2 Model Description
The model presented here is the latest version of a series of models to describe the propagation of Alfvén waves in magnetospheric geometries. Previous versions of this model have been used in the magnetosphere of Earth, where it has been used to describe the structure of field line resonances and cavity modes in response to the propagation of Pi2 pulsations (Lysak et al., 2015; Takahashi et al., 2022), interplanetary shock impacts (Takahashi et al., 2018) and the excitation of quarter-wave modes due to interhemispheric asymmetry in the ionosphere (Lysak et al., 2020). Versions of this model have also been used to study field line resonances at Jupiter (Lysak & Song, 2020) and the effect of the ionospheric Alfvén resonator at Jupiter (Lysak et al., 2021). The model is cast into magnetic dipolar coordinates, defined in terms of magnetic spherical coordinates by
Here ν is the negative inverse of the L-shell parameter and points outwards (we will use L-shell in this paper rather than the M-shell used when non-dipolar fields are important to indicate that we are using a dipolar geometry, which is a reasonable approximation at the orbit of Io), φ is the magnetic east longitude, defined so that it corresponds with System III Right-Handed coordinates at the points where Juno crosses the magnetic equator, and μ is a field-aligned coordinate, proportional to the magnetic scalar potential of the dipole field. This coordinate increases from south to north, opposite the direction of the Jovian magnetic field. Note that we will use east longitudes throughout this paper, although west longitudes are often used by Earth-based observers since the west longitude increases with time as seen from Earth as Jupiter rotates. For simplicity, we consider only the shear Alfvén mode; however, we can include the effect of parallel electric fields due to electron inertia or kinetic effects. In this case the relevant electrodynamic equations are
Here we have introduced the scale factors , and , whereRJ = 71492 km is the equatorial radius of Jupiter, BJ = 417.7 μT is the equatorial dipole magnetic field at the surface of Jupiter according to JRM33 (this value is 426.4 μT for the VIP4 model and 419.9 μT for JRM09). We use where is the non-relativistic Alfvén speed to denote the Alfvén speed including the displacement current. The coefficient ν* is a parameter to characterize the kinetic effects (such as the development of double layers) that can lead to parallel electric fields in strong current regions. This parameter will be set to zero until the last section of this manuscript.