Driven Field Line Resonance Polarization in a General Magnetospheric Geometry

The question of interest in this study is: Given the external driving of monochromatic MHD fast mode waves into the magnetosphere, what determines the location and polarization of (fundamental mode) field line resonances (FLRs), in the general case of a non-axisymmetric Alfvén speed and magnetic field topology? This is of particular interest for considering the role of FLRs, or Shear Alfvén Wave (SAW) eigenfunctions, in radiation belt and ring current particle energization, transport and loss by resonant wave-particle interactions, since the efficiency of coupling is dependent on the SAW polarization.

In particular, we seek to determine whether or not the SAW polarization direction in externally driven FLRs remains constant as a function of position along a given field line with respect to neighbouring field lines, as has been assumed in previous studies. In addressing this question we seek to extend and unify the works of: a) Wright et al., (Astrophys. J., 2016, J. Geophys. Res. 2022), which considered the case of a non-axisymmetric Alfvén speed in a dipole and compressed dipole magnetic fields (making the above assumption); and b) Rankin et al. (Adv. Space Res., 2006) and Kabin et al., (Ann. Geophys., 2007), which considered an arbitrary magnetic geometry, but made no constraints SAW polarization.

A new formulation based on vector Sturm Liouville theory for driven SAW eigenfunctions in the Resonant Zone (Wright et al., 2016) is proposed, in which the unconstrained vector eigenfunctions of Kabin et al. (2007) form a complete basis under background conditions without field-aligned currents (FACs). Based on the results of our coupled 3D MHD model for ULF waves, we show that only a very limited number of these eigenfunctions are required to represent the MHD waves in the vicinity of an FLR with reasonable accuracy. Using this as an assumption, we can find solutions for fundamental mode SAWs within the Resonant Zone (described as a linear combination of these basis functions) with eigenfrequencies that match an external driving frequency – essentially producing resonance maps for FLRs similar to those of Wright et al (2016), but without any assumption on the polarization. We further generalize our approach by considering the additional effect of the addition of background FACs on the SAW eigenfunction solutions. In this case the vector ODE equation for SAWs is no longer self-adjoint, however we show that a basis can still be defined by a biorthogonality condition using the adjoint differential operator. This allows a similar spectral method to calculate resonance maps for a given driving frequency.