||Simulation of the Formation of the Ion Ring Velocity Distributions
||Assistant, Institute of Space, Astrophysical and Plasma Sciences(ISAPS)
Ion Ring Velocity Distribution
Ion ring velocity distributions are commonly observed near the magnetic equator in the evening side of the radiation belts. In this study, we would like to explore magnetotail ions as a source of the ion ring velocity distributions due to their drift from the magnetotail to the radiation belts; these ions have increasing velocity perpendicular to the magnetic field velocity perpendicular to the magnetic field as they drift toward the Earth because their magnetic moment is an invariant. When their perpendicular velocity is significantly larger than their local drift velocity, the majority of their perpendicular motion goes into gyration rather than drift. The variation of their gyration phase may give rise to the ring velocity distribution.
In our simulations of the ion’s motions, we combine the Tsyganenko model T01 with the dipole field model to form the magnetic field, and impose a dawn-to-dusk electric field based on solar wind observations. We set an observation site near the magnetic equator in the evening side of the radiation belts to record the velocities of H+ ions that reach there. To emphasize the importance of the gyration phase in the theory, we use 20 different ions with different gyration phases to represent any given pair of parallel and perpendicular velocities in the initial conditions of the particle simulations. By tracking the motions of H+ ions from a specific source region in the magnetotail, we show that their perpendicular velocities at the observation site of our simulations constitute the form of a ring in the velocity distribution. Using the idea of continuity, we discuss how to calculate the density contribution at the observation site due to the ions from the magnetotail source.
LIST OF FIGURES V
1. INTRODUCTION 1
2. Mathematical Methods & Analysis Procedure 3
2.1 Magnetic Field Model 3
2.2 Fourth-Order Runge-Kutta Method 5
2.3 Unit Integration, Time Step and Coordinate Transformation 7
3. Simulations with an Arbitrary Magnetotail Source Location 10
4. Simulations Back In Time from Observation Site to the Magnetotail 15
5. Simulations of Ion from The Magnetotail to the Observation Site (0,5,0) 19
6. Summary and Future Work 37
Appendix 1 40
Appendix 2 45
LIST OF FIGURES
Figure 1.1 Ion ring velocity distribution observed by the GEOS satellite 2
Figure 2.1 Earth’s field lines generated by the Tsyganenko T01 model 5
Figure 2.2 Illustration of the idea of the Fouth-order Runge-Kutta method 6
Figure 3.1 A test H+ ion particle trajectory in x-y-z three-dimensional space 10
Figure 3.2 A test H+ ion trajectory in the two-dimensional x-y space 11
Figure 3.3 Twenty H+ ion trajectories in the x-y-z space 13
Figure 3.4 Twenty H+ ion trajectories in the x-y space 14
Figure 4.1 The scattered plot of the locations of the H+ ions in the R=15RE plane when traced back from the observation site (0,5,0). The color indicates the speed of the ions 17
Figure 4.2 The counts of the distribution of the R=15RE plane, which is divided into 2。by 2。 in the plane 18
Figure 5.1 Schematic diagram of one of the 20 H+ ions a gyroradius rL away from the guiding center in double-primed coordinates 20
Figure 5.2 Illustration of the relationship between the locations and velocities of an ion and its guiding center in double-primed coordinates 24
Figure 5.3 Scattered plot of the three-dimensional velocities of the H+ ions in double-primed coordinates, as observed at the simulation observation point 29
Figure 5.4 Scattered plots of the velocities of the H+ ions projected onto the plane as observed at the simulation observation point 30
Figure 5.5 The counts in each grid of the two dimensional velocities ( ) of the observed H+ ions 31
Figure 5.6 Demonstrate the four ion particles around the observed particles at the magnetotail 33
Figure 5.7 Two possible ways of forming the cross section for A2 34
Figure 5.8 Scattered plot the two-dimensional velocities of the observed H+ ions in double-primed coordinates, with the color indicating the amount of density n2 carried by each ion 35
Figure 5.9 The total n2 in each grid of the two-dimensional velocity distributions of the observed H+ ions in double-primed coordinates 36
Figure A1.1 Coordinate transformations from GSM to double prime. Dotted blue lines are the projection of the magnetic field vector onto the x-y plane, and the z-component of the magnetic field vector 40
Figure A1.2 Illustration of coordinate transformation processing from a three-dimensional point of view 41
Figure A1.3 Coordinate transformation from (x,y,z) to ( ) as viewed from the z-axis: fix the z-axis and rotate in the x-y plane 42
Figure A1.4 Coordinate transformation from ( ) to ( ) as viewed from the : fix the and rotate in the x-y plane 43
Figure A2.1 Wall of the tube is formed by the trajectories of the ions initially around point 1 45
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