WAVE DYNAMICS OF REGULAR AND CHAOTIC RAYS

The method of geometrical optics has become a popular tool in the analysis of short wavelength wave propagation in inhomogeneous plasmas. Recent numerical ray tracing studies for the purposes of radio frequency heating have indicated that in many plasma profiles with two-dimensional nonuniformity the ray trajectories can become chaotic. In addition, other investigations have shown that bound ray systems may also exhibit varying degrees of ergodic ray behavior. Indeed, there is reason to believe that chaotic rays are a characteristic of most wave/ray systems with more than one degree of freedom.;In order to investigate general relationships between waves and rays in chaotic systems, I study the eigenfunctions and spectrum of a simple model, the two-dimensional Helmholtz equation in a stadium boundary, for which the rays are ergodic. Statistical measurements are performed so that the apparent "randomness" of the stadium modes can be quantitatively contrasted with the familiar regularities observed for the modes in a circular boundary (with integrable rays). The local spatial autocorrelation of the eigenfunctions is constructed in order to indirectly test theoretical predictions for the nature of the Wigner distribution corresponding to chaotic waves. A portion of the large-eigenvalue spectrum is computed and reported in an Appendix; the probability distribution of successive level spacings is analyzed and compared with theoretical predictions. The two principal conclusions are: (1) Waves associated with chaotic rays may exhibit randomly situated localized regions of high intensity; (2) The Wigner function for these waves may depart significantly from being uniformly distributed over the surface of constant frequency in the ray phase space.;These results suggest that a phase space representation of a wave (such as the Wigner function) is crucial to the understanding of the correspondence between rays and waves. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author.) UMI.;In this work I concentrate on the relationship between waves and rays, and specifically how this relationship is affected when the rays are chaotic. For the case of well-behaved (integrable) ray trajectories, modern eikonal theory and the Einstein-Brillouin-Keller method of quantization (for normal modes) have provided the correspondence between properties of the waves and certain structures in the ray phase space. These theories and associations fail, however, for the case of nonintegrable or chaotic rays.