A simple quantitative model of plasma flows and currents in Saturn s polar ionosphere

A simple quantitative model of plasma flows and currents in Saturn s polar ionosphere S.W.H. Cowley*, E.J. Bunce, and J.M. O Rourke Department of Physics & Astronomy, University of Leicester, Leicester LE1 7RH, UK Draft A: Submitted to J. Geophys. Res., December 2003 *Corresponding author: Tel.: 0044 116 223 1331; Fax: 0044 116 252 3555; E-mail: swhc1@ion.le.ac.uk

Abstract. We propose a simple illustrative axi-symmetric model of the plasma flows and currents that occur in Saturn s polar ionosphere which are due both to internal magnetospheric plasma processes and the solar wind interaction. The features of the model are based on previous physical discussion, guided quantitatively both by Voyager plasma observations on closed field lines and remotesensing IR Doppler observations on open field lines. With increasing latitude, the flow features represented include a region poleward of ~25 co-latitude where the angular velocities decrease continuously from rigid corotation to ~60% of rigid corotation due to plasma production from internal sources in the central magnetosphere, a narrow band of higher but still sub-corotating angular velocities mapping to Dungey-cycle return flow and Vasyliunas-cycle flow regions in the outer closed-field magnetosphere, and finally a region of low angular velocities, ~30% of rigid corotation, on open field lines in the polar cap. We show that these flows require a four-region pattern of field-aligned currents. With increasing latitude, these consist of regions of upward and downward current on closed field lines peaking at a few 10 s of na m -2 (for an effective ionospheric Pedersen conductivity of 1 mho), a narrow ring of upward field-aligned current across the openclosed field line boundary of order 100 na m -2, and distributed downward currents on open field lines of order 10 na m -2. Of the upward currents, only that at the open-closed field line boundary is of sufficient intensity to require significant acceleration of magnetospheric electrons, resulting in total precipitating electron powers of ~0.03 TW, together with auroral UV emissions of a few 10s of kr. The latter emissions occur in a ring of a few hundred km latitudinal width at ~13 colatitude, which we thus associate with Saturn s main auroral oval. However, by far the most important energy input to the polar upper atmosphere is due to Joule heating by the ionospheric Pedersen currents, which we estimate as typically several mw m -2 poleward of ~20 co-latitude. The overall Joule powers are estimated to be ~2 TW on both open and closed field lines in each hemisphere, thus representing a very significant energy input to Saturn s thermosphere, more than an order of magnitude larger than the globally-averaged solar input. Joule heating is thus likely to make a significant contribution to an explanation of why Saturn s thermosphere is observed to be hot, ~400-600 K, compared with less than ~200 K expected on the basis of solar heating alone. 2

1. Introduction Recent work on Saturn s coupled solar wind-magnetosphere-ionosphere-thermosphere system has been stimulated both by the recent availability of new information on auroral emissions using data from both space- and ground-based telescopes, and also by the prospect of new in situ data from the Cassini space mission, due for orbit insertion at Saturn in July 2004. Following the first detections of ultra violet (UV) auroral emissions in Saturn s polar regions during spacecraft fly-by missions [Judge et al., 1980; Broadfoot et al., 1981; Sandel and Broadfoot, 1981; Sandel et al., 1982; Shemansky and Ajello, 1983], and in data from the IUE spacecraft [Clarke et al., 1981; McGrath and Clarke, 1992], images of increasing resolution and sensitivity have been obtained by a series instruments on the Hubble Space Telescope (HST) [Gérard et al., 1995; Trauger et al., 1998; Cowley et al., 2004]. These observations show that the FUV auroral emissions, which are due to kev electron impact on Saturn s hydrogen atmosphere, generally take the form of a narrow ring around each pole located at ~15 co-latitude, whose intensity varies with both universal time and local time over the range ~1 kr (the threshold of the most recent observations) to ~100 kr. Intensities appear to peak in the pre-noon sector, and are generally stronger at dawn than at dusk. Infra red (IR) emissions from the ionospheric H + 3 ion have also been detected using ground-based telescopes [Geballe et al., 1993], and have been shown to be mainly auroral in nature [Stallard et al., 1999]. Most recently, Stallard et al. [2004] have used the Doppler shift of these emissions to show that the polar ionosphere significantly sub-corotates relative to the planet, with angular velocities of ~20-40% of the planetary angular velocity. In parallel with these observational developments, theoretical work has also been in progress considering the origins of the auroral emissions, and their implications for the nature of Saturn s plasma environment. It is generally assumed from their bright, discrete, and variable nature that the UV emissions are associated with a system of field-aligned currents which flow between the magnetosphere and ionosphere, specifically with regions of upward-directed field-aligned current where accelerated magnetospheric electrons precipitate into the ionosphere. Such current systems flow when plasma momentum is transferred between these plasma regions, resulting from their coupling through the planetary magnetic field. Two basic systems of momentum transfer and current flow may be envisaged. The first is that produced through the interaction between the solar wind and the magnetosphere, as at Earth (see e.g. Paschmann et al. [2003] and references therein). The second is associated with the pick-up and radial transport of plasma produced internally from moon surfaces and ring material within the magnetosphere [Johnson et al., 1989; Pospieszalska and Johnson, 1991; Richardson, 1992; Richardson et al., 1998], leading to significant sub-corotation of 3

the plasma, as at Jupiter (see e.g. Cowley and Bunce [2001] and Cowley et al. [2003], and references therein). No dynamical models of plasma production, transport, and sub-corotation have yet been developed for Saturn as they have for Jupiter [e.g. Hill, 1979, 2001; Pontius, 1997; Cowley et al., 2002; Nichols and Cowley, 2004], but plasma angular velocity observations are available from the Voyager Saturn fly-bys which provide an empirical guide [Richardson, 1986, 1995; Richardson and Sittler, 1990]. Cowley and Bunce [2003] have used these observations to derive an empirical model of the plasma angular velocity in Saturn s inner and central equatorial magnetosphere, and have mapped these flows into Saturn s ionosphere along model magnetic field lines to derive the pattern of field-aligned currents which are associated with sub-corotation of the plasma on closed magnetospheric field lines. The calculation employed a value of the effective ionospheric Pedersen conductivity (see later) of ~1 mho, estimated by Bunce et al. [2003] from an analysis of Voyager fly-by azimuthal magnetic field data. The calculation showed that the region of upward current associated with plasma sub-corotation is both too weak (peaking at ~10 na m 2 at ionospheric heights) and occurs at too large a co-latitude (~20 ) to account for the observed UV emission. They thus suggested that the alternative solar wind explanation should be explored more fully. This suggestion has been taken up by Cowley et al. [2004], whose semi-quantitative discussion led to the picture of equatorial magnetospheric and corresponding ionospheric flows in Saturn s system shown in Fig. 1. Figure 1a shows the proposed pattern of equatorial flows, consisting of three basic regions. In the inner and central regions, plasma production and transport from internal sources causes the plasma angular velocity to fall away from near-rigid corotation close to the planet, to values of ~50 60% of rigid corotation at equatorial distances of ~10-15 R S, as observed by the Voyager spacecraft (the sub-corotating region in Fig. 1a). R S here is Saturn s equatorial radius, taken to have the conventional value 60,330 km, and we note for purposes of scale in Fig. 1a that the distance to the subsolar magnetopause is ~20 R S. This fall from near-rigid corotation in the inner magnetosphere to significant sub-corotation in the central region, is the behaviour whose consequences were modelled by Cowley and Bunce [2003]. Beyond this in the equatorial plane we then hypothesise the existence of a region where the planetary plasma is lost by plasmoid-ejection down the dusk tail, as first discussed for Jupiter by Vasyliunas [1983] (the Vasyliunas cycle flow). At largest distances adjacent to the magnetopause, we then have plasma flows driven by the solar wind interaction, specifically by reconnection at the magnetopause and in the tail, as first discussed for Earth by Dungey [1961] (the Dungey-cycle flow). In the latter process, flux tubes reconnect with interplanetary field lines at the dayside magnetopause, are carried over the pole (out of the plane of the equatorial diagram) by the solar wind flow, and are stretched into a long magnetic tail. 4

In Saturn s magnetosphere we hypothesise that they reconnect in the tail again and flow back to the dayside preferentially in the dawn magnetosphere, as shown in Fig. 1a, both due to the Vasyliunascycle outflow into the dusk sector of the tail, and to the action of the corotational atmospheric torque resulting from ion-neutral collisions at the feet of the field lines. Cowley et al. [2004] suggested that the plasma angular velocity in the equatorial Dungey-cycle return flow and Vasyliunas-cycle region may exceed those in the outer sub-corotating region, particularly on the dawn side, due to the loss of planetary plasma on these field lines in the reconnection processes involved. It is noted that the Voyager plasma data are indeed suggestive of an increase in plasma angular velocity in a layer several R S wide adjacent to the dayside magnetopause. However, this effect was not included in the model calculations presented by Cowley and Bunce [2003]. Figure 1b shows the corresponding flows mapped to the ionosphere, together with the corresponding pattern of field-aligned currents. In addition to the regions mapped from the equatorial plane on closed field lines in Fig. 1a, this diagram also shows the central polar region of open field lines mapping to the tail lobes ( Dungey-cycle open flux ). In this region Cowley et al. [2004] suggested that the flow consists of a slow ~200 m s 1 anti-sunward motion driven by the solar wind, combined with a strongly sub-corotational circulation (~30% of rigid corotation) driven by the atmospheric torque, which tends to twist the tail lobe flux tubes [Isbell et al., 1984]. This suggestion is thus entirely consistent with the observed sub-corotation in the polar region reported by Stallard et al. [2004]. With increasing latitude, therefore, the plasma angular velocity thus first drops from near-rigid corotation at the outer boundary of Fig. 1b (which is intended to correspond to ~30 co-latitude, mapping to ~3 R S in the equatorial plane) to ~50-60% of rigid corotation at ~15-20, as modelled by Cowley and Bunce [2003]. It then increases again to perhaps ~80% of rigid corotation in the Vasyliunas-cycle ring and Dungey-cycle return flows, particularly on the dawn side, before falling again to perhaps ~30% of rigid corotation on open field lines in the polar cap. As outlined by Cowley et al. [2004], these flows then give rise to a four-region pattern of fieldaligned currents, shown in Fig. 1b by the circled dots and crosses, which indicate upward and downward field-aligned currents respectively. These currents are determined by consideration of the divergence of the horizontal ionospheric Pedersen current driven by the flow, in which it is recalled that the electric field driving the current is that specifically in the rest frame of the neutral atmosphere, and that the neutral atmosphere itself is rotating at a greater angular velocity than the plasma throughout the region shown. A Pedersen current is thus driven towards the equator in each hemisphere, whose strength is proportional to the departure of the plasma from rigid corotation with 5

the planet, and to the distance from the axis of rotation. With this in mind, it follows that with increasing latitude from the outer boundary in Fig. 1b, an upward current flows in the region where the plasma angular velocity increasingly departs from rigid corotation at lower latitudes, as shown by the lower latitude region of circled dots, and that this then reverses to downward current at higher latitudes particularly where the angular velocity increases in the Vasyliunas-cycle and Dungey-cycle return flows, as shown by the ring of circled crosses. A narrow region of strong upward field-aligned current then marks the transition into the polar cap, where the plasma angular velocity falls precipitately once more, followed by a region of distributed downward currents which flow in the region of open field lines in the polar cap. Cowley et al. [2004] suggested that it is the latter region of upward field-aligned currents, flowing at the open-closed field line boundary, that results in the formation of Saturn s main auroral oval. As indicated above, of the latter regions and currents, only the lowest-latitude region of subcorotating plasma and associated upward field-aligned current was modelled quantitatively by Cowley and Bunce [2003]. The purpose of this paper is then to extend this illustrative model to cover the high-latitude ionosphere at all latitudes up to the pole. The model is based on the semiquantitative physical discussion of Cowley et al. [2004], but also using Voyager plasma data as an empirical guide on closed field lines, and the results of Stallard et al. [2004] on open field lines. 2. Theoretical background In order to quantify the plasma flows, currents, and related auroral precipitation in Saturn s polar ionosphere, four theoretical ingredients are required. The first is a model of the magnetospheric plasma flows on open and closed field lines, as discussed above. The second is a model of the magnetic field required to map these flows (e.g. those observed in the near-equatorial magnetosphere by the Voyager spacecraft) into the polar ionosphere. The third is a measure of the ionospheric conductivity, possibly modified by the presence of winds in the neutral upper atmosphere induced by ion-neutral collisions, required to calculate the field-perpendicular ionospheric currents, and from their divergence the field-aligned currents. The fourth is a theoretical model which allows the calculation of the properties of precipitating accelerated electrons in regions of upward field-aligned current. In this section we discuss the last three of these topics. The first, modelling of the magnetospheric plasma flows, will be discussed in the Section 3. 6

The principal simplification made in the calculations presented here is that the magnetic field and plasma flows are taken to be symmetric about the spin axis of the planet. As we will show below, this is a sufficient approximation to capture the basic variations of the plasma flow, currents, and precipitation with latitude, thus allowing these properties to be quantified. However, some features anticipated in Fig. 1 will not then be represented in the model calculations, specifically the variation of the co-latitude of the boundary of open and closed field lines with local time, which reflects both day-night and dawn-dusk asymmetries in the magnetospheric magnetic field, and the local time asymmetries associated with the Vasyliunas- and Dungey-cycle flows. The model presented here is thus configured primarily to reflect conditions in the noon and dayside sector, though it should be illustrative of the basic behaviour at all local times. In addition, given that the gradients in physical quantities are expected to be primarily latitudinal at all local times, the parameters of the model presented here can also be modified to represent conditions in other local time sectors as well. Given the axi-symmetric approximation, the development of the underlying theory then follows that presented previously for the lower-latitude sub-corotating regime by Cowley and Bunce [2003], such that only a brief outline of the essential features is needed here. We begin with the model for the magnetic field, which is required to map the plasma angular velocities observed by the Voyager spacecraft from their point of observation in the near-equatorial magnetosphere into the ionosphere. This is taken to consist of two components. The first is the spin axis-symmetric SPV model of the planet s internal field [Davis and Smith, 1990], which consists of dipole, quadrupole, and octupole terms. The second is the field of a similarly axi-symmetric magnetospheric ring current, flowing between 8 and 15.5 R S in the equatorial plane, obtained from a fit to Voyager magnetic field data by Connerney et al. [1983] (see also Bunce and Cowley [2003]). The combined field is thus poloidal and axi-symmetric about the planet s spin axis, and can be represented by a flux function Fr,θ ( ), related to the field components by B = ( 1/ rsinθ) F ˆ ϕ. Here we use spherical polar co-ordinates, where r is the distance from the planet s centre, θ is the co-latitude angle measured from the northern pole, and ϕ is the azimuth angle. The surface F = constant then defines an axi-symmetric shell of field lines passing from the northern to the southern ionosphere, via the equator. In order to map from some magnetospheric point to the ionosphere, therefore, we simply compute the flux function at that point, and solve F = F i, where F i is the flux function in the ionosphere. In Fig. 2 we plot the model flux function versus radial distance in the equatorial plane r e (normalised to R S ) (Fig. 2a), and versus co-latitude in the ionosphere θ i (Fig. 2b), taken from Cowley and Bunce [2003]. The absolute value of F is specified by taking F = 0 on the axis of 7

symmetry. In Fig. 2a the solid line shows the flux function of the total field, while the dot-dashed line shows that of the planetary field alone, essentially that of the dipole over the range displayed. The enhanced values of F at a given radial distance in the full model reflect the radial extension of the field lines due to the ring current, which flows in the region between the dotted vertical lines in the figure. Observed dayside magnetopause positions on the three fly-bys which have occurred to date, by Pioneer 11 and Voyager 1 and 2, vary in radial distance between ~17 and ~24 R S [Smith et al., 1980; Behannon et al., 1983]. Here we will therefore take a subsolar magnetopause distance of 20 R S, corresponding to a value of the flux function of ~1526 nt R 2 S. In Fig. 2b we then show the flux function in the ionosphere for the northern (solid line) and southern hemispheres (dashed line) respectively, plotted versus co-latitude from their respective axes. Here the effect of the ring current field is negligible, such that only the flux function of the planetary field is shown. The difference between north and south is due to the quadrupole term, which results in the field strength being larger in the north than in the south. The planet s ionosphere, or more exactly the Pedersenconducting layer thereof, has been taken to lie at a radial distance 1000 km above the 1 bar reference spheroid of the planet, corresponding approximately to the altitude where the Pedersen conductivity is expected to maximise [e.g. Atreya et al., 1984; Moses and Bass, 2000]. The position of the dayside magnetopause mapped to the ionosphere, corresponding to the boundary between open and closed field lines, then occurs where the flux function reaches a value of FOC 1526 nt R 2 S in the two hemispheres. This corresponds to a co-latitude of 12.8 in the north and 14.1 in the south (for a dipole field alone the value would be 15.0 ). We note that these values are comparable to the co-latitude of the dayside UV aurora observed using the HST, which lie typically in the range ~12-15 [Trauger et al., 1998; Cowley et al., 2004]. As an independent check, the location of the boundary can also be estimated by considering the amount of open flux in the lobes of the magnetic tail. Using nightside outbound data from the Voyager 1 spacecraft, Ness et al. [1981] derived co-latitudes of ~11-15 on this basis, consistent with the estimates made here. The angular velocity of the magnetospheric plasma can then be mapped into the ionosphere using the magnetic model, from which the electric field in the neutral atmosphere rest frame, E i, and the height integrated Pedersen current intensity, i P, can be calculated. The latter is given by i P * ( θ i ) = Σ P E i ( θ i ) = Σ P ρ i ( Ω S ω) B i, (1) where Σ P is the height-integrated Pedersen conductivity of the ionosphere, ρ i is the perpendicular distance of the conducting layer from the magnetic (spin) axis, Ω * S is the angular velocity of the 8

neutral atmosphere in the Pedersen layer (possibly reduced from rigid corotation at the planetary angular velocity Ω S due to ion-neutral frictional drag), ω is the angular velocity of the plasma, and B i is the magnetic field strength in the Pedersen layer. Since we expect * Ω S to lie at a value intermediate between ω and Ω S (see e.g. Huang and Hill [1989] in the jovian context), we may write for some 0 < k <1 (whose value is at present uncertain) * ( Ω S Ω S )= k( Ω S ω), (2) such that the Pedersen current can be written as i P ω Ω S * ( θi ) = Σ PρiΩ S 1 Bi, (3) where Σ P * is the effective value of the height-integrated Pedersen conductivity Σ P * = ( 1 k)σ P, (4) reduced from the true value Σ P by the factor ( 1 k), representing the slippage of the neutral atmosphere from rigid corotation. Neither the values of k nor Σ P are well known at present. However, Bunce et al. [2003] have made an empirical estimate specifically of the effective conductivity value Σ P *, by examining the azimuthal perturbation fields observed in Saturn s magnetospheric magnetic field that are produced by the observed sub-corotation of the magnetospheric plasma. They showed that the observed fields are consistent with an effective Pedersen conductivity of ~1-2 mho. In this paper we shall therefore take a constant value of 1 mho as a simple assumption, such that the currents we derive can be simply linearly scaled for other choices. From the Pedersen current intensity we can also derive the Joule heating rate per unit area in the Pedersen layer, p J, which is potentially important for the thermal structure of the upper atmosphere. From Eqs. (1)-(4), this is given by p J = ( θ ) ( θ ) 2 2 ip i ip i i P.E i = = ( 1 k). (5) * Σ P Σ P 9

It can thus be seen that the Joule power per unit area depends not only on the parameters defined above, but also explicitly upon the essentially unknown parameter k, defined by Eq. (2). In the results below, therefore, we simply show plots of the Joule heating rate parameter * p J, defined by p * J = ( k) ( θ ) 2 pj ip i = * 1 Σ P, (6) which is thus larger than the true heating rate by the factor ( 1 ) 1 k. Parameter p * J should nevertheless provide a reasonable estimate of the Joule heating rate provided the value of k is not too close to unity, i.e. provided the slippage of the neutral atmosphere relative to rigid corotation is not by too large a factor. We note that if the expression for i P from Eq. (3) is substituted into Eq. (6), we find that * p J is linearly proportional to Σ * P for a given flow model. We also wish to calculate the field-aligned currents which connect the ionospheric and magnetospheric currents. To do this we first multiply the field-perpendicular Pedersen current intensity given in Eq. (3) by a factor of order unity to account for the fact that the ionospheric magnetic field is not exactly perpendicular to the conducting layer, thus forming the horizontal Pedersen current intensity i hp ( θ ) i ( θ ) i cosα P i =, (7) i where α i is the angle of the planetary magnetic field to the local vertical in the Pedersen layer. Factor ( cos ) 1 α increases slowly with co-latitude from unity at the pole where the field is vertical, i to ~1.03 at 15 and ~1.3 at 30, the value being slightly higher in the north than the south. The Pedersen current intensity is thus modified only by a modest factor due to this effect. We then integrate this horizontal ionospheric current in azimuth to find the total equatorward current flowing at a given latitude I hp ( θ ) 2πρ i ( θ ) i =, (8) i hp i and then differentiate this with respect to co-latitude to determine the field-aligned current density just above the ionosphere 10

j 1 di hp i =, (9) 2πρi cosα i dsi where ds i is an element of path length in the ionosphere along a line of given longitude from the pole towards the equator (see Cowley and Bunce [2003] for further details). Positive values of then correspond to upward currents, and negative values to downward currents. With regard to the Hall currents, we note that in our axi-symmetric approximation, these flow eastward around the pole and close wholly in the ionosphere, thus making no contribution to the field-aligned current density. j i We now suppose that the upward currents are carried by precipitating magnetospheric electrons, and consider the conditions under which acceleration by field-aligned electric fields are required. The maximum current density that can be carried by magnetospheric electrons without field-aligned acceleration is j i0 W 2 th = en π me 1 2, (10) where the magnetospheric electron population has been assumed to be an isotropic Maxwellian of density N and thermal energy W th (= kt), and e and m e are the electron charge and mass respectively. Equation (10) corresponds to the case of a full downward-going loss-cone, and an empty upward-going loss-cone. The corresponding precipitating electron energy flux is W E f 0 = 2NW th th 2π m e 12. (11) If the upward current required by Eq. (9) is larger than j i0 given by Eq. (10), then a field-aligned voltage must be present to accelerate the magnetospheric electrons into the ionosphere. According to kinetic theory due to Knight [1973], the minimum field-aligned voltage which must be present is given by W j th i Φ = 1, (12) e j i0 11

this value being appropriate if the top of the voltage drop is located at a radial distance well above the minimum value given by r Ri min j j i i0 1/ 3, (13) where R i is the radial distance of the ionospheric current layer. In Eq. (13) we have assumed as a sufficient approximation that the field strength drops as the inverse cube of the radial distance along the polar field lines, corresponding to the planetary dipole field. The enhanced precipitating electron energy flux corresponding to Eq. (12) is then 2 E f 0 j i E = + 1 f, (14) 2 j i0 following Lundin and Sandahl [1978]. Equations (12)-(14) will be used here to estimate the acceleration conditions and precipitating electron energy fluxes required in regions of upward current. 3. Plasma angular velocity profile, currents, and electron precipitation 3.1 Plasma angular velocity model As indicated above, the form of the angular velocity model employed here is motivated by the physical ideas summarised in Fig. 1, and is guided quantitatively by Voyager plasma observations on closed magnetic field lines and IR Doppler observations on open field lines. In Fig. 3 we show the angular velocity of the plasma determined from Voyager data by Richardson [1986] and Richardson and Sittler [1990], plotted versus the flux function of our model magnetic field evaluated at the point of observation. Solid dots and crosses show the inbound data from Voyager-1 and -2, respectively, obtained at similar post-noon local times (LT) between ~13 and ~15 h. Open circles also show partial outbound data from Voyager-1 at ~02 h LT. Data from other segments of the spacecraft trajectories are not available, either because the flow speed dropped 12

below detectability by the Voyager instruments in the innermost region, or because of unfavourable instrument orientation with respect to the flow on portions of the outbound passes. The plot spans the range of F values between zero and 5500 nt R 2 S, where the open-closed field line boundary at FOC 1526 nt R 2 S is indicated by the vertical dotted line. Values of F to the left of this line then correspond in our model to open field lines which map through the dayside magnetopause and into the tail lobes, while those to the right correspond to closed field lines which thread the equatorial plane. The maximum flux function value shown, 5500 nt R 2 S, corresponds to a radial distance of ~3.9 R S in the equatorial plane, and to ionospheric co-latitudes of ~26 in the north and ~28 in the south (see Fig. 2). It can be seen that the inbound Voyager-2 data extend up to the model (and actual) magnetopause, while the inbound Voyager-1 data begin somewhat inside, mapping to the equatorial plane at ~17 R S. The Voyager-1 data between the magnetopause and the latter point were omitted in the above publications due to uncertainties in the identity of the ion species in the outer magnetosphere in this case [Richardson, 1986]. The inbound data on both passes then extend inward to an equatorial radial distance of ~4 R S ( F 5000 nt R 2 S ). The Voyager-1 outbound data span mapped equatorial distances between ~4 and ~8 R S. It can be seen from Fig. 3 that the angular velocities are generally near to rigid corotation for large values of F, as expected. With decreasing F the angular velocity then falls, reaching ~60% of rigid corotation at 2 F 2500nT R S, corresponding to an equatorial radial distance of ~12 R S, and an ionospheric co-latitude of ~17 in the north and ~18 in the south (see Fig. 2). This is the subcorotation behaviour whose effects were modelled previously by Cowley and Bunce [2003]. Beyond this point, however, at smaller F on closed field lines, the angular velocities tend to increase again, to perhaps ~80% of rigid corotation near the magnetopause, though the Voyager-2 values in particular show significant scatter. In conformity with the discussion in the previous section, we suggest that these increased angular velocities may be the signature of the flux layer adjacent to the magnetopause which corresponds to the Dungey-cycle return flow and Vasyliunascycle flow (see Fig. 1a). To represent this behaviour on closed field lines we use a similar model to that employed by Cowley and Bunce [2003], but modified to include the increase in angular velocity at small F values inside the magnetopause. The chosen function is ω Ω S Closed ω = ΩS ω Ω 1 + 1 n1 ( 1 ( F F ) ) + + Ω S ΩS + ( F F ) 1 1 C0 S C1 C C0 C1 ω ω 1 n2 ( ) C 2. (15) 13

This expression is such that ( Ω S ) ( Ω S ) 0 ω ω when F 0. For suitable values of the parameters it then initially falls with increasing F in the vicinity of F F C 1, then increases again in the vicinity of F F C 2, before approaching unity for F between F C1 and F C 2 C. It thus possesses a minimum located. The function adopted previously by Cowley and Bunce [2003] was this expression but with F put equal to infinity in the second term, such that the angular velocity monotonically increased from ( S ) C0 ( S ) C1 ω Ω ω Ω when F 0, to unity for F. Two sets of function parameters were then chosen by Cowley and Bunce [2003] to represent the Voyager-1 and -2 data separately. Here, however, we have instead sought one set of parameters which are motivated by the physics which we wish to represent, as outlined in Section 2, and which also give a reasonable overall representation of the main features of the data from both Voyager fly-bys. The chosen parameters are ( ω ) 8, ( ) 2 Ω S 0 = 0. ω Ω = 0. S, 1 F = 2200 C1 nt R S, 2 F = 3600 C 2 nt R S, n = 50, and n = 8 1 2. This function is shown by the dot-dashed line in Fig. 3, which is visible to the left of the open-closed field line boundary (the vertical dashed line), but which is overlain to the right of the boundary by the solid line to be described below. It has a minimum value of ( ) 0. 62 ω at Ω S F 2404 2 nt R S, corresponding to an equatorial radial distance of ~13.0 R S, and an ionospheric co-latitude of ~16.2 in the north and ~17.9 in the south. The RMS deviation per point between this line and the Voyager data is ( ) ~ 0. 19 δω. Ω S 2 It can be seen in Fig. (3), however, that Eq. (15) does not represent the dips in angular velocity observed in Voyager-1 data (but not in Voyager-2 data), which occur in the vicinity of F 3250 and 2700 nt R 2 S, corresponding to distances of ~7 and ~10 R S in the equatorial plane. As previously pointed out by Richardson [1986], these dips are located just outside the orbits of the moons Dione and Rhea, which orbit in the equatorial plane at distances of 6.25 and 8.74 R S. As in Cowley and Bunce [2003], these features have been represented by adding small-scale Gaussian terms to Eq. (15) of the form ω Ω S Dips = n ω Ω S Dn F F exp FDn Dn 2. (16) The dotted line shown in Fig. 3 represents the effect of adding two such terms, with ( Ω ) 25 ω = 0. S, D1 2 F = 3250 nt R D1 S (7.2 R S in the equatorial plane), and 2 F = 210 nt R D1 S, representing the dip outside Dione s orbit, and ( Ω ) 3 ω = 0. S, D2 2 F = 2700 nt R D2 S (10.4 R S in 14

the equatorial plane), and 2 F =120 nt R D2 S, representing the dip outside Rhea s orbit. Below we show results both with and without inclusion of these dips, since they appear to be an intermittent feature of the flow. Equations (15) and (16) thus form our description of the plasma flow on closed field lines. To complete our model we need also to represent the flow on open field lines, that is to say for F values less than FOC 1526 nt R 2 S. As discussed above, the flow on open field lines is expected to consist of two components, a sub-corotational twisting of the open tail lines produced by the atmospheric torque, combined with anti-solar Dungey-cycle flow. Given some assumptions of simplified geometry, Isbell et al. [1984] have shown that the former of these flows takes place at a uniform angular velocity given by * ω µ oσ PV = * Ω S 1+ µ oσ PV SW SW, (17) where µ o is the permeability of free space, and V SW is the speed of the solar wind. If we then take an effective ionospheric conductivity of 1 mho, as discussed above, and a solar wind speed of 400 km s -1 *, we find Σ 0. 50 µ, such that Eq. (17) gives ( ω ) 0. 33 o P V SW Ω S. This value is in excellent agreement with the IR Doppler observations reported by Stallard et al. [2004], which indicate angular velocities in the polar regions between ~20 and ~40% of rigid corotation, averaging to ~34%. At the boundary of the polar cap, at ~13-14 co-latitude, the average angular velocity corresponds to azimuthal plasma velocities of ~800 m s -1 ( Ω S 1.638 10 4 rads 1 ). By comparison, Cowley et al. [2004] estimated the transpolar flow speed associated with the Dungey cycle to be typically ~200 m s -1. The principal flow within the polar region is thus expected to be the sub-corotational flow first discussed by Isbell et al. [1984], given by Eq. (17). As discussed by Cowley et al. [2004], the Dungey-cycle flow on open field lines then effectively modulates the subcorotational flow, modestly increasing the plasma velocities on the dusk side of the polar cap compared with those estimated from Eq. (17), and modestly reducing them on the dawn side. Here, however, for simplicity we will ignore this effect, and just take a sub-corotational flow of 30% of rigid corotation within the polar cap, as a round value which is consistent both with Eq. (17) and the results of Stallard et al. [2004]. It will be noted from the dot-dashed line plotted in Fig. (3) that Eq. (15) produces an essentially constant angular velocity within the open field region (to the left of the vertical dotted line) of 15

( ω Ω ) = ( ω Ω ) 8 S S C 0 = 0. this an angular velocity ( Ω ) 0. 5. To represent the polar flows we thus need essentially to subtract from ω within the open field region, and then to revert to Eq. (15) S within the region of closed field lines. This behaviour has then been represented by adding the following term to Eq. (15) ω Ω S Open 1 ω = 2 Ω S O F F 1 tanh FO O. (18) This expression is such that, for small values of FO, the value of ( Ω S ) Open ω is very nearly equal ω for F smaller than F O, and switches rapidly to zero for F larger than F O. When Ω S to ( ) O combined with Eq. (15), this results in nearly constant angular velocities given by ( Ω S ) C ( ω Ω S ) O ω 0 for F smaller than F O, representing the flow on open field lines, switching essentially to the value given by Eq. (15) alone for F larger than F O, as required. With ( ω Ω ) = 0. 8 as indicated above, we thus also employ ( ω Ω ) = 0. 5 C0 S S O, as just discussed. We also use F O = F OC 1526 nt R 2 S, such that the switch in angular velocity is centred on the model open-closed field line boundary. The value of FO describing the width of the boundary region is not well constrained by theory, but an indication is given by the width of Saturn s auroral oval, if the auroras are indeed associated with the upward currents at the open-closed field line boundary as suggested above. The HST data presented by Cowley et al. [2004] indicate a latitudinal width of the auroras of typically ~300-600 km at ionospheric heights, corresponding to ~0.3-0.6 of latitude. Here, therefore, we have correspondingly chosen 2 F O = 50 nt R S, which results in a flow shear of appropriate width in the ionosphere. The overall angular velocity model, obtained by combining together Eqs. (15) and (18), is then shown by the solid line in Fig. 3, with the effect of the dips given by Eq. (16) being indicated, as indicated above, by the dotted lines. In Fig. 4 we show our plasma angular velocity model mapped both into the equatorial plane and into the ionosphere. In the equatorial mapping in Fig. 4a, the solid line shows the angular velocity given by Eq. (15) alone, plotted versus equatorial radial distance r e (normalised to R S ) out to the magnetopause at 20 R S. Near-rigid corotation is maintained in the equatorial plane to a radial distance of ~4 R S, beyond which the angular velocity falls monotonically to ~60% of rigid corotation at ~13 R S. The angular velocity then increases again, plateauing at ~80% of rigid corotation in a layer ~5 R S wide inside the magnetopause. We note that the transition to somewhat higher angular velocities in the outer magnetosphere coincides approximately with the outer edge of 16

the ring current region, which extends to 15.5 R S in the equatorial plane in the Connerney et al. [1983] model determined from Voyager magnetic field data. The short-dashed line then shows the model with the inclusion of the dips, where the vertical dotted lines show for comparison the radial distances of the orbits of Dione and Rhea. These dips are ~1.5-2 R S wide in the equatorial plane, and reach local minima of ~47% of rigid corotation at ~7.2 R S outside of Dione s orbit, and ~34% of rigid corotation at ~10.4 R S outside of Rhea s orbit. In Fig. 4b we similarly show the plasma angular velocity mapped along field lines into the northern and southern ionospheres, and plotted versus co-latitude angle θ i with respect to the corresponding pole. The solid and long-dashed lines show the angular velocities in the northern and southern hemispheres, respectively, given by summing Eqs. (15) and (18), while the shortdashed line and the dotted line show the effect of adding the dips in each hemisphere, given by Eq. (16). It should be noted that, in conformity with the respective flux function profiles shown in Fig. 2b, features in the southern hemisphere are displaced equatorward of corresponding features in the north by ~1 at ~10 co-latitude, increasing to ~3 at ~25 co-latitude. It can be seen that the model angular velocity remains at an almost fixed value of 30% of rigid corotation within the region of open field lines, essentially up to the model open-closed field line boundary at 12.8 in the northern hemisphere and 14.1 in the south. It then undergoes a rapid increase to ~80% of rigid corotation on closed field lines in a region ~2 wide, corresponding to the band of flows associated with the Dungey-cycle return flow and Vasyliunas-cycle flow, before falling once more to a local minimum of ~60% of rigid corotation at ~16 in the north and ~18 in the south. The angular velocity in the model without dips (Eq. (15)) then increases once more to approach near-rigid corotation at co-latitudes of ~25 in the north and ~28 in the south. The inclusion of the dips results in the occurrence of narrow minima in the angular velocity profile, ~1 latitude wide, centred at co-latitudes 17.3 and 19.1 in the north (the Rhea and Dione dips, respectively), and 19.1 and 21.1 in the south. The Dione dip in the north thus occurs at essentially the same colatitude as the Rhea dip in the south (with respect to the southern pole). These are the model ionospheric angular velocity profiles whose consequences for currents and particle precipitation will now be investigated. 3.2 Pedersen current and Joule heating Given the angular velocity models shown in Fig. 4b, we can now use the theory outlined in Section 2 to calculate the Pedersen current flowing in the ionosphere, from which we can also 17

estimate the Joule heating rate. In Fig. 5a we thus show co-latitude profiles of the equatorwarddirected height-integrated Pedersen current intensity given by Eq. (3), using the same line format as in Fig. 4b. It can be seen that within the open field line polar cap, the current intensity increases nearly linearly with the distance from the pole. This is due to the fact that at essentially constant angular velocity, the actual velocity of the plasma relative to that of the neutral atmosphere is proportional to distance from the magnetic (and spin) axis, hence so is the electric field and current. For the chosen effective Pedersen conductivity of 1 mho, the current intensity peaks near the open-closed field line boundary at ~0.097 A m -1 in the north and ~0.087 A m -1 in the south. Across the boundary in the region of closed field lines the current intensity then falls by a factor of about three to ~0.030 A m -1 in the north and ~0.027 A m -1 in the south, due to the increase in plasma angular velocity closer to rigid corotation in the region we associate with the Dungey-cycle return flow and Vasyliunas-cycle flow. It then increases again to peak in the sub-corotating region on closed field lines, at ~0.068 A m -1 at ~16.7 co-latitude in the north, and at ~0.061 A m -1 at ~18.4 in the south, before declining continuously to small values at larger co-latitudes, as the plasma angular velocities increase towards rigid co-rotation. The effect of the angular velocity dips is then to superimpose structured peaks in the current across the latter broad maximum of upward current on closed field lines. The current intensity in the Dione and Rhea dips peak at ~0.11 and ~0.12 A m -1, respectively, in the north, and at ~0.095 and ~0.11 A m -1 in the south. In Fig. (5b) we show the corresponding Joule heating rate. Specifically we show parameter * p J defined by Eq. (6), which is larger than the true Joule heating rate 18 p by the factor ( ) 1 J 1 k, where 0<k<1 is related to the slippage of the neutral atmosphere from rigid corotation due to ion-neutral collisions (Eq. (2)). The value of k for Saturn s upper atmosphere is unknown at present, but recent modelling for Jupiter indicates values k ~ 0.5 [G. Millward, personal communication, 2003]. If similar values apply also to Saturn, then value. It can be seen that in the open field region * p J could be up to a factor of two larger than the true * p J grows approximately as the square of the distance from the axis, reaching a peak of ~9.3 mw m -2 near the open-closed field line boundary in the north, and ~7.6 mw m -2 in the south. There is then a local minimum on closed field lines in the band of elevated flows corresponding to the Dungey-cycle return flow and the Vasyliunas-cycle flow, followed by a second maximum (in the model without the dips ), peaking at ~4.6 mw m -2 in the north and ~3.8 mw m -2 in the south. The inclusion of the dips then again results in structured peaks being superimposed in the latter region, reaching values locally up to ~15 mw m -2 (specifically for the Rhea dip in the northern hemisphere). Overall, integrating over the whole region of open field lines yields a total p * J power of ~2.5 TW in each hemisphere. Similarly,

integrating over the closed field region yields a total * p J power of ~2.3 TW per hemisphere for the model without the dips, increasing to ~4.0 TW per hemisphere when the dips are included. These values compare with a solar input to the upper atmosphere, integrated over the whole sunward-facing disc of the planet, of between ~0.2 TW for solar minimum conditions and ~0.5 TW for solar maximum conditions [Müller-Wodarg et al., 2004]. It thus seems inevitable that Joule heating provides a major source of heat energy to Saturn s thermosphere, thus at least contributing to an explanation of why the observed thermospheric temperature is considerably in excess of values expected for solar heating alone (see Miller et al. [2000] and references therein). Solar heating produces temperatures below ~200 K, compared with observed temperatures between ~400 and 600 K. Müller-Wodarg et al. [2004] have shown that an additional heat source, comparable to those estimated here, is required to reproduce the observed values. 3.3 Total horizontal current and field-aligned current We now discuss the overall Pedersen current budget in the ionosphere associated with our flow model, and the consequences for the field-aligned current flow required by its divergence. In Fig. 6a we first examine the latitude profile of the azimuth-integrated or total horizontal Pedersen current I hp defined in Eq. (8), noting that it is the latitude gradient of this specific quantity which appears in the expression for the field-aligned current density in Eq. (9). It can be seen that in the region of open field lines, the total equatorward horizontal Pedersen current grows approximately as the square of the distance from the pole to reach a value of ~7.4 MA at the open-closed field line boundary in both hemispheres. This requires the presence of a downward-directed field-aligned current throughout this region, as indicated above in the discussion of Fig. 1a. The total current then falls to ~2.5 MA just inside the region of closed field lines, such that ~4.9 MA flows up the field lines into the magnetosphere in this boundary, as also depicted in Fig. 1a. The total current then grows again within the region of closed field lines, reaching a maximum of ~7.3 MA at a colatitude of ~17.7 in the north (for the basic model without dips ), and at ~19.5 in the south. Thus a downward current of ~4.8 MA flows into the ionosphere in this model between the open-closed field line boundary and the maximum on closed field lines. The whole ~7.3 MA then flows out of the ionosphere again at co-latitudes beyond the maximum as the plasma angular velocity increases towards rigid corotation. The effect of the angular velocity dips is again to superpose structured peaks on the latter maximum, the current in these peaking at ~13.3 MA in the Rhea dip, and at ~12.7 MA in the Dione dip. The variations of these horizontal currents thus also require 19

downward then upward field-aligned currents to flow with increasing co-latitude, the total currents carried being ~6 MA. The corresponding field-aligned current density is shown versus co-latitude in Fig. (6b), derived from Eq. (9). Positive values indicate upward-directed currents, and negative values downwarddirected currents (in both hemispheres). As indicated above, the currents are downward-directed throughout the region of open field lines, with a nearly constant current density of ~17 na m -2 in the north and ~14 na m -2 in the south. The smaller value in the southern hemisphere is a consequence of the reduced ionospheric field strength in the south compared with the north, resulting from the quadrupole term of the internal field. However, the southern polar cap is also somewhat larger than that in the north as a consequence of the same effect, leading to almost the same values of the total Pedersen current flowing at the northern and southern polar cap boundaries, as seen in Fig. 6a. A spike of upward-directed current then flows at the open-closed field line boundary, peaking at ~168 na m -2 in the north and ~137 na m -2 in the south. The latitudinal width of this feature is marginally wider in the south than in the north for reasons just given, being ~0.37 FWHM in the north and ~0.41 in the south (corresponding to north-south distances of ~400 km). This factor, combined with the larger radius of the current ring in the south compared with the north again results in very similar total upward currents flowing in the two boundaries (~4.9 MA as indicated above). Equatorward of the spike, in the region of closed field lines, the current then reverses sign once more to become downward-directed (negative) in the region mapping to the Dungey-cycle return flow and Vasyliunas-cycle flow, peaking in magnitude at the equatorward edge of this band of elevated flows, where the angular velocities decrease once more towards values characteristic of the central sub-corotating closed field region (Fig. 4b). In the north, the downward current peaks at ~62 na m -2 at ~15.5, while in the south it peaks at ~50 na m -2 at ~17.1. The field-aligned current density (for the model without dips) then finally reverses sense to upward (positive) for colatitudes greater than 17.7 in the north and 19.5 in the south, where it rises to a maximum value of ~13 na m -2 at ~20.5 in the north and ~11 na m -2 at ~22.6 in the south, before falling to small values again at large co-latitudes. The upward current densities associated with sub-corotation on closed field lines are thus estimated to be more than an order of magnitude smaller than those which occur at the open-closed field line boundary, despite the fact that the total upward current carried in the closed field regime is larger, ~7.3 MA compared with ~4.9 MA. This results from the fact that the area of the ionosphere carrying the former currents is much larger than that carrying the latter. Inclusion of the angular velocity dips then produces large-amplitude bipolar pairs of field-aligned currents centred on each dip, each current region being ~0.5 wide, with downward currents lying poleward of corresponding upward currents. As can be seen in Fig. (6b), the current densities and 20