3.1. Results With Nc Independent of
 We now present the results obtained using the model described in section 2. We first show in Figure 4 the structures of the magnetic field (black contours) and total azimuthal current density (colors) as computed using three values for the quotient (ΣP*/) = 10−5, 10−4, and 5 × 10−4 mho s kg−1. These roughly bracket both the value assumed by Caudal  and the revised value of Hill , as discussed in section 2. Note that here we keep the cold plasma density independent of the plasma mass outflow rate. Hill  showed that higher values of (ΣP*/) result in higher plasma angular velocity values, and it is apparent that higher values of this quotient result in a more stretched magnetic field structure with a thinner, more intense current sheet, particularly in the region outward of ∼20 RJ. Specifically, the half width of the current sheet in the middle magnetosphere is typically ∼8–10, ∼6–8, and ∼3–5 RJ for (ΣP*/) = 10−5, 10−4, and 5 × 10−4 mho s kg−1, respectively. These are all somewhat larger than the value of 2.5 RJ employed in the empirical “Voyager-1/Pioneer-10” (“CAN”) current sheet field model of Connerney et al. , with the (ΣP*/) = 5 × 10−4 mho s kg−1 result being most consistent with the latter. The increased azimuthal current for higher values of (ΣP*/) is required to balance the elevated centrifugal force imparted by the faster rotating equatorial plasma for higher values of (ΣP*/).
 This can be further appreciated from Figure 5, in which we show various parameters associated with the magnetodisc model and M-I coupling current system for each of the above values of (ΣP*/) and where for the purposes of the M-I coupling current calculations we take the canonical value of = 1000 kg s−1. Specifically, we show, from top to bottom, the magnitude of the north-south magnetic field threading the equatorial plane |Bze| in nT, the ionospheric colatitude to which the magnetic field maps θi in degrees, the equatorial plasma angular velocity normalized to the planet's rotation rate (ω/ΩJ), the ratio of the equatorial azimuthal current density associated with the cold plasma centrifugal force to that of the hot plasma pressure , the cold plasma pressure Pc° in Pa, the azimuthally integrated equatorial radial current Iρ in MA, and finally the field-aligned current density at the top of the ionosphere in μA m−2, all versus equatorial radial distance in RJ. Note that the solid colored lines indicate results from model runs which converged, while long-dashed lines indicate results from model runs which have reached a quasi-steady state as discussed in section 2.3. Starting with the equatorial magnetic field strength |Bze| shown in Figure 5a, it is first evident that all three model results are similar out to ∼15 RJ, beyond which they diverge. Also shown in Figure 5a for comparison are the magnetic field strengths given by the pure planetary dipole (dashed black line), given by
and the “CAN-KK” current sheet magnetic field model of Nichols and Cowley  (dot-dashed line), given by
where B°′ = 3.335 × 105 nT, ρe* = 14.501 RJ, B° = 5.4 × 104 nT, and m = 2.71. This form closely approximates the field model used by Cowley and Bunce  and Cowley et al. [2002, 2003], who employed the CAN field model of Connerney et al.  in the inner region and the Voyager 1 (“KK”) outbound pass model of Khurana and Kivelson  in the outer region. In the model results obtained in this study, the |Bze| values are less than those for the dipole in the inner region owing to the radial distention of the field by the current sheet. All three results are reasonably consistent with the CAN-KK model to distances of ∼20 RJ, but (ΣP*/) = 5 × 10−4 mho s kg−1 again gives the best agreement, roughly tracking the CAN-KK model values out to ∼40 RJ. We note that the slight jitter in the latter |Bze| profile between ∼40 and ∼60 RJ is representative of the spontaneous instability, which prohibits models runs with higher values of (ΣP*/) from truly converging and which, as mentioned by Caudal , can lead to the formation of neutral points for more stretched magnetodiscs. In the outer region, the |Bze| values become greater than those for the dipole, with the transitions occurring at ∼38, ∼42, and ∼52 RJ for (ΣP*/) = 10−5, 10−4, and 5 × 10−4 mho s kg−1, respectively. This transition, also originally noted by Caudal , is due to the outer fringing fields of the current sheet, and the outermost values of |Bze| of ∼10–20 nT are consistent with the values of ∼16 nT observed just inside the magnetopause [Acuña et al., 1983]. Note that in contrast, in the radial range of Figure 5 the CAN-KK values are always less than those of the dipole, indicating that the current sheet in the predawn region of the Voyager 1 outbound pass was evidently extended because of the distant (∼160 RJ) magnetopause in this region [Acuña et al., 1983].
Figure 5. Plot showing magnetodisc and current system parameters computed for (ΣP*/) = 10−5 (blue), 10−4 (green), and 5 × 10−4 (red) mho s kg−1, plotted versus equatorial radial distance. Parameters shown are (a) the magnitude of the north-south magnetic field threading the equatorial plane |Bze| (in nT), (b) the ionospheric colatitude to which the magnetic field maps θi (in degrees), (c) the equatorial plasma angular velocity normalized to the planet's rotation rate (ω/ΩJ), (d) the ratio of the equatorial azimuthal current density associated with the cold plasma centrifugal force to that of the hot plasma pressure , (e) the cold plasma pressure Pc° (in Pa), (f) the azimuthally integrated equatorial radial current Iρ (in MA), and (g) the field-aligned current density at the top of the ionosphere (in μA m−2). In Figure 5a and 5b the dashed black lines show the planetary dipole values, and the dot-dashed black lines show the values using the model of Nichols and Cowley . In Figure 5c the horizontal dotted line indicates rigid corotation, and the dashed black line indicates the profile taken by Caudal . In Figure 5d the horizontal dotted line indicates where the azimuthal current densities associated with the cold plasma centrifugal force and the hot plasma pressure are equal. In Figure 5e the dashed and dot-dashed black lines show the power laws given by Frank et al. . Note that both the positive and negative values in Figure 5g are plotted on logarithmic scales, such that the horizontal line is at ±0.0001, and the resulting apparent discontinuities at the transition points are simply artifacts of the plotting scale. In Figures 5f and 5g the canonical value of = 1000 kg s−1 is used. In all plots the solid colored lines indicate results from model runs that converged, while long-dashed lines indicate results from model runs that have reached a quasi-steady state, as discussed in section 2.3. In Figures 5c, 5f, and 5g the colored dot-dashed lines indicate results obtained using the fixed magnetic field model of Nichols and Cowley  for comparison.
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 Figure 5b shows the ionospheric colatitude to which the magnetic field maps, calculated using equations (2) and (9). Also shown are the values for the planetary dipole (dashed black line), for which
and the CAN-KK field model (dot-dashed line), for which
where F∞ ≈ 2.841 × 104 nT RJ2 is the value at infinity, and Γ(a, z) = ∫z∞ta−1e−t dt is the incomplete gamma function. It is apparent that for each value of (ΣP*/) used the field line mapping is more consistent with that of the CAN-KK field model than the dipole, although the elevated values of |Bze| in the outer region relative to the CAN-KK values results in a broadening of the ionospheric latitudinal band to which the outer magnetosphere maps. It is evident, however, that for increased values of (ΣP*/) the middle magnetosphere field lines map to a modestly more equatorward and thinner latitudinal band in the ionosphere. For example, field lines threading the equatorial plane between 20 and 60 RJ map to between ∼12.6° and 16.6°, ∼14.0° and 16.9°, and ∼15.6° and 17.1° for (ΣP*/) = 10−5, 10−4, and 5 × 10−4 mho s kg−1, respectively, with the latter result being most consistent with the CAN-KK model. In addition, it is worth noting that in this model the ionospheric colatitudes of the last closed field line are ∼7.9°, ∼9.0°, and ∼11.0° for (ΣP*/) = 10−5, 10−4, and 5 × 10−4 mho s kg−1, respectively. The latter value is in excellent agreement with the value of ∼11° recently determined by Vogt et al. , and is also consistent with the value of 10.25° used by Cowley et al.  in their global model of Jupiter's polar ionospheric flows.
 The effect of the plasma angular velocity on the azimuthal current is shown in Figure 5d, in which we plot the ratio of the equatorial azimuthal current density associated with the centrifugal force to that of the hot plasma pressure , giving an indication as to which of these two components of the azimuthal current is dominant. Caudal  concluded that the latitude-integrated current associated with the hot plasma pressure dominates both the cold plasma pressure current and the centrifugal force current over the whole of the magnetosphere. This result was supported by Achilleos et al. , although these authors also showed that in the original Caudal  model the effect of the centrifugal force strongly peaks near ∼27 RJ, such that equatorial current densities associated with the hot plasma pressure and centrifugal force become comparable between ∼20–30 RJ, a concern which was originally raised by Mauk and Krimigis  on the basis that it apparently contradicts observation [McNutt, 1983, 1984]. It is therefore worth noting that the revised cold plasma input parameters employed in our model eliminate this effect here, and considering first the current ratio profile for (ΣP*/) = 10−5 profile, it is apparent that the hot plasma current is significantly larger than that of the centrifugal force over essentially all the magnetosphere. However, this is not the case for the higher values of (ΣP*/), for which the centrifugal force current exceeds the hot plasma pressure current outward of ∼54 and ∼34 RJ for (ΣP*/) = 10−4 and 5 × 10−4 mho s kg−1, respectively. It should be noted that this does not contradict the conclusions of McNutt [1983, 1984] and Mauk and Krimigis , which were based on Voyager data that were obtained at current sheet crossings within 40 RJ and that are somewhat sparse beyond ∼30 RJ [see, e.g., McNutt, 1983, Figure 2]. McNutt  and Mauk and Krimigis  computed the ratio of the rotational kinetic energy density to magnetic energy density, which can be thought of a “plasma beta for bulk rotation,” comparable to the traditional plasma beta β = (P/PB), where PB = B2/2μ° is the magnetic energy density (note they termed this quantity M2, since it is equal to the square of the Alfvénic Mach number). Achilleos et al.  pointed out that the plasma beta for bulk rotation is given by βcent = (βcρ2/2l2), and thus confirmed that in Caudal's  model the hot plasma beta βh dominates the bulk rotation beta βcent beyond ∼40 RJ. We have calculated the ratio (βcent/βh) using our model results, and although for clarity we have not plotted the profiles in Figure 5, we note that they are very similar to those for . Achilleos et al.  showed that in the original model of Caudal , βcent peaks near ∼25 RJ at ∼16, whereas McNutt  obtained values of ∼3 near 25 RJ. In our results βcent = 0.75, 6.63, and 24.96 at 25 RJ for (ΣP*/) = 10−5, 10−4, and 5 × 10−4 mho s kg−1, respectively, with the value for (ΣP*/) = 10−4 mho s kg−1 thus being in most agreement with observations.
 The cold plasma pressure computed in this model is plotted in Figure 5e, along with power laws fitted by Frank et al.  to the pressure values as measured by the Galileo spacecraft. These fits are given by
shown by the dot-dashed and dashed black lines, respectively, although we note that in Figure 8 of Frank et al. , the scatter in the measured values is generally at least an order of magnitude, and in the region 20 < (ρ°/RJ) < 50 the points generally lie between the two power laws. All three cold plasma pressure profiles are similar out to distances of ∼15 RJ, beyond which the profiles for (ΣP*/) = 10−4 and 5 × 10−4 mho s kg−1 are in best agreement with the observed profile.
 Figure 5f shows the azimuth-integrated equatorial radial current computed from the plasma angular velocity and magnetic field profiles using equation (7), where we note that for the M-I coupling current equations we explicitly take the canonical value of = 1000 kg s−1, such that ΣP* = 0.01, 0.1, and 0.5 mho. The solid lines show the results obtained using the magnetic field model discussed here, while the dot-dashed lines show the profiles obtained using the empirical CAN-KK magnetic field model employed in previous studies for comparison. It is evident that the current profiles are similar to the results for the CAN-KK model out to ∼40–50 RJ, beyond which they reduce to smaller values owing to the lower values of Fe in the outer region relative to the CAN-KK values because of the current sheet outer fringing field. Nichols and Cowley  used the midnight sector Galileo Bϕ data of Khurana  to show that the observed values of Iρ increase rapidly in the inner region, between ∼15 and 25 RJ, before plateauing at ∼100 MA at distances beyond, out to ∼100 RJ (see, e.g., their Figure 12). It is worth noting that the Khurana  data obtained at midnight are unconstrained by an assumed magnetopause distance of 85 RJ, such that it is not surprising that the decrease in the outer region is not evident in those data. This aside, the current profile which best fits this pattern is that for ΣP* = 0.5 mho.
 The resulting field-aligned current at the top of the ionosphere computed using equation (8) is then plotted in Figure 5g. In the inner region the currents are upward and peak at similar values to those obtained using the CAN-KK field model, at radial distances of ∼22, 28, and 33 RJ for ΣP* = 0.01, 0.1, and 0.5 mho, respectively. However, the decreasing values of Iρ in the outer region result in a reversal of the field-aligned current at ∼40–60 RJ, such that the current is then downward in the region beyond. Note that the oscillation in the ΣP* = 0.5 mho profile between ∼40–60 RJ is due to the instability in the magnetic field model as discussed above. While we note that such layering of upward and downward field-aligned current has been observed in Jupiter's middle magnetosphere by Mauk and Saur , we do not wish to infer too much from the structure in our results, and simply note that the overall structure is that of consistent upward current inward of ∼40 RJ and downward current outward of ∼60 RJ. This confinement of the upward field-aligned current to the region inward of ∼40–60 RJ, depending on (ΣP*/), is consistent with the results of Vogt et al. , who showed using flux equivalence calculations that the poleward boundary of the main auroral oval maps to ∼30–60 RJ depending on local time, and we also note that Khurana  showed using Galileo data that the main oval field-aligned currents flow inward of 30 RJ. Inclusion of local time asymmetry is not possible in our axisymmetric model, but the overall results are broadly consistent with the observations of Khurana  and Vogt et al. . The downward current in the region outward of ∼40–60 RJ thus corresponds to the dark polar region just poleward of main oval, which typically exists on the dawn side but sometimes extends to all local times [Grodent et al., 2003b; Nichols et al., 2009]. Note that while Nichols and Cowley  showed that the modulation of the ionospheric Pedersen conductivity by auroral electron precipitation concentrates the peak field-aligned current in the ∼20–40 RJ region, in their model the field-aligned current was still upward throughout the magnetosphere, albeit at low values in the outer region. In addition, while Cowley et al.  included a region of downward current in the outer magnetosphere by design of their specified plasma velocity profiles, the results presented here are the first to self-consistently produce this downward current region. The latter authors also showed that a second sheet of upward field-aligned current should exist, associated with the ionospheric flow shear at the boundary between open and closed field lines, and indeed it is thought that Saturn's main auroral oval is due to such a layer between the outer edge of the ring current and the open-closed field line boundary [Badman et al., 2006; Bunce et al., 2008]. Since our model only includes closed field lines, we do not consider this second layer of upward current and simply note that it will act to modify the field-aligned current profiles in the very outer region from those computed here.
 Overall, then, it is apparent that the results for (ΣP*/) = 10−4 and 5 × 10−4 mho s kg−1 provide the best agreement with various sets of observations, with the magnetic field (both |Bze| and Bϕ) most consistent with the latter, and plasma data (angular velocity and pressure) in best agreement with the former. It is, however, instructive to examine how the M-I coupling currents vary in peak magnitude and location over a range of values of (ΣP*/), which we thus show in Figure 6, again taking here = 1000 kg s−1 (note that we consider the effect of changing in section 3.2). From top to bottom, the joined crosses in Figure 6 show the maximum azimuth-integrated equatorial radial current, the maximum upward field-aligned current density at the top of the ionosphere, the equatorial radial distance of the peak upward field-aligned current, and finally, the ionospheric colatitude of the peak upward field-aligned current. Also shown for comparison by the dashed lines in Figure 6 are results calculated using the analytical solution of the Hill-Pontius equation (equation (3)) obtained by Nichols and Cowley  for a power law current sheet magnetic field which maps to a thin latitude band in the ionosphere, such as that given by the second term in equation (34), thus appropriate for the Jovian middle magnetosphere. The analytic power law field result for the maximum azimuth-integrated radial current is given by
where F° is the value of Fe at the location of the latitude band, taken by Nichols and Cowley  to be F° = Fe(70 RJ) ≃ 3.22 × 104 nT RJ2, a representative value for the middle magnetosphere current sheet. The maximum value for the power law field strictly occurs at ρe = ∞, while at large but finite distances in the numeric solution using the full empirical field model (e.g., beyond 1000 RJ for ΣP* = 0.5 mho). It is apparent that the maximum radial current computed using the model employed here increases less quickly with ΣP* than for the power law field, i.e. from ∼5 MA at ΣP* = 0.01 mho to ∼242 MA at ΣP* = 2 mho. This occurs since, for a power law current sheet field the total azimuthal current increases monotonically toward the maximum value given by equation (38) at a rate determined solely by the corotation breakdown distance, given by ρH for a dipole field and by
for the power law current sheet field. The radial current profiles obtained using the model presented here, however, are also constrained by the assumed magnetopause distance and thus drop away from the power law profiles at distances increasingly small relative to ρHcs as (ΣP*/) increases, such that the peak current rises less quickly with ΣP* than for the power law field.
Figure 6. Plots showing (a) the maximum azimuth-integrated radial current, (b) the maximum field-aligned current density at the top of the ionosphere, (c) the equatorial distance of the maximum field-aligned current, and (d) the ionospheric colatitudes of the maximum field-aligned current (joined crosses) and Ganymede footprint (joined asterisks), all versus ionospheric Pedersen conductance. Also shown by the dashed lines are the results for a power law current sheet magnetic field.
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 The maximum field-aligned current density at the top of the ionosphere is plotted in Figure 6b, alongside the result for the power law current sheet field, shown by Nichols and Cowley  to be
It is evident that the maximum field-aligned current density computed here increases with ΣP* similarly as does the result for power law field, i.e. from ∼0.004 μA m−2 at ΣP* = 0.01 mho to ∼4.6 μA m−2 at ΣP* = 2 mho, such that the latter is a reasonable approximation for the results obtained here.
 The same is not true, however, for the equatorial radial distance of the peak field-aligned current density, shown in Figure 6c, in which the distance for the power law field, given by
rises much more quickly than do the results here, which increase from 22 RJ at ΣP* = 0.01 mho to ∼44 RJ at ΣP* = 2 mho. This is again due to constraint by the finite magnetopause distance in the current sheet field model employed here, rather than the power law field which simply decreases monotonically toward ρe = ∞. The ionospheric colatitudes of these peak field-aligned current locations are shown in Figure 6d by the joined crosses, along with the mapped location of Ganymede's orbit at 15 RJ, shown by the joined asterisks. Again, shown by the dashed line for comparison is the location of the peak field-aligned current for the power law field, given by
which indicates that in this case the peak current shifts poleward as the equatorial radial distance increases, although, as shown in Figure 5b, the colatitude is only weakly dependent on the radial distance because of the stretching of the middle magnetosphere field lines. However, although the equatorial radial distance of the peak field-aligned current in the results presented here increases with ΣP*, above ΣP* = 0.05 mho, the ionospheric colatitude actually increases slowly with ΣP*, moving from ∼16.3° for ΣP* = 0.01 mho to ∼16.8° for ΣP* = 2 mho. This arises since the outward movement of peak field-aligned current with increasing ΣP* is offset by the modified mapping of the increasingly stretched magnetic field. This can be appreciated by examination of Figures 5b and 5g, in which the peak field-aligned current moves outward for the blue, green and red profiles, respectively, while the associated ionospheric mapping profiles also move equatorward, counteracting the outward shift. Considering now the colatitude of the Ganymede footprint, it is evident that this is also only very weakly dependent on ΣP*, moving from ∼17.6° for ΣP* = 0.01 mho to ∼18.0° for ΣP* = 2 mho. This is simply due to the fact that in these runs the magnetic field model is relatively insensitive to changes inside ∼15 RJ, and, as can be seen from Figure 5b, the mapped ionospheric colatitudes of ρe = 15 RJ are very similar for all values of ΣP*. Thus, on the basis of these results, it is unlikely that a change of ionospheric conductance is responsible for the ∼3° and ∼2° shifts in latitude of the main oval and Ganymede footprint, respectively, reported by Grodent et al. . In section 3.2 we therefore examine the effect of changing cold plasma number density.
3.2. Comparison With Results Taking Nc Proportional to
 We now compare results for which the cold plasma density is assumed constant, such that the outward transport rate is proportional to , with results for which the cold plasma density is taken to be given by equation (32), such that in this case the outward transport rate is assumed constant. Here we take ΣP* = 0.1 mho, and = 500, 1000, and 2000 kg s−1, typical of the range of values determined by various studies [e.g., Hill, 1980; Khurana and Kivelson, 1993; Delamere and Bagenal, 2003]. Figure 7 thus shows the magnetodisc and M-I coupling current system parameters for constant plasma density in the same format as for Figure 5, while Figure 8 shows the results taking the cold plasma density to be given by equation (32). It is first evident from Figures 7a and 8a that taking Nc ∝ acts to suppress the divergence of the |Bze| profiles in the middle magnetosphere beyond ∼20 RJ. It is, however, just apparent that for the case with Nc ∝ , the higher value of leads to slightly lower equatorial magnetic field strengths in the region inside ∼40 RJ, i.e., the opposite behavior to the case with constant Nc. This is more evident in Figures 7b and 8b, where in the former the field maps to lower colatitudes for higher mass outflow rates, indicating a less stretched field, while in the latter case the field maps to higher colatitudes, indicating a more stretched field. It is also worth noting that for the case in 8b the difference in field mapping is larger at all radial distances than for 7b, in which the divergence is only significant outward of ∼15 RJ. This indicates the nature of the centrifugal force acting on the plasma in the two cases, which we now discuss.
 Figures 7c and 8c show that the plasma angular velocities in the two cases are very similar, with perhaps modestly increased values in Figure 7c over those in Figure 8c. The plasma angular velocity is related to the centrifugal force acting on the rotating plasma, which we recall from equation (11) is proportional to dω2, such that if these two parameters are dependent on , the centrifugal force is proportional to some power of , i.e. γ. An understanding of the difference in behavior between the two cases can then be obtained if we consider the power law magnetic field approximations of Nichols and Cowley . In this approximation, the plasma angular velocity scales with the current sheet Hill distance ρHcs given by equation (39), such that ω ∝ −1/m. Hence, if the plasma density is independent of the mass outflow rate, we have γ = −2/m, while if the plasma density is proportional to the mass outflow rate we have γ = 1 − 2/m. Therefore, in the former case γ < 0 for all positive values of m (i.e., for fields which decrease in magnitude with distance), such that the centrifugal force decreases with increasing . On the other hand, for the latter case we have γ < 0 for m < 2, such that the centrifugal force decreases with increasing , and 0 < γ < 1 for m > 2, such that the centrifugal force increases with increasing in this case. Thus, examination of Figure 7d, for which γ = −2/m, indicates that the centrifugal force is lower for increasing (note that the hot plasma pressure current does not differ significantly between the different cases). In Figure 8d, on the other hand, in the inner region where the field strength decreases quickly, the centrifugal force is larger for higher values of , while in the outer region, where the field is very weakly dependent on ρe, the centrifugal force is somewhat lower for higher values of . Physically, the competing effects of increasing , i.e., increased plasma density but decreased angular velocity, mutually counteract in the middle magnetosphere, such that the magnetic field in this region becomes relatively insensitive to the value of . It is important to note that the power law approximation is not perfectly applicable to the model results obtained here; for example, in the outer region, the field strength increases slowly with radial distance, a situation not considered by Nichols and Cowley , and for which the power law approximations were not designed. Second in the outer region, the field does not map to a narrow band in the ionosphere, such that the approximation conditions do not strictly hold in this region. Hence, while caution should be used when comparing with the power law approximation, it nevertheless gives a reasonable insight into the behavior of the system. The profiles shown in Figures 7d and 8d also indicate why the ionospheric mapping differs between the two cases. In the former case, the centrifugal force is solely dependent on the plasma angular velocity, which inside 15 RJ is not particularly sensitive to , such that in this region the azimuthal current profiles, and thus the field mapping, do not differ greatly. In the latter case, the centrifugal force also depends on the plasma density, such that the azimuthal current, and thus the field mapping, in the inner region varies significantly with .
 The azimuth-integrated radial current profiles are shown in Figures 7f and 8f, while the field-aligned current profiles are shown in Figures 7g and 8g. Both sets of current profiles in Figures 7 and 8 are reasonably similar, differing most significantly in the degree to which they track the CAN-KK results, leading to different peak current values as will be discussed further below. As Nichols and Cowley  pointed out, for the CAN-KK field model the radial and field-aligned currents both tend to values dependent only on in the inner region, and the radial current tends to a value dependent on ΣP* at large distances. In Figures 7 and 8 the currents thus exhibit three distinct profiles in the inner region, in contrast to Figures 5f and 5g, although the radial current profiles do not converge on a single value in the outer region because of the decrease in current intensity owing to the increased field strength over the CAN-KK model in this region. In both cases the field-aligned current reverses from upward to downward between ∼45–50 RJ, decreasing with radial distance for higher values of .
 Turning now to Figure 9, we show the magnitudes and locations of the peak currents versus in the same format as for Figure 6, except that here the points joined by the dotted lines show results taking the cold plasma density to be independent of , while those joined by the solid lines show those taking it to be given by equation (32). Figure 9a shows that the peak azimuth-integrated radial current increases with for both cases, from ∼15 to ∼40 MA and ∼5 to ∼50 MA for constant Nc and Nc ∝ , respectively, as goes from 100 to 4000 kg s−1, i.e., somewhat quicker for Nc ∝ . As is evident from Figure 7, this arises since, as increases while Nc is constant, the magnetic field becomes less stretched because of the lower plasma angular velocity, such that the Iρ profiles fall away from the CAN-KK results at closer distances. Thus, the peak currents increase slowly with . On the other hand, for Nc ∝ the reverse is true, i.e. the field is more stretched for higher , such that the current profiles follow the CAN-KK results further, increasing the rate at which the peak current increases with . This behavior also accounts for the difference in the field-aligned current profiles shown in Figure 9b, in which changes from ∼0.2 to 0.05 μA m−2 and ∼0.01 to 0.1 μA m−2 for constant Nc and Nc ∝ , respectively, as goes from 100 to 4000 kg s−1.
Figure 9. As for Figure 6, except that here the results are plotted versus , and the points joined by the dotted lines assume the cold plasma density is independent of , while those joined by the solid lines assume it is given by equation (32).
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 The equatorial distance of the peak field-aligned current shown in Figure 9c changes similarly for both cases, i.e., decreasing from ∼35 to ∼23 RJ and from ∼31 to ∼24 RJ for constant Nc and Nc ∝ , respectively, as goes from 100 to 4000 kg s−1. However, the difference in the field mapping results in the ionospheric colatitudes plotted in Figure 9d varying differently with for the two cases. First, as for Figure 6d, for constant Nc the radial motion of the peak field-aligned current is offset by the changing field mapping such that the ionospheric colatitude of the peak current is only weakly dependent on , changing from ∼16.8° for = 100 kg s−1 to ∼16.3° for = 4000 kg s−1. On the other hand, the reverse behavior of the field mapping for Nc ∝ reinforces the radial motion in this case, such that the colatitude increases more rapidly than for the CAN-KK field, moving from ∼15.1° for = 100 kg s−1 to ∼17.5° for = 4000 kg s−1. Similarly, the colatitudes of the Ganymede footprint change from ∼18.0° to ∼17.7° and ∼17.4° to ∼18.5° for constant Nc and Nc ∝ , respectively, as goes from 100 to 4000 kg s−1. Thus, while it is difficult to generate the ∼3° and ∼2° shifts in latitude of the main oval and Ganymede footprint reported by Grodent et al. , these results suggest that a significant change in the iogenic plasma mass outflow rate, combined with an associated variation in the cold plasma density in the magnetosphere, possibly as a result of changing volcanic activity on Io, is the best candidate for explaining the shift in these auroral features. In this case then, the blue image in Figure 2 corresponds to an epoch of low volcanic activity, and the red image corresponds to an interval of high activity.