Barbara Thompson's 1995 AGU Spring Meeting Poster

Alfven Wave Interactions

Basic Introduction - What is an Alfven wave?

An Alfven wave is like a wave travelling along a stretched string. The magnetic field line tension is analogous to string tension, and when the magnetic field is "plucked" by a perturbation, the disturbance propagates along the field line. At auroral altitudes, an Alfven wave typically has frequencies near a few Hertz. This corresponds to wavelengths reaching an Earth radius - this long-scale coherence, coupled with the notion that the wave is carried by ions and is capable of transporting significant energy in the form of Poynting flux towards the earth, indicates that Alfven waves may play a significant role in magnetosphere-ionosphere coupling.

Derivation of Wave Equations (sort of)

In addition to a standard Alfven wave, the effect of electron inertia is included (see equation on Electron Inertia page in poster). For conditions in auroral regions, the inertial term reduces to a simple expression for the parallel electric field. The current is related to the vector potential through the magnetic field, resulting in an expression for the parallel electric field in terms of the time derivative of the vector potential. The overall electric field is related to the vector and scalar potentials by Faraday's law (the gradient of the scalar potential plus the time derivative of the vector potential).

The Alfven wave is expressed in terms of its vector and scalar potentials. This is accomplished by treating it as a wave in a dielectric medium, with the dielectric constant and the magnetic permeability expressed in terms of the plasma density and the background magnetic field. Since the parallel electric field due to the electron inertial effect becomes another term depending on the vector potential, the equations continue to depend on the vector and scalar potentials, and the two functions are integrated together.

Dielectric treatment of Alfven waves

The dielectric constant of an Alfven wave is e = 1 + c^2/VA^2, where VA is the Alfven speed and c is the speed of light. This relates the electric field and displacement field, E and D. The magnetic field and magnetic inductance have a similar relationship. After fourier transforming in the direction perpendicular to the geomagnetic field (assuming that most of the variation is along the field lines) and examining the equations, it becomes apparent that a factor u = 1 + Kperp^2lambda^2 exists where the magnetic permeability would be in Maxwell's equations (Kperp is the perpendicular wave number and lambda is the electron inertial length, c/w, where w is the electron plasma frequency.) Since u = 1 + x, where x is the magnetic susceptibility, we conclude that x = Kperp^2lambda^2 is the effective magnetic susceptibility of an inertial Alfven wave. The factor u = 1 + Kperp^2lambda^2, like the dielectric constant, is also a function of the background geomagnetic field and plasma density. As e = 1 + c^2/VA^2 relates the fields E and D, u = 1 + Kperp^2lambda^2 relates B and H.

Because we have expressions for both the magnetic permeability and the dielectric constant, we can solve for the index of refraction: n = sqrt(ue). This is equal to the speed of light divided by the phase speed of an electromagnetic wave in the medium. Solving for the phase speed, we see that electron inertia slows an Alfven wave down by a factor of 1/sqrt(u). Further derivations reveal that u reduced field line tension, and affects the equations for Poynting flux and diffusion.

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Other Sections

System properties and physical behavior

Wave integration - fluid motion

Electron integration - particle motion

Electron effects and commonly observed distributions

Conserving energy - numerical scheme

Chaotic behavior of system

References - please see poster