I would like to acknowledge financial support in the form of scholarships, summer scholarships. In Part II, series expansions of time-dependent solutions are presented for the magnetic piston problem, which describes waves propagating in a cold collisionless quasi-neutral plasma arising from magnetic perturbations at the plasma boundary. We assume that the trajectories for electrons and ions in the shock frame have a non-overlapping loop with neighboring loops.
Quasi-neutrality of the plasma requires that loops in the electron orbits must be accompanied by corresponding loops in the ionic orbits. So in the region x < x 1 there is only one current for each fluid, henceforth signed 1. We limit our attention to the case where the trajectories lie in the x-y plane, the magnetic field B lies in the z direction and the electric field has x-component E and y-component F.
At the edges of the loop, two streams of ions and electrons have zero velocities in the x direction; therefore their particle density is infinite. In the loop region, it is the time for a particle in the flow k to cross its trajectory.
2 FOR ELECTRONS
PART II
INTRODUCTION
This initial electric field is part of the initial conditions that provide the initial acceleration of the charged particles. The same results can be obtained by formally expanding the solution in a virtual parameter, which serves the sole purpose of combining the various terms in the expansion. For large perturbations, the partial sum of the first few terms of the set is a good approximation of the sum of the set.
In Chapter 4, we extend the method to the general case in which the initial magnetic field is at an arbitrary angle to the direction of perturbation propagation. The initial electric field immediately created by the magnetic disturbance at the plasma boundary initiates the motion of the charged particles from rest. The governing equations of plasma motion are the continuity equations, momentum equations, and Maxwell's equations.
The quasi-neutrality approximation of the plasma makes possible a fluid-equivalent description of the plasma in terms of the mass density p defined by p = p++ p_, the velocity i. Assuming that Pref and Bref are typical values of mass density and magnetic field magnitude in the problem, we can use them as reference values for mass and magnetic density. We use the initial values of mass density and magnetic field magnitude as reference values.
The criterion for the accuracy of the approximation is that the last preserved term should be small compared to the preceding terms.
THE GENERAL CASE
When the magnetic field has a nonzero longitudinal component, the transverse component cannot be of plane polarization. The same results using this iteration scheme can be obtained by substituting the following formal expansions. This dummy parameter only serves in the grouping of various terms in the expansions and finally us.
We conclude that the break will occur approximately at the first zero of the following expression. Saffman, 1961 Propagation of a solitary wave along a magnetic field in a cold collisionless plasma, Journal of Fluid Mechanics 11, 16. Ferraro, 1956 Hydromagnetic waves in a rarefied ionizing gas and galactic magnetic fields, Proceedings of the Royal London.
Allen, 1958 Large Amplitude Hydromagnetic Disturbances in a Plasma, Proceedings of the United Nations Second International Conference on the Peaceful Uses of Atomic Energy. Rossow, 1965 Graphical results for large-amplitude unsteady one-dimensional waves in magnetized collisionless plasmas of discrete structure, National Aeronautics and Space Administration Technical Note D-2536.
PART III
I N TRODUCTION
Within the limits of linear theory, two waveforms exist in a cold quasi-neutral plasma: slow waves and fast waves. characterized by different distribution relationships. However, when they are in the same mode and their frequencies differ only slightly, the coupling between them can be strong enough to cause a large energy transfer from the main wave to the disturbance. waves resulting in instability of the main wave. In Chapter 2 we show that waves with a permanent shape in the Eulerian coordinates also appear as waves with a permanent shape in the Lagrangian coordinates and vice versa.
It is easier to perform the calculations in the Lagrangian coordinates, because in those coordinates the equations of plasma motion can be reduced to those for the inverse mass density and the magnetic field only. The contact transformation applied between the Eulerian coordinates and the Lagrangian coordinates, together with the continuity equation, gives such a kinematic property that waves with a permanent shape in the Eulerian coordinates also appear as waves with a permanent shape in the Lagrangian coordinates and vice versa. Suppose a wave with a permanent shape in the Eulerian coordinates is described in terms of a phase variable. where W and K' are the frequency and wavenumber in the Eulerian coordinates.
The continuity equation in Eulerian coordinates. in which U is the value of u at the place where p is equ•. 2), d'se= 0 gives the phase velocity in Eulerian coordinates. Conversely, we assume that a standing wave in Lagrangian coordinates is described by a phase variable. Thus, the phase velocities in the two coordinates differ by U which can be identified as the plasma flow velocity.
If we identify the frequency in Lagrangian coordinates with the frequency in Euler coordinates. We will show that the equations of motion of the plasma also allow a solution in the form of Stokes expansion in amplitude, which represents a periodic wave propagating at an oblique angle to the magnetic field. As reference values for normalization, we will use stationary values of mass density and magnitude of the magnetic field.
Periodicity of period 2TI excludes the appearance of either cos v¢ or sin v¢ in the solution. The self-interaction of the component of wavenumber kn will produce components of wavenumber whose amplitudes have a magnitude of order O(aN). Equation (27) determines the evolution of the disturbance as time passes from its initial value.
In the case that D is negative, the main wave is stable, the energy flows back and forth bet~ They are the main wave and the disturbance waves, the amplitude of the dis-. It is clear that D is negative when the disturbance is in a mode different from that of the main wave · or when the frequency of the disturbance wave is very different from that of the main Nav, so f is of order 0(~) while f2, .
In mks units, the cutoff lifetime number is approximately equal to -1 divided by the geometric mean of the. We conclude that slow waves are always unstable to disturbances while low frequency fast ~Taves are stable but. When the propagation angle is not very close to 90°, the cutoff frequency is about one fourth of the electron's gyro frequency in the magnetic field.
When the propagation angle is very close to 90°, the cutoff frequency is so high that waves of all frequencies are stable.