MHD field line resonances and global modes in three‐dimensional magnetic fields

By assuming a general isotropic pressure distribution P = P (ψ, α), where ψ and α are three‐dimensional scalar functions labeling the field lines with B = ∇ψ × ∇α, we have derived a set of MHD eigenmode equations for both global MHD modes and field line resonances (FLR). Past MHD theories are restricted to isotropic pressures with P = P (ψ) only. The present formulation also allows the plasma mass density to vary along the field line. The linearized ideal MHD equations are cast into a set of global differential equations from which the field line resonance equations of the shear Alfvén waves and slow magnetosonic modes are naturally obtained for general three‐dimensional magnetic field geometries with flux surfaces. Several new terms associated with ∂ P /∂α are obtained. In the FLR equations a new term is found in the shear Alfvén FLR equation due to the geodesic curvature and the pressure gradient in the poloidal flux surface. The coupling between the shear Alfvén waves and the magnetosonic waves is through the combined effects of geodesic magnetic field curvature (κ s = 2 κ · B × ∇ ψ/ B 2 ) and plasma pressure as previously derived. The properties of the FLR eigenfunctions at the resonance field lines are investigated, and the behavior of the FLR wave solutions near the FLR surface are derived. Numerical solutions of the FLR equations for three‐dimensional magnetospheric fields in equilibrium with high plasma pressure will be presented in a future publication.