Date of Award
Dr. Robert Hamilton
Dr. Steve Hennagin
Dr. Isaac Mwase
Our solar system contains more activity and complexity than can be seen through a telescope. One such "invisible" phenomenon is the solar wind, created by a steady stream of particles blasted away from the sun in all directions. The sun's donut-shaped magnetic field lines channel this stream. Particles moving along the field lines perform an intricate helical dance, with ions winding one way and electrons the other.
The solar wind shapes and is shaped by the magnetic fields of the planets and the sun. If left undisturbed by outside influences, the earth's magnetic field, like the sun's, would resemble a donut surrounding the earth. However, the solar wind constantly streams around the earth, and its pressure compresses the earth's magnetic field in front and elongates it in back, much like a comet's tale. This is known as a planetary bow shock.
Physicists have found that certain types of electromagnetic waves, called Alfvén waves, are formed in planetary bow shocks, near comets, and even within the solar wind itself. The waves, which are nonlinear, travel along the solar wind's magnetic field lines. If the field lines are represented by a stretched rubber band, Alfvén waves are represented by the wave that travels the length of the rubber band when it is plucked.
Since about 1968, scientists have sought a way to describe these waves mathematically. They have tried several equations, one family of which, the Derivative Nonlinear Schrodinger (DNLS) equation and its offshoots, has proven especially useful.
This paper will explain the origin of the DNLS equation in plasma physics, show its relationship to simpler wave equations, discuss the solution to the DNLS equation, and explain the numerical techniques used to find its eigenvalues. Eigenvalues are mathematical constructs which correspond to physically significant obervables, such as a wave's velocity or amplitude. All of these topics will then be brought together to explain the new mechanism for phase steepening that was discovered in the course of the research by examining the DNLS equation's eigenvalues. The type of phase steepening explored by this research may serve to explain why only one of two different types of Alfvén waves have been observed in satellite data.
Granade, Stephen R., "Dynamic Phase Steepening in Alfven Waves" (1995). Honors Theses. 105.