On the consequences of the interaction between the auroral plasma and the geomagnetic field

ON THE CONSEQUENCES AURORAL PLASMA

OF THE l[NTERACTION BETWEEN AND THE GEOMAGNEZTK FIELD WALTER

Department

THE

LICNi’IAR~QNf

of Plasma Physics, Royal Institute of Technology, S-10044 Stockholm 70, Sweden (Received in final form 6 September 1979)

A&tract-This report is the second phase of an attempt to find a unified description of aurora1 electric fields based on the interaction between a hot plasma and the Earth’s magnetic field. It is shown that a large electrostatic potential difference must exist along the magnetic field lines under very general conditions, as a direct consequence of this interaction. This does not require “anomalous resistivity”, and is not a consequence of anisotropic pitch-angle distributions, but the electric Geld aIways produces anisotropy. In general, the electric geld has a time-varying funstablef spatial structure. An important source of temporal ~uctuaiions is charge trapping in the combined electric and magnetic fields, which is an inevitable effect of scattering. A necessary condition for a stable electric field structure to form is that this trapping saturates. The conditions for this are most favorable in a thin “double layer” structure, in particular one that is aligned along the magnetic field with IEL/ >rIE ,, 1. A secondary effect of charge trapping is a collimation of the electron precipitation flux. Another effect consists of increasing the transverse energy of cold ions (where IV, * El 2 IV,l * El). This effect is particularly strong for oxygen, resulting in a “conical” pitch-angle distribution.

1. lNTRoDu~oN

A great many observations over the past several years have been interpreted as evidence of electrostatic acceleration of the aurora1 electrons along the Earth’s magnetic field lines (cf. Evans, 1976). Along with the progress of observations several different theoretical models have been put forward in the literature to describe the various aspects of amoral electric fields with a component along the magnetic field direction, commonly referred to as “parallel electric fields”. Some of these models have been reviewed by Block and ~lthammar (1976) and slthammar (1977). In brief, most of these models may be placed into one of the foflowing four categories. (A) Models of a scatter-free plasma in a quasineutral steady state. If constants of the motion can be derived for all charged particles, it is found that a given set of velocity distributions of the “source” particles may cause the number densities n, and ni of electrons and positive ions, respectively, to vary in different fashions along the IEarth’s magnetic field lines in the absence of a parallel electric field. This logical contradiction may, in certain cases, be resolved in a formal sense by introducing a paratlel electric field which changes the particle dynamics so as to make n, = n, everywhere. This is commonly *Present Address: Lockheed Palo Alto Research Laboratories. Department 52-12, Building 205, 3251 Hanover Street, Palo Alto, CA 94304, U.S.A.

referred to as “the effect of anisotropic pitch-angle dist~butions”. For specific examples of this type of model, see Alfven and F~lthammar (1963, pp. 162167), Persson (1966), Whipple (1977) and Chiu and Schuftz (1978). A recent study of “oblique shocks” by Swift (1976) may also be placed in this category, although the parallel electric field in his model is required to compensate for a charge separating effect of the inhomogeneous transverse electric field component, whereas the magnetic field is assumed homogeneous. (B) Models of steady scatter-free “double layer’” or “electrostatic shock” structures, in which n, # n, and the space charges appear in two adjacent layers of opposite polarity. Structures of this kind, often quite well defined and stable, are known to appear in laboratory plasmas, when a sufficiently large current is driven through the plasma by an external current-voltage source, and to have spatial dimensions of a few to several Debye lengths in the direction along the electric field vector (see e.g. Block, 1975). It was suggested several years ago that such structures may also appear in the magnetospheric plasma, in regions of strong fieldaligned currents (see Block, 1972, and references therein). The fact that aurora1 structures are often quite thin is per se suggestive of a large divergence of the transverse electric field, and has been the motivation for studies centered on structures with IV’, * El 3 \V,, * El (e.g. Swift, 1975). (C) Models of the current-voftage characteristics

135

136

W.

LENNARTSSON

of a plasma with strong scattering, driven in part or entirely by the current itself. These models lead to the concept of “anomalous resistivity”. See, for instance, Coroniti and Kennel (1972) and the review by Papadopolous (1977). (D) Models of the current-voltage characteristics of a scatter-free plasma in a magnetic-mirror configuration. Even in the absence of scattering the magnetospheric plasma has a finite “resistance” to field-aligned currents flowing upwards (eiectrons moving downwards), as a consequence of the magnetic mirroring of energetic charge carriers. Studies in this category have been made by, among others, Knight (1973), Lemaire and Scherer (1974), and Lennartsson (1976, 1977). These four types of models differ not only quantitatively but are descriptions of fund~entally different and partially inconsistent physical processes. As such they reflect our lack of a qualitative understanding of amoral electric fields. For the future development of a unified quantitative theory it is therefore of prime importance to determine which type of model, or which combination of these models, is best fitted to improve our qualitative understanding of the observed phenomena. The present report is a follow-up of the studies by Lennartsson (1976, 1977) and has the purpose of pursuing the idea that a model of type (0) is the natural basis for a unified physical description. In a formalistic sense type (D) may seem a subset of type (A). There is, however, an important difference in their physical implications. While a type (A) model eliminates Poisson’s equation, by presuming a unique dependence of the particle densities on both the spatial coordinate and the potential, a model of type (D) only sets a boundary condition for Poisson’s equation. As a consequence, a model of type (D) can be more easily generalized to a plasma with scattering. In this report the role of the geomagnetic field is analyzed in stricter physical terms by taking into account scattering and other interactions between the various plasma components. Several of the results obtained here were not foreseen in the previous studies, others are generalizations of previous results. This study is not a “model” but a qualitative analysis. The only assumption made a priori is that the charge separating mechanism ultimately responsible for setting up a magnetosphereionosphere current loop is entirely due to a drift across magnetic field lines of hot particles. Although this can also be well justified on physical grounds (see e.g. Lennartsson, 1977, and references therein} it is taken as an ad hoc assumption

here. The emphasis of this study is on the physical mechanisms and the mathematical treatment is accordingly, brief. The essential results are the following: (1) Because of the magnetic mirroring of hot electrons, a large potential drop AV must exist along the magnetic field lines in regions of upward current under very general conditions. Specifically, if the electrons have an isotropic Maxwellian dist~bution with thermal energy kT, increasing the precipitation flux by a factor of @, as compared with the field-free precipitation, requires a net (time-averaged) potential drop AV 2 (@ - 1) kT’/e. This result is valid without any scattering, as well as in the presence of random elastic pitch-angle scattering of arbitrary strength. The electric field does not necessarily have a time-invariant spatial structure. (2) This result is obtained without invoking any kind of “anomalous resistivity” nor any particular pitch-angle anisotropy. Of course, the electric field will affect the pitch-angle distribution of hot ions and electrons by changing the “loss cones”. (3) The cold electrons may carry a finite current for an extended time, but their motion is always hampered by charge balance requirements at high altitudes. This current is indirectly a function of AV, such that a larger AV allows a more rapid depletion of the cold electrons. (4) A time-invariant electric field cannot be in a stable equilibrium with a plasma that is quasineutral everywhere. As a consequence, the electric field must be at least partially concentrated into “double layer” structures, or else it has a time fluctuating spatial structure, or both. (5) The plasma is unstable with respect to trapping of hot electrons between a growing electric field, with an upward parallel component, and the magnetic mirror below. (6) No “double layer” structure can be stable until this charge trapping has saturated. The conditions for saturation are most favourable in a thin structure with IV, - El >>/V11+El. (7) This trapping easily increases the “fieldalignment” and decreases the intensity of precipitation fluxes at a given AV. To maintain the current density AV has to increase. (8) No electric field structure with IV, * E/Z IV,, =E/ can be in a stable equilibrium with a cold plasma on the positive side without breaking down the gyrotropy of the ions and accelerating these in the transverse direction. This effect is expressed by IV, - E]?w,B, where wg is the ion gyrofrequency and B is the magnetic field strength. The resulting

Aurora1 plasma-geomagnetic pitch-angle distribution of the accelerated ions readily becomes “conical”, in particular that of oxygen ions (low u,) and in particular at high altitude (low B). (9) Transient, or fluctuating, electric field structures with IV, - Ef z w,B and with upward directed E ,I, appearing randomly over large altitude intervals continuously increase, on the average, the transverse energy of ions moving upwards. (10) The last two results suggest that observed ion cyclotron waves may be a secondary effect of transverse acceleration of ions, in qualitative agreement with the fact that observed wave-fields have an energy density that is orders of magnitude below the energy density of the energized ions. One particular feature of aurora1 displays which was addressed in the previous studies (Lennartsson, 1976, 1977), namely the formation of aurora1 ray structures, is not discussed here, but is deferred to forthcoming studies. 2. CURRENT-VOLTAGE

2. I. ~ecipitution

RELATIONS

137

field interaction

;hd) FIG. 1. A MAGN!ZTOSPHERE-IONOSPHERE CURRENT SYSTEM.

The dynamo current P,-P4 is assumed to be caused by the differential drift of hot protons and electrons. The downward parallel current P4-P3 is carried mainly by escaping ionospheric electrons, while the upward parallel current P2 - P, is carried to a large extent by downflowing hot electrons. Point Pa is at a high positive potential with respect to point P,, which enables the downflowingelectrons to overcome the magnetic mirror. The current P, P2 is a Pedersen current (from Lennartsson, 1976).

current

In a discussion of particle pitch-angle distributions which are subject to scattering, as well as to time-independent electric and magnetic forces, it is useful to introduce the concept of “nearly isotropic” (see e.g. Persson, 1966). A ‘nearly isotropic” velocity distribution f(v) is such that at any given energy, f(v)=$(u) in certain pitch-angle regions, and f(v)=0 in others. In many cases with random pitch-angle scattering a velocity distribution may be better approximated by a “nearly isotropic” distribution than by an isotropic one, because of the continuous loss of particles in a “loss cone”, for instance. Also in the absence of scattering this concept is useful because “near isotropy” is conserved under far more general conditions than is isotropy. Consider Fig. 1. This is a schematic illustration of a magnetosphere-ionosphere current loop and has been discussed previously (Lennartsson, 1976). For the present it is sufficient to consider the upward field-aligned current from P2 to Pr. Suppose the distribution function fe of electrons moving downward and the distribution function f: of electrons moving upward at a given point are both “nearly isotropic”. Then the net upward fieldaligned current density iTt may be expressed as *_ .-a‘ ,IXK) cos a da {feCv,J

i’=fyj b

where K is the kinetic energy K = $m,v2,

ar is the pitch angle, and fit(K) 5 212 and R’(K) 5 2a are the respective solid angles occupied by electrons of energy K. 2.1.1. The scatter-free state. Suppose Pr is at a negative potential with respect to Pz and suppose the downflowing electrons at P, have an isotropic distribution fel throughout the entire downward half sphere @=2x) in velocity space. As long as the phase space density, magnetic moment and total energy are conserved, that is

fewma

cos CYdR K dK, I

(I)

=f.,(G),

lip= KJB

(3)

= K,JB,

(4)

and K-eV=K,-eVi,

(5)

the integral in (1) yields the following upper limit on the field-aligned electron current density at P2:

x(K+eAV)dK where

n’(K)

'f,(v)

(2)

Ko=-

Bl eAV, Bz-Bz

I

,

(6)

138

W. LENNARTSSON

and AV=V,-V,

(8)

is the totd potenti& difference between PI and Pz. The upper limit in (6) is reached if and only if the “equivalent potential” &3 -eV (e.g. Whipple, 1977), has no local maximum between P, and Pz, that is if and only if

throughout the interval 8,~ B I &. This rehation is illustrated by the solid contour in Fig. 2 being everywhere above or on the dotted straight line. With (9) being satisfied the first integral on the right hand side of (6) represents the “field-aligned” portion [sZ(K)~2n] of the precipitation at P, whereas the second integral represents the isotropic part [n(K) = 2~1. Whenever the potential distribution V = V(B) drops below the dotted line in Fig 2, as iliustrated by the dashed contour, the current decreases as a result of removal of electrons with large pitch angles i-w*) and the precipation flux becomes “field-aligned”. If the source electrons at P, have a Maxwellian distribution, that is

where n, refers to complete isotropy (n, is twice the density of downflowing electrons), the relation (6) may be replaced by

(lla)

The right-hand member of this relation is equal to the current density derived and evaluated numerically by Knight (1973) for upward field-aligned currents, with B1 being the equatorial field strength. A more practical form of (lla) is

where equality is reached asymptotically when (9) is satisfied and (B,/(B,-B,))eAV/kT,,~O. The product outside of the parenthesis is approximately

prG. 2. ANY D)STRYBLlTION BETWEEN

v,

STRAIGHT

LINE

RENT AND B,,

AND v

v= WHICH

a B PRODUCES

MAXIMUM GIVEN

v,,

ISOTROPY

THE PoTENTlAL

v(B) IS

OF THE ABOVE

MAXIMUM OF: THE

POTENTIAL

OR

ON

ELECTRON

ELECTRON

DB%ERENU3

v,

-

THE? CUR-

FLUX

AT

v, .

Any distribution which falh below, like the dashed graph, produces less current and more cdtimatian of the flux. equal to 2,7X lo-l4 n,@?~ with n, in nt-’ and kT,, in eV. 2.1.2. Elastic ~~~d~~ scatter&g. When random scattering occurs (3) and (4) are no longer true. However, as long as the scattering is elastic (5) is stili true and therefore the two expressions (6) and (lib) still define the upper limit on the possible current density at a given tatal potential difference AV. That is, the scattering can only reduce the current by transferring electrons into the otherwise empty “toss cone” in the return fhrx from below. 2.1.3. Inelastic and non-random scattering. In principle, inelastic and non-random scattering may increase the current density at given AV, but the effect may also be the opposite, depending on the nature of the scattering process. A systematic decrease of the magnetic moment of the electrons wifl increase the current, but a ““frictional force” acting primarily on the parallel velocity component will not.

The total current density at P2 in Fig. 1 consists of three major components

i (12= i412+iillz+ i4i2 (trans),

(12)

where i4,2 is due to precipitating hot electrons, i’llz is due to the outflux of positive ions from the ionosphere and i;l2 ftrans) is due to the depletion of cold electrons from high altitudes. The ion current density iillz. depends to some extent on the depth of penetration of the electric field but is otherwise independent of AV for AVZ 1OV (which corresponds to the gravitational potential at infinity for oxygen), in which case the ion

Aurora1 plasma-geomagnetic field interaction current is saturated (cf. Lennartsson, 1978). As long as the parallel electric field is within or above the topside ionosphere i’ttZ is limited to what can be supplied by thermal emission from the topside and i’il.-10-7-10-6Am-2 (cf. for instance Lemaire and Scherer, 1974). In view of the quite low plasma density seen at several thousand km altitude in the aurora1 regions (see e.g. Mozer et al., 1977), the lower figure may be a typical value. The negative contribution from precipitating positive ions and escaping backscattered electrons are assumed negligible in (12). The transient current density, iilZ (trans), is subject to somewhat more complex limitations. If cold (1 eV) electrons were free to drift downward at their thermal speed and n,(cold) - 10cme3 at an altitude of lR,, this current density might reach lo-’ Am-* at ionospheric altitude (and the cold electrons would retreat at a speed on the order of lR, per 10s). However, as the cold electrons become depleted at high altitude the cold-ion density has to adjust to the densities of the remaining hot particle components. n, (cold) = ne(hot) - r+(hot).

(13a)

If the cold electrons drift at a speed of ued, then

x(v,,,, fv,,

+m)-r

n+,,

(13b)

where “ > ” accounts for cold ions trapped above the electric field, B is the local magnetic field strength, uah the ion thermal speed, AV the total potential difference and B. and q, refer to some lower altitude, say lR,. This follows from conservation of flux in a frame of reference moving with the cold electrons. Hence, (13a) is strongly affected by v,+ The difference between ved= v~),,), and v,, =5 uith may be more than a factor of 40 in ~(cold). If (13a) cannot be satisfied, the electric field, of course, reverses and the cold electrons stop moving. Therefore, it has no meaning to speak of the local “resistivity” at a certain point; the entire flux tube must be considered. This is placed in its appropriate context below.

3. THE ELECTRIC FIELLB

It follows from (lib) and (12) that the current path from P2 to Pi in Fig. 1 has a finite real impedance associated with the magnetic mirroring of negative charges from PI.

139

Hence, if (14) a nonzero parallel electric field component must exist somewhere between Pi and Pz, having a net integral AV that satisfies (llb). More precisely, in order to precipitate a large enough number of electrons, electrostatic energy amounting to eAV has to be fed into the paraliel velocity component of certain electrons to barely compensate for the transferring of parallel to perpendicular kinetic energy by the converging magnetic field lines. The same amount of energy is given to other precipitating electrons, as well, resulting in excess parallel kinetic energy. The most efficient transfer of electrostatic to kinetic energy occurs when (9) is satisfied, resulting in a “quasi-linear” relation between iilZ and AV. When the electric field has a time varying spatial structure, the current densities in (llb), (12) and (14) may be defined as time averages over a typical travel time of the hot electrons and AV consequently becomes the net potential difference seen by the “average electron”. The only exception occurs when transverse kinetic energy is systematically removed from the electrons by some waveparticle interaction. This is oniy a boundary condition for the electric field, however. The actual spatial and temporal structure of the electric field is defined via a complex interaction with all charged particles along the current path, both positive and negative, and (llb) can be satisfied with an infinite set of electric fields. For the following discussion it is necessary to define “steady” and ‘“quasi-steady” states. Definition 1. In a “steady” state, the temporal changes are negligible during a complete transit of any ion from P2 to P, in Fig. 1. Definition 2. In a “quasi-steady” state, the temporal changes are negligible during a transit of any ion through any spatial substructure under consideration. If the external voltage source (between P1 and P4 in Fig. 1) is of a steady-state nature, the electric field along the current path can, in principle, be of either one of the following three types: (El) a steady-state field which allows quasineutrality of the plasma at all points between P, and P2; (E2) a steady-state field which violates quasineutrality in thin spatial regions; (E3) a non-steady field with or withoutsteady or quasi-steady structures embedded.

140

W. LENNARTSSON

It is the purpose of the following section to exclude (El) as being a realistic solution. Solutions of type (E2) may be more realistic and require formation of steady space charge structures (“double layers”, cf. e.g. Block, 1972) of various geometry. Type (E3) represents the most general possible solution and includes various plasma wave modes. 3.1. Local space charge effects The electrostatic potential V is defined by Poisson’s equation, which for singly charged particles may be written V * V V = .so-‘e( n, - q). For the present purposes this is rewritten as 62V-e~o-‘A2(n,-~)~1.8x10-8A2(pl,-~),

(15)

where S2V is the second order spatial change of V and A is a measure of the smallest spatial dimension of the region of excess charge of one polarity. Typically, A is a distance along the electric field vector. The individual number densities, n, and nZevidently cannot be defined by (15) only, but are subject to other conditions as well. At any given point I? the density n, of a certain particle species 1, can be written as the integral over velocity space of the phase space density f”

member is more than 8 orders of magnitude smaller than the right hand member. If the plasma were scatter-free and the potential dist~bution were characterized by a spatial scale A -R,, it then followed from (15) that (17) where q,, denotes the charge (+e or -e). This condition has no physical meaning however, unless it represents a stable state. For a simple ilfustration of the opposite state the cold plasma components may be considered. In order to do that it is necessary to consider separately those regions where lV1l* El >>/V, - El (parallel electric field) and those where /V, - El 2 IV, * El (oblique electric field). Suppose the electric field has an upward parallel component and the ions and electrons entering from below both have isotropic Maxwellian source distributions with the same temperature T (the ions filling a solid angle of 27r, the electrons a solid angle of 4~r). NOW. where IV,.EI >>/C,-El the transverse acceleration can be neglected in the frame of reference moving with the local EXE? drift. In this frame, uIZ/B= constant and the velocity distribution remains gyrotropic, f(ur, u2, Q)= f(v,, q). It then follows from Liouville’s theorem that fy is “nearly isotropic”, and hence

JJJ

where v refers to particles of a certain group, like “hot electrons”, etc. If each particle at P can be traced backwards through velocity space to some other point then f, can be constructed from the phase space density at each such point by means of Liouvitle’s theorem. If the potential distribution is time-invariant, it then follows that n,, is a unique function of the potential V and the spatial coordinate r=(ri, rZ, r3), and hence $$+

$ $ ~~~-“ i1 t i

For any realistic velocity distribution dn,, -I <
II

(16)

where the radius of the Earth, RE, is a characteristic spatial scale length of both the Earth’s magnetic field and the plasma distribution. For instance, if n,, is the density of a Maxwellian distribution of temperature kT, which is being reflected by the electric field the left hand member is equal to kT/(eq), or less. With kT= 1 keV and n = 1 crnm3the left hand

n, =

J JJfud’u =2rJYv(u) 0 x(cos cx,,,in - cos a,,,)~~ du,

(1%

where f”(u) is independent of 1~ in the interval a,,(u)5 (Y5 q,.,,,(u) and zero elsewhere, and cos cy =e/v. By setting uL a constant of the motion (implying cx,,,_ = 180” for downward B), assuming B = constant, and by neglecting the gravitational force (5 10v6 eVm-‘), the ion and electron number densities, n, and n,, are seen to vary with V as illustrated numerically in Fig. 3. Here the particle sources are to the right. In Fig. 3, the deviation from charge neutrality is approximated by n, - n, - 0.1 x n, (0). By comparison with (15) it follows that A is of the order of a few Debye lengths, A - 3 x J~okZ’,~~2n~-’ - 3 x AD, and the electric field is approximated by

With n, = lOem-” and kT= 1 eV (cf. Mozer et al., 1977), this means EZO.1 Vm-‘. The important point here is that the potential gradient in Fig. 3 only spans a few volts. Yet the associated electric field is strong and is not limited

AuroraI plasma-geomagnetic fieid interaction n/n,

,E t

\r IkTiel FIG.

3.

i%JhfFJER

ELECTRONS, sUM~NG

Iii

THAT

DENSiTY

AND THE

?l,, IN

DEXRIBUTlONS A

POTENTLAL

GYRATIONAL

OF

IONS

GRADlFXT.

ENERGY

t?lVL2/2

AhXl ASIS

PRESERVED. partides enter from the positive side with an isotropic Mazweliianvelocity distribution.The dashed graph shows ni(V) fur ions which alI enter with paratlel vetocity ody. The small negative charge near V = 0 disappears . when the electrons drift slowly (v,, i= vith) m the directron of the applied electric force (towards the right). The ion density ni and, hence, the spatial gradient of V increase in magnitude with increasing electron drift speed.

The

in space to this region. It must extend to the left as far as necessary to find a matching negative charge, and the total potential difference can only be limited by external conditions. The smafi potential difference in Fig. 3 makes it impossibb for any particle component other than the cold components to neutralize the net positive space charge in a stable configuration. Since, by definition, the other components, including backscattered electrons, have a different energy distribution, their number densities are not sensitive to a potential change of only a few volts. Therefore, (17) can only be satisfied via the spatial dependence of the other components (cf. for instance, Chiu and Schulz, 1978). This is a highly unstable arrangement, however. For if h in (1.5) is much larger than a Debye length of the cold electrons, ho, even a very smaff displacement of the cold electrons in the direction of the applied electric force (downwards) teads to a large enhancement of the eiectric field and, consequently, to a further increase of n, -n, by reflexion of cold electrons. This is equivalent to a lowering of the potential energy of the cold electrons and absorbs energy from the external voItage source. Where /Vi * El 5: \V,,- El (oblique electric field) the ion number density is modified as a function of V. As long as f(v,, I.I~,u,)=f(uL, u,,) the change in E, over a single gyration is small and, hence, the magnetic moment remains an adiabatic constant of the motion in a cur&free and time-invariant electric:

141

field. With B *constant this means uL2= constant. However, as IV, * El//V,,- El increases the parallel energy at a given potential decreases, and an increasing fraction of the energy goes into E XB motion. Since the phase space density is preserved, the net result is an increase in n, at a given V [cf. (IS)]. The electron density, on the other hand, remains nearly the same function of V because the electrons are reflected and their EX B drift has negligible energy. This means that the net space charge is still characterized by A -AD and, hence, gyrotropy must break down since A,5 a cold-ion gyroradius pi for parameters of interest. This break down occurS when the spatial change of eE, over an ion gyroradius is comparable to, or larger than, the Lorenz force eziL,B, that is when pi@‘, * E,\ z u,$?, Since pi = v,,f~,~ this can be written as /v, . v, vi 22w.&

(194

where wsi is the angular gyrofrequency. This relation implies E >O.l Vm-’ over a few cold hydrogen gyroradii at an altitude of II?,. Note, the total &V associated with (19a) can only be limited by external conditions. Where IV, . El CCIV,,- El the appropriate formula is still IV,,. V,,VI - k T,/kb*~.

(19b)

As a resutt of this consideration, it must be concluded that solutions of type (El) are unstable. The cold electrons can lower their potential energy by moving collectively downwards until either (19a) or (19b) occurs. Whether or not this leads to a stable state is a much more complicated problem. When (19a) occurs it has a profound effect on the ion pitch-angle distribution, which wit1 be discussed briefly below. Since only a limited aspect has been considered, it cannot be excluded that solutions of type (E2) exist. However, (171 is an arbitrary condition. ft does not follow from Poisson’s equation (IS). This latter equation merely implies that the spatial dimension of a space charge region be small enough that e6V 1 the maximum kinetic energies involved. The most general solution of (15), subject to the boundary condition (lfb), is therefore a nonsteady, fluctuating, electric field, everywhere characterized by A <
(20)

which can have any direction with respect to the magnetic field vector at a given point. Even if the electric field is of this general nature, it may contain quasi-steady sub-structures of large

142

W. LENNARlSSON

e-

w

(hot)

$1

‘II

(b)

I

e- fcdd!

Fro. 4. (a) AN FIELD FROM

FIELD AND

ARE AND

EXPEL

REGION

SCATS’ERED

TRAPPED

DIRECTED

REMOVES

A PITCH-ANGLE

WHICH COME

UPWARD

COMPGNENT

AROUND

THE MAGNETIC

A PGSITION

90”. (b)

MIRROR FROM

ELECTRIC

WITHIN

BELOW,

A COLD

J3J3xRK EL,ECTRONS

INTO TKfS PITCH-ANGLE

B ETWEEN

ELECTRONS

P ARALLEL

DOWNFJ_OWING

ELECTRONS REGION

BE-

THE EIECllUC

UNLESS

PLASMA

a,,,
BELOW

THE

RELD.

This process corresponds to an upward current with a non-zero divergence. amplitude. This is indicated by observations (e.g. Mozer et al. 1977). In the formation of such structures, charge trapping must play a crucial role, as outlined below.

Consider Fig. 4. The upper half of part (a) depicts a contour of constant phase space density f for hot electrons from the magnetospheric source (at P, in Fig. 1) at a given kinetic energy K. It is assumed that f is isotropic, except for the empty loss cone in the return flux from below. The lower half of part (a) depicts a contour of the same constant f at energy K +eSV after passage through a localized upward directed electric field with elEii >KK’[VBI (-10-s-10-3Vm-‘). Immediately after passage of the electric field, f is no longer isotropic around a! = 90”, but is zero witbin a ‘Lforbiddeny’ pitch-angle interval o,
(21)

This pitch-angle interval is accessible only to electrons that are scattered below the electric field. Any electron that is scattered into this “forbidden”

interval becomes trapped between a position within the electric field and the magnetic mirror below. The counterpart of (a) in coordinate space is illustrated by part (b). It is assumed here that a cold plasma of high “conductivity” is present below the electric field, at least. The solid curve labelled e(hot) depicts the trajectory of a “hot” electron that is scattered into the “forbidden” velocity region in part (a). In order to maintain net internal charge neutrality, the cold plasma must expel a “cold” electron to the ionosphere and rearrange the charges as depicted by the solid and dashed line labelled e- (cold). These charge motions carry a net upward current (from PZ to P, in Fig. 1) which is discontinuous, that is, a charging current. The important qualitative point to be made here is that the trapping necessarily leads to an increased electric field. The accumulation of trapped electrons within the electric field, of course, leads to a rearranging of the other charged particle components, but only as a result of the increased field strength. In the simplest config~ation the number density of the other components at a point within the electric field structure is a function only of the potential differences between that point and the adjacent plasma on either side. The direct effect of trapping is therefore a net increase of the electron density at a given potential, expressed as n, = n,(SV, V, - V), say. Since n, > n, on the negative side to start with, the result is an increase in the electric field strength, either in the form of an increased 6V or in the form of a decreased thickness d of the electric field structure. It is noted that the cold plasma. in particular, is unstable with respect to trapping of hot electrons, because if an initial 6V of only a few volts is applied along the magnetic field (by a wave field, for instance) within the cold plasma, the rearranging of charges is such as to enlarge the electric field (as can be realized from Fig. 3 by interchanging the roles of ions and electrons). As trapping of hot electrons then occurs the electric field grows further. For a rough estimate of the electrodynamic effect of this trapping, suppose the region of strong electric field in Fig. 4(b) is a plane slab with a thickness d, small compared to the transverse dimensions and with an angle Jf between the electric field vector and the magnetic field line. If H is the altitude of point P’ above the atmosphere (the latter being defined by the 100 km altitude layer, say), the trapped electron spends at least a fraction d(cos ~}-‘/(~~d(~s +)-I) of its bouncing period within the electric field. Hence, the capacitance C per unit area of current cross-section is, to the

Aurora1 plasma-geomagnetic field interaction order of magnitude, limited by C = (i9V/aQ (trapped)))’ C: so(H + &OS @)d-*. (22) This is a quite small capacitance except where 9 r= 90”. For example, if H = 104 km, d is as small as 10 km, say, and cos Q 2 d/H the capacitance is still only about low3 PAS mm*kV‘-‘, or less. For most trapped electrons the capacitance is much less, because they mirror far above the atmosphere. If the number density of hot electrons is 1 cm-” and the average energy - 1 keV, the downward flux density is about 1 x 10” mm2s-‘. With Clo-’ PAS m-‘kV_‘, only about lo-’ of these electrons need to be scattered into the “forbidden” velocity region to increase the potential by 1 kV over a period of 1 s. Alternatively, out of the entire population of hot electrons (from P,) below an altitude of lo4 km, only a fraction of lo-” needs to become a trapped population to increase the potential by 1 kV over an arbitrary period of time. By setting K = eSV = 1 keV in (21) it is realized that the “forbidden” velocity region is capable of holding much more than this fraction, however. The number of trapped etectrons must continue increasing at each energy until either the “forbidden” region becomes filled, or a,(K) becomes smaller than the half-angle a, of the local loss cone. The first condition must be satisfied above some energy Kc, the second below this energy. The second condition is equivalent to a collimation of the precipitation flux at P2 in Fig. 1. Hence, if no other potential drops exist below point P’ the precipitation flux at P2 is collimated within the energy interval eAVsK
143

are most favourable when the capacitance in (22) is large, that is when d is small (a thin structure) and 4 -90” (E, >>E& This is apparently consistent with observations (cf. Mozer et al., 1977). At this point it is important to recall (13a) and (13b). As the cold electrons become depleted at high altitude an upper boundary of the cold plasma must form. In order for ib (trans) in (12) to be non-zero it is necessary that this boundary moves downwards at some speed u,+ In passing through this boundary, the cold ions have to adjust their density to satisfy (13a). By setting B = Bo, AV = SV and u,,! = ud = uelh in (13b) it is realized that changing the cold-ion density by only an order of magnitude, it is necessary that eSV - 100 x (m,/2)czt,, 180 keV. In other words, the cold electrons contribute a large transient current only if AV( > SV) is large. A typical value of vd during quasi-steady conditions may be ud-with corresponding to ii2 (trans) -10m7-10-6 A mm2 and SVS a few keV. 4. CONCLUSIONS

The most immediate conclusion that can be drawn from (1 lb) is that the hot magnetospheric plasma is a poor conductor of field-aligned current. Increasing the electron precipitation flux by a factor of Cp, as compared to the field-free precipitation, requires a parallel electric field with an integral AV satisfying AV=-(CD- 1) kT,,/e.

(24)

The same conclusion, of course, holds for any magnetically confined plasma and implies that “frozen-in” magnetic field lines are a concept with strong Iimitations in cosmic physics. It is true that the magnetospheric plasma also has a cold component of ionospheric origin but this component has a quite limited “conductivity” with respect to outward field-aligned currents, as seen from (12) and (13a and bf. The cold electrons alone cannot carry such a current for any length of time in the same flux tube if the “dynamo” current P, -PA in Fig. 1 is carried by hot particles. Moreover, the cold transient component in (12) is large only if AV is large, because of (13a) and (13b). The physical meaning of this is quite simple: whenever a magnetic flux tube is receiving more net negative charge by particle drift than it is losing via naturat electron precipitation and neutralization by upflowing ions, a parallel electric field develops to increase the precipitation. This may be concluded without invoking any kind of “anomalous

144

W. LJZNNARTSSON

resistivity” or “current-driven” wave-particle interaction. As a qualitative conclusion, this must be true also in the presence of scattering and other nonadiabatic processes. In particular, (11 b) remains valid in the presence of random elastic scattering of arbitrary strength. A certain rate of scattering is actually required to provide the isotropic electron source flux assumed in (llb). Only with the help of an inelastic scattering process that systematically removes the transverse kinetic energy of the electrons is it possible to invalidate (llb). Apart from this case, (llb) is independent of the temporal and spatial structure of the electric field, if by AV is meant the net potential drop seen by an average precipitating electron. This is a generalization from Lennartsson (1976, 1977) and offers a qualitative explanation of the observed energy spectrum of aurora1 electrons, at least of the basic and most typical features. In the typical case the spectrum does show the signature of electrostatic acceleration with eAV5 several times the apparent thermal energy (e.g. Evans, 1974; Lundin 1976). According to (llb) this is to be expected when the net influx of negative charge at high altitude requires an increase of the precipitation by several times. The only assumption required is that the negative charge entering the lh.rx tube at high altitude be carried solely by energetic electrons. Other features of the observed phenomena, like the frequent collimation of the electron fluxes along the magnetic field lines, certainly cannot be explained by this simple scheme, but are related to the structure of the electric field. However, a proper consideration of the effect of magnetic mirroring again proves fruitful, when it is recognized that a time-invariant electric field and a quasi-neutral plasma represents an energetically unstable system. When no scattering occurs q and n,, and hence, the space charge e(s -ne,), are uniquely determined by constants of the motion over spatial scales of h -RE- Because the potential V is so sensitive a function of the charge deviation, this leads to overspecification of n, and n,, as expressed by (15), (16) and (17). As a result, quasi-neutrality breaks down and scattering starts via etectrostatic fluctuations, as expressed by (20). This is entirely equivalent to having the entropy increase, which is characteristic of any thermodynamic system. Since the charged particles do have a certain inertia, in particular the ions, any rapid fluctuation in the electric field must involve oscillating components. The fact that electrostatic “noise” with frequen-

ties in the neighborhood of the local ion gyrofrequencies is commonly observed in aurora1 current structures (e.g. Gurnett and Frank, 1977) thus may not be surprising. This “noise” does not per se imply “anomalous resistivity”. On the contrary, the relatively low energy density (“pressure”) of the wave field as compared to the kinetic energy density (“pressure”) of the hot electrons strongly suggests that the electrostatic fluctuations have little or no direct effect on the “‘resistivity”. The “intense noise” observed by Gurnett and Frank (1977), for instance, appears to have an energy density of less than 100 keV mm3(see Fig. 6 of their paper.) This is more than four orders of magnitude less than a typical energy density of the hot primary electrons (- 5 keV x lo6 m-‘). Even the cold plasma has a higher “pressure” than this, by two orders of magnitude or so (- 1 eV x 10’ m-“). Scattering inevitably leads to trapping of hot electrons between a localized upward electric field and the magnetic mirror below. No stable electric field structure can exist until this trapping has saturated. Since this is most easily reached in a thin structure with the electric field nearly perpendicular to the magnetic field [in a structure with large capacitance, see (22)] it may be understood qualitatively, why such structures, sometimes referred to as “paired shocks” (Mozer et al., 1977), are observed imbedded in regions of electrostatic “noise” and strong electron fluxes (Mozer et al., 1977). It should be noted that this inte~retation does not invoke any current “pinching”, but does invoke a combined thermodynamic (scattering) and electrostatic (trapping) effect. When the trapping has saturated it is very possible that the energy K, in (23b) has reached such a value that a strong colIimation can be observed in the precipitation flux at low altitude. In fact, K, will be larger than the thermal energy of the source electrons if eW = eAV 2 (B2 - B’) kT, JB’. Such a case is illustrated schematically by the dashed contour in Fig. 2. One of the first attempts towards a quantitative treatment of the field-plasma distributions within a thin electric field structure with IV, - El 2 (Vi1-El was made by Swift (1975 and 1976), who refers to these structures as “oblique electrostatic shocks”. The thinness of the “shock”, a few km, is taken as an assumption in these calculations, based on observational evidence of fast E XB drift in aurora1 structures, and no attempt is made to explain the formation process. It is taken for granted in these calculations (as is done in most other similar models) that gyrotropy is conserved, f(u,, ~2, 03) zf(n~. ~II),

145

Auroral plasma--geomagnetic field interaction 5K-3 AK,-n,

keV

AK-1

213 eV~kin@,~ x

A KrlWeV

kOS@)

keV

a isinil12x

K

KL -500 eV (if Jt ? 45’1

Ii+

o+

(costi)

213

x(sin

$I2

H+ O+

The transverse electric field component, E,, is assumed to have a transverse distribution as shown in the bottom graphs. The magnetic field strength is chosen to be representative of an altitude of 1R, within the aurora1 regions. The increments in the transverse kinetic energy, AK,, as shown at the top, are rough approximations only. The exact analytical expressions are rather complex. for all particles, even though the magnetic moment of the ions is not. Therefore, the solutions obtained are difficult to interpret physically. One of the results of the present study is that gyrotropy must always break down for the ions in a stable electric field structure of this kind, including the cold ions. As a consequence, the ions tend to obtain a wide pitch-angle distribution which may readily take the form of a “conical” distribution, as observed by Sharp et al. (1977). This is highly interesting in itself, and deserves a separate comment. It is outside of the scope of this report to analyze the consequences in detail, however. Consider Fig. 5. It depicts schematically two thin eiectric field structures at l& altitude tilted with respect to the magnetic field. The angle between the electric field vector and the upward magnetic field-aligned direction is 9. The transverse electric field E, varies across these structures as shown by the bottom graphs. This distribution of E, is compatible with measurements of “quasi-steady” structures by Mozer et al. (1977). It is assumed here that (19a) is fulfilled for H’ on the bottom side of the structures, within a depth as indicated. For 0’ the relation (19a) is

satisfied by a wide margin throughout the right structure. If ions with energies -1 eV enter from below, they can be seen to gain total AK and transverse AK, kinetic energies as shown in the upper part of the figure. The important point here is not the exact numbers, but rather the fact that no ions passing through these structures from below have a pitch angle of 180”. In particular, in going through the right structure 0’ acquires a pitch angle (Y=i: 180” -Vr. It makes no difference whether P is such that the particles reach within the structures during many gyrations after acceleration (Y-900), because the electric field is repelling. The crucial effect of the electric field is the initial acceleration, which destroys the magnetic moment and the gyrotropy of the cold ions from below. Structures of this kind may well exist for only a few ion gyroperiods. If zV=90° to the left in Fig. 5 and the life time of the structure is - a gyroperiod of 0’ it follows that 0’ will gain much more energy than H’. If the O’-ions encounter several of these short-lived structures they continue to gain transverse energy while moving upwards. As a consequence, 0’ may generally appear with higher

146

W. LENNARTSSON

energy than H’ in regions of fluctuating electric fields in accordance with observations (Ghielmetti et al., 1977). It is recognized that several mechanisms have been suggested in the literature to account for “conical” and generally wide ion pitch-angle distributions in terms of “heating” by ion cyclotron waves (e.g. Lysak et al., 1978; Ungstrup et al., 1979; Kintner et at., 1979). It is also recognized that ion cyclotron waves are observed associated with these ion events (e.g. Kintner et at., 1979). It is noted, however, that gyrating ions are a prerequisite for this kind of waves to appear in nature, so there is a definite possibility that the waves are a secondary effect, draining energy from the ions. This possibility is even consistent with the fact that reported “conical” ion distributions (c.f. Sharp et aL1977) may have an energy density on the order Of 10” eV rnT3, or more, which is three or more orders of magnitude larger than the electric energy density of the most “intense” cyclotron waves (e.g. Gurnett and Frank. 1977; Kintner et al., 1979). According to the reported data (e.g. Sharp et al., 3977; Kintner et al., 1979) the “conical” ion distributions occur in regions of intense precipitation of energetic electrons, preferentially at altitudes above 5000 km. This would indeed be expected if the “tonics” were produced by strongly inhomogeneous El-fields, resulting from charge trapping, as sketched above. A given number density of trapped hot electrons will more easily match (19a) at 5000 km altitude than at 1000 km, say. The electric field structure does not have to survive very long for (19a) to be satisfied, only long enough for the cold electrons to move - a cold-ion gyroradius relative to the structure (lowering their potential energy). On the other hand, the structure has to survive for a time - an ion gyroperiod in order for any appreciable energy to be transferred to the ions. If these structures appear in a random fashion over a large altitude interval they continue to increase the transverse energy of ions moving upwards, on the average, because the probability of an ion traversing a structure from the positive to the negative side is always larger than that of opposite traversal, if the ion is moving upwards and El1 is upward. That is, E, then has the right “phase” for acceleration, on the average, irrespective of direction within the plane of E,. Any temporal changes of E, over an ion gyroperiod would cause a gyrophase modulation of the ion fluxes and hence, cause ion cyclotron waves. The probability of detecting these waves by a spacecraft would presumably be much larger than that of detecting

shortlived and spatially localized structures of strong E,. Besides, these structures might not appear drastically different from large-amplitude cyclotron waves to a fast moving spacecraft. For a brief summary of these conclusions, see the introduction. A more stringent study of charge trapping and its effects on the pitch-angle distributions of electrons and ions is underway and will be reported soon. AcknowledgementsThis work was supported in part by the Swedish board for Space Activities. The work at Lockheed was supported by the Office of Naval Research under Contract N00014-78-C-0479. RFFEmNCES

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Aurora1 plasma-geomagnetic Lennartsson, W. (1977). On the role of magnetic mirroring in the amoral phenomena. Astrophys. Space Sci. 51, 461. Lennartsson, W. (1978). On the parallel electric field associated with magnetic mirroring of aurora1 electrons-some basic physical properties. Technical Report TRITA-EPP-78-08, the Royal Inst. of Technology, S-10044 Stockholm 70. Lundin, R. (1976). Rocket observations of electron spectral and angular characteristics in an “inverted-V” event. Planet. Space Sci. 24, 499. Lysak, R. L., Hudson, M. K. and Temerin, M. (1978). Enhanced ion heating by coherent electrostatic ion cyclotron waves. Trans. Am. geophys. Union 59,1155. Mozer, F. S., Carlson, C. N., Hudson, M. K., Torbert, R. B., Paraday, B., Yatteau, J. and Kelley, M. C. (1977). Observations of paired electrostatic shocks in the polar magnetosphere. Phys. Rev. Lett. 38, 292. Papadopoulos, K. (1977). A review of anomalous resistivity for the ionosphere. Rev. geophys. Space Phys. 15, 113.

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Persson, H. (1966). Electric field parallel to the magnetic field in a low-density plasma. Phys. Fluids 9, 1090. Sharp, R. D., Johnson, R. G. and Shelley, E. G. (1977). Observation of an ionospheric acceleration mechanism producing energetic (keV) ions primarily normal to the geomagnetic field direction. .I. geophys. Rex 82, 3324. Swift, D. W. (1975). On the formation of auroral arcs and acceleration of aurora1 electrons. .I. geophys. Res. 80, 2096. Swift, D. W. (1976). An equipotential model for aurora1 arcs--II. Numerical solutions. .I. geophys. Res. 81, 3935. Ungstrup, E., Klumpar, D. M. and Heikkila, W. J. (1979). Heating of ions to superthermal energies in the topisde ionosphere by electrostatic ion cyclotron waves. J. geophys. Res. 84,4289. Whipple, E. C., Jr. (1977). The signature of parallel electric fields in a collisionless plasma. .I. geophys. Res. 82, 1525.