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Atmospheric wave-induced instability in the nighttime E-region
Press.Printed inNorthern Ireland Journal of Atmospheric andTerrestrial Phyaia,1072.Vol.84.pp.2026-2043.Pergamon
Atmospheric wave-induced instability in the nighttime E-region TOM BEER* and D. R. MOORCROFT Physics Department, University of Western Ontario, London, Canada (Received 3 November1971; ti revieedform 16 March 1972)
Abstract--This article examines the perturbed continuity equation when the perturbations are the result of an internal atmosphericgravity wave in the E-region. The transient responseof the ionization ia interpretedas the gradient instability and the values of the vertical and horizontal wavenumbers that will induce it are plotted for various heights. Only in the presence of westward directed electric fields, which are believed to occur only at night, will the gravity waves induce the gradient instability. Approximate analytic expressionsare obtained for the permitted wavenumbers as well ae for the instability growth times. In the course of this andysis it ia shown that in the D-region all irregularities,even those that are field-aligned, will tend to move with the ion velocity.
1. INTRODUCTION IN 1963, A. Simon and I?. C. Hoh independently published papers in which they considered the laboratory applications of a new type of plasma instability. This particular instability occurs in a non-uniform plasma that contains gradients in ionization density, under the influence of perpendicular electric and magnetic fields. The ionization moves up or down due to the influence of space charge electric fields that are set up by the background field’s separation of the oppositely charged particles. Because of these conditions this mechanism has been called a ‘cross field instability’, ‘gradient instability’, and ‘drift instability’. Ionospheric workers were quick to utilize this ‘gradient in&ability’ in a myriad of ways. Some used the instability a,s a source mechanism for irregularities in the P-region (REID, 1968; CUNNOLD, 1969; WILLIAMS and WEINSTOCK, 1970), or to explain the striations in artificial plasma clouds formed by rocket released barium (LINSON and WORKMAN, 1970; REID, 1970; RAO and RUSSELL, 1971). MEADA et aZ. (1963) and KNOX (1964) attempted to explain the irregularities in the equatorial electrojet by means of the gradient instability, though it is nowadays believed that these are manifestations of another plasma instability called the two stream instability. Nevertheless, there are so-called Type II irregularities which cannot be explained in terms of the two stream instability. ROGISTER and D’ANOELO (1970) have proposed a high frequency counterpart of the gradient instability as an explanation. WHITEHEAD (1971b) has also proposed a gradient instability based mechanism. TSUDA et al. (1966) were the first to use the gradient instability in relation to midlatitude sporadic-E (E,), and were followed by WHITEHEAD (1967, 1968, 1970) and SATO et al. (1968). This series of papers demonstrated that the gradient in&ability mechanism was unable to produce thin layers, but it is still believed (TSUDA et al., 1969) that it could be a mechanism for the non-blanketing diffuse type of E,. In most of these cases no explanation was offered as to the mechanism that produces the initial non-uniformity in the plasma. WILLIAMS and WEMSTOCK (1970) and FARLEY et al. (1970) point out that a background noise level of weak irregularities due to * Present address: Physics Department, University of Ghana, Legon, Ghana. 2026
2026
TOM BEER and D.
R. MOORGROIQ
random thermal ~uctuations will always exist. CHXXONAS (1969) assumes small scale perturbations to the background and shows that in the presence of a constant background wind, field aligned Earn.patches could form under the action of the gradient instability. In this article we find the resulting instab~itios when gravity waves perturb the ionization through neutral-charged particle collisions (HINES,1955,1956; GEORGES 1967) and variations in recombination and photoionization rate (HOOKE, 1968, 19704 b, c). Occasionally various authors have pointed out that a gravity wave could produce resonances with the ionization. HINES (1955, 1956) tried to use them to explain travelling disturbances in the P-region but mistakenly neglected the background electric fields. KATO et cd. (1970) showed that a gravity wave produces two irregularity waves. One moves with the same velocity and period as the gravity wave and the other, which can only exist in very non-uniform plasmas, travels with the ion veIo&y along the non-u~ormity, HINES and HOOKE(1970) considered the case when the ionization irregularities affect the gravity wave, though this only happens to very long period waves. WHITEHEAD (1971%)considered a spatial resonance when the irregularities drift with the gravity wave velocity. He suggested that this may explain the exceptionally high electron densities often observed in wind-shear E,layers. CHIMONAS (1971) extended the spatial resonance concept by allowing for ion trapping and downward trausport in height varying tidal winds. Thus he claimed that an interaction between the tidal winds and gravity wave winds produces E8. In a subsequent article we shall examine the elects of the gravity wave induced gradient drifts in the E-region. 2. THEORY 0~ THE GRADIENT INSTABILITY ~~t~~~t~~
This section deals with the mathematical framework of the gradient instability. It encompasses the work of the above-mentioned authors but extends the analysis in four irnpor~t ways: (1)It allows for complex wavenumbers in the horizontal and vertical directions; (2) The effects of atmospheric gravity waves are examined, by assuming that the horizontal and vertical wavenumbers of the gravity waves and of the ionization perturbations are identical and by including a sinusoidal gravity wave in the equations of motions; (3) The amplitudes (Nt- and N-) of the sinusoidal perturbations in the electron and the ion concentrations (nl+ and n,-), are taken as being unequal. This leads to both a low frequency and a high frequency case of the gradient instability; and (4) Perturbations (cI’) in the recombination coefficient (a) during the passa’ge of a gravity wave are not neglected. The first part of the section follows the previous developments of the theory and considers the gradient instability as zLneffect upon pre-existent ionization perturbations. The second part of this section presents an original attempt to describe the effects of gravity waves upon the io~zation in the presence of both background electric fields and space-charge fields set up by movement of the induced io~ation perturbations. It is shown that the perturbed continuity equation may be linearized
Atmospheric wave-induced instability in the nighttime
E-region
2027
and treated in a manner analogous to that used in problems involving mechanical vibrations. The ‘transient’ response is interpreted as the cross field gradient drift (the gradient instability in the unstable case) and the ‘steady state’ response is discussed elsewhere (BEER, 1971, 1972a, b). To begin, it may be instructive to consider a greatly simplified model of the gradient instability mechanism. Once the charged particle concentration has been perturbed by a sinusoidal variation, the drifts of the ions and electrons wiIl not be identical, In the case of upward baok~ound electric fields this is a consequence of the difIeriug Hall mobilities. In the case of horizontal electric fields it arises from the opposite directions of the Pedersen mobilities. This limits the instability to the E-region because in the F-region collisions are so infrequent that the Hall mobilities are identical and the Pedersen mobilities are negligible. As an example, consider the case when n,, the background ionization eoncentration increases upwards. Once the ions and electrons have been separated, perturbation electric fields appear. These small scale fields produce El x B drifts which will then carry the enhanced regions downwards and the depleted regions upwards so that they will both appear to grow against the background density. Obviously if either the electric field or the steady density gradient were reversed in direction, these probations would tend to disappear and we would have a stable situation. If both are reversed the situation is once again unstable. In the subsequent analysis, the magnetic field is wholly in they direction whilst the z-axis which is perpendicular to the Bfield, is directed upwards, thex-axisis horizontal to the east and 13is the dip angle in the y-z plane. v and h will be used as subscripts and variables to denote the components of variables in the vertical and horizontal directions respectively. For wave motions in the ionosphere that are not of a high frequency, the inertia and gravitational effects are negligible. In this case the charged particle velocity V* with respect to the neutral wind U (measured relative to the ground) in the presence of an electric field E and a magnetic field B is given by
PP&
0
(1)
0 po”
0
ip~_” 0 (see, for example, RISHBETHand GARRIOTT, 1969, Chap. IV) where in the case of double signs, the topmost is applicable to ions and the bottom one to electrons. D represents the diffusion coefficient and ,uthe mobility. The subscripts P and H refer to the Pedersen and Hall effects respectively where
(1 (DH’) BP* PP
PH
=
=
(vff2
(v&J2+ (%fP WfVf
Do *
(1 (1 po Do
(v”J2+ (Wff*)2 po
*
2028
Tolvr B~EIXand D. R. Mooausow
lel B/r?&”is the gyrofrequency. given by WE
=
The longitudinal transport coefficients are
and In computing the transport coefficients (Appendix I) the CIRA (1966)model atmosphere was used in conjunction with CEAPMAN’S(1965)expressions for the ion collision frequency Y+ and the electron collision frequency Y-. The electric field in the ionosphere may be estimated by using the dynamo theory of atmospheric tides. MATSUSHITA (1969)has prepared charts that give the direction and mag~tude of the electric field in the dynamo region as a function of local time and dip angle. He shows that at latitudes less than 45" the horizontal fields are eastward from 0400 to 1700local time and westward during the night with maxima at around noon and midnight. 1 mVfm represents a typical horizontal field strength though it increases markedly towards the poles. The E-field in the z direction (per~n~c~ar to the B field in the upward direction) is more latitude de~ndent, but its magnitude is approximately the same as that of the horizontal field. The Sq currents are larger in magnitude during summer than in winter or in the equinoctial months and this seasonal variation becomes more marked as the dip angle increases (WEDA, 1957).The ionospheric electric field perpendicular to B remains reasonably constant with altitude (HAERENDEL et aE., 1967; PO~REBNOYand FATKULLIN,1969) and the electric fields parallel to the magnetic field are negligible.
The particular background ionization profile that is used is tabulated in Appendix I and represents the 0140 local time profile obtained by BLUMLEet al. (1965) over Peru. This is representative of mid-latitude ~ghttime profiles and contains the characteristic nighttime E-region valley between 100 and 150 km. This profile was also used by REID (1968) so that our results may be directly compared with his. There were a number of reasons for choosing the nighttime case, Firstly, it simplifies the mathematics by allowing the production term to be neglected. Secondly, as there are larger gradients at night than during the day one would expect the gradient instab~ity to be more effective during the night and thirdly, once all the relevant calculations had been done, it was found that only at night will the electric field point in such a direction as to permit gradient instabilities to be induced by gravity waves.
Consider the perturbations in 71,E and U such that n = n, + n,; E = F& f E,; TJ = U’; where U is the neutral wind velocity. Take *1 f =N*
exp[@t-
K$-Ksy--
Kg)]
(2)
and assume that the perturbation electric field is derivable from a scalar potential ‘v” so that
Atmospheric wave-induced instibility
in the nighttime E-region
2029
where, in terms of the vertical wavenumber K, and the wavenumber in the horizontal north-south direction KNs K, =KNssin6’+Kecos8 K, = KNs COB0 - K,, sin 8. In general, Q and K will be used when it is possible for the frequency and the wavenumber to be complex and o and k will be used for the real part of these quantities. Previous workers have all assumed U’ = 0. Though this seems surprising at first, it will be shown later in this section that even in the case of ionization perturbation produced by gravity waves when it is possible to assume an analytic expression for U’, the gradient instability condition is independent of U’. So let us consider the situation when U’ = 0. The neglect of the gradient of s2is formally incorrect. This could be included by substituting K, - tiXl/& for K, and Q - dK,/& for R in the derivations. However as a linearized analysis is strictly applicable only for an instant of time then the inclusion of the above terms becomes unnecessary (especially at t = 0) if it is realized that the linearized results obtained can only provide information as to the degree of stability or instability possessed by the ionization. The linearized analysis can only be used as an indicator and is inapplicable for exact quantitative results. The perturbed continuity equation for the nighttime E-region is y
+
v * (nov,)*
+
v * (nlV,)*
= -
wdn,+ + n13
which may be written, by using (l), (2) and (3), as (X! + A*)N*
+ B*V
+ a,n,(NF
-
IV*) = 0
(4)
where n,+ = n,- = n, is necessary to maintain charge neutrality, and
A* = 2a,% -t %*K2 --iK&b+*%,
+ Kz21+ D,,*Kv2 +
tz[
f,@E,,
-PIS*E,,I --iK,[fpp*Eo,
+ pH*E,,J
(5a)
+ ,uH*E,J
and V represents the scalar potential. Also,
pp*noK2 + Kz2) f n,pa*K,2.
(5b)
The perturbation electric field will satisfy Gauss’s law and defining e = absolute value of the electron charge e0 = permittivity of free space then by using equations (3) and (5) it can easily be shown tha.t V = e(N+ - N-)/E,,(K,~ + Kv2 + Kz2).
(6)
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TOMBEERand D. R. MOORCROPT
Now equations (4) and (6) provide three linear equations in three unknowns: Nf, N- and V. By eliminating N- one obtains (Xl + A+)iV+ + (B+ - aon,.5,K2/e)V = 0 (iSI + A-)N+
+ (B- -
(ifi + A- -
aon&,K2fe)V
(7) = 0
(9)
where K2 = KS2 + Kg2 + Kz2. Non-zero solutions to this may easily be found by setting the determinant composed of the coefficients of N+ and V equal to zero. Using 11= e/a,K2 this results in a dispersion relation that is quadratic in w. The roots of this are given by 2iQ = q(B- - Bf) - (A+ + A-) +
‘(A+ -
+ 2a,n,
A-)2 + q2(B-
J
-
2q(A- - A+)(B+ + B-) + 4?&n,(B- - B+) + 4Q%,2
B+)2 -
(9)
which represents two solutions : a high frequency case and a low frequency case. By evaluating A* and B* it may be shown that A* is always very much smaller than yB*. Therefore, in the low frequency case (iR + A-)/q < B- and so equation (9) reduces to Q = --i(A+B-
-
A-B+)/(B+
-
B-).
PO)
When Im (Q) = CC)~ < 0, then equation (10) indicates an instability condition. These solutions were the ones examined by REID (1968), CUNNOLD(1969), WHITEHEAD (1967, 1968, 1970) and TSUDA et al. (1966, 1969). We plan to investigate equation (10) in the case when the wavenumbers and cc)= Re (a) are determined by gravity wave criteria. In the high frequency case iQ > A *. In this case our treatment is no longer valid because the neglect of the inertia terms in the equations of motion is no longer justified. This high frequency case appears to be the same as that already studied by RO~ISTERand D’ANOELO(1970) in an effort to explain certain irregularities in the equatorial electrojet. Gravity wave induced perturbations
The existence of internal atmospheric gravity waves in the atmosphere at ionospheric heights is generally acknowledged. Consequently, one would expect these perturbations of the neutral atmosphere to influence the charged particles through two processes : collisions and heating. The collisional effects are obvious, namely that the neutral-electron and neutral-ion collisions impart the periodic gravity wave structure to the ions and electrons. There is an excellent treatment of this in GEORGE&(1967) report, and for our purposes this interaction between the gravity wave and the ionosphere may be described by the use of equation (1). The heating effects, however, will produce variations in the recombination coefficient and the production rate (HOORE, 1969) which will not be quite in phase with the gravity wave. Nevertheless, these two effects-collisions and heating-will produce some initial perturbations of the electrons and ions (viz. equation (2)) and the possibility of the cross-field instability’s occurrence will now be examined after
Atmospheric wave-induced instability in the nighttime E-region
2031
discussing some of the assumptions used. It is assumed that the wavelengths and period of the gravity wave induced variation in the charged particle density are the same as the wavelengths and period of the gravity wave itself. The conditions for this to be a valid assumption are analyzed by KATOet al. (1970) who find that it is strictly valid only for a uniform plasma. Nevertheless, if the medium does not vary appreciably over 42~ then this assumption will not lead to large errors. The data in Appendix I indicates that the plasma does not vary significantly over this distance. Other ~sumptions about the ionospheric parameters are the same as those used in the wind shear theory of E, (AXFORDand CUNNOLD, 1966). The need for the assumption that n1 has the same upward amplification as the gravity wave is not obvious. Experimental data by CONRAD(1969) appears to suggest that it is, however, reasonable to assume this. Let us now turn to the gravity wave parameters that will be used in equation (10). The components of the perturbation wind velocity produced by a gravity wave of amplitude A, are Uh’/X = U,‘lZ = A, exp i(ot - KS
(11)
- K,y - K,x).
Assuming that K, and K NS are real quantities (the perturbations are not damped or amplified in the horizontal direction) then X and 2 are given by the pola~zation relations defined in HINES (1960). However, as U,‘/X = U,‘/Z it should be possible to calculate the effect of a single gravity wave passage by evaluating the divergence term of the continuity equation in terms of the polarization terms and mobilities. We will now do this. The first order perturbed nighttime continuity equation during the passage of a gravity wave is
an,* at
+
V . (n’V)l* = - a’no2 - ao(S1+ + nl-).
(12)
Now if it is assumed that nl* = ~*(~) exp [-i(Ks
+ Kvg + K,z)]
then by using equation (6) we have two coupled first order differential equations for N+(t) and N-(t). Restricting interest to the equation for N+(t) it may be shown after some algebraic manipulation that li;+ + [r,@3+ - B-) + A+ + A- - ~cQ,R,,]$+ +[q(B+A-
- A+B-)
= A,, exp (iwt)[(a,,no - B+)(f-X where
+ A+A-
- aort,(A+ + A-)]N+
+ g) - (io + A- - aon, - .Z?-)(j’+X + g)] (13)
Owing to the large wavelengths of gravity waves, in general the diffusion terms in Af may be neglected.
2032
TOM BEEB and
D. R. MOORCROET
Following HOOKE (1969), a’, the perturbation in the recombination coefficient, will not be taken as zero, but will be found from the temperature dependence of the unperturbed recombination coefficient a’/aO = --6T’/T,
where T = T, + T’ is the electron temperature and it was assumed that it equals that of the neutrals and ions. Then by using the ideal gas equation T’/T,
= p’/p, -
p’/p,, = U,‘(P
-
R)/X
as U,,‘/X = U,‘lZ
= p’/p,P
= p’/p,,R = A,, exp i(wt - K,h -
K,v)
where ti and h are coordinates in the vertical direction and the direction of horizontal gravity wave propagation respectively. Experimental results indicate that 6 = l-2 f 0.2 (BIONDI, 1967), however, it should be realized that as the altitude increases the recombination coefficient changes form and the dissociative recombination coefficient gives place to the attachment like coefficient /?, which has a different temperature dependence: 6 = 1 according to HOOKE (197Oc). Thus, g = -a”o&z,2( P -
R)
during the night. We have assumed a = 10-13(T/300)-1.2 m3 s-l. Equation (13) may be solved in the same way as problems involving mechanical vibrations. The solution is N* = C, exp (iQ,t) + C, exp (iQ,t) + C, exp (if&) where c, =
aon AJ-(
-
B+)(f X + g) - (io + A- - q,no - B+)(f+X + g)]
q(B+A- - A+&) + A+A- - aono(A+ + A-) + io[l;l(B+ - B-) + A+ + A- - 2aono]
o2 I
C, and C, are arbitrary constants. 51, and s1, represent the low and high frequency solutions of IRgiven in equation (9). Transient response of the ionization
The ‘transient’ response of the ionization may be seen to be equivalent to the gradient drift effect in both the stable or unstable case, and we shall continue to use the word transient in analogy with mechanical vibrations, even in the unstable case when the response is not really transient. The resulting irregularities will be greatest when their phase velocities match the phase velocity of the inducing gravity wave. As we know gravity waves are long period waves, we are interested only in the low frequency case given by equation (10). In this case (equivalent to A,, = 0), the nature of the perturbation in u’ is immaterial. It is possible to use the continuity equation for neutral particles, the equations of motion and the adiabatic equation of state to obtain a dispersion relation for the atmospheric waves.
Atmospheric
wave-induced
instability
in the nighttime E-region
2033
It is given by (HIKES, 1960): w* - (t’%a(K~~+ J&z) + (r -l)gaKJ&2 + ~~$~Z~~ = 0
(14)
where o is the angular frequency of the gravity wave. It will be assumed that o = Re(sZ) and that the gravity wavenumbers are the same as the wavenumbers of the ionization perturbations. y is the ratio of specific heats, g is the acceleration due to gravity and c is the speed of sound. Equation (14) when broken into real and imaginary parts gives, for k, # 0: cd -
dc2(kh2 + k,” + L/4H2) + (y -
l)g2kh2= 0
(15)
and Im (KJ = yg[2c2 = 1/2_Ei
(16)
where Lf is defined by equation (16) as the scale height of the neutral atmosphere. What we now proceed to do is to find values of kh and k, that satisfy equation (15) and also the real part of equation (10). These values can then be substituted into the imaginary part of equation (10) to solve for the oi of the ionization. When w1 < 0 it represents an instability in the ionization and corresponds to the gradient instability. For simplicity only eastwest propagation waves will be considered. Though the computations will be done in the equatorial case of zero dip angle, the effect of latitude variations will be discussed in a later section. Approximate relations can be found for internal gravity waves. Provided Ice2 > 1/4ris, then by defining wp2= (y - l)g2/c2. The dispersion relation becomes oat;:Y2 = (@,a - w2)k,,2. With the added condition that w < g/c (i.e. at low frequencies) w2k‘D 2 = og2kb2.
(17)
3. RESULTS: GRAVITYWAVE-INDUCEDEFFECTS HINES(1960,1964) and PITTEWAYand HIXES (1963) point out that all the values of k, and h+,are not possible. Viscous damping dictates a minimum vertical wavelength for internal gravity waves. Another atmospheric feature that will limit the allowed values of k, and kh is gravity wave reflections due to the variations of temperature with height. RIXES (1960) has shown that gravity waves with periods less than 8 mm are reflected at an altitude of 54 km and that gravity waves with kv < 9 x 1O-5 m-l are reflected at 79 km. As it seems most likely that the gravity waves found in the E- and lower F-regions have had their energy propagate up from lower heights (GEORCES,1967; HINES, 1960,1963) these parameters will also supply limits on the allowed values of k_ and k*. Furthermore, the sign of k, is dependent on the direction from which the energy that rna~ta~ the gravity wave is supplied. Therefore, if the group velocity-which represents the direction of energy propagation-is upward (positive) then the phase velocity is downward and k, is negative, although it should be noted that the horizontal components of energy and phase velocity are in the same direction.
2034
Toad BEER and D. R. MOORCRO~
Equation (10) which represents the transient solutions to equation (13), was solved for heights from 80 to 160 km in the case when k, < 0. This was done by firstly taking the maximum allowed magnitude of k, based on viscous dissipation criteria. Now as equation (10)is a function of km,k, and the electric fields-when the electric fields are held constant then there are various values of L, that produce the same (u when the set (&, k,) is substituted into both equation (15) and the real part of equation (10).Of these values of & the only ones of interest are those falling in the region of allowed gravity wavenumbers. This procedure is repeated by decreasing the magnitude of k, and finding the ic* value that satisfies equation (Ifif-which is necessary if the perturbation is due to a gravity wave-and equation (10). These values of k, and k, are depicted in Figs. 1 and 2 in the case of westward propagating waves (k, < 0) and in Fig. 3 for eastward propagating waves. For each (k$, k,) set there corresponds a unique value of CL+ This value is found from the imaginary part of equation (10) and is dependent on the altitude and the electric fields as well as on the wavenumbers. So each point on the lines of Figs. l-3 represents a unique value of wi. If oi -C 0 it represents an instability in the ionization, and it will be shown that it corresponds to the gradient instability. To arrive at the solutions, depicted by the lines in Figs. l-3,four sample field configurations were tried : (1)E,, = (2)&m = (3) J%, = (4) JL! =
lmV/m (eastward) - 1mV/m (westward) 0 0
E,, = E,, = E,, = E@‘,,=
0 0 lmV/m (upward and northward) lmV/m (downward and southward)
and it was found that only in the case of the westward directed field did the solutions to equation (10) fall in the permitted region of the k, - k, diagram. As the electric field is westward only during the night, gravity waves will be unable to generate the -Id’ED,=-0.001V/m Ea, = 0 -I
0-Z -
Fig. 1. Wavelen* which produce the gravity wave resonanceeffect in the case of westward and downwaxd directed gravity wave phtlse propagation. The area representedby the box is expanded in greater detail in Fig. 2. The star represents the dominant gravity wavelength at 90 km. All wavenumbers are in m-l.
Atmospheric wave-induced instability in the nighttime E-region
2036
--2x1
-t
-10-S
-10-4
-lo-’
kx
Fig. 2. The same results a~ shown in Fig. 1 but on a larger scale. The 100 km results are also now included. Reflection levels are shown aa broken lines and wavelengths below and to the right of these do not exist in the E-region.
Fig. 3. Wavelengths which produce the gravity w&ve resonanceeffect in the case of eastward landdownward directed gravity wave phase prop~&tion.
instability during the day. Thus, this mechanism can provide a theory for the formation of nighttime E,.
gradient
Discussion
To obtain a ‘feel’ for the resuIts from equation (10) let us first of all consider the simplest quantitative ease. This occurs at heights above 130 km when the ion and electron Hall mobilities are almost the same and more than an order of magnitude greater than the Pedersen mobilities. In this case it is reasonable to start by taking A+ = A- so that equation (10) becomes Q = iA*. Now from the definition of A*
2036
TON BEER and D. R. MOORCROPT
This expression was obtained by neglecting the Pedersen mobilities (which are smaller than the Hall mobilities), the diffusion terms (which are small in the case of large wavelengths) and the variations of the velocities with altitude (which is a reasonable fLrst approximation). Then because lk,l > Ilc,l only the recombination term and the --ik,puaE, term remain significant and these are the terms in equation (18). For east-west propagating gravity waves, o = k,j+E,,
= k,pHEO, cos 8
by using equations (10) and (US), and an expression for a+ can also be obtained. The validity of these expressions can be checked by comparing the numerical results obtained through their use with the curves of Figs. 2 and 3 which give both w and oi values for each value of k,. Typical values of oi are tabulated in Table 1 where the parameter [ = (2a,n, - oJ/~E,I, These values show that equation (18) is not accurate enough to use as an approximation for wi. Now equation (17) provides a useful approximation for o when k, 2 ,> 1/4H2,so that in the low frequency case one may use (18) and (17) as an approximation for w in the east-west geometry used for the computations. o = o,kJk, = k,/_+E,,, cos 8.
(19)
In the general case this becomes w = ogk,,/k,,= ,uaE,,,(k, cos 8 + kNs sin 0) where ~2 = (y - l)g2/c2. Equation (19) g ives the allowed values of kmand k,. It is a quadratic in k, which agrees with the shape of the curves of Figs. l-3 at the dip equator. Above 130 km equation (19) as it stands provides an excellent approximation to these curves. At lower heights when ,u~+ # ,u~- effective values of ,uH can be obtained by using the curves of Figs. 1 and 2 in conjunction with equation (19). These effective values of puRare listed in Table 2. Each curve in the figures is characterized by values of (~1~)~~~ and T, the atmospheric temperature. T was obtained from the CIRA 1965 Standard Atmosphere. Special note should be made that since o/k, < 0, only westward directed fields will satisfy (19) regardless of the direction of Ic,. Interpretation Now that an analytic expression for o has been obtained, it would be advantageous to have one for mtas well. In this case we need to study Ah more closely. It may be recalled from Section 2 that A* represents the coefficients of N+ in the perturbed continuity equation. The N* terms arise from the perturbation electron the density n1 so that by examining coefficients of n, in the perturbed continuity equation, Ah may be considered as being A* = 2a,n,, + VO*. Vn,/n, + V . V,,+ where the V . V,,* term represents wind shear effects and will be dealt with later on. Now by using an exponential form for the perturbation electron density Vn,/n, = K, the wavenumber vector. So dropping the V . VO*term from the expression for A*
Atmospheric wave-induced in&ability in the nighttime E-region
2037
Table 1. V~Iuea of [ Height (km) 80 90 100 110 120 130 140 160
-3
x 10-2
Vertical wavenumber (m-1) x lo-3 -1.9 x 10-S -4.7 x 10-a
-7.5
-0.6
-0.006 3.43
0.025 1.10 4.7 -0.9 0.34 0.92 0.61 -0.20
0.024 1.67
-1.8
x lo-’ 0.023 -2.09 2.6 1.15 0.83 0.67 0.61 0.6
Weatwaxd phase propagation k, < 0 80 90 100 110 120 130 140 150
-0v5
-0.008 -2.90
0.023 -0.17
0.026 0.43 0.9 6.76 2.82 1.64 0.61 1.09 Eaatwaxd phase propagation k, > 0
0.023 0.48 1.86 6.34 1.51 0.88 0.61 1.70
Table 2 Altitude (km)
(PI?z/%?ff
80 90 100
63.26 2.025 x IO3 -
1()2.88 104.44
-
110 120 130 140 150 160
1.83 2.11 2.18 2.29 2.15 2.05
106.46
105.60 105*@ 105.74 105.74 105.74
equation (10)may be rewritten as 32 = -2a,n,
(Pa)ett
+ i
.
IT,--B+ B+-
x x x x x x
IO4 104 IO4 IO4 104 104
- . K
B-
V$B1 ’
By considering the magnitude of each term in the defining equation of B* it may be shown that
B+ - B-M (pp++ pp-)no(k2 + k.7 + pu,-@,2 which is always a real quantity. The terms for B+ and B- were obtained by neglecting any variation in p in the z
TOM BEER and D. R. MOORCROPT
2038
direction. This seems reasonable for a tist approximation. Then as Im (K,,) = 0 and Re (K,) > Im (K,) the definition of B* leads directly to expressions for B*. To get the expression for B + - B- a few more approximations were used. It was assumed that the pHf and ,uH- were about equal and that ,ur* was negligibly small (this follows the procedure used for A* earlier). It was also assumed that This may readily be seen to be true when rewritten as n0kS2 > Im (K,) &,/&. kZ2 > (scale height of neutral atmosphere) x (scale height of the ionization). Finally one needs to use the fact that p,,- > p 0+. Thus, the growth rate is given by o. __ 2cr n + Im (K) I 0 0 +
&(VO,- Im (B+) -
. L-
Re P+) - Im W,)L+ B+ - B-
voZf Im (B-)) + k,(V,,W-B-
For simplicity, consider large altitudes TIo*+M V,,--. In terms of vertical coordinates wi
=
Re VW
Im (B+) -
where
VoZ+ Im (B-))
1
. (20)
,uH+ = ,uH- and ,u~-- = 0, then
2a,n, + co9 OVo,*. Im (K,)
[Ic,. V,- +
Vo+)z + k, cos O(Vo- -
Vo+)J ;
,+*(kz2
PO-kv2 s%, 8
+ kv2 COST 0)
+
2
j&k;
cos 8
(21)
Then E,, < 0 The Im (K,) . V,, l term represents an E x B drift of ionization. will drift the ionization down and if the perturbations increase in magnitude upwards, as they do in the case of gravity wave induced perturbations, then uoi < 0 because to an observer on the ground it will appear as if the perturbation density at a particular height is increasing. The inclusion of this apparent instability due to the drift motion has led WHITEHEAD (1970) to doubt whether LC)~ < 0 really implies instability if the amplitude of the perturbation varies spatially. He provides an example of this by a damped electromagnetic oscillation whose amplitude decreases in the positive x direction. This is known to be a stable system. When this is viewed by someone moving towards the negative x direction, the observer sees an instability, yet the system is stable. Now the gravity wave perturbations will be something like this. In fact, what happens is that the viscous dissipation leads to a height at which the amplitude of a gravity wave with a given k, and k, is a maximum. Below this height the amplitude increases as exp (v/2H), above it, it is damped through viscous interaction. Therefore, below this particular height a downward drift of the ionization as a whole will appear to a ground based observer as if the irregularities at any one height are growing. Whether this is a ‘real’ instability or not is a moot point. After all, even in the case of the temporally varying perturbations a non-linear analysis reveals that they only grow for a certain length of time (called the stabilization time) and then start to fade (TSUDA et al., 1969; WILLIAMS and WEINSTOCK, 1970), yet we are certainly justified in calling them unstable. We, therefore, believe that because the Im (K,) . V 0a* term represents an effect that will appear to a ground based observer as if it were a growing irregularity at a particular height it should be included in the analysis and this is what has been done.
Atmospheriowave-inducedinstabilityin the nighttimeE-region
2039
The last term in equations (20) and (21) represents the gradient instability. By last term we mean the sum that is in square brackets and the coefficient by which this sum is multiplied. This coefficient includes expressions for the horizontal and vertical wavelengths and for the gradients in ionization density. The k,z(‘V,,-- - Vo+)a = -k,2(pclp+ + pp-)E,, term describes the simple situation outlined in Section 2 and it assumes there are no upward phase variations, For westward directed fields this term will always be positive (inducing stability) for upward gradients of ionization and negative (aiding the instability) for ionization that decreases upwards. Let us now isolate the situation represented by the other part of the last term of (21) which arises because k, # 0 and is characterized by
When a~*~~~ > 0 and E,,, < 0 then if &/lEz< 0 it represents a stable solution and if l&/i& > 0 then it denotes instability. But as k, < 0, then for k% < 0 (westward propagation) there is a manifestation of the gradient instability when the ionization increases upwards. The mode of operation for this instability may be visualized by means of Fig. 4.
+
7
trough
0
Fig. 4. One mode of the gradient instability.
Double headed arrows
show the
drift direction.
The greater Hall mobility of the electrons will drift them downwards faster than the ions and this will set up pe~urbation electric fields that drift the ionization crests down and the troughs up. The Pedersen effects that are described in Section 2 will tend to oppose this, but provided that I& cos e(pu,- - /++)I > I&(/++ + ,+-)I (22) then the z direction drifts of (21) are dominant and instability may occur in regions of upward increasing backgro~d io~zation. Equation (22) is true in general except at very high altitudes (above 160 km) and at large dip angles. As oi may be represented by 0,. = 2a 0n0 - 5 jE,,I where wind shear effects are neglected, Table 1 lists the computed values of E from Figs. l-3 where Im (K,) = 1/2H.
2040
TOM BEEB and D. R. MOORCROFT
Field-aligned and non-aligned iwegularitiea Below 100 km the Pedersen mobility of the electrons is greater than the Pedersen mobility of the ions. Seeing as (,u~),,~~+ p,+ it appears that in the D-region, even when there exist field aligned irregularities it will be the electrons--moving across the field lines in the field aligned case-that move so as to attempt to neutralize space charge effects. This leaves the ions free to move under external influences and it appears that in the D-region even field aligned irregularities move with the ion velocity. Eflect of E,, In an earlier section of this paper it was pointed out that the gravity wave induced gradient instability could only operate in the presence of westward directed electric fields. To simplify the graphs presented in Figs. 1-3 the effect of the electric field in the x direction was ignored because it plays a subordinate role to the east-west directed field. This is because the gradient instability is a mechanism primarily dependent on the realignment of the ionization in the vertical direction. As the electron Hall mobility is greater at all heights than any other transport coefficient, it is clear that for all field-aligned cases in which the irregularities move with the electron drift velocity, it would take an exceptionally strong electric field in the z direction to exceed the Hall effects of the E,, field. A similar situation applies when the irregularities are non-aligned except at D-region heights near the geomagnetic poles. In fact, it is possible to obtain analytic estimates of the effect of non-zero E,, upon both real and imaginary parts of the angular frequency. 1. Effect on W. As o M k . VO-Re (I?+) - k . V,,+Re (B-)/(B+ = (P,%-&~
-
+ (~H+b%
+ rUp+WL
PUp-vhz
-
/+-b%,
+ ~p+b%,
-
B-)
-
~~-k-G,z)Re
(B+)
-
kz~n+Eol)Re
(B-)
at high heights the pH*k eE ,-,*term dominates and E,, will affect the graphs of Figs. l-3 when kJ$,, > k,E,. At lower heights E,, becomes important if
-,++k&d~up+PP&~~) 2 -ruck.&&~+nok~2 +~o+nok,2) +ruo+@3 1 I[+hd2%z(~p+n -/++k2U+-@,2
[
+ ~n,lc,~)
at low dip angles this means C/+-kz +
m-lc,lE,z 3 [rup-k - PH-WL
has to be quite large However, the pH-k$,,$ term will once again dominate and EOLl to affect the aforementioned results. 2. Effects on q. By multiplying all the relevant terms out it may be shown that when E,, # 0 then wi = 2a,n, - E IE,,l Im (U%,(~~+~0-
- rup-ruo+)lc,2 - 2Eozk&.(/+z-/++ +
+ (I,++
+
rUp-P,2
+
PO-
. k,
1 an, P~+PP-1
;
z 0
Atmospheric wave-induced instability in the nighttime E-region
2041
at low dip angles this becomes
and so exceptionally large gradients or very large values of E,, are required to affect cot. However, at large dip angles the E,, may make a substantial contribution to coi. Seeing that at low heights r-sPC M
Cl*+*
rrsp- *
ii cco-
and
,u~- > pa+:
But as Im (K,) = yg/2c8 > 0, in order to aid the instability E,, must be negativei.e. downwards and southwards. It should be realized that pa+ increases with height, and so the E,, term would probably have the greatest effect around 120 km altitude. However, the contribution from E, is still the greatest unless E,, is very large. 4. COx?CLUSION
We have analyzed the ~rm~ible wavenumbers of the gravity waves that can produce the gradient ~stab~ity and their resulting growth times. However the effects of this instability will be discussed elsewhere (BEER and MOORCROFT, 1972) as will the effects of the steady state response which produces B-region partial reflections (BEER, 1972a) and could, under special circumstances, produce a spatial resonance mechanism (WHITEHEAD,1971a; BEER, 1972b) in the ionosphere. AokwZedgemW--‘J!hi work was supported by the National Research Council of Canada. And also in part, by N.A.S.A. Grant NGR-05-009-076.
REFERENCES W.1. and C!UNNOLD D.M. T. BEER T. BEER T. and MOOIWROXT D. R. BLUTYILE L. J. et&.
1972 1965
CHAPMAN&
1966
CHIMONAS G. CHXONASGT. CIRA CUNNOLD D.M. FABLEY D. T. et d.
1969 1971 1965 1969 1970
AXPORD BEER
J. Atmo8& Tew. Phys. s4, 2046. Proo. 2nd. Int. Symp. Equatorial Aeronomy (Edited by F. DE MENDONCA). C.N. Pq-CNAE-LAFE, Sao Paulo, Brazil. NWVO Cimento Swppl. Ser. 10, 4, 1385. J. geophys. Res. 74,4091. J.gvhy8. Res. 76,4578. North-Holland, Amsterdam. J. ge~hy8. Re8.74, 5709. J, g~hy8. Res. 75,719Q.
2042 HAERENDEL
TOM BEER and D. R. MOORCRO~
G.etal.
1967
HiNES C. 0.
1955
HINES C. 0.
1956
HINES C. 0.
1960
HINES C. 0.
1963
HINES C. 0.
1964
HINES C. 0. and HOOBE W. H.
1970
HOE F. C.
1963
HOOKE W. H.
1968
HOOKE W. H.
1969
HOOKE W. H.
1970e
HOOKE W. H.
1970b
HOOKE W. H. KAISER T. R. et al. KATO S. KATO S. and MATSUSHITA S. KATO S. et aZ. KNOX F. B. LINSON L. M. and WORK J. B. MAEDA H. MAZDA K. et al. MATSUSHITA S. P~EWAY M. L. and HINES C. 0. PO~REBNOY V. N. and FATKULLZN M. N. RAO P. B. and RUSSELL W. H. REID G. C. REP G. C. RISHBETH H. and GARRIOTT 0. K.
19700 1969 1965 1969 1970 1964 1970 1957 1963 1969 1969 1969 1971 1968 1970 1969
RO~ISTER A. and D’ANOELO N. SATO T. et al. SIMON A. TSEDELINA YE. TSUDA T. et al. TSUDA T. et al. WHITEHEAD J. D. WHITEHEAD J. D. WHITEHEAD J. D.
1970 1968 1963 1965 1965 1969 1967 1968 1970
WHITEHEAD J. D.
1971a
Pl4md. Space sci. 15, 1. J. Atmmph. Terr. Phys. 7, 14. J. Atwwsph. Terr. Phys. 9, 56. Can. J. Phys. 38, 1441. Q. J. R. met. SOG.89,1. Can. J. Phys. 48, 1424. J. geophya. Rea. 75, 2563. Phy8. Fluids 6, 1184. J. Atnwsph. Tern. Phya. 30, 795. Planet. Spe Sci. 17,749. J. geophy8. Res. 75,5535. J. geophys. Res. ‘95, 7229. J. geophys. Rea. 75,7239. Planet. Space Sci. 17,519. Spe Sci. Rev. 4,223. J. Atmoaph. Terr. Phys. 31, 193. J. geophys. Res. 75,2540. J. Atmosph. Terr. Phy8. 26, 239. J. geophys. Res. 75,3211. J. Geomag. Geoekct. 9,86. Phys. Rev. Lett. 11,406. Radio Sci. 4, 771. Can. J. Phys. 41, 1935. Geomagra. & Aeron. 9, 456. Radio Sci. 6, 221. J. geophys. Res. 78, 1627. Planet. Space Sci. 18, 1105. Ionospheric Phyaks. Academic Press, New York.
WHITEHEAD J. D.
1971b
WILLIAMS R. H. and WEINST~CX J.
1970
J. geophys. Re8. 75, 3879. Radio Sci. 3, 529. Phy8. Fluids 6, 382. Geowqn. & Aeron. 5, 525. Radio Sci 1,212. J. geophys. Res. 74, 2923. J. Atmoaph. Terr. Phys. 29, 1285. J. Atmosph. Terr. Phys. 36, 1663. J. Atmosph. Tern. Phys. 32, 1283. J. geophys. Res. 76, 238. J. geophys. Rea. 76, 3116. J. geophys. Rea. 75, 7217.
Reference is also made to the following unpublished material BEER T.
1971
Ph.D.
Thesis,
University
of Western
Ontario, London Canada. BIONDI M. A.
1967
DASA Reaction Rate Hand Information
and
CONRAD J. C.
1969
Santa Barbara. Cornell University
GEOR~ES T. M.
1967
IER-67/ITSA-54. Report.
U.S.
ook. DASA
Andy&
Center,
CRSR 348. ESSA
Government
Office, Washington,
D.C.
Technical Printing
Atmospheric wave-induced instability in the nighttime E-region AFPENDIX
2043
I
Ionospheric transport coefficients(m.k.s. units)
PH-80
85 90 95 100 105 110 116 120 125 130 135 140 145 150
l.OE8 8.538 4.OE9 l.OElO 1.8ElO 1.2ElO l.OElO 5.OE9 3.OE9 2.OE9 1.2E9 l.lE9 l.OE9 l.lE9 1.5E9
1.80E5 7.5134 3.llE4 I.2234 5.2833 2.3233 1.07E3 5.1232 2.7732 I.5832 1.OOE2 6.7931 4.8931 3.5231 2.71El
2.7536 1.14E6 4.7335 1.91E5 8.4534 3.8734 1.8734 9.7333 5.6733 3.4833 2.3433 1.6633 I.2433 9.1832 7.2832
1*84El 4.41El 1.07E2 2.7432 6.4032 I.4733 3.1533 6.1233 9.0733 9.9233 8.4633 6.5133 4.9233 3.6233 2.8033
5.7033 2.5633 1.07E3 4.3632 1.9232 8.81El 4.2531 2.22El 1.29El 7.9230 5.3230 3.77 2.82 2.09 1.66
1.69E - 2 9.71E - 2 5.68E - 1 3.75 2.05El 1.08E2 5.0932 2.0933 5.8033 1.13E4 1.5334 1.7634 1.8734 1.9334 I.9634
1.8234 1.9734 1.9934 2*00E4 2.00E4 2.00E4 2*00E4 2.00E4 2.00E4 2.OOE4 2.00E4 2.00E4 2.00E4 2.00E4 2.00E4
DP+ 2.953 - 1 7.07E - 1 1.71 4.68 1.15El 2.90E1 6.8231 1*60E2 2.7832 3.5732 3.4832 2.9532 2.4332 1.91E2 1.5732
DP9.14E + 4.llE 1.72E 7.45 3.45 1.74 9.21E 5.79E 3.953 2.853 2.19E 1.71E 1.39E l.llE 9.323 -
1 1 1
1 1 1 1 1 1 1 1 2
Atmospheric wave-induced instability in the nighttime E-region TOM BEER* and D. R. MOORCROFT Physics Department, University of Western Ontario, London, Canada (Received 3 November1971; ti revieedform 16 March 1972)
Abstract--This article examines the perturbed continuity equation when the perturbations are the result of an internal atmosphericgravity wave in the E-region. The transient responseof the ionization ia interpretedas the gradient instability and the values of the vertical and horizontal wavenumbers that will induce it are plotted for various heights. Only in the presence of westward directed electric fields, which are believed to occur only at night, will the gravity waves induce the gradient instability. Approximate analytic expressionsare obtained for the permitted wavenumbers as well ae for the instability growth times. In the course of this andysis it ia shown that in the D-region all irregularities,even those that are field-aligned, will tend to move with the ion velocity.
1. INTRODUCTION IN 1963, A. Simon and I?. C. Hoh independently published papers in which they considered the laboratory applications of a new type of plasma instability. This particular instability occurs in a non-uniform plasma that contains gradients in ionization density, under the influence of perpendicular electric and magnetic fields. The ionization moves up or down due to the influence of space charge electric fields that are set up by the background field’s separation of the oppositely charged particles. Because of these conditions this mechanism has been called a ‘cross field instability’, ‘gradient instability’, and ‘drift instability’. Ionospheric workers were quick to utilize this ‘gradient in&ability’ in a myriad of ways. Some used the instability a,s a source mechanism for irregularities in the P-region (REID, 1968; CUNNOLD, 1969; WILLIAMS and WEINSTOCK, 1970), or to explain the striations in artificial plasma clouds formed by rocket released barium (LINSON and WORKMAN, 1970; REID, 1970; RAO and RUSSELL, 1971). MEADA et aZ. (1963) and KNOX (1964) attempted to explain the irregularities in the equatorial electrojet by means of the gradient instability, though it is nowadays believed that these are manifestations of another plasma instability called the two stream instability. Nevertheless, there are so-called Type II irregularities which cannot be explained in terms of the two stream instability. ROGISTER and D’ANOELO (1970) have proposed a high frequency counterpart of the gradient instability as an explanation. WHITEHEAD (1971b) has also proposed a gradient instability based mechanism. TSUDA et al. (1966) were the first to use the gradient instability in relation to midlatitude sporadic-E (E,), and were followed by WHITEHEAD (1967, 1968, 1970) and SATO et al. (1968). This series of papers demonstrated that the gradient in&ability mechanism was unable to produce thin layers, but it is still believed (TSUDA et al., 1969) that it could be a mechanism for the non-blanketing diffuse type of E,. In most of these cases no explanation was offered as to the mechanism that produces the initial non-uniformity in the plasma. WILLIAMS and WEMSTOCK (1970) and FARLEY et al. (1970) point out that a background noise level of weak irregularities due to * Present address: Physics Department, University of Ghana, Legon, Ghana. 2026
2026
TOM BEER and D.
R. MOORGROIQ
random thermal ~uctuations will always exist. CHXXONAS (1969) assumes small scale perturbations to the background and shows that in the presence of a constant background wind, field aligned Earn.patches could form under the action of the gradient instability. In this article we find the resulting instab~itios when gravity waves perturb the ionization through neutral-charged particle collisions (HINES,1955,1956; GEORGES 1967) and variations in recombination and photoionization rate (HOOKE, 1968, 19704 b, c). Occasionally various authors have pointed out that a gravity wave could produce resonances with the ionization. HINES (1955, 1956) tried to use them to explain travelling disturbances in the P-region but mistakenly neglected the background electric fields. KATO et cd. (1970) showed that a gravity wave produces two irregularity waves. One moves with the same velocity and period as the gravity wave and the other, which can only exist in very non-uniform plasmas, travels with the ion veIo&y along the non-u~ormity, HINES and HOOKE(1970) considered the case when the ionization irregularities affect the gravity wave, though this only happens to very long period waves. WHITEHEAD (1971%)considered a spatial resonance when the irregularities drift with the gravity wave velocity. He suggested that this may explain the exceptionally high electron densities often observed in wind-shear E,layers. CHIMONAS (1971) extended the spatial resonance concept by allowing for ion trapping and downward trausport in height varying tidal winds. Thus he claimed that an interaction between the tidal winds and gravity wave winds produces E8. In a subsequent article we shall examine the elects of the gravity wave induced gradient drifts in the E-region. 2. THEORY 0~ THE GRADIENT INSTABILITY ~~t~~~t~~
This section deals with the mathematical framework of the gradient instability. It encompasses the work of the above-mentioned authors but extends the analysis in four irnpor~t ways: (1)It allows for complex wavenumbers in the horizontal and vertical directions; (2) The effects of atmospheric gravity waves are examined, by assuming that the horizontal and vertical wavenumbers of the gravity waves and of the ionization perturbations are identical and by including a sinusoidal gravity wave in the equations of motions; (3) The amplitudes (Nt- and N-) of the sinusoidal perturbations in the electron and the ion concentrations (nl+ and n,-), are taken as being unequal. This leads to both a low frequency and a high frequency case of the gradient instability; and (4) Perturbations (cI’) in the recombination coefficient (a) during the passa’ge of a gravity wave are not neglected. The first part of the section follows the previous developments of the theory and considers the gradient instability as zLneffect upon pre-existent ionization perturbations. The second part of this section presents an original attempt to describe the effects of gravity waves upon the io~zation in the presence of both background electric fields and space-charge fields set up by movement of the induced io~ation perturbations. It is shown that the perturbed continuity equation may be linearized
Atmospheric wave-induced instability in the nighttime
E-region
2027
and treated in a manner analogous to that used in problems involving mechanical vibrations. The ‘transient’ response is interpreted as the cross field gradient drift (the gradient instability in the unstable case) and the ‘steady state’ response is discussed elsewhere (BEER, 1971, 1972a, b). To begin, it may be instructive to consider a greatly simplified model of the gradient instability mechanism. Once the charged particle concentration has been perturbed by a sinusoidal variation, the drifts of the ions and electrons wiIl not be identical, In the case of upward baok~ound electric fields this is a consequence of the difIeriug Hall mobilities. In the case of horizontal electric fields it arises from the opposite directions of the Pedersen mobilities. This limits the instability to the E-region because in the F-region collisions are so infrequent that the Hall mobilities are identical and the Pedersen mobilities are negligible. As an example, consider the case when n,, the background ionization eoncentration increases upwards. Once the ions and electrons have been separated, perturbation electric fields appear. These small scale fields produce El x B drifts which will then carry the enhanced regions downwards and the depleted regions upwards so that they will both appear to grow against the background density. Obviously if either the electric field or the steady density gradient were reversed in direction, these probations would tend to disappear and we would have a stable situation. If both are reversed the situation is once again unstable. In the subsequent analysis, the magnetic field is wholly in they direction whilst the z-axis which is perpendicular to the Bfield, is directed upwards, thex-axisis horizontal to the east and 13is the dip angle in the y-z plane. v and h will be used as subscripts and variables to denote the components of variables in the vertical and horizontal directions respectively. For wave motions in the ionosphere that are not of a high frequency, the inertia and gravitational effects are negligible. In this case the charged particle velocity V* with respect to the neutral wind U (measured relative to the ground) in the presence of an electric field E and a magnetic field B is given by
PP&
0
(1)
0 po”
0
ip~_” 0 (see, for example, RISHBETHand GARRIOTT, 1969, Chap. IV) where in the case of double signs, the topmost is applicable to ions and the bottom one to electrons. D represents the diffusion coefficient and ,uthe mobility. The subscripts P and H refer to the Pedersen and Hall effects respectively where
(1 (DH’) BP* PP
PH
=
=
(vff2
(v&J2+ (%fP WfVf
Do *
(1 (1 po Do
(v”J2+ (Wff*)2 po
*
2028
Tolvr B~EIXand D. R. Mooausow
lel B/r?&”is the gyrofrequency. given by WE
=
The longitudinal transport coefficients are
and In computing the transport coefficients (Appendix I) the CIRA (1966)model atmosphere was used in conjunction with CEAPMAN’S(1965)expressions for the ion collision frequency Y+ and the electron collision frequency Y-. The electric field in the ionosphere may be estimated by using the dynamo theory of atmospheric tides. MATSUSHITA (1969)has prepared charts that give the direction and mag~tude of the electric field in the dynamo region as a function of local time and dip angle. He shows that at latitudes less than 45" the horizontal fields are eastward from 0400 to 1700local time and westward during the night with maxima at around noon and midnight. 1 mVfm represents a typical horizontal field strength though it increases markedly towards the poles. The E-field in the z direction (per~n~c~ar to the B field in the upward direction) is more latitude de~ndent, but its magnitude is approximately the same as that of the horizontal field. The Sq currents are larger in magnitude during summer than in winter or in the equinoctial months and this seasonal variation becomes more marked as the dip angle increases (WEDA, 1957).The ionospheric electric field perpendicular to B remains reasonably constant with altitude (HAERENDEL et aE., 1967; PO~REBNOYand FATKULLIN,1969) and the electric fields parallel to the magnetic field are negligible.
The particular background ionization profile that is used is tabulated in Appendix I and represents the 0140 local time profile obtained by BLUMLEet al. (1965) over Peru. This is representative of mid-latitude ~ghttime profiles and contains the characteristic nighttime E-region valley between 100 and 150 km. This profile was also used by REID (1968) so that our results may be directly compared with his. There were a number of reasons for choosing the nighttime case, Firstly, it simplifies the mathematics by allowing the production term to be neglected. Secondly, as there are larger gradients at night than during the day one would expect the gradient instab~ity to be more effective during the night and thirdly, once all the relevant calculations had been done, it was found that only at night will the electric field point in such a direction as to permit gradient instabilities to be induced by gravity waves.
Consider the perturbations in 71,E and U such that n = n, + n,; E = F& f E,; TJ = U’; where U is the neutral wind velocity. Take *1 f =N*
exp[@t-
K$-Ksy--
Kg)]
(2)
and assume that the perturbation electric field is derivable from a scalar potential ‘v” so that
Atmospheric wave-induced instibility
in the nighttime E-region
2029
where, in terms of the vertical wavenumber K, and the wavenumber in the horizontal north-south direction KNs K, =KNssin6’+Kecos8 K, = KNs COB0 - K,, sin 8. In general, Q and K will be used when it is possible for the frequency and the wavenumber to be complex and o and k will be used for the real part of these quantities. Previous workers have all assumed U’ = 0. Though this seems surprising at first, it will be shown later in this section that even in the case of ionization perturbation produced by gravity waves when it is possible to assume an analytic expression for U’, the gradient instability condition is independent of U’. So let us consider the situation when U’ = 0. The neglect of the gradient of s2is formally incorrect. This could be included by substituting K, - tiXl/& for K, and Q - dK,/& for R in the derivations. However as a linearized analysis is strictly applicable only for an instant of time then the inclusion of the above terms becomes unnecessary (especially at t = 0) if it is realized that the linearized results obtained can only provide information as to the degree of stability or instability possessed by the ionization. The linearized analysis can only be used as an indicator and is inapplicable for exact quantitative results. The perturbed continuity equation for the nighttime E-region is y
+
v * (nov,)*
+
v * (nlV,)*
= -
wdn,+ + n13
which may be written, by using (l), (2) and (3), as (X! + A*)N*
+ B*V
+ a,n,(NF
-
IV*) = 0
(4)
where n,+ = n,- = n, is necessary to maintain charge neutrality, and
A* = 2a,% -t %*K2 --iK&b+*%,
+ Kz21+ D,,*Kv2 +
tz[
f,@E,,
-PIS*E,,I --iK,[fpp*Eo,
+ pH*E,,J
(5a)
+ ,uH*E,J
and V represents the scalar potential. Also,
pp*noK2 + Kz2) f n,pa*K,2.
(5b)
The perturbation electric field will satisfy Gauss’s law and defining e = absolute value of the electron charge e0 = permittivity of free space then by using equations (3) and (5) it can easily be shown tha.t V = e(N+ - N-)/E,,(K,~ + Kv2 + Kz2).
(6)
2030
TOMBEERand D. R. MOORCROPT
Now equations (4) and (6) provide three linear equations in three unknowns: Nf, N- and V. By eliminating N- one obtains (Xl + A+)iV+ + (B+ - aon,.5,K2/e)V = 0 (iSI + A-)N+
+ (B- -
(ifi + A- -
aon&,K2fe)V
(7) = 0
(9)
where K2 = KS2 + Kg2 + Kz2. Non-zero solutions to this may easily be found by setting the determinant composed of the coefficients of N+ and V equal to zero. Using 11= e/a,K2 this results in a dispersion relation that is quadratic in w. The roots of this are given by 2iQ = q(B- - Bf) - (A+ + A-) +
‘(A+ -
+ 2a,n,
A-)2 + q2(B-
J
-
2q(A- - A+)(B+ + B-) + 4?&n,(B- - B+) + 4Q%,2
B+)2 -
(9)
which represents two solutions : a high frequency case and a low frequency case. By evaluating A* and B* it may be shown that A* is always very much smaller than yB*. Therefore, in the low frequency case (iR + A-)/q < B- and so equation (9) reduces to Q = --i(A+B-
-
A-B+)/(B+
-
B-).
PO)
When Im (Q) = CC)~ < 0, then equation (10) indicates an instability condition. These solutions were the ones examined by REID (1968), CUNNOLD(1969), WHITEHEAD (1967, 1968, 1970) and TSUDA et al. (1966, 1969). We plan to investigate equation (10) in the case when the wavenumbers and cc)= Re (a) are determined by gravity wave criteria. In the high frequency case iQ > A *. In this case our treatment is no longer valid because the neglect of the inertia terms in the equations of motion is no longer justified. This high frequency case appears to be the same as that already studied by RO~ISTERand D’ANOELO(1970) in an effort to explain certain irregularities in the equatorial electrojet. Gravity wave induced perturbations
The existence of internal atmospheric gravity waves in the atmosphere at ionospheric heights is generally acknowledged. Consequently, one would expect these perturbations of the neutral atmosphere to influence the charged particles through two processes : collisions and heating. The collisional effects are obvious, namely that the neutral-electron and neutral-ion collisions impart the periodic gravity wave structure to the ions and electrons. There is an excellent treatment of this in GEORGE&(1967) report, and for our purposes this interaction between the gravity wave and the ionosphere may be described by the use of equation (1). The heating effects, however, will produce variations in the recombination coefficient and the production rate (HOORE, 1969) which will not be quite in phase with the gravity wave. Nevertheless, these two effects-collisions and heating-will produce some initial perturbations of the electrons and ions (viz. equation (2)) and the possibility of the cross-field instability’s occurrence will now be examined after
Atmospheric wave-induced instability in the nighttime E-region
2031
discussing some of the assumptions used. It is assumed that the wavelengths and period of the gravity wave induced variation in the charged particle density are the same as the wavelengths and period of the gravity wave itself. The conditions for this to be a valid assumption are analyzed by KATOet al. (1970) who find that it is strictly valid only for a uniform plasma. Nevertheless, if the medium does not vary appreciably over 42~ then this assumption will not lead to large errors. The data in Appendix I indicates that the plasma does not vary significantly over this distance. Other ~sumptions about the ionospheric parameters are the same as those used in the wind shear theory of E, (AXFORDand CUNNOLD, 1966). The need for the assumption that n1 has the same upward amplification as the gravity wave is not obvious. Experimental data by CONRAD(1969) appears to suggest that it is, however, reasonable to assume this. Let us now turn to the gravity wave parameters that will be used in equation (10). The components of the perturbation wind velocity produced by a gravity wave of amplitude A, are Uh’/X = U,‘lZ = A, exp i(ot - KS
(11)
- K,y - K,x).
Assuming that K, and K NS are real quantities (the perturbations are not damped or amplified in the horizontal direction) then X and 2 are given by the pola~zation relations defined in HINES (1960). However, as U,‘/X = U,‘/Z it should be possible to calculate the effect of a single gravity wave passage by evaluating the divergence term of the continuity equation in terms of the polarization terms and mobilities. We will now do this. The first order perturbed nighttime continuity equation during the passage of a gravity wave is
an,* at
+
V . (n’V)l* = - a’no2 - ao(S1+ + nl-).
(12)
Now if it is assumed that nl* = ~*(~) exp [-i(Ks
+ Kvg + K,z)]
then by using equation (6) we have two coupled first order differential equations for N+(t) and N-(t). Restricting interest to the equation for N+(t) it may be shown after some algebraic manipulation that li;+ + [r,@3+ - B-) + A+ + A- - ~cQ,R,,]$+ +[q(B+A-
- A+B-)
= A,, exp (iwt)[(a,,no - B+)(f-X where
+ A+A-
- aort,(A+ + A-)]N+
+ g) - (io + A- - aon, - .Z?-)(j’+X + g)] (13)
Owing to the large wavelengths of gravity waves, in general the diffusion terms in Af may be neglected.
2032
TOM BEEB and
D. R. MOORCROET
Following HOOKE (1969), a’, the perturbation in the recombination coefficient, will not be taken as zero, but will be found from the temperature dependence of the unperturbed recombination coefficient a’/aO = --6T’/T,
where T = T, + T’ is the electron temperature and it was assumed that it equals that of the neutrals and ions. Then by using the ideal gas equation T’/T,
= p’/p, -
p’/p,, = U,‘(P
-
R)/X
as U,,‘/X = U,‘lZ
= p’/p,P
= p’/p,,R = A,, exp i(wt - K,h -
K,v)
where ti and h are coordinates in the vertical direction and the direction of horizontal gravity wave propagation respectively. Experimental results indicate that 6 = l-2 f 0.2 (BIONDI, 1967), however, it should be realized that as the altitude increases the recombination coefficient changes form and the dissociative recombination coefficient gives place to the attachment like coefficient /?, which has a different temperature dependence: 6 = 1 according to HOOKE (197Oc). Thus, g = -a”o&z,2( P -
R)
during the night. We have assumed a = 10-13(T/300)-1.2 m3 s-l. Equation (13) may be solved in the same way as problems involving mechanical vibrations. The solution is N* = C, exp (iQ,t) + C, exp (iQ,t) + C, exp (if&) where c, =
aon AJ-(
-
B+)(f X + g) - (io + A- - q,no - B+)(f+X + g)]
q(B+A- - A+&) + A+A- - aono(A+ + A-) + io[l;l(B+ - B-) + A+ + A- - 2aono]
o2 I
C, and C, are arbitrary constants. 51, and s1, represent the low and high frequency solutions of IRgiven in equation (9). Transient response of the ionization
The ‘transient’ response of the ionization may be seen to be equivalent to the gradient drift effect in both the stable or unstable case, and we shall continue to use the word transient in analogy with mechanical vibrations, even in the unstable case when the response is not really transient. The resulting irregularities will be greatest when their phase velocities match the phase velocity of the inducing gravity wave. As we know gravity waves are long period waves, we are interested only in the low frequency case given by equation (10). In this case (equivalent to A,, = 0), the nature of the perturbation in u’ is immaterial. It is possible to use the continuity equation for neutral particles, the equations of motion and the adiabatic equation of state to obtain a dispersion relation for the atmospheric waves.
Atmospheric
wave-induced
instability
in the nighttime E-region
2033
It is given by (HIKES, 1960): w* - (t’%a(K~~+ J&z) + (r -l)gaKJ&2 + ~~$~Z~~ = 0
(14)
where o is the angular frequency of the gravity wave. It will be assumed that o = Re(sZ) and that the gravity wavenumbers are the same as the wavenumbers of the ionization perturbations. y is the ratio of specific heats, g is the acceleration due to gravity and c is the speed of sound. Equation (14) when broken into real and imaginary parts gives, for k, # 0: cd -
dc2(kh2 + k,” + L/4H2) + (y -
l)g2kh2= 0
(15)
and Im (KJ = yg[2c2 = 1/2_Ei
(16)
where Lf is defined by equation (16) as the scale height of the neutral atmosphere. What we now proceed to do is to find values of kh and k, that satisfy equation (15) and also the real part of equation (10). These values can then be substituted into the imaginary part of equation (10) to solve for the oi of the ionization. When w1 < 0 it represents an instability in the ionization and corresponds to the gradient instability. For simplicity only eastwest propagation waves will be considered. Though the computations will be done in the equatorial case of zero dip angle, the effect of latitude variations will be discussed in a later section. Approximate relations can be found for internal gravity waves. Provided Ice2 > 1/4ris, then by defining wp2= (y - l)g2/c2. The dispersion relation becomes oat;:Y2 = (@,a - w2)k,,2. With the added condition that w < g/c (i.e. at low frequencies) w2k‘D 2 = og2kb2.
(17)
3. RESULTS: GRAVITYWAVE-INDUCEDEFFECTS HINES(1960,1964) and PITTEWAYand HIXES (1963) point out that all the values of k, and h+,are not possible. Viscous damping dictates a minimum vertical wavelength for internal gravity waves. Another atmospheric feature that will limit the allowed values of k, and kh is gravity wave reflections due to the variations of temperature with height. RIXES (1960) has shown that gravity waves with periods less than 8 mm are reflected at an altitude of 54 km and that gravity waves with kv < 9 x 1O-5 m-l are reflected at 79 km. As it seems most likely that the gravity waves found in the E- and lower F-regions have had their energy propagate up from lower heights (GEORCES,1967; HINES, 1960,1963) these parameters will also supply limits on the allowed values of k_ and k*. Furthermore, the sign of k, is dependent on the direction from which the energy that rna~ta~ the gravity wave is supplied. Therefore, if the group velocity-which represents the direction of energy propagation-is upward (positive) then the phase velocity is downward and k, is negative, although it should be noted that the horizontal components of energy and phase velocity are in the same direction.
2034
Toad BEER and D. R. MOORCRO~
Equation (10) which represents the transient solutions to equation (13), was solved for heights from 80 to 160 km in the case when k, < 0. This was done by firstly taking the maximum allowed magnitude of k, based on viscous dissipation criteria. Now as equation (10)is a function of km,k, and the electric fields-when the electric fields are held constant then there are various values of L, that produce the same (u when the set (&, k,) is substituted into both equation (15) and the real part of equation (10).Of these values of & the only ones of interest are those falling in the region of allowed gravity wavenumbers. This procedure is repeated by decreasing the magnitude of k, and finding the ic* value that satisfies equation (Ifif-which is necessary if the perturbation is due to a gravity wave-and equation (10). These values of k, and k, are depicted in Figs. 1 and 2 in the case of westward propagating waves (k, < 0) and in Fig. 3 for eastward propagating waves. For each (k$, k,) set there corresponds a unique value of CL+ This value is found from the imaginary part of equation (10) and is dependent on the altitude and the electric fields as well as on the wavenumbers. So each point on the lines of Figs. l-3 represents a unique value of wi. If oi -C 0 it represents an instability in the ionization, and it will be shown that it corresponds to the gradient instability. To arrive at the solutions, depicted by the lines in Figs. l-3,four sample field configurations were tried : (1)E,, = (2)&m = (3) J%, = (4) JL! =
lmV/m (eastward) - 1mV/m (westward) 0 0
E,, = E,, = E,, = E@‘,,=
0 0 lmV/m (upward and northward) lmV/m (downward and southward)
and it was found that only in the case of the westward directed field did the solutions to equation (10) fall in the permitted region of the k, - k, diagram. As the electric field is westward only during the night, gravity waves will be unable to generate the -Id’ED,=-0.001V/m Ea, = 0 -I
0-Z -
Fig. 1. Wavelen* which produce the gravity wave resonanceeffect in the case of westward and downwaxd directed gravity wave phtlse propagation. The area representedby the box is expanded in greater detail in Fig. 2. The star represents the dominant gravity wavelength at 90 km. All wavenumbers are in m-l.
Atmospheric wave-induced instability in the nighttime E-region
2036
--2x1
-t
-10-S
-10-4
-lo-’
kx
Fig. 2. The same results a~ shown in Fig. 1 but on a larger scale. The 100 km results are also now included. Reflection levels are shown aa broken lines and wavelengths below and to the right of these do not exist in the E-region.
Fig. 3. Wavelengths which produce the gravity w&ve resonanceeffect in the case of eastward landdownward directed gravity wave phase prop~&tion.
instability during the day. Thus, this mechanism can provide a theory for the formation of nighttime E,.
gradient
Discussion
To obtain a ‘feel’ for the resuIts from equation (10) let us first of all consider the simplest quantitative ease. This occurs at heights above 130 km when the ion and electron Hall mobilities are almost the same and more than an order of magnitude greater than the Pedersen mobilities. In this case it is reasonable to start by taking A+ = A- so that equation (10) becomes Q = iA*. Now from the definition of A*
2036
TON BEER and D. R. MOORCROPT
This expression was obtained by neglecting the Pedersen mobilities (which are smaller than the Hall mobilities), the diffusion terms (which are small in the case of large wavelengths) and the variations of the velocities with altitude (which is a reasonable fLrst approximation). Then because lk,l > Ilc,l only the recombination term and the --ik,puaE, term remain significant and these are the terms in equation (18). For east-west propagating gravity waves, o = k,j+E,,
= k,pHEO, cos 8
by using equations (10) and (US), and an expression for a+ can also be obtained. The validity of these expressions can be checked by comparing the numerical results obtained through their use with the curves of Figs. 2 and 3 which give both w and oi values for each value of k,. Typical values of oi are tabulated in Table 1 where the parameter [ = (2a,n, - oJ/~E,I, These values show that equation (18) is not accurate enough to use as an approximation for wi. Now equation (17) provides a useful approximation for o when k, 2 ,> 1/4H2,so that in the low frequency case one may use (18) and (17) as an approximation for w in the east-west geometry used for the computations. o = o,kJk, = k,/_+E,,, cos 8.
(19)
In the general case this becomes w = ogk,,/k,,= ,uaE,,,(k, cos 8 + kNs sin 0) where ~2 = (y - l)g2/c2. Equation (19) g ives the allowed values of kmand k,. It is a quadratic in k, which agrees with the shape of the curves of Figs. l-3 at the dip equator. Above 130 km equation (19) as it stands provides an excellent approximation to these curves. At lower heights when ,u~+ # ,u~- effective values of ,uH can be obtained by using the curves of Figs. 1 and 2 in conjunction with equation (19). These effective values of puRare listed in Table 2. Each curve in the figures is characterized by values of (~1~)~~~ and T, the atmospheric temperature. T was obtained from the CIRA 1965 Standard Atmosphere. Special note should be made that since o/k, < 0, only westward directed fields will satisfy (19) regardless of the direction of Ic,. Interpretation Now that an analytic expression for o has been obtained, it would be advantageous to have one for mtas well. In this case we need to study Ah more closely. It may be recalled from Section 2 that A* represents the coefficients of N+ in the perturbed continuity equation. The N* terms arise from the perturbation electron the density n1 so that by examining coefficients of n, in the perturbed continuity equation, Ah may be considered as being A* = 2a,n,, + VO*. Vn,/n, + V . V,,+ where the V . V,,* term represents wind shear effects and will be dealt with later on. Now by using an exponential form for the perturbation electron density Vn,/n, = K, the wavenumber vector. So dropping the V . VO*term from the expression for A*
Atmospheric wave-induced in&ability in the nighttime E-region
2037
Table 1. V~Iuea of [ Height (km) 80 90 100 110 120 130 140 160
-3
x 10-2
Vertical wavenumber (m-1) x lo-3 -1.9 x 10-S -4.7 x 10-a
-7.5
-0.6
-0.006 3.43
0.025 1.10 4.7 -0.9 0.34 0.92 0.61 -0.20
0.024 1.67
-1.8
x lo-’ 0.023 -2.09 2.6 1.15 0.83 0.67 0.61 0.6
Weatwaxd phase propagation k, < 0 80 90 100 110 120 130 140 150
-0v5
-0.008 -2.90
0.023 -0.17
0.026 0.43 0.9 6.76 2.82 1.64 0.61 1.09 Eaatwaxd phase propagation k, > 0
0.023 0.48 1.86 6.34 1.51 0.88 0.61 1.70
Table 2 Altitude (km)
(PI?z/%?ff
80 90 100
63.26 2.025 x IO3 -
1()2.88 104.44
-
110 120 130 140 150 160
1.83 2.11 2.18 2.29 2.15 2.05
106.46
105.60 105*@ 105.74 105.74 105.74
equation (10)may be rewritten as 32 = -2a,n,
(Pa)ett
+ i
.
IT,--B+ B+-
x x x x x x
IO4 104 IO4 IO4 104 104
- . K
B-
V$B1 ’
By considering the magnitude of each term in the defining equation of B* it may be shown that
B+ - B-M (pp++ pp-)no(k2 + k.7 + pu,-@,2 which is always a real quantity. The terms for B+ and B- were obtained by neglecting any variation in p in the z
TOM BEER and D. R. MOORCROPT
2038
direction. This seems reasonable for a tist approximation. Then as Im (K,,) = 0 and Re (K,) > Im (K,) the definition of B* leads directly to expressions for B*. To get the expression for B + - B- a few more approximations were used. It was assumed that the pHf and ,uH- were about equal and that ,ur* was negligibly small (this follows the procedure used for A* earlier). It was also assumed that This may readily be seen to be true when rewritten as n0kS2 > Im (K,) &,/&. kZ2 > (scale height of neutral atmosphere) x (scale height of the ionization). Finally one needs to use the fact that p,,- > p 0+. Thus, the growth rate is given by o. __ 2cr n + Im (K) I 0 0 +
&(VO,- Im (B+) -
. L-
Re P+) - Im W,)L+ B+ - B-
voZf Im (B-)) + k,(V,,W-B-
For simplicity, consider large altitudes TIo*+M V,,--. In terms of vertical coordinates wi
=
Re VW
Im (B+) -
where
VoZ+ Im (B-))
1
. (20)
,uH+ = ,uH- and ,u~-- = 0, then
2a,n, + co9 OVo,*. Im (K,)
[Ic,. V,- +
Vo+)z + k, cos O(Vo- -
Vo+)J ;
,+*(kz2
PO-kv2 s%, 8
+ kv2 COST 0)
+
2
j&k;
cos 8
(21)
Then E,, < 0 The Im (K,) . V,, l term represents an E x B drift of ionization. will drift the ionization down and if the perturbations increase in magnitude upwards, as they do in the case of gravity wave induced perturbations, then uoi < 0 because to an observer on the ground it will appear as if the perturbation density at a particular height is increasing. The inclusion of this apparent instability due to the drift motion has led WHITEHEAD (1970) to doubt whether LC)~ < 0 really implies instability if the amplitude of the perturbation varies spatially. He provides an example of this by a damped electromagnetic oscillation whose amplitude decreases in the positive x direction. This is known to be a stable system. When this is viewed by someone moving towards the negative x direction, the observer sees an instability, yet the system is stable. Now the gravity wave perturbations will be something like this. In fact, what happens is that the viscous dissipation leads to a height at which the amplitude of a gravity wave with a given k, and k, is a maximum. Below this height the amplitude increases as exp (v/2H), above it, it is damped through viscous interaction. Therefore, below this particular height a downward drift of the ionization as a whole will appear to a ground based observer as if the irregularities at any one height are growing. Whether this is a ‘real’ instability or not is a moot point. After all, even in the case of the temporally varying perturbations a non-linear analysis reveals that they only grow for a certain length of time (called the stabilization time) and then start to fade (TSUDA et al., 1969; WILLIAMS and WEINSTOCK, 1970), yet we are certainly justified in calling them unstable. We, therefore, believe that because the Im (K,) . V 0a* term represents an effect that will appear to a ground based observer as if it were a growing irregularity at a particular height it should be included in the analysis and this is what has been done.
Atmospheriowave-inducedinstabilityin the nighttimeE-region
2039
The last term in equations (20) and (21) represents the gradient instability. By last term we mean the sum that is in square brackets and the coefficient by which this sum is multiplied. This coefficient includes expressions for the horizontal and vertical wavelengths and for the gradients in ionization density. The k,z(‘V,,-- - Vo+)a = -k,2(pclp+ + pp-)E,, term describes the simple situation outlined in Section 2 and it assumes there are no upward phase variations, For westward directed fields this term will always be positive (inducing stability) for upward gradients of ionization and negative (aiding the instability) for ionization that decreases upwards. Let us now isolate the situation represented by the other part of the last term of (21) which arises because k, # 0 and is characterized by
When a~*~~~ > 0 and E,,, < 0 then if &/lEz< 0 it represents a stable solution and if l&/i& > 0 then it denotes instability. But as k, < 0, then for k% < 0 (westward propagation) there is a manifestation of the gradient instability when the ionization increases upwards. The mode of operation for this instability may be visualized by means of Fig. 4.
+
7
trough
0
Fig. 4. One mode of the gradient instability.
Double headed arrows
show the
drift direction.
The greater Hall mobility of the electrons will drift them downwards faster than the ions and this will set up pe~urbation electric fields that drift the ionization crests down and the troughs up. The Pedersen effects that are described in Section 2 will tend to oppose this, but provided that I& cos e(pu,- - /++)I > I&(/++ + ,+-)I (22) then the z direction drifts of (21) are dominant and instability may occur in regions of upward increasing backgro~d io~zation. Equation (22) is true in general except at very high altitudes (above 160 km) and at large dip angles. As oi may be represented by 0,. = 2a 0n0 - 5 jE,,I where wind shear effects are neglected, Table 1 lists the computed values of E from Figs. l-3 where Im (K,) = 1/2H.
2040
TOM BEEB and D. R. MOORCROFT
Field-aligned and non-aligned iwegularitiea Below 100 km the Pedersen mobility of the electrons is greater than the Pedersen mobility of the ions. Seeing as (,u~),,~~+ p,+ it appears that in the D-region, even when there exist field aligned irregularities it will be the electrons--moving across the field lines in the field aligned case-that move so as to attempt to neutralize space charge effects. This leaves the ions free to move under external influences and it appears that in the D-region even field aligned irregularities move with the ion velocity. Eflect of E,, In an earlier section of this paper it was pointed out that the gravity wave induced gradient instability could only operate in the presence of westward directed electric fields. To simplify the graphs presented in Figs. 1-3 the effect of the electric field in the x direction was ignored because it plays a subordinate role to the east-west directed field. This is because the gradient instability is a mechanism primarily dependent on the realignment of the ionization in the vertical direction. As the electron Hall mobility is greater at all heights than any other transport coefficient, it is clear that for all field-aligned cases in which the irregularities move with the electron drift velocity, it would take an exceptionally strong electric field in the z direction to exceed the Hall effects of the E,, field. A similar situation applies when the irregularities are non-aligned except at D-region heights near the geomagnetic poles. In fact, it is possible to obtain analytic estimates of the effect of non-zero E,, upon both real and imaginary parts of the angular frequency. 1. Effect on W. As o M k . VO-Re (I?+) - k . V,,+Re (B-)/(B+ = (P,%-&~
-
+ (~H+b%
+ rUp+WL
PUp-vhz
-
/+-b%,
+ ~p+b%,
-
B-)
-
~~-k-G,z)Re
(B+)
-
kz~n+Eol)Re
(B-)
at high heights the pH*k eE ,-,*term dominates and E,, will affect the graphs of Figs. l-3 when kJ$,, > k,E,. At lower heights E,, becomes important if
-,++k&d~up+PP&~~) 2 -ruck.&&~+nok~2 +~o+nok,2) +ruo+@3 1 I[+hd2%z(~p+n -/++k2U+-@,2
[
+ ~n,lc,~)
at low dip angles this means C/+-kz +
m-lc,lE,z 3 [rup-k - PH-WL
has to be quite large However, the pH-k$,,$ term will once again dominate and EOLl to affect the aforementioned results. 2. Effects on q. By multiplying all the relevant terms out it may be shown that when E,, # 0 then wi = 2a,n, - E IE,,l Im (U%,(~~+~0-
- rup-ruo+)lc,2 - 2Eozk&.(/+z-/++ +
+ (I,++
+
rUp-P,2
+
PO-
. k,
1 an, P~+PP-1
;
z 0
Atmospheric wave-induced instability in the nighttime E-region
2041
at low dip angles this becomes
and so exceptionally large gradients or very large values of E,, are required to affect cot. However, at large dip angles the E,, may make a substantial contribution to coi. Seeing that at low heights r-sPC M
Cl*+*
rrsp- *
ii cco-
and
,u~- > pa+:
But as Im (K,) = yg/2c8 > 0, in order to aid the instability E,, must be negativei.e. downwards and southwards. It should be realized that pa+ increases with height, and so the E,, term would probably have the greatest effect around 120 km altitude. However, the contribution from E, is still the greatest unless E,, is very large. 4. COx?CLUSION
We have analyzed the ~rm~ible wavenumbers of the gravity waves that can produce the gradient ~stab~ity and their resulting growth times. However the effects of this instability will be discussed elsewhere (BEER and MOORCROFT, 1972) as will the effects of the steady state response which produces B-region partial reflections (BEER, 1972a) and could, under special circumstances, produce a spatial resonance mechanism (WHITEHEAD,1971a; BEER, 1972b) in the ionosphere. AokwZedgemW--‘J!hi work was supported by the National Research Council of Canada. And also in part, by N.A.S.A. Grant NGR-05-009-076.
REFERENCES W.1. and C!UNNOLD D.M. T. BEER T. BEER T. and MOOIWROXT D. R. BLUTYILE L. J. et&.
1972 1965
CHAPMAN&
1966
CHIMONAS G. CHXONASGT. CIRA CUNNOLD D.M. FABLEY D. T. et d.
1969 1971 1965 1969 1970
AXPORD BEER
J. Atmo8& Tew. Phys. s4, 2046. Proo. 2nd. Int. Symp. Equatorial Aeronomy (Edited by F. DE MENDONCA). C.N. Pq-CNAE-LAFE, Sao Paulo, Brazil. NWVO Cimento Swppl. Ser. 10, 4, 1385. J. geophys. Res. 74,4091. J.gvhy8. Res. 76,4578. North-Holland, Amsterdam. J. ge~hy8. Re8.74, 5709. J, g~hy8. Res. 75,719Q.
2042 HAERENDEL
TOM BEER and D. R. MOORCRO~
G.etal.
1967
HiNES C. 0.
1955
HINES C. 0.
1956
HINES C. 0.
1960
HINES C. 0.
1963
HINES C. 0.
1964
HINES C. 0. and HOOBE W. H.
1970
HOE F. C.
1963
HOOKE W. H.
1968
HOOKE W. H.
1969
HOOKE W. H.
1970e
HOOKE W. H.
1970b
HOOKE W. H. KAISER T. R. et al. KATO S. KATO S. and MATSUSHITA S. KATO S. et aZ. KNOX F. B. LINSON L. M. and WORK J. B. MAEDA H. MAZDA K. et al. MATSUSHITA S. P~EWAY M. L. and HINES C. 0. PO~REBNOY V. N. and FATKULLZN M. N. RAO P. B. and RUSSELL W. H. REID G. C. REP G. C. RISHBETH H. and GARRIOTT 0. K.
19700 1969 1965 1969 1970 1964 1970 1957 1963 1969 1969 1969 1971 1968 1970 1969
RO~ISTER A. and D’ANOELO N. SATO T. et al. SIMON A. TSEDELINA YE. TSUDA T. et al. TSUDA T. et al. WHITEHEAD J. D. WHITEHEAD J. D. WHITEHEAD J. D.
1970 1968 1963 1965 1965 1969 1967 1968 1970
WHITEHEAD J. D.
1971a
Pl4md. Space sci. 15, 1. J. Atmmph. Terr. Phys. 7, 14. J. Atwwsph. Terr. Phys. 9, 56. Can. J. Phys. 38, 1441. Q. J. R. met. SOG.89,1. Can. J. Phys. 48, 1424. J. geophya. Rea. 75, 2563. Phy8. Fluids 6, 1184. J. Atnwsph. Tern. Phya. 30, 795. Planet. Spe Sci. 17,749. J. geophy8. Res. 75,5535. J. geophys. Res. ‘95, 7229. J. geophys. Rea. 75,7239. Planet. Space Sci. 17,519. Spe Sci. Rev. 4,223. J. Atmoaph. Terr. Phys. 31, 193. J. geophys. Res. 75,2540. J. Atmosph. Terr. Phy8. 26, 239. J. geophys. Res. 75,3211. J. Geomag. Geoekct. 9,86. Phys. Rev. Lett. 11,406. Radio Sci. 4, 771. Can. J. Phys. 41, 1935. Geomagra. & Aeron. 9, 456. Radio Sci. 6, 221. J. geophys. Res. 78, 1627. Planet. Space Sci. 18, 1105. Ionospheric Phyaks. Academic Press, New York.
WHITEHEAD J. D.
1971b
WILLIAMS R. H. and WEINST~CX J.
1970
J. geophys. Re8. 75, 3879. Radio Sci. 3, 529. Phy8. Fluids 6, 382. Geowqn. & Aeron. 5, 525. Radio Sci 1,212. J. geophys. Res. 74, 2923. J. Atmoaph. Terr. Phys. 29, 1285. J. Atmosph. Terr. Phys. 36, 1663. J. Atmosph. Tern. Phys. 32, 1283. J. geophys. Res. 76, 238. J. geophys. Rea. 76, 3116. J. geophys. Rea. 75, 7217.
Reference is also made to the following unpublished material BEER T.
1971
Ph.D.
Thesis,
University
of Western
Ontario, London Canada. BIONDI M. A.
1967
DASA Reaction Rate Hand Information
and
CONRAD J. C.
1969
Santa Barbara. Cornell University
GEOR~ES T. M.
1967
IER-67/ITSA-54. Report.
U.S.
ook. DASA
Andy&
Center,
CRSR 348. ESSA
Government
Office, Washington,
D.C.
Technical Printing
Atmospheric wave-induced instability in the nighttime E-region AFPENDIX
2043
I
Ionospheric transport coefficients(m.k.s. units)
PH-80
85 90 95 100 105 110 116 120 125 130 135 140 145 150
l.OE8 8.538 4.OE9 l.OElO 1.8ElO 1.2ElO l.OElO 5.OE9 3.OE9 2.OE9 1.2E9 l.lE9 l.OE9 l.lE9 1.5E9
1.80E5 7.5134 3.llE4 I.2234 5.2833 2.3233 1.07E3 5.1232 2.7732 I.5832 1.OOE2 6.7931 4.8931 3.5231 2.71El
2.7536 1.14E6 4.7335 1.91E5 8.4534 3.8734 1.8734 9.7333 5.6733 3.4833 2.3433 1.6633 I.2433 9.1832 7.2832
1*84El 4.41El 1.07E2 2.7432 6.4032 I.4733 3.1533 6.1233 9.0733 9.9233 8.4633 6.5133 4.9233 3.6233 2.8033
5.7033 2.5633 1.07E3 4.3632 1.9232 8.81El 4.2531 2.22El 1.29El 7.9230 5.3230 3.77 2.82 2.09 1.66
1.69E - 2 9.71E - 2 5.68E - 1 3.75 2.05El 1.08E2 5.0932 2.0933 5.8033 1.13E4 1.5334 1.7634 1.8734 1.9334 I.9634
1.8234 1.9734 1.9934 2*00E4 2.00E4 2.00E4 2*00E4 2.00E4 2.00E4 2.OOE4 2.00E4 2.00E4 2.00E4 2.00E4 2.00E4
DP+ 2.953 - 1 7.07E - 1 1.71 4.68 1.15El 2.90E1 6.8231 1*60E2 2.7832 3.5732 3.4832 2.9532 2.4332 1.91E2 1.5732
DP9.14E + 4.llE 1.72E 7.45 3.45 1.74 9.21E 5.79E 3.953 2.853 2.19E 1.71E 1.39E l.llE 9.323 -
1 1 1
1 1 1 1 1 1 1 1 2