~
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1.’ c

ATMOSPHERIC OSCILLATIONS by A . J. Lineberger and H. D. Edwards

Georgia Tech P r o j e c t A-652-001
Contract NO. m19(628)-393

GPO PRICE $ OTS PRICE(S) $
Microfiche (MF)

I
I 1 I
‘d

Prepared f o r National Aeronautics and Space Administration Washington 25, D. C.

IACCtSSION NUMBER)

ITHRUI
L (CODE)
ICATEdOORY)

A p r i l 1965
Engin e ering E xp eriment Station
GEORGIA INSTITUTE OF TECHNOLOGY
Atlanta, Georgia

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U.S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES WASHINGTON 25, D. C .'I

.
*
ATMOSPHFRIC OSCILLATIONS by
A . J . Lineberger and H . D. Edwards
Georgia Tech P r o j e c t A-652-001
Prepared for National Aeronautics and Space Administration
Washington 25, D. C.
Contract No. NsG 304-63
April 1965
* The s t u d i e s reported here were a l s o supported by t h e A i r Force Cambridge Research Laboratories under
Contract n19( 628)-393.

.
ATMOSPHERIC OSCILLATIONS

C'

Aileen J. Lineberger and Howard D. Edwards

Space Sciences Laboratory

Georgia I n s t i t u t e of Technology

Atlanta, Georgia

ABSTRACT

The development of present theories o f atmospheric o s c i l l a t i o n s i s outlined

i n the following report with special emphasis being given t o points of i n t e r e s t

t o persons studying upper atmospheric motions. The general mathematical a t t a c k

has been summrized and references t o complete derivations have been included.

Current research on atmospheric o s c i l l a t i o n s has been r e l a t e d t o s t u d i e s of atmos-

pheric phenomena conducted i n t h e Georgia Tech Space Sciences Laboratory. P a r t i -

c u l a r a t t e n t i o n has been given t o t h e r e l a t i o n of postulated g r a v i t y waves t o

observed wind motion with reference t o t h e following: a downward propagation of

.

g r a v i t y wave phase velocity, a phase change i n the region of a negative tenrpera-

ture gradient, and the energy flux from the lower atmosphere t o the upper atmos-

phere.

1

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I .'
I-

INTRODUCTION Evidence of periodic changes i n the atmosphere w a s f i r s t obtained from barometric observations m d e i n the 18th century. In 1882 Kelvin w a s able t o demons t r a t e t h a t t h e f l u c t u a t i o n of barometric pressure through the day was the sum
of Fourier components with 24, 12, and 8 hour periods. He noted t h a t i n high
l a t i t u d e s the 12 hour component was l a r g e r than t h e 24 hour component. The reverse of t h i s observation would be expected i f t h e f o r c i n g f u n c t i o n f o r t h e o s c i l l a t i o n was s o l a r d i u r n a l heating. Kelvin attempted t o explain t h i s e f f e c t by a "resonance theory" i n which he postulated a f r e e period of the atmosphere close t o 12 hours. Wilkes [19491 noted t h a t t h e maximum of t h e pressure v a r i a t i o n occurred a t approximately 10 a.m. and 10 p.m.
The argument for a resonance of the atmosphere w a s based on t h e following. If t h e g r a v i t a t i o n a l f o r c e s of the sun and moon dominated t h e pressure v a r i a t i o n , then t h e lunar force, having almost twice t h e e f f e c t i v e force on t h e earth, should be the primary cause of o s c i l l a t i o n . Consequently one should observe changes i n p r e s s u r e t o have a period of 12.5 hours. However, t h e period of semidiurnal
*
o s c i l l a t i o n w a s found t o be much closer t o 1 2 hours than 12.5 hours. Thus, it may be concluded t h a t the influence of t h e sun must be stronger than t h a t of the moon. For t h i s t o be the case, the temperature e f f e c t must be l a r g e r than t h e g r a v i t a t i o n a l e f f e c t . The telrperature v a r i a t i o n i s diurnal, however. Therefore, t h e r e must be a strong resonance of approximately I 2 hours, such t h a t the 12 hour
component of temperature v a r i a t i o n would be l a r g e r than t h e 24 hour component.
Pr.ioiq t o :<eIvin' G i c v e s t . i g s t i ons, Laplace had worked out, under simplifying assumptions, t h e equations of o s c i l l a t i o n of a homogeneous ocean of uniform depth. He w a s a b l e t o apply h i s r e s u l t s t o t i d e s of a uniform isothermal atmos-
* Chapman [1941] quoted Hough as computing t h a t the f r e e period must be within 2 or 3 minutes of t h e 12 hours observed. 2

phere, i f he m d e the assumption t h a t the s c a l e height of t h e atmosphere was the

equivalent depth f o r which t h e atmosphere would obey the ocean approximtion. The

scale height,

H,

2 equals c/yg,

where c i s t h e speed of sound,

y is the r a t i o of

t h e s p e c i f i c h e a t s and g is t h e acceleration of gravity. Iamb [1932] l a t e r assumed

t h a t pressure changes i n t h e atmosphere occurred a d i a b a t i c a l l y and came t o t h e con-

clusion that the equivalent depth of the atmosphere w a s equal t o the scale height,

which substantiates Laplace's assumption. For the semidiurnal v a r i a t i o n t o be

predominant, i , e . f o r a 12 hour period, Iamb [1932]computed t h e equivalent depth

t o be approximately 26,000 f e e t ,
Later evidence showed a f r e e o s c i l l a t i o n period of 10.5 hours which seemed

t o c o n t r a d i c t Lamb's v u r k . 'Ihe ,ei-itd,ct uf' 10.3 hsurs vas computed from t h e time

that w a s required for t h e waves generated by a p o i n t pulse t o t r a v e l around t h e

e a r t h . The p o i n t p u l s e s which were large enough t o be observed were t h e e r u p t i o n

C'

of t h e volcano Krackatoa i n 1883, t h e Great Siberian Meteor i n 1908, and s e v e r a l

Soviet megaton nuclear explosions i n 1951 t o 1962. These pulses were analyzed

r e s p e c t i v e l y by Pekeris [1939], Donn and G i g [1962l, and Press and Harkrider
[I962 3

The e a r l y t h e o r i e s of atmospheric o s c i l l a t i o n s were based on Lamb's work.
I n 1936 Taylor used t h e m t h e m a t i c a l device of approximating t h e depth of t h e

e a r t h ' s atmosphere by i t s equivalent depth as an ocean. He approximated t h e

Lernperat,ure as a function of a l t i t u d e , the v e l o c i t y and pressure a s functions of

t h e a l t i t u d e and l a t i t u d e , and t h e v a r i a t i o n s of v e l o c i t y end pressure 8 6 f u n c t i o n s

oi e i'O

+ '@) ~ F x r e2 ; ; / c is t h e perinrl nf o s c i l l a t i o n , (0 i s t h e longitude, t i s

t h e t i m e , and s i s a constant. H e then explained the lO.5 hour f r e e period

observed i n terms of t h e f r e e period of an o c c ~ ~onf cyuivnlenL d e p t h .

The 10.5 and 1 2 hour p e r i o d s were explained by I'ekeris [lc)37J by assuming

3

a layered atmosphere with s e v e r a l equivalent depths. He approximated the tempera-

t u r e by a function of a l t i t u d e i l l u s t r a t e d i n Figure 1. The changes i n t h e tem-

perature gradient from negative t o zero (points A and D) i n t h e temperature versus

a l t i t u d e curve gave two equivalent depths. Pekeris a l s o found t h a t o s c i l l a t i o n s

t r a v e l i n g upward would experience phase s h i f t s a t p o i n t s A and D. The o s c i l l a t i o n s

would a l s o be amplified due t o t h e decreasing d e n s i t y and pressure by a f a c t o r of

100 a t 100 km. In a paper i n 1939 Pekeris examined the records of pressure

f l u c t u a t i o n s excited by t h e Krackatoa e r u p t i o n s t o a s c e r t a i n i f modes of the 12

hour component could be detected. He had computed the r a t i o of t h e 10.5 hour com-

ponent t o t h e 12 hour component t o be 5 : 2 . I n t h e barographic records t h e f l u c t u a -

t i o n s caused by the 12 hour component were too small t o be p o s i t i v e l y i d e n t i f i e d ,

but t h e r e w a s no evidence t o contradict t h e existence of a 12 hour component.
The next s i g n i f i c a n t s t e p was mde when Weeks and Wilkes [19471 organized

t h e theory developed up t o that time and analyzed t h e energy trapped i n a c e r t a i n

I'

region of the atmosphere, between a temperature minimum and the e a r t h . They used

a d i f f e r e n t i a l analyzer t o study the free oscillations f o r different given tempera-

t u r e d i s t r i b u t i o n s . They assumed t h a t most of the energy supplied t o the atmosphere

enters the lower atmosphere i n the more dense regions as gravitational energy.

Later Wilkes [19511 extended the mathemtical a n a l y s i s t o include s o l a r thermal

input.

The energy forcing function (thermal o r g r a v i t a t i o n a l ) i s -understood t o

e x c i t e a series of modes of o s c i l l a t i o n which depend on l a t i t u d e and longitude.

The energy f o r each mode i s introduced a t low a l t i t u d e and spreads as a s p h e r i c a l

wave f r o n t i n t h e atmosphere. 'Yhe motion of tile aii- paLrtizlz hzs cmp0Eent.s h0t.h
c'
p a r a l l e l t o and perpendicular t o t h e d i r e c t i o n of propagation. One w i l l r e c a l l

t h a t a sound wave i s considered t o be a compression and r a r e f a c t i o n longitudinal

t o t h e d i r e c t i o n of propagation.

4

.

According t o Weeks and Wilkes [19471 t h e d i u r n a l mode of o s c i l l a t i o n i s

.

damped out by v i s c o s i t y a t 100 t o 300 km. The semidiurnal o s c i l l a t i o n s , because

of t h e i r periods, a r e r e f l e c t e d i n the 50-100 km region by the temperature minimum

and negative temperature gradient. Modes with periods of t h e order of those o f t h e

semidiurnal modes w i l l be trapped and multiply r e f l e c t e d between t h e e a r t h and the

temperature minima a t 30 and 80 km. The multiple r e f l e c t i o n allowed pressure

o s c i l l a t i o n s caused by Krackatoa and similar sources t o propagate around the

e a r t h s e v e r a l times. With each r e f l e c t i o n some f r a c t i o n of t h e energy w a s t r a n s -

mitted and m i g h t then be observed i n the upper atmosphere.

M. L. White [1955, 1956, 1960aI f u r t h e r developed the theory t o cover o s c i l -

l a t i o n s caused by g r a v i t a t i o n a l forces a t low a l t i t u d e s and thermal input a t a l l
a l t i t u d e s . Recently White [1960b 1 combined thermally and g r a v i t a t i o n a l l y excited

o s c i l l a t i o n s with t h e ionospheric dynamo e f f e c t f o r an e l e c t r o n and p o s i t i v e ion

gas i n an imposed s t a t i c magnetic f i e l d .

Recently enough d a t a from r a d i o wave r e f l e c t i o n techniques of E-region d r i f t

have been collected t o imply that the o s c i l l a t i o n phase i n the a l t i t u d e region

95-115 km i s c o n s i s t e n t with the phase observed a t the ground. Studies from meteor trails show t h a t a phase r e v e r s a l e x i s t s a t 85 km a s would be expected

i n conjunction with the temperature minimum. The temperature v a r i a t i o n s would

a f f e c t t h e phase angle and amplitude. The region of t h e m 1 input would a l t e r

t h e rate of change of amplitude and of phase with height.

Superposed on t h e p e r i o d i c p a t t e r n of o s c i l l a t i o n s are seemingly random

o s c i l l a t i o n s . The random o s c i l l a t i o n s may be grouped i n t o acoustic and g r a v i t y

.

waves according t o t h e i r frequency. ,Tne a c o u s t i c aiid gavit.; mves IT^ derived

from dynamical equations and a r e governed by g r a v i t a t i o n a l and compressional

forces. These o s c i l l a t i o n s w i l l be described l a t e r mathemtically. Depending

on frequency, these random modes may be r e f l e c t e d or transmitted a t c e r t a i n

5

a l t i t u d e s under the same conditions a s the periodic modes. Thus, random a s w e l l as periodic o s c i l l a t i o n s should be observed i n the upper atmosphere.
Gossard [19621 observed g r a v i t y waves i n t h e troposphere which p e r s i s t e d
f o r 10 t o 12 hours. H e attempted t o show t h a t g r a v i t y waves generated i n the
troposphere m i g h t propagate i n t o the upper atmosphere. Gossard 119621 l i s t e d
t h r e e p r i n c i p a l mechanisms for generating random i n t e r n a l g r a v i t y waves i n the troposphere. F i r s t , i n t e r n a l g r a v i t y waves may be generated as standing waves i n t h e l e e of topographic f e a t u r e s . Second, i n t e r n a l g r a v i t y waves may be produced by t h e motion of a boundary between two c e l l s of a i r of d i f f e r e n t d e n s i t i e s inverted with respect t o density. I n t h i s second case a very regular, sinusoidal
g r a v i t y wave t r a i n m y be generated a s t h e wake, i f t h e v e l o c i t y of the boundary i s of the proper magnitude r e l a t i v e t o the height and i n t e n s i t y of the inversion and t o the slope of t h e boundary. Third, l a r g e t r o p o s m e r i c storms and l a r g e s c a l e f e a t u r e s associated with s t a b l e l a y e r s i n t h e lower atmosphere will produce oscillations of long duration.
I n some r a r e instances the gravity waves may be v i s i b l e i n the lower atmosphere as layering i n cloud f o r m t i o n s . Gossard has photographs of t h e waves on
page 747 of h i s 1962 a r t i c l e . Hines a l s o mentions that noctilucent clouds
occasionally r e v e a l t h e g r a v i t y wave p a t t e r n . The noctilucent clouds occasional-
l y form i n long p a r a l l e l bands 9 k m a p a r t .
One, then, should observe continuous periodic motion i n the upper atmosphere from t h e transmitted d i u r n a l and semidiurnal modes as w e l l a s random o s c i l l a t i o n s . The wavelength of t h e random modes of o s c i l l a t i o n should be roughly t h e same mgnitude a s t h e periodic modes or o s c i l l a t i o n , since r e f l e z t i c x by the t.hem.1 g r a d i e n t and d i s s i p a t i o n by viscous, eddy, and kinematic e f f e c t s remove a l l but certain wavelengths a t high altitudes.

E s s e n t i a l l y a l l of t h e energy of t h e atmosphere comes from r a d i a t i o n or g r a v i t a t i o n a l forces with the motion being caused by the conversion of t h i s energy t o k i n e t i c energy. There i s l i t t l e generation of entropy. The n e t heating of the atmosphere i s due t o the d i f f e r e n c e between solar r a d i a t i o n absorbed and infrared r a d i a t i o n emitted by t h e atmosphere. The next s e c t i o n describes the governing equations f o r t h i s motion which i s found t o be o s c i l l a t o r y i n many considerations. The o s c i l l a t o r y motion i s broken i n t o i n t e r n a l g r a v i t y wave motion and acoustic wave motion by most authors.
7

DYNAMICS OF THE ATMOSPHERF:
Equations describing o s c i l l a t i o n s of the atmosphere were f i r s t obtained by adapting hydrodynamic equations of nonviscous, compressible f l u i d s , i .e. gases. Laplace performed t h e f i r s t major work i n t h i s a r e a by r e l a t i n g t i d e s of an ocean t o an atmosphere of an equivalent depth.
I a m b ’ s book, Hydrodynamics [1932]i s a c l a s s i c i n t h i s f i e l d and i s t h e b a s i s
f o r t h e t h e o r e t i c a l work of Taylor, Pekeris, Wilkes, and others. Lamb r e l a t e d hydrodynamic equations t o atmospheric t i d a l o s c i l l a t i o n s f o r a number of s p e c i a l c a s e s . He made t h e j u s t i f i a b l e approximation t h a t , f o r changes i n t h e atmosphere as a whole, v i s c o s i t y and nonadiabatic l o s s e s my be neglected. Only i n a highly turbulent region i s t h i s approximation poor. This approximation i s used i n a l l of t h e work considered unless stated otherwise.
The mathematical manipulations were c a r r i e d out i n e i t h e r rectangular o r spherical coordinates. For a viscous, compressible f l u i d undergoing changes a d i a b a t i c a l l y , one may obtain the equations governing the motion of t h e atmosphere from t h e following t h r e e equations. The equations are 1, t h e equation of motion;
2, t h e equation of continuity, and 3, t h e equation of a d i a b a t i c state.
I n t h e above equations -v = u i + v-j + w&, i s t h e velocity, p i s t h e density,
p i s t h e pressure, c i s the speed of sound, CD i s t h e angular v e l o c i t y of t h e
8

earth, and Fr i s the f r i c t i o n a l f o r c e . Also, g i s t h e "observed g r a v i t a t i o n a l force" which i s the sum of - @ and w2R, t h e g r a v i t a t i o n a l p o t e n t i a l and t h e

centrifugal force.

To solve t h e above s e t of equations one commonly uses a p e r t u r b a t i o n

a n a l y s i s and l i n e a r i z e s t h e r e s u l t i n g equations. One considers the s t a t i o n a r y

, values, po, p To, and v f o r density, pressure, temperature, and velocity,

0

-0

and l e t s p', p', T', and -v t be the perturbation of these q u a n t i t i e s . The l i n e a r

approximation i s f a i r l y good below 100 km where f l u c t u a t i o n s i n the density a r e

l e s s than 10 per cent. According t o Hines [19601 the d e n s i t y m y f l u c t u a t e as

much a s 20 p e r c e n t above 100 km. I n t h e regions above 100 km the l i n e a r

approximation i s not as good.

x, If one replaces p, p, and

i n equations 1, 2,

and 3,

by

p
0

+

p',

and v + v' and s i m p l i f i e s one obtains t h e following: -3-

po + p t ,

iw aw
at =

aZ

+ g p' - 2 w ( - cos a)u'

(44

1%. * v =a t+ v - . v p o = c2

- v

where a i s the l a t i t u d e , and v i s s e t equal t o zero. -0 One may now solve t h e equations as they a r e w r i t t e n above, as Wilkes [19491
o u t l i n e s . An a l t e r n a t i v e i s t o f u r t h e r simplify the equations by m k i n g approxi-
9

mations on both t h e equations and t h e model of t h e atmosphere described. The simplified equations w i l l be discussed f i r s t ; then t h e more general approach w i l l be described.
A model frequently used i s t h a t of a f l a t , nonrotating earth. The temperat u r e i s assumed e i t h e r t o be constant,to increase o r decrease monotonically with a l t i t u d e , or t o be s t r a t i f i e d . Gravity i s u s u a l l y considered t o be constant. Density and pressure a r e usually considered t o vary exponentially with a l t i t u d e .
The most one can p r o f i t a b l y simplify t h e problem i s t o consider an isothermal atmosphere, plane l e v e l surfaces, and a nonrotating earth. This case has been handled by Eckart [1960], Lamb [1932], and Hines [l96O]. The s i m p l i f i c a t i o n i s not v a l i d f o r small e f f e c t s , but general, l a r g e e f f e c t s m y be described and discussed. Hines t r i e d with apparent success t o r e l a t e his r e s u l t s t o e f f e c t s observed experimentally. Eckart went over n e a r l y t h e same d e r i v a t i o n as Hines b u t included more d e t a i l . However, Hines used n o t a t i o n that i s mbre p h y s i c a l l y meaningful. Both used l i n e a r i z e d equations f o r small perturbations on a s t a t i o n a r y system. Eckart used entropy concepts, while Hines used t h e approximation of a n a d i a b a t i c s t a t e . Both found a high and low s e t of allowed frequencies separated
by a region of forbidden frequencies. Waves with frequencies below the forbidden
region were c a l l e d g r a v i t y waves and waves with frequencies above t h e forbidden region were called acoustic waves.
I n p a r t i c u l a r Hines assumed wave s o l u t i o n s f o r density, pressure, v e r t i c a l ,
- - and h o r i z o n t a l v e l o c i t y t o be of t h e form C exp i (-a t Kx X KZ 2 ) . He j
- s u b s t i t u t e d t h i s i n t o t h e equations 4, 5 , and 6, neglected w x -v terms, and
o"vtaiiied 2 d i s p z r o i m r e l z t i m
where w i s t h e frequency of o s c i l l a t i o n , y i s t h e r a t i o of s p e c i f i c heats, and
10

K and K a r e wave numbers given by ~ T Tt i m e s inverse wave l e n g t h s . To i n t e r p r e t

X

Z

t h e d i s p e r s i o n r e l a t i o n Hines assumed t h a t K i s r e a l = k and, therefore, K i s

X

X

Z

- p u r e l y

imaginary

or

is

=

k
Z

+

2iy,cg2

where k i s r e a l .
Z

a l t e r n a t i v e t o allow f o r v e r t i c a l phase propagation.

Hines chose the second A s a r e s u l t of t h i s assum-p-

t i o n Hines was able t o i n t e r p r e t t h e phase change i n the o s c i l l a t i o n s of t h e

upper atmosphere as g r a v i t y waves. He noted t h a t i n t h e absence of g r a v i t y t h e
dispersion r e l a t i o n becomes w2 = ( k z + Kz2 ) c2 which i s t h e familiar equation
f o r sound propagation. Then f o r simple sound waves Kx and K would be r e a l . Z When Hines solved the dispersion r e l a t i o n under t h e condition KZ = kZ + k/2H

he found t h a t 4 m has two p o s i t i v e roots, and i s double valued f o r r e a l wave

number p a i r s ( k k Z ) . He designated t h e two choices of w as corresponding t o
*
acoustic or g r a v i t y waves. The frequencies f o r acoustic waves a r e g r e a t e r than

wa = yg/2c and t h e frequencies f o r i n t e r n a l g r a v i t y waves are less than
. w = ( 7 - 1)1/2 g/c. Since y< 2 then wa > w There i s a gap of forbidden

g

g

frequencies w such t h a t w < wi< wa. Recently Pitteway and Hines [19631 extended

i

g

t h e i r model t o include viscous damping of atmospheric g r a v i t y waves.

Eckart [l960]went through a second d e r i v a t i o n i n which the e f f e c t of the

e a r t h ' s r o t a t i o n was included. The other conditions a r e the same as the f i r s t

case discussed. He again found t h a t c e r t a i n frequencies a r e not allowed and the

acoustic and g r a v i t y waves a r e similar t o the ones already described. Figures

2 and 3 show these allowed frequencies versus wave number i n t h e cases of a non-

r o t a t i n g and a r o t a t i n g e a r t h . The unshaded a r e a represents an imaginary propa-

gation surface.
*
There e x i s t s some ambiguity i n tne use of t'ne Leiziii gi-&\-iQ VZ'V'S for vzrious xxediz> i . e . l i q u i d s or gases. A surface gravity wave must be distinguished from a n int e r n a l g r a v i t y wave with which w e are concerned. Also, d i f f e r e n t terms may appear
i n t h e equations of motion of gravity waves depending upon t h e assumptions made and t h e media described. It appears t o be usual, however, t o c a l l the s e t of waves with lower frequency, of t h e two s e t s of allowed frequencies, g r a v i t y waves.

11

The g r a v i t y wave propagates energy upward i n modes whose phase progression i s downward, while acoustic wave energy propagates i n nearly t h e same d i r e c t i o n a s the phase.
Acoustic and g r a v i t y waves are governed by compressional and g r a v i t a t i o n a l forces; the r o t a t i o n a l force modifies but does not change the type of wave which i s found. Eckart described one important d i f f e r e n c e between a c o u s t i c and g r a v i t y waves. On page 120, Eckart [1960l discussed the idea t h a t g r a v i t y waves with s h o r t wave lengths "have one outstanding c h a r a c t e r i s t i c which d i s t i n g u i s h e s
*
them from sound waves. I n the l a t t e r , the r a t i o of p a r t i c l e v e l o c i t y t o pressure amplitude i s very small--on the order of mgnitude of l/pc. I n the g r a v i t y waves t h i s r a t i o becomes much l a r g e r and approaches i n f i n i t y f o r s h o r t wave lengths. This i s a l s o a c h a r a c t e r i s t i c of t h e f l u c t u a t i o n s i n wind v e l o c i t y t h a t occur without marked pressure f l u c t u a t i o n s . One may therefore make a t e n t a t i v e i d e n t i f i c a t i o n of the g r a v i t y waves with the f l u c t u a t i n g component of the wind."
A more general approach uses spherical coordinates. Wilkes 119491 outlined
the basic mathematical equations i n h i s book, and ramifications were developed
i n papers by Wilkes [1951] and by White [1955, 1956, 1960a, 1960bl. The basic
equations of motion a r e taken t o be, i n t h e linearized, perturbation form,

-aaut - 2 w v c 0 s e =

(t - -ai - aae

+n)

a -aavt + 2 w v c o s e

=

-

1 a

s

i

n

8

( 2 +n) P,

*
Eckart used the term sound wave i n the sense i n which t h i s paper uses acoustic wave.

where a i s t h e radius of t h e earth, cois t h e angular v e l o c i t y of t h e earth, 8 i s the l a t i t u d e , 4 i s the longitude, z i s the height above the e a r t h ' s surface, u i s t h e southward component of a i r v e l o c i t y a t ( z , 8, 4), v i s t h e eastward component, w i s t h e v e r t i c a l l y upward component, c i s t h e v e l o c i t y of sound a t height z, and R i s the t i d e producing potential, gravitational i n origin. I n the above t h e e a r t h i s considered t o be spherical, and t h e v a r i a t i o n of r a d i u s vector,
gravity, and an with height are neglected. Also, t h e v e r t i c a l acceleration i s
considered t o be n e g l i g i b l e . Temperature, density, and pressure are f u n c t i o n s of t h e a l t i t u d e . The equation of continuity becomes
The a d i a b a t i c gas l a w i s
i f t h e thermal f o r c i n g f u n c t i o n i s ignored. If one considers a thermal f o r c i n g function, Q, one must use
where Q = Q ( z , 8, 4 ) . Note that i n t h i s a n a l y s i s t h e forcing functions are considered, while Hines simply looked f o r allowed motions under c e r t a i n condit i o n s . Wilkes considered only t h e g r a v i t a t i o n a l rviiciiig f i i i i c t l o n ; Sz:: zzd Vkite
119551 considered thermal and g r a v i t a t i o n a l f o r c i n g functions a c t i n g a t ground l e v e l . White [19561 extended t h e theory t o include a ground l e v e l g r a v i t a t i o n a l
forcing function and a thermal forcing function which varies with a l t i t u d e .
13

Solutions t o these d i f f e r e n t i a l equations a r e worked out i n the papers referenced.
The N S wind v e l o c i t y w a s found by White [19561 t o be

When the r a t i o of N S t o Ew v e l o c i t y i s formed t h e dependence on a l t i t u d e cancels. The r a t i o of t h e N S t o EW v e l o c i t y components i s

V

10," - - e i

(

- C

OS
f

a
de

+

S
sin e'

iut (0) e

where (J = 27r/period of o s c i l l a t i o n , s i s a constant, r i s t h e component of o s c i l l a t i o n considered, we = angular v e l o c i t y of e a r t h , f = (J/2 we, and 8 = co-
0 l a t i t u d e . Also, 2 = P22 ( e ) - B P24 ( e ) f o r the s o l a r semidiurnal o s c i l l a t i o n .
Pf i s t h e associated Legendre function and B i s a constant determined empiri-
c a l l y from experimental data. These equations may be used t o make approximations t o wind motions.
Pekeris solved the governing equations f o r the case of a purely gravitational f o r c i n g function. In h i s s o l u t i o n s he derived an expression for t h e pressure.
, For c h a r a c t e r i s t i c values of t h e period, 7 27- he showed t h a t t h e amplitude becomes
i n f i n i t e , and a f r e e period, o r resonance occurs. Lower boundary c o n d i t i o n s are u s u a l l y s e t by s p e c i f y i n g that the v e r t i c a l
v e l o c i t y must be zero a t the e a r t h ' s surface. To s e t t h e upper boundary condition

14

it i s u s u a l t o consider the r a t e of flow of energy i n a column of a i r of constant cross s e c t i o n . One considers the horizontal energy flow t o be constant and assumes the energy t o decrease vertically, going t o zero a t i n f i n i t y . Since energy i s assumed t o e n t e r a t t h e low a l t i t u d e s it i s apparent t h a t a t some high a l t i t u d e t h e energy must be flowing outward only, which j u s t i f i e s the assumption
t h a t t h e energy w i l l go t o zero a t i n f i n i t y . Wilkes E19491 on page 49 of h i s
book obtained t h e r e f r a c t i v e index for atmospheric waves by m k i n g an analogy t o electromagnetic waves. He found t h e r e f r a c t i v e index p t o be given by

v2 =

1

Y

If

2 p,

is

negative

at

c e r t a i n a l t i t u d e s some of

the

energy w i l l be

transmitted

and some w i l l be r e f l e c t e d . Low temperatures and negative temperature gradients

may

cause

2
p

to

become

negative.

For various values of h, which i s a function

of t h e mode of o s c i l l a t i o n and arises as a separation constant i n t h e d i f f e r e n t i a l

equation, some waves will be r e f l e c t e d and some transmitted.

CORREZATION WITH EXPERIMENTAL OBSERVATION
Several authors have conducted theoretical studies which can be correlated with experimental observations carried out i n our laboratory. Motions characteri s t i c of gravity waves were evident i n our d a t a .
Several analyses w i l l be discussed, but the one described by Hines [1960] was t h e most successful i n r e l a t i n g experimental observations t o a model. Gossard
[1954, 19621 r e l a t e d t h e energy f l u x from t h e troposphere i n t o the upper atmosphere
t o g r a v i t y waves. White [l960b] expanded t h e theory t o cover the dynamo e f f e c t and has graphically related t h e theory of semidiurnal t i d a l components t o experimental observation.
Hines l i s t e d s i x observed properties which he correlated with a simplified model of t h e atmosphere. The p o i n t s were (1)wide v a r i a t i o n s i n t h e wind component with a l t i t u d e , ( 2 ) persistence of a wind p a t t e r n f o r time i n t e r v a l s as l a r g e
a s 100 minutes, ( 3 ) a r a t i o of horizontal s c a l e s i z e t o v e r t i c a l s c a l e s i z e of 20 t o 1, ( 4 ) dominant h o r i z o n t a l motions and n e g l i g i b l e v e r t i c a l wind accelerations, ( 5 ) increasing speed of dominant i r r e g u l a r winds with a l t i t u d e , ( 6 ) smallest
v e r t i c a l structure s i z e increasingwith a l t i t u d e . Hines obtained these properties
from experimental observations made before 1959. These p r o p e r t i e s a r e c o n s i s t e n t
with our data, and it i s then reasonable t o assume t h a t h i s model w i l l hold f o r t h e winds observed by t h i s laboratory-.
Upon analyzing a dispersion relation, Hines f i n d s t h a t t h e r e e x i s t two sets of allowed frequencies, g r a v i t y waves and acoustic waves. A c h a r a c t e r i s t i c of t h e g r a v i t y waves i s t h a t while energy i s c a r r i e d upward the phase propagates downward with time.
This laboratory has attempted t o demonstrate t h e existence of g r a v i t y waves i n t h e following manner. Our observations show t h a t t h e wind vector viewed from above performs clockwise r o t a t i o n with increasing a l t i t u d e a t a given time, and
16

performs clockwise r o t a t i o n with time a t a given a l t i t u d e . More than 75% of t h e wind data show a n t i c y c l o n i c motion between 100 and 115 km, and over 90% of t h e
wind data show a n t i c y c l o n i c motion between 110 and 112 km. One may r e l a t e t h e two observed r o t a t i o n s of t h e wind vector by assuming t h a t g r a v i t y and t i d a l waves were propagating upward with an a t t e n d a n t downward propagation of phase i n t h e region under observation.
Under t h e above assumptions t h e phase v e l o c i t y was computed f o r two s e t s of sodium r e l e a s e data obtained from rocket f l i g h t s over Eglin A i r Force Base,
Florida. For t h e f i r s t s e t released on 3 December 1962 a t l7:2O, 1 8 ~ 0 1 ,21:45,
and 22:45 CST, t h e r a t e of r o t a t i o n of t h e wind vector a t a given a l t i t u d e as a function of t i m e , and a t a given time as a function of a l t i t u d e was computed.
Averaged between 98 and 113 km t h e wind vector was found t o r o t a t e 15' per km
change i n a l t i t u d e and 0.4' p e r minute a t a given a l t i t u d e . Over t h i s a l t i t u d e range the wind vectors c o n s i s t e n t l y moved clockwise with increasing a l t i t u d e and with increasing time. Upon dividing one obtains a v e r t i c a l phase v e l o c i t y of 0.03 km/min or approximately 0.44 m/s. A similar a n a l y s i s w a s performed on t h e
f o u r r e l e a s e s on 17 May 1963 a t 1 9 ~ 0 6and 22:19 CST and on 18 May 1963 a t 0 2 ~ 5 6 and 04:06 CST. Averaged between 106 and 113 km the wind vector w a s found t o
r o t a t e 10' p e r km and 0.52' p e r minute.
The phase v e l o c i t y i n t h i s case was 0.8 m / s . One my assume t h a t t h e wind
p a t t e r n i s descending a t t h e above r a t e s and compare t h e wind component curves
f o r t h e two s e t s of f o u r wind determinations. I n Figures 4 and 5, each of the
wind curves has been s h i f t e d up along t h e z a x i s a distance corresponding t o i t s computed descent i n the elapsed time between wind measurements. A s one observes t h e r e i s d e f i n i t e l y a c o r r e l a t i o n i n the two s e t s of f o u r wind p a t t e r n s . I n
Figure 4 the t o t a l descent of the wind p a t t e r n between t h e 1 9 ~ 0 6wind determination and t h e 04:06 wind determination w a s 27 km. I n Figure 5 t h e t o t a l descent of the
17

wind p a t t e r n between l7:20 and 22:45 was 8.5 km.
A downward s h i f t of the wind p a t t e r n has been discussed i n the paper by Rosenberg and Edwards [19641. A study of t i m e and s p a t i a l v a r i a t i o n s of winds was r e c e n t l y m d e by Rosenberg, Edwards, and J u s t u s [ i n p r e p a r a t i o n ] . The s i n g l e
sodium t r a i l release on 17 May 1963 a t l 9 : 0 6 CST w a s observed t o e x h i b i t the same
r o t a t i o n previously discussed and t o reveal a wind p a t t e r n with a downward motion
of 1 . 3 m/s over an observed period of approximately 15 minutes. The downward v e l o c i t y of t h i s s i n g l e release of the 17 May 1963 s e r i e s i s l a r g e r than the
average phase v e l o c i t y computed f o r a l l f o u r r e l e a s e s . The phase v e l o c i t i e s observed seem t o vary over a f a i r l y narrow range f o r the winds observed thus f a r . The v a r i a t i o n m i g h t be explained a s the changing superposition of a number of gravity waves.
Gossard [l9&1 observes t h a t f l u c t u a t i o n s of pressure due t o random g r a v i t y waves i n t h e lower atmosphere a r e seen t o p e r s i s t a s long as 10 t o 12 hours. Since random o s c i l l a t i o n s a r e superposed on d i u r n a l and semidiurnal wind motion, i t would seem t h a t one should observe b e t t e r c o r r e l a t i o n between wind p a t t e r n s measured a t closely spaced i n t e r v a l s than widely spaced i n t e r v a l s but there should s t i l l be observable c o r r e l a t i o n throughout t h e day. The determination of the rot a t i o n of t h e wind v e c t o r with time a t a given a l t i t u d e f o r the two sets of f o u r rocket r e l e a s e s averages t o approximately 0.5 0/min or a p p r o x i m t e l y two revolutions per day. Apparently t h i s rotation i s predominantly a semidiurnal effect.
The wind motion i s considered t o be t h e sum of a general d r i f t , a p e r i o d i c o s c i l l a t i o n and a random component. No e f f o r t has been m d e y e t t o separate these mnt.ic?ns in cnnnect.inn w i t h t h e Fhase v e l o c i t y computed here.
One m y make a comparison between the energy which would be c a r r i e d by t h e g r a v i t y waves from the troposphere t o t h e ionosphere and the energy d i s s i p a t e d by turbulence i n the ionosphere. Gossard [l962]considers a n e g l i g i b l y viscous
18

atmosphere and neglects energy reflected by thermal barriers. H e notes t h a t t h e l a r g e r waves become nonlinear above c e r t a i n a l t i t u d e s and deposit some of t h e i r energy i n t h e turbulence spectrum. For several d i f f e r e n t observations he computes
the t o t a l energy d e n s i t y of t h e gravity wave t o range between 0.73 ergs/cm3 and
. 3.2 ergs/cm 3 On days of high gravity wave a c t i v i t y the mximum energy f l u x i s
on t h e order of s e v e r a l hundred ergs/cm2 s e e . If one t a k e s t h e energy d e n s i t y
of t h e g r a v i t y wave t o be approximately 1 erg/cm 3 and the energy f l u x t o be
approximately 100 ergs/cm 2 see, then 100 e r g s must be c a r r i e d through a cubic centimeter i n one second. This implies t h a t the v e l o c i t y of the energy being transported i s 100 cm/s.
J u s t u s and Edwards [NASA Technical Note i n P r e s s ] have shown t h a t a t 100 km
. the energy d i s s i p a t i o n i s approximtely 0.1 j/kg sec From t h i s value one may
compute the energy d i s s i p a t i o n per u n i t volume p e r u n i t time t o be 4.97 x lo-"
ergs/cm3 see. The energy f l u x which Gossard shows m y leave the troposphere i s seen t o be much l a r g e r than the d i s s i p a t i o n due t o turbulence i n the ionosphere. Turbulent d i s s i p a t i o n i s low between the troposphere and the ionosphere. As Gossard mentions, energy w i l l be l o s t due t o r e f l e c t i o n and turbulence. The amount of energy d i s s i p a t e d by turbulence decreases from t h e ionosphere t o the upper troposphere according t o t h e limited data a v a i l a b l e i n t h e study by Justus and Edwards. Energy d i s s i p a t i o n i n c r e a s e s q u i t e rapidly, however, i n t h e region above 100 km. Reflection w i l l probably be t h e p r i m r y mechanism which keeps energy from t h e troposphere from reaching the ionosphere.
Gossard [1962l a l s o computed the amplification of t h e g r a v i t y waves which reach t h e ionosphere. The v e r t i c a l wave lengths and wind v e l o c i t y perturbations which Gossard computed a r e the same mgnitude a s the wave lengths which we observed.
Another method. f o r determining phase v e l o c i t i e s i s suggested i n a paper by
Axford [19633 i n which the Dungey process f o r t h e f o r m t i o n of sporadic E i s de-

scribed. This process i s simply t h a t the component of t h e e l e c t r i c a l l y n e u t r a l

wind p a r a l l e l t o t h e magnetic f i e l d drives t h e f r e e ions and e l e c t r o n s along

f i e l d l i n e s and the perpendicular component d i s t o r t s t h e f i e l d s l i g h t l y . Thus,

i f t h e wind p r o f i l e i s s i n u s o i d a l along t h e v e r t i c a l a x i s , t h e f r e e i o n s and

e l e c t r o n s w i l l be forced t o t h e p o i n t where the v e l o c i t y i s zero u n t i l t h e i r

p a r t i a l pressure gradient balances the f o r c e exerted by t h e n e u t r a l p a r t i c l e s .

Then, i f t h e phase v e l o c i t y i s downward, and t h e p o i n t s of zero v e l o c i t y move

downward with time, the ionization w i l l tend t o move down with t h e zero p o i n t s .

Then, l a y e r s of sporadic E separated by h a l f the wave length of t h e g r a v i t y wave

should move down with a v e l o c i t y equal t o t h e phase v e l o c i t y of t h e g r a v i t y wave.

A layering of sporadic E has been observed. A t the present no systematic review

of the l i t e r a t u r e has been undertaken t o c l a r i f y the motion of the layers.
Axf'ord [19631 presents a table which includes the following "typical" values.

Altitude

km

145

120

100

Horizontal velocity

m/s

Phase velocity

m/s

50

50

30

1

1

0 -5

Wave length

km

12

10

5

The nagnitude of h i s "typical" phase v e l o c i t y i s very close t o t h e phase v e l o c i t y

a t t h e corresponding a l t i t u d e s and horizontal v e l o c i t i e s .

20

CONCLUSIONS The s t u d i e s presented here indicate t h a t winds observed i n t h e upper atmosphere may be composed of d i u r n a l and semidiurnal motions upon which a r e superposed random g r a v i t y waves. If t h e i n t e r p r e t a t i o n given i n t h i s paper r e l a t i v e t o phase v e l o c i t y i s correct, then the propagation of t h e phase downward and energy upward m i g h t be r e l a t e d by means of t h e observed r o t a t i o n of t h e wind vector and used t o demonstrate the existence of gravity waves. Further study m y r e l a t e t h e rate of change of the phase of t h e wind vector t o temperature gradients. I n addition, one m i g h t be able t o r e l a t e t h e energy d i s s i p a t e d i n turbulence i n t h e ionosphere t o t h e energy f l u x which is generated from t h e lower atmosphere and c a r r i e d t o t h e upper atmosphere. Characteristics of the motion of sporadic E may a l s o be r e l a t e d t o t h e g r a v i t y waves.
21

ACKNOWLEDGMENTS We a r e indebted t o P r o f e s s o r C . 0. Hines of t h e University of Chicago f o r reviewing t h e d r a f t and f o r o f f e r i n g many h e l p f u l suggestions. Much c r e d i t i s due our colleague, C . G . J u s t u s f o r discussions during t h e study. Financial support f o r the work has been supplied by the National Aeronautics and Space Administration under Grant NsG-304-63 and by the A i r Force Cambridge
Research Laboratories under Contract AF l9(628) -393.
22

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46. Wexler, H., - J. Geophys. L Res J-967 3875 (1962)

47. White, M. L., - J. Geophys. 2 Res ,-6.'1 489 (1956)

48. White, M. L., - J. Atmos. - and Terrest. 9-Phys -717 220 (1960a)

65, 49. White, M. L., - J. Geophys. Res.,

1-53 (1960b)

50 Wilkes, M. V., O s c i l l a t i o n s - o f -the E a r t h ' s Atmosphere, Cambridge University P r e s s (1949)

51 Wilkes, M. V., - Proc. Roy. SOC. London, 9-207 358 (1951)
-, - 52. Woodbridge, D. D., ffIonosphericWinds," J. Geophys. 2 Res , 67 4221 (1962)

.
25

Figure 1. The Assumed Temperature Variation as a Function of A l t i t u t e .
.
26

.

Figure 2. Wave Number, k, as a Function of Frequency, o, for an Isothermal Atmosphere.

t w

I-

Figure 3. Wave Number, k, as a Function of Frequency, o, for an
Isothermal Atmosphere Rotating About a Vertical Axis

~

~ 5 t Phzg1da.r Velocity 66.

118

I

116

I\

17, 18 MAY 1963

114

112

110

108 106

19:06
-- 22:19
----- 02:56 --- 04:06

104

102
= 100
5
W 5 98
t
<I96

94

92

90

88

86

84

82

80

/

I

7-8 10

I -120

I -100

I -80

I -60

I -40

\I/ -20

1 1 1 1 1 20 40 60 80 100

NORTH-SOUTH VELOCITY (misj

4

Figure h. The North-South Components of Wind Velocity Shifted
Along the Ordinate Relative t o t h e 04:06 Release.
North i s Taken as Positive.

28

.

118

116

114

17, 18 MAY 1963

112

110

108 106

19:06
-- 22:19 -- --- 02:56 --- 04:06

104

-- --.

102

2 100
s

5w 98

k

I-

J
a

96

94

92
90
(
88

86

84

(’

82

\

80

\

78

I I

/

- 0 -80 -60 -40 -20 0

~

~~

20 40 60 80 100 120 140 160

EAST-WEST VELOCITY (m/s)

_c

Figure 4b. The East-West Components of Wind Velocity Shifted Along the Ordinate Relative t o t h e 04:06 Release.
East i s Taken as Positive.

29

.

126 124 122 120 118 116 114 112 110 108 ti.
-
<I104 102 100 98 96 94 92 90 88

NORTH-SOUTH VELOCITY (rn/s)

--c

Figure 5a. The North-South Components of Wind Velocity Shifted
Along t h e Ordinate Relative t o t h e 22:45 Release. North i s Taken as Positive.

.

126

124

122 3 DECEMBER 1962
120

118

116

-- 17:20

-------

18:Ol 21:45 22:45

I i
I I

\ \ \
\
\ \
\ \ \
\
\ \
\
\ \
I
i
/

/ I
/*
,/’

I

I

I

I

I

1

I

20 40 60 80 100 120 140

Figure 5b. The East-West Component o f Wind Velocity S h i f t e d
Along the Ordinate Relative t o the 22:45 Release. East i s Taken a s P o s i t i v e .