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.