Effective Mode Number of ULF Waves Generated by Interplanetary Shocks: A Cross-Wavelet Analysis of GOES Observations

Interplanetary shocks compress the magnetosphere and launch fast-mode waves propagating in the inner magnetosphere, which can excite field line resonance in the ultra-low frequency (ULF) range. The resulting phase delay due to wave propagation between two observation points manifests observationally as a phase difference, which is commonly used to calculate the azimuthal wavenumber. In this paper, we suggest this wavenumber to be named as effective azimuthal mode number (m*) to distinguish it from the azimuthal mode number derived from solving the MHD wave equation in a dipole magnetic field, which is ∼0 for the toroidal component and infinite for the poloidal component. Utilizing magnetic field measurements from Geostationary Operational Environmental Satellites after interplanetary shocks spanning from January 2011 to March 2024, we perform a statistical analysis of shock-induced ULF wave characteristics based on cross-wavelet analysis. Our results for dayside events reveal that the magnitude of m*-values derived from observations, denoted as (Formula presented.), for compressional, poloidal and toroidal components are predominantly less than 3. All three wave components are characterized by anti-sunward propagation. The (Formula presented.) -values for the two transverse components are slightly higher than that for the compressional component. These signatures are consistent with the theory proposed in this study, providing a generalized estimation of the m*-value of shock-induced ULF waves for the analysis of radial diffusion coefficients in the radiation belts.