Cascade-Dissipation Balance in Astrophysical Plasmas: Insights from the Terrestrial Magnetosheath

The differential heating of electrons and ions by turbulence in weakly collisional magnetized plasmas and the scales at which such energy dissipation is most effective are still debated. Using a large data sample measured in Earth's magnetosheath by the magnetospheric multiscale mission and the coarse-grained energy equations derived from the Vlasov-Maxwell system, we find evidence of a balance over two decades in scales between the energy cascade and dissipation rates. The decline of the cascade rate at kinetic scales (in contrast with a constant one in the inertial range), is balanced by an increasing ion and electron heating rates, estimated via the pressure strain. Ion scales are found to contribute most effectively to ion heating, while electron heating originates from both ion and electron scales. These results can potentially impact the current understanding of particle heating in turbulent magnetized plasmas as well as their theoretical and numerical modeling.

Exact law for compressible pressure-anisotropic magnetohydrodynamic turbulence: Toward linking energy cascade and instabilities

We derive an exact law for compressible pressure-anisotropic magnetohydrodynamic turbulence. For a gyrotropic pressure tensor, we study the double-adiabatic case and show the presence of new flux and source terms in the exact law, reminiscent of the plasma instability conditions due to pressure anisotropy. The Hall term is shown to bring ion-scale corrections to the exact law without affecting explicitly the pressure terms. In the pressure isotropy limit we recover all known results obtained for isothermal and polytropic closures. The incompressible limit of the gyrotropic system leads to a generalization of the Politano and Pouquet's law where a new incompressible source term is revealed and reflects exchanges of the magnetic and kinetic energies with the no-longer-conserved internal energy. We highlight the possibilities offered by the new laws to investigate potential links between turbulence cascade and instabilities widely observed in laboratory and astrophysical plasmas.

Proof of the zeroth law of turbulence in one-dimensional compressible magnetohydrodynamics and shock heating

The zeroth law is one of the oldest conjectures in turbulence that is still unproven. Here, we consider weak solutions of one-dimensional compressible magnetohydrodynamics and demonstrate that the lack of smoothness of the fields introduces a dissipative term, named inertial dissipation, into the expression of energy conservation that is neither viscous nor resistive in nature. We propose exact solutions assuming that the kinematic viscosity and the magnetic diffusivity are equal, and we demonstrate that the associated inertial dissipation is positive and equal on average to the mean viscous dissipation rate in the limit of small viscosity, proving the conjecture of the zeroth law of turbulence and the existence of an anomalous dissipation. As an illustration, we evaluate the shock heating produced by discontinuities detected by Voyager in the solar wind around 5 AU. We deduce a heating rate of ∼10^{-18}Jm^{-3}s^{-1}, which is significantly higher than the value obtained from the turbulent fluctuations. This suggests that collisionless shocks can be a dominant source of heating in the outer solar wind.

In Situ Observation of Hall Magnetohydrodynamic Cascade in Space Plasma

We present estimates of the turbulent energy-cascade rate derived from a Hall-magnetohydrodynamic (MHD) third-order law. We compute the contribution from the Hall term and the MHD term to the energy flux. Magnetospheric Multiscale (MMS) data accumulated in the magnetosheath and the solar wind are compared with previously established simulation results. Consistent with the simulations, we find that at large (MHD) scales, the MMS observations exhibit a clear inertial range dominated by the MHD flux. In the subion range, the cascade continues at a diminished level via the Hall term, and the change becomes more pronounced as the plasma beta increases. Additionally, the MHD contribution to interscale energy transfer remains important at smaller scales than previously thought. Possible reasons are offered for this unanticipated result.

Scale-to-scale energy transfer rate in compressible two-fluid plasma turbulence

We derive the exact relation for the energy transfer in three-dimensional compressible two-fluid plasma turbulence. In the long-time limit, we obtain an exact law which expresses the scale-to-scale average energy flux rate in terms of two point increments of the fluid variables of each species, electric and magnetic field and current density, and puts a strong constraint on the turbulent dynamics. The incompressible single fluid and two-fluid limits and the compressible single fluid limit are recovered under appropriate assumption. In the single fluid limits, analyses are done with and without neglecting the electron mass thereby making the exact relation suitable for a broader range of application. In the compressible two-fluid regime, the total energy flux rate, unlike the single fluid case, is found to be unaltered by the presence of a background magnetic field. The exact relation provides a way to test whether a range of scales in a plasma is inertial or dissipative and is essential to understand the nonlinear nature of both space and dilute astrophysical plasmas.

Energy Cascade Rate Measured in a Collisionless Space Plasma with MMS Data and Compressible Hall Magnetohydrodynamic Turbulence Theory

The first complete estimation of the compressible energy cascade rate |ϵ_{C}| at magnetohydrodynamic (MHD) and subion scales is obtained in Earth's magnetosheath using Magnetospheric MultiScale spacecraft data and an exact law derived recently for compressible Hall MHD turbulence. A multispacecraft technique is used to compute the velocity and magnetic gradients, and then all the correlation functions involved in the exact relation. It is shown that when the density fluctuations are relatively small, |ϵ_{C}| identifies well with its incompressible analog |ϵ_{I}| at MHD scales but becomes much larger than |ϵ_{I}| at subion scales. For larger density fluctuations, |ϵ_{C}| is larger than |ϵ_{I}| at every scale with a value significantly higher than for smaller density fluctuations. Our study reveals also that for both small and large density fluctuations, the nonflux terms remain always negligible with respect to the flux terms and that the major contribution to |ϵ_{C}| at subion scales comes from the compressible Hall flux.

Turbulence-Driven Ion Beams in the Magnetospheric Kelvin-Helmholtz Instability

The description of the local turbulent energy transfer and the high-resolution ion distributions measured by the Magnetospheric Multiscale mission together provide a formidable tool to explore the cross-scale connection between the fluid-scale energy cascade and plasma processes at subion scales. When the small-scale energy transfer is dominated by Alfvénic, correlated velocity, and magnetic field fluctuations, beams of accelerated particles are more likely observed. Here, for the first time, we report observations suggesting the nonlinear wave-particle interaction as one possible mechanism for the energy dissipation in space plasmas.

Passive Advection of a Vector Field by Compressible Turbulent Flow: Renormalizations Group Analysis near d = 4

The renormalization group approach and the operator product expansion technique are applied to the model of a passively advected vector field by a turbulent velocity field. The latter is governed by the stochastic Navier-Stokes equation for a compressible fluid. The model is considered in the vicinity of space dimension d = 4 and the perturbation theory is constructed within a double expansion scheme in y and ε = 4 − d , where y describes scaling behaviour of the random force that enters the Navier-Stokes equation. The properties of the correlation functions are investigated, and anomalous scaling and multifractal behaviour are established. All calculations are performed in the leading order of y, ε expansion (one-loop approximation).

Compressible Magnetohydrodynamic Turbulence in the Earth's Magnetosheath: Estimation of the Energy Cascade Rate Using in situ Spacecraft Data

The first estimation of the energy cascade rate |ε_{C}| of magnetosheath turbulence is obtained using the Cluster and THEMIS spacecraft data and an exact law of compressible isothermal magnetohydrodynamics turbulence. The mean value of |ε_{C}| is found to be close to 10^{-13}  J m^{-3} s^{-1}, at least 2 orders of magnitude larger than its value in the solar wind (∼10^{-16}  J m^{-3} s^{-1} in the fast wind). Two types of turbulence are evidenced and shown to be dominated either by incompressible Alfvénic or compressible magnetosoniclike fluctuations. Density fluctuations are shown to amplify the cascade rate and its spatial anisotropy in comparison with incompressible Alfvénic turbulence. Furthermore, for compressible magnetosonic fluctuations, large cascade rates are found to lie mostly near the linear kinetic instability of the mirror mode. New empirical power-laws relating |ε_{C}| to the turbulent Mach number and to the internal energy are evidenced. These new findings have potential applications in distant astrophysical plasmas that are not accessible to in situ measurements.

Energy transfer in compressible magnetohydrodynamic turbulence for isothermal self-gravitating fluids

Three-dimensional, compressible, magnetohydrodynamic turbulence of an isothermal, self-gravitating fluid is analyzed using two-point statistics in the asymptotic limit of large Reynolds numbers (both kinetic and magnetic). Following an alternative formulation proposed by Banerjee and Galtier [Phys. Rev. E 93, 033120 (2016)2470-004510.1103/PhysRevE.93.033120; J. Phys. A: Math. Theor. 50, 015501 (2017)1751-811310.1088/1751-8113/50/1/015501], an exact relation has been derived for the total energy transfer. This approach results in a simpler relation expressed entirely in terms of mixed second-order structure functions. The kinetic, thermodynamic, magnetic, and gravitational contributions to the energy transfer rate can be easily separated in the present form. By construction, the new formalism includes such additional effects as global rotation, the Hall term in the induction equation, etc. The analysis shows that solid-body rotation cannot alter the energy flux rate of compressible turbulence. However, the contribution of a uniform background magnetic field to the flux is shown to be nontrivial unlike in the incompressible case. Finally, the compressible, turbulent energy flux rate does not vanish completely due to simple alignments, which leads to a zero turbulent energy flux rate in the incompressible case.

Exact law for homogeneous compressible Hall magnetohydrodynamics turbulence

We derive an exact law for three-dimensional (3D) homogeneous compressible isothermal Hall magnetohydrodynamic turbulence, without the assumption of isotropy. The Hall current is shown to introduce new flux and source terms that act at the small scales (comparable or smaller than the ion skin depth) to significantly impact the turbulence dynamics. The law provides an accurate means to estimate the energy cascade rate over a broad range of scales covering the magnetohydrodynamic inertial range and the sub-ion dispersive range in 3D numerical simulations and in in situ spacecraft observations of compressible turbulence. This work is particularly relevant to astrophysical flows in which small-scale density fluctuations cannot be ignored such as the solar wind, planetary magnetospheres, and the interstellar medium.

Alternative derivation of exact law for compressible and isothermal magnetohydrodynamics turbulence

The exact law for fully developed homogeneous compressible magnetohydrodynamics (CMHD) turbulence is derived. For an isothermal plasma, without the assumption of isotropy, the exact law is expressed as a function of the plasma velocity field, the compressible Alfvén velocity, and the scalar density, instead of the Elsasser variables used in previous works. The theoretical results show four different types of terms that are involved in the nonlinear cascade of the total energy in the inertial range. Each category is examined in detail, in particular, those that can be written either as source or flux terms. Finally, the role of the background magnetic field B_{0} is highlighted and a comparison with the incompressible MHD (IMHD) model is discussed. This point is particularly important when testing this exact law on numerical simulations and in situ observations in space plasmas.