The Theoretical Foundation of 3d Alfvén Resonances: Normal Modes

Abstract

We consider the resonant coupling of fast and Alfvén magnetohydrodynamic (MHD) waves in a 3D equilibrium. Numerical solutions to normal modes (∝ exp(−iωt)) are presented along with a theoretical framework to interpret them. The solutions we find are fundamentally different to those in 1D and 2D. In 3D there exists an infinite number of possible resonant solutions within a "Resonant Zone," and we show how boundary conditions and locally 2D regions can favour particular solutions. A unique feature of the resonance in 3D is switching between different permissible solutions when the boundary of the Resonant Zone is encountered. The theoretical foundation we develop relies upon recognising that in 3D the orientation of the resonant surface will not align in a simple fashion with an equilibrium coordinate. We present a method for generating the Alfvén wave natural frequencies for an arbitrarily oriented Alfvén wave, which requires a careful treatment of scale factors describing the background magnetic field geometry.

Simulations of MHD Wave Propagation and Coupling in a 3‐D Magnetosphere

“…Thus, Alfvén waves near noon on the outer dashed circle ( ) at lie on the blue ridge that moves to the inner dashed circle ( ) at later MLT. Moreover, Wright and Elsden () show that the Alfvén wave plasma displacement lies along (i.e., is tangential to) these ridges.…”

“…In a 2‐D dipole geometry Wright and Elsden () exploited the fact that, on a chosen field line, the exact path length could be expressed as a function of the scalar potential. They then used this path length as a new field‐aligned coordinate, which afforded a more computationally efficient grid.…”

“…In an effort to retain analyticity Kageyama et al () show how a particular function of the potential can be used as a suitable field‐aligned coordinate. Our approach is similar in spirit to theirs, and we implement it in two stages to provide extra flexibility: First, we determine a coordinate that is equal to path length along on a chosen field line (see Wright & Elsden, for a 2‐D dipole). This essentially flattens the coordinate so equal increments correspond to equal increments in real space.…”

“…However, the representation of a realistic‐shaped magnetopause as a coordinate surface in a simulation employing field‐aligned coordinates remains elusive. Simulations in the above studies and those of Degeling et al (), Wright and Elsden (), and Elsden and Wright () simply take the magnetopause boundary to coincide with a coordinate surface, which has limited realism, but makes the application of boundary conditions straightforward. In this paper we adopt an equatorial profile for the magnetopause based upon the model given by Shue et al ().…”

The Effect of Fast Normal Mode Structure and Magnetopause Forcing on FLRs in a 3‐D Waveguide

“…Further, the inclusion of azimuthal magnetic and density asymmetries has also been used to consider the effect of a plasmaspheric plume on ULF wave propagation (Degeling et al, ). A key aspect of the current work is considering the excitation of FLRs in 3‐D (Elsden & Wright, , ; Wright & Elsden, ), where resonances can exist at any given polarization angle between toroidal and poloidal. Elsden and Wright () showed that the important quantity for efficient resonant excitation is the magnetic pressure gradient along the direction of polarization of the Alfvén wave.…”