Generation of gravity waves by inhomogeneous heating of the atmosphere

Ioumalof Atmospheric andTemsaial Physics,Vol. 42,~~.877-883 Pergamon Press Ltd.1980. Printed in Northern Ireland

Generation

of gravity waves by inhomogeneous C. COT andH. Service d’ACronomie (Receioed

du CNRS,

28 December

heating of the atmosphere

TEITELBAUM 91370

1979;

Verrieres

le Buisson,

France

in revised form 8 May 1980)

Abstract-Local variation of atmospheric heating which might occur in inhomogeneities of various constituents such as ozone or molecular oxygen may generate gravity waves. These perturbations are

induced by the terminator crossing constituent inhomogeneities of short lifetime. The quasi-point heating model developed here shows that the largest amplitude must appear vertically above the source, where the perturbation frequency is close to the Vaisala-Brunt tions not band limited in the frequency suggest several characteristics 1. INTEODUCFION

occurrence of irregularities of different kinds in the lower thermosphere and in the region of the solar terminator has been reported by several authors. It was generally interpreted in terms of an increase of the occurrence of gravity waves at sunset and sunrise as suggested by REES et al. (1972) to explain the irregularities of wind profiles measured at sunrise. HERRON and DONN (1973) measured gravity waves (Doppler radio sounder measurements) in the F-region and found an increase at sunset and sunrise. TEITELBAUM and BLAMONT (1977) have reached a similar conclusion from their estimation of the variation of the mean turbopause height during the night. TEITELBAUM and PETITDIDIER (1978) have also shown that there is an increase of short fluctuations in the green line emission at sunset and sunrise which was essentially attributed to a larger occurrence of gravity waves at this time of the day. From meteoritic radar data, SIDI and TEITELBAUM (1978) have made a statistical study of the echo pair which show a velocity difference exceeding 40 ms-’ for a 1 km altitude difference. The histogram shows maxima between 6 and 9 h. CHERNVSHJWA et al. (1977) show the influence of this phenomenon on the appearance of the sporadic layer Es in the ionosphere. This increase of gravity waves has been attributed to the supersonic motion of the solar terminator by BEER (1973), DATA (1974), SOMSIKOV and TROITSKIY (1974) by comparison with the induction of gravity waves during a solar eclipse The

(CHIMONAS

frequency. Numerical of the perturbation.

calcula-

The problem of gravity wave generation has been extensively studied. The effects of nuclear detonation and earthquakes (Row, 1967), of aurora1 currents (CHIMONAS, 1970) of moving sources (KATO et al., 1977), of the terminator passage BEER (1978) have been studied. The model which we will develop mathematically in Section 2 will lead to the study of the consequences of a space localised heating, for a time period not exceeding a day. Compared to previous works, this study is not band limited in the frequency domain, which allows us to find new characteristics in the atmospheric response to a space localised heating disturbance. In addition the existence in our model of a thermal energy localised input source allows us to extend its application to other phenomena such as for example particle precipitation in aurora1 regions. In Section 3, we will describe the characteristics of the perturbation and in Section 4 we will estimate the consequences of the perturbation in the atmosphere.

FOEMULATION OF THE HEATING MODEL AND ITS DEVELOPMENT

2. MATDEMATICAL

As previously mentioned this model describes a space localised thermal heating from sunrise to sunset. The response is studied in an isothermal atmosphere with no viscosity and the amplitude of the phenomenon is supposed small enough to be in agreement with the linear approximation. The symbols used in the mathematical derivation are as follows:

1970).

We will show in this paper that a local variation of atmospheric heating which might happen on inhomogeneities on various constituents such as ozone or molecular oxygen (above 90 km) may also generate gravity waves.

x, y, z = Cartesian coordinates, x, y being horizontal and z vertically upward. t = time u, U, w = gas velocity in the x, y, z directions respectively 877

C. COT and H.

878 g = p= PO= p = pa = H = y= w= k, 1, m = Y0 = K, =

gravitational acceleration mass density unperturbed mass density pressure unperturbed pressure scale height = RT/g specific heat ratio between constant volume and pressure angular frequency wave number, in the x, y, .z directions respectively Bessel function modified Bessel function

TERELBAUM

J can be written

as

/t/c: T

(6)

J= J,S(x)8(~)8(z) 0 otherwise

where 8 is the Dirac function. The different steps on the following calculation will be: to derive P(w, k, I, m), amplitudes of the Fourier component exp i (ot- kx - ly - mz), for three regimes of w; to invert them by Fourier transformation and give the pressure perturbation P(r, x, y, z). We will use the following

f(r) =

Fourier

transform:

= F(o) exp (id) dw ++ F(o) I -m f(t)exp(-iwt)dt m

The atmospheric motions can be described by the equations of conservation of momentum (l), (2), (3), the equation of energy (4) and the equation of conservation of mass (5)

gb, Y72) =

III

G(k, I, m) exp [ - i( kx + ly + mz)]

-cc

xdk dl dm t, G(k, I, m) ,Z 1 ?Zg(x, y, z) exp [i(kx + ly + (2n)3

(1)

mz)l

-m

a6 aP

xdxdydz

-+-_=o

at

ay

a+ aP 1 at+Z-2HP+gR=0

exp (- ikx) dx (3) With these definitions becomes

aP ,,+g(y-1)0-c

dR ti ^ A ^ _-_+au+av+aw=O~ at 2H ax ay a2

the Fourier Jo

transform

of J, j

sin (wT)

8=o’7L-

(8)

(5)

J is the heat input per unit mass per unit time. We will consider quasi-point sources i.e. sources which can be considered as points at the distance from which the perturbation is observed. The variation of J with respect to time is a function of the cosine of the zenith angle, but for mathematical simplification we have chosen for the dependence of J with respect to time a rectangular pulse, this choice is justified at the high altitude of the stratospheric ozone layer. At lower altitude a gradual heat change at sunset and sunrise [for instance a cosine dependence of J(t)] does not affect significantly the final results: short period gravity waves are always generated.

and using (7), (8) and the set of equations (1) to (5) the Fourier transform of P, P can be derived as p=-i

(y - 1) sin (wT) 7427$c*(w2-

cog’) (w*-g/2H-igm)Jo

X

k2+l”+L wZ- wsZ where oI = Vaisala-Brunt

frequency

0, = sonic cutoff frequency

= [(y - l)“‘/C]g = (7/2C)g

P will then be given by taking the Fourier transform of P. First its expression will be derived

879

Generation of gravity waves by inhomogeneous heating of the atmosphere analytically as the space inverse Fourier transform with respect to k, 1, m and the last inverse transformation with respect to f will be computed numerically. For convenience P is split into two different factors: 0(w)= -i

(y - 1) sin (we.&

(10)

7T(27T)3c2(W2- 0,“)

the Fourier

transform

with respect to m of (15)

(OBERHETTINGER, 1957) is for 0 < w, P(w, x, y, 2) =2(w)

exp (-;“J&;-wzJ

X

[i

wZ-giZH+gd

--

for w>o,

(lea)

X

In a first step the term igm will be omitted and will be considered further as a derivative with respect to z and we define

&2?-

w2-

-2(w)

x, y, z)=

w>w,P(w.

(11)

[(

sin

wZ-gi2H+gd dz

R x&J”o,‘)/(w,‘-

w”)1 (lob)

(b) w,
w*z

which leads to

(12) A>0

*=g* w

-w*

of equation (ll), three different cases have to be considered comparing o to wS and w,. (a) w < us. < w, which leads to and

B
and

Am’-B>o

To perform the transformation

A<0

)I

dz > R &.o,“-o?)/(w,z-a?)

and

Am’-Bi-I’>O.

with respect to k leads to

The Fourier transform

exp(-Ixl~1’-tAm’-B)

P(w, n, E,m) = 77

’

y’l’+Arn-B

B>O

(17) (1) if B2- 12-Am’<

0 the integration

in the

complex plane gives

The Fourier

with respect to l leads to

transform

(OBERHETITNGER, 1957)

P(w, x, I, m) = 7r

exp (- 1x1Jl” -(B - Am2

(2) if B2 - 1’- Am’> 0 the same integration P(w,x, 1, m)= -T

sin (1x1dB - Am’JB-Am’-P

1’)

’

P(w, x, y, m) = d&,(~~)

(13)

Jf*(B-AmZ) gives (14)

The Fourier transform with respect to 1 of (13) and (14) leads then to (OBERHETTINGER, 1957)

(18)

and finally the Fourier transform with respect to m leads to (OBERHETITNGER, 1957) P(0, x, y, 2) =;

X

li

e(w)

o’-g/2H+g5

exp i

-.!J:Q

dz 1 Rd( co2- Wc2)/(WZ - w,“, 1

P(w,x,y,m)= -&*Y,(

Jz-1

(19) B/(-A)+mZ1/-A*(x’+y’)).

(15)

(c) w > w, which leads to

Finally defining w, = w,Z/R

R=

Jx2+Y2+~Z

Q_wa;-w: w*

-0

A>0

and

B
Two cases will again be considered: (1”) B-Am’>0 B-Am’--12>Ot

880

C. COT and H.

where

the Fourier

transform

P(W, x, I, m) = - 7r

with respect

sin (1x1JB -Am*

to k is

TEITELBAuM

The Fourier transform with respect to I of (16a), (16b), (19) and (25) which leads to the determination of P(t, x, y, z) has been computed numerically and the mathematical transformations required for the computation are described in the Appendix.

- 1’ ) (20)

J~-Am2-1* B-Am’-l’
3. CHARACPERISTICS OF THE PERTURBATION

m)=rrex~(-~xlJ~*-(B-Am2)

>

p(o,x

l

;

3

From the result of the computation of P(t, x, y, z) we can now describe the spatial characteristics of the perturbation and its temporal evolution. If the perturbation is observed from a fixed point a quasisinusoidal wave is found with an approximate frequency o, = o,(Z/R) which depends on the angle between the observation point, the source and the horizontal line drawn from the source (Figs. 1,2,3) and w, is usually called the characteristics frequency. The amplitude of the perturbation is a maximum at time t = R/C where the time origin is taken either at the beginning (sunrise) or the end (sunset) of the heating. Our analysis shows that the oscillation is not purely sinusoidal: the two components of the perturbation exist at the frequencies wg and W, and the amplitude of the perturbation increases with the observation angle. This explains the beating between these two frequencies and w, observed in the oscillation when w, is close either to o1 or w,. Also when the observer is close to the vertical, o, = wI so that the beating between these two frequencies cannot be observed [its period 27~/(w, - w,) is then too large], whereas the beating between o, and o, is observed (Fig. 1).

(21)

JF-(B-A&)

(2”) if B-Am’<0

The Fourier transform with respect to 1 of (20) and (21) leads to (OBERHE~TINGER, 1957)

p(w, x, y, m) = -;

Y&h=%

m)

(23)

and of (22) I’(,,

x, y, m) = ~K,(hii??J)

and finally the transform and (24) leads to

P(0, x, y, 2) =

X

(24)

with respect

to m of (23)


w’-gl2H+gd dz

R ‘/(o’-

'I u,~)/(o* - a,‘)

.

(25)

5mn

T

Time,

Fig. 1. Amplitude of the pressure perturbation at the following coordinates

h

P as a function of time. The perturbation is observed (with respect to the source) x = 1 km, y = 1 km, z = 30 km. The amplitude is normalised for JO = I ergs g- ’ SK’.

Generation of gravity waves by inhomogeneous heating of the atmosphere

1.

-T

881

’

T lime,

h

Fig. 2. Same as Fig. 1 but for x = 50 km, y = 50 km, z = 20 km.

On Fig. 4, two calculated vertical profiles of the relative pressure perturbation are shown corresponding to two different times separated by a time interval of 5 mn. The wavelength (i.e. the frequency) varies so that the phase velocity of the perturbation increases with altitude. This variation is essentially sensitive to the proximity of the source, and can be used as a test in order to estimate the distance between the observation point and the source.

4. ESTLMA’ITON OF THB AMP-E PEBTU&BATfON

As the SOUPX of the perturbation is the thermal heating produced by spatial ozone concentration irregularities, we have to estimate the order of magnitude of these irregularities in order to compute the amplitude of the perturbation. Measurements of ozone concentration (RANDAWA, 1971) show important variations from day to day at about 50 km altitude. Measurements made by B.I.J.V.

r IO’;;;‘, 5-

-T Time,

OF THE

h

Fig. 3. !&me as Fig. 1 but for x = 50 km, y = 50 km, z = 50 km.

C. COT and H. TEEITELBAUM

882

and time for at least a few days cannot produce tides but only short-lived perturbations. Hence making the hypothesis of a turbulent layer with a vertical thickness of 2 km and with a horizontal extensionof a few tens of kilometersproducing a 20% variation of the ozone concentration at about 28 km altitude, the heating thus produced can be estimated using a previous analysis (TEITELBAUM and COT, 1979). This leads to an increase of the heating of the order of 5.103 ergs g-is-’ at noon. Taking this value to compute the relative induced pressure variation at 110 km altitude and in the proximity of the vertical line drawn from the source a relative pressure variation of 10 -3 is obtained, the induced horizontal wind speed are then of the order of 1 m s-i. 5.

Fig. 4. Amplitude of the pressure perturbation P along a vertical located at x = 50 km, y = 50 km, and for two different times, ---t, = t,; ~ t, = t,, + 5 min.

experiments on NIMBUS IV show also important daily variations of zonal mean ozone concentration above 10 mb and above 4 mb (HEATH, 1974). Vertical soundings of ozone concentration. show 30% variations between 25 and 30 km altitude and in a 48 h interval (Observation d’ozone-Belgique, 1970). At about 50 km turbulent layers may produce ozone concentration irregularities which disappear in a few hours. Between 25 and 30 km irregularities which may also be produced by turbulent layers can move horizontally with the mean flow. It is well known that the passage of the terminator over the ozone layer is explicitly part of tidal forces, but irregularities that are not stationary both in space

CONCLUSION

We have suggested a mechanism in order to explain some dynamical perturbations which appear in the lower thermosphere at dusk and dawn. These perturbations are induced by the therminator crossing ozone inhomogeneities of short lifetime. The quasi-point heating model developed here shows that the largest amplitude must appear at the vertical of the source i.e. where the perturbation frequency is close to VaisaIla-Brunt frequency. As it is difficult from our model to locate precisely the altitude of the source, we can only suggest an altitude around 30 km for inhomogeneities of the ozone concentration. At higher altitude such inhomogeneities are less likely. Nevertheless, one has also to consider that above 50 km the atmosphere becomes less stable and that turbulent layers might be able to create such inhomogeneities in the ozone concentration. These kinds of inhomogeneities can last only for a few hours (BANKS and KOCKARTS, 1973) but their occurrence at the time of the terminator passage will be sufficient to induce the described phenomenon.

REFERENCFS

BANKS P. M. and Kocx~~rs, G. BEER T. CHERNYSHEVA S. P., SHEFIU V. M. and

1973 1978 1977

Aeronomy, Academic Press New York. Planet. Space Sci. 25, 185. Geomag. & Aeronomy 17, 633.

CHIMONASG. DAITA R. N. HEATH D. F.

1970 1974 1974

HERRON T. J. and DONN W. L. ROYAL ME-~OROLOGIQUEDE BELGIQUE KATO S., KAWAKAMIT. and ST JONES D.

1973 1970 1977

.I. geophys. Rex 75, 5545. J. aeouhvs. Res. 79, 1583. Recent hdvances in satellite observations of solar uariability and global atmospheric ozone. Proceedings of the international conference on structure, composition and general circulation of the upper and lower atmospheres and possible anthropogenic perturbations, Melbourne. J. atmos. terr. Phys. 35, 2163. Observation d’ozone. J. atmos. terr. Phys. 39, 581.

SHCHARENSKAYA E. G.

INSITIWT

Generation

of gravity

waves by inhomogeneous

heating

883

of the atmosphere

OBERHETTINGERF.

1957

RANDAwA J. S. R&ES D., ROPER R. G., LLOYD K. H.

1971 1972

Transformation, Mathematisthen Wissenshaften Band xc, Springer-Verlag. Berlin. _J. geophys. Res. 76, 8139. Phil. Trans. R. SOC.London A271, 631.

1967 1978 1974 1977 1979 1978

.l. geophys. Res. 72, 1599. J. atmos. terr. Phys. 40, 529. Geomag. & Aeron. 15, 625. Planet. space Sci. 25, 723. J. annos. krr. Phys. 41, 33. J. armos. terr. Phys. 40, 223.

and Low C. H. Row R. V. SIDI C. and TE~UAUM H. SOMSIKOV V. N. and TRO~KW B. V. TEI-IELBAUMH. and BLAMONT J. E. TEITEUAUM H. and COT C. TE~L~AUM H. and F%WDIDIER M.

Equations (16b), (19) require some transformations in order to be computed numerically. The Fourier transform of (16b) with respect to w for w>O is

Tabellen zur Fourier

From

(25) we can see 6J2 ~ sin (UT) R wz - wsz

P(x, y. z. t) =-_

7r2

-

i(y-

leads to a divergent

“I sin (UT) ~

1)

(*Al)

(w~-cIJ,~)/(w~-+_~)

large

2 n(27T)3CZ I we (wZ-wgZ)

integral when w + a because A(1) is equivalent to

w equation

sin (wT) cos

for

R - w iC J (-42)

R

We can point The term sin wT is the only one which is odd in the integrand. If we integrate over both negative and positive frequencies, the contribution to the integral of the product sin (UT) [cos (wT) + i sin (wt)] is due to the product of sin (UT’) sin (ot) the only one which is even. So the integral to be calculated is P(x, Y>z, t)

g X(1 2 --)

m

w.4 F(w) dw +

F,(w) dw I0.4

[o(t + T)])

dw.

If we take into account exp (id) dw is proportional

to

F(w) dw =

P(x, y, 2, t)

I_ %I ?r/z =-cos[w,

=(F(w) - FA(o)) dw.

l/R E sin (or) cos to 6 (t + T tt :) then

((R/c)w)

= (F(w) - F/,(w)) dw - I”* F(w) dw b

0

+fi(t*TZ).

{cos[w, sin 4(t - T)l I Arcsin

do

I,

RJKLQ

w by og sin 4 leads

-FA(ti) d =

sin [fk&q2, C

2H

w&o,

sin+(t+T)]}(ti2sin2+&)

sin (:&w) X

R dw2 - w,’

dw

and this integral is convergent. Using the same method in (19) but substituting w, sin 4 leads to P(x, Y, z, t) ITi2 {cos [w, sin &(t - T)] - cos[w,

I0

we have

vqz

=-I

Substituting

if we write (Al) = F(w) (-42) = F,(o)

I0 w.(Cos[dt - T)l-cos

out that

sin +(t + T)]}

w by

6(t + Ti-R/c) and iZi(t- T+ R/c) are excluded by (26) but S(t - T-R/c) s(r + T-R/c) represent a strong perturbation which grows at the beginning and at the end of the heating and which moves from the source with the velocity of sound. As P(x, y, z, t) is a casual function, we can integrate it in the complex plane and write the exponent of the function in the semicircle as exp (- (R/c)q) with wi = imaginary part of w. If we take into account the factor sin (UT) then the exponent becomes oi(t * T-R/c). Then when t *T-R/c O the contour must be closed in the upper half of the o plane, this implies that the response appears at a time R/C after the beginning of the heating and at a time R/C after the end of the heating.