c European Geosciences Union 2003
Annales Geophysicae (2003) 21: 1419–1441 Annales
Geophysicae
Magnetosphere-ionosphere coupling currents in Jupiter’s middle
magnetosphere: dependence on the effective ionospheric Pedersen
conductivity and iogenic plasma mass outflow rate
J. D. Nichols1 and S. W. H. Cowley1
1 Department
of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK
Received: 27 August 2002 – Revised: 9 December 2002 – Accepted: 20 February 2003
Abstract. The amplitude and spatial distribution of the coupling currents that flow between Jupiter’s ionosphere and
middle magnetosphere, which enforce partial corotation on
outward-flowing iogenic plasma, depend on the values of the
effective Pedersen conductivity of the jovian ionosphere and
the mass outflow rate of iogenic plasma. The values of these
parameters are, however, very uncertain. Here we determine how the solutions for the plasma angular velocity and
current components depend on these parameters over wide
ranges. We consider two models of the poloidal magnetospheric magnetic field, namely the planetary dipole alone,
and an empirical current sheet field based on Voyager data.
Following work by Hill (2001), we obtain a complete normalized analytic solution for the dipole field, which shows in
compact form how the plasma angular velocity and current
components scale in space and in amplitude with the system parameters in this case. We then obtain an approximate
analytic solution in similar form for a current sheet field in
which the equatorial field strength varies with radial distance
as a power law. A key feature of the model is that the current sheet field lines map to a narrow latitudinal strip in the
ionosphere, at ≈ 15◦ co-latitude. The approximate current
sheet solutions are compared with the results of numerical
integrations using the full field model, for which a power law
applies beyond ≈ 20 RJ , and are found to agree very well
within their regime of applicability. A major distinction between the solutions for the dipole field and the current sheet
concerns the behaviour of the field-aligned current. In the
dipole model the direction of the current reverses at moderate equatorial distances, and the current system wholly closes
if the model is extended to infinity in the equatorial plane
and to the pole in the ionosphere. In the approximate current sheet model, however, the field-aligned current is unidirectional, flowing consistently from the ionosphere to the
current sheet for the sense of the jovian magnetic field. Current closure must then occur at higher latitudes, on field lines
outside the region described by the model. The amplitudes
Correspondence to: J. D. Nichols: (jdn@ion.le.ac.uk)
of the currents in the two models are found to scale with the
system parameters in similar ways, though the scaling is with
a somewhat higher power of the conductivity for the current
sheet model than for the dipole, and with a somewhat lower
power of the plasma mass outflow rate. The absolute values
of the currents are also higher for the current sheet model
than for the dipole for given parameters, by factors of ≈ 4 for
the field-perpendicular current intensities, ≈ 10 for the total
current flowing in the circuit, and ≈ 25 for the field-aligned
current densities, factors which do not vary greatly with the
system parameters. These results thus confirm that the conclusions drawn previously from a small number of numerical
integrations using spot values of the system parameters are
generally valid over wide ranges of the parameter values.
Key words. Magnetospheric physics (current systems,
magnetosphere-ionosphere interactions, planetary magnetospheres)
1
Introduction
The dynamics of Jupiter’s middle magnetosphere are dominated by the processes that couple angular momentum between the planet’s atmosphere and the equatorial plasma that
flows outwards from the Io source at ≈ 6 RJ (Jupiter’s radius, RJ , is ≈ 71 373 km) (Hill, 1979; Siscoe and Summers,
1981; Hill et al., 1983; Belcher, 1983; Vasyliunas, 1983;
Bagenal, 1994). The field and plasma structures envisaged
are sketched in Fig. 1, where the arrowed solid lines indicate
magnetic field lines, while the dots indicate the region occupied by dense rotating iogenic plasma. The region of flux
tubes threading this plasma disc constitutes Jupiter’s middle
magnetosphere, where the field lines are characteristically
distended outward by azimuthal plasma currents associated
with radial stress balance. In the inner region, the iogenic
plasma approximately corotates with the planet, but as it
moves outward its angular velocity falls, as the inverse square
of the distance if no torques act. However, when the angular
1420
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
WJ
w
Bj
WJ*
Io
Bj
Fig. 1. Sketch of a meridian cross section through Jupiter’s inner
1
and middle magnetosphere, showing the principal physicalFig
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).
velocity of the plasma and field lines (ω in Fig. 1) falls below that of the planet (J ), or more specifically, below that
of the neutral upper atmosphere (∗J ), a frictional torque is
imposed on the feet of the field lines due to ion-neutral collisions in the Pedersen-conducting layer of the ionosphere.
This torque acts to spin the flux tubes and equatorial plasma
back up towards rigid corotation with the planet, so that in
the steady state the plasma angular velocity falls less quickly
with distance than as the inverse square. At the same time,
the equal and opposite torque on the neutral atmosphere results in atmospheric sub-corotation (“slippage”) in the Pedersen layer, so that ω < ∗J < J (Huang and Hill, 1989). The
spin-up torque on the plasma is communicated to the equatorial region by the magnetic field, which becomes bent out
of meridian planes into a “lagging” configuration, associated
with the azimuthal fields Bϕ shown in the figure. The corresponding magnetosphere-ionosphere coupling current system, of primary interest here, is shown by the arrowed dashed
lines in Fig. 1. The current flows radially outward across
the field in the equatorial plane, such that the torque associated with the j × B force accelerates the plasma in the sense
of Jupiter’s rotation. In the ionosphere the Pedersen current
flows equatorward in both hemispheres, producing an equal
and opposite torque which balances the torque due to ionneutral collisions. The current circuit is closed by a system
of field-aligned currents which flow from the ionosphere to
the equator in the inner part of the system, and return in the
outer part.
The steady-state angular velocity profile of the out-flowing
equatorial plasma was first calculated on the above basis by
Hill (1979). In this study the poloidal field was taken to be
that of the planetary dipole alone, such that the radial distension of the middle magnetosphere field lines shown in Fig. 1
was not taken into account. This restriction was later removed by Pontius (1997), who introduced empirical poloidal
field models into the calculations. He found that the solutions for the plasma angular velocity profile are remarkably
insensitive to the field model employed. Although consid-
ered implicitly, the properties of the coupling currents were
not calculated in these studies. Recently, however, attention
has focussed directly on the currents, it being suggested by a
number of authors that the ring of upward field-aligned current surrounding each magnetic pole is associated with the
“main oval” observed in Jupiter’s auroras (Bunce and Cowley, 2001; Cowley and Bunce, 2001; Hill, 2001; Southwood
and Kivelson, 2001). Hill (2001) considered the currents in
his original dipole problem, while Cowley and Bunce (2001)
calculated the currents for both a dipole field and an empirical current sheet field. The latter authors found that the fieldaligned current densities are an order of magnitude larger for
the current sheet model than for the dipole. The physical origins of this effect have been discussed further by Cowley et
al. (2002, 2003), who show that it relates to the fact that the
field lines on which corotation breaks down, while mapping
to similar distances in the equatorial plane, map in the ionosphere to a narrower range of latitudes further from the pole
for a current sheet field than for the dipole.
The solutions for the plasma angular velocity and the current depend on two system parameters, the “effective” value
of the height-integrated ionospheric Pedersen conductivity
(possibly reduced from the true value by the atmospheric
“slippage” mentioned above), and the plasma mass outflow
rate from the Io torus. However, neither of these parameters
is well determined at present. Estimates of the conductivity
have ranged from ≈ 0.1 to ≈ 10 mho (Strobel and Atreya,
1983; Bunce and Cowley, 2001), while estimates of the
mass outflow rate have ranged from ≈ 500 to ≈ 2000 kg s−1
(Broadfoot et al., 1981; Hill et al., 1983; Vasyliunas, 1983;
Khurana and Kivelson, 1993; Bagenal, 1997). The purpose
of the present paper is to examine how the solutions for the
plasma angular velocity and coupling currents depend on
these two parameters for both dipole and current sheet field
models. Some general results for the dipole model have been
presented previously by Hill (2001). For the current sheet
field model, however, only a few numerical solutions using
“reasonable” spot values of the system parameters have been
published to date (Cowley and Bunce, 2001; Cowley et al.,
2002, 2003). In this paper we first provide a complete solution for the dipole field, before going on to examine related
results for a current sheet model in which the equatorial field
is taken to vary with distance as a power law. The parameter
ranges considered are 0.1–10 mho for the “effective” conductivity, and 100–10 000 kg s−1 for the mass outflow rate. The
work presented here thus shows how the coupling current
system depends on the system parameters over a wide range
of values, here taken as constant quantities in a given solution. This knowledge should provide valuable background
for more complex future studies in which the system parameters are taken to vary in time and/or space, as may more
generally be the case.
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
2
Basic equations
In this section we first summarise the basic equations governing the magnetosphere-ionosphere coupling current system
depicted in Fig. 1, and then discuss the nature of the solutions
at small and large distances. Derivations were given earlier
by Hill (1979), Vasyliunas (1983) and Pontius (1997), and
have most recently been discussed by Hill (2001), Cowley
and Bunce (2001), and Cowley et al. (2003). Only the central results will, therefore, be stated here, together with an
outline of the assumptions and approximations which have
been made.
We first assume that the magnetic field is axisymmetric, such that the poloidal components can be specified by
a flux function F (ρ, z) related to the field components by
B = (1/ρ)∇F × ϕ̂, 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.
In this case F =constant defines a flux shell, such that magnetic mapping between the equatorial plane (subscript “e”)
and the ionosphere (subscript “i”), as required here, is simply achieved by writing Fe = Fi . Neglecting non-dipole
planetary fields and the small perturbations due to magnetospheric currents, the flux function in the ionosphere is taken
to be
Fi = BJ ρi2 = BJ RJ2 sin2 θi ,
(1)
where ρi is the perpendicular distance from the magnetic
axis, θi the magnetic co-latitude, and BJ the dipole equatorial magnetic field strength (taken to be 4.28 × 105 nT in
conformity with the VIP 4 internal field model of Connerney
et al., 1998). The absolute value of F has been fixed by taking F = 0 on the magnetic axis. The flux function in the
equatorial plane is then obtained by integrating
Bze
1 dFe
=
,
ρe dρe
(2)
where Bze (ρe ) is the north-south field threading the current
sheet.
Assuming for simplicity that the magnetic and planetary
spin axes are co-aligned, the equatorward-directed heightintegrated Pedersen current in the ionosphere is given by
ω iP = 26P∗ BJ J ρi 1 −
J
s
Fi
ω = 26P∗ BJ J
1−
,
(3)
BJ
J
where we have taken the polar magnetic field to be nearvertical and equal to 2BJ in strength (an approximation valid
to within ≈ 5% in our region of interest). In this expression,
6p∗ is the “effective” height-integrated ionospheric Pedersen
conductivity, reduced from the true value 6p = 6p∗ / (1-k) by
neutral atmosphere “slippage” as mentioned above, where
parameter k (whose value 0 < k < 1 is also uncertain at
present) is related to the angular velocities by (J − ∗J ) =
1421
k(J − ω) (Huang and Hill, 1989). The radial current intensity in the equatorial plane iρ , integrated through the full
thickness of the sheet, then follows from the current continuity requirement ρe iρ = 2ρi iP (assuming symmetry between
the two hemispheres)
46P∗ J Fe ω 1−
,
(4a)
iρ =
ρe
J
so that the total equatorial radial current, integrated in azimuth, is
ω Iρ = 2πρe iρ = 8π6P∗ J Fe 1 −
,
(4b)
J
equal, of course, to twice the azimuth-integrated total Pedersen current IP = 2 πρi iP flowing in each conjugate ionosphere. The field-aligned current density follows from the divergence of either of these field-perpendicular currents. Differentiating the equatorial current gives
j dIρ
1
k
=
=
B
4πρe |Bze | dρe
F d ω ω e
−26P∗ J
+ 1−
, (5a)
ρe |Bze | dρe J
J
where we have put Bze = −|Bze |, since the jovian field is
negative (points south) at the equator, 6P∗ has been taken to
be constant, and the sign of jk is appropriate to the Northern Hemisphere (as employed throughout). The quantity
(jk /B) is constant along field lines between the equator and
the ionosphere in the assumed absence of significant fieldperpendicular currents in the intervening region. The fieldaligned current density just above the ionosphere is then
given by
j k
jki = 2BJ
,
(5b)
B
using the same approximation for the polar field as indicated
above.
The analysis is completed by determining the steady state
angular velocity profile of the equatorial plasma. Following
Hill (1979, 2001), Pontius (1997), and Cowley et al. (2002),
Newton’s second law applied to a steady outflow of plasma
from the Io torus gives
2πρe2 iρ |Bze |
d
(ρe2 ω(ρe )) =
,
dρe
Ṁ
(6a)
where Ṁ is the plasma mass outflow rate. Expanding and
substituting Eq. (4a), we find
ω ρe d ω ω 4π 6P∗ Fe |Bze | +
=
1−
,
2 dρe J
J
J
Ṁ
(6b)
an equation we refer to as the Hill-Pontius equation (though
here slightly simplified, as in Hill (2001), by taking the ionospheric field strength to be equal to 2BJ ). It is a first order
linear equation for ω that can be solved with the use of one
boundary condition. We note with Hill (2001) that if the angular velocity profile obeys this equation, the derivative may
1422
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
be substituted directly into Eq. (5a) to yield a form for the
field-aligned current which involves (ω/ J ) only.
It is an important general property of the physically interesting solutions of the above equations that at small radial
distances the currents depend on Ṁ and not on 6P∗ , while
at large radial distances they depend on 6P∗ and not on Ṁ.
The small-ρe approximations follow from a series solution
of Eq. (6b) for the case in which (ω/ J )=1 at ρe = 0 (such
that the plasma rigidly corotates at small distances). Taking
(Ṁ/6P∗ ) as the formal expansion parameter, we write
n
∞
X
Ṁ
ω
=
an
,
J
6P∗
n=0
(7a)
and substituting into Eq. (6b) and equating coefficients of
powers of (Ṁ/6P∗ ) we find that a0 = 1, and that for n ≥ 1
the an are determined by the recurrence relation
an+1 = −
1
d
(ρ 2 an ).
8πρe Fe |Bze | dρe e
(7b)
Thus, the leading term describing the breakdown of rigid
corotation in the inner region which we take as our smallρe (‘S’) approximation is
Ṁ
ω
=1−
,
(8)
J S
4π 6P∗ Fe |Bze |
as given previously (but not derived in this manner) by Cowley et al. (2003). We note that the departure from rigid corotation is proportional to Ṁ and inversely proportional to 6P∗ .
When substituted into Eqs. (3)–(5) to find the corresponding
approximations for the currents, we then find that the currents on a given field line depend only on Ṁ and not on 6P∗
iP S =
ṀJ
,
2πρi |Bze |
iρ S =
ṀJ
,
πρe |Bze |
j k
B
=−
S
Iρ S =
ṀJ
2πρe |Bze
ṀJ
,
|Bze |
(9a, b)
2ṀJ
,
|Bze |
(9c, d)
IP S =
|3
d|Bze |
,
dρe
(9e)
and
jki S = −
ṀJ BJ d|Bze |
.
πρe |Bze |3 dρe
(9f)
These expressions can also be derived directly by substituting ω = J into the left side of Eq. (6a), i.e. they are just
the currents required to maintain near-rigid corotation in the
inner region.
The large-ρe (‘L’) approximations are simply obtained by
putting (ω/J )L = 0 into Eqs. (3)–(5) to find
iP L = 26P∗ BJ J ρi ,
(10a)
IP L = 4π 6P∗ BJ J ρi2 ,
(10b)
iρ L =
46P∗ J Fe
ρe
(10c)
Iρ L = 8π 6P∗ J Fe ,
j k
B
L
= −26P∗ J ,
(10d)
and
jki L = −46P∗ BJ J ,
(10e)
(10f)
which thus depend only on 6P∗ and not on Ṁ.
3
Plasma angular velocity and coupling current system
for a dipole magnetic field model
Following the earlier work of Hill (1979, 2001) and Cowley
and Bunce (2001, 2003), in this section we provide a complete analytic solution for the case where the poloidal field is
taken to be the planetary dipole alone, showing how the solutions for the angular velocity and current components scale
in space and in amplitude with the system parameters. These
results form a useful introduction to, and point of comparison
with, the results for the current sheet field to be presented in
the Sect. 4. Using Eq. (2), for the dipole field we have
R 3
J
Bze dip = −BJ
and hence
(11a)
ρe
Fe dip =
BJ RJ3
.
ρe
(11b)
Substituting these into Eq. (6b), the Hill-Pontius equation for
the dipole field is
ω
ω
RDe 4
ω
ρe d
+
=2
1−
,
(12)
2 dρe J
J
ρe
J
where RDe is the equatorial “Hill distance” for the dipole
field (subscript ‘D’), given by
2π6P∗ BJ2 RJ2 1/4
RDe
=
.
(13)
RJ
Ṁ
It can thus be seen that the angular velocity in this case is
a function only of (ρe /RDe ), so that the solutions scale with
equatorial distance as RDe and hence with the system parameters as (6P∗ /Ṁ)1/4 . The general solution of Eq. (12) can be
obtained by the integration factor method (Hill, 1979)
!2 "
!4 #
√
RDe
ω
RDe
= π
exp
×
J
ρe
ρe
"
"
!2 #
#
RDe
× erfc
+K ,
(14)
ρe
where erfc(z) is the complementary error function (related to
the error function erf(z) by erfc(z) = 1 - erf(z)), and K is a
constant of integration. All solutions diverge at the origin
except the solution with K = 0. This special solution rigidly
corotates (i.e. (ω/ J ) = 1) when (ρe /RDe ) goes to zero,
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
1423
wêWJ
and is the solution derived previously by Hill (1979, 2001).
Mapping to the ionosphere is achieved by equating the flux
functions given by Eq. (1) and (11b), such that
s
ρi
RJ
sin θi =
=
.
(15)
RJ
ρe
1wêWJ
1
0.8
0.8
0.6
0.6
0.4
Introducing an ionospheric counterpart of the “Hill distance”
given by
s
!1/8
RJ
Ṁ
RJ ,
(16)
RDi =
RJ =
RDe
2π 6P∗ BJ2 RJ2
we then find that the angular velocity mapped to the ionosphere is a function only of (ρi /RDi ), where
s
ρi
RDe
=
,
(17)
RDi
ρe
such that the solutions scale with distance from the magnetic
pole as (Ṁ/6P∗ )1/8 . With regard to physical units, introduction of the constant quantities given above yields the following values for the equatorial and ionospheric scale lengths
!1/4
6P∗ (mho)
RDe
≈ 49.21
(18a)
RJ
Ṁ(103 kg s−1 )
and
RDi
≈ 0.1426
RJ
Ṁ(103 kg s−1 )
6P∗ (mho)
!1/8
,
(18b)
such that for system parameters at the centre of the ranges
mentioned in the Introduction, i.e. 6P∗ = 1 mho and Ṁ =
1000 kg s−1 , we find RDe ≈ 49.2 RJ and RDi ≈ 0.14 RJ
(corresponding to a co-latitude of 8.2◦ ). If we fix Ṁ at this
value and allow 6P∗ to increase from 0.1 to 10 mho, we find
that RDe increases from 27.7 to 87.5 RJ , while RDi decreases
from 0.19 to 0.11 RJ (co-latitudes between 11.0◦ and 6.1◦ ).
Similarly, if we fix 6P∗ at 1 mho and allow Ṁ to increase
from 100 to 10 000 kg s−1 , we find that RDe and RDi vary
over the same ranges but in the reversed sense. Thus because RDe and RDi depend on the system parameters only
as the quarter and eighth powers, respectively, they change
only by modest factors as the system parameters vary widely.
We note that the equatorial scales are comparable to the radial scale of the jovian middle magnetosphere, which extends
to distances between ∼40 and ∼100 RJ , depending on local
time and the state of the magnetosphere.
The solid lines in Figs. 2a and b show the normalised angular velocity solution with K = 0 plotted versus (ρe /RDe )
in the equatorial plane, and versus (ρi /RDi ) in the ionosphere, respectively. Near-rigid corotation is maintained to
(ρe /RDe ) ≈ 0.5, beyond which (ω/ J ) decreases rapidly,
reaching 0.5 when (ρe /RDe ) ≈ 1.52. We also note that solutions started with non-zero K within (ρe /RDe ) ≤ 0.5 converge very rapidly onto this solution at larger distances, such
that the solutions are only weakly dependent on the choice
0.4
0.2
0.2
re êRDe
re êRDe
1
2
3
4
5
1
2
3
4
5
0.5
1
1.5
2
0.5
1
1.5
2
ri êRDi
2.5 Fig 2b
wêWJ
(a)
1wêWJ
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2.5
ri êRDi
(b)
2b
Fig. 2. Plots of the steady-state plasma angular velocity profileFigfor
a
dipole magnetic field, shown (a) versus normalised radial distance
in the equatorial plane (ρe /RDe ), and (b) versus normalised distance from the magnetic axis in the ionosphere (ρi /RDi ) . The
solid line in each case shows the full solution obtained from Eq. (14)
with K = 0, such that the plasma rigidly corotates at small radial distances. The long-dashed lines show the small ρe form
given by Eq. (19), while the corresponding large ρe form is just
(ω/J )L = 0. The downward-pointing tick marks indicate the
limits of validity of both these approximations, as defined in the
text. The short-dashed lines show the higher-order large ρe form
given by Eq. (20), whose limit of validity (as also defined in the text)
is indicated by the upward-pointing tick mark. The horizontal dotted lines indicate the condition for rigid corotation, (ω/ J ) = 1.
of boundary condition in this case (Cowley et al., 2003). The
dashed lines in Fig. 2 show some approximate forms, with
the tick marks indicating their regimes of validity. The longdashed lines show the small-ρe (‘S’) approximation given by
Eq. (8), which in normalised form becomes
ω
J
!
S
1 ρe
=1−
2 RDe
!4
1 RDi
=1−
2 ρi
!8
.
(19)
(We note that the series generated by Eq. (7) is the same as
that obtained by asymptotic expansion of the error function
in Eq. (14) for large argument.) The approximate form falls
away from rigid corotation more quickly than the full solution, and reaches zero, equal to the large-ρ
e (‘L’) approxima√
tion (ω/ J )L = 0 at (ρe /RDe ) = 4 2 ∼ 1.189. We define
1424
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
Table 1. Principal features of the plasma angular velocity and coupling current system for a dipole field in normalised units
ρe
RDe
Feature
Maximum
upward field-aligned
current
density
(jk /B)/(jk /B)D
= jki /jkiD
≈ 0.6111
max
max
ρi
RDi
ω
J
1.0203
0.9900
0.7467
Maximum
sheet-integrated equatorial radial current
iρ /iρD
≈ 0.9809
max
Plasma angular velocity falls to ω = 0.5
1.1034
0.9520
0.7014
1.5201
0.8111
0.5
Maximum
total current
azimuth-integrated
Iρ /IρD
= 2 IP /IP D
≈ 8.404
1.7409
0.7579
0.4178
2.5674
0.6241
0.2284
∞
0
0
J
max
max
Field-aligned current passes through zero
Maximum
ionospheric Pedersen
height-integrated
current iP /iP D
≈ 0.9631
max
Maximum
downward
current
density
field-aligned
(jk /B)/(jk /B)D
= jki /jkiD
= –2
min
min
the limits of validity of these approximations as being the
points where (1 − (ω/ J )S,L ) = 1.1(1 − (ω/ J )), such
that the departure from rigid corotation given by the approximate form exceeds that of the full solution by 10% of the
latter. These limiting positions are shown by the downwardpointing tick marks in Fig. 2. The short-dashed lines in the
figure also show a higher-order large-ρe form (‘L0 ’), in which
the plasma angular velocity falls with distance as ρe−2 , due
to negligible ionospheric torque. Noting that both the exponential and the error functions go to unity in Eq. (14) as
ρe → ∞, we find with Hill (1979) that for K = 0
!
!2
√
ω
RDe
= π
.
(20)
J 0
ρe
L
The limit of validity of this approximation is similarly defined as the point where (1−(ω/ J )L0 ) = 0.9(1−(ω/ J )),
and is marked by the upward-pointing tick marks in Fig. 2.
The normalised solutions for the current components then
follow from Eqs. (3)–(5), giving
iP
ρi
ω
=2
1−
,
where
iP D
RDi
J
!1/8
!3/4
6P∗ 7 Ṁ
iP D =
BJ RJ
J ,
(21a)
2π
!2
!
IP
ω
ρi
= 4π
1−
,
where
IP D
RDi
J
!1/4
!1/2
6P∗ 3 Ṁ
3
IP D =
BJ RJ
J ,
(21b)
2π
iρ
RDe
=4
iρ D
ρe
6P∗ Ṁ
2π
iρD =
!2
where
!1/2
6P∗ 3 Ṁ
2π
(21c)
J ,
Iρ
RDe
= 8π
IρD
ρe
IρD =
!
ω
1−
,
J
!
!
ω
1−
,
J
!1/4
where
!1/2
BJ R J 3
J ,
(21d)
"
!
!4 !
!#
jk /B
ω
RDe
ω
=2 2
− 1+4
1−
,
J
ρe
J
jk /B D
where
jk /B
D
= 6P∗ J ,
(21e)
and
"
!
!8 !
!#
jki
ω
ρi
ω
=2 2
− 1+4
1−
,
jkiD
J
RDi
J
where
jkiD = 26P∗ BJ J .
(21f)
We note that Eqs. (21b) and (21d) are equivalent to Hill’s
(2001) Eq. (A13), while Eqs. (21e) and (21f) are the same as
Hill’s Eq. (A12). These normalised forms are plotted as solid
1
1
0.75
0.75
0.5
0.5
J. D. Nichols
and S. W. H. Cowley: Magnetosphere-ionosphere0.25
coupling currents in Jupiter’s middle magnetosphere
0.25
PD
IPiPêIêiPD
0.5
1
1.5
2
2.5
riêRDi
IPIFig
êIPD
r êI rD
3a
2
10
1.75
10
20
1.5
8
8
15
1.25
1
1.5
2
2.5
riêRDi
Fig 3a
6
61
10
4
0.75
4
0.5
2
0.25
IPiIrêI
PD
rêiêIrD
rD
0.5
0.5
11
1.5
1.5
riêRDi
2.5 riêRDi
2.5
22
25
(a)
êi rD3a
rêI
IirFig
rD
Figj3b
H j3»» êBLêH
»» êBLD
3
10
20
2.5
20
2.5 1
8
15
2
0.5
152
6
1.5
10
4
1
0.5
1
i rêi rD
H j3»» êBLêH j»» êBLD
0.51
1
12
2
1.53
3
2
4
riêRDi
2.5 re êRDe
5
(c)
0.5
1
2
3
4
5
re êRDe
1
1
1
0.5
1
-0.5
-0.5
riêRDi
2.5 re êRDe (b)
5
2
4
2
3
4
2
2
3
3
1
2
1.5
3
4
4
2
4
5
re êRDe
re êRDe
5 re êRDe
5
(d)
Fig
Fig 3c
3e
Fig 3d
ri êRDi
2.5 re êRDe
5
-1
-1
0.5
-1.5
j»»i ê j»»iD
-2
Fig 3c
3b3d
j»»iFig
êFig
j»»iD
H j»» êBLêH
j»» êBLD
1
1
0.5
0.5
-0.5
1
-1
-2
1.5
3
1
-1
5
0.5
-1.5
2.5 1
1.5
1
2
Fig
Fig3b
3d
1.5
10
-0.5
25
0.5
2
0.5
1425
1
2
3
4
5
re êRDe
1
-1.5
-1.5
(e)
-2
-2
Fig
Fig3c3e
j»»i ê j»»iD
(f)
Fig 3e
Fig 3f
1
Fig. 3. Plots of normalised steady-state current components for a dipole magnetic field, plotted versus normalised equatorial radial distance
(ρe /RDe ), or0.5normalised distance from the magnetic axis (ρi /RDi )0.5in the ionosphere, as appropriate. The plots show (a) the heightintegrated ionospheric Pedersen current intensity, (b) the azimuthri êRDi and height-integrated total ionospheric Pedersen current, (c) the current
ri êRDi
0.5
1
1.5
2
2.5
sheet-integrated equatorial
radial current
intensity,
(d)
the azimuthand current sheet-integrated
total1.5equatorial2 radial current,
(e) the equa0.5
1
2.5
-0.5
torial field-aligned current density per unit magnetic field strength, and
-0.5(f) the field-aligned current density just above the ionosphere. The
corresponding-1normalization constants are given by Eq. (21). The solid lines show the full solutions given by Eqs. (14) and (21), while √
the
4
-1
long-dashed lines show the small√
and large ρe forms (the ‘S’ and ‘L’ approximations),
shown to their point of intersection at (ρe /RDe ) = 2
-1.5
8
(or equivalently (ρi /RDi ) = 1/ 2), where (ω/J )S = (ω/J )L =
0. The downward-pointing tick marks show the limits of validity of
-1.5
these approximate
forms, as shown in Fig. 2. The short-dashed lines show the higher-order large ρe form (the ‘L0 ’ approximation) obtained
-2
-2
from Eq. (20), whose limit of validity is indicated by the upward-pointing
tick mark, as also shown in Fig. 2.
Fig 3f
Fig 3f
lines in Fig. 3 versus either (ρe /RDe ) or (ρi /RDi ) as appropriate. The values and positions of principal features are also
tabulated in Table 1 in normalised units, and in Table 2 in
physical units. The dashed lines and tick marks in Fig. 3 then
show approximate forms in the same format as Fig. 2 for the
angular velocity. Specifically, for small ρe the long-dashed
lines show the currents obtained by introducing Eq. (19) (the
‘S’ approximation into Eq. (21)). These currents are the same
as Eq. (9) for a dipole field, when expressed in normalised
form. This ‘S’ approximation is drawn to the point where
(ω / J )S falls to zero. Beyond this we draw the currents
obtained by introducing (ω/ J )L = 0 into Eq. (21) (the
‘L’ approximation), which are the same as Eq. (10) for the
dipole, when expressed in normalised form. The long-dashed
lines thus represent the current profiles that would be driven
by an angular velocity profile given by the ‘S’ approximation to the point where falls to zero, with zero being taken
beyond. The short-dashed lines then show the profiles obtained by introducing Eq. (20) (the ‘L0 ’ approximation) into
Eq. (21).
The normalised solutions given above show how the form
and amplitude of the plasma angular velocity and currents
vary with the system parameters for a dipole field. Specifically, Eqs. (13) and (16) show that the solutions scale
spatially in the equatorial plane and in the ionosphere as
ρe ∝ (6P∗ /Ṁ)1/4 and ρi ∝ (Ṁ/6P∗ )1/8 , respectively, while
1426
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
Table 2. Principal features of the plasma angular velocity and coupling current system for a dipole field in physical units
Feature
ρe
RJ
sin θi = Rρi
J
Maximum upward field-aligned current density
(jk /B)max ≈ 0.10766P∗ (mho) pA m−2 nT−1
6P∗ (mho)
Ṁ(103 kg s−1 )
jkimax ≈ 0.092066P∗ (mho)µA m−2
50.21
Maximum sheet-integrated equatorial radial current
1/2
iρmax ≈ 2.178 6P∗ (mho)Ṁ(103 kg s−1 )
mA m−1
6P∗ (mho)
54.30
Ṁ(103 kg s−1 )
Plasma angular velocity falls to
ω
J
= 0.5
!1/4
0.1411
!1/4
74.80
6P∗ (mho)
Ṁ(103 kg s−1 )
85.67
6P∗ (mho)
Ṁ(103 kg s−1 )
Ṁ(103 kg s−1 )
6P∗ (mho)
Ṁ(103 kg s−1 )
0.1357
6P∗ (mho)
!1/4
!1/8
!1/8
0.1156
Ṁ(103 kg s−1 )
6P∗ (mho)
0.1080
Ṁ(103 kg s−1 )
6P∗ (mho)
0.0890
Ṁ(103 kg s−1 )
6P∗ (mho)
!1/8
Maximum azimuth-integrated total current
1/4
Iρ max = 2 IP max ≈ 65.54 6P∗ 3 (mho)Ṁ(103 kg s−1 )
MA
Field-aligned current passes through zero
Maximum ionospheric Pedersen current
1/8
∗
7
3
−1
iP max ≈ 0.7381 6P (mho)Ṁ(10 kg s )
A m−1
126.34
Maximum downward field-aligned current density
(jk /B)min ≈ −0.3526P∗ (mho) pA m−2 nT−1
jki min ≈ −0.30136P∗ (mho)µA m−2
i ∝ 6P∗
(1+γ )
2
Ṁ
(1−γ )
2
(22)
,
where γ is equal to zero for the sheet-integrated equatorial radial current, 1/2 for the azimuth-integrated total fieldperpendicular current, 3/4 for the height-integrated ionospheric Pedersen current, and 1 for the field-aligned current
density. The fact that these spatial and amplitude scales combine to produce a linear dependence of the current on Ṁ for
small ρe , as given by Eq. (9), and a linear dependence on
6P∗ at large ρe , as given by Eq. (10), implies that the currents grow with a specific power of the distance in the inner
region, and decline with a specific power of the distance at
large distances. It is easy to show that at small distances the
currents grow as
2(1+γ )
iS ∝ Ṁρe
∝
Ṁ
4(1+γ )
(23a)
,
ρi
while at large distances they decline as
iL ∝
6P∗
2(1−γ )
ρe
4(1−γ )
∝ 6P∗ ρi
,
6P∗ (mho)
Ṁ(103 kg s−1 )
∞
Eq. (21) shows that the amplitude of each component of the
current system scales as some power of 6P∗ and Ṁ of the
form
(23b)
!1/4
!1/4
!1/8
!1/8
0
as may be readily verified by substituting the appropriate
form for the angular velocity (i.e. the ‘S’ or ‘L’ approximations) into Eq. (21). Thus, in summary, the currents
grow in the inner region according to Eq. (23a), and depart
from this behaviour at an equatorial distance proportional to
(6P∗ /Ṁ)1/4 (as shown by the ‘inner’ downward tick marks
in Fig. 3), where the current value depends on 6P∗ and Ṁ
according to Eq. (22). Similarly, the currents decline in the
outer region according to Eq. (23b), starting at an equatorial distance proportional to (6P∗ /Ṁ)1/4 (as shown by the
‘outer’ downward tick marks in Fig. 3), where the current
value again depends on 6P∗ and Ṁ according to Eq. (22).
4
Plasma angular velocity and coupling current system
for a current sheet magnetic model
The solution for the coupling currents for a dipole field represents an important paradigm case. Nevertheless, the model
is unrealistic in its application to Jupiter because the middle magnetosphere field lines are not quasi-dipolar, but are
significantly distorted outward from the planet by azimuthal
currents flowing in the equatorial plasma, as shown in Fig. 1.
Thus field lines at a given radial distance in the equatorial
plane map to a significantly lower latitude in the ionosphere
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
»Bze
ȐnT
zeȐnT
»B
1000
1000
100
100
10
10
11
0.1
0.1
Feeze
ênT
RJJ 22
»B
ȐnTR
F
ênT
20
20
40
40
60
60
80
80
100
100
120
120
êRJJ
rreeêR
(a)
Fig 4a
4a
Fig
80000
80000
1000
60000
60000
100
40000
40000
10
20000
20000
1
0.1
êdeg
2
qqFiiêdeg
e ênT RJ
25
25
20
20
20
40
40
40
60
60
60
80
80
80
100
100
100
êRJJ
rreeêR
120 re êRJ
120
120
(b)
1427
Bze versus equatorial radial distance (the actual values are, of
course, all negative), while the dashed line shows the dipole
field for comparison. The model field departs from the dipole
in the inner part of the middle magnetosphere, and remains
significantly lower in magnitude throughout the region of interest, reflecting the outward distension of the current sheet
field lines. The model employs the “Voyager-1/Pioneer-10
model” of Connerney et al. (1981) (the “CAN” model) out
to a certain radial distance (using the analytic approximations derived by Edwards et al., 2001), and the empirical
Voyager-1 outbound model of Khurana and Kivelson (1993)
(the “KK” model) beyond. The radial distance at which these
models are joined, ρe∗ ∼ 21.78 RJ , indicated by the “kink” in
the solid curve in Fig. 4a, is determined from the intersection
of the two model curves, such that there is no discontinuity in
the field magnitude at this point (only in the first derivative).
The expression for the field in the “CAN” region has been
given previously by Cowley and Bunce (2001) and Cowley
et al. (2002), and will not be repeated here. The expression
for the field in the “KK” region, however, is simply a power
law given by
Fig4b
4a
Fig
4b
Fig
80000
20
20
Bze = −B0
60000
15
15
40000
10
10
20000
55
qi êdeg
25
2020
20
4040
40
6060
60
8080
80
100
100
100
êR
rreerêR
e êR
JJ J
120
120
120
(c)
Fig 4c
4c
Fig
Fig 4b
Fig. 4.20 Plots showing parameters of the current sheet field model
employed in Sect. 4 (solid lines), compared with values for the planetary dipole
field alone (dashed lines). Plot (a) is a log-linear plot
15
of the modulus of the north-south equatorial magnetic field |Bze |
threading
the equatorial plane, shown versus jovicentric equatorial
10
radial distance ρe . This field component is actually negative (i.e.
points 5south) in both cases. The kink in the curve at ∼21.78 RJ in
the current sheet model marks the point where we switch from the
re êRJ Plot (b)
‘CAN’ model to
the ‘KK’
model,
as discussed
in the120
text.
20
40
60
80
100
similarly shows the equatorial flux function Fe of the model fields
Fig 4c
versus jovicentric equatorial radial distance ρe . Plot (c) shows the
mapping of the field lines between the equatorial plane and the ionosphere, determined from Eq. (1). The ionospheric dipole co-latitude
of the field line is plotted versus jovicentric equatorial radial distance ρe .
than for a dipole, thereby increasing the electric field and current for a given departure of the plasma from rigid corotation.
In their previous investigations, Cowley and Bunce (2001)
and Cowley et al. (2002) employed an empirical model of
the equatorial field based on Voyager magnetic data. This
model will also be used here, its properties being illustrated
in Fig. 4. The solid line in Fig. 4a shows a log-linear plot
of the modulus of the equatorial north-south magnetic field
RJ
ρe
m
,
(24)
where B0 = 5.4 × 104 nT and m = 2.71. This function was
determined from a fit to outbound Voyager-1 data over the
range from ∼20 to ∼100 RJ , corresponding approximately
to the range over which we employ it here.
The equatorial flux function Fe satisfying Eq. (2) is shown
by the solid line in Fig. 4b, where the dashed line again shows
the dipole value (Eq. 11b). The value of Fe at ρe∗ is set by the
CAN model, equal to ∼3.70×104 nT R2J , while beyond this,
in the “KK” region, integration of Eq. (2) using Eq. (24) for
Bze yields
!m−2
B0 RJ2
RJ
Fe (ρe ) = F∞ +
,
(25)
(m − 2) ρe
where F∞ , the model value of Fe at infinity, is
∼2.85×104 nT R2J . It can be seen in Fig. 4b that the value
of Fe for the current sheet model is much larger than for the
dipole at a given equatorial distance, and varies over only
a narrow range in the outer middle magnetosphere. Since
the value of Fe is directly related to the magnetic co-latitude
where the field lines map to the ionosphere through Eq. (1),
the implication is that the current sheet field lines map to significantly larger co-latitudes than for the dipole, and also to
a very narrow co-latitude range. This is shown explicitly in
Fig. 4c, where we plot the co-latitude of the field lines in the
ionosphere versus equatorial radial distance. In the current
sheet model (solid line), the ionospheric mapping varies from
a co-latitude of ∼16.7◦ at 30 RJ to ∼15.6◦ at 120 RJ , a range
of only ∼1.1◦ . Even the current sheet field line from infinity,
should the model (unrealistically) be taken to extend that far,
only maps to ∼15.0◦ . Field lines at smaller co-latitudes then
1428
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
do not thread the current sheet in this model, but must map
instead to the outer magnetosphere and magnetic tail, regions
which are not described by the present theory. In the dipole
model, by contrast, equatorial field lines in the range 30 to
120 RJ map between ∼10.5◦ and ∼5.2◦ , a range of ∼5.3◦ ,
and go to the pole, of course, at infinity.
Solutions for such a “current sheet” field must generally
be computed numerically for specific values of the system
parameters, and results have been presented to date by Cowley et al. (2002, 2003) for a few spot values. Here we present
an approximate analytic solution which applies to the region
beyond ρe∗ ∼21.78 RJ , where the field varies with distance
as a power law, which previous work has shown to be the
main current-carrying region. However, we must first enquire how solutions in the power law region depend on conditions inside the region, where the dipole field is dominant and
the transition to the power law takes place. We commented
previously for the dipole problem that solutions of the HillPontius equation which are started at an arbitrary angular velocity well within the “Hill distance” converge rapidly onto
the solution which rigidly corotates at small distances, such
that the behaviour at larger distances is very insensitive to the
choice of initial condition. Numerical investigation shows
that the solutions for the current sheet field exhibit the same
property (Cowley and Bunce, 2002). The implication for the
present problem is that, provided the effective “Hill distance”
is larger than ρe∗ ∼20 RJ (i.e. provided the value of (6P∗ /Ṁ)
is not too low), the solutions in the “power law” region will
be very insensitive to conditions in the interior region. In
this case, we can simply take the power law field to be valid
over all distances, but apply the results only to the region
outside of ρe∗ . The validity of this statement may be judged
from Fig. 5, where we show solutions for the plasma angular velocity in the inner part of the system spanning ρe∗ ,
for three values of (6P∗ /Ṁ) covering our range of interest,
i.e. 10−4 , 10−3 and 10−2 mho s kg−1 . The solid lines show
numerical solutions using the full current sheet field model
shown in Fig. 4, while the dashed lines similarly show numerical solutions using the power law field over the whole
range. (The dot-dashed lines show the analytic approximation to be derived below.) Both numerical solutions were
initialised by imposing the near-rigid corotation approximation given by Eq. (8) at ρe = 5 RJ . The position of ρe∗ is
indicated by the tick mark in each plot, such that both models use the same power law field at larger distances. It can be
seen that the two numerical solutions converge rapidly beyond this distance, the convergence becoming increasingly
rapid as (6P∗ /Ṁ) increases. Thus, in the parameter range
of interest, the solutions in the power law field region can
be approximated by taking the power law field to be valid at
all distances. We note that the values of the “Hill distance”
corresponding to the values of (6P∗ /Ṁ) shown in the figure
are 27.7, 49.2, and 87.5 RJ , thus exceeding ρe∗ ∼21.78 RJ in
each case, though only just so at the lower limit. Convergence of the two solutions is found to break down for lower
values of (6P∗ /Ṁ) ∼ 10−5 mho s kg−1 , corresponding to a
“Hill distance” of 15.6 RJ , at the limit of the parameter range
considered here.
We thus consider solutions for the case in which the equatorial field is taken to be given by Eq. (24) at all distances.
To obtain an analytic result we also make the further approximation that the flux function is taken to be a constant in the
Hill-Pontius equation. Thus, over the region of interest, the
equatorial field is taken to map in the ionosphere to a narrow range of distances from the magnetic axis, an approximation shown to be well satisfied for the empirical current
sheet model discussed above. While this approximation will
generally be valid for “current sheet” fields, it is clearly not
valid for quasi-dipolar fields. We thus note that the solutions
obtained here do not reduce to the dipole case in the limit
that we choose m = 3 in Eq. (24). With this “current sheet”
approximation, then, Eq. (6b) becomes
ω
ω
RCSe m
ω
ρe d
+
=2
1−
, (26)
2 dρe J
J
ρe
J
where RCSe is the equatorial “Hill distance” for the power
law current sheet field (subscript ‘CS’)
!1/m
2π 6P∗ B0 F0
RCSe
=
.
(27)
RJ
Ṁ
Here we have put Fe = F0 , a constant, into Eq. (6b),
such that the field lines are taken to map in the ionosphere to a fixed
q distance from the magnetic axis given by
(ρi0 /RJ ) = (F0 /BJ RJ2 ) (Eq. 1). The value of F0 could
be taken for example, to be equal to F∞ in Eq. (25) (in
which case (ρi0 /RJ ) = (ρi∞ /RJ ) where (ρi∞ /RJ ) =
√
2
(F∞ /BJ RJ ) ≈0.258, corresponding to a co-latitude of
◦
∼14.95 ), or to some nearby (larger) value representative
of the field lines in the region of interest. Equation (26) is
then of the same form as Eq. (12) for the dipole field (they
are identical when m = 4), from which it is clear that the
solutions are functions only of ρe /RCSe , and hence scale
with distance as RCSe and with the system parameters as
(6P∗ /Ṁ)1/m . With m = 2.71, therefore, as used throughout here, the scale length varies somewhat more rapidly with
the system parameters than for the dipole, which varies as
(6P∗ /Ṁ)1/4 . The general solution of Eq. (26) can again be
found by the integration factor method
" 2
#
ω
4 m RCSe 2
4 RCSe m
=
exp
×
J
m
ρe
m
ρe
" #
2 4 RCSe m
× 0 1− ,
+K ,
m m
ρe
where 0(a, z) is the incomplete gamma function
Z ∞
0(a, z) =
e−t t a−1 dt.
(28a)
(28b)
z
The solutions again diverge at the origin except for the special solution with K = 0, which rigidly corotates for small
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
HwêWJ LL
HwêW
JJ
11
function only of
-4
-1
°° L = 10-4
HSP***êM
êM
mho ss kg
kg-1
-1
HS
L = 10 -4 mho
PP
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
HwêWJ LL
HwêW
JJ
11
10
10
20
20
30
30
40
40
êRJ
rreêR
50 ee JJ
50
(a)
Fig 5a
5a
Fig
-1
-3
°° L = 10-3
HSP***êM
êM
mho ss kg
kg-1
-1
HS
L = 10 -3 mho
PP
0.95
0.95
1ρi
RCSi
=
RCSe
ρe
m−2
(30)
,
where 1ρi = (ρi − ρi∞ ), and ρi∞ is the distance from the
magnetic axis of the field line from infinity as given above.
In deriving Eq. (30) we have assumed that 1ρi is small compared with ρi∞ , in keeping with the “current sheet” approximation introduced above. Numerically, for the above power
law field we find
! 1
2.71
6P∗ (mho)
RCSe
≈ 56.38
RJ
Ṁ 1000kg s−1
and
0.9
0.9
0.85
0.85
Ṁ 1000kg s−1
RCSi
≈ 0.0197
RJ
6P∗ mho
0.8
0.8
HwêWJ LL
HwêW
JJ
11
1429
10
10
20
20
30
30
40
40
êRJ
rreêR
50 ee JJ
50
(b)
-2
-1
°° L = 10-2
HSP***êM
êM
mho ss kg
kg-1
-1
HS
L = 10 -2 mho
PP
0.995
0.995
Fig 5b
5b
Fig
0.99
0.99
0.985
0.985
0.98
0.98
0.975
0.975
0.97
0.97
10
10
20
20
30
30
40
40
êRJ
rreêR
50 ee JJ
50
(c)
Fig 5c
5c
Fig
Fig. 5. Plots showing plasma angular velocity profiles versus equatorial radial distance for (6P∗ /Ṁ) equal to (a) 10−4 , (b) 10−3 , and
(c) 10−2 mho s kg−1 . The solid lines show the solution obtained by
numerical integration of Eq. (6b) using the full ‘current sheet’ magnetic model shown in Fig. 4, starting from the near-rigid corotation
approximation Eq. (8) at ρe = 5 RJ . The tick marks show the point
(ρe∗ ≈21.78 RJ ) where the magnetic field switches from the ‘CAN’
model to the power-law ‘KK’ model. The dashed lines then show
the numerical solution obtained by employing the ‘KK’ power law
field (given by Eqs. 24 and 25) over the full distance range, the solutions again being initialised using the appropriate form of Eq. (8)
at ρe = 5 RJ . The dot-dashed lines show the approximate analytic
solution using the ‘KK’ power law field, given by Eq. (28) with
K = 0 and m = 2.71. Note that the vertical scale has been tailored
to the plot in each case.
ρe . To map the solution to the ionosphere we equate Eqs. (1)
and (25), and define an ionospheric scaling distance
s
1− 2
m
BJ RJ2
B0
Ṁ
RCSi
1
= ,(29)
∗
RJ
F∞ 2π 6P B0 F0
2 m − 2 BJ
such that the angular velocity mapped to the ionosphere is a
! 0.71
2.71
,
(31)
where
we
have
chosen
to
put
F0 = Fe (70 RJ ) ≈ 3.22 × 104 nT RJ2 , a representative value
in the middle magnetosphere current sheet. Thus, as
(6P∗ /Ṁ) varies over the range of interest from 10−4 to
10−2 mho s kg−1 , we find that RCSe varies between 24.1 and
131.9 RJ (compared with 27.7 to 87.5 RJ for RDe for the
dipole), while RCSi varies between 0.046 and 0.0084 RJ
(compared with 0.19 to 0.11 RJ for RDi for the dipole).
Consequently, since (ρi∞ /RJ ) ≈ 0.258, as indicated above,
we will indeed have 1ρi small compared with ρi∞ for
values 1ρi ∼ RCSi .
The solid lines in Figs. 6a and b show the normalised angular velocity solution given by Eq. (28) with K = 0 and
m = 2.71, plotted versus (ρe /RCSe ) in the equatorial plane,
and versus (1ρi /RCSi ) in the ionosphere, respectively. The
form is similar to that for the dipole, though falling away
from rigid corotation more quickly (in normalised units) in
the inner region, and less quickly in the outer region. This solution is also shown in un-normalised form (with the above
value of F0 ) by the dot-dashed lines in Fig. 5, where it is
compared with the results of numerical integration of the full
solution (solid and dashed lines as described above). It can
be seen that the analytic solution forms a very close approximation to the numerical solutions for ρe > ρe∗ under all
conditions of interest here, a result we have confirmed by
a wider comparative study not illustrated here. The dashed
lines and tick marks in Fig. 6 show normalised approximate
forms and their regimes of validity, in the same format as
Fig. 2. Specifically, the long-dashed lines show the smallρe (‘S’) approximation given by Eq. (8)
m
ω
1 RCSi m−2
1
ρe m
=1−
=1−
,
(32)
J S
2 RCSe
2 1ρi
where we again note that the series generated by Eq. (7)
(of which Eq. (32) is the leading term) is the same as that
obtained by asymptotic expansion of the gamma function
in Eq. (28) for large argument. The short-dashed lines
1430
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
F0
IP CS =
6P∗ BJ RJ2 J ,
(34b)
BJ RJ2
iρ
RCSe
ω
=4
1−
,
where
iρCS
ρe
J
wêWJ
wêW
1 J
1
0.8
0.8
0.6
0.6
0.4
iρCS =
0.2
0.2
1
2
3
4
5
6
1
2
3
4
5
6
wêWJ
re êRCSe
reFig
êRCSe
6a
(a)
Fig 6a
1 J
wêW
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.5
1
1.5
2
2.5
3
3.5
0.5
1
1.5
2
2.5
3
3.5
Dri êRCSi
(b)
Dri êRCSi
Fig 6b
Fig. 6. Plots of the approximate analytic plasma angular velocity
profile for the current sheet power law magnetic field model, given
by Eq. (24) with m = 2.71, shown (a) versus (ρe /RCse ) in the
equatorial plane, and (b) versus (1ρi /RCsi ) in the ionosphere. The
solid line in each case shows the full solution obtained from Eq. (28)
with K = 0, such that the plasma rigidly corotates at small radial
distances. The long-dashed lines show the small-ρe (‘S’) form given
by Eq. (32), while the large-ρe (‘L’) form is just (ω/J )L = 0.
The downward-pointing tick marks indicate the limits of validity of
both these approximations as defined for the dipole case shown in
Fig. 2. The short-dashed lines show the higher-order large-ρe (‘L0 ’)
form given by Eq. (33), whose limit of validity (defined as for the
dipole case) is indicated by the upward-pointing tick mark. The
horizontal dotted line indicates the condition for rigid corotation,
(ω/J ) = 1.
then show the higher-order large-ρe (‘L0 ’) approximation, obtained from Eq. (28) as
ω
J
=
L0
4
m
2 m
2
RCSe 2
0 1−
.
m
ρe
(33)
The lower-order large-ρe (‘L’) approximation is again simply
(ω/J )L = 0.
The normalised approximate solutions for the currents
then follow from Eqs. (3)–(5)
iP
ω
=2 1−
,
where
iP CS
J
s
iP CS =
IP
IP CS
F0
6P∗ BJ RJ J ,
2
BJ RJ
ω
= 4π 1 −
,
J
where
(jk /B)
ρe m−2
=4
×
(jk /B)CS
RCSe
"
#
ω
RCse m
ω
×
−2
1−
,
J
ρe
J
Fig 6b
1/m
F0
Ṁ
6P∗ BJ RJ J ,
2π6P∗ B0 F0
BJ RJ2
Iρ
ω
= 8π 1 −
,
where
IρCS
J
F0
IρCS =
6P∗ BJ RJ2 J ,
BJ RJ2
0.4
(34a)
(jk /B)CS =
×
BJ
B0
2π6P∗ B0 F0
jki
jki CS
"
×
(34d)
where
F0
×
BJ RJ2
1− 2
m
6P∗ J ,
Ṁ
RCSi
=4
×
1ρi
ω
J
(34c)
1ρi
−2
RCSi
m
m−2
and
ω
1−
J
#
,
(34e)
where
BJ
F0
jk iCS = 2
×
B0
BJ RJ2
×
2π6P∗ B0 F0
Ṁ
1− 2
m
6P∗ BJ J .
(34f)
These forms are shown by the solid lines in Fig. 7, plotted
versus either (ρe /RCsi ) or (ρi /RCsi ) as appropriate. The
dashed lines show approximate forms based on the ‘S’, ‘L’,
and ‘L0 ’ approximations for the angular velocity, in the same
format as Fig. 3 for the dipole. Comparison with Fig. 3
shows similarities, but also major differences with the currents for the dipole field. The differences arise from the fact
that the current sheet field lines reach the ionosphere in a narrow band at a finite co-latitude, rather than continuously approaching the pole with increasing radial distance, as for the
dipole. The ionospheric Pedersen current (Eq. 34a), while
being proportional
q to the displacement of the band from the
magnetic axis F0 /BJ RJ2 , then varies with co-latitude only
through the variation of the plasma angular velocity. As seen
in Fig. 7a, the Pedersen current, therefore, peaks at the poleward edge of the band, where the angular velocity is zero,
0.5
0.5
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
IiPPêIêiPCS
PCS
0.5
1
1.5
2
2.5
3
3.5
Dri êRCSi
I rêI rCS
IPêIPCS
25 Fig 7a
2
12
1
1.5
2
2.5
3
3.5
1431
Fig 7a
12
20
10
10
1.5
8
15
8
1
6
6
10
4
0.5
4
5
2
2
I rêI rCS
IPi rêIêiPCS
rCS
253.5
0.5
Dri êRCSi
0.5
0.5
11
1.5
1.5
22
2.5
2.5
33
3.5
3.5
Dr
DriiêR
êRCSi
CSi
(a)
12
3
2010
2.5
1
0.5
HIirjrêI
j
êBL
»»êiêBLêH
»»
CS
rCS
rCS
Fig
4 7a7b
3.5 Fig
25
3.5
3
20
2.5
2
1
3
1.5
2
4
2.5
3
4
44
5
55
4
5
3
5
3.5
re êRCSe
Dr
6 i êRCSi
(b)
Fig 7d
Fig 7b
3
2.5
15 82
152
6
1.5
10
4
1
1.5
1.5
10
1 1
5
2
0.5
5
0.5 0.5
i rêi rCS
H j »» êBLêH j »» êBLCS
3.5
4
0.5
1 1
122
1.5 3 3
2
44 2.5
553
3.5
66
Dr
êRCSe
CSi
rreeiêR
êR
CSe
2
Fig 7c
»»i ê jFig
»»iCS
H j »»jêBLêH
j 7b
êBLCS
Fig
»» 7d
24
11
(c)
3
3.5
1.75
3.5
2.53
1.5 3
2.5
2
1.25
2.5
2
1.5
12
1.5
1
1
0.5
0.5
1.5
0.75
1
2
22
33
re êRCSe
rreeêR
êRCSe
CSe
66 6
(d)
Fig Fig
7c 7e
Fig 7d
0.5 1
1
1
j»»i ê j»»iCS
2
2
2
3
3
4
4
5
5
reêRCSe
6 re êRCSe
6
(e)
0.5
0.25
Fig 7c 0.5
j»»i ê j»»iCS
Fig 7e
2
1
1
2
1.5
3
2
2.5
3
reDr
êRCSe
i êRCSi
6
3.5
(f)
FigFig
7e 7f
Fig. 7. Plots of normalised steady-state current components for a power
1.75law equatorial magnetic field (Eq. (24) with m = 2.71) and ‘current
1.75
sheet’ approximate mapping to the ionosphere, plotted versus (ρe /RCSe
) or (1ρi /RCSi ) as appropriate. As in Fig. 3 for the dipole, the
1.5
1.5 show (a) the height-integrated ionospheric Pedersen current intensity, (b) the azimuth- and height-integrated total ionospheric Pedersen
plots
1.25 (d) the azimuth- and current sheet-integrated total equatorial radial
current,
(c) the current sheet-integrated equatorial radial current intensity,
1.25
current, (e) the equatorial field- aligned current density per unit magnetic1field strength, and (f) the field-aligned current density just above the
1
ionosphere. The normalization constants are given by Eq. (34). The solid
lines show the full approximate solutions given by Eqs. (28) and
0.75
0.75
(34), while the long-dashed lines show the small
and
large-ρ
forms
derived
from (ω/J )S given by Eq. (32) and (ω/ J )L = 0, drawn to
e
√
m
0.5
their
2 . The downward-pointing
tic-marks show the limits of validity of these approximate forms as
0.5 point of intersection at (ρe /RCSe ) =
0.25large-ρe form given by Eq. (33), whose limit of validity is indicated
defined for the dipole case. The short-dashed lines show the higher-order
by0.25
the upward-pointing tick mark, again defined as for the dipole case.
Dr êRCSi
0.5
1
1.5
2
2.5
3
3.5
Dri êRCSi
and falls monotonically with distance from the boundary as
the angular velocity approaches rigid corotation (Fig. 6b).
This implies that the azimuth-integrated total current also
varies monotonically with distance, the total equatorial current (Eq. 34d) thus rising with increasing equatorial distance
towards 8π F0 6P∗ J (strictly, 8π F∞ 6P∗ J ) at infinity, as
seen in Fig. 7d. This behaviour also implies that the radial
current intensity (Eq. 34c) rises to a peak value with increas-
0.5
Fig 7f
1
1.5
2
2.5
3
3.5
i
ing distance, and then falls as ρe−1 at large distances, as seen
in Fig. 7c. The further implication of a monotonically increasing total current is that the field-aligned current is unidirectional, flowing consistently from the ionosphere to the
equatorial current sheet, as shown in Figs. 7e and f. Closure of the current system must then occur outside the region
described by the model, on field lines mapping between the
ionosphere at higher latitudes and the outer magnetosphere
Fig 7f
1432
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
Table 3. Principal features of the plasma angular velocity and coupling current system for the power law current sheet field in normalised
units, obtained (with m = 2.71) from the approximate analytic solution of Sect. 4
Feature
Maximum sheet-integrated equatorial radial current
(iρ /iρCS )max ≈ 1.2109
Plasma angular velocity falls to ω =0.5
J
Maximum
upward field-aligned
current density
(jk /B)/(jk /B)CS
= jki /jkiCS
≈ 1.5274
max
max
ρe
RCse
= 2 IP /IP CS
max
max
1ρi
RCsi
ω
J
1.6142
0.7118
0.5113
1.6521
0.7002
0.5
2.3777
0.5407
0.3339
∞
0
0
Maximum height-integrated ionospheric Pedersen
current (iP /iP CS )max = 2
Maximumazimuth-integrated
total
current
Iρ /IρCS
= 8π ≈ 25.133
Field-aligned current goes to zero
Table 4. Principal features of the plasma angular velocity and coupling current system for the power law current sheet field in physical units,
obtained (using B0 = 5.4 × 104 nT, m = 2.71, F∞ ≈ 2.85 × 104 nT R2j and F0 ≈ 3.22 × 104 nT R2j ) from the approximate analytic solution
of Sect. 4
Feature
Maximum sheet-integrated equatorial radial current
1
2.71
iρ max ≈ 8.690 6P∗ (mho)1.71 Ṁ(103 kg s−1 )
mA m−1
Plasma angular velocity falls to
ω
J
Maximum upward field-aligned current density
(jk /B)max ≈
1
2.71
2.808 6P∗ (mho)3.42 Ṁ(103 kg s−1 )−0.71
pA m−2 nT−1
∗
1/2.71
6P (mho)
91.02
−1
3
Ṁ(10 kg s
= 0.5
ρe
RJ
93.15
6P∗ (mho)
Ṁ(103 kg s−1 )
134.06
)
!1/2.71
6P∗ (mho)
Ṁ(103 kg s−1 )
1ρi
RJ
0.71
Ṁ(103 kg s−1 ) 2.71
0.01400
6 ∗ (mho)
P
0.71
Ṁ(103 kg s−1 ) 2.71
0.01377
6 ∗ (mho)
P
1/4
0.71
Ṁ(103 kg s−1 ) 2.71
0.01063
6 ∗ (mho)
P
jk i max ≈ 2.404(6P∗ (mho)3.42 Ṁ(103 kg s−1 )−0.71 µA m−2
Maximum height-integrated ionospheric Pedersen current
iP max ≈ 2.9506P∗ (mho) A m−1
Maximum azimuth-integrated total current
Iρ max = 2IP max ≈ 725.86P∗ (mho) MA
Field-aligned current goes to zero
and magnetic tail. The values and positions of principal features of the solution are again tabulated in Table 3 in normalised units, and in Table 4 in physical units.
The behaviour of these approximate solutions thus reflects
the results presented previously by Cowley and Bunce (2001)
and Cowley et al. (2002, 2003), using the full current sheet
field model shown in Fig. 4. One minor difference is that in
the numerical solutions the total equatorial (and ionospheric)
current rises with increasing distance to a maximum value
slightly above 8πF∞ 6P∗ J before falling with decreasing F
∞
0
to the latter value at infinity, rather than following the strictly
monotonically rising behaviour of the approximation. Correspondingly, the field-aligned current in the numerical solutions reverses to small negative values (given by Eqs. 10e
and f) at large radial distances (and hence, close to the poleward boundary in the ionosphere), rather than going to zero
as in the approximation (Fig. 7f), though the net current closure is small. However, for the range of system parameters
considered here, the maximum in the total current and the
concurrent reversal of the field-aligned current typically take
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
place at equatorial distances of several hundred to several
thousand RJ , far beyond the limit of physical applicability
of the model. Within the region of applicability, the agreement between the numerical and approximate analytic results
is found to be very good.
In like manner to the dipole results, the normalised solutions given above show how the form and amplitude of
the plasma angular velocity and currents vary with the system parameters in the case of a power law current sheet
field. Equations (27) and (29) show that the solutions scale
spatially in the equatorial plane and in the ionosphere as
ρe ∝ (6P∗ /Ṁ)1/m and 1ρi ∝ (6P∗ /Ṁ)1−2/m , respectively,
while Eq. (34) shows that the amplitude of each component of the current system scales as some power of 6P∗
and Ṁ of the same form as Eq. (22), but with γ equal to
1–2/m for the equatorial radial current, 1 for the Pedersen
current and azimuth-integrated total field-perpendicular current (such that these currents scale linearly with 6P∗ and are
independent of Ṁ), and 3–4/m for the field-aligned current
density. Since these values of γ are consistently higher for
a given current component for the current sheet model than
for the dipole (at least for m > 2, as investigated here), the
implication is that the currents scale as a somewhat higher
power of the conductivity for the current sheet model than for
the dipole, and as a somewhat lower power of the mass outflow rate. The corresponding behaviours at small and large
distances, such that the solutions obey Eqs. (9) and (10), are
m
iS ∝ Ṁρe2
(1+γ )
∝
Ṁ
(m(1+γ )/2(m−2))
1ρi
(35a)
and
6P∗
(m(1−γ )/2(m−2))
∝ 6P∗ 1ρi
,
(35b)
m(1−γ )/2
ρe
as can be verified by substituting the appropriate approximations (‘S’ and ‘L’) for the angular velocity into Eq. (34).
Thus, for example, with γ = 1–2/m, the equatorial radial
current increases as ρem−1 in the inner region and falls as
ρe−1 at large distances, while the field-aligned current with
γ = 3–4/m grows as ρe2(m−1) in the equatorial plane in the inner region and approaches zero in this approximation at large
distances, as indicated above (Figs. 7e, f). The spatial variation of the field-aligned current in the large-distance limit
may then be obtained from the higher-order large-distance
approximation (Eq. 33), from which it is found that the
current varies as ρe−(4−m) in the equatorial plane, and as
(4−m)/(m−2)
1ρi
in the ionosphere.
iL ∝
5 Comparison of system behaviour for the dipole and
current sheet field models
In this section we finally provide a summary and comparison
of how the major features of the plasma flow and coupling
current system vary with 6P∗ and Ṁ for the dipole and current sheet field models. Specifically, we consider the location of corotation breakdown, the magnitudes and locations
1433
of the peak values of the various current components, and the
latitudinal width in the ionosphere of the region of upwarddirected field-aligned current. With the exception of the latter
parameter, in effect we here provide plots showing how the
quantities in Tables 2 (for the dipole) and 4 (for the power
law field approximation) vary with 6P∗ and Ṁ. We also compare the approximate results for the current sheet field with
spot values obtained by numerical integration using the full
current sheet field.
Figure 8 shows how the spatial scale on which plasma
corotation breaks down depends on 6P∗ and Ṁ for the two
models. Specifically, we show the position where (ω/J ) =
0.5, as previously given in Tables 1–4. In Fig. 8a the equatorial distance is plotted versus 6P∗ in log-log format for
Ṁ = 100, 1000 and 10 000 kg s−1 , while in Fig. 8b it is
plotted versus Ṁ in similar format for 6P∗ = 0.1, 1 and
10 mho. Solid lines give results for the dipole field obtained from Eqs. (13) and (14), showing that the distance in∗1/4
creases with the conductivity as 6P , and decreases with
the mass outflow rate as Ṁ −1/4 . The dashed lines show
corresponding results obtained from the power law field approximate solutions Eqs. (27) and (28) (with m = 2.71 and
F0 ≈3.22×104 nT R2J as above), which, of course, are not
applicable to the full field model at distances smaller than
ρe∗ ≈21.78 RJ . These increase more rapidly with 6P∗ and
∗1/2.71
decrease more rapidly with Ṁ, as 6P
and Ṁ −1/2.71 ,
respectively. Overall, however, the equatorial distances of
corotation breakdown are similar for the dipole and current
sheet fields as noted above, but are generally somewhat larger
for the current sheet model than for the dipole, particularly
for larger values of 6P∗ and smaller values of Ṁ. The solid
dots in the figures provide spot values obtained by numerical
integration of the full current sheet solution, their close association with the dashed lines clearly indicating the values
of Ṁ (in Fig. 8a) and 6P∗ (in Fig. 8b) employed. This close
association also confirms that the analytic solutions provide
good approximations to the numerical results in the power
law regime over essentially the whole parameter range considered here. The only notable deviations occur at small 6P∗
and large Ṁ, where corotation breakdown occurs at equatorial distances approaching the radial limit of the power law
field region at ρe∗ ≈21.78 RJ . In this case the numerical results give somewhat larger distances than the analytic approximation, as also seen in Fig. 5. Corresponding results
projected to the ionosphere are shown in Figs. 8c and 8d, in
a similar format. The horizontal dotted line at θi ≈14.95◦
shows the co-latitude of the current sheet field line from infinity (the corresponding limit for the dipole being, of course,
the pole at θi = 0◦ ). These plots again emphasise the significantly larger distance from the magnetic axis at which
plasma corotation breaks down in the ionosphere for the current sheet field than for dipole, despite the similarity of the
equatorial results. They also display the relative lack of response of this distance to the system parameters in the current
sheet model, this forming the basis of the “current sheet” approximation Fe ≈ F0 employed to obtain the analytic results
° ° -1 -1
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere
coupling currents in Jupiter’s middle magnetosphere
MHkg
s sL L
MHkg
re HwêW
êR êR
re HwêW
J = 0.5L
J = 0.5LJ J
500
500
1434
° 10
° 10
MHkg
s-1sL-1 L
MHkg
2 2
re HwêW
êRJêRJ
re HwêW
J = 0.5L
J = 0.5L
500500
10221022
1010
103103
200
200
re HwêW
êRJêRJ
re HwêW
J = J0.5L
= 0.5L
500500
re HwêW
êRJêRJ
re HwêW
J = 0.5L
J = 0.5L
500500
200200
10231032
10310 3
10
10
104104
10341043
104104
104104
200200
100
100
100100
5050
50 50
2020
2020
0.10.1 0.20.2
0.10.1 0.20.2
qi HwêW
êdeg
qi HwêW
êdeg
J = J0.5L
= 0.5L
0.50.5
1 1
2 2
5 5
* *
S PS
HmhoL
P HmhoL
1010
1010
1.0
101.0
10
1.01.0
20 20
200200
8a8a
* Fig
Fig
SP SHmhoL
P HmhoL
200200
10 10
(a)
êdeg
° ° -1 -1Figqi8a
HwêW
qi HwêW
êdeg
J = J0.5L
= 0.5L
MHkg
s sL L Fig 8a
MHkg
500500 1000
1000 2000
2000
0.50.5
1 1
2 2
5 5
104104
3
° 10
° 310-1
-1
MHkg
2s 2 sL L
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10Hkg
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15 15
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107 10
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2 2
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* *
S PS
HmhoL
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(c)
M 8b
Hkg
s sL-1 L
FigFig
8b
M Hkg
5000
500010000
10000
(b)
.
-1
* *
SPFig
HmhoL
Fig
S8b
HmhoL
P 8b
0.10.1
1.01.0
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*
SP10
SHmhoL
P HmhoL
0.1
15 15
1010
5 5
3 3
500500 1000
1000 2000
2000
qi HwêW
= 0.5L êdeg
qi HwêW
êdeg
15J 15J = 0.5L
10321023
1.00.1
0.1
1.0
1.01.0
0.10.1
0.10.1
. .
M Hkg
s-1sL-1 L
M Hkg
0.1
0.1
5000
500010000
10000
.
*
qi HwêW
êdeg
qi HwêW
êdeg
J = 0.5L
= 0.5L
15
15J
32 3
2
0.10.1 0.20.2
10 10
* *
SP10
SHmhoL
HmhoL
P10
100100
5050
50 50
20 20
*
SP *SHmhoL
P HmhoL
200200
100100
0.1
0.1
1.00.1
1.0
10 10
1.0
0.11.0
0.1
10 10
1.0
1.0
1010
5 5
3 3
23 23
M Hkg
s-1sL-1 L
M Hkg
5000
500010000
10000
(d)
.
200200
500500 1000
1000 2000
2000
.
. .
Fig
8c8c 2 2
* Fig
FigFig
M8d
Hkg
s-1sL-1 L
2 2
SP *SHmhoL
M8d
Hkg
P HmhoL
°
°
200
500
1000
2000
5000
10000
-1 -1
200
500
1000
2000
5000
10000
0.10.1
0.2
0.5
1
2
5
10
0.2
0.5
1
2
5
10
Hkg
s sL L
MHkg
* on the system
-1
SP *SHmhoL
Fig.iP8.max
showing
the location of plasma corotationMbreakdown,
specifically
êAêA
m-1
iP max
êAêA
m-1
iPPlots
m
. .
P HmhoL
iP max
m-1 where (ω/ J ) = 0.5, and its dependence
max
Fig
8c
∗
∗
Fig
8c
All
M
Fig
8d
parameters 6P and Ṁ. Plot (a) shows the equatorial distance
=
100, 1000 and
Fig
AllatMwhich (ω/J ) = 0.5 versus 6P in log-log format for Ṁ
10
108d
° ° -1 -1
∗
−1-1, while
MHkg
s versus
MHkg
sL L Ṁ for 6
* * for the dipole
-1
-1 -1= 0.1, 1 and 10 mho. Solid lines give results
10 000
s
(b)
shows
this
distance
similarly
plotted
S
HmhoL
iP max
êA
m
S
HmhoL
iPkg
êA
m
P P
iP max
êA êA
mm
P
iP max
max
. .
field obtained from Eqs. (13) and (14), while the dashedAll
lines
field
M Mshow corresponding results derived from the power law10
All
10 approximate
1010
10
10
solutions for the current sheet model Eqs. (27) and (28). The
dots provide spot values obtained from numerical integration
of the full
4
10 10
104310solid
∗ (in (b)) values
10103are obvious from their close association with the corresponding dashed lines.
current 5sheet
solution,
whose
Ṁ
(in
(a))
and
6
5 5
5
P
102102
∗
10 10
10 10
Corresponding
plots of the ionospheric co-latitude at which10(ω/
horizontal dotted
4 4 J ) = 0.5 are shown versus 6P and Ṁ in (c) and (d). The
1.0
101.0
10
10
3
3
line shows
the
latitude
of
the
field
line
from
infinity
in
the
current
sheet
field
model.
10
10
5 5
5
5
2 2
10 10
1.01.0
1.0
1.0
1 1
1 1
in Sect. 4.
ing F0 = Fe (ρe = 70 RJ ) in the approximation, as above.
0.50.5
1.01.0 if we had
Very
close agreement would have been obtained
1 1
0.10.1
instead
taken
F
=
F
.
These
plots
also
show
that for given
∞
0
the behaviour
of the peak Pedersen current, plotted in a 0.50.5
0.50.5
system
parameters
the
peak
Pedersen
current
for
0.10.1 the current
similar format to Fig. 8. The magnitude of the peak cur0.10.1
0.1
sheet
model
exceeds
that
for
the
dipole
by
relatively
constant
0.10.1
0.1
rent, plotted versus 6P∗ and Ṁ in Figs. 9a and 9b, respecfactors
of
∼3
to
∼5
(typically
∼4).
This
difference
°
° s-1 L-1 arises
* *
tively, shows that for the dipole (solid lines) the peakSPPederM0.1
Hkg
0.1
M Hkg
s L
SHmhoL
P HmhoL
from
the
different
ionospheric
mappings
of
corotation
break200
500
1000
2000
5000
10000
0.20.2
0.50.5 1 1
2 2
5 5 ∗7/8
0.10.10.1 increases
200
500 1000 2000
5000 10000
sen current
with
the conductivity
as 6P1010, while 0.10.1
down,
as
shown
in
Figs.
8c
and
8d.
Figures
9c
and
9d
° ° -1 -1 show
*Ṁ*1/8
also increasing weakly with the mass outflow rate as
M Hkg
s sL L
M Hkg
SFig
HmhoL
PS
P9aHmhoL
Fig
9a
the
co-latitude
of
the
peak
Pedersen
current,
which
Fig
9b9b for the
Fig
200200
500500 1000
1000 2000
2000 5000
500010000
10000
0.10.1 0.20.2
0.50.5 1 1
2 2
5 5 10 10
0.50.5
1 1 now to the current components, in Fig. 9 we show
Turning
(Eq. (21a)), while for the current sheet (dashed line) approximations the peak current varies linearly with 6P∗ but is indeFigFig
9a 9a
pendent of Ṁ (Eq. (34a)). The modestly lower numericallydetermined spot values in the latter case result from our tak-
dipole field lies typically at ∼5◦ and is such that the distance
from the magnetic axis varies with the system
parameters
as
FigFig
9b9b
∗−1/8
1/8
6P
and Ṁ
(Eq. 16), while for the current sheet ap-
3 3
3 3
*
SP *SHmhoL
P HmhoL
2 2
M Hkg
s-1sL-1 L
M Hkg
.
2 2
.
1000
200 500in
500Jupiter’s
10002000
2000 5000
500010000
10000
0.10.1 and
0.20.2
1 1 Magnetosphere-ionosphere
2 2
5 5 10 10
J. D. Nichols
S. W.0.5
H.0.5Cowley:
coupling200
currents
middle
magnetosphere
FigFig
8c 8c
° ° -1 -1
MHkg
s sL L
MHkg
iP max
êA êA
m-1m-1
iP max
. .
AllAll
MM
iP max
êA êA m
iP max
m-1 -1
10 10
5 5
FigFig
8d 8d
*
SP *SHmhoL
P HmhoL
10 10
10 10
1043104
10 103
102102
1435
10 10
5 5
1.01.0
1 1
1 1
0.50.5
0.50.5
1.01.0
0.10.1
0.10.1
0.10.1
0.10.1
0.10.1 0.20.2
qi HiPqmaxL
êdeg
i HiP maxL êdeg
0.50.5
1 1
2 2
5 5
*
SP *SHmhoL
P HmhoL
200200
10 10
(a)
qi HiPqimaxL
êdeg
HiP maxL êdeg
FigFig
9a 9a
500500 1000
1000 2000
2000
° ° -1 -1
M Hkg
s sL L
M Hkg
5000
500010000
10000
(b)
FigFig
9b 9b
15 15
15 15
10 10
10 10
0.10.1
° ° -1 -1
MHkg
s sL L
MHkg
7 7
1.01.0
104104
5 5
10 10
7 7
5 5
*
SP *SHmhoL
P HmhoL
103103
3 3
3 3
102102
2 2
0.10.1 0.20.2
0.50.5
1 1
2 2
5 5
*
2 2
SP *SHmhoL
P HmhoL
10 10
(c)
200200
FigFig
9c 9c
° ° -1 -1
M Hkg
s sL L
M Hkg
500500 1000
10002000
2000 5000
500010000
10000
(d)
FigFig
9d 9d
*
êmA
êmAêmA m Pedersen current intensity in the ionosphere,
SP *SHmhoL
êmAshowing
m
Fig.i rmax
9.i rmax
Plots
the magnitude and location of M
the
i r max
Hkg
s-1 sL -1height-integrated
P HmhoL and their
Mpeak
Hkg
L i r max
∗
∗
100
4
100
100 on 6 and Ṁ. Plot (a) shows the magnitude10
4 peak
100
dependence
of
the
Pedersen
current
plotted
versus
6
in
log-log
format
10
10
10 for Ṁ = 100,
P
P
−1 , while plot (c) show the ionospheric location of the peak. Plots (b) and (d) similarly show the magnitude and
1000 and
10
000
kg
s
50 50
50 50
3 ∗3
106
10P = 0.1, 1 and 10 mho. Solid lines give results for the dipole field obtained
location of the peak Pedersen current plotted versus Ṁ for
10 10
from Eq. (21a), while the dashed lines show corresponding
derived from the approximate solutions for the power law current sheet
104results
104
2 2
field given by Eq. (34a). The solid dots show spot values 10
obtained
from numerical integration using the full current sheet
solution. For the
1.01.0
10
case of10the
current
sheet
approximation
the
peak
current
depends
only10on106P∗ and not on Ṁ, so that only one dashed line is shown in plot
10
(a), valid for all Ṁ. The peak current in this case always occurs
as1.0
indicated by the
103103 at the poleward boundary of the current sheet field lines,1.0
5 5dotted lines at ∼14.95◦ in plots (c) and (d).
5 5
horizontal
0.10.1
m-1 -1
.
.
m-1 -1
0.10.1
102102
proximation it is located consistently at the poleward boundmagnitude of the azimuthal magnetic field outside of the cur1 1
1 1
rent sheet (Fig. 1) (Bϕ (nT) ≈0.63iρ (mA m−1 )). Figures 10a
ary of the
current sheet field lines at ∼14.95◦ , (dotted line)
where0.5
the
and b show that for the dipole field the peak current varies
0.5plasma angular velocity falls to zero. The numeri- 0.50.5
∗1/2
cally computed positions are located at a slightly higher coas 6P and Ṁ 1/2 (Eq. (21c)), while for the current sheet
. .
latitude, typically by ∼0.1◦ , like the total field-perpendicular
*
approximation it varies more strongly with Mthe
conductivity
Hkg
s-1sL-1 L
SP *SHmhoL
M Hkg
P HmhoL
200200 500500 1000
∗1.71/2.71
10002000
2000 5000
500010000
10000
0.50.5In1practical
5 5 10the
0.1 0.20.2 above.
1 2 2 application
10 peak
current 0.1
mentioned
as 6P
, and less strongly with the mass outflow rate
10a
10a
current will thus be limited instead by the radial extentFigofFig
the
1/2.71
Fignumerical
10b10b
Fig
as Ṁ
(Eq. 34c). The values given by the
in.
.
-1
-1
region
to
which
the
model
is
taken
to
apply,
with
the
peak
M
Hkg
s
L
M
Hkg
s
L
tegrations
are
in
close
agreement
with
the
latter.
The
current
re HirremaxL
êRJêRJ
r
êR
r
êR
e
Hir
maxL
J
Hir maxL
e
Hir
maxL
J
value500
occurring
at its outer (poleward) boundary. 104104
500500
500
sheet values are again higher than the dipole values by factors
Figure 10 similarly provides results for the peak equatorial radial current, a parameter which relates directly to the
200200
100
103103
104104
3
10 10
3
of ∼3 to ∼5 (typically ∼4), for reasons given above. Fig200200
100
*
SP *SHmhoL
P HmhoL
3 3
2 2
0.1
0.20.2
0.50.5 1 1
2 2
5
2 20.1
0.20.2
0.50.5
1 W.
5
1436 0.10.1J. D.
Nichols
and S.
1 H.2 Cowley:
2
i rmax
êmA
m-1m-1
i rmax
êmA
100êmA
100 m-1 -1
i rmax
i rmax êmA
m
100100
50 50
5050
10 10
1010
5 5
5 5
1 1
1 1
0.50.5
0.50.5
0.10.1 0.20.2
0.50.5
1 1
2 2
5
0.10.1 0.20.2
0.50.5
1 1
2 2
5
re Hir
êRJêRJ
remaxL
Hir maxL
500
500
re Hir
êR
re maxL
Hir maxLJêRJ
500500
3 3
2
2
1010
° ° -1
* *
2 2
M Hkg
s sL-1 L
SP S
HmhoL
M Hkg
P HmhoL
200
500
1000
2000
5000
10000
° ° -1
10
200
500 1000 2000
5000 10000
5
10
*
2
M
Hkg
s sL-1 L
*
SPSHmhoL
2
M Hkg
P HmhoL
200
500
1000
200
500
1000in2000
2000 5000
500010000
10000
9c 9c
Magnetosphere-ionosphere
coupling
currents
Jupiter’s
middle
magnetosphere
5 1010FigFig
FigFig
9d9d
. . Fig 9c
* * 9d
-1 -1
Fig
-1
Fig
9c
i
êmA
m
S
HmhoL
9d
PSPFig
MHkg
s sL-1 L r max
i r max êmA m
HmhoL
MHkg
.
*
-1
100100
. 4 -1
4
10
10
*
-1
êmA
m
S
HmhoL
P
10
10
-1 i r max
MHkg
s
L
SP HmhoL
MHkg s L i r max êmA m
4
100
4
1010
1010
100
50
50
3 3
10 10
1010
5050
34
34
10
1010
1010
10421042
1.01.0
10
10
2
2
1010
1010
1.01.0
103103
1.01.0
1010
5 5
3
0.1
3
1010
1.00.1
1.0
5 5
0.10.1
2 2
0.1
10 10
0.1
2 2
0.10.1
1010
1 1
1 1
0.50.5
0.50.5
. .
* *
M Hkg
s-1sL-1 L
SP S
HmhoL
M Hkg
P HmhoL
. .
200
500
1000
2000
5000
10000
10
200
500
1000
2000
5000
10000
5
10 *
MM
HkgHkg
s-1sL-1 L
*
SPSHmhoL
HmhoL
P10a
FigFig
200
500
1000
2000
5000
10000
10a
10
Fig
10b
200
500
1000
2000
5000
10000
(a)
Fig 10b (b)
5
10
-1
10a
MHkg
sFig
MFig
Hkg
sL-110a
L r
êRJêRJ
e Hir
remaxL
. . 4 4
Hir maxL
-1 -1
10
10
MHkg
s sL L 500500
MHkg
re r
êR
Hire maxL
4 4
Hir maxL JêRJ
1010
500
500
.
.
103103
10341034
1010
4 4
1010
103103
1032102
3
1010
2 2
1010
2 2
10 10
200200
200200
100100
100100
50 50
200200
* *
SPS
HmhoL
P HmhoL
200
200
*
SPS* HmhoL
10P10HmhoL
100100
100
100
2 2
1010
5050
FigFig
10b
10b
1010
1010
5050
101.0
1.0
10
1.01.0
1.01.0
1.00.1
1.0
0.1
5050
2020
20 20
2020
*
SP SHmhoL
P HmhoL
2020
*
0.10.1 0.20.2
0.10.1 0.20.2
0.50.5
0.50.5
1 1
1 1
2 2
2 2
5 5
5 5
10 10 *
(c)
*
SPSHmhoL
P HmhoL
1010
FigFig
10c10c
200200
500
500 1000
1000 2000
2000
200
200
500
500 1000
1000 2000
2000
0.10.1
0.1
. .
0.1
M0.1
Hkg
s-1sL-1 L
M Hkg
5000
. .
500010000
10000
(d)
MM
HkgHkg
s-1sL-1 L
5000
10000
5000 10000
FigFig
10d
10d
Fig. 10. Plots showing the magnitude and location of the peak
sheet-integrated
equatorial radial current intensity, and their dependence on
FigFig
10c
10c
Fig 10d
6P∗ and Ṁ. Plot (a) shows the magnitude of the peak current versus 6P∗ in log-log format for Ṁ = 100, 1000 and 10 000 kgFig
s−110d
, while plot
(c) shows the corresponding equatorial location of the peak in a similar format. Plots (b) and (d) similarly show the magnitude and location
of the peak current versus Ṁ for 6P∗ = 0.1, 1 and 10 mho. Solid lines give results for the dipole field obtained from Eq. (21c), while the
dashed lines show corresponding results derived from the approximate solutions for the power law current sheet model given by Eq. (34c).
The solid dots show spot values obtained from numerical integration of the full current sheet solution.
ures 10c and d show that the equatorial distance of the peak
is typically located at ∼50 RJ for the dipole model, vary∗1/4
ing with the system parameters as 6P and Ṁ −1/4 , while
for the current sheet approximation it is generally located at
somewhat larger distances ∼90 RJ , and varies more strongly
∗1/2.71
as 6P
and Ṁ −1/2.71 . The positions given by the numerical integrations are again in close agreement with the
latter, except for small 6P∗ and large Ṁ, where the position
of the peak approaches ρe∗ ≈ 21.78 RJ . In fact for small 6P∗
and large Ṁ, the peak current in the numerical solutions lies
consistently at ρe∗ , where the field models are joined. Such
points are omitted from the plots. At large distances the ra-
dial regime of applicability is again limited, such that the
peak radial currents will actually occur at the outer boundary of the region for sufficiently large 6P∗ and/or sufficiently
small Ṁ, as can be determined from the position of the peak
in Figs. 10c and d.
Results for the magnitude and location of the peak
azimuth-integrated total equatorial radial current, equal, of
course, to twice the peak azimuth-integrated total Pedersen
current in each conjugate ionosphere, are shown in Fig. 11.
Figures 11a and b show that the magnitude of the peak current for the dipole field increases with the conductivity as
∗3/4
6P , and less strongly with the mass outflow rate as Ṁ 1/4
Ir max
Ir êMA
max êMA
I
êMA
10000
.
Ir êMA
.
Ir max
max êMA
AllAll
M M 10000
10000
.
Ir max
.
IrêMA
.
max êMA
. M-1
MAll
Hkg
s M
MAll
Hkg
sL -1 L10000
10000
4 4
1000
1000
10
. 10
.
3s-13Ls-1 L
M
MHkg
10Hkg
10
1000
4104
1000
. 10
. 2 -12
10
103Ls-1 L
MHkg
s10
3Hkg
M
10
1000
1000
104 1024
100100
103210 3
10 10
100100
102 102
SP SHmhoL
.
r max
Ir max êMA
P HmhoL
AllAll
M M 10000
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere
coupling
currents in Jupiter’s middle magnetosphere
.
Ir max
êMA
Ir max
êMA
Ir max
IrêMA
max êMA
1000
1000
1000
1000
1000
1000
100100
100100
100100
10 10
100100
10 10
10 10
10 10
10 0.1
100.1 0.20.2
0.50.5
1 1
0.10.1 0.20.2
0.50.5 1 1
re HIr
êRJêRJ
remaxL
HIr maxL
0.1 0.1 0.2 0.2
0.5 0.5 1 1
1000
1000
re HIr
êRJêRJ . . 3
remaxL
HIr maxL
-1 -1
M=
kg3 skg
M10
= 10
s
1000
1000
re HIrremaxL
êR
J êRJ . .
HIr maxL
500500
3 3 -1 -1
M=
kg kg
s s
M10
= 10
1000
1000
500500
.
.
3
kg3 s-1 -1
M =M10= 10 kg s
500500
200200
200200
100100
200200
100100
50 50
100100
50 50
*
SP *SHmhoL
P HmhoL 10 10
200200
2 2
5 5 10 10
*
SP *SHmhoL
HmhoL
P
FigFig
11a11a
200200
2 2
5 5 10 10
*
r
êR
SP * S
HmhoL
e
HIr
maxL
J
J
P HmhoL
11a11are HIr maxL êR
200
200
2 2
5 5 10 10 FigFig
(a)
1000
1000
r11a
êRJ êRJ
remaxL
e HIr
HIr maxL
FigFig
11a
1000
1000
104104
re500
êRJ êRJ
HIr
remaxL
HIr maxL
500
4 4
1000
1000
10 10
500500
102102
104 104
500500
200200
102102
103103
2
2
200200
10 10
103103
100100
4 4
10 10
200200
3
10 103
100100
104104
50 50
100100
104 104
50 50
*
0.50.5
1 1
2 2
0.10.1 0.20.2
qHIP qmaxL
êdegêdeg
HIP maxL
0.1 0.1 0.2 0.2
qHIPqmaxL
êdeg
êdeg
HIP maxL
15 15
qHIP maxL
êdegêdeg
qHIP maxL
15 15
10 10
15 15
10 10
7 7
10 10
7 7
5 5
7 7
5 5
0.50.5
1 1
2
0.5 0.5 1 1
2
3 3
2 2
3 0.130.1 0.20.2
0.50.5 1 1 2
2 2
0.10.1 0.20.2
0.50.5 1 1
2
2 2
Plots0.1
showing
0.5 0.5 1 1 and
2
0.1 0.2 0.2the magnitude
*
5 5
1437
* 10
SP10
HmhoL
SP * HmhoL
10 10
1.01.0
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and their
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Fig 11e11e
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11f11f
Pedersen current) versus 6P∗ in log-log format for Ṁ = 100,Fig1000
and 10 000 kg s−1 , while plots (c) and (e)
show
the corresponding
equatorial and ionospheric locations of the peak in a similar format. Plots (b), (d) and (f) similarly show the magnitude and location of
the azimuth-integrated peak total current versus Ṁ for 6P∗ = 0.1, 1 and 10 mho. Solid lines give results for the dipole field obtained from
Eqs. (21b, d), while the dashed lines show corresponding results derived from the approximate solutions for the power law current sheet
model given by Eqs. (34b, d). The peak total current in the latter model is independent of Ṁ, so that only one dashed line is shown in (a). It
occurs at infinity in the equatorial plane so that no dashed lines are shown in (c) and (d), or equivalently at the poleward boundary of current
sheet field lines in the ionosphere at ∼14.95◦ as shown in (e) and (f) (dotted line). The solid dots show spot values obtained from numerical
integration of the full current sheet solution. In this case the peak values occur at large but finite distances such that only the closest of them
are included in (c) and (d).
1438
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
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Fig. 12. Plots showing the magnitude, location and half-width of the peak upward-directed field-aligned current density, and their dependence
on 6P∗ and Ṁ. Plot (a) shows the magnitude of the peak total current density versus 6P∗ in log-log format for Ṁ = 100, 1000 and
10 000 kg s−1 , where the left-hand scale shows the peak current density in the ionosphere, while the right-hand scale shows the peak (jk /B)
value, simply related to the latter via Eq. (5b). Solid lines give results for the dipole field obtained from Eqs. (21e ,f), while the dashed lines
show corresponding results derived from the approximate solutions for the power law current sheet model given by Eqs. (34e, f). The solid
dots show spot values obtained from numerical integration of the full current sheet solution. Plot (c) shows the corresponding location of the
peak (jk /B) in the equatorial plane in a similar format, while plot (e) shows the conjugate location of the peak field-aligned current in the
ionosphere. Plots (b), (d), and (f) similarly show the magnitude and equatorial and ionospheric locations of the peak current density versus
Ṁ for 6P∗ = 0.1, 1 and 10 mho. Plots (g) and (h) show the latitudinal width of the upward field-aligned current region in the ionosphere in
a similar format, defined as the full width at half maximum.
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
(Eqs. 21b, d), while for the current sheet approximation it is
linearly proportional to 6P∗ and independent of Ṁ (Eqs. 34b,
d). The numerical values shown by the dots in the latter case
are a little lower than the dashed line approximation for reasons given above for the Pedersen current. The value of the
peak current is a factor of ∼5 to ∼20 (typically ∼10) larger
for the current sheet than for the dipole. The location of the
peak in the equatorial plane (where the field-aligned current
passes through zero), is shown in Figs. 11c and d. It is located
∗1/4
typically at ∼90 RJ for the dipole field, scaling as 6P and
Ṁ −1/4 , but occurs at infinity for the current sheet approximation (such that no dashed lines are shown in Figs. 11c and
d), or, in other words, at the outer boundary of the relevant
region in practical application. The peak value in the numerical curves, shown by the dots, occurs at large but finite radial distance, as mentioned above, typically well beyond the
region of physical applicability (∼500 to ∼5000 RJ ). Only
the closest of them (for small 6P∗ and large Ṁ) are included
in Figs. 11c and d. The corresponding location of the peak
azimuth-integrated Pedersen current in the conjugate ionosphere is shown in Figs. 11e and f. It is located typically
∗−1/8
at ∼6◦ for the dipole field, scaling as 6P
and Ṁ 1/8 as
before, but for the current sheet it is located consistently at
(for the approximation) or near (for the numerical values) the
poleward boundary of the current sheet field lines at ∼14.95◦
(dotted line).
Figure 12 shows results for the upward-directed fieldaligned current density, a parameter of relevance to the origins of the jovian auroras. The magnitude of the peak upward
current is shown in Figs. 12a and 12b in a similar format to
the above, where, since jki and (jk /B) are simply related
through the constant factor 2 BJ in the approximation for the
ionospheric magnetic field employed here (Eq. 5b), one plot
serves the purpose of both parameters according to the leftand right-hand scales. These plots show that for the dipole
field the peak upward current density depends linearly on 6P∗
and is independent of Ṁ (Eqs. 21e, f), while for the current
sheet approximation it increases somewhat more rapidly with
∗3.42/2.71
the conductivity as 6P
(i.e. as ∼ 6P∗1.26 ), while decreasing slowly with the mass outflow rate as Ṁ −0.71/2.71
(Eqs. 34e, f). The latter values agree well with those obtained from numerical integration, and exceed those obtained
for the dipole field by factors of ∼10 to ∼50 (typically by
∼25). The position of the peak value of (jk /B) in the equatorial plane is shown in Figs. 12c and d. It lies typically at
∗1/4
distances of ∼50 RJ for the dipole field and varies as 6P
−1/4
and Ṁ
, while lying at larger typical distances beyond
∗1/2.71
∼100 RJ for the current sheet model and varies as 6P
and Ṁ −1/2.71 . The position of the peak field-aligned current
density in the ionosphere is shown in Figs. 12e and f. For the
dipole it lies typically at a co-latitude of ∼8◦ and scales in
∗−1/8
distance from the magnetic axis as 6P
and Ṁ 1/8 , while
for the current sheet model it lies just equatorward of the
boundary of current sheet field lines, with variations which
are in the same sense as for the dipole, but with amplitudes
1439
which are much smaller. In Figs. 12g and h we finally show a
measure of the latitudinal width of the region of upward fieldaligned current, potentially related to the latitudinal width of
associated jovian auroras, plotted versus 6P∗ and Ṁ, respectively. The width of the upward current given here is the full
width at half maximum. The solid and dashed lines show
results for the dipole and power law current sheet approximation, respectively. The results were derived from the fact
that in the equatorial plane the value of (jk /B) reaches half
its peak positive value for the dipole field at normalised radial distances (ρe /RDe ) of ∼0.629 and ∼1.470 (see Fig. 3e),
while for the power law current sheet approximation the corresponding values of (ρe /RCSe ) are ∼0.940 and ∼8.337 (see
Fig. 7e). The dots again show spot values obtained numerically using the full current sheet field. It can be seen that
the width for the dipole field is typically ∼3◦ –5◦ , decreasing
modestly with increasing 6P∗ and increasing modestly with
∗−1/8
increasing Ṁ (as 6P
and Ṁ 1/8 , respectively). For the
current sheet model the thickness is reduced to ∼0.5◦ –1.5◦
(less if the system is limited in radial distance), varying in
the above manner more strongly with the system parameters
(as ∼ 6P∗−0.26 and ∼ Ṁ 0.26 ).
6
Summary
In this paper we have considered the steady-state properties of the magnetosphere-ionosphere coupling current system that flows in Jupiter’s middle magnetosphere, associated with the enforcement of partial corotation on outwardflowing plasma from the Io torus. The solutions depend on
the values of two parameters, the effective Pedersen conductivity of the jovian ionosphere 6P∗ , and the mass outflow rate of iogenic plasma Ṁ, these being taken to be constants. However, their values remain uncertain at present,
thus prompting the study presented here of how the solutions
depend on these parameters over wide ranges of the latter.
We have also focussed on two models of the magnetospheric
poloidal field, taken for simplicity to be axisymmetric. The
first is the planetary dipole alone, which constitutes an instructive paradigm. Some general results for this case have
previously been given by Hill (1979, 2001). Here we have
provided a complete analytic solution for this case, showing how the plasma angular velocity and current components
scale in space and in amplitude with 6P∗ and Ṁ. We find that
the plasma angular velocity and current components scale
in equatorial radial distance as (6P∗ /Ṁ)1/4 , as found previously by Hill (and correspondingly as (6P∗ /Ṁ)1/8 ) in the
ionosphere), while each current component scales in ampli∗(1+γ )/2 (1−γ )/2
tude as (6P
Ṁ
, where γ has a particular value
for each component. The scales in space and amplitude then
combine to produce current values which depend only on Ṁ
at a fixed position at small radial distances, and only on 6P∗ at
a fixed position at large radial distances, these dependencies
then requiring current variations at small and large distances
1440
J. D. Nichols and S. W. H. Cowley: Magnetosphere-ionosphere coupling currents in Jupiter’s middle magnetosphere
2(1+γ )
with particular powers of the distance, as ρe
at small
−2(1−γ )
distances, and as ρe
at large distances.
These results provide useful background for the second,
more realistic field model, based on Voyager data, in which
the equatorial field strength is significantly less than for the
dipole field due to the radial distension of the middle magnetosphere field lines, and is taken to vary with distance as a
power law ρe−m . Solutions for a few spot values of 6P∗ and
Ṁ have previously been presented by Cowley et al. (2002,
2003), obtained by numerical integration of the corresponding Hill-Pontius equation. Here we have derived an analytic
approximation, applicable to the power law regime, which
shows how the plasma angular velocity and current components scale with 6P∗ and Ṁ in this case. We find that these
solutions provide accurate approximations to the full numerical results within the power law regime (roughly ρe > 20 RJ )
over very wide ranges of the system parameters, provided
(6P∗ /Ṁ) is not too small (∼10−4 mho s kg−1 or larger). The
results show that the conclusions concerning the nature of the
current sheet solutions, and their relation to the dipole solutions, which were drawn previously on the basis of a limited number of numerical investigations are generally valid
over wide ranges of the parameters. In particular, it has been
shown that in the current sheet model the field-aligned current flows unidirectionally outward from the ionosphere into
the current sheet over the whole current sheet, in all cases
of interest. The closure of this current must then occur on
field lines at higher latitudes which map to the outer magnetosphere and tail, which are not described by the present
theory. This situation contrasts with the dipole model, in
which (at least in principle) all the flux in the system is described by the theory, such that complete current closure occurs between the equator and the pole. The results for the
power law current sheet show that the plasma angular veloc1/m
ity and currents now scale in radial distance as (6P∗ /Ṁ) ,
while each current component again scales in amplitude as
∗(1+γ )/2 (1−γ )/2
Ṁ
, where the values of γ for each compo6P
nent exceed those of the corresponding component for the
dipole field (at least for m > 2 as considered here). The current components thus scale as a somewhat higher power of
6P∗ for the current sheet than for the dipole, and as a somewhat lower power of Ṁ. These scales in space and amplitude
again combine to produce current values which depend only
on Ṁ at a fixed position at small radial distances, and only
on at a fixed position at large radial distances (both being
general properties of the solutions), these dependencies then
m(1+γ )/2
requiring current variations as ρe
at small distances,
−m(1−γ )/2
and as ρe
at large distances. The absolute values of
the currents are also higher for the current sheet model than
for the dipole, by a factor of ∼4 for the Pedersen and equatorial currents, ∼10 for the total current flowing in the circuit,
and ∼25 for the field-aligned current densities. These factors do not vary greatly over the range of system parameters
considered here.
Acknowledgements. JDN was supported during the course of this
study by a PPARC Quota Studentship, and SWHC by PPARC Senior Fellowship PPA/N/S/2000/00197.
Topical Editor T. Pulkkinen thanks V. M. Vasyliunas and another
referee for their help in evaluating this paper.
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