Project:

Cosmic-ray modulation studies deal with charged particles in the heliosphere, which is the region of space in which plasma and magnetic field originating at the Sun, dominate interstellar fields and plasma. Cosmic rays are actually mainly charged particles, the largest fraction of which are protons. The name is historic and comes from the early 1900s when scientists became aware of some form of radiation that penetrated heavily isolated containers. The assumption was that it was likely electromagnetic in nature and had to come from somewhere in the cosmos, hence the name cosmic ray.
It is (usually) a simple exercise to calculate the trajectory of a charged particle in a magnetic field that varies smoothly and slowly in time and space. We know however that the heliospheric magnetic field that cosmic rays interact with inside of the heliosphere is anything but smooth. It contains fluctuations at all scales relevant to particle transport. Such a field is referred to as turbulent. Fluctuations in the field are characterized by calculating the power spectral density (energy density) as function of frequency.
Single-spacecraft measurements can only yield frequency spectra since observations are obviously only made at a single point in space. However, diffusion coefficients that appear in the transport equation for cosmic rays require wavenumber spectra. This is further complicated by the structure of the turbulence. A widely-used description is the slab/2D model. In a nutshell, wave vectors for slab turbulence are parallel to the background magnetic field (think Alfvén waves) and those for 2D turbulence, in planes (this is where the 2D comes from) perpendicular to it. Bieber and co-workers devised a method to derive slab- and 2D wavenumber spectra from frequency spectra.
The aim of this project is to repeat their calculations and check the accuracy of their results (up to Eqs. 16 and 17 of the attached paper). It is purely theoretical and requires a sound knowledge of Fourier transforms. Note that if you get exactly the same results that they do, you are doing something wrong.

Requirements for students to address:

Student should have a sound knowledge of Fourier transforms and complex functions in general