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 speci c 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 di erent energies collide with the walls of the detector and are absorbed. Therefore using di erent 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 di erent geometric factors. This means that they are optimized for di erent ranges of electron uxes and energies mak-ing them suitable for di erent environments. LEEA is the Low Energy Electron Analyzer and its geometric factor makes it appropriate for higher uxes and rela-tively 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 magnetosphere and the magnetotail. The two PEACE sensors are placed on op-posite sides of the spacecraft.
Each of the two top hat analyzers has 12 anodes covering 15o each in the polar direction, so that a combined eld of view of 180o is provided. As illustrated in gure 3.10. They are placed perpendicular to the spacecraft body allowing a cov-erage of all 180o elevation angles with respect to the spin axis of the spacecraft. For each measurement, a sweep through the appropriate energy channels is done simultaneously for all the anodes. Hence, as the spacecraft rotates, di erent az-imuthal directions are scanned. Each di erent measurement scans a part of the sky with a 180o eld of view in elevation and a slice of a few degrees in the azimuthal direction. Every 4s the spacecraft completes a full spin and both sensors are able to complete a scan of the whole sky. The resolution of this scan in terms of az-imuth angle and energy channels depends on the mode of the instrument. Higher angular resolution comes at the expense of reduced energy resolution, given that each anode has a de ned geometry and requires a su cient accumulation time to complete a meaningful measurement.
There are three modes of operation of the instrument in terms of angular reso-lution, scanning over a di erent number of energy channels: LAR (250ms sweep time) , MAR (125ms sweep time) and HAR (62:5ms sweep time), which stand for Low, Medium and High angular resolution respectively. So, in MAR mode for example the energy sweep lasts 125ms, performing 32 sweeps per spin, with each spin covering 11:25o in azimuth and 180o in elevation. The sensors have a nominal energy range from 0:6eV to 26; 460eV , which is divided into 88 channels spaced linearly up to 9:5eV and logarithmically above that value. The sensor sweeps through these energy channels from high energies to low energies. The number of energy channels covered by each sweep depends on the instrument mode. The accumulation time for each energy channel is equal. For the three modes of opera-tion, LAR, MAR and HAR, the number of energy bins is 60, 30 and 15 respectively. That yields the di erent time it takes for a full sweep in each mode. The time it takes for the voltage to be reset for the next sweep ( » y-back time ») varies ac-cording to the mode. The data collected during the y-back time is discarded and the corresponding bins are empty in the nal data product. The nal measure-ment has 64 energy bins for the LAR mode (with 4 being empty corresponding to the y-back time), 32 for the MAR (with 2 for the y-back time) and 16 for the HAR mode (with 1 for the y-back). There are always 12 polar angle bins. The number of the azimuth angle slices varies also with respect to how many energy sweeps the sensor has the time to do in each spin. Hence, in LAR mode there are 16 azimuth bins, 32 in MAR and 64 in HAR mode. With this con guration the sensor returns always 11,520 values per spin. This is the basic data product with the best resolution available. However, it must be noted that both energy and angular resolution at the nal data products vary to conform with telemetry limitations.
While the sensor operates most of the time in MAR mode, the resolution of the data transmitted to Earth depends on the available telemetry. The most com-monly available data product is the moments of the electron distribution function, namely the density, the velocity and the temperature at the spin resolution of 4s, calculated on board or on the ground. The other spin-resolution products are the P IT CH SP IN and P AD. They provide two measurements per spin at the two moments during the spin where the eld of view of the sensor is aligned with the direction of magnetic eld vector. That o ers full pitch angle coverage provided that the magnetic eld is relatively stable during the time interval of the spin.
Figure 3.10: Schematic of Cluster spacecraft. Placement of the HEEA and LEEA sensors of the PEACE instrument. The eld of view of each sensor is a fan arranged looking radially out from the spacecraft body. Courtesy of MSSL.
There are a number of sub-spin data products which provide some of the infor-mation of the individual measurements at di erent azimuth angles which make up the nal distribution from which the moments are calculated. The data prod-ucts P IT CH F U LL,P IT CH 3D and 3DR o er resolution that depends on the available telemetry. The data products 3DX and 3DXP have the best resolution the instrument can provide, and is usually always available when the spacecraft operates in Burst Mode.
There are a number of known issues and caveats regarding PEACE measurements. They are issues that concern top-hat analysers and particle detectors in general and are always a cause of concern in space missions with in situ plasma instruments. The most important of them are outlined below.
The rst issue is the e ect of the spacecraft potential on the measured electrons. A spacecraft orbiting in space, in the environments that Cluster operates in, has usually a small positive electric charge. It ranges from a few eV to tens of eV depending on the technique and the strategy of potential control. The charged particles that end up in the detectors travel through that potential and are in u-enced by it. In practice, that means that the electrons are accelerated by a few eV and that the measured velocity distribution is shifted accordingly in velocity space. While this is not a problem for high energies, it has a signi cant impact on lower energy electrons. In the case where the bulk of the thermal population is in the range of the spacecraft potential, such e ect must be taken into account. This issue requires careful calculation of the potential around the spacecraft. It must be noted that, while the spacecraft surface is conductive, there are still minor spatial and temporal anisotropies of the potential around the spacecraft. This means that electrons are not a ected equally and in the same way in all directions. In the case of fast rotating spacecraft like Cluster this is mostly due to the di erences between the sun-lit and the dark side of the spacecraft. However, usually this e ect is small and can be safely ignored or averaged out.
The second issue is the e ect of the photo-electrons that are emitted by the sun-light that hits the spacecraft. Due to the positive charge of the spacecraft, the photo-electrons that have energies below the spacecraft potential will return to it, with some of them being captured by the detectors. Although the photo-electron ux can be signi cant, the bulk of this problem can be mitigated by discarding measurements below the energy of the spacecraft potential. It must be noted however that the actual energy cut-o of the photo-electron population is not al-ways clear and can partially contaminate higher energies. Additionally, given the complex and varying structure of the spacecraft potential, the photo-electron ux is slightly anisotropic. Those e ects are usually not very signi cant and rather complex. A cut-o value at the energy corresponding to the spacecraft poten-tial remains the most widely used approach. However, it must be noted that in some cases where the electron temperature is relatively low and the potential of the spacecraft relatively high (e.g. due to limited potential control), the presence of photo-electrons can dominate the entire thermal part of the electron popula-tion. In such cases the estimation of moments of the electron distribution function becomes less reliable. This is frequently the case in the solar wind.
A nal issue that is present in most top hat analyzers, including PEACE, and should be mentioned here, is UV-photon produced electrons inside the sensors. Such electrons are produced when UV photons from the Sun (or to a lesser extent from other luminous objects), hit an anode. For that reason the interior of the analyzer plates is covered by appropriate absorbing coating and has a grating surface. However, such protection does not completely mitigate the e ect. When the Sun is directly in the eld of view of an anode, the e ect will be measurable. It results in a sharp spike at low energies at a speci c azimuthal direction once for every spin. It is limited to a single azimuth bin and usually does not have a dramatic impact on the data.
In this chapter, I present the methods used in the thesis to analyze the in situ data as well as the relevant diagnostics that were performed to test the robustness of such methods.
The rst set of methods concerns how to detect regions of strong electric current in turbulent space plasma, usually referred to as current sheets, since these are the regions where reconnection is expected to occur. As it follows from Ampere’s law, current sheets are associated with a sharp change of the magnetic eld in terms of its magnitude and direction. Three di erent methods have been used, namely the Partial Variance of Increments (PVI), the magnetic shear angle and the direct computation of current through four-point magnetic eld measurements by the curlometer technique.
A second set of methods concerns how to determine the orientation and motion of current sheets in the hypothesis that they are planar structures, that is, their local reference frame and velocity. The frame was estimated using the Minimum Variance Analysis. The timing method was used to estimate the velocity of the current sheet in the normal direction to the current sheet plane. This method also provides the normal direction that has been compared with that from minimum variance for consistency checks.
The last set of methods are related to the estimation of electron temperature from partial distribution functions measured at sub-spin time resolution. These estimates are necessary since the full distribution function are available on spinning spacecraft at spin resolution ( 4s for Cluster), while the typical duration of current sheet crossings in the turbulent plasma studied here is of the order of 1s or below. In this chapter I also discuss in detail the diagnostics performed to check the robustness of the sub-spin data analysis, as well as the limitations and caveats associaetd to this analysis. The derived estimates of the electron temperature have been used to study the electron heating associated to the current sheets, as discussed in section 5.
The tests and the diagnostics presented here for the di erent methods were done by using the same dataset presented in Chapter 5, that consists of 1h300 of Burst Mode data in the Earth’s magnetosheath downstream of the quasi-parallel bow shock. The Cluster data was provided by the Cluster Science Archive. The data analy-sis tools used in this work were implemented in MATLAB.The IRFU-MATLAB toolbox (https://github.com/irfu/irfu-matlab) was used to construct the sub-spin distribution functions as well as for data visualization.
Methods for the detection of current sheets
Partial Variance of Increments
The Partial Variance of Increments (PVI) method has been described in  and  and has been implemented for the detection and study of intermittent struc-tures such as current sheets in simulations  and in solar wind observations . It consists of calculating the change of the magnetic eld vector over a time lag :
This computation produces a time series of the PVI index for a given time lag. An example of a PVI analysis in simulation data by  is shown in Figure 4.1. Current sheets are associated with sharp gradients of the magnetic eld and therefore are expected to appear as large values of the PVI index. The selection of current sheets of di erent intensity is then obtained by imposing a threshold
N on the equation 4.1.1, P V I (t) > N, where N = 1; 2; 3; ::: typically indicates a multiple of standard deviations of magnetic eld increments. Previous work has established a link between high values of the PVI index and non-Gaussian tails associated with intermittent structures such as current sheets [47, 48, 104]. Speci cally, it has been proposed from empirical results, that a PVI index lower than 1 is associated with low intensity Gaussian uctuations, a PVI index between 1 and 3 is likely to be associated with intermediate structures such as magnetic islands, whereas highly intermittent structures such as reconnecting current sheets are signi cantly more likely to have a PVI index larger than 3. This aspect will be discussed in more detail in Section 5.1.1, where speci c thresholds and selection criteria will be applied to Cluster data. Figure 4.1 also shows the e ect of using averaging intervals of di erent lengths for the normalization of the PVI index. Speci cally, the signal becomes weaker as the averaging interval is reduced. It is important to notice here that, for a meaningful PVI analysis, the length of the normalization interval should be much larger than the typical correlation length of the turbulence, that roughly corresponds to the size of the biggest structure in the turbulence. This is expected to provide a stable value of the average as discussed by  and also in Section 5.1.1.
It must be noted that the PVI method has been used so far in single spacecraft observations. In those cases, however, there is no information regarding the size and the velocity of the observed current sheets. Using data from multiple space-craft allows us to observe directly structures of size comparable to the separation between the spacecraft. In this thesis, I have implemented the PVI technique for multi-spacecraft data and used this multi-spacecraft technique to detect the cur-rent sheets studied in Chapter 5. The results and the comparison with the single spacecraft approach used in previous studies is discussed in detail in Section 5.1.1.
Figure 4.1: PVI index with di erent averaging intervals obtained from simu-lation results by . The signal becomes weaker as the averaging interval l decreases. C denotes the turbulence correlation length and d the dissipation length as de ned in . From .
Magnetic Shear Angle
A number of previous studies have focused on the rotation of the magnetic eld across current sheets in turbulent plasma [20, 97, 111, 161]. The value of the magnetic shear angle has been used both as an identi cation method and as an important element for the physical processes occurring within the current sheet, e.g. magnetic reconnection or other instabilities. In these studies, the magnetic shear angle is calculated from single spacecraft measurements of the magnetic eld by using the formula:
Figure 4.2 shows an example of current sheet detection in the solar wind by using the magnetic shear angle performed by Miao et al. .
Figure 4.2: Example of the calculation of the magnetic shear angle for a current sheet in the solar wind using data from the Ulysses spacecraft . The time lag in the gure corresponds to the lag discussed in the text .
The gure shows that, for di erent values of , increases gradually to a maxi-mum value that corresponds to the rotation of the magnetic eld across the current sheet (shear angle). Once this maximum value is reached, further increase of the values of leads to the peak becoming wider, but the maximum value does not change. The time 0 it takes to reach the maximum value corresponds to the time it takes for the spacecraft to cross the current sheet. If the velocity of the current sheet is known, then this crossing time can be converted to the actual current sheet thickness.
It should be noticed, however, that if the chosen for the analysis is smaller than 0, then does not reach its maximum value and the magnetic shear as well as the duration of current sheet is underestimated. This could lead to underestimate the thickness of the current sheet. However, as for the case of the PVI index, the magnetic shear angle can also be computed using simultaneous measurements from multiple spacecraft, as discussed in detail in Section 5.1.1. While this limits the analysis to a given scale set by the spacecraft separation, it allows us to avoid the issue discussed above related to undersampling the current sheet.
The curlometer technique is a multi-spacecraft technique that allows us to directly compute the electric current at the barycenter of a tetrahedron of spacecraft such as Cluster , by using four-point magnetic eld measurements. The main as-sumption of the method is that the variations of the magnetic eld are linear inside the tetrahedron so that non-linear terms of B are neglected and the current is assumed uniform over the tetrahedron. Under this assumption, the current is computed through Ampere’s law 0J = r B in the following way. First, the reciprocal vectors of the tetrahedron k are computed:
where < r > is the average separation between the spacecraft, < B > is the average magnetic eld respectively, r and B are the measurement errors of the separation and the magnetic eld respectively, and FS and FB are de ned empirically in .
As mentioned above, the curlometer is based on the assumption that gradients are linear. This is not always the case for thin current sheets, that can have a thick-ness smaller than the spacecraft separation. The curlometer technique provides an average current over the space of the tetrahedron formed by the four space-craft. Therefore, sharp gradients at scales smaller than the scale of the separation between the spacecraft tend to be smoothed out, leading to an underestimation of the electric current. This is illustrated in gure 4.3 where one component of the current is estimated for one current sheet using both the curlometer and a single spacecraft estimation of the current through Ampere’s, law assuming a planar cur-rent sheet moving with constant velocity. The use of the curlometer for detecting thin current sheets and its relationship with other detection methods is discussed in detail in Section 5.1.1.
Orientation and motion of current sheets
In this section, I will discuss few standard methods to determine the orientation and motion of current sheets. The most important assumption below such methods is that the current sheet is a planar structure where variations occur only in the direction normal to the current sheet plane, as sketched in Figure 4.4. Other assumptions, namely time stationarity for the timing method, are discussed below.
Table of contents :
2 Theoretical Background
2.1 Magnetic Reconnection
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
4.3.3 Instrumental Limitations
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
6.2 Future Work