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Effect of the acceleration current on the centrifugal interchange instability
Effect of the acceleration current on the centrifugal interchange
instability
T. W. Hill
1
Received 26 July 2005; revised 5 December 2005; accepted 12 December 2005; published 17 March 2006.
[
1
]
The centrifugal interchange instability is the primary driver of radial plasma
transport in the magnetospheres of Jupiter and Saturn. Most previous theoretical
treatments of this instability have ignored the role of the acceleration current and have
assumed that the divergence of the centrifugal drift current in the magnetosphere is closed
by Pedersen currents in the underlying ionosphere, through the connection provided by
Birkeland (magnetic-field-aligned) currents. This is a reasonable approximation when
the radial transport speed is small compared with the rotation speed. However, the
exponential growth of the instability inevitably leads to the eventual violation of this
condition. I analyze a simplified model of the Io plasma torus to show that when the radial
transport speed becomes comparable to the rotation speed, the acceleration current
becomes the primary mechanism for closure (actually local cancellation) of the centrifugal
drift current, and the connection to the planetary ionosphere therefore becomes
irrelevant. An immediate consequence is that the growth rate of the instability does not
exceed the planetary rotation rate.
Citation:
Hill, T. W. (2006), Effect of the acceleration current on the centrifugal interchange instability, J. Geophys. Res., 111,
A03214, doi:10.1029/2005JA011338.
1.
Introduction
[
2
] The magnetospheres of Jupiter and Saturn have
rapid rotation rates and strong internal plasma sources,
the two main ingredients for producing the centrifugal
interchange instability. Interchange motions, as originally
defined by Gold [1959], do not alter the magnetic-field
configuration, so their stability is determined solely by
the radial distribution of plasma mass density and
pressure. In the region outside a source, where the
gradient of flux-tube content is inward, the distribution
is unstable. Subsequent work has generalized the inter-
change concept to include the effects of magnetic-field
variations, which can become important when
b (ratio of
plasma to magnetic field pressure) is not
1 [Cheng,
1985; Southwood and Kivelson, 1987; Ferrie`re et al.,
1999]. The principal focus of this paper is the Io plasma
torus, where
b
1 and the simple Gold criterion is
sufficient.
[
3
] In gasdynamic terms, the mass distribution produced
by the internal source is unstable to interchange motions
because the centrifugal force of corotation exceeds the
inward force of planetary gravity [e.g., Gold, 1959; Siscoe
and Summers, 1981; Southwood and Kivelson, 1987]. In
electrodynamic terms [e.g., Siscoe and Summers, 1981;
Huang and Hill, 1991; Yang et al., 1994; Pontius et al.,
1998], any azimuthal variation of plasma mass density
r is
amplified because it produces a divergence of the centrifu-
gal drift current
j
cent
¼ rW
2
r=B ^
j
ð1Þ
which is closed by ionospheric Pedersen currents. Current
closure requires an electric field whose E
B drift
amplifies the radial displacement that led to the
divergence. Thus both gasdynamic and electrodynamic
arguments indicate that the mass distributions in the
magnetospheres of Jupiter and Saturn are centrifugally
unstable. In (1), (r,
j, z) is a cylindrical coordinate
system aligned with the planetary spin vector
6, and B is
the strength of the magnetic field in the equatorial plane,
assumed to be in the
z direction.
[
4
] Inclusion of the plasma thermal energy density in the
gasdynamic analysis, or, equivalently, of the gradient-
curvature drift current in the electrodynamic analysis,
reinforces this conclusion. The plasma distribution from
an internal source is still unstable outside the source, but the
instability is describable, in part, as a pressure-driven
(flute) instability rather than a centrifugal instability.
For Jupiter, in the neighborhood of the Io plasma torus,
the thermal energy density is perhaps 30% of the
centrifugal potential energy density [e.g., Huang and Hill,
1991]. In this paper I will neglect the thermal energy
density and hence the gradient-curvature drift currents;
their inclusion would modify the results quantitatively but
not qualitatively.
[
5
] In a rotation-dominated magnetosphere, the magneto-
spheric plasma is centrifugally confined near the equatorial
plane [Hill and Michel, 1976; Siscoe, 1977; Vasyliunas,
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, A03214, doi:10.1029/2005JA011338, 2006
1
Physics and Astronomy Department, Rice University, Houston, Texas,
USA.
Copyright 2006 by the American Geophysical Union.
0148-0227/06/2005JA011338$09.00
A03214
1 of 6 1983], and the equation of motion, integrated across the
equatorial plasma sheet, is
h dv
dt ¼ hW
2
r
2
hW^
z
v
þ ^z
J
ð2Þ
where
h
Z rdz
B
ð3Þ
is the mass per unit magnetic flux and J is the current
density integrated across the sheet. The velocity v is
measured in the corotating frame of reference; thus the three
terms on the right side of (2) are the centrifugal, Coriolis,
and Lorentz forces, all per unit magnetic flux. The cross
product of ^
z with (2) gives
J
¼ hW
2
r^
j
þ 2hWv
h^
z
dv
dt
ð4Þ
where the three terms on the right can be identified as the
centrifugal current (1) integrated across the sheet, the
Coriolis current, and the acceleration current. Hill [1983]
has noted that these three terms are of zeroth, first, and
second order, respectively, in the ratio v/
Wr (where the last
part of the statement follows from the ansatz dv/dt
v
.
r
v).
[
6
] It is therefore justifiable to neglect the last two terms,
provided that attention is restricted to situations for which
v
Wr. This has been done in most previous treatments
of the centrifugal interchange instability, both analytical
[Siscoe and Summers, 1981; Southwood and Kivelson,
1987; Huang and Hill, 1991] and numerical [Yang et al.,
1992, 1994; Pontius et al., 1998]. However, recent numer-
ical simulations with the RCM-J (Rice Convection Model
for Jupiter) indicate that this condition is quickly violated,
even within the confines of the Io plasma torus, because the
dominant scale size of the interchange cells is very small,
corresponding to very large linear growth rates [Spiro et al.,
2000; Goldstein et al., 2001, 2002]. We are thus motivated
to consider the effects of the Coriolis and acceleration
currents on the development of the centrifugal interchange
instability within the Io plasma torus.
[
7
] Work is in progress to include these effects within the
RCM-J formalism. The Coriolis effect can be included
relatively simply by the use of an effective Hall conduc-
tance in the Jovian ionosphere. Inclusion of the acceleration
current, however, is more complicated and requires a major
reworking of the numerical code. In the meantime, as a
prelude to this numerical work, it seems useful to consider
the effects of the acceleration current through a simple,
analytically tractable model, and that is the purpose of this
paper.
[
8
] There are two notable exceptions to the above rule
(neglect of the Coriolis and acceleration terms in the
equation of motion). Vasyliunas [1994] has elucidated
the role of the acceleration timescale
t
a
=
h/SB, i.e., the
timescale required for the Jovian ionosphere to impose its
motion on the magnetospheric plasma, where
S is the
height-integrated ionospheric Pedersen conductivity. He
shows that when
Wt
a
>
1 (generally true in the Jovian
magnetosphere beyond r
10 R
J
), the growth rate of the
centrifugal interchange instability is limited to values g <
W. Pontius [1997] has generalized this result to include
the effect of the Coriolis acceleration, which reduces the
effective value of
W for outflowing plasma. He finds a more
stringent limit on the growth rate, particularly for values of
Wt
a
1. Both of these papers simplify the ionosphere-
magnetosphere coupling equation by assuming that the
ionospheric current is equal in magnitude and opposite in
direction to the magnetospheric current when mapped along
the field. Here I relax that assumption and recover essen-
tially the same result, that the growth rate is limited to
values <
W. This limitation becomes important even within
the Io torus (r
6 R
J
) for sufficiently small scale sizes.
2.
Model
[
9
] In order to obtain an analytically tractable model, I
adopt the simplest possible representation of a plasma torus
that retains the essential physics of the centrifugal inter-
change instability. Specifically, I assume a step-function
distribution of the plasma mass content per unit magnetic
flux:
h L; j; t
ð
Þ ¼
h
1
;
L < L
1
j; t
ð
Þ
0; L > L
1
j; t
ð
Þ
8
<
:
9
=
;
ð5Þ
where L = r/R
J
and L
1
(
j, t) is the outer edge position. I
assume that the outer edge is perturbed by a simple
sinusoidal ripple:
L
1
j; t
ð
Þ ¼ L
0
þ dL
0
sin m
j
ð
Þe
gt
ð6Þ
where L
0
( 6) and
dL
0
are given constants, m is an integer,
and g(m) is a real growth rate to be determined.
[
10
] I assume a spin-aligned dipole magnetic field and a
simple ionosphere with a uniform height-integrated Peder-
sen conductivity
S in each hemisphere. The height-inte-
grated Hall conductivity is also assumed to be uniform, in
which case its value d