Shocks, solitons and the plasmapause

Journal o/Atmospheric and TerrestrialPhysics, Vol. 38, pp. 1055-1060. _Pergamon Press, 1976. Printed in Nonhero Ireland

Shocks, solitons and the plasmapause H. KIKUCHI Nihon University,Collegeof Scienceand Technology, Tokyo and Nagoya University,Instituteof Plasma Physics,Nagoya, Japan Abstract--According to existing magnetospheric models, the plasmapause is thought to be the intersection of the solar wind induced convective flow with the Earth's corotational flow (AxFoRD, 1969). This fluid model is intuitive and acceptable, at least for the first sight. However, no further discussions are available as yet for the plasmapause formation, specifically from the plasma physics point of view. This paper, first of all, intends to indicate a mechanism for the formation of a global plasmasphere very briefly and then to focus our attention to the local formation of the equatorial nightside plasmapause which is supposed to provide the onset of its global formation. Figure 1 shows schematically an equatorial and a noon-midnight meridian cross section of the magnetosphere on the left and on the right panel, respectively. When the solar wind induced convective flow in the magnetosphere is impinging toward the Earth, the onset of the plasmapause formation is assumed to begin near a midnight equatorial region designated by a solid boundary line, where electrostatic shocks are thought to be formed nearly perpendicular to the up-stream convective flow lines, thereby causing the plasmapause. These plasma discontinuities or electrostatic shocks tend to be developed and linked to the dayside as well as polewards along a geomagnetic field line with the aid of both the Earth's corotational flow and the convective flow controlled geomagnetically, as indicated by the solid arrows, thus forming a global torus-like plasmasphere. The shaded region with a limited extent in longitude just inside the plasmapause indicates elongated field-aligued irregularities or ducts which are most likely formed in the recovery phase of a magnetic storm and have been locally observed by the O G O satellites (KIKUCHI,1971; KIKUCHI and TAYLOR,1972). The formation of these irregularities or ducts will be discussed as well on the basis of a model being proposed. We are now interested in the local formation of the nightside equatorial plasmapause for which an ideal but plausible model is presented, thus making a mathematical treatment possible. Figure 2 illustrates how one can reach a rather simple model where the plasmapause is considered essentially as an electrostatic laminar shock with a plane boundary. The basic idea is that the thermal plasma inside the plasmapause is of terrestrial origin, while the plasma beyond the plasmapause is basically of solar

wind origin. When the plasma produced by the UV ionization in the ionosphere diffuses upwards in the topside ionosphere, the electrons go ahead of the ions toward a plasmaspheric wall, i.e. the plasmapause, giving the wall a negative charge and leaving the interior close to the wall positive, thus forming an electrostatic double layer (sheath) at the plasmapause. On the other hand, the convection electric field drives a convective flow whose velocity is determined by a drift velocity t ) = ( E × B ) / B 2. Suppose the steady-state situation in which the convective flow is incident perpendicular to the plane boundary layer with a polarization. Then, the incident ions are decelerated and deflected westwards within the boundary layer by the polarization electric field, generating a thermal proton ring current parallel to the boundary layer. Indeed, the outward polarization electric field is balanced by the inward Lorentz force u × B . The convective flow electrons, however, are accelerated and bent around eastwards in the direction opposite to the proton ring current in such a way that ihe polarization electric force exerted on the electrons balances the outward Lorentz force on them. The orbits of the particles and the direction of current flows are illustrated schematically in Fig. 2. Consequently, a total net ring current flows westwards parallel to the boundary layer, thus making a plasmapause formation stable and self-consistent. Figure 2 also shows the distribution of excess terrestrial charges, resulting from the quicker d i t ~ sion of the terrestrial electrons, and the neutralizing distribution of convective flow charges. Indeed, the terrestrial plasma plays a primary role in plasmasphere-plasmapause formation. Thus the convective flow electrons tend to penetrate more deeply into the plasmapause layer than the ions,

1055

H. K~uc]ax

1056

because of the electric field produced by polarization charges of the background terrestrial plasma. It is thus seen that in the presence of the external electric fields, convection and polarization, the geomagnetic field produces the convective flow in the plasma trough and the ring current in the plasmapause layer but its direct influences in the plasmapause formation are considered to be rather secondary at least in its zeroth order plasma structure. In fact, satellite data have demonstrated no zeroth order magnetic field discontinuities across the plasmapause in contrast to the region of the magnetopause. Therefore, one can take a purely electrostatic model, where the plasma is composed of cold ions and hot electrons, as shown in Fig. 3. There are some indications from O G O data that the ion temperature is rather less than the electron temperature in the nightside plasmasphere. It should be noted that the problem is restricted to a low energy thermal plasma and does not take into account energetic plasmas, because their interactions are thought to be secondary. Further, it should be added that the effects of the Earth's corotational flow are ignored because they are so small in the local formation near the midnight hours. Starting with the fluid equations in a cold ion plasma and the Boltzmann equation with a collision term in a hot electron plasma together with Poisson's equation, it turns out that the model can eventually be described approximately in terms of a combined form of the KOaTZWEG and DE VRIES (1895) and the BURGERS (1940) equation:

equivalent to a nonlinear oscillator with damping. When the dissipation is large compared with the dispersion, i.e. for tz2>4ac (c =shock velocity in the stretched space-time frame), an ordinary monotonic shock is formed in a stationary state, as shown in Fig. 4 and corresponds to the prestorm or quiet plasma profile of the plasmapause. If one sets the electron temperature equal to T_ = 2.5 X 10 4 °K, K T _ = 2.32 eV, the ion sound velocity cs = 20.3 km/sec which is fairly close to the shock velocity or the up-stream convective flow velocity. This is just one check that the equatorial nightside plasmapause be a manifestation of electrostatic shocks. SOLAR

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Fig. 1. Cross-sectional view of the magnetosphere. (a) An equatorial cross section of the magnetosphere. The shaded region just inside the plasmapause indicates plasmapause-associated irregularities. Note that the solar wind induced convective flow is continued to a surfacelike thermal ring current at the plasmapause. (b) A noonmidnight meridian cross section of the magnetosphere. The shaded region just inside the plasmapause again indicates elongated field-aligned irregularities or ducts which are most likely formed in the recovery phase of a magnetic storm. Note that plasma discontinuities or electrostatic shocks formed near the magnetic equator tend to be developed and linked polewards along a geomagnetic field line.

1057

Shocks, solitons and the plasmapause

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With decreasing dispersion or a, the K - d V Burgers equation goes to the Burgers equation and the half thickness of the shock or the plasmapause 8 can be written as ~ = 2lx/c. With increasing shock velocity c, ~ becomes smaller and the shock front or the plasmapause is steepened, practically being displaced toward the Earth for a greater up-stream or convective flow velocity during the main phase of a magnetic storm. For a smaller & ix, i.e. dissipa-

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Fig. 2. A schematic illustration of the trajectories of convective flow ions and electrons incident normally on a plane plasmapause layer on the equatorial nightside.

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Fig. 3. Sketch of the variation of the potential ~k(x) through a monotonic shock transition or a plasmapause boundary layer. For a higher dispersion and/or lower dissipation, i.e. under a dispersion-dominated situation /~=< 4ac, the shock possesses a distinct oscillating structure behind the shock or inside the plasmapause because of negative plasma dispersion, as illustrated schematically in Fig. 5. Indeed, this happens to occur in the recovery phase of a magnetic storm.

The so-called plasmapause-associated irregularities or ducts may be a manifestation of this oscillating .structure with predominant dispersion. Nightside O G O plasma data have also revealed such plasma irregularities just inside the plasmapause. This is another evidence that the plasmapause may be a manifestation of electrostatic shocks. With further increasing dispersion and/or decreasing dissipation, the first waves of the structure behind the shock tend to solitons but with oscillating tail. In this case, the leading solitons may form the double or multiple plasmapause which has been observed occasionally during the post-storm recovery or on the dusk side. For a limit #---, 0, the K - d V - B u r g e r s equation tends to the K - d V equation and we eventually have some periodic solitons with a maximum amplitude 3c and a half-width A = 2 , f ~ , one of which is shown in Fig. 6.

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-2% . . . . . . . . . . . . MONOTONIC SHOCK (,~Z)4aC): Fig. 4. Schematic diagram of a monotonic shock corresponding to a prestorm or quiet plasma profile for the dissipation-dominant case. The lower panel illustrates the phase plane diagram with a nodal point. Note that q( O) = q ( x - cO = 4J(x - ct).

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SHOCK WITHOSCILLATORYSTRUCTURE (u2<4aC) Fig. 5. Schematic diagram of a dispersiveshock with oscillatory structure corresponding to a recovery profile for the dispersion-dominantcase. The lower panel shows the phase plane diagram with a spiral point.

Shocks, soKtons and the plasmapause

Figure 7 represents a comparison of the present theory with O G O satellite data for a specific magnetic storm event with a sequence of prestorm, main phase and poststorm recovery. Figure 7(a) shows a prestorm, midnight ion plasma profile obtained from an O G O inbound pass on 23 June 1966. This is a typical undisturbed profile with no significant structure besides the plasmapause which corresponds to a monotonic shock with a thickness of 2 g = 0 . 1 L shown in a dashed curve. Figure 7(b) shows a main phase midnight profile obtained from an O G O inbound pass on 25 June and a corresponding theoretical profile in a dashed curve. The shock front or the plasmapause is steepened considerably, compared with that of the prestorm plasmapause, having a plasmapause thickness 28 = 0.026L because of increasing convective flow velocity. The observed structured plasma in the plasmasphere indicates a growing phase of the storm rapidly varying in a non-stationary state, because the quasi-stationary main phase profile obtained from the same O G O - 3 during the July 1966, storm event does not exhibit structured profile, although the plasmapause is very much compressed and steepened. Figure 7(c) represents a recovery midnight profile

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Fig. 7. Comparison of the present model with OGO satellite data for the 1966 June storm event. The dashed curve refers to the present theory. (a) A prestorm midnight profile of thermal ion plasma and a monotonic shock. (b) A main phase midnight profile and a critical shock in the transition from a monotonic to an oscillating structure. (c) A recovery midnight profile and a soliton-like shock with oscillating structure.

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obtained from an inbound pass on 27 June with a corresponding theoretical profile. Both profiles possess oscillating wave-like structure inside the plasmapause, whose scale sizes of small irregularities are similar and are observed to be 0 . 1 - - 0 . 2 L . In summary, it can be stated that the present model is rather crude and may be restricted to a particular limit of the problem in an exact sense but nevertheless this model does explain plasma struc-

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tures for various magnetic phases. Consequently, the model appears to provide a mechanism for the formation of the plasmapause and its associated irregularities or ducts in the simplest form.

Acknowledgements--The author expresses his sincere thanks to Professor N. MARCUVITZwho led my interest to nonlinear wave phenomena while he was with New York University, Bronx, New York.

REFERENCES

AXFORD W. I. KIKUCHI H.

IOKUCHZ H. and TAYLOR H. A. KORTEWEG D. J. and DE WRIES G. BURGERS J. M.

1969 1971 1972 1895 1940

Rev. Geophys. 7, 421. Nature, Lond. 229, 79. J. geophys. Res. 77, 131. Phil. Mag. 39, 442. Proc. Acad. Sci. Amsterdam 43, 2.