Hydrostatic equilibrium and convective stability in the plasmasphere

Journal of Atmospheric and Solar-Terrestrial Physics 61 (1999) 867±878

Hydrostatic equilibrium and convective stability in the plasmasphere J.F. Lemaire* Center for Atmospheric and Space Sciences, Utah State University, Logan, Utah 84322-4405, USA Received 18 August 1998; received in revised form 24 June 1999; accepted 30 June 1999

Abstract It has been shown that no hydrostatic/barometric model for a rotating plasmasphere is able to ®t the equatorial electron density distributions observed by the ISEE satellite for L < 8, following prolonged periods of quite magnetic conditions. Indeed, it has been found that all these saturated plasma density pro®les are characterized by scale heights which are independent of L, while those corresponding to any hydrostatic/barometric models are always increasing functions of L. Furthermore, all calculated barometric equatorial density pro®les have a minimum value at an equatorial distance given by Lo ˆ …GME =O2 R3E †1=3 ˆ 6:6…OE =O†2=3 , depending on the value of the angular rotational speed, O: Lo R 6.6 for O/OE r 1. The position of this minimum is independent of the particle velocity distribution function (VDF) assumed (i.e. on the energy spectrum of the ions and electrons); the value of the minimum density depends on the temperature and density at the low altitude reference, as well as on the kappa index of the Lorentzian VDF assumed to calculate the barometric model. There is no evidence for such a minimum value in the observed equatorial density pro®les of ISEE. This is considered as a second indication that the plasmasphere is not in hydrostatic/barometric equilibrium. Next, it is shown that for L > Lo all barometric models are convectively unstable with respect to interchange and quasi-interchange. Therefore, it is not surprising that observed density pro®les do not ®t any of the pro®les corresponding to corotating barometric models. A ®nal indication that the plasmasphere is not in hydrostatic equilibrium but in a state of continuous hydrodynamic expansion comes from the fact that the total plasma pressure at the outer edge of the magnetosphere is lower than the kinetic pressure predicted by barometric models at large radial distances. As a consequence of this pressure unbalance, the plasmasphere is expected to expand continuously like the solar corona. Evidence for such a plasmaspheric wind had already been inferred earlier from the study of the ion density pro®les obtained from the OGO-5 mission. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction Several characteristic features of plasmaspheric density pro®les observed before, during and after sub-

* Corresponding author. Institut d'AeÂronomie Spatiale de Belgique, 3 Ave Circulaire, B-1180 Bruxelles, Belgium. Tel.: +32-2-373-81-21; fax: +32-2-373-84-23. E-mail address: [email protected] (J.F. Lemaire)

storm associated magnetic disturbances are illustrated in Fig. 1, adapted from Carpenter and Anderson (1992). Fig. 1 shows four typical equatorial electron density distributions observed with the Sweep Frequency Receiver (SFR), in the post-dawn to postnoon local time sectors. The density is determined along the ISEE orbit every 32 s; this rather high time resolution resolves small irregularities with scale sizes of 150 km. The orbital period of ISEE is 57 h, i.e. 2.375 days.

1364-6826/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 4 - 6 8 2 6 ( 9 9 ) 0 0 0 4 4 - 9

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2. Is the saturated plasmasphere in hydrostatic/ barometric equilibrium?

Fig. 1. A sequence of four ISEE equatorial electron density pro®les illustrating a recovery phase starting on day 215. A well-de®ned plasmapause is seen near L = 3. This sequence ends on days 217, 219 with an extended, saturated plasmasphere. Following a new disturbance on 12 August 1983 (day 224), a well-de®ned plasmapause was again formed near L = 3.3. These were outbound passes in the period 3±12 August 1983, in a typical case spanning several local time hours from postdawn to postnoon. The straight line corresponds to the average equatorial density pro®le observed after several days of low geomagnetic activity as determined by Carpenter and Anderson (1992).

The sequence shown in Fig. 1 begins on day 215 with a well developed plasmapause discontinuity near L = 3; it had been formed a day earlier during the magnetic substorm of 2 August 1983. The re®lling of the plasmatrough illustrated by the pro®le of day 217, lasted during four days when the geomagnetic activity level was reduced (Kp < 2+). The ®nal ``saturated'' plasmasphere extending smoothly beyond L = 8 is shown by the equatorial electron distribution of day 219. Following another magnetic disturbance which occurred on 12 August (day 224) when Kp increased to a maximum of 5, a new plasmapause ``knee'' developed near L = 3.2. Similar sequences of density pro®les have been observed earlier with satellites and determined from ground-based whistler observations; they have been reported by Chappell et al. (1970), Corcu€ et al. (1972) and Park (1973, 1974) (see also Carpenter and Park, 1973). For further reference, the characteristic features and the detailed evolution of the plasmasphere and plasmapause during magnetic substorms are described in Appendix A. In the next section hydrostatic/barometric density distributions are compared to the ``saturated'' density pro®le of day 219, which had followed a rather long series of days without signi®cant geomagnetic activity. The stability of these equatorial density pro®les with respect to interchange and quasi-interchange will be examined and discussed.

After a prolonged period of quiet conditions, ¯ux tubes corotating inside the plasmasphere ®ll up and eventually become saturated like that of day 219. It is generally postulated that, after several days of quiet conditions, hydrostatic/barometric equilibrium is eventually reached inside all corotating ¯ux tubes, and in the whole corotating plasmasphere. If this is the case, the saturated plasma density pro®le of day 219 should then ®t that corresponding to a protonosphere in hydrostatic/barometric equilibrium or di€usive equilibrium when there are several di€erent ionic species. For further reference, the properties of corotating protonospheres in hydrostatic/barometric are recalled in Appendix B where the e€ects of the centrifugal acceleration are especially emphasized. Two classes of barometric models are considered and described. In the ®rst the velocity distributions of the electrons are isotropic and Maxwellian in a corotating frame of reference: this class corresponds to the classical barometric model described in standard textbooks on planetary atmospheres. In this type of model the temperature is uniform, i.e. independent of the position in space. The plasma density (B1) decreases with altitude along the magnetic ®eld lines. The slope of the density distribution determines the density scale height which is given by Eq. (B4) in a Maxwellian barometric model. It can be shown that this ®eld-aligned density distribution reaches a minimum value in the equatorial plane for all corotating ¯ux tubes whose equatorial distance is smaller than Lc ˆ

2GME 3O2 R3E

!1=3

 2=3 OE ˆ 5:78 O

…1†

where ME, RE, OE are the Earth's mass, radius and angular rotational speed; O is the actual angular rotational speed of the plasmasphere, and G the gravitational constant. Indeed for any L0. Lc=5.78 for O=OE and Lc=2.78 when O/OE=3. Lc is the equatorial distance (in RE) where the Roche limit surface intersects the equatorial plane. This cylindrical symmetric surface is de®ned in Appendix B. First introduced by Lemaire (1974), it has also been called the ``Zero-ParallelForce'' surface (Lemaire, 1985) since it is the locus of points where the ®eld-aligned components of the gravitational acceleration and pseudo-centrifugal acceleration balance each other, i.e. where gk, the e€ective acceleration of a particle, vanishes. From Eq. (B5) it can be seen that the ®eld-aligned density scale height, Hn, becomes in®nitely large where

J.F. Lemaire / Journal of Atmospheric and Solar-Terrestrial Physics 61 (1999) 867±878

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Fig. 2. Distribution of plasma density in a rotating hydrostatic/barometric model of a protonosphere when the particle velocity distribution of the protons and electrons is an isotropic Maxwellian function. The temperature (4000 K) and H+ ion density (103 cmÿ3) at the reference altitude (2000 km) are assumed to be independent of latitude. Several dipole magnetic ®eld lines corresponding to L = 9, 6.6, 3 and 2.78 are shown. The dashed-dotted line corresponds to the meridional section of the Roche limit surface when the angular rotational speed is three times larger than that of the Earth.

a ¯ux tube crosses the Roche limit surface i.e. where gk=0. At this critical surface, an isolated plasma density enhancement (a plasma ``droplet'' or ``plasmoid'' with an excess density), at rest in the corotating frame of reference, tends to break into two pieces, one falling toward the Earth in the gravitational potential well, while the other will move up the ®eld line and ``fall'' into the equatorial potential well created by the centrifugal acceleration. Along the segment of a ®eld line beyond the Zero-Parallel-Force surface, the barometric plasma density distribution increases with altitude and has a maximum in the equatorial plane, instead of a minimum as in the case when L
barometric density distribution of electrons and H+ ions in a meridional plane in the case when O/OE=3. The velocity distribution function (VDF) of the particles is assumed to be Maxwellian with an uniform temperature of 4000 K, and a density of 103 ions/cm3 at a reference altitude of 2000 km. This exobase density as well as O are assumed to be independent of latitude. The meridional section of the Roche limit surface is shown by the dashed-dotted line in Fig. 2, where several ®eld lines are also shown including that corresponding to L ˆ Lc ˆ 2:78 which is tangent to this surface in the equatorial plane. The dashed curve in Fig. 3 shows the equatorial density pro®le for this rapidly rotating O=3OE protonosphere. It can be seen that this curve has a minimum value at an equatorial distance, Lo, which is a factor (3/2)1/3 larger than Lc; Lo=6.6 for O/OE=1 and Lo=3.17 when O/OE=3.

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Fig. 3. Equatorial density distribution for di€erent hydrostatic/barometric models. The dotted-dashed line corresponds to a non-rotating protonosphere when the particle velocity distribution is a Lorentzian function whose kappa index is equal to 4. The temperature (4000 K) and plasma density (3890 cmÿ3) at the reference level (: 1RE) are assumed to be independent of latitude. The dotted line is for a similar particle velocity distribution function (k=4), and identical low altitude boundary conditions, but an angular rotational speed equal to three times that of the Earth. The dashed curve corresponds to a rotating hydrostatic/barometric model similar to the previous one except that the particle velocity distribution function is assumed to be a classical isotropic Maxwellian function with the same boundary condition at the reference altitude. Note that both density distributions for a protonosphere have a minimum value at an equatorial distance Lo=3.17 when O=3OE; in the case of corotation, Lo=6.6. The solid line corresponds to the empirical model obtained by Carpenter and Anderson (1992) from ISEE observations for saturated equatorial density pro®les. Note that the latter has no minimum for L < 8; the observed density scale height is equal to 1.38 RE and is independent of L.

An alternative family of hydrostatic/barometric models has been constructed and is also illustrated in Fig. 3. In this new class and non-classic barometric models, the velocity distributions of particles are not Maxwellian but are isotropic and Lorentzian. The Lorentzian VDF is characterized by three parameters: the temperature, the density and an index kappa whose value can always be adjusted to ®t the energy spectrum of suprathermal particles to a power law whose index is precisely equal to kappa. Decreasing the value of the index k corresponds to increasing the hardness of the energy spectrum; when k=1 the Lorentzian VDF degenerates into the Maxwellian with the corresponding temperature and density.

The equatorial density pro®le corresponding to a barometric Lorentzian model for k=4 is shown by the dotted curve in Fig. 3; at the reference altitude, the temperature (4000 K) and electron and ion density (3900 cmÿ3) are all assumed to be independent of the latitude. This curve has a minimum at the same equatorial distance (Lo=3.17) as the dashed one: in both cases O/OE=3; the position of this minimum does not depend on the type of VDF, but only on the value of the ratio O/OE. The same conclusion applies for the ®eld-aligned density distributions. It can also be seen that the value of the minimum density is larger in the Lorentzian model than in the standard Maxwellian case. The density from the Lorentzian model is enhanced relative to that of the Maxwellian model up to about 8RE (cf. Fig. 3) since the population of suprathermal particles is relatively larger for a Lorentzian than Maxwellian VDF. As a matter of consequence there is a larger number of suprathermal particles that can reach higher altitudes. Beyond 8RE the Maxwellian barometric density becomes larger than that of the Lorentzian model. The dashed-dotted line corresponds to a non-rotating Lorentzian model for k=4. In this case Lo=1, and the density decreases asymptotically to a constant value for r 4 1 and L 4 1. Comparing this latest and dotted curves, it can be seen that the total mass of plasma accumulated above the ionosphere is signi®cantly larger in a rotating barometric model than in a non-rotating one. Increasing the angular rotational speed increases the capability to store more plasma in the outer regions of the plasmasphere and magnetosphere. The solid curve in Fig. 3 represents the equatorial density pro®le given by Eq. (A1). It corresponds to the empirical model proposed by Carpenter and Anderson (1992) to ®t saturated equatorial density pro®les like that of day 219 and observed by ISEE within the plasmasphere after several days of quiet geomagnetic conditions. Note that the values of the density at the reference level (L = 1) have been adjusted in the three previous barometric models to coincide precisely with 3890 cmÿ3, the value of the density of the empirical model of Carpenter and Anderson (1992) extrapolated at this same reference level. As indicated in Appendix A, the density scale height of the observed equatorial pro®le (solid curve) is independent of L: Hn ˆ 1:38RE : This typical feature of the observed pro®les is unexpectedly di€erent from the prediction of all three barometric models discussed above, and for which Hn is an increasing function of altitude. We consider this di€erence as a ®rst indication that the saturated plasmasphere density distribution is not in barometric equilibrium whatever the type of VDF (Maxwellian or Lorentzian) and whatever the angular rotational speed.

J.F. Lemaire / Journal of Atmospheric and Solar-Terrestrial Physics 61 (1999) 867±878

The slope of the observed density pro®le is smaller than in the theoretical models for low altitude i.e. L < 1.06. This is hard to see due to the linear density scale used in Fig. 3. This di€erence could be removed by increasing arbitrarily the reference level temperature from 4000 K to 5520 K, or else by assuming that the H+ density at the reference altitude increases as a function of the invariant latitude (i.e. with L ). However, neither of these two hypotheses is e€ectively supported by observations. The addition of a signi®cant concentration of heavier O+ ions and He+ at the reference altitude would not improve the ®t either. Indeed, on the contrary, the presence of a signi®cant fraction of these heavier ions would steepen the slope of the electron density pro®le at low altitude where O+ is the dominant species. This e€ect was ®rst outlined by Mange (1960), Bauer (1962) and Angerami and Thomas (1964). A quantitative (semi-empirical) multi-ionic model of the ion and electron density distributions of a corotating plasmasphere in barometric Maxwellian equilibrium has been produced by Rycroft and Alexander (1969), but neither their Winter-Night (WN) nor their Summer-Day (SD) equatorial density pro®les resemble that observed by Carpenter and Anderson (1992). The density slopes in both models WN and SD are steeper at low altitudes than that of the extrapolated density model of Carpenter and Anderson (1992). This additional di€erence between the observed equatorial density pro®le and those of rotating plasma in di€usive or barometric equilibrium is a second indication that the plasmasphere is not in hydrostatic/barometric equilibrium at any time, not even after a prolonged period of quiet geomagnetic conditions. The third indication supporting this conclusion comes from the fact that there is no evidence of a minimum value in any of the observed ``saturated'' density pro®les for L < 8 unlike in all rotating barometric models. To remove this extremum from any barometric model a di€erential angular velocity with an angular rotational speed decreasing to zero faster than L ÿ2 would have to be assumed. However, whistler observations and magnetospheric electric ®eld measurements in the magnetosphere do not support such an assumption, on the contrary, in the post-midnight local time sector the azimuthal component of the magnetospheric convection velocity increases faster than L (McIlwain, 1986). As a matter of consequence O(L ) is not a decreasing but, on the contrary, an increasing function of radial distance. Therefore, the absence of an extremum in the observed equatorial density pro®le, as a hypothetical consequence of some suitable di€erential angular rotational speed tending to zero as L 4 1, can also be ruled out. As a result of this series of arguments we are led to conclude that the

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saturated plasmasphere is never in hydrostatic/barometric equilibrium. In the next section additional arguments are o€ered along this line. Indeed, we show that all hydrostatic/ barometric models are convectively unstable with respect to pure interchange and quasi-interchange displacements. The de®nitions of interchange and quasiinterchange are given in the following section. 3. Convective stability criteria In a historical article Gold (1959) was the ®rst to apply the theory of convective or thermal instability in magnetospheric physics. In his article he assumed that magnetic stresses ``rigidize'' the interchanging ¯ux tubes and prevent them from bending. This class of plasma displacements has been called interchange motion; it has been intensively studied and applied in magnetospheric physics (Tserkovnikov, 1960a, 1960b; Sonnerup and Laird, 1963; Vinas and Madden, 1986; Huang et al., 1990; Richmond, 1973; Cheng, 1985; Southwood and Kivelson, 1989). Let us consider that the direction of the magnetic ®eld B(z ) is parallel to the horizontal x axis, and the gravitational acceleration is along the vertical z axis. All classes of displacements, u, can be Fourier analyzed into normal modes in which iux, iuy and uz are real functions of z multiplied by exp …ikx ‡ ily†: The pure interchange motions correspond to the special case where k = 0; the horizontal wavelength (2p/k ) is then in®nitely long and neighbouring ¯ux tubes interchange their positions without being bent. A plasma in hydrostatic equilibrium in a vertical gravitational ®eld, g, and horizontal magnetic ®eld, B(z ), is stable against ``pure interchange'' when the following inequality is satis®ed in the plasma ÿ

d ln n rg? >  H ÿ1 1 dz gp ‡ B 2 =mo

…2a†

or Hn
…2b†

where g is the adiabatic index: g=2 is the appropriate value for adiabatic compression of a collisionless plasma in directions transverse to B; r and p are, respectively, the mass density and kinetic pressure of the plasma; Hn is the actual density scale height de®ned by: Hn=ÿd ln m/ dz; H1 is the critical scale height for which pure interchange becomes unstable. This local stability criterium was introduced by Tserkovnikov (1960a, 1960b) and again later by Cheng (1985). A global, or integral, stability criterion is often used in the case when the plasma is con®ned along curved magnetic ¯ux tubes, instead of straight ones (e.g.

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Southwood and Kivelson, 1987). The limitations of such global stability criteria should, however, be pointed out when the value of g_ changes sign along ¯ux tubes as is the case at the Roche limit surface. Besides pure interchange there is an additional class of ``quasi-interchange'' displacements, u, which is generally overlooked in magnetospheric physics. Quasiinterchanges are displacements for which k $ 0; they have been studied by Newcomb (1961), and can be regarded as the interchange of ®nite segments of ®eld lines instead of the whole ¯ux tube. A plasma equilibrium is stable against ``quasi-interchange'' when ÿ

d ln n rg? >  H ÿ1 2 dz gp

…3a†

or Hn
…3b†

where H2 is the threshold density scale height for which quasi-interchange motions become unstable. Note that H2 RH1 ; the equality holds when B = 0, i.e. in the case of unmagnetized plasma. The inequalities (3a) or (3b) correspond to the convective stability criterion derived by Schwarzschild (1906) for non-conducting compressible ¯uids; they agree also with those derived by Chandrasekhar (1960) for magnetized plasmas. For a recent review and discussion of the stability criteria (2) and (3), see Lemaire (1999). In the following section we apply these stability criteria to the theoretical plasma distributions discussed in the previous section; the aim is to test the stability of the hydrostatic/barometric models vs pure interchange and quasi-interchange.

4. Application of the convective stability criteria to hydrostatic/barometric models The right hand sides of Eqs. (2a), (2b), (3a) and (3b) are functions of L which depend on the density and kinetic pressure in the model; they depend also on the distributions of g(r), B(r). For the sake of simplicity we assume that the magnetic ®eld is a centred dipole, i.e. that B…L†0Lÿ3 : The ®eld-aligned component of g(r) changes sign at the Roche Limit surface while g_, the transverse component of g, changes sign at the equatorial distance Lo. The density distribution, n…r†, is given by Eqs. (B1) or (B2) respectively, for a Maxwellian or a Lorentzian protonosphere in hydrostatic/barometric equilibrium. The distributions of the kinetic pressure, p(r), can be obtained from Eq. (B7) for any value of the kappa index. The solid line in Fig. 4 corresponds to the density slope, ÿd ln n=d L ˆ RE =Hn , for the Lorentzian model

Fig. 4. Equatorial distribution of the logarithmic density slope (left scale) and plasma density scale height (right hand side scale) for a corotating (O=OE) hydrostatic/barometric protonosphere, when the particle velocity distribution is an isotropic Lorentzian function for which the kappa index is equal to 4. Beyond Lo=6.6 the slope of density becomes positive in all corotating barometric models. The dotted line is the threshold density slope below which ¯ux tubes in a barometric model becomes convectively unstable with respect to interchange displacements (i.e. without bending of the ¯ux tubes). The dashed line is the threshold density below which ¯ux tubes in this same corotating barometric model become convectively unstable with respect to quasi-interchange displacements (i.e. when magnetic ®eld lines are bent as in longitudinal AlfveÁn waves). Note that both modes become convectively unstable beyond Lo in all rotating barometric models. The slope of the observed equatorial electron density, ÿd ln ne/ dL = 0.724, is out of the frame of this ®gure. It exceeds everywhere that of the theoretical hydrostatic/barometric model, for L>1.06.

k=4 in hydrostatic/barometric equilibrium. The dotted and dashed lines in Fig. 4 correspond, respectively, correspond to the threshold values ÿ…d ln n=d L†1 or RE =H1 (Eq. 2), and of …d ln n=d L†2 or RE =H2 (Eq. 3) for that same Lorentzian model (k=4) when O=OE (corotation). The slope of the observed electron density of Carpenter and Anderson (1992) is independent of L: ÿd ln ne =d L ˆ 0:724 (see Eq. A1). The corresponding density scale height, H ˆ 1:38RE , maps as a horizontal line out of the frame of Fig. 4. This implies that the observed density slope is larger than the theoretical density slope of our protonosphere model in hydrostatic equilibrium, except at very low altitude for L < 1.06 where H
J.F. Lemaire / Journal of Atmospheric and Solar-Terrestrial Physics 61 (1999) 867±878

tance any hydrostatic/barometric model has a negative density scale height and becomes convectively unstable with respect to pure interchange and even more critically with respect to quasi-interchange; indeed the inequalities (2b) and (3b) are not satis®ed for L > Lo : Similar results hold for the Maxwellian model, i.e. when k 4 1. This conclusion constitutes an additional argument indicating that saturated density pro®les like that of day 219 cannot be in hydrostatic/barometric equilibrium. Indeed such a model would not be convectively stable beyond L = 6.6 while the observed density pro®le appears to be stable for several days in a row. 5. The excess kinetic pressure at large distances There is one more argument supporting this conclusion. It is the analogue of that used by Parker (1958) to argue that the solar corona cannot be maintained in (stable) hydrostatic/barometric equilibrium. His physical argument was that the kinetic pressure at large radial distance in hydrostatic models of the corona dramatically exceeds that existing in the interstellar medium. The total plasma pressure at the surface of the magnetosphere is of the order of 10ÿ11 dyne cmÿ2. When extrapolated to L = 10 the kinetic pressure in a corotating barometric Maxwellian model is greater than 5  10ÿ10 dyne cmÿ2. The pressure unbalance at large radial distance is even higher in the case of a Lorentzian barometric model and/or when O/ OE>1. This implies that the plasmasphere cannot be in hydrostatic/barometric equilibrium, but must be in a state of hydrodynamic equilibrium like the solar corona. A slow (subsonic) continuous expansion of the plasmasphere would result in density scale heights closer to those exhibited by the observations of ISEE. Evidence for such an outward expansion of the plasmasphere has already been found by Lemaire and Schunk (1992, 1994) based on ion density measurements along the OGO-5 orbit (Chappell, 1972). This radial plasma ¯ow could be called ``plasmaspheric wind'' by analogy with the coronal expansion at equatorial heliospheric latitudes Ð the dense and slow solar wind. A kinetic model of such a plasmaspheric wind does not yet exist; it can be built as an extension of the exospheric polar wind models of Lemaire and Scherer (1974) or Pierrard and Lemaire (1996) for the cases of corotating-interchanging magnetic ¯ux tubes which are ®lled with plasma which has either an anisotropic Maxwellian VDF or Lorentzian VDF. 6. Conclusion Equatorial electron density pro®les determined from

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the Sweep Frequency Receiver of ISEE have been used and discussed. It has been found that the density scale height, Hn, is independent of L, for saturated plasma density pro®les observed following several days of quiet geomagnetic conditions. This feature can be considered as a ®rst indication that the saturated plasmasphere is not in hydrostatic/barometric equilibrium. Indeed, the equatorial density scale height is an increasing function of altitude for all hydrostatic/barometric Maxwellian or Lorentzian models, whether or not they are rotating faster than the Earth. Furthermore, the observed density slope extrapolated at very small L (L < 1.06) is less steep than those predicted by hydrostatic/barometric model unless extreme exobase temperatures are assumed at the low altitude reference level. It has also been pointed out that any of these theoretical models has a minimum density at an equatorial distance Lo which is equal to 6.6 (geostationary orbit) for corotation, and at a smaller radial distance when the angular rotational speed exceeds that of the Earth. Since there is no evidence for such a minimum in the observations, it is concluded that the quiet time plasmasphere does not settle in a state of hydrostatic/barometric equilibrium whatever its angular rotational speed may be. After having recalled the di€erence between ``pure interchange'' (Tserkovnikov, 1960a) and ``quasi-interchange'' stability (Newcomb, 1961), it has been shown that, for L > Lo , all hydrostatic/barometric models of a corotating plasmasphere are convectively unstable with respect to interchange and even more drastically with respect to quasi-interchange. Therefore, it should not be surprising that the observed equatorial density pro®les do not ®t those predicted by corotating hydrostatic/barometric models, since the latter are anyway convectively unstable. Finally, it is shown that, at large radial distances, the kinetic pressure in hydrostatic/ barometric models exceeds the total plasma pressure in the outermost region of the magnetosphere. This pressure unbalance is larger for a Lorentzian than for a Maxwellian model. Therefore, since the actual total pressure at the frontier of the magnetosphere is so low, hydrostatic/ barometric models cannot be in mechanical equilibrium; they must expand continuously, like the equatorial solar corona which expands radially into the interplanetary medium for a similar reason (Parker, 1958). A stationary radial expansion with a subsonic bulk speed in the dense regions of the plasmasphere had already been inferred from earlier OGO-5 ion density pro®les (Lemaire and Schunk, 1992, 1994). It was then given the name of ``plasmaspheric wind''. The same conclusion is reached here from the analysis of the electron density pro®les determined by the SFR experiment onboard of ISEE.

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Acknowledgements I wish to acknowledge useful comments by N. Meyer-Vernet and M.G. Kivelson, as well as V. Pierrard, K. Issautier and J.M. Vandenberghe for their help in preparing some of the ®gures. The useful remarks and suggestions of both referees have been appreciated by the author; they led to a reorganisation of this paper and of the forthcoming one. R.W. Schunk, CASS, Utah State University, Logan, is acknowledged for his kind hospitality, when I visited the Center for Atmospheric and Space Sciences and worked as a CoInvestigator of his ``Theory project'', NASA Grant NAG5-1484.

Appendix A Characteristic features of density pro®les in the plasmasphere before and after a magnetic substorm Fig. 1 shows radial distributions of the electron density observed by the ISEE sweep frequency receiver between L = 2 and 10 in the equatorial region and in the local time interval 00-15 MLT. A sequence of four outbound ISEE passes have been selected in the period 3±12 August 1983 following the substorm of day 214 until that which occurred on day 224. In this appendix we outline the characteristic features of the plasmasphere that are illustrated in Fig. 1. Feature 1 Ð the sharpness of the plasmapause immediately after substorm The ®rst characteristic feature evident in Fig. 1 is the steepness of the newly formed plasmapause or ``knee''. When the magnetic substorm developed on day 214 (2 August 1983), the equatorial electron density dropped suddenly by a factor of 25, beyond L = 3.5. The cold plasma distribution drops over a distance of 0.2 RE, or 01200 km. Several other examples of steep plasmapause pro®les from ISEE observations were presented in Fig. 8 of Carpenter and Anderson (1992). To quantify this steepness a ``scale width parameter'', Dpp, has been introduced by these authors. This is the distance in L (or in km) over which the electron density varies by a factor of 10: Dpp ˆ DLpp=D log 10 ne : All values of Dpp fell in the range 250±1250 km for a sample of 62 well de®ned plasmapause pro®les examined by Carpenter and Anderson (1992): thus, Dpp<0:2RE : This scale width parameter does not appear to vary strongly with Lpp, and hence with the strength of the magnetic storm, i.e. with the intensity of the convection electric ®eld or velocity enhancements associated with the formation of a new

plasmapause, nor with their depth of penetration in the night side magnetosphere. Furthermore, the ISEE observations con®rm that nightside plasmapauses are generally steeper than those observed on the dayside (see also Gringauz and Bezrukikh, 1976; Horwitz, 1983; Nagai et al., 1985). The same ISEE data show that the density jump varies from an average factor of 15, in the post-midnight sector, to less than a factor of 10 in the afternoon sector. The rapid formation of a steep ramp, like those of days 215 and 224, can hardly be explained by a gradual steepening of a pre-existing density slope in the nightside sector as considered by Rasmussen (1992). According to this scenario an extended pre-existing density pro®le would be convected/pushed anti-sunward, and would be compressed by the enhanced dawn±dusk electric ®eld component. The initially smooth and extended density slope should upheave as a consequence of the compression of the nightside plasmasphere. Feature 2 Ð the folds in the density slopes The well developed folds/breaks observed in the slope of the density pro®les at the inner and outer edges of newly formed plasmapause regions, identi®ed in the passes of days 215 and 224, also militate against this early scenario for the formation new plasmapauses. Indeed, the earthward shift and compression of a pre-existing smooth density pro®le like that of day 219 can hardly produce the observed breaks in the plasma density gradients. Feature 3 Ð the reduction in plasmaspheric density after substorm onset Further evidence against this ideal-MHD model for the formation of the plasmapause results from the decrease of the plasma density within the plasmasphere after a new plasmapause has formed: the density level inside the plasmasphere is lower on day 224 than on day 219. Whistler observations also indicate that the night-time density in the outer plasmasphere is generally reduced immediately after the substorm, i.e. after the formation of a new plasmapause. This reduction can exceed a factor 3 according to Park and Carpenter (1970) (see also Carpenter and Lemaire, 1997). According to Rasmussen's scenario, however, sunward convection of a pre-existing smooth density pro®le should lead to compression of the plasma, i.e. result in an increase instead of the observed decrease inside the plasmasphere. Feature 4 Ð the gradual plasmasphere re®lling Another characteristic feature shown in Fig. 1 is the gradual evolution of the equatorial density pro®le from day 217 to day 219 during the recovery phase. Snapshots of the plasmasphere densities are shown

J.F. Lemaire / Journal of Atmospheric and Solar-Terrestrial Physics 61 (1999) 867±878

during its re®lling over a period of four quiet days. The ramp formed in the inner magnetosphere on day 214 gradually moves radially outward during the following quiet period, and at the same time it becomes less steep. These unique in-situ ISEE observations nicely con®rm ground-based whistler results of the day-to-day re®lling of plasmaspheric ¯ux tubes during extended periods of low magnetic activity following isolated substorms (see Corcu€ et al., 1972; Park, 1973, 1974; Carpenter and Park, 1973). In all cases reported in the literature saturation of magnetic ¯ux tubes is achieved in about two days for L < 4, and less than four days for L < 7, provided that the geomagnetic activity remained low for a suciently long time before the next substorm occurs. Feature 5 Ð the absence of a plasmapause ``knee'' in saturated plasmasphere density pro®les. There is another characteristic feature that is clearly illustrated in Fig. 1: the ``saturated'' density pro®le of day 219 in the postdawn sector does not show evidence of a plasmapause ``knee'' within L < 8. If the plasmapause would coincide with the ``last closed equipotential (LCE)'' surface of the quiet time convection electric ®eld, the ``stagnation point'' of the ``teardrop'' E-®eld model would have to be located way beyond L = 8 in the dusk local time sector on day 219. Indeed, the LCE of the ``teardrop'' electric ®eld model is located at L = 5 in the pre-noon sector, when this mathematical singularity is at L = 8 in the dusk sector (Grebowsky, 1970; Doe et al., 1992). Feature 6 Ð the exponential density pro®le of the saturated plasmasphere Fig. 1 shows that the saturated density pro®le of day 219 varies exponentially with L as ne …L†0eÿ0:87L : Carpenter and Anderson (1992) show another plot with 11 other similar ISEE ``saturated'' density pro®les all extending beyond L = 5 on the dayside. All these pro®les were observed for very quiet magnetic conditions and appear to be well approximated by a linear relation between log10ne and L. The best least squares ®t to these data is given by ne …L† ˆ 10…ÿ0:3145L‡3:9043† ˆ 8022eÿ0:724L

…A1†

if its density would be in hydrostatic/barometric equilibrium. Furthermore the density scale height, Hn ˆ kT=mg…L†, would then be an increasing function of L. The mathematical expressions for the equatorial density pro®les in corotating magnetospheres/protonospheres are recalled in Appendix B for di€erent types of electron and ion velocity distributions functions. In all cases the slope of the hydrostatic density pro®le between L = 3 and L = 8 is less steep than that determined from whistler observations or from the satellite observations like those illustrated in Fig. 1. Appendix B Hydrostatic/barometric plasma distributions in rotating magnetospheres Introduction The model of hydrostatic or barometric equilibrium is the simplest and was the ®rst one used to describe planetary and stellar atmospheres and ionospheres. Hydrostatic equilibrium distribution of H+ and O+ ions in the Earth's ionosphere was studied by Mange (1960) who draws attention to the fundamental importance of the Pannekoek±Rosseland ambipolar electric ®eld which is required to balance the density of electrons and ions, i.e. to maintain local quasi-neutrality of the plasma in the ionosphere and plasmasphere (Pannekoek, 1922; Rosseland, 1924). The e€ects of corotation, of non-uniform temperature distributions, of anisotropic and non-Maxwellian velocity distribution functions on the hydrostatic equilibrium density distribution in a rotating protonosphere and exosphere were subsequently developed by Bauer (1962, 1963), Angerami and Thomas (1964), Eviatar et al. (1964), Gledhill (1967), Lemaire (1974, 1976), Hill and Michel (1976), Huang and Birmingham (1992), and Pierrard and Lemaire (1996). Maxwellian & Lorentzian distributions When a plasma is in hydrostatic/barometric equilibrium, the ®eld-aligned electron density distribution is given by n…L, l† ˆ no …L, lo † exp‰ÿmH C…L, l†=

3

where ne is expressed in electrons/cm . The slope of this empirical density pro®le is d ln ne =d L ˆ ÿ0:724; it corresponds to a density scale height H ˆ 1:38RE : Park et al. (1978) deduced a more gradual value …d ln ne =d L ˆ ÿ0:826† from a month of Whistler data, but it was averaged over a wide range of magnetic activity levels. This slope of density corresponds to a slightly smaller value of the scale height: H ˆ 1:15RE : Much smaller slopes are expected in a plasmasphere

875

k…To, e …L, lo † ‡ To, p …L, lo ††Š

…B1†

or n…L, l† ˆ  no …L, lo † 1 ‡

mH C…L, l† kk…To, e …L, lo † ‡ To, p …L, lo ††

ÿk‡1=2 (B2)

respectively, when the velocity distribution of the elec-

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J.F. Lemaire / Journal of Atmospheric and Solar-Terrestrial Physics 61 (1999) 867±878

trons is isotropic and Maxwellian or Lorentzian. In these expressions no …L, lo † and To …L, lo † are, respectively, the density and temperature of the electrons at a low altitude reference level ho, and at the dipole latitude, lo. These equations give the densities along a geomagnetic ®eld line crossing the equatorial plane at the radial distance LRE; k is the kappa index of the generalised Lorentzian velocity distribution function (VDF) (see Pierrard and Lemaire, 1996). C…r† ˆ f …fg …r† ÿ 12 O2 r2 cos 2 l† ‡ constant

…B3†

is the potential of the electrons along the ®eld line at the dipole latitude l; fg(r) is the gravitational potential; O is the angular velocity of the exosphere; C(L,lo) is set equal to zero at the low altitude reference level. When the value of k equals 1 the Lorentzian velocity distribution tends to the Maxwellian one, and the barometric density given by Eq. (B2) tends toward that given by Eq. (B1). Similar expressions are applicable for the ion densities. More complicated formulae are obtained in the case of hydrostatic/exospheric models when the VDFs are truncated so that a class of trapped particles is missing in their pitch angle distributions (see Lemaire, 1976; Pierrard and Lemaire, 1996). In hydrostatic/exospheric models the density decreases much faster with r and L than in hydrostatic/barometric models (Lemaire and Scherer, 1974). Eq. (B1) has been generalized by Huang and Birmingham (1992) in the special case when the velocity distributions of the electrons and protons are anisotropic and bi-Maxwellian, i.e. when g=Tok/To_ is not equal to 1 as was assumed to obtain (B1). Density scale height The ®eld-aligned density distributions decrease with altitude. The density scale height, Hn ˆ ÿ…d ln n=d h†ÿ1 , is determined by the gravitational ®eld and the ambipolar electric ®eld. When the protonosphere is in barometric equilibrium and when the VDF is Maxwellian and isothermal, the density scale height can be written as Hn ˆ k…To, e ‡ To, ion †=…mp ‡ me † j g6 j

…B4†

where gk is the combined gravitational and centrifugal acceleration. When the electron and ion velocity distributions are Lorentzian and if the kappa index is the same for the electron and ion VDFs, k=ke=kion, the corresponding density scale height is given by

  3 k…Te ‡ Tion † kÿ 2  Hn ˆ  1 kÿ …mp ‡ me † j gjj j 2

…B5†

where the electron and ion temperatures are functions of L given below by Eq. (B8). It can be seen that in a Lorentzian protonosphere the density scale height is a factor (kÿ3/2)/(kÿ1/2) larger than in the Maxwellian case corresponding to k=1. The Roche limit At high altitude the centrifugal e€ect signi®cantly reduces the e€ective gravitational acceleration. There may be a point where the ®eld-aligned components of the gravitational and centrifugal accelerations balance each other. This point is part of what has been called the Roche limit surface (Lemaire, 1974) or the ``ZeroParallel Force'' surface (Lemaire, 1985). At this surface a corotating droplet of plasma tends to split into two parts, one falling in the gravitational potential well, while the other tends to move away in the opposite direction due to the centrifugal acceleration. Within this surface br…mC† ˆ 0

…B6†

i.e. the gradient of the total potential energy of the particles has a vanishing component parallel to the magnetic ®eld direction b. Both density scale heights (B4) and (B5) become in®nitely large at the Roche limit surface where gk=0 and vHkCv=0. As a consequence, the ®eld-aligned barometric density distributions (B1) and (B2) have a minimum value at the places where the magnetic ®eld lines traverse the Roche limit surface. Beyond that surface the total potential C(L,l ) decreases with altitude, and the ®eldaligned density distribution has a maximum value in the equatorial plane. Pressure and temperature distributions When the VDF is isotropic and Lorentzian the pressure tensor is also isotropic and given by p…L, l† ˆ no …L, lo †kTo

 ÿk‡3=2 k mH C…L, l† 1‡ : k ÿ 3=2 kk…To ‡ To, p †

…B7†

Pierrard and Lemaire (1996) have shown that the temperature distribution in such a protonosphere model is given by:   k mH C…L, l† T…L, l† ˆ To …L, lo † 1‡ …B8† k ÿ 3=2 kk…To ‡ To, p †

J.F. Lemaire / Journal of Atmospheric and Solar-Terrestrial Physics 61 (1999) 867±878

When C(L,l ) is an increasing function of altitude, the temperature given by Eq. (B8) increases as a function of altitude along the magnetic ®eld lines, for any value of k between 3/2 and 1. This interesting property of the Lorentzian VDF has been pointed out by Pierrard and Lemaire (1996) to explain that the plasma temperature at high altitude in the plasmasphere could indeed be larger than at lower altitudes in the ionosphere when the VDF has an enhanced tail of suprathermal particles. Note also that, when k41, T…L, l†4To …L, lo †: the plasma then becomes isothermal. More complicated expressions for the components of the pressure tensor are obtained in the case of hydrostatic/exospheric models when the VDFs are truncated so that the class of trapped particles is missing in the pitch angle distributions (see Lemaire, 1976; Pierrard and Lemaire, 1996). In this case the parallel pressure and temperature are larger than the perpendicular ones.

References Angerami, J.J., Thomas, J.O., 1964. Studies of planetary atmospheres. 1. The distribution of electrons and ions in the earth's exosphere. Journal of Geophysical Research 69, 4537±4560. Bauer, S.J., 1962. On the structure of the topside ionosphere. Journal of Atmospheric Science 19, 276±278. Carpenter, D.L., Anderson, R.R., 1992. An ISEE/Whistler model of equatorial electron density in the magnetosphere. Journal of Geophysical Research 97, 1097±1108. Carpenter, D.L., Lemaire, J., 1997. Erosion and recovery of the plasmosphere in the plasmapause region. Space Sciences Reviews 80, 153±179. Carpenter, D.L., Park, C.G., 1973. On what ionosphere workers should know about the plasmapause±plasmasphere. Reviews of Geophysics and Space Physics 11, 133± 154. Chandrasekhar, S., 1960. In: Plasma Physics. University of Chicago Press, Chicago, IL, pp. 85±96. Chappell, C.R., 1972. Recent satellite measurements of the morphology and dynamics of the plasmasphere. Reviews of Geophysics and Space Physics 10, 951±979. Chappell, C.R., Harris, K.K., Sharp, G.W., 1970. A study of the in¯uence of magnetic activity on the location of the plasmapause as measured by OGO5. Journal of Geophysical Research 75, 50±56. Cheng, A.F., 1985. Magnetospheric interchange instability. Journal of Geophysical Research 90, 9900±9904. Corcu€, P., Corcu€, Y., Carpenter, D.L., Chappell, C.R., Vigneron, J., Kleimenova, N., 1972. La plasmaspheÁre en peÂriode de recouvrement magneÂtique, Etude combineÂe des donneÂes des satellites OGO 4, OGO 5 et des si‚ements rec° us au sol. Annales Geophysicae 28, 679±695. Doe, R.A., Moldwin, M.B., Mendillo, M., 1992. Plasmapause morphology determined from an empirical ionospheric

877

convection model. Journal of Geophysical Research 97, 1151±1156. Eviatar, A., Lenchek, A.M., Singer, S.F., 1964. Distribution of density in an ion-exosphere of a nonrotating planet. Physics of Fluids 7, 1775±1779. Gledhill, J.A., 1967. The structure of Jupiter's magnetosphere and the e€ect of Io on its decametric radio emission. NASA Techn Memo, TMX-555980. Gold, T., 1959. Motions in the magnetosphere of the Earth. Journal of Geophysical Research 64, 1219±1224. Grebowsky, J.M., 1970. Model study of plasmapause motion. Journal of Geophysical Research 75, 4329±4333. Gringauz, K.I., Bezrukikh, V.V., 1976. Asymmetry of the Earth's Plasmasphere in the direction noon±midnight from PROGNOZ-1 and PROGNOZ-2 data. Journal of Atmospheric and Solar-Terrestrial Physics 38, 1071±1076. Hill, T.W., Michel, F.C., 1976. Heavy ions from the Galilean satellites and the centrifugal distortion of the Jovian magnetosphere. Journal of Geophysical Research 81, 4561± 4565. Horwitz, J.L., 1983. Plasmapause di€usion. Journal of Geophysical Research 88, 4950±4952. Huang, T.S., Birmingham, T.J., 1992. The polarization electric ®eld and its e€ects in an anisotropic rotating magnetospheric plasma. Journal of Geophysical Research 97, 1511±1519. Huang, T.S., Wolf, R.A., Hill, T.W., 1990. Interchange instability of the Earth's plasmapause. Journal of Geophysical Research 95, 17,187±17,198. Lemaire, J., 1974. The ``Roche-limit'' of ionospheric plasma and the formation of the plasmapause. Planetary and Space Science 22, 757±766. Lemaire, J., 1976. Rotating ion-exospheres. Planetary and Space Science 24, 975±985. Lemaire, J., 1985. Frontiers of the Plasmasphere (Theoretical Aspects). 264 pp. Universite Catholique de Louvain, Faculte des Sciences, Editions Cabay, Louvain-la-Neuve. Lemaire, J., 1999. Convective instability in the outer plasmasphere (submitted). Lemaire, J., Scherer, M., 1974. Exospheric models of the topside ionosphere. Space Science Reviews 15, 591±640. Lemaire, J., Schunk, R.W., 1992. Plasmaspheric wind. Journal of Atmospheric and Solar-Terrestrial Physics 54, 467±477. Lemaire, J., Schunk, R.W., 1994. Plasmaspheric convection with non-closed streamlines. Journal of Atmospheric and Solar-Terrestrial Physics 56, 1629±1633. McIlwain, C.E., 1986. A Kp dependent equatorial electric ®eld model. Advances in Space Research 6 (3), 187±197. Mange, P., 1960. The distribution of minor ions in electrostatic equilibrium in the high atmosphere. Journal of Geophysical Research 65, 3833±3834. Nagai, T., Horwitz, J.L., Anderson, R.R., Chappell, C.R., 1985. Structure of the plasmapause from ISEE 1 low energy ion and plasma wave observations. Journal of Geophysical Research 90, 6622±6626. Newcomb, W.A., 1961. Convective instability induced by gravity in a plasma with a frozen-in magnetic ®eld. Physics of Fluids 4, 391±396. Pannekoek, A., 1922. Ionization in stellar atmospheres. Bulletin Astronomical Institute Mathematics 1, 107±118.

878

J.F. Lemaire / Journal of Atmospheric and Solar-Terrestrial Physics 61 (1999) 867±878

Park, C.G., 1973. Whistler observations of the depletion of the plasmasphere during a magnetospheric substorm. Journal of Geophysical Research 78, 672±683. Park, C.G., 1974. A morphological study of substorm-associated disturbances in the ionosphere. Journal of Geophysical Research 79, 2821±2827. Park, C.G., Carpenter, D.L., 1970. Whistler evidence of largescale electron-density irregularities in the plasmasphere. Journal of Geophysical Research 75, 3825±3836. Park, C.G., Carpenter, D.L., Wiggin, D.B., 1978. Electron density in the plasmasphere: whistler data on the solar cycle, annual and diurnal variations. Journal of Geophysical Research 83, 3137±3144. Parker, E.N., 1958. Dynamics of the interplanetary gas and magnetic ®eld. Astrophysical Journal 128, 664±676. Pierrard, V., Lemaire, J., 1996. Lorentzian ion exosphere model. Journal of Geophysical Research 101, 7923±7934. Rasmussen, C.E., 1992. The plasmasphere. In: Chang, T., Jasperse, J.R. (Eds.), Physics of Space Plasmas (1992). Scienti®c Publishers Inc, Cambridge, Massachusetts, pp. 279±302. Richmond, A.D., 1973. Self-induced motions of thermal plasma in the magnetosphere and the stability of the plasmapause. Radio Science 8, 1019±1027. Rosseland, S., 1924. Electrical state of a star. Monthly Notices Royal Astronomical Society 84, 720±728.

Rycroft, M.J., Alexander, P.D., 1969. Model of hydrogen and helium ion concentrations in the plasmasphere. In: Proceedings of the COSPAR Meeting, Prague, Czechoslovakia [Paper A.3.11]. Schwarzschild, K., 1906. Ueber das Gleichgewicht der SonnenatmosphaÈre. Nachrichten K. Gesellschaft der Wissenschaften zu GoÈttingen, 41±53. Sonnerup, B.U.O., Laird, M.J., 1963. On magnetospheric interchange instability. Journal of Geophysical Research 68, 131±139. Southwood, D.J., Kivelson, M.G., 1987. Magnetospheric interchange instability. Journal of Geophysical Research 92, 109±116. Southwood, D.J., Kivelson, M.G., 1989. Magnetospheric interchange motions. Journal of Geophysical Research 94, 299±308. Tserkovnikov, Yu.A., 1960a. Convective instability of a rare®ed plasma. Division of theoretical physics, Mathematical Institute of V.A. Steklov, Academy of Sciences of USSR, Moscow, Report No. T-2. Tserkovnikov, Yu.A., 1960b. The question of convectional instability of a plasma. Soviet Phys. Doklady 5, 87±90. Vinas, A.F., Madden, T.R., 1986. Shear ¯ow-ballooning instability as a possible mechanism for hydromagnetic ¯uctuations. Journal of Geophysical Research 91, 1519± 1528.