogorov wins a competition with Landau for the free energy in a collisionless turbulent plasma. This reaffirms the universality of Kolmogorov’s picture of turbulence

and explains, for example, why broad Kolmogorov-like spectra of density fluctuations are observed in the collisionless plasma of the solar wind.

and Landau’s damping, which transfers energy to small scales in velocity space via “phase mixing”, also leading to dissipation and entropy production. We show that, in a magnetised plasma,

another textbook process, the plasma echo, will bring back energy from phase space and on average cancel the effect of phase mixing. The energy cascades essentially as it would in a fluid system and thus Kolmogorov wins a competition with Landau for the free energy in a collisionless turbulent plasma. This reaffirms the universality of Kolmogorov’s picture of turbulence

and explains, for example, why broad Kolmogorov-like spectra of density fluctuations are observed in the collisionless plasma of the solar wind.