Hydromagnetic Waves in a Compressed Dipole Field via Field-Aligned Klein-Gordon Equations

TL;DR

This paper introduces a novel method using Klein-Gordon equations to analyze ultra low frequency hydromagnetic waves in Earth's magnetosphere, providing insights into wave behavior in a compressed dipole magnetic field.

Contribution

It develops a Klein-Gordon equation approach for modeling toroidal and poloidal Alfvén waves in a compressed dipole field, aligning theoretical results with spacecraft observations.

Findings

Eigenfrequency solutions distinguish propagation and evanescent wave types.

The model's wave period and amplitude ratios match spacecraft measurements.

Eigenvalue solutions for compressional modes in the dipole field are presented.

Abstract

Hydromagnetic waves, especially those of frequencies in the range of a few milli-Hz to a few Hz observed in the Earth's magnetosphere, are categorized as Ultra Low Frequency (ULF) waves or pulsations. They have been extensively studied due to their importance in the interaction with radiation belt particles and in probing the structures of the magnetosphere. We developed an approach in examining the toroidal standing Aflv\'{e}n waves in a background magnetic field by recasting the wave equation into a Klein-Gordon (KG) form along individual field lines. The eigenvalue solutions to the system are characteristic of a propagation type when the corresponding eigen-frequency is greater than a cut-off frequency and an evanescent type otherwise. We apply the approach to a compressed dipole magnetic field model of the inner magnetosphere, and obtain the spatial profiles of relevant parameters and the spatial wave forms of harmonic oscillations. We further extend the approach to poloidal mode standing Alfv\'{e}n waves along field lines. In particular, we present a quantitative comparison with a recent spacecraft observation of a poloidal standing Alfv\'{e}n wave in the Earth's magnetosphere. Our analysis based on KG equation yields consistent results which agree with the spacecraft measurements of the wave period and the amplitude ratio between the magnetic field and electric field perturbations. We also present computational results of eigenvalue solutions to the compressional poloidal mode waves in the compressed dipole magnetic field geometry.