# Kinetic investigation of the impulsive penetration of 2D plasma elements into the Earth's magnetosphere

Abstract (Summary)

In this thesis I investigate the dynamics of charged particles and plasma
into non-uniform distributions of the electric and magnetic fields.
In the first part attention is focused on the motion of test particles. The
interaction between particles as well as the perturbations they might produce
to the external charge and current density are neglected. I investigate a distribution
of the magnetic field that depends on only one spatial coordinate, x,
with the Bx component of the magnetic field being equal to zero everywhere,
like in tangential discontinuities. The magnetic vector, B, can rotate across
the discontinuity by an angle ? ?" [00, 1800]. In addition to the B-field distribution
I assumed different distributions of the electric field, E, with Ex = 0.
I have considered three cases: (A) a uniform electric field; (B) a non-uniform
electric field perpendicular everywhere to B and conserving the zero order
drift, and (C) a non-uniform electric field, perpendicular everywhere to B
and conserving the magnetic moment of the drifting particles. The particles
are drifting into these steady state electromagnetic field distributions; their
orbits together with the path of the first order guiding center are integrated
numerically.
The numerical results show that the ”antiparallel” distribution of the
magnetic field (obtained when ? = 1800) with B = 0 at x = 0 does not produce
anomalous acceleration of the test-particle as assumed in some steady
state reconnection models. Although the zero and first order guiding center
approximations diverge where B = 0, the exact equation of motion is not
singular, it can be integrated throughout the integration time. The mathematical
singularity of the approximative solutions does not correspond to
a “true” (physical) singularity of the exact equation of motion. When the
magnetic field is sheared with a non-zero By-component, and B can rotate
with respect to E (case A), the particle orbit is confined into a sheath centered
onto the x-position where B becomes parallel to E. Partial or total
penetration of the test-particle is equally possible, as demonstrated for the
E-field distributions of case B and case C. In case C the distance of penetration
depends on the initial total energy of the test particles. Except for one
of six different configurations considered, the reversal point of Bz does not
correspond to a point of particle acceleration in the direction normal to B
nor is the stopping point of the particle's motion in the direction normal to
B. Indeed, it is the relative orientation between E and B, together with the
vi
total initial energy of the particle that determine the distance of penetration
across the sheared magnetic field distribution. Penetration into the region
of non-uniform magnetic field produces separation of charges. Particles with
the highest energy are deflected the most.
In the second part of the thesis I treat the dynamics of an ”ensemble” of
electrons and protons forming a plasma stream. The plasma flow is spatially
two-dimensional. In this case the plasma ”internal” contribution to the external
fields is evaluated and self-consistently computed. The method adopted
is the kinetic theory approximation of plasma physics instead of one-fluid
magnetohydrodynamic (MHD) approximation or the Particle-In-Cell (PIC)
generally used. Both the ensembles of electrons and protons are described
by their velocity distribution function (VDF) that has to satisfy the Vlasov
equation derived from the general Liouville theorem for a collisionless plasma.
The VDFs are given in terms of the two constants of mechanical motion, the
total energy, H, and one canonical momentum, px. The first adiabatic invariant,
µ - the magnetic moment which is almost conserved when the Alfven
conditions are satisfied, approximates a third constant of motion. I have
found a velocity distribution function that describes a plasma moving in the
Ox direction with a two-dimensional bulk velocity Vx(y, z) depending both on
y and z. The moments of the VDFs of electrons and ions were computed analytically.
The self-consistent electromagnetic potentials are found by solving
the Maxwell equations and the plasma quasineutrality equation. The partial
current densities, jx(y, z), determined by the first order moments of the
VDFs were introduced into Ampere's equation in order to compute Ax(y, z),
the component of the magnetic vector potential. The charge densities of
the component species, q?n?, determined by the zero order moments of the
VDFs have been introduced into the quasineutrality equation, ? q?n? = 0,
from which the distribution of the electric potential, ?(y, z), is computed.
The solutions for the electromagnetic potentials are found numerically.
I have obtained a kinetic model that describes a two-dimensional plasma
stream whose perpendicular bulk velocity varies (or is sheared) both in the
direction normal to the magnetic field (perpendicular shear) and parallel to
the magnetic field (parallel shear ). The parallel shear of velocity has never
been modeled before using kinetic equations. On the other hand the two
dimensional models proposed till now for the dynamics of magnetospheric
plasma did not consider differential (or sheared) plasma motion across magnetic
field lines.
Several kinetic solutions are given for two-dimensional plasma flows and
for different values of asymptotic densities, temperatures and bulk velocity.
The key-feature of these numerical models is the existence of a parallel component
of the electric field, Eparallel. It is shown that the parallel electric field
vii
is sustained by the parallel shear of the perpendicular plasma velocity. The
amplitude of the parallel electric field depends on the value of the magnetic-
field-aligned gradient of the perpendicular plasma velocity and also on the
relative density and temperature of the moving stream with respect to the
background, stagnant plasma. This is a new mechanism to generate parallel
electric fields that adds to the ones already described in the literature and
that are discussed in part 2 of this Thesis.
In the kinetic models presented in the second part I have adopted a set of
plasma densities and temperatures typical for the terrestrial magnetopause
region. A parallel gradient of the density or electronic pressure enhances
the intensity of the parallel electric field. The scale length of the boundary
layer of transition from moving to stagnant regime can be of the order of
the electron Larmor radius (“electron layer”) or the proton Larmor radius
(“proton layer”). The scaling of the boundary layer is determined by the relative
orientation of the magnetic field and the plasma bulk velocity. Eparallel
is stronger in the case of Parallel Sheared Electron Layer than in the case of
Parallel Sheared Proton Layer.
The existence of a parallel component of the electric field invalidates
the MHD approximation. In the case of the two-dimensional plasma flow
studied in this Thesis the MHD convection velocity, UE = E ×B/B2 is not
a satisfying approximation of the plasma bulk velocity, V . I illustrate the
differences between UE (assigned in MHD approximations to a “frozen-in”
motion of B-field lines) and V obtained by the kinetic models described in
part 2. It is shown that the “de-freezing” is produced in those regions where
a non-vanishing parallel electric field component was determined.
The kinetic treatment of the plasma dynamics adopted in this Thesis
evidence kinetic effects disregarded in the one-fluid approximations: finite
Larmor radius effects that are illustrated in Part I and non-MHD parallel
electric fields that are described in Part II. These effects play an important
role in the processes taking place at the magnetopause, the interface region
between the solar wind and the terrestrial magnetosphere.
Bibliographical Information:

Advisor:

School:Université catholique de Louvain

School Location:Belgium

Source Type:Master's Thesis

Keywords:théorie cinétique ecoulement scisaillé du plasma champs électriques parallèles physique de la magnétosphère

ISBN:

Date of Publication:07/05/2004