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.