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Edited by: Antonella Greco, Department of Physics, University of Calabria, Italy

Reviewed by: Yan Yang, Southern University of Science and Technology, China; Anton Artemyev, Space Research Institute (RAS), Russia

This article was submitted to Space Physics, a section of the journal Frontiers in Astronomy and Space Sciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The process of conversion or dissipation of energy in nearly collisionless turbulent space plasma, is yet to be fully understood. The existing models offer different energy dissipation mechanisms which are based on wave particle interactions or non-resonant stochastic heating. There are other mechanisms of irreversible processes in space. For example, turbulence generated coherent structures, e.g., current sheets are ubiquitous in the solar wind and quasi-parallel magnetosheath. Reconnecting current sheets in plasma turbulence are converting magnetic energy to kinetic and thermal energy. It is important to understand how the multiple (reconnecting) current sheets contribute to spatial distribution of turbulent dissipation. However, detailed studies of such complex structures have been possible mainly via event studies in proper coordinate systems, in which the local inflow/outflow, electric and magnetic field directions, and gradients could be studied. Here we statistically investigate different energy exchange/dissipation (EED) measures defined in local magnetic field-aligned coordinates, as well as frame-independent scalars. The presented statistical comparisons based on the unique high-resolution MMS data contribute to better understanding of the plasma heating problem in turbulent space plasmas.

Turbulence represents an unsolved problem in classical physics of continuous media (e.g., fluids) characterized by velocity shears, intermittent distribution of kinetic energy over multiple spatial, and temporal scales involving strong non-linear interactions and many (possibly infinite) degrees of freedom (e.g., Frisch,

In this paper, using high-resolution field and plasma data from the Magnetospheric Multi-Scale (MMS) mission we investigate derived energy exchange/dissipation measures at (reconnecting) current sheets in the turbulent quasi-parallel magnetosheath. The paper is organized as follows. Section 2 explains the data and instrumentation and section 3 introduces the EED measures and their time evolution during a more than 4 min long time interval in the turbulent magnetosheath. Section 4 presents a conditional statistics of averaged EED measures for normalized current densities. Section 5 contains the summary and conclusions.

We consider the time interval between 00:21:45 and 00:26:15 UT on November 30, 2015, when the MMS spacecraft were in the strongly compressed quasi-parallel magnetosheath. The MMS fleet was at the GSE position (9, –3, –0.5) _{E} in tetrahedron configuration with inner probe separation between 4 and 22 km comparable to the electron and ion inertial lengths of ~1 and 20 km, respectively. During the selected time interval the ion and electron moments with time resolution of 150 and 30 ms, respectively, are available from Fast Plasma Investigation (FPI) instrument (Pollock et al.,

_{ij} obtained between spacecraft pairs MMS2-1, 3-1, and 4-1.

Overview of the event. _{X}, _{Y}, _{Z}); _{X}, _{Y}, _{Z}) _{X}, _{Y}, _{Z}); _{e}), parallel (_{e‖}) and perpendicular (_{e⊥}) to magnetic field electron temperatures; _{X}, _{Y}, _{Z}); _{i}), parallel (_{i‖}) and perpendicular (_{i⊥}) to magnetic field ion temperatures; _{X}, _{Y}, _{Z}).

On the X axis of _{ij}, curvB,

The electron momentum equation in a two-fluid collisionless plasma can be expressed in the form (Gurnett and Bhattacharjee,

where _{e} is the electron pressure tensor, _{e}), the electron inertia term, is negligible when the spatial scale lengths are greater than the electron inertial length. Since collisionless reconnection is associated with multi-scale physics, the ion and electron scales are important in describing the electric fields and currents. In this paper we neglect the last term in Equation (1) and we consider the two remaining terms for constructing the EED measures:

and

which correspond to the rate of work done by non-ideal part of electric field on plasma particles. Here ∇.ℙ_{e} was again calculated by using the four-point techniques (Chanteur,

which is similar to Equation (2), however, Equation (4) contains the additional second term on the right side corresponding to the work associated with the transport of net charge. For the time interval considered here the mean value of this term is ~ 0 nW/m^{3} with dispersion of ~ 0.7 nW/m^{3} (not shown). We note that the average values of electron and ion densities are rather high and approximately equal to ~ 100 1/cm^{3}, therefore the plasma moments are well determined.

_{X}, _{Y}, _{Z}); _{⊥1}, _{⊥2}, _{‖}); _{∇.Pe}) in FAC coordinates; _{e} (Equation 4).

The goal of the paper is to determine how the local dissipation depends on the strength of the current density in turbulence. To this end we calculated the time averaged EED measures conditioned on current density and normalized to the time averages of the same measure over the whole time interval. In other words, the relative local enhancements of EED measures for certain values of current densities are estimated relative to the background fluctuations of EED measures, for examples < _{rms} ≡ _{rms}, where _{rms} is the root mean square. For each EED measure and MMS1-4 spacecraft the parallel (triangles) and the perpendicular components (circles) of the dot products are shown in _{e}|

_{rms} (rms = root mean square) versus conditional temporal averages of normalized EED measures calculated by conditioning on the values of current density

In _{rms} and there are significant differences between components and spacecraft. Actually, _{rms} ~ 0.6 μ^{2} and for e.g., 4_{rms} the threshold for ^{2} which corresponds to only a few current sheets in _{rms} = 3.6 μ^{2} it is only one current sheet (event 2 in

In this study more than 4 min of high resolution field and plasma data from the MMS spacecraft was analyzed. Although longer time intervals of magnetosheath data were available from the previous missions, the time resolution of the plasma data was not sufficient to study the thin structures generated by turbulence. The measures corresponding to the work done by electric fields (_{e}), were estimated. The statistical analysis of the temporally and spatially averaged and normalized measures has shown that there is a net irreversible work done by electric fields at current sheets. The averaged < _{e}|_{rms} are considered for statistical analysis.

The relative importance of the terms in Equation (1) and of the EED measures in Equations (2–4) have been studied both numerically and experimentally at reconnecting current sheets (Hesse et al., _{e}), is much larger than the second inertial term. However, near the reconnection X-line the inertial term can reach half of the pressure divergence term (Genestreti et al., _{e} is narrower than the distribution of _{e} is slightly larger than

It was also found that during the analyzed time interval dissipation occurred preferentially in parallel direction to the magnetic field. This is seen in _{e‖} > _{e⊥}, but mainly in

Although our understanding of the energy conversion mechanisms at current sheets has improved over the past years, we are far from seeing the complete picture of the associated turbulent dissipation. We mention here two limiting factors. First, the generating mechanisms of current sheets and the role of velocity gradients needs to be understood better. Second, reconnecting current sheets in 3D turbulence can be associated with electron scale coherent structures, for example, interacting extended flux ropes (Daughton et al.,

Certainly, further numerical simulations, event studies and statistical analysis of current sheets will be needed to understand better the role of coherent structures in kinetic energy conversions in collisionless turbulent plasmas and their contribution to the total heating of larger plasma volumes.

The datasets analyzed for this study can be found in the MMS science data archive

ZV and EY analyzed the data and drafted the manuscript. YK, AV, and YN contributed to the interpretation of the analysis and general improvements in the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are grateful for numerous discussions on this topic with R. Nakamura and O. W. Roberts, both from Space Research Institute, Graz.