AMS WORDS

VOL. 20 NO. 2 (1997) 367-374

ACOUSTIC-GRAVITY

WAVES IN

A VISCOUS

AND

THERMALLY CONDUCTING ISOTHERMAL

ATMOSPHERE

(Part

II1: For ArbitraryPrandtl

Number)

HADIYAHYAALKAHBY

DepartmentofMathematics andComputerScience DillardUniversity

New Orleans,LA 70122

(ReceivedDecember 13, 1994andin revisedform July14,1995)

ABSTRACT. Inthispaperwe willinvestigate thecombinedeffect ofNewtoniancooling, viscosity and thermalcondition onupward propagatingacoustic waves in anisothermalatmosphere. Inpart oneofthis

seres

^weconsidered the caseof largePrandtlnumber,while in parttwoweinvestigatedthe case of small Prandtlnumber Inthosepartswe examined onlythelimiting cases, e. the casesofsmalland large Prandtlnumber,andit is moreinterestingtoconsiderthecaseof arbitrary Prandtlnumber,which isthe subject ofthis paper, because it is abetter representative model. It is shownthat if the Newtonian coolingcoefficient issmall comparedtothefrequencyofthewave, the effect ofthethermalconduction is dominatedbythatof the viscosity.Moreover, the solution can be written as a linear combinationofan upward andadownward propagatingwave withequalwavelengths and equal damping factors On the other hand if Newtoniancoolingislarge comparedtothe frequency of thewavethe effect of thermal conduction willbeeliminatedcompletely and the atmospherewillbe transformed fromthe adiabaticform toanisothermal. Inaddition, allthe linear relationsamongtheperturbations quantitieswillbe modified.

It follows from the aboveconclusionsand thoseof thefirst two parts,that when theeffect ofNewtonian coolingisnegligible thermalconductioninfluencesthepropagation of thewaveonlyin the caseofsmall Prandtlnumber.

KEY WORDSANDPIIRASES: Acoustic-GravityWaves,AtmosphericWaves,Wave Propagation 1991AMSSUBJECT CLASSIFICATION CODES: 76N, 76Q.

1. INTRODUCTION

It is well known that upward propagating acoustic waves of small amplitude may be reflected downwardif theBrunt-Vdlsdld frequencyvarieswith altitude. However,evenwhen theBrunt-Vgz’sld

frequencyis constant, additional reflection ispossible because oftheexponentialdecreaseofthedensity withheight. Thistype ofreflectionismost importantwhen thewavelengthis large comparedtothe density scale height.

The reflectionproperties ofaviscous isothermalatmosphere wereexamined byYanowitch [27], Alkahbyand Yanowitch[3], Campos

[14].

Itwasshown that the viscositycreates a transitionregion, whichconnecttwodistinctregionsandactslike anabsorbingandreflectingbarrier Inthe lowerregion theeffectofthe kinematicviscosityisnegligibleand the solution can bewritten, forfrequenciesgreater thanthe adiabaticcutoff frequency, as a linearcombinationofanupward andadownward propagating

wave In the upper region the effect ofthe kinematic viscosity is large and the solution decays exponentiallywith altitudetoaconstantvalue.

Thepresence of thermalconductionalso producesareflecting layer,with different mechanismfrom that of the viscosity. The exponential increase of the thermal diffusivity with height creates a semitransparentlayerallowing part of theenergytopropagate upward Asaresult,thereflecting layer separates two distinctregionswith differentsoundspeeds,becausethe signals propagatewith Newtonian soundspeedinthe isothermalregion. Consequently,thewavelengthsin thetworegionsaredifferentand this will account forthe reflection (Alkahby[7], Alkahbyand Yanowitch [3,4], Lyonsand Yanowitch [18]).

The combined effect ofNewtonian cooling, viscosity and thermal conduction, for large Prandtl number, is investigated in Alkahby [8]. It was shown that the effect of thermal conduction can be excluded and the solution, abovethereflectinglayerthat iscreated bytheviscosity, decays exponentially with altitudebeforeit isinfluencedby the effect ofthermal conduction.

Moreover,

whentheNewtonian coolingcoefficient islargecomparedtothefrequencyof thewavethelowerregionwillbe transformed from the adiabatic form to the isothermal one. Theeffectof Newtoniancooling, viscosity andthermal conduction, for small Prandtl number, on upward propagating acoustic waves in an isothermal atmosphereisinvestigatedby Alkahby

[9].

^Itwas shown that when theNewtoniancoolingcoefficient is smallcomparedtothefrequencyof thewavetheatmosphere maybedivided intothree distinct regions connected bytwodifferentreflectinglayers Inthelower region the oscillatory processisapproximately adiabatic, it is isothermal in the middle region and in the upper region the solution will decay exponentiallywith altitude. Onthe otherhandif the Newtoniancoolingcoefficientislargecomparedto thefrequency of thewavetheoscillatoryprocessin the lowerregionwillbe transformedtoan isothermal one Asaresult,the twolower regions becomeonebecausethereflectinglayer, which iscreated by thermal conduction,willbeeliminated.

In the abovetwo limiting cases, which were discussed in part and part II of this series, our conclusions are presumed and it is more important^to consider the case ofthe effect of Newtonian cooling, viscosity and thermal conduction for arbitrary Prandtl number because this case is a more representative model forthe reflectionand dissipation ofacousticwavesin anisothermalatmosphere. It is shown that the atmospherecan be divided into two distinct regions connectedby areflecting and absorbing layer, whentheNewtoniancooling coefficientissmall comparedtothe frequency ofthewave, the oscillatoryprocessinthe lower regionis adiabaticandabove the reflectingbarrierthesolution will decay exponentially with altitude. Whenthe Newtonian coolingoefficient is large comparedtothe frequency of the wave, theatmospherewillbe transformedto anisothermal oneandtheeffect ofthermal conduction will be eliminated completely while the influence ofthe viscosity will remain the same.

the case of small Prandtl number and the effect ofNewtonian cooling is negligible. Moreover, the atmosphere can only be divided into three distinct regions in the case of small Prandtl number and negligible effect ofNewtonian cooling. Itis shown that ifthe Newtonian cooling coefficientis large comparedtothe adiabaticcutofffrequency, itwillactdirectlytoeliminatethe temperature perturbation quantityassociated with the wave in a time which issmallcomparedtothe period of oscillations ^Since Newtoniancooling adds an additionaltermtothe linearizedequation of theenergy,damping modifies all linear relationsamong perturbationquantities. Inparticular,it causes attenuation inthe amplitude ofthe waveand thereby theenergyfluxaswell Alsothe attenuation inthe amplitude ofthe wave will vanish not only when Newtonian cooling is eliminated but also when Newtonian cooling becomes large compared to the adiabatic cutofffrequency. The reflection coefficient and the damping factors are obtainedand the conclusionsarediscussed in connection with theheating of the solar atmosphere

2. MATHEMATICALFORMUlaTIONOF THE PROBLEM

Inthis section we will indicatethemainsteps oftheformulation of theproblemand the details canbe found in Part or Part II. Suppose that an isothermal atmosphere, which is viscous and thermally conducting,andoccupiestheupperhalf-spacez

>

⁰ Wewillinvestigatetheproblemof smallvertical oscillations about equilibrium, i.e. oscillations which depend only on the time t and on the vertical coordinatez.

Lettheequilibriumpressure,densityandtemperaturebe denotedby P0,Po,and

To,

where

P0

^and

To

satisfy the gas law

Po RToPo

^{and the} hydrostatic equation

P ⁺

^{gPo O. Here}

^R

^{isthe gas}

constant, g isthe gravitationalaccelerationandtheprimedenotesdifferentiationwith respecttoz The equilibrium pressure and density,

Po(z)

P0(O)exp(-

1/H),

po(z) p0(O)exp(-

z/H),

where

H

RTo/gisthe density scale height

Letiv,p,w, and

T

be the perturbationsin thepressure,density,verticalvelocity, and temperature.

Thelinearizedequations ofmotion(conservation^ofmomentumand mass, the heat flow equation and the gaslaw)are

powt

+

Pz

+

^gP

(4/3)/zw,

(2 I)

Pt

+

(p0w)z 0, (2.2)

po(cv(T

+

^qT)

+

^gHw,)

T,,,,

(2.3)

p R(poT

+ Top).

^(2.4)

Here

cv

^{is thespecificheatatconstant}volume,/zisthedynamic viscosity coefficient, q is the Newtonian cooling coefficientwhichrefers^tothe heat exchange and is thethermal conductivity, all assumedto be constants. The subscript zand denotedifferentiation withrespectto zand t respectively Equation (2.4)includes the heatfluxterm

cvpoqT,

whichcomesfromthe lineafizedform oftheStefan-Boltzmann law Wewill consider solutions which are harmonic intime,i.e

w(z,t)

W(z0exp(-/),

T(z,) T(z)

exp(

-/),

(25) wherewdenotesthefrequencyofthe wave.

It is more convenient to rewrite the equations in dimensionless form; z"

z/H,

w,, c/2g,W*

w/c,

^w"

w/w,,

^t* tw,, ^," 2/cvcHpo(O),T*

T/2"yTo,

q" q/wa, wherec

v/7RTo vgH

^{istheadiabatic}sound speed, and

w

^isthe adiabatic cutofffrequency. The primescanbe omitted,sinceallvariables willbewritten in dimensionlessform fromnow on

Onecan eliminate p, wandpfrom equation

(2.1)

bydifferentiatingit withrespectto

,

^{then with}

respectto z andsubstituting equations(2.2-2.5)toobtainasingle fourth-orderdifferentialequation for

T(z)

only:

[(D ^D + TW2/4) i(/rn)d’D2(D + D + 7w2/4)

iP,.m’r(,/m)eD(D + 1) (’yP,.rn)(e/m)2D2(D + 1)(D + 2)IT(z)

^{0. (2.6)}

whereT

")’(w +

iq)/(Tw

.+

iq) "y(w

+

iq)/m,^rn "yw

+

^iq,D

d/dz

^andP,.

BOUNDARY CONDITIONS: To complete the formulation of the problem certain boundary conditions mustbe imposedtoensure auniquesolution. Physically relevant solutions mustsatisfy the followingtwoconditions(Alkahby[7],Alkahby andYanowitch[4], Lyonsand Yanowitch

18])

IWldz ^<

^{oo, (27)}

Thefirstof theseisthedissipation(DC),whichfollowsfrom thefinitenessoftheenergy dissipationrate in a column of fluid ofa unit cross-section. The second one, the entropy condition (EC), is a consequenceofthefiniteness ofthe entropy growthratein acolumn of fluidofa unitcross section

Boundaryconditions arerequiredatz 0,and weshall adopt thelowerboundarycondition(LBC) In a fixed interval 0

<

^z

<

zo, the solutionof the differential equation(2.10) should approach some solutionofthelimiting differentialequation

(

^{0 and}#

0),

^i.e the solution canbewritten in the form

T(z)

Const.

[exp[(l ^{+ V/l-} -)z/2] ^{+ K, exp[(v/l-} "rz/2)]],

^(2.9>

where

Kq

is a constant. Considering the lower boundary conditionissimplerthanprescribing

T(z)

^and

W(z)

at z 0because we avoid the computation of the boundary layerwhich hasno effecton the reflection anddissipation processesthattakeplaceathighaltitudes.

3. THE EFFECTOFNEWTONIAN COOLING ALONE

Inthis sectionwe will reviewthe effect of Newtoniancooling alone on the wavepropagationto make thepapermoreselfcontained. Also the results of thissection areneededfortheresults andthe analysis ofsection

(4).

^Forthiscase, thedifferentialequationcanbe obtained by setting # 0 in thedifferentialequation(2.10). Theresultingdifferentialequationis

[D

^D

+ -a/4] T(z)

O. (3.1)

where r 7(w+iq)/m, and m 7w+iq. The solution ofthe differemial equation (3.1) can be writtenin thefollowingform

T(z)

^c,xp

[(1 + V/1 "rw2)z/2] +

^c,

exp[(1 V/i -- "rw,)z/2],

(3.2) where_cland_c2are constantsand theywillbedeterminedfrom the boundarycondition Toexaminethe effectofNewtoniancoolingon the wavepropagation and dissipation, let

(V/1 TO’)/2

^-4-

(--

^d(q,w)

+ i)).

(3.3) Toinvestigatethe behaviorofd(q,

w)

and/3,wehave tostudythefollowingcases

(N1)

When q 0 and w

>

1, the solution of the differential

,equation

^{(3.1), defined in equation} (3.2), ^{becomes a linear} combination of an upward and a downward propagating wave with equal adiabatic wave

number/3a (V/u 1)/2

^andd(q,

w)

O.

(N2) Whenw/q^--,0andw

> 1/V/’,

^{equation(3.2)}which definesthesolutionofthe differentialq (3.1) can be written as a linear combinationof anupward and downward traveling wave withequal isothermalwavenumber/3,

(,v/’Tw2 1)/2

^andd(q,

w)

0. This caneasily beseenfromlimitofras q oo.

(N3)

^Whenq/w

<<

^{1thesolutionof theproblemcanbedescribedas}follows: thefirst term on the right of equation(3.2)willbeanupward propagatingwavedecaying exponentiallylikeexp(- d(q,w)z) andthe secondtermisadownward traveling wave decayinginthesame rate

(N4) As aresult of(N1)and (N2), the dampingfactor, d(q,w)becomes zeronotonly whenthe effect ofNewtoniancoolingiseliminatedbut alsowhentheNewtoniancoolingcoefficientbecomes large comparedtothe adiabaticcutofffrequencyof thewave. Also thewavenumber/3^{increasesfromthe} adiabatic value /3, to the isothermal/3,. Atthe same time the oscillatory process changes from the adiabaticformto theisothermal one.

(N5) Asaresult of the abovediscussion wehave threeranges for thefrequencyof thewave. above the adiabaticcutofffrequencyoa, below the isothermalcutofffrequencyw,andbetween

oa

^and

(N6) When thefrequencyofthe wave is greater than the adiabaticcutoff frequencythedamping factor

d(q,w)

is positive and equals zero at the extreme limits, e. when the Newtonian cooling coefficientequalszeroandwhen it islarge comparedtothe adiabatic cutoff frequency Thedamping factor increasestoits maximumvalue,d(q,w) 0.1,when(q/w)

O(1)

anddecaystozero as q 0 4. SOLUTIONOFTHE PROBLEM

Inthis section we willinvestigate the singular perturbation boundary valueproblemforthefollowing differentialequation

[(D

^D

+ TW2/4) i(/rn)eZD2(D +

^D

+ ,w2/4)

iP,.mT(/m)eD(D + 1) (P,.m)(e’/m)2D2(D + 1)(D + 2)]T(z)

^{0, (4.1)}

whereT

( +

^iq)/(7w

+

iq)

7( +

iq)/mandrn 7w

+

^{iq,subjectedtothe}boundarycondition (2.7), (2.8), and the lowerboundarycondition. Atthe outset wehavetoindicate that theparameters#

and aresufficiently small and proportionaltothe valuesatz 0 of thekinematicviscosity and thermal diffusivity Prandtl number

Pr

^{canbewritten as}

P,-

#/

(tg/Po)/(/PO) I(/mpo)/(/mpo)[. (4.2)

Itiscl[earthat Prandtl number

P

^measuresthe relativestrength oftheviscositywithrespecttothermal conduction. Asaresult, smallPrandtl numbermeans thatthermalconduction dominatestheoscillatory processand largePrandtlnumber indicates that theviscosity dominatesthemotion.ForsmallPrandtl number theatmosphere maybedividedintothreedistinctregions because thermalconductioncreatesa semitransparentreflectinglayer. Inthecaseof large Prandtl number the atmospheremaybedivided into twodifferentregionsconnectedbyanabsorbingandreflectinglayer. ForarbitraryPrandtl numberthe reflectionand dissipation process depends mainlyontheviscosity andNewtoniancooling. Toobtain the solutionofthe differential equation(4.1)it is convenienttointroduce a newindependentdimensionless variable definedby

exp(

z)/(i/m) exp[-

^z log[/m[

+

^iO,

+ 3ri/2],

(43) where0, arg(m). As^aresult,the differential equation(4.1)becomes

[2(o + o + /4) o2( e + /4)

mP(O e) + 7Pme2(o 1)(e 2)]T()

^{0. (4 4)}

where 0

dld.

^{The point 0 is a}regular singular point ofthis differentialq

(4.4). Consequently,there arefour linearly independent solutions,which inthe neighborhood of 0 canbe written in thefollowing form

TI() E ^{an (el),n+.,, T2 ()} E ^aS (e2)"+e’ + T1

^()log(),

T3() E an(e3)"+e" T4() E

^a,

^(e4)

^’+e

⁺

^{T3()log(), (4 5)}

whereel 2,_e2 1, _{e3 e4} 0 The primedenotes differentiationofa, and the sums are takenfrom n 0 to n oo. The coefficients

o(e,)

are determined from the following three termrecursion formula

po(n

+

²

+ e)an+2 +

Pl

(n +

¹

+ e)an+l +

^p2(n

+ e)an

0, (4.6) where

Po(Y) 7PrmY2(Y-

1)(y-

2),

Pl(Y)

mTPry(y

1) y2(y2

y

+ 7w2/4),

(v) (v + v + .2/).

( 7)

Following the sameprocedureasinPartII(Alkahby[9]),the solution of the differential equation,which satisfiesthe prescribed boundary conditionscanbewritten inthe following form

T(z) clTI(z) + c2T3(z).

(4$)

Todeterminethe linear combinationof

T(z)

inequation(4.8),thebehavior

ofT1 ^(z)

^and

^T3(z)

^forsmall

z must be found Since small z corresponds to large

](]

with arg()=

3r/2 +

^{O,, the asymptotic}

expansions

ofT1 ()

^and

T3()

about infinity should be found. Thedifferentialequation(4$)is similar to equation(26)inAlkahby [10]. Sincethecalculations are similartothoseinAlkahby [10],^wewill omit themandmerelyindicatethe results Althoughtheseproblemsaremathematically similar, the physical conclusionsarecompletely different. In Alkahby[$,9], thedifferentialequation(4 4)issolved, only for small and for large Prandtl number, by matching inner and outer approximations in an overlapping domain. Thematchingprocedure reduces the three termsrcursionformula, equation(4.6),to atwo term one. This will simplify the computations for obtaining the asymptotic behavior ofthe solution defined in equation

(4.8).

^For arbitrary Prandtl number the solution of the problem by Laplace integration (Alkahby

[10]).

^It follows thatthe solutionof thedifferential equation

(4.8)

for arbitrary Prandtl numbercanbewritten inthe following form

T(z)

Const.

[exp[(1/2

d(q,w)

+ i5)z] + ^{K exp[(1/2} +

^d(q,w)

i5)z]],

^{(4 9)}

where

K

denotes thereflectioncoefficiemand definedby

g U(d(q,w, iS,5,)exp[L

iL2],

(4 10)

[r(a

^a2

^{)r (2 + )r(2}

^;h

^{)r(,2 &}

u Lr--= ^{)r-( + 1)F(al )F(a2 )}

L + 2d(q,w(ln(/m})

2BSm,

L2 ^2n(]/m[) +

rd(q,w)

+

^2d(q,w)8,

a l/2

d(q,w)

+

^i, a2

-1/2 +

^{d(q,w) i,}

-1/2 +

i,,

-1/2

en Neo

coolingiselnatedwe

r

the resultsof kby 10].

Fromthe

ave

results d

disssio

^{wehavethefollong}conclusions

[I] lthe conclusionsofseion(3)cbe restatedin

tMs

sion. Inaddition,we indicate the follongobseations.

[II] Equation(4.9)representsthebehaor of thesolutionbelowthe refling layer, hindicates ttthe

oillato

^{process,below}

e

reflexinglayer,dependsonthe eff ofNeocooling dit ll be chged^to isoth if the

Neo

cooling cfficientislge comped^to

e

adiabatic cutofffrequencyofthe wave.

[III] If

e

Neocling coefficientislge comped^totheadiabaticcutoffffuencyof

e

wave, it ll act directlyto elinate the temperature prbation qutiWin a time wMchis sml compedtoapmod of oiRations. TMs ilybe sfrom equation

(2.3)

^where

T/w

O(1/q).

us,

q mthetemperature peculation vmshes,^r _{7 d}the equation forW

(z)

reducedto

(1 i6exp(z))D2W(z) DW(z) + (Tw2/4)W(z)

^O, (4.11) where 6

27w>/3cHpo(O).

the equation for

T

issupeffiuoussince 0. Moreover,the equation for

W(z)

^cbe trsfoedtothe hypergeometficdifferentiMequation wMch h

o

linly independent

solutions. Onesolution willbeeliminatedby the dissipationconditionbecauseit increaseslinearlywithz Thesecondsolution has thefollowing asymptotic form

W(z) [exp[(1/2 +K

1

+

^i,)z

+ K ^{exp[(1/2 ift,)z]],}

^{(4 12)}

where

Ku

^denotesthe reflection coefficient and definedby

K

^exp(

^i,)[cos

^O

^{+ isinO],}

^{(4 12)}

0 2argr(2iD,) 4argl"(1/2

+ iO.

It isclear that the magnitude of thereflection coefficient

]K,]

^{exp( i3)}

^Moreover,

^{if q 0the}

magnitude ofthereflectioncoefficient willbeexp(

i3).

[IV] Itfollows from[III]and theresults,whichwere obtained inAlkahby[8,9], that the effectof thermalconduction onthereflectionanddissipation of thewave willbeeliminated ifthe heatexchange between the hotter and cooler region in the atmosphere is intense and the oscillatory process ^is transformedfrom theadiabaticformtotheisothermalone.

[V]

^Thereflected wave, fromthereflectinglayer,willbereflected upwardat z 0. the reflection ofthe waves and the dissipation of the energywillcontinue until the energyof the wavesdissipates completely. Thedissipatedenergyof theacousticwavesmaycontribute to theprocess oftheheating of the solaratmosphere.

ACKNOWLEDGMENT. would liketoexpress mysincerethankstoProfessorMichael Yanowitch forhissupport andinvaluable criticismduringthepreparationof thiswork. amalsogratefultoAdelphi University and in particularits Department ofMathematics and Computer Sciences for its generous supportandencouragement.

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