Ionospheric irregularities produced by internal atmospheric gravity waves

Journal of Atmosphericand TerrestriaIPhysics, 1968, Vol. S0, pp. 795-823, Pergamon Press. Printed in Northern Ireland

Ionospheric irregularities produced by internal atmospheric gravity waves W. H . HOOKE* Department of Geophysical Sciences, University of Chicago, Chicago, Illinois 60637

(Received 15 August 1967) Abstract---A perturbation treatment is used to determine the nature and magnitude of the effects of internal atmospheric gravity waves on the ambient rates of production, chemical loss, and motion of the ionization. The relative and absolute importance of these effects in the creation

of ionospheric irregularities are assessed. This assessment yields several conclusions of particular interest. Firstly, in the F2-region the dominant effect of the gravity waves is that of imparting the motion of the neutral gas parallel to the magnetic field to the ionization through collisional interaction. Secondly, at heights at or below the height of the Fl-ledge, chemical effects, in particular the effect of gravity waves on the rate of photoionization, are quite important. Thirdly, gravity waves affect the rate of photoionization at a given point by changing both the neutral gas number density and the ionizing radiation flux at that point, and this latter effect, hitherto ignored, is in some respects the more important of the two. Fourthly, as a result of the interplay of a number of factors, certain Fourier components of that portion of the gravity-wave spectrum permitted at ionospheric heights are more successful than others in creating observable disturbances. Finally, gravity waves creating neutral gas velocities of the order of 20 m s e c - : seem capable under the right conditions of creating TID's of the largest magnitudes observed. 1. INTRODUCTION

SINCE about 1948, a number of investigators have studied traveling ionospheric disturbances (TID's) using a variety of techniques. The interested reader is referred to C~AN and VILLARD (1962) and FRIEDMAN (1966), Table I, for partial summaries of the results of T I D observations and listsof references. T h e m o s t successful i n t e r p r e t a t i o n of t h e s e o b s e r v a t i o n s is t h a t w h i c h a t t r i b u t e s T I D ' s t o c o r r e s p o n d i n g d i s t u r b a n c e s of t h e n e u t r a l g a s a s s o c i a t e d w i t h t h e p a s s a g e of i n t e r n a l a t m o s p h e r i c g r a v i t y w a v e s (MxRTYN, 1950; HrN~,S, 1960, 1963, 1964a, 1964b; PITTEWAY a n d HINES, 1963, 1965; FRIEDMAN, 1966; H l ~ E S a n d REDDY, 1967). T h i s i n t e r p r e t a t i o n is c o n s i s t e n t w i t h T I D o b s e r v a t i o n s in so m a n y r e s p e c t s as t o l e a v e little d o u b t as t o its b a s i c c o r r e c t n e s s . A t p r e s e n t , h o w e v e r , t h e r e is n o t a v a i l a b l e in t h e l i t e r a t u r e a n y d e s c r i p t i o n of t h e precise n a t u r e o f t h e c a u s a l c o n n e c t i o n b e t w e e n t h e g r a v i t y - w a v e p e r t u r b a t i o n s of t h e n e u t r a l g a s a n d t h e r e s u l t i n g i o n o s p h e r i c d i s t u r b a n c e s w h i c h does n o t suffer f r o m m a j o r deficiencies. N o n e of t h e a v a i l a b l e d e s c r i p t i o n s gives e x p l i c i t c o n s i d e r a t i o n t o all of t h e n u m e r o u s effects o f g r a v i t y w a v e s on a m b i e n t i o n i z a t i o n . M o r e o v e r , t h e s e d e s c r i p t i o n s all suffer a d d i t i o n a l l y f r o m special a s s u m p t i o n s of t h e a u t h o r s . S o m e s t u d i e s of t h e n a t u r e of t h i s c o n n e c t i o n h a v e b e e n m a d e . M ~ T Y N (1950) a n d HINES (1955, 1956) b o t h a d o p t e d as t h e e s s e n t i a l m e c h a n i s m t h e collisional i n t e r a c t i o n b y w h i c h t h e n e u t r a l g a s m o t i o n s i n d u c e m o t i o n s o f t h e ionization. CHESNUT a n d WICKE~SEAM (1966) c o n s i d e r e d t h e role o f i o n o s p h e r i c c h e m i s t r y in d e t e r m i n i n g t h e c h a n g e s in i o n o s p h e r i c s t r u c t u r e c a u s e d b y a c o u s t i c g r a v i t y w a v e s . Present address: Institute for Telecommunication Sciences ESSA Research Laboratories Boulder Colorado 80302. 795

796

W. tI. H o e =

Gershman and Grigor'yev considered both the effect of the gravity waves on the ionization (1966) and the effect of the ionization on the internal gravity waves (1965). The relevance and indeed correctness of many of the detailed conclusions of the above works are questionable, however, because of special assumptions made by the authors. The deficiencies in blXRTYN'S (1950) interpretation are discussed by HINES (1960). HINES (1955, 1956) initially assumed that TID's were the result of resonant response of the ionosphere to certain Fourier components of the gravitywave spectrum. It now seems likely that such resonance concepts are unnecessary to interpretation of the observations, for reasons discussed in Appendix A. HINES (1960) himself has expressed doubts about the pertinence of this earlier work, and no longer espouses it. GERSHMAN and GRIGOR'YEV (1965, 1966) have misused results obtained by Hines in such a way as to render uncertain the relevance of some of their results (HINES, 1967a). None of the above authors either took chemical effects into account or explicitly justified their neglect. CHESNUT and WICKERSHAM (1966) have taken such effects into account. They have assumed, however, that the motion of the ionization is the same as that of the neutral gas. They have also assumed that the photoionization rate at any point, proportional to both the neutral gas number density and the local ionizing radiation flux, varies during gravity-wave passage only because of variations in the local value of the neutral gas number density. Because at F-region heights motions of the ionization are constrained to follow field lines, and because variations in the distribution of the neutral gas cause significant variations in the distribution of the ionizing radiation flux (as is shown in Section 4) the validity of their conclusions is questionable. HINES (1967a) has raised further doubts about this work. The theory as it stands, therefore, provides no correct description of the relationship between the magnitude and phase of disturbances of the neutral gas caused by gravity waves and the magnitude and phase of the resulting TID's. I t is this problem which is investigated below. First, a perturbation approach is outlined and applied to the governing equations of continuity (Section 2). Further development requires determination of the changes induced by gravity waves in the rates of ionization motions (Section 3), photoionization (Section 4), and chemical loss rates (Section 5). The results of these sections are combined to yield theoretical estimates of the net variations of free electron number density resulting from any given gravity wave (Section 6) and these in turn are compared with the variations observed in TID's (Section 7). The conclusions are then briefly observed in TID's (Section 7). The conclusions are then briefly summarized (Section 8). Two appendices complete the work. 2. F-REGION IONOSPHERIC CHEMISTRY

In the F-region of the Earth's ionosphere, the time rate of change a/at of the free electron number density N, is related to the photoionization rate q, the rate of free electron loss by chemical means L(N,), and the mean velocity of the free electrons u, by the equation ON, at + v . (N,u,)

=

q

-

L(N.)

(1)

Ionospheric irregularities produced by internal gravity waves

797

I n considering diurnal and other long-period behaviour, one usuallyassumes t h a t the value of N~ adjusts to changes in the values of a n y of the system parameters explicit or implicit in equation (1) quasi-statically, i.e. in a time small compared to the time scale of the system changes. I n such cases, the term 8N~[at m a y be legitimately neglected, and the value of IV~ at any given m o m e n t can be determined to good approximation b y the values of the relevant system parameters at t h a t moment: V . (N~oU~o) - - go - - L 0 ( N ~ o )

(2)

Here, as in what follows, quantities with zero subscripts are u n p e r t u r b e d or equilibrium values. The basis for the neglect of the term ~N~/at is that chemical reactions and other important system processes have time scales generally small compared to times of the order of a day. T I D observations are observations of changes in the value of N~ of a few tens of percent or less in times of the order of half an hour. The value of N~ does not in general adjust quasi-statically to changes in the values of system parameters which occur in such short times, so t h a t in general the term O N d a t is not negligible in theoretical descriptions of such disturbances. The departures of the value of hr~ from its u n p e r t u r b e d value ~r~0 are slight, however: [ ( N ~ - N~0)/_N~oI ~ 1, except in the more extreme cases. One can therefore hope t h a t the departures of the values of the other quantities from their equilibrium values are also small, in the sense that a perturbation t r e a t m e n t can be applied and terms of second or higher order in the perturbation quantities can be

neglected. In such an approximation, aN~'

a-t- + V . (N/u~0) = q' - - L ' ( N ~ ) -- V . (N~0u/)

(3)

where the primed quantities are the departures from the equilibrium values of these quantities. For instance, N~ -----N~0 4- N / , etc. One can thus determine No' as a function of position and time from equation (3) if one knows the solution of equation (2) and if one can compute the gravity-wave-induced changes in g, L ( N ~ ) , and u~ as functions of position and time. A similar philosophy can be applied to the equations relevant to the determination of q', L ' ( N ~ ) , and u,', etc., and this is in fact the approach adopted here. The reaction scheme listed below, due to RATCLZFF~ (1956), is the simplest that retains the essential feature t h a t the free electron loss rate is recombinative at low altitudes and attachment-like at the higher levels: Reaction

Rate (crn8 sec-1)

0 + hv~O + + e 0 + 4- XY--* (rate ~/(n(XY))--* X Y + 4- 0 X Y + 4- e ----*(rate co)--* X + Y

~ / n ( X Y ) ,~ 10-12

c~~ 10-~

Temperature Dependence

Unspecified Unspecified

Here X Y denotes a molecular constituent and X Y + denotes a molecular ion. n (0+), n (X Y+), and n (X Y) are n u m b e r densities of 0 +, X Y+, and X Y respectively.

798

w.H.

HOOKE

In the unperturbed state, V . (no(O+)uo(O+)) = qo - flono(O+)

(4)

V . (no(X Y+)uo(X Y+)) -----/3one(O+) -- %no(X Y +)Neo

(5)

N,o = no(O +) + no(XY +)

(6)

while during gravity-wave passage, an'(0+____~) + V . (no(0+)u'(0+) + n'(0+)Uo(0+)) = q' - ~'no(0+) - flon'(O+) • at

(7) an' (X :Y+) a~ + v . (no(XY+)u'(XY +) + n ' ( X Y + ) u o ( X Y + ) )

(8)

= fl'no(O +) +flon'(0 +) -- oJno(XY+)N,o -- % n ' ( X Y + ) N , o -- % n o ( X Y + ) N ,' N~' = n ' ( 0 +) + n ' ( X Y +)

(9)

Letting a/Ot = iw, where co is the angular frequency of a single gravity wave, one finds from equations (4-9) t h a t (fie + i°J)n'(0+) + V . (n'(O+)uo(O+)) = q' -- fl'no(O +) -- V . (no(O+)u'(O+)) (lO) (o~oN~o + ~ono(XY+) + i~o)(flo + io~)N,' + (%N~o + fie + ioJ)V . (n'(O+)uo(O+)) + (fl0 + io))V. ( n ' ( X Y + ) u o ( X Y + ) ) T

= (~o~ ~o + ~o + ~ ) q ' '

_~ F~ ~oN,o + 'L

/30

~' _

me

--

t

"r

O I

1

[~ to~oo + _ (~o + ~,,~)j qo L /30 me 7

(ao + io,)j v . (no(O+)uo(O+))

(11)

+ - - (rio + ico)V. (no(XY+)uo(XY+)) O~o

-

(~05:e0 + /~0 + i o ~ ) V .

(no(O+)u'(O+))

-- (fie + ico)V. ( n o ( X Y + ) u ' ( X Y + ) ) n ' ( X y + ) = _hro' -- n ' ( 0 +)

(12)

Equations (10-12) relate gravity-wave-induced changes in the values of the photoionization rate, chemical lose rates, and divergence of ionization to the resulting changes in the values of n(O+), n ( X Y + ) , and N,. The complexity in the form of this system of equations is due to the generality of the model adopted. This model probably gives a reasonably good description of the F-region ionosphere over a wide altitude range, from the base of the F-region (about 140 km) to heights more t h a n several neutral gas scale heights above the height of the F2-peak. Some of the ionospheric processes in this model are in fact important only over part of this range. Consequently, m a n y of the terms of equations (10-12) m a y be neglected with little sacrifice in accuracy in descriptions of the F-region ionosphere

Ionospheric irregularities produced by internal gravity waves

799

in certain height subranges included in this height range. Approximations to these equations which accurately describe the ionosphere in these height subranges are developed in Section 6. 3. G R A V I T Y

WAVES

AND

MOTION

OF T H E IONIZATION

3a. Motion of the ionization in the unperturbed 8rate. The equation of motion of the ions in the unperturbed state is N,omi(Uto .

V)uio ---- N~oei(Eo + U~o × Be) + Niom~i.o(Uo -- U/o)

+ Niom~,o(U,o -- U~o) -- VN~oKT~o -}- Niom~g

(13)

where N~o is the ion n u m b e r density, m t is the ion mass, e~ is the ion charge, ~,,o is the ion-neutral frictional frequency, ~,o is the ion-electron frictional frequency, u~0, u,0, and u o are the mean velocities of the ions, the electrons, and the neutral gas respectively, E 0 is the electric field, Be is the magnetic induction, K is Boltzmann's constant, T~o is the ion temperature, and g is the local acceleration of gravity. Quantities with zero subscripts are the unperturbed values of these quantities. A smaller equation holds for the electrons. Several terms of this equation are of negligible importance in present considerations. The t e r m N~omi(U~o. V)u~o is negligible for motions with spatial scales the order of a neutral gas scale height (about 50 k m at the height of the F2-peak) or greater such as are being considered here unless Uio is the order of the local speed of sound (about 700 m sec -1 at F-region heights). It is assumed here t h a t the neutral atmosphere is stationary in the absence of gravity-wave perturbations, i.e. t h a t u o = 0. I n the lower F-region, collisions of charged particles with neutral particles are m u c h more frequent t h a n collisions between charged particles, and the ionelectron collision term m a y be legitimately ignored. Inclusion of this t e r m in cases where the two collision terms are of the same order does not change the later conclusions of this section, namely, t h a t the neutral gas imparts the component of its motion parallel to Be to both the ions and the electrons through collisional interactions. Since the neutrals t e n d to set both the ions and electrons into motion along the field lines at the same speed, the differential ion-electron motion tends to vanish and the ion-electron collisional interaction t h e n exchanges no m o m e n t u m between the species. Subject to these approximations, equation (13) reduces to 0 ---- _N~0ei(Eo + ui0 x Be) -- Niomi~,oU~o -- V~VioKTi 0 W -Niomig

(14)

The ion motions in the unperturbed state m a y be written as the sum of two parts: Uio = UiemO+ U~dO (15) Here ui,mo is the electromagnetic drift velocity. I t is related to Eo and Be b y Eo × Be Ui6ra 0 ~

(16)

Be ~

U~ao is t h a t part of U~o due to plasma diffusion. At F-region heights, U~ao is nearly parallel to Be (except possibly at the magnetic equator, where its magnitude

800

W.H.

Hoox~

should in any event be relatively small) and the magnitude of this component is given b y Uiaob - -

KT

- -

IN, aN~°

m~vi,o__o ab

1 aT~o sin I~ -t-T~ ° ab q - - - ~ - J

(17)

(See, for example, RISEBETH and BARRON, 1960.) Here H is the scale height of the neutral gas, I is the magnetic dip-angle, and b is a coordinate directed parallel to B. 3b. Motion of the ions in the perturbed state. I n the perturbed state, the equation of motion for the ions is 0 ---- N~0ei(E ' + u / x Bo) + Ni'et(Eo + U~o× Bo) q- NiomiVino(U' -- u~,) -- N~'m~v~noUio -- Niomi~in'U~o -- V . (N/KT~o + N~oKT/) + N/mig

(18)

where the primed quantities are the departures of these quantities from their unperturbed values. Terms of second order or higher of the primed quantities are neglected. The inertial t e r m of this equation is also neglected. This neglect is legitimate here because the time scale of gravity-wave-associated changes in the system ( > 10 min) is large compared to the ion gyroperiod (,-~0.06 see), and the time interval between collisions suffered by an ion with molecules of the neutral gas (v~ o decreases with increasing height, but 1/v~ o is < 100 sec at heights below about 500 kin). I f one defines U / ~ Ui1' ~ Ui2'

(19)

and further defines u~2' b y

0 = N~oe~Ui2'x Bo + N/ei(Eo ÷ U~o + Bo) -- Niom~Vi~oUi2' -- N/miVinoUio -- Niomi~i~'U~o -- V ( N / K T i o + NioKTi ') --b N/m~g

(20)

t h e n it follows from equations (18-20) t h a t 0 -----N~oei(E' q- Uil' x Bo) q- NiomiVino(U -- uil')

(21)

Inspection of equations (14) and (20) reveals t h a t U~o -t- u~2' is the velocity t h a t the ions would have if the ambient ionization n u m b e r density, N~o, were replaced b y N~o -t- N / , the ambient ion temperature, T~o, were replaced by T~o -t- T / , and the ambient ion-neutral frictional frequency, V~o, where replaced by v~,o ÷ v~n'. u~', which like U~aois nearly parallel to Bo, can thus be determined by writing the perturbation equation corresponding to equation (17):

,

_VKT/

KT,ov,,'lF 1 ~N,o

KTio F1 ON, lOT_ ~ s i n q ' miVinOk~ - - 0b -~- T~ 0b -t- 2H J

1 ~T,o

sinq

(22)

I t is clear from equation (22) t h a t variations in Ni, T~, and vi~ of only a few percent

change ulao by only a few percent or less as long as the spatial scales of the variations are equal to or greater t h a n a neutral gas scale height ( ~ 5 0 km). The observations indicate t h a t T I D spatial scales are indeed this great.

Ionospheric irregularities produced by internal gravity waves

801

E q u a t i o n (21) m a y be written in the form u~

,

q- 2 ~no

e~ u × B o ~in0 -~- got2 mt

=u

+

e~ (E'. 13)lB miVino 1

+

e~2 (u x Bo) x Bo o 1 ~i~0 -]- go2 mi2

~i.o et (E' × 1 B) x 13 v~,,o + go2 mi

(23)

e~-~ E ~ x B 0

+ ~'~,,o+ go2 m 2 -

eiBo

(See, for example, HINES, 1953; DUNG~.Y, 1959.) Here w/ -is the ion gyrofrequency and 13 is a unit vector in the direction of Bo. ~n/ I t is shown in Appendix A t h a t the gravity-wave-induced electric field terms of equation (23) are negligible, so t h a t

u~l , " - - u +

~ vino

e/ u x B 0 +

~inO -~- goi2 ~ni

2

e/2 (u x Bo) x B o

1

(24)

~inO -~ wi 2 ~1~i2

The circumstances under which u~2' m a y be ignored in comparison with thl' are determined in Appendix B. The conclusions of this appendix are stated here: u~2' m a y be legitimately ignored in comparison with u ~ ' when /c~K T i

1

and

~i nogomi

leg

--

~ 1

(25)

~inOgo

where go and ]c are the gravity-wave angular frequency and angular wave n u m b e r respectively. I f the first of these conditions is not satisfied, the viscous term in the equation of motion of the neutral gas is comparable to the other terms of t h a t equation, and the detailed mathematical description of the neutral gas motion given b y H ~ s (1960) is not applicable, although the more general description of PITTEWAY and HINES (1963) m a y be. Waves satisfying the first of the above restrictions automatically satisfy the second if/oH ~ 1. The analysis below is limited in applicability to cases in which conditions (25) are satisfied. 30. Motion of the ionization in the F-region. In the F-region, ~t ~ go/, and equation (24) simplifies to - U x 1B Uil ' -~ (U . 1B)I B -I- -~/ gO/

(26)

The resulting divergence of ionization is then

/ u xgo/ l~) V . (N~ouil)' -- V . ((N~ou. 1B)13) + V . ( ~N~o

(27)

since Nio -- N~o to extremely good approximation. The term on the far right of this equation is small compared to the other terms unless u is within a few degrees of perfect perpendicularity to Bo, or unless there is no variation o f N , o(U. ln) along the field lines. In most cases, then, V . (N~oui~') -- ((VN~o. 1B)(u • 13)) -}- N~oV. ((u. I~)IB) 10

(28)

802

W. It. H o o x ~

Equations (26-28) show clearly several well-known features of the nature of gravity-wave-induced ion motions in the F-region. First, at F-region heights, the neutral gas is relatively unsuccessful in moving the ionization in directions perpendicular to Bo, but it does impart its motion parallel to the field to the ionization. Second, the divergence of ionization given by equation (28) has two origins. The variation of the neutral gas motion along the field lines, V . ((u. 1B)IB), results in actual compression or rarefaction of the ionization. Such variations lead to convergence or divergence of ionization even at points where the ambient free electron number density profile along a given field line is locally constant. In addition, the neutral gas motion parallel to the field lines results in shifting of the ionization at any given point to neighboring points on the same field line. If the ambient value of the free electron number density is different at these neighboring points, the result of the neutral gas motion is the replacement of ionization at one point by ionization of different number density from a neighboring point. This is the contribution of the ((VN~0.1B)(u. 13) ) term. The vector VNo0 is directed nearly vertically except at the heights of layer peaks and at times of local sunrise. Thus ]VNe0 • 13[ is relatively small at all heights at the magnetic equator, since the field lines are horizontal there, and it is small at the height of peak ionization number density at all latitudes. Third, variation of the values of the ionic mobilities with height is of little importance in the F-region, since here these mobilities have nearly attained their limiting values. 4. GRAVITY W A V E S AND PHOTOIONIZATION RATES In the analysisbelow it is assumed that the photoionization rate q at any point in the atmosphere can be determined through the general approach due to CHAPMAN (1931a, 1931b, 1939). Chapman considered only the case of an undisturbed, stationary atmosphere; here the effect of gravity-wave perturbations of the atmospheric density on q is shown explicitly. The photoionization rate at any point is given by

where nj is the number density of the j t h constituent of the neutral atmosphere, ~j(2) is the photoionization absorption cross section of t h e j t h constituent for radiation of wave-length 2, and s'(2) d2 is the radiation flux in the wavelength band d~. l is a coordinate directed along the path followed by the radiation. It is assumed that absorption of radiation is due only to photoionization. The integral is over all the ionizing ~t. Gravity waves cause departures of neutral gas number density from the ambient, and therefore change the value ofq in two ways. :First, they change the local number densities nj. Second, they change the local value of S'(2) d2, since they change the amount of absorption undergone by the radiation during its passage through the atmosphere to the point in question. This latter effect has been completely ignored in past studies, but it turns out that it is quite important, especially at altitudes low relative to the height of maximum rate of photoionization.

Ionospheric irregularities produced by internal gravity waves

803

Consider an unperturbed plane atmosphere of constant temperature and composition, consisting of a single ionizable constituent, whosephotoionizationabsorption cross section is independent of 4. I n such an atmosphere, n0(~) = n . e x p ( - ~ / / / )

(30)

where n, is the unperturbed number density of the ionizable constituent at some reference level, n0(~) is the unperturbed number density of this constituent at a height ~ above this reference level, and H is the scale height of this constituent, constant under the above assumptions. When such an atmosphere is perturbed b y a gravity wave, the perturbed number density is given to first order by (Hr~v.s, 1960) n(}, 7, ~, t) = n0(~)[1 ~- A_R exp i(wt -- ]Q} -- ]c,~ -- ]c~)]

(31)

Here } and ,] are horizontal coordinates, directed so t h a t together with ~ t h e y form a right-handed coordinate system. ~o is the angular frequency of the gravity wave. I t is assumed to be constant, real, and positive, k~, k~, and k~ are the components of the wave vector k . ]c~ and k, are assumed to be constant and real; k~ is assumed to be constant but in general complex. I n the absence of dissipation, k~, the imaginary part of ]% is equal to 1/2H, and in such a case the fractional number density fluctuations increase in magnitude exponentially with increasing height. The magnitude of the perturbation term is IA_RI . lexp ( k ~ ) l , and it must have a value <1 for the linear perturbation t r e a t m e n t used by H r s ~ s (1960) to be valid. Since a is assumed to be independent of 4, equation (29) reduces to aS q ----n~S = -- a-1

(32)

where S ------j" #(2) d4 is the total ionizing radiation flux. To compute q, one proceeds just as in the unperturbed case, by first computing S from the second equality in equation (32). The radiation flux at the point (x, y, z) is then given by

ln S ( X , y , z , t )

~.(~.v,z)

(

~1

× [1 -}- A R exp i(o~t -- k ~ -- k ~ -- k¢~)] dl

(33)

where the line integral is taken along the path followed by the incident radiation. S~ is the radiation flux incident on the Earth's atmosphere, dl is an element of path length, x and y are horizontal coordinates, replacing ~ and '2 respectively, while z is the vertical coordinate, replacing ~. The integral is easy to evaluate with the aid of the substitutions ~(~) ----x -}- (~ -- z) tan ;~ cos ¢ ~(~) = y -}- (~ -- z) tan Z sin ¢

:(34)

dl = --csc z d~ where Z is the local solar zenith angle, and ¢ is the angle between the ~-axis and the projection of the unit vector 1~ (directed toward the Sun) on the ~-~ plane.

804

W. It. ttoo~_E

(See Fig. 1.) One finds S(x, y, z, t) = S~

AR

exp

i(o~t--l¢~x--k~y--k,z)]] i Hkrl x

(1--Hk~)+

(35)

JJ

cosz

where k~r and lc~ are the real and imaginary parts of k~ respectively, and kr ---(kx, k~, kzr). Since kzi has the value 1/2H in the absence of dissipation, and smaller

Fig. 1. The relative orientation of the vectors k~ and 1z. The angles 7., 0, and ¢. values when dissipation is non-zero, the quantity (1 -- Hkzi ) has a value greater than or equal to one half. In the absence of gravity-wave-induced perturbations, the height z,n of maximum photoionization rate is given b y exp ( H )

= ng~H sec Z

(36)

and the unperturbed radiation flux has a value equal to

So(X; Y, Z) = Soo exp (--exp ( Z ~ )

)

(37)

During gravity-wave passage, then,

S(x, y, z, t) = See exp

( ox ( )ii+

AR

i(wt--k~X--k~y--k~z!J I (1 -- Hk~) -~ iHkrlz

exp

cos Z

(as)

Ionospheric irregularities produced by internal gravity w a v e s

805

The neutral gas scale height is ~-~50 k m at heights near the height of the $'2-peak, and ~-~10 km at E-region heights. The gravity waves required to produce TID's of the scales observed must have wavelengths of the order of 60-600 kin, while gravity waves responsible for the irregular structure of wind profiles at E-region heights must have vertical wavelengths of the order of 10 km (Hr~Es, 1960). H lkr] therefore has a magnitude of roughly 0.5-10 in these cases. The magnitude of the perturbation term of equation (38) thus depends signiRcantly on the value of cos 0 =

kr • iz

[kr----F ' the

cosine of the angle between the real part of the gravity-wave wave vector and the vector antiparallel to the radiation flux, taking its largest value when cos 0 = 0. I n this case the Sun's rays lie in surfaces of constant wave phase. This dependence on 0 is relatively marked for large values of H ]kr[ and/or for small values of cos Z, and less marked for small values of H [kr[ and/or large values of cos Z. These statements are summarized graphically in Fig. 2, which shows how the magnitude of the perturbation term in equation (38) varies with 0 for extreme values of H [krl and X. I.¢

~ ~ 4 4 H

o, %

z k,2=a -..~= oo

~

0.6

H 2 k2,= I

Q~

-

g N

0.4

H2 k2=Joo ~>

0.2

~

4

H

;~ k~= }00 X = GO°

8 75"

I o

60"

45"

~1 0.4i I o.6I I 0.2

30" Jl

o.s

Lo

cos 8

Fig. 2. The dependence of the value of the perturbation of the radiation flux on the values of the parameters H, kr, Z, and 0. Since IAR exp (kziz)l is assumed much smaller t h a n unity, I(S -- So)/Sol is quite small compared to u n i t y at heights at and above z m. Because of the factor

however, the ratio I(S -- So)/Sol m a y be quite large at heights below z,,. For example, with kr • 1 z = 0, 2A_R exp i(o~t -- k=x -- k~y -- kz(z~ -- 3//)) = --0.08, and k,~ = S ( x , y, Zm -- 3H) ~ 5. W i t h 2A.R exp i(cot -- k ~ -- k~y -N0(x, y, z,~ -- 3//) k~(zm -- 3H)) = 0-08 and the same choice of values for the other parameters, one 1/2//,

one

obtains

806

W.H.

Hoo~:~

S ( x , y, zm ~ 3H) finds S ~ : y : z ~ : 3-H-) ~" 1/5. Thus the ionizing radiation flux at a given point well

below the height of maximum photoionization rate may vary during gravity-wave passage by a factor of the order of 25, even though the neutral gas number density at this point may remain constant to within 4 percent. Using equations (31), (32), and (38), one finds that in general, q(x, y, z, t) ---- no(1 -~ A R exp i ( w t -- k~x -- lc~y -- Ic~z))aS~ A R .exP(1. --i(wt--lc~--Ic~Y+kzZ])Hlc~,) . . . ~i~-r:-l~

cos 2[ (39) while in cases where ](S -- So)/So] ~ 1, (x, y, z, t) -- q0(x, y, z, t) q qo(X, y, z, t) "-- A I ~ exp i(o~t -- k~x -- k~y -- k~z)

x 1-

(4o) (1

-

Hkzi) + iHkr cos

Equations (39) and (40) are the mathematical statement of the earlier assertion that gravity waves influence photoionization rates in two ways, through changes in the local neutral gas number density and through changes in the amount of absorption undergone by the ionizing radiation in its passage through the atmosphere. Because of the factor exp (z---~-S) which appears in the terms representing the changes in the ionizing radiation flux, it is the local neutral gas number density fluctuations which predominate in determining changes in q at heights several scale heights above z~; q'/qo - - n'/no. On the other hand, again because of this factor, it may be the local fluctuations in the value of the ionizing radiation flux which predominate in determining changes in q at heights at or below zm; q'/qo - - S ' / S o . The factor (i/Tkr • 1z/cos :~) appears in the denominators of the terms representing changes in the value of the ionizing radiation flux, and these changes therefore increase in magnitude with decreasing values of the magnitude of this factor. This factor is a measure of both the ratio of the neutral gas scale height to the gravitywave wavelength and of the degree to which the radiation flux vector lies in planes of constant gravity-wave phase. The above statements.have the following physical significance. At heights well above zm, the ionizing radiation has undergone relatively little absorption. Small changes in the amount of this absorption change the value of the radiation flux at these heights relatively little. At heights well below zm, however, the ionizing radiation has been in large part absorbed. Only a small residual flux remains. When li'Hkr 1Jcos 2[I41, i.e. when the radiation flux vector lies in the planes of the wavefronts and/or when the neutral gas scale height is much less than a gravity-wave wavelength, there are significant changes during gravity-wave passage in the amount

Ionospheric irregularities produced by internal gravity waves

807

of attenuation of the relatively large flux at heights near zm which result in very large fractional changes of the small residual flux at the lower levels. Conversely, when liHkr, lz/cos ~l >~ 1, the changes in the amount of absorption suffered by the radiation in its passage through any level of the atmosphere are somewhat cancelled by changes in the amount of absorption of the opposite sense at the neighboring levels, and the resulting changes in the value of the residual flux at the lower levels are relatively small. It should be noted that the two types of gravity-wave-induced changes in ? do not in general act in such a way as to purely cancel each other. In general they are related in phase to each other in a complicated way. Only in the case kr. I x ---- 0 do they act in exact opposition. In this case points on a surface of constant wave phase which is a surface of maximum (minimum) neutral gas number density experience a minimum (maximum) ionizing radiation flux. Equations (39) and (40) should necessarily be quantitatively correct only when k is constant and when X is less than about 75 ° (since it is assumed in the quantitative development that the atmosphere is plane). The above qualitative statements still apply, however, even in cases in which the quantitative statements fail. The large gravity-wave-induced changes in the value of the photoionization rate at heights at and below the height of peak production may be quite important to proper interpretation of some of the observed fine structure of the various ionospheric layers and ill particular to the theory of magnetoshear sporadic-E (HooKE, 1968). Because of the importance of gravity-wave-induced changes in the value of the ionizing radiation flux to resulting changes in the value of q, and because of the dependence of the magnitude of these changes on the relative orientation of the gravity-wave wave vector and the radiation flux vector, one might expect to see indications of a solar control in the observed direction of travel of TID's at low Fregion heights. This point is pursued further in Section 7. 5. GRAWTr WAVES AND C~.~rrCAL Loss RATES The rates of the ion-atom interchange reactions important to F-region chemistry are apparently temperature-independent (SwxDwR, 1965). I t is therefore assumed here that changes fl' in fl are due only to changes in the number density of the molecular constituents of the neutral gas which occur during gravity-wave passage: = n o fie -~ fie A I ~

exp i ( w t - - k x x - - ]c~y - - k~z)

(41)

The rates of the dissociative recombination reactions important to F-region chemistry are apparently dependent upon the value of the electron temperature To although there is considerable uncertainty about the exact extent of this dependence (Swn)ER, 1965). Determination of the gravity-wave-induced changes in the value of T, is a major problem in itself and it is not attempted here. Instead it is assumed that the rates of the dissociative recombination reactions are independent of the value of T~, so that ~' ~ 0 (42) Such an assumption may be justifiable in view of present uncertainties in the estimates of the temperature dependence of these reaction rates.

808

W.H.

ttooxE

6. DETEI%MINKTION OF THE V A L U E OF N,' Du~n~G G R A V I T Y - W A V E I~ASSAGE

In Sections 3, 4, and 5 the changes in the values of the photoionization rate, chemical loss rate, and divergence of ionization due to gravity-wave passage are determined as functions of gravity-wave and system parameters and geometry. These results m a y now be used in conjunction with equations (I0-12) of Section 2 to determine N / . Figure 3 shows the magnitudes of ~oN~o, rio, and o~and their variation with height. The figure shows t h a t at heights well above z~ (z~ m 180-200 km), both rio, w < %N,o. This fact implies t h a t N~o ~ no(0+ ) >>no(XY+ ) (RATCr,IFFE, 1956). I n Section 4 it is shown t h a t since gravity-wave-induced changes in the radiation flux are small at these heights, q'/qo "-- n'/no. In Section 5 it is shown t h a t fl'/flo --'--n'/no, and it is assumed t h a t ~' -- 0. Thus

(%N.o + %no(XY+ ) + io~)(flo + iw)N~" -- %N~o(flo + ioJ)N e' (%N~o + 3o + i ~ ) v .

(n'(O+)uo(O+)) -

~oYooV. (n'(O+)uo(O+))

(%N~o + flo + ion)q" -- %N~oqo(n'no)

fl----o- +

-

-

(fie + ice) qo -- °~oN,o(n'/no)qo

(43)

~0

r~ ' tXoNeo 0¢' 7 ~' •= + - (~o + ~ ) j v . (no(O+)uo(O+)) -- ~oNeo -- V . (no(O+)Uo(O+)) L. 3o ~o no 0[/

- - ( f l o + i o ) Y . (no(XY+)uo(XY+)) = 0 0Co

(~#Lo + 3o + / o ~ ) v . (no(O+)u'(O+)) - %NeoV. (no(O+)u'(O+)) At the heights under consideration, qo ~'~ floN,o and V . (no(0+)Uo(0+)) ~ floN,o

(RIs]~BETH and BA_~RON, 1960). Thus for motions of spatial scales comparable to a neutral gas scale height, %Y,oV. (n'(0+)Uo(0+)) ~ flo%N,'/N,o

{xoNeo

-

-

Tt0

V.

(no(O+)Uo(O+)) ~ ~0~O.~'e0 2 - -

no

°¢oN~o

n0

(44)

qo ~ flo%N~o2 --

n0

I n Section 3 it is shown t h a t lu'(0+)l ~ lul e x c e p t in cases where the neutral gas motions are nearly perpendicular to B o. Thus %N.oV • (no(0+)u'(O+)) --~ %N,o~'ku

(45)

The gravity-wave-induced motions and number density fluctuations satisfy the order of magnitude relation (HINES, 1960) ln'[,no_..~--UC

(46)

Ionospheric irregularities produced b y internal gravity waves

6co

809

"\\\[\~'

~L..U,>:/I [ I0 -'~

I0 - 5

I0 - z

lO-'

I

SeC - I

Fig. 3. The values of the parameters %Nee, rio, and co as functions of height. The value of fie is uncertain, b u t it should probably lie somewhere near, if not within, the shaded area. A given gravity wave has an angular frequency to which corresponds to a vertical line somewhere within the singly-hatched area. The dashed vertical line corresponds to a wave of 20-rain period, the mean of T I D observations. The value ofthe parameter %Ne0, which is also uncertain, a n d s u b j e c t to a great deal of temporal variation, should probably correspond to a curve lying somewhere within the cross-hatched area, to the right of the dotted line during the day, and to the left at night.

so that ¢.oN,o~k u , ~ %N~o 2 n'

(47)

Ck > %N,o ~

where the inequality follows because ¢o/k < C (HI~ES, 1960). At heights at or above the height of the F2-peak, w >~ rio, and equations (43-47) and the fact t h a t no(X Y+) ~ no(0+ ) -- N~o imply t h a t equation (11) reduces to iroN,' - - -- [(u. ln)((VN~o). I n) 4- N,o v . ((u. 1B)ln)]

(4s)

I f z is a vertical coordinate, directed upwards, and h is a horizontal coordinate, then [aNeo/OZI is typically much greater than lOhr~o/Ohl, except possibly at sunrise. I t is assumed here t h a t ONce/Oh =~ O. Thus N , ' "-- i w -z

ONce

ub ~

sin I

-

-

ilcbN,oUb

]

(49)

where I is the magnetic dip angle, ub -- u . Is, and kb = k . Is. kb is in general complex, and it can be written k b ---- kbr 4- ikb~, where both kbr and kb~ are real. I f one assumes t h a t only k z, the vertical component of k, is complex, then kb~ = kz~ sin I, and

r,

N . ' - - iN,ou~O) -1 L\N~ ° 2z

)

4- k,~ sin I -- ikb

(50)

810

W.H.

Hoo~E

One form of this equation which is especially convenient for computational purposes is N . ' ( x , y, z, t ) = N~o(z)Ub(zo)sin I exp (kzi(Z - z°)c°-l~/( 1---]~eo ~Ne°~z -q- kzi_)2+ \si-n-i] ( k b r ~2.

exp i wt -- k ~ -- k~y -- k,rz ~- ~

--

tan -a

-

\sin/

1

kbr

aN,0

_h~o az

(51)

fi-kz~

wimre z0 is an arbitrary reference h e i g h t and u b ---- Ub(zo) exp (k~i(z -- %)) exp i(o)t -- k ~ -- kvy -- ]c~ z)

(52)

Figs. 5, 6, and 7 illustrate typical patterns of the isoionic contours one obtains by using equation (51) with various values of the parameters and by adopting the height profile of the ambient free electron n u m b e r density N,0 given in Fig. 4. This N,0 profile is chosen from a n u m b e r of theoretical profiles obtained by RmBe.TH and B~a~RON (1960); it should be representative of ionospheric ~Ve0 profiles in general. Figure 5 shows the disturbance of (otherwise horizontal) isoionie contours at temperate latitudes (I = 45 °) resulting during passage through the ionosphere of Fig. 4 of an internal atmospheric gravity wave which is propagating meridionally, toward the Equator, and phase downward. The surfaces of constant phase are taken to coincide with the direction of the Earth's magnetic field (]Cbr -----0). The relative orientation of kr and B 0, and the values of the parameters used in equation (51) in the construction of this figure, are shown in the caption. The entire pattern shown moves from left to right with increasing time. Note the inclination of the T I D frontal surfaces, which is in agreement in sense with t h a t always observed. Note also t h a t the wave results in the formation of ripples in the isoionic contours at the lower levels, while at levels near the height of the F2-peak, the wave results in the formation of large regions or 'eyes' of enhanced free electron n u m b e r density. The formation of these'eyes' is due to the variation of the neutral gas velocity along the field lines, which in this special case is due entirely to the increase of wave amplitude with height. Figure 5 shows t h a t with this particular choice of system and wave parameters, an internal atmospheric gravity wave of amplitude such t h a t it causes neutral gas motions the order of 100 m see -a, and therefore by equation (46), neutral gas n u m b e r density fluctuations the order of 10 per cent (the local speed of sound i n t h e F-regionis ~--700 m see-~), causes fractional perturbations in the value of the free electron n u m b e r density of q- 20 per cent (about the largest ever observed). Waves of the same spatial and temporal scales and of the same orientation b u t with smaller amplitudes of course result in proportionately smaller fractional perturbations in the free electron n u m b e r density, since equation (51) shows t h a t Ne' is proportional to U b. Such wave amplitudes are indeed small in the sense t h a t waves of these amplitudes should be adequately described by linearized gravity wave theory available. Figure 6 shows the disturbance of isoionic contours resulting at more northerly temperature latitudes (I = 60 °) during passage of an internal gravity wave of larger spatial scales and a somewhat longer period t h a n t h a t of Fig. 5. I n this ease, the surfaces of constant wave phase do not coincide with the direction of the Earth's

I o n o s p h e r i c irregularities p r o d u c e d b y i n t e r n a l g r a v i t y w a v e s

500

--

E =: 40C

3O(

I

0.5

i

I

I.O

I. 5

Free electron number density,

I

I

2.5 X IO6

2.0 el/cm 3

Fig. 4, T h e h e i g h t profile of Neo a d o p t e d in t h e c o n s t r u c t i o n of Figs. 5-7 (after RISHBETH and BAaRON, 1960).

~F 350

24" "~ 30C--

25C -

i

o

I

I

IOO

Horizon,ol d i s , o n c e ,

20o

km

Fig. 5. A p a t t e r n of isoionic contours d e t e r m i n e d f r o m e q u a t i o n (51) w i t h eo = 5 x 10 -S sec -z, kbr = 0, k=~ = 10 -5 m -1, a n d U5(250 kin) = 75 m see -1. B 0 a n d kr lie in t h e plane of t h e p a p e r a n d h a v e t h e r e l a t i v e o r i e n t a t i o n shown. T h e u n d i s t u r b e d ionosphere has t h e ~reo profile of Fig. 4, N u m b e r s on t h e c o n t o u r s are t h e free electron n u m b e r densities (el c m - s x 10-e).

811

812

"W. ]~. HooY,~

so/. 3,5O-

-

I. 8

300

25o

[

I

1,

200

40O

600

HorizonfQI d i s t a n c e ,

km

Fig. 6. A pattern of isoionic contours determined from equation (51) with oJ = 4.37 x 10-3 sec-1, /¢br = --1"57 X 10-5m-1, kzi = 10-sin-1, and Ub(310 kin) = 40 m sec-1. B o and kr lie in the plane of the paper and have the relative orientation shown. The undisturbed ionosphere has the Nee profile of Fig. 4. Numbers on the contours are the free electron number densities (el em-3 x 10-e). m a g n e t i c field. T h e w a v e is p r o p a g a t i n g m e r i d i o n a l l y a n d p h a s e d o w n w a r d . T h e resulting ionospheric d i s t u r b a n c e is q u a I i t a t i v e l y similar t o t h a t s h o w n in Fig. 5. Here, however, t h e d i s t u r b a n c e o f the isoionic c o n t o u r s results in p a r t f r o m t h e f a c t t h a t t h e surfaces of c o n s t a n t w a v e p h a s e are n o t coincident w i t h t h e direction of t h e E a r t h ' s m a g n e t i c field. T h e g r a v i t y w a v e o f Fig. 6 has a n a m p l i t u d e s u c h t h a t it causes n e u t r a l gas m o t i o n s t h e order o f 50 m see -z, a n d it causes f r a c t i o n a l p e r t u r b a t i o n s in t h e v a l u e of t h e free electron n u m b e r d e n s i t y of :t:15 per cent. F i g u r e 7 shows t h e d i s t u r b a n c e o f isoionic c o n t o u r s resulting a t t h e m a g n e t i c e q u a t o r ( I = 0 °) d u r i n g passage o f a n i n t e r n a l a t m o s p h e r i c g r a v i t y w a v e w i t h t h e same p a r a m e t e r s as t h a t o f Fig. 6. A g a i n t h e r e are q u a l i t a t i v e similarities, b u t B

25C

l

I. 2

]

I

200 400 Horizontal d i s t a n c e , km

I

600

Fig. 7. A pattern of isoionie contours determined from equation (51) with oJ = 4 - 3 7 x 10-Ssec - 1 , kbr ~ 1"57 x 10-Sm-i, kz~ = 10- S m -i, Ub(310 km) = 40 m see-z. B 0 and kr lie in the plane of the paper and have the relative orientation shown. The undisturbed ionosphere has the Nee profile of Fig. 4. Numbers on the contours are the floe electron number densities (el era-3 × 10-6).

Ionospheric irregularities produced b y internal gravity waves

813

quantitative differences, here due to the difference in the orientation of the magnetic field lines at the two locations. I n the lower part of the F-region, where #o ~ aoNeo, or flo >~ aoNeo, equation (11) m a y not be greatly simplified. I t should be pointed out t h a t at these heights flo "~ o~; this implies that, at such heights, the magnitudes of the photoionization terms are comparable to the magnitude of the divergence terms. I t should also be noted t h a t at the same heights the molecular ions are relatively numerous, and therefore it is at these heights t h a t gravity-wave-induced changes in the value of ~, the dissociative recombination rate, m a y assume their greatest relative importance. Uncertainties in our knowledge of the dependence of ~ on T~ and in our knowledge of the gravitywave-induced changes in T~ lead to similar uncertainties in our understanding of the nature of gravity-wave production of ionospheric irregularities at these levels; no detailed calculations of the disturbances in the isoionic contours at these heights resulting from gravity-wave passage are made here. 7. COMPARISON OF THEORY W I T H OBSERVATION

7a. The F2-region. BOWMAN (1960a, 1960b) deduced t h a t patterns of isoionic contours such as shown in Figs. 8 and 9 could account for temperate-latitude spreadF observations. Comparison of these rough results with Figs. 5 and 6 reveals a good /~0 40C --

30(-0

!

1

IO0

Horizontol distonce,

200

krn

Fig. 8. Model of the ionization distribution at a resolved range-spreading irregularity (after Bow~A~, 1960b). Numbers on the contours are the free electron n u m b e r densities (el cm -3 × 10-5).

deal of qualitative similarity. There are of course quantitative differenees. No a t t e m p t is made in Section 6 to choose an ambient N~0 profile which closely approximates t h a t in existence at the time and place of Bowman's observations, nor is there any a t t e m p t to match wave spatial scales or TID amplitudes to the values which correspond to Figs. 8 and 9. MUNRO and HEISLER (1956) arbitrarily created the pattern of isoionie contours shown in Fig. 10 to provide a point of reference for discussion of their observations, and this figure is also qualitatively similar to Figs. 5 and 6 of the present work. The disturbance t h e y postulate has the form of a pulse rather t h a n a continuous wave train, but it is similar in all other respects. Note the ripples in the lower isoionie

~V. H. H o o x E

814

40C

~

-

-

4

.

5

~

E

-r 30(

I

I

iO0

2uO

Horizontal dlsfance,

km

Fig. 9. Model of t h e ionization d i s t r i b u t i o n at a r e s o l v e d f r e q u e n c y - s p r e a d i n g irregulaxity (after B o ~ , 1960b). N u m b e r s on t h e contours are t h e free electron n u m b e r densities (el cm - s x 10-s).

contours and the 'eyes' of enhanced free electron number density at the height of the layer peak. Figure 11 is after STERLING (1967); it is a measurement of the temporal fluctuations of the heights of isoionie contours at the magnetic equator. This figure is qualitatively similar to Fig. 7. (Note, however, that the disturbance of Fig. 7 moves from left to right with increasing time. Thus the abscissas of the two figures are opposite in sense; this accounts for the apparent difference in the sense of the inclination of the TID frontal surfaces in the two cases.) The amphtudes of the fractional free electron number density perturbations of Fig. 11, typical of what Plosma

2

frequency,

-Mc/s

4

8

6

I0

IOC

I

0

I

I00

I

[

200

Distance,

300

J

400

~krn

Fig. 10. Model of distortion of isoionic contours b y a T I D (after Mu~u~o a n d H E r o , R , 1956). N u m b e r s on t h e contours are t h e local p l a s m a frequencies of t h e order of Me s e e - 4

Ionospheric irregularities produced by internal gravity waves

815

400

E 4-

300

I

2oc

r

1400

I

1600

1800

Locol time

Fig. i 1. A typical temporal variation of the heights of the isoionic c o n t o u ~ at the magnetic equator (after Sm]~RLINO, 1967). Numbers on the contours are the free electron number densities (ol cm -a x 10-8).

Sterling observes, are much smaller than those of Fig. 7, indicating that the gravity waves responsible typically have amplitudes much smaller than the amplitude adopted in making the calculations that led to Fig. 7. Waves associated with neutral gas motions of only a few m sec -1 should create disturbances of the amplitude he observes. Sterling's data typically indicate an upturn of T I D fronts at gre~t heights; H I ~ . s (1967b) has shown that viscosity acts in a way such as to cause an upturn. MuNt~o (1950) deduced the pattern of isoionic contours shown as dashed curves in Fig. 12 as a result of his observation of 5 July, 19~8. Using a spaced-receiver

28C- \\

,.o

/

.J.6-~--~

I //~///~\ IIII \"

---

22C 0.8~

2oc

-

O.

I

0

I

200

I

400

Horizontol dis~once,

I

600

km

Fig. 12. T I D of 5 July, 1948 (dashed curves) (after M u ~ o , 1950). Theoretical curves computed by use of equation (51), w i t h co = 1.75 × 10 -s sec- l , ~ = 600 kin, kzi = O, kbr - - - 2.94 x 10 -5 m -1, a n d Ub(270 kxrl) = 11 m see -1 (solid curves).

Numbers on the contours are the free electron number densities (el em -3 x lO-e).

816

W.H.

Hoo~.

network, he also determined the rate of travel of this disturbance to be 10 km min -1 (167 m see-l). The results of an a t t e m p t to determine the state of motion of the neutral gas from the pattern of isoionic contours shown in Fig. 12 is discussed below. The method used is that of making several choices for the values of the parameters of equation (51) and determining wlfich choice results in the best fit. One such fit is shown as the pattern of solid curves of Fig. 12. Discrepancies between the theoretical and experimental plots probably have several origins. Firstly, if one computes the value of the parameter (k2KTJ~Vi,oCO) for this case, one finds that this parameter has a value of about 0.04 at 200 km, b u t a value of about 0.4 at 300 km. At the greater heights, then, equation (51) and the theory behind it cannot be expected to describe adequately the perturbations of free electron number density observed. Agreement between the theory and the data might also be improved if there were an upturn of the theoretical contours at heights above about 250 kin. As mentioned above, HINES (1967b) has asserted that the effect of viscosity should be such as to cause an upturn. This is another indication t h a t viscous effects are important at the greater heights. Secondly, photoiolfization and other chemical effects should be important at the lower heights, and these effects are not included in the foregoing calculation. Thirdly, the determination of the ambient Ne0 profile is arbitrary to some extent since the ionosphere is disturbed throughout the period shown. Other estimates of this profile might lead to better a ~ e e m e n t between theory and experiment. Fourthly, it is assumed in the calculation that the local Brunt period is 15 min. An error in this estimate results in errors in the estimate of the vertical wavelength, and subsequently in errors in the estimates of kr and kor. Fifthly, the observed disturbance resembled not a wave train, b u t a pulse; theoretical estimates describe the disturbance due only to a single Fourier component of this pulse. Finally, as MUNRO and HEISLER (1956) point out, the raw data used to construct Fig. 12 are ionogram virtual-height data, and the true-height analysis used in reduction of the data is inaccurate when the isoionic contours are not strictly horizontal. I t would be of interest to see whether ray tracing analyses would show that these inaccuracies in the experimental contours are such that their elimination would lead to better agreement between present theory and observation. In any case, the above analysis is a preliminary one and it is intended to be purely illustrative. I t merely outlines the type of procedure that one may use in analyzing data such as that of any of Figs. 8-12 to obtain information about the state of motion of the neutral gas. 7b. The Fl-region. I t is quite likely that many of Munro's fixed-frequency observations are observations of disturbances at heights below 200 km (HEISLEI~, 1958). As mentioned in Section 6, photoionization processes are quite important to the gravity-wave production of ionospheric irregularities at these heights. In Section 4 it is shown that at heights comparable to z~ changes in q are quite large if induced b y a gravity wave the real part of whose wave vector kr satisfies the relation kr • 1 z = 0. This condition is satisfied if the Suns' rays lie in surfaces of constant wave phase. Figure 13 shows the geometry of the orientation of the wave vector kr and the surfaces of constant wave phase. (It is a peculiar property of these waves that when the energy flow is upward, the phase propagation is downward.) When the Sun is in position A, the condition krl z = 0 is satisfied. When the Sun is in

817

Ionospheric irregularities produced by internal gravity waves

Raudi~~ ~

~luadxi fi°n~ a ~ JO,.ec,,o° a, .ave

I

I

/

I

I

I

/

I

/

/

/

I

I

/

/

/

/

/

/

/

/ //

,

/

,

Fig. 13. Two of many possible geometries of the wave-sun system. W h e n the sun is in position A, there are relatively large changes during gravity-wave passage in the value of the radiation flux at any given point. When the sun is in position B, these changes are smaller. position B, this condition is not satisfied. It should be clear that the condition k r . 1 z ---- 0 is satisfied not only when the Sun is in position A, but also when it is outside the plane of the paper in any position in which its rays are perpendicular to k. MunRo (1950) observes that TID's propagate primarily northward in the winter and eastward and westward in the summer (in Australia). Figure 14 shows the extent of this tendency. Mm~mo (1958) also finds t h a t there is normally considerable scatter in the directions of travel on any given day, but that the great majority of TID's

oD

August I•

31 to

September

20 3O

•• =~

. .:'."

;..%

• •o e

t.=

o• ~

e

e%

o•

. . •"

I0

October

!#.

og

.'"~ .

.." . ; . . . . - . -

.

2O 50

•

•

..;

~:::

...

I0

November

2O

303jo 0o •

•

• 00 • 60 °

o• • • 120 °

•

•

• •

] 180 °

240 °

• 300 °

Directions of travel (degrees east of north)

Fig. 14. Plot of all the directions of TID movement observed by Munro during Augusf~-November 1948 (after MEzzo, 1950). 11

818

W. IT. H o o r ~

observed traveling toward the east are observed in the mornlng, while the great majority of TID's observed traveling toward the west are observed in the afternoon. This tendency is shown in Fig. 15. There is also a trend on m a n y days for the direction of travel to change from east to west during the day. The reader should be able to convince himself with the aid of Fig. 13 t h a t all of these observational findings are consistent with the idea t h a t photoionization effects, whose magnitudes are controlled by the sun-wave geometry in the manner described above, are important to the production of TID's at low F-region heights.

IiI IO

~

Mornlnq

"6

~.~AH'ernoon

z

i O

I O

• O

• O

e O

• O

• O

e O

• O

I

I

I

I

[

I

I

I

I

I •

• • 0 0 I

I

e O I

I

¢ O I

•

• O I

I

Directio(dnegeas eastofnorth) --

N

N

N

N

N

I~'~

¢ O

o O

I

I

I~'1

f")

Fig. 15. Number of TID'B propagating in any given azimuthal direction in the morning and in the afternoon (after Muffle, 1958). • H~nes has made the alternative suggestion t h a t directional filtering of the gravitywave spectrum due to tidal wind shears m a y be responsible for this diurnal variation in the observed directions of travel (Hn~.s, 1963; H r ~ s and R~.DD¥, 1967), and this is certainly a possibility. The present interpretation has the advantage t h a t it predicts the sense of the changes in the direction of propagation which occur, b u t there is some evidence t h a t similar changes in the observed direction of travel occur during the night (1Kum~o, 1958), and the interpretation of Hines should be more relevant here. I n any case, the two interpretations are complementary, not mutually exclusive, and it is probable t h a t both processes are at work. Finally, it should be noted t h a t the solar control exhibited in Figs. 14-15 is not perfect. This gives rise to no inconsistency since gravity waves create ionospheric disturbances at the height of the _~l-ledge through more t h a n one mechanism. ~urthermore, the tuning, of the wave to the condition kr • I z ---- 0 need not be infinitely sharp, as Fig. 2 suggests. 8. S U M ~ y This paper represents an a t t e m p t to determine the effects of gravity-wave passage on the rates of ionospheric dynamical processes, photoionization processes, and chemical loss processes, and to assess the relative importance of each of these effects in the production of ionospheric irregularities by internal atmospheric gravity waves.

Ionospheric irregularities produced b y internal g r a v i t y waves

819

8a. Dynamical processes. The principal effect of gravity-wave motions of the neutral gas at F-region heights is that of imparting the motion of the neutral gas parallel to the magnetic field lines to the ions through coUisional interaction. This statement holds to the extent that the gravity-wave motions are themselves correctly described by the linearized theory of H I ~ . s (1960). The effect of gravity waves on ionization movement is the most important mechanism through which gravity waves produce ionospheric irregularities at heights well above the height of the Fl-ledge. Waves of moderate amplitude (inducing motions of the neutral gas the order of 20 m scc -z) are capable of producing TID's of the largest magnitudes observed, in favorable cases. 8b. Photoionization processes. Gravity waves produce changes in the value of the photoionization rate at any point in two ways. Firstly, they cause changes in the local value of the neutral gas number density. Secondly, they cause changes in the local value of the ionizing radiation flux, because they change the amount of absorption undergone by the radiation in its passage through the atmosphere to the point in question. Changes in the rate of photoionization, and the solar control of these changes, seem to play a significant role in the daytime gravity-wave production of F-region irregularities at the height of the/~l-ledge. 8c. Chemical loss processes. At heights well above the height of the /Vl-ledge, where particle number densities are low and ionospheric chemistry is slow, it can be assumed to good approximation for present purposes that the lifetime of the dominant ionic constituent, 0 +, is infinite, Gravity-wave-induced changes in the values of ~ and 8, if small, have little effect. At lower heights, however, such changes may be quite important to the production of ionospheric irregularities. Uncertainties in the estimates of the magnitude of these changes lead to corresponding uncertainties in our understanding of the nature of gravity-wave production of ionospheric irregularities at these heights. The uncertainties in present estimates of the dependence of the value of ~ on the value of T~ and of the nature and extent of gravitywave-induced changes in the value of T~ are the major source of this uncertainty. Further work remains, both in the determination of the precise temperaturedependence of the value of ~ and in the determination of the gravity-wave-induced changes in the value of T~.

Acknowledgement----This work was supported b y N A S A Grant NsG 467 Research. I wish to t h a n k Professor C. O. H/~-ES of the University of Chicago b o t h for suggesting t h a t I work on this problem and for generously providing guidance and encouragement along the way. I also wish to t h a n k Mr. D. L. STERLZ~O of the J i c a m a r c a R a d a r Observatory for permitting m y use of his results in advance of publication.

AP~EI~DIX A I t is asserted in Section 3 t h a t the electric field terms of equation (23) are negligible. This appendix is devoted to justification of t h a t assertion. The gravity-wave-induced electric field is related to the current donsity J" b y

al V x ( V X E') = --/~o "~" = - - i m ~ o ~

(A1)

in MKSA units; 8/8~ ~ ira, whero m is the gravity-wave angular frequoncy. In the F-region, J N (v~n0/m~)Ne0e~lU~ + E'~/B], where quantities with the subscript r denote the components

820

W.H.

HOOK~

of these quantities transverse to the magnetic field lines, v<~/oj~ < 0-1, a n d -hre0 < 101~n-8. T~tfing co N 3 x 10-3 sec-1 a n d IV] N 10-5 m -1, one finds that

IE'I ~; 10-'-]UI. IBI,

(A2)

implying that the electric field is negligible except possibly in rather special circ,,m~tanees. One cannot a pr/or/exclude the possibility t h a t such special instances might be important to proper interpretation of the observations. The gravity-wave spectrum incident on the upper atmosphere m a y be quite broad. I f the ionosphere were to respond resonantly to a narrow b a n d of that spectrum in such a way as to create enhanced fluctuations of the free electron n u m b e r density, t h a t b a n d of the spectrum could dominate the T I D observations. Pursuing the problem along these lines, HINES (1955) showed t h a t in the case of a homogeneous, pl~-qma, the electric field is indeed negligible except in two special eircnrnatances: (1) ( k . 1B)z = 0, where k is the gravity-wave vector, here assumed to be a complex constant, and 1B is a u n i t vector parallel to the magnetic field lines. (2) ( k . k) - 0, the condition ofhydromagnetic resonance discussed b y ] : [ r ~ S (1955). Since k is complex, this condition does not imply t h a t k = 0. Hines showed t h a t the hydromagnetic resonance interpretation of T I D observations is consistent with the data only if y, the ratio of specific heats of the neutral gas in the F-region, has the value 1.1 instead of the more generally accepted value of 1.4. This interpretation of T I D observations has other shortcomings as well. I f it is correct, the effect of the large fields created b y waves satisfying the resonance condition should be such as to enhance the free electron n u m b e r density perturbations which are created relative to those created nonresonantly b y other waves of the same amplitudes. This does n o t seem to be the case. The three scalar equations (A6) of Hr~ES (1955) m a y be used to solve for the components of E ' in terms of the components of U. I n the resonant case, the field components are given b y

A E g = --qo(ql z + qzZ)BoUn + qo[qlk~ + q2k,]k~BoU~ + qo(ql]% -- q21c~)knBoU, + (f12 + q2~)k~SBoUn AE.7. = qo(ql~ + q ~)BoU e _ (f12 + q22)k~2BoU~ _ qo(flk~ + qg]cn)k~BoU~ -- qok~(qxk.~ -- q~k~)BoU. 7 A E ( = (qx~ + q~2)lc~(knBoU ~ -- keBoU,), (A3) where A ------qo(ql ~ + q2~) + qoql(k~2). The notation here is t h a t of ]:[ines (1955). The coordinates (~,7/,~) are oriented in such a fashion t h a t ~ is parallel to B o, ~ is horizontal, directed west, and ~ makes the system Cartesian and right-handed, qo.L2 =-- ie°~ao.L~ respectively, where a0.L~ are the longitudinal, Pedersen, a n d Hall conductivities respectively. The divergence of ionization in this case is Ni0k. u a ' , and from equation (23) of the present work one finds

k.

Uil' =

k . u + ~no~i.o + w,~ [(k~UnB o -- knU~Bo) + (k~.E~' + I%En')] e~a

I

+ -

-

?)inO ~- 60i 2 m i 2

+ •

et

[(--/c~U~Bo~ --k~UnBo ~) + (k~En'B o --k,~E~'Bo)]

(A4)

]c~E(.

H ~ m s (1955) gives the following estimates for the order of magnitude of the quantities %. qp q2. and k~~ in the F-region:

I%1"~ 10-7m-~ Since q~ ~ ql, one finds

[qxl " ~

10-12Trb--$

Iq21 "-~ lO-lam-Z

q~ (k~UnB o -- knU~Bo) q~ + k~ ~ q2 k ~2 k~En" -- knE~'Bo "-- Bo(Ic~BoU~ + knBoU n) + q~ q- - ~ - ~

l/¢¢2[ ~ 10-'m-z

k~E¢" + I%En" "-

k~E~" --

qoq~2 + qoqlk~ 2

(I%BoU ~ -- k~BoUn).

(knBoU ¢ -- k~BoUn)

(A5)

Ionospheric irregularities produced b y internal gravity waves

821

Substitution of equations (A5) into equation (A4) shows t h a t if b{ = 0, then k . Utl' -~--k . u, while i f / ~ ~: 0, then k . u~' ----k . u ~- terms ( 4 ]kl.[u]). I n the resonant case, ]k] ----~/2~o~(7--1)g (HINES, 1955). T I D observations indicate t h a t co < 10-~ see-~, so t h a t if 7 ~ 1-4, Ikl < 3 x 10--~m-1. I f in fact ~o = 5 x 10- s sec-1, the mean value of T I D observations, then [k[ = 7.5 × 10-'~m-1. I n the non-resonant case, k . u~" - - b~u~. (See Section 3c.) I n general, b£ ~ 3 × 10-~ m -~. I t is only in the case k . 1~ ~ ]k] that b~ is much less t h a n this value. I t therefore seems t h a t gravity waves satisfying the resonance condition and associated with neutral gas motions of a given amplitude typically cause free electron n u m b e r density fluctuations of a smaller amplitude t h a n do gravity waves of a similar amplitude failing to satisfy the resonance condition. Finally, gravity waves satisfying the resonance condition have no variation of phase with height (Hri~-Es, 1955). T I D ' s frequently have inclined fronts, a n d the resonance theory would have to account for the observed frontal inclinations in some relatively ad hoc manner.

~LPPENDIX B It is sta~d in Section 3 that gravity-waveJnduced changes in the magnitudes of the pressuregradient and gravitational-acceleration terms in the equation of motion for the ions can be legithnately neglected if

k~K~ ~i~inO CO

.~ 1

and

kg

- -

~tnO CO

(B1)

~ 1,

where co and/c are the gravity-wave angular frequency a n d angular wave n u m b e r respectively. This assertion is substantiated below. The argument closely parallels in approach an argument given b y H~rEs (1956), who made a slrnilar b u t more detailed calculation for the electrons. Note t h a t gravity waves satisfying the first of conditions (B1) satisfy the second if/oH ~ 1. Before proceeding further, it should be pointed out that the linearized theory developed b y I-I_u~s (1960) is not a valid description of the neutral gas motions unless the first of these conditions is met (although the more general description of Pr~-x~wAY a n d H.rN~S (1963) m a y be). The equation of motion for the neutral gas m a y be written

au Po-~- = Pg - - V P + ( V . ~ V ) u + V(/~V.u)/3 + (V/~ × V) x u + ~

×B,

(B2)

where p is the neutral gas mass density, P0 is the unperturbed value of this mass density, 1~ is the neutral gas pressure, u is the neutral gas velocity, J is the current density, B is the magnetic induction, a n d / ~ is the coefficient of viscosity. The theory has been developed only for the case in which the ~ × B term is negligible. I - I x ~ s (1955) has shown t h a t during the daytime hydromagnetic effects are certainly n o t negligible for waves with period greater t h a n an hour. The development of H I ~ S (1960) is valid only when the viscous terms are negligible, i.e. when k2/~/ogp0 ~ 1. The coefficient of viscosity ~ is given by/~ ----PoT where ~ is the kinematic viscosity. ~ has a magnitude the order of KT/mv where m is the mass of a molecule of the neutral gas a n d v is the neutral-neutral collision frequency. Since ~n N m t and T N T t so t h a t v N ~n0, the conditions (kz~u/wp0) ~ i a n d (k2KTdm~.oW) ~ 1 are roughly equivalent. The ratios of the magnitudes of the pressure-gradient and gravitational terms of equation (18 ) of the text, to the term N~0rn~v~n0u'of t h a t equation, are of the form

~,~o~V~,,olU[

and

N~o~V.,olUl,

respectively. These ratios m u s t be small compared to u n i t y if the pressure-gradient a n d gravitational-acceleration terms are to be negligible. I n the absence of the dissipative effects of plasma diffusion, the fractional perturbations of free electron n u m b e r density satisfy the order of magnitude relation [ N t ' ] N t o [ ~ W~--I[u[

(S4)

822

W.H.

HOOKE

except in cases in which u is nearly perpendicular to B 0. (See equation (48) of the text.) Substituting this relation into the ratios (B3) one finds t h a t the requirement t h a t these ratios be small compared to u n i t y is precisely t h a t of equation (B1). I n words, gravity waves which are not themselves subject to viscous dissipation and which satisfy the conditions/oH > I induce motions of the ionization which result in the formation of ionospheric irregularities of small amplitude, small in the sense that plasma diffusion does not dissipate them to a n y significant extent. I n a similar way it can be shown t h a t all the other terms of equation (20) are small compared to the large terms of equation (21), since the gravity-wave-induced changes in the values of T~ and v~n are also small. The ions should remain in thermal equilibrium with the neutral gas, which experience fractional temperature fluctuations of the same order as its fractional density fluctuations, given b y equation (48). The magnitude of the change lu~9'l of the value of lu~a01 satisfies the relation

lu,,'l ,'-- (N,'/N,o) lu,,~ol '-.' k lu,,,ol lul cg)

(Bs)

if conditions (BI) and (B4) apply. Since [Uiao] is much less than the gravity-wave phase speed lue'[ < [u,x"I.

(B6)

jr. ¢N,oU,~')l ~ k~,o lu,.;I < k~.,o l u . ' l .---. Iv. (N.;oUa'l

(B7)

I t follows t h a t

I f the linearized theory of H_rm~s (1960) is to be valid, lul < oa//~, and it follows from equation (B4) t h a t / 7 1 / N i e < 1. Hence it follows t h a t

W. (N,'U,ao)l ,-, kN,' lU~aol < k~-~o lu~'] ~ W . (N,ou;~')l.

(B8)

~EFERENCES Bow'~-~TG . G . CHA~ K. L. and VIL~A.RD O . G . CHAP~.A-~ S. C~ Av~.~w S. C~ Av~.A~ S. DU2¢GEY J . W . FRI~D~A-~ J.P. GERSB~AW B. 1~. and GRIOOR'Y~V G . I . Gv.RSHMX~ B. 1~. and GRIO0~'YEV G . I . HEmLER L . H . H n ~ s C. 0. Hn,rES C. 0. l=h:s-Es C.O. H . u ~ s C.O. Hrs-ES C. 0. H n ~ s C. 0.

1960a 1960b 1962 1931a 1931b 1939 1959 1966 1965 1966 1958 1953 1955 1956 1960 1963 1964a

Planet. Space Sci. 2, 133. Planet. Space Sci. 2, 150. J. Geophys. Ices. 67, 973. Prec. Phys. Soc. 43, 26. Prec. Phys. Soc. 43, 483. Prec. Phys. Soc. 51, 93. J. Geophys. Ices. 64, 2188. J. Geophys. Ices. 71, 1033. Geomagn. Aer. 5, 656. Geomagn. Aer. 6, 193. Aust. J. Phys. 11, 79. Prec. Camb. Phil. Soc. 49, 299. J. Atmosph. Terr. Phys. 7, 14. J. Atmosph. Terr. Phys. 9, 56. Can. J. Phys. 38, 1441. Quart. J. Roy. Meteorol. Soc. 89, 1. Icesearoh in Geophysics, Vol. I. 299 (M LT

1964b 1967 1950 1950 1958 1956

Can. J. Phys. 42, 1424. J. Geophys. ICes. 72, 1015. Prec. Icoy. Soc. London A201, 216. Pro& Icoy. Soc. London A202, 208. Aust. J. Phys. 11, 91. Aust. J. Phys. 9, 359.

Press). Hn,rES C. 0. Hirers C. 0. and P~EDDY C.A. MX~Tr~ D . F . MUNRO G . H . Mm,mo G. H. and HEISLER L . H .

Ionospheric irregularities produced b y internal g r a v i t y waves PI~I'~wAY M. L. V. and HrtrEs C. 0. RATCZa_FFEJ . A . RISHBETH H. a n d BARROlV D . W . SwIDER W'. JR.

1963 1965 1956 1960 1965

823

Can. J. _Phys. 41, 1935. Can. J. Phys. 48, 2222. J. Atmosph. Terr. Phys. 8, 260. J. Atmoaph. Terr. Phys. 18, 234. J. Geophys. 2~ea. 70, 4859.

Reference is also made to the following unpublished material CKESNUT W. G. and WICK:ERS:HA~WA . F . JR. HrNES C . O . I4_rtrES C . O . HOOKE W . i . STERLING D . L .

1966 1967a 1967b 1968 1967

National Meeting, U.R.S.I., Washington, D.C. (April, 1966). P r i v a t e communication. Paper in preparation. P a p e r in preparation. P r i v a t e communication