Magnetosphere-ionosphere coupling currents in Jupiter's middle magnetosphere: effect of magnetosphere-ionosphere decoupling by field-aligned auroral voltages

We consider the effect of field-aligned voltages on the magnetosphere-ionosphere coupling current system associated with the breakdown of rigid corotation of equatorial plasma in Jupiter's middle magnetosphere. Previous analyses have assumed perfect mapping of the electric field and flow along equipotential field lines between the equatorial plane and the ionosphere, whereas it has been shown that substantial field-aligned voltages must exist to drive the field-aligned currents associated with the main auroral oval. The effect of these field-aligned voltages is to decouple the flow of the equatorial and ionospheric plasma, such that their angular velocities are in general different from each other. In this paper we self-consistently include the field-aligned voltages in computing the plasma flows and currents in the system. A third order differential equation is derived for the ionospheric plasma angular velocity, and a power series solution obtained which reduces to previous solutions in the limit that the field-aligned voltage is small. Results are obtained to second order in the power series, and are compared to the original zeroth order results with no parallel voltage. We find that for system parameters appropriate to Jupiter the effect of the field-aligned voltages on the solutions is small, thus validating the results of previously-published analyses.


Introduction
The physics of Jupiter's middle magnetosphere is dominated by the source of sulphur and oxygen plasma originating from the moon Io (e.g.Siscoe and Summers, 1981;Hill et al., 1983;Belcher, 1983;Vasyliunas, 1983;Bagenal, 1994;Delamere and Bagenal, 2003).Particle pick-up and centrifugally-driven outflow result in sub-corotation of Correspondence to: J. D. Nichols (jdn@ion.le.ac.uk) this plasma, which bends the field lines out of magnetic meridian planes, and sets up the large-scale current system shown in Fig. 1 (Hill, 1979;Huang and Hill, 1989;Dougherty et al., 1993;Pontius, 1997;Bunce and Cowley, 2001;Khurana, 2001).This current system communicates atmospheric torque to the magnetospheric plasma, and is directed equatorward in the ionosphere, radially outward in the equatorial plane, and closes via field-aligned currents flowing outward in the inner region and inward in the outer region.Current calculations were performed for a dipole field by Hill (2001), and for both a dipole and current sheet field by Cowley and Bunce (2001), Cowley et al. (2002Cowley et al. ( , 2003)), and Nichols and Cowley (2003).For plasma mass outflow rates of ∼1000 kg s −1 and ionospheric Pedersen conductivities of a few tenths of a mho, the peak upward field-aligned currents in the ionosphere were calculated to be a few tenths of a µA m −2 , in agreement with empirical estimates based on magnetospheric magnetic field data (Bunce and Cowley, 2001;Khurana, 2001).Bunce and Cowley (2001) and Cowley and Bunce (2001) also showed that in the tenuous high-latitude Jovian environment such fieldaligned currents exceed those that can be carried by precipitating hot magnetospheric electrons, thus requiring the existence of field-aligned voltages.These were computed using Knight's (1973) kinetic theory, with electron source plasma parameters derived from Voyager data.The calculations of Cowley and Bunce (2001) and Cowley et al. (2002Cowley et al. ( , 2003) ) yielded field-aligned voltages of ∼25-100 kV for typical parameters, sufficient to produce precipitating electron energy fluxes of up to several tens of mW m −2 , thus leading to 'main oval' auroras of up to several hundred kR in brightness, comparable to observed intensities (e.g.Satoh et al., 1996;Prangé et al., 1998;Clarke et al., 1998;Vasavada et al., 1999;Pallier and Prangé, 2001;Grodent et al., 2003).Most recently, Nichols and Cowley (2004) have also self-consistently included the effect of precipitation-induced enhancement of the ionospheric Pedersen conductivity.
One key feature of all these calculations, however, is the assumption of perfect mapping of the electric field and

Io
Fig. 1 Fig. 1.Sketch of a meridian cross section through Jupiter's inner and middle magnetosphere, showing the principal physical features involved.The arrowed solid lines indicate magnetic field lines, the arrowed dashed lines the magnetosphere-ionosphere coupling current system, and the dotted region the rotating disc of out-flowing iogenic plasma (From Cowley and Bunce, 2001).plasma flow along equipotential field lines between the equatorial plane and the ionosphere.The associated field-aligned voltages were then calculated from the field-aligned current using Knight's (1973) theory.In principle, however, the field-aligned voltage modifies the mapping of the electric field, and hence the plasma flow, between the two regions.This lack of internal consistency was discussed briefly by Cowley and Bunce (2001), who argued that the effect is likely to be small, since field-aligned voltages of ∼25-100 kV are small compared with field-perpendicular voltages across the middle magnetosphere of the order of several MV.However, to date no self-consistent calculation has been performed which quantifies the effect of the parallel voltage on the flow and current system, such that this argument has remained untested.In view of the significance of these processes for Jovian middle magnetosphere physics, it is important to quantify these effects within a self-consistent calculation, and hence to determine whether previously-presented results are indeed valid.In this paper we derive a theory which, for the first time (to our knowledge), self-consistently incorporates the field-aligned voltage into the calculation.This theory is then applied to the Jovian middle magnetosphere, and results are compared with those previously derived.

Governing equations
In this section we summarise the equations which govern the system, the main new feature being the selfconsistent inclusion of the field-aligned voltage calculated from Knight's (1973) kinetic theory.In other aspects, however, the analysis follows those given previously by Hill (1979Hill ( , 2001)), Pontius (1997), Cowley et al. (2002Cowley et al. ( , 2003)), and Nichols and Cowley (2003), such that only the main results will be outlined, together with the approximations and assumptions made.We begin with a description of the magnetic field model which serves as the essential background to the problem.

Magnetic field model
The magnetic field model is that used previously by Nichols and Cowley (2004).The field is assumed to be axisymmetric, as appropriate to Jupiter's middle magnetosphere, such that the poloidal components can be described by a flux function F (ρ, z), related to the magnetic field by where ρ is the perpendicular distance from the magnetic axis, z is the distance along this axis from the magnetic equator, and ϕ is the azimuthal angle.Function F is then constant along a given field line, such that mapping between the equatorial plane and the ionosphere is obtained simply from writing F e =F i .Assuming the ionospheric field is purely dipolar and neglecting any small effects due to magnetospheric currents, the ionospheric flux function is where B J is the dipole equatorial magnetic field strength (equal to 426 400 nT in conformity with the VIP 4 internal field model of Connerney et al., 1998), R J is Jupiter's radius (71 323 km), ρ i is the perpendicular distance from the magnetic axis, and θ i is the magnetic co-latitude.The absolute value of F is fixed by setting F =0 on the magnetic axis.The flux function in the equatorial plane is found by integrating where B ze is the north-south magnetic field threading the equatorial plane where B o =3.335×10 5 nT, ρ eo =14.501R J , A=5.4×10 4 nT, and m=2.71.This field is close to that employed by Cowley and Bunce (2001) and Cowley et al. (2002Cowley et al. ( , 2003)), who used the 'Voyager-1/Pioneer-10' model of Connerney et al. (1981) (the 'CAN' model) in the inner region, and the Voyager-1 model of Khurana and Kivelson (1993) (the 'KK' model) at large distances.The second term in Eq. ( 4a) is simply the latter model, while the first is a modified dipole field.The corresponding equatorial flux function is where J is the value of the flux function at infinity, and (a,z) is the incomplete gamma function The ionospheric mapping between the equatorial plane and the ionosphere is then given from Eq. ( 2) by such that the field line passing through the equatorial plane at the outer edge of the model at 100 R J maps to ∼15.7 • in the ionosphere.Figure 2 shows |B ze |, F e , and θ i versus ρ e , over the range 0 to 100 R J .The solid lines show the above model, while the long-dashed lines show the dipole values for comparison.The dotted lines in panel (a) also show the values for the CAN/KK models, which are projected beyond their intersection for ease of visibility.The horizontal dotted lines in panels (b) and (c) show the asymptotic values of F e and θ i at large distances.
2.2 Magnetosphere-ionosphere decoupling by fieldaligned voltages The primary new feature of this calculation is the selfconsistent inclusion of the field-aligned voltage in the mapping of the electric field and flow between the magnetosphere and ionosphere.It is convenient to use the flux function F as the spatial coordinate, such that the equatorial and ionospheric plasma angular velocities and the field-aligned voltage are given by the functions ω e (F ), ω i (F ) and (F ), respectively.We assume a steady flow, such that the electric field E=−v×B can be described by a scalar potential through E=−∇ .Using Eq. ( 1) we then find that and Taking the field-aligned voltage to be positive when the ionosphere has a higher potential than the equator (the case for  Plots showing the parameters of the current sheet field model employed in this paper (solid lines) compared with values for the planetary dipole field alone (dashed lines).Plot (a) is a log-linear plot of the modulus of the north-south component of the equatorial magnetic field |B ze | in nT threading the equatorial plane, shown versus Jovicentric equatorial radial distance ρ e .We note that the actual values are negative (i.e. the field points south).The solid line shows the field model employed in this paper, given by Eq. (4a), which is based on the CAN-KK model of previous papers.The dotted lines show the CAN and KK models themselves, plotted beyond their intersection for ease of visibility.Plot (b) similarly shows the equatorial flux function of the model field F e in nT R 2 J versus Jovicentric equatorial radial distance ρ e , given by Eq. (4b).The horizontal dotted line shows the value of the flux function at infinity F ∞ .Plot (c) shows the mapping of the field lines between the equatorial plane and the ionosphere, determined from Eq. ( 4).The ionospheric co-latitude of the field line θ i is plotted versus Jovicentric equatorial radial distance ρ e .The horizontal dotted line shows the ionospheric co-latitude of the field line which maps to infinity in the equatorial plane for the current sheet field model.upward-directed electric field and downward precipitating electrons), then Differentiating with respect to F and combining with Eqs.(6a, b) yields This is the equation which relates the ionospheric and equatorial angular velocities in the presence of a field-aligned voltage, and which we refer to as the magnetosphereionosphere decoupling equation.We must also specify how depends on the conditions present in the magnetosphere.Here, in common with previous work, we use Knight's (1973) kinetic theory.This gives the field-aligned voltage required to drive a field-aligned current j i that exceeds the maximum value j io , that can be carried by unaccelerated precipitating magnetospheric electrons alone.For an isotropic Maxwellian electron source population of density N and thermal energy W th (equal to kT e ), j io is Under the usual simplifying assumptions (e.g.Cowley and Bunce, 2001), the minimum field-aligned voltage required to drive a current greater than j io is then In principle, will vary with F on differing flux shells due to variations in the source parameters N and W th .However, in the absence of detailed models, here we employ constant values based on Voyager data, i.e.N =0.01 cm −3 and W th =2.5 keV, as used in our previous papers (Scudder et al., 1981).In this case varies with F due to variations in j i only.Substitution of Eq. (10) into Eq.( 8) then gives This equation is strictly valid only for j i ≥j io .However, this condition is met essentially everywhere in the middle magnetosphere, except in the innermost region where the field-aligned current drops to small values.Here, therefore, we assume that Eq. ( 11) is valid for all j i >0, i.e. throughout the middle magnetosphere.This is equivalent to making the approximation in Eq. ( 10).We also note that the corresponding precipitating energy flux of accelerated electrons is a result due to Lundin and Sandahl (1978).In this expression E f o is the precipitating energy flux of unaccelerated magnetospheric electrons corresponding to field-aligned current j io , given by

Current circuit equations
We now outline the calculation of the current system components illustrated in Fig. 1.This is essentially the same as that given previously by Hill (2001), Cowley and Bunce (2001), Cowley et al. (2002Cowley et al. ( , 2003) ) and Nichols andCowley (2003, 2004), except that we now specifically use the ionospheric plasma angular velocity ω i to derive the ionospheric electric field in the rest frame of the neutral atmosphere, and hence the Pedersen and field-aligned currents.The equatorwarddirected height-integrated Pedersen current is then ) where * J is the angular velocity in the inertial frame of the neutral atmosphere in the Pedersen layer, which is reduced from the planet's angular velocity J (1.76×10 −4 rad s −1 ) due to ion-neutral collisional drag.This slippage can be parameterised by the factor k defined by as introduced by Huang and Hill (1989).The value of k is not well known at present, but recent modelling suggests k≈0.5 under Jovian auroral conditions (Millward et al., 2004).Parameter * P in Eq. ( 15) is the effective value of the heightintegrated Pedersen conductivity, related to the true value P by * P = (1−k) P .In deriving Eq. ( 15) we have also assumed that the polar magnetic field is vertical and equal to 2B J in strength.
Current continuity in the circuit shown in Fig. 1 requires ρ e i ρ =2ρ i i P , taking into account both northern and southern hemispheres, such that the equatorial radial current i ρ is where we have used F =B J ρ 2 i on a flux shell from Eq. (2).We hence find that the total radial current, integrated in azimuth, is which is equal, of course, to twice the azimuth-integrated Pedersen current in each conjugate ionosphere I P .The fieldaligned current density is then calculated from the divergence of either I ρ or I P .Using the former, we have which is the parallel current to be substituted into Eq.( 11).Note that in deriving Eq. ( 19) we have assumed for simplicity that * P is a constant quantity.

Conservation of angular momentum (the Hill-Pontius equation)
The analysis is completed by consideration of conservation of angular momentum of the equatorial plasma.Following Hill (1979) and Pontius (1997), this is described by where Ṁ is the iogenic plasma mass outflow rate, also assumed to be a constant.Substitution of Eq. ( 18) into Eq.( 20) then yields the modified 'Hill-Pontius' equation where we note that the LHS now specifically contains ω e , the angular velocity of the equatorial plasma, while the RHS, representing the ionospheric torque on the equatorial plasma, contains ω i .

Governing equation of the self-consistent problem
There are three equations to be solved; the decoupling equation incorporating Knight's (1973) theory Eq. ( 11), Eq. ( 19) for the parallel current, and the Hill-Pontius equation Eq. ( 21).Substitution of Eq. ( 19) into Eq.( 11) yields We thus introduce the dimensionless parameter ε given by which for typical Jovian parameters * P =0.1 mho, j io ≈0.01 µA m −2 , and W th =2.5 keV, for example, has the value ε≈1.5×10 −5 .Since the first term in the differential vanishes, Eq. ( 22) then becomes where we note that all previous papers cited above have employed the limit ε→0, such that ω e →ω i .Substitution of Eq. ( 24) into the Hill-Pontius equation Eq. ( 21) finally yields the governing equation for ω This is a third order linear inhomogeneous equation for ω i , from which ω e can be obtained from Eq. ( 24), and the current system and field-aligned voltage from Eqs. ( 18), ( 19) and ( 12).

Series solution of the governing equation
The general solution of Eq. ( 25) is the sum of a complementary function which solves the homogeneous equation and contains three arbitrary constants, plus some particular integral.The physical solution which we require here, however, is the particular integral which reduces to our previous solutions in the limit ε→0.This solution may be obtained as a power series in ε where each coefficient (ω i / J ) (n) is a function of F .Substitution of Eq. ( 26) into Eq.( 25) gives from which the required functions are found by equating terms of the same power of ε.For the zeroth order n=0, we have which is just the Hill-Pontius equation solved in previous papers.The solution required is the particular integral for which (ω i / J ) →1 as ρ e →0, i.e. for which the plasma rigidly corotates at small distances.For n≥1 we have which is a first order linear inhomogeneous equation for (ω i / J ) (n) , in which the inhomogeneous term contains the derivative of the solution of the previous order, (ω i / J ) (n−1) .The solutions required of these equations are the particular integrals which satisfy (ω i / J ) (n) →0 as ρ e →0 for all n≥1.In principle, we can then solve Eqs. ( 28) and ( 29) in sequence to any desired order in ε.Here we will obtain solutions up to second order, n=2.
The other parameters of interest are obtained by substitution of Eq. ( 26) into the appropriate equation and equating powers of ε.Thus, for ω e we obtain from Eq. ( 24) and for n ≥1 . (30b) The total radial and field-aligned currents follow from Eqs. ( 18) and ( 19), respectively.The former is given by the power series where  and for n≥1 The field-aligned current is similarly given by where and for n≥1 Since from Eqs. ( 12) and ( 19) we can write in the approximation employed here, we have on substituting Eq. ( 26) for (ω i / J ) i.e. to lowest order the parallel voltage is zero as in previous published solutions, while and for n≥2 Hence, if we evaluate (ω i / J ) and (ω e / J ) to second order, for example, we can determine the parallel voltage to third order, etc.With regard to the precipitating energy flux, we have from Eqs. ( 12) and ( 13) Thus we express E f to the same order as , such that if we determine the plasma flows to a given order, we can compute the precipitating energy flux to the next highest order.

Results
We now present the results of numerical evaluation of the equations in Sect.2, in order to assess the significance of the effects of field-aligned voltages under typical Jovian conditions.We have used typical values of the system parameters * P =0.1 mho and Ṁ=1000 kg s −1 , along with magnetospheric electron source parameters N =0.01 cm −3 and W th =2.5 keV.For these values we find ε≈1.5×10−5 , as indicated above, which is small, such that a power series solution in ε seems appropriate.In Fig. 3 we show the sum of first and second order terms of the series solution for the plasma angular velocity, the azimuth-integrated equatorial radial current, and the ionospheric field-aligned current, all plotted versus distance in the equatorial plane (ionospheric quantities being mapped along the field lines).These represent the amounts by which the previously-published zeroth order solutions are modified by the inclusion of field-aligned voltages (hence the " " notation on the vertical axis labels).The solid line in Fig. 3a shows the change in ionospheric angular velocity  ( ω i / J ) calculated from Eq. ( 29), while the long-dashed line similarly shows the change in the equatorial plasma angular velocity ( ω e / J ) obtained from Eq. (30b).It can be seen that the ionospheric plasma angular velocity will be modestly reduced in the region within ∼35 R J , and elevated thereafter, while the angular velocity of the equatorial plasma will be raised over most of the region.Using techniques introduced by Nichols and Cowley (2003) (see their Eq.7), it is possible to show that to a first order in ε the solutions for small ρ e are and That is, to a first order the ionospheric angular velocity, and hence the current system, is unaffected by the field-aligned voltages.This is because to the lowest order the currents in the inner region are just such as to maintain rigid corotation of the equatorial plasma, and may be deduced by putting (ω e / J ) =1 in the LHS of the Hill-Pontius equation, Eq. ( 20).The equatorial angular velocity is modified to a first order in ε, however, falling less rapidly with distance than when the effect of the field-aligned voltages are neglected.The main point we wish to emphasise with regard to Fig. 3a, however, is that for typical Jovian parameters, the changes in the normalised angular velocity are small compared with unity.The difference in angular velocity between the equatorial plane and the ionosphere due to field-aligned voltage decoupling is typically a few thousandths of the planetary angular velocity.Figure 3b similarly shows the sum of first and second order terms for the azimuth-integrated equatorial radial current I ρ given by Eq. (31c).The profile essentially mirrors that of the ionospheric angular velocity as expected, such that the current is slightly raised within ∼35 R J and decreased beyond.Figure 3c shows the sum of first and second order terms for the field-aligned current j i given by Eq. (32c), mapped to the equatorial plane.It can be seen that the field-aligned current is reduced over most of the middle magnetosphere.We note that the rapid variations of these profiles in the inner region results from the B ze model used, which exhibits rather sharp behaviour at ∼20 R J in the transition region between the dipolar form and the power law (Fig. 2a).Such variations do not occur if a simple dipole model is used, though the overall nature of the results remains similar.We also note that the quantities shown in Fig. 3 are dominated by the first order term, which for all parameters is typically an order of magnitude larger than the second order term.
The effect of including these terms on the overall solution for the angular velocity and currents is shown in Figs. 4  and 5. Figure 4 shows these parameters plotted versus radial distance in the equatorial plane, while Fig. 5 shows profiles mapped along field lines to the ionosphere and plotted versus co-latitude.The solid lines in the figures show the sums of the zeroth, first, and second order terms for the angular velocity and current parameters (i.e. the zeroth order terms summed with the profiles shown in Fig. 3), while the shortdashed lines show the zeroth order terms alone for comparison.Figure 4 shows (a) the equatorial plasma angular velocity (ω e / J ), (b) the azimuth-integrated equatorial radial current I ρ , (c) the ionospheric field-aligned current j i , and (d) a log-linear plot of the equatorial electrostatic potential ⊥e computed from the equatorial plasma angular velocity profiles shown in Fig. 4a using where the arbitrary zero of potential is taken to be at the outer edge of the solution at 100 R J .Also shown in Fig. 4d is the field-aligned voltage computed from the respective orders of the plasma angular velocity using Eqs.(33c, d), such that this is shown to first and third orders by the dashed and solid lines respectively (recalling that the zeroth order field-aligned voltage is zero).It can be seen directly from these plots that the effect of the self-consistent inclusion of the field-aligned voltage in the solution is small.The equatorial angular velocity profiles (and hence the equatorial electrostatic potential profiles) are closely similar, such that their difference cannot be distinguished on this scale, while the magnitudes of the perpendicular and parallel currents are slightly reduced in the outer region, corresponding to the negative values of I ρ and j i in Fig. 3.We also note from Fig. 4d that the drop in ⊥e across the middle magnetosphere, between ∼20 R J and the outer edge of the model at 100 R J , is ∼3 MV.This is approximately two orders of magnitude larger than .This formed the basis of Cowley and Bunce's (2001) conjecture that the field-aligned voltage would have little effect on the solutions, as is now confirmed quantitatively here.
Figure 5 similarly shows the system parameters plotted in the ionosphere versus co-latitude.Specifically, we show (a) the ionospheric plasma angular velocity (ω i / J ), (b) the ionospheric field-aligned current j i , (c) the field-aligned voltage (also shown by the 'lower' line in Fig. 4d), and (d) the precipitating energy flux E f calculated from the fieldaligned voltages shown in Fig. 5c using Eq. ( 34).As with the equatorial parameters, the effect of the higher order terms is seen to be small.The difference between the ionospheric angular velocity profiles is almost indistinguishable, while the peaks in Figs.5b, c and d are reduced in amplitude by values approximately two orders of magnitude below the zeroth order results, and are shifted equatorward by ∼0.1 • , which is small with respect to the ∼2 • width of the peaks.

Summary and conclusion
In this paper we have considered the magnetosphereionosphere coupling current system that flows in Jupiter's middle magnetosphere, which is believed to be associated with the Jovian main auroral oval.Previous analyses have assumed a perfect mapping of the electric field and flow along equipotential field lines between the ionosphere and the magnetosphere, while it is known that substantial field-aligned voltages must exist to drive the currents responsible for the main oval auroras.Cowley and Bunce (2001) suggested that the effect of the field-aligned voltages on the solutions would not be great, but did not compute quantitative results.In this paper we have self-consistently incorporated the fieldaligned voltages into the analysis of the system, such that the plasma angular velocities in the magnetosphere and the ionosphere are in general different, with consequences for the currents flowing in the system.The field-aligned voltages were incorporated using Knight's (1973) kinetic theory, and a third order linear inhomogeneous equation for the ionospheric plasma angular velocity was derived (Eq.25) that can be solved as a power series under appropriate conditions.The zeroth order solution corresponds to those which have been obtained previously, in which there is no decoupling between the ionosphere and magnetosphere.Higher orders then introduce decoupling due to field-aligned voltages, such that the ionospheric and equatorial plasma angular velocity profiles are modified, as are the resulting current profiles.Here the solution has been taken to a second order.The results of numerical evaluation show that, for parameters which are representative of Jovian middle magnetosphere conditions, the decoupling effect of the field-aligned voltages is small.The equatorial and ionospheric plasma angular velocity profiles differ by only a few thousandths of the planetary angular velocity, while the currents, and hence the auroral parameters, are slightly reduced in magnitude by up to a few percent.Our most important conclusion, however, is that our calculations have confirmed the essential validity of previously-published results that did not self-consistently include the decoupling effect of field-aligned voltages in the Jovian middle magnetosphere. Fig.2

Fig
Fig. 2.Plots showing the parameters of the current sheet field model employed in this paper (solid lines) compared with values for the planetary dipole field alone (dashed lines).Plot (a) is a log-linear plot of the modulus of the north-south component of the equatorial magnetic field |B ze | in nT threading the equatorial plane, shown versus Jovicentric equatorial radial distance ρ e .We note that the actual values are negative (i.e. the field points south).The solid line shows the field model employed in this paper, given by Eq. (4a), which is based on the CAN-KK model of previous papers.The dotted lines show the CAN and KK models themselves, plotted beyond their intersection for ease of visibility.Plot (b) similarly shows the equatorial flux function of the model field F e in nT R 2 J versus Jovicentric equatorial radial distance ρ e , given by Eq. (4b).The horizontal dotted line shows the value of the flux function at infinity F ∞ .Plot (c) shows the mapping of the field lines between the equatorial plane and the ionosphere, determined from Eq. (4).The ionospheric co-latitude of the field line θ i is plotted versus Jovicentric equatorial radial distance ρ e .The horizontal dotted line shows the ionospheric co-latitude of the field line which maps to infinity in the equatorial plane for the current sheet field model. Fig.3

Fig. 3 .
Fig.3.Plots of the sums of the first and second order terms in the series solutions for (a) the normalised equatorial (dashed line) and ionospheric (solid line) plasma angular velocities, (b) the total azimuth-integrated equatorial radial current in MA, and (c) the ionospheric field aligned current in µA m −2 , all plotted versus equatorial radial distance (with ionospheric parameters being mapped along field lines).The system parameters employed were * P =0.1 mho and Ṁ=1000 kg s −1 , with magnetospheric electron source parameters N=0.01 cm −3 and W th =2.5 keV.

Fig. 5 .
Fig. 5. Plots of the sums of the zeroth, first, and second (and third in the case of the field-aligned voltage) order terms for the system parameters.All parameters are plotted versus ionospheric co-latitude θ i .For comparison, the zeroth order solution is shown by the shortdashed lines.Panel (a) shows the normalised ionospheric plasma angular velocity, (b) the field-aligned current density in µA m −2 , (c) the field-aligned voltage in kV, and (d) the precipitating energy flux (calculated from the field-aligned voltage profile shown in panel c) in mW m −2 .