INTRODUCTION AMS

VOL. 19 NO. 3

(1996)

587-594 587

ON RESISTIVE DISSIPATION OF ALFVIN WAVES IN AN ISOTHERMAL ATMOSPHERE

H. Y.

ALKAHBY

Department

ofMathematics and

Computer

Science Dillard University

New Orleans, LA 70122, U S.A.

Received

August 6, 1994)

Abstract:-

In

this paperwewillexaminethe reflection and dissipation ofAlfv6nwaves,resulting froma uniform vertical magneticfield, in aninviscid, resistiveand isothermalatmosphere.

An

equation forthe damping length distance that wave cantravelat Alfvnspeed ^isderived. This equation shows that the dampinglengthisproportionalto thewave numberand the density scale height andit isvalid not onlyfor Alfvn wavesbut also for anywavethat travelsat Alfvnspeed.

Moreover,

^{it is}shown that the atmospheremay be dividedinto twodistinct regions connectedbyanabsorbingand reflecting transition region.

In

thelower region the solution canberepresented as alinear combinationof two,incident and reflected, propagatingwaveswiththe same wavelengths and thesamedissipativefactors.

In

the upper region the effect of the ^resistivediffusivityand Alfvnspeed islargeand thesolution,which satisfies the prescribed boundary conditions, eitherdecays with altitudeor behaves as aconstant.

In

the transition region the reflection, dissipation and absorption of the magnetic energy ofthe waves take place. The reflectioncoefficient,the dissipativefactors,whichareproportionaltothedampinglength,^aredetermined andthe conclusionsarediscussedinconnection withheating ofthesolar atmosphere.

KEY WORDS:

Alfvn

Waves,

Atmospheric

Waves, Wave

Propagation AMS

SUBJECT CLASSIFICATION CODES,

76N, 76Q

1 INTRODUCTION

The mainchallenging goal ofthe theory of the formation of the solarchromosphere and corona isthe specification ofthe solarheatingmechanism.

Many

modelshavebeen suggested and investigated forthe specification ofthe heating process ofthe solar atmosphere Alkahby 1993a,

1993b,

1993c, 1994a

],

Alkahbyand Yanowitch 1989,

1991],

Susse

[1970,

¹⁹⁷⁵

], Campos

1983a,

19835]

Parker

[1979],

^Priest

[1984],

^Soward

[1986],

^Webb

1980],

^Yanowitch

[1967, 1979],

^Zhuzghdaand Dzhlalilov

[1986]).

^{The old}

ideafor coronalheatingwasthat soundwavesgeneratedin theconvection zonecouldpropagatethrough thesolarchromospheresteeping^intoshocks togive heating. Theheating^ofthe solaratmospherebysound waveshas been ruled out because of their lowgroupvelocity, whichmeansthat they cannotsupply the necessaryenergy.

However,

thisideaisstillunder investigation^{because of}the coupling ofthe soundwaves and magnetohydrodynamic^{waves into}slow andfastwaves

(see

^Priest

[1984],

^Parker

[1979]

^for

references).

On

the otherhand theimportance of the magnetic fieldand the dissipationofAlfvnwavesinthe heating

process of the solaratmosphere ^is beingincreasingly recognized see Alkahby 1993a, 1993b, 1993c

],

Alkahby and Yanowitch 1989,

1991],

^Eltayeb

1970],

^Moffatt

[197} ],

^Priest 1984

],

Robert

[1968]

Webb

[1980 ],

^Zhughdaand Dzhalilov

[1986

for

references).

Infact recent investigation emphasizes the influence of the vertical magnetic field and its role in the heating process of the solar atmosphere.

One

of the dissipative mechanisms ofAlfvnwaves, inanisothermalatmosphere,is Ohmic dissipation, whichisthe subject of this paper.

In

this articlewewill investigateupwardpropagating Alfvnwaves, resultingfromauniform magnetic field, inaresistive and isothermal atmosphere. It is shown that if the effect of the resistive diffusivity dominates the oscillatory process, theatmospheremay be divided intotwodistinct regions.

In

the lower regionthe effect of the resistive diffusivity and Alfvn speed isnegligibleand in it the solutioncan be writtenas alinear combinationofan upward andadownwardpropagatingwave. Thewavelengths and the dissipativefactors of the incident and reflected wavesare equal.

In

the upper region theeffect of Alfvnwavesandthe resistive diffusivityislargeand thesolution,which satisfies theprescribed boundary condition, either increases exponentially with altitude or behaves as a constant. The lower and upper regionsareconnected byatransition region, whichacts like areflectingand absorbing barrier.

In

the transitionregion reflection of Alfvnwaves, dissipation of the magnetic energy and modification of the waves, frompropagatingtostanding,takeplace. Thereflected wave from the transition region, will be reflected upward again. The process of reflection and dissipation will continue until the energy of the vaves dissipatecompletely. The dissipation of theenergytakesplaceastheAlfvnwavespropagate upward and downwardbecause the dissipative factorsarefunctions of Ohmic electricalconductivity.

An

equation for the dampinglength the distance thatwavestravel at Alfvnspeed isderived. This equation indicates that thedamping lengthisproportionaltothewavenumberandit isvalid notonly for Alfvnwavesbut also for anywave that travels at Alfvnspeed.

As

a

result,

the dampinglength isproportional tothe dissipativefactorbecause thedissipative factorand thewave areequal.

It

follows that alarger damping lengthmeans moremagneticenergywillbereleasedasthewavepropagatesinthe solaratmosphere. The reflection coefficient and the dissipative factorsaredetermined. This problemisanalysed inconnection with theheatingofthesolaratmosphere.

Thisproblemleads toasingularperturbationproblemand itisinterestingmathematically because it canbetransformedtothe hypergeometricequation.

2 MATHEMATICAL FORMULATION OF TIlE PROBLEM

Suppose

anisothermal atmosphere, which is resistive and thermally non-conducting, and occupies the upper half-space ^z

>

^0.

^It

^{willbe assumed that the gas is under the influence ofa}uniform vertical magnetic field.

We

willinvestigate the problem of smalloscillations about equilibrium, i.eoscillations whichdepend only^ontimet,^onthe verticalcoordinatez.

Let

the equilibriumpressure, density,temperatureand magneticfieldstrengthbe denoted by

Po(z),

po(z), To,

^and

B0= (0, 0,B0),

^where

Po(z), po(z)

and

To

satisfy thegaslaw

Po(z) RTopo(z)

and the hydrostatic equation

P(z) + ^gpo(z)

^0.

^Here

^1isthegasconstant, g 0,

0,-g)

is

the.

gravitational acceleration and the prime denotes differentiation of the pressure with respect to z. The equilibrium

pressure and density,

Po(z) Po(O)ezp(-z/K), po(z) po(O)ezp(-z/K),

(2.1)

whereK RTo/gisthe density scale height.

Let

p(z,t), p(z,t), V(z,t),

and

h(z,t)

^be the perturbations quantities in the pressure, density, ve- locity, and the magnetic fieldstrength. The non-linearform of theequations of motion induction and conservationofmomentumequations are:

OH

c

+ x(HxV)-

7

x[(4-V)vxHl,

^(2.2)

0---

0v #

[nx(xH)]

0, ( 3)

,o[-- ^{+ (v.v)v] + v}

^g

⁺

where

H(,z,t)= B0 + h(z, t), v

^isthe differential operator

(nabla), V(z,t) (U(z,t),

0,

0), and/

^isthe permeability of the magneticfield.

Here,

c denotes thespeed of lightin a vacuum and a isthe Ohmic electrical conductivity.

Alfv6n wavesareincompressible because they havemotions transverse tothe magneticfield,i.ethey donotcouple withthe sloworfastmagnetohydrodynamicswavesin anhomogenousmedium.

As

a

result,

theycanbe describedonly bythe induction andmomentumequations and thedissipationof linearwaves isnot affected by thermalconduction orradiation. The induction equation

(2.2)

balances magnetic field oscillation,velocitytransportalong^themagneticfield lines andcoxnpressibilityagainstresistivedissipation by Ohm

effect,

the Hall effectbeingomitted. Themomentumequation

(2.3)

balances the inertia force and pressuregradient againstweight,magnetic andviscous forces.

Inthisarticlewewill considerthecasewhere theverticalmagnetic^field

B0

and the electricaldiffusivity r/= areconstants. Itfollows from equation

(2.1)

that theAlfv6n speedcanbewritteninthefollowing forln

,,(.) a(O),/,

²

.

wherea.4

v/tz/4rpo(O)Bo. Moreover,

the linear forms of equations

(2.2)and (2.3)

^are:

Dthx(z,t)- BoDU(z,t)= TD2h(z,t),

^(2.5)

DtV(z,t) (a(z)/So)D,hx(z,t),

(2.6)

where

h:(z, t)

^denotesthex-component ofthemagneticfield perturbation.

In

addition,the magnetic field perturbation

h,(z,t)

can be eliminatedtoobtainanequation for

U(z,t)

only. Thiscan beaccomplished by differentiatingequations

(2.5)

^and

(2.6)

^{withrespect to and} using equation

(2.6).

^{The resulting}

differential equationis

DttU(z,t) a(z)DU(z,t) la(z)Da-(z)D,U(z,t)

^{O. (2.7)}

We

will consider solutions ofthe following forms

uC,t) u(,-)e(-),

^(2.9)

then the differential equation

(2.7)

^canbe simplifiedtothefollowingform

[(1 ie-’/a)D + 2ie-/aD (1 + ie)ie-’/a]U(z,t)

0, ^(2.9)

where

D

d/dz, ag/w,i, wll2/rl,

^z

z/K.

BOUNDAKY CONDITIONS:

To completetheproblemformulationcertainboundaryconditions mustbe mposedtoensure auniquesolution. Sincethe gas^isresistivethe dissipation condition will be necessary and sufficient,as anupperboundarycondition,to ensureauniquesolution. Thedissipationcondition requires the finiteness of therateof the energy dissipationin aninfinite column of fluid ofaunitcross-section. This implies,

olU,.(z,t)12dz

<

c, (2.to)

I’hs boundary condition will notbeapplicableifa O,butitwill be applicable only ifa O.

Moreover,

aboundary conditionisalso,requiredat z

O,

and^weshallset

U(O)=

1, (2.11)

by suitably normalizing

U(z, t).

Itwill beseenthat theseboundaryconditionswillensure aunique solution to withinamultiplicativeconstant.

SOLUTION OF THE PROBLEM

To

solve the differential equation

(2.9)

it is convenient to introduce a new independent dimensionless variable

,

^definedby

ie-Z/,

then the differential equation

(2.9)

^{will bewritten inthefollowingform}

[/(1 ()D + (1 3/)D (1 + ze)]U(z,t) ^O,

where

D (d/d(.

It

isclear that the differential equation

(3.2)

^{is a}specialcaseofthehypergeometric equation

with

[((1 ()D + ^(c (1 +

^a

+ b)()D ab](I,([)

0,

(3.1)

(3.2)

(3.3)

c 1, a

+

^{b 2, ab}

(1 + if.).

(3.4)

Moreover,

equation

(3.2)

^{has three}regular singular points, 0, 1, and

[

^oo. Theintermediate

regular

singularpoint correpondstothereflecting layer. Solving forthe dimensionlessparametersa andbwehave

a=1-/+i/, ^b- l+/-i/, ^(3.5)

where

fl . ^{For II <I,}

^thehypergeometricequation

(3.2)

^{hastwolinearly}independentsolutions ofthefollowing^form

() F(a,b,2,),

^(3.6)

1) + (n)]

^(3.7)

I2(X)-- l(X)lw,

"JI-

(!)2

ATMOSPHERE 591 where

F

isthehypergeometricequation and definedby

FCa,

b,

2,) (a)nCb),

=o

(c).

n!

r(,:) r(,, + n)r(b+n) ("

_(3.)

r(,lr(t,) r(,: +,.,1 ^’j

For I1 >

^and

laTg(-)l < ,

the solutionof equation

(3.2)

^canbe written in the followingform

,baCOn) (- )-a F(

a, c

+

^{a, b}

+

^a,

-1),

3,⁹⁾

’bb() (-)-bF(b,

^c

+

b, ^a

+ b,-).

(3.10)

The second solution

’I’2()

^{will beeliminated bythe boundary}condition

(2.10)

^because it increases to tnfinityasz oo.

As

aresult,the solution of the differential equation

(3.2),

which satisfies the dissipation condition,is amultiple of

(bl()

^i.e

,() C’i(() CF(a,b,c,(),

^(3.!)

where

C

isaconstantwhichcanbe determinedfromtheboundarycondition

(2.11). For I1 ^> , larg(-)l <

7r,the analytic continuationofthe solutionofthe differential equation

(3.2)

^canbe written in thefollowing form

r()r(- ) ,() c[

r(t,)r(,: ) (-)-)f(, + , + , -’)

^3.)

r()r(=- )

+ _ b)(-()-bF(b,

^c

⁺

^{b, a}

^{+ b,(-)].}

For >

and

larg(-)l <

^r the asymptotic behaviour of thesolution, defined in equation

(3.12),

^as

0canbeobtainedbyretaining themostsignificanttermsin equation

(3.12),

the resulting equationis

c r()r(t ^{) r()r( )} .

(() r()r( ) (-)-" +

r(=)r( )(-)-1

4 MAGNITUDE OF THE REFLECTION COEFFICIENT

Introducingthe variablezbymeanof

(3.1)

^{andretainingthe}significantterms inequation

(3.13)

^wehave

v(.) c ^{r(’::)r(t’} a),=r,[(’,, +

r(,)r(,:: ,,) [,,:r,[C:t + ,),.l + ,=[(. + it),ll

^ca.,)

where

R.,

theratioof the amplitudesofthereflectedtothe incidentwaves,denotes the reflectioncoefficient, whichisdefinedby

r(a )rc)r(, )

=[(9.Z(to + =) + (=

r(a)r(- a)r(a )

The constant

C

canbe determinedby usingtheboundary^condition

(2.11).

^{It follows}

r()r(- )

/ + to)a]

C rc)r( f R)[-(

and thesolution canbewrittenin thefollowingform

U(z) + ^{R [ezp[(1 + i)z] + R ezp[(1} + ifl)zl].

finMlythemagnitude ofthe reflection coefficientis

r(a )r()r(- )

IRI _{r(a)r( )r(} )[(Z(to ^{+ )].}

(4.2)

(4.4)

(4.5)

5 DISCUSSION AND CONCLUSIONS

It .

well known that the solaratmosphereisextremelyhot;typical temperaturesare

10K,

comparedwith 5xl0a _{at the} photosphere. Thermal energy must be continually suppliedto maintain this temperature

against radiative cooling timescale-

1day)

In fact, recent investigation enphazises ^the influence of the vertical nagnctic field ⁱⁿ generation, propagation and dssipation of the waves that may heat the ciromosphcre and corona.

Many

models have been establishedtoanswerthefollowingtwoquestions: how is magneticenergy suppliedtothecorona,and howis it&ssipated Asafurtherdichotomy, mostheating theories can be classified as either wave theories, where Alfvn wavescarry energy into the corona; or current dssipation theories, where energy is released from thebackground magneticfield. Furthermore, the corona s a low-beta plasma. This means that magnetic forces dominateover fluid pressure. The opposite holds below the photosphere, where betais high and knetic forces dominate.

Moreover,

field linesdonotpass througheach

other,

coronalmotionspreserve thetopologyof the field.

In

this articleweareinterested in the Ohmic dissipation of Alfvnwavesandit isimportant tohave someinformatonsabout the timeand thelengthof the dissipation.

For

awavewith )wavelength the timescale forOhmic dissipationis7"

A/,

^where/isthe magneticdiffusivity. As aresult,the distance thata wavecantavelat Alfvnspeed, beforeitdssipates called dampinglength ^is

Ld aA=/7-

5.

In

termsof thefrequencyof thewavevthe dampinglengthcanbewrittenlike

Ld os/z/w :.

^(5.2)

Let vaH

then intermsof thewavenumber

f

^wehave

: ^./2 (waHS)/(2H) Ld/2U.

^(5.3)

It

follows that the dampinglength canbewritten inthefollowingform

L ^2,H.

^(5.4)

Equation

(5.4)

shows that thedampinglength^isproportionaltothewavenumber.

It

isalsoproportional tothedissipative

factor,

becausethedissipativefactor and thewavenumberareequal. Furthermore, the damping lengthisvalidnotonlyforAlfv4nwavesbutalso for anywavesthattravelwithAlfvnspeed.

Asa consequence of the above results and discussionwe havethefollowing conclusions:- A Equation 4.1 represents the behaviourof the solutionofthe boundaryvalue problem, defined bythe differential equation 2.9

),

in the lower region. The first ^{term on}theright representsanupward propagatingwavedecayinglike

exp(-z)

and thesecondterm representsadownward propagatingwave decaying^atthesamerate.

In

the upperregionthe solution that satisfies theprescribedboundaryconditions of theproblemwilldecay exponentiallywith altitude.

It

isclear that thesolution isalinear combination ofanincident andareflectedwavewith thesamewavelength andthesamedissipative factors.

ISOTHEL ATIIOSPHERE 593

B

The upper and the lower regions are connected by atransition

regi.on

^{in which the}dissipation of the magnetic energy and the reflection of Alfvdnwavestakeplace. Also the electric diffusivity and Alfvdn speed changefrom smalltolargevalues.

C

Itisclear that the dissipative factorsarefunctions of^aandthedamping length depends^onthewave number

3.

^Asaresult,Alfvdnwaves oranywavetravelsatAlfvdnspeed,travelslongdistancesforlarge/3 and dissipates part of theenergyⁱⁿthe lower region. This indicatesthatthe energyofthewavedissipates notonlyinthetransitionregion but alsointhe lower regionasthewavepropagates.

D

Thereflected wave,from thetransitionregion, will be reflectedupwardat z 0. The reflection and dissipationsof thewaveswillcontinueuntiltheenergyof thewavedissipatescompletely. Thedissipated magnetic energy may contribute to the heating of the solaratmosphere.

E

Thedependenceof thedamping lengthandthe dissipative factorsonthe electrical diffusivity indicates the importance of the resistivedissipationof Alfvdnwavesthat mayheatthechromosphere andcorona.

ACKNOWLEDGEMENT

would liketo express my thanks to Professor Michael Yanowitch ^for hissupport and invaluable criticismduringthe preparationof this work. amalso

grateful

toAdelphi University andinparticularits

Department

of Mathematics and

Computer

Science foritsgenerous support andencouragement.

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