Electron temperature and electron density in the F-region of the ionosphere. II. The role of atomic oxygen and molecular nitrogen

Joumal of Atmospheric and Terrestrtil Great Britain.

Printedin

Physics,

Vol.49,

No. 9, pp. 873-887,

OOZI-9169/87 f3.00+ .OO 0 1987PergamonJournalsLtd.

1987.

Electron temperature and electron density in the F-region of the ionosphere. II. The role of atomic oxygen and molecular nitrogen P. J. S. WILLIAMS and J. N. MCDONALD Department of Physics, University College of Wales (Coleg Prifysgol Cymru), Aberystwyth, U.K. (Received in final form 3 1 October 1986)

Abstrad-Atomic oxygen and molecular nitrogen play several important roles in the processes that control the thermal balance of the electron population near the F-region peak. Both 0 and N2 are ion&d by EUV radiation and produce photoelectrons. Ion exchange with Nt contributes to the recombination of O+ and in this way controls the equilibrium electron density. In the transfer of photoelectron energy to thermal electrons competition arises from ionisation and excitation of oxygen and nitrogen. Finally, of the processes by which thermal electrons lose energy, the three most important are Coulomb collision with ionised oxygen, excitation of the fine-structure states of neutral oxygen and vibrational cooling by molecular nitrogen. A simple model expresses all these processes and accurately describes the observed relationship between electron temperature, electron density and solar flux.

INTRODUCTION An anti-correlation between electron temperature T, and electron density N in the F-region of the ionosphere has been recognised for many years (e.g. LEJEUNEand WALDTEUFEL,1970 ; WALDTEU~L, 1971; TAYLORand RISK, 1974). In a previous paper M&ONALD and WILLIAMS (1986) studied ionospheric data from three incoherent scatter stations for all daylight hours over a wide range of solar conditions and established the following empirical relationship between T,, N and solar flux at 10.7cm wavelength S,,,, : T, = A-B(N-5x

lO”)+C(S,,,-750),

(1)

where N was in units of m- 3 and S,, , in kJy. This provided a satisfactory expression for all data taken at St. Santin and Malvern over the height range 250-350 km between 0800 and 1600 LT. At Arecibo, however, the simple relationship broke down in summer, especially at greater heights. Clearly, the origin of the relationship lies in the thermal balance of the F2-region. However, this is affected by many factors, including: ionisation and recombination ; the heating effect of photoelectrons ; the loss of heat from the electron gas through collisions with ions and neutral molecules; diffusion, conduction and transport under the influence of electric fields and neutral winds. A full model of these processes is extremely complicated, but the problem is simplified by the fact that atomic oxygen and molecular nitrogen dominate most of the processes involved.

In the F-region of the ionosphere solar radiation ionises both atomic oxygen and molecular nitrogen to give O+ and N: , providing at the same time a flux of photoelectrons. Equilibrium is reached when ionisation is balanced by recombination. For N: ,recombination and the reaction with atomic oxygen are both very rapid, so the ionisation of molecular nitrogen does not contribute significantly to the equilibrium value of N, but it does contribute to the flux of photoelectrons. For O+, on the other hand, recombination is much slower. It occurs via ion exchange and dissociative recombination and the overall rate of recombination is controlled by the densities of molecular nitrogen and oxygen, [N,] and [O,]. If we consider a constant solar flux and a reference height determined by a fixed value of the density of atomic oxygen [0], and if the atmosphere above this height is optically thin, then the most important remaining variable will be [N2]. For example, if [NJ increases, the ionisation cross-section per unit volume of the atmosphere increases and more photoelectrons are produced. The extra photoelectrons distribute some of their energy among the population of ambient electrons, so that under normal conditions, when [N,] increases T, also increases. At the same time, any increase in [N,] accelerates the rate of recombination, so that N decreases. It follows that for a fixed value of [0] any variation in [N2] causes a correlated variation in T, and an anti-correlated variation in N. The photoelectrons distribute their energy in a variety of ways. For energies greater than 20eV secondary ionisation and excitation of the ‘D state

873

874

P. J. S. WILLIAMSand J. N. MCDONALD

of atomic oxygen are the main processes and energy transfer to thermal electrons is negligible. For energies between 5 and 20eV the main competition is between the excitation of fine-structure states of atomic oxygen and vibrational states of molecular nitrogen, but a small proportion of energy is transferred to the thermal electrons. As the photoelectrons lose their last 5 eV it is clear that energy transfer to thermal electrons becomes more important, but the crucial question is how important? If at this stage this were the only loss process, all remaining energy would be transferred to the thermal electrons. The total energy transferred per photoelectron would then be independent of the rate of transfer, for with a reduced rate of transfer there would be a corresponding ‘pile-up’ of photoelectrons in the appropriate energy range. In other words, the energy transferred to the thermal electrons per photoelectron, or the ‘photoelectron heating efficiency’ (E), would be approximately constant. If, however, there were serious competition, for example from continuing transfer of energy to the neutral atmosphere, then (E) would be smaller and proportional to the rate of transfer shown by BUTLER and BUCKINGHAM (1962) to be N const. N/,/T,. Finally, we must consider the mechanisms by which the thermal electrons themselves lose heat. Coulomb losses to O+ and fine-structure cooling by atomic oxygen are usually the most important processes below 300 km at mid-latitudes (LEJEUNE, 1972). Taken together these factors cause a strong anti-correlation between T,and N, but the relationship is sharply nonlinear, due, in particular, to the N * dependence of the Coulomb loss. If these were the only loss mechanisms, this non-linearity would be specially noticeable at low electron densities, where the predicted value of T, would increase very rapidly for smaller N. However, molecular nitrogen plays yet another role in the thermal balance of the electron gas. Vibrational cooling of electrons through collision with nitrogen is itself a process which varies with T, in a very nonlinear way and the cooling becomes far more efficient for T,--T, > 1000K. As a result, any rapid increase in T, for small N is counteracted in the presence of nitrogen. This provides a qualitative explanation for the difference in behaviour between St. Santin and Malvern on the one hand and Arecibo on the other. In the measurements reported in MCDONALD and WILLIAMS (1986) electron temperatures were higher at St. Santin and Malvern and so was the N2 concentration predicted by the MSIS83 atmospheric model (HEDIN, 1983). As a result, excitation of the vibrational states of nitrogen was a significant cooling process limiting

any excessive rise in T, at low values of N, so that the final relationship was approximately linear. At Arecibo, in contrast, the values of T,were lower and not normally high enough for vibrational cooling to be important. The concentration of N, was also lower and hence a non-linear relationship was observed between T,and N. The prime aim of the present paper is to demonstrate the overall role of molecular nitrogen and atomic oxygen in controlling the relationship, rather than to establish a complete model of thermal balance in the ionosphere. In particular it explains : (i) why the basic relationship between T,and N can follow a linear form so closely over such a wide range of electron density ; (ii) why the relationship is modified for low electron densities in a way that varies with height and latitude. Moreover, when this qualitative explanation is matched with a simple quantitative model it turns out that the observed relationship is described with surprising accuracy by this model.

QUANTITATIVEMODEL OF RELATIONSHIP In assembling a model of the relationship we must consider in turn (a) ionisation, (b) recombination, (c) transfer of energy from photoelectrons, (d) direct loss of energy from thermal electrons to ions or neutral molecules and (e) transport processes. Due to uncertainties at several stages it is impossible to complete such a model with total confidence, but by using reasonable assumptions a simple model can be constructed which clearly demonstrates the way in which the overall relationship depends critically on the ratio R = W2Wl. (a) Zonisation If we take a typical flux of photons I in the wavelength band 8-103 nm then the rate of production of photoelectrons P is given by

P = (0.49 x lo-2’[0]+

1.07 x lo-2’[0*] +0.94x

lo-*‘[N2])I

(2)

using values of ionisation cross-section given by TORR and TORR (1979). This expression can be simplified further by using the fact that the ratio of [O,] to [Nd is small. MSIS83 shows that for the three stations over the height range 270-350 km during the period covered the ratio varied between 0.03 and 0.05 and, although seasonal variation may be greater during high solar activity (FON-

F-region

electron

temperature

TANARIet al., 1982), the ratio may be given a constant value of 0.04 without serious error. Thus the expression for P can now be re-written as P=0.49x10~2’(1+2.0Z?)[O]Zm-3s-‘.

to

and electron

density

875

thermal electrons in a fully ionised plasma dE dt=

(3)

Hence the rate of production of oxygen ions Q is given by

where m, is the mass of the electron, up is the velocity of the photoelectron, v, is the thermal velocity of the F is a function of v,/v, and A Q = 0.49 x 10-2’[O]Zm~3 s-‘. (4) electrons =,/(2kT,lm,), is the ratio of the Debye length to the impact parUnfortunately, no continuous measurements of EUV ameter. flux at the top of the atmosphere were available for This result was modified for high energy photothe period covered by the observations in MCDONALD electrons to take account of other effects, including and WILLIAMS(1986). However, HINTEREGGER (1979) Cerenkov radiation @CHUNKand HAYS, 1971), but has shown that over this wavelength range there is a for electrons with energies of a few electron volts the very high correlation between EUV flux and S,0.7, original formula is unchanged. the flux of solar radio emission at 10.7cm, which is Over the whole range of conditions covered in the measured continuously. On the basis of measurements observations FlnA varies by less than lo%, so that made during the period covered in MCDONALDand for the purpose of the present model equation (9) can WILLIAMS (1986) and quoted by Hinteregger, an be written as approximate relationship between Z and S,,,, can be derived and the effects of incident solar radiation at dE N -_= the top of the ionosphere can be summarised as dt -‘lT,lii) P=

3.3x 10-‘“S,0,,(l+2.0R)[O]m~3s-‘,

Q = 3.3x 10~‘“S,0.,[0]m-3s-‘.

(5) (6)

(b) Recombination As the recombination of N: (and 0:) is so rapid, the equilibrium value of N is controlled by L, the rate of recombination of O+, given by L =

(~,N1+k,[WN

(7)

where k, and k2 are coefficients whose values were given as functions of neutral temperature T,, by TORR and TORR (1979). Typically, both terms in the expression for L are about the same size, and whereas k, increases with T,, between lOOOK and 1400K, k2 decreases at about the same rate. Therefore, for all observations in the height range 275-325 km the loss rate can be represented accurately by L = 1.04 x lo-‘* [N2] N.

where c, can be assumed constant. However, in the ionosphere the transfer of heat to the thermal electrons has to compete with excitation of low energy states of nitrogen and oxygen. In estimating the effect of this, SWARTZand NISBET(1971) gave a suitable weighting to the different neutral constituents and represented the overall effect in terms of the parameter N/([N,] + [O,] + 0.1 [O]). In the present model we have reconsidered the division of photoelectron energy between electron and neutral constituents and obtained the following expression for the heating efficiency of each photoelectron

(6) =

cwV& C3([N21+[021+0.1

(8)

(c) Heating of thermal electrons by photoelectrons This is the least certain stage in the whole model. At higher energies the photoelectrons lose most of their energy by excitation or ionisation of the far more numerous neutral atoms and molecules. It is only when their energy has fallen below the excitation thresholds of the various neutral species that the photoelectrons lose a substantial part of their remaining energy to the ambient electrons. BUTLERand BUCKINGHAM(1962) used a classical treatment to determine the rate of transfer of energy

Pl)+c,N/& c2

=

l+(c,/c,)([N21+[021+0.1

PIh/WN’

(11)

where c2 and c3 are constants. If we take c2 to be 7.9eV and the ratio cJc, to be 6.5 x 10-6K- I’*, then for average atmospheric conditions during the observations described in MCDONALD and WILLIAMS(1986), as predicted by MSIS83, the expression gives the photoelectron heating efficiency as a function of N in a very similar form to the function proposed by Swartz and Nisbet (see Fig. 1). The following equation will therefore be used to model the transfer of energy from photoelectrons to

876

P. J. S. WILLIAMS and J. N.

MCI)ONALD

derived an expression for this term which, in the F2region, can be written as CL=

-4.8x

10~‘3N2(Te-~j)~~3~2eVm~3s~‘. (13)

Fig. 1. Relationship between photoelectron heating efficiency and electron density for ([NJ + [OJ + 0.1[0]) = 0.2 x 10” me3. Te = 175OK.

(ii) Fine structure e~citution of oxygen. The tine structure levels of atomic oxygen consist of a triplet state with energies at the ground level, + 0.02 eV and +0.028 eV. DALGARNO and DEGGES(1966) have formulated an expression for the electron cooling rate due to excitation of the oxygen fine-structure. More rigorous calcuIations of the excitation cross-section followed and HOEGY(1976) used these to derive a new expression for the cooling rate. If we assume that the temperature of the excited states is equal to T,,, this can be summarised as FsL = NID] (T, - I’,) e(T,, TJ,

where e(T,, T,) is a very complicated expression. The following approximation is equivalent to the full

the thermal electrons

expression

7.9 eV

‘a) = 1+6.5x

(14)

lo-“([N,]+[O,]+o.l

with an accuracy

better than 6% over the

[O]),/T,/N’

(12) (d) Cooling of thermal electrons The cooling of the electron population in a fixed volume of plasma can be brought about by direct heat loss or by transport. Direct heat loss can occur through elastic collisions with O+, 0, O2 or N,, excitation of the rotational and vibrational states of O2 and N, and excitation of the ‘D and tine-structure states of 0. Transport losses occur through conduction or diffusion. In total 12 different cooling terms should be included in a complete expression for heat loss. Estimates of all components of direct heat ioss were made to cover the whole range of conditions experienced during the observations reported in MCDONALD and WILLIAMS (1986), using the appropriate atmospheric parameters predicted by MSIS83. The average values of each term are listed in Tables la, b and c for all three stations in winter and summer. It is clear that Coulomb loss to O+, fine-structure cooling of atomic oxygen and cooling via the vibrational states of molecular nitrogen are the most important processes in each case, except at Arecibo, where vibrational cooling is very small. We shall discuss these three cont~butions to direct heat loss in turn. (i) Coulomb loss to ionised uxygen. Under most circumstances the major loss term arises from elastic collisions between electrons and O+. BANKS (1966)

Table la. The main FZ-region heat loss terms at Malvern Height (km) Loss term

Winter N2 elastic Nz rotational N2 vibrational 0, elastic O2 rotational Oz vibrational 0 elastic 0 fine structure 0 ‘D excitation 0+ elastic Conduction Diffusion Nz elastic N2 rotational N2 vibrational 0, elastic 0, rotational O2 vibrational 0 elastic 0 fine structure 0 ‘D excitation O+ elastic Conduction Diffusion

270

300

0.6 1.7 4.6 0.0 0.2 0.0 2.0 12.9 0.1 42.2

0.3 0.7 3.4 0.0 0.1 0.0 1.3 8.8 0.3 31.4 (-2.2) (-0.4)

1.5 3.0 24.9 0.0 0.5 0.0 2.1 6.4 2.5 16.0

0.8

1.5 22.4 0.0 0.2 0.0 1.3 4.7 3.5 14.8 (-0.9) (-1.4)

All terms quoted in units of 10’ eV m- 3s- ‘.

340 0.1

0.2 2.4 0.0 0.0 0.0 0.6 4.8 0.3 16.5

0.3 0.6 15.8 0.0 0.1 0.0 0.7 3.3 3.2 11.4

F-region

electron

temperature

whole range of conditions encountered in the data. FsL = 5.32 x lo- 16(T,,- T,)N[O] T,JT,(9-

1010/T,)

(15)

.

heat loss terms at St. Santin Height

Loss term Winier N, elastic N, rotational N, vibrational O2 elastic O2 rotational O2 vibrational 0 elastic 0 fine structure 0 ‘D excitation 0+ elastic Conduction Diffusion Summer N, elastic N2 rotational N, vibrational 0, elastic O2 rotational 0, vibrational 0 elastic 0 fine structure 0 ‘D excitation 0+ elastic Conduction Diffusion All terms quoted

275

300

0.3 0.6 3.8 0.0 0.1 0.0 1.0 11.7 0.5 17.3

0.1 0.2 2.2 0.0 0.0 0.0 0.5 6.2 0.5 10.8

877

The full expression for this loss term takes the form VC = 3 x lo- “N[NJ

x

[exp{-g(T,-T,)/T,.r,)-ll,

Table lc. The main FZ-region

0.3 0.6 5.2 0.0 0.1 0.0 0.5 4.6 0.9 11.7 (0.9) (0.0)

in units of 10’ eV m-3 s-l.

heat loss terms at Arecibo Height

Loss term

0.0 0.1 2.2 0.0 0.0 0.0 0.3 3.5 0.4 7.2

0.1 0.2 2.9 0.0 0.0 0.0 0.3 2.7 0.7 8.4

(16)

where f and g are functions of T, (SCHUNKand NAGY, 1978). RICHARDSet al. (1986) have recently re-examined the overall relationship between vibrationally excited N2 and thermal electrons. They report that the crosssection for the interaction between electrons and moiecular nitrogen is still uncertain and it has been suggested (SCHULTZ, 1976) that the normally accepted value should be increased by a factor of 2 or more. At the same time, however, enhanced vibrational excitation of N2 can provide a source of energy for thermal electrons and the net cooling rate of the electrons could be reduced by a factor of 3! As the two factors would largely cancel out, we have continued to use the published value, but in the

(km)

(0.1) (0.0)

0.6 1.2 11.5 0.0 0.2 0.0 0.7 6.9 1.1 12.3

density

x exp {f( T, - 2000)/( T, *2000))

There is some evidence (e.g. GRO&WANN and OFFERMANN,1978 ; CARUON and MANTAS,1982) that Hoegy’s result may need to be adjusted by a factor much larger than 6%, so in order to discuss the physical role that fine-structure cooling plays in the thermal balance of the ionosphere we have used the approximation, at the same time paying due regard to the uncertainty in the values of FSL derived. (iii) Vibrational cooling by nitrogen. The initial calculations of direct heat loss through exciting vibrational states of nitrogen were presented by REES et al. (1967) and DALGARNOet al. (1968), assuming that superelastic collisions only occurred with the first excited vibrational level and that the vibrational temperature was equal to T,. Later work added small corrections to allow for the possibility of further excitation from the first four excited levels.

Table lb. The main FZregion

and electron

Winter N, elastic N, rotational N, vibrational O2 elastic O2 rotational O2 vibrational 0 elastic 0 fine structure 0 ‘D excitation 0+ elastic Conduction Diffusion Summer N, elastic N, rotational N, vibrational O2 elastic 0, rotational O2 vibrational 0 elastic 0 fine structure 0 ‘D excitation 0+ elastic Conduction Diffusion All terms quoted

(km)

270

300

340

0.1 0.2 0.6 0.0 0.0 0.0 0.4 5.6 0.0 16.5

0.0 0.1 0.2 0.0 0.0 0.0 0.2 2.5 0.0 10.5 (-0.6) (-0.2)

0.0 0.0 0.1 0.0 0.0 0.0 0.1 1.1 0.0 6.5

0.1 0.3 0.4 0.0 0.0 0.0 0.2 4.5 0.0 22.8

0.0 0.1 0.1 0.0 0.0 0.0 0.1 2.3 0.0 20.2

0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 14.3

in units of 10’ eV mm3 s-‘.

P. J. S.

878

WILLIAMS and

J. N.

MCDONALD

light of the uncertainty in VC the following approximation has been used, which is equal to the full expression with an accuracy better than 7% over the whole range of conditions covered by the data. VC=

6.5x 10~28N[N,](T,-310)2(T,-T,) x exp {O.O023(T,- T,)).

(17)

Equation (17) illustrates the sharp non-linearity of this cooling term, as shown in Fig. 2. For values of T,- T, less than SOOK this cooling term can be ignored, but as soon as T,-- Tfi exceeds IOOOKit can act as a tem~rature limiter in the ionosphere, preventing the sharp rise in electron temperature that would otherwise occur. This is clearly demonstrated in Fig. 3, where the three direct heat loss terms are introduced in turn for a height in the atmosphere where [0] = 1.5 x lOI md3. Curve (a) shows the hypothetical relationship between T, and N that would occur if Coulomb collision was the only direct loss term. In curve (b) both Coulomb loss and fine-structure excitation are considered. In both cases the electron temperature increases sharply for low electron density. In curve (c), however, all three cooling terms are introduced and the tem~rature limiting effect of the N2 vibrational excitation is clearly shown. Moreover, this simple calculation predicts a relationship between T, and N that is approximately linear. (e) Transport terms In addition to the direct heat loss, heat can be lost by conduction and diffusion. At F-region heights, where the gyrofrequency is far greater than the collision frequency, electrons are

Electron

l

N

- s IO” m-J

?O-

rNzl.

i e

*-

: E >,

40-

“0 ;; 0 >

IOOOK

TX

ao-

0.4 lOI4m-’

!m-

30aOIO&O

A

I I800 2x)0 2600 3oco 34m

Fig. 2. Heat loss from thermal electrons due to excitation of the vibrational states of molecular nitrogen as a function of electron temperature.

f IO” me3

Fig. 3. Hypothetical relationship between electron temperature and electron density if heat loss from the thermal electrons is due to (a) Coulomb loss only, (b) Coulomb loss plus excitation of the fine-structure states of atomic oxygen and (c) Coulomb loss plus excitation of the fine-structure states of atomic oxygen plus excitation of the vibrational states of molecular nitrogen.

responsible for conducting heat along the magnetic field lines with a conductivity K’ given by E;“= -7.7~1O’T,‘~~eVm~‘s-‘K-‘,

(18)

so that the vertical flux of heat is given by H = - K” sin’ ?aT,jah = -2.2x

107sin2 IaT,7’2/aheVm-2s-‘,

(19)

where h is the height above the ground and I is the angle of inclination of the magnetic field lines. The divergence of this flux represents the main transport term in the thermal balance of the ionosphere, equal to TL = -2.2x

go-

density

10’sin2~(~2T~!~~ah~)eVm-3s-‘,

(20)

and BAILEY et af. (1981) have shown how at greater heights this conduction term becomes relatively more important. As incoherent scatter measurements provide height profiles of electron temperature it would ideally be possible to determine the heat loss due to conduction directly. However, the random errors in measuring T,. create large errors in determining aT,./ah and accurate estimates of G72T~l’/ah2are out of the question for a single set of data. As a result, it is impossible to estimate the heat loss due to conduction on a day-to-day basis. Instead, for each station and each season in turn values of Tp7i2can be averaged over the whole data set and these average values plotted against height, as in Fig. 4. A least squares method is then used to fit a smooth parabolic curve of the form r + sh + t/z* to the data, where r, s and t are constants.

F-region electron temperature and electron density

lolvern

: Winter

879

l

: Summer 0

: Summer0

recibo

: Winter

l

: Summer o

Height/km

Fig. 4. The variation of TJ’? with height at Malvern, St. San& and Arecibo in winter and summer.

The heat loss per unit volume over the height range covered then equals -4.4 x 10’ sin’ I t eV m- ’ s- ‘. When averaged in this way the data suggest that the heat loss due to conduction over the height range 270-350 km is in every case less than 8% of the direct loss due to Coulomb collision, fine-structure excitation, vibrational cooling, etc., and in most cases much less. Moreover, the smallest estimates of the average loss due to conduction correspond to the most accurate measurements of T,, those made at St. Santin. In the absence of more accurate data these small average values are the only terms due to conduction that can be included in the present model, but it must be remembered that such ‘average’ values may fail to represent a considerable day-to-day variation in the cont~bution of conduction to the thermal balance, either as a source or as a sink.

Diffusion along the magnetic field line is another transport term that contributes to the heat loss in the F-region. The rate of diffusion also depends on the gradient of T, and once again the actual heat loss per unit volume is given by the divergence of the heat flux due to diffusion. Fortunately, estimates of diffusion between 270 and 350 km for conditions during the measurements reported in MCDONALDand WILLIAMS (1986) show that the magnitude of heat loss due to diffusion is even smaller than that due to conduction and can be ignored. A SIMPLE MODELOF THE

THERMAL

BALANCE

OF

F-REGION ELECTRONS

The components discussed in the previous section can now be assembled into a simpte model of thermal

P. J. S. WILLIAAB and J. N. MCDONALD

880

balance of the ionosphere, assumptions.

making

the following

(i) P, the rate of production of photoelectrons, is related to Q, the rate of production of oxygen ions, by the expression P=(I+Z.OxR)Q.

(21)

(ii) Q can, as a first approximation, be equated to the local rate of recombination so that 0.49x 10-2’[O]Z-

1.04x IO-‘*[Nz]N

(22)

N 3.2 x lO*&,,/N.

(23)

and hence R N 4.7 x l0-4I/N

For constant solar flux and a fixed level of [0] this represents the anti-correlation between nitrogen concentration and electron density that is a crucial feature of the present study. A more detailed model of the ionosphere near the F-region peak (RISHBETH,1986) takes full account of diffusion, which is controlled by O-O+ ion exchange, and this leads to a relationship of the form In N,,, = 1.3 In [0] -0.7ln([N,]

=

7.9eV/{1+6.5

(25)

x 10-6([N2)+[02] fO.1 IOl),h%).

(27)

where A is a separate constant for winter and summer. If we take fixed values of solar flux &,’ and atomic oxygen density [0] we can equate the rate of energy input via photoelectrons (= P(E)) to the rate of direct heat loss (= CL+FSL + VC-/- TL) and hence derive an expression relating electron temperature T, to electron density N, with R as the independent variable. A very good approximation to the full expression is given by the following equation, which represents the three most important loss terms and assumes that T, N T,, T, = A+~&,x

(0.37+[1+2.OR]/

([N+3.0 x lo-6[O]Jrp] x[1.76x

10-4NT;3’2[0]-i

$1.95~

lo-‘T,-IT;‘.’

x (T,-310)Zexp

In practice it turns out that the form of the relationship between Te and N is almost the same whether equation (23) or (25) is employed, but of course the range of values of R corresponding to the observed range of N is larger if equation (25) is used. (iii) Below 35Okm the photoelectrons lose their energy locally and
x S,0.7,

lo-“R

fO.O023(T,- TJ)])),

(24)

For a fixed value of [0] this also represents an anti-correlation between nitrogen concentration and electron density and leads to a relationship of the form R = 7.1 x lo”,!? 10.7 N-‘.37 Inax .

T, = A+0.37

x(9-1010/T,)-‘i-2.38x

+[O&+const.

= 0.6ln[O]-0.7lnR+const.

height range 270-350 km and, on the basis of MSIS83, can be represented by the expression

(26)

The actual rate of energy transfer from a photoelectron population to the thermal electrons is still a matter of discussion (STAMNESand REES, 1983), but for the values of N/([N,l + [O,] +O. l[O]) which apply in the present case (E) is almost constant and not very sensitive to the actual rate of transfer. (iv) The loss of heat from the thermal electrons is mainly accounted for by the direct loss to Of, 0 and N, and transport terms are ignored over the height range 270-350 km for the reasons explained above. (v) The neutral temperature is constant over the

(28)

where S,0.7is measured in kJy. For fixed values of S,,, and [0] the crucial variable is l&l. When [N2] is relatively low, the rate of production of photoelectrons and the rate of recombination are also low, so that T, is low and N is high, which further reduces T, through the N2 dependence of Coulomb loss. When [Nz] is relatively high, the rate of production of photoelectrons and the rate of recombination are high, so that T, is high and N is low. However, the high values of [N2] and T, - T, combine to bring into play vibrational cooling. This calculation can be repeated for different values of &3.7 to model the variation of T, with solar flux. At a quick look, the linear variation with S,,, seems obvious and this is indeed the case for high values of N, where T, and Ti both tend to T,, which over the range of solar conditions covered in MCDONALDand WILLIAMS(1986) has an almost linear relationship with S,0.7. For low values of N it is the balance between solar heating and vibrational cooling which eventually determines the value of C in equation (1). For different reasons, the slope of T, versus S10.7is very similar for both high and low values of N and this dictates the fairly constant value of C actually observed at all values of N, at all heights in the range 270-350 km and at all seasons. Finally, it is possible to repeat the calculation for

881

F-region electron temperature and electron density different heights by choosing a different initial value of [O]. Once again, it is the parameter R which determines the basic relationship. As height increases, [O] decreases, but so does R, i.e. fNZ]decreases at an even greater rate. As a result, vibrational cooling by N, is less effective and the relationship between T, and N becomes non-linear, as demonstrated in MCDONALD and WILLIAMS (1986). COMPARISON OF THE OBSERVED DATA WITH THE MODEL

In comparing this model with the observed data at a given height the first step is to take a suitable value

of&4,

and, using the corresponding predictions of determine the average value of [0] for that height. With these values of S,,, and [OJ as input the model then predicts a relationship between T, and N for a series of values of R. Such a relationship is plotted in Figs. 5,6 and 7 as a continuous line. Figures 5a and b compare the relationship predicted for S,,, = 1600 kJy and [OJ = 1.2x 10” me3, 0.8 x 10” rn-’ and 0.4x IO” me3 in winter and 0.9 x lOi md3, 0.6 x 10” m-3 and 0.35 x lOI mm3 in summer with the corresponding measurements made at Malvern at 270, 300 and 340 km, respectively. The agreement is remarkably good, especially when we remember that the parameters of the model are all MSIS83,

.

34Okm EOY* 0.40

-

*I --+i+

IO” m-j

_

l

;

27Okm

l

-

I

5

I

1

IO Electron density /IO”

Fig. 5a. A comparison of the

103 =I.20

I5

10’5m-3

1 20

mm3

measurements of T,and Nmade at Malvern during winter with the relationship

predicted by the model for different values of oxygen ~n~ntmtion ~rr~pondjng 300 and 340 km. SiO7 = 1600 kJy.

to approximately 270,

882

P. J. S. WILLIAM and J. N. MCDONALD

. --w+__

-

270 km

CO3=0.90 lO'smS3

2ooo ‘\ -----

1000

I 0

1

I 5

I IO ELectron density/IO”

I I5

I

20

rn-’

Fig. 5b. A comparison of the meas~ements of T,and Nmade at Malvem during winter with the re~ations~p predicted by the model for different values of oxygen concentration corresponding to approximately 270, 300 and 340 km. &,, = 1600 kJy.

derived from MSIS83 or from theory and no attempt

has been made to fit the model to the data. In the case of Malvern the deviation of the observed points from the predicted curve are all comparable with the large random error in measurement. Figures 6a and b make a similar ~mpa~son for St. Santin using measurements made at 275, 300 and 325 km. In this case the measurements of T, are more accurate and this seems to be matched by even better agreement between the observed data and the model.

It is especially satisfying that the model for summer observations confirms the remarkable linear relationship observed between T,- C(S,,,, - 750) and N that first attracted our interest to the subject. Finally, in Figs. 7a and b the different behaviour observed at Arecibo is correctly echoed by the model, with the slope steepening for I? < 5 x 10” m-‘, especially at greater heights in summer, although the detailed agreement shown in Fig. 7a is not as good as in the other cases.

883

Fregion electron temperature and electron density

300 km

I

l

0

1

I

t

5

IO

ELectron density/

275km

I

15

I

20

10” rn-’

haa. A comparkcrn of the rn~asur~men~~ of T, and N made at St. Santin during winter with the relationship predicted by &t: model for different values of oxygen concentration corresponding to approximately 275,300 and 325km. S,,, = 1600 kJy.

Fig.

CONCLUSIONS

The prime purpose of the model was to obtain a relationship between electron temperature and electron density which would illustrate the rule of atomic oxygen and molecular nitrogen and hence explain : (a) the quasi-linear relationship between T, and N over a wide range of electron densities at Malvern and St. Sank, where the concentration of N2

at low electron densities was adequate for vibrational cooling to restrict the increase in r,; (b) the steepening of the relationship for low electron densities at greater heights, where the proportion of Nz was smaller and vibrational cooling Iess effective ; (c) the similar steepening of the relationship for low electron densities at lower latitudes, where the proportion of N2 was smaller and the electron temperature was also lower.

P. J. S. WILLIAMS and J. N. MCDONALD

884

l

-

l

I

l

0.60

IO” me3

275km

co1- 0.90 lOisC3

.

+\

‘.

1000

0

tOI

-

I

li

300 km

--

I

I

IO

5

---_ I

I5

I 20

Electron density / IO” rn-’

Fig. 6b. A comparison of the measurements of T, and N made at St. Santin during summer with the relationship predicted by the model for different values of oxygen concentration corresponding to approximately 275, 300 and 325 km. S,,, = 1600 kJy.

Consideration of the various roles played by processes involving molecular nitrogen and atomic oxy gen led to a model in which the ratio of the concentration of these constituents was an important parameter. This model was successful in explaining the general features of the relationship qualitatively. In addition, Figs. 5 and 6 show a remarkably good quantitative relationship for Malvern and St. Santin and this agreement has been reached without any attempt to fit the model to the data. This tends to

confirm the assumptions made in deriving the simple model. In particular, it suggests that at heights between 270 and 350 km photoelectron heating is localised and, on average, conduction makes a relatively small contribution, although the scatter of observed data could be partly due to divergence of heat conduction on a day-to-day basis. The good agreement also tends to confirm the currently accepted form of the expression for energy loss due to vibrational cooling. On the other hand, almost

885

Region electron temperature and electron density l

-

35okm co1 8 0.20

l

-

-

320km CO1 *

*

l0’S me3

0.30 IO” rn-’

2SOkm CO3 *OS0

1

I

I

5

IO

15

1O’s mm3

J

20

Electron density / IO” mS3 Fig. 7a. A comparison of the measurements of T, and Nmade at Arecibo during winter with the relationship predicted by the model for different values of oxygen concentration corresponding to approximately 290, 320 and 350 km. S,,, = 750 kly.

all the predicted values are unaffected by the exact form of the expression for tine-structure cooling, except in the case of measurements at Arecibo with electron density lying between 5 and 7 x 10” m-‘. In this case a reduction in the value of FSL, as suggested by Grossmann and Offermann, would lead to a small increase in the predicted vaiues of T, and hence reduce the small discrepancy between the observed values and the model as seen in Figs. ?a and b. Even better agreement between the observed data

and the model could also be obtained by introducing a legitimate element of ‘fitting’, but for a fair comparison it was thought better to keep the model and the measurements totally independent. For example, T,, in the model is derived directly from MSIS83. If, instead, it were based on the observed values of T, the agreement would be closer for N > 10x 10” rnb3 in every case. The formula for photoelectron heating efficiency could also be adjusted to give a better fit, but the values of c2 and CJC, were in fact chosen to

P. J. S. WILLIAMS and J. I?. MCDONALD

886

l

-

co3 =0.20 IO” ma3

* -

~-

5

IO Electron density/

350km

320 km CO3- 0.30

15

IO” mb3

20

IO” me3

Fig. 7b. A comparison of the measu~ments of ‘T,and N made at Areeibo during summer with the relationship predicted by the model for different values of oxygen concentration corresponding approximately to 290,320 and 350 km. Z& = 750kJy.

give the best agreement with Swartz and Nisbet for the average conditions encountered. As it happens, for most values of N the result is not very sensitive to C&l. Finally, the constant of proportionality between the EUV fIux and &,, is uncertain. In the way the model is constructed this has very little effect on the shape of each curve : it does alter the relationship between R and hr. An important feature of the model is that for fixed vaiues of S,,, and [0] the ratio of molecular

nitrogen to atomic oxygen R varies considerably about the mean value predicted by MSIS83, covering a range of values of at least 4 : 1. However, there is strong evidence that this actually occurs, derived both from measurements and from theoretical models. The results presented by FONTANAFG et al. (1982) suggest that, in addition to a marked seasonal variation in R, there is considerable variation from day-to-day, while RISHBETHet al. (1985) have used a thermospheric mode1 to demonstrate how R can vary considerably at

F-region electron temperature and electron density

mid-latitudes following an aurora1 disturbance. When complete sets of incoherent scatter data and simultaneous satellite data are available a full study of the relationship between T,, N, R and EW flux on a day-to-day basis, including the role of transport processes, may be possible. It is suggested that the simple model described in such a study.

here will be a useful first step

887

authors are deeply indebted to the Royal Signals and Radar Establishment for the data from Malvern, to the Centre National de Reeherche Scientifique for the data from St. Santin, and to the Are&o Ionospheric Observatory for the data from Arecibo. We are especially indebted to Dr G. N. TAYL.QR,Dr M. BLANCand Dr J. HAGEN.We are also indebted to Professor R. MOFFETIand Dr H. RISHBETH for valuable discussions. Acknowledgements-The

REmRENCES BAILEYG. J., FOOTITTR. J. and MOFFETTR. J. BANKSP. M. BUTLERS. T. and BUCKINGHAM, M. J. CARL~~NH. C. and MANTASG. P. DALGARNO A. and DECXXST. P. DALGARNO A., MCELROYM. B., REESM. H. and WALKERJ. C. G. FONTANARI J., ALCAYDB D. and BAUERP. GRO%?.MANN K. ANDOFFERMANN D. HP.DINA. E. HI~REGGER H. E. HINTEREGGER H. E. HOEGYW. R. LEIEUNEG. LEJEUNEG. and WALDTEUFEL P. MCDONALDJ. N. and WILLIAMS P. J. S. REEs M. H., WALKERJ. C. G. and DALGARNO A. RICHARDSP. G., TORRD. G. and AB~U W. A. RISHBETH H. RISHBEXX H., GORDOE R., REESD. and Fu~~ER-ROLL T. J. SCHULZG. J.

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1976

Principles of Laser Plasmas,

p. 33, John Wiley, New

York. SCHUNKR. W. and HAYSP. B. SCHUNKR. W. and NAGYA. F. STAR K. and REESM. H. SWARTZW. E. and NISBETJ. S. TAYLORG. N. and RISK R. 3. TORRD. G. and TORRM. R. WALUTEUFEL P.

1971 1978 1983 1972 1974 1979 1971

Planet. Space Sci. 19, 113. Rev. Geophys. Space Sci. 16, 3.55. Geophys. Res. Lett. 10, 309. J. geophys. Res. 77,6259. J. atmos. terr. Phvs. 36, 1427. J. atmos. terr. Phys. 41,797. Ann& Giophys. 26,167.