Allowance for thermal flux variations in the model of ionosphere-plasmasphere interactions

Plmef. Space Sci., Vol. 39, No. 6. PP. 847457, Printed in Great Bntaio

0032-0633/91 s3.ot+o.ml Pergamon Press plc

1991

ALLOWANCE FOR THERMAL FLUX VARIATIONS THE MODEL OF IONOSPHERE-PLASMASPHERE INTERACTIONS

IN

0. A. GORBACHEV and 1. M. SIDOROV Irkutsk Polytechnical Institute, 664074, Irkutsk, U.S.S.R.

YU. V. KONlKOV Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation of the Academy of Sciences of the U.S.S.R. (IZMIRAN), 142092, Troitsk, Moscow Region, U.S.S.R.

G. V. KHAZANOV Altai State University, 656099, Bamaul, U.S.S.R. (Received infinal form 19 November 1990) Abstract-This paper presents a numerical model of ionosphere-plasmasphere interactions. Plasma parameters within the framework of the model are calculated on the basis of a complete system of equations (including the transfer equation for thermal fluxes) and by use of a standard approach (the thermal flux is expressed in terms of the temperature gradient). The equations are integrated along the entire length of a geomagnetic field line between the conjugate ionospheres, together with the kinetic equation for photoelectron spectra. A comparison is made of space-and-time variations of macroscopic plasma characteristics calculated on the basis of a complete and standard system of equations. The most pronounced difference for electron temperature is observed during the morning and evening hours and is attributed to

the thermal flux non-stationarity. Non-stationarity effects also lead to the faster heating and cooling of electrons, respectively, at dawn and dusk. At plasmaspheric heights the ion temperature calculated by taking account of thermal flux variations, exceedsthat obtained based on a standard approach throughout the 24 h of the day. In the topside ionosphere the character of the ion temperature behaviour is similar to the electron temperature behaviour. The differences in height distributions of plasma densities are caused by those in electron temperature and are, therefore, largest during the morning and evening hours. The above differences in the calculated plasma parameters increase with increasing height and number of the L-shell. It is pointed out that a plasma heating caused by additional sources of magnetospheric origin must lead to an enhancement of the effects considered in this paper.

1. INTRODUCTION

One of the methods of investigation of near-terrestrial space implies developing mathematical models of ionosphere-plasmasphere coupling. Such models are based on the numerical solution of hydrodynamical equations for multi-component plasma which represent the laws of conservation of mass, momentum and energy. These models make it possible to obtain the distribution of macroscopic plasma parameters and to investigate their space-and-time variations, depending on different heliogeophysical conditions. At present there exist a large number of models to describe the “ionosphere-plasmasphere” system (see, for example, Briunelli and Namgaladze, 1988 ; Krinberg and Tashchilin, 1984; Oraevsky et al., 1985 ; Schunk, 1983, 1988, and references therein). They differ by the number of modelling equations and their form, by the extent to which external factors are taken

into account, the diversity of the processes described, the validity range in height and latitude, and by the allowance for other geophysical parameters. It is appropriate to single out a group of models in which plasma heating sources are calculated consistently

with the hydrodynamical equations. Thus, for example, in models of the mid-latitude ionosphere and plasmasphere (Polyakov et al., 1975, 1982; Krinberg and Tashchilin, 1980 ; Young et al., 1980 ; Konikov et al., 1982, 1983 ; Khazanov et al., 1984) the kinetic equation for the spectrum of photoelectrons which provide the main source of plasma heating in this geomagnetospheric region, is solved simultaneously. Such an approach permits the ionosphere and the plasmasphere to be treated as an integral dynamicand-energetic system, thus making it possible to adequately describe (with a reasonably full account of the external factors) the space-time distribution of plasma parameters. 847

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848

GORBACHEV

As shown by calculations made in the papers cited above, when modelling the ionosphere-plasmasphere interactions, one of the important factors is the study of the thermal regime of plasma components, which is accounted for by the presence of a strong interrelationship between the character of the behaviour of temperatures and of other plasma characteristics. Plasma parameters in the above models are calculated by numerical integration of the standard system of transfer equations (Schunk and Watkins, 1979). In this case it is supposed that the thermal energy flux S, involved in the heat conduction equations obeys the Fourier law S, = -rc,VT,,

(1)

where K, is the heat conduction coefficient, and T, is the temperature of charged particles of type tl, tl = e, i. The relationship (1) is valid if the macroscopic conditions 1, <
2, << to

(2)

are satisfied. Here 1, and r, are the mean free path length and time of particles of type tl, respectively, and to and L,, are the typical time-scale and longitudinal (with respect to the geomagnetic field) spacescale of the problem, respectively. Strong inequalities (2) do not always hold in the near-terrestrial plasma conditions and in such a case the use of the approximation (1), when calculating the temperatures, become incorrect. In this case the system of modelling equations should incorporate, instead of relationship (1), the transfer equation for the thermal flux. Within the framework of the Grad method (Grad, 1949; Schunk, 1977) used in this paper this corresponds to treating the thermal flux as an independent macroparameter, whose space-and-time variations are represented by their own differential equation. Note that from this equation, provided that the conditions (2) are satisfied, the relationship (1) follows.+ It is of interest to compare results of calculations

t It should be noted that a variation of one of the conditions (2) still does not imply mapplicabihty of a hydrodynamical description within the framework of the Grad method used in this paper. The inequalities (2) are only sufficient, rather than necessary, conditions for rapid convergence of the series for the distribution function of particles in Hermite polynomials in the Grad method (for further details see Kogan, 1967; Oraevsky et al., 1985). At the same trme their fulfilment guarantees not only the smallness of the corresponding expansion terms of the distribution function but also allows simple linear relationships [such as (I)] to be constructed, which relate non-equilibrium macroscopic quantities (thermal flux, in this case) to gradients of the main thermodynamical parameters.

et

al.

on the basis of a standard system of equations [where the thermal flux is described by relationship (I)] and of a complete system of equations involving the transfer equation of the thermal flux. Schunk and Watkins (1979) were the first to consider this question. They compared results of calculations of ionospheric and plasmaspheric plasma parameters obtained on the basis of a standard and a complete system of equations. The equations were integrated over a limited range of plasmaspheric heights, the stationary approximation was considered and the magnetic field was considered radial. This paper presents a theoretical model of ionospheric-plasmaspheric interactions, in which plasma heating due to photoelectrons is calculated consistently and the transfer equations for thermal fluxes of electrons and ions are taken into account. Within the framework of this model a comparison is made of the space-time distributions of plasma parameters calculated from a standard and a complete system of equations. Unlike Schunk and Watkins (1979), nonstationarity effects were taken into account, plasma heating was calculated consistently, and a dipole approximation of the geomagnetic field was used. Besides, integration in coordinates was performed throughout the length of the geomagnetic field line. This paper is organized as follows. Section 2 presents a system of modelling equations and discusses briefly the technique of numerical solution. Input parameters of the model are also given here. Section 3 presents results of calculations of plasma parameters in the ionosphere-plasmasphere system of equations. A comparison of them is also made here, and factors responsible for the observed differences are discussed. The main conclusions of this paper are formulated in the Conclusions. 2. THEORETICAL

FORMULATION

The model of ionosphere-plasmasphere interactions is based on the numerical solution of a system of hydrodynamical equations for multi-component plasma consisting of O+ and H+ ions and electrons. We shall not take into account the transverse (with respect to the geomagnetic field) plasma transfer caused by external electric fields. In this case the averaged motion of charged particles and the thermal energy transfer occur along geomagnetic field lines and, therefore, all macro-characteristics of plasma will depend only on time t and on coordinate s along the direction of the geomagnetic field. In this paper a system of equations in a 13-moment approximation is used. By disregarding the internal electric field and neglecting the inertia of electrons and viscosity effects,

Thermal flux variations and ionosphere-plasmasphere

the system of modelling equations can be represented as

dV, a w a m,N,dt+dS(N'T;)+N,s(N"Te) _m,N,gli

=

2

(4)

849

interaction model

electrons, thus causing their heating during the daytime. The thus heated thermal electrons in turn heat the ion component through collisions with them. In this connation the collisional sum on the righthand side of equation (5) for electrons can be represented as 6TJ& = Qo-L,, where L, represents energy losses of electrons when they are scattered due to ions and during inelastic collisions with neutral particles, and is the source of heating due to photoelectrons. Q = A-N,

$dEdR,

(7)

s (5)

where dJdt = apt-k ~={a/a~), in = B/B, is the crosssection of the geoma~etic flux tube equal to 1 cm2 at its footpoint, (B = B,), m, N, V are, respectively, the mass, density, hydrodynamical velocity along the geomagnetic field B, and g,, is free fall acceleration projected onto the field line. When writing equations (3) and (4), the suppositions about plasma quasineutrality and conservation of the electric charge were used. The sums on the right-hand sides of equations (3)-(6) represent the variation of the corresponding macroscopic plasma parameters owing to collisions of charged particles with each other and with neutrals. The model takes the following collisional processes into account. For electrons : Coulomb collisions, elastic collisions with neutral particles as well as inelastic collisions associated with excitation of rotational and oscillatory degrees of freedom of molecules and sublevels of an oxygen atom in state 3P (fine structure levels). For ions, allowance was made for Coulomb collisions and scattering due to neutral particles as a result of the polarization interaction (the gas approximation of Maxwellian molecules) and resonant charge exchange (collisions of ions with neutral particles of the same sort). Also, the contribution from the charge exchange reaction O+ + II e 0 f H +, which is a resonant one, was taken into account. The expressions for relevant collisional sums involved in the righthand sides of equations (3)-(6) for the above-mentioned processes were taken from Schunk (1983, 1988). This paper takes into account only one plasma heating source due to photoe&ctrons. Photoelectrons during their collisions transfer their energy to thermal

where 4 = 2EfJm: is the spectrum of photoelectrons, f is their distribution function. The spectrum of photoel~trons is defined such that the quantity Cp(s, E, 8, p) dEdS;t is the density of electron flux with energies from E to E+dEand with the velocity direction inside the solid angle dQ = sin 6 d6 dq at a point with coordinate s on the magnetic field line. Here 8 is the pitch-angle, cp is the azimuthal angle, A = 2ne2 In I, e is the charge of an electron, and In A is the Coulomb logarithm. For a consistent calculation of the thermal plasma heating, together with equations (3)-(6) we solved a kinetic equation for the spectrum of electrons in a drift approximation (Khazanov, 1979)

[s

1 -%,,+C&+d #; II Zn

X

0

q(E, m

cos (;o)dp

dE

Here : p = cos 0, II, is the density of neutral particles ; ‘se”, ze’, are the differential cross-sections of elastic scattering and ionization for neutral particles of type a$ are the total cross-sections of inelastic n;&;uc,,; scattering (k is the type of excitation for a neutral particle of type n), of elastic and ionizing collisions,

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GORBACHEV et al.

respectively ; Ek, EA are threshold energies of excitation and ionization. ,At the ionospheric height h < lo3 km the solution of (8) was sought in the form of a series in Legendre polynomials (Khazanov, 1979) +(s,Ep)

= f &(s,E)P,(p) n-0

(9)

that transforms (8) to a system of equations for coefficients of expansion of (9)-4,. Depending on illumination conditions in magnetically conjugate ionospheric regions, the number of expansion terms in (9) varies from two to four. The ionospheric solution was then joined with a plasmaspheric one, this being derived by solving a kinetic equation, averaged over oscillations between the reflection points of the plasmaspheric height range (h > lo3 km) (Polyakov et al., 1979)

Here f is the distribution function avera ed over the oscillation period, p0 = ,/m is the equatorial pitch-angle cosine, and $, and $2 are respectively determined as

where +s,,(p) is the reflection point, and a(s) is the cross-section of geomagnetic field tubes. The above method is discussed in greater detail by Polyakov et al. (1979). Apart from atomic O+ and H+ ions, the model included molecular NO+ and 0: ions, whose densities were calculated in the photochemical approximation. The pertinent schemes of reactions and their constants are given in reviews by Schunk (1983,1988). The solution of the system (3)-(6) was compared with results of calculations according to a standard system of equations, differing from (3) to (6) by the replacement of the equation for thermal flux (6) with the relationship (1). Let us consider briefly the method of numerical realization of the system (3)-(6). The equations were integrated using a two-dimensional difference “coordinate s-time t” network. In the first stage we calculated densities and velocities of particles. For that purpose, equations (3) and (4) were reduced to a single second-order differential equation for a flux of particles of a given sort. After the replacement of the derivatives with their difference analogs with a four-

point approximation according to an implicit divergence scheme, we arrive at a system of non-linear algebraic equations with coefficients which depend, in a complicated manner, on the desired functions, their derivatives and on a number of other ionospheric parameters (Samarskii and Nikolaev, 1978). The difference equations were linearized by calculating the coefficients from values of unknown functions taken from the preceding temporal layer, or from the previous iteration. The systems of difference equations were solved by the method of passage, and steps in coordinate and time were chosen from the condition of passage stability and monotonicity. The solution of equations for molecular ion densities was searched by the method of successive approximations. The obtained values of densities and velocities of charged particles were used in the simultaneous solution of the kinetic equation for the spectrum of photoelectrons and of equations (5) and (6). The simultaneous solution of equations (5) and (6) was performed by the method of matrix passage (Samarskii and Nikolaev, 1978). When integrating the standard system of equations (3)-(5) and (l), the equation of thermal balance (5) [together with relationship (l)], was solved by the same method as equations (3) and (4). Iterations were made for each time layer until the solutions were obtained, with the required degree of accuracy. The system of equations was integrated along the entire length of a geomagnetic field line. Steps in coordinates were chosen proportionally to the plasma scale of heights. The step in time was varied from N 10 min in the morning and evening periods of l-2 h during the daytime and at night. The boundary conditions needed to solve the equations were specified at the ends of field lines in the magneto-conjugate ionospheres at height h,, = 100 km. At these heights the transport processes can be neglected, and values of the desired quantities can be determined from the local equilibrium conditions : V,(so, t) = 0, T,(so, t) = T,(q,, t), and SZ(.sO,t) = 0, where T,, is the temperature of the neutral atmosphere, and is a coordinate along the field line corresponding to height ho. Time integration is performed until a steady-state periodic solution with a period of 24 h is obtained. The calculations have used a neutral atmosphere model (Hedin et al., 1977), spectra of solar U.V.emission (Hinteregger, 1970) and ionization and absorption cross-sections of neutral particles (Stolarsky and Jonson, 1972). 3. RESULTS OF CALCULATIONS

AND THEIR

DISCUSSION

Based on the calculations performed here we have inter-compared the space-and-time distributions of

851

Thermal flux variations and ionosphere-plasmasphere interaction model

plasma parameters in the ionosphere and plasmasphere for a standard (3)-(9, (1) and a complete (3)-(6) system of equations under different helio~eophysi~l condi~ons. A plasrn~~he~~ region with L = Z-5 was considered. In what follows, we present the results of the calculations, corresponding to the equinox conditions (symmetrical conditions of illumination in the magneto-conjugate ionospheres) and to moderate solar activity, Flo 7 = 150. Height profiles of the calculated parameters are given from the lower boundary of the field line to the equatorial plane. 3.1. Electrons The results of the calculations have shown that, for the steady-state solution analyzed here, V CCI_+, where uTC= ,/G is the thermal velocity of electrons and, therefore, the cont~bution of the terms containing the directed velocity in equation (6) is small. In this connection the differences in values of fluxes calculated by equations (6) and (l), will be determined by the presence of the non-stationary term. The infIuence of thermal flux non-stationa~ty, when the electron temperature is calculated, must most conspicuously be manifested in periods of temporal rearrangement associated with sunrise and sunset. This is confirmed by the calculations performed, according to which the greatest differences between values of eiectron temperature obtained by taking equation (6) into account and with the help of the standard equation of heat conduction, respectively, correspond to the morning and evening periods. The height distribution of temperatures along different geomagnetic field lines for the morning and evening moments of time are shown in Fig. 1, and the diurnal variation for L = 4 and L = 5 is presented in Fig. 2. It is quite evident that the largest differences correspond to the morning hours and to the plasmaspheric height region. For the conditions considered here they reach significant values: 25% for L = 5. In the evening period the differences do not exceed 15% and at night they are as small as several per cent. In the morning hours the electron temperature Te calculated by equations (5) and (6), exceeds the temperature cb obtained by solving the standard equation of heat conduction l(5) together with (l)]. In the evening and nocturnal periods, on the contrary, T. c TJ. Such a character of the behaviour of electron temperatures is accounted for by the influence of nonstationarity of the thermal flux. Namely, as a result of the plasma heating due to photoelectrons produced after sunrise, there is an increase in thermal energy flux from the plasmasphere to the ionosphere (see Fig.

06.15LT

19.30 LT

RO

Cc)

Electron tempemtm (I@ K)

FIG. 1. TEE?CALCULATED PERATURE AND

DlSTRIBUTlONS

ALONG G~~AG~~C

EvENINe

hf0mNl-s

OF

OF ELECTRON

TEM-

FIELD LINES FOR THE MORNING TIME:

L=3

(a); L.=4

(h);

L = 5 (c). Here and in all of the other figures the solid line represents the plasma parameters calculated on the basis of the standard system of equations (3)-(5), (1) ; the dashed line indicates the same values obtained on the basis of the complete system of equations (3)-(6).

3). This means that the derivative &$/at > 0. The appearance of a positive non-stationa~ty term in equation (6) leads to an increase (in absolute value) of the term proportional to the temperature gradient as compared with its value which follows from the Fourier law (1). This, in turn, corresponds to an increase of T, and S, as compared with their values e and Sz obtained in terms of a standard approach. In the evening period the situation is the opposite. After sunset, the photoelectron-induced heating source is “switched-off” and the electrons begin to cool, as a result of which the heat flux decreases: c?S,/CV-C 0 (Fig. 3). That the time derivative of the thermal flux changes its sign from positive to negative, means a decrease of the term containing the tem-

0.

852

A.

GORBACHEVet al.

0

L

0

I

I

I

4

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3

6

9

12

15

18

21

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24

(b)

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1

I

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~

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P w L 0

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I 15

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LT FIG.~.THE DIURNALVARIATIONOFTHEELECTRONHEATFLUX FOR L = 4 (a) AND L = 5 (b) AT THE 1000 km HEIGHT. I 0

I

I

I

I

I

I

I

3

6

9

12

15

18

21

24

LT FIG. 2. THE DILJRNALVARIATIONOFELECTRON FOR L = 4 (a) AND L = 5 (b) AT THE 1000 INTHEEQUAToRIAL PLANE.

TEMPERATURE HEIGHTAND

km

perature gradient on the left-hand side of equation (6) which leads to a decrease of values of electron temperature and thermal flux. Besides, the thermal flux non-stationarity, as is evident from Fig. 2, is responsible for the enhancement of time variations of T, and S, : the more rapid heating during sunrise and the more rapid cooling during the evening hours. During the daytime the derivative &S,/& is small, and differences are virtually absent. Appreciable differences between values of T, and T:’ occur in the topside ionosphere approximately at heights h 3 800 km, where they amount to ~10% and increase with increasing height. This is because at heights of the topside ionosphere and plasmasphere the decisive influence upon the electron temperature distribution is exerted by thermal flux. Since its value is sensitive

to temperature variations (the thermal flux increases rapidly with increasing T,), then, according to (6), the differences between T, and c,’ will be enhanced with increasing electron temperature and, consequently, with height as well. This effect is also one of the factors responsible for the growth of the differences between T, and T”: with L-shell number at the same moments of time (see Fig. 1). As an illustration of the differences in thermal fluxes of electrons calculated using a standard and a complete system of equations, Fig. 4 gives the height profiles of S, and Sz*. The differences of these values in the morning period are larger compared with the evening and are, at 1000 km, 40% for L = 4 and about 55% for L = 5. Figure 5 gives the dependence of electron temperature on the L-shell number in the equatorial plane. With increasing L, the differences between T, and TsJ increase and, for all the L-shells considered, T, > ct in the morning hours and rS: > T, in the evening hours. Maximum differences are observed during the morning hours. This gives support to the previous conclusions as regards the behaviour of T,

Thermal flux variations and ionospher~plasmasphere interaction model

853

Ehxtron heat flow (loSeV, cm-*. 8)

\

06.15LT

\ \

0

t

3

6

i

I

1

1

9

12

3

6

Electmn heat flow (lO%V. a+.

s-‘)

-I

FKi. 4. THEHElCm’r PROFILES OF ELECTRON HEAT FLUX ALONG A G~~G~lC FIELD LINE WtTH AND L = 5 (b) AT THE MORNING AND EVENING MOMENTSOF TIME.

and Pt based on Figs 1 and 2. The decrease of values of Pi in the morning period with increasing L-shell number in the region L > 3 (Fig. Sb) is attributable to the following factor. This model includes only one plasma heating source, i.e. photoelectrons, and the value of electron temperature depends on the energy corresponding to the unit volume of the plasmaspheric reservoir. Since the power of the photoelectron

L =

4 (a)

source increases more siowly with latitude as compared with the volume of the geomagnetic flux tube (o;L’), the density of the input energy decreases. Therefore, the larger the L-shells are, the smaller c is. For the electron temperature obtained from the complete system of equations when L > 3, there is only a decrease in the growth rate with increasing L. In this case the decrease in density of the photoelectron

854

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(a) 2x1@

GORBACHEV

et al.

(4

‘F ==?I

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2

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A.

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3

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5

L

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Iontempem~(l0-'K)

b) 19.3OLT

II

L FIG. 5. THE ELECTRON TEMPERATURE DISTRIBUTIONS AS A FUNCTION OF THE &Z-IELL NUMBER IN THE GEOMAGNETIC EQUATORIAL PLANE AT MOMENTS OF TIME 06:15 L.T. (b) AND

19:30 L.T. (a).

source in the plasmasphere is compensated for by the increase in the heating rate of thermal electrons due to the non-stationarity effect of the thermal flux considered above. In the post-sunset period (Fig. Sa) the decrease in temperatures is due to the cooling caused by the’ thermal energy transfer from the plasmasphere into the ionosphere. For T,, an additional cause of the decrease is the thermal flux non-stationarity causing T, to decrease, starting from smaller L-shells than with Tz’.

3.2. Ions In order to simplify the procedure of numerical integration of the system of equations, it was assumed that H+ and O+ ions have the same temperature: TH+ = To+ = T,.

The calculations we performed have shown that the difference of values of ion temperatures calculated from the complete system of equations (T,) and from

I 1

I 2

I 3

lontempe-

(l@K)

FIG. 6. THEHEIGHT DISTRIBUTIONS OF THE ION T!?MPERATUREZ FOR A FIELD LINEWITH L = 5 AT MOMENTSOFTIME 06:15 L.T. (a) AND 19:30 L.T. (b).

the standard system of equations (r’), reaches appreciable values on geomagnetic field lines with large L. Therefore, we shall present the results of the calculations for L = 5. Figure 6 gives profiles of ion temperatures at the morning and evening moments of time. The diurnal variations of ion temperature at a height of lo3 km and in the equatorial plane are presented in Fig. 7. It should be noted at once that, unlike the corresponding electron temperature profiles (Figs lc and 2b), values of T, exceed c’ in the plasmasphere throughout the day and appreciable differences occur at high altitudes (h > 30004000 km). The greatest differences are observed in the post-sunset period and make up N 5%. Such a character of the behaviour of ion tem-

Thermal flux variations and ionospher+plasmasphere interaction model

=

component, this is caused by the thermal flux nonstationarity effects. Figure 8 presents the height profiles of thermal fluxes of ions for the moving and evening moments of time. Tt is evident that on the plasmaspheric portion Si is larger than S:’ at all moments of time. Unlike the electron component, the greatest differences here correspond to the evening period and make up * 4.0% in the plasmasphere.

H=WJOkm

2-

d

$1

I

I

,

3

6

9

,

12

855

/

IS

/

18

21

I

24

LT

FIG. 7. THEDIURNAL VARIATION OF ION TEMPERATUREAT THE 1OOOkmHEIGHTANDINTHEEQLJATORIALPLANEFORL = 5.

peratures in the plasmasphere is explained by the fact that in this height region the value of thermal flux q calculated by equation (6), is higher as compared with thermal flux ST, defined by expression (l), throughout the day (see Fig. 8). This, in turn, does indeed explain ,the excess of I; over v. The character of the time distribution of ion temperature at the topside ionospheric heights (Fig. 7) is similar to the corresponding diurnal variations of T;, (Fig. 2b) : in the morning period T, > rt, and in the evening period Ti c rt. As in the case of the electron

3.3. T&zdensity The above variation in distributions of plasma temperatures must have an effect on the electron density profiles being calculated. The calculations carried out in this paper have confirmed this supposition, The difference in the temperature dist~bution leads to a change in the plasma scale of heights, which, in turn, influences the character of the electron density behaviour calculated on the basis of the complete and standard systems of equations (N, and N:). Within the framework of the assumptions about the plasma heating adopted in this paper, the differences between N, and N:* are determined mainly by the difference between T, and c’ and are insignificant. The calculations have shown that, for the range of L-shells considered here, maximum deviations are observed on L = 5 in the plasmaspheric height range where differences between T, and c,” are maximal. Figure 9 presents the height profiles of density on L = 5 for the morning and evening moments of time for h > loo0 km. At lower altitudes, differences are absent. In the morning period N, > NC and in the evening period, on the contrary N, c N”:, which corresponds to the character of the electron temperature behaviour at these moments of time (see Fig. lc). Deviations of N, from N$ increase with height and reach maximum values of N 6% at the equator in the morning hours. 4. CONCLUSIONS

Ion heat flow (106cV. cm?. d) FIG. 8. THE HEIGHT DISTRIBUTIONS OF THE ION HEAT FLUX ALoNGAmELDLIM:WITH~=~ATMO~'FSOFTIME06:1~ L.T. (a) AND 19~30L.T. (b).

In this paper we have presented a consistent model of ionosphe~plasmasphe~ interactions based on a complete system of equations in the Ifmoment Grad approximation, involving the transfer equation for thermal flux. The equations are integrated in the nonstationary approximation along the entire length of the field line between the conjugate ionospheres, together with the solution of the kinetic equation for the spectrum of photoelectrons. A comparison has been made of the space-time distributions of plasma parameters obtained within the framework of this model, with their values calculated on the basis of a standard system of equations. These calculations lead us to draw the following conclusions.

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IVIA Density(cm”)

W

_

104-

Ia-

Y 3

a

Density (crn.l) FIG. 9. THE CALCULATED ELECTRON DENSITY DISTRIBUTIONS FOR THE PLASMASPHERIC PoRTiON OF A GEOMAGNETIC FIELD LINE WITH L = 5 AT THE MORNING (a) AND EVENING (b) MOMENTSOFTIME.

(1) The differences between electron temperature calculated on the basis of the complete system of equations (3)-(6), and temperature YJ which is the solution of the standard equation of heat conduction [equation (5) together with (I)], is most conspicuous during sunrise and sunset when the thermal flux nonstationarity effects exert an appreciable influence upon the behaviour of temperatures. In this case in the morning hours T, > Pet and in the evening hours T, < [ri. Similar relationships also occur for heat fluxes. (2) Thermal flux non-s~tiona~ty leads to a more rapid heating and cooling of electrons, respectively, during the morning and evening hours compared with the standard model. (3) With increasing height and L-shell number, the T,,

et al.

differences between T, and Tz’ increase. For the conditions considered in this paper, the deviations reach 25% for temperature and 55% for heat flux. (4) The character of the behaviour of the differences of ion temperatures T, and Tf” in the plasmasphere differs from that of the electrons: during the 24 h of the day T, > r’. At the topside ionospheric heights the relationships between T, and r’ are the same as for electron tem~ratures: in the morning period T, > Ts’ and in the evening period T, < y’ ; however, the amount of the differences is smaller than the electron component. (5) Appreciable differences in height distributions of N, and Ni’ densities, calculated on the basis of the complete and standard systems of equations, respectively, are observed only at large L-shells (L - 5) in the morning and evening hours at plasmaspheric heights. They increase with increasing height and display the character of the behaviour of the differences in electron temperature: during the sunrise period N, > Nzl and during the sunset period N, < Nz’. It should be noted that the calculated differences in the values of plasma parameters described on the basis of the complete system of equations and within the framework of the standard approach, represent only minimum values in the plasmaspheric region with L > 3. This is because the model under consideration includes consistently only one source of plasma heating, i.e. photoelectrons. For L > 3, a substantial contribution to the heating can be made by energy sources of magnetospheric origin ; for example, plasma heating during an interaction of the ring current with the plasmasphere (Cornwall et al., 1971; Galeev, 1975; Gorbachev et al., 1988; Konikov et al., 1989). The increase of plasma temperatures caused by this heating (Decreau et al., 1982; Gringauz, 1983 ; Olsen et al., 1987) will contribute to an increase in differences between T, and Et, T, and cl, and N, and Ng’. An additional influence in these plasmaspheric regions can also be exerted by effects associated with the action of magnetospheric substorms. However, quantitative analysis of the influence of the above factors upon the calculation of macroscopic characteristics is beyond the scope of the consistent model of the “ionosphere-plasmasphere” system used here. One can only limit oneself to a remark of a qualitative character, that the inclusion of additional energy sources wouid contribute to an enhancement of the differences between the values calculated on the basis of the complete and standard systems of equations. REFERENCES

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