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Field-aligned currents and auroral acceleration by non-linear MHD waves
Planet. Spme SC,., Vol. 32. No Printed in Great Britain
FIELD-ALIGNED A~CELERATrO~
Institute
0032&0633/8453.00 +O.oO Pergamon Press Ltd.
12, pp. 1497 -1503. 1984
CURRENTS AND AURORAL BY NON-LINEAR MHD WAVES
M. S. TIWARP and G. ROSTOKER of Earth and Planetary Physics, and Department of Physics, Edmonton, Alberta, Canada T6G 251
(Received
13 February
University
of Alberta,
1984)
Abstract-This paper examines the consequences of the assumption that substorm-associated growth of magnetosphere-ionosphere current systems is triggered by the incidence, on the ionosphere, of a large amplitude Alfven wave generated in the distant magnetotail. It is pointed out that there is a large body of evidence suggesting that, in the acceleration region near 1 R,, one is likely to find a major discontinuity in mass density. Following the approach of Cohen and Kulsrud (1974) who studied the steepening of large amplitude hydromagnetic waves into shocks, we demonstrate that the character of the background plasma and magnetic field in the aurora1 acceleration region near 1 R, can be idea1 for the generation of MHD shocks and that these shocks can lead to the acceleration of ions and electrons as reported by investigators using S3-3 satellite data.
1. INTRODUCTION
In the recent
past various theories have been proposed to explore the question of field-aligned currents and acceleration of charge particles precipitating in the aurora1 ionosphere, but none have been completely successful in explaining the various phenomena observed simultaneously by rocket and satellite and ground-based instruments. Amongst the various theories, two major classes have been proposed to describe some of the phenomena. The first are the timeindependent models based on the single particle approach (Knight, 1973; Lemaire and Scherer, 1974a,b; Lyons et al., 1979; Chiu and Cornwall, 1980; Fridman and Lemaire, 1980; Chiu et al., 1981 and Lyons, 1980, I 98 1a,b). These models explain large-scale phenomena but have not yet been extended for microturbulence generally observed in the acceleration region. Drawbacks in the applicability of the doubie layer/electrostatic shock model (Block, i 972 ; Swift, 1975 ; Hudson et al., 1983) and anomalous resistivity (Papadopoulos, 1977 ; Dum, 1981) are indicated by Lyons (1981b) and the applicability of the single particle theory is emphasized. The second kind of approach considers transient phenomena and explains the aurora1 particle acceleration in terms of linear AlfvCn waves propagating paraIle1 to or obliquely with respect to the ambient magnetic field (Scholer, 1970; Mallinckrodt and Carlson, 1978; Goertz and Boswell, 1979; Miura and Sato, 1980 ; Lysak and Carlson, 198 1; Lysak and Dum, 1983). These theories mainly describe the field-aligned * Department India.
of Physics,
Sagar
University,
Sagar (M.P.),
1497
current and parallelelectricfield associated with kinetic AlfvCn waves which have originated in some magnetospheric generator region via impressed perpendicular and time varying electric fields. The field-aligned current associated with kinetic Alfv& waves is considered to be sufficient for the excitation of electrostaticion cyclotron waves and hence anomalous resistivity. While this theory is growing in popularity, it is still incomplete in dealing with the non-linear phenomena for large amplitude waves. It does not locate the region where the enhancement ofthe currents actually should take place. This depends upon the background plasma and the parallel potential drop may occur anywhere between the magnetosphere and the ionosphere. Recent observations, however, have defined the region around 1 R, and above where the acceleration actually takes place. For a detailed analysis of the existing theories the reader is referred to reviews by Shawhan et al. (1978), Mozer et al. (1980), Kan (1982) and articles in Akasofu and Kan (1981). In the recent past several observations are reported which are not explained by these existing theories. Electron and ion acceleration are sometimes observed simultaneously and in the same direction as advocated by Heikkila (1981) and experimentally reported by Bryantetal.(1977),GurnettandFrank(1973),Krimigis and Sarris (1978) and Aggson et al. (1983) near the aurora1 ionosphere and in the magnetotail regions. Heppner et al. (198 I), for their Cameo Barium release experiments, have reported both accelerating and decelerating parallel electric fields above 1400 km exhibiting a periodicity of 334 min and, because of an apparent wave behaviour in the PC 5 band, an association with hydromagnetic waves was suggested.
1498
M. S.
TIWARI and G. ROSTOKER
However, they exclude the hydromagnetic wave theory because of high-p medium over the polar cap. Acceleration of electrons to 6 keV above 15,800 km has also been reported by Heppner et al. (1981). Recently Aggson et al. (1983) reported a strong inductive electric field observed on the ISEE- satellite at substorm time around midnight in the L = 7.5 region of the near Earth magnetospheric tail and used that observation in proposing an explanation of the observed correlation between electric and magnetic fields by the AlfvCn or fast magnetosonic mode. Further, they suggested a theory to explain the relative effectiveness of these fields in accelerating particles for both the plasma population and individual particles. They also indicated the importance of using Faraday’s law for these observed electric fields instead of using solely the divergence of the field as used by Mozer (1981) to explain the electrostatic shock for a similar event. It is generally believed that field-aligned currents are driven by Alfvtn waves. Magnetic fields detected by the satellites with magnetometers on board have reported large magnetic field perturbations of several hundred nT (Zmuda and Armstrong, 1974 ; Iijima and Potemra, 1976a,b; Shuman et al., 1981; Burke et al., 1983) which are comparable to local ambient magnetic fields. Under these circumstances the analysis of field-aligned current using linear Alfven waves may be inappropriate and the existing theory should be extended to the nonlinear regime. The wave-associated centrifugal force -p(V*V)V and the wave-associated Lorentz force J x B/c (p is the mass density, V is the velocity, J is the current density, B is the magnetic field and c is the speed of light) may play a significant role in determining the various properties ofthe system. Burke et al. (1983) have also indicated the significance oflarge amplitude Alfven waves in their analysis of S3-2 data; however they excluded them from their theoretical considerations because they deduced what they contended was an unreasonably large perpendicular electric field from the E = (I/,!+) relation. We are motivated to further explore the ramitications of the incidence of a large amplitude AIfvCn wave on the region of space - 1-3 R, above the Earth’s surface on aurora1 oval field lines for the following reasons. To begin with, it would appear that the onset of a substorm expansive phase is linked to a sudden change in the tail electric current system and hance in the NAD generator responsible for that current flow. Such a sudden change would lead to propagation of information from the tail to the ionosphere in the form of a kinetic Alfven wave. It this wave was of small amplitude, it would probably not play any role in the acceleration process for amoral particles. If, on the other hand, the wave amplitude is of the order > 0.1 B,
(B, being the background magnetic field), one is justified in examining what is essentially a non-linear problem. The important observational feature which further stimulates our approach relates to the character of the topside ionospheric plasma into which the large amplitude Alfven wave would propagate. In particular Rostoker et al. (1976) have reported enhanced concentrations of thermal plasma in the topside ionosphere above the region of the aurora1 electrojets where the aurora1 acceleration mechanism is expected to be operative. Based on mechanisms which Lead to the polar wind (Banks and Holzer, 1969), one would expect a strong gradient in O+ concentration along the magnetic field lines in the altitude range where the acceleration is known to be operative (Lemaire and Scherer, 1974). Their observation is complemented by the paper of Calvert (1981) which reports a region of density depletion in the region where the acceleration mechanism is supposed to be operative; similar observations have also been reported by Mozer et al. (1977) and by Heppner et al. (198 1). We conclude from the behaviour of 0 ’ at high altitude and the region of density depletion statistically colocated with the O+ enhancement, that there is a marked density discontinuity in the region where aurora1 particles are accelerated. Calvert (1981) reports density jumps from can be 1 to 100cm-3 and changes in O+ concentration expected to occur over an altitude range of - I500 km near 5000 km altitude along typical aurora1 field lines (Lemaire and Sherer, 1974). Based on the above evidence, we shall explore in this paper the steepening of large amplitude Alfven waves when they encounter the density discontinuity (reported above) as they propagate into the topside ionosphere. The theory of steepening and MHD shock formation for non-linear Alfven waves has been reported some time ago (Montgomery, 1959; Parker, 1958) and the theory is well developed for the solar wind environment (Cohen and Kulsrud, 1974; Barnes and Hollweg, 1974; Steinolfson, 1981 ; Hollweg et al., 1982a,b). Almost no attention has been paid to the phenon~enon by amoral physicists for the purpose of explaining acceleration of aurora1 particles at altitudes of - 1 R,. It is shown independently by Montgomery (1959) and Cohen and Kulsrud (1974)that a non-linear wave in which the magnitude ofthe magnetic field is not constant steepens after a certain time which they calculated. In our study we follow closely the mathematical model developed by Cohen and Kuisrud (1974) to explain the shock formation process and we further utilize the method of characteristics developed for non-linear waves (Courant and Friedricks, 1948; Lighthill, 1956; Jeffrey and Taniuti, 1964; Lax, 1972; Whitham, 1974).
Field-aligned 2. MATHEMATICAL
currents
and aurora1 acceleration
1499
MHD waves
with the relations b, = b,(z+ VAT, z) and v1 = vr(z + I/,T, Z) where V, = i?,/(4np,)‘/2 is the AlfvCn wave phase velocity, T is the time period and z involves the times series expansion in second order such that
MODEL
The basic equations leading to the characteristic are the ideal magnetohydrodynamic equation equations :
g+v.(pv)=o
by non-linear
(1)
dV 1 pdt=;(JxB)-VP dB -=Vx(VxB)
at
; (P,p’) = 0. Here P is the pressure and y represents the ratio of two specific heats. Following the approach of Cohen and Kulsrud (1974) by using plane wave analysis assuming all the quantities are functions of z and t, these equations can be written as
Here a/&,, is of the order (b/B,)” and we adopt the notation 1, E T, t, e z for further discussions. The magnetized plasma we consider can support both AlfvCn wave and sound wave [c, = (yP,/p,)“*] propagation, however the plasma ,8( =cz/Vi) is sufficiently small such that mode coupling between sound waves and Alfvtn waves need not be considered. In the first order it is assumed that p1 = u, = 0. In the second order the velocity u2 and the density perturbation p2 are defined by : (13)
1
““=j(1-j)--
PO
b:-(b:)
(14)
(5)
(6) dV
ih
c!b
_
+
at
u ;!
db
B,
Pz+vuz-
4n iiz +
b f!!
_
B,
$
-0
(7)
=
()
(8)
c;Z
where the quantities in angle brackets represent the average values, i.e. (p) = o0 is the ambient plasma density. It follows from equations (13) and (14) that large amplitude Alfven waves locally lead to a flux of particles along the magnetic field in direction of propagation, the magnitude ofwhich is proportional to the square of the AlfvCn wave amplitude. In the third order we have the expression describing the evolution of the transverse magnetic field ‘b’ as :
(9) In equations (5))(9) II and v are the longitudinal and transverse components of velocity V, and B, and b are the ambient and transverse components of the magnetic field B. The velocity u and ambient magnetic field B, are directed along the z-axis and are positive towards the ionosphere. Using a multiple time series expansion (Davidson, 1972) Cohen and Kulsrud in first order derived the Alfven wave equation quantities v1 and b, as
1
VA
- ~ ----~? 4 (1 -[j)@
(72
[(bf-(b:))b,]
which describes the characteristic lines of constant magnitude and constant direction as : = cc(3bZ-(b’))
(16)
and = (b2 - (b*))
(17)
(10)
(11) or more explicitly (12)
(15)
where i = Z- VAT, a = V,/[4(1 - /I)Bi] and the subscript on “b,” has been dropped. Thus equations (16) and (17) describe the characteristic velocities in the Alfven wave frame of reference. Integration of (16) gives to(T)
=
3(T-Tr,)rb2-a
’ (b2)
I ro
dt+Z
(18)
1500
M. S. TIWARI and G.
which are the straight (but not parallel) lines in characteristic plane and Z = &,(zJ. A shock is formed when two characteristics intersect at any time z which has been calculated from (18) by Lax (1972); viz.
Z+(4-Z-(4 z--zo
= 3[c(_bZ -cc+b:].
Note that Z, and Z_ delinate the two sides of the discontinuity and 3 [cr- b? -a + b$] represents the characteristic velocity. The subscripts + and represent the upstream and downstream quantities respectively. If the perturbation magnetic field is of the same order of magnitude as the background magnetic field, then steepening will occur even if the unperturbed medium is uniform (Montgomery, 1959; Cohen and Kulsrud, 1974). However, even if the perturbation magnetic field has a magnitude somewhat less than that ofthe background field, breaking will arise immediately if the incoming wave encounters a discontinuous step for which the characteristic velocity behind the discontinuity is greater than that shead (Whitham, 1974). The multivalued region starts right at the discontinuity and is bounded by the two characteristics originating from two sides of the discontinuity. However, ifthe discontinuity has a finite thickness, then breaking starts after the time which is given by the ratio of the thickness to the difference in characteristic speed at the two ends of the discontinuity. Thus a continuous wave breaks and leads to a shock if and only if the propagation velocity decreases as z increases. The shock is formed much closer to the discontinuity as the thickness approaches to a step function.
DISCUSSION
We now discuss the shock formation in the acceleration region based on the presence of a mass density discontinuity in that altitude range. The characteristic velocity 31/, c=4(l--p)
b2-b: Bi
depends both on the AlfvCn velocity and on the amplitude of the hydromagnetic wave. The wave will steepen into a shock when it encounters the mass density discontinuity which stems from both a change in number density and a change in mass composition. One may expect rather extreme changes in number density across the discontinuity. For example Hoffman et al. (1974) and Taylor (1974) used Isis 2, and 0G04 and OG06 data respectively to study O+ concentrations and reported number density enhancements up to - lo3 cm 3 during active periods in regions where the
ROSTOKER
normal concentrations during quiet times were below their detection threshold of - 1 cmm3. One might expect this on the basis of the expected change in the scale height of O+ when the topside ionosphere is heated by aurora1 particle precipitation (Whitteker, 1976). For our calculations we will use the density jump of 1 to 100 cme3 reported by Calvert (1981). Since the acceleration region is reported to lie between - 1 and 3 R, we shall calculate our characteristic velocity at 2R, where B, N 4300 nT. We assume our plasma B N 0 and an incoming wave amplitude of bm - 700 nT. As we expect the wave energy to be given to the particles during the acceleration process, we expect the wave amplitude to decrease during passage through the acceleration region and we will consider a case where the downstream wave has an amplitude b, - 500 nT. (Azimuthal perturbation magnetic fields in excess of 500 nT are often observed in polar orbiter magnetometer data taken near 1000 km altitude.) Based on the observations of Calvert (1981) quoted above for number density, an ambient magnetic field of - 4300 nT and values of b + and b _ as discussed above, we calculated the steepening time.z-to from equation (19)tobe -5.4x 10-‘(z+-z_)swhere(z+-z_)isthe discontinuity thickness in m. During this steepening time, the wave will have propagated (T - zo)V,+ = 1.26(z+ -z_)m. For a number density jump thickness from 1to 100 cm 3 of - 0.3 R,, the wave may be expected to steepen after travelling -0.4 R, which suggests that, for the mass density discontinuity starting near 2 R, the wave will have steepened into a shock before it reaches an altitude of 1.5 R,. It should be emphasized that the steepening time is quite sensitive to wave amplitude and discontinuity thickness instead of the size of the mass density jump. A large amplitude wave may be expected to steepen around the discontinuity whereas small amplitude waves will not steepen and will propagate, without significant energy loss, until it reflects from the ionosphere below - 500 km altitude. The magnetic field lines in the vicinity of the forming shock will be distorted due to currents associated with the jump in transverse magnetic field over the shock thickness (Greenstadt and Fredricks, 1979) as reported by Burke et al. (1983) in their S3-2 data analysis. The perpendicular electric field may couple to a parallel one through the time rate of change of magnetic field db/at and a burst of electrons and ions should be created along the field lines by the induced parallel electric field. Using Maxwell’s equation VxE=
-~-
1 ab c
at
the parallel electric field may be given by the relation
Field-aligned currents and aurora1 acceleration by non-linear MHU waves
where x is northward, yis eastward and z downward. In this case the field lines are no longer rigid but would be slightly bent with a parallel potential developing by the bending of magnetic field lines which couple to the perpendicular electric field (Kadomtsev, 1965). In case the transverse component of the electric field is curl free (Kadomtsev, 1965) the parallel potential will be developed by induction. However, a more detailed microscopic treatment is needed as the scale size along the field lines becomes comparable to ion gyroradius. The parallel electric field develops inside the shock as it propagates along the magnetic field and particles can be accelerated up to the Alfvin speed by this mechanism. However, many other acceleration mechanisms are also probably applicable by which energy contained in the shock is dissipated. In the case of a curl free perpendicular electric field or in the region where E, 2 0 we have El, N VA (Ab,/Az)* Ax. For the typical values of Ax = 100 m (thickness of the current sheet) AZ rr 1000 m (shock thickness of the order of the ion gyroradius) VA = 2.4 x 106ms-‘andAb,-2OOnT,wefindE,, =48mVm-‘. Thus for acceleration of electrons to energies of -5 keV,
1501
plasma flow is observed in which positive ions and electrons are accelerated in opposite directions (Hultqvist et al., 1971; Bryant et al., 1977 ; Aggson et al., 1983). The observation (Krimigis and Sarris, 1979) of the acceleration of plasma sheet particles to energies greater than the electrostatic potential across the magnetosphere may confirm this generalization. The circularly polarized AlfvCn waves do not steepen into shocks; however they evolve as rotational discontinuities (highly crested AlfvCn waves) in which transverse velocity and magnetic field r&ate through the 180” without changing their magnitude. An aurora1 arc may develop in the form of rotational discontinuity (Atkinson, 1982) but this mechanism will not give any acceleration. In case the shock develops through the alternate cycles of a linearly polarized wave, it may give acceleration and deceleration alternately and the electric field may be rotated through 180” and a steep gradient in electric and magnetic fields should be observed at the shock front. This may explain some of the experimentally reported observations (Heppner et al., 1981; Shuman et al., 1981 and Burke et al., 1983). Thus it seems reasonable that some attention should be paid to the MHD waves (Heppner et al., 1981) and acceleration by an MHD shock. Acknowledgement-This Sciences and Engineering
work was supported by the Natural Research Council of Canada.
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1502
M. S. TIWARI and G. ROSTOKER
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currents
and amoral
acceleration
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by non-linear
MHD waves
1503
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FIELD-ALIGNED A~CELERATrO~
Institute
0032&0633/8453.00 +O.oO Pergamon Press Ltd.
12, pp. 1497 -1503. 1984
CURRENTS AND AURORAL BY NON-LINEAR MHD WAVES
M. S. TIWARP and G. ROSTOKER of Earth and Planetary Physics, and Department of Physics, Edmonton, Alberta, Canada T6G 251
(Received
13 February
University
of Alberta,
1984)
Abstract-This paper examines the consequences of the assumption that substorm-associated growth of magnetosphere-ionosphere current systems is triggered by the incidence, on the ionosphere, of a large amplitude Alfven wave generated in the distant magnetotail. It is pointed out that there is a large body of evidence suggesting that, in the acceleration region near 1 R,, one is likely to find a major discontinuity in mass density. Following the approach of Cohen and Kulsrud (1974) who studied the steepening of large amplitude hydromagnetic waves into shocks, we demonstrate that the character of the background plasma and magnetic field in the aurora1 acceleration region near 1 R, can be idea1 for the generation of MHD shocks and that these shocks can lead to the acceleration of ions and electrons as reported by investigators using S3-3 satellite data.
1. INTRODUCTION
In the recent
past various theories have been proposed to explore the question of field-aligned currents and acceleration of charge particles precipitating in the aurora1 ionosphere, but none have been completely successful in explaining the various phenomena observed simultaneously by rocket and satellite and ground-based instruments. Amongst the various theories, two major classes have been proposed to describe some of the phenomena. The first are the timeindependent models based on the single particle approach (Knight, 1973; Lemaire and Scherer, 1974a,b; Lyons et al., 1979; Chiu and Cornwall, 1980; Fridman and Lemaire, 1980; Chiu et al., 1981 and Lyons, 1980, I 98 1a,b). These models explain large-scale phenomena but have not yet been extended for microturbulence generally observed in the acceleration region. Drawbacks in the applicability of the doubie layer/electrostatic shock model (Block, i 972 ; Swift, 1975 ; Hudson et al., 1983) and anomalous resistivity (Papadopoulos, 1977 ; Dum, 1981) are indicated by Lyons (1981b) and the applicability of the single particle theory is emphasized. The second kind of approach considers transient phenomena and explains the aurora1 particle acceleration in terms of linear AlfvCn waves propagating paraIle1 to or obliquely with respect to the ambient magnetic field (Scholer, 1970; Mallinckrodt and Carlson, 1978; Goertz and Boswell, 1979; Miura and Sato, 1980 ; Lysak and Carlson, 198 1; Lysak and Dum, 1983). These theories mainly describe the field-aligned * Department India.
of Physics,
Sagar
University,
Sagar (M.P.),
1497
current and parallelelectricfield associated with kinetic AlfvCn waves which have originated in some magnetospheric generator region via impressed perpendicular and time varying electric fields. The field-aligned current associated with kinetic Alfv& waves is considered to be sufficient for the excitation of electrostaticion cyclotron waves and hence anomalous resistivity. While this theory is growing in popularity, it is still incomplete in dealing with the non-linear phenomena for large amplitude waves. It does not locate the region where the enhancement ofthe currents actually should take place. This depends upon the background plasma and the parallel potential drop may occur anywhere between the magnetosphere and the ionosphere. Recent observations, however, have defined the region around 1 R, and above where the acceleration actually takes place. For a detailed analysis of the existing theories the reader is referred to reviews by Shawhan et al. (1978), Mozer et al. (1980), Kan (1982) and articles in Akasofu and Kan (1981). In the recent past several observations are reported which are not explained by these existing theories. Electron and ion acceleration are sometimes observed simultaneously and in the same direction as advocated by Heikkila (1981) and experimentally reported by Bryantetal.(1977),GurnettandFrank(1973),Krimigis and Sarris (1978) and Aggson et al. (1983) near the aurora1 ionosphere and in the magnetotail regions. Heppner et al. (198 I), for their Cameo Barium release experiments, have reported both accelerating and decelerating parallel electric fields above 1400 km exhibiting a periodicity of 334 min and, because of an apparent wave behaviour in the PC 5 band, an association with hydromagnetic waves was suggested.
1498
M. S.
TIWARI and G. ROSTOKER
However, they exclude the hydromagnetic wave theory because of high-p medium over the polar cap. Acceleration of electrons to 6 keV above 15,800 km has also been reported by Heppner et al. (1981). Recently Aggson et al. (1983) reported a strong inductive electric field observed on the ISEE- satellite at substorm time around midnight in the L = 7.5 region of the near Earth magnetospheric tail and used that observation in proposing an explanation of the observed correlation between electric and magnetic fields by the AlfvCn or fast magnetosonic mode. Further, they suggested a theory to explain the relative effectiveness of these fields in accelerating particles for both the plasma population and individual particles. They also indicated the importance of using Faraday’s law for these observed electric fields instead of using solely the divergence of the field as used by Mozer (1981) to explain the electrostatic shock for a similar event. It is generally believed that field-aligned currents are driven by Alfvtn waves. Magnetic fields detected by the satellites with magnetometers on board have reported large magnetic field perturbations of several hundred nT (Zmuda and Armstrong, 1974 ; Iijima and Potemra, 1976a,b; Shuman et al., 1981; Burke et al., 1983) which are comparable to local ambient magnetic fields. Under these circumstances the analysis of field-aligned current using linear Alfven waves may be inappropriate and the existing theory should be extended to the nonlinear regime. The wave-associated centrifugal force -p(V*V)V and the wave-associated Lorentz force J x B/c (p is the mass density, V is the velocity, J is the current density, B is the magnetic field and c is the speed of light) may play a significant role in determining the various properties ofthe system. Burke et al. (1983) have also indicated the significance oflarge amplitude Alfven waves in their analysis of S3-2 data; however they excluded them from their theoretical considerations because they deduced what they contended was an unreasonably large perpendicular electric field from the E = (I/,!+) relation. We are motivated to further explore the ramitications of the incidence of a large amplitude AIfvCn wave on the region of space - 1-3 R, above the Earth’s surface on aurora1 oval field lines for the following reasons. To begin with, it would appear that the onset of a substorm expansive phase is linked to a sudden change in the tail electric current system and hance in the NAD generator responsible for that current flow. Such a sudden change would lead to propagation of information from the tail to the ionosphere in the form of a kinetic Alfven wave. It this wave was of small amplitude, it would probably not play any role in the acceleration process for amoral particles. If, on the other hand, the wave amplitude is of the order > 0.1 B,
(B, being the background magnetic field), one is justified in examining what is essentially a non-linear problem. The important observational feature which further stimulates our approach relates to the character of the topside ionospheric plasma into which the large amplitude Alfven wave would propagate. In particular Rostoker et al. (1976) have reported enhanced concentrations of thermal plasma in the topside ionosphere above the region of the aurora1 electrojets where the aurora1 acceleration mechanism is expected to be operative. Based on mechanisms which Lead to the polar wind (Banks and Holzer, 1969), one would expect a strong gradient in O+ concentration along the magnetic field lines in the altitude range where the acceleration is known to be operative (Lemaire and Scherer, 1974). Their observation is complemented by the paper of Calvert (1981) which reports a region of density depletion in the region where the acceleration mechanism is supposed to be operative; similar observations have also been reported by Mozer et al. (1977) and by Heppner et al. (198 1). We conclude from the behaviour of 0 ’ at high altitude and the region of density depletion statistically colocated with the O+ enhancement, that there is a marked density discontinuity in the region where aurora1 particles are accelerated. Calvert (1981) reports density jumps from can be 1 to 100cm-3 and changes in O+ concentration expected to occur over an altitude range of - I500 km near 5000 km altitude along typical aurora1 field lines (Lemaire and Sherer, 1974). Based on the above evidence, we shall explore in this paper the steepening of large amplitude Alfven waves when they encounter the density discontinuity (reported above) as they propagate into the topside ionosphere. The theory of steepening and MHD shock formation for non-linear Alfven waves has been reported some time ago (Montgomery, 1959; Parker, 1958) and the theory is well developed for the solar wind environment (Cohen and Kulsrud, 1974; Barnes and Hollweg, 1974; Steinolfson, 1981 ; Hollweg et al., 1982a,b). Almost no attention has been paid to the phenon~enon by amoral physicists for the purpose of explaining acceleration of aurora1 particles at altitudes of - 1 R,. It is shown independently by Montgomery (1959) and Cohen and Kulsrud (1974)that a non-linear wave in which the magnitude ofthe magnetic field is not constant steepens after a certain time which they calculated. In our study we follow closely the mathematical model developed by Cohen and Kuisrud (1974) to explain the shock formation process and we further utilize the method of characteristics developed for non-linear waves (Courant and Friedricks, 1948; Lighthill, 1956; Jeffrey and Taniuti, 1964; Lax, 1972; Whitham, 1974).
Field-aligned 2. MATHEMATICAL
currents
and aurora1 acceleration
1499
MHD waves
with the relations b, = b,(z+ VAT, z) and v1 = vr(z + I/,T, Z) where V, = i?,/(4np,)‘/2 is the AlfvCn wave phase velocity, T is the time period and z involves the times series expansion in second order such that
MODEL
The basic equations leading to the characteristic are the ideal magnetohydrodynamic equation equations :
g+v.(pv)=o
by non-linear
(1)
dV 1 pdt=;(JxB)-VP dB -=Vx(VxB)
at
; (P,p’) = 0. Here P is the pressure and y represents the ratio of two specific heats. Following the approach of Cohen and Kulsrud (1974) by using plane wave analysis assuming all the quantities are functions of z and t, these equations can be written as
Here a/&,, is of the order (b/B,)” and we adopt the notation 1, E T, t, e z for further discussions. The magnetized plasma we consider can support both AlfvCn wave and sound wave [c, = (yP,/p,)“*] propagation, however the plasma ,8( =cz/Vi) is sufficiently small such that mode coupling between sound waves and Alfvtn waves need not be considered. In the first order it is assumed that p1 = u, = 0. In the second order the velocity u2 and the density perturbation p2 are defined by : (13)
1
““=j(1-j)--
PO
b:-(b:)
(14)
(5)
(6) dV
ih
c!b
_
+
at
u ;!
db
B,
Pz+vuz-
4n iiz +
b f!!
_
B,
$
-0
(7)
=
()
(8)
c;Z
where the quantities in angle brackets represent the average values, i.e. (p) = o0 is the ambient plasma density. It follows from equations (13) and (14) that large amplitude Alfven waves locally lead to a flux of particles along the magnetic field in direction of propagation, the magnitude ofwhich is proportional to the square of the AlfvCn wave amplitude. In the third order we have the expression describing the evolution of the transverse magnetic field ‘b’ as :
(9) In equations (5))(9) II and v are the longitudinal and transverse components of velocity V, and B, and b are the ambient and transverse components of the magnetic field B. The velocity u and ambient magnetic field B, are directed along the z-axis and are positive towards the ionosphere. Using a multiple time series expansion (Davidson, 1972) Cohen and Kulsrud in first order derived the Alfven wave equation quantities v1 and b, as
1
VA
- ~ ----~? 4 (1 -[j)@
(72
[(bf-(b:))b,]
which describes the characteristic lines of constant magnitude and constant direction as : = cc(3bZ-(b’))
(16)
and = (b2 - (b*))
(17)
(10)
(11) or more explicitly (12)
(15)
where i = Z- VAT, a = V,/[4(1 - /I)Bi] and the subscript on “b,” has been dropped. Thus equations (16) and (17) describe the characteristic velocities in the Alfven wave frame of reference. Integration of (16) gives to(T)
=
3(T-Tr,)rb2-a
’ (b2)
I ro
dt+Z
(18)
1500
M. S. TIWARI and G.
which are the straight (but not parallel) lines in characteristic plane and Z = &,(zJ. A shock is formed when two characteristics intersect at any time z which has been calculated from (18) by Lax (1972); viz.
Z+(4-Z-(4 z--zo
= 3[c(_bZ -cc+b:].
Note that Z, and Z_ delinate the two sides of the discontinuity and 3 [cr- b? -a + b$] represents the characteristic velocity. The subscripts + and represent the upstream and downstream quantities respectively. If the perturbation magnetic field is of the same order of magnitude as the background magnetic field, then steepening will occur even if the unperturbed medium is uniform (Montgomery, 1959; Cohen and Kulsrud, 1974). However, even if the perturbation magnetic field has a magnitude somewhat less than that ofthe background field, breaking will arise immediately if the incoming wave encounters a discontinuous step for which the characteristic velocity behind the discontinuity is greater than that shead (Whitham, 1974). The multivalued region starts right at the discontinuity and is bounded by the two characteristics originating from two sides of the discontinuity. However, ifthe discontinuity has a finite thickness, then breaking starts after the time which is given by the ratio of the thickness to the difference in characteristic speed at the two ends of the discontinuity. Thus a continuous wave breaks and leads to a shock if and only if the propagation velocity decreases as z increases. The shock is formed much closer to the discontinuity as the thickness approaches to a step function.
DISCUSSION
We now discuss the shock formation in the acceleration region based on the presence of a mass density discontinuity in that altitude range. The characteristic velocity 31/, c=4(l--p)
b2-b: Bi
depends both on the AlfvCn velocity and on the amplitude of the hydromagnetic wave. The wave will steepen into a shock when it encounters the mass density discontinuity which stems from both a change in number density and a change in mass composition. One may expect rather extreme changes in number density across the discontinuity. For example Hoffman et al. (1974) and Taylor (1974) used Isis 2, and 0G04 and OG06 data respectively to study O+ concentrations and reported number density enhancements up to - lo3 cm 3 during active periods in regions where the
ROSTOKER
normal concentrations during quiet times were below their detection threshold of - 1 cmm3. One might expect this on the basis of the expected change in the scale height of O+ when the topside ionosphere is heated by aurora1 particle precipitation (Whitteker, 1976). For our calculations we will use the density jump of 1 to 100 cme3 reported by Calvert (1981). Since the acceleration region is reported to lie between - 1 and 3 R, we shall calculate our characteristic velocity at 2R, where B, N 4300 nT. We assume our plasma B N 0 and an incoming wave amplitude of bm - 700 nT. As we expect the wave energy to be given to the particles during the acceleration process, we expect the wave amplitude to decrease during passage through the acceleration region and we will consider a case where the downstream wave has an amplitude b, - 500 nT. (Azimuthal perturbation magnetic fields in excess of 500 nT are often observed in polar orbiter magnetometer data taken near 1000 km altitude.) Based on the observations of Calvert (1981) quoted above for number density, an ambient magnetic field of - 4300 nT and values of b + and b _ as discussed above, we calculated the steepening time.z-to from equation (19)tobe -5.4x 10-‘(z+-z_)swhere(z+-z_)isthe discontinuity thickness in m. During this steepening time, the wave will have propagated (T - zo)V,+ = 1.26(z+ -z_)m. For a number density jump thickness from 1to 100 cm 3 of - 0.3 R,, the wave may be expected to steepen after travelling -0.4 R, which suggests that, for the mass density discontinuity starting near 2 R, the wave will have steepened into a shock before it reaches an altitude of 1.5 R,. It should be emphasized that the steepening time is quite sensitive to wave amplitude and discontinuity thickness instead of the size of the mass density jump. A large amplitude wave may be expected to steepen around the discontinuity whereas small amplitude waves will not steepen and will propagate, without significant energy loss, until it reflects from the ionosphere below - 500 km altitude. The magnetic field lines in the vicinity of the forming shock will be distorted due to currents associated with the jump in transverse magnetic field over the shock thickness (Greenstadt and Fredricks, 1979) as reported by Burke et al. (1983) in their S3-2 data analysis. The perpendicular electric field may couple to a parallel one through the time rate of change of magnetic field db/at and a burst of electrons and ions should be created along the field lines by the induced parallel electric field. Using Maxwell’s equation VxE=
-~-
1 ab c
at
the parallel electric field may be given by the relation
Field-aligned currents and aurora1 acceleration by non-linear MHU waves
where x is northward, yis eastward and z downward. In this case the field lines are no longer rigid but would be slightly bent with a parallel potential developing by the bending of magnetic field lines which couple to the perpendicular electric field (Kadomtsev, 1965). In case the transverse component of the electric field is curl free (Kadomtsev, 1965) the parallel potential will be developed by induction. However, a more detailed microscopic treatment is needed as the scale size along the field lines becomes comparable to ion gyroradius. The parallel electric field develops inside the shock as it propagates along the magnetic field and particles can be accelerated up to the Alfvin speed by this mechanism. However, many other acceleration mechanisms are also probably applicable by which energy contained in the shock is dissipated. In the case of a curl free perpendicular electric field or in the region where E, 2 0 we have El, N VA (Ab,/Az)* Ax. For the typical values of Ax = 100 m (thickness of the current sheet) AZ rr 1000 m (shock thickness of the order of the ion gyroradius) VA = 2.4 x 106ms-‘andAb,-2OOnT,wefindE,, =48mVm-‘. Thus for acceleration of electrons to energies of -5 keV,
1501
plasma flow is observed in which positive ions and electrons are accelerated in opposite directions (Hultqvist et al., 1971; Bryant et al., 1977 ; Aggson et al., 1983). The observation (Krimigis and Sarris, 1979) of the acceleration of plasma sheet particles to energies greater than the electrostatic potential across the magnetosphere may confirm this generalization. The circularly polarized AlfvCn waves do not steepen into shocks; however they evolve as rotational discontinuities (highly crested AlfvCn waves) in which transverse velocity and magnetic field r&ate through the 180” without changing their magnitude. An aurora1 arc may develop in the form of rotational discontinuity (Atkinson, 1982) but this mechanism will not give any acceleration. In case the shock develops through the alternate cycles of a linearly polarized wave, it may give acceleration and deceleration alternately and the electric field may be rotated through 180” and a steep gradient in electric and magnetic fields should be observed at the shock front. This may explain some of the experimentally reported observations (Heppner et al., 1981; Shuman et al., 1981 and Burke et al., 1983). Thus it seems reasonable that some attention should be paid to the MHD waves (Heppner et al., 1981) and acceleration by an MHD shock. Acknowledgement-This Sciences and Engineering
work was supported by the Natural Research Council of Canada.
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1502
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currents
and amoral
acceleration
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by non-linear
MHD waves
1503
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