Indian Journal or Radio & Space PhysicsVol. II, April 1982, pp. 45-63

Interaction of Medium Scale Gravity Waves with Ionization & Its Applicationto Electron Content Measurements of TIDs

A SEN GUPTA' & 0 P NAGPAL

Department or Physics & Astrophysics, University or Delhi, Delhi 110007

Received 23 January 1982

The perturbations in the total electron content caused by atmospheric gravity waves propagating in a realisticthermosphere have been derived. Standard but latest models or the neutral thermosphere, ionospheric electron density andneutral winds have been incorporated. Numerical computations have been made with the satellite ATS-6 to Delhi ray-path forsome typical gravity wave parameters representative or the observed medium scale TlDs. It is observed that the electroncontent perturbations show an anisotropic behaviour with the azimuth or propagation or the gravity wave. This arises due tothe combined effects or three factors, namely, (i) anisotropic dissipation of waves due to the presence or neutral winds, (ii) phase-cancellation effects and (iii) the nature of the gravity wave-ionization interaction. The electron content response is round to varymarkedly with the time or day mainly because or a diurnal variation or the neutral wind vector. The theoretical computationsare in good agreement with the experimental observation or TlDs observed through ATS-6 and ATS-3 satellites.

1 IntroductionThe geostationary satellites permit continuous

monitoring of the total electron content (TEC) along afixed ray-path from the receiver to the satellite. Amongother things, these measurements provide a veryconvenient means of studying the travellingionospheric disturbances (TIOs) based on thesimultaneous recordings of the TEe at three or morereceiving ground stations! -4. Quantitative in-terpretation of such TIO data in terms of atmosphericgravity waves is, however, not very straightforwardbecause of the complicated nature of the interactionbetween gravity waves and the electron content. Inrecent times considerable progress has been made indeveloping theoretical models for gravity wave-induced perturbations in TEe with varying degree ofsophistication. The first of these theoretical studies wasmade by Georges and Hooke" in 1970. By consideringa very simple model of interaction between theatmospheric gravity wave and the electron content,they demonstrated that integrated ionosphericresponse to a white spectrum of atmospheric gravitywaves is highly anisotropic. One important conclusionof these studies is that the TEe technique is unable toreveal the presence of waves of certain periods at agiven geomagnetic location. Although their modelpredicted the correct order of magnitude for theobserved TIO amplitudes, it had several deficienciesparticularly in terms of atmospheric wave model used.

Importance of using fairly complete and realisticmodels in determining the response of electron content* Present address: Time & Frequency Section, National PhysicalLaboratory, New Delhi 110012.

to an atmospheric gravity wave was realized by Davis".He considered, on the one hand, the dissipative effects(viscosity and thermal conduction) which become veryimportant above 250 km altitude and, on the otherhand, the effects of photochemical loss and diffusion(besides the dynamical effects) in determining the wave-ionization interaction. Provision was also made forheight variation of some of the wave parameters suchas vertical velocity amplitude. However, the effects ofbackground neutral winds which deeply affect thepropagation of medium scale gravity waves wereneglected. More recently Bertel et al.' following asimilar approach as that of Davis" included the steadyneutral winds in their analysis. However, thepolarization relation which relates the horizontal andvertical components of wave velocity (Vh, vz) was solvedfor a fixed height of 250 km. This is not a validapproximation as it can alter the magnitudes of Vh andVz substantially (which in turn determine the amplitudeof TEe perturbations) and, therefore, the polarizationrelation needs to be solved for every height startingfrom a certain base value. In this paper we havepresented a general treatment of gravity wave-electroncontent interaction using realistic and recentatmospheric models.

2 Propagation of Gravity Waves in RealisticAtmosphereFor studying the gravity wave-ionization in-

teraction, one needs to know the propagationcharacteristics of gravity waves in the neutralatmosphere. This is usually done by solving thehydrodynamic equations which govern the pro-

4S

INDIAN J RADIO & SPACE PHYS, VOL II, APRIL 1982

pagation of these waves. The work on this subject wasinitiated by Hines" for an idealized isothermal andloss less atmosphere and has been subsequentlydeveloped for more realistic conditions by a number ofworkers? -15. In describing realistic atmosphericconditions we must take into account (i) the non-isothermality which is quite significant in the lower F-region; (ii) the various dissipative processes whichcause damping of the gravity wave energy, namely,molecular viscosity, thermal conduction and ion drag,and (iii) the thermospheric neutral winds which havebeen shown to have magnitudes comparable to thephase velocities of the medium scale TIDs.

The basic equations which govern the propagationof atmospheric gravity waves are the followingIO•13.

The equation of continuity:

... (1)

The equation of momentum conservation:

aVn .: (V V)PTt+p(Vn.V)Vn=pg-Vp+V. S -VinPi n- i

- 2p(Q x Vn) ..• (2)

The equation of energy conservation:

pK [a t; ]-+ Vn.VTn = Q +V.(AVTn)- p(V,Vn)(y-1)m at

... (3)

The equation of state:

pKTnp=--

m... (4)

where

P Neutral gas densityt Time

Vn Neutral gas velocity consisting of two partsnamely, the ambient thermospheric wind Vno

and the wave induced perturbation Vn

g Acceleration due to gravityp Pressure of the neutral gas

Vin Ion-neutral collision frequencyVi Ion velocityn Angular velocity of the earthK Boltzmann constant

y Ratio of specific heats (CpICv)m Mean molecular massT; Temperature of the neutral gasQ Heat input to the mediumA Thermal conductivity coefficient

46

s: (0 Vni a Vnj 2 )Su=u -+---c5V V1) aXj OXi 3 1) • n

Viscous stress tensoru Molecular viscosity coefficient

For solving the set of Eqs. (1) to (4) a linearperturbation method is applied. The perturbation isconsidered to be a plane harmonic-free internal wavewith a single constant frequency wand wave numberk(kx, ky, kz).

The form of the steady state solution adopted is ofthe type

Tn - Tno = P - Po P - PoTno.T PoR PoP

w=expj[wt-kx.x-ky-y-kz'z] ... (5)

where T, R, P, U and Ware the polarization constants.The subscript 0 stands for the zero order state in theabsence of perturbation. The details of the method ofobtaining the dispersion and polarization relations bysolving Eqs (1)-(4) have been described by Clark et a/.3

Here only the main assumptions used and the finalresults obtained are described.

The waves are assumed to be propagating in themodel atmosphere of Jacchia!". The atmosphere isassumed to be horizontally stratified with a steadyhorizontal neutral wind blowing through it. Forpurposes of computing wave propagation, the wavenumber in the horizontal direction, krh(kx, ky), isconsidered to be purely real and constant with altitude.In solving the hydrodynamic equations the viscosityterm is ignored, but its effect is taken into accountapproximately by increasing the coefficient of thermalconduction by a factor of 1.8. For evaluating the iondrag term, the wave induced perturbations in theionization density Pi are neglected and only its ambientvalue has been considered. The atmosphere is assumedto be slowly varying so that WKB solutions are valid.The work of Clark et a/.13 contains detailed account inthis regard.

The effect of ambient thermospheric winds is mainlyto introduce a Doppler shift in the gravity wavefrequency. In order that the dispersion relation byClark et a/. 13 be valid, the wind model must satisfythe following constraints.

(i) The vertical wind shears should be small. This isindeed valid for altitudes higher than about 180 km.

(ii) The term ( V,J2 ~ C 2 (C = speed of sound). Thiscondition is generally valid in the F-region.

SEN GUPTA & NAGPAL: MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEe

__ Y_. '" _ A.TN . K _ k-xvcosDsin DYl - Y _ l' - jw' Po' 1- W' - jvsin2 D

W' k2 k2C - __ -x Y_.

1 - gH w' - jvsin2 D w' - jv'

C2 = Yl + "'(k; + k;)

w'(w' - jv)

C3 =(' .. 2 )W - JVSlO D

(iii) We assume that the Doppler shifted wavefrequency does not tend to zero or to a negative valuewhich results in a condition for asymptotic trapping 1 7.

Such a condition indeed arises on some occasions as weshall see in Sec. 4.1. The effects of dissipation in such

.cases become very large and the wave gets completelydissipated.

(iv) The magnetic field vector 80 is assumed to bealong the X-axis (towards north) making an angle Dwith the horizontal.

Making use of the assumptions outlined above onearrives at the following dispersion and the polarizationrelations.

v = (PiViN)/P

Ii =dH/dz

Neutral-ion collision frequency

Vertical gradient of scale height

R = P - T ... (9)

... (12)

... (13)

k: +d.k; +d2 =0

The roots of this equation are of the form

The choice of negative sign outside and positive signwithin the bracket gives the root appropriate to theupgoing gravity wave12. Thus for numerical.solution ofthe dispersion relation, the value of kz is firstdetermined using the Eq. (13) with the signs alreadystated. This is followed by an iterative process to solvethe full Eq. (6). Five iterations have been found to besufficient for this purpose. The value of kz thusdetermined is introduced in the polarization relations'to yield the values of U and W. Hence VNh and VNZ aredetermined.

It may be mentioned that the' dispersion andpolarization relations have also been derived byThome and Rao18 for a viscous medium excludingthermal conduction and ion drag. It has been shown byDavis6 that their relations give values of kz which are infairly good agreement with those obtained by usingEqs (6)-(11)when the neutral winds are not taken intoaccount.

and P, T, R, U and Ware, respectively, the polarizationconstants for pressure, temperature, density, horizon~tal and vertical velocity perturbations.

The dispersion relation given by Eq. (6) is of thefourth order in kz and hence gives four roots. The rootscorrespond to (a) upgoing and (b)down coming gravitywaves and (c) upgoing and (d) down coming thermalconduction waves.

We are interested in only the upgoing gravity wavemode. In order to select the root (a) we followVolland 12. Putting v = Ii = 0 (i.e.,neglecting ion dragand temperature gradients) Eq. (6) reduces to a formsimilar to equation of Volland's paper which is givenby... (6)

... (7)

+Y1Ii( I_+K + C1g)=0H 2H 1 w'

Polarization relation:

P = W'2(y -1)[(kz - iH -jK1)

x {C2 +"'(k; +4~2)}+j(l +:1Ii)J

T=W'(Y-l{jc1g(1 +y.Ii) +W'(kz- iH- jK.) J

... (8)

where

w' = W - (k-x V nxO + ky VNyO)

Doppler shifted wave frequency

Dispersion relation:

"'k: -2j.K1 "'.k: + k;{ C2+ "'C~2 - Kl - C1 C3)}

1

U=(, .. 2D){k-xgHP+vcosDsinDW} ... (10)W - Jvsm

W = W'2(y -1)[ C1gH{ C2+ ",(k; + 4~2)} -w]... (11)

47

<.---..-- ..---

INDIAN J RADIO & SPACE PHYS, VOL II, APRIL 1982

3 Gravity Wave-Ionization InteractionThe gravity wave-induced perturbations in the

electron density are determined by the solution of thecontinuity equation for the electrons given as follows.

N=No

q=qo

1=/o=PNo

V = VdO + VdrO

where N, q, I and V are the concentration, productionrate, linear loss rate (applicable to heights above 150km) and the velocity of the electrons, respectively.

In absence of waves in the medium all the quantitieshave their respective ambient values, viz.

The divergence terms involving VdrO is left outfollowing Thome and RaolB and Klostermeyer14,Ignoring terms involving VN' and V2 N', which havebeen shown by Testud and Francois20 to be muchsmaller than N', we get

r

... (16)

... (17)

~ [~ (V No) ~ ](v". 80) (k. 80) +j ---,;;-;.80

w - j(P - v. VdO)

N'

For a horizontally stratified atmosphere thevariations of the ambient parameters are only in thevertical direction, so that Eq. (16) reduces to

~ = (Vll.Bo{(k. Bo)+ j(z.Bo) ~(ln No)]

No [ d ]w - j P - dz ( VdO)

... (14)oN =q-l-V.(NV)ot

... (15)

where P is the linear loss coefficient, VdO is theambipolar diffusion velocity· and VdrO is the steadydrift velocity due to neutral winds and electric fields.

The presence of a gravity wave introducesperturbations in all the above quantities, so that wehave

N=No+N'

q = qo + q'1= 10+ I' = pNo + PN'

V = VdO + V~o+ VdrO + Vw

where, Vw = (v".Bo)Bo is the wave induced plasmavelocity. This form of expression is valid in the regionabove 150km where ion gyrofrequency, WHi ~ Vi" (Ref.19).

The perturbation q' is made up of contributions dueto two factors. These are perturbations in the neutralconcentration and altered intensity of radiationreaching a particular level. Oavis6 shows that q' is verysmall for waves with amplitudes considered thereinand also here. The perturbation I' is essentially due tothe perturbation N'. It has been shown by Thome andRao 18 and Clark et al.13 that V~ ~ vwand hence can beneglected.

Linearizing Eq. (14) for the perturbed terms andnoting that for a steady state solution N' must havesimilar temporal variation as the wave, we get

jwN' = PN' - V. (No vw) - V.(N'Vdo)

- V. (N' VdrO)

where T _ Ti + T.p- -2

Eq. (17) is the expression for the fractionalperturbation in the electron density caused by gravitywaves. The numerator of the right-hand side ofEq. (17)is a product of two factors. The first is the component(vll'Bo) parallel to the geomagnetic field lines. Thesecond is a sum of two parts, i.e. (k. Bo), which is thevariation of this motion parallel to the field lines and

j(i. Do)~ (In No) which arises due to a gradient in the

electron density profile. The presence of the term j[P

- dVdO/dz] in the denominator arises due to theinclusion of the photochemical loss and the diffusionterms. It has the effect of reducing the electron densityperturbation. Since P falls off exponentially withheight, the effect of this term is greater at the lowerheights of interest. On the other hand the diffusionvelocity increases with height and this term becomesimportant at heights above the F-region. The effect ofboth the terms increases in importance as the wavefrequency is reduced.

4 Results of Numerical Calculations

Here we present, as illustrative examples, someresults of numerical calculations based on theformulations described in sections 2 and 3. We havediscussed (i) the models used for the ambientatmosphere and the ionosphere, (ii) the wave modelschosen for study and some propagation characteristicsof gravity waves in the neutral atmosphere, (iii) theinteraction between gravity wave and ionization andsome calculations of the electron density perturbationsand (iv) the electron content perturbations. The effectsof phase cancellation and varying the electron densityprofile are discussed. Variations of the electron contentresponse with the azimuth of propagation for sometypical waves are shown for various times of the day.

48

""'1 I !lli I' I It II I' fI i 1l1,lliI 'i "'I! II rill 1 i~

SEN GUPTA & NAGPAL: MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEC

P(Z)=P(za)exp[- rz ~JJZ.HN2(Z)where

P(za) = 1O-18n[N2]z.

The collision frequency between ions and neutrals istaken from Chapman23 as -

ViII(Z)= 1.5~5 x 10 -15 p,,(z)/m (z)1.5

and y varies between 1.4 and 1.67 over the height rangeof interest. However, it has been shown by Davis6 that

the effect of taking a height varying or a constant y

makes little difference on the computations.The coefficient of thermal conductivity is taken from

Dalgarno and Smith24 as

A(Z) = 6.71 X 10-4 T,,(Z)O,71

In order to compensate for the neglect of molecular

viscosity coefficient, J.L, we multiply the coefficient, A, by1.8.

The models for the thermospheric neutral winds are

taken from the theoretical calculations by Lum:b and

Setty25 who obtain the wind profiles by solving acomplete set of the coupled continuity equations forthe ions and the neutrals. The wind profiles for the lowsolar activity period and for winter and sum:mer

seasons were supplied by Lumb26. A typical heightprofile for the neutral wind for winter months is shownin Fig. 1. The eastward (WSPE) and the northward

SPEED OF SOUNOICI.m sec-1:lOO 400 500 600 700 800

300

-,

4.1 Ambient Atmospheric and Ionospheric Models

The calculations are performed assuming a flat earthin a rectangular coordinate system where the X ~axispoints to the north, Y-axis to the east and the Z-axis to

the vertical. Such an assumption is generally valid for aspherical earth if the distances along the X-and the Y_

axes are interpreted as actual distances on the curvedsurface of the earth21.

The magnetic meridian is assumed to coincide withthe geographic meridian which is true in theneighbourhood of our location where the declination is

small. Thus the magnetic field has components cos Dand sinD along the X- and Z-axes, respectively. The

dip angle is considered constant over the entire rangeof satellite ray-path.

The temperature profile for the neutral gas is takento be of the form22

T,,(z) = Too - [Too - T,,(z..)]exp[ - s(z - za)]

S = 0.0265exp[ - O.5(Too - 8oo)/{750+ 1.722 x 10-4

x (Too - 8oo)2)]

where Too is the exospheric temperature, Za the lowerboundary (usually 120 km, T,,(za) = 355°K) and s aconstant depending on Too on!y. The constant, s, hasbeen modified by putting the factor 0.0265 instead of

0.029 so that the temperature profile closely matcheswith the Jacchia16 model. The electron and ion

temperatures are calculated from the followingempirical relations assumed by Clark et al.13

TAz) = T,,(z) + (Teoo - T oo)exp [ - 80/(z - 119)]Ti(z) = T,,(z) for z < 300 km

1j(z) = TAz) + [T,,(z) - Te(z)]exp(1 - z/3OO)

for z~ 300km

where Teoois the exospheric electron temperature. Theneutral gas scale-height is given by the expression

) 8.31 T,,(z)H(z =--­m(z).g(z)

where the mean molecular mass m(z) is taken from theJacchia16 model and g(z) is given by

g(z) = 9.8

WSPE (1400 hrsJI I

WSPN(1400hrsll I\/i1/

}''1I,. Il I

.I II I:I: / :/ ,./ I/

,.,'".•......

", .•..

-100

The neutral gas density follows a barometric law, viz.

T,,(za) [[~Jp(z) = p(za)' T,,(z)' exp - z. H (z)

As mentioned in Sec. 3 we have considered only theattachment type linear loss process. The linear loss

coefficient P which is dependent mainly on the densityof N 2 falls off exponentially with a scale-height of N 2 as

so

-100 -so 0 50 100

S,W WIND SPEEO,m see' N.E

Fig. I-Height profilesfor the neutral wind and the speed of sound (C)

assumed in the model calculations [The eastward (WSPE) and thenorthward (WSPN) components of the neutral wind are shown for 1400

hrs,]

49

INDIAN J RADIO & SPACE PHYS, VOL II, APRIL 1982

N

m sec-1

300 WINTERLOW SUNSPOT

200 ACTIVITY

300

Fig. 2-Locus of neutral wind velocity vector at 300 km

corresponding to the middle of the observation time

groups

(WSPN) components of the neutral wind are shown for1400 hrs. Fig. 2 shows the diurnal plot of the neutralwind vector at a height of 300km.

The electron density profiles are assumed to ha ve theshape of a composite Chapman function with a scale­height H, below and Hu above the peak of the F-Iayer.

No(z) = NmexpHl- xexp( - X)]

wherez-z

X = H, m for z < Zm,

z-zX= __ m for z>zm

Hu

and, Nm and Zm are, respectively, the peak electrondensity and its height. The scale-heights H, and Hu aresuitably adjusted so that the profiles match closely withthose of Rawer et al.27 for low latitude and low solaractivity conditions.

4.2 Propagation Characteristics of Gravity Waves

In the following, calculations for the upgoing gravitywave mode have been made using the ray theoryapproach based on the WKB approximation2B• Theatmosphere is divided into thin layers of 5km thicknessand within each layer the solution of the upgoing waveparameters is assumed to be of the form

Vnhm Vnzm [{Um = Wm = Aoexp -j wt-kx·x-ky·Y

where m refers to the layer, Ao is an arbitrary realconstant usually taken as unity and z, is the lower

50

boundary from where the calculations are started. Theparameter z, has been taken as 150km. The boundaryconditions for matching the solutions at the interfacebetween layers are the constancy of w, kx, ky and thecontinuity of the vertical displacement given by(vnz/jw') (Ref. 29). However, since in the F-region,above 150km, the vertical shears of the neutral windsare generally quite weak, the changes in w' in thesuccessive 5km thick layers are very small Heru.;e theinterfacial condition can be taken as the continuity oflIlIz• This would also be consistent with the WKBapproximation since we are not considering reflectionat the interfaces. The vertical wave number kz and thepolarization relations in each slab are obtained fromEqs. (6)-(11).

For the purpose of illustrative calculations we havechosen three waves which are thought to berepresentative of medium scale TIDs in accordancewith the numerous determinations by various workersin the past. These are given in Table 1.

In Figs. 3 and 4 we have plotted the real part of thevertical wave number, k", and the amplitude factor,

A [ = exp f (kzi + 2~ ) dz

(where kzi = imaginary part of kz)J

for two typical waves of 20 and 45 min period.The plotted quantities represent the vertical structureof the wave; kzr is related to the vertical wavelength A.z

by the relation A.z = 2n/kzr and A, the amplitude, showshow the losses due to the various dissipative processesoffset the exp(1/2H) growth with height. The windvector for 1400hrs is along an azimu th of 26° and has amagnitude l04msec-1 at F-region heights (300km).In the cases with the wind included, three azimuths ofwave propagation are considered, namely 0, 120 and180°, which represent roughly the propagation along,transverse and opposite to the wind direction,respectively. Considering first the situation without thewind, we notice that the value of kzr is smaller for thewave with 45-min period than that with 20-min periodwhich implies a longer ;,z for the former. It has alsobeen noticed that there is a steady decrease in kzr with

Table I-Gravity Wave Parameters Used in the NumericalComputations

Period HorizontalHorizontal

minwavelengthphase speed

km

msec-\

45

550200

30

330160

20

170140

" ,. I ,it: 1;1--

SEN GUPTA cl NAGPAL. MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEC

... (18)

where the various quantities have the usual meanings.Clearly for downwind propagation the reduction of w'increases the damping term. Although kz increases insuch a case this increase is smaller than the decrease inw'.

(ii) Considering next the propagation of the wave ina direction opposite to the wind (qJ =180°) we find thatthe wavefrequency is Doppler shifted to a higher value.The value of kzr in this case increases from that in thewindleSs situation while again the steady decreasingtrend with height is maintained. The decrease in kzr isfound to be smaller for the longer wave period of 45min. It is clear that in the limit of very high wind speedkzr would become very small and Az would tend toinfinity. This leads to a condition known as "criticalcoupling" and implies possible mode reflection!J.Strictly speaking, in the limit kzr -+ 0, the ray theoryassumption is not valid, since under such conditionconsiderable amount of coupling occurs between thegravity wave modes and viscosity and thermalconduction wave modes. We have, however, notconsidered such a case in detail, since it is felt that withthe wave parameters assumed and model wind profiles,the above limiting condition is never reached.

be greater for the lower wave period of 20min. Weobserve that with increasing height kzr does notdecrease as smoothly as in the windless case. This isbecause of the height variation of the magnitude anddirection of the wind vector in the assumed windprofile. An interesting limiting condition arises whenthe wind speed along the wave azimuth is very high sothat w' -+ 0 and kzr becomes very large. It can be shownthat in such a case the group velocity is almosthorizontal and very little vertical energy transporttakes place. This is the case of "asymptotic trapping" asdiscussed by Cowling et a/.! 7

Dissipative effects on the amplitude increasemarkedly in the case of downwind propagation. Figs 3and 4 show a rapid fall in amplitude with heightwithout the initial increase as in the windless case. Theenhancement in the rate of decrease of the amplitude isobserved to be relatively greater for the lower waveperiod of 20 min. '

The increase in the damping due to winds can beappreciated also from the simplified analysis ofPitteway and Hines9. Eqs (19) and (39) of the work ofPitteway and Hines9 can be combined to giveapproximately the expression for the imaginary part ofkz due to viscosity and thermal conduction responsiblefor damping, as

j(1 +~~(y - Wg2k~~Jlkz+ ~).kz =

10

0,5 1.0

AMPLITUDE

WINTER 1400 hrs

T z: 20 min Vph=140 m uc-'

WINTER 1400 hrs

T= 45 min Vph = 200 m sec-1

\,\,.</>zo' 'iNO WIND

\ .,4>zl20II\

T =45 min

10-5 10-40 1.0 20

Kzr, m-1 AMPLITUDE

Fig. 4-Same as Fig. 3 but for the wave of 45-min period

15010-6

200

15010-6

550

soo

4SO'

200

250

w:r 300

E

~400•...

:rC> 350

250

'" 3001:I:

Fig. ~Height variation of the propagation parameters k•• andamplitude for the waves of 20-min period (The dashed curves representcalculations without neutral wind. The continuous curves are those

which include a daytime wind model for 1400 hrs and are labelled by the

respective wave azimuths.)

height for both the wave periods. This decrease in kzr

occurs because of the increasing effectof the dissipation, due to molecular viscosity and thermal conduction

with increasing height, as has been shown by HinesJo.Decreasing kzr implies increasing Az and consequentlyslower vertical phase propagation as we move up inheight. The rate of decrease in kzr is observed to befaster for the 20-min wave period, implying greatereffect of dissipation on it.

In the presence of neutral winds, the propagationcharacteristics become anisotropic with respect to theazimuth of propagation. The three cases ofpropagation directions are discussed in turn and thewind filtering mechanisms are illustrated.

(i) For propagation along the wind direction (qJ

= 0°), the wave frequency w is Doppler-shifted to alower value. Figs 3 and 4 show that kzr increases fromits value as that in the windless case, although, thedecreasing trend with height due to dissipativeprocesses is maintained. The increase in kzr is found to

-,

51

INDIAN J RADIO & SPACE PHYS, VOL II, APRIL 19X2

250

450

200

... (19)(Vn• 80{(k. 80)+j(z. 80)~(ln No)]OJ

Nr

No

and this Eq. (19)h~s been used by many workers in thepast5, 14,19,35- 38.

Fig. 6 [(a) and (b)] shows two typical plots of N' / N 0

as a function of height for the two waves of 20 and 45­min period using both Eqs. (17) (solid curves) and (19)(dashed curves). The two curves match very wellaround 250-400 km range but differ at lower andhigher heights. The discrepancy is greater for thelonger wave period. The reason for these discrepancies

4.3 Perturbation in Electron Density

The electron density perturbation due to gravitywaves is given by the Eq. (17) which includes the effectofthe ambipolar diffusion and perturbation in the lossrate. It is, however, of interest to compare our resultsusing Eq. (17) with those obtained when both these areneglected and only the wave induced plasma velocity isincluded. The Eq. (17) then reduces to

wave period. For downwind propagation of the wave

the period gets Doppler shifted to a higher value andhence the ion drag effect is enhanced, whereas forupwind propagation Doppler shifting of the period to alower value reduces the ion drag. Fig. 5 shows the effectof the ion drag on the amplitude of a wave with 45-minperiod, in the presence of daytime wind (at 1400 hrs).Two azimuths cp = 0°, and cp = 180\ corresponding,respectively, to downwind and upwind propagations,are considered. We observe that for cp = 0\ at a heightof 300 km the amplitude reduction due to ion drag is33 % compared to 18.6% in the windless case. On theother hand for upwind propagation (cp = 180°), thereduction is only 1.6% compared to 3.4 % in thewindless case. Thus in presence of the daytime wind,the azimuthal anisotropy of the ion drag effectenhances. It is, however, clear from Fig. 5 that the effectofthe wind on the wave amplitude is very much greaterthan that of the ion drag.

Thus it appears that the directional anisotropy of thegravity wave dissipation caused by the neutral windscan be sufficiently marked to give preferred direction ofpropagation. The predominance of experimentalobservations suggesting equatorward propagation ofTIDs33,34 would be consistent, at least during thedaytime, with a poleward component of the neutralwind. The role played by the ion drag, which is alsoanisotropic, is found to be relatively small compared tothat of the wind for the medium scale TIDs. This wouldmodify the contentions of Clark et al.13 and Nagpa132,who maintain that the ion drag itself can explain thepreferred direction of wave propagation.

500

</>=0' T= 45min ( 1400hr. WIND)5501- I I

I -- ION DRAG INCLUDEDI -- WITHOUT ION DRAGII\II,I\\\\\\\\\\ \,,,,"

E:£. 400

f-~ 350

WI 300

'500---~-0.5 - 1.0 1.5 2.0 2.5 10AMPLITUDE

Fig. 5-Height variation of 1he dissipative effect of ion drag in thepresence of typical daytime neutral wind for a wave period of 45 min(The broken and continuous curves are, respectively, obtained by

neglecting and including the ion drag effect.)

Amplitude curves in Figs 3 and 4 reveal Ihat thedissipative effects on the amplitude are reducedconsiderably for the upwind propagation than for thewindless situation. Reduction of dissipative effectsallows the growth of the wave amplitude to continueup to a much greater height. Evidently the propagationof the waves in this case is favoured since they canattain much greater amplitudes and penetrate deepinto greater heights. The reduction of the dissipativeeffects for the upwind propagation can also beexplained qualitatively by Eq. (18) in a similar manner.

(iii) For propagation approximately transverse tothe wind direction (cp = 1200)we notice that both kzr

and the amplitude are almost unaffected, This result isintuitively obvious since in this case the Dopplershifting of the wave frequency is very small, and thewind has hardly any effect on the wave propagation.

The preceeding discussion highlights the importanceof inclusion of neutral winds in the gravity wavepropagation, particularly, in the presence of dissipativeprocesses. Propagation against the wind is favouredmuch more than that along the wind due toconsiderable reduction of the dissipation in the formercase. These results supplement the findings of Bertel etal.?

Another feature which is apparent from Eq. (18) isthe dependence of the damping on the horizontal wavenumber kh• The larger the value of kh or smaller thehorizontal wavelength Ah' the greater is the damping.

Ion drag is a significant loss mechanism for gravitywaves. The dissipation due to ion drag31.32 is directlyproportional to the ambient ionization density,increases with increasing wave period, and varies withthe azimuth of propagation of the wave in such a waythat minimum drag occurs when the particle orbits arebest aligned with the geomagnetic field.

In the presence of neutral winds the ion drag effectgets modified once again due to Doppler shifting of the

52

I'

SEN GUPTA & NAGPAL: MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEC

6

T=45min (1400hrsWINO)

T = 20 min ( 1400 hrs WIND)450

Fig. 7-Height variaiton of N'/No for different azimuths of wave

propagation for the wave periods of: (a) 20 min and (b) 45 min fA

daytime wind corresponding to 1400 hrs is included,)

anisotropy. Fig. 7(a) shows the effect of the varyingazimuth on N'/No, for a wave of 20-minperiod. Foreach of the curves we take an arbitrary constant gravitywave velocity amplitude Vnz at the lower boundary of150km. A winter daytime wind for 1400hrs is included.Fig. 7(a) shows that the N'/No response is highlyanisotropic with respect to the wave azimuth. Thisanisotropy arises due to the following two reaSQns.

(i) The interaction Eq. (17) contains terms (vn• Bo)

and (k .Bo), (k = k"" k" kzr) which are anisotropic withrespect to wave azimuth, since they depend on therelative geometry involving the wave parameters andthe magnetic field.

(ii) The velocity amplitude Vn suffers anisotropicattenuation due to the winds and ion drag as discussedin Sec. 4'.2.

Thus, comparing the responses for q> = 0° and q>

= 180°,it is seen that at 150km, where we start off withthe same gravity wave amplitude Vnz' N' / Na is greaterfor q> = 0° than for 180°. This is evidently aconsequence ofthe geometric terms (vn• Bo) and (k . Bo)being more favourable for q> = 0° than for q> = 180°.However, as we go up in height the wave amplitude

NEGLECTING LOSS

'-'.t & DIFFN.'" -.•...•.•...

T,45min (1400hrs WIND)

<1>=0'

Vnz(150km)= 3m sec-1

600I-- T, 20min (1400hrs WINe»

4>:180'

550~ Vnz(150km)=10msec-'

250

300

200

E

-'"'.. 450l-I 400I!>

4J 350I

.6 8 0 2 4 6' 8 10 12 14 16

N'/No,'}''''Fig. 6-Curves showing the effect of diffusion and perturbation in the

loss rate on the electron density perturbations (N'I No) for wave periods

(a) 20 min and (b) 45 min (The continuous curves include all the terms

while the dashed curves are drawn neglecting the effects of diffusion and

perturbation in the loss rate.)

can be easily explained in terms of neglect of the lossand diffusion terms. At lower heights (<200 km) thevalue of P is high (~ 10- 2 sec -I) and is comparablewith the wave frequency w. Thus, the inclusion of thisterm reduces N' considerably. Since P falls offexponentially with height, .its effect also gets reducedaccordingly and above about 250 km the contributionof this term is negligible. The effect of inclusion of thediffusion term, on the other hand, increases with heightand the discrepancy due to the neglect of this termbecomes important only above about 400 km. It is,however, clear from Fig. 6 [(a) and (b)] that the effectofthe diffusion term is much less important than that ofthe P loss term.

In gravity wave studies using measurements ofcritical frequency tluctuations38•39 or group heightoscilIations37•4o, a limited range of heights (about 250­400 km) is involved and as such equation Eq. (19) canbe used. However, in the present work, since theintegration of N' is to be carried out over a large heightrange, a fairly complete model for the gravity wave­ionization interaction is necessary.

In a detailed study Hookel9•36 has studied theanisotropy of the ionization response at F-regionheights, using a simplified gravity wave model ofHines8 and the interaction model given by Eq. (19). Inthe present work such a detailed study is not possibledue to the complexity of the wave model and theinteraction processes; indeed, such a study would bebeyond the scope of the present work as we are mainlyinterested in the electron content perturbations. It is,however, useful to consider a few illustrative cases foran understanding of the physical basis of the

-,

-.

53

INDIAN J RAD10 & SPACE PHYS, VOL 11, APRIL 1982

Fig. S--Curves showing (a) the geometry involving the wave vectork(k" k", k.,) and the magnetic field 80 for a wave propagating along <p

= 180°,OkB being the angle between k and Do and (b)height variaiton ofOkS for a wave period of 45 min

as

l' = fZu N' dr ... (20)Zl

where dr is an element of path along the satellite toobserver ray. In the present work we have confined ourattention to only transmission from geostationary sate­llites. Such a restriction gives considerable simplificationsince we need only to consider a fixed magnetic fieldgeometry. In particular we have taken for ourcalculations the satellite ATS-6 stationed at 35°E over

the equator and an observer located at Delhi. Theradio ray-path is considered to be a straight linewithout refractive bending. In solving Eq. (20) we takeintegration steps of 5 km within which N' is consideredto be constant. It is also necessary to specify the lowerand the upper limits of integration Z, and Zu, defined byN' = O. For all the wave periods we take the lower limitof integration at 150 km. The effect of moving thislower boundary by ± 20 km is found to give amaximum error of less than 0.5 % in I'.The upper limitof integraiton depends on how high the wave can

penetrate with substantial amplitude. We take an

90 120

8k B, deg

400

E 350

X~l-Il!> 300-WI 250

200150

60

s

(0)gets very rapidly attenuated for qJ = 0° while there is

considerable amplification for qJ = 180° (see Fig. 3,amplitude curve). Hence, at higher heights above 250km the value of N' / N 0 for cp = 0° falls off to a muchsmaller value than for cp = 180°. For cp = 45° thesituation is almost the same as for cp = 0°. The effect ofthe wind is small for cp = 90° and 135°, these directions

being roughly transverse to the wind. However, due tothe unfavourable geometry between k or Vn and Bo thevalues of N' j N 0 are relatively smaller. In the case ofpropagation along cp = 135°, we see that N'/No has adouble peaked structure w'ith a minimum around

225 km. This behaviour can be explained in thefollowing way 7. On examining Eq. (17) we see that thedominant term in the numerator within the

parentheses is (k. Bo), which varies with height due tothe variation of kz,. This factor is positive below225 km, negative above it and passes through zero atthis height. This gives rise to a minimum in N' j No at aheight of 225 km.

In Fig. 7(b) is shown the effect of varying the azimuthof propagation on N'jNo for the 45-min wave. Here

again at the lower boundary we take an arbitaryconstant wave amplitude. From Fig. 7(b) we note thatthe magnitudes of N'jNo remain substantial up tomuch higher heights than for the 20-min wave shown inFig. 7(a). This is clearly due to the ability of the longerwave periods to penetrate to higher height with greateramplitude as discussed in Sec. 4.2. The nature ofazimuthal variation of N'jNo can be interpreted in asimilar way as in the previous case for the 20-min wave.Fig. 7(b) shows a double peaked height variation ofN'jNo, with an intermediate minimum for cp = 135°and 180°. This minimum is very prominent for cp

= 180° and occurs at a height of 280 km. The geometryinvolving k and Bo is fairly simple to visualize in thiscase and is shown in Fig. 8(a). The vector k makes anangle (}kB with Bo, and r [ = tan - l(kz,jkh)] with thehorizontal. From the geometry, (}kB = 180° - (r + D),where D is the angle of dip. Now since kz, has beencalculated over the entire height range (Fig. 4)and kh and D are known, ()kB can be evaluated.Variation of (}kB with height is shown in Fig. 8(b). It isfound that (}kB gradually increases with height. Itcrosses the value of 90° at the height of 280 km,implying thereby that k. Bo vanishes at this height, ispositive below and negative above it. This incidentallyalso confirms the explanation given for the doublepeaked height variation of N'jNo in Fig. 7(a).

4.4 Perturbation in the Electron Content

Having obtained the electron density perturbationover an extended range of heights in the ionosphere it issimple to integrate it along the satellite observer ray­path to get the perturbation in the electron content, 1',

54

"I'll I 111111 II i I~ 'I.".. "..~.~l~,:.:J!~l,,"d.1~~";" 1!lIUIIIIIK 'l!l 'II! I I'ld 'I '11,-~-,'''"" -,,-~..... -"-"-'- --,. ,.. '.'~"' .."-"",,,

SEN GUPTA &: NAGPAL: MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEC

upper limit of 500 km for the 20 and 30-min periodwaves and 600km for the larger scale waves of 45-minperiod. Once again shifting this boundary by ± 20 kmgives less than 0.5 % error in 1'.

Several workers in the past have calculatedtheoretically the spectrum of the expected response ofeither the electron density or electron contentperturbations as a function ofthe wave period38•41 -44.

The theoretical spectra have been compared with theexperimentally observed ones in order to verify someaspects of the gravity wave-ionization interaction. Thecalculations of such theoretical spectra, however,involve certain assumptions which are somewhatstringent. Some of these assumptions are given belowas examples.

(i) The direction of propagation of the wavesis assumed to be constant and generally equatorward,based on the bulk of experimental observations (Ref.34and references therein). Propagations off theequatorward direction are found to occur frequently.

(ii) The horizontal wave number kh is generallyassumed to be constant over the entire spectrum. Thisis quite an unreasonable assumption if we areconsidering the entire spectrum ranging from 15 to 80min.

(iii) The amplitudes are assumed to be constant forall wave periods. This assumption suffers from twoproblems. First, we do not know anything about thenature of emission spectrum of the gravity wavesources and, secondly, the different wave periods wouldsuffer unequal attenuation in the vertical direction.

Apart from the above mentioned difficulties, theinclusion of neutral winds poses a new problemin thatthe shape of the theoretical spectra should vary withtime because of diurnal variation of the wind vector.

Thus in the present study we have not attempted toconstruct the theoretical spectra of I'versus period.Instead, we have considered the three medium scalewaves of20, 30 and 45-min period as described in Table1,and studied the perturbation I' produced by each oneof them as we vary the azimuths of propagation. Theparameter I' is calculated with and without consideringthe winds. Before going on to study the azimuthalvariation of 1', we discuss some factors, namely, thephase cancellation and the ambient electron densityprofile, which are likely to affect 1'.

4.4.1 Effect of Phase Cancellation-Theparameter I' is obtained by integrating theinstantaneous values of N' along the satellite observerradio ray-path. Now in general, the phase of N', beingcaused by a propagating wave, is not the same over theentire ray-path but may vary regularly. Thus, onintegration, the contributions from some portions ofthe ray-path would tend to cancel out those from some

other portions oscillating in the opposite phase. Thiseffect is known as phase cancellation and introducesbiases in 1', reducing its value anisotropically in theazimuthal plane depending on the relative geometry ofthe radio ray-path and the phase front of the gravitywave. The value of I' is naturally greater for waves forwhich the phase cancellation is small. An understand­ing of this phenomenon is thus worthwhile for anefficient interpretation of the response of the electroncontent to gravity waves.

We can make the following basic observationsregarding the phase cancellation effect.

(i) The effect of cancellation is minimum when thephase of N' is the same at all points on the radio ray­path. This would obviously happen when the ray-pathlies in the phase front of the wave.

(ii) The cancellation is maximum when the phasechange of N' along the radio ray-path is most rapid.This would happen when the phase front isperpendicular to the ray-path, i.e. when the wavevector k is along the ray-path.

(iii) Since the contribution to I' reduces as we moveaway from the peak, the cancellation effect shouldreduce if either we make the electron density profilenarrower or increase the wavelength of the gravitywave relativeto the scale height of the electron densityprofile. In the limit when the wavelength is very large,e.g. for a large scale TIO, the cancellation effect shouldbe small.

Georges and Hookes were the first to investigate thephase cancellation effect analytically by using idealizedmodel of constant amplitude transverse gravity wavesin an isothermal atmosphere and several modelionization profiles. Their analysis, although simplified,brought out clearly some basic physics behind themechanism of the cancellation effect, as noted in thethree points mentioned above. In a realisticatmosphere there are several departures from theidealized behaviour. For instance, we have seen in Sec.4.2 that the wavefront tilt varies with height in arealistic thermosphere. Thus the condition ofcoincidence or perpendicularity ofthe ray-path and thewavefront can never be satisfied at all heights.However, we expect the phase cancellation to bedetermined effectively by the angle between the ray­path and the wavefront over the height range where N'is largest.

In the present study, we do not attempt toinvestigate the phase «ancellation effect under allpossible ray-path geometries and wave parameters, asit would be very difficult due.to the complications in thenumerical approach of the solution. We shall restrictourselves to showing some illustrative calculations forthe fixed ray-path geometry involving ATS-6 to Delhiray-path. On examining Eq. (l7)for N' we find that the

55

INDIAN J RADIO & SPACE PHYS, VOL II, APRIL 1982

parameters and the interaction geometry. Thus ignoring

this phase change its effect is discussed briefly in Sec. 4.4.3

while explaining some of the features of the azimuthalvariation of I'.

Figs 11 and 12 show some typical illustrations of phasecancellation effect for two waves of 20- and 45-min

period, respectively. The parameter I~ ( = rr N' dr, i.e. theelectron content perturbation integrated along the ray­

path up to a certain height z) versus height z is plotted forseveral azimuths of propagation. The I~curves are seen to

rise and fall periodically indicating clearly the partialcancellation of the contribution from adjacent regionsoscillating in opposite phase. At great heights, as N' tendsto zero, all the I~curves tend to a constant value equal to1'. Evidently, the greater the phase cancellation effect, themore prominent are the dips in the I~curve. Consideringthe wave with 20-min period (Fig. 11), we notice thefollowing features.

(i) For propagation azimuth opposite to the ray-pathazimuth (<p =60°), the phase front makes the largest anglewith the ray-path. The geometry of the ray-path and thewavefront for this azimuth shows 0=69° at 250 km

(where N' is largest). Phase cancellation effect for thisazimuth is very severe as shown in Fig. 11. Near the F­region peak where N' is large, almost total cancellationresults as shown by the sharp dips in the I~curve.

In order to assess quantitatively the reduction of l' dueto phase cancellation we adopt the following procedure.We calculate r assuming that N' has the same phase at allpoints on the ray-path (so that only the magnitudes of N'in each slab get just added up on integration). This gives avalue of l' = 4.6 x 1014 el. m - 2. Fig. 11, however, shows

that l' = 0.17 X 1014 el. m - 2, implying a reduction by a

90 0 90

PHASE SHIFT,deg

Fig. IO-Height variation of the phase shift between N'and Vn for rp= 210' for a wave period of 20 min

phase variation along the ray-path consists of two

parts.

(i) The first is the phase variation due to the neutral

wave itself, i.e. the term (Vn• Bo). This term has a space­time dependence of the form expj(wt - k. r). This

spatial variaiton arises from (k. r) (r is the radio ray­path vector). As observed earlier, the effectiveness of

phase cancellation depends on how rapidly the phaseof N' is changing along the radio ray-path. Thus for agiven magnitude ofk, the smaller the angle 0kr between

k and r, the greater will be the cancellation. Fig. 9shows the geometry involving k, the wave front and r

for a general case of propagation of the wave along anazimuth <po The angle ()between the phase front and r isgiven by

()= sin -1 [sinE~sinr - cos Escos rcos(As - <p)] ... (21)

where Es and As are, respectively, the elevation and theazimuths of rand r is the tilt of the wavefront with the

vertical. We note that () is the complement of 0kr so thatthe closer it is to 90° the greater is the phase cancellationeffect. This point will be demonstrated in our laterdiscussions.

(ii) The second cause of phase variaiton arises due tovariation with height of the phase shift between theneutral wave Vn and the ionization response N.Normally, the height variation of this phase shift is notvery much, as shown by Clark et al.31 They foundvariations of only about nl2 over the entire height range200-600km in most of the cases. However, for some

azimuths of propagation this phase variation can beconsiderable. Fig. 10 shows, for a wave period of 20 min,the height variation of the phase shift between Nand Vn>

for <p =210° (the same variation would hold good for <p

= 150° also, because of meridional symmetry). Around200-300 km there is a phase change of nearly n radians. Itis difficult to include this second source of phase variationin a general way for the calculation of phase cancellationeffect, since it is a complicated function of the wave

l ,v ERTICAL

TO SATELLITE,

I

II

X,NORTH

Y, EAST

Fig. 9-Geome1ry involving the wave vector k(kx• ky, ku)' the phasefront and the radio ray, path r (Here (I is the angle between the phase

fron1 and r.)

56

450

400

E 350~~

~:r: 300l')

W:r: 250

200

150

T=20 mln (NO WINO)

•,

,~ ! Ilil Ii II" I 'j" l·'r-

SEN GUPTA & NAGPAL: MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEC

450

4.4.2 Effect of Ambient Electron Density Profile­The electron density perturbation, N', given by Eq. (17)is directly proportional to the ambient electron density,No. Now in the electron content perturbation [Eq. (20)],the maximum contribution comes from the height rangewhere N' is maximum. The maximization of N' arises dueto the combination of two factors, viz. (a) the fractional

perturbation, N'/N0, which is a function of the amplitudeof the gravity wave and the relative geometry between thewave parameters and the magnetic field; and (b) the valueof No which has a maximum at the F-Iayer peak. Thus

changes in the No profile are likely to alter the value of thefractional electron content perturbation 1'/10'

In order to illustrate the effect of variation of the No

profile on 1'/10 we have considered the following twosimple cases.

(i) In the first case, keeping the shape of profile and theheight of the peak, hm, constant, we vary the peak electrondensity N m' The only way in which such a change wouldaffect N' /N 0 is through a change in the amplitude of vn

due to altered ion drag effect. However, we have noted in

Sec. 4.2 that the ion drag effect and its variations are

relatively small compared with those due to otherfaciors,e.g. the neutral winds. Thus changes in the ion drag would

(iii) For propagation azimuth transverse to the radioray-path (qJ = 150°), the phase front is oriented at a smallangle 9 = 15S at 250 km. This results in almost nocancellation as indicated by the monotonically increasing

I~curve. The reduciton in I' in this case is calculated to be

only by a factor of 1.13.(iv) An interesting case is the propagation along qJ

= 330° which is diametrically opposite to qJ = 150°. Assuch (J should have the same value as for both azimuths

because of the geometrical symmetry. However,due tothe presence of the neutral winds, the wavefront tilt isdifferent for the two azimuths and this results in differentvalues of (J. The calculations show the value of (J = 30.1°for this azimuth, at 250 km. Fig. 11 also reveals a

significantly greater phase cancellation, by a factor of 3;5,in this case.

Phase cancellation is less effective for the 45-min wave

period as shown in Fig. 12. As before the effect ismaximum for qJ =60° for which we get 9 = 510. Similarcalculations as before show that the phase cancellationreduction for this azimuth is by a factor of 4.1, which issmaller than that for the 20-min wave period. The reasonfor this relatively smaller effect for the larger wave periodis its larger wavelength. From Fig. 12 it is clear that theseparation between the successive dips in the I~curves aremuch larger. Unlike for the 20-min wave period, thephase front is better aligned with the ray-path for qJ

=240° (9 = 8°) than for qJ = 150° (9 = 26°). Fig. 12 alsoshows greater cancellation for qJ = 150° than that for qJ

=240°.

4>= 1500

4>= 3300

4>= 2400

T=45min (1400 hrs WIND)

200

1=20 min (1400 hrs WIND)

250

150

1014 1015 1016

I, I -2Z,e .m

Fig. 12-Same as Fig. 11 but for the wave period of 45 mm

factor of 27 due to phase cancellation. This is a very largereduction amounting to an almost total cancellation for

all practical purposes.(ii) For propagation azimuth same as the ray-path

azimuth (qJ = 240°) the phase front is oriented at an angle9 = 30S at 250 km. The cancellation effect in this case is

thus less than that for qJ = 60° but is still considerable.Similar calculations as in (i) show a reduction by a factor

of 4 due to phase cancellation.

'0'4 '0'5I 2

Iz,el. m-

Fig. II-Variation with height of the parameter l~fordifferentvaluesofthe wave azimuth for the wave period of 20 min at 1400 hrs

600

550500, E

~ 450-t-:I:

c.!) 400lJ.J:I: 350

300250200150

1013

57

INDIAN J RADIO & SPACE PHYS, VOL II, APRIL 1982

10

1.5

1> =180'(1400 hrs WIND)

90'

0.6

0.8 1110, '/,

N

1.0

,/TO SAT.

T(min) Vnz(150km)-- 205m sec'!--- 302m sec'!--- 45 1 m 6ec-1

WINTERNO WIND

270'

o225 250 215 300 325 350 315

PEAK HEIGHT, km

180'

Fig. 14--Variation of f'j1 0 with the azimuth of wave propagation in theabsence of neutral wind for the wave periods of 20, 30 and 45 min

Fig. l3-Curves showing Ihe variaiton of 1'/10 with the

height of F -layer peak for the wa ve periods of 20 min and45 min

give rise to relatively small changes in Vn and hence in

N'/No· Changing the value of Nm would change both I'and loin a similar way. Thus the overall effect on 1'/10 dueto just changing Nm are expected to be small. It is found ina typical computation that for day (at 1400 hrs) electrondensity profile (Nm = 5.25 x 1011 el. m - 3), changing Nm

by ±25 ~~changes 1'/10 by less than ±2 % for the 20-minwave period and by less than ± 6 ~~for the 45-min wave

period. The larger variation for the longer wave period isevidently due to a greater influence of the ion drag effect.

(ii) In the second case, keeping the shape of the profileand Nm constant, we vary the height of the F-Iayer peak,

hm• Such a variation has a substantial effect, since varyinghm may take the peak of the No profile to regions withgreater or smaller values of N'INo resulting incorresponding increase or decrease in 1'/10' In Fig. 13 weshow two typical illustrations of the variation of 1'/10 withhm· For the 20-min wave period, 1'/10 initially shows anincrease with increasing hm, till it starts decreasing for hnf

> 250 km. The nature of variation in 1'/10 is easilyunderstood if we refer to Fig. 7(a) showing the heightvariation of N'INo for the 20-min wave. We find that

N'INo also shows a peak around 250km above which it

decreases rapidly. For the 45-min wave period, 1'/10

continues to increase with increasing hm up to about375 km. This increase is also explained by referring to thenature of variation of N'INo shown in Fig. 7(b).

We have not illustrated the effect of changes in theshape of the No profile. However, it is expected that suchchanges would introduce complicated changes in 1'/10

due to altered weightage of the regions with differentvalues of N'jN o.

Thus, it is clear that the ambient electron densityprofile, especially with regards to its peak height, hm,

would have substantial effect on 1'110' The choice of the

electron density profile has to be given due attention inany quantitiative calculation of 1'/10'

4.4.3 Electron Content Perturbations without

Neutral Winds-In Fig. 14 are shown the vari­ation of 1'110 with the azimuth of propagation forthree waves of 20, 30 and 45-min periods in theabsence of neutral winds. The ambient atmosphereand the ionosphere models are typical of winternoontime low solar activity conditions. The curvesare drawn assuming a certain fixed value of Unz for eachwave at the lower boundary, i.e. 150km. The value of Unz

is, however, different for the three waves and a direct

comparison of the magnitudes of I'll 0 is not possiblebetween the three curves. The azimuthal variation of 1'110

for all the waves is generally complicated and anisotropic.It is small in all the cases in the range 30-120° as a result ofexcessive phase cancellation. The variation of () for thethree waves (without wind) shown in Fig. 15 reveals highvalues over this range with a maximum at <p = 60°.

Relatively larger value of 1'/10 for the 45-min wave overthe range <p = 30-120° reflects the relative ineffectivenessof phase cancellation for the larger period. In the rest ofthe azimuth range J' is larger. All the curves show apeaked structure with a first peak around 160-180° (thispeak is very prominent for the 20-min wave period), asecond peak around 240° and a third peak around 330­345° (Fig. 14).The third peak is almost absent for the 20­min wave period. The kind of complicated response arisesfrom the behaviour of the magnitude of N' and theeffectiveness of phase cancellation that oppose each other.The smooth azimuthal variation of () shown in Fig. 15(and consequently the phase cancellation effect), however,does not explain some of the fine structure of the I'll 0

curves. For instance, for the 20-min wave the values of ()for <p = 150° and 210° are, respectively, 25 and - 2.3°. Thevalues of N' are same for both the azimuths because of the

58

IIH I II

SEN GUPTA & NAGPAL: MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEC

T=20min (NO WIND)

450

Fig. 15-Curves showing the azimuthal variaiton of (J, in the absence of

winds, for the wave periods of 20, 30 and 45 min

150

E.>£

~ 310--:I:

~ 300W:I:

ZlO

4.4.4 Electron Content Perturbations IncludingNeutral Winds-Presence of the background neutralwind modifies 1'/10 considerably from that in thewindless case. Fig. 18[(a)-(c)] shows the azimuthalvariation of 1'/10 respectively for the three mediumscale waves of 20, 30 and 45-min period, inthe presence of neutral winds for various times of theday. The times chosen are 0200, 0600, 1000, 1400,1800 and 2200 hrs LT. The atmospheric model and thewind proftles are typical of winter season and of low solaractivity period. For each curve we take, as before, fixedvalues of VlIZ at the lower boundary of 150 km. The gravitywave amplitude (Sec. 4.2) undergoes selectiveamplification or attenuation depending on whether thepropagation has a component against or along theneutral wind direction. Consequently, I' also becomesselective, with greater response in a direction roughlyopposite to the wind direction. The influence of the phasecancellation is also of considerable importance in shapingthe response curves especially for the smaller periods. Fig.18[(a) and (b)] shows the following features.

During the nighttime at 2200 and 0200 hrs, the wind isvery strong and has a large southward component. The fresponse in the southern sector is thus very small. Largevalues of 1'/10 occur over the band of azimuths of about240-360° with a peak around 330-345°. As expected, thewidth of the allowed range in the response is largest forthe 45-min wave period due to smaller phase cancellationeffect on it.

During the early morning hours (0600 hrs) the wind istowards southwest and is very strong. However, 1'/10,

cannot peak along the azimuth which is diametrically

200

400

IOU 1014 II/'l'z,ft. m-2

Fig. 17-Height variation of I~ for the 2o..min _veperiod for the two azimuths 150 and 2100

results that complicated and highly anisotropic 1'/10

responses, like those shown in Fig. 14, occur when thesatellite ray-path azimuth is very much away from theobserver's meridian.

200

2S0

360

180 ISO

270180

AZIMUTH, deg

90

<D

-45

-90o

E'000~:I:

~QOO,~a: 800,..:«VI'.700

VlIIIo600

C)Zo~ SOOUJU~400--Vl

0300QO

/",/

-0-- QOPHASE,deg

Fig. 16-Curves showing the variation along the satellite observer ray­path of the phase of N' (continuous curves) and of v. (dashed curves) for

the two azimuths 150 and 2100

meridional symmetry. Thus we expect greater responsefor cp =210° than for 150°, while the curve in Fig. 14shows just the opposite. This discrepancy is explainableby considering the phase shift (between N' and v,,)variation with height as mentioned in Sec. 4.4.1. The neteffect of phase cancellation is determined by the jointaction of the phase variation of N' due to v. alone and theN' - v. phase shift, along the ray-path. Now dependingon whether these two phase variations add up in the samesense or cancel out, the effect of phase cancellation isenhanced or reduced from that expected by taking v.phase variation alone. This point is illustrated in Fig. 16.The dashed curves show the phase variation of v. alonewhile the continuous curves show the phase variation of

N' due to both v. and the variation of (N' - v,,) phase shift(shown in Fig. 10). We fmd that the total variation of thephase of N' is indeed greater for (p = 210° than for 150°.Fig. 17 shows the variation of I~for the two azimuths 150and 210°. We notice a somewhat sharp dip in the I~for cp

=210° as a consequence of the rapid phase changeshown in Fig. 16. It is, however, found that such drastic(N' - v,.) phase variations occur only for very limitedazimuth ranges.

In an earlier study Davis6 calculated the azimuthalvariation, in the absence of winds, of 1'/10 for differentlocations and satellite geometries. It is apparent from his

59

INDIAN J RADIO & SPACE PHYS, VOL II, APRIL 1982

la)lb)

Ie)

•90

,

.• I / W2200 hrs·~.A1800 hrs

N WINTERT =45 min -I

NEuTRALI100m see-Il Vnz l150kml :Im seeWIND 08I'/Io.'I.

0200 hrs

•270

270·

270·

OOOhrS'...•""

200 hIS 'tN WINTERT=30min _I

08 Vnz 1150km)=2m seeI'/Io • ."

NEUTRAL I100 m see-I4 WIND-INEUTRALIIOO m see0'4 WIND

N Wlm-ER

'·2 T: 20 min _IVnz 1150km) :5m see

08 II Io •'/,

•270

270· .< i.W0600hrs,// . 'r0600hrs..... ,/'/' W0200hrs

TO SAtw,ooOhrsIO'4O~ 1"1400 hrs

....

'To SAT.

o180

Fig.i8-Azimuthal variation ofI'jlo for the wave period of: (a) 20 min, (b) 30 min and (c) 45 min with Vn• = 5,2 and I msec- "respectively (The

wind velocity vectors at the height of 300km are indicated. The small straight arrow heads on the diagram represent the experimentallyobserved velocity vectors of TIDs.)

opposite (cp =60°), because of the large phasecancellation effect along that direction. The responsecurve thus shows two small peaks, one along 0-15° andanother along 150-165°.

During the daytime (1000 and 1400 hrs) the wind hasmainly a poleward component. Consequently I' is verysmall in the northern sector. Very large response occursover the band of azimuths 120-190° with a peak around165°. A somewhat smaller peak also occurs around 220­300°. The slight rotation ofthe response curves from 1000to 1400 hrs with the rotation of the wind vector is evident.

In the evening (1800 hrs) the wind direction has astrong eastward component. The 1'/10 value for the 20­min wave period shows two peaks along 180 and 300°.The deep minimum in between the two peaks occursbecause of the excessive phase cancellation along thisdirection. However, for the 30 and 45-min periodsappreciable response occurs over the entire band 150­345°.

Thus with a complete 360° rotation of the wind vectorover a whole day, the response of I' also goes through asimilar 360° rotation. The rotation of the response curvesis, however, not very smooth because of the constraintsimposed by the phase cancellation and the anisotropicnature of lV'.

Similar calculations performed for the summer timewind and atmospheric models give qualitatively similarresponse patterns. Fig. 19 shows the response ofl'/lo for atypical nighttime (??oo hrs) and daytime (1200 hrs)conditions for the wave periods of 20, 30,45 and 60 min

(The 6O-min wave has Vph=200msec-1 and Ah = 720km). A notable feature of the nigh time conditions in Fig.19 is that the 1'/10 response for the lower period (20min) isconsiderably suppressed compared to that during thedaytime as well as those for the higher periods. Thisfeature is explainable on the basis of the considerable rise

of the nighttime F-Iayer and consequent reduction in 1'/10

(see, for example Fig. 13). A similar suppression of 1'/10 isevident for the 20-min wave during winter nighttime alsobut it is less marked.

Thus it is necessary to mention the limitations inherentin some of the assumptions made in the modelcalculations described in this and the previous sections.These are as follows.

(i) The satellite-observer radio ray-path is tilted at anangle, considerably away from the vertical. So the portionof it intercepted by the ionosphere covers a few degrees oflatitudes. We have, however, not taken any latitudinalvariation of either the dip angle or the ambient electrondensity profile. These assumptions may be somewhat

60

,.,,;q IlilI I" I' .'~ ,j·l -

-----------------~-_._~~------.

SEN GUPTA &: NAGPAL: MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEC

Fig. 19--Azimuthal variation of 1'{10 computed for the different waveperiods for typical daytime (1200 hrs) and nighttime (??oo hrs) insummer (Small arrowheads for the experimental TID velocity vectors

are also shown.)

5 Com ••••••• with Observations

In Figs 18 and 19 we have shown by small straightarrowheads, the phase velocity vectors of medium scaleTIOs obtained by using the multistation ATS-6 Faradayrotation data recorded at several Indian stations4S• For

each TIO the arrowhead is shown alongwith theappropriate theoretical curve which corresponds to the

questionable since it is known that both these parametershave finite latitudinal gradients at low latitudes.

(ii) In constructing the response curves we take thewind profde for fIxed times. In actual observations,however, the wave, especially the large period ones, maylast for a few hours. Over this much interval of time the

wind vector may change appreciably near the earlymorning and evening hours ..

(iii) We have taken waves with the same amplitude forall the azimuths at the lower boundary of 150 km inconstructing the response curves. Fairly high speed windsare, however, known to exist below this level and can

result in considerable anisotropic attenuation of the waveamplitude. Inclusion of these underlaying winds willprobably make the response curves more selective thanshown in Figs 18 and 19.

wave parameters and time of the day closest to those ofthe observations. We fInd that in all the cases the TIOazimuths are along directions for which the theoreticall'response is substantial. This provides a confIrmation ofthe validity of theoretical response curves.

We note, however, that our present observations arenot very extensive, so that a thorough comparison is notpossible for various times of the day. A much moreextensive set of measurements of the velocities of medium

scale TIDs has been reported by Cowling et al.17 Theirmeasurements are for two summer months of 1968 at

Urbana, and cover almost the whole day. The TIOvelocity vectors observed by Cowling et al.17 are shownin Fig. 20. The grouping of the TIO observation timeS:isthe same as done by Cowling et al.17 In each of thesefigures we have also plotted the theoretical 1'/10 responsecurves for two typical medium scale TIOs with periods27.5 and 47.5 min and horizontal phase velocity of ISO

msec -1. The parameters for the ionosphere and neutralatmosphere have been chosen for midlatitude, summer,moderate solar activity conditions. The satellite-observerradio ray-path azimuth and elevation are, respectively,1300and 310(corresponding to ATS-3-Urbana ray-path~The wind model is that given by Cho and Yeh46• For eachof the curves in Fig. 20, the wind velocity is chosencorresponding to the midpoint of the time group andtaken constant with height (since we do not have theheight profile of the wind).

Examination of Fig. 20 shows clearly the gradualrotation of the theoretical response curves of 1'110 due torotation of the neutral wind vector. The cluster of the

observed TID azimuths in different time groups alsoshow some tendency of rotation in a similar way. As aresult we observe a very good agreement between the

theoretically allowed bands of propagation azimuthswith those of the observed TIOs. Indeed, there are only 6TlDs out of a total of 45 which seem to be moving alongazimuths for which the theoretically expected l' responseis small.

In their report Cowling et al.17 have also sought toexplain the diurnal trends in the TID azimuths in ~rms ofthe directional filtering of gravity waves due to winds. Thenature of the fIltering considered by them involveddiffering propagation times, in reaching up to theionosphere, of waves with different azimuths. This isunlike the directional f1ltering considered in the presentstudy which is due to anisotropic dissipative effectscaused by the winds. The anisotro}1y introduced due tothe gravity wave-ionization interaction and the effects ofphase cancellation were not considered by Cowling etal.17 In the present work we have taken into account boththe above factors and thus the theory is much more

complete. The good agreement between the experimentand theory shown in Fig. 20, is also indicative of this.

I -1!OOmKe

10·

10·

52

2

1

Vn,l150km I-1

m5tC

SUMMER NIGHT{OOCO hIS I

SUMMER DAY

(1200 hIS I~ o·e

/''''','I V( //""

.....-:/~ \

// \ \TO SAT \ 1. \ :

\ \. \

-,) \T-t.5min\ \\ \

\ \

\ ,.,

110·

170·

, '

61

INDIAN J RADIO & SPACE PHYS, VOL II, APRIL 1982

II

(b)

T=27·5 mln

"/ ,,/ "-

TO SATELLITE

(d)

o~ 0'011+100..•...

TO SATELLITE

(c)

""--0

250

'~m/stcTO SATELLITE

N

0'02-+200

N

r0·02

200 (a)0,02 200T

/~mm;'0·015 150l~

0'015150~0

E o ••H

:0-... H'j...j 0'01 100til ::;:;- 0·01

WIND VECTOR

1 I294 m1sec 310 m/stc

Fig. 20-Comparison between the experimental observations of the velocity vectors of medium scale TIDs reported by Cowling et al. 1 7 andthe theoretical computations of the azimuthal variation of I'/lo for two representative waves with periods 27.5 min (continuous curves) and47.5 min (dashed curves) and horizontal phase velocities 150m sec -1 in respect of observation time groups of: (a)0600-1200 hrs; (b) 1000-1400

hrs; (c) 1200-1800 hrs; and (d) 1800-2400 hrs

We notice in Fig. 20(d) that there is an appreciablereduction in the magnitude of l' during nightime (at 2100hrs.).This is reflected in the relatively smaller probabilityof occurrence of TlDs at night as compared with thedaytime observed at Urbana47•

6 Conclusion

In the present paper we have developed a fairlycomplete model for the gravity wave induced

perturbations in total electron content. The results ofcomputations bring out, among other things, theimportant role played by the neutral winds, dissipativeprocesses and phase cancellation effects in determiningthe TEe response. A general feature of all the responsecurves (Figs 14-19)is the marked azimuthal anisotropy asdescribed earlier by Davis6 and Bertel et al.7• This, apartfrom being due to the nature of gravity wave-ionizationinteraction, is also caused by two additional factors,

62

"1 '111 tl'l I ! I "', .' II ~:~J!l!tl. ":~"'!.!j..~,~l!,~t!ll,~l!j,I!.. .;:,.I1,~.,..J)~l.I,,,U1.,,;,;,, J ,l;.LI..:oiI.i.l I ,J..,",L,.

,,; SEN GUPTA &; NAGPAL: MODEL FOR GRAVITY WAVE INDUCED PERTURBATIONS IN TEe

t.

namely, the phase cancellations and the directionalfiltering due to neutral winds. The phase cancellation is (i)directly related to the angle, 0, between the wavefront ofthe gravity wave and satellite observer ray-path, and (ii)inversely related to wave period or the wavelength of the

gravity wave. For the waves considered here (Table 1),theangle 0, and consequently the phase cancellation ismaximum when the wave azimuth is opposite to thesatellite ray-path direction. This is reflected in all the 1'/l0curves also in the form of reduced response in the azimuth

range near around 60°, which is opposite to the ray-pathazimuth of 240°. The diurnal variation in 1'/10 responsecurves is caused by a diurnal variation of the neutral windvector. Thus with a complete 360° rotation of the windvector over a whole day, the response of I' also goesthrough a similar 360° rotation. Satisfactory agreementwith the TIOs observations obtained through ATS-6 and

ATS-3 satellites provides a fair test of the modelcalculations.

References

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(1975) 785.

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during June, 1972.44 Yeh K C, DuBroff R E & Nagpal 0 P, Ann Geophys (France~ 32

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63