Electron heating and acceleration mechanisms

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Magnetic Reconnection in Turbulence

Magnetic reconnection in turbulent plasmas takes place in small-scale current sheets that form spontaneously in the turbulence [90, 91, 134, 135]. An illustration of turbulent reconnection can be seen in Figure 2.9, that shows the results of numerical simulations. In this gure we observe a large number of thin current sheets forming in the turbulence. A subset of those current sheets are sites of ongoing magnetic reconnection. Such reconnection has been observed for the rst time in the terrestrial magnetosheath downstream of the quasi-parallel shock [124] as shown in Figure 2.8, and later also in the solar wind [20, 44]. Although it has been evoked in other astrophysical environments such as solar ares to account for the observed energetic particles [76, 153], observational evidence remains relatively scarce due to the limited availability of in situ data. Recent statistical observations [105] have indicated a link between the structures formed in the turbulence and reconnection regions although measurements could not specically address kinetic
scales due to limitations in the resolution of plasma measurements. These observations strongly support magnetic reconnection as a major mechanism of energy dissipation in turbulent plasma at kinetic scales, as indicated by [141]. However, how signicant is the contribution of magnetic reconnection as a dissipation mechanism at kinetic scales remains largely an open question. Turbulent reconnection can have dierent characteristics from that occurring in laminar boundaries such as the magnetopause and the magnetotail. One important aspect is related to the reconnection rate, specically if turbulent reconnection can occur with similar rates as laminar reconnection (rate 10%) or even with faster rates. A number of studies have investigated the properties of turbulent reconnection in theory and numerical simulations [69, 74, 80, 90, 134] and discussed this aspect. These studies have demonstrated that turbulent reconnection has a much weaker dependence to the Lundquist number compared to the predictions of the Sweet-Parker model and therefore can be fast. Figure 2.10 shows a cartoon of turbulent reconnection, where the introduction of small-scale magnetic eld uctuations in the in ow region can make reconnection faster by locally reducing the elongation of the diusion region, that is otherwise much larger than its thickness in the Sweet-Parker model. The fact that turbulent reconnection is fast has been veried by in-situ observations during which a rate of 10% was measured [124].
Recent simulations [134] also suggest that turbulence can signicantly enhance the reconnection rate beyond the typical value of 10% typical of Hall reconnection, but such expectations has not yet been experimentally veried.

Search-coil magnetometer

STAFF consists of a magnetic waveform unit (STAFF-SC) and a spectrum analyzer (STAFF-SA). The rst measures uctuations of the magnetic eld from 0:1Hz up to 4kHz while the second computes spectra of the magnetic and electric eld from 8Hz up to 4kHz. It consists of a coil wrapped around a ferromagnetic core. In the presence of a varying external magnetic eld, the variation of the magnetic ux induced by the eld is proportional to the voltage induced in the coil. The STAFF instrument has a tri-axial search-coil sensor operating at frequencies between 0:1Hz and 4kHz. Given the transfer function of the instrument noise, the measurements are less sensitive than FGM below 1Hz. STAFF-SC provides waveforms of the 3 components of the magnetic eld with a sampling frequency of up to 450Hz in Burst Mode. STAFF-SA provides spectral matrices and power spectral densities for each of the 3 components of the magnetic eld (Bx;By;Bz) and for two components of the electric eld (Ex;Ey) at dierent time resolutions for frequencies from 8Hzup to 4kHz. Those quantities are calculated on board using measurements by the magnetic eld sensors (STAFF) and electric eld probes (EFW). In Burst mode, spectral matrices are provided in the frequency range of 64Hz to 4kHz with 1s resolution, while the power spectral densities are provided
for the same frequency range but at higher time resolution (0:125s or 0:25s).

Electric Field Measurements

The electric eld is measured by the EFW instrument [50] using four spherical probes located at the end of thin wire booms extending outwards from the spacecraft in the spacecraft spin plane and phased by 90o in that plane [110]. Two probes in each pair are kept at a distance of about 88m from each other by the rotation of the spacecraft around its axis. The requirement for a relatively long distance between the probes and the spacecraft comes from the necessity to overcome the eects of the Debye shielding as well as the photoelectron cloud around the spacecraft. Additionally, the larger distance between the probes results in a larger potential dierence, which is easier to measure. The potential dierence between two opposed probes yields a measurement of the electric eld in the direction along the axis dened by the two probes (Figure 3.7). The use of the  probes allows an estimation of two orthogonal components of the electric eld in the plane of the spacecraft spin. There is no antenna to measure the component of the electric eld along the spin axis, that can only be estimated from the two components measured in the plane. Such third component is deduced by using the assumption ~E ~B = 0. This assumption leads to signicant errors when the angle between the magnetic and the electric eld is small and cannot be used at guration. Courtesy of IRF. all when they are aligned since it degenerates. In the case of EFW the third component is usually not calculated if the angle between the magnetic and the electric eld is below 15o. More recent missions such as MMS are equipped with a rigid antenna and directly measure the component of the electric eld along the spin axis. For Cluster, the probe to probe potential and the subsequent electric eld measurement are given at a sampling rate of up to 450Hz in when operating in Burst Mode.

Electron Measurements

The two instruments aboard Cluster that measure the properties of the electrons are PEACE and RAPID. PEACE consists of two top-hat electrostatic analyzers (shown in gure 3.8) that measure 3-dimensional electron distribution functions at energies up to 30keV . RAPID uses pin-hole detectors to measure high energy electrons in the range from 39keV to 406keV .
As illustrated in gure 3.9, the electrostatic analyzers used by PEACE operate by applying a specic voltage between two plates that are shaped as part of a circle. Electrons that enter with energy corresponding to that voltage follow the path down to the detector, whereas electrons with dierent energies collide with the walls of the detector and are absorbed. Therefore using dierent values of the voltage, the distribution function is sampled.
The PEACE instrument has two sensors, designated HEEA and LEEA. Both are top hat electrostatic analyzers but have dierent geometric factors. This means that they are optimized for dierent ranges of electron uxes and energies making them suitable for dierent environments. LEEA is the Low Energy Electron Analyzer and its geometric factor makes it appropriate for higher uxes and relatively low energies, such as those found in the magnetosheath and the solar wind. HEEA (High Energy Electron Analyzer) is more suitable for environments where the electrons have lower density and higher energy such as those found the outer PEACE there is simultaneous coverage of a 180o in elevation angle in 12 equal parts. Adapted from [39].
magnetosphere and the magnetotail. The two PEACE sensors are placed on opposite sides of the spacecraft.

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Minimum Variance Analysis

As mentioned above, during the crossing of a current sheet by the spacecraft, we expect to have a variation of the magnetic eld along the normal to the current sheet plane. Such direction, as well as the other two directions in the current sheet plane, can be obtained through the minimum variance analysis of the magnetic eld [43] and can be used to dene a local reference frame for the current sheet.
The normal direction is dened as the direction along which the magnetic eld has the smallest variation across the current sheet (minimum variance), as follows directly by integrating the equation r B = 0 across the current sheet. The direction of the component of maximum variance corresponds to the direction in the current sheet plane along which the magnetic eld changes the most. The intermediate variance direction is the second component in the current sheet plane and coincides with the direction of the electric current.

Estimation of electron temperature at subspin time resolution

As described in the previous chapter, the two sensors LEEA and HEEA of the PEACE instrument on board Cluster scan the full sky as the spacecraft rotates, covering all directions once every spin which lasts about 4s, see Figures 3.9 and 3.10. Each sensor measure electron counts N(E; ; ) in a given energy range E and two angles (elevation angle from the spin plane) and (azimuthal angle inthe spin plane). The phase space density is then obtained as: f(v; ; ) = 2N(E; ; ) tav4G(E).

Electron heating within reconnecting current sheet

The rst part of the results, discussed in Section 5.1, was focused on providing evidence of electron heating within thin current sheets in turbulence. Magnetic reconnection is suggested to be the dominant mechanism for high PVI current sheets, while for low PVI current sheets other mechanisms dierent from reconnection, such as heating within small-scale shocks or wave-particle interaction, could also occur. Such other mechanisms are the subject of future studies, as discussed in chapter 6. Despite of the fact that reconnection is accepted as dominant dissipation and heating mechanisms for reconnection at kinetic scales, the exact way particles are heated and accelerated during reconnection is not yet fully understood both from theoretical and observational point of view. In this study, we focus on the heating of electrons. Several possibilities exist, e.g. heating due to the reconnection parallel electric eld, to adiabatic Fermi and betatron mechanisms and to resonant interaction with dierent wave modes within current sheets. The relative importance of such mechanisms is not fully understood, as well the dependence of dierent boundary conditions such as guide eld as well as magnetic eld and density asymmetries. In this second part of the thesis, I have studied in detail one reconnecting current sheet having symmetric density, symmetric magnetic eld and a very small guide eld. Observations are done in the diusion region very close to the X-line. For such current sheet, I have focused on the detailed observations of the electron distribution functions and I have attempted to evaluate the role of dierent heating mechanisms. The results of this part are still preliminary, in particular regarding the analysis of dierent wave modes in the current sheet and of their role for possible electron heating, and the study is still ongoing. The analysis of other current sheets with dierent values of guide eld and asymmetries and the comparison of the heating mechanisms with the current sheet studied here has been left for future studies.

Table of contents :

Acknowledgements
Abstract
Resume
List of Figures
List of Tables
1 Introduction 
2 Theoretical Background 
2.1 Magnetic Reconnection
2.2 Turbulence
2.3 Magnetic Reconnection in Turbulence
3 Instruments and Data Products 
3.1 Cluster Mission Overview
3.2 Magnetic Field Measurements
3.2.1 Flux-gate magnetometer
3.2.2 Search-coil magnetometer
3.3 Electric Field Measurements
3.4 Spacecraft Potential
3.5 Electron Measurements
3.5.1 Instrument Operation
3.5.2 Data Products
3.5.3 Main issues
4 Methods of data analysis 
4.1 Methods for the detection of current sheets
4.1.1 Partial Variance of Increments
4.1.2 Magnetic Shear Angle
4.1.3 Curlometer technique
4.2 Orientation and motion of current sheets
4.2.1 Minimum Variance Analysis
4.2.2 Timing Analysis
4.3 Estimation of electron temperature at sub-spin time resolution
4.3.1 Implementation
4.3.2 Diagnostics
4.3.3 Instrumental Limitations
5 Results 
5.1 Statistics of thin current sheets
5.1.1 Detection of current sheets
5.1.2 Properties of current sheets
5.1.3 Electron heating
5.1.4 Energy partition
5.2 Electron heating within reconnecting current sheet
5.2.1 Electron heating and acceleration mechanisms
5.2.2 Evidence of reconnection
5.2.3 Electron distributions and heating
5.2.4 Wave measurements
6 Conclusion and future work 
6.1 Conclusion
6.2 Future Work
Bibliography

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