A time-varying model of the ionospheric F2-layer

JournalofAtmosphericand Terrestrial Physics,1964,Vol.26, pp. 657to 6~5.PergamonPressLtd. Printed in NorthernIreland

A time-varying model o2 the ionospheric F2-1ayer H. RISHBETH Central Radio Propagation Laboratory, National Bureau of Standards, Boulder Laboratories, Boulder, Colorado (Received 30 December 1963)

Abstraet--A model of the quiet F2-1ayer at mid-latitudes is presented, which combines timcdependent solutions of the F2-1ayer continuity equation in a static atmosphere (RISHBETIq:, 1963) with the time-varying atmospheric models of HA~n~s and PRIESTER (1962). The analysis makes use of the fact that the diurnal variations of the photoionization rate, and of the loss and diffusion coefficients, become relatively simple when considered in terms of fixed pressure-levels. The rates used for this model provide a good representation of the F2-1ayer, especially by (lay. Several published values of production and loss rates at 300 km, for different levels of solar activity, are compared and found to be re~sonably consistent, ttowever, these rates are regarded as being determined only to within a factor of two. Not all F2-1ayer phenomena can be explained by the model. Other processes may be important; in particular, there are difficulties in accounting for the behaviour of the F2-1ayer at night, and these may indicate that the layer is maintained by slow downward diffusion of exospheric ionization or by corpuscular ionization. 1. I:)ROCESSES IN THE F-REGION ]. 1 I n t r o d u c t i o n

DURING t h e t h i r t y y e a r s or so in w h i c h ionospheric t h e o r y has been a c t i v e l y studied, it has b e c o m e clear t h a t t h e b e h a v i o u r of the F - l a y e r is controlled b y m a n y p h y s i c a l processes. B e c a u s e of the c o m p l e x i t y of the problem, it has p r o v e d difficult to d e v e l o p even a basic t h e o r y to a c c o u n t for the m a j o r features of the F - l a y e r , a n d a c o m p l e t e e x p l a n a t i o n of the o b s e r v e d p h e n o m e n a is n o t y e t in sight. T h r e e m a i n questions are considered in this paper. First, h o w would an idealized F 2 - 1 a y e r b e h a v e , in which o n l y t h e processes of p h o t o i o n i z a t i o n , recombiuatior~ a n d diffusion are o p e r a t i v e ? Secondly, w h a t n u m e r i c a l values can be assigned to t h e p a r a m e t e r s of this m o d e l b y c o m p a r i s o n with o b s e r v a t i o n a l d a t a ! T h i r d l y , w h a t o t h e r processes m u s t be i n v o k e d to a c c o u n t for o t h e r features of the a c t u a l F2-1ayer? The p a p e r deals w i t h t h e large-scale s t r u c t u r e of the u n d i s t u r b e d F2-1ayer a t t e m p e r a t e latitudes. Small-scale irregularities a n d short-lived f l u c t u a t i o n s of electron d e n s i t y are n o t considered at all. M a n y of the ideas d e v e l o p e d should a p p l y also to the F - r e g i o n a t e q u a t o r i a l , a u r o r a l a n d p o l a r l a t i t u d e s a n d to F region storms, b u t these are n o t c o n s i d e r e d in detail. N o r can a n y detailed analysis of the c o m p l e x p h o t o c h e m i s t r y o f the lower F - r e g i o n be a t t e m p t e d in t e r m s of the idealized m o d e l used. 657

658

H . RISHBETH

1.2 Photochemical and transport processes Ionospheric processes are divided into two broad categories, "photochemical" and "transport." The F-region is complicated because these two categories are comparable in importance, in contrast to the situation in the D- and E-regions where the electron distribution is determined mainly by photochemical processes, and in the exosphere where transport processes, such as diffusion, are dominant. Although there is no assurance that all the processes which determine F2-1ayer electron densities are known, those in the following list seem likely to be the most important. The four processes included in the model described in this paper are marked by asterisks: it is convenient to include corpuscular ionization in the "photochemical" list. Photochemical processes *Photoionization by solar extreme ultraviolet (EUV) radiation Corpuscular ionization *Loss via ion-atom interchange and dissociative recombination reactions

Transport processes *Ambipolar (or plasma) diffusion Electromagnetic drifts, winds Flow of plasma between exosphere and ionosphere, along magnetic field lines *Thermal expansion and contraction of atmosphere and plasma

None of these processes can be adequately discussed without consideration of the structure of the neutral atmosphere; lack of detailed knowledge of many atmospheric parameters is one of the principal limitations of present F-region theory. Another limitation is the uncertainty concerning such key parameters as solar E U V fluxes, rate coefficients of photochemical reactions, diffusion coefficients and the like. In principle, values of these quantities m a y be deduced from the ionospheric data, b u t it is always difficult to isolate the effects of any one process and the results of such analysis are liable to be influenced by other processes. The last entry in the above list of processes refers to the influence of large systematic changes of the neutral atmosphere, notably the diurnal variations, on the distribution of ionization in the F-region. The analysis presented in Section 2 of this paper takes account of such effects in an approximate way. 1 . 3 0 v t l i n e of the paper Following a statement of the assumptions to be adopted, the method used to obtain theoretical electron density distributions in a time-varying model atmosphere is described in Section 2. The method is based on analogue solutions of the continuity equation (RISHBETH, 1963), and on principles described b y GARRIOTT and RISItBETH (1963). Further theoretical analysis, presented in Appendix A, enables the effects of changes in the model to be studied semiquantitatively, without solving the equations in full for the altered conditions. Section 3 is

A time-varying model of the ionospheric F2-1ayer

659

devoted to comparison between the model and data on the actual F-region, and discusses some outstanding problems connected with the night F-layer. Appendix B contains a detailed comparison between the numerical rates used in this paper, and those suggested by other authors. The main conclusions of the paper are recapitulated in Section 4. 2. SOLVING THE CONTINUITY EQUATION

2.1 Assumption, s In the theoretical models, the electron density N is determined, as a function of height h and time t, by solving the equation of continuity. The assumptions used in the model are listed below: (i) The neutral atmosphere consists of atomic oxygen and molecular nitrogen, whose scale heights H(O) and H(N2) are in the ratio 1.75/1. It is assumed t h a t the height at which diffusive separation begins is constant; and t h a t then neutral gas concentrations, denoted by [O] and [N2], are fixed at this height which is taken as about 120 km. (ii) Atomic oxygen is ionized by solar EUV radiation, producing 0 ÷ ions and electrons at a rate q per unit volume. Molecular nitrogen absorbs part of the EUV radiation but does not contribute to the observable ionization. At noon, the peak of production lies at the level of the Fl-ledge. (iii) 0 + ions are lost by an i o n - a t o m :interchange reaction with N 2 molecules (rate coefficient y) and the resulting NO + ions undergo dissociative recombination with electrons (rate coefficient ~). (iv) In the F2-1ayer the i o n - a t o m interchange reaction proceeds more slowly than the dissociative recombination, so t h a t it determines the rate of loss which is f i n ~ y[N2]N per unit volume. In the lower F-region the second process determines the rate of loss, which is eN 2 per unit volume. (v) O t- ions and electrons diffuse together through the neutral gas, the plasma diffusion coefficient D being given by D = b/{[O] ÷ [N2]}. At great heights, D is very large and the ionization takes up the vertical distribution corresponding to gravitational and electrostatic equilibrium, N ~ exp (--h/2H~). Because the ions are 0 +, the "scale height of the ionizable gas," Hi, is equal

to H(O). (vi) For the present, the parameters y and b introduced in (iii) and (v) above are assumed to vary as the square root of the gas temperature T. Some consequences of assuming different temperature dependences are discussed in Appendix A.2. (vii) At this stage, no account is taken of differences between the electron and ion temperatures, T~ and T i. It can be shown t h a t if T~ j= T i the diffusion terms in the continuity equation (Section 2.2) are modified in such a way t h a t the diffusion coefficient D is multiplied by ½(1 + T~/T~) and the scale height H~ is replaced by ½H~(1 ~ T,,/T~). 2.2 The equation of continuity r[he continuity equation satisfied by the electron d e n s i t y / ¥ is written ON/Ot - - q -- f i n + Dfl)N

(1)

660

H. RISHBETH

in which the "diffusion o p e r a t o r " ~ contains the operators 02~Oh 2 and 3/ah, a n d the scale height H~ of the ionizable gas. The basic param et ers q, fi and D depend on the structure of the neutral atmosphere, and so are in general functions of height and time. To evaluate q, let I ~ be the flux of E U V photons incident at the top of the atmosphere, and A(O) and A(N2) the cross-sections for ionization of 0 and absorption b y N 2. I t is convenient to set P® = I ~ A ( O ) ; this is the probability per unit time for ionization of an oxygen at om at zero optical depth. If 70 is the optical d ep th for overhead sun (X = 0) then, except near sunrise and sunset when the solar zenith angle g is large, q = P~ [ O ] . exp ( - - % see X) r o = [O]. A ( O ) t t ( O )

(2a)

+ [N2]. A(N2)H(Nz)

(2b)

q~here are several reasons for believing t h a t the general character of the comp u ted electron distributions does not depend critically on the assumptions of Section 2.1. First it is i m p o r t a n t to note t h a t at the F2-peak at noon, the optical d ep th % see X ~ 0.1 and the at t enuat i on of the incident radiation is therefole small. Un d er these circumstances, q(h) ~: [0] (el. equation (2a)) so t h a t q depends v er y little on the poorly-known [O]/[Nz] ratio at this height. Moreover, the wavelength dependence of % is u n i m p o r t a n t so t h a t it is unnecessary to express q as an integral with respect to wavelength, as is required at lower levels. Again, an accurate formula for fi should also contain a t erm denoting the contribution of O z. B u t because the scale heights of O z and N e are similar, the simple expression for fl is adequate: since the rate coefficient y is here t reat ed as a disposable constant, the uncertainties in the absolute concentrations of N~ and O~ are immaterial. I t seems t h a t omission of minor constituents, such as Oe or the light gases, can have no great effect on the present F2-1ayer model. 2.3 Solving the c o n t i n u i t y equation f o r a t i m e - v a r y i n g atmosphere The procedure for solving the time-varying continuity equation is based on the principle described by RATCLIFF:E and WEEKES (1960, p. 390) and by SHIMAZAgI (1957), who discuss the motion of a "cell" of air subjected to t e m p e r a t u r e changes. Within any cell the pressure p remains constant, and the cell can be assigned a fixed value of " r e duc e d height" z which is a function of pressure only. I t is found most convenient to define the reduced height z in terms of the partial pressure p~ of the ionizable gas (atomic oxygen), so t h a t z(h) = In (Pil/Pih) =-

fhh d h / H i

(3a)

1

h(z) -- h a ~

;

(dh/dz) dz --o

f:

H i dz

(3b)

in which Pim is the pressure at the height ha, the lower b o u n d a r y of the part of the atmosphere subject to diurnal t e m p e r a t u r e change. (In other contexts. however, z ~ 0 is often defined as the level of peak production for Z = 0.) I f the composition at height h 1 is fixed, t hen it will be independent of t e m p e r a t u r e at an y other given reduced height z. The height h I m a y be taken to lie at about

A time-varying model of the ionospheric F2-1ayer

661

]00 km, where the composition is unlikely to vary diurnally, though it might vary seasonally. It is convenient to use symbols N h and ~"z to denote electron densities at constant real heights and constant reduced heights respectively; and ~Tm for the peak electron density. When the time variation of IV at fixed h is considered, tile term "N(t) curve" is used, as is customary in the literature. The procedures for obtaining the electron distribution _N(h, t) m a y be described as follows: (A) Obtain a solution of the time-varying continuity equation (1) for a model in which the neutral atmosphere does not vary with time. Results obtained previously, by means of an analogue computer (RISHBETH, 1963), are used for this purpose. (B) Express the electron density N as a function of atmospheric pressure p (,r reduced height z, instead of real height h, for each hour of local time t. (C) Adopt a time-varying model atmosphere, and compute for each hour the real height h corresponding to fixed values of p, from equation (3b). For this, the models of HARRIS and PRIESTER (1962) are used. (D) Apply scaling laws to the values of Nz, to take account of the explicit dependence of the functions q(z), fl(z) and D(z) on temperature T. These are developed from the analysis of GARRIOTT and RISHBETtt (]963). Hence the solution of equation (1) for a given height in the time-independent model atmosphere is now taken to apply to a given "cell" (or given pressure-level) in the varying atmosphere. This assumes that, if the temperature increases, the neutral air expands vertically upwards and carries the ionization with it: this should be substantially correct except at low magnetic latitudes where the geomagnetic field prevents such motion of the ionization. 2.4 Approximate scaling laws .for electron densities GARRIOTT and RIStIBETH (1963) discussed the effect of temperature changes on the continuity equation (1). Using the concept of "reduced height" z, they found the dependence of the functions q(z), fi(z) and D(z) on temperature T to be very simple. In particular, b y assuming both of the parameters 7 and b to vary as T ~/2 (Section 2.1 (vi)), they found that N~ ~= T -1/2 at equilibrium. In the present work, this result is used for the time-varying situation. This m a y be a good approximation during most of the day, when aN/Or is small compared to the other terms in (1). At other times, ON/at is not small but other approximations can be made. For instance, just after sunrise the loss and diffusion terms are small, and at night photoionization is neglected. The arguments used by GARRIOTT and RISHB~TK can easily be extended to derive the approximations (4a) and (4c) appropriate to these times, in addition to the result (4b) actually given in that paper: Morning (post-sunrise): (aN~/Ot ~ q) aN~/0t o¢ T - 1 (4a) Midday and afternoon: (aN~/at ~ O) N~ oc T -1/2 (4b) Night : (q = O) fl' oc T -~/2 (4c) I~ (4e), fl' stands for the "effective" decay coefficient of the ionization at night

662

H . ]=~ISHBIgTH

(e.g. DUXGEY, 1956): it is defined by ON=/Ot = --fl'N=, and is independent of z. I f the assumption about the temperature dependences of 7 and b were altered, somewhat different results would be found (Appendix A.3). At times of day when the temperature is changing rapidly, it might be more appropriate to use the the perfect gas law p = N k T , which implies that N~ oc T ~, instead of equations (4). An important property of the relations (41)) and (4c) is t h a t at any given time, changes of T do not affect the shape of the N~ profile and hence the reduced heigt~t z,,~ of the electron pe~k is independent of T. Consequently, the diurnal wtriati(m of z m is unaffected by the diurnal temperature variation, though of course ~he real height h m is very sensitive to 5r'. In the calculations, the approximations given in equations (4) arc used as follows. The "equilibrium" formula (4b) is used from 0700 hours L.T. (by which hour the magnitude of the loss term, fiN, already exceeds a:V/~t) until F-layer sunset near 1900 hours L.T. At this latter hour, O,\'/at is not small and mine of the approximations is really applicable; it has been thought best to obtain ~:(z, t) by using equation (4b) up to 1900 hours L.T., and equation (4c) t'~r the nighttime period from 1900 hours L.T. until F-layer sunrise shortly before 05O0 hours L.T. Sinee little attention is paid to the sunrise period in this paper. ~lo a t t e m p t has been made to apply equation (4a), and the computed electron densities for 0500-0700 hours L.T. are very rough. Once the formulas (4) have been applied to each X(z) profile, the X(h) profile can be determined by applying equation (3b). It is then possible to draw X(t) curves for fixed real heights t~. The calculations are quite straightforward, but are obviously rather approximate. Moreover, it has not been possible to make all the assumptions strictly consistent with those originally made in the, design of the analogue computer used to solve equation (1). For instance, the form ulas (4) are strictly applicable only to models which are isothermal, at any given time, so t h a t the existence of appreciable temperature gradients below 300 km may introduce second-order errors. Nevertheless, it is believed t h a t the approximati(ms made do not affect the results in any essential way. 2.5 Details of the models The basic data for the calculations are solutions of equation (1) obtained by means of an analogue computer (BRIGGS and RISHBETH, 1961). The two N(h, t) distributions used for this purpose are based on the Model 4D atmosphere (NlcoLrn'. 1961, Table XLa) and refer to equinox at latitude 52°; they are described in a second paper (RISHBETH, 1963) in which they are denoted S1 and $2. The diffusion coefficients used in these two models are in the ratio 10 : 1 and they are accordingly named "fast diffusion" and "slow diffusion" models respectively; the values of q and l~ in the two models are identical. These N(h, t) distributions have been scaled to the time-varying model atmospheres of HARRIS and PRIESTER (1962; abbreviated below as " H & P"), which were obtained by solving the thermospheric heat-conduction equations, using boundary conditions consistent with satellite drag data. Harris and [?riester extrapolated their calculations to sunspot minimum, and have published tables applicable to several discrete values of solar 10.7 em flux density S, which serves

A time-varying model of the ionospheric F2-1ayer

663

as a c o n v e n i e n t i n d e x of solar a c t i v i t y . F o r p r e s e n t p u r p o s e s , t h e t a b l e s for S = 70 a n d S = 200 flux u n i t s are t a k e n to r e p r e s e n t s u n s p o t m i n i m u m a n d s u n s p o t m a x i m u m r e s p e c t i v e l y ( r o u g h l y e q u i v a l e n t to m e a n Ztirich s u n s p o t n u m b e r s R = 0 a n d R = 200): t h e y are c o m b i n e d with the a n a l o g u e S1 a n d $2 models to give four models w h i c h are referred to as " v a r y i n g - t e m p e r a t u r e " m o d e l s in c o n t r a s t to the " f i x e d - t e m p e r a t u r e " models S1 a n d $2, in which the t e m p e r a t u r e of the n e u t r a l a t m o s p h e r e is a s s u m e d i n d e p e n d e n t of time of day. The f o u r models are identified as follows: LI: L2: HI: H2:

L o w solar L o w solar H i g h solar H i g h solar

activity, activity, activity, activity,

R R R R

-= -=

0; 0; 200; 200;

H H H H

& P " S - - 70;" analogue & P ~"S = 70; " analogue & P " S = 200;" a n a l o g u e & P " S = 200;" a n a l o g u e

"fast "slow "fast "slow

diffusion" diffusion" diffusion" diffusion"

S1 $2 SI $2

These s y m b o l s a n d n a m e s are used t h r o u g h o u t the paper. N u m e r i c a l values assigned to the q u a n t i t i e s used in t h e calculations are listed in T a b l e 1, b u t no justification of these can be a t t e m p t e d until a c t u a l F - r e g i o n d a t a are discussed, as in Section 3. F r o m the d a t a of Table l, values of q, fi a n d D can be calculated. R e s u l t s are s h o w n in T a b l e 2, b o t h for 300 k m a l t i t u d e a n d for the level of the F 2 - p e a k . I t is later suggested t h a t the q u o t e d rates q, fl a n d D can p r o v i d e a good re,p r e s e n t a t i o n of the a c t u a l F 2 - l a y e r even if some of the a s s u m p t i o n s m a d e in Section 2.1 are i n a c c u r a t e . Because the F 2 - p e a k r e m a i n s at a fixed pressure-level (or fixed r e d u c e d h e i g h t z) w h e n t h e t e m p e r a t u r e is varied, the values of fi a n d D a t the peak v a r y m u c h less with t e m p e r a t u r e t h a n do the values at a n y fixed height h (Table 2). On the o t h e r h a n d , the values of q a n d fl at h)wer heights, such as the level of the peak o f Table 1. Values of parameters Name of atmospheric model Name of ionospheric model Number of ionospheric model Mean Zfirich sunspot No. R Solar 10.7 cm flux density S Noon 300 km temperature T(12) Midnight 300 km temperature T(00) Ionization cross-section A(O) Absorption cross-section A (N~) Incident EUV photon flux Im Ionization probability P~ (product A(0) I v ) Rate coefficient for / ~ ion-atom interchange J (value at 1250°K) Diffusion parameter b (value at 1250°K) "Fast diffusion" S1/L1/H1 "Slow diffusion" $2/L2/tt2

Nicolet

Harris & Priester

"4D"

" S - 70 .... S -- 200"

FixedTemp. S1/S2

Varying-Temp. "Low" "High" L1/L2 H1/H2

- 1236 1236 1-3 2-1 2.6 3.4 1

0 70 837 579

200 200 1614 1175

1.3

1.3

2.1

2.1 8.5 ll.0

1.5

2-0 1

1

lYnits 10-22 w m-2(c/s) -1 c,K °K 10-17 cm2 10--17 cm 2 101o cm-2 see-1 1 0 - 7 see-1 10-12 em ~ see-1 (assumed oc ~/~) (assumed ~ ~/~)

2.4 0.24

2.4 0.24

2.4 0.24

1019 c m I see 1 1019 cm-1 see-1

664

H . RISHBETH

Table 2. Production, loss and diffusion rates Atmospheric model (10.7 cm flux) Ionospheric model Sunspot number R Rates at 300 km Noon production rate Noon loss coefficient Midnight loss coefficient Noon diffusion coefficient* Midnight diffusion coefficient* ]Rates at noon F2-peak Height h.m Production rate q(hm) Loss coeff, fl(hm) Diffusion coeff. D(h,~)

q fl fi D D

S = 70 L1/L2 0

S = 200 H1/H2 200

Unit

44 0-35 0.056 73 145

660 6.8 3.6 20 22

cm 3 see-1 l0 a sec * l0 -a sec 1 109 cm 2 see 1

Fast diffusion L1 HI S = 70 S = 200 232 341 180 465 4.7 3-2 11.4 36.1

Slow diffusion L2 H2 S = 70 S -- 200 256 392 110 290 1.8 1-3 2.4 7.0

10 9 tin 2 see -1

km cm -3 se(! -1

10-a s e c t l09 cm 2 sec *

* VMues are for "fast diffusion" models; those for "slow diffusion" models are smaller by a factor of ten. p r e d u c t i o n , are v e r y sensitive to the a s s u m p t i o n s m a d e (such as the [O]/[N2] ratio of the model a t m o s p h e r e ) b u t it is e m p h a s i z e d t h a t these u n c e r t a i n t i e s arc n o t of g r e a t i m p o r t a n c e at F2-1ayer heights. 3. COMPARISON WITH OBSERVATIONAL DATA

3.1 The computed electron distributions Figures 1 to 8 show the diurnal v a r i a t i o n s of several q u a n t i t i e s for the v a r y i n g t e m p e r a t u r e model ionospheres, including the p e a k electron densities N,,, (Figs. 1 a n d 2), the heights h m of the p e a k (Figs. 3 a n d 4), a n d electron densities at several fixed heights on the b o t t o m s i d e a n d topside of the layers (Figs. 5-8). Also s h o w n in Figs. 1-4 are the p a r a m e t e r s Nm a n d h m for the f i x e d - t e m p e r a t u r e models Sl a n d $2 of ]:~ISHBETtt (1963); the g r a p h s of h,, are p a r t i c u l a r l y interesting, because. b y c o m p a r i n g t h e m w i t h the g r a p h s for the v a r y i n g - t e m p e r a t u r e models, the effects of t e m p e r a t u r e changes on the height of the F 2 - p e a k m a y be readily seen. T h e rise of h,~F2 in the evening, w h i c h occurs in the f i x e d - t e m p e r a t u r e models, is c o u n t e r a c t e d in the v a r y i n g - t e m p e r a t u r e models b y the t h e r m a l c o n t r a c t i o n of the a t m o s p h e r e . As a result, n o o n a n d m i d n i g h t h~F2 are a l m o s t equal. Qualitatively, the b o t t o m s i d e N(t) curves resemble those for m i d - l a t i t u d e stations, an e x a m p l e for m o d e r a t e solar a c t i v i t y (Slough, S e p t e m b e r 1950: R - 5 l) being s h o w n in Fig. 9. Too detailed c o m p a r i s o n s s h o u l d be a v o i d e d at this stage, b u t ( a p a r t f r o m fluctuations) the o b s e r v e d a n d c o m p u t e d N(t) curves are b r o a d l y similar b y day. So far, t h e n u m e r i c a l values of p a r a m e t e r s used in the calculation of the N(h, t) d i s t r i b u t i o n s h a v e been chosen more or less arbitrarily. This applies especially to the values o f the incident flux, I~, which is r e g a r d e d as a " f a c t o r of p r o p o r t i o n a l i t y . " I n o r d e r to j u s t i f y the values used, statistical analyses of F2 critical f r e q u e n c y d a t a are used in Section 3.2 to " n o r m a l i z e " the models to the

A time-varying model of the ionospheric _F2-layer t

i

1

i

I

665

i

18, 15 i

HI

:

-"s •9¢"

o'~" s

I

,

'~. ~"~'~'~.

I

sl

i

00

I

I

12

1 "~'--'~-18

24

0

00

12

IB

24

LOCAL TIME (HOURS) Figs. 1 and 2. Diurnal variation of peak electron density N m for the "fast diffusion" models (Fig. 1) and "slow diffusion" models (Fig. 2). The broken lines refer to models S1 and $2 of RISHBETH (1963), in which the neutral atmosphere has no time variation. The continuous lines refer to the models with time-varying neutral atmospheres; L1 and L2, low solar a c t i v i t y (Harris and Priester "S =70"), H1 and H2, high solar activity (Harris and Priester "S -- 200"). Dotted portions after sunrise are approximate. LOCAL

500,

i

L

TIME (HOURS)

I

i

I

500[

,

,

j

I

;

i

) 18

/ 4O(

i gO0 L2 El ~0(

I

I 06

I LOCAL

I 12

i

I 18

I

24

2%

TIME (HOURS)

,

I 06

~

LOCAL

I 12

i

24

TIME (HOURS)

Figs. 3 and 4. Diurnal variation of height of F2-peak, h m, for the "fast diffusion" (Fig. 3) and "slow diffusion" models (Fig. 4); identified as for Figs. 1 and 2. F o r the "fast diffusion" models in Fig. 3, hn, is not shown between midnight and sunrise as N m is vemy small. a c t u a l F 2 - 1 a y e r . O b s e r v a t i o n a l d a t a o n hmF2 a n d t h i c k n e s s p a r a m e t e r s a r e d i s c u s s e d i n S e c t i o n 3.3. I t w o u l d b e d e s i r a b l e t o c o m p a r e t h e m o d e l w i t h d a t a oil t h e t o t a l e l e c t r o n c o n t e n t o f t h e F - r e g i o n , S+N dh, a n d o n t h e a i r g l o w [O I ] 6 3 0 0 / k l i n e w h o s e i n t e n s i t y s h o u l d b e r e l a t e d t o t h e c o l u m n a r r a t e o f loss o f i o n i z a t i o n , .f~flN dh, b u t d e t a i l e d d i s c u s s i o n o f t h e s e t o p i c s d o e s n o t s e e m w a r r a n t e d a t present. Since the nighttime F2-1ayer presents special difficulties of interpretation. a f u l l e r d i s c u s s i o n is g i v e n in S e c t i o n s 3.4 a n d 3.5. 3.2 Normalization of the models against N,nF2 F o r " n o r m a l i z a t i o n " o f t h e m o d e l s , t h e a n a l y s e s b y YO~EZAWA a n d AI~IMA (1959) a n d BAZZARD a n d MI~]<~s (1961) a r e u s e f u l . T h e m o d e l s r e f e r t o e q u i n o x a n d w i l l b e c o m p a r e d w i t h a v e r a g e n o o n v a l u e s o f N,,F2; a n n u a l a n d s e m i a n n u a l

666

I f . RISHBETH

i

L

I

~

I

i

L

L

I aL

E -

Nm

~220

--210

.

z

/1

,~../

0!

•"

[//

\,

.-""/- \+0-~

I

06

12

18

24

LOCAL TIME (HOURS)

LOCAL TIME (HOURS)

Fig. 6. Slow diffusion, l o w s o l a r activity, m o d e l L2.

Fig. 5. Fast diffusion, l o w s o l a r activity, m o d e l L1 j

i

I

i

~

18

| _

15 'E 12

=

Nm

o

z

.-"

o

320

//500 / /

:"

9 Z

/

"

" 3 -

••

/ / / ..i/ / 6 ~ 0

280

/

" ~ . ,

" ' ~ ~, \

-" !

0,

06

t2

18

LOCAL TIME (HOURS)

Fig. 7. Fast diffusion, high solar activity, model HI.

24

0

06

12

18

24

LOCAL TIME (HOURS)

Fig. 8. Slow diffusion, high solar activity, model H2.

Figs. 5-8. D i u r n a l v a r i a t i o n s o f p e a k e l e c t r o n d e n s i t y (heavy line), and of elect r o n d e n s i t i e s at s e v e r a l f i x e d h e i g h t s b e l o w t h e p e a k ( t h i n c o n t i n u o u s lines) a n d a b o v e t h e p e a k ( d a s h e d lines). T h e d o t t e d p o r t i o n s o f t h e c u r v e for N , c after sunrise, are approximate. H e i g h t s a r e given in kilometres.

c o m p o n e n t s in the o b s e r v a t i o n a l d a t a can be disregarded. U n % r t u n a t e l y , three different indices of solar a c t i v i t y are i n v o l v e d , since t h e Harris and Priester m o d e l s use t h e 10.7 c m flux d e n s i t y S, Y o n e z a w a a n d A r i m a use the m o n t h l y m e a n s u n s p o t n u m b e r R, and B a z z a r d a n d Minnis use t h e i o n o s p h e r i c index IF2 w h o s e scale r e s e m b l e s t h a t of R. F o r present purposes, the relations b e t w e e n t h e s e indices are a s s u m e d linear, and t h e f o l l o w i n g v a l u e s are used: Sunspot Minimum. S = 70, R =: 0, IF2 0 (Models L1 and L2). F o r l a t i t u d e s o f a b o u t 50°N, Y o n e z a w a a n d A r i m a give t h e " a v e r a g e n o o n N,,F2 as 3.7 x 105 c m -3, c o n s i s t e n t w i t h t h e v a l u e d e d u c e d f r o m t h e c o n t o u r m a p s o f B a z z a r d a n d Minnis. T h e " i o n i z a t i o n p r o b a b i l i t y " P~_ ( e q u a t i o n (2a)) is t h e n c h o s e n to be

A time-varying model of the ionospheric F2-1ayer

667

2 >,~ 10 -7 sec -1, which gives n o o n N m F 2 as 3.0 x 105 cm -a a n d 5.0 × 105 cm -a for models L1 a n d L2 respectively. S u n s p o t m a x i m u m . S = 200, R = 200, IF2 = 200 (Models H1 a n d H2). The, Y o n e z a w a a n d A r i m a analysis refers o n l y to R = 0 a n d R = 100, b u t a linear e x t r a p o l a t i o n to R -- 200 gives n o o n N , ~ F 2 as 15.5 × 105 cm -a, in g o o d a g r e e m e n t with the v a l u e d e d u c e d f r o m the B a z z a r d a n d Minnis data. T a k i n g Po~ -- l l >: 10 - T s e c -1, n o o n N m F 2 is 11.4 × 1 0 5 c m -a for model H1 a n d 19.4 × 10 ~ c m -3 tbr model H2. 8

~4

SLOUGHSEPTEMBER~950

240

.~

I

i

I 24

LOCAL TIME (HOURS)

Fig. 9. Mean quiet N(t) curves for Slough, September 1950 (R - - 5 1 ) (After CROOM,]:~OBBINSand THOMAS 1959). Heights are given in kilometres.

I f A(O) -- 1.3 x 10 -17 cm 2, as a s s u m e d in Table 1, the chosen values of P< at the t w o epochs S = 70 a n d S = 200 c o r r e s p o n d to E U V p h o t o n fluxes I ~ of 1-5 x 10 l° cm -2 sec -1 a n d 8.5 x l01° cm -e sec -1, a n d to e n e r g y fluxes of 0.5 and 2.9 erg cm -2 sec -1 if the m e a n w a v e l e n g t h is 600/~. L i n e a r i n t e r p o l a t i o n between these values of I ~ gives I ~ = 3.4 × 101° cm -2 sec -1 for S = 106, a p p r o p r i a t e to the m o n t h of A u g u s t 1961 in w h i c h the r o c k e t e x p e r i m e n t of HINTEREGGER and WATAbTABE (1962) was carried out. The; d a t a of H i n t e r e g g e r a n d W a t a n a b e , when s u m m e d over the r a n g e f r o m 280 A to 911 A, give I ~ = 4.0 x 101° cm -e s e c - L The values of I ~ used in this p a p e r are t h u s in s u b s t a n t i a l a g r e e m e n t with the r o c k e t data, a n d w o u l d agree even b e t t e r if the a d o p t e d cross-section ,4 (O) were r e d u c e d b y 15 per cent. Some f u r t h e r discussion is given in the A p p e n d i x

3.3 Comparison with observed parameters of the F2-pealc I n this section the models are c o m p a r e d w i t h d a t a for e q u i n o x m o n t h s , listed in Table 3, for six m i d - l a t i t u d e stations. O n l y n o o n a n d m i d n i g h t values are used. The p a r a m e t e r s used are t h e p e a k electron d e n s i t y N,~F2, the h e i g h t hmF2, a n d a p a r a m e t e r hTF2 or " S c a t " (scale h e i g h t at the p e a k ; cf. WRIGHT, 1962a) which describes the t h i c k n e s s of t h e l a y e r just below the peak. The thickness ]~7 is defined as the vertical distance below h m in which :\; decreases to 0.7 N,~: a c o r r e s p o n d i n g " r e d u c e d t h i c k n e s s " z 7 can be defined b y using the scale height H i, as the the following e q u a t i o n s :

H i . z~ = h 7 -- h m F 2 -- h(0.7 Nm).

(5).

F o r a " C h a p m a n a l p h a " layer, h 7 a n d " S c a t " are equal, a n d z v = 1.00 scale height.

668

H. RISHBETH

201

/5

S (FLUX UNITS) 150

I00

'

~

200

iJ

H

L2 50

o

I

I

I

ioo

150

2o0

25o

MEAN SUNSPOTNUMBERR Fig. 10. Peak F2 electron densities ~ m for "mean quiet day's" in equinox months at six mid-latitude stations (see Table 3). Open circles refer to noon, filled circles to midnight. The ordinates are the m o n t h l y mean Ziirich sunspot n u m b e r R and the solar 10.7 cm flux density S, between which a linear relation is assumed. The points * at R = 0 and R -- 200 refer to the "fast diffusion" models L1 and H I (for which midnight values of N m are too small to be shown) and "slow diffusion" models L2 and }{2. Noon values are joined by continuous lines, midnight values by broken lines.

S (FLUX UNITS) 40070

I00

150

I

•

200

•

I

•

s~+

mF H2

°

350--

,+"~ acE

. "' J

250

20¢

~

o

o ,,s~,,~ ' ~

:>" o

--

o

'

50

i

I

100 150 ZOO MEAN SUNSPOTNUMBERR

250

Fig. l l. Heights of F2-peak, using observational data as for Fig. 10. Open circles refer to noon, filled circles to midnight. The points * refer to the computed models; noon values are joined by continuous lines, midnight values by broken lines. F o r t h e s t a t i o n s St. J o h n ' s , F o r t M o n m o u t h a n d W h i t e S a n d s t h e v a l u e s o f

NmF2, h,,,F2 a n d " S c a t " h a v e b e e n r e a d f r o m t h e c o n t o u r d i a g r a m s g i v e n b y WRIGHT (1962a); for t h e o t h e r s , d a t a for s e v e r a l q u i e t d a y s h a v e b e e n a v e r a g e d t o o b t a i n r e p r e s e n t a t i v e v a l u e s o f N,nF2, hmF2, a n d hTF2. The d a t a are p l o t t e d a g a i n s t m e a n s u n s p o t n u m b e r in Figs. 10-12, b u t no

A t i m e - v a r y i n g m o d e l of t h e i o n o s p h e r i c F2-1ayer

669

S (FLUX UNITS)

?0

I00

150

200

~"~

511

io"~

I 50

0

I 100 MEAN

~ 150

SUNSPOT NUMBER

: 200

250

R

Fig. 12. T h i c k n e s s e s of F 2 - l a y e r below p e a k , g i v e n b y h 7 ( e q u a t i o n (5)) or " S c a t ;" u s i n g o b s e r v a t i o n a l d a t a as for :Fig. 10. O p e n circles refer to n o o n , filled circles t o m i d n i g h t . C o m p u t e d t h i c k n e s s e s for t h e " f a s t " a n d " s l o w " diffusion m o d e l s a r e v e r y similar, so t h a t p o i n t s * a t R =- 0 ( m a r k e d L) refer t o t h e m e a n of L1 a n d L 2 ; a n d a t R -- 200 ( m a r k e d H) r e f e r t o t h e m e a n of H 1 a n d H2. N o o n v a l u e s are j o i n e d b y c o n t i n u o u s lines, m i d n i g h t v a l u e s b y b r o k e n lines.

distinction between the different stations is made. Starred points are plotted at R ~ 0 for the computed models L1 and L2 and at R ---- 200 for models H1 and H2, and are joined by straight lines (though the relationships are not necessarily linear). One of the continuous lines joins the noon values for the "fast diffusion" models L1 and H1, and the other refers to the "slow diffusion" models L2 and H2. Similarly, midnight values are joined by dashed lines, but N,~ is not shown for the "fast diffusion" models at midnight, since it has decreased to very small values by this time. In :Fig. 10, nearly all the points for noon NmF2 fall between the lines for the " f a s t " and "slow" diffusion models. This of course confirms the validity of the normalization made in Section 3.2, and also shows t h a t if the " f a s t " model, say, were found to be preferable the values of q should be increased somewhat. Since the midnight values of N~ for the "fast diffusion" models are too small to be plotted on the graph, it might seem t h a t the "slow diffusion" model is better in T a b l e 3. M i d - l a t i t u d e i o n o s p h e r i c s t a t i o n s

Station Slough St. J o h n Ft. Monmouth White Sands Watheroo Christchurch

Geog. lat.

Mag. dip.

Ref.

M o n t h s used a n d m e a n Ziirich s u n s p o t n u m b e r s

52 48 40 32 - 30 -- 44

68 72 72 60 -- 64 -- 68

THR ~V ~,V W CRT K

Sep. Sep. Sep. Sep. Sep. Eqx.

1953 1959 1959 1959 1954 54/5

(19) (142) (142) (142) (2) (5)

Sep. Mar. Mar. Mar. Sep. Eqx.

1950 1960 1960 1960 1950 55/6

(51) Sep. 1957 (235) (104) (104) (104) (51) Sep. 1948 (143) (80) E q x . 56/7 (200)

R e f e r e n c e s : W" WRIGHT (1962a); THR--THOMAS, HASELGROVE a n d ROBBINS (1957); C R T - - C R o o ~ , ROBBINS a n d THOMAS (1959); K - - - p r i v a t e c o m m u n i c a t i o n f r o m G. A. M. KIN(; ( d a t a are t a k e n f r o m a few m o n t h s of s i m i l a r solar a c t i v i t y ) .

670

H. RISHBETH

this respect; but in the next section alternative interpretations of the nighttime data are discussed. Again, the nighttime values of hmF2 present further difficulties, to be discussed shortly. The noon values are consistent with the "fast diffusion" models: however, if the points for individual stations were identified on the figure, they would show a decrease of h m with increasing latitude which m a y well be due to variations of' temperature, not considered in the Harris and Priester model. There is satisfactory agreement between the observed and computed thicknesses hTF2 and " S c a t " (Fig. 12). At noon, particularly at low sunspot numbers. the observed values are increased by the presence of a well-developed Fl-ledge, of which no account is taken in the model. Calculated values of h 7 for the " f a s t " and "slow" diffusion models are so similar t h a t only the average of the two is shown. Noon values exceed midnight values on account of the greater scale height by day, but the "reduced thickness" z 7 (equation (5)) is actually smaller at Hoon than at midnight. 3.4 Decay of the F2-1ayer at night Although the midnight F2-1ayer data show" some agreement with the "slow diffusion" models, a more detailed investigation is called for in which the "effective decay coefficient" fi' is estimated from observational N(h, t) data. According to the theory developed earlier in this paper, fi' should correspond to the rate of decay of N~ rather t h a n the rate of decay of Nh, which is influenced greatly by the thermal contraction of the atmosphere (Appendix A.4). Of course, if strong electromagnetic drifts are present, neither N m nor N h can be relied on to give values of fl', but this possibility is not considered here. Values of fl' have been computed from the rate of decay of N,, during the few hours following F-layer sunset, on International Quiet Days during a number of equinox months. The values obtained on different nights in a month, and for successive hours on the same night naturally show a considerable scatter, and the numbers shown in Table 4 are medians. The scatter is such t h a t the quartiles generally differ from the medians by 30-40 per cent. Data for only two stations are shown, since no exhaustive survey is a t t e m p t e d here; but further values deduced from the contours of :¥~ (WRIGHT, 1962a) are generally eonsistel~t with those given. The second value of fl', which appears beside most of the values in Table 4. is derived in a different way. Inspection of actual N(t) curves, such as those in Fig. 9, suggests t h a t Nm decays exponentially only during the first few hours of the night. After this, electron densities become relatively constant and sometimes even increase. This behaviour cannot possibly be explained in terms of the processes so far considered in this paper. I t suggests t h a t there exists a "base level" of ionization maintained by some other agency. I f this hypothesis is adopted, theu fl' is obtained from the equation

-~'

=

(a/at) In (N~

-

N~)

(~)

in which N ~ is the "base level" of the peak electron density N m. The values of fl' found from this equation, using values of Nmb estimated from the night N(t)

A time-varying model of the ionospheric F2-1ayer

671

Table 4. Effective decay coefficients after sunsef

fl', Unit l0 -~ see-1 A. Models: "Fast diffusion" "Slow diffusion" B. Observations:

L1 (R = 0) 3.0 L2 (R =- 0) 0.77 Smnmer R

Slough (52°N)

22 91

~Vatheroo (30°S)

28 54 138

~'

0.6 0.3 0.4 0-4 0.3

(2) (*) (2) (2) (*)

(R = 200) 2-0 H2 (R = 200) 0.55 Hi

Equinox R

19 51 2 51 143

~'

0-8 0.8 0-7 0.7 0.4

(1½) (2) (2) (2) (11 )

Winter ~' 2 1.2 (3) 102 1.1 (3) 0 1.3 (3) 84 1.3 (3) 168 1.1 (3) R

These data refer to months for which N(h, t) distributions were computed by THOMASet al. (1957) and CRoo-~ et al. (1959). Bracketed values of fl' are derived by using equation (6), the "base level" -'¥mb (generally about 105 cm-a) being estimated from the N(h, t) data. No estimate could be made in cases marked * curves, are s h o w n b r a c k e t e d in T a b l e 4 a n d are larger t h a n those of the original calc u l a t i o n s in which it w a s a s s u m e d t h a t Nm~, = 0. G e n e r a l l y , N,, b is e s t i m a t e d to be a b o u t 1 × 1 0 5 c m -a. I f t h e " b a s e l e v e l " h y p o t h e s i s is n o t a d o p t e d , t h e n T a b l e 4 shows t h a t t h e n i g h t t i m e loss r a t e s c o r r e s p o n d b e s t to t h e " s l o w diffusion" models. T h e "levelling:" of t h e N(t) curves in t h e l a t t e r p a r t of t h e n i g h t r e m a i n s to be explained. B u t if t h e h y p o t h e s i s is a d o p t e d , t w o a d v a n t a g e s result. First, it is possible to a d o p t t h e " f a s t diffusion" m o d e l s w h i c h are p r e f e r a b l e as far as t h e d a y t i m e F 2 - 1 a y e r is concerned. Second, t h e a p p a r e n t seasonal v a r i a t i o n in fl', which h a s been note,] b y WRIGHT (1962b), is r e d u c e d a n d (in v i e w of t h e u n c e r t a i n t i e s i n v o l v e d in e s t i m a t i n g fl') m a y e v e n d i s a p p e a r c o m p l e t e l y . O f course, t h e existence of a " b a s e level" poses a p r o b l e m in itself, which m u s t n o w be discussed. 3.5 Further discussion of the night F-layer T h e m a j o r difficulties concerning t h e n i g h t F 2 - 1 a y e r are t h e m a i n t e n a n c e of t h e i o n i z a t i o n in the l a t t e r p a r t of t h e night, especially in winter; a n d t h e rise of h,~F2 a f t e r sunset w h i c h o f t e n a c c o m p a n i e s large f l u c t u a t i o n s of electron d e n s i t y , a n d w h i c h is a s s o c i a t e d w i t h t h e difference b e t w e e n n o o n a n d m i d n i g h t v a l u e s of hmF2 (Fig. 11). A m o r e d e t a i l e d s t u d y of d a t a , such as t h e c o n t o u r s of hmF2 g i v e n b y WRIGHT (1962a), shows t h a t this difference increases w i t h i n c r e a s i n / latitude. F i g u r e 11 shows t h a t t h e n o o n v a l u e s of h,,F2 are c o n s i s t e n t w i t h t h e " f a s t diffusion" models, b u t t h a t m i d n i g h t v a l u e s m o s t l y exceed t h o s e of t h e " s l o w diffusion" models, especially a t s u n s p o t m i n i m u m . I t also shows t h a t for either model, n o o n a n d m i d n i g h t hmF2 are a l m o s t e q u a l (cf. Section 3.1). Possible reasons for t h e d i s c r e p a n c y include: (a) T h e d a y - t o - n i g h t t e m p e r a t u r e a n d d e n s i t y v a r i a t i o n s of t h e H a r r i s a n d P r i e s t e r m o d e l a t m o s p h e r e s are seriously in error, p a r t i c u l a r l y a t s u n s p o t m i n i m u m . Discussion of this p o s s i b i l i t y is b e y o n d t h e scope of this p a p e r .

672

]:I. I~ISIIBETtt

(b) The diffusion or loss coefficients undergo diurnal changes arising from large changes in neutral composition at any fixed z; or in the parameters b or ~. I t is important to distinguish these from the variations of D and fl associated with the diurnal expansion and contraction of the atmosphere, which are already included in the analysis. (c) The night F-layer is controlled by other processes, such as electromagnetic movements or nocturnal sources of ionization. As regards (b), it should be noted t h a t the discrepancy cannot be removed just by scaling the diffusion and loss parameters, b and y, by any numerical factor. Any such change would alter the reduced height of the peak, Zm, by a certain a m o u n t (Appendix A.1), but the resulting change of h m would be much the same by day and by night. The hypothesis t h a t large decreases of fi (at any given z) occur at night is unattractive, largely because such a decrease would bring about a further drop in hmF2. The behaviour of h,~F2 might be consistent with a large nocturnal decrease of D(z), but this is an ad hoc hypothesis. The d a y - t o - n i g h t change of h m could be increased, though probably to no great extent, by assuming t h a t the diffusion parameter b and the i o n - a t o m interchange coefficient y have different temperature dependences (Appendix A.3). Another factor is t h a t the unequal ion and electron temperatures lead to all increase of the diffusion coefficient by day, as ill (vii) of Section 2.1. These suggestions are probably inadequate to explain the discrepancy, so other processes must be considered, as in (c). Discussion of a "nocturnal source" is linked to the question of the "base level" of the F-layer proposed in Section 3.4 to account for the "levelling" of observed N(t) curves late at night. Often NmF2 is observed to increase at night, and if it could be proved t h a t such increases are not just due to redistribution of existing ionization, the case for a nocturnal source of ionization would be very strong. ~[he nocturnal source might take various forms. One possibility is photoionization by scattered solar EUV, but this seems to be ruled out by the observations of BYRAI~I et al. (1961). Production of ionization by energetic electrons is a plausible mechanism (ANToNOVA and IVANOV-KHOLODNII, 1961) but quantitative data is lacking. Another suggestion is t h a t tim exosphere acts as a reservoir, from which plasma diffuses downward to maintain the "base level" of the night F2-1ayer This explanation depends on the existence of a diffusive barrier, such as t h a t described by HANSON and ORTE~BURGER (1961), tO limit the rate of flow from the exosphere I f there were no such barrier, the theory of the "shape-preserving" distribution would apply to the exosphere just as much as to the F2-1ayer; the electron density in both regions would decay with the same time constant (set by the value of fl at the F2-peak) and there could be no question of maintaining a "base level." The flux required to maintain a "base level" Nmb = 1 × 105 cm -3, consistent with observations, can be found from the analogue calculations described in Section 5.2 of RISI~IBETI~I (1963): it is 5 × 10s cm -2 see -1 for the "fast diffusion" model S1 and 1.3 × 10 s em -2 see -1 for the "slow diffusion" model $2. But it is doubtful whether downward fluxes exceeding 10 s em -2 see -1 can be sustained througb~out the night (HAnsON and PATTERSON, 1963), SO t h a t the "base level"

A time-varying model of the ionospheric F2-1ayer

673

m a y be largely maintained by corpuscular ionization. The difficulty about the nighttime level of hmF2 might be resolved if it were shown t h a t the peak of a layer maintained by exospheric flux or corpuscular ionization were higher t h a n the peak of a normal " s t a t i o n a r y layer." Detailed calculations would be required to establish this. The possibility t h a t electromagnetic drifts play an important part in the night F-layer is neither established nor ruled out. Drifts might well be responsible for some irregularities in N(t) curves, particularly those observed near sunset. They might be responsible for the discrepancy between day and night values of h,,~F2, but it is difficult to see how t h e y could maintain a "base level" of ionization. 4. DISCUSSION

4.1 The idealized models It is now possible to give some answer to the questions raised in the Introduction. The first of these concerned the behaviour of an idealized F2-1ayer controlled only by the rates of photoionization (q), linear recombination (fl) and plasma diffusion (D). In order to take account of the diurnal temperature changes which are an essential feature of the F-region, a method has been developed which modifies the solutions N(h, t) of the continuity equation originally obtained for a fixed-temperature model atmosphere. This is accomplished by assuming t h a t the ionization takes part in the thermal expansion and contraction of the atmosphere: the variation of N at a fixed pressure level, or fixed reduced height z, is easily calculated--subject to certain assumptions and approximations--because the diurnal variation of q, fl and D at any constant z are much simpler than their variations at a n y real height h. Although the analogue method used to compute the original electron distributions was applied to only two examples, termed "fast diffusion" and "slow diffusion" models, the analysis in Appendix A of this paper enables the effects of changes of numerical values, and modifications c,f certain assumptions, to be studied qualitatively. Figures 1-8 demonstrate some of the features of the computed models; Figures 1-4 also compare the diurnal variations of the F2-peak with those of models which do not take account of the diurnal temperature variations. For the latter, the interaction of production, loss and diffusion processes was discussed in Section 7 of RISHnETH (1963), and very similar considerations apply to the varyingtemperature models presented in the present paper. Variations of temperature can greatly influence the variation of electron density at fixed heights, but have smaller effect on NmF2 and thus seem unlikely to be the cause of major anomalies in this parameter or of effects during storms (cf. Appendix A.3). 4.2 Comparison with mid-latitude F2-layer data The computed electron distributions are consistent with F-region data for mid-latitude stations in the following respects. Except where reference is made to an Appendix, more detailed discussion has been given in Section 3. (i) I f the models are "normalized" for varying sunspot numbers against observed values of hmF2 for equinox noon, the deduced solar EUV flux is consistent with the data of ~IINTEI~EGGER and WATANABE (1962). (But see A.2.)

674

H. ~ISHBETH

(iX) In the models, hmF2 falls rapidly at sunrise and then rises slowly during the day. Noon values of hmF2 for the "fast diffusion" model are in good agreement with observational data. (iii) The observed thickness of the F2-1aycr below the peak, as measured by the parameters hTF2 or "Scat", agree well with the models both at noon and at midnight. (iv) Both hmF2 and hTF2 increase with increasing sunspot number, on account of the increase of atmospheric temperature. (v) The shape of daytime N(t) curves at fixed heights, and in particular the morning maxima of N, are explained qualitatively by the diurnal temperature variations, though increases of N near or after sunset are not explained (Appendix A.4). (vi) The "effective" decay coefficient fl', deduced from the rate of decay of N,,F2 after sunset, agrees with the "slow diffusion" models. (vii) If however, a nighttime "base level" of ionization exists, as the behaviour of the night F2-1ayer indicates, then the values of fl' favour the "fast diffusion" models. (iii) Values of q and fl at 300 km are consistent, to within a factor of two, with those derived in other papers (Appendix B.1, B.2). In principle, the model m a y also be made consistent with models of the lower F-region (See Appendix B.4). 4.3 Comments on some diJflculties r[here are, of course, many features of F-region behaviour that the model does not a t t e m p t to reproduce. Outstanding amongst these are the seasonal anomaly, which m a y well require explanation in terms of structural changes in the atmosphere; and the equatorial anomaly, which m a y partly be due to electromagnetic drifts. Of the difficulties that have become apparent in the discussion of this mode], one is that the solar-cycle variation of E U V flux required to account for F-region ionization is greater than that used in current atmospheric models, such as the Harris and Priester model actually employed in the calculations of this paper. The other major problems concern the night F-layer: how it is maintained in the latter part of the night, and w h y the F2-peak at mid-latitudes is much higher by night than b y day. The latter observation might indicate large d a y - t o - n i g h t changes in diffusion or loss coefficients, quite different from the diurnal changes which are expected to result from the thermal expansion and contraction of the neutral atmosphere. Alternatively, these difficulties might be resolved if the existence of a nocturnal source, capable of maintaining a "base level" of ionization in the F-region, were established. Two processes were mentioned which might contribute to this source; namely, corpuscular ionization and the downward diffusion of plasma from the exosphere. There does not seem to be any compelling evidence that electromagnetic drifts play a major role in the mid-latitude F2-1ayer, but neither can the possibility be excluded. It is also questionable (Appendix A.3) whether temperature changes can cause anomalies of NmF2, such as storm effects.

A time-varying model of the ionospheric F2-1ayer

675

4.4 Numerical values of rates The remaining question asked in the I n t r o d u c t i o n concerns the magnitude of the various rates. I t must be recognized t h a t because of the diurnal and solarcycle variations of the neutral atmosphere, the rates q, fl and D cannot be regarded as unique functions of height; it is essential to specify additional parameters, such as atmospheric t e m p e r a t u r e which depends on time of day and solar activity. A corollary is t h a t calculations of rates from N(t) curves at fixed heights m a y not give reliable results at times when the t e m p e r a t u r e is changing rapidly. The "fast diffusion" model used in this paper has been found quite successful in explaining the behaviour of the d a y t i m e F2-1ayer. I t can also be applied to the nighttime layer, provided the "base level" hypothesis is adopted. Noon values Noon rates at 300 km Photoionization rate q(cm-3 sec-1) Loss coefficient

Sunspotminimum R - 0, S = 70, T ~850°K 50

Sunspot maximum (R 200,S := 200, T~1700°K) 750

0-4

7

7

2

fi(10 - 4 see -1)

Diffusion coefficient D(10TM cm2 sec-1)

of the rates at 300 km, t aken from Table 2, are summarized again here; the numbers have been rounded to some extent, and the values of q slightly increased since this was found desirable after detailed comparison with observational data. Values for different times and other levels of solar activity can be computed with the aid of an appropriate atmospheric model. Once a model atmosphere has been adopted, the equations given in Sections 2.1 and 2.2 can be used to derive other quantities, such as the solar flux, i o n - a t o m interchange rate coefficient and diffusion parameter, from these data. I n Appendix A.2, it is suggested t h a t the values of fi and D are upper limits, and could be reduced by factors not exceeding two. Some adj ust m ent to values of q would then be necessary, and f ur ther changes would be required if it were desired to include other processes, such as electromagnetic drifts or corpuscular ionization. Hence it cannot y e t be claimed t h a t a n y of the rates in the F2-1ayer are well known. Progress in this field m a y be expected from detailed studies of the p h o t o c h e m i s t r y of the lower F-region; comparison of ionospheric and airglow data: and f ur t he r direct observations of solar ionizing radiation. APPENDIX A:

FURTHER THEORETICAL ANALYSIS

This appendix is intended to supplement the dat a obtained from the analogue co mp u ter with further semiquantitative analysis. The effects of changing the diffusion and loss parameters are derived in Appendix A.], and applied in A.2 to the question of how far the values of parameters used in the com put ed models might be varied. Section A.3 deals with the effects of changing the assumptions

676

I-I. R~S~B~T~

made in Section 2.1 regarding the temperature dependences of the diffusion and loss coefficients, and the factors which govern the shape of the diurnal N(t) curves are considered in A.4. A.1 D e p e n d e n c e of the 2'2 p e a k on d i f f u s i o n a n d loss p a r a m e t e r s

The behaviour of the model F2=layer depends on three basic parameters q, fl and D, of which q m a y be regarded as a "factor of proportionality" in the daytime electron density. The effects of changes of fl and D are more complicated but m a y be studied semiquantitatively by applying the theory of the "equilibrium layer" (RISHBETH and BAR~ON, 1.960) to daytime conditions and the "shapepreserving layer" (DuNcAN, 1956; DUNGE¥, 1956) to the nighttime situation. Vertical variations of temperature are not considered here because the assumption of a simple isothermal atmosphere is quite adequate for present purposes. The formulas are best expressed in terms of reduced height z (Section 2.3). In the F2-1ayer, q oc e-z approximately. I f H i / k is the scale height of the molecular gas involved in the loss process, then fl = D0e-~:z; and another parameter j is introduced for the height dependence of D , so t h a t D = D0e+Jz. In numerical examples it is assumed t h a t k = 1.75 (as for N2) and t h a t atomic oxygen plays a major part in determining D, so t h a t j = 1. The relevant results in the papers cited above m a y be summarized thus: (a) The F2-peak is situated at a level z,~ where the ratio L = fiH[~/D takes specific values, L~ for the " d a y equilibrium" layer and L~ for the "night s t a t i o n a r y " or "shape-preserving" layer. L~ > L, so t h a t z m is greater by night than by (lay. I f L 0 is the value of L at the reference level z = 0 (whose precise position is immaterial, but which m a y be assumed to lie well below the F2-peak) then for the d a y t i m e layer In (floHi2/Do) = In L o = In L~ ÷ (k ÷ j ) z m (7a) with a similar equation (with L~ instead of L~) at night. (b) The equilibrium peak electron density by day is

N~

= u'q(~m)/~(~,m)

(Tb)

(c) At night, N m decreases with an "effective" coefficient t~' = ~"~(~,,,)

(7c)

The quantities L~, L~, u' and u" are numbers of order unity, and m a y be treated as constants for given values of k and j. Their actual values are immaterial. By combining these equations, the parameters % , N,n ~ and/3' can be written as functions of/3 o and Do, which are differentiated to give the following equations for finite differences in z~, N~, and /3' resulting from finite changes in the magnitudes of /3o and D 0. The symbol i = (j + /~)-1 is introduced for brevity. Az m = i A(ln/30) -- i A(ln Do)

A(ln Nm~) = - - i ( j + 1) A(ln/3o) -- i(k -- 1) A(ln Do) A(ln/3') = ij A(ln/30) ÷ ik A(ln Do)

(Sa) (Sb) (Sc)

A time-varying model of the ionospheric F2-1ayer

677

I n s e r t i o n of the specimen v a l u e s /c = 1.75, j = 1, i = 0.36 gives Az~

+ 0 . 3 6 A(ln/30) -- 0.36 A(ln Do)

(ga)

A(ln Nm~ ) = --0-73 A(ln/30) -- 0.27 A(ln Do)

(gb)

A(ln/3') = + 0 " 3 6 A(ln rio) + 0-64 A(ln Do)

(9c)

~[hese e q u a t i o n s show t h a t an increase of either /30 or D o leads to r e d u c e d electron d e n s i t y b y d a y a n d an increased r a t e of d e c a y at night. I t is interesting t h a t , while a given p e r c e n t a g e increase' of/30 reduces N ~ more t h a n does an equal p e r c e n t a g e change of D o, the reverse is t r u e as regards the n i g h t t i m e d e c a y r a t e fl'. B o t h b y d a y a n d b y night, z~ is raised b y increasing fie or reducing D o. T h e e q u a t i o n s can n o w be used to discuss in w h a t respects the diffusion a n d loss coefficients p r o p o s e d in this p a p e r m i g h t be i m p r o v e d .

4.2 Possible changes in the suggested rates F o r the p u r p o s e of t h e following discussion, the " f a s t diffusion" models L1, I-t 1 are a d o p t e d as " s t a n d a r d . " F i g u r e 11 shows t h a t t h e y give good a g r e e m e n t with o b s e r v e d n o o n values of hmF2. F r o m Fig. 10 it is seen t h a t an increase of N,,, b y a f a c t o r of 1.3 from the " f a s t diffusion" values would i m p r o v e the agreem e n t w i t h the data, b u t an increase b y a f a c t o r exceeding a b o u t 1-7 is u n d e s i r a b l e H e n c e the values of q m i g h t be increased b y a b o u t 30 per cent. As n o t e d in Sectim~ 3.2, t h e y are in good a g r e e m e n t with the results of HINTEI~EGGER and WATANABE (1962), a l t h o u g h t h e solar-cycle v a r i a t i o n of E U V flux has n o t y e t been experi= m e n t a l l y d e t e r m i n e d . A difficulty exists in t h a t the ratio of the a d o p t e d values for I ~ at the epochs S = 70 a n d S ~ 200 is 5.5, as c o m p a r e d to a ratio of 2-s for the Harris a n d P r i e s t e r models in which the E U V flux is assumed to v a r y as the 10.7 cm flux d e n s i t y S. The ratio of 2.8 seems inconsistent with the observed solar-cycle v a r i a t i o n of NmF2, b u t this d i s c r e p a n c y c a n n o t p r o f i t a b l y be discussed until more c o m p l e t e d a t a for sunspot m i n i m u m b e c o m e available. I f q is n o w c o n s i d e r e d to be fixed, the effects of changing fl a n d D can be discussed, with reference to e q u a t i o n s (8) of A p p e n d i x A.1 and to the d a t a on N,,,F2, h,,~ a n d fi' . As n o t e d in Section 3.4, the values of ~' are v e r y u n c e r t a i n , b u t it is d o u b t f u l w h e t h e r fl' could be m u c h larger t h a n is given b y the " f a s t diffusion" model, t h o u g h it could well be smaller. H e n c e the " f a s t diffusion" values of D and/~ could be reduced, and the following t h r e e cases d e m o n s t r a t e how this could be done: (a) I f D is r e d u c e d b y a f a c t o r of 3, t h e n N .... is increased b y a f a c t o r of 1.35 and z~ is raised b y 0.4 scale height, c o r r e s p o n d i n g to an increase of 20 to 30 km in h,,F2: fi' is r e d u c e d b y a f a c t o r of 2.0. (b) I f D remains fixed b u t fi is r e d u c e d b y a f a c t o r of 2, t h e n N~, is increased 1)3" a f a c t o r of 1.6 a n d z m is lowered b y 0.25 scale height; fl' is r e d u c e d b y a f a c t o r <>f 1.3. (c) I f D a n d fl are b o t h r e d u c e d b y a f a c t o r of 1-5, N .... is increased and fi' r e d u c e d b y this same factor; z~ is u n a l t e r e d . '[hese changes could be t o l e r a t e d , b u t a n y larger decreases in t h e values of fl or D would a p p e a r to be undesirable on a c c o u n t of the resulting changes of ~"me"

678

H. RISHBETH

Moreover, it is d o u b t f u l w h e t h e r the shape of the d a y t i m e N(t) curves would fit the o b s e r v a t i o n s so well if fl were m u c h reduced. N a t u r a l l y , t h e values of q, fl and D would h a v e to be reconsidered if o t h e r processes, such as e l e c t r o m a g n e t i c drift, were f o u n d to p l a y a large p a r t in the d a y t i m e F2-1ayer. I t is unprofitable to speculate f u r t h e r on this topic w i t h o u t more definite i n f o r m a t i o n . A.3 Temperature dependences of diffusion and loss parameters T h e calculations of Section 2 i n v o l v e d the assumptions t h a t the i o n - a t o m i n t e r c h a n g e r a t e y a n d the diffusion p a r a m e t e r b v a r y as T 1/2. Since these m a y not be correct, it is necessary to e x a m i n e w h a t h a p p e n s if different a s s u m p t i o n s are made. I t will n o w be assumed t h a t 7 oc T'I a n d b oc T'2, where t 1 a n d t 2 are constants. According to FERRARO (1957), it seems p r o b a b l e t h a t ~ < t 2 -< 1 for the F-region. T h e r e seems to be no e x p e r i m e n t a l evidence for a n y strong t e m p e r a t u r e d e p e n d e n c e of 7, so t h a t t 1 m a y be small. I n the special case l I ~ 12 ~ l , the ratio L = flH~2/D is i n d e p e n d e n t of T. Consequently, the diurnal v a r i a t i o n of z~ is just the same as in the models of Section 2, b u t the electron densities h a v e to be calculated f r o m the following rules: b y d a y N m a n d N , b o t h v a r y as T-', a n d b y night fl' oc T '-1. These are generalizations of the e q u a t i o n s (4b) a n d (4c) which a p p l y to the case of t = ½. I f h o w e v e r h :~ t2, t h e n L oc T'I-'~ so t h a t the relative i m p o r t a n c e of loss a n d diffusion at a n y z is no longer i n d e p e n d e n t of T. T h e t e m p e r a t u r e d e p e n d e n c e s of z .... ~V.... a n d fl' are fairly complicated; t h e y m a y be shown to d e p e n d on the fractional t e m p e r a t u r e change A T / T as follows, the q u a n t i t i e s i, j, ~: being as defined in Section A . l : Az m = i(t~ -- t 2 ) ( A T / T ) (10a) aNme/Nme

~-- i ( t 2 - -

I 1 -- kt 2 --jt~)(AT/T)

Afl'/fi' = i(j h + kt~)(AT/T)

(10b) (10c)

I f 11 = t 2 = ½ t h e n (10b) a n d (10c) reduce to (4b) a n d (4c). E q u a t i o n (10a) shows t h a t z~ is insensitive to changes of t e m p e r a t u r e , since it is i m p r o b a b l e t h a t It 2 - - tl] exceeds u n i t y . If, as before, i = 0-36, t h e n e v e n if t 2 t 1 = 1 the entire diurnal v a r i a t i o n of T o v e r a ratio of 1.5/1 would only cause a v a r i a t i o n of 0-15 scale height in z~, or a b o u t 10 k m in h m. I t m u s t be e m p h a s i z e d t h a t this figure o n l y applies to the v a r i a t i o n of h m associated with i n e q u a l i t y of the indices h a n d t2, a n d is small c o m p a r e d with the diurnal v a r i a t i o n s of h m arising f r o m the causes a l r e a d y considered in Section 2. As regards the t e m p e r a t u r e d e p e n d e n c e s of N ~ and fl', detailed discussion1 is n o t w a r r a n t e d w i t h o u t i n f o r m a t i o n a b o u t t 1 a n d t~. H o w e v e r , it seems likely t h a t the a s s u m p t i o n m a d e in Section 2.1, t h a t b o t h q u a n t i t i e s v a r y as T -~/2, m a y n o t be too inaccurate. I f this is so, it is w o r t h n o t i n g t h a t t e m p e r a t u r e v a r i a t i o n s are unlikely to be responsible for a n y m a j o r anomalies of N,,~F2, or for its depression d u r i n g storms. A.4 Temperature effects in the shape of N(t) curves T h e o t h e r topic to be considered in detail is the occurrence of maxima, of N,~ and of N(t) curves at fixed heights. T h e model N(t) curves for heights below the

A time-varying model of the ionospheric F2-1ayer

679

peak, in Figs. 5-8, generally show m o r n i n g m a x i m a : in real N(t) curves, additional[ m a x i m a are often o b s e r v e d in t h e late a f t e r n o o n a n d evening, and it is interesting to e x a m i n e w h e t h e r these m i g h t be due to the t h e r m a l c o n t r a c t i o n of the atmosphere, as suggested b y LOWAN (1955). According to t h e a p p r o a c h a d o p t e d in this paper, N m a y be r e g a r d e d as a f u n c t i o n of t h r e e variables z, t a n d T. At a n y fixed height h, changes of T affect N in two ways: i m p l i c i t l y because the f u n c t i o n z(h, t) depends on T, a n d e x p l i c i t l y t h r o u g h the relations given b y e q u a t i o n s (4). T h e t o t a l d e r i v a t i v e of N, with respect to time, is

dN dt

ON ON de ON d T + ..... + - - Ot Oz dt OT dt.

(11)

Now T a n d z are b o t h functions of h a n d t, b u t only the partial derivatives Oz/at a n d OT/Ot are r e q u i r e d to e v a l u a t e dN/dt at a fixed h. I t is c o n v e n i e n t to use a dot (-) to d e n o t e (O/at). N o w in the H a r r i s and P r i e s t e r models, it is v e r y n e a r l y t r u e t h a t the fractional r a t e of d e c a y of T, n a m e l y T I T or 0(ln T)/Ot, is independ-ent of height at a n y i n s t a n t d u r i n g the cooling of the a t m o s p h e r e : in this case, Oz/Ot can be e v a l u a t e d b y using the relation ~'/T = I'tJH~ and the definition of z (equation (3a)):

az afdl,

(I'Iflh

at-at

j H,.

-

Tfdh,

Tz

P

T

The lower limit of i n t e g r a t i o n is the lowest height at which the diurnal v a r i a t i o n of T becomes appreciable. H a r r i s a n d P r i e s t e r t a k e this to be 120 kin, in which ease z ~ 4 at the F 2 - p e a k . The der v a t i v e ON/Oz is also required. At heights some distance below the peak, the electron profile is a p p r o x i m a t e l y e x p o n e n t i a l . I f it is r e p r e s e n t e d as iV oc e ~', t h e n ON/Oz = p N ; it m a y be e s t i m a t e d f r o m o b s e r v a t i o n a l d a t a t h a t p ~ 1 in the late a f t e r n o o n a n d evening. As for ON/O!I' in e q u a t i o n (11), the relation A" oc T -1/2 has been a s s u m e d to hold during the day, b u t at sunset when St' is large t h e " p e r f e c t gas" relationship N oc T -1 m i g h t be preferable (Seetioll 2.4). Lastly, --ON/Ot ~_ f i n just a f t e r sunset, at heights below the peak; b u t one(, the night " s t a t i o n a r y " l a y e r becomes established the ':effective" coefficient fi' is applicable instead of ft. I n s e r t i o n of all these results into e q u a t i o n (11), and division t h r o u g h o u t b y N, gives

--d(ln 5")~dr

I~ + (TIT)(pz + 1).

(~:~i,

Now at levels a b o u t a scale height below tt.,, p --- 1 a n d z = 3 a p p r o x i m a t e l y , st, t h a t pz L 1 ~ 4. H e n c e NI, increases at sunset if - - 4 T / T > ft. B u t at the peak OA;/Oz = O, so t h a t N,, increases o n l y if - - T I T > ft. I n L o w a n ' s model, --~]'/T 2 × 10 a sec-~ just a f t e r sunset, b u t in the H a r r i s and P r i e s t e r "S -- 200" mode] ~'/T ~ 2 × 10 -s see -1 and is e v e n smaller in the " S == 70" model. T h e v a l u e of fi at the F 2 - p e a k a t 1900 h o u r s L.T. ranges from 5 × 10 -5 sec 1 for mode] H2 to 3 × 10 -4 see -1 for model L1. H e n c e no increase ~f N m due to cooling can t)e e x p e c t e d , unless p e r h a p s if the v e r y rapid decrease of t e m p e r a t u r e suggested 1)v L o w a n is correct. Much of the same conclusion applies r e g a r d i n g N h at fixed

680

H . RISHBETH

heights below the peak: in this case the t erm "pz" is included in the equation, but is largely offset by the larger values of/3 at these lower heights. After sunset, [T/T I decreases rapidly with time and no increases of N h can be expected later in the night. However, the cooling of the atmosphere certainly reduces the rate of decrease of N h, as shown by the N(t) curves in Figs. 5-8. Well before sunset, however, there does exist a possibility t h a t dN/dt might change sign; because [ON/Ot[ ~ t3N during the d a y so t h a t quite small decreases of t e m p e r a t u r e might suffice to increase N~. This might explain the double maxima of N h sometimes observed during the day, and which appear to be more common at sunspot minimum t h a n at sunspot maximum. This would be consistent with the Harris and Priester models, in which the time of m a x i m u m 7' becomes earlier with decreasing solar activity. It is concluded t h a t temt,erature changes could cause double m a x i m a of YT, during the day; at sunset they reduce the rate at which N h decreases, but are unlikely to cause actual m axi m a of N h at this time of day. Somewhat similar analysis could in principle be applied to the variation of N~, with latitude, once the latitudinal variations of atmospheric t e m p e r a t u r e become known: it could not be expected to hold near the magnetic equat:or, but might provide a qualitative description in middle and higher latitudes. APPENDIX

]3.1

]~:

COMPARISON

Production rate~' at

WITH

OTHER

MODELS

OF

THE

F-REGION

300 kin, q(300)

I t is simplest to compare the rates suggested by different authors by using a fixed height, such as 300 kin. U n f o r t u n a t e l y the available data refer to different times of day, and moreover different indices of solar activity are used in the literature. Consequently, comparisons between different models can seldom be made directly, but involve subsidiary calculations in which certain assumptions have to be made. I t is therefore necessary to describe in some detail the procedure t h a t has been followed, in order to show all the data on one graph, Fig. 13, which gives q at 300 km as a function of m o n t h l y mean solar 10.7 cm flux density S. The continuous line in the figure refers to the model presented in Section 2 of this paper. I t passes t hr ough five points computed by means of equation (2a), Section 2.2, for the Harris and Priester noon models for S = 70, 100, 150, 200 and 250. The points at S = 70 and S = 200 nat ural l y correspond to the values of q tab u lated in Table 2. Although it is assumed t h a t the ionization param et er P ~ varies linearly with S, the variation of q(300) with S is nonlinear because of the complicated way in which the concentration [O] and optical depth ~ at a fixed height depend on temperature. ~[he dots with vertical bars in Fig. 13 refer to the C h a p m a n - t y p e production formula proposed by R SST (1956), which in equations (8) and (18) of t h a t paper is defined as a function of mean sunspot n u m b e r R, solar zenith angle Z and reduced height z. For present purposes the values of z at 300 km can be estimated from the Harris and Priester models. Since this involves yet more assumptions, especially regarding the height h o where z = 0 (here t aken as the production peak when Z = 0), the two values of q(300) are plotted with error bars which correspond

A time-varying model of the ionospheric F2-1ayer

681

to the s t a t e d u n c e r t a i n t i e s in z(300). Values used for Fig. 13 are: S = 70, R = i~, h 0 = 1 6 0 k m , z(300) = 3-3 ± 0 . 5 ; a n d s = 2 0 0 , R = 200, h 0 = 1 8 0 k i n , z(300) == 1.5 -+- 0.25.

The p o i n t Y is c a l c u l a t e d f r o m the p r o d u c t i o n r a t e given in Fig. 1 of YONEZAWA a n d TAKAHASHI (1960); t h a t figure o n l y shows q u p to 220 k m altitude, so it has been n e c e s s a r y to use the a t m o s p h e r i c model of YOXEZAWA (1960) to e s t i m a t e :~ value of q(300). T h e Yonez&wa a n d T a k a h a s h i v a l u e of q, which is s t a t e d to refer to a " m o d e r a t e l y h i g h " level o f solar a c t i v i t y , has been p l o t t e d in Fig. 13 at S = 100 as this seems to be consistent with the t e m p e r a t u r e at 300 k m in the ¥ONEZAWA (1960) model a t m o s p h e r e . ~lhe t h r e e r e m a i n i n g points in Fig. 13 are d e r i v e d as follows. T h e circled dot is c o m p u t e d f r o m the d a t a given b y NOt~TOX, VAN ZANDT and DENNmO~~ (1963): since these a u t h o r s c o m p a r e t h e i r model with the solar E U V fluxes m e a s u r e d b v H i n t e r e g g e r and W a t a n a b e during August 1961, the point is p l o t t e d at the v a l u e S = 106 a p p r o p r i a t e to t h a t m o n t h . T h e solid circle is the v a l u e d e t e r m i n e d from a solar eclipse in O c t o b e r 1958 (S = 225) by VAN ZANDT, ~ORTON and STONE~OCKE~ (1960). Finally, the open circle is t a k e n f r o m the e q u a t o r i a l F - l a y e r model of NORTON a n d VAN ZANDT (1964), which represents conditions d u r i n g S e p t e m b e r 1957 (S = 266). T h e values of Z a n d local time t for these t h r e e point.s are detailed in the caption to Fig. 13. 1000

I

I

I

l

i

I O

800

--

600

,.n i

E £3 ~: 400

200

10

] ioo

I 150

I 160 S

I 190

1 220

{ 250

280

(FLUX UNITS)

F i g . 13. R a t e o f p h o t o i o n i z a t i o n a t 300 kin, q(300), as a f u n c t i o n o f s o l a r 10.7 e m f l u x d e n s i t y S. T h e c o n t i n u o u s line is c o m p u t e d f o r t h e a u t h o r ' s m o d e l ( n o o n , Z -- 52°), a n d i n c l u d e s t h e v a l u e s q u o t e d in T a b l e 2 f o r S -- 70 a n d S = 200. F u r t h e r d e t a i l s f o r t h e f o l l o w i n g p o i n t s a r e g i v e n in t h e t e x t : D o t s w i t h v e r t i c a l b a r s : t~ATCLIFFE et al. ( R S S T , 1956), c o m p u t e d f o r Z - 52° a n d f o r z = 3.3 ._4_.0.5 a t S -- 70 (R = 0) a n d z = 1.5 ~ 0.25 a t S = 200

(R = 200) Y: o : o : • :

YONEZAWA a n d TAKAtIASItI (1960), Z = 9°, m i d - l a t i t u d e F - l a y e r m o d e l NORTON a n d VAN ZANDT (1964), Z -- 0°, e q u a t o r i a l F 2 - 1 a y e r m o d e l NORTON et al. (1963), Z = 34°, m o d e l o f E - a n d aW-layers VAN ZANDT et al. (1960), y. = 46 °, eclipse o b s e r v a t i o n .

682

H. RISttBETI{

I n principle, all these values of q could be r e d u c e d to some s t a n d a r d v a l u e of a n d t, with the aid of a suitable a t m o s p h e r i c model. H o w e v e r , r o u g h calculations suggest t h a t n o n e of t h e m would be altered b y m o r e t h a n 10 per cent if t h e y were a d j u s t e d to t h e values X = 52° a n d 1200 hours L.T., used for the c o n t i n u o u s line, a n d no such a d j u s t m e n t has been m a d e in plotting the values. B.2 Loss coeJficients at 300 kin, fl(300) T h e c o m p a r i s o n of values of fl is m o r e s t r a i g h t f o r w a r d t h a n the c o m p a r i s o n of values of q, b u t nevertheless involves some u n c e r t a i n t y on a c c o u n t of the differing indices of a c t i v i t y which h a v e been used. Values of fi(300) are p l o t t e d in Fig. 14 as a f u n c t i o n of a t m o s p h e r i c t e m p e r a t u r e T at 300 km, a n d in Fig. 15 as a f u n c t i o n of solar 10.7 cm flux S. T h e H a r r i s a n d P r i e s t e r models are used for the i n t e r c o n v e r s i o n of values of S a n d T w h e n e v e r required, to enable all the d a t a to be p l o t t e d on b o t h graphs. T h e s t a r r e d points r e p r e s e n t t h e v a l u e s given in T a b l e 2, for n o o n a n d m i d n i g h t a n d are joined b y c o n t i n u o u s curw~s which also include points calculated for the H a r r i s and P r i e s t e r models for S -- 100, S -- 150 a n d S = 250. F o r a n y given level of S, the n o o n values of fi(300) c o n s i d e r a b l y e x c e e d the m i d n i g h t values because the N 2 c o n c e n t r a t i o n is g r e a t e r at noon. W h e n p l o t t e d as functions of T, as in Fig. 14, the n o o n a n d m i d n i g h t values are different on a c c o u n t of the different t e m p e r a t u r e profiles at these times. The triangles refer to the values d e d u c e d b y NISBET a n d QuIN~~ (1963) f r o m o b s e r v a t i o n s of the d e c a y of the F - l a y e r at night, a n d q u o t e d for t h r e e values of T. As in Fig. 13, t h e p o i n t Y c o m p u t e d for the YONEZAWA a n d TAKAHASHI: (1960) model has been p l o t t e d at S = 100, which is consistent with Y o n e z a w a ' s value of ~/' at 300 km. T h e values of fi given b y VA>~ ZANDT el al. (1960) and b y NORTO>- et al. (1963) relate to times o t h e r t h a n local noon; b u t it is n o t difficult to e s t i m a t e the corr e s p o n d i n g values for n o o n w i t h the aid of a p p r o p r i a t e H a r r i s a n d P r i e s t e r models, a n d so two points are shown in b o t h cases. T h e assigned values of S are the same as those used in Fig. 13 for the c o r r e s p o n d i n g values of q. F o r the p u r p o s e of Fig. 14, it is possible to derive values of T(300) f r o m i n f o r m a t i o n supplied in the p a p e r s b y NORTON et al. (1963) a n d :NORTON a n d VAN ZANDT (1964); b u t h)r the eclipse v a l u e of/~ (VAN ZA>-DT et al., 1960), T m u s t be d e d u c e d from the value of S. The h o r i z o n t a l d a s h e d line in Figs. 14 a n d 15 is d r a w n at the v a l u e of fi(300) p r o p o s e d b y R S S T (1956), for which no p a r t i c u l a r level of solar a c t i v i t y is specified. U n f o r t u n a t e l y t h e r e is no i n d e p e n d e n t check on the v a l u e of fi(300), because n e i t h e r the i o n - a t o m i n t e r c h a n g e r a t e coefficients nor the molecular c o n c e n t r a t i o n s at 300 k m are a c c u r a t e l y k n o w n . B.3 1)iff~sion coeJficients at 300 km, D(300) A l t h o u g h the " f a s t diffusion" models fit the d a t a on the d a y t i m e F 2 - p e a k fairly well, t h e y are o p e n to the t h e o r e t i c a l o b j e c t i o n t h a t t h e diffusion coefficients are too large. T h e v a l u e of b q u o t e d in T a b l e 1, 2.4 x 1019 cm -1 see -1 at 1250°K, is o b t a i n e d f r o m e q u a t i o n s (10) and (11) of FERRARO (1945) b y t a k i n g the masses of ions and n e u t r a l a t o m s to be 16 a.m.u., and doubling the resulting v a l u e of tile

A t i m e - v a r y i n g m o d e l of t h e i o n o s p h e r i c F2-1ayer

683

ionic diffusion coefficient to obtain the ambipolar diffusion coefficient (of. RATCLIFFE and W E E K E S , 1960, p. 391). The result is larger t h a n Ferraro's value of b, which is 1.2 × l0 ~9 cm -1 sec -1 at 1250°K. Neither of these values takes account of charge-exchange between 0 + ions and oxygen atoms, which m a y reduce the diffnsion coefficient (DALGARNO, 1958). Thus a smaller value of b might be preferable on theoretical grounds. 2

~

I

/

2V

y

|

'O 10-4

/

/

RSST

]

5500 =

/

/

/

I0-4~-- /

qg_

I05

:J

¥

1

/

/

/

j"

- ........RSST

=-

/

/ 1000

1500 T (°K]

280

2000 S (FLUX UNITS)

Figs. 14 a n d 15. Loss coefficient a t 300 kin, /~(300), as a f u n c t i o n of t e m p e r a t u r e T (Fig. 14) a n d 10.7 c m solar flux d e n s i t y S (Fig. 15). P o i n t s * a r e t a k e n f r o m t h e a u t h o r ' s m o d e l ( T a b l e 2) a n d j o i n e d b y c o n t i n u o u s lines c o m p u t e d w i t h t h e aid of t h e H a r r i s a n d P r i e s t e r m o d e l s for o t h e r v a l u e s of S : n o o n a n d m i d n i g h t v a l u e s are s h o w n . F u r t h e r e x p l a n a t i o n s for t h e o t h e r p o i n t s are g i v e n in t h e t e x t . N o p a r t i c u l a r level of a c t i v i t y was specified b y I~SST (1956) for t h e i r v a l u e , fl(300) = 10 4 s e e - L Y : YONEZAWA a n d TAKAHASHI (1960), m i d - l a t i t u d e _F-layer m o d e l ± : NISBI~T a n d QuINt,= (1963), o b s e r v a t i o n s of n i g h t F - l a y e r ~) : NORTON a n d VAs= ZANDW (1963), e q u a t o r i a l F 2 - 1 a y e r m o d e l , t -- 1200 h o u r s © : NORTON et al. (1964), m o d e l of E.. a n d F - l a y e r s , t -- 1000 h o u r s (below) a n d 1200 h o u r s (above) • : VA~+ ZANDT et al. (1960), eclipse o b s e r v a t i o n , t = 0900 h o u r s (below') a n d 1200 h o u r s (above).

F r o m an analysis of Puerto R i c o N(h, t) data for March 1959 (S ---- 228), SHIMAZA~:I (1964) finds D(300) = 5 × 109 cm ~ sec - t at midnight. This m a y be c o m p a r e d w i t h the m i d n i g h t v a l u e for the "S = 200" m o d e l of Tables 1 and 2 which is 22 × 109 cm 2 see -1 for model H I (it m a y be estimated, b y referring to the Harris and Priester models, t h a t the v a l u e w o u l d be r o u g h l y 20 per cent smaller at N = 228 t h a n at S -= 200). If D were reduced b y a factor of three from

the "fast diffusion" value (this being the greatest "t ol erance" indicated in Appen-dix A.2), it would nearly agree with Shimazaki's result but would not suit the d a y t i m e d a t a so well. This is of course the same d a y - t o - n i g h t discrepancy as was discussed in Section 3.4, where the existence of a nocturnal source of i o n i z a t i o n was offered as a possible interpretation.

684

H . RISHBETH

B.4 Relation between the F2-layer model and the lower F-region Even though the procedures used in Sections B.1 and B.2 might be open to criticism, the comparisons made in Figs. 13-15 are felt to establish t h a t the values of q and/? at 300 km suggested in this paper are consistent with other determinations, to within a factor of two. No a t t e m p t has been made to compare the scale heights of either fi or q proposed by different authors, since this would lead t() further complexity. Although there is order-of-magnitude agreement between the rates at 30() km proposed by different authors, there exists some controversy about the rates in the lower F-region. This concerns especially the peak production rate % t'()r X = 0, and the height h 0 at which it is found, and the magnitude of the linear and quadratic loss coefficients, ~ and fi, at this height (NORTON c4 al.. 1963). Clearly, it would be desirable to relate the F2-layer model presented in this 1)aper to models of the lower F-region, particularly in connection with the current theory of the F1/F2 "transition region" in which the F1 "ledge" is identified with the daytime production peak (RATCLIFFE, 1956). Proper study of this topic requires a much more detailed photochemical model of the F-region than is needed for the consideration of the F2-1ayer alone, so t h a t the simple assumptions made in Section 2.1 no longer suffice. Although no detailed calculations are shown here, it is believed that the F2layer model given in this paper can be fitted on to a variety of models of the h)wer F-region, so t h a t the uncertainties concerning the latter have little bearing on the previous discussion of rates at 300 km. This is largely attributed to two factors. One is the small optical depth of the atmosphere at F2-1ayer heights, on account of which the production rate q(300) depends principally on the i)roduet P ~ of the incident flux I ~ and the ionization cross-section A (0) (cf. equation (2a)). Thus if I ~ and A(O) are varied but their product kept constant, q(300) is scarcely affected but the height variation of q in the lower F-region m a y be pro[bundly affected--the more so because both these quantities are in reality functions of EUV wavelength. The other factor is t h a t the linear loss coefficient [~ plays a minor role in the lower F-region; whereas the quadratic recombination coefficients for different ions, on which the N(h) distribution in the lower F-region largely depends, are of very minor importance at 300 km.

Acknowledgement--The author is grateful to Mr. R. B. Norton and Dr. T. E. Van Zandt for m a n y helpful discussions during the preparation of this paper. This work was carried out during the tenure of a consultancy at the ('entral Radio Propagation Laboratory, National Bureau of Standards. t~.EFERENCES ANTONOVA L. A. a n d [VANOVI~HOLODNII Cr. S. BAZZARD G. H . a n d MINNIS C. M. BRIGGS g . H . a n d RISHBETH H . BYRAlV~ E . T., CHUBB T. A. a n d FRIEDMAN H.

1961

Geomagnetism and Aerono~ny E d i t i o n ) 1, 149.

1961 1961 1961

J. Atmosph. Terr. Phys. 22, 192. Proc. Phys. Soc. Lond. 78, 409. J. Geophys. Res. 116, 2095.

(English

A t i m e - v a r y i n g m o d e l of t h e i o n o s p h e r i c F2-1ayer

685

CaOOM S. A., }{OBBINS A. IR. a n d THOMAS J . O. DALGARNO A. I)UNCA~" R. A. [)(~N(;EY J . A. FERRARO V. C. A. FERRARO V. C. A. (~ARRIOTT (). K. and I~IS]~BETH H . HANSON ~V. B. a n d ORTENBURGER I. B. HANSON "W. B. a n d PATTERSON T. N. L. HAIU~S [. a n d PRIESTER "~V. [H & P ] H[NTEREt4(;j~21¢ H. E. a n d ~VATANABE K. LOWAN A. N. N ICOLET 5[.

1959

Tables of Ionosphere Electron Density,

1958 1956 1956 1945 1957 1963 1961 1963

J. Atmosph. Terr. Phys. 12, 219. Aust. J. t)hys. 9, 436. J. Atmospt~. Terr. Phys. 9, 90. Terr. z]/lagn. Atmos. Elect. 50, 215. J. Atmosph. Terr. Phys. 11, 296. Planet. Space Sci. 11, 587. J. Geophys. Res. 66, 1425. Planet. ~S'pace. ,S'ci. 11, 1035.

1962 1962 1955 1961

J. Geophys. Res. 67, 3373. J. Geophys. Res. 60, 421. Handbuch der PAys|k, V o h 49. S p r i n g e r -

NISBET .[. S. a n d QuINN T. P. NORTON }1. B. a n d VAa" ZANDT T. E.

1963 1964

Cavendish Laboratory, Cambridge.

N A S A T e c h n i c a l N o t e 1)-1444.

Vcrlag, Berlin.

J. Geophys. Res. t18, 1031. P a p e r p r e s e n t e d at U R S I Symposium on Ionospheric Soundings in the IGY/IGC, h e h | a t Nice 11-16 D e c e m b e r , 196]. Pergamon Press, Oxford (to bo published).

Proe. Int. Conf. on the Io~osphere, p. 26.

NORTON ]Z[. B., VAN ZANDT T. E. a n d I)ENISON J. S.

1963

t{ATCLIFFE J . A. RATCLIFFE J . A. a n d \VEEKES K.

1956 19(;0

J. Atmosph. Terr. Phys. 8, 260. Physics of the Upper Atmosphere (edited

1956

b y J . A. RATCLIFFE), C h a p t e r A c a d e m i c Press, N e w Y o r k . Phil. Trans. A248, 621.

1960 1963 1957 1964 1957

J. Atmosph. Terr. Phys. 18, 234. Proc. Phys. Soc. Lond. 81, 65. J. Radio Research Labs. Japan. 4, 309. J. Geophys. Res. 69. I n press. Tables of Ionospheric Electron Density.

1960

J. Geopbys. Res. (}5, 2003.

1962a 1962b 1959 1960 1960

N B S Technical Note 40-13. J. Res. NB, b' 66D, 297. J. Radio Research Labs. Japan 6, 293. J. Radio Research Labs. Japan 7, 69. J. Radio Research Labs. Japan 7, 335.

]~ATCLIFFE J . A., SOg~ERLING E. IR., NETTY C. S. G. K. a n d THObIAS J . O. [RSST] ICLSHBETH H. a n d BARRON D. ~V. P~[SHBETH H . SI[IMAZAXI T. S HIMAZAKI T. TtIOMAS ,]-. O. HASELGROVE J . a n d }~.OBBINS A. II. V~br ZANDT T. E., NORTObr II. B. a n d STONEHOCKER G. H . WRIGHT J . W. ~¥RIGHT J. W .

YONEZAVCA T. a n d ARIMA Y. YONEZAWA T. YONEZA'WA T. a n d TAKAHASHI H.

I n s t i t u t e of P h y s i c s Society, L o n d o n .

and

Physical

Cavendish Laboratory, Cambridge.

9.