The formation of the daytime peak of the ionospheric F2-layer

Journalof Atmospheric and!l!errest,rial Physics,1962,Vol.21,pp.503to519.Pergamon PressLtd. Printed in Northern Ireland

The formation of the daytime peak of the ionospheric 1;2dayer* S. A. BOWHILL Ionosphere Research Laboratory, The Pennsylvania Universit,y Park, Pennsylvania (Received

19 December

State Universit,y,

1961)

Abstract-The physical processes occurring in the PP-layer of the ionosphere are discussed. A simplified model, in which the loss region for the ionization has a sharp upper boundary, is shown to give a layer, having a deflned maximum, due to production and diffusive transport alone. The complete continuity equation for ionization in an isothermal atmosphere, nearly transparent to the ionizing radiation, is solved explicitly in terms of Lommel functions. Comparison of the complete solution with the simplified model mentioned above, indicates that the simplified model gives sufficiently accurate results for heights at and above the layer maximum. 1. INTRODUCTION

it has been realized for some time that the peak of the ionospheric F2-layer is produced by some type of diffusion process, the exact mechanism is not yet fully understood. This is the more remarkable in that the continuity equation for the ionization has been known for some time and appears to have within it the potential Efforts to solve this rather to explain most of the features of HZ-layer behaviour. difficult equation have proceeded along three lines: direct analytical solution for one particular case (YONEZAWA, 1956, 1958); numerical solution by means of digital computers (RISHBETH and BARRON, 1960; GLIDDON and KENDALL, 1960); and comparison of the equation with parameters measured directly from rocket measurements of FZ-layer profiles (BDWHILL, 1961). The purpose of this paper is to give an explicit solution to the continuity equation for ionization in the FZ-layer, for pseudo-equilibrium conditions, and in The solution is developed the absence of important vertical drifts of the ionization. in a step-by-step manner, paying considerable attention at all points to the rather complicated physical processes. Because of the variety of assumptions made by various workers on this type of problem, a special section is devoted to a consideration of these assumptions. Specifically excluded from this investigation are the processes, occurring near the peak of electron production in the F-layer, responsible for the appearance of the Fl-layer. This is intentional, since the shape of the layer immediately below, at and above the peak is gaining increasing importance for the estimation of atmospheric parameters from rocket and satellite ionosphere sounding; in particular, the forthcoming topside sounder results will provide a hitherto unprecedented abundance of data in this region. It has been found expedient to develop a solution for the continuity equation in which the integration is carried downward from a point far above the layer peak. WHILE

* The research described in this paper was supported by the National Science Foundation under Grant G-18852. ,503

S. A. BOWHILL

504

This solution is treated in Section 3. For compactness only one example of the analytic solution is given, in Section 4. No additional problems arise from the general case. Because of the interest in useful approximate solutions at and above the layer peak, these are discussed specifically in Section 5. 2. PHYSICAL ASSUMPTIONS As in all idealized investigations of ionospheric theory, it is necessary to form some theoretical atmospheric model. The key assumptions made in this paper are that the atmosphere is isothermal in the height range considered and that the While these assumptions are still neutral constituents are in diffusive equilibrium. open to question, present evidence appears to show that the daytime atmosphere is isothermal within the limits of measurement above a height of about 250 km, and that diffusive equilibrium for the major neutral constituents exists at all heights above about 110 km. It is convenient to define the height variation of the various atmospheric parameters by four constants, a,, u2, a3 and a,; these are defined exactly 5~sin a previous paper (BOWHILL, 1961). The number density ~(0) of atomic oxygen is given as a function of height x above some reference level x = 0 (indicated by the subscript O} by the equation ~(0) = B,(O) exp (-aiz). This constituent is assumed to be ionized by radiation X-ray region, according to the reaction

in the far ultra-violet

0 + hv + O+ + e. This gives a variation

of the electron production

and

(1)

q with height of the form

4 = q. exp (yv)

(2)

that the optical depth for the ionizing radiation is small compared with unity in the height range considered. This assumption has been found by rocket measurements to apply at heights above about 250 km (HINTEREGGER, 1961). provided

Recombination of the atomic oxygen ions is assumed to proceed in two steps: ion-atom interchange with a molecular species XY (which might be molecular nitrogen or molecular oxygen) according to the reaction 0’ + XY followed

by

dissociative

recombination

--+ x0+ of

X0+ + e --+ X’

+ Y’ the

molecular

+ 0’.

(3) ion

X0+, (4)

From rocket measurements using ion mass spectrometers it is now well established that O+ ions predominate in the height range above 200 km; it therefore follows that, for heights well above this, the electron density N may be equated t.o the Consequently, since the distribution of the density of atomic oxygen ions. molecular species XY is exponential, given by n(XY)

= ?ao(XY) exp ( -azz)

The formation of the daytime peak of the ionospheric FL’-layer

the loss coefficient

B for the ionization

505

has a similar height variation,

16 = 16, exp (-v).

(5)

The ambipolar diffusion coefficient for the ionization depends, according simple theory, on the number density n(M) of the total atmosphere, defined

to by

n(M) = m,(M) exp ( -a3z). Two difficulties arise in estimating this quantity: the relative composition of the atmosphere, and therefore u3, may vary with height; and oxygen atoms may need to be considered specifically, in view of the suggestion of DALGARNO (1958) that the collision cross-section of atoms and atomic ions of oxygen may exceed the classical value by a considerable factor. It will be assumed, therefore, that suitable allowance has been made for these two effects in computing the value of a3. With this reservation, then, the diffusion coefficient D is given by the relation D = D, exp (a+).

(6)

The fourth constant a4 pertains to the height distribution that would be adopted by the oxygen ions and their accompanying electrons under conditions of diffusive equilibrium. This distribution is defined as 72(0+) = n&O+) exp ( -a4x).

(7)

Under normal circumstances u1 = 2aq, but the two quantities a, and a4 have been preserved separately in the ensuing discussion to clarify the physical processes. With these assumptions the continuity equation of the ionization (ROWHILL, 1961) becomes

ax

-=q--_N+D at

g+(a3+a,)g+ a,a,N 1.

As mentioned above, only the equilibrium condition is considered in this paper. It is reasonable to suppose that this condition applies near noon under normal conditions. Setting aN/& = 0, assuming N to be a function of z only, and denoting derivatives with respect to z by primes, the equation N” + (a3 + a,)N’

+ a,a,N

+Jzl - b%lD,J~ex~ [-(a,+ %kl = 0 (9)

+ (q,,/D,) exp [ -(a1

+

is obtained. Various properties of the solutions of this equation are investigated in detail in this paper. The boundary conditions are crucially important in this problem. The downward flux G of ionization at any height is given by G = D[N’

+ a,N].

(10)

Since the loss coefficient ,4 decreases very rapidly with height, most of the oxygen ions produced at high altitudes diffuse downward before recombining, leading to a Since all the ions are supposed net downward flux everywhere in the atmosphere. to arise by photoionization of atomic oxygen, it follows that the downward flux must tend to zero at infinite height, where the concentration of oxygen atoms is

S. A. BOWHILL

506

it is equivalent to saying that the small. This gives the first boundary condition; electron density at great heights behaves like exp (+x4x). The second boundary condition, to be applied below the layer, involves the fact that the two last terms in equation (9) become much larger than the first three as z -+ - co, and must therefore be nearly equal. This leads to N oc exp (a2 -

al)2 -+ 0

as

z +

- co

(II)

since a2 > a, for all practical cases. These two boundary conditions will be referred to subsequently as the upper and lower boundary conditions, respectively. Any solution to equation (9) must satisfy both in order to accord with the physical assumptions made in this section. In illustrating features of the solution, reference will be made to four types of atmospheric model, each having particular values of a,, u2, a3 and a*. These values are given below in Table 1; each quantity is given as a ratio to ad. Table

A B C D

2 2 2 2

1

>l 4 3.5 2k

2 2 2 2

All four models apply to atmospheres composed predominantly of atomic oxygen. The values of a2 correspond to the molecular species, being oxygen in case B and nitrogen in case C. Type A is a special model, used for purposes of illustration in Section 3.1, in which the molecular species forms a “sink” for oxygen ions at a defined height in the atmosphere, there being no loss above that height. The model studied by YONEZAWA (1956, 1958) was effectively type B. Those studied by RISRBETH and BARRON (1960) corresponded to type D, using five values of k ranging from I.0 to 2.6. Both sets of workers used a Chapman form for the production function rather than the exponential form given by equation (2). Recent measurements of optical depth in the far ultra-violet (HINTEREGGER, 1961) appear to show that the exponential assumption is justified at heights somewhat above 200 km. 3. SOLUTION OF THE CONTINUITY EQUATION FROM ABOVE Most of the difficulty in solving equation (9) arises from the difficulty of satisfying both the upper and lower boundary conditions simultaneously. RISHBETH and BARRON (1960) proceeded by integrating downward numerically, starting with a solution satisfying the upper boundary condition. Typically, such a solution was found to diverge rapidly as the lower boundary was approached; by successive computer trials the correct solution was selected, intermediate between solutions which diverged in the positive and negative N direction at the lower boundary. They pointed out that a change of one part in lo4 at the upper boundary was sufficient to change from a solution diverging in the positive direction to one diverging in the negative direction. This raises the question of whether an approach from

507

The formation of the daytime peak of the ionospheric P2-layer

the approach from above is treated in this paper, but above or below is preferable; it can be shown that the asymptotic form of the functions giving the solutions from above is equivalent to a solution from below. To shed some light on the physics of the problem, an idealized atmosphere of Referring to Fig. 1, the loss is confined to the region type A will first be considered. z < 0,the only processes occurring for z 2 0 being ion production and diffusive transport. Since molecular gases are supposed to be abundant in the region x < 0, the lower boundary condition is just N = 0, when z = 0.

DIFFUSIVE

EQUILIBRIUM REGION

t 2

PRODUCTION

-TRANSPORT

REGION

0

“SINK”

ELECTRON

Fig. 1. The formation of an Fd-layer in a type A atmosphere, with electron loss confined to the “sink” at z < 0. A downward flux of electrons exists throughout the production-transport region. 3.1.

Lossless l?24ayer

In the production-transport is governed by the equation N” + (a-, + a&N This equation

evidently

region z 2 0 in Fig. 1, the ionization

+ a,a,N

+ (pa/D,,) exp [-(a1

has a particular

[(al + G2

-

= 0.

integral of the form

A exp [-(a1

A being given by the relation

+ c&l

+

MI

(aa + aJ(a;l + a& +

w,lA

+ ~$0 = 0.

distribution

(12)

S. A. BOWHILL

508 The particular

integral therefore

Nl =

has the form cl0 exp

7

Do%@,

+

a3 -

[-(a,

+

a3)4.

(13)

%I

This variation of N with z represents the distribution required to sustain a downward transport of ionization, at any height, equal to the total production above that height. Obviously, it satisfies the upper boundary condition but not the lower. The homogeneous equation N” + (a3 + a,)N’ has the complementary

function

+ a,u,N

= 0

(14)

solution

N, = B exp ( -adz)

+ C exp ( -u3z).

(15)

The upper boundary condition may be satisfied by substituting (10): G = D[ -Cu, exp ( -u3x) + Cu, exp ( -u3z)] G =

--CD,(u,

-

N, into equation

uJ.

(16)

Evidently, therefore, the upper boundary condition can be met only if C = 0. The complete solution is then the combination (NI + NJ of equations (13)and (15): N = Bexp

(-adz)

-

40 Boa,@,

The constant B can then be determined when z = 0:

+ a3 - %)

exp

[-(a,

from the lower boundary

B = ~ol[Bo%(%

+ a3 -

+

f33)zl. condition,

%)I

(17) X = 0

(18)

and the final solution is, for z 2 0, N=

Dou,(u, yu3 _ u4IIexPC-w) - exp [-(a,+ f~,bl}.

(19)

The electron density given by this expression has a peak value, and displays many of the features which characterize the full solution of equation (9). As indicated above, the second term in the solution, which is negative in sign, represents the ionization distribution necessary to sustain the downward flow; the first term, which has a distribution corresponding to diffusive equilibrium, involves no flow and is present in just sufficient amount to make N 2 0 over the entire production-transport region. This is a feature also of the full solution of equation (9). The peak of the layer is determined primarily by production-transport considerations, but its location is determined by the amount of diffusive equilibrium distribution present, which in turn depends on the precise form and location of the loss region. The height z, at which the maximum ionization density N, is formed is given bY 2, = (a, + a3 - a,)-lln [(al +

a3)h1

(20)

509

The formation of the daytime peak of the ionospheric F?-layer

and the maximum

density by a,l(a,+lz-a,).

N,

(21)

=

It is notable that the value of x, is not affected by the diffusion coe@icient or tfhe production rate. For the particular example of the type A atmosphere, these two quantities have the values z?n = (3a,)-l In 4 = 0.462~,-~ N, 3.2.

(22)

1

= 21’3qo/16Doa,2

with loss

FB-Layer

With the insight acquired from solution of the simpler problem in the previous section, the equation corresponding to an F24ayer with loss processes distributed through it will now be investigated. The equation to be solved is N” + (a3 + a,)N’ It is found variable z :

u3kl - (P,/D,P exp [-(a,i-a3121 = 0. (9)

+ a3a4N + (qo/oo)

expedient

exp [-(a,

to make the following x = exp [ -(a2

Equation x~N” +

+

substitution

for the independent

+ a3)2].

(2.7)

(9) then becomes 1 -

a3

~

+

a4

a2

+

a3

xN’

a3a4

+ (a,

zN

PO

+

a3)2

N

-

Do@2

+

a312

Qo

zz-

Do@2

+

$9

+ %M%

+ a,)

(24)

a3J2

The homogeneous form of this equation, with the right-hand side replaced by zero, is MalmstBn’s generalization of Bessel’s equation, and has the solutions (KAMKE, 1944)

N, = Ez”ZJ2il/(bx)], where E is an arbitrary

constant,

(25)

and a3 tc

=

y=p

+

2(a,

a4 +

a31

a3

-

a4

a2

+

a3

(26)

PO

b= Do(a2

+

a3j2

The function N, is to be regarded as the complementary function part of the solution, analogous to N, of equation (15). As in the previous section, the procedure will be to identify the solution N, which obeys the upper boundary condition, and combine it with a particular integral solution N,, also obeying the upper boundary

510

S. A.BOWHILL

condition, in such a way as to satisfy the lower boundary condition with the complete solution. In terms of t’he Bessel function of imaginary argument, I,, equation (25) can be written N, = Ex” exp (&~i)I,[2z/(bx)] or

(2’)

Ex” exp ( - &mi)l_,[2 I]

It is now necessary to decide between these two solutions. The appropriate solution to satisfy the upper boundary condition can be found from the series expansion

the solution then becomes

Using the suffix 00 to represent the limiting form of the solution for infinite x, one need take only the leading term of the power series in equation (29): N3m = From the relations

(30)

xx**v.

(26), _

cr.*@=

Combining

Eb**” exp (&yriv) (*v)! a3

a2 +

or a3

(31) with (23), the two limiting

N,,

p.

a4

a2

+

(31) a3

forms

oc exp ( -a3z)

or

exp ( -a4z)

(32)

are obtained. These are exactly analogous to the solutions of the homogeneous equation (14) in the previous section. As before, the upper boundary condition, that the net downward transport shall become zero at high altitudes, indicates that the second solution is the appropriate one in this case; giving I_,[2z/(bs)] as the form of the solution, together with an undetermined integration coefficient, E. Since E is an arbitrary constant, a new arbitrary constant P may be defined without loss of generality: F = Eb@” exp (-&m-i)

(33)

(-v)! giving,

as the complementary N,

function =

solution of equation

Fx%lh+ad

cr, (bx)‘( -v)!

2

r=O r!(r -

Y)! .

(X4), (34)

Though the particular integral N, of the equation (24) could be found by the method of variation of parameters, it is more convenient to solve (24) in power series. Comparison with the previous section suggests that the limiting form N,, of the particular integral solution should be N,,

cc exp [ -(a,

+ a3)z] = xY

(35)

The formation

defining the quantity

of the daytime

peak of the ionospheric

F2-layer

511

y by a1 + Y=

a3

a,

(36)

*

This also implies Such a solution will certainly obey the upper boundary condition. that the power series expansion for N, must have x’ as its leading term. The type of expansion used will therefore be N, = x:’ 2 c,xs. s=o Substituting

the series (37) in equation

(37)

(24), the coefficient

of xY gives the equation (35)

with the substitution d=

The coefficient

(39)

q. D&2

+

a3)2 *

co of the leading term gives, after some rearrangement, w,

c o=

-~

%(%

+

+

a3)2

a3 -

(40)

a4).

Thus, the leading term of the power series is determined only by the production term, and the loss term does not enter into it, On the other hand, all subsequent terms of the series are connected by the regression relation a3

r+s--

a2 +

a4

r+s----

a2 +

a3

c, = a3

bc,_l

which involves the loss term b rather than the production of xs in the series is then given by

c, =

Substituting

(8 > 0)

(41)

term d. The coefficient

cobs

(42)

in equation

(37), the particular

integral N, becomes

Y-N

(43)

4

This is a power series in the quantity (bx), just as was equation The limiting form N,, is given by the leading term: N

_ 4m -

_

which is identical

@2

+

a3)2x’

40

=Do%(%

with equation

+

a3 -

(13) of the previous

a4)

(34).

exp [- (a, + a3bl

section.

It therefore

(44)

appears

512

S.

A. BOWHILL

that the form of the particular integral solution N,, at the upper boundary is independent of the loss mechanism. The only remaining unknown in the electron density distribution (Ns + NJ is the constant F in equation (34). This is determined by the lower boundary condition; it< can be evaluated analytically by solution of equation (24) in terms of associated Lommel functions. This solution is carried out in the Appendix, and the value of F is found as

p = (& + av - g)!(+p - gv - &)! &-3’44.

(AZO)

The quantities p and v in this expression are defined, respectively, by equations (A4) and (26). This constant F has a special significance, since it represents the electron distribution above the peak of the layer, which determines the total ionospheric column con%&. Values of the various nondimensional quantities, appearing in equations (34), (43) and (A19), are given in Table 2 for the models described in Table 1; values of the parameter A used in the Appendix are also given. Table

R f’

116

213

Z/f1

s/11

n

1

2

3(1 + E)

4. EXAMPLE

OF

2

1 + 12

- If6 -l/11 3 - 2k 2(1 + k)

l/2 6jll 31%

-_ 1 +- k

SOLUTION OF CONTDWITY EQUATION FROM ABOVE

To compare with the results of other workers, an example will now be given of the solution of the continuity equation for an atmosphere of type B, in Table 1. This will then be compared with the solutions of YoNEzAwA (1956) and of BIsnnxrH: and BARRON (1960), both of whom studied this type of atmosphere. Since both the power series of equations (34) and (43) involve powers of the term (hz), it is convenient to make a substitution to eliminate b in the power series. First, the combined result for N from the Appendix is taken; and using the results for a type B atmosphere from Table 2, equation (AZI) becomes 8

= 6&Z/3

Expressing

(~)!(1~3)! Z* (1~6)!(~~)~ l/(bz) m=o(m - l/S)!%!

_

(~)!(1~3)!(~~)~ _ 2 m=o(~~, -i_ &)!(m +m

the first few terms of these series in numerical

iV = 6dx213

36bx

(36bx)2

1+5.6+5.6.11.12+.** +

Making the substitution u = 36bx

(36W’ 8.9.14.15

form,

1

.

(45)

one obtains

i ..- - (46)

-L

The formation

of the daytime

peak of the ionospheric

PP-layer

513

-3-

-2-

-IT

2-

3-

t

4I I $3’ 0

f

/I

:

0.1

0.2

0.3

0.4

0.5

0.6

0.7

W)

Fig. 9. The PB-layer in a type B atmosphere. The upper and lower broken represent diffusive equilibrium and photoequilibrium, respectively.

where u is a nondimensional variable, particularly suitable for computation, N

=

p/3

($NwP~“6

&_2’3

[

(-l/6)!

_-u-113

1

25 + 2,3

say;

the series (46) can be expressed in the form,

u

+

i

lines

U2

-6.6+5.G.II.12 942 I

2.3.8.9

+...

+...

=

1 --P/3

dPT(U)

(48)

and

The double series T(u) of equation (48) has been calculated for values of log,,zc increasing from -4 to +3 by steps of O-2, and is shown plotted in Fig. 2 as the full line. For values of u less than 1O-2, only the first term of the series need be taken; this extension is indicated by the upper broken line on Fig. 2. The lower broken

514

f3.d. BOWHILL

line shows the height equation (A8);

variation

of the leading

N = ?exp

[(as -

term

a&]

of the asymptotic

series,

= q/p

PO

from the definitions of q and /3. This corresponds to setting the diffusion term equal to zero in the continuity equation, and therefore is the photoequilibrium condition. With the definitions of the quantities b and CE,the electron density becomes

$($Dj”3T(-&j

N =

4

where q, fl and D are the values of these quantities at the height considered. It should be noted that the coefficient of T, which has the form q~--2/3D-1~3, is independent of height for the assumptions made in this model. The maximum value of N coincides with the maximum of T, which can be found by inverse numerical interpolation as T(23641)

= 0.568558.

(52)

5. DISCUSSION

An important consequence of the solution derived in Section 3 arises when we consider the situation at and immediately above the layer peak. Equat’ion (A22) can be written bx

lThe condition t,hat

(1 + 1)(1+

2 -

I

v) + . . . .

(53)

all terms, other than the first, of each power series is

for neglecting

bx
bx< of which justified,

the former

condition

is the more stringent;

db-./ N = A(A This function

has a maximum

v)

‘!(;_;)

;)! (bx)“-” _

value of the electron

LA!@ -

q-v)(y

(bz)‘] .

is

(55)

:“2

v)!

( -v)!

1

density being

&-?’

N

if this approximation

at a height where

y (bx), = [ the maximum

v)]

y

-A)

[

-

y

il A!(2 -

(-v)!

?‘I

v)! 1

.

(57)

915

The form&ion of the daytime peak of the ionospheric BP-layer Substituting

for a type

D atmosphere

from

Table

2, these relations

where Y = y/4 = 113 = l/(Z + SC). For the particular example of the type B atmosphere (bx),

become

this results in

= 0.03072

compared with a value of 0.06289 derived from the series of equation case formula (57) gives N, w 1.7656 db2/3

(4s).

For this (59)

compared with 1.7166 db213 for the accurate series solution given by equation (52). For this case the approximation (57) gives within 3 per cent of the correct value. It therefore appears that the approximate solution, equation (55), is a good representation of the density of the ionization at and above the maximum of the layer, compared with the complete series solution. However, it should now be noted that this solution is identical with that of equation (17), derived neglecting loss in this region altogether; it follows, therefore, that the simple productiontransport mechanism can explain the formation of the peak of the .P’Z-layer. This result should not be interpreted as meaning that the loss mechanism is unimportant in determining the magnitude of the peak electron density. Its effect, however, is confined to setting the value of P of equation (34), which multiplies a term having the form of a diffusive equilibrium distribution for heights at and above the maximum [as evidenced by the form of the leading term of equation (34)l. Inspection ofFig. 2 of RISHBETH and ~~~~o~(l960), and Fig. 1 of YONEZAWA (1956) indicates consistency with the results of this paper, as regards both the height and the magnitude of the maximum electron density. It is obviously important to discuss the effects of magnetic latitude, and other parameters, on the solution found for this equation; however, this will be carried out in subsequent publications. REFERENCES BOWHILL 8. A. DALMRNO A. GLIDDON J. E. C. and KENDALL P. C. HINTERECGISRH. E. KAMKE E.

1961 1958

RISHBETW H. and BARRON D. W. WATSON G. N.

1960

YONEZAWA

T.

YONEZAWA T.

1960 1961 1944

J. Atmsph. Tew. Phys. 21, 272. J. .A~?r&oq3h.Terr. P&s. 12,219. J. Geophys. Res. 65, 2279. J. Geophys. Res. 68, 2367. D~~erentialgleichunge~ (3rdEd,), P. 440. Leipzig. J. Atmosph. Terr. Phys. 18, 234.

Bd

1,

1952

Theory of Bessel Fumtions (2nd Ed.), p. 345. CambridgeUniversity Press.

1956 1958

J. Radio J. Radio

Res. Lab. 3, 1. Res. Lab. fi, 165.

S. A. BOWHILL

516

APPENDIX INTEGRATION OF THE CONTINUITY EQUATION With the notation of equations can be written in the form x2N” + (1 -

(26), (36) and (39), the continuity

2a)xN’

+ (a2 -

4~“)~~ -

equation

bzN = dx:‘.

(24)

(Al)

A particular solution of this equation can be obtained in terms of Lommel functions. While these are less convenient for calculation than the power series of equation (43), they permit a direct derivation of the asymptotic form of the solution of equation (Al) for large x: that is, for heights well below the layer maximum. This enables an analytical formulation of the lower boundary condition. resulting in a determination of the unknown constant of integration, F, in equation (34). The power series expansion of the analytical solution of equation (Al) is then compared with N, and N, of Section 3.2. Equation (Al) can be put in the familiar Bessel form by the substitutions N = Px” x = - 2

1

to give y2P”

+ yP’

+ (y2 -

(A2)

i J

= 5: exp (G)

v2)P =fy”+l

(A3)

where f = 4d(4b)a-“exp

For arbitrary

[&(y - a)]

(A4)

p=22y-2a-1

1

values of ,u and v, this equation

has a solution

pi = f&&“(Y) = f%,V(Y) + f@-“(Y) where 8 and s are associated g = 2-1

h

=

_2/‘_1

Lommel

(&A + $v -

functions, i)!(ip

-

+ @J”(Y)

+v-

iv -

i)! cos

(

$)!(&u -

(As)

and

sin vrr

(Qp +

(WATSON, 1952):

iv -

sin VT

Q)!

ru2 * ?

cos

1:

Iu+v 2 (

?7

L

1 1 J

(A6)

It will next be shown that X,,,(y) satisfies the lower boundary condition and that sJy) satisfies the upper boundary condition. The leading terms of the asymptotic expansion of S,,(Y) are (WATSON, 1952): S(,,,y) -

y”-1 -

[(p -

1)s -

v2]yP-3 + . . .

since IYI is large at the lower boundary and larg yI < r. term only, equations (A2) and (A7) give N 1 px” N fxaY”-l

Now,

(A7) taking

the first

The formation

and on substituting

of the daytime

from equations N -

peak of the ionospheric

P3-layer

517

(A2), (A4), (26), (36) and (39),

40 exp [(a2 -

a&]

-+

0

as

z -

- co

(As)

PO

and ;j’,,JY) therefore satisfies the lower boundary condition. The ascending power series for s,,,,(Y) is (WATSON, 1952): P+l ~,,.A!/)=

(p +$

P+3 -. Y2 -

[(/A +

1)2 &(p

+ 3)2 -

Y2]

. **.

(A9)

Although this quantity is not defined when either of the numbers (,u f v) is an odd negative integer, these cases can be shown to have the necessary (but not sufficient) conditJions, respectively, al + a3 I a4 a, I 0 which are certainly not fulfilled for any of the cases discussed in Table 1. At the upper boundary, where y is small, only the first term need be considered; by appropriate substitution, it is found that N cc exp [-(a,

+

a&l

(Alo)

for this term, which therefore satisfies the upper boundary condition. The solution, P, to equation (Al) is given by PI of equation (A5), together with a (yet undetermined) complementary function P,, of the form p,

=

LJ-v(Y) + MJAY)

where L and M are constants; P, is the general solution of the homogeneous form of equation (Al). This complementary function is subject to two constraints: (a) since PI obeys the lower boundary condition, P, must also; (b) PI + P, must fulfill the upper boundary condition. Condition (a) may be met by considering the leading terms of the asymptotic expansion of J_,(y) and J,(Y) : J_,(y)

-

JAY) From equation

(A2), arg y =

(Y + &Jr)

J(-J? 1 =Y

cos (y -

&a-).

-(TT/~), and so these forms become

J-v(Y)

-

J”(Y) -

exp (iy) exp (+ri) 2/(2nY 1

(All)

exp (iy) exp ( -&~i) d(2TY)

J

In order to meet condition (a), it is evident that PI and P, must have the same coefficient of y-” exp (iy); or, from equations (All),

51s

s. 9.

BOWHILL

iIf = -_L exp (Y7rif.

(Ala)

The complete solution then becomes P = PI -+ P,

= ~S~~,~(~) + (fg --t &J_**(y) t ffh - .L exp (~)~~~(~).

(A231

Now it is shown in Section 3.2 that J_,(y) satisfies the upper boundary condition but that J,(y) does not. Further, it was shown above that s,,*(g) satisfies this condition. It follows that the coefficient of J&J) in equation (Al.31 must be zero; hence L = f;h exp (-+7&j. Equation (A13) then becomes p = &&I)

+ f&I + h exp f --~41J-“fvf

and from equations (Ati), g + A exp (-vni)

= 2”-1(&/i + Qv - &)!(&u - Jp - ft-)! exp

. (AH)

The solution of equation (Al) is therefore compIcted; for comparison with Section 3.2, however, the solution yill now be expressed as an ascending power series in x. Using the forms

and the substit~tjon~ from equation (A%), of

(A17) substituting these series in equation (Aldf, and using equations (A25), (A2)> (A4), (2Bfi (36) and (39), the solution is obtained as:

The formation of the daytime peak of the ionospheric FL’-layer

519

Comparison with equations (34) and (43) indicates that the first and second terms of (A18) are equal, respectively, to N, and N,. The unknown constant F of equation (34) has therefore been determined as F = (*#/A +

frv-

Using the notation

i)!(.$p

-

&J -

4)!

-(a,7~a,-a,m,i

PO

cl0

Do@, + a312[ Do(a, + aJ21

a, )

(919)

of Section 3.2, this may be written

F = (+

+ +, _ .$)!(aP _ Qv _ +)!(__d)@-““-“.

(A%)

With the further substitution *p + $,! + 3 = 2 equation

(A18)

ax:’ s

= @* -

(A21)

becomes (A!(1 v) I

(bxy

v)! 2

(-v)!(bx)m

nh=o (m -

v)!m!

_

2

A!@. -

?,I=” (I + m)!(ll

v)!(bx)” -

\

v + m)!\’

(AZ)