Three-Dimensional Hybrid Simulation Of Dayside Magnetic Reconnection
by
Binying Tan
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
May 9, 2011
Keywords: magnetic reconnection, magnetospheric physics, hybrid simulation
Copyright 2011 by Binying Tan
Approved by
Yu Lin, Chair, Professor of Physics
Satoshi Hinata , Professor of Physics
Stephen Knowlton, Professor of Physics
J. D. Perez, Professor of Physics
Abstract
With a 3-D global hybrid simulation model, we investigate magnetopause reconnection
and the energy spectra of cusp precipitating ions under di erent IMF conditions.
First, the magnetic con guration and evolution of Flux Transfer Events (FTE) and the
associated ion density and ion velocity distribution at various locations on the magnetopause
are investigated under a purely southward IMF. The results reveal: (1) Multiple X lines are
formed during the magnetopause reconnection, which lead to both FTEs and quasi-steady
type reconnection under a steady solar wind condition. The resulting bipolar signature of
local normal magnetic eld of FTEs is consistent with satellite observations. (2) A plasma
temperature rise is seen at the center of an FTE, compared to that of the upstream plasma in
the magnetosheath. The temperature enhancement is mainly in the direction parallel to the
magnetic eld due to the mixing of ion beams. (3) Flux ropes that lead to FTEs form between
X lines of nite lengths and evolve relatively independently. The ion density is enhanced
within FTE ux ropes due to the trapped particles, leading to a lamentary global density.
(4) Di erent from the previous understanding based on the asymmetric density across the
magnetopause, a quadrupole magnetic eld signature associated with the Hall e ects is found
to be present around FTEs. (5) A combination of patchy reconnection and multiple X-line
reconnection leads to the formation of reconnected eld lines from the magnetosphere to
IMF, as well as the closed eld lines from the magnetosphere to the magnetosphere in the
magnetopause boundary layer.
Secondly, both the spatial and temporal energy spectra of cusp precipitating ions are
computed by tracing trajectories of the transmitted magnetosheath ions under a southward
IMF. The spatial spectrum shows a dispersive feature consistent with satellite observations,
with higher energy particles at lower latitudes and lower energy particles at higher latitudes.
ii
The simulation reveals (1) how and where particles are transmitted from the solar wind into
the magnetosphere via direct magnetic reconnection on the dayside; (2) how the features
of the spectra are related to ongoing magnetic FTEs; (3) how the motion of the cusp,
particularly the latitudinal variation of the open/closed eld line boundary, is correlated
with the dayside reconnection and re ected in the spectra, energy ux due to precipitating
ions as a function of time.
Third, the energy spectra of cusp precipitating ions and magnetopause reconnection
under an IMF of a nite By component are investigated. It is found that component recon-
nection is the reconnection process at the dayside magnetopause. Dispersive feature is also
shown in spatial spectra for precipitating ions in the cusp, compared to that under a purely
southward IMF. When IMF clock angle is larger than 180 , the heaviest precipitation shifts
to the dawn side.
iii
Acknowledgments
I want to thank my supervisor Dr. Lin whose advice and encouragement guided me in
the process of research and writing of this thesis. At the Physics Department, I owe a debt
of gratitude to all the professors and sta . Thanks in particular to Dr. J. D. Perez and Dr.
Xueyi Wang for invaluable discussions, and to Dr. Ming-kuo Lee, Dr. Stephen Knowlton,
Dr. Satoshi Hinata for taking the time to review my thesis.
I would also like to thank my family and all my dear friends for their precious support
through this.
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 An Introduction to Magnetopause Reconnection . . . . . . . . . . . . . . . . . . 1
1.1 A Brief Overview of Magnetic Reconnection . . . . . . . . . . . . . . . . . . 1
1.1.1 Basics of the Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Magnetospheric Con guration and Reconnection at the Magnetopause . . . . 11
1.3 Types of Simulation Model in Space Plasma Physics . . . . . . . . . . . . . . 26
1.4 Objectives and Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . 27
2 Global-Scale Hybrid Simulation Model . . . . . . . . . . . . . . . . . . . . . . . 29
2.1 The Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Simulation Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Data Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Dayside Magnetic Reconnection Under a Purely Southward IMF: Structure and
Evolution of Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Magnetic Field Line Con guration under Southward IMF . . . . . . 44
4.2.2 Structure and Evolution of Flux Transfer Events (FTEs) . . . . . . . 49
4.2.3 Magnetic Field Signature and Ion Velocity Distributions in FTEs . . 55
v
4.2.4 Wal en Test of Rotational Discontinuity in a Quasi-Steady Reconnection 58
4.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Global Hybrid Simulation of Dayside Magnetic Reconnection Under a Purely
Southward IMF: Cusp Precipitating Ions Associated With Magnetopause Recon-
nection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.1 Reconnection Events and Spatial Energy Spectrum . . . . . . . . . . 65
5.3 Precipitating Ions at Low-energy Cuto . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 Precipitating Ions at Low-energy Cuto Associated with Region A . . 71
5.3.2 Precipitating Ions at Low-energy Cuto Associated with Region B . . 73
5.3.3 Precipitating Ions at Low-energy Cuto Associated with Region C . . 75
5.3.4 Energy Flux due to Precipitating Ions as a Function of Time . . . . . 76
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 Dayside Magnetic Reconnection under an IMF with a Finite Guide Field By . . 79
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A Fundamental Plasma Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B Maxwell?s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C Sample Subroutine in MATLAB: GUI of Finetuning Field Line . . . . . . . . . 104
D Sample Subroutine in MATLAB: Finetuning Field Line . . . . . . . . . . . . . . 105
E Sample Subroutine in MATLAB: 2-D Contours Plotting . . . . . . . . . . . . . 108
vi
List of Figures
1.1 Frozen-in theorem of ideal MHD: a closed loop of plasma elements l(t) at time t
moves to l(t+dt) at time t+dt . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Illustration of magnetic reconnection in the x z plane. Magnetic eld strength
and current density variation along the z axis, corresponding to the current sheet
shown in the upper right plot (upper left); Initial conditions of magnetic recon-
nection, which consists of a current sheet in the y direction, the in ow of magnetic
ux as well as the plasma towards the z = 0 plane (upper right); Illustration of
magnetic reconnection occurrence near an X line (lower panel). Adapted from
Hughes [1995] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 An event of emerging coronal matter ejection. Adapted from Patsourakos and
Vourlidas [2011] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Geometry of a ow (or ux) tube de ned by radial streamlines, adapted from
Hundhausen [1995] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 (Upper left) Magnetic eld lines drawn by the radially expanding solar wind,
assuming the Sun is not rotating; (Upper right) The spiral eld lines illustrate
the impact of the rotation of the Sun on the magnetic eld geometry. Vw is the
radial velocity of the uid. (Bottom) the Earth?s orbit in the spiral magnetic
eld lines from the Sun. This gure is adapted from [Meyer-Vernet, 2007] . . . 9
1.6 Illustration of the interaction of fast and slow solar wind as the substructure of
solar wind, adapted from Pizzo [1985] . . . . . . . . . . . . . . . . . . . . . . . 10
vii
1.7 Deformation of the current sheet (the shadowed surface) by a tilt angle of the
Sun?s magnetic axis (M) with respect to the rotation axis ( ) with a constant
solar wind radial velocity, adapted from Meyer-Vernet [2007] . . . . . . . . . . 11
1.8 Magnetospheric con guration in the noon-midnght meridian plane under a south-
ward IMF, showing the convection of plasma within the magnetosphere driven
by the magnetic reconnection. The eld lines are numbered to show the succes-
sion of eld line con guration. The inset (lower right) shows the position of the
footprint of the numbered eld lines in the northern high latitude ionosphere.
Adapted from Hughes [1995]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 Magnetic eld during a clear example of an FTE, where the magnetic eld is
expressed in boundary normal coordinates with Bn outward along the bound-
ary normal, Bl along the projection of the eld in the magnetosphere and Bm
completing a right-handed set. Adapted from Russell et al. [1995] . . . . . . . 14
1.10 Schematic diagram of the structure of an FTE, proposed by Russell and Elphic
[1978]. On the left a bundle of ux tubes is shown having become connected
between the magnetosheath (foreground) and magnetosphere (background), and
on the right the bulge in the magnetosheath and magnetospheric elds is illus-
trated. A virtual satellite passing as marked with the red line leads to the bipolar
signature of Bn. Adapted from Russell and Elphic [1978] . . . . . . . . . . . . 15
1.11 Six possible patterns of eld lines con guration of the magnetic ux rope that
can generate signatures of FTEs. The heavy bars denote the reconnection X
lines. The solid and dashed lines are used to distinguish eld lines on di erent
side of the current sheet. Adapted from Lee et al. [1993] . . . . . . . . . . . . 17
viii
1.12 (a) Petschek?s symmetric reconnection model which consists of the in ow region,
the out ow region, and the small central di usion region as shown in the dark
area. (b) Magnetic eld lines and stream lines of plasma ow in a MHD simula-
tion [Yan et al., 1992]. Adapted from Lin and Lee [1994] . . . . . . . . . . . . 18
1.13 Asymmetric reconnection model by Levy et al. [1964]. A rotational discontinuity
and a slow expansion wave are presented in the reconnection layer. Note that the
plasma density decreases slowly to zero at right hand side of the current sheet.
Adapted from Lin and Lee [1994] . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.14 A series of simultaneous snapshots of 2-D ion (above) and electron (below) ve-
locity distribution functions during an outward traversal of the earthward edge
of the LLBL on August 12, 1978. The distribution are shown as contours of con-
stant phase-space density separated logarithmically, (two contours per decade).
Numbers on the dotted circles indicate the velocity scale in km/s. Vectors drawn
represent the projection of the magnetic eld onto the xy plane. Adapted from
Gosling et al. [1990] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.15 Sketch of the magnetopause region for quasi-stationary reconnection. Adapted
from Gosling et al. [1990] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.16 Illustration of velocity space distributions of ions expected just inside the mag-
netopause in the magnetosphere for a quasi-steady single X-line reconnection
case. The contours are of constant distribution functions marked with value of
log10f(s3m 6)), the accelerated D-shaped distributions are simply taken to be a
mirror image of the in owing distributions, re ected in the plane Vk = VF, where
the de Ho mann-Teller velocity VF has been taken to be 200kms 1:Adapted from
Cowley [1982] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
ix
1.17 Example that for the same event, two groups inferred the reconnection sites to
di erent locations. Adapted from Fuselier et al. [2000] and Russell et al. [2000] 25
2.1 A coordinates cell showing the position and orientation of curvilinear components
of the magnetic and electric elds. Adatped from [Swift, 1996] . . . . . . . . . 33
2.2 Simulation domain in the GSM system . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 Graphic user interface of the visualization package . . . . . . . . . . . . . . . . 37
3.2 An example of 3-D data visualization: The contours are the total magnetic eld in
the noon meridian and equatorial planes, supposed with the isosurface of B = 4:5
in the simulation units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Slice plotting integrated the 3-D data visualization package. . . . . . . . . . . . 39
3.4 Isosurface plotting integrated the 3-D data visualization package. . . . . . . . . 40
3.5 Graphic user interface of ne tuning magnetic eld lines. . . . . . . . . . . . . . 40
4.1 Magnetic eld line con guration in a global view obtained in case 1 at t = 5
(top left), t = 15 (top right), t = 25 (bottom left), and t = 35 (bottom right),
respectively. The closed dipole eld lines are in black. Yellow lines are open eld
lines before magnetic reconnection. Field lines in other colors are reconnected
eld lines between the IMF and dipole eld in di erent regions. Contours in the
equatorial plane show the ion density. . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 (Left) Four eld lines of di erent topologies in case 1 at t = 55. (Right) Illustra-
tion of how the patchy reconnection and multiple reconnection can explain the
coexistence of the four eld lines in the left plot [Lee et al., 1993]. . . . . . . . 47
x
4.3 Three dimensional plots ( rst row) illustrate how X line is de ned, with Vz con-
tours in the noon meridian plane. Two dimensional intensity plots of By (second
row), ion density (third row), total magnetic eld (fourth row), parallel temper-
ature ( fth row) and perpendicular temperature (bottom row) are also in the
noon-midnight meridional plane, zoomed around the dayside magnetopause at
t = 15, 25, 35, 65, respectively (from left to right). . . . . . . . . . . . . . . . . 50
4.4 Ion density laments inside FTEs of case 1. The upper panel shows ion density
contours at x = 9:5 and t = 80, superposed onto a eld-line plot, and the lower
panel is a close-up of the same plane. . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Ion density laments inside FTEs of case 1. The upper panel shows ion density
contours at x = 9:5 and t = 80, superposed onto a eld-line plot, and the lower
panel is a close-up of the same plane. . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Top left, top right, bottom left and bottom right show the ion velocity distri-
butions at four chosen locations centered at D1, D2, D3 and D4 in Figure 4.5,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.7 Wal en test in case 2 in the northern hemisphere at t = 100. The top left contour
plot shows the ion ow velocity Vz in the noon meridian plane, in which E1-E2
is a line segment in the r direction across a rotational discontinuity. Spatial cuts
of eld components Bl, Bm, Bn and ion ow velocities Vl, Vm, Vn along E1-E2
are shown in the top right plot. The result of Wal en test is shown in the bottom
left. The ion velocity distribution at location D5 between E1 and E2 is shown
in the bottom right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
xi
5.1 (Upper) The 3-D plot in the left shows reconnection X lines in blue dots and red
line segment at t = 15 with ion bulk ow contours in the noon meridian plane.
The contours in the right show By component in the noon meridian planes and
projected magnetic eld lines at t = 15;25;35, respectively. (Lower) Spatial
energy spectrum of cusp precipitating ions in the logarithmic scale showing a
dispersive feature. The black, white, and red dots indicate energies of some
typical ions at low-cuto energies of parts A, B, and C, respectively, which are
related to three reconnection events. . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Particle speed as a function of time for typical ions associated with reconnection
A, B, and C, shown by the red, orange, and blue curves. . . . . . . . . . . . . . 69
5.3 The trajectories in colored tubes of particles at low-energy cuto from region
A and the magnetic eld con guration in black lines at t = 5 (upper left), 25
(upper right), 35 (bottom left) and 40 (bottom right) in the GSM system. Axis
direction is shown in the left of either row. The trajectories are color-coded with
their current kinetic energy. Also shown are the contours of ion density N . . . 70
5.4 The trajectories in colored tubes of particles at low-energy cuto from region
B and the magnetic eld con guration in black lines at t = 5 (upper left), 25
(upper right), 35 (bottom left) and 40 (bottom right) in the GSM system. Axis
direction is shown in the left of either row. The trajectories are color-coded with
their nal energy. Starting from 52:5 , particles at low-energy latitude in the part
B of spectrum have nal energies of 12:37;4:65;2:47;1:47 and 0:82, respectively.
Also shown are the contours of ion density N. . . . . . . . . . . . . . . . . . . . 74
5.5 (Top) Particle energy spectrum in the logarithmic scale as a function of time at a
xed position of r = 7:5 RE and latitude of 52:5 . (Bottom) latitudinal position
of the dayside open/close eld boundary, at r = 7:5 RE, as a function of time. . 77
xii
6.1 The con guration of magnetic eld just inside the virtual magnetopause (small
black arrows), the reconnecting component magnitude (color scale), and resul-
tant X line (XL) when integrated away from the point of maximal reconnecting
component. Boundary layer ow (white and black vectors) for eld lines rooted
in each hemisphere, on each side of the XL, as projected on the plane normal to
a view from the Sun. Individual plots represent the results for various interplan-
etary magnetic eld clock angles according to their labels. Adapted from [Moore
et al., 2002] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Solid black lines shows magnetic merging line of the compound eld [Hu et al.,
2009], superposed y the Earth?s dipole eld and the IMF of strength 10nT, pro-
jected in the 0 planes for several typical IMF clock angles, where 0 stands
for the longitude so that 0 = 0 corresponds to the noon meridian plane and
for the latitude. By symmetry, the merging line for a clock angle of 225 , the
clock angle in our simulation is expected as the red dashed line. . . . . . . . . . 82
6.3 Illustration of the location of component magnetic reconnection at the dayside
magnetopause when the IMF is Bx0 = 0,By0 = 0:707 and Bz = 0:707 as indi-
cated by the red arrow. The red eld lines are reconnected eld lines. The eld
lines of the Earth?s dipole and the IMF are shown as black lines. The contours
of ion density are shown in three planes, the equatorial plane, the noon meridian
plane and the maridian plane of 0 = 20 , or = 70 . The two boundary
regions featuring sharp ion density are the bow shock and magnetopause. . . . 84
6.4 Typical spatial energy spectrum of cusp precipitating ions in the logarithmic scale
for r = 7:5RE, Latitude = 57:5 and t = 40, recorded in the plane of 0 = 20
(upper) and 0 = 0 (lower) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
xiii
6.5 Spatial energy spectrum of cusp precipitating ions of r = 7:5RE and t = 40 in
the plane 0 = 20 under a IMF of Bx0 = 0,By0 = 0:707 and Bz = 0:707.
The low energy cut-o in the logarithmic scale indicated by a black line . . . . . 85
xiv
List of Abbreviations
1-D One Dimensional
2-D Two Dimensional
3-D Three Dimensional
ASCII American Standard Code for Information Interchange
AU Astronomical Unit
CME Coronal Mass Ejection
FTE Flux Transfer Event
HDF Hierarchical Data Format
LLBL Low Latitude Boundary Layer
MHD Magnetohydrodynamic
PIC Particle-in-cell
xv
Chapter 1
An Introduction to Magnetopause Reconnection
The purpose of this chapter is to provide background knowledge to prepare readers for
the following chapters by trying to make the thesis a self-contained piece.
Magnetopause reconnection is referred to magnetic reconnection that occurs near the
outermost boundary of the Earth?s magnetic eld in this thesis. In the next few sections,
we rst give a brief overview of magnetic reconnection and some basics of solar wind as the
external plasma source for reconnection, then we will review current progress on dayside
magnetopause reconnection, e. g., ux transfer events (FTEs). The last section of this
chapter compares di erent simulation models used in space plasma physics.
1.1 A Brief Overview of Magnetic Reconnection
Magnetic reconnection, namely, is the process that two magnetic eld lines bend towards
each other and fuse to create new eld lines. The concept of motion of magnetic eld lines is
closely connected with the magnetohydrodynamic (MHD) description of plasmas as follows
[Priest and Forbes, 2000a].
As we all know, Faraday?s law gives
@B
@t = r E; (1.1)
where B is the magnetic eld, E is the electric eld and t is time. With the displacement
current omitted, Amp ere?s law gives
r B = 0J; (1.2)
1
where B is the magnetic eld, J is the current density. The Ohm?s law in the resistive MHD
can be written as,
E + V B = eJ; (1.3)
where e represents the scalar electrical resistivity. Combine Equation 1.1, 1.2 and 1.3, the
magnetic induction equation is obtained as
@B
@t =r (V B) + r
2B; (1.4)
where = e 0 is the magnetic di usivity. The rst term on the right hand side of Equation
1.4 is the contribution to @B@t from eld convection, and the second term is the contribution
from eld di usion.
The ratio of the two terms, the magnetic Reynolds number, is de ned as
Rm = L0V0 ; (1.5)
where L0, and V0 are typical length and velocity scales, respectively. If Rm 1 , the
convection dominates over the di usion and Equation 1.4 becomes
@B
@t =r (V B); (1.6)
which leads to the ?frozen-in? theorem of the ideal MHD as describe below.
In Figure 1.1, a closed loop of plasma elements l(t) at time t moves to l(t+dt) at time
t+dt, where t+dt and t are two successive instants. During this process, the change of the
magnetic ux enclosed by the closed loop of plasma at t can be written as
d
dt
Z
B dS =
Z @B
@t dS +
I
B (V dl); (1.7)
2
where S is the vector area enclosed by the loop. The rst term on the right hand side of
Equation 1.9 is due to the time variation of the magnetic eld, and the second term is due
to the motion of the plasma loop. Since the vector identity gives
B (V dl) = (B V) dl; (1.8)
Equation 1.7 can be written as
d
dt
Z
B dS =
Z @B
@t dS +
I
(B V) dl: (1.9)
Recall Equation 1.6, we have
d
dt
Z
B dS =
Z
r (V B) dS
I
(V B) dl: (1.10)
Applying Stokes? theorem, we nd that the right side of Equation 1.10 is identically equal to
zero. In another word, the magnetic eld and the plasma elements are ?frozen-together? so
that the plasma elements move with magnetic eld lines and the plasma within a magnetic
ux tube always stays inside the same ux tube. If the ?frozen-in? condition always holds,
magnetic reconnection never happens, which is de ned below.
Let us consider another limit. When Rm 1, the di usion dominates over the convec-
tion, and Equation 1.4 becomes
@B
@t = r
2B; (1.11)
which is actually a di usion equation that implies the magnetic eld variation di uses away
the same as the heat di uses when a temperature gradient exists.
However, we see a more complicated behavior of magnetic eld than a simple di usion
process when the ?frozen-in? condition no longer holds, where eld lines are no longer frozen-in
the plasma elements.
3
Figure 1.1: Frozen-in theorem of ideal MHD: a closed loop of plasma elements l(t) at time t
moves to l(t+dt) at time t+dt
4
Figure 1.2: Illustration of magnetic reconnection in the x z plane. Magnetic eld strength
and current density variation along the z axis, corresponding to the current sheet shown in
the upper right plot (upper left); Initial conditions of magnetic reconnection, which consists
of a current sheet in the y direction, the in ow of magnetic ux as well as the plasma towards
the z = 0 plane (upper right); Illustration of magnetic reconnection occurrence near an X
line (lower panel). Adapted from Hughes [1995]
5
Figure 1.2 illustrates the concept of magnetic reconnection in the Cartesian coordinate
system, where the magnetic eld possess anti-parallel components across the current sheet.
As shown in the upper right plot, the plane of the current sheet lies in the y direction at
z = 0 while the in ow of magnetic ux as well as plasma towards the z = 0 plane. The
upper left plot gives the variation of the corresponding magnetic eld strength and current
density along the z axis.
The lower panel of Figure 1.2 illustrates the occurrence of magnetic reconnection. The
plasma and magnetic eld convect in from the top and bottom uniform plasma region and
towards the z = 0 plane. At the center of the current sheet, there is a di usion region, where
Rm < 1 and plasma elements are not tied to eld lines. Note that Rm is proportional to
L0 so that the thinner the current sheet is, the smaller Rm is. Because electrons have much
smaller mass than ions do, the di usion region of electrons during reconnection is of smaller
size than the di usion region of ions. At the center of the di usion region, the intersection
of the X-type separatrices is the X point in the xy plane. The X point is the projection of
a singular line (X line) in the 3-D space onto the xy place, near which reconnection occurs
and the magnetic eld has an X-type con guration. A more rigorous de nition will be given
in Chapter 4.
So qualitatively, the reconnection process is such that magnetic eld lines from di erent
magnetic domains are spliced to one another, changing their patterns of connectivity with
respect to the sources. The in ow plasma slide across the eld lines in the di usion region,
leading to the formation of an out ow region where the plasma ows out towards the left
and right. In the MHD description, magnetic reconnection occurs on timescales intermediate
between the slow resistive di usion of the magnetic eld and the fast Alfv enic timescale.
During magnetic reconnection, stored magnetic energy is released in the form of thermal
energy and kinetic energy of plasm particles.
Magnetic reconnection is a fundamental process in in both space and laboratory plasmas.
In solar physics, it is believed that magnetic reconnection explains the ejection of coronal
6
Figure 1.3: An event of emerging coronal matter ejection. Adapted from Patsourakos and
Vourlidas [2011]
matter from the Sun [Antiochos et al., 1999]. Figure 1.3 shows an event of emerging coronal
matter ejection. In magnetospheric physics, reconnection is believed to be closely connected
with the aurora intensi cation under some circumstances, Flux Transfer Events (FTEs) in
the current sheet at the dayside magnetopause [Russell and Elphic, 1978; Lee and Fu, 1985;
Russell et al., 1995; Scholer et al., 2003; Raeder, 2006; Dorelli and Bhattacharjee, 2009] and
in the magnetotail plasma sheet. For laboratory plasmas, reconnection is important to the
science of controlled nuclear fusion because it may cause the failure of magnetic con nement.
1.1.1 Basics of the Solar Wind
The rst source of plasma, solar wind, is a fully ionized plasma that carries the inter-
planetary magnetic eld (IMF) and streams continuously outward from the Sun into the
solar system. In this section, some basics of the solar wind are introduced.
7
Figure 1.4: Geometry of a ow (or ux) tube de ned by radial streamlines, adapted from
Hundhausen [1995]
The Sun can be seen as a magnetohydrodynamic dynamo, which generates magnetic
eld and plasmas ow outwardly from the base of the corona sphere. If we do not consider
the rotation of the Sun rst, the classical picture of an outwardly moving solar wind should
be the one shown in Figure 1.4. According to the frozen-in eld line condition, the magnetic
structure is consistent with the plasma ow. In another word, if a plasma ux tube exists,
the corresponding identical magnetic ux tube exists too.
The simple picture in Figure 1.4 becomes more complicated as the Sun rotates with an
average period of 25:4 days. The upper left plot of Figure 1.5 illustrates the magnetic eld
con guration around the sun if there is no rotation, while the upper right plot of Figure
1.5 illustrates the magnetic eld con guration around the sun, considering the rotation of
the Sun. The lower panel of Figure 1.5 shows the Earth?s orbit in the spiral magnetic eld
lines from the Sun. Also shown in the lower panel of Figure 1.5, the magnetic eld B is
8
Figure 1.5: (Upper left) Magnetic eld lines drawn by the radially expanding solar wind,
assuming the Sun is not rotating; (Upper right) The spiral eld lines illustrate the impact
of the rotation of the Sun on the magnetic eld geometry. Vw is the radial velocity of the
uid. (Bottom) the Earth?s orbit in the spiral magnetic eld lines from the Sun. This gure
is adapted from [Meyer-Vernet, 2007]
9
Figure 1.6: Illustration of the interaction of fast and slow solar wind as the substructure of
solar wind, adapted from Pizzo [1985]
approximately radial at small distances while at large distances, the azimuthal component
of B becomes dominant.
Across the equatorial plane, B changes direction dramatically and gives rise to a current
sheet. Note that in reality the solar dipole eld is not aligned to the rotation axis, which
causes the deformation of the current sheet so that the current sheet has a similar shape to
a twirled ballerina skirt as shown in Figure 1.7.
As a result, the Earth moves above and below the warped current sheet alternatively
at a distance to the Sun of 1 astronomical unit (AU). Note that 1 AU is approximately the
mean Earth-Sun distance. In another words, the IMF at the Earth could possibly be in any
directions, including both ?northward? and ?southward?.
The speed of the solar wind usually measures as low as 300 400km s 1 at sector
boundaries while the fast solar wind measures as high as 700km s 1. The interaction of
fast and slow solar wind is shown in Figure 1.6, which is one of the major perturbations in
the solar wind. Other important perturbations in the solar wind include shocks, which form
when the solar wind is compressed at speeds greater than the phase speed of the related
waves, and coronal mass ejections (CME) .
10
Figure 1.7: Deformation of the current sheet (the shadowed surface) by a tilt angle of the
Sun?s magnetic axis (M) with respect to the rotation axis ( ) with a constant solar wind
radial velocity, adapted from Meyer-Vernet [2007]
Hydrodgen is not the only element in the solar wind. Helium is the second most abun-
dant element. The ratio na=np is about 0:04 0:05, where na is the number of helium ions
per unit volume and np is number density of protons [Meyer-Vernet, 2007].
1.2 Magnetospheric Con guration and Reconnection at the Magnetopause
The rst concept of the Earth?s magnetosphere was deduced by Sydney Chapman and
his student Vincenzo Ferraro in the early 1930s [Akasofu, 1974]. The rst concepts of
open magnetosphere and magnetic reconnection were proposed by Dungey [1961], which
explained why the convection pattern of plasma in the high latitude is dependent on the
direction of the IMF.
Figure 1.8 shows the magnetospheric con guration in the noon-midnght meridian plane
under a southward IMF, implying the convection of plasma within the magnetosphere driven
by the magnetic reconnection. The Earth?s intrinsic eld is approximately a magnetic dipole
in the rst order. As solar wind ows against the dipole eld, the bow shock and magne-
topause shown in dashed lines are the two boundaries. Note that the bow shock is the curved,
11
stationary shock wave where the super sonic solar wind past the Earth. The magnetopause
is the boundary between the solar wind and the geomagnetic eld. Inside the magnetopause,
the region is called magnetosphere, while the region between the magetopause and the bow
shock is the magnetosheath. The magnetosheath is the shocked solar wind.
In Figure 1.8, the red cross near the subsolar region of the magnetopause current sheet
indicates the location of the eld neutral point at the dayside magnetopause under a south-
ward IMF, where favors magnetic reconnection. Some eld lines are numbered to show
the succession of eld line con guration. Under the southward IMF, a new eld line (2)
forms after reconnection between eld line (1) and (1?) at the dayside magnetopause. The
newly reconnected eld lines convect tailward. As the reconnection proceeds, more originally
"closed" geomagnetic eld lines become "open". There is another reconnection site where
eld line (6) and (6?) reconnects in the tail. Magnetic reconnection at the dayside primarily
leads to magnetic energy storage rather than the covert of magnetic energy into kinetic and
thermal energy as it does at the night side [Cowley, 1980].
The inset shows the foot print of the numbered eld lines in the norther high-latitude
ionosphere. The solid line with arrows in the inset indicate high-latitude plasma ows
correspondingly antisunward in the polar cap and returns to the dayside at lower latitude.
Now, magnetic reconnection at the magnetopause is believed to be a fundamental pro-
cess that leads to the transfer of the solar wind mass, momentum, and energy into the
Earth?s magnetosphere [Cowley, 1996]. Satellite observations have identi ed both transient
and quasi-steady reconnection at various locations on the magnetopause under various IMF
conditions [Russell and Elphic, 1978; Paschmann et al., 1982; Sonnerup et al., 1981; Fuselier
et al., 1991; Lockwood and Smith, 1992; Scholer, 1989a; Phan et al., 2005].
A ux transfer event (FTE) is a characteristic set of perturbations in the magnetic
eld observed by spacecraft in the vicinity of the terrestrial magnetopause , which was rst
observed by the International Sun-Earth Explorer-1 and -2 satellites in 1978 [Russell and
Elphic, 1978]. Because of their regular appearance and easy identi cation, FTEs attract
12
Figure 1.8: Magnetospheric con guration in the noon-midnght meridian plane under a south-
ward IMF, showing the convection of plasma within the magnetosphere driven by the mag-
netic reconnection. The eld lines are numbered to show the succession of eld line con g-
uration. The inset (lower right) shows the position of the footprint of the numbered eld
lines in the northern high latitude ionosphere. Adapted from Hughes [1995].
13
Figure 1.9: Magnetic eld during a clear example of an FTE, where the magnetic eld is
expressed in boundary normal coordinates with Bn outward along the boundary normal, Bl
along the projection of the eld in the magnetosphere and Bm completing a right-handed
set. Adapted from Russell et al. [1995]
a lot of attention. Figure 1.9 shows the magnetic eld during a clear example of an FTE
[Russell et al., 1995], where the magnetic eld is expressed in boundary normal coordinates
with Bn outward along the boundary normal, Bl along the projection of the eld in the
magnetosphere and Bm completing a right-handed set. The signatures of an FTE include
a bipolar variation in the magnetic eld component normal to the magnetopause Bn and
simultaneous de ection in the tangential components [Uberoi, 2003].
An FTE is limited in space and the quasi-period of the reoccurrence is around 8 minutes
[Rijnbeek et al., 1984] . So far, an FTE is thought to be a typical transient magnetic
reconnection occurrence because of its strong control by the direction of the IMF [Russell,
1995]. Once the IMF turns southward, the occurrence rate of FTEs increases sharply.
Figure 1.10 shows a simple schematic model proposed by Russell and Elphic [1978],
which explains the main magnetic signatures of FTEs. FTEs were thought to be ux tubes
14
Figure 1.10: Schematic diagram of the structure of an FTE, proposed by Russell and Elphic
[1978]. On the left a bundle of ux tubes is shown having become connected between the
magnetosheath (foreground) and magnetosphere (background), and on the right the bulge
in the magnetosheath and magnetospheric elds is illustrated. A virtual satellite passing as
marked with the red line leads to the bipolar signature of Bn. Adapted from Russell and
Elphic [1978]
15
that connect geomagnetic eld and the IMF. If the satellite ies through the ux tube with a
path in red lines in Figure 1.10, the variation of the normal component of the magnetic eld
in the local normal coordinates has a bipolar signature. Later, Lee et al. [1993] argued that
FTEs could be generated in the process of multiple X-line reconnection(MXR). As shown in
Figure 1.11, the ux rope that generates the signatures of FTEs is formed between X lines
denoted with heavy bars, which have nite length and in parallel with the current sheet.
There are six possible patterns of eld lines con guration of the magnetic ux rope.
And even today, people have not stopped asking questions about FTEs since then. Is
there any possibility that an FTE is not caused by reconnection? Can we divide FTEs into
di erent categories? What is the internal structure of FTEs? What trigger FTEs? What
is the relationship between FTEs and the separatrices of magnetic reconnection? How do
FTEs evolve? Do reconnection rate vary during the life cycle of an FTE?
Overall, FTEs are transient reconnection occurrences. Another category of reconnection
events have a much longer time scale, which are considered to be steady, or quasi-steady.
Except for the time scale, they also have a magnetic eld con guration as they are usually
associated with a single X line.
P. A. Sweet and E. N. Partker in 1950s [Priest and Forbes, 2000a] were the rst to
develop a simple MHD model of steady-state reconnection in a current sheet formed at a
null point, which is often referred to as a model for slow reconnection. Petschek [Petschek,
1964] later developed an alternative model with a current sheet whose length is many order
of magnitude smaller than the Sweet-Parker model. Petschek?s reconnection model can
predict a faster reconnection rate than that predicted by the Sweet-Parker model, also for
steady-state reconnection with a single X line.
As shown in the upper plot of Figure 1.12, Petschek?s [Petschek, 1964] reconnection
models a symmetric case with equal plasma density, equal magnetic strength, and antiparallel
magnetic elds on the two sides of the current sheet. Plasma particles are accelerated through
16
Figure 1.11: Six possible patterns of eld lines con guration of the magnetic ux rope that
can generate signatures of FTEs. The heavy bars denote the reconnection X lines. The solid
and dashed lines are used to distinguish eld lines on di erent side of the current sheet.
Adapted from Lee et al. [1993]
17
Figure 1.12: (a) Petschek?s symmetric reconnection model which consists of the in ow region,
the out ow region, and the small central di usion region as shown in the dark area. (b)
Magnetic eld lines and stream lines of plasma ow in a MHD simulation [Yan et al., 1992].
Adapted from Lin and Lee [1994]
18
the two slow shocks in the current layers. Note that the slow shocks are located downstream
of the sparatrices as indicated in the lower panel of Figure 1.12.
The steady reconnection at the dayside magnetopause is an asymmetric case as rst
modeled by Levy et al. [1964]. In this model, the magnetic eld strength on one side is
larger than that on the other side, and the plasma density is set to be zero on the side of a
higher eld. The result is shown in Figure 1.13, where the two slow shocks are replaced by
a rotational discontinuity and a slow expansion wave. The magnetic eld changes direction
cross the rotational discontinuity (an MHD discontinuity) , and high speed plasma ow along
the current layer. Note that an MHD discontinuity is a thin transition region, across which
the plasma density, magnetic eld, ow velocity etc. satisfy a set of jump conditions [Lin
and Lee, 1994].
Besides theoretical work for steady reconnection, observations [Gosling et al., 1990]
pointed out that quasi-steady reconnection indeed occurs at the dayside magnetopause.
Figure 1.14 shows a series of simultaneous 2-s snapshots of 2-D ion (above) and electron
(below) velocity distribution functions during an outward traversal of the earthward edge
of the low latitude boundary layer (LLBL) from the magnetosphere, which is identi ed as
a quasi-steady reconnection event. As shown in the lower row, the evolution of the hotter
population is consistent with their escape into the magnetosheath on newly reconnected eld
lines with the most eld-aligned electron escaping rst. Note that the vector (black arrow)
shows the projection of the magnetic eld onto the xy plane. More analysis on this event
shows that the electron edge of the LLBL lies closer to the separatrix of the corresponding
reconnection site that does the ion edge, which is consistent with the fact that the electron
di usion region of reconnection is samller than ion di usion region. Thus, the author drew
a diagram, showing the magnetopause region for quasi-stationary reconnection as in Figure
1.15
For a quasi-steady single X-line reconnection case, the velocity space distributions of
ions just inside the magnetopause in the magnetosphere are expected to have a D-shaped
19
Figure 1.13: Asymmetric reconnection model by Levy et al. [1964]. A rotational discontinu-
ity and a slow expansion wave are presented in the reconnection layer. Note that the plasma
density decreases slowly to zero at right hand side of the current sheet. Adapted from Lin
and Lee [1994]
20
Figure 1.14: A series of simultaneous snapshots of 2-D ion (above) and electron (below)
velocity distribution functions during an outward traversal of the earthward edge of the
LLBL on August 12, 1978. The distribution are shown as contours of constant phase-space
density separated logarithmically, (two contours per decade). Numbers on the dotted circles
indicate the velocity scale in km/s. Vectors drawn represent the projection of the magnetic
eld onto the xy plane. Adapted from Gosling et al. [1990]
21
Figure 1.15: Sketch of the magnetopause region for quasi-stationary reconnection. Adapted
from Gosling et al. [1990]
22
Figure 1.16: Illustration of velocity space distributions of ions expected just inside the mag-
netopause in the magnetosphere for a quasi-steady single X-line reconnection case. The
contours are of constant distribution functions marked with value of log10f(s3m 6)), the
accelerated D-shaped distributions are simply taken to be a mirror image of the in owing
distributions, re ected in the plane Vk = VF, where the de Ho mann-Teller velocity VF has
been taken to be 200kms 1:Adapted from Cowley [1982]
23
distribution for magnetosheath transmitted ions, which is illustrated in Figure 1.16. Since
only the particles with parallel velocity toward the magnetosphere in the reference frame of
eld line convection can penetrate to the Earth, the accelerated D-shaped distributions have
a low Vk cut-o at Vk = VF, where VF is the de Ho mann-Teller velocity. Except for the
magnetosheath sourced ions, ion populations of cold magnetospheric ion of ionosphere origin
is also one of the main ion populations [Cowley, 1982] at the location as shown in Figure
1.16.
As shown above, satellite data supports the argument that reconnection could be a
very important process in magnetospheric physics. But in most cases, the satellite crossings
not through the electron di usion region of reconnection directly. Thus, although various
observations have implied reconnection sites under various IMF and solar wind conditions
[Marcucci and Polk, 2000; Trattner et al., 2007] based on certain models/methods, there
has been few observations simultaneously in both the di usion and the out ow regions
that justify the implied location of reconnection. Plus, observations of the magnetopause
reconnection have often generated contradictory opinions [Fuselier et al., 2000; Russell et al.,
2000]. Figure 1.17 shows an example that for the same event, Fuselier et al. [2000] connected
the event with component reconnection equatorward of the cusp and considered a negative
cut-o velocity as an artifact at location M while [Russell et al., 2000] treated the negative
cut-o velocity as valid data, which indicated a slow-drifting of plasmas outside the current
sheet. As a result, two groups inferred the reconnection sites to di erent locations as shown in
the cartoons of Figure 1.17. The controversy is whether component reconnection occurred
equatorward of the cusp. For complex plasma process in the magnetosphere, computer
simulation are often used to help scientist to understand the process of reconnection at the
magnetopause.
24
Figure 1.17: Example that for the same event, two groups inferred the reconnection sites to
di erent locations. Adapted from Fuselier et al. [2000] and Russell et al. [2000]
25
1.3 Types of Simulation Model in Space Plasma Physics
Today computers are utilized to solve equations to address problems in space plasma
physics, which deal with multi-scale, multi-species, complex systems of plasmas. Generally,
simulation models in space plasma physics fall into several categories: magnetohydrodynamic
(MHD) models, 2- uid or hall MHD models, hybrid models and fully kinetic particle-in-cell
(PIC) models. In a kinetic model, a distribution function f(x;v;t) of the kinetic species is
introduced, where the variables x,v and t are position, velocity and time, respectively. Ki-
netic models work in a way that the charges and currents are determined by the distribution
functions self-consistently via Maxwell?s equations.
In an MHD model, all plasma species are treated as a single or multiple uid. MHD
simulations have been carried out for the uid properties and eld line topology associated
with reconnection [Scholer, 1989a,b; Shi and Lee, 1990; Lin and Lee, 1999]. Since the kinetic
e ects have been fully neglected, the behavior of kinetic waves generated in reconnection
[Chaston et al., 2005; Lin and Lee, 1994; Kraussvarban and Omidi, 1995], signature of non-
Maxwellian particle transports as observed by satellites at the dayside magnetopause can not
be explained by MHD models. The upside of using MHD models for global scale simulation
is that MHD requires less computing resources, and thus can be used to investigate large
scale structure of the magnetosphere.
In a PIC model, electrons and ions are treated as individual particles that move in
imposed electric and magnetic eld, where no approximations are made to the basic laws of
mechanics, electricity and magnetism. The PIC models have been utilized to investigate the
triggering mechanism of reconnection process [Terasawa, 1981; Price et al., 1986; Hoshino,
1987; Shay et al., 2001; Hesse and Winske, 1998]. It is found that the reconnection rate
is insensitive to the mechanisms that initiate the reconnection at the X line [Drake et al.,
2003; Birn et al., 2005; Birn and Hesse, 2001; Shay et al., 2001; Pritchett, 2001; Drake
et al., 2005]. Due to the limited computing resources, most PIC models address reconnection
problem in a way that the simulation domain is at the vicinity of the reconnection X line
26
and boundary conditions are applied locally. This may not be correct when it comes to the
magnetopause reconnection as it is of a multi-scale, long-term evolution of the global system.
Another concern of using PIC models for magnetopause reconnection is that some artifacts
are introduced to expedite the simulation in PIC models. For example, proton to electron
mass ratio much less than 1836 is adopted.
Hybrid models are between MHD and PIC models, in which one or more plasma species
are treated as a single or multiple uids, while the remaining species are treated kinetically
as particles. The hybrid model in this thesis treats ions as particles, while electrons are
treated as massless uid. Sometimes electron inertial and non-gyrotropic pressure e ects
are also included in hybrid models [Kuznetsova et al., 2000]. In general, the hybrid models
are valid in studying the physics with wave frequency ! i, and have been successful in
explaining the structure of magnetic eld, kinetic waves and the ion signatures observed in
reconnection at the magnetopause [Gosling et al., 1986]. 1-D and 2-D hybrid simulations
have shown the presence of slow shocks and rotational discontinuities [Lin and Lee, 1994;
Lin, 2001; Lin and Swift, 1996; Nakamura and Scholer, 2000] in collisionless reconnection
and the structure of FTEs [Karimabadi et al., 2005; Omidi and Sibeck, 2007].
1.4 Objectives and Outline of the Thesis
As shown in previous sections, magnetic reconnection is very important physical pro-
cess at the dayside magnetopause when the magnetic shear across the magnetopause is large
enough. The challenge to the understanding of FTEs in the magnetopause reconnection is
that they are of 3-D nature, embedded in a multi-scale solar wind-magnetosphere global
system, in which the ion physics is a very important. Inclusion of plasma kinetic physics is
necessary for the modeling of the generation and evolution of magnetopause reconnection
as well as the understanding of their internal structure. Although local simulations provide
better resolution for processes which involve small scales, the 3-D global approach accom-
modates the interaction of di erent parts of the magnetosphere. So far, there has been no
27
such a model in the investigation of magnetic reconnection in the magnetosphere. Given
the importance of magnetic reconnection in the global transport of the magnetosphere, it is
expected that such a study will push forward our understanding of the transport process.
By using a global-scale hybrid model, we aim to investigate:
(1) The magnetic eld con guration under a southward IMF with or without the guide
eld By.
(2) The evolution and internal structure of FTEs caused by reconnection process after
their onset.
(3) The quasi-steady reconnection under a purely southward IMF.
(4) The spectra of cusp precipitating ions associated with the dayside magnetopause
reconnection.
In Chapter 2, our simulation model, initial and boundary conditions are explained in
detail. In Chapter 3, the package for data visualization is introduced brie y. Starting from
Chapter 4, we presents the simulation results from di erent IMF conditions. In Chapter 4, we
represent the simulation results from 2 cases with a purely southward IMF, focusing on the
eld con guration and the evolution and internal structure of FTEs caused by reconnection.
In Chapter 5, we explains the dayside magnetopause reconnection impact on the spectra of
cusp precipitating ions under a purely southward IMF. In Chapter 6, we report preliminary
simulation results from a case with a nite guide eld By. In Chapter 7, we summarize and
give a glimpse of future work.
28
Chapter 2
Global-Scale Hybrid Simulation Model
The global hybrid simulation scheme is described by Swift [1996] and implemented
by Lin and Wang [2005] for 3-D simulations of the dayside magnetosphere. Note that our
simulation addresses the reconnection processes within the dayside convection time scale and
not for the entire global magnetosphere.
2.1 The Scheme
Except for the inner magnetosphere of r < 7RE, the ions (protons) are fully kinetic
particles, and the equation in the simulation units for ion motion, is given by
dvi
dt = E + vi B (Vi Ve); (2.1)
where vi is the ion particle velocity, E is the electric eld in units of ion acceleration, B
is the magnetic eld in units of the ion gyrofrequency [Swift, 1996] while Ve is the bulk
ow velocities of the electrons and Vi is the particle ion bulk ow velocity. A small ad hoc
current-dependent collision frequency, ?0:01 J=J0, is imposed in order to model ad-hoc
anomalous resistivity and trigger magnetic reconnection in the simulation, where is the
local ion gyrofrequency and J0 = B0= 0 0 (here 0 is the ion inertial length of the solar
wind). A uid approximation is used to model the inner magnetosphere of r < 7RE, given
that the uid plasma in this region is not expected to a ect the kinetics in the region of the
magnetopause.
29
The electrons are treated as a massless uid, and quasi charge neutrality is assumed in
the calculation. The electric eld is determined by the electron momentum equation
E = Ve B (Ve Vti) rPe=N; (2.2)
where Ve is the electron bulk ow velocity, Vti is the total bulk ow velocity, Pe is the
thermal pressure of electrons and N is the electron number density.
The total ion bulk ow velocity Vti in Equation 2.2 is given by
Vti = nin Vi + nfn Vf; (2.3)
where the subscripts i an f stand for discrete particle and uid, respectively and the n?s are
the densities. Note that at the radial distance r 7 and as r increases, the particle ion bulk
ow velocity Viis the dominant component for total ion bulk ow velocity.
The equation for uid ion velocity is given by
dVf
dt = E + Vf B (Vf Ve) (2.4)
The electron ow speed is evaluated from Ampere?s law
Ve = Vti r B n ; (2.5)
where a constant = (4 e2=mic2) is introduced. Note that ( n) 12 is the ion inertial length.
The magnetic eld is updated by Faraday?s law
@B
@t = r E: (2.6)
The equations above then are written in a form more convenient for time-stepping, where
magnetic eld, particle motion and uid term are kept while electric eld E is eliminated.
30
The particle equations are advanced with one time step dt and the uid and magnetic eld
equations are advance using 0:1dt, one tenth of the time step used for the particle motion.
Note that magnetic eld B is expressed as
B = B0 + B1; (2.7)
where B0 is a time independent and curl-free portion and only B1 is updated.
As for the time-stepping algorithm, the velocities are known at the half time step, and
the positions and elds are at the even time step. The fourth order Ronger-Kutta Method
and the leapfrog technique are the two options for the subcycle update of the magnetic
elds. Interpolation has been used for the update of both elds and particles. Second-order
accuracy have been obtained in the discretization.
2.2 Curvilinear Coordinates
This section describes the use of the curvilinear coordinates. The corresponding Carte-
sian positions of the curvilinear grid points can be calculated through a table of geometrical
coe cients. Figure 2.1 shows a curvilinear coordinate cell, of which the center is at the grid
point i, j, k, while the coner point is at i+ 12, j+ 12, k+ 12 and i+ 12, j+ 12, k 12, etc. Corre-
spondingly, a dual cell is de ned as a cell with centers at the half grid points and corners at
the whole grid points, i, j, k. The magnetic eld components are shown as vectors on cell
faces with the components pointing normal to the faces while the electric eld resides on the
cell edges. The magnetic eld components can be presented by taking the scaler product of
B with the unit vector normal to the cell surfaces. In another word, the magnetic eld can
be presented in contraviant components.
B = l1^B1 + l2^B2 + l3^B3; (2.8)
31
where l1, l2,and l3 are tangent vectors, while ^B1, ^B2, and ^B3 are contravarient tensor
components. Note that a set of reciprocal basis vectors !?s can also be de ned so that the
magnetic eld can be expressed as
B = !1^B1 +!2^B2 +!3^B3; (2.9)
where !i lj = ij. Applying this to the right-facing face of the parallelepiped cell, which is
centered at (i+ 12;j;k), the discretization of Farady?s law gives
[(
^B1)n+1 ( ^B1)n
t
l1
l1 A
1]
i+12;j;k =(^E
n+12 l3)
i+12;j 12;k (^E
n+12 l3)
i+12;j+12;k
+ (^En+12 l2)i+1
2;j;k+
1
2
(^En+12 l2)i+1
2;j;k
1
2
;
(2.10)
where l2;3 are lengths of the cell edges and reside at the center of the edges. A1 = l2 l3 is
the area of the correspoing cell surface and l1 is the correspoding dual cell tangent vector.
The length of tangent vectors (l?s) are the di erence between coordinates points speci ed
in the coe cient table as mentioned previously. I.e.,
(l2)i+1
2;j;k+
1
2
= (r)i+1
2;j+
1
2;k+
1
2
(r)i+1
2;j
1
2;k+
1
2
; (2.11)
where the r?s are the position of the coordinate points.
The particle position is updated from
(qi)n+1 = (qi)n + tMi vn+12; (2.12)
where the vector tensor Mi convert v into particle position qi in a contravariant form. Note
that the positions of particles are also position dependent and so they are interpolated from
the grid to the particle position using PIC weighting.
32
Figure 2.1: A coordinates cell showing the position and orientation of curvilinear components
of the magnetic and electric elds. Adatped from [Swift, 1996]
2.3 Simulation Domain
The simulation domain contains the system of the bow shock, magnetosheath, and
magnetosphere in the dayside region with x > 0 as shown in Figure 2.2 and a geocentric
distance 4 r 24:5. The Earth is located at the origin (x;y;z) = (0;0;0). Out ow
boundary conditions are utilized at x = 0, while in ow boundary conditions of the solar
wind are applied at r = 24:5. The inner boundary at r = 4 is perfectly conducting.
Spherical coordinates are used in the simulation. The polar angle is measured from
the positive z axis, and the azimuthal (longitudinal) angle from the negative y axis.
2.4 Initial and Boundary Conditions
Initially, a geomagnetic dipole eld plus a mirror dipole is assumed in r < 10RE [Lin
and Wang, 2005], and a uniform solar wind with the IMF B0 of Bx0 = 0, By0 = 0 and
Bz0 = 1 is imposed for r > 10RE. The mirror dipole in the initial setup is to speed up
the formation of the bow shock and magnetopause. The bow shock, magnetosphere and
magnetosheath are formed by interaction between the solar wind and the dipole eld.
33
Figure 2.2: Simulation domain in the GSM system
34
We choose a uniform solar wind with i = e = 0:5 and an Alfv en Mach number
MA = 5. The solar wind ows into the system along the x direction with an isotropic
drifting-Maxwellian distribution. The ion number density in the solar wind is set to be
N0 = 11;000R 3E for macro particles in this kinetic simulation, and a total of 4 108
particles are used in a run. For a typical ion gyrofrequency of 0:1 s 1, the corresponding
IMF is ?10 nT.
Non-uniform grid spacing r is used to produce a higher resolution near the magne-
topause, where r ? 0:09RE. A total grid of 160 104 130 is used. The time step to
advance the positions of ions is 0:05 10 , where 10 is the inverse of the solar wind ion
gyrofrequency ( 10 ).
In the presentation below, the magnetic eld B is scaled by the IMF B0; the ion number
density N by the solar wind density N0; the time t by the inverse of the solar wind ion
gyrofrequency ( 10 ); the ow velocity V by the solar wind Alfv en speed VA0; the temperature
by V2A0; the length in units of the Earth?s radius RE.
In order to accommodate to the available computing resources, a larger-than-reality
ion inertial length 0 = 0:1RE of the solar wind is chosen in the simulation. Note that
VA0 = 0 0. In an e ort to examine the e ects of various values of 0, we have also run a
case with 0 = 0:05RE, although still about 3 times larger than that in reality. The resulting
structures of the magnetopause reconnection are qualitatively the same as that shown in this
thesis.
Three cases are presented in this thesis. In case 1, the dipole axis is tilted sunward by
15 , so that the northern cusp region is well within the simulation domain. Case 2 is similar
to case 1, except that the dipole tilt angle is equal to zero. Case 3 is also similar to case 1,
except that the IMF is Bx0 = 0, By0 = 0:707 and Bz0 = 0:707.
35
Chapter 3
Data Visualization
The raw data generated from supercomputers are processed and visualized, using desk-
top computers. As a joint e ort with Dr. Wang Xueyi, the visualization tool of 3-D data is
developed in MATLAB, which is a numerical computing environment and fourth-generation
programming language. Note that a certain fraction of open source code mainly from the
website of Mathworks is also included in the package. Figure 3.1 shows the main graphic
user interface of the visualization package.
As seen in Figure 3.1, the data processing consists of two parts. The rst part is data
conversion and data reading, as shown in the upper part of the main interface. There are
several options for data format, including ?BOX-DATA?, ?RAW-DATA? and ?COV-DATA?.
?RAW-DATA? is the ASCII (American Standard Code for Information Interchange) out-
put from Fortran 90 program, which is in spherical coordinates. The icon ?RAW-DATA
CONVERSION? provides a function to convert the raw data to a HDF (Hierarchical Data
Format) format, namely, the format of ?COV-DATA? as shown in the interface. Note that
the ?COV-DATA? is in the Cartesian coordinate system.
The lower part of the main interface are the main menu for plotting. Most of our
data are 3-D. After reading data, clicking icon ?3D PLOT? leads to the 3-D data plotting
menu. Figure 3.2 showcases an example of 3-D data visualization. The contours are the total
magnetic eld in the noon meridian and equatorial planes, supposed with the isosurface of
B = 4:5 in the simulation units.
Figure 3.3 shows the capability of slice plotting integrated in the the 3-D data visual-
ization package. Not only planes of a constant x;y or z in the GSM system but also planes
with a rotation around the three principle axes can be chosen to plot contours.
36
Figure 3.1: Graphic user interface of the visualization package
Figure 3.3 shows the capability of isosurface plotting integrated in the the 3-D data
visualization package. Lighting and transparency can be adjusted as needed.
Both scaler data sets such ion density, temperature and vector data sets such as velocity
and magnetic eld can be visualized in the 3-D frame. For magnetic eld, eld lines can be
plotted interactively. In the eld line plotting mode, clicking a point inside a plotted slice
within the 3-D visualization frame will enable the package to plot a eld line through that
point. Figure 3.5 is the graphic user interface of ne tuning magnetic eld lines. Clicking
the icon ?x-? changes the position of the point through which the visualization package plots
the magnetic eld line by a small step, e.g., 0:1, in the negative x direction.
In addition to the general 3-D plotting capability, new graphic objects such as particle
trajectories can be plotted within the same 3-D gure. The package also has relatively
independent functions for one dimensional (1-D) plotting and 2-D (two dimensional) plotting
as indicated in the main interface.
37
Figure 3.2: An example of 3-D data visualization: The contours are the total magnetic eld
in the noon meridian and equatorial planes, supposed with the isosurface of B = 4:5 in the
simulation units.
38
Figure 3.3: Slice plotting integrated the 3-D data visualization package.
39
Figure 3.4: Isosurface plotting integrated the 3-D data visualization package.
Figure 3.5: Graphic user interface of ne tuning magnetic eld lines.
40
The sample source code in the appendix is the function in MATLAB for contours plotting
and ne tuning eld lines, which contain frequently used technique in the package.
41
Chapter 4
Dayside Magnetic Reconnection Under a Purely Southward IMF: Structure and Evolution
of Reconnection
4.1 Introduction
As we discussed in Chapter 1, magnetic reconnection is believed to play an important
role in geomagnetic and magnetospheric plasma processes. Without magnetic reconnection,
the magnetosphere of the Earth is a relatively ?isolated? space from the solar wind. With
the magnetospheric eld lines opened up by the reconnection process, solar wind plasma is
able to penetrate through the magnetopause; momentum and energy can also be transferred
from the solar wind and interplanetary magnetic eld (IMF) into the magnetosphere as the
connectivity of the magnetic eld lines changes.
One of the important questions about magnetic reconnection is the magnetic con g-
uration of connectivity associated with it, i.e., whether the reconnection occurs through a
single X line or multiple X lines [Winglee et al., 2008]. According to the time scales of the
in-situ observation results, magnetic reconnection events fall into two categories: (1) quasi-
stationary magnetic reconnection; (2) transient magnetic reconnection. A quasi-stationary
magnetic reconnection might have a time scale of hours [Gosling et al., 1982], which is
dominated by a single X-line. The transient counterpart usually has a quasi-period around
8 minutes [Rijnbeek et al., 1984]. Flux transfer events (FTEs), which are widely considered
to be associated with transient magnetic reconnection nearby[e.g., Hasegawa et al., 2006;
Kuznetsova et al., 2009], were rst discovered by Russell and Elphic [1978]. Initially, FTEs
were thought to be ux tubes that are the products of single-X-line, patchy reconnection
[Russell and Elphic, 1978]. Due to complicated geometries and the multi-scale nature of re-
connection, numerical simulations have been utilized to investigate the reconnection physics.
42
Lee and Fu [1985] suggested that FTEs are multiple-X-line ux ropes with helical internal
structures, in which a magnetic eld line is reconnected at two or more reconnection sites.
Alternative models were also proposed to account for the formation of FTEs. For example,
Scholer [1988] suggested that the variation of reconnection rate could give rise to a loop-like
eld lines based on the single X-line reconnection.
Since local simulation models have strong dependence on the boundary conditions, three-
dimensional (3-D) global simulations using modern computers have become a powerful tool
to study the reconnection and the associated structure of the dayside magnetopause. The
latest work includes global magnetohydrodynamic (MHD) simulations by Fedder et al.
[2002], Raeder [2006], Dorelli and Bhattacharjee [2009], Kuznetsova et al. [2009] and
Winglee et al. [2008]. Dorelli and Bhattacharjee [2009] point out that it is likely that the
resistive MHD Ohm?s law may fail to capture much of the physics relevant to the FTE
generation when the model assumption is no longer valid under certain circumstances.
The challenge to the understanding of FTEs in the magnetopause reconnection is that
they are of 3-D nature, embedded in a multi-scale solar wind-magnetosphere global system,
in which the ion physics is a very important. Inclusion of plasma kinetic physics is necessary
for the modeling of the generation and evolution of magnetopause reconnection as well as
the understanding of their internal structure. For the rst time, here we present a numerical
simulation of dayside reconnection using a 3-D global scale hybrid model, in which fully-
kinetic ion physics is solved in a self-consistent electromagnetic eld [Lin and Wang, 2005].
The reconnection events are identi ed by the connectivity change of magnetic eld lines
and supported by the presence of reconnection jets away from the reconnection sites. Both
multiple X-line and single X-line type structures are examined, where the term ?X lines?
is de ned in the context of local 3-D eld con guration [e.g., Priest and Forbes, 2000;].
The detailed de nition will be given in section 3.2. Note that our de nition of X line is
di erent from the one used by Dorelli and Bhattacharjee [2009], which de nes the X line
as a separator line, the intersection of two north pole, and closed eld lines. Based on a
43
global MHD simulation, Dorelli and Bhattacharjee [2009] have found that the instability
associated with FTEs is triggered by a movement of the ow stagnation point away from
the magnetic separator, which modi es the subsolar stagnation point ow.
In this chapter, two cases with a purely southward IMF are investigated. In the presence
of a strong southward IMF, the Earth?s dipole eld is anti-parallel to the IMF around the
equatorial plane, leading to the occurrence of magnetic reconnection at the low latitude mag-
netopause. Statistical studies show that majority of FTEs are observed during a southward
IMF [Rijnbeek et al., 1984; Sibeck and Lin, 2010].
4.2 Simulation Results
4.2.1 Magnetic Field Line Con guration under Southward IMF
We rst present results from case 1, with a 15 dipole tilt angle and a purely southward
IMF. As the solar wind ions convect earthward carrying the IMF, the bow shock, magne-
tosheath, and magnetopause form gradually in a self-consistent manner. Most magnetic
reconnection events originate in the equatorial region, which are related to X-lines of nearly
anti-parallel magnetic eld reconnection. Flux ropes are generated above and below the
equator and reconnected magnetic eld lines are approximately symmetric about the noon
meridian plane.
Figure 4.1 shows the magnetic eld line con guration obtained from case 1 in a global
view, emphasizing eld lines in the northern hemisphere at t = 5, 15, 25 and 35. The
blue sphere at the origin represents the Earth. The contours in the equatorial plane show
the ion density. Outward from the Earth, the boundary region with a sharp ion density
increase is the magnetopause, at a stando distance of r?9:5 10. The bow shock is the
boundary region with a sharp density decrease. The black lines are the closed eld lines of
the geomagnetic dipole eld. The yellow lines are open eld lines of the shocked IMF before
magnetic reconnection. Field lines in other colors are reconnected eld lines with one end
44
Figure 4.1: Magnetic eld line con guration in a global view obtained in case 1 at t = 5
(top left), t = 15 (top right), t = 25 (bottom left), and t = 35 (bottom right), respectively.
The closed dipole eld lines are in black. Yellow lines are open eld lines before magnetic
reconnection. Field lines in other colors are reconnected eld lines between the IMF and
dipole eld in di erent regions. Contours in the equatorial plane show the ion density.
45
connecting to IMF and the other end to the magnetosphere, where the colors are used to
distinguish di erent regions of magnetic reconnection as in the description below.
The upper left plot of Figure 4.1 shows the magnetic con guration of the initial phase
at t = 5, or t?25 1 with 1 being the local ion gyrofrequency. There appears to be no
signi cant evidence of magnetic reconnection at this time.
At t = 15 (Figure 4.1, upper right), looped ux ropes (FTEs), form in between two
neighboring X lines, while the X lines of nite length lie in the dawn-dusk direction. The
axial extent of the ux ropes is limited as seen in the plot. Around the equator, there are
both looped ux ropes and adjacent single X-line magnetic reconnection shown by the red
eld lines. FTEs are found to be localized looped ux ropes corresponding to multiple X-line
reconnection (MXR) [Raeder, 2006; Hasegawa et al., 2006] with helical internal structures,
which generate signatures consistent with in situ observations. The bipolar FTE signature of
Bn, the local normal component of magnetic eld, will be discussed below. A Wal en test will
also be performed for the single X-line reconnection to investigate the presence of relevant
MHD discontinuities, using the data from case 2.
The blue eld lines in the mid latitudes show another layer of looped ux ropes (FTEs),
while the green eld lines in the northern high latitudes show the occurrence of a single
X-line-type reconnection with no obvious looped eld lines.
At t = 25 (Figure 4.1, lower left), or t?125 1, the looped ux ropes illustrated by the
red eld lines of t = 15 have moved poleward with a noticeable expansion of their azimuthal
size. The blue FTEs at t = 15 have also propagated towards the cusp and tailward, and
meanwhile toward dawn/dusk side.
At t = 35 (Figure 4.1, lower right), part of the red ux ropes around the equator at
t = 15 have moved to the middle latitudes, so have segments of X lines adjacent to the
FTEs. The blue FTEs in the middle latitudes at t = 15 also continue moving tailward and
poleward, and part of them have disintegrated during the interaction with the cusp while
the rest have convected past the pole.
46
Figure 4.2: (Left) Four eld lines of di erent topologies in case 1 at t = 55. (Right) Illustra-
tion of how the patchy reconnection and multiple reconnection can explain the coexistence
of the four eld lines in the left plot [Lee et al., 1993].
47
In our simulation, the magnetopause reconnection is found to generate not only the
reconnected eld lines connecting the IMF to the dipole eld, but also other topologies
due to 3-D e ects, including both purely closed reconnected eld lines as well as purely
open reconnected eld lines. The color plots in Figure 4.2 show four eld lines of case 1
at t = 55, in which the ux ropes are mainly located at x ? 9:5-10, y ? 0:6- 1:8 and
z? 0:5-1. In addition to the blue and black eld lines that are between the magnetosphere
and the magnetosheath, there also exist the green closed eld lines that are connected from
the magnetospheric eld to the magnetospheric eld, threading from the northern cusp
to the southern cusp, as well as the red open eld line from the magnetosheath to the
magnetosheath. The rectangular contour slice shows ion density at z = 0:5 around the
reconnection site. Note that the sliver of green density on the slice is adjacent and on the
sunward side of the ux ropes.
The green and red eld lines in Figure 4.2 are associated with the FTE ux ropes, and
thus are di erent from the unperturbed closed geomagnetic eld lines and open solar wind
eld lines. A previous 3-D MHD simulation of Lee et al. [1993] shows that due to the
non-uniformity along the plane of the current sheet, four possible topologies of FTE eld
lines can be generated. They are the eld lines connected (1) from IMF to the Earth?s dipole
eld ( eld lines between solar wind and north pole); (2) from the Earth?s dipole eld to the
Earth?s dipole eld; (3) from the Earth?s dipole eld to IMF ( eld lines between solar wind
and south pole); and (4) from IMF to IMF. The right plots (a)-(d) in Figure 4.2 ( Right)
[Lee et al., 1993] can be used to illustrate how the patchy, multiple reconnection events can
explain the coexistence of the four types of eld lines obtained from our 3-D simulation as
shown in the color plots of Figure 4.2. The patchy reconnection indicates that the size of
the reconnection region has a fairly limited extent in space [Kan, 1988; Pinnock et al., 1995].
At the initial stage (a), ux tubes AA0 and BB0 are two bundles of magnetosheath elds
and CC 0 and DD0 are of magnetospheric elds. At the second stage (b), reconnection takes
place at two patches. At the third stage (c), the reconnected ux tubes BD0 and CA0 move
48
toward each other so that at the fourth stage (d) a re-reconnection has occurred, leading
to the formation of closed ux tube CD0. Flux tubes CD0 and BA0 correspond to the green
(closed) and red (open) eld lines in the color plots of Figure 4.2. Since ux ropes form
between multiple X lines, a position shift of neighboring X lines relevant to ux ropes can
also play a critical role in determining the connectivity of eld line [Lee et al., 1993].
The existence of reconnected eld lines from the magnetosphere to the magnetosphere
or from IMF to the IMF, in addition to the opened magnetospheric eld lines and the
unperturbed magnetosheath eld lines, has also been suggested by Lui et al. [2008] and
Kuznetsova et al. [2009]. We note that our study is the rst global simulation that shows
these four types of eld lines. The reason why a 3-D global hybrid simulation ?shows? but
a 3-D global MHD simulation does not deserves further investigation. In our simulation,
localized ux ropes at low latitudes with dominant purely closed eld lines do not occur
often and can not survive more than 20 10 . Further investigation is needed to address how
long their life time is for more general cases with a nite IMF By component.
4.2.2 Structure and Evolution of Flux Transfer Events (FTEs)
The structure and evolution of FTEs are illustrated in Figure 4.3 around the dayside
magnetopause. Here, we give our de nition of the reconnection X line before the discussion
on physical quantities associated with 3-D reconnection. We adopt a procedure like the one
described by Priest and Forbes [2000b], which de nes reconnection in a general way.
We seek a set of singular lines, near which the magnetic eld has an X-type con guration.
The rst row of Figure 4.3 illustrates how a singular line, or in another word, "X line" is
de ned, which is the intersection of two surfaces that separate distinct eld lines of four
di erent local regions (earthward side of the magnetopause current sheet, sunward side of
the current sheet, from the sunward side to the earthward side, and from the earthward side
to the sunward side following the magnetic eld direction). The black tubes are eld lines
traced in the 3-D space. The X-type con guration is determined by mapping the 3-D eld
49
Figure 4.3: Three dimensional plots ( rst row) illustrate how X line is de ned, with Vz
contours in the noon meridian plane. Two dimensional intensity plots of By (second row),
ion density (third row), total magnetic eld (fourth row), parallel temperature ( fth row)
and perpendicular temperature (bottom row) are also in the noon-midnight meridional plane,
zoomed around the dayside magnetopause at t = 15, 25, 35, 65, respectively (from left to
right).
50
lines into an x-z reference plane. An X point of the X line that separates local areas of
four di erent magnetic connectivity types is found, shown as a blue dot. By connecting the
X points in a series of such planes, an X line segment naturally forms. Other supporting
evidence of the X line, such as the existence of opposite ow jets and the quadrupole magnetic
eld structure in the guide eld, is given below. The FTEs to be discussed in this chapter
are of an O-line that requires two X-lines.
As shown in the left plot of the rst row, there are two X-line segments as in blue dots
at t = 15. There are also two X lines at t = 65. But the previous two at t = 15 are not
seen at t = 65 as they have moved northward. The northern X line segment at t = 65 was
below the plane of z = 1 at t = 15 before it moves to its present location in this case with
a tilted dipole axis.
In order to illustrate the presence of reconnection jets from the X lines, the contours
in the rst row of Figure 4.3 show Vz, the z component of ion bulk ow velocity in the
noon-midnight meridional plane obtained from case 1. Our de nition of the 3-D magnetic
reconnection is consistent with observations that plasma jets are moving away from X lines
due to reconnection. At t = 65, two opposite jets with Vz of opposite sign occur inside
the FTE ux ropes between the two X lines. The Vz of magnetosheath plasmas near FTEs
increases as the latitude increases. Although two opposite jets are expected from an X line,
the southward jet at t = 15 appears missing, due to the the non-zero northward background
Vz of the ambient magnetosheath plasmas. Meanwhile, strengths of the two opposite jets
from two adjacent X lines inside one FTE are not necessarily equal to each other. As in the
case at t = 15 shown in the top left plot of Figure 4.3, one direction of ion acceleration is
dominant inside the FTE between the two X lines.
The intensity plots in Figure 4.3 show By (second row), ion density N (third row), mag-
netic eld strength B (fourth row), parallel ion temperature Tk ( fth row), and perpendicular
ion temperature T? (bottom row) in the noon-midnight meridional plane at t = 15;25;35
and 65. The black lines superposed on the contours are two dimensional (2-D) eld lines
51
projected onto the noon meridian plane. Note that the density of 2-D eld lines shown here
does not represent the magnetic eld strength. The locations of FTEs can be identi ed from
the magnetic islands traced by the eld lines in the contour plots. The center of a highlighted
FTE island is marked with arrows in the plots from t = 15-35.
The FTE marked in Figure 4.3 forms around t = 15 at the subsolar region, and moves
poleward as time proceeds. The FTE speeds up as it moves away from the equator. Att = 65,
another FTE forms near the subsolar region, which re ects the quasi-periodic generation of
FTEs. Between t = 35 and t = 65, there is a relative quiescent period, and magnetic
reconnection remains single X-line-like in the region.
Perturbations in By are obtained in the vicinity of X lines, as indicated by the black
rectangles in Figure 4.3. Because the initial By in the solar wind is set to be zero, the intensity
of By in the plot can be viewed as a perturbation. The By pattern is consistent with the
Hall e ects due to the ion kinetic e ects [Sonnerup, 1979; Terasawa, 1983; Pritchett, 2001;
Shay et al., 2001]. In a simple 2-D reconnection model, plasma ows into the vicinity of an X
line. Electrons are frozen-in to the eld lines and ions lag behind due to their larger inertia,
which produces a net current and a corresponding By perturbation. This leads to a negative
By above and a positive By below the X line on the magnetosheath side, with a negative
By below and a positive By above the X line on the magnetospheric side. For the dayside
magnetopause, it is expected that the polarity on the magnetosheath side dominates the Hall
pattern due to the much larger density on the magnetosheath side, which is di erent from
the quadrupole structure of By for a nearly symmetric current sheet [Karimabadi et al.,
1999; Pritchett, 2001; Birn et al., 2008].
In contrast to a dominant polarity on the magnetosheath side, By perturbations with
near equal strength are seen in the multiple X-line reconnection at t = 25, 35 and 65 in our
simulation, as shown in Figure 4.3. The quadrupole By perturbations are within a boundary
layer, of which the sunward thickness is 0:5RE 1:0RE. The plasma density level in
the boundary layer adjacent to the magnetosphere is about 1.0, comparable to 2.0, in the
52
ambient magnetosheath. The local plasma density adjacent to the magnetosphere may be
enhanced by the trapped ions around the nearby O line of an FTE. The presence of near
quadrupole By may be due to the thick boundary layer around the FTEs.
The third row of Figure 4.3 reveals that the FTEs are associated with ion density
enhancements at the core. Note that the low density region at t = 15, x = 10-11 and z = 5-6
are in the solar wind. Corresponding to the density enhancement, the magnetic eld strength
is found to dip in the core of the FTE as seen in the fourth row of Figure 4.3. Such results are
consistent with the 2-D hybrid simulation results of Omidi and Sibeck [2007] for a similar
case of a purely southward IMF, in which the reconnection is mainly of anti-parallel eld
type. It has been suggested [e.g., Hasegawa et al., 2006; Scholer et al., 2003] that whether
an FTE possesses a strong core eld may be associated with whether the reconnection is an
anti-parallel or component merging.
In the upper panel of Figure 4.4, ion density contours at x = 9:5 at t = 80 are shown as
well as eld lines, while a close-up plot around three FTEs at the magnetopause is shown in
the lower panel. The ux ropes are seen to wrap around a lament of relatively higher ion
density. The peak density is about 2 times that of the ambient plasma inside the boundary
layer. Our results indicate that the density at the FTE core may be larger than that near
the edge, and that the spatial pro le of the density along a path through the core may be
very di erent from that through the edge.
Ion heating is found in the FTEs, as shown in the bottom two rows of Figure 4.3.
Stronger enhancement in the parallel ion temperature Tk is seen inside FTEs, while the
perpendicular temperature T? shows a mild increase compared with that in the ambient
magnetosheath. The enhancement of Tk=T? in magnetic reconnection has also been reported
in satellite observations[Klumpar et al., 1990] and numerical simulations[Birn and Hesse,
2001]. Note that the fading of the ion temperature in the closed eld line region between
t = 15 and t = 65 is due to the loss of ion particles in the magnetosphere because the returned
53
Figure 4.4: Ion density laments inside FTEs of case 1. The upper panel shows ion density
contours at x = 9:5 and t = 80, superposed onto a eld-line plot, and the lower panel is a
close-up of the same plane.
54
Figure 4.5: Ion density laments inside FTEs of case 1. The upper panel shows ion density
contours at x = 9:5 and t = 80, superposed onto a eld-line plot, and the lower panel is a
close-up of the same plane.
ions from the magnetotail are not included in the simulation model. Only the transmitted
ions from the magnetosheath are emphasized in the simulated magnetopause reconnection.
4.2.3 Magnetic Field Signature and Ion Velocity Distributions in FTEs
Bipolar signature of the normal component of magnetic eld, Bn, has been considered
a typical signature of the observed FTEs [e.g., Russell and Elphic, 1978; Dorelli and Bhat-
tacharjee, 2009]. Viewed in the local normal coordinates, satellites traveling through an
FTE along the magnetopause usually observe a transient magnetic eld structure in which
Bn changes either from + to or from to +.
The ux ropes obtained in our simulation indeed produce the bipolar signature similar
to satellite observations. The rightmost plot in Figure 4.5 shows the spatial variation of
the magnetic eld and ion density along a virtual satellite path through the magnetospheric
edge of an FTE at t = 40 in the simulation of case 1. In the contour plots of Figure 4.5, the
path is from point P1 to point P2 along the magnetopause, illustrated by the thick black
line. The contours are of N, By and Vz in the noon meridian plane, with 2-D black eld
lines superposed on.
The top three panels in the rightmost plot of Figure 4.5 show the spatial variations
of Bn, Bl, and Bm components of magnetic eld, where the local normal direction ^n of the
55
lmn local coordinate system is determined by the minimum variance method [Sonnerup and
Cahill, 1967]. In this coordinate system, three directions^l, ^m and ^n complete a right-handed
orthogonal system with ^l de ned as (^z ^n(^n ^z))=j^z ^n(^n ^z)jand ^m as the vector product
^n ^l.
The local normal direction ^n is found to be (0:8259;0:0139;0:5636) and ^m is nearly
^y in the GSM system. As shown in the top panel, Bn changes from positive near P1
to negative as the virtual satellite ? ies? toward P2, consistent with the typical bipolar
magnetic eld signature of FTE. The Bl component remains a positive magnetospheric value
during the crossing. The magnitude of Bm is approximately equal to By. The two bipolar
enhancements of Bm are due to two parts of the adjacent Hall eld perturbations from two
X-lines as illustrated in the contours of By. Near the center of the FTE, the ion density goes
up as the virtual satellite cuts through the density lament inside the ux ropes as shown in
the contours of N. The bottom panel shows the eld strength B, which exhibits a pattern
that is not of a simple anti-phase relationship with ion density.
To investigate the properties of ion particles around FTEs, we ?probe? in Figure 4.6
the ion velocity distributions at the speci c FTE highlighted with white dots in Figure 4.5.
Figure 4.6 plots the parallel velocities (relative to the local magnetic eld) vik of ion particles
versus one of the two perpendicular ion velocities, vi?1. Here, the perpendicular direction
^e?1 is chosen to be in the direction of B ^y. The top left, top right, bottom left and bottom
right plots of Figure 4.6 show the distributions at four chosen locations centered at D1, D2,
D3 and D4, respectively, which are marked in the contours of Figure 4.5.
The top left of Figure 4.6 corresponds to locationD1 centered at (x;y;z) = (8:08;0;2:62).
Among the four locations, D1 is the closest to the magnetosphere but still on open eld lines.
The velocity distribution features a tenuous ion population, and the majority of ions pos-
sess near zero bulk velocity. The bottom left plot of Figure 4.6 corresponds to location D3
centered at (x;y;z) = (9:48;0;4:02) in the magnetosheath outside the FTE. This dense ion
population convects northward with the bulk Vik< 0, opposite to the southward IMF.
56
Figure 4.6: Top left, top right, bottom left and bottom right show the ion velocity distribu-
tions at four chosen locations centered at D1, D2, D3 and D4 in Figure 4.5, respectively.
57
The top right plot of Figure 4.6 corresponds to location D2, centered at (x;y;z) =
(8:69;0;3:03), which is well inside the ux ropes. The bottom right plot of Figure 4.6
corresponds to location D4, centered at (x;y;z) = (8:83;0;3:27), which is near the center of
the ux ropes. Mixtures of multiple ion beams transmitted from the magnetosheath are seen
in the vik-vi?1 plane. The presence of multiple ion beams in FTEs has also been reported
from satellite observations [e.g., Hasegawa et al., 2006]. The velocity distributions at D2,
D4 feature the highest ion temperatures among the selected locations, with Tk > T? as
indicated by the larger extent of the contours in vik than that in the perpendicular velocity
space of vi?1 and vi?2 (not shown). The majority of ions possess a large positive velocity
vik at D2 while the majority of ions at D4 possess a large negative velocity vi?1. Positive
vik at D2 and negative vi?1 at D4 both indicate that these ions are accelerated northward
away from the X line south to the FTE that the four locations are associated with. Several
other locations to the north of D2 and D3 and inside the FTE were "probed" and the results
show that ions are also accelerated northward away, which is consistent with the positive Vz
shown in the contour plot.
4.2.4 Wal en Test of Rotational Discontinuity in a Quasi-Steady Reconnection
To evaluate the in uence of the tilt angle of the Earth?s dipole eld, we have run case
2 with the same parameters except that the tilt angle is chosen to be zero. It is found that
under the new condition, X lines also form and move poleward. Although the time scale of
the reconnection in case 2 is of the same order as in case 1, the average time of reoccurence
of FTEs from the subsolar region in case 2 is longer. The top left plot of Figure 4.7 shows an
example of case 2 at t = 100, where that the magnetic reconnection structure north to the
subsolar looped ux ropes is ?single-X-line-like?. The semi-transparent contour plot shows
Vz, the z component of the ion ow velocity in the noon meridian plane, illustrating plasma
jets away from X lines. As the plot in fact is a 3-D one with a view from dawn to dusk, the
black tubes that sometimes pass through the noon meridian plane are true 3-D eld lines.
58
Figure 4.7: Wal en test in case 2 in the northern hemisphere at t = 100. The top left contour
plot shows the ion ow velocity Vz in the noon meridian plane, in which E1-E2 is a line
segment in the r direction across a rotational discontinuity. Spatial cuts of eld components
Bl, Bm, Bn and ion ow velocities Vl, Vm, Vn along E1-E2 are shown in the top right plot.
The result of Wal en test is shown in the bottom left. The ion velocity distribution at location
D5 between E1 and E2 is shown in the bottom right.
59
While the FTE ux ropes are forming at the subsolar region in the northern hemisphere, a
rotational discontinuity is found north to the subsolar looped ux ropes, associated with a
single X line. Note that in case 1 with a dipole tilt of 15 , the magnetopause is dominated by
FTEs as also found in the 3-D global MHD simulation of Raeder [2006]. No clear rotational
discontinuities are found in case 1.
Both theoretical models [e.g., Lin and Lee, 1994], and observations [Phan et al., 1994]
show that a large amplitude rotational discontinuity, an intermediate-mode MHD disconti-
nuity, may exist in out ow regions of quasi-steady reconnection at the magnetopause. Here,
we examine the existence of a rotational discontinuity in the magnetopause boundary layer
in our 3-D global scale hybrid simulation. The Wal en relation is applied to identify the
rotational discontinuity [Sonnerup et al., 1981], which states that across the rotational dis-
continuity the tangential plasma ow velocity Vt changes, as
Vt = VAt0 = VAt[1 ( k ?)=2]1=2; (4.1)
where Vt is the change of the tangential ow velocity across the rotational disconti-
nuity and VAt0 is the corresponding change in the tangential Alfv en velocity corrected by
a temperature anisotropy factor. The plus (minus) sign is applied to discontinuities with
a normal component of the upstream in ow velocity parallel (anti-parallel) to the normal
component of the magnetic eld [Paschmann et al., 1986]. In satellite observations [Son-
nerup et al., 1981; Phan et al., 1996] and a previous hybrid simulation [Lin, 2001], equation
(4) is not exactly satis ed. Instead, Wal en ratio A j Vtj=j( VAt[1 ( k ?)=2]1=2)j
usually ranges from 0.6 to 0.9 for ions, and for electrons the Wal en relation can be nearly
satis ed [Scudder et al., 1999].
Figure 4.7 shows an example of Wal en test around a single X line reconnection site in case
2 in the northern hemisphere at t = 100. The contour plot on the upper left side of Figure
4.7 shows the Vz component of ion bulk ow velocity in the noon meridian plane, in which
60
the black tubes are 3-D eld lines. The magnetic eld lines in this plot sometimes cross the
noon meridian plane, indicating a non-zero By component. E1-E2 is a line segment in the r
direction across the discontinuity to be studied below, while E1 is at (x;y;z) = (7:926;0;7:3)
and E2 at (x;y;z) = (7:059;0;6:55). Point D5 is centered at (r; ; ) = (9:8;47 ;90 ), also
on the path E1-E2.
A rotational discontinuity is identi ed along the line cut, of which the local normal
direction to the discontinuity, ^n, is determined by the minimum variance method used in
Section 3.3. The ^l, ^m, ^n are ( 0:6643; 0:2307;0:7110), (0:3281; 0:9446; 0:0000) and
(0:6716;0:2333;0:7032) in the GSM coordinate system, respectively. Spatial cuts of magnetic
eld components Bl, Bm, Bn and ion ow components Vl, Vm, Vn along E1-E2 are shown
in the top right plot Figure 4.7. Whereas the rotational discontinuity is identi ed between
the two blue vertical lines, across the magnetopause current layer. From E1 to E2, the
normal component of magnetic eld remains nearly constant while Bl changes sign. The Bm
component, although possessing a sign change too, is dominated by Bm > 0, or By < 0,
unlike the quadrupole By pattern shown around the case 1 FTEs (Figure 4.3). Through
the sharp kinks of eld lines from E1 to E2 across the magnetopause, the dominant ow
component Vl is accelerated from 2:5 in the magnetosheath to 5:3 in the boundary layer, a
change of ?1:46 local Alv en speed, by the eld tension force.
The bottom left plot of Figure 4.7 shows the result of the Wal en test. Throughout
the points from E1 to the right blue vertical line along the path E1-E2, Vt is obtained
by calculating the di erence between local Vt and that at E1 on the upstream side of the
discontinuity, and the change in Alfv?en velocity V0At is calculated similarly [Paschmann
et al., 1986]. The blue, green and red hexagrams represent the change ofx, y andz component
of electron tangential ow velocity versus the change of V0Atx, V0Aty and V0Atz, respectively.
The star-shaped markers show the data for ions. Linear tting based on the method of least
squares is performed to obtain the Wal en ratio. The red line is the line tting result for
ions while the black line for electrons. Both slopes are positive because the in ow velocity is
61
parallel to the normal component of magnetic eld. The Wal en ratio for the electron uid
is about 0:83, shown as the slope of black line, and for ions the number is 0:80, shown as
the slope of the red line. This rotational discontinuity is not fully developed as expected for
discontinuities not far enough from the X line [Lin, 2001]. As a result of our simulation,
the perturbation in By is small and the slopes of ions and electrons are not well separated
as predicted by the two- uid theory [Wu and Lee, 2000].
The bottom right plot of Figure 4.7 shows the ion velocity distributions at location
D5 in the magnetopause boundary layer. The main population of ions have a positive
parallel velocity with a fairly clear cut along a constant minimum vik, and the distributions
in the vik-vi?1 plane appears to be a D-shaped distribution [Cowley, 1982; Fuselier et al.,
1991]. A small fraction (5:2%) of ions, which possess near zero average velocities, are the
cold ions initially loaded in the magnetosphere. In contrast, no clearly D-shaped velocity
distributions of transmitted magnetosheath ions are found at locations around the FTEs in
case 1. Along the path E1-E2, the perpendicular ion temperature T? is larger than the
parallel ion temperature Tk on the magnetosheath side while Tk again, increases signi cantly
in the boudary layer near D5.
4.2.5 Summary
The main results of this 3-D self-consistent global scale hybrid simulation for cases under
a steady, purely southward IMF are summarized below.
1. As a result of magnetic reconnection, magnetic eld line con guration in the case
with a 15 dipole tilt angle exhibits multiple reconnection sites around the equator and mid-
latitude. Flux ropes form spontaneously in between multiple X lines of nite length, which
are able to generate clear bipolar signatures of the local normal magnetic eld, which has
been used to identify FTEs in observations. Around the noon meridian plane, contours of
the By component show a nearly quadrupole Hall signature near FTEs due to ion kinetic
e ects. There usually appears an ion density enhancement of plasma core inside the ux
62
ropes, leading to a lamentary density structure along the reconnected ux tube. Heating
and multiple beams of magnetosheath ions are found inside FTEs.
2. Four types of topologies of reconnected magnetic eld lines (magnetosphere-to-IMF,
IMF-to magnetosphere, IMF-to-IMF, magnetosphere-to-magnetosphere) are obtained in the
simulation, which can be explained by combinations of patchy single reconnection and mul-
tiple X-line reconnection.
3. In the case in which the dipole tilt angle is 0 , single X line reconnection coexists with
multiple-X-line reconnection, while the single X-line process produces 1-D like structures.
A Wa len test is performed to con rm the existence of a rotational discontinuity, which is
expected for a quasi-steady like reconnection. A D-shaped ion velocity distribution with a
cuto at minimum vik is obtained, whereas no clear D-shaped distributions are developed in
the region trailing an FTE closely.
Finally, it should be noted that the scale length 0 used in our simulation is about
6 times larger than that in reality. The solar wind convection speed in the simulation is
thus 6 times faster than the typical value in reality due to the larger Al fen speed used in
the simulation. The larger convection speed is expected to lead to a faster magnetic ux
removal and a shorter reoccurrence period of FTEs at the magnetopause. In the simulation,
the average time of reoccurrence of FTEs at the subsolar region is found to be 60 10 ,
where 10 1s is the ion gyroperiod in the solar wind. Considering the 6 times di erence of
the convective from reality, the recurrence period of FTEs is estimated to be 6 60s 6min,
which is comparable to that inferred from magnetosphere observations.
63
Chapter 5
Global Hybrid Simulation of Dayside Magnetic Reconnection Under a Purely Southward
IMF: Cusp Precipitating Ions Associated With Magnetopause Reconnection
5.1 Introduction
The dayside cusp is the region where downward precipitating ions re ect magnetic
reconnection at the magnetopause. The precipitating cusp particles as observed by satellites
inside the magnetosphere often show an energy-latitude dispersion, i.e, decreasing energy
with increasing latitude [Rosenbauer et al., 1975; Rei et al., 1977]. The observed, dispersive
structure exhibits ?stepped? ion signatures with variations in ux levels and sudden changes
in the energy [Newell et al., 1991]. Two-dimensional models have associated the observed
dispersive energy spectra [Onsager et al., 1993; Lockwood et al., 1994] with ongoing magnetic
reconnection at the magnetopause. And statistical research shows that the cusp precipitation
depends on the magnetopause merging rate in both quantitative and qualitative ways [Newell
et al., 2007]. Although researchers have made a lot of e orts to understand them, in some
cases ambiguity cannot be eliminated in the interpretation of observed energy spectra of
precipitating ions because of limited spatial coverage of spacecrafts [Onsager et al., 1995;
Trattner et al., 2007]. Thus, a simulation study will provides a new perspective as it shows
direct and clear connection between the large scale magnetic eld con guration and local
structures of the cusp ion signatures due to magnetopause reconnection.
In Chapter 4, we presented a numerical simulation case of dayside magnetic reconnection
during southward interplanetary magnetic eld (IMF) with the dipole tilt angle of 15 ,
focusing on the magnetic con guration and evolution of FTEs. A 3-D global hybrid model
was utilized in the simulation, in which the fully-kinetic ion physics is solved in the self-
consistent electromagnetic eld. In this chapter, we analyze the precipitating ions in the
64
cusp region from that simulation. In section 2, we brie y describe the simulation model, and
the results are presented in section 3, followed by a summary in section 4.
5.2 Simulation Results
5.2.1 Reconnection Events and Spatial Energy Spectrum
Although the frequency of FTEs generation in the northern hemisphere are not exactly
the same as that in the southern hemisphere, it is found that the 15 degree sunward dipole
tilt does not introduce qualitative interhemispheric asymmetry. In this chapter, we continue
to focus on the magnetic reconnection in the northern hemisphere. As the solar wind ions
convect carrying the IMF, the bow shock, magnetosheath, and magnetopause gradually
form approximately t > 10 in a self-consistent manner. The reconnection events discussed
in Chapter 4 as well as in this chapter are identi ed by the connectivity change of magnetic
eld lines.
The upper panel of Figure 5.1 is basically part of Figure 4.3 in Chapter 4, which illus-
trates the magnetic eld con guration around the magnetopause in the northern hemisphere
before t = 40. The left 3-D plot shows the magnetic eld con guration at t = 15. The
black lines are magnetic eld lines. Note that the magnetic eld lines are approximately
symmetric about the noon meridian plane. There are two X-line segments shown as in blue
dots. Note that other supporting evidence of X lines include the existence of opposite ow
jets and the quadrupole magnetic eld structure in the guide eld, which is discussed in
detail as in Chapter 4. Under the purely southward IMF, X line segments are approximately
parallel to the equatorial plane along the dayside magnetopause.
In addition to the two X lines illustrated by blue dots, there is another X line below all
the FTEs in the 3-D gure and illustrated by red line segments. The reconnection events
that we will discuss in this chapter are associated with these three X lines.
The contours in the upper panel of Figure 5.1 show Vz, the z component of ion bulk ow
velocity in the noon-midnight meridional plane obtained from case 1. Although two opposite
65
Figure 5.1: (Upper) The 3-D plot in the left shows reconnection X lines in blue dots and red
line segment at t = 15 with ion bulk ow contours in the noon meridian plane. The contours
in the right show By component in the noon meridian planes and projected magnetic eld
lines at t = 15;25;35, respectively. (Lower) Spatial energy spectrum of cusp precipitating
ions in the logarithmic scale showing a dispersive feature. The black, white, and red dots
indicate energies of some typical ions at low-cuto energies of parts A, B, and C, respectively,
which are related to three reconnection events.
66
jets are expected from an X line, the southward jet at t = 15 appears missing as seen in the
gure for the ux ropes in the middle, due to the the non-zero northward background Vz of
the ambient magnetosheath plasmas. The ux ropes that can generate the bipolar signature
of FTEs are of an O-line that requires two X lines. The two X lines seen in the 3-D plot of
Figure 5.1 move northward as well as the FTEs, which are illustrated by the looped black
eld lines, as time proceeds.
The three contour plots in the upper panel of Figure 5.1 correspond to By components
in the noon meridian plane at t = 15;25;35. The black lines are projected eld lines. The
boundary region with a dramatic eld direction change is the magnetopause. The arrows
indicate that the same FTE at the lowest latitude in the 3-D gure of Figure 5.1 is moving
northward.
The lower panel of Figure 5.1 shows the typical energy ux spectrum of the precipitating
transmitted magnetosheath ions obtained from case 1 during the magnetopause reconnection,
where the latitudes are related to the spherical polar angle. Near eld-aligned particles with
pitch angle less than 10:0 are shown for r = 7:5 RE. The critical pitch angle is chosen so
that it is as close as possible to zero degree while the recorded spectrum includes enough
particles. The energy gaps at constant latitudes in the energy spectrum are due to the
limited number of ?particles? at the position where ion uxes are recorded.
It is found that three parts of the spectrum, A, B, and C, shown in the lower panel of
Figure 5.1, are related to the entrance areas of the cusp precipitating particles associated
with the three reconnection X lines on the magnetopause, as indicated with three lines along
the low-energy cuto , i.e., the minimum energy at each latitude in the spatial spectrum. To
interpret observation data, the low-velocity cuto has been used to infer the reconnection
sites on the magnetopause [Trattner et al., 2007] by tracing particles at the low-energy cuto
back along the magnetic eld lines. Overall, the resulting particle spectrum in each of the
three parts roughly replicates the dispersive feature in spacecraft data under similar IMF
conditions [Onsager et al., 1995] that shows higher energy particles at lower latitudes and
67
lower energy particles at higher latitudes, under IMF conditions similar to the ones in the
case shown here. Notice the overlap and ?step? between the low-energy cuto parts B and C
in the spectrum, between latitudes 60 - 70 .
5.3 Precipitating Ions at Low-energy Cuto
In this subsection, we will explain how and where precipitating ions at low-energy cuto
are transmitted from the solar wind into the magnetosphere via direct magnetic reconnection
related to the three X lines shown in Figure 5.1, by tracing the trajectory of the ions. One
reconnection region is associated with the X line shown in red line segments, which we will
refer to as region A as it is the source region of the ions at the low-energy cuto in part A
of the spectrum we show in Figure 5.1. Similarly, ions at the low-energy cuto in part B
of the spectrum come from the reconnection region associated with the X line in the blue
dots at middle latitudes in Figure 5.1. This region is de ned as region B. As for ions at the
low-energy cuto in part C of the spectrum, they are found to be from reconnection region
associated with the X line in blue dots at high latitudes in Figure 5.1, which is de ned as
region C.
Figure 5.2 shows the total speed, not the parallel speed only, of precipitating ions as a
function of time for typical precipitating ions associated with the reconnection in region A
(red curve), region B (orange curve), and region C (blue curve). As indicated by the y axis
label, the speed is normalized to the magnitude of Al ven velocity in the solar wind jVA0j.
In the following subsections, we will describe magnetic reconnection in each region as
well as their impact on the spatial spectrum shown in Figure 5.1, and explain in Figure
5.2 how the particles at low-energy cuto accelerate and decelerate during the reconnection
process.
68
Figure 5.2: Particle speed as a function of time for typical ions associated with reconnection
A, B, and C, shown by the red, orange, and blue curves.
69
Figure 5.3: The trajectories in colored tubes of particles at low-energy cuto from region A
and the magnetic eld con guration in black lines at t = 5 (upper left), 25 (upper right), 35
(bottom left) and 40 (bottom right) in the GSM system. Axis direction is shown in the left
of either row. The trajectories are color-coded with their current kinetic energy. Also shown
are the contours of ion density N
70
5.3.1 Precipitating Ions at Low-energy Cuto Associated with Region A
Figure 5.3 shows in the GSM system particle trajectories of the two particles at low-
energy cuto from reconnection in region A, which possess the highest nal energies at
t = 40 and reach the lowest latitudes of 47:5 and 50 at t = 40. The trajectories are
colored tubes coded logarithmically with the kinetic energies of the particles at each time
step. The starting positions of particles are marked with a star. The current positions of
particles are marked by colored balls along the trajectories while the black balls indicate
the nal positions. The kink of a particle trajectory that is the closest to the nal position,
marked with a yellow pyramid, is where the corresponding particle enters the magnetopause
due to magnetic reconnection. The entry region of these particles is well localized near the
noon-midnight meridian. The black eld lines illustrate magnetic eld con guration near
the two particles at t = 5;25;35 and 40. The contours show ion density in the noon meridian
and equatorial planes.
The particle ending at the latitude of 47:5 has a nal kinetic energy of 64:3, while that
ending at the latitude of 50 has a nal kinetic energy of 32:4. Note that an ion possessing a
speed ofjVA0jhas kinetic energy of 0:5 in the simulation units. Particles at low-energy cuto
tend to be at locations close to the X lines when magnetic reconnection occurs because a
short distance of ight compensates a low parallel speed.
At t = 5, both particles are on open eld lines, shown in the upper left plot of Figure
5.3. The red curve in Figure 5.2 corresponds the ion at the low-energy cuto that ends at
the latitude of 47:5 in Figure 5.3, has a speed of vp 4:5 around t = 5 before entering the
magnetopause boundary layer. In the upper left plot of Figure 5.3, the trajectory of the ?red
curve? particle has been marked with a black arrow.
At t?11, the ?red curve? particle undergoes gyro-motion plus drift motion as seen from
the wiggles in Figure 5.2. It is slightly accelerated to vp 7 at t?16 when being trapped
in the ux rope shown in the upper right plot of Figure 5.3. The ux rope forms between
71
the two X lines we mentioned previously. Note that the two X lines have moved northward,
compared to their location at t = 15 shown in the upper panel of Figure 5.1.
At t ? 35 and later, the ?red curve? particle is gradually accelerated to vp 12:5 at
t ? 35 as seen in Figure 5.2 when passing the entry point on the magnetopause marked
by the yellow pyramid in Figure 5.3. The entry point is near the X line that is previously
shown as red line segments. The speed is increased by nearly twice the local Alfv en speed
while reconnected eld lines convect with an average speed of 0:9 local Alfv en speed
of 1:7, as expected for acceleration in Alfv en type large-amplitude waves in magnetic
reconnection [Lin and Lee, 1994]. Over all, near the entry point at the magnetopause, the
particles are accelerated, while deceleration occurs when they come into the magnetosphere.
The acceleration in the FTE is caused by the E B motion, where the reconnection electric
eld E = Ve B is nearly in the +y direction. Note that ions become magnetized again
after they leave the reconnection sites.
The particle then remains at nearly the same speed while traveling in the boundary
layer toward high latitudes until it is slightly decelerated to vp 11:4 at t = 40 in the
magnetosphere earthward of the kinked eld line region of the boundary layer. Because the
ion picks up speed mainly in the direction of E B force, which is perpendicular to the
magnetic eld, it is decelerated when the eld line convection speed (plasma perpendicular
speed) drops in the cusp at the nal position of the ion.
The average speed of this particle after the entry point at t = 34 is about 12.3, which is
larger than an average speed of 9.3 for the other particle at the low-energy cuto and ending
at the latitude of 50 . The particle ending at the latitude of 50 enters the magnetopause
at t? 31, a slightly earlier time than the particle ending latitude 47:5 . A longer time of
ight for the slower particle compensates an energy di erence of 32.4, which is consistent
with the time of ight or velocity lter e ect suggested by Shelley et al. [1976] and Rei
et al. [1977]. Note that only particles moving with a positive parallel velocity relative to the
poleward convecting eld lines can enter the magnetosphere [Cowley, 1982], which should
72
lead to a D-shaped ion velocity distribution of the transmitted magnetosheath ions if the
eld line convection speed is nearly a constant, as in quasi-steady reconnection [Tan et al.,
2010].
5.3.2 Precipitating Ions at Low-energy Cuto Associated with Region B
In this subsection, we discuss the precipitating ions at low-energy cuto associated with
region B, of which the trajectories are shown in Figure 5.4. The format of Figure 5.4 is
similar with Figure 5.3, except that the trajectories are color coded with the nal energy of
each particle. Particles precipitating at these latitudes have been accelerated at the middle
latitudes in the pre-noon ank of the magnetopause. The ion ending at a latitude of 60
and associated with reconnection B is used as an example in Figure 5.4 to illustrate how the
precipitating ions at low-energy cuto associated with region B enter the magnetopause and
reach their nal position, of which the speed is shown as the ?orange curve?. In the upper
left plot of Figure 5.4, the trajectory of the ?orange curve? particle has been marked with a
black arrow.
At t = 5, all the particles are on open eld lines, shown in the upper left plot of Figure
5.4. As in Figure 5.2, the ?orange curve? ion has an initial speed about the same as that of the
?red curve? particle from region A. Di erent from the ?red curve? particle, the ?orange curve?
particle is originally moving tailward with the magnetosheath bulk ow but then dragged
sunward by a reconnected eld line on the ank side of the magnetopause, resulting in a
deceleration at t?10, as shown in Figure 5.3.
At t = 15, one of the particles from region B has passed the entry point and been inside
the magnetopause as seen in the upper right plot of Figure 5.4. The ?orange curve? particle
enters the magnetopause at t? 25 and is then accelerated by a much smaller factor than
that of region A due to the weaker kink in the eld line and thicker boundary in the higher
latitude, reaching only vp 6 around t? 31, as shown in Figure 5.2. The motion is then
followed by a deceleration in the magnetosphere when the particle loses the perpendicular
73
Figure 5.4: The trajectories in colored tubes of particles at low-energy cuto from region B
and the magnetic eld con guration in black lines at t = 5 (upper left), 25 (upper right),
35 (bottom left) and 40 (bottom right) in the GSM system. Axis direction is shown in
the left of either row. The trajectories are color-coded with their nal energy. Starting
from 52:5 , particles at low-energy latitude in the part B of spectrum have nal energies of
12:37;4:65;2:47;1:47 and 0:82, respectively. Also shown are the contours of ion density N.
74
speed due to a reduced eld line convection speed. It is also found that the site in the cusp
where the number of precipitating particles peaks is correlated with region B.
Over all, the mechanism for part B of the dispersive spectrum appears to be quite
di erent from that for part A. The particles at low-energy cuto enter from region B with
a smaller average speed of 4:4, compared to the average speed of 10:8 for region A. The
standard deviation of their average energy after entering is less than 5:0, compared with the
energy range of 32:4 for region A as we mentioned earlier. Thus, the energy di erence is
not enough to account for the latitudinal separation in the corresponding cusp spectrum.
Rather, particles ending at di erent latitudes over the spectral range enter the magnetosphere
at di erent times from t = 15 to t = 31, as the X line moves poleward and sunward and
thus the entry points of particles appear to have shifted accordingly. Note that the X line
corresponding to region A is located near the equator, of which the poleward movement is
less signi cant than that for the X line of region B.
Particles ending at lower latitudes enter the magnetopause at later times so that by
t = 40 the corresponding eld lines have convected shorter latitudinal distances. The energies
of these particles, therefore, must be higher in order to reach the same cusp altitude within
the shorter time intervals. Moreover, the foot point of the latest-reconnected eld line shifts
equatorward with time due to the erosion of magnetospheric eld lines within one generation
quasi-period of FTEs at the low latitudes under the purely southward IMf as we will show
in Figure 5.5, which also contributes to the appearance of higher low-energy cuto at lower
latitudes.
5.3.3 Precipitating Ions at Low-energy Cuto Associated with Region C
Particles associated with reconnection region C have entry points localized in the pre-
noon ank of the high-latitude magnetopause. The trajectories for the precipitating ions at
the low-energy cuto from region C are not shown because they are similar with that from
75
region B while the typical speed change is shown in Figure 5.2 as the blue curve, which
corresponds to the particle that reaches a nal position at a latitude of 70 .
Similar to the ?orange curve? particle for region B, the ?blue curve? particle also undergoes
a deceleration and then acceleration in the magnetopause, followed by a nal deceleration
in the magnetosphere. Since the particle enters the magnetosphere from a higher latitude,
it is accelerated in an even wider layer at the magnetopause. The particles in part C of
the spectrum are around the highest latitudes in the cusp. The eld line convection speed
(perpendicular plasma speed) becomes small in the northern cusp at the nal position of the
region C ions, where the radial distance is smaller and the eld lines are close to 90 degree
of geomagnetic latitude, compared to that at the dayside subsolar region. Thus the energy
of the ?blue? particle thus quickly drops back to its original value before the acceleration
as the acceleration are mainly perpendicular to the magnetic eld due to E B motion.
The fact that three part of the ion spectrum are associated with the three independent X
line segments, as in our discussion for Figure 5.1 previously, explains the independency of
each part of the spectrum. During the time dependent reconnection, di erent locations and
extensions of the X line segment cause "energy plateaus" and "energy step" between the
three parts in the spectrum shown in the lower panel of Figure 5.1.
5.3.4 Energy Flux due to Precipitating Ions as a Function of Time
For satellite observations, energy spectrum is a combination of both spatial and temporal
e ects. In this section, we focus on the impact of temporal e ect on energy spectrum of the
cusp precipitation. The upper panel of Figure 5.5 plots energy spectrum as a function of
time while our virtual satellite is stationed at the xed location of r = 7:5 RE and latitude
of 52:5 in the noon meridian plane. The y axis is the particle energy on a logarithmic scale,
while the color represents the energy ux levels, also on a logarithmic scale. The reoccurred
of precipitating ions and the trend of the low-energy cuto in the spectrum are found to be
associated with the oscillation of the open/close eld line boundary.
76
Figure 5.5: (Top) Particle energy spectrum in the logarithmic scale as a function of time at
a xed position of r = 7:5 RE and latitude of 52:5 . (Bottom) latitudinal position of the
dayside open/close eld boundary, at r = 7:5 RE, as a function of time.
The lower panel of Figure 5.5 plots the latitudinal position of the dayside open/close
eld line boundary in the noon meridian plane at r = 7:5 as a function of time, from t = 0 to
t = 80. The open/closed eld line boundary is determined by the reconnection at the lowest
latitude (region A) and oscillates with time due to the time variation of the reconnection rate
there. Our analysis focuses on the spectrum after t = 15 because the spectrum collected in
the ?cusp? at earlier stage might be associated with the initialization. From t = 15 to t = 40,
the probe location is well inside the cusp while the open/close eld boundary is located at
41 latitude, and a continuous temporal spectrum of ion precipitation is observed. When
the open/close eld boundary of the low-latitude cusp moves to a higher latitude around
t = 50, the ion precipitation becomes scarce and then disappears at the speci c location.
This is because as the cusp moved poleward of the stationary "virtual satellite" so that no
precipitating ions are recorded at that location following the poleward motion of the cusp.
Later at t = 70, the eld-aligned ion ux recurs as the open/close eld boundary retreats to
lower latitudes again.
The low-energy cuto at t> 15, after the formation of the bow shock, magnetosheath,
and magnetopause, clearly trends down in the temporal spectrum when the open/closed eld
boundary of the cusp moves to lower latitudes, and up when the boundary moves to higher
77
latitudes. Our simulation shows that as the subsolar reconnection (region A) gets weaker,
which results in a high-latitude shift of the open/closed eld boundary, the magnetopause
reconnection is dominated by events at higher latitudes. As reconnection site moves to the
higher latitudes, it corresponds to a shorter time of ight for particles ending at the speci c
cusp latitude, leading to an increased low-energy cuto of particles.
5.4 Summary
In summary, cusp ion injections associated with magnetopause reconnection are investi-
gated with 3-D hybrid simulation. Under a purely southward IMF, both spatial and temporal
energy spectra of cusp precipitating ions are obtained.
The spatial spectrum of eld-aligned particles at a constant geocentric radius in the
cusp replicates the observed dispersive feature of particles. Multiple parts of the spectrum,
however are found to be associated with multiple reconnection events at the dayside mag-
netopause. Both the multiple X-line reconnection with ux ropes and single-X line type
reconnection events result in similar dispersive ion spectra.
In the temporal energy spectrum recorded at a certain location, the occurrence and
disappearance of ion precipitation re ect the latitudinal oscillation of the open/closed eld
line boundary of the cusp. Data from satellite crossings of the cusp are expected to be a
combination of the spatial and temporal e ects.
The simulations reported here show that the dispersive ion spectra of particles entering
the cusp observed by satellites is more involved than the 2-D pictures based upon time of
ight e ects. It has been shown that (1) the reconnection responsible for the particle entry
is neither steady in time nor localized in space; (2) the particle acceleration does not occur
at a single point in space and time as the particle crosses a thin magnetopause distinguished
by a kink in the eld line, but is a much more involved process; and (3) di erent degrees of
acceleration for particles entering the magnetopause at di erent latitudes also contribute to
the latitudinal energy dispersion.
78
Chapter 6
Dayside Magnetic Reconnection under an IMF with a Finite Guide Field By
6.1 Introduction
It is well accepted that the antiparallel reconnection may occur at or near the locus
at the dayside magnetopause along which magnetosheath and magnetospheric elds are
antiparallel. For purely southward IMF condition, we have shown in Chapter 4 that in
our simulation antiparallel reconnection is the dominant reconnection process at the dayside
magnetopause, where the reconnecting component of the magnetic eld is maximal. Besides
anti-parallel reconnection, the component reconnection (or, component merging) hypothesis
states that that reconnection can occur at the locations where the shear angle of the magnetic
eld across the magnetopause is less than 180 .
Recently, observations pointed to that the component reconnection is a competitive
process for non-zero By conditions. Chandler et al. [1999] presented an reconnection event
where the spacecraft detected D-shaped ion distributions, which has been considered as a
signature of reconnection. In this event, the absence of ions in the antiparallel direction to the
upstream magnetic eld is taken as the evidence that component reconnection was occurring
at a location southward of the spacecraft and equatorward of the Southern Hemisphere cusp
under a northward IMF. But so far, when and where the component reconnection occurs at
the dayside magnetopause is still not completely understood.
To model the location of reconnection under an IMF of a nite By, Moore et al. [2002]
developed a method to calculate the dayside reconnection X line under di erent IMF direc-
tions. Their hypothesis is that reconnection occurs along a locus determined by integrating
the local X line direction away from that region with the largest reconnecting eld mag-
nitude. They developed a description of the magnetospheric and magnetosheath magnetic
79
elds at the magnetopause, compute the angle between the sheath and boundary layer elds,
the reconnecting component, and the orientation of the local reconnection X line everywhere
on the magnetopause. The direction of local X line is the direction normal to which the two
reconnecting elds have equal and opposite component. The last step of Moore?s method is
to integrate the local X line across the magnetopause from a starting point or points where
the reconnecting component is maximal. In this step, to trace the local X line from that
point or points is as one would trace the streamlines of any vector eld.
Moore?s result is shown in Figure 6.1, where the white curve represents the X line on
the dayside Magentopause. Note that the clock angle of the IMF is de ned as the angle
from z direction in the GSM system to the direction of the IMF while the x component of
the IMF is zero. Thus, the purely northward IMF has a clock angle of 0 and an IMF of
Bx0 = 0,By0 = 0:707 and Bz = 0:707 has a clock angle of 135 .
A more recent work regarding to the location of dayside reconnection is done by [Hu
et al., 2009] through a global MHD simulation. The merging line (black solid line) for
di erent IMF clock angles are projected in the 0 planes as in Figure 6.2, where is the
latitude, and 0 is related to (de ned in Chapter 2) by 0 = 90 .
Fiugre 6.1 and Figure 6.2 show that the merging line always pass through the subsolar for
a non-zero IMF By. This indicates that both work support that component reconnection is
the reconnection process under a non-zero IMF By as the magnetic elds across the subsolar
magnetopause are not anti-parallel to each other. For a clock angle of 135 , both Moore
and Hu infer that the merging line is located south to the equator plane in the dawn side
and north to the equator plane in the dusk side. But over all, the merging line inferred by
Moore?s group signi cantly deviated from Hu?s result. Under a northward IMF with the
clock angle of 45 , the merging lines calculated by Moore have a di erent shape, compared
with Hu?s calculation. Under a southward IMF when the clock angle is not 180 , the merging
line are in di erent locations at the ank of the magnetopause in the two models.
80
Figure 6.1: The con guration of magnetic eld just inside the virtual magnetopause (small
black arrows), the reconnecting component magnitude (color scale), and resultant X line
(XL) when integrated away from the point of maximal reconnecting component. Boundary
layer ow (white and black vectors) for eld lines rooted in each hemisphere, on each side of
the XL, as projected on the plane normal to a view from the Sun. Individual plots represent
the results for various interplanetary magnetic eld clock angles according to their labels.
Adapted from [Moore et al., 2002]
81
Figure 6.2: Solid black lines shows magnetic merging line of the compound eld [Hu et al.,
2009], superposed y the Earth?s dipole eld and the IMF of strength 10nT, projected in the
0 planes for several typical IMF clock angles, where 0 stands for the longitude so that
0 = 0 corresponds to the noon meridian plane and for the latitude. By symmetry, the
merging line for a clock angle of 225 , the clock angle in our simulation is expected as the
red dashed line.
Since the location of reconnection under a nite By at the dayside magnetopause is still
an unsolved issue, in this chapter a simulation for a case under a nite By is carried out to
investigate the location of the reconnection. The related spectra of precipitating ions at the
high latitudes are also obtained and compared with that for a purely southward IMF.
6.2 Simulation Results
As the third case (case 3) presented in this thesis, the initial condition of the simulation
is the same as case 1, except that the IMF is Bx0 = 0, By0 = 0:707 and Bz = 0:707. The
corresponding IMF clock angle is 225 .
Figure 6.3 illustrates the typical magnetic eld topology near the dayside magnetopause
under the given IMF condition. The red eld lines are reconnected eld lines. The contours
of ion density are shown in three planes, the equatorial plane, the noon meridian plane
and the meridian plane of 0 = 20 , or = 70 . It is found that component magnetic
reconnection occurs at a location that magnetic eld has a shear angle less than 180 at the
82
dayside magnetopause, indicated by the red arrow. Furthermore, the dominant location of
component reconnection is on the dawn side of the dayside magnetopause (around the plane
of 0 = 20 ) in the north hemisphere, near the red reconnected eld lines. This nding is
qualitatively consistent with the merging line at the subsolar magnetopause found by Hu
et al. [2009] and Moore et al. [2002] under the current IMF clock angle.
To understand the ion dynamics associated with reconnection in case 3, we also com-
puted the energy spectra of precipitating cusp ions as a function of time in di erent planes
of 0 = constant. Figure 6.4 shows the temporal spectra in a format similar to that in the
upper pannel of Figure 5.5 in the two planes of 0 = 20 and 0 = 20 .
As shown in Figure 6.4 in the northern hemisphere, ion precipitation is much heavier
in the plane of 0 = 20 than that of 0 = 20 . We argue that this can be explained with
the location of merging line, which is located in the northern hemisphere in the dawn side
magnetopause ( 0 < 0 ) and in the southern hemisphere in the dusk side magnetopause
( 0> 0 ). The out ow ions from the reconnection sites in one hemisphere are much easier to
reach the cusp in the same hemisphere than the opposite hemisphere, given that the plasma
convection speed increases as the latitude increases in both hemispheres. Compared with
a typical energy spectrum as a function of time in the upper panel of Figure 5.5, the low
energy cut-o of cusp precipitating ions in the By case is generally lower than that in case
1, a case of purely southward IMF.
As for spatial energy spectrum of cusp precipitating ions, we choose a virtual satellite
path atr = 7:5RE andt = 40 in the meridian plane 0 = 20 . As in Chapter 5, precipitating
ions with a pitch less than 10 are used to compute the spectrum in Figure 6.5. It is found
that the ions at the low energy cut-o in the logarithmic scale indicated by a black line may
also exhibit a dispersive feature.
83
Figure 6.3: Illustration of the location of component magnetic reconnection at the dayside
magnetopause when the IMF is Bx0 = 0,By0 = 0:707 and Bz = 0:707 as indicated by
the red arrow. The red eld lines are reconnected eld lines. The eld lines of the Earth?s
dipole and the IMF are shown as black lines. The contours of ion density are shown in three
planes, the equatorial plane, the noon meridian plane and the maridian plane of 0 = 20 ,
or = 70 . The two boundary regions featuring sharp ion density are the bow shock and
magnetopause.
84
Figure 6.4: Typical spatial energy spectrum of cusp precipitating ions in the logarithmic
scale for r = 7:5RE, Latitude = 57:5 and t = 40, recorded in the plane of 0 = 20
(upper) and 0 = 0 (lower)
Figure 6.5: Spatial energy spectrum of cusp precipitating ions of r = 7:5RE and t = 40 in
the plane 0 = 20 under a IMF of Bx0 = 0,By0 = 0:707 and Bz = 0:707. The low
energy cut-o in the logarithmic scale indicated by a black line
85
6.3 Summary
The investigation of the dayside magnetopause reconnection under the IMF of Bx0 =
0,By0 = 0:707 and Bz = 0:707 reveals that the component reconnection is the dominant
reconnection process, especially near the noon meridian plane, which is con rmed by the
location shift of cusp precipitating ions. When the IMF clock angle is larger than 180 , the
heaviest precipitation shifts to the dawn side of the dayside magnetopause in the northern
hemisphere and to the dusk side of the dayside magnetopause in the southern hemisphere.
As for spatial spectra, dispersive feature is also shown in for precipitating ions in the cusp.
86
Chapter 7
Summary and Future Work
In this work, 3-D global hybrid simulations have been carried out to investigate mag-
netic reconnection at the magnetopause, using massively parallel computation. Substantial
visualization package has been developed to illustrate the 3-D results.
(1) For a case under a purely southward IMF and a tilt angle of 15 , it is found that:
a. FTEs are dominant in a case with a tilt angle of 15 degree. In this case, FTEs can
be generated from multiple X line reconnection.
b. There are four types of reconnected eld lines, which is con rmed by a recent
observation.
c. Quadrapole-like signature of By perturbation exists near X lines.
d. There is ion density and parallel ion temperature enhancement inside FTEs. Heating
and multiple beams of magnetosheath ions are found inside FTEs.
(2) For a case under a purely southward IMF and a tilt angle of 0 , it is found that:
a. Single X line reconnection is found to be quasi-steady.
b. Rotational Discontinuities (RDs) is identi ed in quasi-steady reconnection with
Walen relation satis ed in RDs.
c. D-shaped ion velocity distributions exists in single X line reconnection, but not in
FTEs.
d. The presence of D-shaped and RDs is consistent with observations and previous local
hybrid simulations
(3) Energy spectra of cusp precipitating ions are investigated in a case with a purely
southward IMF and a tilt angle of 15 and it is found that:
87
a. Dispersive feature in the low energy cuto for spatial energy spectra consistent with
observations.
b. Di erent parts in spatial spatial spectra could be associated with di erent reconnec-
tion events and time of ight e ect still hold in the 3-D reconnection process, but particle
acceleration does not occur at a single point in space/time as the particle crosses the kinked
eld lines, but is a much more involved process.
c. Di erent degrees of acceleration for particles entering the magnetopause at di erent
latitudes also contribute to the latitudinal energy dispersion.
(4) IMF By e ects:
a. The reconnection under an steady southward IMF of a nite By is dominated by
component reconnection.
b. When the IMF clock angle is larger than 180 , the heaviest precipitation shift to the
dawn side of the dayside magnetopause in the northern hemisphere.
c. Dispersive feature is also shown in spatial spectra for precipitating ions in the cusp.
As seen in the previous chapters, the current work only address the impact of the
uniform IMF conditions to the magnetosphere. There are a still a lot of questions that
demand answers in magnetospheric physics. The following lists a few that we can do.
1. Understand how perturbations in the solar wind such as shocks interacts with the
magnetosphere.
2. Time-dependent structures of FTEs: how FTEs decay, where those trapped particles
go, how to distinguish a newly formed FTE from an old one?
3. Wave behavior: Alfv en waves? compressional waves?
4. Extend the global scale model to the night side of the Earth. This improvement will
enable the model to tackle the problems such as reconnection at the tail, and mechanism of
substorms and etc.
5. In the long run, explore the possibility of new models, or incorporate new physics
into the current model. Couple the current model to the model that describe the ionosphere
88
(inside the current domain) and the model that describe the solar wind (outside the current
domain), which will expand the applications. Ultimately, the goal could be forecasting the
space weather around the Earth as the incoming solar wind varies.
89
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Bibliography
100
Appendices
101
Appendix A
Fundamental Plasma Parameters
Physical Quantity Symbol De nition
Alfv en velocity VA B4 nimi
Debye length D ( kT4 ne2 )0:5
ion gyrofrequency !i ZeB/mic
ion gyroradius ri vTi/!i
ion plasma frequency !pi (4 niZ2e2=mi)12
ion thermal velocity vTi (kTimi )0:5
ion inertial length i c=!pi
magnetic mach number M V/VA
Table A.1: Fundamental plasma parameters, adapted from Huba [2009]. (All quantities are
in Gaussian cgs units except that temperature expressed in eV and ion mass in units of the
proton mass.)
102
Appendix B
Maxwell?s Equations
Name SI Gaussian
Faraday?s law r E = @B@t r E = 1c @B@t
Ampere?s law r H = @D@t + J r H = 1c @D@t + 4 c J
Poisson equation r D = r D = 4
[Absence of magnetic monopoles] r B = 0 r B = 0
Lorentz force on charge q q(E + v B) q(E + 1cv B)
Constitutive relations D = E D = E
B = H B = H
Table B.1: Maxwell?s equations, adapted from Huba [2009].
In a plasma 0 = 4 10 7Hm 1 (Gaussian units: 1). The permittivity satis es
0 = 8:8542 10 12Fm 1 (Gaussian units: 1)
103
Appendix C
Sample Subroutine in MATLAB: GUI of Finetuning Field Line
function MyGui(action) global Field;
global param;
global myplt;
global Handle0 Handle1 Handle2 Handle3;
% Generates GUI menu popup index = get(Handle1.popupmenu, ?Value?);
switch popup index
case 1
% Call the subroutine of Field Line Finetuning
BlineFineTuneGui;
case 2
minimum variance;
case 3
return;
end
104
Appendix D
Sample Subroutine in MATLAB: Finetuning Field Line
function BlineFineTune(step,direction)
global Field;
global param;
global myplt;
global c info;
global V3D HANDLES;
global Handle0 Handle1 Handle2 Handle3;
gure handle=V3D HANDLES. gure handle;
axis handle=V3D HANDLES.axis handle;
handles=V3D HANDLES.handles;
gure( gure handle) ;
gure(handles. gure1);
hold on;
if (myplt.Bline.sx==0 && myplt.Bline.sy==0 && myplt.Bline.sz==0) ,
myplt.Bline.sx=15;
myplt.Bline.sy=0;
myplt.Bline.sz=0;
return;
else
switch direction
case 1
myplt.Bline.sx=myplt.Bline.sx+step;
myplt.Bline.sy=myplt.Bline.sy;
myplt.Bline.sz=myplt.Bline.sz;
case 2
myplt.Bline.sx=myplt.Bline.sx;
myplt.Bline.sy=myplt.Bline.sy+step;
myplt.Bline.sz=myplt.Bline.sz;
case 3
myplt.Bline.sx=myplt.Bline.sx;
myplt.Bline.sy=myplt.Bline.sy;
myplt.Bline.sz=myplt.Bline.sz+step;
end;
end;
105
[n1,n2]=size(myplt.Bline.numexist);
if n2>=1 ;
delete(myplt.Bline.h1(myplt.Bline.numexist(1)));
end;
%[sx, sy, sz]: the starting point where we begin to trace the eld lines.
sx=myplt.Bline.sx;
sy=myplt.Bline.sy;
sz=myplt.Bline.sz;
x=myplt.plot3D.x3;
y=myplt.plot3D.y3;
z=myplt.plot3D.z3;
u=myplt.plot3D.u;
v=myplt.plot3D.v;
w=myplt.plot3D.w;
verts1 = stream3(x,y,z,u,v,w,sx,sy,sz);
verts2 = stream3(x,y,z,-u,-v,-w,sx,sy,sz);
w=myplt.tubewidth;
verts3=[];
aa=verts11,1?;
nn=size(aa);
aa(:,1:nn(1,2))=aa(:,nn(1,2):-1:1);
verts3=[verts3 aa];
verts3=[verts3 verts21,1?];
verts3=verts3?;
nn=size(verts3);
verts = mat2cell(verts3, nn(1,1), nn(1,2));
h1=streamtube(verts,w);
colorBline = str2num(get(Handle1.editHandle77, ?string?));
if(colorBline > 1);
colorBline=1;
end;
color= param.fmin+colorBline*(param.fmax-param.fmin);
newcolor=get(h1,?CData?);
cmin=min(min(newcolor));
cmax=max(max(newcolor));
if(colorBline >= 0);
cnew=newcolor.*0+color;
end;
if(colorBline < 0);
cnew=param.fmin+(newcolor-cmin)/(cmax-cmin)*(param.fmax-param.fmin);
end;
if(colorBline <=-1);
cnew=ones([size(newcolor) 3]);
value = get(Handle1.popupmenuHandle7, ?Value?);
colorselect = get(Handle1.popupmenuHandle7, ?String?);
106
colorselect = colorselectvalue;
switch colorselect;
case ?red?, cnew(:,:,1) = 1; cnew(:,:,2) = 0; cnew(:,:,3) = 0;
% [1 0 0]; red
case ?green?, cnew(:,:,1) = 0; cnew(:,:,2) = 1; cnew(:,:,3) = 0;
% [0 1 0]; green
case ?blue?, cnew(:,:,1) = 0; cnew(:,:,2) = 0; cnew(:,:,3) = 1;
% [0 0 1]; blue
case ?yellow?, cnew(:,:,1) = 1; cnew(:,:,2) = 1; cnew(:,:,3) = 0;
% [1 1 0]; yellow
case ?cyan?, cnew(:,:,1) = 0; cnew(:,:,2) = 1; cnew(:,:,3) = 1;
% [0 1 1]; cyan
case ?magenta?, cnew(:,:,1) = 1; cnew(:,:,2) = 0; cnew(:,:,3) = 1;
% = [1 0 1]; magenta
case ?pink?, cnew(:,:,1) = 1; cnew(:,:,2) = 0; cnew(:,:,3) = 0.5;
% = [1 0 .5]; pink
case ?orange?, cnew(:,:,1) = 1; cnew(:,:,2) = 0.5; cnew(:,:,3) = 0;
% [1 .5 0]; orange
case ?lime green?, cnew(:,:,1) = 0.5; cnew(:,:,2) = 1; cnew(:,:,3) = 0;
%[.5 1 0]; lime green
case ?aquamarine?, cnew(:,:,1) = 0; cnew(:,:,2) = 1; cnew(:,:,3) = 0.5;
%[0 1 .5]; aquamarine
case ?sky blue?, cnew(:,:,1) = 0; cnew(:,:,2) = 0.5; cnew(:,:,3) = 1;
%[0 .5 1]; sky blue
case ?violet?, cnew(:,:,1) = 0.5; cnew(:,:,2) = 0; cnew(:,:,3) = 1;
%[.5 0 1]; violet
case ?black?, cnew(:,:,1) = 0; cnew(:,:,2) = 0; cnew(:,:,3) = 0;
%[.5 0 1]; violet
case ?white?, cnew(:,:,1) = 1; cnew(:,:,2) = 1; cnew(:,:,3) = 1;
%[.5 0 1]; violet
end;
end;
set(h1,?CData?,cnew,?FaceColor?,?interp?,...
?EdgeColor?,?none? ,...
?FaceLighting?,?phong?,...
?BackFaceLighting?,?lit?,?visible?,?on?);
h2=h1;
myplt.Bline.h1(myplt.Bline.numexist(1))= h1;
myplt.Bline.h2(myplt.Bline.numexist(1))= h2;
myplt.Bline.start(myplt.Bline.numexist(1),:)=[myplt.Bline.sx myplt.Bline.sy
myplt.Bline.sz];
return;
107
Appendix E
Sample Subroutine in MATLAB: 2-D Contours Plotting
function diag2Dplot global Field;
global param;
global myplt;
global Handle0 Handle1 Handle2 Handle3;
global gure handle;
global V3D HANDLES;
streamline option=1;
movie2d=struct(?cdata?,[],?colormap?,[]);;
for jj=1:param.nt
titlestring= get(Handle1.popupmenuHandle2, ?Value?);
if param.nt==1
a le=param. lenames
else
a le=param. lenames(1,jj) ;
end
a lename=[param.pathname,char(a le)];
plotName=char(a le);
if(jj = 1);
%case ?checkDATAbox?
value= get(Handle1.checkboxHandle1, ?Value?);
if value ==1
param.dataType =1;
set(Handle1.checkboxHandle2, ?Value?, [0]);
set(Handle1.checkboxHandle3, ?Value?, [0]);
%read Boxdata text le(a lename)
read hy3d box data(0,jj)
end
%case ?checkDATAcov?
value= get(Handle1.checkboxHandle3, ?Value?);
if value ==1
param.dataType =3;
set(Handle1.checkboxHandle1, ?Value?, [0]);
set(Handle1.checkboxHandle2, ?Value?, [0]);
read hy3d convert data(a lename)
108
end
end;
reset plot3D p Handle2. gureHandle1= gure( ...
?Name?,?2?, ?NumberTitle?,?o ?, ...
?Visible?,?o ?, ?BackingStore?,?o ?);
gure(Handle2. gureHandle1);
set(Handle2. gureHandle1,?Units?,?centimeters?,...
?PaperPositionMode?,?Auto?,?Position?,[2, 5 ,12, 12]);
reset 2d param
if streamline option==1;
x=myplt.plot3D.x3;
y=myplt.plot3D.y3;
z=myplt.plot3D.z3;
u=myplt.plot3D.u;
v=myplt.plot3D.v;
w=myplt.plot3D.w;
daspect([1 1 1])
my streamslice=streamslice(x,y,z,u,v,w,[],[0],[],0.25);
set(my streamslice,?Color?,?k?);
hold all;
my slice=slice(x,y,z,myplt.plot3D.p,[],[0],[]);
caxis([-5 5]);
set(my slice, ?EdgeColor?,?None?,?FaceColor?,?Interp?)
temp= ndobj(gca,?type?,?surface?);
my x=get(temp(1,1),?xdata?);
my y=get(temp(1,1),?ydata?);
my z=get(temp(1,1),?zdata?);
my c=get(temp(1,1),?cdata?);
my x=interp2(my x,3);
my y=interp2(my y,3);
my z=interp2(my z,3);
my c=interp2(my c,3);
set(temp(1,1),?xdata?,my x,?ydata?,my y,?zdata?,my z,...
?cdata?,my c);
this axis= ndobj(gca,?type?,?axes?);
xlim(this axis,[7.5 10.5]);
zlim(this axis,[0 6]);
view(0,0);
set(gcf,?color?,[1,1,1])
set(gcf,?outerposition?,[1, 3 ,10.9,10.9]);
set(gca,?FontWeight?,?bold?,?FontSize?,14,...
?FontName?,?Arial?,...
?FontAngle?,?italic?);
zlabel(?Z?,?FontWeight?,?bold?,?FontSize?,14,...
?FontName?,?Arial?,?FontAngle?,?italic?);
109
xlabel(?X?,?FontWeight?,?bold?,?FontSize?,14,...
?FontName?,?Arial?,?FontAngle?,?italic?);
colorbar(?location?, ?eastoutside?);
colorbar(?delete?);
else
colormap jet; contourf(myplt.plot2D.xaix,myplt.plot2D.yaix,...
myplt.plot2D.p,20,?LineColor?,?none?);
caxis([-1 1]);
colorbar;
shading at;
title(titlestring);
end
end
lename=?movie2d.avi?;
d=fopen( lename,?w?);
if d < 0
warndlg([?Selected le, "?, lename, ...
? ", cannot be created or exists alreadey ...
and is used by another program.?]);
else
movie2avi(movie2d, lename,?fps?,4,...
?compression?,?none?,?quality?,100);
try
fclose( d);
catch;
end;
end;
% to add black regtangular option=0;
option1 tag=25;
if option==1,
switch option1 tag
case 15
width=1;
length=2;
angle=15;
rectangle xi=[0 width width 0 0]
rectangle zi=[0 0 length length 0]
rectangle xf=cos(angle*pi/180).*rectangle xi...
-sin(angle*pi/180).*rectangle zi
rectangle zf=sin(angle*pi/180).*rectangle xi...
+rectangle zi*cos(angle*pi/180)
line(rectangle xf+9.5,rectangle zi*0,...
rectangle zf,?Linewidth?,2,?Color?,?k?);
case 25
width=1;
110
length=3;
angle=20;
rectangle xi=[0 width width 0 0]
rectangle zi=[0 0 length length 0]
rectangle xf=cos(angle*pi/180).*rectangle xi...
-sin(angle*pi/180).*rectangle zi
rectangle zf=sin(angle*pi/180).*rectangle xi...
+rectangle zi*cos(angle*pi/180)
line(rectangle xf+9.18,rectangle zi*0,...
rectangle zf+1.1,?Linewidth?,2,?Color?,?k?);
case 35
width=1;
length=3;
angle=0;
rectangle xi=[0 width width 0 0]
rectangle zi=[0 0 length length 0]
rectangle xf=cos(angle*pi/180).*rectangle xi...
-sin(angle*pi/180).*rectangle zi
rectangle zf=sin(angle*pi/180).*rectangle xi...
+rectangle zi*cos(angle*pi/180)
line(rectangle xf+9,rectangle zi*0,...
rectangle zf-1.5,?Linewidth?,2,?Color?,?k?);
case 65
width=1;
length=3;
angle=20;
rectangle xi=[0 width width 0 0]
rectangle zi=[0 0 length length 0]
rectangle xf=cos(angle*pi/180).*rectangle xi...
-sin(angle*pi/180).*rectangle zi
rectangle zf=sin(angle*pi/180).*rectangle xi...
+rectangle zi*cos(angle*pi/180)
line(rectangle xf+9.2,rectangle zi*0,...
rectangle zf+1,?Linewidth?,2,?Color?,?k?);
end
end
return
111