E-region ionospheric irregularities produced by internal atmospheric gravity waves

Planet.SpaceSci.1969,Vol.17,pp. 749 to 765. PergamonPress. Printedin NorthernIreland

E-REGION IONOSPHERIC IRREGULARITIES BY INTERNAL ATMOSPHERIC GRAVITY

PRODUCED WAVES

WILLIAM H. H O O K E

Institute for TelecommunicationSciences, Environmental Science Services Administration, Boulder, Colo. 80302, U.S.A. (Received 8 November 1968)

Abstraet--A linearized perturbation treatment is developed for study of the roles of both photochemical and dynamical effects in the gravity-waveproduction of E-region ionospheric irregularities. It is found that the two effectsmay be comparable in magnitude. It is also found that the small wave-associatedvertical motions of the neutral gas may play a significantrole in producing ion convergence. Results of numerical calculations are presented to illustrate these statements. 1. INTRODUCTION The magnetoshear theory of sporadic-E, as it now stands in the literature, appears to give a satisfactory description of the behavior of night-time sporadic-E-layers (Es-layers) composed of metallic ions. For example, several authors have shown that the observed heights of these layers are usually in good agreement with the heights predicted on the basis of the theory and the simultaneously observed neutral wind structure (e.g. MacLeod, 1966; Wright et al., 1967). Prediction of other Es-layer parameters, however, such as the maximum free electron number densities of the layers, is much more tentative because of uncertainties in present estimates of the values of reaction rates, diffusion coefficients, the number of metallic ions available, etc. and comparison of theory with observation on such points has not yet been attempted. There is a distinct possibility because of the large number of idealizations implicit in the magnetoshear theory, that it in fact inadequately describes such weak layers of enhanced free electron and molecular-ion number density as may be present during the daylight hours. In particular, the emphasis on the dynamical means by which neutral-gas motions result in the formation of layers of enhanced free electron number density, to the exclusion of the photochemical effects associated with such motions, severely restricts the applicability of the theory. For instance, it is invariably assumed in the theory that the photoionization rate q (which depends on neutral-gas density) and the recombination coefficient 0~ (which is temperature dependent (Biondi, 1967)) are constant both in position and in time. This assumption restricts theoretical consideration to wind profiles that are unaccompanied by perturbations in neutral-gas density and temperature. It is quite likely, however, that much if not all of the neutral-gas wind shear observed at E-region heights is due to tidal and other gravitationally controlled motions (Hines, 1960, 1963, 1966a, 1966b, 1966c; Rosenberg, 1966; Rosenberg and Justus, 1966; Woodrum and Justus, 1968). The theory of such motions shows that they are in fact necessarily accompanied by such fluctuations in both neutral-gas number density and temperature (Hines, 1960, 1965; Lindzen, 1967; Siebert, 1961). Since it is fast photochemistry that results in limiting the intensity of Es-layers composed of the common molecular ions, one must expect that wave-associated variations in the rate 749

"750

WILLIAM 1t. HOOKE

of this chemistry make an important contribution to the form of wave-produced l:~-tayers. This point was first noted in passing by Hines (1960); it is examined quantitatively here, In the next five sections a linearized perturbation treatment is developed which permits study of all the wave-associated contributions, both dynamical and photochemicai, t,,~ Es-layer production. A linearized treatment is justifiable here for several reasons. I:irst, it is expected on theoretical grounds that Es-layers composed of the common molecular ions exhibit peak densities only a few tens of per cent above that of the ambient. Secund, the analysis includes consideration of several processes. The mathematical difficult? involved in providing a completely general treatment is not justifiable at this point. Third, a linearized treatment is in fact preferable since it permits separation of the various photochemical and dynamical effects responsible for E,-layer formation and thereby allows for the separate assessment of the importance of each. In Section 7 the results of numerical calculations made for an isothermal model atmosphere are presented and discussed. The calculations themselves are described briefly in the Appendix. 2. E-REGION I O N O S P H E R I C CHEMISTRY IN THE UNPERTURBED STATE A N D D U R I N G WAVE PASSAGE

In the E-region of the Earth's ionosphere, the time rate of change O/Ot of the free electron number density N~ is related to the photoionization rate q, the recombination coefficient and the velocity of the free electrons us by the continuity equation

OAr"__x_ V. (Neu~) == q -- ~N~'z.

('2.1)

at The common E-region molecular ions have lifetimes that are small compared with a day. For this reason, in considering diurnal and other long-period behavior of the E-region, one usually assumes that the value of No adjusts to changes in the values of any of the system parameters explicit or implicit in Equation (2.1) quasi-statically, i.e., in a time small compared with the time scale of the system changes. One usually assumes further in E-region studies that during the day the E-region is in a state of photochemical equilibrium: q0 = ~0N~o~-

(2.2)

Here, as in what follows, quantities with zero subscripts are unperturbed or equilibrium values. We know from the magnetoshear theory of sporadic-E that when the neutral atmosphere is not motionless, but rather in a state of motion which exhibits strong vertical shears in the wind profile, the divergence term of Equation (2.1) is no longer negligible (Whitehead, 1961). Observations indicate that this wind shear pattern itself typically has a lifetime that is large compared with the important chemical reaction or diffusion time scales (see, for example, Wright et aL, 1967), so that even in this perturbed state, the value of Are adjusts quasistaticaUy to temporal variations in the wind shear pattern: V. (Neu,) = q -- ~N. 9'.

(2.3)

While Equation (2.3) is a good approximation to Equation (2.1) when the wind shear is of tidal origin, it need not be a good approximation when the wind shear is caused by a gravity wave of short period. Hence the full expression (2.1) is used here.

E - R E G I O N I R R E G U L A R I T I E S P R O D U C E D BY GRAVITY WAVES

751

When a single gravity wave of small amplitude passes through the ionosphere, the values of q, ~ and us depart by amounts q', ~' and uo' respectively from their unperturbed values qo, ~o and ue0 (U~ois here taken to be zero). As a result, the value of No also departs from its unperturbed value N~0 by an amount Ne'. The primed quantities are assumed small so that products of the primed quantities can be ignored in the equation of continuity (2.1). Taking O]Ot =- i~o, where co is the angular frequency of the gravity wave, one finds (io~ +

2otoN~o)N~'

=

q'

- - ~ ' Neo ~ - - V .

(N~0uo).

(2.4)

The changes uc', q' and ~' that occur during gravity-wave passage are determined in the three succeeding sections. In Section 6, the results are used in conjunction with Equation (2.4) to yield a relation determining the change N~' resulting during passage through an isothermal E-region ionosphere of any given gravity wave. 3. M O T I O N S O F E-REGION I O N I Z A T I O N RESULTING F R O M GRAVITY WAVES

The spatial and temporal scales of the neutral-gas motions believed to be responsible for the production of temperate latitude sporadic-E can be determined from both neutral wind measurements and observations of the E~-layers themselves. All observations to date indicate that these motions have vertical spatial scales the order of a few kilometers or tens of kilometers, horizontal spatial scales the order of 100 km and temporal scales typically the order of 1 hr or more. Hines (1960) has shown that such E-region motions are of the internal gravity wave type. The motions of the E-region ions under the action of neutral-gas motions of these spatial and temporal scales are given by (MacLeod, 1966) 1 ui --

-1- --% - - pi ~ [pi~u + piu × F .% (u. F)r].

(3.1)

Here ui is the ion velocity, u is the neutral-gas velocity, pi = vi/co~ is the ratio of the ionneutral collision frequency vx to the ion gyrofrequency coi and I" is a unit vector directed parallel to the local direction of the Earth's magnetic field. The unperturbed plasma is electrically neutral, and the induced plasma motions give rise to currents that are divergence free, so that N~o : Ni0

(3.2)

V . (Neou~') = V . (Nio. ui') = V . (N~o. ui')

(3.3)

and where Ni0 is the unperturbed value of the ion number density. Passage of a gravity wave results in neutral-gas density and temperature variations and hence variations p~' in the value of pi: Pi = Pi0 .% Pi'.

(3.4)

The wave is taken to be of perturbation amplitude, however, and substituting Equation (3.4) into Equation (3.1) and ignoring products of p~' and u one finds ui' --

1 ~1 .% Pio [Pi°~u .% P i ° U × r -% (u. r')r].

Hence it suffices in this analysis to consider only the unperturbed value of Pi.

(3.5)

752

WILLIAM H. HOOKE 4. THE GRAVITY-WAVE PERTURBATION OF E-REGION PHOTOIONq[ZATION RATES

Let the atmosphere be of constant temperature and composition, consisting of a single ionizable constituent whose photoionization absorption cross-section is independent of the wavelength of the ionizing radiation. In such an atmosphere (Chapman, 1931) ndz) .... n, exp ( - - z / H )

i4, !

and qo(z)

-~

qo0 exp

I1

"-

zo ~z

see Z exp \----H---l J"

(4.2.)

Here ng is the unperturbed number density of the ionizable constituent at some reference level; no(Z) is the unperturbed number density of this constituent at a height z above this reference level; H is the scale height of this constituent; Z is the solar zenith angle; zo is the height of maximum photoionization rate qoo at the subsolar point (Z = 0°) • Hines (1960) has shown that when such an atmosphere is perturbed by a gravity wave, the perturbed number density, ,'7 == n 0 -+- n' (4.3) is given by n(x, y , z, t) == no(Z ) [1 I A R exp (z/2H) exp i(cot -- k~x -- kv)' .... k~z)]. (14.4) Here x and y are horizontal coordinates, directed so that together with z they form a righthanded coordinate system; ~o is the angular frequency of the gravity wave, taken to be real, constant and positive; A and R are complex constants; k~, k~ and ks, the components of the wave vector k, are constant and real• The magnitude of the perturbation term is [AR[ exp ( z / 2 H ) which must have a value HI for the linear perturbation treatment used by Hines (1960) to be valid. Hooke (1968) has shown that during perturbation of the atmosphere by such a gravity wave, the photoionization rate departs from its unperturbed value q0 by an amount q', which is related to q0 by

q' _ n' I 1

2 exp

sec Z 2iHk. I z

¢4.5)

COS 7

Here 1x is a unit vector directed parallel to the solar ionizing radiation flux. In Equations (4.4) and (4.5) n' and q' are complex. In the physical applications, the real parts of these variables should be used. This result and its significance have been discussed in some detail elsewhere (Hooke, 1968), but a few comments are worth repeating here. The photoionization rate q at any point is proportional to both the neutral-gas number density and the ionizing radiation flux at that point. When the neutral-gas distribution is altered, both these latter quantities change. The first term in the brackets represents the contribution to the change in q of the change in the local value of the neutral-gas number density. The second term represents the contribution of the change in the local value of the ionizing radiation flux, which is relatively large at the lower heights (z < zo), at the higher zenith angles (large sec Z),

E - R E G I O N I R R E G U L A R I T I E S P R O D U C E D BY G R A V I T Y WAVES

753

Xx (rn)

-10-2

106

105

104

i

I

I

l0s

aoNeo:SX10 -2 sec-i I =-25 ° P~o:3 -10-S --

/

R:O-4

/ //

0.6

/

/

/

/~

/ /

~'~/

--104

0.8

-10-4

/

//I// [i /

/

'

/// 105

J /

_to-5

10-6

1

10-5

10-4

10-3

kx(m "l ) FIG. 1. CONTOURS OF CONSTANT

R

IN THE k DOMAIN FOR THE MODEL ATMOSPHERE DESCRIBED IN THE APPENDIX.

The solid curves are curves of constant R. The dotted curves are curves of constant wave period; the periods in minutes are indicated in the boxes. Wave phase propagation is eastward and downward. I ~ --25°; pi0 = 3; :toNe0= 5 x 10-~sec -x. and in cases where k . 1 x = 0. This last condition requires that the Sun's rays lie in surfaces o f constant wave phase. Since H k ,'~ 1-5 in the cases under discussion, the magnitude of this second term is quite sensitive to the degree to which the condition k . 1 z = 0 is met. 5. T H E EFFECT O F GRAVITY WAVES O N E-REGION ELECTRON T E M P E R A T U R E S AND REACTION RATES

Let

c~(To)=g(300OK)

(

T~ ~_e

• \300OK/

(s.l)

where To, the electron temperature, is measured in °K, and ~(300°K) and 6 are constants. D u r i n g gravity-wave passage, To departs slightly, by an a m o u n t To', f r o m its unperturbed value Too, and this small departure is related to the small departure c~' of ~ from its unperturbed value % by 0t

_ 0~0

t

=

_ ~ __T".

(5.2)

~e0

Gleeson and Axford (1967) have shown that to g o o d approximation, Tc - - T~ = [q]{KT°t + [c¢(N~ -- Neo ~)]~KTe s - - [otNcZ]~KTo, Le,,*N~

(5.3)

754

WILLIAM H. HOOKE x im}

10 ~

i0 ~

;0 ~

-iO g F ..............................................

[

1

/

{~o Neo :5 x FO 2 sec

I :, -60 ° ,~':o:: 5

/

/

/"

-3 ~,

F0

J

104 /

~

~"

I

-Jo-4\

/

///

/

t If05

_0s

\//\i i0-6

\l

J

:\

t

10-5

,

10-4

10-3

k×{rr,,". FIG. 2. CONTOURS OF CONSTANT R IN THE k DOMAIN FOR THE MODEL ATMOSPHERE DESCRIBED IN THE APPENDIX.

The solid curves are curves of constant R. The dotted curves are curves of constant wave period; the periods in minutes are indicated in the boxes. Wave phase propagation is eastward and downward. I ~ --60°; p~0 = 3; ~oNeo= 5 × 10-2sec-1. in the E-region, where Tn is the neutral-gas temperature, Tot is an 'injection temperature' of energetic photoelectrons (which Gleeson and Axford take to be equal to 8000°K), and K is Boltzmann's constant. L ~ * -= 1"55 × 10-15n(N2) + (5 X 10-1STo -- 1.0 x 10-15)n(O2)

(5.4)

where n(Nz) and n(Oz) are the number densities of the nitrogen and oxygen molecules respectively, measured in cm -z. In the E-region, T~ ~-~ 400°K and n(OD ~-~ 0"2n(N~), and the temperature dependence of L~n* is weak. Furthermore, I(No -- Noo)/Noof < 1 and T~ ~ T/~, so that Equation (5.3) reduces approximately to

7;-

:In = [ql]KTot/(l'55 x lO-lSn(N2)Ne)

(5.5)

which is the same as a result obtained by Cole and Norton (1966). In Equations (5.3) and (5.5) the equilibration time for the electrons is assumed much shorter than the time scale of the dynamical changes in the system, an assumption that should be good here (see Hooke, companion paper). During gravity-wave passage, q, n(N~), No, To and :inn, but not Zet , depart from their unperturbed values; Tot is a measure of the energy to which the energy of a newly created photoelectron falls through inelastic collisions with the ions and neutral molecules before it begins to lose energy primarily through elastic collisions with the ambient free electrons.

E-REGION IRREGULARITIES PRODUCED BY GRAVITY WAVES

755

Xy(m)

_10-2

l

a o

106

105

104

'

'

I1

N e o = S X l O -2 see -~ I = - 60 ° P~o =

3

L

_

R=t

/

103

/"

2

r20///

°31 v

2

-10-4 _

ii

/

Io s

/ / /

_10-6

l

_10-5

_10-4

_10-3

k y ( m -I ) FIG. 3. CONTOURS OF CONSTANT R IN THE k DOMAIN FOR "IHE MODEL A'IMOSPPIERE DESCRIBED IN THE APPENDIX.

The solid curves are curves of constant R. The dotted curves are curves of constant wave period; the periods in minutes are indicated in the boxes. Wave phase propagation is equatorward and downward. I = --60°; pt0 = 3; ~ 0 N e o = 5 × 10-2sec-a. This energy is determined primarily by the excitation energies of the atmospheric constituents. It does not change significantly with small changes in the relative or absolute concentrations of these constituents, because this energy is reached in a time very short compared with the time required for the electron to lose the remainder of its excess energy to the ambient electrons, and very short compared with the average free electron lifetime. The perturbed equation corresponding to Equation (5.5) is To' T~o

1 T.'+ 1 + e -Tno

e rq' 1 + eLqo

n'(N2) Ne'] no(N2) -- ~-~j,

(5.6)

where the constant e is defined by e =--

[qo]~KTe t ,

Le.oN~oT.o

.

(5.7)

Combining Equations (5.2) and (5.6) gives ~o

~

+ 1 +

e Tno

. ~

Here n'(N2)/no(N2) is taken to be equal to n'/n o.

no

(5.8)

756

WILLIAM H. HOOKE iO~

0~

.¢;~

t0 4

+

i : -25 °

/

P,s

/

//

/I

i

_10-3.

_i0-4

_10-5

i0-6

( /' / ,Ij t'/,/I

/

10-5

-e'o

10-4

0-3

kx(m -I FIG. 4. CONTOURS OF CONSTANT D IN THE k DOMAIN FOR THE MODEL ATMOSPHI~RE DESCRIBED IN THE APPENDIX.

The solid curves are curves of constant D. The dotted curves are curves of constant wave period; the periods in minutes are indicated in the boxes. Wave phase propagation is eastward and downward. I -- - 2 5 ° ; pto ----3. Because the value of To depends upon Ne, and vice versa, it is difficult to determine these two variables simultaneously. Gleeson and Axford (1967) avoid this problem by postulating a very intense sporadic-E layer (NdNoo -': 10) and noting that variations in To of a few tens of percent leave the major features of the layer relatively unchanged. In this study the problem is avoided by linearizing the equations. 6. THE LINEARIZED CONTINUITY EQUATION Combining Equations (2.4), (3.5), (4.5) and (5.8), one finds that

I

( io~ +

) 2 ~ 1 ~e + ~

- V.

1 %N~o

Noo ~

n'

2 sec Z exp (~-~---z) -

N + , ' = =n- o

1 --

2iHk. 1z l q - - cos Z

~oN+o~

}1 sec exp( ) t

[mo'ZU+ ,O~oU× 1" + (u. 1")I']

1

T~'

2e 1 +en

n' o

l+

~o~

/

.

(6.1)

E-REGION IRREGULARITIES PRODUCED BY GRAVITY WAVES

757

The linearized internal gravity wave theory relates the perturbations in neutral-gas number density to the motions of the neutral gas by the formula (Hines, 1960)

n' [co2k~+_ i(~ = 1)g(k~2 +_ kuz) = i~gcoZ/2C2] no = ,ok~C2[k~ -- i(l -- y/2)g/C 2] 3 " u~

(6.2)

I¢,2k~ + i ( y - 1)g(k~2 q- ku2) -- iygco2/2C2~ where ~ is the ratio of specific heats, g is the acceleration of gravity, C is the speed of sound, and u~ and u~ are the x and y components of the neutral-gas motion u. Hines' results may also be used to show that the fluctuations in neutral-gas temperature are related to the motions of the neutral gas by T,,'

I(y -- l)o~2k~- i ( y -

1)g(k~2 q- k~2) q- iyco2(y -- l)g/2C 2-} ~ok--~C--~[~ : -T(l -----~2 )--g/C-ffj ... u¢

T.,, - _ __ [-(y

--

l)¢oek~-

i(y

-- 1)g(k~2 q- k~2) q-

(6.3)

i7~0'(2 -- 1)g/2C2q

(Hines, 1965; Hooke, companion paper to the present work.) With the system of Equations (6.1)-(6.3) together with the dispersion equation (Hines, 1960) co4 - o~2C~(k~2 + k~2 + k~2) q- (y -- l)g2(k~2 + k~2) -- y2g2~o"'/4C2 = 0

(6.4)

one can determine the perturbations No' that result during passage of a single gravity wave through a given isothermal atmosphere. Only k¢, ku and the wave amplitude need be specified. The results of some numerical calculations are discussed in the next section, and the procedure for making the calculations is discussed briefly in the Appendix. 7. RESULTS OF THE NUMERICAL CALCULATIONS AND DISCUSSION One question of great interest is whether the photochemical and divergence terms of Equation (6.1) are ever in fact of comparable magnitude. A general order-of-magnitude estimate of their ratio gives, for the specific case of Pi0 ~'~ 3 (corresponding to an altitude of about 108 km (see Fig. 2 of MacLeod, 1966)) and small dip angle ( l I [ < 30°), R ~ amplitude of photoionization-induced variations in Ne amplitude of divergence-induced variations in No

(7. l)

/1 ~

,.~_ %N~o/O'3ku tl 0

where k is the wave number. Choosing k~, ku < 0.1k~ ~-~ k ~ 6 × 10-4 m -t, in agreement with E-region observations, one finds n'/n o ,-~ 0.6u/C (see Hines, t960). Taking C ~ 300 m/see one finds R ~ 10%Ne0

(7.2)

in this special case. The quantity l[%N~o is the ion lifetime. When this lifetime is short (%N~0 ~ 5 × 10-2 see-t), the ratio of Equation (7.2) is ~-~0.5. When this lifetime is long (%N~o < 5 × 10-z see-t), the ratio is <0.05.

758

'~ILLIAM H. HOOKE

Broadly speaking then, during the day, when the ionospheric chemistry is tast, the photoionization terms of Equation (6.1) are comparable in magnitude to the divergence terms. At night, however, when the chemistry is slow, the photoionization terms arc negligible. Again, generally speaking, the larger the wavelength of the ~ave, the smaller the resulting ionization divergence, and the greater the relative importance of the phot~ionization terms. Finally, the higher the latitude (the greater the magnetic dip), the more ineffective is the divergence process, anti the greater is the relative importance ,~,~ the photoionization terms. The magnitude of the temperature-dependent recombination-induced variations m N,~ depends on the value of O. Obviously, if 6 is zero (no temperature dependence of the reaction rates) then the recombination terms are also zero. If [d[ ~ 1, however, these terms are of the same order of magnitude as the photoionization terms. The above estimates apply only to a specific case. As mentioned before, the value of R is sensitive to the magnetic dip angle. It is also sensitive to the direction of wave propagation, as illustrated in more detail by Figs. 1, 2 and 3. These figures show contours of constant R superimposed on wave dispersion diagrams. Figure 1 shows R contours for waves propagating due East at a location where the magnetic dip angle is ....25" (the minus sign indicates the Northern Hemisphere). A different figure, not presented here, would apply for waves propagating due West, for reasons discussed below. Figure 2 shows a similar plot for waves propagating due East at a location where the dip angle is 60 ", and Fig. 3 shows such a plot for waves propagating equatorward at a location where the dip angle is --60 °. All three figures are for ~oNoo- 5 . 10 -3 sec ~. To find the values of R corresponding to other values of ~.oN~0, simply divide the new value of 70N,,o by 5 :~: 10 ~ sec -~ and multiply the value of R shown on the contours by this fraction. The value of Pi0 also is important in determining the value of R, although this point is not illustrated here. Another question of interest is the role of the small vertical neutral-gas motions in producing ion convergence. Consider the case of waves for which 4kz 2 ~ 1/H ~ and k~ >~ k n where k~, is the horizontal component of the wavevector. If the vertical and horizontal components of u are u~ and u~ respectively, then (Hines, 1960) kh - - ~ u,,

t7.3)

amplitude of N e variations due to vertical motions alone amplitude of N e variations due to horizontal motions alone

(7.4)

u~ =

and D ~

kh Pi0 ~ for East-West horizontal motions kn ~-~ pl0~ .-- for North-South horizontal motions, kz as can be seen from Equation (3.5). Although the ratio kh/k z may be small, the ratio D may be appreciable if Pio is sufficiently large. This order-of-magnitude result is illustrated more generally in Figs. 4 and 5 (for the case P i 0 - 3). Both figures exhibit contours of constant ratio D superimposed on wave dispersion diagrams, for eastward- and equatorward-propagating waves respectively. Equation (7.3) holds approximately for points to the left of the 60-min dashed curve and for ks > 10-4 m -1 in each figure.

E-REGION IRREGULARITIES PRODUCED

BY G R A V I T Y WAVES

by(m)

I0~

_10-2

i05 I

104 ----l0 s

I :

-25 °

/

-10-3~

-- 104 / _,¢4

,_.o/ ~ " - 7 " " - ~ /

s° ~ " " ' ~ J /

/

/

l0 s

/

I//

_10-_10_ G5

-IT s

,

-10-4

-I 3-s

ky(m -I ) FIG. 5. CONTOURS OF CONSTANT D IN THE k D O ~ I N FOR THE MODEL ATMOSPHERE DESCRmED IN THE APPENDIX.

The solid curves are curves of constant D. The dotted curves are curves of constant wave period; the periods in minutes are indicated in the boxes. Wave phase propagation is equatorward and downward. 1 = --25°; pl0 = 3.

Eastv,,ard-Propagatir',gWave I

Westward-PropaaatingWave

s Layer" J

Fast

' '\,,

West

FIG. 6. THE DIFFERENCE BETWEEN Es-LAYERS PRODUCED BY EASTWARD- AND WESTWARDPROPAGATING GRAVITY WAVES.

Their wavevectors are shown. The vertical scale is exaggerated. The magnetic field B is directed out o f the paper and downward, as indicated. The Es-layer in b o t h cases is f o r m e d according to the usual criterion, in a region of wind shear with westward-directed wind overlying eastward-directed wind. The horizontal and vertical c o m p o n e n t s o f the winds are indicated. The vertical neutral-gas motions associated with the eastward-propagating wave act through collisions with the ions to enhance convergence of the ionization; those associated with the westward-propagating wave do the opposite.

759

76o

WILLIAM H. H O O K E LJ~ Jr 5~C;

-

~. NeE

/

~P6

1

~ I ~ 1i! ~/ . -

i

"'"Neo, -...

(\

1(]8

"} " ],

~o6i ,0~i

...7

i

-'J"

/t ~z,,/

,02} ,,/.-. 1 i~',

IO0

//

.-

, /

-

0'6

0,7

0"8

j

1

I

09

1,0

I

I.I

1.2

Free Electron Number Density

I 3 x ]0

N e (cm -3)

FIG. 7. A N Ne HEIGHT PROFILE FOR AN AMBIENT IONOSPHERE, AND FOR THE CASE OF PERTURBATIONS OF THAT IONOSPHERE AT A GIVEN MOMENT BY AN EASTWARD-PROPAGATING WAVE (NEE) OR A WESTWARD-PROPAGATING WAVE (NEW).

The E-Wwind profile is the same in both cases as illustrated, the direction toward which the wind is blowing being indicated. The model atmosphere is that described in the Appendix. Here ~5 is taken to be zero.

i

)

II4 - / . / . . . ~ / /

"..

J

~ - ~ t

oo I 06

/-E.~'- ~ 07 0~

wa rd- Propagating Wove/

l 09

I I.o

~ I-I

Free Electron Number Density

FIG. 8. TIlE ATe, NEDYNH,

AND

f 1.2

/ s 1.3 x I0

N e (cm -3)

NEDYN PROFILES ASSOCIATED PROPAGATING WAVE.

WITH

THE EASTWARD-

E-REGION IRREGULARITIES PRODUCED BY GRAVITY WAVES

761

The role of the vertical motions assumes added interest when it is noted that these motions may act to enhance the convergence of ionization induced by horizontal motions or to destroy it, depending on the direction of wave propagation. The vertical motions of eastward-propagating waves energized from below enhance the ion convergence, while in westward-propagating waves energized from below they tend to reduce it. The reason for this is illustrated in Fig. 6. When Equation (7.3) holds, the main contribution of the vertical neutral-gas motions is what MacLeod (1966) terms the 'collision wind-shear contribution', while the main contribution of the horizontal neutral-gas motions is what

EoYNHJ 120

114

II0

,o4

/

.f/"

0-1

0.8

0-9

I-0

FI

I-2

1.5 x 105

Free Electron Number Density N e (cm -3) FIG. 9. T~E Ne, N E D Y N H ,

AND N E D Y N

PROFILES ASSOC~TED WITH "rUE WESTWARD-

PROPAGATING WAVE.

he terms the 'mixed wind-shear contribution'. As can be seen from Fig. 6, the 'mixed wind-shear contribution' is in the same sense in the two cases, while the 'collision wind-shear contribution' is different in sense in the two cases. It should be noted in this connection that in the case of the eastward-propagating wave, the Es-layer surface is also a surface of maximum neutral-gas density and hence maximum photoionization rate (except when the wave-induced variations in radiation flux are important). In the case of the westward-propagating wave, however, the E,-layer surface is a surface of minimum neutral-gas density and hence minimum photoionization rate (subject to the above proviso). Figures 7-9 illustrate these statements in another manner. Figure 7 shows the ambient Neo profile, as well as the N e profile (NEE)generated at a given moment during perturbation of the ambient ionosphere by an eastward-propagating wave (2n = 100 km, 2, = 10 km). It also shows the Ne profile (No I4/) generated during perturbation by a westward-propagating wave of the same parameters. The horizontal wind profile existing at the moment is the same in both cases, and it is also shown. Note that the direction of wave propagation has a marked effect on the magnitude of the resulting ionospheric disturbance. Figure 8 shows 12

762

WILLIAM tt. HOOKE

!

;

- .\ \

\

ilo!

-~",11

!

.-1,'/

1

!

.i///

it

...$7/

,o~r 102

r

.//

I "/

t -

t

1.2 I-3 x 105 Ambient Free Electron Number Density Neo (cm -3 )

'~0.6

07

0-8

0-9

t-0

H

FIG. 10. THE Neo HEIGHT PROFILE FOR EACH OF FOUR VALUES OF (~.

The model atmosphere is that described in the Appendix.

120 /

r

/

~18I-

.

r " ' " ~ ; - ~ ~ J .

.

i

.

.

I .

- ~ ' ~

104-

a~l,

-o

i

Te 2

. • ";'>f'"/"/

.-'--"

...

I

102 ."//'/ 1°°(~.5

0.6

:',~

0-7

/

v" O~

Free E l e c t r o n

Eastward-Propagating

~

0.9

t

1.0

Number

I

I-I Density

t,

1.2 Ne

t

1.3

Wave |

I

/

1.4 x 105

(cm-3)"

FIG. ~1. PERTURBATION OF EACH OF THE FOUR IONOSPHERES OF FIG. ] 0 BY AN EASTWARDPROPAGATING WAVE ('~h ~ 100 kin, g, = 10 kin).

The wind profile is that of Fig. 7.

E-REGION IRREGULARITIES PRODUCED BY GRAVITY WAVES

763

again the Neo profile and NeE profiles of Fig. 7, along with the Ne profile (NEDYNH) which would have resulted if only the dynamical effects of the horizontal winds had been active. The NEDYN curve is the curve resulting from the total dynamical effect of the wave. The difference between the Ne profile and the NEDYN profile is caused by the wave-associated chemical effects (in these examples, 6 : 0, so that only the photoionization rate variations are important). Note that here the dynamical effects of the horizontal motions and of the vertical motions and the photoionization rate variations add constructively to /20

m

118 Ct a: Te 't - - ~ , . ~

t16

J

/

~ C I

(::(Te-2

tl4

E 2

If2

"."..- ~ \- - . . . . ~

If0

re-1

IDn

•

z

N

108

,o6

,//i,

/J

J

I

/-'"

J

102 -

100!~

••

4

I

:""/""

06

I 0'8

07

Westward-PropagatingWave i 0-9

I I' 0

I t.I

Free E l e c t r o n N u m b e r Density

I

i2

Ne

1.13

I 1.4 x 10:

J

(cm -3)

FIG. 12. PERTURBATION OF EACH OF THE FOUR ioNosPrm~Es OF FIc. 10 I~v A WESTWARDPROPAGATING WAW (2h == 100 kin, 2, = 10 km).

The wind profile is that of Fig. 7.

disturb the ionosphere. Similar profiles for the westward-propagating wave are shown in Fig. 9. Note that here the dynamical effect of the vertical motions and the chemical effect of the photoionization variations add destructively to the dynamical effect of the horizontal motions, and the resulting disturbance is relatively small. Of interest, finally, is the effect of different temperature dependences o f , on the shape of the resulting disturbances. Figure 10 shows an ambient ionospheric profile for each of four ~ temperature-dependences 6. Figure 11 shows the resulting No profiles during passage through these ionospheres of an eastward-propagating gravity wave (2 h = 100 km, 2~ = 10 km). Figure 12 shows a similar ph)t for a westward-propagating gravity wave of the same parameters. The horizontal wind profile in both cases is that of Fig. 7. The

764

WILLIAM H. HOOKE

author does not mean to imply that our knowledge o f the temperature dependence o~ these reaction rates is in fact so uncertain; it now appears that %'o~',, ~.~-! o :

i T~,

(5 c~z 1) :~: 10 -7 ~ 1 \300 K

~-,t.~±0.,-,)

cm3/sec

"7.5)

(Biondi, 1967). Figures 10-12 merely serve to indicate the role played by the temperature dependence o f c~ in the gravity-wave production of E-region ionospheric irregularities. The quantitative conclusions presented here m a y not be directly applicable to the actual ionosphere because of the large n u m b e r o f idealizations implicit in their development. Also, atmospheric tides rather than internal gravity wave motions seem to be d o m i n a n t in the production of E~-Iayers. Nevertheless, atmospheric tides and internal gravity wave motions are similar in m a n y respects, differing primarily in the ease with which they may be studied quantitatively. The conclusion that tidal- and wave-associated photochemical effects are important contributors to tidal- and wave-generated E~-layers seems inescapable. Acknowledgement--The author is grateful to Dr. T. M. Georges and Mr. J. W. Wright of the ESSA Research Laboratories for their many helpful comments and suggestions.

REFERENCES BIONDI,M. A. (1967). Recombination processes (chargedparticles), DASA reaction rate handbook, 11.1 -~11.2I. DASA Information and Analysis Center, Santa Barbara, California. CHAPMAN,S. (1931). The absorption and dissociative or ionizing effect of monochromatic radiation in an atmosphere on a rotating Earth. Proc. phys. Soc. 43, 26-45,483-501. COLE, K. D. and NORTON, R. B. (1966). Some problems associated with midlatitude sporadic-E. Radio Sci. 1 (New series), 235-241. GLEESON, L. J. and AXFORD,W. I. (1967). Electron and ion temperature variations in temperate zone sporadic-E-layers. Planet. Space Sci. 15, 123-136. HINES,C. O. (1960). Internal atmospheric gravity waves at ionospheric heights. Can. J. Phys. 38, 1441-1481. HINES, C. O. (1963). The upper atmosphere in motion. Q. JIR. met. Soc. 89, 1-42. HINES, C. O. (1965). Dynamical heating of the upper atmosphere. J. geophys. Res. 70, 177-183. HINES, C. O. (1966a). Prevailing and tidal wind shears in the E-region. Radio Sci. 1, (New series), 69. HINES, C. O. (1966b). Diurnal tide in the upper atmosphere, J. geophys. Res. 71, 1453-1459. HINES, C. O. (1966c). Ionospheric dynamics, with emphasis on the E-region. URSI Meet., September 1966, Munich. HOOKE, W. H. (1968). Ionospheric irregularities produced by internal atmospheric gravity waves. J. atmos. terr. Phys. 30, 795-823. LINDZEN, R. S. (1967). Thermally driven diurnal tide in the atmosphere, Q. Jl. R. met. Soc. 93, 18-42. MACLEOO, M. A. (1966). Sporadic-E theory. 1. collision-geomagnetic equilibrium. J. atmos. Sci. 23, 96-109. ROSENBERG,N. W. (1966). Summary and conclusions from the Estes Park Sporadic-E Seminar. 3. ionospheric wind patterns. Radio Sci. 1 (New series), 246-247. ROSENBERG,N. W. and JUSTUS, C. G. (1966). Space and time correlations of ionospheric winds. Radio Sci. 1 (New series), 149-155. SIEBERT,M. (1961). Atmospheric tides. Adv. Geophys. 7, 105-187. SMITa, L.G. (1966). Summary and conclusions from the Estes Park Sporadic-E Seminar. 2. Rocket measurements. Radio Sci. 1 (New series), 244-245. WmTEHEAD, J. D. (1961). The formation of the sporadic-E-layer in the temperate zones. J. atmos, terr. Phys. 20, 49-58. WOOORUM,A. and JUSTUS, C. G. (1968). Atmospheric tides in the height region 90-120 kin. J. geophys Res. 73, 467-478. WRIGHT,J. W., MURPHY,C. H. and BULL,G. V. (1967). Sporadic-Eand the wind structure of the E-region J. geophys. Res. 72, 1443-1460. APPENDIX The numerical calculations'

The atmosphere is determined by specification of ),, g, H , qoo, zo, Z, Pio(zo), no(N2)(z0), e(300°k), T~0, 6 and the magnetic dip angle L Pi0 and n0(Ng.) are allowed to vary as

E-REGION IRREGULARITIES PRODUCED BY GRAVITY WAVES

exp

765

(--z/H). qo is calculated from Equation (4.2). Then a trial solution for Neo is obtained: Ne°(1)

qo °~(300°K)- - T~

71/~ -~ •

~] Juu

/

~./

(A.l)

j

Next e is calculated f r o m Equation (5.7) by using the trial solution N~o. Then s o is determined : %

=

~(300 o K)

no

(1 - - e b ) .

(A.2)

Finally, N~o is calculated:

Neo(Z) = [qo(z)l%(z)] li~.

(A.3)

This procedure is reasonably accurate since in the cases considered, e ~ 1. It is repeated for a n u m b e r of heights in the height range of interest, and the ambient ionosphere is thus determined. The perturbed ionosphere is then determined by specifying the wave p a r a m eters (k~, ku, k, and the wave amplitude) and by using the equations of Section 6. In the calculations referred to in the text, the following values of the parameters were used: H ---- 8 km, 7 ---- 1"4, g = 9.5 m/sec, qoo ---- 5 × 103 cm -3 sec -1, Tno = 300°K, Zo = 108 k m , n(N2)(z0) = 2 × 1012 cm -~, pio(Zo) ---- 3, I = --25 °, Z ---- 0°, ~(300°K) = 5 × 10 -7 c m 3 see -1.