Integral Equation Approach for Electrostatic Waves in a Dipole Magnetic Field

We provide an integral equation framework for the analysis of electrostatic waves

along the midplane of a dipole magnetic field. The creation of this model starts with

the wave equation for cold plasma fluid model, and then assumes a periodic solution in

time. The wave equation then takes the form of div(K dot grad ϕ), which is then treated

as a boundary value problem. The boundary condition of most relevance is a radiation

boundary condition at a distance away from the magnet. We then compose the solution

of the wave equation as a Fredholm Integral Equation of the Second Kind (FIE2) with a

Green’s Function as its kernel. Two cases of the FIE2 are shown with their numerical

solutions. The first solution includes just the case of there being a magnetic field

gradient. Next, the second solution includes density and temperature gradients shown

in experiment. We comment on future work on this approach, including hot plasma

effects.