5 Parallel Electric Fields: Current limitation
The models described above only included the effects of electron inertia
on the formation of parallel electric fields. Inertial electric fields
are favored when the perpendicular wavelength is comparable to the
electron inertial length. In the low-density model, the lowest density
is 0.1 cm−3, corresponding to an inertial length of 17
km; Io’s radius of 1820 km mapped to the ionosphere is about 140 km,
much larger than the inertial length. Therefore, the inertial fields are
small, with the parallel fields integrated along the field line being
the order of 500 volts. However, the resulting field-aligned currents
can be very large, 15-20 μA/m2 as can be seen in
Figures 4 and 6. Kinetic modeling of the Io flux tube (Ray et al., 2009)
indicates that such large currents cannot be supported without the
formation of large potential drops along the field lines. Thus, the
effect of these parallel potential drops should be taken into account.
One difficulty in including this type of potential drop in the present
model is that the Ray et al. (2009) formulation relates the
field-aligned current to the total potential drop on the field line,
while the present model includes the parallel electric field at each
point along the field line. However, we can model this effect by
including a current-limiting term in the equations. The maximum current
that can be carried by a Maxwellian distribution with no bulk
acceleration is for the case of a totally empty loss cone, i.e., a
half-Maxwellian distribution with only one sign of the parallel
velocity. In this case, the effective drift velocity is , where the
electron temperature is given in electron volts. Thus, the maximum
current is jmax = nevd , which is
3.39 μA/m2 for a density of 1 cm−3and a temperature of 1 keV.
The parallel electric field can then be modeled by introducing the ν*
term in the equation for the field-aligned current in equation . This
term should be zero when the current is less thanjmax and increase rapidly as the current
increases above this value. For this model, we choose the form
The parameter ν* is set to zero for weaker currents. The constant
ν0 can be estimated from the linear form of the Knight
(1973) relation, j = K Φ, where . The effective parallel
conductivity is then σ = KL , where L is the distance over
which the parallel field is distributed. Then we can write
Where L is in Jovian radii and we have again taken n = 1
cm−3 and Te = 1keV. Thus, if we
take this scale length to be about half a Jovian radius, we can
conveniently set ν0 = 1 s−1.