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Raytracing

Introduction

Raytracing is one of the first theories of light propagation to ever be developed. A ray, conceptually, represents the flow of power or the path taken by "packet" of electromagnetic energy through a system. Raytracing, more formally, is a reduction of Maxwell's equations to a much simpler and more computationally tractable form, under fairly modest assumptions of smooth variation.

Raytracing has been used to study wave propagation and phenomena in the ionosphere and magnetosphere for over 50 years [Kimura(1966), Haselgrove(1955)]. Initially, the primary motivator was computational tractability. The cost in computing a ray path in the magnetosphere is relatively small compared to the cost of a global solution like that obtained using finite difference or finite element methods. Initial investigations were even carried out graphically [Maeda and Kimura(1956)].

In a modern context, however, raytracing is still a very important computational tool for investigating magnetospheric phenomena. Due to the enormous length scales (many megameters) and timescales (10s of seconds), full simulations of wave propagation over the entire magnetosphere are still difficult if not intractable on modern computers.

Modeling efforts

The Stanford VLF group's raytracing modeling efforts include:
  • Flexible raytracing simulation: The Stanford 3D ray tracer can accommodate any arbitrary magnetic field or plasma density function, opening up the possibility to study the role of large-scale inhomogeneities, like plumes or notches, in the propagation of VLF waves.
  • Modeling of loss and growth processes: Accurate modeling of loss processes, like Landau and cyclotron damping, is essential to accurate interpretation of raytracing results. The Stanford VLF ray tracer can calculate the losses for any arbitrary distribution function distributed over space, including observation-based models.
  • Physically-based and observation-based models: We have integrated the IGRF, Tsyganenko, GCPM, and our own radial diffusion model in our raytracer.

VLF Raytracing demonstrations

The VLF group at Stanford uses a 3D magnetospheric raytracer to study many different wave propagation effects in the inner magnetosphere. Ray paths in HF (as shown in Figure 1) show characteristic total internal reflection off the ionosphere, an effect that is used for long-distance HF communication.

Figure 1: Ray paths (white) taken by an HF source at 15 MHz, showing reflection off the ionosphere. For reference, we also show the log of the electron number density.
Image hfraytracing

At VLF frequencies, however, the phenomena encountered are much richer - rays can reflect off specular boundaries, magnetospherically reflect, or even significantly drift in longitude, as shown in Figures 2 and 3.

Figure 2: Ray path (white) taken by a single VLF ray at 2 kHz using the GCPM plasmasphere model and the IGRF+Tsyganenko-05 magnetic field model, showing multiple magnetospheric reflections. The sun is at the right (positive x direction).
Image vlfraytracing1

Figure 3: Left, poleward view of the path taken by a VLF ray at 150 Hz launched from sunside in the GCPM+IGRF+Tsyganenko-05 model, showing significant longitudinal drift. Right, the same path from a different view. In both views, the sun is to the right (positive x direction).
Image vlfraytracing2

A short movie here illustrates how VLF waves can propagate and reflect in the magnetosphere, in real time:

The panel on the left shows the refractive index surface, which is a plot of the refractive index as a function of direction. The arrow in blue on this plot shows the group velocity, or the direction of the ray. On the right, we show the actual ray path superimposed on a model-based electron density distribution with a very sharp plasmasphere "knee". The red arrow shows the direction of the wavenormal, which in a plasma is not, in general, coincident with the group velocity. A few interesting features can be seen clearly in this visualization. Four of the reflections are termed MR (magnetospheric reflections) and occur because the refractive index surface's topology changes from opened to closed, permitting an abrupt change in the group velocity for a relatively small change in the wavenormal at near-perpendicular propagation. The remaining two are pseudo-specular reflections at the sharp plasmasphere boundary.

Bibliography

Haselgrove(1955)
Haselgrove, J. (1955), Ray Theory and a New Method for Ray Tracing, in Physics of the Ionosphere, pp. 355-+.

Kimura(1966)
Kimura, I. (1966), Effects of Ions on Whistler-Mode Ray Tracing, Radio Science, pp. 269-283.

Maeda and Kimura(1956)
Maeda, K., and I. Kimura (1956), A Theoretical Investigation on the Propagation Path of the Whistling Atmospherics, Rept. Ionosphere Res., pp. 105-123.