# Global frequency distributions of pulsations driven by sharp decrease of solar wind dynamic pressure

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Adiabatic invariants for charged particles in the electromagnetic field

An ion or an electron moving through the Earth’s magnetosphere executes three motions: (1) cyclotron motion about the magnetic-field line, (2) bounce motion along the field line, and (3) drift perpendicular to magnetic field line. For charged particles in electromagnetic fields, an adiabatic invariant is associated with each of the three types of motions the particles can perform. This can be clearly seen in Fig.2.1.

A remarkable feature of the motion of charged particles in collisionless plasmas is that even though the energy changes, there is a quantity that will remain constant if the field changes slowly enough. By ‘slowly enough’ we mean that the field changes encountered by the particle within a single gyration orbit will be small compared with the initial field. If this condition is satisfied, then the particle’s ‘magnetic moment’ will remain constant.
Note that if μ remains constant as the particle moves across the field into regions of different field magnitudes, some acceleration is required. The quantity μ is also called the first adiabatic invariant. Here, ‘adiabatic’ refers to the requirement that μ may not remain invariant or unchanged unless the parameters of the system, such as its field strength and direction, change slowly [Kivelson and Russell, 1995].

The longitudinal invariant is associated with the V‖ motion. If the field has a mirror symmetry where the field lines converge on both sides as in a dipole field, there is the possibility for a second adiabatic invariant J. A particle moving in such a converging field will be reflected from the region of strong magnetic field and can oscillate in the field at a certain bounce frequency ωb. The longitudinal invariant is defined by where V‖ is the parallel particle velocity, ds is an element of the guiding center path and the integral is taken over a full oscillation between the mirror points.
For electromagnetic variations with frequencies w ≪ wb, the longitudinal invariant is a constant, irrespective of weak changes in the path of the particle and its mirror points due to slow changes in the fields [Baumjohann and Treumann, 1996].

Nonadiabatic acceleration of ions by spatial variations of the magnetic field

The energization of plasma sheet ions is a fundamental subject of the dynamics of magnetotail. Theoretical and observational studies indicate that in the plasma sheet, the ions can be non-adiabatically energized when the first adiabatic invariant is violated [e.g. Chen, 1992].

Nonadiabatic acceleration of ions by temporal variations of the magnetic field

Second, it is the temporal variation of the magnetic field that leads to the violation of the first adiabatic invariant. In this case, the magnetic field variation period should be comparable to or shorter than the cyclotron period of charged particles. It occurs frequently during magnetospheric dipolarization, and always leads to significant energy gains. Delcourt et al. [1990] examined the nonadiabatic behavior of ions during magnetic field dipolarization by means of three-dimensional particle codes, and demonstrated that the large electric fields induced by the dipolarization can lead to dramatic energization and pitch angle diffusion for the midtail populations. Delcourt et al. [1997] more specifically examined the behavior of O+ ions during dipolarization. Their results of numerical trajectory calculations and analytical analyses show that due to the effect of the inductive electric field, a three-branch pattern of magnetic moment jump was obtained, which depends upon the initial energy of the particles before accelerations.

### Simulation results for parallel motion of the particles

In order to bring out key parameters in the particle orbit, the parallel and the perpendicular motions will be considered separately. Ions that are initially aligned with the magnetic field will be examined first.
For the temporal variation of the magnetic field, Delcourt et al. [1990] examined the nonadiabatic behavior during magnetic field dipolarization by means of three-dimensional particle codes, and demonstrated that it can lead to dramatic energization and pitch angle diffusion.
Fig.2.2 Simulation results for initially field-aligned O+: (a) trajectory projections in the GSM XZ plane, (b) kinetic energy, (c) electric field, and (d) pitch angle versus time. The particles are injected with 100 eV energy and 0° pitch angle, at 9 RE geodistance, 00 MLT and symmetrical positions about the equator: -5° (trajectory 1) and 5° (trajectory 2) latitude. In Fig.2.2a, dots indicate the time in 10 second steps (taken from Fig.2 of [Delcourt et al. 1990]).
The storm time trajectory calculations were carried out using the magnetic field model of Mead and Fairfield [1975]. Fig.2.2 presents the trajectory results of two O+ ions initially aligned with the magnetic field and located at symmetrical positions about the center plane. Fig.2.2a shows the trajectory projection in the midnight meridian plane, while the respective energy and pitch angle variations are indicated in Fig.2.2b and Fig.2.2d. Also, the electric fields ‘viewed’ during transport are displayed in Fig.2.2c. With such zero initial pitch angle, both particles are expected to move from south to north, which is indeed the case for the southern hemisphere originating ion (trajectory 1). For this O+, Fig.2.2b shows a net acceleration of the order of 3 keV.
Fig.2.2d reveals a substantial pitch angle scattering for this ion, up to 30° at the end of the collapse.
A drastically different behavior is noticeable for the O+ initiated above the equator (trajectory 2) since, instead of traveling northward, it is rapidly reflected toward the opposite hemisphere (Fig.2.2a). This creation of a high-altitude mirror point is further apparent from Fig.2.2d which depicts a final pitch angle of the order of 130°. Also, it can be seen in Fig.2.2b that this latter O+ globally experiences a weaker energization (of the order of 300 eV), though both particles experience similar electric fields during transport (Fig.2.2c), up to 6 mV/m at half-collapse (note that similar but inverted patterns are obtained using 180° initial pitch angle). These features can be qualitatively understood by examining the parallel equation of motion (details in [Delcourt et al. 1990]).
To further confirm these patterns, Fig.2.3 presents the results of the simulations for protons injected with velocities identical to those of O+ (Fig.2.2), yielding an initial energy of 6 eV. Fig.2.3a indeed displays H+ orbits similar to those of O+, with creation of a high-altitude mirror point for the northern hemisphere originating particle. And the final pitch angle can develop to be as high as 180° for the H+ initiated above the equator (trajectory 2, in Fig.2.3d).

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#### Simulation results for perpendicular motion of the particles

Now we discuss the particle’s motion in the plane perpendicular to the magnetic field. Delcourt et al. [1997] investigate the dynamics of near-Earth plasma sheet ions during storm time dipolarization of the magnetospheric field lines. They more specifically examine the behavior of O+ ions that are trapped in the equatorial vicinity. It is showed that during field lines dipolarization, these particles may be transported in a nonadiabatic manner and experience large magnetic moment enhancement together with prominent ‘bunching in gyration phase’ which was first observed in the solar wind at the bow shock [Gurgiolo et al., 1981; Eastman et al., 1981; Fuselier et al., 1986]. The nonadiabatic acceleration and the gyration phase bunching effect occur below some threshold energy at the dipolarization onset, this threshold energy being controlled by the amplitude of the magnetic transition and by the injection depth in the magnetotail. In the near-Earth tail, the phase-bunched particles possibly experience intense (up to the hundred of keV range) nonadiabatic energization.
In this study, instead of the Mead and Fairfield [1975] model which was used in [Delcourt et al., 1990], the trajectory calculations during storm time dipolarization were carried out using the magnetic field model of Tsyganenko [1989] (referred to be T-89), which provides an accurate average description of the various magnetospheric current contributions.
Fig.2.4 Computed trajectories of equatorially trapped O+ ions in the time-dependent T-89 model: (a) trajectory projection in the equatorial plane, (b) magnetic moment (normalized to the initial value) versus time, (c) gyration phase versus time. The ions are launched from a guiding center position at 7.3 RE with distinct gyration phases (from 0° to 360° by steps of 45°) and distinct energies: (from left to right) 100 eV, 1 keV, and 10 keV. In Fig.2.4c, the time is normalized to the dipolarization timescale (taken from Fig.2 of [Delcourt et al., 1997]).

Chapter 1 The magnetosphere and the space instruments used in this thesis
1.1 The Earth’s magnetosphere
1.2 Plasma in the near-Earth magnetotail
1.2.1 Tail Lobes
1.2.2 Plasma Sheet Boundary Layer
1.2.3 Plasma Sheet
1.3 The Cluster mission
1.3.1 The CIS instrument
1.3.2 The FGM instrument
1.3.3 The EFW instrument
1.3.4 The RAPID instrument
1.4 The Double Star Program
1.5 The cl software
Chapter 2 Nonadiabatic acceleration of ions
2.1 Adiabatic invariants for charged particles in the electromagnetic field
2.2 Nonadiabatic acceleration of ions by spatial variations of the magnetic field
2.3 Nonadiabatic acceleration of ions by temporal variations of the magnetic field
2.3.1 Simulation results for parallel motion of the particles
2.3.2 Simulation results for perpendicular motion of the particles
2.3.3 Observations and statistical results
Chapter 3 Specific energy flux structures formed by the nonadiabatic accelerations of ions
3.1 Observations by CIS instrument
3.2 Observations by EFW and RAPID instruments
3.3 Wavelet analysis of the magnetic and electric field
3.4 Discussion about the formation of the ‘energy flux holes’
3.4.1 Possibility of nonadiabatic acceleration by the spatial variation of the magnetic field
3.4.2 Possibility of nonadiabatic acceleration by the temporal variation of the magnetic field
3.6 Comparisons between the observed and the simulated spectrum
3.7 Similar events observed by Cluster in the south of the plasma sheet
3.8 Similar events observed by Cluster and Double Star TC-1 in the ring current
3.9 Conclusions
Chapter 4 Global frequency distributions of pulsations driven by sharp decrease of solar wind dynamic pressure
4.1 Introduction
4.1.1 Interplanetary shocks and the associated geomagnetic pulsations
4.1.2 The theory of Field Line Resonance (FLR)
4.1.3 The theory of cavity resonance mode
4.1.4 Twin-vortex current system in the ionosphere
4.2 Observations
4.2.1 Observations by spacecraft
4.2.2 Geomagnetic response at subauroral and auroral latitudes
4.2.3 Global characteristics of pulsations at subauroral and auroral latitudes
4.2.4 The characteristics of pulsations at different latitudes around the same local times
4.2.5 Comparison between the pulsations driven by sharp increase and sharp decrease of Psw
4.3 Discussions
4.4 Conclusions
Conclusions et perspectives (en français)
Conclusions and perspectives (in English)

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