coupled rotational dynamics of jupiter’s thermosphere and ......meteorology and atmospheric...

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Ann. Geophys., 27, 199–230, 2009 www.ann-geophys.net/27/199/2009/ © Author(s) 2009. This work is distributed under the Creative Commons Attribution 3.0 License. Annales Geophysicae Coupled rotational dynamics of Jupiter’s thermosphere and magnetosphere C. G. A. Smith 1,* and A. D. Aylward 1 1 Atmospheric Physics Laboratory, Department of Physics and Astronomy, University College, London, UK * now at: Teesdale School, Barnard Castle, County Durham, UK Received: 16 March 2008 – Revised: 13 September 2008 – Accepted: 2 December 2008 – Published: 13 January 2009 Abstract. We describe an axisymmetric model of the cou- pled rotational dynamics of the thermosphere and magne- tosphere of Jupiter that incorporates self-consistent physical descriptions of angular momentum transfer in both systems. The thermospheric component of the model is a numerical general circulation model. The middle magnetosphere is de- scribed by a simple physical model of angular momentum transfer that incorporates self-consistently the effects of vari- ations in the ionospheric conductivity. The outer magneto- sphere is described by a model that assumes the existence of a Dungey cycle type interaction with the solar wind, pro- ducing at the planet a largely stagnant plasma flow pole- ward of the main auroral oval. We neglect any decoupling between the plasma flows in the magnetosphere and iono- sphere due to the formation of parallel electric fields in the magnetosphere. The model shows that the principle mech- anism by which angular momentum is supplied to the po- lar thermosphere is meridional advection and that mean-field Joule heating and ion drag at high latitudes are not responsi- ble for the high thermospheric temperatures at low latitudes on Jupiter. The rotational dynamics of the magnetosphere at radial distances beyond 30 R J in the equatorial plane are qualitatively unaffected by including the detailed dynam- ics of the thermosphere, but within this radial distance the rotation of the magnetosphere is very sensitive to the rota- tion velocity of the thermosphere and the value of the Peder- sen conductivity. In particular, the thermosphere connected to the inner magnetosphere is found to super-corotate, such that true Pedersen conductivities smaller than previously pre- dicted are required to enforce the observed rotation of the magnetosphere within 30 R J . We find that increasing the Joule heating at high latitudes by adding a component due to rapidly fluctuating electric fields is unable to explain the Correspondence to: C. G. A. Smith ([email protected]) high equatorial temperatures. Adding a component of Joule heating due to fluctuations at low latitudes is able to explain the high equatorial temperatures, but the thermospheric wind systems generated by this heating cause super-corotation of the inner magnetosphere in contradiction to the observations. We conclude that the coupled model is a particularly useful tool for study of the thermosphere as it allows us to constrain the plausibility of predicted thermospheric structures using existing observations of the magnetosphere. Keywords. Magnetospheric physics (Magnetosphere- ionosphere interactions; Planetary magnetospheres) Meteorology and atmospheric dynamics (Thermospheric dynamics) 1 Introduction The magnetospheres of Jupiter and Saturn are dominated by the influence of the planets’ rapid rotation frequencies. The plasma in the magnetospheres of both planets exhibit partial corotation (McNutt et al., 1979; Richardson, 1986), indicat- ing that angular momentum has been transferred from the planet. This angular momentum transfer occurs because of the presence of a conducting layer in the atmosphere, with which the magnetosphere may interact. This conducting re- gion occurs within the ionosphere, which is colocated with the (neutral) thermosphere, and it is the rotation of this region of the neutral atmosphere, not the deep rotation velocity, that directly controls the magnetosphere. In the case of Jupiter, on which we focus here, this corresponds to pressures lower than 2 microbar (altitudes greater than 300 km above the 1 bar level). The importance of the thermospheric rotation velocity, as distinct from the deep rotation velocity, has long been recog- nised in magnetospheric studies. There are very few mea- surements of thermospheric winds at either Jupiter or Saturn Published by Copernicus Publications on behalf of the European Geosciences Union.

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  • Ann. Geophys., 27, 199–230, 2009www.ann-geophys.net/27/199/2009/© Author(s) 2009. This work is distributed underthe Creative Commons Attribution 3.0 License.

    AnnalesGeophysicae

    Coupled rotational dynamics of Jupiter’s thermosphere andmagnetosphere

    C. G. A. Smith1,* and A. D. Aylward1

    1Atmospheric Physics Laboratory, Department of Physics and Astronomy, University College, London, UK* now at: Teesdale School, Barnard Castle, County Durham, UK

    Received: 16 March 2008 – Revised: 13 September 2008 – Accepted: 2 December 2008 – Published: 13 January 2009

    Abstract. We describe an axisymmetric model of the cou-pled rotational dynamics of the thermosphere and magne-tosphere of Jupiter that incorporates self-consistent physicaldescriptions of angular momentum transfer in both systems.The thermospheric component of the model is a numericalgeneral circulation model. The middle magnetosphere is de-scribed by a simple physical model of angular momentumtransfer that incorporates self-consistently the effects of vari-ations in the ionospheric conductivity. The outer magneto-sphere is described by a model that assumes the existence ofa Dungey cycle type interaction with the solar wind, pro-ducing at the planet a largely stagnant plasma flow pole-ward of the main auroral oval. We neglect any decouplingbetween the plasma flows in the magnetosphere and iono-sphere due to the formation of parallel electric fields in themagnetosphere. The model shows that the principle mech-anism by which angular momentum is supplied to the po-lar thermosphere is meridional advection and that mean-fieldJoule heating and ion drag at high latitudes are not responsi-ble for the high thermospheric temperatures at low latitudeson Jupiter. The rotational dynamics of the magnetosphereat radial distances beyond∼30RJ in the equatorial planeare qualitatively unaffected by including the detailed dynam-ics of the thermosphere, but within this radial distance therotation of the magnetosphere is very sensitive to the rota-tion velocity of the thermosphere and the value of the Peder-sen conductivity. In particular, the thermosphere connectedto the inner magnetosphere is found to super-corotate, suchthat true Pedersen conductivities smaller than previously pre-dicted are required to enforce the observed rotation of themagnetosphere within∼30RJ . We find that increasing theJoule heating at high latitudes by adding a component dueto rapidly fluctuating electric fields is unable to explain the

    Correspondence to:C. G. A. Smith([email protected])

    high equatorial temperatures. Adding a component of Jouleheating due to fluctuations at low latitudes is able to explainthe high equatorial temperatures, but the thermospheric windsystems generated by this heating cause super-corotation ofthe inner magnetosphere in contradiction to the observations.We conclude that the coupled model is a particularly usefultool for study of the thermosphere as it allows us to constrainthe plausibility of predicted thermospheric structures usingexisting observations of the magnetosphere.

    Keywords. Magnetospheric physics (Magnetosphere-ionosphere interactions; Planetary magnetospheres) –Meteorology and atmospheric dynamics (Thermosphericdynamics)

    1 Introduction

    The magnetospheres of Jupiter and Saturn are dominated bythe influence of the planets’ rapid rotation frequencies. Theplasma in the magnetospheres of both planets exhibit partialcorotation (McNutt et al., 1979; Richardson, 1986), indicat-ing that angular momentum has been transferred from theplanet. This angular momentum transfer occurs because ofthe presence of a conducting layer in the atmosphere, withwhich the magnetosphere may interact. This conducting re-gion occurs within the ionosphere, which is colocated withthe (neutral) thermosphere, and it is the rotation of this regionof the neutral atmosphere, not the deep rotation velocity, thatdirectly controls the magnetosphere. In the case of Jupiter,on which we focus here, this corresponds to pressures lowerthan∼2 microbar (altitudes greater than∼300 km above the1 bar level).

    The importance of the thermospheric rotation velocity, asdistinct from the deep rotation velocity, has long been recog-nised in magnetospheric studies. There are very few mea-surements of thermospheric winds at either Jupiter or Saturn

    Published by Copernicus Publications on behalf of the European Geosciences Union.

    http://creativecommons.org/licenses/by/3.0/

  • 200 C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling

    (e.g.Gladstone et al., 2005), which are insufficient to empir-ically determine the appropriate rotation velocity. To copewith this lack of information, a simple theoretical modelof magnetosphere-thermosphere coupling was developed forJupiter byHuang and Hill (1989) and later improved byPontius(1995). This model assumed that the principal pro-cess by which angular momentum is transported within thethermosphere was vertical viscous transport. This yieldeda linear relationship between the thermospheric and magne-tospheric rotation velocities that allowed the unknown dy-namics of the thermosphere to be simply parameterised usingan “effective” ionospheric conductivity. Although developedfor Jupiter, the effective conductivity model has also beenadopted for the Saturn case (e.g.Cowley and Bunce, 2003;Saur et al., 2004).

    Recently, this model has been critically analysed for thecase of Saturn bySmith and Aylward(2008). This studyreached four main conclusions:

    1. Meridional advection, not vertical viscous transport, isthe principal mechanism for transporting angular mo-mentum to the high latitude thermosphere.

    2. As a result, the effective conductivity model ofHuangand Hill (1989) is a poor parameterisation of the ther-mospheric rotation velocity.

    3. Meridional advection of angular momentum producesmeridionally smoothed structures in the thermosphericrotation velocity which feed back on the rotationalstructure of the magnetosphere.

    4. Super-corotation of the neutral atmosphere arises at lat-itudes coupled to the inner magnetosphere. This maylead to super-corotation of the inner magnetosphere it-self.

    These conclusions represent a new perspective on thethermosphere-magnetosphere interaction, in which merid-ional advection of angular momentum within the thermo-sphere is as important for the rotational structure as radialdiffusion of angular momentum in the magnetosphere. Asa result rotational structures are influenced by the complex-ity of both systems, and develop mutually, rather than themagnetosphere imprinting its rotational structure on an es-sentially passive thermosphere.

    While the basic physics of angular momentum transferis essentially identical at both Jupiter and Saturn, there areseveral important differences that influence the character ofmagnetosphere-thermosphere coupling. Jupiter’s magneticfield is an order of magnitude stronger than Saturn’s, re-sulting in a much larger magnetosphere. At Jupiter inter-nal mass-loading is dominated by a single source, Io, whichinjects material at a rate of approximately 1000 kg/s (e.g.Delamere and Bagenal, 2003). At Saturn internal mass-loading is distributed between a number of moons and rings,resulting in a complicated plasma distribution and a lower

    rate of mass-loading of the order of∼40 kg/s (Richardsonet al., 1998). The greater size and rate of mass-loadingof the Jovian magnetosphere means that magnetosphere-thermosphere coupling currents must be stronger in the sensethat they transfer a greater quantity of angular momentum.

    The region of Jupiter’s thermosphere from which thisgreater quantity of angular momentum is extracted is ex-pected to have a higher column mass than the equivalentregion at Saturn. This is because the peak of the Peder-sen conductivity corresponds approximately to the altitudewhere the ion-neutral collision frequency and ion gyrofre-quency are equal. The higher magnetic field at Jupiter (bya factor of∼10) means a higher gyrofrequency, so, sincethe collision frequency is proportional to neutral density, thisequality occurs in a proportionately denser layer of the atmo-sphere. This effect is compensated for by the higher gravi-tational field strength at Jupiter’s surface (by a factor of∼2)which reduces the scale height, so that the vertical width ofthe peak of the Pedersen conductivity is proportionately re-duced. Overall, therefore, we can estimate that the columnmass of thermosphere from which angular momentum is ex-tracted is approximately 5 times greater at Jupiter than at Sat-urn.

    This means that, all other things being equal, a smallerproportion of the angular momentum present in that layerof the thermosphere will be removed per second, and atmo-spheric advection and viscosity should be able to replace itmore rapidly. However, as commented above, the rate of an-gular momentum transfer is much greater at Jupiter than atSaturn, in which case the thermosphere at Jupiter is expectedto be able to replace extracted angular momentum relativelyless efficiently. While these simple considerations illustratesome of the possible differences between the response of thethermosphere to magnetospheric forcing at Jupiter and Sat-urn, many other factors must be included to understand thefull picture, including the precise distribution of conductiv-ity, the mapping of field lines between the thermosphere andmagnetosphere, and the relative magnitude of the plasmaflows in the magnetosphere. It is to include all these fac-tors simultaneously that we require a numerical model. Notethat the purpose of this paper is to examine magnetosphere-thermosphere coupling at Jupiter only; we hope to present acomparison of the two planets in a future study.

    A related problem at both planets is the high thermo-spheric temperature. At Jupiter, the observed equatorial neu-tral temperatures of∼900 K (Seiff et al., 1998) are well inexcess of those expected if absorption of solar EUV is theprimary energy source. Spectroscopic measurements of thetemperature of the H+3 molecular ion have shown that thehigh latitude upper atmosphere temperatures are also high, inthe region of∼700–1250 K (Lam et al., 1997; Stallard et al.,2002). The source of the energy required to produce hightemperatures at low latitides remains a mystery. It has beenproposed that these globally high temperatures may be ex-plained by the injection of energy from the magnetosphere at

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  • C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling 201

    high latitudes that is subsequently redistributed globally byequatorward winds (Waite et al., 1983; Atreya, 1986; Milleret al., 2000; Bougher et al., 2005; Smith et al., 2005a; Melinet al., 2006).

    Similar high temperatures are present in Saturn’s thermo-sphere, and recent studies have attempted to explain thesemeasurements by the redistribution of high latitude heating.Smith et al.(2005b) found that an arbitrary high latitudesource of thermal energy did generate sufficient redistributivewinds to reproduce the observed temperatures, but the moresophisticated calculations ofSmith et al.(2007) showed thatwhen high-latitude energy inputs of Joule heating and iondrag were included, poleward meridional winds were gener-ated that cooled, rather than heated, low latitudes.

    The results ofSmith et al.(2007) andSmith and Aylward(2008), pertaining to thermospheric temperature and angularmomentum transfer, respectively, were both generated usingthe same simplified model of Saturn’s thermosphere. In thispaper we describe an application of the same modelling tech-niques to Jupiter, with the advantage that we are able to makeuse of an existing physical model of Jupiter’s middle magne-tosphere (Nichols and Cowley, 2004), allowing us to self-consistently couple models of the thermosphere and magne-tosphere.

    We also investigate a possible additional thermosphericenergy source that has been raised for Earth (Codrescu et al.,1995) and more recently for Jupiter and Saturn (Smith et al.,2005a) but not yet tested: the possibility that small scale fluc-tuations in the electric field may increase the total Joule heat-ing and thus account for the energy deficit. Our model notonly allows us to examine this question – it also allows usto test the consistency of such a situation with existing mea-surements of the magnetosphere.

    In Sect.2 we describe the background observations andtheory necessary to understand our model; in Sects.3–6 wedescribe respectively our magnetosphere, thermosphere andionosphere models and how they are coupled together. Ourinitial results are described in Sect.7 and the results of forc-ing the model with extra Joule heating from electric fieldfluctuations in Sect.8. In Sects.9 and 10 we discuss ourresults and conclude.

    2 Theoretical background

    2.1 Magnetosphere-thermosphere coupling at Jupiter

    In Jupiter’s magnetosphere the dominant internal source ofplasma is volcanism on the moon Io, which orbits Jupiterevery∼40 h at approximately 6RJ . Clouds of sulphur andoxygen ejected from Io form a vast torus of neutral gas closeto the equatorial plane. Some of this neutral gas subse-quently becomes ionised and is then under the influence ofthe planet’s magnetic field.

    Fig. 1. Schematic diagram of Jupiter’s inner and middle magne-tosphere (adapted fromCowley and Bunce, 2001). Solid lines aremagnetic field lines; dashed lines are the corotation enforcementcurrents.

    The newly created plasma in the Io torus feels a torque ex-erted on it by the planet’s upper atmosphere which acceler-ates it towards corotation with the upper atmosphere. The up-per atmosphere feels an equal and opposite anti-corotationaltorque. In steady state it is supposed that the upper atmo-sphere is viscously and convectively coupled to the deep at-mosphere of Jupiter, and that this coupling supplies sufficientangular momentum to balance the anti-corotational torquefrom the plasma disk.

    In the Io torus, this ionospheric torque is adequate to en-force almost perfect corotation of the plasma. However, theplasma is known to diffuse radially outwards, driven by cen-trifugal “interchange instabilities” (e.gSiscoe and Summers,1981). This outward diffusion leads to the formation of aplasma disk in the equatorial plane of the magnetosphere(Fig. 1). At larger radial distances the angular momentumrequired to enforce corotation of the plasma disk is muchgreater than that required in the Io torus, and ultimately theionosphere is unable to supply sufficient torque to enforcecorotation.

    The result is that the plasma disk in the middle magneto-sphere and the magnetically connected upper atmosphere areboth expected to sub-corotate with respect to the deep plan-etary angular velocityJ , at angular velocities ofM andT respectively, where, initially, we expectM≤T ≤J .The plasma in the disk continues to diffuse radially outwardsand is eventually lost from the magnetosphere, by processeswhich are not well understood. Thus the plasma disk slowlyextracts angular momentum from the planetary rotation.

    Hill (1979) constructed a simple physical model of thissituation, assuming a constant rate of plasma outflow, a dipo-lar field and a uniform ionospheric conductivity. This modelwas later developed to include thermospheric sub-corotation(Huang and Hill, 1989; Pontius, 1995), to take account ofa non-dipolar magnetospheric field (Pontius, 1997), and tocalculate the possible association between the plasma diskand Jupiter’s main oval auroras (Hill , 2001; Cowley andBunce, 2001). More recently,Nichols and Cowley(2004)

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  • 202 C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling

    have examined the effect of enhancements to the ionosphericconductivity associated with the main oval auroras. It is thismodel of the middle magnetosphere that we employ here. Allof the studies mentioned above neglected the development ofparallel electric fields in the magnetosphere that may decou-ple the plasma angular velocities in the upper atmosphere andmagnetosphere. This possibility is discussed in Sect.3.3.

    2.2 Magnetosphere-thermosphere coupling currents

    Throughout this study we simplify our modelling by assum-ing axisymmetry of the entire magnetosphere-atmospheresystem. If we further assume during this initial discussionthat the conducting region of the thermosphere-ionosphere isa thin sheet with a uniform rotational velocityT at each co-latitude and the connected region of the magnetosphere has auniform rotational velocityM , then the appropriate electricfield in the ionosphere is:

    E∗θ = ρi(T − M)Bi (1)

    whereρi is the perpendicular off-axis distance in the iono-sphere, and we have assumed that the ionospheric magneticfield Bi is vertical. This implies an equatorward-directedPedersen currentJθ :

    Jθ = 6P E∗θ = 6P ρi(T − M)Bi (2)

    where6P is the height-integrated Pedersen conductivity ofthe ionosphere.

    As shown in Fig.1, to ensure continuity this current, whichflows in both the Northern and Southern Hemispheres, mustclose in the equatorial plane of the magnetosphere, such thatthere is a radial currentJρ flowing away from the planet. Theionospheric currentJθ exerts an anticorotational (clockwiseviewed from above the north pole)J×B torque on the iono-sphere, to be discussed further below. An equal and oppositecorotational torque acts on the plasma in the equatorial mag-netosphere.

    2.3 Hall current

    In addition to the meridional Pedersen current we must alsoconsider the meridional Hall current, which may be drivendirectly by meridional winds in the thermosphere, as dis-cussed bySmith and Aylward(2008). This can be incor-porated if we appropriately defineT :

    ρiT = ρiJ + Uφ +6H

    6PUθ (3)

    whereUφ andUθ are the eastward and southward thermo-spheric wind speeds respectively. HereJ is the deep rota-tion velocity of the planet which is also the rotation veloc-ity of the reference frame with respect to which the thermo-spheric wind speeds are defined.

    If this definition ofT is used in Eq. (2) then the contri-butions from Pedersen and Hall currents are accounted for.

    ThusT is not strictly the rotation velocity of the neutrals –it is an effective rotation velocity to which meridional windsmay contribute. If the Hall conductivity is bigger than thePedersen conductivity then the contribution from meridionalwinds may dominate. A more detailed discussion is given bySmith and Aylward(2008).

    2.4 Vertically extended ionosphere

    The above definition ofT assumed that the ionosphere wasa thin sheet. In practice, the ionosphere has vertical structure,such that each layer has a local effective rotation velocityωTwhich is completely analogous to that defined in Eq. (3)

    ρiωT = ρiJ + uφ +σH

    σPuθ (4)

    whereσP , σH , uφ and uθ are local values of the conduc-tivites and neutral wind speeds respectively. To find the totalcurrent we just add the various layers in parallel, resulting inthe following definition forT :

    6P T =

    ∫σP ωT dz (5)

    wherez is altitude andσP is the local Pedersen conductivityat each layer so that

    6P =

    ∫σP dz (6)

    ThusT represents a weighted average of the effective ro-tation velocity throughout the ionosphere; it is possible thatno level of the thermosphere that is coupled to the magneto-sphere actually physically rotates at this velocity.

    2.5 Effective conductivity

    Huang and Hill(1989) showed that if the angular momen-tum extracted from the thermosphere was replaced primar-ily by vertical viscous transfer, then for a given atmosphericstructure the corotation lag of the thermosphere was a fixedproportionK (our notation) of the corotation lag of the con-nected magnetosphere:

    J − T = K(J − M) (7)

    which can be rearranged to give

    T − M = (1 − K)(J − M) (8)

    Substituting this into (Eq.2), we have

    Jθ = 6P ρi(T − M)Bi= 6P ρi(1 − K)(J − M)Bi= 6∗P ρi(J − M)Bi (9)

    where we have defined the effective conductivity

    6∗P = 6P (1 − K) (10)

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  • C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling 203

    The RHS of Eq. (9) does not containT , such that any ex-plicit reference to the thermospheric rotation can be elim-inated from calculations that refer to the magnetospherealone. Note that to clearly distinguish the conductivity6P from the effective conductivity6∗P we will refer to itthroughout this paper as the “true” conductivity.

    It is tempting to interpret the effective conductivity as aharmless mathematical trick that combines the two unknownparameters6P andT into a single unknown6∗P . How-ever, it is important to emphasise that its usefulness is largelydependent on the special properties of theHuang and Hillmodel, in particular that the corotation lags of the thermo-sphere and magnetosphere are proportional (Eq.7). If thismodel is not valid, then the quantity defined by Eq. (10) haslimited physical meaning; we shall see in Sects.7and8 that itbehaves strangely if the inequalityM≤T ≤J is violated.

    3 Magnetosphere model

    The plasma flows in the Jovian magnetosphere are complexand not fully understood (see reviews byKhurana et al.,2004; Krupp et al., 2004). In particular, there is consider-able debate concerning the structure of the outer regions ofthe magnetosphere, which couple to the solar wind and areconnected to high magnetic latitudes at the planet (Kivelsonand Southwood, 2005; McComas and Bagenal, 2007; Cow-ley et al., 2008; McComas and Bagenal, 2008).

    Our model of the plasma flows in Jupiter’s magnetosphereis a combination of the simple model of the whole magne-tosphere presented byCowley et al.(2005) and the more so-phisticated model of the middle magnetosphere ofNicholsand Cowley(2004). In outline, the model assumes that therotation of the middle magnetosphere is controlled by out-ward diffusion of iogenic plasma balanced by angular mo-mentum transfer from the planet (Hill , 1979). The motionof plasma at very large radial distances is described basedon a solar wind interaction model first outlined byCowleyet al. (2003). This proposes that there is a significant quan-tity of open flux involved in a Dungey type interaction withthe solar wind, mapping to a region of stagnant plasma flowof radius∼10◦ colatitude at the centre of the polar cap. Theflow in the outer magnetosphere – lying between the two re-gions just described – is modelled as closed flux involved inthe Vasylīunas cycle and the return flow of the Dungey cycle.

    Other models have been proposed for the structure of theouter jovian magnetosphere that describe a much weakerDungey type interaction with the solar wind (Kivelson andSouthwood, 2005; McComas and Bagenal, 2007). Both thesepapers suggest that the polar cap region of open flux is muchsmaller than that described by the Cowley model, but do notmake alternative concrete quantitative statements about theplasma flows in the polar cap. We do not have a strong prefer-ence for the overall interpretation of plasma flow representedby the Cowley model: however we choose to employ it for

    Fig. 2. Plasma velocity model for Jupiter, based onNichols andCowley (2004) and Cowley et al.(2005), mapped into the polarionosphere (solid line). The shaded area denotes region B. The pro-file of plasma velocity in regions A and B is fixed; the profile inregions C and D is calculated by the model. The profile shown isthat calculated by our baseline model. Also shown are the height-integrated Pedersen conductivities (dashed line) and neutral rota-tion velocities (dotted line) calculated by the baseline model (seeSect.7).

    the polar cap plasma flows because it provides a simple quan-titative formulation that is easily integrated with our thermo-sphere model. In Sect.9.1we briefly discuss how our resultsmight change if we used a model with a less stagnant (morerapidly rotating) plasma flow in the polar cap region.

    The model assumes for simplicity that the magnetic fieldis axisymmetric, aligned with the planet’s rotation axis, andnorth-south symmetric. The behaviour of the magnetosphereis then described in terms of the rotation velocities of ax-isymmetric shells of magnetic field lines. Each shell inter-sects the ionosphere at some colatitudeθ and axial distanceρi=RJ sinθ and the equatorial plane at some axial distanceρe, independent of longitude (φ).

    The Cowley model may be split, conceptually, into fourregions A–D. For regions A and B we useCowley et al.(2005) and for regions C and D we useNichols and Cowley(2004) (Fig. 2). These represent respectively the regions al-ready discussed: regions that are open to the solar wind (A);regions of the outer magnetosphere involved in the Dungeyand Vasylīunas cycles (B); the sub-corotating middle mag-netosphere (C) and the corotating inner magnetosphere (D).These are the principal flow regions described byCowleyet al. (2005); we label them using the same letters as em-ployed bySmith and Aylward(2008) in the context of Saturn.Note that the boundary between regions C and D is not well-defined, since the transition to perfect corotation is gradual.

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  • 204 C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling

    Fig. 3. Relation between ionospheric colatitude and magneto-spheric radius implied by the Jovian magnetic field model.

    3.1 Open field and outer magnetosphere

    As already mentioned, we use the empirical model ofCowleyet al. (2005) to represent the plasma flows in regions A andB of the magnetosphere. This model is specified accordingto co-latitudeθ in the ionosphere:

    M(θ) = ωA +1

    2

    [1 + tanh

    (θ − θAB

    1θAB

    )](ωB − ωA) (11)

    Region A represents the region of “open” field lines associ-ated with the tailward flow of the Dungey cycle. The colat-itudinal extent of this region, which extends from the poleto θAB=10.25◦, was determined byCowley et al.(2005)through consideration of the amount of “open” flux expectedin the system. A rotation velocity ofωA=0.091J is cho-sen. This gives a rotation velocity close to zero, which isconsistent with the theory ofIsbell et al.(1984) if the effec-tive conductivity is 0.2mho (we will discuss our adopted con-ductivities in more detail below). This “stagnant” behaviouris also consistent with the IR Doppler observations ofStal-lard et al.(2003). Region B is analagous to that describedfor Saturn, representing the Dungey cycle return flow andVasyliūnas cycle. We use a value ofωB=0.25 mho, which,in the formulation ofCowley et al.(2005), represents “ex-panded” conditions in which the magnetosphere is not overlycompressed by the dynamic pressure of the solar wind.

    Smith and Aylward(2008) found, in their study of Saturn,that using a fixed magnetosphere model similar to this pro-duced anomalous behaviour and modified their plasma flowmodel such that, instead of the absolute plasma rotation ve-locity M being fixed in each region, the ratio of the plasmaand neutral velocitiesχ=M/T was fixed instead. Sinceour main focus in this study is the structure of the physicallyself-consistent middle magnetosphere model, we will not im-plement such a modification here.

    3.2 Middle magnetosphere model

    3.2.1 Summary of model

    Our middle magnetosphere model is closely based on thework of Nichols and Cowley(2004). In this section wefirstly summarise the equations that constitute theNicholsand Cowley(2004) model and state our modifications. Fur-ther details of the model, in particular the magnetic fieldmodel and the origin of Eqs. (15) and (16) are given in Ap-pendixA.

    The relative simplicity of the model is underpinned by aseries of simplifying assumptions. Firstly, we assume ax-isymmetry and north-south symmetry of both the planet andmagnetosphere. For the planet, we believe this is justified asa first approximation. This will be discussed in Sect.4.2. Forthe middle magnetosphere, this is a good approximation tothe observed geometry (Nichols and Cowley, 2004).

    Secondly, we assume sphericity of the conducting layerin the polar upper atmosphere and that it is permeated by aconstant vertical polar B-field of magnitudeBi=2BJ , whereBJ =426 400 nT. This value was determined byNichols andCowley (2004) based on the VIP 4 internal field model ofConnerney et al.(1998). Both of these assumptions aregood first-order approximations to the observed geometryand magnetic field. Finally, we assume negligible field-aligned potential drops in the magnetosphere. We will dis-cuss this assumption in Sect.3.3.

    Since we are describing a physical model of the equatorialmagnetosphere itself, it must be specified in terms of radialdistanceρe in the equatorial magnetosphere. We thus need amethod of mapping rotation velocities along field lines be-tween the equatorial magnetosphere and the high latitudeionosphere.

    This is achieved using a “flux function”F . In this formu-lation, axially symmetric shells of magnetic field lines are de-fined by surfaces across whichF is constant. By separatelydefining this function in the ionosphere (Fi) and magneto-sphere (Fe) it is then possible to map field lines along theseshells by settingFe=Fi . We employ identical flux functionsto those ofNichols and Cowley(2004), as described in Ap-pendixA; for further details the reader is referred to this pa-per and the references therein.

    The resulting relationship between ionospheric colatitudeand magnetospheric radius is plotted in Fig.3. The mostsignificant consequence of this mapping is that the region ofthe magnetosphere in the range 20–100RJ maps to just 2◦ ofcolatitude at the planet.

    The plasma flow model itself is summarised by three equa-tions. The first, whichNichols and Cowley(2004) refer to asthe Hill-Pontius equation, is derived by balancing torques inthe magnetosphere due to the outward diffusion of plasmawith torques due to the magnetosphere-ionosphere coupling

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  • C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling 205

    currents:1

    ρe

    d

    dρe

    (ρ2e M

    )=

    8π6∗P Fe|Bze|

    Ṁ(J − M) (12)

    Hereρe is the radial distance from the centre of the planetin the equatorial plane of the middle magnetosphere;M isthe angular rotation velocity of the plasma at that radius;6∗Pis the effective Pedersen conductivity of the ionosphere (tobe discussed further below);Fe andBze are the flux functionand magnetic field in the magnetosphere respectively (fullydefined in AppendixA); Ṁ is the mass outflow rate andJis the deep rotation velocity of the planet.

    The second equation describes the magnitude of the up-ward field-aligned current in the ionospherej‖i :

    j‖i =4BJ

    ρe|Bze|

    d

    dρe

    [6∗P Fe (J − M)

    ](13)

    In isolation, the field-aligned current specified by this secondequation has no effect on the solutions of the first equation.However, in reality the field-aligned current should influencethe angular rotation velocities since any particle precipitationrelated to large values of the upward field-aligned current isexpected to enhance the effective ionospheric conductivity6∗P . The third equation thus expresses this relationship:

    6∗P = 6∗

    P (j‖i) (14)

    The functional form that is specified for this relationship in-fluences the solutions of the first two equations, as discussedin some depth byNichols and Cowley(2004). Details of ouradopted functional form are given in Sect.5.2.

    Our principal modification to the model is to self-consistently specify the neutral rotation velocityT usinga model of the thermosphere. To include this, we must re-move the effective conductivity6∗P and reintroduceT . Wethus substitute everywhere6P (T −M) for 6∗P (J −M)to reintroduceT into Eqs. (12) and (13). We then replaceEq. (14) with a function that specifies the true conductivity6P in terms of the field-aligned current. Finally, we intro-duce a fourth equation which formally represents the ther-mosphere model by specifyingT in terms ofM and6P :

    1

    ρe

    d

    dρe

    (ρ2e M

    )=

    8π6P Fe|Bze|

    Ṁ(T − M) (15)

    j‖i =4BJ

    ρe|Bze|

    d

    dρe

    [6P Fe (T − M)

    ](16)

    6P = 6P (j‖i) (17)

    T = T (M , 6P ) (18)

    As we shall see below,T is not a simple function ofM and6P , but is calculated by a complex time-dependent numeri-cal model. However, for a given distribution ofM and6Pthere presumably exists at least one steady state of the ther-mosphere model. Ideally, we wish to run the thermospheremodel towards steady state so that all four of the equationsabove represent steady state conditions. Our procedure forapproaching steady state is described in Sect.6.

    3.2.2 Method of obtaining solutions

    Our method of solution closely follows that employed byNichols and Cowley(2004). The main difference is thatwe calculate a profile ofT with radial distance based onthe current state of our thermosphere model and employthis in Eqs. (15) and (16). We specify as our outer bound-ary condition at 100RJ an azimuth-integrated radial currentI=100 MA, approximately equal to that determined fromGalileo data (Khurana, 2001). We then select a value of theplasma rotation velocityM at the outer boundary and to-gether with Eqs. (A4) and (17) this allows us to also calcu-late j‖i at the outer boundary. We then integrate Eqs. (15),(16) and (17) inwards from the outer boundary. Typicallythis initial choice diverges to very large negative or positiverotation velocities at small radial distances. We thus iterateour chosen value ofM at the outer boundary until we ob-tain a solution that converges to the required inner boundarycondition of near-rigid corotation at small radial distances.

    The integration is specified using a double-precision FOR-TRAN subroutine, such that the magnetosphere model caneasily be integrated with the existing FORTRAN thermo-sphere model. In practice, it is not possible to specifyMwith sufficient precision at the outer boundary to determinea solution that near-rigidly corotates at∼4RJ . Instead, fol-lowing Nichols and Cowley(2004), we employ an approxi-mation in the inner region. This is based on the observationthat at small radial distances the plasma is very close to rigidcorotation such thatM≈T .

    Firstly, we assume perfect corotationM=T within4RJ . At the planet, this corresponds to colatitudes greaterthan 30◦. We then substitute the relationM≈T intoEq. (15) to yield the following approximation forM at ra-dial distances slightly greater than 4RJ :

    M ' T

    [1 −

    4π6P Fe|Bze|

    (1 +

    ρe

    2T

    dT

    dρe

    )](19)

    which is equivalent to Eq. (24) ofNichols and Cowley(2004)but with an extra term added to take account of the inde-pendent variability ofT with radial distance. We use thisformula to calculate an approximate value ofM at a radialdistance of 4RJ .

    A similar expression can be determined for the field-aligned current. We do not quote this (complicated) expres-sion because its only practical use is the determination of6P in the inner region. When6P is calculated with thisexpression across the range 4–12RJ is found to be alwaysvery close to the background value6P0 (to be defined inSect. 5.2). Given this approximate profile of6P in the range4–12RJ and our approximate value ofM at 4RJ , we inte-grate the exact formulation of Eq. (15) outwards with respectto radial distance to determine an approximate solution forM in the range 4–12RJ .

    This approximate profile in the inner region then providesthe convergence criteria for the integration from the outer

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  • 206 C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling

    boundary. When this integration reaches 12RJ , we com-pare the calculated value ofM and its radial gradient to theequivalent values calculated from the inner region approx-imation at this radius. If the absolute values match within1% of the inner region value and the gradients within 50%of the inner region value then we are satisfied that the in-tegration has sufficiently converged. In practice, the lattercondition is probably not necessary, since, if our inner re-gion approximation is sufficiently accurate, a solution thatmatches within 1% ofM would be expected to match thegradient within a narrower margin than 50%. This is indeedthe case: if the curves shown in Figs.10and11are examinedat a radius of 12RJ a small kink is apparent where the innerand outer region solutions join. It is clear that the gradientsat this join, although not perfectly matched, differ by muchless than 50%.

    We have validated this method by finding solutions corre-sponding to the assumptions ofNichols and Cowley(2004)and comparing these solutions to their results. We find thatour method produces solutions closely matching those of theearlier study.

    3.3 Parallel electric fields

    The model of the middle magnetosphere that we haveadopted fromNichols and Cowley(2004) makes the impor-tant simplifying assumption that plasma flows in the equato-rial plane of the magnetosphere map exactly to magneticallyconnected plasma flows in the ionosphere. In reality this as-sumption partially breaks down because in order for the field-aligned currents implied by the model to flow it is necessaryfor parallel electric fields to form along the magnetic fieldlines connecting the thermosphere and magnetosphere (e.gMauk et al., 2002). The presence of these parallel electricfields means that the magnetic field lines are not equipoten-tials and the plasma flows in the equatorial magnetosphereand ionosphere become partially decoupled. These fieldsform because at high altitudes there are insufficient currentcarriers such that thermal currents cannot provide the neces-sary current density (Su et al., 2003).

    The practical consequence of this decoupling is that theflux is not frozen into the plasma in the region of paral-lel electric fields. The flux tubes are able to “slip” rela-tive to each other, so that the plasma angular velocity in theionosphere is closer to corotation than that in the magneto-sphere. This reduces the magnitude of the magnetosphere-thermosphere coupling currents and hence the rate at whichangular momentum is transferred from the planet. Accu-rately modelling this effect is beyond the scope of this study:our results should therefore be considered to be a baselineapproximation in which this aspect of the interaction is ne-glected.

    4 Thermosphere model

    4.1 Summary of thermosphere model

    To calculateT self-consistently as a function of the valuesof M and6P calculated from the magnetosphere model, weemploy a global, three-dimensional numerical model of thethermosphere. The model employed for this study is largelyidentical to that described bySmith and Aylward(2008) fortheir study of Saturn’s thermosphere. In particular, the corethermospheric equations and our formulation of Joule heat-ing and ion drag are completely unchanged. We will de-scribe here only those features that it has been necessaryto alter. We have incorporated elements from two existingJupiter models – the JIM global three-dimensional model ofthe thermosphere and ionosphereAchilleos et al.(1998) andthe Grodent et al.(2001) one-dimensional model of the au-roral thermosphere and ionosphere.

    We have, of course, changed the core parameters of thethermosphere model to those appropriate for Jupiter. Themost of important of these is the planetary rotation frequencywhich we have set toJ =1.76×10−4 rad s−1, consistentwith Nichols and Cowley(2004). We place the base ofthe model at a pressurep0=2µbar (300 km above the 1 barlevel) consistent with the JIM model (Achilleos et al., 1998).The temperature at the base of the model is set to a constanttemperature of 262 K, the temperature at 2µbar in the dif-fuse auroral model ofGrodent et al.(2001). The winds at thebase of the model are set to zero: thus the base of the modelcorotates with the planetary angular velocityJ .

    The eddy coefficient in our baseline model – requiredin order to calculate eddy conduction and viscosity –is set consistent with that ofGrodent et al.(2001) atKτ=1.4×102 m2/s, placing the methane homopause at∼1µbar, just above the base of our model. Note that in ourmodel Kτ does not vary with altitude. The mixing ratiosof H, H2 and He are taken from the diffuse auroral modelof Grodent et al.(2001). These mixing ratios are fixed as afunction of pressure. We do not calculate changes in thesemixing ratios self-consistently because, since H2 is the dom-inant component throughout most of the thermosphere, theinfluence of composition changes upon the dynamics is rela-tively unimportant.

    There is some flexibility in the boundary conditions thatwe specify. However, locating the lower boundary below thehomopause, where vertical diffusive transport is dominatedby eddy processes and hydrocarbon species are abundant, al-lows a double simplification. Firstly, strong eddy diffusionshould keep the horizontal winds tightly coupled to thoseat lower altitudes. There are, to our knowledge, no mea-surements of significant horizontal winds in the mesosphere.However, the observed eastward speeds of zonal jets in thelower atmosphere are never greater than∼150 m/s (Ingersollet al., 2004) globally and are less than∼50 m/s poleward of45◦ colatitude. Compared to the zonal plasma flow velocities

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  • C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling 207

    of order 1000 m/s close to the main auroral oval, these valuesare negligible and in any case we lack sufficient informa-tion to construct a reliable model of zonal winds at the lowerboundary.

    The second advantage of a lower boundary below the ho-mopause is that infrared radiative cooling becomes impor-tant due to the high hydrocarbon densities. This processmoderates the temperature, ensuring that the assumption ofa relatively cool (262 K) fixed temperature lower boundaryis reasonable. Again, there may be structure in the lowerboundary temperature. For example, we would expect it tobe somewhat warmer in regions subject to particle precipita-tion. However, we would need explicit models of radiativecooling and particle precipitation to accurately model suchstructures. The details of these processes are not the mainfocus of this study, so for simplicity it seems reasonable toassume a fixed temperature lower boundary.

    In parallel with the magnetosphere model, the thermo-sphere model further assumes axial and north-south symme-try of the planet. These are much poorer assumptions forJupiter than for Saturn because Jupiter’s magnetic dipole isconsiderably tilted with respect to the rotation axis and off-set with respect to the centre of the planet. This means thatthe structures of the magnetic north and south polar regionsare very different and far from axisymmetric. However, ax-isymmetry is a useful first approximation that allows us touse a relatively high latitudinal resolution in our thermo-sphere model. This allows us to resolve the thermosphericbehaviour close to the auroral oval in acceptable detail whilemaintaining manageable runtimes for our model. Possibleconsequences of our axisymmetry assumption are discussedin Sect.4.2. Note that the assumption of axisymmetry es-sentially reduces the model from a three-dimensional modelto a two-dimensional model in that the winds are assumed tobe identical at all longitudes. However, the model still calcu-lates three-component winds.

    The horizontal resolution of our model is 0.2◦ in latitude.Vertically, we use 30 pressure levels with a resolution of 0.4pressure scale heights. We use a timestep of 3.0 s.

    4.2 Symmetry assumptions

    As already mentioned, a major assumption of our thermo-sphere model is axisymmetry. The relatively small effectof solar forcing on the thermal structure and dynamics sug-gests that this component of axial asymmetry is not a majorsource of error in our calculations. However, the magneticfield of Jupiter is strongly asymmetric about the rotation axis,such that we must consider whether our symmetry assump-tions seriously affect our calculations of magnetosphere-thermosphere coupling.

    The observed location of the main auroral oval is a clearindication of the magnetic field asymmetry. The referenceovals provided byGrodent et al.(2003), based on UV imag-ing, show that the northern main oval does not normally en-

    Fig. 4. Cosine of co-latitude plotted against longitude for the refer-ence ovals ofGrodent et al.(2003).

    circle the rotation pole, lying wholly in the sector betweensystem III longitudes of∼130–300◦. Within this range itforms an irregular oval shape that extends from just southof the rotational pole down to approximately 35◦ colatitude.The southern oval is less irregular, forming an almost circularoval that encloses the southern rotational pole. It is, however,asymmetric about the pole, its colatitude varying in the range8–14◦.

    To build an accurate model that incorporated these com-plexities, we would require a detailed magnetic field modelthat allowed us to map points in the middle magnetospheredirectly onto these irregular ovals. An intermediate step thatwould capture most of the physics of the irregular ovals whilemaintaining simplicity of computation would be to use anoffset tilted dipole model of the magnetic field. This is thetype of magnetic field model employed by the JIM three-dimensional model of the coupled thermosphere and iono-sphere (Achilleos et al., 1998).

    The resultant ovals in such a model are very nearly circularand offset from the rotational poles to a similar degree to theobserved ovals. If we assume that the middle magnetosphereis axisymmetric then the plasma flow velocity along theseovals will be roughly constant in magnetic longitude. Thus,to first order, the principal difference between an axisymmet-ric and a non-axisymmetric model is that the circular regionto which Joule heating and ion drag is applied is shifted awayfrom the rotational pole. Thus the magnetospheric forcing ofthe thermosphere is essentially unchanged – only its locationon the surface of the planet is altered.

    The thermosphere responds to this forcing through a num-ber of processes. Most of these, including effects due tothermal conduction, viscosity, advection, pressure gradientsand the curvature of the planet (assuming perfect sphericity)are independent of latitude and longitude. The only processthat is affected by the offset of the region of magnetosphericforcing is the Coriolis force. This is directly proportional tocosθ whereθ is the colatitude. Thus, in the northern oval,a thermospheric wind driven by ion drag in the region of the

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  • 208 C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling

    auroral oval close to the pole will experience a greater Cori-olis force than a thermospheric wind driven by an identicalion drag at 35◦ colatitude.

    To first order, the cosθ dependence of the Coriolis forceis the only parameter that changes with the location of theregion of magnetospheric forcing. Clearly, if the auroral ovalwas located at the equator, this would significantly alter theresponse of the thermosphere since the Coriolis force wouldbe close to zero. However, within 35◦ of the pole, the cosθfactor reduces the Coriolis force by less than 20%. Figure4shows this factor plotted against longitude for the referenceovals provided byGrodent et al.(2003). It is clear that in thenorth, the Coriolis force varies by no more than∼20% withinthe colatitude range observed and in the south it varies by lessthan∼10%. These are therefore the order of magnitude er-rors that we expect the assumption of axisymmetry to intro-duce into our calculations of thermospheric winds. Given thenumerous other simplifications and approximations involvedin our modelling, these errors are not great and we expectthat a non-axisymmetric model would produce very similarresults to those presented here.

    There are several caveats to the analysis above. Firstly, wehave made the approximation of a perfectly spherical planet.This is a reasonable approximation, since even if the smalloblateness of Jupiter were globally important, the polar re-gions themselves could be accurately approximated as partof a perfectly spherical surface.

    Secondly, and more importantly, we have assumed that theonly forcing is that due to magnetosphere-thermosphere cou-pling. There is of course a small forcing from absorption ofsolar radiation. The structures driven by solar radiation willinteract with those driven by the magnetosphere. However,since solar driven effects are unable to explain the high ther-mospheric temperatures at Jupiter it seems certain that theyhave a relatively small influence on the thermal structure andtherefore dynamics of the upper atmosphere, especially inthe polar regions. It is also possible that strong zonal windsat the lower boundary or forcing by gravity waves may im-pose zonal structures from the lower and middle atmosphereson the thermosphere that interact with the magnetosphericforcing. In this case we might expect significantly differ-ent results in the non-axisymmetric case. However, thereis no evidence that such forcing from below is of compa-rable magnitude to the magnetospheric forcing in the polarregions. There is some evidence for gravity waves in thethermosphere from the Galileo probe (Seiff et al., 1998), butthis applies to the equatorial regions and the extent to whichthese gravity waves might transfer momentum to the upperatmosphere is not clear.

    Thirdly, the predicted thermal and dynamical structure atthe rotational equator clearly will be affected by an offsettilted dipole, since the tilt pushes part of both auroral ovalscloser to the rotational equator. However, since the dipole istilted by only∼15◦, the magnetic equator lies within±15◦

    of the rotational equator. Since the magnetic equator is ap-

    proximately equidistant from the auroral ovals in the offsettilted dipole model, we would expect it to correspond ap-proximately to the equator in our axisymmetric model, with∼20% modelling errors as discussed above. Thus, in our ax-isymmetric model, we expect the rotational equator to corre-spond to latitudes

  • C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling 209

    transfer models such as those ofHuang and Hill(1989) andPontius(1995) to include a more sophisticated model of thethermosphere. It thus seems appropriate to maintain the latterauthors’ approach of a simple fixed model of the ionosphericconductivity.

    We globally employ a conductivity model that is derivedfrom a model of the auroral ionosphere. The principal rea-son for this is that the solution of the coupled equations thatconstitute the magnetosphere model is simplified consider-ably if the shape of the vertical profile of conductivity isconstant globally. By “shape” we mean that, comparing twopressure levels, the relative conductivities are the same at alllatitudes, even if the absolute conductivities vary with lat-itude. This simplification is discussed further in Sect.5.3.We use an auroral conductivity model because most of thecoupling with the magnetosphere occurs in regions with en-hanced conductivity resulting from auroral processes. Thevertical profile of this conductivity model and the horizontalprofile of height-integrated conductivity are determined sep-arately, as described below.

    Finally, we also simplify our calculations of ionosphericconductivity by assuming a vertical magnetic field at all lati-tudes. This assumption is discussed further in Sect.5.3.

    5.1 Vertical distribution

    For the vertical distribution we have chosen to use the 1-Dauroral ionosphere model ofGrodent et al.(2001) at all lat-itudes. TheGrodent et al.model calculates auroral ion andelectron densities and temperatures using a two-stream elec-tron transport model. They present two versions of theirmodel – a “diffuse” model intended to represent unstruc-tured auroras in the polar cap and the afternoon sector ofthe main oval and a “discrete” model intended to representthe brighter, more structured aurora observed in the morn-ing sector of the main oval. The models differ in the choseninput electron energy distributions which are determined sothat the temperatures and emission signatures predicted bythe model closely match a range of observational constraintsderived from measurements of UV and IR emissions.

    Both the diffuse and discrete models incorporate a dou-ble Maxwellian distribution with characteristic energies of100 eV and 3 keV. This component is largely responsible forheating and ionisation above the homopause. This is the re-gion that contributes most significantly to the conductivity,since below the homopause the ionosphere is significantlydepleted through charge exchange reactions with hydrocar-bons. The models differ in the form of the high energy com-ponent, which has its greatest effect below the homopause, aregion that has little influence on the conductivity. Thus, forour purposes, the two versions of the model yield very simi-lar results. We choose to employ the diffuse model becausewe wish to represent the conductivity reasonably accuratelyacross the whole of the main oval and polar cap: the diffuse

    Fig. 5. Vertical distribution of conductivity. The left hand plotshows the local conductivitiesσP (solid line) andσH (dashed line)as a function of pressure calculated using theGrodent et al.(2001)H+3 densities and background atmosphere. The right hand plotshows the conductivities per unit masssP andsH , also calculatedfrom the Grodent et al. model, using the same line formats. Thealtitude scale on the right hand side shows the altitude mapping inthe Grodent et al. model; this does not apply to the results of ourthermosphere model, which has a variable thermal structure.

    model is clearly applicable over a greater proportion of thisarea.

    TheGrodent et al.model provides us with a single staticprofile of neutral densities and temperatures and H+3 andelectron densities. To include this 1-D fixed ionospheremodel in our 2-D time-variable thermosphere, our first step issimply to calculate values ofσP andσH using the output oftheGrodent et al.(2001) model. The expressions used to cal-culate these conductivities are given in AppendixB. The ver-tical conductivity distributions so calculated are shown in theleft panel of Fig.5. The pressure range shown corresponds tothat covered by our thermosphere model. It is clear that thePedersen conductivity is much more important than the Hallconductivity at almost all pressure levels. In particular, thepeak Pedersen conductivity is several times greater than thepeak Hall conductivity, such that the Pedersen dominates in aheight-integrated sense. Thus the contribution of meridionalwinds toT , as described by Eq. (3), is likely to be minimal.

    The question then arises as to how one should apply theseconductivity profiles, calculated using the specific thermalstructure from theGrodent et al.model, to a thermospheremodel that exhibits variable thermal structure. Our solu-tion is to calculate, as a function of pressure, the quantitiessP =σP /ρ andsH =σH /ρ, whereρ is the neutral mass den-sity. We then use the same profiles ofsP andsH , as a functionof pressure, at each latitude. The advantage of these quanti-ties is that the height-integrated conductivities then dependonly on sP andsH , not on the thermal structure, since, for

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  • 210 C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling

    Fig. 6. Relation between ionospheric field-aligned current and trueheight-integrated conductivity. Solid line: true conductivity modelused in this study; dashed line: effective conductivity model usedby Nichols and Cowley(2004) for Ṁ=1000 kg s−1. Note that fornegative values of the field-aligned current both models take a con-stant value of 0.0275 mho.

    example:

    6P =

    ∫ z1z0

    σP dz =

    ∫ z1z0

    sP ρdz =

    ∫ p0p1

    sP gdp (20)

    where the last step follows from hydrostatic equilibriumdp/dz=−ρg, whereg is the acceleration due to gravity. Ifg is constant with height (a good approximation since thevertical extent of the thermosphere is small compared to theradius of the planet) then6P depends only on the profileof sP with respect to pressure, not on the thermal structure.Similar arguments apply for6H andT .

    Thus employing the quantitiessP andsH – which we mayusefully refer to as conductivities “per unit mass” – allowsus to control the height-integrated values of the conductivityand the neutral rotation velocity independent of changes inthe thermal structure of the upper atmosphere. The verticalprofiles ofsP andsH are plotted in the right hand panel ofFig. 5. The general shape of these profiles is the same asthat described for the conventional Pedersen and Hall con-ductivities shown in the left hand panel. However, dividingby the mass densityρ (which of course decreases approxi-mately exponentially with altitude) has the effect of respec-tively strengthening and weakening the high and low altitude“tails” of the conductivity distributions. Thus, there is a veryrapid decrease at altitudes below the peak, but a much slowerdecrease with increasing altitudes, such thatsP decreases byonly one order of magnitude between the peak and the topof our model. This indicates that effects due to Joule heat-ing and ion drag are likely to be significant in determiningthe energy balance and kinematics at high altitudes, even if

    this region is relatively insignificant in terms of the overallenergy and momentum budget of the thermosphere.

    The above fully specifies the shape of the vertical profileof conductivity. The magnitude of the conductivitiessP andsH at a particular point in the model is determined by scalingthe entire vertical profile to obtain the desired value of6P atthat latitude, as described below. The conductivitiesσP andσH can then be calculated using the values ofρ calculatedfrom the local thermal structure of the model.

    5.2 Horizontal distribution

    The horizontal distribution of true height-integrated conduc-tivity in regions C and D is determined directly from the mag-netosphere model in terms of the field-aligned currentj‖i , asdescribed by Eq. (17). We now expand on the precise formof this function:

    6P (j‖i) = 6P0 + 6Pj (j‖i) (21)

    where the two components are, respectively, a “background”conductivity due to solar-produced conductivity, and an au-roral enhancement that depends onj‖i .

    For the “background” conductivity,Nichols and Cowleyuse a value of6∗P0=0.0275 mho. This is derived fromthe results ofHill (1980), who compared his theoreticalmodel with the observed plasma rotation velocity (McNuttet al., 1979). This comparison fixed the ratio between theeffective conductivity and the mass outflow rate. TakingṀ=1000 kg s−1, this implies6∗P0=0.0275 mho. For con-sistency withNichols and Cowleywe also adopt this value.However, we specify the true conductivity6P0. Since we donot know in advance the factor of(1−K) required to convertbetween these two quantities, we initially assume that in theinner region of the magnetosphere to which this backgroundconductivity appliesK∼0 and set the background conductiv-ity 6P0=0.0275 mho.

    The auroral enhancement is described using the samefunction asNichols and Cowley(2004). This function wasdeveloped through detailed manipulation of the modellingresults ofMillward et al. (2002). In order to maximise ourconsistency with their model, we adopt the same form:

    6Pj (j‖i) = 0.16j‖i +

    {2.45

    [(j‖i/0.075)2

    1 + (j‖i/0.075)2

    ]

    ×1[

    1 + exp(−(j‖i − 0.22)/0.12)]} (22)

    where6P is in mho andj‖i is in µAm−2. Note that weuse this function to specify the true conductivity, and explic-itly calculate any neutral winds that may reduce the effectiveconductivity. Nichols and CowleyassumeK=0.5 in orderto specify an enhancement in the effective conductivity suchthat their6∗Pj (j‖i)=0.56Pj (j‖i).

    The solid line in Fig.6 shows the functional form specifiedby Eqs. (21) and (22), taking, as discussed, a true background

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  • C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling 211

    conductivity of6P0=0.0275. The dashed line shows the ef-fective conductivity model ofNichols and Cowley(2004).In this case the contribution from the auroral enhancement isreduced by a factor ofK=0.5 while the background effec-tive conductivity is6∗P0=0.0275, as discussed above. Thusfor negative values ofj‖i our true conductivity model andthe Nichols and Cowleyeffective conductivity model takeidentical constant values; for positive values ofj‖i the en-hancement to our true conductivity model is twice that to theNichols and Cowleyeffective conductivity model. This lattermodel – for the effective conductivity – is used to calculatethe reference model describe in Sect.7.3.

    For regions A and B we expect the conductivity to be en-hanced over the background level, since diffuse UV and IRemission is observed in these regions, indicating some levelof particle precipitation. The observed emission is rathercomplex (e.g.Stallard et al., 2003), but for simplicity weassume a constant conductivity across this region. We fol-low Cowley et al.(2005) in setting the conductivity here to6P =0.2 mho (although, again, we note that they specified6∗P =0.2 mho). Note that at the boundary between regions Aand B we expect a sheet of upward field-aligned current,which, according to Eq. (22) may enhance the conductivity(Cowley et al., 2005). For simplicity we neglect any such en-hancement at this boundary; our major focus is understand-ing the influence of the neutral atmosphere on the middlemagnetosphere (region C).

    5.3 Problems with our approach

    One problem with our approach is that we use a conduc-tivity model derived from an auroral model at all latitudes.Thus at low latitudes our “background” conductivity, whichshould represent conductivity due to solar-produced ionisa-tion, has the same profile as the enhanced conductivity in theauroral zones. This, of course, is incorrect. However, if ourbackground conductivity were of a different vertical form tothe enhanced conductivity, then both the magnitude and thevertical distribution of the conductivity would vary withj‖i .For example, if there was no precipitation, we would have asolar-produced conductivity profile, which would be likely tohave a broader vertical distribution and a less intense peak;whereas in the regions where there was significant precipi-tation the profile would be dominated by the sharply peakedauroral profile.

    The value ofT depends on the vertical distribution ofconductivity through Eq. (5) – thus if the profile dependedonj‖i , T would become a function of the field-aligned cur-rent. This considerably reduces the tractability of solvingEqs. (15–17) simultaneously. While one might envisage thisdependence ofT onj‖i producing some interesting effects,it seems an unnecessary complication for this initial study.Hence we tolerate some inaccuracy in the mid-latitude con-ductivity profile for the sake of simplicity.

    Note also that the only role of the conductivity in ther-mospheric structure is to determine the magnitude of Jouleheating and ion drag, which we expect to be most importantin the polar regions where the relative velocities of plasmaand neutrals are largest. In these regions the conductivity isalso likely to be largely auroral in origin, so it makes sense topick a model of the auroral ionosphere to calculate our con-ductivity profile. We expect pressure gradients to be a moreimportant driver of dynamics at mid-latitudes – if so an inac-curate conductivity profile should not significantly affect ourresults.

    A secondary source of inaccuracy at mid-latitudes is ourassumption that the magnetic field is vertical at all latitudes.This is a good approximation at high latitudes and is nec-essary for consistency with theNichols and Cowley(2004)model, which makes the same assumption. Of course, a ver-tical magnetic field is a poor assumption at mid latitudes andan extremely poor assumption at the magnetic equator wherethe field is horizontal. However, we find that the magnitudesof the Joule heating and ion drag terms in the equatorial re-gion are relatively so small that we believe the impact on ourresults to be negligible. The exception to this is the experi-ment described in Sect.8 in which the Joule heating at lowlatitudes is specifically enhanced; this will be discussed inmore detail in Sect.8.

    6 Coupled model

    We have described three separate component models of themagnetosphere, thermosphere and ionospheric conductivityrespectively. In order to couple these models together wehave ensured that shared physical quantities and assumptionsare as consistent as possible:

    1. We assume Jupiter to be a sphere with a radius ofRJ =71 492 km. Nichols and Cowley(2004) used avalue ofRNCJ =71 323 km, a difference of∼0.2%. Westill useRNCJ to calculate the value ofFe as a function ofradial distance. Our value,RJ , is employed in all subse-quent calculations, introducing very small (∼0.2%) in-consistencies between our model andNichols and Cow-ley (2004). Note that some ambiguity in radial dis-tance is in any case inevitable since our ionosphericconducting layer varies in altitude according to the ther-mal structure, while the magnetosphere model assumesa spherical conducting layer.

    2. We assume a constant vertical magnetic fieldBi=2BJ ,whereBJ =426 400 nT, across the whole planet. This isa simplifying assumption of the magnetosphere modelin the polar regions. Identically applying this assump-tion to our ionosphere and thermosphere models ensuresthat our calculations of ion drag and subsequent angu-lar momentum exchange with the thermosphere modelare consistent with the calculated angular momentum

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    transfer in the magnetosphere model. Vertical magneticfield is a poor assumption at low latitudes, but is ex-pected to have a small effect on our results as discussedin Sect.5.3

    3. Parallel electric fields in the magnetosphere are ne-glected such that we can mapM unchanged along fieldlines between the magnetosphere and thermosphere.This ensures that the quantityM discussed in the con-text of the magnetosphere in Sect.3 is the same as thatdiscussed in the context of ionospheric electric fieldsand currents in Sect.2.

    These assumptions allow us to alter theNichols and Cowley(2004) model as little as possible so that we can clearly ex-amine the influence of thermospheric dynamics on the mag-netosphere.

    To couple the three models together we use an iterativeapproach. We first solve Eqs. (15), (16) and (17) simultane-ously, using the procedure outlined in Sect.3.2.2, assumingthatT =J at all latitudes.

    The resultant Pedersen conductivities6P are used toscale the vertical ionospheric conductivity model (Sect.5.1)at each latitude according to the formulation discussed inSect.5.2. The plasma rotation velocitiesM are then usedtogether with the ionospheric conductivities and the existingthermospheric wind speeds to calculate Joule heating and iondrag. The thermosphere model is then stepped forward intime, driven by these values of Joule heating and ion drag,which are updated each timestep as the thermospheric windspeed evolves while the values ofM and6P are kept fixed.After the thermosphere model has been run for one tenth ofa planetary rotation, we calculate a new magnetosphere, im-plying new profiles ofM and6P , using the values ofTgenerated by the thermosphere model. We repeat this processiteratively.

    Thus we run the thermosphere model continuously, calcu-lating a new steady state magnetosphere 10 times per plane-tary rotation, until the thermosphere is close to a steady state.We find that this is reached reliably after 200 Jovian rota-tions, and adopt this runtime for all of the results shown inthis study.

    7 Response of baseline model

    7.1 General thermospheric behaviour

    The response of the baseline model is illustrated in Fig.2in terms of the rotation velocities of the plasma and neutrals(solid and dotted lines) and true height-integrated conduc-tivity (dashed line). Regions A and B exhibit fixed plasmavelocities as required by the model. Regions C and D, forwhich the plasma velocity is explicitly calculated, exhibitgreater structure, which will be discussed in detail below.The conductivity shows a sharp peak at the poleward edge

    of region C; this is the conductivity enhancement due to theparticle precipitation that forms the main auroral oval.

    The neutrals show corotation of∼50% in regions A and B.Just equatorward of the main oval, in region C, is a region ofsuper-corotation. Similar super-corotation was observed bySmith and Aylward(2008) in the case of Saturn. Further to-wards the equator the neutral velocity returns to the planetaryrotation velocity.

    A more detailed view of the thermospheric dynamics isshown in Fig.7, which shows the temperatures (top), winds(middle) and Pedersen conductivities, Joule heating and iondrag energy (bottom). The temperature structure is, in out-line, identical to that described for Saturn bySmith et al.(2007), with a hotspot at the pole and rather inefficient re-distribution of thermal energy to regions equatorward of themain oval. While the polar temperatures peak at∼700 K,which approaches the H+3 temperature measurements in thisregion (Lam et al., 1997; Stallard et al., 2002), the low lat-itudes retain cool temperatures close to those generated byabsorption of sunlight.

    The winds also exhibit similar structures. The zonal windsexhibit a single broad sub-corotating jet in regions A and B;this terminates rather sharply at the boundary with region C.This is associated with the sudden change in the plasma ve-locity in the region of the main oval. Most importantly, thepoleward flow at low altitudes that cools mid latitudes (Smithet al., 2007) is clearly present, indicating that this dynamicalprocess is a feature common to Jupiter and Saturn.

    It is worth briefly summarising the analysis of this pole-ward flow presented bySmith et al.(2007). The westwardwinds generated by ion drag are acted on by Coriolis forceswhich generate strong poleward flows throughout the regioncoupled to sub-corotating plasma. At the boundary betweencorotating and sub-corotating plasma – in this case corre-sponding approximately to the location of the main auro-ral oval – there is a divergence in this poleward flow whichdrives upwelling from lower altitudes. The upwelling gascools adiabatically, producing a region just equatorward ofthe main auroral oval that is cooler than the lower boundarytemperature. The poleward pressure gradient on the equa-torward edge of this cool region drives gas towards the pole.Since there is negligible westward ion drag in this region,Coriolis forces drive this gas into super-corotation. One over-all effect of sub-corotational ion drag at the pole is thereforea cool, super-corotating region just equatorward of the mainoval.

    The distribution of energy inputs is also interesting. Re-gions A and B exhibit significant Joule heating and ion dragpowers at low altitudes. This is simply due to the action ofthe magnetospheric frictional drag in the region that has thehighest conductivity. At high altitudes there are areas of neg-ative ion drag energy in both of regions B and C. Both ofthese regions of ion drag represent extraction of kinetic en-ergy stored in the thermally-driven high-altitude winds thatare sub-corotating relative to the plasma. Some of the K.E.

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    from these regions is thermalised – generating a region ofJoule heating at high altitudes in region C – and some is re-distributed, either to the magnetosphere or to lower altitudes.

    The high altitude region of Joule heating in region C is alsointeresting because it is partly a conductivity enhancementeffect. Thermally-driven winds blowing through the mainauroral oval encounter resistance from the enhanced conduc-tivity in this region, which slows the winds and extracts ki-netic energy as heat. Were the conductivity not enhanced, theJoule heating in this region would be relatively unimportant.

    Indeed, the most interesting aspect of the energy input dis-tributions shown is that the Joule heating due to the main oval– i.e. due to the middle magnetosphere-thermosphere cou-pling currents – is much less significant than that due to thegreater sub-corotation rates over a much greater area that arepresent inside the polar cap. This indicates that the main ovalitself may be relatively unimportant in terms of the thermalstructure.

    7.2 Momentum balance

    We now summarise the height-integrated momentum balanceof the thermosphere. We calculate a height-averaged veloc-ity which represents the total momentum per unit mass of acolumn of thermosphere at a given location:

    ūφ =

    ∫ρ(z)uφ(z)dz∫

    ρ(z)dz(23)

    It should be emphasised that this weighted height-average isof a different nature to the weighted height-average used tocalculateT (Eq.5). In this case, we are weighting the ther-mospheric velocity by mass in order to summarise the overallmomentum budget of the thermosphere. In the case ofT weweight the thermospheric velocities according to the Peder-sen and Hall conductivities in order to summarise the specificprocess of momentum exchange with the magnetosphere.

    We can also calculate height-averaged momentum terms,analagous tōuφ , which represent the total column rate ofchange of momentum per unit mass due to the various termsin the momentum equation. Figure8 shows the plasma ve-locity vφ and height-averaged neutral velocityūφ in the up-per panel and the corresponding height-averaged zonal mo-mentum terms in the lower panel. The upper panel has beenshaded to give an impression of the true height-integratedPedersen conductivity at each co-latitude. Thus the darkestshaded region corresponds to the conductivity enhancementdue to the main auroral oval. We group together the mo-mentum terms into advection (horizontal and vertical) vis-cous drag (horizontal and vertical), inertial terms (Coriolisand curvature) and ion drag.

    This analysis shows that the behaviour is very similarto that described for Saturn bySmith and Aylward(2008).Viscous drag is insignificant. Everywhere ion drag actsto increase the westward velocity and this is opposed al-most everywhere by the Coriolis force. Across most of the

    Fig. 7. Temperatures, winds and energy inputs for our baselinemodel. Top: temperatures in K are shown by the contours. Solidcontours are plotted every 100 K and dotted contours at 20 K in-tervals inbetween. The highest temperature solid contour, on theleft of the plot, is at 600 K; the solid contour at the far right handside is at 300 K. Grey shading shows areas that are cooler than thelower boundary temperature of 262 K. Middle: winds. Grey shad-ing shows westward winds, the darker shading indicating greaterspeeds. Eastward winds are in general of much smaller magnitude;these regions are not shaded. The solid contour represents zerozonal wind speed, thus dividing regions of westward and eastwardwinds. Arrows show the combined meridional and vertical circu-lation. The thickness of the arrows is indicative of the combinedmeridional and vertical wind speed. Bottom: Conductivity distri-bution and energy inputs. The grey shading shows the distributionof Pedersen conductivityσP . The darkest region represents con-ductivities greater than 10−6 mho/m; the next darkest region con-ductivities greater than 10−7 mho/m and so on until the unshadedregion represents conductivities less than 10−10mho/m. Diagonalhatching indicates Joule heating in excess of 2 W/kg. Dashed con-tours enclose regions in which ion drag inputs kinetic energy at arate in excess of 2 W/kg; dotted contours enclose regions in whichion drag extracts kinetic energy at a rate in excess of 2 W/kg.

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    Fig. 8. Height-averaged zonal velocities and momentum terms for our baseline model. Upper plot: height-integrated zonal neutral velocity(solid line) and zonal plasma velocity (dashed line). Shading represents the value of the true height-integrated Pedersen conductivity. Lowerplot: height-averaged zonal momentum terms, grouped into ion drag (solid line), advection (short dashed line), Coriolis and curvature (dottedline) and viscous drag (long dashed line). Note that the latter line, representing viscous drag, is almost contiguous with the x-axis for theentire range shown.

    shown region advection acts in the same direction as iondrag, i.e. towards sub-corotation. This is due to the sub-corotational zonal velocity decreasing towards the pole, theprevailing meridional wind being poleward. In the regionof the main oval advection becomes important in support-ing the flow against sub-corotation, since in this region thesub-corotational zonal velocity increases towards the pole.Gas that is advected poleward across the main oval supportsthe flow against sub-corotation because it has arrived from aregion where both the plasma and neutrals almost corotate.There is a lag before it acquires the greater sub-corotationtypical of the polar regions A and B.

    Just equatorward of the main oval Coriolis forces againbecome more important than advection, and, as at Saturn, itis this that generates the small region of super-corotation inthe neutral velocity profile.

    7.3 Response of magnetosphere

    Whereas the thermospheric response is clearly similar to thatcalculated for Saturn (Smith et al., 2007; Smith and Aylward,2008), the use of a physical model of the middle magneto-sphere allows us to directly assess the influence of the ther-mospheric winds on the plasma flows in this region. In thefollowing, we will always plot our results alongside a ref-erence model corresponding to the assumptions ofNicholsand Cowley(2004). Specifically, our reference model cor-responds to the results shown in Fig. 15 of their paper, fora mass-loading ratėM=1000 kg s−1. The calculation of the

    reference model was used to validate our method of obtain-ing solutions for the middle magnetosphere. In general weplot the results of the reference model as a dashed line andthose of our full model as a solid line.

    The reference model is calculated using an almost identi-cal model of the true ionospheric conductivity, but with theneutral rotation velocity calculated assuming thatK=0.5.The only small difference between the conductivity modelused for the reference model and that used for our full modelruns, is that the reference model assumes an effective back-ground conductivity of6∗P =0.0275 mho (6P =0.055 mho,K=0.5) whereas the full model uses a true background con-ductivity of 6P =0.0275 mho, the effective conductivity6∗Pthen following from the value ofK that is implied by the ther-mosphere model (see discussion in Sect.5). We will discussthe effect of our assumed background conductivity furtherbelow.

    Figure9 shows the standard format in which we presentour results. The results are plotted as a function of radial dis-tance in the magnetosphere. Our model extends from 4RJ to100RJ ; within 4RJ we assumeM=T . The left hand col-umn represents parameters associated with the rotation ve-locities of the plasma and neutrals, and the right hand columnparameters associated with currents and conductivities.

    Figure9a is the effective thermospheric rotation velocity(T ). The dashed line shows the thermospheric rotationvelocity implied by the reference model. Since the refer-ence model assumesK=0.5, this is calculated by halving theplasma corotation lag. Figure9b shows the plasma angular

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    Fig. 9. Magnetospheric parameters for our baseline model. Solid lines: full model. Dashed lines: reference model.(a) T , (b) M , (c) K,(d) χ , (e) j‖i , (f) 6P , (g) 6∗P (dotted line shows6P /2 for the full model),(h) Iρ (dotted line shows values deduced from the data ofKhurana, 2001).

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  • 216 C. G. A. Smith and A. D. Aylward: Thermosphere-magnetosphere coupling

    velocity M . Again, the reference model is represented bya dashed line. Figure9c shows our calculated value ofKmapped into the magnetosphere. The horizontal dashed linerepresents the reference model assumption thatK=0.5. Fig-ure9d shows the ratioχ=M/T . This parameter is usefulin comparing the relative plasma and neutral rotation veloci-ties; it was also used bySmith and Aylward(2008) to specifythe plasma rotation velocities as a function of the neutral ro-tation velocities.

    Figure9e–h shows, respectively, the field-aligned currentin the ionosphere (mapped to the magnetosphere), the result-ing true height-integrated conductivity, the effective height-integrated conductivity, and the radial current in the magne-tosphere. Figure9g also includes a dotted line which indi-cates the effective conductivity that we would calculate if wehalved our true conductivity calculated with the full model.This gives an indication of the features in the effective con-ductivity that are due to structure inK rather than structurein the true conductivity.

    Finally, the dotted line in Fig.9h shows values of the ra-dial current deduced from Galileo magnetometer data (Khu-rana, 2001). These are the same values shown byNicholsand Cowley(2004); for further details of the origin of thesevalues, the reader is referred to both of these papers.

    Initial inspection of Fig.9 indicates that the introductionof coupling to the neutrals has only a small effect on the ma-jority of the magnetospheric parameters. In particular, theplasma angular velocities calculated using the full model areeffectively identical to the reference model beyond∼30RJ .At smaller radial distances our plasma angular velocity isconsiderably greater than that in the reference model, ly-ing much closer to corotation. These two regimes dividethe model naturally into “outer” and “inner” regions, respec-tively. The region within 30RJ is shaded grey to representthis division.

    Looking first at the “outer” region, beyond 30RJ , wecan see that although the plasma angular velocity is almostunchanged, each of the other parameters differs, at leastmarginally, from the reference model. These differencescan be traced to the differing behaviour of the neutral atmo-sphere, which rotates slightly more slowly than in the refer-ence model between 30RJ and 55RJ , and then slightly morequickly beyond 55RJ . This slight difference in the neutralrotation velocities is because our model does not generate ra-dially constant profiles ofK, and it is thus unsurprising thatwe do not exactly reproduce the reference model. However,the model does produce values ofK in the range 0.35–0.7throughout the whole outer region. This suggests that theassumptionK∼0.5 is reasonable in this part of the magneto-sphere. This is presumably because, as is clear from Fig.3,most of the magnetosphere beyond 20RJ maps to a very con-fined range of latitudes in the thermosphere, implying a fairlyhomogeneous behaviour. It must be emphasised, though, thatalthough a numerical value ofK∼0.5 is a good match in this

    region, this does not mean that the physical meaning ofKimplied byHuang and Hill(1989) is valid (see Sect.2.5).

    In the outer region our currents are distributed very slightlydifferently from the reference model. The auroral oval mapsto a slightly larger radius, and consequently the radial currentis concentrated very slightly more in the outer regions of themagnetosphere. However, these small changes do not seemparticularly significant, and are simply minor consequencesof K not being exactly equal to 0.5.

    The inner region is considerably more interesting. To aidour discussion of this region, Fig.10 shows the same pa-rameters as Fig.9, but across the range 0–40RJ only. Thisregion maps to latitudes just equatorward of the main auro-ral oval, which exhibit super-corotation via the mechanismdescribed in Paper 1. Super-corotation was not envisagedwhen the parameterK and the concept of effective conduc-tivity were introduced (Huang and Hill, 1989), so the con-sequences of this behaviour are somewhat peculiar, as antic-ipated in Sect.2. Firstly K becomes strongly negative, in-dicating that the neutral atmosphere is deviating from coro-tation in the opposite sense to the plasma, and to a muchgreater degree – i.e. it is super-corotating much more thanthe plasma is sub-corotating. There are two deep troughs inK in this region, each of which corresponds to peaks in theplasma angular velocity. As the plasma angular velocity ap-proaches corotation, the factor(J −M) in the formula forK (Eq. 7) approaches zero, while the left hand side is neg-ative. This generates strongly negative values ofK, whoseonly physical significance is that the plasma angular velocityis close to corotation. This structure inK is similar to thatcalculated for Saturn bySmith and Aylward(2008), since itis ultimately a consequence ofT behaving partly indepen-dently of the profile ofM , behaviour that the constantKmodel cannot account for.

    The effect of these negative values ofK on the effectiveconductivity is to enhance it significantly, since the factor(1−K) becomes strongly positive. This explains the twopeaks in Fig.9g. Since the effective conductivity representsthe effective ability of the thermosphere to enforce corota-tion, it is not surprising that when the thermosphere super-corotates it is able to enforce corotation more effectively.

    This interpretation is borne out by the behaviour of thesub-corotation parameterχ , which lies very close to that cal-culated from the reference model. This indicates that thesub-corotation of the plasma relative to the neutrals is rel-atively model-independent: introducing internal dynamicsof the thermosphere merely shifts the absolute value of theplasma angular velocity.

    The negativeK values also enhance our effective conduc-tivity in the inner region above that in the reference model,even though we assume a lower true background conductiv-ity. The effective conductivity is the empirically constrainedparameter in this region, since this almost directly determinesthe plasma rotation velocity (McNutt et al., 1979; Hill , 1980),

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    Fig. 10. Magnetospheric parameters for the baseline model in the inner region, in the same format as Fig.9.

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    and we are currently overestimating both. This suggests thatwe may require a still lower true background conductivity.

    To investigate this, the model has been run with thetrue background conductivity reduced by a factor of two to6P0=0.01375 mho, the results of which are shown, for theinner region, in Fig.11. This reduction in the conductivityhas an almost imperceptible effect on the super-corotatingthermospheri