Field line resonances in quiet and disturbed time three‐dimensional magnetospheres

Numerical solutions for field line resonances (FLRs) in the magnetosphere are presented for three‐dimensional equilibrium magnetic fields represented by two Euler potentials as B = ∇ ψ × ∇ α, where ψ is the poloidal flux and α is a toroidal angle‐like variable. The linearized ideal MHD equations for FLR harmonics of shear Alfvén waves and slow magnetosonic modes are solved for plasmas with the pressure assumed to be isotropic and constant along a field line. The coupling between the shear Alfvén waves and the slow magnetosonic waves is via the combined effects of geodesic magnetic field curvature and plasma pressure. Numerical solutions of the FLR equations are obtained for a quiet time magnetosphere as well as a disturbed time magnetosphere with a thin current sheet in the near‐Earth region. The FLR frequency spectra in the equatorial plane as well as in the auroral latitude are presented. The field line length, magnetic field intensity, plasma beta, geodesic curvature, and pressure gradient in the poloidal flux surface are important in determining the FLR frequencies. In general, the computed shear Alfvén FLR frequency based on the full MHD model is larger than that based on the commonly adopted cold plasma model in the β eq > 1 region. For the quiet time magnetosphere, the shear Alfvén resonance frequency decreases monotonically with the equatorial field line distance, which reasonably explains the harmonically structured continuous spectrum of the azimuthal magnetic field oscillations as a function of the dipole L shell in the L ≤ 8.8 R E region. However, the FLR frequency spectrum for the disturbed time magnetosphere with a near‐Earth thin current sheet is substantially different from that for the quiet time magnetosphere for R > 6 R E , mainly due to shorter field line length due to magnetic field compression by solar wind, reduced magnetic field intensity in the high β current sheet region, azimuthal pressure gradient and geodesic magnetic field curvature.