ACOUSTIC-GRAVITY WAVES IN A VISCOUS AND THERMALLY CONDUCTING ISOTHERMAL ATMOSPHERE

Internat. ^,J. Math. & Hh. Sci.

VOL. 18 NO, 3 (1995) 579-590

579

ACOUSTIC-GRAVITY WAVES IN A VISCOUS AND THERMALLY CONDUCTING ISOTHERMAL ATMOSPHERE

(Part ^I1: For Small Prandtl Number)

HADIY.ALKAHBY

[)eI)arttent

f hlathenatcs ad (’on,purer

Scerc’e

])llard llnrerstv

New

^{Orleans, I,A 7(1122}

(Received July 27, 1993)

ABSTRACT: In part oneof these series we investigated the effect of Newtonian cooling ^{on acoustic-} gravity^{wavesin an} isothernalatmosphere forlargePrandtl number. Itwasshown that the atmosphere canbe divided intotworegionsconnected byanabsorbing and reflecting layer, created by the exponential increase of the kinematic viscosity with height, and ifNewtonian coolingcoefficient goes to infinity the temperature perturbation associated with the wave will be eliminated.

In

addition all linear relations among theperturbationquantities will be modified.

In

this paperwewill consider the effect of Newtonian cooling^onacoustic-gravitywavesfor small Prandtl number in anisothermalatmosphere. Itisshown that if theNewtonian cooling coefficient issmall compared to the adiabaticcutofffrequency the atmosphere may be divided intothree distinctregions.

In

the lower region the motionisadiabaticand theeffect of the kinematic viscosity andthermal diffusivityarenegligible,while theeffectofthesediffusivitiesis more pronounced in the upper region.

In

the middleregion the effect ofthethermal diffusivityis

large,

while that of the kinematic viscosity is stillnegligible.The twolower regionsareconnected byasemitransparent reflecting layer as a result of the exponential increaseof the thermal diffusivity with height. The two upper regionsarejoinedbyanabsorbingandreflectingbarriercreated but the exponential increase of the kinematic viscosity.If the Newtoniancoolingcoefficientislarge comparedto the adiabaticcutofffrequency, thewavelengths below and abovethelowerreflecting layerwillbe equalized. Consequently thereflection produced bythethermalconductioniseliminatedcompletely. This indicates that in thesolarphotosphere thetemperaturefluctuations may be smoothedbythetransferof radiationbetweenany tworegions with differenttemperatures. Alsotheheattransfer byradiationis moredominantthantheconduction process.

KEY

WORDS: Acoustic-Gravity

Waves,

Atmospheric

Waves, Wave Propagation

AMS

SUBJECT

CLASSIFICATION CODES. 76N,Y6Q

1 INTRODUCTION

Upward propagatingsmallamplitude acoustic-gravity^wavesinanisothermalatmospherewill bereflected downwardifthe gasis viscousorthermally conducting. Thistype ofreflection is mostsignificant when the wavelengthislarge compared tothe density scale height Alkahby and Yanowitch

[1989, 1991], ^Campos [1983a, 198:b],

^Cally

[1984],

Lindzen

[1968, 1970],

^{Webb andRoberts}

[1980], ^Lyons

and Yanowitch

[1974],

Priest

[s41,

^Yanowitch

[1967a,

1967b,

1979],

ZhugzhdaandDzhallov

[1986]).

In

part oneofthis serieswe consideredthe theeffect of Newtoniancoolingon acoustic-gravitywaves in anisotherrnal atmospherefor largePrandtl

number.

It wasshown that for an arbitrary value of the Newtoniancoolingcoetcient theatmospheremaybedivided into two distinct regions, whichareconnected by anabsorbingand reflecting layer produced bythe exponentialincreaseof the kinematic vicosity with

.580 H. Y. ^,\I,KAHB

height In thelowerregionthemotionsadiabaticif the NewtonlancoohngcoefficientIssmallcomparedto the a&abatc cutofffrequency Onthe otherhand,f the Newtomancooling coefficient ssufficientlylarge themotionwillbe driven towardsanisothermalone Consequentlyall linear relations among perturbation quantities wllbe modified

In

particular,it decreases the ampiitude of thewaveand thereby the energy fluxaswell. In the upper regionthesolutionwll^decay exponentially wth alt:tudc beforet:s nfluenced

by theeffect of the thermalconduction

In

ths paper wewill study theeffect of Newtoman coolingon upward propagating acousncgravity waves nanisothermalatmospherefor small Prandtl number. Itisshownthat,if theNewtoniancooling coefficientis smallcompared tothe adiabaticcutofffrequency,the atmospheremay be^dividednto three

&fferent regions.

In

thelower regmn the effect of thermal diffusvity and kinematic viscosityisnegligible, the oscillatory process is a&abatic and for frequencies greater than the adiabatic cutofffrequency the solution can be written as a linear combinationof anupward and a downward travelhngwave.

In

the mddle region the effect of the thermal diffusivityis large whale that of thekinematic viscosity isstill negligible. Consequently,themotioninthe middle regionisisothermaland thesetworegionsareconnected by ^a semitransparent reflecting layer, Mlowing part of the energy to be transmittedupward, while the remaining partisreflected downward. Thereflecting layer^nowseparatestworegions with different sound speeds, and therefore differentwavelengths,which account forthe reflection process.

In

the upper region, where the kinematic viscosity and thermal diffusivityarelarge,the amplitude of the velocity oscillations approaches^a constantvalue. Thetwoupper regions^areconnected by anabsorbing andreflecting layer, throughwhich the kinematic vicositychangesfrom small tolargevalue. The existenceoftworeflecting layer will influence the reflection process in the lower region and the final conclusion depends on their relative locations.

When Newtoniancoolingis large compared ^tothe adiabatic cutoff frequency the oscillatory process in the adiabatic regionis driven towardsan isothermal oneand this will decrease the wavelengthfrom the adiabatictothe isothermal values.

Thus,

thewavelengthsbelow and above the lowerreflecting layer areequalized.

As

a

result,

thereflection, produced by thethermal conduction,is eliminated altogether.

This indicatesthat Newtonian cooling influencesonlythe adiabaticregionsinthe atmosphereand if the heat exchange, due toradiation, is intense the temperature perturbation associatedwith thewavewill be eliminated inatimesmall comparedtothe periodof oscillation. Consequently,the effect ofthermal conduction willbeexcluded.

From

theabove discussionwemay conclude that in the solarphotospherethetemperaturefluctuations associated with verticallypropagating acoustic-gravitywavesmay be evenedoutbythe transferofradiation between any tworegionswith different temperatures.

Also,

highemissivities in the presence ofanopen boundary allow rapidloss of radiation to space.

In

addition thisresultindicatesthattheheat transfer by radiationis moredominantthanthatof the conduction process which is thecasein the hotregionsof the solaratmosphere.

We

conclude by reconsideringthecaseof the effect of Newtoniancoohngalone insection

(3).

^Three

rangesforthe frequency^are

identified,

above theadiabaticcutofffrequency,below the isothermalcutoff frequency andinbetween. Theresultsofsection

(3)

^areused in section

(4).

^{Finallytheproblemin section}

(4)

^isdescribed by afourth-order differential equation which^issolved by matching procedure, in which innerand outer expansionsarematchedinanoverlappingdomain.

AC()LISI’IC-(’RAV1TY WAI’S 1N AN ATNOSPHt’RtC 581

2 MATHEMATICAL FORMULATION OF THE PROBLEM

\Ve wll consider an isothermalatmosphere,which is viscousand thermallyconducting, and occupies the upper half-space z > ⁰ We will investigate the problem of small vertical oscillationsabout eqmlibrium,

1.eoscillations which depend onlyonthetime andon thevertical coordinate

Let theequihbrlum pressure,density and temperature be denoted by Pc,,po. and

7o.

^where

Po

^{and /}}

satisfythegas law

Po RToPo

^{and the}hydrostaticequation

P ⁺

^{gPo 0. Here}

^R

^{isthe gas} constant, g

is the gravitationalaccelerationand the prime denotes differentiation with respect toz. The equilibrium pressure and density,

Po(z) Po(O)exp(-z/H), po(z)- po(O)exp(-z/H),

where

H

RTo/gisthe density scale height

Let p, p,w, and

T

be the perturbationsin the pressure, denmty, vertical velocity, and temperature.

Thelinearizedequations ofmotionare:

pow,

+

^p

+

gp

(4/3)zw=,,

(2.1)

Pt

+ (pow)

0, ^(2.2)

po(cv(Tt + ^qT) + ^gHw=)

^(2.3)

p

R(poT + ^Top).

^2.4

These are, respectively, the equation for the change in the verticalmomentum, ^{the mass} conservation equation, the equation for theratechangeof the x-component of the magneticfield,the heat flow equation and thegaslaw.

Here

cy isthe specific heat at constant volume, qisthe Newtoniancoolingcoefficient whichrefers to the heat exchangeand isthe thermal conductivity, all assumed to beconstants. The subscriptzand denote differentiation with respect^{to z}and respectively. Equation

(4)

^{includesthe heat}

flux term

cvpoqT,

whichcomes from the linearized form of the Stefan-Boltzmanlaw.

We

will consider solutions whichareharmonicintime i.e

w(z,t) W(z)ep(-iwt)

and

T(z,t) T(z)exp(-wt),

wherew denotesthefrequencyof thewave.

It

is moreconvenient torewrite the equations in dimensionlessform;

z’ _{z/H, w=} c/2H, W’ w/c,

w’ w/w=, t’

tw=,

’ 2/cvcHpo(O), T’ T/27To, q’ q/w=,

wherec

x/TRTo x/-gH

is the adiabatic sound speed, and

w=

^is the adiabaticcutofffrequency. The primes can beomitted, since all variableswill be written in dimensionless form

)rom

now on.

Onecan eliminate p, and p from equation

(1)

by differentiatingit withrespect to andsubstituting equations

(2.2-2.5)

^toobtainasystem of differential equations for

W(z)

^and

T(z),

(D ^D + "w/4)W(z) + 7#e=DW(z) + iT(D 1)T(z)

0, ^(2.5)

(7 1)DW(z) 7(iw- q)T(z) + 7ae=DT(z),

^(2.6)

where

D d/dz. If, furthermore, W(z)

^iseliminated from the differential equation

(2.6)

by applying

D

to itandsubstitutingfor

DW(z)

^fromdifferential equation

(2.7)

^{oneobtainsasingle}fourth-order differential equationfor

T(z):

[Tw(D

D

+ wu/4) + zq(D D + 7w/4) zeD(D +

^D

+ 7w2/4)

iT(w+ zq)#eZ(D + D)-ae2=D2(D + 1)(D + 2)iT(z)=

^{0 (2.7)}

582 H. Y. ,\I.KAHBY

Inaddition11 s convenient tontroduce thedmens,nless Prandtl number

P ^tz/n,

which measuresthe relativestrengthof the thevscosltywithrespecttothethermal conduction. Consequently the differential equation

(2 8)

becoznes

.ww’(D D

+

w2/4) zq(D"

D

’w2/4) zeD2(

D

D

--zPq(w- zq)ge’D(D. 1)- wP,(ge*)2D2(D. 1)(D -2)7"(:)

^{0 (2.8)}

Fnallv thefirsttwotermsofequation (2

9)

^{may be}combined togve the followingequatxon,

(D

D

+ /4)- (/,)*D:(D +

D

+ :/4)

zPmr(/m)eD(D + 1)- (q’Pm)(xe/m)2D2(D + 1)(D + ^2)]T(z)=

^{0 (2.9)}

Boundaw

^Condztzons To completethe ormulaton of theproblemcertain boundary conditions must be imposedto ensureaunique solution. Theexact natureof the exciting force neednotbe specifiedsinceour objectstoinvestigate the reflection and dissipatmn of thewaveswhich,for smM1 and

Pr,

takeplaceat ahigh altitude.

Boundaryconditions arerequiredat 0, andweshladoptthe lowerboundarycondition

(LBC): In

afixed anterval0

<

^z

<

z0, the solution of the differential equation

(2.10)

^{shouldapproachsomesolution}

of thelimitingdifferential equation 0 and_#

0),

[D ^D + r=/4]T()=

^{O. 2. 0)}

Consideringthe lowerboundary conditionissimplerthan prescribing

T(z)

^and

W(z)

^{atz 0.}

^To

^first

order the boundary layer has noeffect on the reflection and dissipationprocess, which takesplaceat a high altitude.

Two furtherconditions which refertothe behaviour of the solutions forlargezarerequired andweshM1 call these contions the upperboundaryconditions. The firstoneisthe

Entropy

Condition

(EC),

^which

is deternedby the equation for the rateofchange of the entropy

(see

^Alkahbyand

Yanowitch[1991], Lyons

and

Yanowitch[1974]). From

whichitfollows that

The second contion is the Dissipation Condition

(DC),

which requiresthefiteness of therateofchange of theenergy dissipationⁱⁿaninfinite columnoffluidofunit cross-section Alkahbyand

Yanowitch[1991], Campos[1983a,1983b], Lyons

and

Yanowitch[1974]).

Since thessipationfunctiondependsonthe squares of thevelocity gradients,^thessipationconditionisequivMent to

IW12dz <

^{oo, (2.12)}

Both of the upper boundary conditions arenecessary and sufficient as an upper boundary condition if

/,

>

0. Finally if

z

⁰the Radiation Conditionissufficientto ensure aunique solution.

3 THE EFFECT OF NEWTONIAN COOLING ALONE

In

his section^wewill consider the effect of Newtoniancoolingaloneonacoustic-gravity^{wavesin anideal} atmosphere. There:ultswillbe usedin section

(4). For

thiscase, the differential equation^canbe obtained bysettinga # 0inthe differential equation

(2.10).

The resulting differential equation^is

[D

⁾

+ /4](z)=

^{0. (3.)}

,\COI_;ST[C-(;tLAVI’iY WAVES IN AN ATMOSPHERE 583 whereT

’( +

zq)/m,^and m /w

-+

^{zq The}solution ofthe above dfferential equationcan bewritten

inthefollowingform

T(z) clexp[(1

/

v/1 -a,’2)z/2.

^/

c2exp[(1 v/1 "rw2)z/2l,

(3.2) where _cl and c2 are constantsand theywill be deterrnned from theboundary ^condition. Toinvestigate thenatureof the effect ofNewtomancoolingonthewavepropagatmnn anisothermal atmosphere,it is convementto considerthe followingtwolimitingcases

CASE A:when q 0, the parameter

"

^reducesto and for frequenciesgreaterthan the adiabatic cutoff frequency

w

^1,equation

(14)

has thefollowing form

T(z) cexp[(1/2 + zka)z] + c:exp[(1/2- zka)z],

^(3.3)

where

2k= v/w

isthe adiabaticwavenumber.

CASE

B:

when q oo, the parameter _{V and} for frequencies greater than the isothermM cutoff frequency

w 1/,

the solution of the differenti equation

(13)

^canbe written like

T(z) cezp[(1/2 + ^zk,)z] + caezp[(1/2- ik,)z],

^(3.4) where

2k ?w2-

^isthe isothermMwavenumber.

To investigate^theeffect of Newtoniancoolingonthe behaviour ofthewavepropagationforanarbitrary

vMue

of q,it isconvenienttowrite inthefollowingform

, _{[(7 + 1)- ( 1)(cos20g ism2eq)]/2}

(3.5)

1 / (-4q) + ik,),

^(.6

where

s(q)

^istheattenuationfactor. Consequentlythe solution in equation

(3.2)

^becomes

T(z) [(/- (q) + i,)z] + [(1/ + (q) ik,)z],

^(.

To

obtNnthebehaviourof

s(q),

^itisconvenient towrite

rw

[(1 -(7 + 1)w/2] + [(7 1)(co20, isin20,)]/2,

^(3.8)

It

isclear that

equation(3.8)represents

asecircle in thecomplex planewith center at

( + ^1)w/2

and radius

( 1)w/2

^as

0

^{variesfrom 0 to}

^/2.

It

followsfrom equation

(3.7)

^{that thesolution canbe}described in the followingway: the first term on the right represents^anupward

travelSng

wave,^itsamplitude decayingwith Mtitude like

exp(-s(q)z),

whilethe second term is adownwardtravelSngwavedecayingatthesamerte.

We

havetoindicate the the upperboundaryconditions

(2.11)

^and

(2.12)

cannotbe applied because 0.

A

unique solution canbe determinedby theradiation condition which requires

c

0. Also there arethreeranges for the frequencyw. The first one is forw

>

w, 1,the secondonefor

1/

^w,

^<

^w

^<

^w., and the thirdone forw

<

w,. They are

denoted,

respectively,by

R, R

^and

From

this observation and the three ranges of thefrequencywehave thefollowingconclusions.

A

Whenwbelongsto

Rm

^theattenuationfactor

s(q)

ispositive and

equMs

to zero at theextreme limits q 0 d q

.

^Itincreases to its mimumvalue,

s(q)

0.1, when

(q/w) O(1)

^anddecys to zero asq

.

58 H.Y. ALKAttBY

B When the frequency n

R2

^{and for smM1 value of q, a decaying wave exists and changes to}

undamped travelling^{wave as}qo

C If02belongs to

R3

^{and for}small value of q,aweakdampedwaveexists Asq oc thetravelling wavechanges^toastandingone

(D)

In

(A), (B),

^and

(C)

^{the wavenumber ncreases}monotically from ts adiabatic value k,, to the isothermal one k,, as q ec, because of the changeof the sound speed from its adtabatic valueto the isothermalone. Thus the attenuatmnfactorremains positive,

s(q) _>

0,for all values ofq. At the same timethe oscillatory processis transformed from the adiabatic formtotheisothermalone.

EFFECTS OF THERMAL CONDUCTION, VISCOSITY AND

NEWTONIAN COOLING

In

this section we will investigate the sngular perturbation boundary value problem for the following differential equation

[(D ^D

^4-

-r022/4)- z(n/m)eD2(D +

^D

+ "),022/4)

-zPmr(/rn)eD(D+ 1)-(TPm)(e/rn)2D2(D+ 1)(D+ 2)JT(z)

0, (4.1) where ^r

"(w + ^,q)/(Tw + ^{,q) "(w} + iq)/m

and m -02

+

^{zq, subjectedto the boundary condition}

(2.11), (2.12),

and the lowerboundary condition. Attheoutset wehavetoindicate that theparameters# and aresufficiently small and proportional to the values atz=0of the kinematic viscosity and thermal diffusivity. Prandtl number

P,

canbe written like

P,- I/ (lz/po)/(’c/po) I(.lmpo)(,qmpo)l

^(4.2)

Thus for small

Pr

^wehave#e

<<

^e

. ^As

^aresult,for

I/mle <<

^{and small}

vMues

ofq theatmosphere may be dividedintothree distinctregionsconnectedbytwodifferent transitionlayersin which the reflection andthewavemodification takeplace.

In

thelower region, 0

<

z

<< z ^-loglrnl,

^{the motionisadiabatic,}

because the effect of thermaldiffusivityisnegligible,and the solution of the differential equation

(4.1)

^can

beapproximated bythe solution ofthe followingdifferential equation

[D D + -w2/4]T(z)--

^{0, (4.3)}

the solution of which is investigatedin section

(3). ^In

^{the middle region,}

z <

^z

<

^{z. -log#, the}

oscillatory process is an isothermal, because the influence of the thermal diffusivity ^is large, and the solution of thedifferential equation

(4.1)

^canbe approximatedbythe solution of

[D2(D

^q-

^D + ?022/4)]T(z)=

^{0. (4.4)}

The twolower regionsare connected by asemitransparent reflecting layerin the vicinity of

z. ^In

^the

upperregion the oscillatory process isinfluenced by the combined effect of thethermalconduction and the viscosity and the solution of the differential equation

(4.1)

^can be approximatedbythe solution of the followingdifferential equation

[D2(D + 1)(O + 2)]T()

^{0. (4.5)}

Thetwoupperregionsarejoined byanabsorbing and reflecting transitionregion in the vicinityof_z2

-log(/#),

aboveitthe solution which satisfies the upper boundary conditionsmust behaveasaconstant

Toobtainthe solution of the differential equation

(4.1)

it is convenient tointroduceanewdimensionless

ACOUSTIC-GRAVITY WAVES IN AN ATMOSPHERE 585 variable

(

defined by

ezp(-z)/(,/rn)= exp[-z

loglc/rn

+

,8,,

+ 3r,/2],

(4.6) where

m

^arg

(m),

which transforms the differential equation

(4.1)

^into

[( + + /4)- (( + /4)

mP,.’r(O -0) + "P,.mO2(O- 1)(0- 2)]T()

^0. (4.7) where O

d/d.

It isclear that the point 0 correspondsto linz oo, the point

0

^exp

[-

logic/m]

+ i( ,+ 3r/2)]

to z 0and the segment connectingthese points in thecomplex -planeto z

>

0.

As I/rnl

0 the point

0

^{tends too.}

Itisclearthatthe point 0is aregular singularpoint ofthisdifferential equation

(4.7). Consequently,

therearefour linearly independent

solutions,

which in theneighbourhood of 0can bewritten inthe following form

T3(() a’’e.t 31’t"+" T,() a:’(e,)

^"+’’

+ ^T()log(),

^(4.8)

where_el 2,

e

^1,e3 e4 0. Theprimedenotes differentiationof andthesumsaretaken fromn 0 ton

.

The coecients

(e,)

^aredeterned fromthefollowing threeterm recursionformul

o( +

²

+ e)+ + ^,( + + e)+, + ( + e) o,

(.)

where

,o(=) -P,,,,="(= )(= ), Pl(=) -m’rP,.m,(m,- 1)- z(z =-

^m

⁺

(=) (= + = ^{+ /4).}

To

determine which ofthesolutions defined in equation

(4.8)

satisfiesthe upperboundarycontions

(2.11)

and

(2.12)

^for

large

z,the solutions must be transformed^tothe riable z by meansof

(4.6).

Thus for

]/m

>0 and

P,

>0wehave

T(z) O(e-2"), T:(z) O(e-’), T3C ^z) O(1), T4(z) ocz).

^(4.10)

It

isclearthat

T4(z)

^{is the}onlysolutionwhich does notsatisfythe entropy condition

(2.11).

To

applythedissipationcondition

(2.12),

^equation

(2.7)

must be used^todetermine thethe amplitudes of the velocitycorrespondingtothe solutions defined in

(4.8).

Asaresultwehave

DW(z) O(e-*), DW2(z)= O(1), DW3(z) O(e-’), DW4(z) O(z).

^(4.11)

Itisclear that

W2(z)

^and

DW4(z)

do not satisfy the boundary condition

(2.12).

Asaresult,weobtain

T() T()+ ,T() W() ’W() + ’W(). .

To

determine the linear combination of

T(z)

ⁱⁿ

equation(4.8),

the behaviour of

T(z)

^and

T3(z)

^forsmallz

mustbefound. Recallthat smallzcorrespondstolarge

I1

^with

^{arg() 3/2} + ,.

Thus theasymptotic expansions of

TI()

^and

T3()

about infinity should be found.

The point o is an irregular singular point of the differential

equation(4.7),

and there arefour linearly independent solutionswhoseasymptotic behaviour,tothe firstorder,^isgovernedby

586 H. Y. ALKAHBY

T() =’[i +

h:(

- ^{+ h} - ^{+ ],}

^{(4. ]3)}

T() ’[i + h2 - ^{+ h22}

^-2

^{+ ],}

^(4.14)

T(() (-/4[1 + ^h(

^-/

+ ]ezp(-(/),

^{(4. ]5)}

T() -1/411 + h41

^-1/2

+ ]exp(l/2),

^(4.16)

where

a -1/2 s(q) +

^,k,,

^a -1/2 + s(q)-

k,. Reintroducingthedimensionlessvariabh zbymeansof

(23)

wehave

T;(z) e[(/ + (q)- .)z], T(z) [(/ (q) + .)],

T(z) e[-(]/,)/], T(z) [(/)/].

As

aresult of that

T(z)

^{represents a} downward propagating wave, its amplitude decaying lik

exp(-s(q)z),

while

T(z)is

^aupward propagatingwavedecayingatthesame rate. Also

T(z)

^correspond

tothe boundary layerterm.

It

is important onlynearz 0 for anysmM1 fixed

vMue

ofz andit decay withzlike

ezp[-(m]/4)/2z].

Itfollows that, forfixed

vMue

ofz, Newtonian coolingreduces the widt of the boundary lyer. Thusit is more convenient to use the lowerboundary condition, from which followsthat the solution of the fferentiM equation

(4.7),

^{when 0andsmM1}

vue

of qshouldbehave asymptoticMlylikea

nnear

combinationof

T()

^nd

T(),

^i.e

The determination of the asymptotic behaviour of

T()

^and

T3()

from thesolution

(4.8)is

^difficult

because the coefficients

(e,)

^{aredeterminedfrom}three termsrecursionformula

(4.9).

^{Insteadwewill dc}

itby

matcng

procedure,ⁱⁿwhich theinnerand the outer expansions for the solutions willbe matched inanoverlappingdomn.

To

findthe inner appromation,assumethereestsaregularperturbation expansion of the form

T() I() + PI() + O(P).

^(4.

Substituting

(4.17)

^intothe differentiM equation

(4.7)

and setting

P

^0.

^We

^{obtn thefollowingdiffer-}

entiM equation

[( + + /4) (

^#

+ /4)]I()

^{0, (4.}

where8

d/d.

The second step of thematchingprocedure beginswith thestretching

transformation,

=P’,

_{(4. 9)}

of thecomplete differentiM_equation

(4.7).

^TheresultingdifferentiMequation is:

[( )(- )- ( + /4)

+ P(( + + /4)- ( ))]T()=

0, (4 .0) where 8

Cd/d. To

obtntheouterappromation,assumethat thereexists asingularperturbation expansionof thefollowing form

T()= ()+ Pbh()+O(P]),

(4.)

substituting

this expansionintothe differentiMequation

(4.7)

^andletting

P

0,wehave

[#( )(- 2) ( + /4)]v()

^{0. (4.)}

ACOUSTIC-GRAVITY WAVES IN AN ATMOSPHERE 587 Thesolutionsofthisdifferential equationwillapproximate the solutions of the differential equation

(4.20)

if(

Pr

^{is small. Since oo for any fixed}

1

^as

^Pr

0, the asymptotic behaviour of

U()

as oo mustbe matched with the asymptotic behavlour of

I(()

^as

(

0.

Hence,

^{themaintask isthe}

determinationof the family of thesolutionsof the singular perturbation differential equation

(4.22),

^which

satisfy the upper boundaryconditions

(2.11)and (2.12).

The differential equations

(4.18)

and

(4.22)

^are

similar tothe differential equation

(4.6)

^and

(4.9)

ⁱⁿ

^Lyons

andYanowitch

[1974].

Theprocedureforfinding theasymptoticbehaviourof thesolution isthesame.

However,

the physicalnatureof the solutionisquite differentin thetwoproblems. Consequentlythe details neednotberepeated,andwemerelyindicate the results.

Nowwewillstudythe behaviourof the solution in theatmosphereforz

_>

0.

It

is convenienttostart withthe upper region.

UPPER REGION:

In

this region the solution of the differential equation

(4.1)

^isapproximatedby the solution of the differential equation

(4.5).

It follows from equation

(4.10)

that the solutionwhich satisfiesthe upperboundaryconditionsmustbehaveas aconstantasz oo.

MIDDLE

REGION:

In

the middle the solution of the differential equation

(4.1)

^{isdescribedbythe}

solution of the differentia/equation

(4.4),

^{whichcanbewrittenlike}

T(z)

^cons*,

ezp[(-1/2 + ^ik,)z] + ^RC, cap[(-1 ik,)z],

^(4.23)

where

RC,

denotes the reflection coefficient in the middleregionand definedby

RCrn ezp[-k, + 2i(0

k,

log(?D))],

^(4.24)

o =(/- i,) + =r(i,) + =r(a/- ,)

(4.2s)

It

isclearthat

IRCI ^ezp(-k,).

Consequentlythetwoupperregionsareconnectedby anabsorbing andreflecting layerinwhichthe kinematic viscositychangesfromasmM1to alarge

vMue

because of the exponentiM decrease of the density withheight. Naddition the^midgeregion will not influenced bythe effect of Newtonian

coo5ng.

LOWERREGION: For

I/m]e <<

1,w

> w=

^{and small}

^vMue

of the Newtoniancoolingcoefficient q, the^solutionof the differentiM equation

(4.1)

^{can bewritten in} the following form

T(z)

^cons$,

ezp[(1/2- s(q) + ik,)z] + RCL ezp[(1/2 + s(q) ik,)z],

^(4.26)

where

RCL

denotes the reflection coefficient definedby

L ^L=ezp[-2k,

^logP

+

^0"

+

^2k,

^O]

RCL ezp[-vka + ^{A, B,]}

^C,

_{La Lezp[-2k,}

_logP

₊

_0"

_{+ 2k,Om]’}

A. 2s(q)logl/m

2k=O,,

B. ^2kalogl/ml + ^rs(q) + ^2s(q)0,, C. r2(1/2 + ^s(q) ,ka)r[s(q)- ,(k, + k.)]r(-2s(q)+ 2iko)r[1 + s(q) + ,(k,-

r2(1/2 s(q) + ^ik)r[-s(q) + i(a, + ka)lr(2s(q) 2ik.)r[1 s(q) i(k, k.)]’

L1 ezp(2rk,) ep(2rk)[cos(2rs(q))+ isi(2rs(q))],

, ,,(,ao)[o(,(q))+ i,(.,(q))]- -,(--,a,),

L3 ea:p(2rk,) exp(-2rk)[cos(2rs(q))- isin(2rs(q))],

,,(-.,o)[o((q))- i,(,(q))]- ,p(,a,),

588 H.Y. ALKAHBY

D. ^(1/2 z/’)[’2( ,)r( + ()- (, + o))r( ()- (,- =)) (1/ + ,k,)r=(-ik,)r( s(q) + ,(k, + k.))r(1 + s(q) + ,(k,- k.))’

O*

argD.

2k, log.

In

additiontothe conclusionsof partone wehave thefollowingobservations

When q 0, we have

s(q) A. 0

^{0 and}

B. 2k=log(n),

and we recover the result obtned in

Lyons

and Yanowitch

[1974]. In

this casethereflection processismorecomplicated because of theexistenceoftworeflecting layersandthe fin conclusiondependsontheir relative locations. The magnitudeof the reflection coefficient is

(II) ^For

^fixed

,

smM1 q and w

> w=

the solution is given by equation

(4.26)

and its behaviour is described in section

(3).

The atmosphere isdivided intothreedistinct regions. The lower region is appromatelyadiabaticandthe middleoneisisothermM.

In

the upperregionthe solutionisinfluenced by the combinedeffects ofthethermM diffusivity and the kinematic viscosity.

Ill When q andw

>

w,one obtns

L L4 s(q) 8 A. ^O, ka k,, C.D.

, L L3

^and

RCL RCm.

Consequently the lowerreflecting layer will be minated because the wavelengthsbelow and above the thislayerbecome

equM. In

additionwehave

/m

^{0and thesolution}

of the differential equation

(4.1)

^canbeapproximatedbythe solutionof the followingdifferenti equation

[(D ^D + ?w/4) ive*(D + D)]T(z)

^O.

the solution of whichisinvestigate in

Compos [1983a, 1983b],

^Yanowitch

[1967]).

IV

The above conclusions indicatethat,in the solarphotospherethetemperaturefluctuationscould be smoothedby the transferof the radiation between any tworegions withdifferent temperatures.

In

additionthe heattransfer bytaxation ismoredominant thantheconduction process.

ACKNOVLEDGEMEN’I

would liketoexpress mysincerethanks toProfessorMichael Yanowitch for hissupportand invaluablecriticismduring the preparation of this work.

REFERENCES

1.

ABIAMOWITZ, M.,& STEGUN, I.,

Handbook of Mathematical Functions, National

Bureau

of Standards,Washington,D.C.1964.

2.

ALKAtIBY, H.Y.,

leflection anddissipation of verticallypropagatingacoustic-gravity wavesinan isothermalatmosphere, Ph.D.Thesis,Adelphi University,

(1988).

3.

ALKAHBY, tt.Y., & YANOWITCH, M.,

The effects of Newtonian coolingon the reflection ofver- tically propagatingacousticwavesinanisothermalatmosphere,

Wave

Motion 11

(1989),

^419-426.

4.

ALKAHBY, H.Y., & YANOWITCH, M.,

R.eflection of vertically propagating^wavesinathermally conductingisothermalatmospherewithahorizontal magneticfield,

G.eophys.

^{Astroph. Fluid}

Dynam.

A

S

^(9,),

5.

ALKAHBY, H.Y.,

^Reflection and dissipation of hydromagnetic ^waves in a viscous and thermally conducting isothermalatmosphere,toappearinGeophys. A.str0ph. Fluid

Dynam.

ACOUSTIC-GRAVITY WAVES IN AN ATNOSPHERE 589

6 AI,KAIIBY, II Y Ac,slit-(;r,ly wavesin vlsusand thermallycond tng sthernal atmo- sphere (part For large Prandl nunber), toappearn lnternat J. Math ^," Matl Sci

7 t/[SSI’;, F II ,’i’hernal nstabilitesnapdly ^rc,tatngsystems, Fluid Mech ^4,1

(1970),.t’t1-’t60

8 BIJSSI’;, F1I, A ^model(}fthe ge{<tynam<>,

Ge,i}hvs,

I Astr. S{>c. 42

(t{}75’,

^t37,t59

9 CALLY,

PS,

Magnet{)hy{Ir,}dynan]c critical levels and radiative damping, Astr Astr{}ph 136

10

CAMPOS,

L M C On ^vis{osand ress{vedsspatonof hydrodynamic and hydr{}magnetc ^waves in atmospheres, J Mec ^2l,eror Appl ’2

(1983b),861-89I

11.

CAMPOS,

L.M.C On magnetoacoustc- grawty waves propagatingor standing vertically in an at- mosphere,J Phys A 16

(1983a),417-437

12 DEBNATII,

L, BASIJ,

IJ.,Capllary-gravitywavesagainstavertcalcliff,Ind. J Math26

(1984),

49

13

EI,TAYEB, A. ROBERTS,

Plt, On the hydroma.gnetcs of rotatingfluids, J. Astrophys. 162

(970)

^9-70

14

IIOLI,WEG, V.,

R.esonanceofcoronM loop,

’2

^{Astrophys 277}

⁽¹⁹⁸⁴⁾

^392-’t03

15.

IONSON,

J.A

Resonant

electrodynamic heating of stellarloops, J. Astrophys. ²⁵⁴

(1982)

318-334 16.

LYONS, P, YANOWITCII, M.,

Verticaloscillationsin viscousandthermally conductingisother-

mala’,mos here,

2

^l;luidMe(:t

6, (197.i),

^273-288

17.

LINDZEN,

R.S. Vertically propagating wavesin an atmospherewith Newtonian cooling nversely proportionaltodensity, Canad.

J.

Phys. 46,

(1968),

1835-1840.

18.

LINDZEN,

R.S Internal gravity waves n atmospheres with realistic dissipation and temperature, Part

I,

GeophysFiudl)ynarncs

(1970),

303-355

19.

MIHALAS, B W. TOOMRE, J.,

Internalgravitywavesin the solaratnosphere part Adiabatic wavesinthe

Chromosphere),

J Astrophys. 249

(1981),

^249-349.

20.

MIHALAS,

B.

W., & TOOMRE, J.,

Internal gravitywavesinthe solar atmosphere partII. Effects ofradiativedamping), J.Astrophys. 263

(1982)

263-286.

21.

MOFFATT,

It

R,

Magnetic field in electrically conducting fluids, Cambridge University

Press, (79).

22

PARKER,

E

N.,

Cosmical Magnetic Vields,Theirorigin andtheir activity,ClarendonPress. Oxford 1979.

23.

PRIEST, E.R.,

SolarMagnetohydrodynamics, D ReidelPub.

Co

1984

24.

ROBERTS, B.It.,

Onthethermal instability ofa.rotating fluidsphere cont.aningheatsources,Phil.

Trans. ^I(. Soc. Lond A263,

(1968)

^93-117

25

SOWARD, A.M.,

Non-lnear marginalconvecIionn rotating magnetic system, Gcophys.

Fluid

Dvna 3 ^(1986),

^329-371

590 H. Y. ALKAHBY

26. SUMMERS, D., Gravity modified sound in conducting strattied atmosphere, Q. J. Mech.

App1. Math 29

(1976),

^17-126.

27.

WEBB,

A.1I

&

ROBEtl’S,B

It.,

Vertical motion in anntense magnetic flux lube part

IV,

Solar Phys,68

(1980),

^71-85.

28.

YANOWITCIt,

M Effect ofvlscosty on vertical oscillations ofanisothermal atmosphere, Canad

_J

^{Phys. 45}

^(1967),

^2003-2008.

29

YANOWITCtI, M,

Effect of vscomty on gravity waves and the upper boundary conditions, J Fluid.Mech. 29

(1967),

209-231.

30

YANOWITCIt, M.,

Vertically propagating hydromagnetic^wavesJn ansothermal atmosphere with ahorizontalmagneticfield, WaveMotion

L ^(1979),

^123-125.

31.

ZttUGZHDA,

YU D

,&: DZIIALILOV,

N.S Magneto-acoustic gravitywavesinahorizontal magnetic field, Geophys. FluidDynamics.

35, (1986)

^131-156.