Mon. Not. R. Astron. Soc. 000, 000–000 (0000)
Printed 1 April 2010
(MN LATEX style file v2.2)
A Saturnian cam current system driven by asymmetric
thermospheric heating
C. G. A. Smith
The Brooksbank School, Victoria Road, Elland, West Yorkshire, HX5 0QG
March 20 2010
ABSTRACT
We show that asymmetric heating of Saturn’s thermosphere can drive a current system consistent with the magnetospheric ‘cam’ proposed by Espinosa
et al. (2003a). A geometrically simple heating distribution is imposed on the
northern hemisphere of a simplified 3D GCM of Saturn’s thermosphere. Currents driven by the resulting winds are calculated using a globally averaged
ionosphere model. Using a simple assumption about how divergences in these
currents close by flowing along dipolar field lines between the northern and
southern hemispheres, we estimate the magnetic field perturbations in the
equatorial plane and show that they are broadly consistent with the proposed
cam fields, showing a roughly uniform field implying radial and azimuthal
components in quadrature. We also identify a small longitudinal phase drift
in the cam current with radial distance as characteristic of a thermospheredriven current system. However, at present our model does not produce magnetic field perturbations of the required magnitude, falling short by a factor
of ∼100, a discrepancy that may be a consequence of an incomplete model of
the ionospheric conductance.
Key words: methods: numerical – planets and satellites: individual: Saturn
– planets and satellites: atmospheres – planets and satellites: magnetic fields
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INTRODUCTION
Planetary-period magnetic field oscillations in Saturn’s magnetosphere were first detected
in Pioneer and Voyager data (Espinosa & Dougherty 2000; Espinosa et al. 2003b). In recent
years these oscillations have been studied in more detail using data from the Cassini mission
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Figure 1. Sketches of the proposed current system, from Southwood & Kivelson (2007). The left-hand figure shows a dipolar
shell of field lines at L = 15RS on which currents flow between the northern and southern ionospheres. The thickness of the
lines represents the magnitude of the interhemispheric current, while solid and dashed lines indicate north-south and southnorth currents respectively. The right-hand figure is a sketch of the approximate magnetic field distribution produced by these
currents in the equatorial plane. In practice this would not be perfectly uniform as shown in the figure. The white dots in the
two figures represent the same location on the L = 15 shell.
(e.g. Cowley et al. 2006; Andrews et al. 2008; Provan et al. 2009) and shown to be correlated
with kilometric radio emissions (Gurnett et al. 2007; Andrews et al. 2008). The most unusual
characteristic of these oscillations is the slow drift in the periodicity, first identified in SKR
emissions by Galopeau & Lecacheux (2000). Smith (2006) suggested that the slowly varying but stable nature of the periodicity of these various phenomena was consistent with a
source in the thermosphere-ionosphere, and suggested in outline how the upper atmosphere
could impose such perturbations on the magnetosphere. Following Espinosa et al. (2003a),
Southwood & Kivelson (2007) separately proposed a model of ‘cam currents’ to explain the
specific phase relations of the observed magnetic field perturbations. The purpose of this
paper is to determine whether such a system of cam currents can indeed be driven by a
source in the thermosphere-ionosphere.
The Southwood & Kivelson (2007) model proposes an interhemispheric current system
flowing between the northern and southern hemispheres on magnetic shells in the range L =
12-15RS . The outline of this current system is shown in Fig. 1. This shows currents flowing
between the northern and southern ionospheres along a shell of dipolar field lines at L=15.
The key characteristic of these currents is that their magnitude and direction is distributed
sinusoidally in longitude, producing an approximately uniform field inside the shell of field
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lines as sketched in the diagram on the right of Fig. 1. In order to match the observations,
the magnitude of this field must be of the order of a few nT (Southwood & Kivelson 2007).
In this study we shall investigate whether asymmetries in the upper atmosphere could
plausibly generate such a system of currents. In Section 2 we describe a model of the
thermosphere-ionosphere and the temperatures and winds it predicts when forced with
asymmetric heating. Sections 3 and 4 describe the resulting current systems and magnetic
perturbations; Section 5 discusses the plausibility and consequences of our results.
2
THERMOSPHERE-IONOSPHERE MODEL
2.1
Details of the model
Our thermosphere-ionosphere model is a global circulation model, essentially identical to
that used by Smith et al. (2007) and Smith & Aylward (2008). Here we describe the modifications that have been made to the model to suit this study.
2.1.1
Resolution of model
The thermospheric model used by Smith et al. (2007) and Smith & Aylward (2008) was a
two-dimensional version, in latitude and altitude, of the three-dimensional model described
by Müller-Wodarg et al. (2006). In the former model, longitudinal symmetry was assumed in
order to study the rotational coupling of the thermosphere-ionosphere and magnetosphere
in detail. For this study we are specifically interested in longitudinal asymmetry, so move
back to a three-dimensional model. This is a trivial step because the full three-dimensional
equations are already represented in the code, as described by Müller-Wodarg et al. (2006).
The increase in dimensionality necessitates a decrease in resolution in order to maintain
manageable runtimes. For this study we use a latitude resolution of 2◦ , a longitude resolution
of 10◦ and 20 pressure levels at a resolution of 0.5 scale heights. The grey lines in Fig. 2 show
how the 2◦ latitude resolution maps to the equatorial magnetosphere along dipole field lines.
The base of the model is at a pressure of 100nb, with the base temperature held at 143K
consistent with the temperature profile presented by Moses et al. (2000). All runs presented
here start ‘cold’ at a global temperature of 143K, and are run forward for 400 planetary
rotations, consistent with the approximate thermal and dynamical equilibrium which was
shown to be established on this timescale by Smith et al. (2005).
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Figure 2. Mapping between ionosphere and magnetosphere using dipole magnetic field. Solid black lines show the latitude
range of our heating distribution. Dotted lines show the approximate 12-15 L-shell range for the cam currents defined by
Southwood & Kivelson (2007). The dashed line shows the location of the flow shear in our simplified model of rotational flow
in the magnetosphere, coinciding with the main auroral oval in the interpretation of Cowley et al. (2004). The thin grey lines
(two of which lie contiguous with the solid black lines already described) show how the latitude points of our thermosphere
model map to the equatorial magnetosphere.
2.1.2
Global heating
The thermal structure of Saturn’s upper atmosphere is not fully explained. The observed
exosphere temperature of ∼400K (Smith et al. 1983) is higher than can be explained by
absorption of solar radiation alone. We are not specifically interested in this question here,
but do wish to represent the thermospheric structure with a reasonable level of accuracy. We
therefore artificially reproduce the observed exosphere temperature with an arbitrary and
globally uniform heat source. This is similar to the approach taken by Müller-Wodarg et al.
(2006) to reproduce the ∼400K exosphere temperature using an arbitrary global distribution
of ‘wave heating’. Our approach is to use the same form of vertical heating distribution as
Smith et al. (2005):
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Figure 3. Left panel: Baseline global temperature profile (solid line) compared to the profiles presented by Hubbard et al.
(1997) (dotted line) and Moses et al. (2000) (dashed line). The thermospheric part of the Moses profile is based on the data of
Smith et al. (1983). Middle panel: Heating profiles per unit volume for the baseline temperature profile in the left panel. Solid
line: global baseline heating rate; dashed line: asymmetric heating rate imposed across a limited longitude-latitude range (see
text). Right panel: Conductivity profiles in mhom−1 for the baseline temperature profile in the left panel. Solid line: Pedersen
conductivity; dotted line: Hall conductivity.
p
qm = qm∞ exp −
p0
!
(1)
where qm is the heating rate per unit mass in Wkg−1 , qm∞ is the limiting value at the top
of the thermosphere, and p0 is the characteristic pressure that determines the peak heating
rate per unit volume. By trial and error we choose parameters of qm∞ = 1.6Wkg−1 and
p0 = 1.6nb to produce a reasonable match to the occultation data of Smith et al. (1983) and
Hubbard et al. (1997), the former being represented by the temperature profile constructed
by Moses et al. (2000). The resulting global temperature profile after 400 planetary rotations
is shown in Fig. 3. Also shown is the heating profile, plotted in units of Wm−3 .
Note that, in principle, we could derive a vertical profile of heating and cooling that
exactly matched any temperature profile. However, we think it preferable to use a simple
analytic heating profile that produces temperatures approximately matching the observations.
2.1.3
Asymmetric heating
In addition to this global heating distribution we introduce an asymmetric heating distribution. Some possible sources of such asymmetric heating of the thermosphere have been
discussed previously by Smith (2006) and will be discussed further in Section 5. However,
at this stage we shall not dwell too much on the possible origin of this asymmetric heating
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— we are principally interested in the hypothetical effects of such a heat source if it does
exist.
We choose to impose the heating using the same functional form of heating distribution
described above (Eqn. 1), but within a limited latitude and longitude range. For our standard
run we add an extra component of heating with qm∞ = 50Wkg−1 and p0 = 1.6nb (see Fig 3)
in the longitude range 180◦ to 360◦ and the latitude range 72◦ N to 78◦ N. Within this region
the asymmetric heating therefore has the effect of increasing the existing heating by a factor
of ∼30 while not changing its dependence on pressure.
The broad longitude range chosen reflects the simplest possible form of longitudinal
asymmetry. This distribution in longitude is equivalent to a square wave plus a constant,
and thus contains a combination of odd harmonics m = 1, 3, 5... and a zero frequency
harmonic. The heating is imposed only in the northern hemisphere in order to be certain
that a hemispheric asymmetry is introduced. Figure 2 shows how the latitude range of the
asymmetric heating maps into the magnetosphere in a dipole field.
The specific latitude range 72-78◦ N is chosen firstly because this introduces the heating in
a region coupled to the field lines that are expected to carry the cam currents. Using a dipolar
model, the range 12-15RS in the magnetosphere maps to latitudes of 73.2◦ N to 75◦ N. Thus
our heating completely encapsulates this range. Furthermore, as described below, we choose
the latitude 75◦ as the approximate boundary between corotational and sub-corotational
flow in our simplified magnetospheric plasma flow model. Thus a second reason for the
location of our asymmetric heating, and the reason for the specific latitude limits of 72◦ and
78◦ , is that it is centred on this important latitude for thermospheric dynamics.
2.1.4
Ionosphere
Thermosphere-ionosphere coupling is implemented using the same simplified scheme described by Smith & Aylward (2008). This entails using data from an ionosphere model to
produce a global fixed map of Pedersen and Hall conductivities and then fixing these in such
a way that the height-integrated conductivities are independent of variations in the thermal structure. We wish to keep our model as simple as possible, so use a globally averaged
profile of conductivities at all latitude and longitudes. We generate this by taking equinox
ionosphere profiles at 70◦ N and 70◦ S from a recent version of the Moore et al. (2004) ionosphere model (Luke Moore, private communication, 2009). We average these in longitude,
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producing average ionospheres for mid-to-high latitudes in both the northern and southern
hemispheres, and then average these two profiles to produce a single representative ionosphere. We then calculate a profile of conductivity, which we fix according to the procedure
described by Smith & Aylward (2008). We choose to use ionosphere profiles from a latitude of 70◦ to provide a reasonable representation of conductivities across the mid-to-high
latitude range in which we are interested.
The resulting profiles of conductivities calculated for our baseline thermal structure are
shown in Fig. 3. This shows that across most of our altitude range Pedersen conductivity is
much more important than Hall conductivity, although Hall conductivity is more important
at the base of the model and may make a dominant contribution to currents at altitudes
that we do not study here. Across our altitude range the height-integrated Pedersen and
Hall conductivities are 0.76mho and 0.55mho respectively.
2.1.5
Rotational coupling to the magnetosphere
Smith & Aylward (2008) presented a detailed analysis of the interaction between the thermosphereionosphere and magnetosphere at flow shears in the rotational profile of the magnetosphere,
as represented by the model of Cowley et al. (2004). Such an analysis is not the purpose
of this study, nor is it possible with the reduced latitudinal resolution necessitated by the
use of a three-dimensional model. We therefore use a very simplified model of the rotational
plasma flows in the magnetosphere which is enough to represent the overall forcing of the
thermosphere-ionosphere by the magnetosphere without introducing unnecessary complications. To this end we assume that the plasma in the magnetosphere rigidly corotates with the
planetary angular velocity (which, for the purposes of the thermosphere model, means rigid
corotation with the lower boundary pressure surface) at colatitudes greater than 15◦ . Polewards of this colatitude we assume 30% of rigid corotation. The location that this colatitude
maps to in a dipole magnetic field is shown in Fig 2 by the dashed line.
These plasma flows then interact with the thermosphere using the same formulations of
Joule heating and ion drag described by Smith & Aylward (2008). We assume a constant
vertical magnetic field at all latitudes, taking a round value of 60,000nT consistent with
the fields observed in Saturn’s polar regions. The advantage of assuming a vertical field is
transparency in linking field-aligned currents to the thermosphere model, since individual
field lines can be correlated uniquely with latitude-longitude grid points. The advantage of
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Figure 4. Excess temperature and winds at pressure level n=6 in the northern polar region after the axially symmetric values
have been subtracted. The plot is shown using an orthographic projection, equivalent to viewing the planet from infinity. The
colour scale indicates the excess temperature, with zero excess temperature indicated by the solid contour. The arrows show the
size and direction of the excess horizontal wind. The length of the arrows represents the total size of the excess horizontal wind:
the longest arrows show a speed of ∼20ms−1 . The dotted lines show circles of constant latitude at 10◦ spacing; the dash-dot
line represents our zero longitude; the dashed circles show the latitudes mapping to ∼ 12 − 15RS in the magnetosphere; and
the triple dot-dash contour shows the region in which our asymmetric heat source is applied.
assuming a constant field is that our model of conductivity can be held fixed at all latitudes
while retaining a reasonable degree of self-consistency. Both assumptions are good at high
latitudes but become progressively weaker moving towards the equator. At latitudes below
about 60◦ at which the assumptions begin to become genuinely poor, our calculations are
unlikely to make an important contribution to the cam currents since these regions map to
radii of less than ∼4RS .
2.2
Results
We now describe the results of one run encapsulating all of the elements describing above.
The overall behaviour of the neutral winds is essentially identical to that described by
Smith et al. (2007), showing a high degree of axial symmetry. Ion drag generates a region
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Figure 5. Fourier decomposition of temperature and winds (at the n=6 pressure level) and current parameters (heightintegrated). The solid, dotted, dashed, dot-dash and triple dot-dash lines respectively show the relative amplitude of the m=1,
2, 3, 4, 5 harmonics of the longitudinal distribution of each parameter as a function of latitude. The thick lines show the
equivalent information for the square wave heating distribution (which by definition does not contain any even harmonics).
of sub-corotating neutral flow at high latitudes, which is forced by Coriolis forces into a
convergent polewards flow that generates heating at the pole itself. This behaviour has
already been comprehensively analysed by Smith et al. (2007) and Smith & Aylward (2008).
Instead we focus here on the axially asymmetric behaviour that is induced by the axially
asymmetric component of heating.
The asymmetric components of the temperatures and horizontal winds are shown in
Fig. 4 for the pressure level n = 6 (at a pressure of ∼8nbar). This altitude coincides with the
peak of the Pedersen conductivity and therefore makes the greatest contribution to coupling
with the magnetosphere. The asymmetric components are calculated by longitude averaging
each quantity and then subtracting this average from the original value at each point. It is
perhaps not surprising that the asymmetric heating has introduced an asymmetry in the
temperatures and winds! What is interesting about these results is that the very simple
and geometrically confined heat source has produced a globally smooth and symmetric
distribution of temperatures and winds which very nearly varies sinusoidally in longitude.
To illustrate this, we have Fourier decomposed the longitudinal distribution of temperatures and winds at each latitude. The upper panels of Fig 5 show the relative amplitudes
of the first five harmonics as a function of latitude for the temperature, meridional wind
and zonal wind. The thick lines show the equivalent decomposition of the square wave
heating distribution. With the exception of the meridional winds in the range 65-75◦ N, the
higher order harmonics are everywhere relatively less significant than they are in the heating
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Figure 6. Sketch of idealised interhemispheric circuit. The northern and southern hemispheres each contain both an ideal
current source, representing the wind generated currents, and a conductance. The vertical wires in the sketch represent the
highly conducting magnetic field lines that connecting the two hemispheres.
distribution. Thus the higher order harmonics in the heating distribution have been very effectively ‘washed out’ by the thermospheric winds to produce a smoothly varying and nearly
sinusoidal pattern of thermospheric flow. It is also notable that the odd harmonics (m = 3, 5)
are both more significant than the next lowest order even harmonics (m = 2, 4) suggesting
that most of the asymmetric structure does indeed originate from the heating distribution,
which contains only odd harmonics. The lower panels in the figure will be discussed later.
We do not show equivalent plots to Figs 4 and 5 for the southern hemisphere because the
asymmetries are negligible, i.e. the asymmetry introduced by heating in the northern hemisphere does not significantly propagate across the equatorial thermosphere. This opens up
the possibility that two independent asymmetric heat sources could exist in each hemisphere,
driving rotating signatures in the magnetosphere at different rates (see Section 5).
3
3.1
INTERHEMISPHERIC CURRENTS
Calculation of currents
Given the distribution of winds, plasma flows, conductivities and magnetic fields discussed
above, we calculate the distribution of horizontal currents in the thermosphere-ionosphere
using the same procedure as Smith & Aylward (2008). Inevitably there are regions where
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these currents are convergent or divergent, and in these circumstances we require that the
current is somehow closed. This can occur either by the local generation of an electric field
that drives currents opposite to the wind-driven currents, or by the current flowing along field
lines into the magnetosphere, where the current can complete the circuit either by flowing
across field lines in the magnetosphere or by flowing across a conductance in a connected
region of the thermosphere-ionosphere. We assume here that none of our current closure is
provided by cross-field line currents in the magnetosphere; all current closure occurs in the
thermosphere-ionosphere.
Since we require our current systems to be quasi-static, we would expect any induced
electric field to be slowly varying and thus to persist for long enough to penetrate along
the magnetic field lines to the opposite hemisphere. Hence any closure current driven in the
ionosphere must flow in both hemispheres. Fig 6 shows the situation as an equivalent circuit
diagram. The winds in both the northern and southern hemisphere act effectively as current
sources providing currents IN and IS which must close through the conductances Σ in the
northern and southern hemispheres (which in this study are assumed to be equal). These
conductances are connected in parallel by ideal wires, representative of the magnetic field,
which carry a current Ik . An identical voltage V must be induced across both conductances
and therefore an identical current IC = ΣV must flow through each. Applying current closure
at each junction it is clear that IC = 0.5(IN + IS ) and Ik = 0.5(IN − IS ). Half of the closure
current for the north flows in the south, and vice versa, and since these currents must flow
along the field in opposite directions, the net interhemispheric current must be half the
difference between the current supplied by each hemisphere.
In reality, the current that is required to close is the divergence of the horizontal current
in each hemisphere. Applying the model described above it is clear that the current jk flowing
along the magnetic field between the hemispheres from north to south is
jk = 0.5(−∇ · JN + ∇ · JS )
(2)
where JN and JS are the horizontal currents in the northern and southern hemispheres
respectively. The minus sign is introduced relative to the above discussion because a positive
divergence in the northern hemisphere implies a current flowing from south to north.
A secondary effect of the induced electric field is energy and momentum transfer between
the hemispheres. In the thermosphere this is manifested as Joule heating and ion drag.
This has the potential to produce feedback effects which may change the nature of the
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Figure 7. Horizontal currents and their divergences in the northern polar region. The colour scale indicates the size of the
divergence. Note that a positive divergence implies a downwards current, corresponding to south-north current in the equatorial
plane. The solid line is a contour of zero divergence, and because the divergences in the southern hemisphere are negligible,
this is, to a very good approximation, magnetically connected to the equivalent solid line in Fig. 8. The arrows show the size
and direction of the height-integrated Pedersen and Hall currents at each location. The longest arrow represents a current of
∼1mAm−1 . The dashed, dotted, dot-dash and triple dot-dash lines have the same meaning as in Fig. 4.
thermospheric flows. To analyse this we would need to explicitly calculate the induced electric
fields: for this first study we simply assume that the interhemispheric current is given by the
formula above and neglect interhemispheric energy and momentum flows.
3.2
Results
Figure 7 shows the asymmetric horizontal currents in the northern hemisphere and the
asymmetric component of their divergence, in a similar format to Fig. 4. This figure shows
only the divergences of the horizontal current in the northern polar region. There are also
divergences in the southern hemisphere, but these are negligible in comparison and are not
shown. We ignore longitudinally symmetric divergences. Although these may be involved in
currents flowing into the magnetosphere they will by definition produce an identical signal
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Figure 8. Interhemispheric currents mapped into the equatorial plane. The colour scale indicates the size and direction of the
interhemispheric currents. The solid line is a contour of zero interhemispheric current. The dashed lines show the L-shells 12
and 15 between which the current system proposed by Southwood & Kivelson (2007) is predicted to flow. These dashed lines
are magnetically connected to the equivalent dashed lines in Fig. 7. The dotted lines shows L-shells at a spacing of 3RS , thus
indicating the locations of the five L-shell ranges used to produce Fig. 9.
at all longitudes, thus not contributing to the cam current system even if they are important
in terms of global angular momentum transfer. However, it is worth noting that since we
are imposing extra heating in the northern hemisphere, we do observe a small north-south
asymmetry in the longitudinally symmetric component of the current divergence.
There are two interesting features of the distribution of current divergence. Firstly, the
regions of positive and negative divergence show a clear spiral structure. This appears to be
a consequence of the action of Coriolis force on thermospheric winds, and will be discussed
further below. Secondly, a Fourier decomposition of the distribution of current divergence
in longitude (Fig. 5) shows that the m=3 and m=5 harmonics are almost as significant
in the distribution of current divergence as in the square wave heating distribution in the
range 70-75◦ N. Since in our simple model of current closure the current divergence directly
determines the interhemispheric current in the magnetosphere, we can therefore expect some
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higher order harmonics to be present in the resulting magnetic field perturbations. It is also
worth noting that these higher order harmonics are much more pronounced in the current
divergence than in the meridional and zonal horizontal currents. This is because taking the
divergence increases the relative amplitude of the higher order harmonics. Thus the situation
that we predict in which the m = 1 harmonic is relatively dominant is only possible because
the action of the thermosphere significantly reduces the relative amplitude of the higher
order harmonics in the horizontal wind and current distributions before the divergence is
calculated. This leads us to the general suggestion that the thermosphere promotes the
formation of current systems with m = 1 symmetry by favouring this mode over higher
order harmonics.
Fig 8 show the resulting field-aligned interhemispheric currents mapped into the equatorial plane of the magnetosphere. We use a dipole field for all calculations in this study as a
sufficient approximation in the inner region of Saturn’s magnetosphere. The greatest current
densities shown are of the order of 0.1pAm−2 at L∼4, inside the region of the predicted cam
currents. The size of the current then decreases rapidly with distance from the planet due
to the inverse cube relationship of the magnetic flux density.
4
MAGNETIC FIELD PERTURBATIONS
4.1
Calculations using Biot-Savart law
To calculate the magnetic field perturbations we treat the magnetic field lines as discrete
wires carrying all of the current flowing out of each latitude-longitude cell in the atmosphere.
Using the relatively low resolution of the thermosphere model this produces poor results, so
we linearly interpolate field-aligned currents onto cells of size 2.5◦ in longitude and 0.25◦ in
latitude. Each magnetic field line originating at the centre of one of these cells is then divided
into approximately 100 sections of equal length spanning the range from the ionosphere to
the equatorial plane. Finally, we apply the Biot-Savart law to these sections, taking into
account both hemispheres, to calculate the induced magnetic field in the equatorial plane.
4.2
4.2.1
Results
General morphology
Figure 9 shows the calculated magnetic field perturbations in the equatorial plane. The first
five panels show the magnetic field induced by the currents flowing in five adjacent L-shell
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Figure 9. Magnetic field perturbations in the equatorial plane. The first five plots show the field perturbations due to currents
flowing on specific ranges of L-values, as marked and indicated by the dashed lines; the final plot, at the bottom right, shows
the total field perturbation due to currents flowing on magnetic shells in the range L = 6 − 21, again indicated by dashed lines.
The colour scale indicates the magnitude of the flux density. The arrows indicate the direction of the perturbation field. The
solid line on each plot indicates the orbit of the hypothetical spacecraft discussed in the text.
ranges (6-9, 9-12, 12-15, 15-18, 18-21), and the final panel shows the total magnetic field
induced by all of these contributions. The total size of the flux density is shown by the
colour contours and the direction of the field by the arrows. The dashed lines show the
L-shell ranges used to calculate the field shown in each plot, and the solid line the orbit of
the hypothetical spacecraft described below.
The magnetic fields induced by each L-shell range are similar in that they are all roughly
uniform within the current-carrying shells, and roughly dipolar beyond them. This is a simple consequence of the asymmetry being dominated by a sinusoidal variation in longitude,
and is essentially identical to the magnetic field that Southwood & Kivelson (2007) described, which would be induced by currents flowing along a narrow range of L-shells with
this type of longitude asymmetry. More interesting than this is that the orientation of the
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Figure 10. Magnetic fields perturbations observed by a spacecraft in orbit at radius 10RS . The thick and thin solid lines show
the azimuthal and radial fields respectively. The dashed and dotted lines show two sinusoids, out of phase by 90◦ with their
phase and amplitude adjusted by eye to fit the model output.
roughly uniform field produced within each L-shell range is different, rotating eastward with
increasing radial distance. The total magnetic field vectors in the final panel also show this
rotation, although the behaviour is slightly more complex when the different contributions
are combined. This effect will be discussed further below.
The observation that the magnetic field induced by this whole range of L-shells is approximately uniform is enough to match the Southwood & Kivelson (2007) model. However,
in order to emphasise this point we show in Figure 10 the magnetic field perturbations that
would be observed by a spacecraft in a circular orbit at 10RS . This shows that the observed
radial and azimuthal fields are approximately out of phase by 90◦ and are a good match to a
sinusoidal model, the azimuthal field appearing closer to a perfect sinusoid. While it is clear
that the morphology of the magnetic field perturbations is a good match to the observations,
the magnitude of the field is not. The typical fields observed by spacecraft are of the order
of a few nT, whereas our predicted fields are two orders of magnitude lower than this, of
the order of 10pT. Some possible mechanisms for increasing the magnitude of the field are
discussed in Section 5.
4.2.2
Phase variation with radial distance
Also of interest is the relative phase of the magnetic field perturbations at different radial
distances. Gurnett et al. (2007) found that within ∼12RS the phase of the magnetic field
perturbations were almost independent of distance, but that beyond this radius a large phase
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Figure 11. Phase shifts of magnetic field perturbations with radial distance. The left and right hand plots respectively show
the variation of the azimuthal and radial field perturbations with longitude for six L-values. The amplitude of each trace has
been normalised. The dashed and dotted lines show a sinusoid fit to the data with a phase varying linearly with L-value. The
sinusoid fits for the azimuthal and radial fields have been determined independently. Note that this plot is similar to Figure
S3 of Gurnett et al. (2007) but they use a west longitude system, such that a left-to-right shift on their figure represents the
opposite phase shift to a left-to-right shift on our figure.
lag of ∼40◦ developed by ∼20RS . In this context ‘phase lag’ implies that the perturbation is
shifted slightly westward at larger radial distances as would be expected if the disturbance
propagates outwards from a source in the inner magnetosphere.
In our results, we find that the opposite situation develops. As has already been commented, the field perturbations induced by different L-shell ranges shown in Fig. 9 rotate
eastward with increasing radial distance. As a result, the magnetic field perturbations at
larger radial distances lead those at smaller radial distances. This is illustrated by Fig. 11,
which shows the normalised azimuthal and radial fields predicted by our model at six radial
distances. In both cases there is a clear eastward phase shift with increasing radial distance.
This phase shift is much clearer in the azimuthal field. Interestingly, the azimuthal field
appears shows most of its phase shift beyond 6RS , while the radial field shows very little
variation beyond this radius.
Fig. 11 also shows sinusoidal fits to the model output. We have fitted sinusoids by eye to
the field perturbations at L=3 and L=18, and then interpolated the phase linearly between
these two extremes. Based on these fits, the azimuthal and radial fields show phase shifts of
80◦ and 30◦ respectively across the range shown. At L=3 the azimuthal field leads the radial
field by 95◦ while at L=18 it leads by only 45◦ . However, it is clear that the sinusoid fit to
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the radial field is poor, so this inferred change in the phase relationship between the two
components should not be considered significant. However, the difference in magnitude of
the phase shifts for the two components is clear and significant and demands an explanation.
The phase shift itself occurs simply because the longitudinal variations of the currents
flowing on each set of L-shells are not precisely in phase. This effect is almost certainly a
consequence of the importance of Coriolis force in the upper atmosphere. Any meridional
transport will tend to be modified by Coriolis forces to produce zonal motions, such that
features at adjacent latitudes are unlikely to be coupled without a longitudinal phase difference being generated by zonal wind shear. Specifically, an equatorwards motion will be
modified westwards and a polewards motion modified eastwards by Coriolis. The effects of
this are visible in Figs. 4, 7 and 8, all of which display clear spiral structures about the pole,
oriented SW-NE at the planet. Although we are only presenting the results of one model
run, we can thus tentatively suggest that phase shifts of this nature — but possibly not of
this magnitude — should be expected to be a universal feature of cam currents driven by
upper atmospheric winds.
The difference in the magnitude of the phase shifts in the azimuthal and radial components can be explained by analysing the distribution of magnetic field perturbations produced
by a specific shell of field-lines. Here, we analyse the magnetic field perturbations produced
by currents on L-shells in the range 12.5-13.5. We choose this range because it lies in the
centre of the postulated region of cam currents, and the radial distance L = 13 maps almost
exactly to a latitude of 74◦ , which coincides with a ring of grid points in the thermosphere
model. This means the currents flowing on this small range of shells are almost exclusively
determined by the divergence of the currents flowing at this specific well-defined latitude.
Fig 12 shows a Fourier analysis of the the amplitude of the first five harmonics of the longitudinal distribution of magnetic field perturbations produced by this range of shells, as a
function of radial distance. The line formats are the same as those in Fig. 5.
The difference between the magnitude of the radial and azimuthal field perturbations
either side of the current sheet is striking. The amplitude of the azimuthal field perturbation
rises by a factor of ∼5 on crossing the current sheet in the direction of the planet, and also
changes sign (which is not shown in the figure) while the radial field remains approximately
constant. Thus, as we move outwards through the magnetosphere, the contribution to the
azimuthal field from each shell that we cross becomes abruptly both small and negative.
Since the phase of the perturbation at any radial distance is a weighted combination of
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Figure 12. Fourier analysis of magnetic field perturbations due to currents flowing in the range L = 12.5 − 13.5RS , shown by
the grey shaded region on the plot. The lines show, using the same formats as Fig. 5, the magnitudes of the first five harmonics
of the longitudinal distributions of azimuthal and radial field perturbations.
the phases of all of the contributions from different magnetic shells at different distances,
the sudden removal of one of these components can significantly alter this combination and
allow the phase of the azimuthal field to change significantly over a relatively short radial
distance. By contrast, the radial field generated by each shell of field lines has a significant
influence on the field at both larger and smaller radii, such that we would expect the radial
field signature to vary more gradually and be a mixture of contributions from a broader
range of radial distances.
The Fourier analysis also allows us to suggest why the azimuthal field is closer to a
perfect sinusoid. As we have just described, most contributions to the azimuthal field are
made inside a current-carrying shell. Figure 12 clearly shows that inside a shell the m = 1
harmonic grows with decreasing radial distance, while the higher order harmonics all decline
in amplitude. The same is true for the radial field inside a shell, but outside the shell the
m = 2 and m = 3 harmonics in the radial field decline at a similar rate to the m = 1
harmonic. Since, as discussed above, a shell makes a significant contribution to the radial
field at both smaller and larger radial distances, this latter effect means that higher order
harmonics in the radial field produced by currents at a given radial distance can persist
to much larger radial distances. This effect is apparent in Fig. 11, in which a single nonsinusoidal structure persists from L = 9 all the way out to L = 18.
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5
5.1
C. G. A. Smith
DISCUSSION
Possible origin of heating
We have shown that asymmetric heating at high latitudes can generate magnetic field perturbations of the appropriate distribution in the magnetosphere. However, our imposed heating
distribution was essentially arbitrary. We now suggest some possible origins of such a heating distribution. Two issues need to be addressed: the location of the rotating feature that
ultimately drives the heating, and the mechanism for generating the heating itself.
The key property of the feature that drives the heating is that it must rotate stably
at a period close to that at which the magnetic field perturbations are observed to rotate.
Since this period is close to the rotation period of the lower atmosphere as observed in
the motion of cloud features, the heating must originate in a layer of the atmosphere that
does not substantially sub-corotate due to drag from the magnetosphere. This would appear
to rule out the high latitude thermosphere that is directly coupled to the sub-corotating
outer magnetosphere. However, it does not rule out regions at lower altitudes or latitudes.
Therefore our heat source could be located anywhere in the lower and middle atmosphere
or at mid-latitudes in the thermosphere.
The existence of stable features rotating with the planet in the lower atmosphere is well
established. Numerous relatively small storm systems are observed at Saturn, and the Great
Red Spot of Jupiter has existed stably for over 150 years (Ingersoll et al. 2004). Although
these features are tropospheric, it is possible that such a large scale vortex exists in the
mid or high latitudes in Saturn’s middle atmosphere, but has not been observed due to
the relative difficulty in making observations of this part of the atmosphere. At Saturn,
the persistence of the northern polar hexagon (first described by Godfrey (1988) and also
a feature of the lower atmosphere) is now well established (e.g. Baines et al. 2009), and a
similar or related feature in the middle atmosphere could also provide the asymmetry that
we require.
It is quite clear that there is no evidence to support the existence of a comparable stable
feature in the mid-latitude thermosphere. However, we can speculate that such a feature
would be most likely to persist at regions of shear in the zonal flow where vortices might be
formed by the Kelvin-Helmholtz instability. Such a flow shear exists at the boundary between
the higher latitude regions subject to sub-corotational ion drag from the outer magnetosphere
and the lower-latitude regions connected to the corotating inner magnetosphere. Smith &
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Aylward (2008) showed that these lower latitude regions actually tend to super-corotate
slightly because the ion drag acting on the higher latitudes induces a polewards flow that
is spun up by Coriolis forces. There is thus a point in this flow shear that rotates at the
correct planetary period, and a vortex forming in this region could thus have the appropriate
properties. In our model such a vortex would lie approximately in the 72-78◦ latitude range in
which we impose our asymmetric heat source. Another possibility is that asymmetric heating
is driven not by a discrete feature but simply by variation in the degree of turbulence in
different longitude sectors. A discrete ‘storm’ localised in longitude could generate such an
asymmetry across a large range of longitudes if it had a turbulent wake.
Any of these possibilities essentially represent asymmetries in the winds in various different regions of the atmosphere. In our model we have represented such an asymmetry by an
increase in thermospheric heating, but in practice any form of asymmetry in thermospheric
forcing will produce an asymmetry that is likely to generate a corresponding asymmetry in
the magnetosphere. A lower or middle atmosphere feature could generate an asymmetry in
thermospheric forcing if there was substantial generation of gravity waves which may break
in the thermosphere and generate asymmetric heating and/or drag. A feature existing directly in the thermosphere would represent the asymmetry itself and would also generate
asymmetries in Joule heating and ion drag.
5.2
Magnitude of field perturbations
As already discussed, our results provide a good match to the observed morphology of the
magnetic field perturbations, but not to their magnitude. For our mechanism to explain the
observations the magnitude of the currents generated must therefore be greater by a factor
of ∼100. There are several possibilities for providing this increase.
Most obviously, any of the atmospheric parameters that directly contribute to the size
of the current may have been underestimated. A greater intensity of heating seems likely to
increase the magnitude of the currents, but a factor of 100 seems unlikely to be achieved
without grossly overheating the thermosphere. Similarly, changing the altitude or vertical
distribution of the asymmetric heat source seems unlikely to produce the required increase.
More plausible is that the conductivities are underestimated. Our ionosphere model, based
on that of Moore et al. (2004), does provide a reasonable match to the observed electron
densities at Saturn. However, there may be localised zones of enhanced electron density, for
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example at or close to the main auroral oval. A latitudinally extended enhancement in the
conductivity may not provide the required enhancement in current density because it will
also lead to increased ion drag, thus slowing the winds that generate the currents. Hence
a latitudinally thin region of enhanced conductivity is probably ideal, providing a large
divergence in the horizontal current density at its edges without significantly impeding the
neutral flow that generates the currents in the first place. However, a further problem with
this might be the formation of almost equal and opposite interhemispheric current sheets at
the northern and southern boundaries of the conductivity enhancement. These would flow
on closely nested magnetic shells, producing almost equal and opposite field perturbations
that would essentially cancel each other out. It is clear that a detailed assessment of the
effect of various plausible conductivity enhancements is required.
A further possibility is that the asymmetric currents required may be generated at lower
altitudes. This study has investigated principally the Pedersen conducting region of the upper
atmosphere. The winds generated by an asymmetry in the Hall conducting region, which lies
largely below our model’s lower boundary at 100nb, may drive more significant currents into
the magnetosphere. However, it is worth noting that there is a limit to the depth at which
such an asymmetry could be coupled to the magnetosphere. The field-aligned conductivity
decreases with increasing neutral density, such that sufficiently deep in the atmosphere any
asymmetric structures will be electrically insulated from the magnetosphere even if there is
sufficient ionospheric density to generate large horizontal currents.
Finally, a limiting factor in the magnitude of the divergences predicted by our model is
the spatial resolution of the thermospheric grid. The latitude resolution of 2◦ and longitude
resolution of 10◦ both correspond to spatial distances of ∼2000km at the latitudes that map
to the cam currents. Structures in the upper atmosphere related to auroral processes are
known to exist on scales smaller than this. For example, the width of the auroral oval is of the
order of 300-600km (Cowley et al. 2004), implying that spatial gradients in parameters such
as the ionospheric conductance and plasma flow speeds must occur on a still smaller scale,
perhaps 100km or smaller, an order of magnitude below our spatial resolution. Divergences
that develop over these scales are not resolved by our model. Thus the lower spatial resolution
of our thermosphere model may partially account for the small size of the calculated magnetic
field perturbations. A related issue is that due to the mapping between the thermosphere
and equatorial magnetosphere, the 2◦ resolution of the model maps to relatively low Lshell resolution in the magnetosphere across our region of interest, as clearly shown by the
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grey lines in Fig. 2. This means that the interhemispheric currents that we calculate are
distributed smoothly over many RS , while in reality they may flow in much more spatially
confined sheets.
5.3
Hemispheric asymmetries
This study has focussed on an asymmetry introduced in the northern hemisphere. This
choice is completely arbitrary. What would be the effect of adding an identical heat source
in the southern hemisphere at the same longitude? The opposite sign of the Coriolis force
in the southern hemisphere would ensure that the winds generated were a mirror image
of those in the north. The opposite sign of the magnetic field in the southern hemisphere
then further ensures that all divergences of horizontal currents would be identical, such that
the currents driven by the two hemispheres would cancel each other out, producing zero
interhemispheric current. However, it seems unlikely that two identical regions of heating
could remain correlated in longitude in this way unless there was an underlying driver much
deeper in the planet.
Furthermore, recent research (Gurnett et al. 2009) has shown a hemispheric asymmetry
in the rotation rates of radio emissions related to high latitude magnetic field lines, with SKR
emission from the northern and southern hemispheres exhibiting periods of about 10.8 and
10.6 hours respectively. If the same mechanism that generates the cam currents is also related
to these radio emissions, then we would require two complementary regions of asymmetric
heating in the north and south rotating at these two slightly different rates. Assuming that
the perturbations in the northern and southern hemispheres were roughly identical in terms
of the magnetic perturbations produced, and that they behaved essentially independently,
we would expect them to beat against each other, sometimes interfering constructively and
sometimes destructively, with a period of approximately 50 Saturn rotations.
Another simplification of our model is that the northern and southern hemispheres have
identical ionospheric conductances, which would only be expected at equinox. We can predict
the effect of seasonally variable ionospheric conductances if we reanalyse the equivalent
circuit in Figure 6 with independent northern and southern conductances ΣN and ΣS . If we
assume to a first approximation that the thermospheric wind distributions are independent
of the conductances, then the currents in the north and south will be proportional to these
conductances, IN = ΣN VN and IS = ΣS VS , where VN and VS are effective wind-generated
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voltages in the northern and southern hemispheres. These modifications imply the following
formula for the closure current Ik :
Ik = (VN − VS )
ΣN ΣS
ΣN + Σ S
(3)
As commented above, this implies that the closure current will fall to zero if there are equivalent thermospheric asymmetries in the northern and southern hemispheres. However, it also
shows a symmetry in the dependence of the closure current on the northern and southern
conductances. For example, the component of the closure current driven by VN has exactly
the same dependence on ΣN has it has on ΣS . This is due to two complementary effects.
Firstly, a larger ΣN means that a larger current is generated by the winds in the northern
hemisphere, implying a commensurately larger closure current. Secondly, a larger ΣS implies that a greater proportion of the closure current flows in the southern hemisphere, also
implying a larger interhemispheric closure current. The combination of these effects means
that for fixed ΣN or ΣS the above expression for Ik maximises when the two conductances
are equal.
A naive interpretation of this result implies that the interhemispheric current should
maximise at equinox. However, the factors influencing conductance are complex, such that
during solstice one conductance may rise much more than the other falls, making it difficult
to predict how the interhemispheric current will change. Nevertheless, it is clear that if either
of the conductances falls to a very small value during solstice (which seems perfectly possible
with the correct combination of winter illumination conditions and ring shadowing) then the
interhemispheric current will also fall almost to zero – either because very little current is
being generated in the winter hemisphere, or because very little current closes across the
winter hemisphere.
All of these considerations make the assumption that solar insolation is the main driver
of ionospheric conductance. While this seems highly likely at low latitudes, the exact balance
between solar and auroral ionisation at mid and high latitudes is open to question. If auroral
ionisation dominates at the latitudes where the cam currents are generated, then we might
expect very little seasonal variation.
Finally, the above discussion assumed that thermospheric winds would remain independent of conductance. In practice, a larger conductance will increase ion drag and may therefore slow thermospheric winds and reduce the effective wind-generated voltages, producing a
non-linear dependence of current on conductance. The effect of this would be to disrupt the
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hemispheric symmetry of Eqn. 3. As commented above – and taking the example of currents
generated by an asymmetry in the north – the two effects that produce the symmetry are
that a rise in the conductance in the north will produce a proportionate rise in the generated
currents, and that a rise in the conductance in the south will produce a proportionate rise
in the interhemispheric closure currents. If the winds that generate the currents are slowed
by a rise in conductance in the north, the former effect would be slightly reduced. Thus a
larger conductance in the south would be most conducive to the existence of a large closure
current generated by an asymmetry in the northern thermosphere. So, in this scenario, we
expect a northern asymmetry to produce the greatest effect during southern summer.
5.4
Radial phase shifts
As described in Section 4.2.2, we find that the phase of the magnetic field perturbations at
larger radial distances lead those at smaller radial distances. This is a specific consequence
of the action of the Coriolis force on structures in the thermosphere, and therefore the
observation of such a phase shift would provide evidence of a thermospheric source for
the cam currents. However, at present there is clear evidence for a large radial phase lag,
increasing with radial distance, beyond L ∼ 12, and indications of a possible small phase lag,
again increasing with radial distance, within this radius (Gurnett et al. 2007). The evidence
for the phase lag in the inner region is marginal, and indeed Andrews et al. (2008) commented
that the order of magnitude of this inner region phase lag was comparable to the scatter in
their data. We thus choose to characterise the observed phase shift as undetermined within
L ∼ 12, and as almost certainly lagging beyond this radius.
Assuming that the Coriolis-driven phase lead is a universal feature of cam currents driven
by the thermosphere, this significant phase lag at large radial distances would at present
appear to rule out direct control of the magnetic field perturbations by the thermosphere
beyond L ∼ 12. However, within this radius the possibility remains open. We can identify
two scenarios in which the characteristics of the thermosphere-driven cam current system
described here would be consistent with the observed phase shifts. Firstly, the thermospheredriven cam currents may be broadly distributed across the region within L ∼ 12. In this
case we would expect our radial phase lead to be observable across this region. Secondly, the
thermosphere-driven cam currents may be confined to a relatively narrow range of shells in
the region of L ∼ 12. This would be consistent with a sheet-like current system produced
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by a sharp latitudinal conductance gradient in the thermosphere, as discussed above. In
this case we would expect to see the radial phase lead only on the current-carrying shells
themselves. Depending on the width of the current-carrying region, such phase leads might
be difficult to detect with certainty.
6
SUMMARY
We have studied the distributions of temperatures, winds and electrical currents driven in
Saturn’s upper atmosphere by a hypothetical asymmetric heat source in the thermosphere
and subsequently examined the related magnetospheric current systems and field perturbations. Our principle conclusions are as follows:
(i) A square-wave distribution of heating is ‘washed out’ by thermospheric winds, such
that the longitudinal distributions of temperature, winds and currents in the ionosphere are
dominated by the m=1 harmonic.
(ii) If divergences in the resulting distribution of currents close via interhemispheric fieldaligned currents, these produce magnetic field perturbations in the equatorial magnetosphere
whose direction matches those predicted by the model of Southwood & Kivelson (2007).
(iii) The size of these magnetic field perturbations are of the order of 10pT, two orders
of magnitude smaller than the observed magnetic field perturbations. This discrepancy may
be explained by the low resolution of our model and/or an incomplete model of ionospheric
conductance.
(iv) The predicted magnetic field perturbations develop a phase lead with increasing radial
distance. This is explained by the importance of the Coriolis force in the development of
structures in the thermosphere. This phase lead is opposite to the phase lag that is observed
(e.g Gurnett et al. 2007). This suggests either that thermospheric cam currents must flow
only at smaller radial distances than those for which a phase lag is observed, or that they
flow only on shells covering a very narrow range of radial distances.
This study clearly demonstrates the plausibility of a thermospheric driver for the rotating
asymmetries in Saturn’s magnetosphere. Further exploration of the many possible distributions of thermospheric forcing and ionospheric conductivity are required to determine
whether this mechanism can reproduce the observations in full.
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ACKNOWLEDGEMENTS
I would like to thank Nick Achilleos, Margaret Kivelson and Tom Stallard for useful discussions, and Luke Moore for providing the model ionospheres.
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