On the coronal heating mechanism by the resonant absorption of Alfven waves

Math. VOL. 16 NO. 4 (1993) 811-816

811

ON THE CORONAL HEATING MECHANISM

BY

THE RESONANT

ABSORPTION OF ALFVEN

WAVES

H.Y.ALKAHBY

Department

ofMathematicsand

Computer

Science

Adelphi University GardenCity,NY 11530

(Received

December 31,

1992)

ABSTRAG’Yl".

In

this paper, we willinvestigate the heating of the solar coronabythe resonant absorption of Alfven waves in a viscous and isothermal atmosphere permeated by a horizontal magneticfield. Itis shown that if the viscosity dominates themotion inahigh

(low)

plasma,

it createsan absorbingand reflectinglayerand theheatingprocess is acoustic

(magneustic).

When the magnetic field dominates the oscillatory process it creates a non-absorbing reflecting

’layer.

Consequently,the heatingprocess ismagnetohydrodynamic.

An

equation forresonanceis derived. It shows that resonances may occur for many values of the frequency and of the

magnetic field if the wavelength is matched with the strength of the magnetic field. At the

resonancefrequencies, magneticand kineticenergies willincreaseto verylargevalueswhichmay account for theheating process. When the motion is dominated bythe combined effects of the viscosity and the magnetic field, the nature of the reflecting layer and the magnitude of the reflection coefficientdependonthe relativestrengthsof the magnetic field and the viscosity.

KEY

WORDS

AND PHRASES.

Alfvenwaves,magnetohydrodynamic,resonance, acoustic.

1991AMSSUBJF,CT CLASSIFICATION CODES. 7fiN, 7fiQ.

1.

INTRODUCTION.

It is well known that the solar corona is extremely hot, typical temperatures are

106K,

compared with 5x

103K

inthephotosphere. Consequently, thermal energy must be continually suppliedtomaintainthistemperature againstradiativecooling. Earlytheories of coronalheating

were essentially based on the dissipation of acoustic waves or shock waves. Recent theories invoke magneticenergydissipationasthesourceofthermal energy. Thus two questions must be

answered: how is magnetic energy supplied to the corona, andhow is it dissipated? Toanswer

these questions, many models and dissipativemechanismsaresuggested

(see

Priest

[10],

chaps4, 5,6;Yanowitch

[12], [13];

Campos

[4], [5];

Roberts

[11];

AlkahbyandYanowitch

[2],

[3]).

Resonance

absorptionwassuggestedas amechanismfor theheatingoffusionplasmas nearly a0 yr go.

Ioo

(IS],

[9])

d

Ho,wg []

both oadd th roc absorption could explain the observed heating in the solar corona. Davila

[6]

calculated theheating rate at the

resonancelayerto determine the energy dissipationintheplasma.

The aim of this paper is to investigate the heating of the solar corona by resonance

propagating magnetoacoustic waves in a viscous and isothermal atmosphere permeated by

horizontalmagneticfield. Itsshown thatifthe viscositydominatesthe oscillatory processfor low

gas pressure

and high

B

plasma( _{magnetic pressure})’ itcreates anabsorbing and reflectingtransitionregion,

in which the reflection and thewaves modificationtakeplace. Belowit themotion is adiabatic

and the effects of the viscosity and the magneticfield arenegligible. Aboveitthemotion willbe

influenced by the combined effects of the viscosity and the magnetic field. Consequently, the heatingmechanism is acousticforlarge

When the magneticfield dominatesthe motion,itgeneratesanon-absorbingreflectinglayer.

This result is expectedbecause of thedissipationless natureofthe magneticfield. As aresult of that, theheatingmechanism ismagnetohydrodynamicand resonancewill occurfor many values of the magnetic field and of the frequency.

At

the resonancefrequency, the magnetic and the

kineticenergiesincrease to verylargevalues whichmay accountfor the heatingprocess.

Finally, if neither the viscositynor the magnetic field dominates the motion, thenature of the transition region and the magnitude of the reflection coefficient depend on the relative strengthsof the viscosity and themagneticfield.

2. FORMULATION OF

THE PROBLEM.

We will consider an isothermal atmosphere, which is viscous andthermally nonconducting, occupies the upper half-space z>0. It will be assumed that the gas is under theinfluenceofa

uniform horizontal magnetic field and that is has infinite electrical conductivity. We will investigate small oscillations about equilibrium which depend only on the time and on the

vertical coordinate z. Let p,p,w, and B be the perturbations in the pressure, density, vertical

velocity, and the magnetic field strength, and

PO,

_Po,

To,

and B₀ are theequilibrium quantities. The equilibriumpressureand density,

PO()/Po(O)

pO(z)/Po(O)

ezp( z/H),

(2.1)

are determined by the gas_{law, P0}

RToPo

and the hydrostatic equation, p’+gP0 0, where R is

the gas constant, g is the gravitational acceleration, and H

RTo/g

is the density scale height. The linearized equations ofmotionare:

POWt

+

Pz

+

gP

+

(BO[4r)Bz

4UWzz/3,

(2.2)

pt+(pOW)z-O,

(2.3)

B

+

BoW

_z 0,

(2.4)

Pt gt’Ow

+

c2/’Owz

o.

(2.5)

These are, respectively, the equation for the change in the vertical momentum, the mass

conservation equation, the equation for the rate of changeof the z-component of the magnetic field,and the pressure equation whichisobtainedfromthe adiabatic equation and the continuity equation andc

V/Tp0/P0

istheadiabatic soundspeed. Here isthe dynamic viscosity coefficient, which is assumed to be constant, and the subscripts z and denote

differentiations with respect to z and respectively. We will consider solutions which are

harmonicintime,i.e., w(z,t) W(z)ezp(- iwt).

It is more convenient to rewrite the equations in dimensionless form; z’=z/H,wa=c/2H,

variables will be written in dimensionless form from now on. Onecan eliminate p,p and B to

haveanequation forW(z)only,by applying to

(2.2)

andsubstituting

(2.3)

(2.5),

(D2 D

+

w2/4)W(z)

+

tleZD2W(z)

0,

(2.6)

whereD d/dz.

Boundary Gon&tions: To complete the formulation of the problem,certain conditions ,nust be

imposed to ensure a unique solution. Physically relevant solutions must satisfy the dissipation condition

(DC),

which requires the finitenessof the rate of the energydissipation in an infinite

column offluidofunit cross-section. Sincethe dissipation function dependsonthesquaresof the velocitygradients, thisimplies

0

IWzl2dz

<

A

boundarycondition is also,requiredat 0andweshallset

(2.7)

w(o)

,

(2.8)

by suitably normalizing w(z). It will beseen that the boundary conditions

(2.7)

and

(2.8)

will determineauniquesolution to withinamultiplicativeconstant.

,3.

KEDUGIOH TO

Tile

HYPERGEOMETRIC EQUATION AND SOLUTIONS.

The differential equation

(2.6)

canbereduced to thehypergeometricequationby introducing

anewdimensionless variable,

-erp(-z)/,#,

(3.1)

then equation

(2.6)

will betransformedinto

[(1 )D2

+

(1 2)D-

w2[4]W()

O,

(3.2)

where D=d/d and arg(-O=-arg(/). Equation

(3.2)

is a special case of the hypergeometric equation

[(1

OD

2+(c-(a+b+1)OD-ab]W(O=0,

(3.3)

with

c=l,a=1/2+s

and

b=1/2-s,

where

s=v/1-w2/2

for w<l, s=0 for w=l and

s

iWrw

2-1/2

ikfor w>1, kis the adiabatic wavenumber andwe

w,

ill be interestedinthe last

case

For

fixed value of _t/] >0, the point =0 corresponds to z=, the point

0

--1/=exp(--logltll +i(x-O)) where O=arg(tl) to z=0, and the segment connecting these pointsinthe complex

-

planetoz>0.

For [[ <1,equation

(3.2)

hastwolinearly independent solutions of the form

WI()-

F(a,b,c,),

W2()

Wl()ln +

E

(a)n(b)n,,

n[(a +

n)- (a)+(b

+

n)- (a)-2(n

+

1)

+

(n)]

n

tn!l

2

where a

1/2+

s,b

1/2-s

and F is the hypergeometric function.

choose

Wa(

-aF(a,a,2a,-

I),

Wb(

-bF(b,b,2b,- 1).

(3.4)

(3.5)

For

I1>

it is convenient to

(3.6)

The secondsolution

w2(

will beruled out by the dissipation condition. Finally, the solution of equation

(3.2)

is

W()

CaWI()

CaF(a,b,c,),

(3.8)

where C_a is a constant which can be determined by the boundary condition W(O)= at

0

=ezp(-loglol +i(-O)). For

I1

> and

we

the analyticcontinuationof w()is

It(b-

a) r(a- b)

]

W()=C

(r2(b)

(_D-aF(a,a,2a,-l)+

ri

a)

(_)-bF(b,b,2b,-1)

(3.9)

Whenw and

I1

>1, theanalyticcontinuation isgiven by

r(.+1/2)

4.

ASYMPTOTIC

ESTIMATE

FOR THE KINETIC AND MAGNETIC ENERGIES.

For a bounded z- interval

-l

=O(r), using equation

(3.1)

and retaining the most significant terms in equation

(3.9),

wehaveasymptoticallyasr-.O

sing

theboundaryconditionW(0)= equation

(4.1)

willbewrittenlike

ik)z

RCezp(-ik)z]}

W(z)-

[(

+

RC)

[eP(

+

+

where the reflectioncfficient RC is definedby

(4.1)

(4.2)

(4.3)

(4.4)

(4.)

The time average ofthe kinetic energy

(KE)

can beevaluated from equation

(4.2)

and one

obtains

KE--PIWI

2=1+

IRCI2+21RClcos(2kz-O1

211 +

RCI

2

(4.6)

It

followsfrom equation

(2.4)

thatthe magneticenergy

(ME)

ME=IWzl

X(KE).

(4.7)

Fromequation

(4.6)

and

(4.7)

wehave the followingobservations

(I)

When the viscositydominates the oscillatory motion(a

:

u), and for largeorsmall #

_..,

t--,

RCI

e:t(- *:), and themaximumand theminimumvalues of the kinetic energyare

andattainedwhen

MaZ(KE)

RCI2

21RC +

+

21RCI +

_2(eoshtkcoshrk

+

+

_eosO

(4.8)

1)

01

2n"

(4.9)

ZM

2k

815

andattainedwhen

01

+(2n

+

1)r

zm

2k

(4.11)

The magnitude of the reflection coefficient can be obtained from the maximum and the

/maz(

K_E) minimumvalues of thekineticenergy. Letd

Vmtn(KE} then

d-I

(4.12)

InCl

_d----<

If

01-.

4-₍

+

_{2n:r) themagnitude}of thereflection coefficientwillbeunchanged.

(II)

When the magnetic field dominates the oscillatory process (/<<a) and for small

Z,0--.0, RCl

,-,i-(KE)--0,

and

If

01-

4-(z-

+

2n’) then

MaZ(ME)’-*Max(KE)--*

_+cosO

(4.13)

(4.14)

MaX(M

E)--,Max(

K

E)--<x).

Consequently,the magnetic andkineticenergieswillbeincreasedtoverylargevalues when

o

_{2 4-}(

+

2)

o

₂+(

+

2.)

In t/l _{2k 4-} A,

(4.15)

where A is the wavelength. We call this equation "the resonance equation". The resonance

equation states that resonance will occur for many values of the frequency or of the magnetic

field ifthewavelengthismatchedwiththestrengthof the magneticfield.

5.

DISCUSSIONS

AND CONCLUSIONS.

It is known that an initiateddisturbance resultsin a soundwave which propagatesradially

away from the source with speed c.

In

the presence of a magnetic field, variations in the atmospheric pressure will causesdisturbancesof the magnetic field -lines. Thus any attempt to initiate a sound wave will result in the variation in the magnetic field. As a conclusion, the sound will not propagate with sound speed c, and the directionality of the magnetic field will

render wave propagation anistropic. As a result of this, the wave speed _WS will be

S

s

<_W

s

<F_{s, where}S

s

isthe slowspeed,F_Sisthe fastspeedandtheyaredefinedby

If follows from our observations in section 4, that for large

/3,S$<_Ws<F

_S, and consequently the heating process is either acoustic or magnetoacoustic. Forsmall we obtain

$s<Ws<CA

and in conclusion the heating mechanism is magnetohydrodynamic. At the

resonance frequency the magnetic and the kinetic energies will increase to very largevalue and thatmayaccount for the heatingprocess.

ACKNOWLEDGEMENT.

would like to express my sincere thanks to Professor Michael

Yanowitch for his continuing encouragement, support, and invaluable criticism during the preparation ofthiswork.

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