On the reflection of Alfvén waves in an ideal magnetoatmosphere

Internat. J. Math. & Math. Sci. VOL. 21 NO. 2 (1998) 381-386

381

ON

THE

REFLECTION OF

ALFVIN

WAVES IN AN

IDEAL MAGNETOATMOSPHERE

HADI YAHYA ALKAHBY

Depanmcm

ofMathematics DillardUniversity

New Orleans, LA70122,U.S.A.

and M.A. MAHROUS

Department

of Mathematics,

Universty ofNewOrleans,

NewOrleas,LA,70148,U.S.A.

(ReceivedMarch 20, 1996)

ABSTRACT: Alineafizedtheory of magnetoatmosphericwaves isdevelopedwherethe restoring forces arethose of compressibilityandmagnetic pressure. Anequation for resonanceisderived. Reflection and tunneling ofupwardpropagating Alfvnwavesinan idealMagnetoatmosphereareconsidered. It

is shown thatthe magneticfieldproducesareflecting nonabsorbingcritical layer. Below thecritical

layer,thesolutionof theproblemcanbewritten as a linear combinationofanupwardandadownward propagatingwave andaboveitthe solutiondecays exponentiallywiththealtitude. Thelocationof the criticallayerandthe magnitudeof the reflection coefficient are determined and the conclusions are discussed in cormeetion withthe heatingmechanismof the solaratmosphere.

KEY

WORDS: Magnetoatmosphere,criticallayer,wavepropagation 1991AMSSUBJECT CLASSIFICATION CODES: 76N,76Q

1

INTRODUCTION

Thedynamics ofthesolaratmosphereiscomplicated bythefact that notonlyis itstrongly stratified, inboth gas density andtemperature, butit isalsopermeated byanon-uniform magnetic field. The solar atmosphereis anexampleof aplasmathatis both structured andstratified. Morespecifically, the sun is acompressible plasma and abletosupport sound waves. The presence ofastrong magnetic field indicatesthatthesolaratmosphereis an elastic medium. Thus,wave motionsofvarioustypeswill occurand becomeasourceof energyinthe solaratmosphere.

In thispaperalinearizedtheory of magnetoatmospheric waves, involving the combined restoring forcesdue tocompressibility and magnetic pressure,isdevelopedfor thecaseofauniform horizontal magnetic field. A general propagation equation is derived for adiabatic perturbations with arbitrary vertical distributionofAlfvtnandsoundspeeds. Anexactanalyticalsolutionofthepropagation equation isobtainedfor theeaseofan isothermalatmospherepermeatedbyauniform horizontalmagneticfield.

382 H. Y. Y AND M. A. MAHROUS

withequal wavelengths. Inthe firstregionthe motion isapproximatelyacousticbecausethemotion isdominatedby the restoring force of compressibility.

In

the second region, where the magneticfield strengthdominates themotion, the solutiondecays exponentially withaltitude. Thefirst and second regionsareconnected bya criticallayerin which thereflection takesplaceand the motionisinfluenced

greatlyby the effect of the magneticfield. Also, thebehaviorof the solution,inboth regions,indicates that the tunnelingisweakasthewaves, of all kinds, propagate between thetworegions and thereflection isvery strong. Weexpect the motion,inboth regionsto continue in itsprescribed form because there is nophysicalmechanismfor dissipation. The reflection coefficient, locationof thetransitionregion, criticallayer,and the conclusionsarepresentedin connectionwiththe heating mechanism of the solar atmosphere.

Finally, in the formulation of theproblem we will be ableto introduce and justify the so-called "magneticenergycondition"in anidealMagnetoatmosphereasanupper boundarycondition toensure aunique solution.

2

PROBLEM FORMULATION

Weconsider an idealmagnetoatmospherewhich is inviscid andthermallynon-conducting, and occu-piestheupper half-spacez

>

0. Itisassumedthat the gasisunder the influenceofauniform horizontal magnetic field and that ithas an infmite electricalconductivity. Weinvestigatetheproblemof small vertical oscillationsabout equilibrium.

Lettheequilibriumpressure, density, magneticfieldintensity and external potential be denoted by

Po(z), (z),

B(z)

ando(Z). Let

P,

p,

B,

andVbetheperturbationsinthepressure,density, magnetic fieldintensity, external potential and velocity. Theequations ofidealmagnetohydrodynamics are those of momentum, induction, isotropy, and continuity:

p

[-

+

V.X;’V -XTp-

Bx

(7x B)-

pX7/,, (2.1)

0B

--

X7x

(V

x

B),

(2.2)

+

V.V

(Pp-")

0, (2.3)

[-+

V.V]

p

-pV

V. ,2.4)

Here₇denotes theratioof the specific heats. These equationsarelinearizedaboutastaticequilibrium definedby

VPo

+ Bo

x

(V

x

Bo)

-Po’o.

(2.5)

In

thisproblem

@o

is considered torepresentauniform gravitationalfield. Thus,

7@o

-g

(0,0,)

and the last term in equations(2.1) and (2.5) is replaced by pog. In terms of the linear Lagrangiandisplacementfield(r,t),thevelocity perturbationis definedby:

V

__0

(2.6)

0t’

andthe’equation ofmotionofidealmagnetohydrodynamicis obtainedby integratingand eliminating allperturbation quantifies except

.

Asaresult,wehave

REFLECTION OF

ALFVE

WAVES 383

where theforce operator

F

()

is definedby

F

()

V

(7

PoV- +

.

VPo)

Bo

(V

R)

R

(V

Bo)

+

V.

(po)

Vo,

(2.8)

where

R--

Vx

(

x

Bo).

(2.9)

The linearLagrangian displacement field isassumedto beof the form (x,y,z,t)and as an integral superposition ofharmonicterms,with andits firstxderivativevanishingatinfinity. Asaresult,

1

e-"’t

(z, k)

e’’’dk,

(2.10)

wherethewavenumberisassumed

m

bek

(k,

0,

0).

Consequently,the horizontalmagneticfield bewrittenas

B,

(B

()

,B

() O)

(:2.11)

Using equation(2.10),equation(2.7)becomes

-pow2

F

(0, k).

(2.12)

Eliminating allvariablesexcept

,,

weobtainthefollowingdifferentialequation: d

A(z,w)

+C(z,w)=O,

dz dz

where thecoefficient

A

(z,w)

andC

(z,w)

have the following form:

(2.13)

A

(z,w)

poAllA12A13 (2.14)

A14AI5

where

C

(z,w)

po

A12

A14A15

po

zz

A14A15

+

C,

AI2

w2

An

ax +

%

A13

2_

a2zk2

A14=td2

l[k2A11

An

’

17

etem

a, d

a

e

e

x dy

comnen

of 6n

sed,

wle denotes

e

eed

of sold.

3

SIMPLIFICATION OF

THE

PROBLEM AND BOUNDARY

CONDITIONS

Forthisproblemtheatmosphereisassumedtobeisothermal(co

=c(z)

constant)andpermeatedby auniformhorizontalmagneticfield

B

(B,

0,

0).

The equilibriumpressure

Po

and constanttemperature

To

satisfy the gas law

Po

RTopo

and the hydrostatic equation

P

(z)

+

gpo 0. Here,

R

denotes thegasconstantandtheprime denotesdifferentiation withrespectwithrespectto z. Asaresult,the equilibriumpressureand densitycanbewritten as:

384 H. Y. ALKAHBYANDM. A. MAIiROUS

where

H

-P0

(z)/Po

(z)

trro isthedensity scale height. Consequently,the differentialequation

(2.13)

becomes

Z C2h;

where

n

2

o

(o)

=o

(o)’

denotes Alfvn speedatz O. Furthermore,we introducethefollowingdimensionlessquantifies and variables:

(z)

W

(x)exp-kz,

z=z0exp(-),

z

D2a

O2o

(0)

Hw

=,

k=Hk,

Zo=k2

a2,

a"

o’=--,C

where the primeonziseliminatedfor simplicity. Thus, thedifferentialequation(3.2)becomes,

(3.3)

(3.4)

(

)

w

_dx

()

+

[c

(a

+

b

+

I)

x]

dW

(x)

abW

(x)

O. (3.5) Itisclearthatthe differentialequation(3.5)is aspecialcaseof thehypergeometricdifferentialequation

a+b c 2k+l, ab

a2+k+(7-1)

k2

(3.6)

9’ 0"2

BoundaryConditions:

Tocomplete the formulation of theproblem,certainconditions mustbe imposed to ensureaunique solution. Ifthe atmosphereis viscous, anappropriate conditionwould be the dissipation condition, which requiresthe finiteness ofthe rote of the energy dissipation in an infinite column offluid of unitcross-section. Since thedissipation functiondependsonthe squares of the velocity gradients,this implies

/o

[w’12a

<o. (3.7)

Inourproblemthe atmosphereisnotviscous,butthe integralin(3.7)isprbportionaltothe magnetic energy inaninfinitecolumnof fluid. This condition is areasonableone toapplysolongasthereis no energyradiation toinfinity,which is true in ourease, andweshall callthis condition"rnagnetie energy condition."Thuswewillrequire(3.7)evenwhen theatmosphereisinviscid. Aboundarycondition is also requiredatx 0, andweshallset

W

(0)

1,

(3.8)

by suitably normalizingW(x). Itwillbeseenthat the boundaryconditions(3.7)and

(3.8)

will determine auniquesolutiontowithinamultiplicativeconstant.

4

SOLUTION OF

THE PROBLEM

Inthis section weinvestigatesolutionsof the following differential equation, x

(1

x)

dWdx

2(x)

+

[c

(a

+

b

+

1)

x]

dWd..._._(x)

abW

(x)

O,

where

a+b=c=2k+l,

ab=a2+k+(7-1)

k2

,,/ 0.2,

(4.1)

REFLECTION OF

ALFVN

WAVES 385

subjecttotheprescribed boundaryconditions. Solvingfor the parameters a andb,we obtain

1 1

a-/k-r,

b=

/k+r,

where

(4.3)

1

k2

2

7- 1k

r

V’rl

r2,

r

-

+

and _r2

+

7

Itisclear that the parameterris areal number, forrl

>

r2, r 0for

r

r2andr iv/r2 rl for

r

< r2.The

differentialequation(4.1)is aspecialcaseofthehypergeometric equationwhich has three regular singular points atx 0, x and x oo. The intermediate regular singular point, x l, correspondstotheexistenceof thecriticallayer,whichhasagreat importance for understandingthe heatingmechanismofthesolaratmosphere. Asaresult,the differential equation(4. l)hastwolinearly independentsolutions which canbewritten inthe following form for

Ix[

<

1.

Wx

(x)

F

(a,

b; c;

x),

(4.4)

W2 (x)

x-CF

(a

c

+

1,b c

+

1;2 c;

x),

(4.5)

r

()

r

(

+

)

r

(

+

)

f

(a,

b;c;

x)

F

(a)

F

(b)

,=0

F

(c

+

n)

n-"[."

(4.6)

Sincek

>

0,then 1-c -2k

<

0,and

x1-c=

[Xo

exp

(-z)]X-c

oo,asz oo. (4.7)

Thus,the solutionof thedifferentialequation(4.1) as definedby equation

(4.5)

willbe eliminated by the magneticenergycondition. Consequently,thegeneralsolutionof thedifferentialequation(4.1)is

W

(x)

AF (a,

b;c;

x)

(4.8)

whereAis anarbitrary constantand canbedetermined by the boundary condition(3.8). Using the asymptoticbehaviorofF

(a,

b; c;

x)

for

Ixl

>

andreintroducingthe variablez,the solutionof

(4.4)

canbewrittenas

W

(z)=

Cons.

[[exp

(

+k+

ir)z]

+

R

exp

(

+k-it)z],

whereRdenotes the reflection coefficient and definedby

(4.9)

n

exp

[i (o

+

og

o)],

and

0 arg

(R)

arg

r

()

r

(b

)

r

(

)

r

(

)

r

(b)

r

(

)

5

GENERAL DISCUSSION

Thestructurednatureofthesolar magneticfieldmeans thatmagnetismisof greater importancein

someregions ofthe sunthaninothers. Asimple guidetotherelativeimportance of magnetic effects isprovided by theplasmabeta,

B,

definedby

=

Po

_2

(__c)’

_(5.1)

386 H. Y. ALKAHBY AND M. A. MAHROUS

where

P

denotesthe magnetic pressure. A

low-3

plasma, such as thecorona,isthusone forwhich theAlfvtnspeed greatlyexeeds the soundspeed,a

>>

c. Wavepropagation, then,involves thetwo

speedscanda. Infact, thesoundspeedexistsand inter into adescription of propagationspeedsonly inthecombination withAlfvCn speed. Asaresult, thewavespeed

W,

issuch,

Ss

<

Ws

< Fs,

where

Fs

and

S

arethefastandslow speeds and def’med by

F

=c

+a

,

S

=c

-

+c

-.

(5.2)

Moreover,

F,

willbe refered to asmagnetoacoustic speed; it is super-sonic and super-Alfv6nic. By

contrast,

S

isbothsub-sonicand sub-Alfv6nic.

Itiseasytoseethatthe maximumof thekineticenergy

Max(K)

o

(1--)"

Asaresult, whenthe reflectioncoefficient/--,_-1,wehave

Max(K)

oo.This occureswhen

0

+

2rlogz0--, 4-

(2rt

+

1)

(5.3)

Wecall thisequation,resonenceequation.

Fromtheabovediscussion and the asymptoticbehaviorofthesolution,expressedin(4.9),wehave

e

followingobservations:

(A) Itisclear that

IRI

1. Asaresult, the magneticfieldproducesanonabsorbingcritical layer, belowitthe solution canbewrittenas a linear combinationofanupwardand adownward propagating wave withthe samewavelengths. Above thecriticallayerthe solutiondecaysexponentiallywith altitude. Thus, thecriticallayer separatestwodistinctregionswithdifferent physical properties.

(B)

In

thecritical layer the reflectionandthe wave modification takeplace. Since

IRI

1, the tunnelingisvery weakwhilethereflection isvery strong.

In

thiscasethetotal energy of thewave is dividedequallybetweenthe incidentand reflectedwaves.

(C)SincetheMagnetoatmosphereisideal,there is aphysicalmechanismfor dissipation. Thus,we expect themotiontocontinue in this form andbecomeoneofthe main sourcesof energyin thesolar

atmosphere.

(D) Asaresultof (B)and

(C)

wesee that the heatingprocessis an acoustic onebelow the critical

layerbecause the compressibility forcedominatesthe oscillatory process. Aswemove fromaregion of weak magnetic field to another one with a strong magnetic field, the

heating

process becomes magnetoacoustic.

References

[1 AL AHBY, H.,

"Reflectionand dissipation of hydromagneticwaves in aviscousandthermally con-ductingisothermalatmosphere,"Geophys.Astroph.Fluid.Dynam., 72,(1993), 197-207.

[2] AL

HBY, H.,

"The dissipation ofmagneto-acoustic

waves,"

ComputerMath.Applic., Vol. 27,(1994),

9-15.

[3]

ALKAHBY,

H.,andYANOWITCH, M.,"Reflectionof vertically propagatingwavesin athermall con-ductingisothermalatmospherewith a horizontalmagnetic field," Geophys.Astroph. Fluid. Dynam.