Propagation of Alfvén waves in multi-layer solar atmosphere model

CHINESE ASTRONOMY AND ASTROPHYSICS PERGAMON

Chinese

Astronomy

and Astrophysics

25 (2001)

446-455

Propagation of Alfven Waves in Multi-layer Solar Atmosphere Model+ * LI Bo Department

WANG Shui

of Earth and Space Sciences,

University

of Science

and Technology

of China, Hefei 230026

Abstract In this paper, we idealize the actual solar atmosphere as a multiisothermal-layer system so as to obtain the energy transmittance of the linear Alfven wave that propagates through such a system in presence of a uniform oblique magnetic filed. The results indicate that the two-layer model is essentially different to the three-layer one. In the two-layer model, the temperature jump acts as a high pass filter. In the three-layer model, resonant transfer will take place and the transmittance undergoes oscillation as the trigonometric function terms dominate its behavior. For actual solar atmosphere,the result reveals that the lower parts of solar atmosphere are more suitable for those Alfven waves

with period Key words:

of seconds

to transfer

solar atmosphere

their energy. -

Alfven

wave ~ coronal

heating

1. INTRODUCTION

The effects of Alfven waves on the solar atmospheric heating and solar wind acceleraBecause Alfven waves are generally considered to origition are widely recognizedl’-31 . nate from the photospheric granular structures, many articles are focused on the properties of the low-frequency Alfven waves with periods near the duration of the photospheric granular structures14,51. While it is hardly possible for the low-frequency(period2 10min) Alfven waves in an open magnetic field to dissipate within the short distance from the sun, in fact they cannot serve as a reasonable mechanism for coronal heating and solar wind accelerationl’l. Axford et a1.161proposed that the small-scale magnetic activity in the chromospheric network may produce high-frequency Alfven waves which will dissipate higher t Supported l

by National Natural Science Foundation and Ministry of Science Received 2000-06-21; revised version 2001-02-25 Vol. 21, No. 2, pp. 97-104, 2001 A translation of Chin. J. Space Science

0275-1062/01/$-see

front

matter

PII: SO2751062(01)00098-4

@ 2001 Elsevier

Science

B. V. All rights

and Technology

reserved.

LI Bo, WANG

up in the atmosphere damping.

Shui / Chinese

Astronomy

near the local proton

This process

and Astrophysics

gyro-frequency

may heat the corona.

25 (2001)

through

The importance

447

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ion-cyclotron-resonance

of these small-scale

magnetic

activity has been supported by observations. Many authors have discussed the propagations of Alfven waves in the solar atmosphere. They adopted different simplifying methods to idealize the actual solar atmosphere and hence derived the profiles of the transmittance and interference of Alfven waves. Usually the solar atmosphere is treated as a multi-isothermal-layer system171, a system of several layers in which the temperature varies linearlylsl, or it is considered a system of several uniform layers denoted by various Alfven velocities lg]. For the lower parts of solar atmosphere, a description of uniform layers may be inappropriate, furthermore, the results can only be applied to the delineation of the wave properties in high-frequency bands. As for the description of multiisothermal-layer, the articles17sl only presented expressions of the energy transmittance in the two-layer model. It will help discuss conveniently the essential difference between the two-layer model and the three-layer model to derive analytical expressions of the energy transmittance in the three-layer model. With the aid of analytical solutions to the linear Alfven wave equation in an isothermal atmosphere subjected to a uniform magnetic field, the expression of energy transmittance for linear Alfven waves propagating through a system of arbitrary isothermal layers is derived. The application of the expression to the two-layer model and the three-layer model is emphasized so that we can discuss the difference between them conveniently. The solution to the wave equation and the expression of energy transmittance for the two-layer model are essentially consistent to the results that have been proposed, whereas the energy transmittance for the three-layer model is substantially different from the counterpart in the two-layer case. Then the expression of energy transmittance is applied to the numerical analysis for the actual solar atmospheric model.

2. MULTI-ISOTHERMAL-LAYER 2.1

Solution

of the

The coordinates

Linear

Alfv&r

are illustrated

Wave

MODELS

Equation

in Fig.1.

A uniform

magnetic

field Bo lies on the x-

z plane, and the angle between Be and z axis is denoted as 0 . Because Alfven wave is transversal, we can assume that the velocity and magnetic field disturbance are along y direction. Likewise, the first order disturbances of the plasma density and pressure are zero, we can just take account of the linearized momentum equation and magnetic induction equation, From these two equations we can obtain the wave equation governing the behavior of linear Alfven waves,

d2v

8% --

“ia

at2 while the magnetic

field disturbance

= 0,

satisfies dh

dv

(2)

dt=a where

sis

the derivative

along

the magnetic

field, i.e., 2

= cose&

+ sine&

h = &in

448

LI Bo, WANG Shui / Chinese

Astronomy

and Astrophysics

25 (2001)

446-455

Fig. 1 Coordinate selection

which B1 denotes

the equilibrium. By employing

the intensity of the disturbed the Fourier

transformation,

magnetic

field, VA is the Alfven

velocity

we have

2r(z, z, t) = J-“, dk J-“, dw{V(z, k,w) exp[i(wt - kz)]} , h(a, z, t) = J-“, dk J-“, o!w{h(z, k,w) exp[i(wt - kz)]} . The disturbance

spectra

in

V(Z, k,w) and h(z, k,w) of the velocity

(3)

and magnetic

field can be

found to satisfy dv d2ii cos’ edzZ - ik sin 20z + (w2vi2 - k2 sin2 0)~ = 0,

h(z,k,w)

= --f-[COSO~

- iksin&?].

For the atmospheric model of a constant temperature T, the density be derived from the hydrostatic equilibrium condition, i.e.,

PO = ~00

hence the AlfvCn velocity

VA

-d-z/W,

is

VA = VA0 exP(z/2H),

where the atmospheric

scale height

(4)

H = RT/g,vAo = Bo/,/=.

(5) distribution

can

LI Bo, WANG

Then functions

Shui / Chinese Astronomy

the solution

of equation

of the first and second

and Astrophysics

(4) can be expressed

25 (2001)

in terms

449

446-455

of the modified

Bessel

kind

V(z,rC,w) = exp(ikztanB)[ciHil)(<)

+ c2HA2)([)],

(6)

where 2wH set 13 exp(-z/2H).

<=

VA0

2.2

Energy

Transmittance

of AlfvCn

Waves

Through

Multi-layer

Atmosphere

We assume that there are m + 1 isothermal layers in total, and each layer is denoted by a constant temperature T,.The interface between adjacent layers is a contact discontinuity corresponding to the height of zi. The physical quantities across the discontinuity satisfy the following match conditions [po,iT,l = 0,

(8)

[v]= [h] = 0)

(9)

where [ ] indicates the jump across the discontinuity. For layer i , we can obtain

PO,~ = Po,~

From equation(8),

we may derive

Ti PO,i+l

=

PO,i

In each layer, the velocity

where [i = *exp(--&-) coefficient for the amplitude may take tl = l,~,+~ = 0. By employing equations

-Ti+l

exp[-.G($

disturbance

V(Z, Ic, W) = exp(ikz

where

exp(-z/Hi) ,

z

spectrum

tan8)[tiH,$l)(
- j&)1, Zfl

can be written + riHi’)(
, while ti, Ti correspond of the velocity disturbance (5),(7)

i = 1,2 ,..., m. as i = 1,2, . . . . m + 1.

(10)

to the transmission and reflection spectrum, respectively. Then we

and (9), it can be derived that at the i-th discontinuity,

450

LI Bo, WANG

Shui / Chinese

Astronomy

and Astrophysics

25 (2001)

446-455

while

52&_1 = &z

= Zi),,$i

Ti (Yi = (-)‘/2, Ti+l ^ii=

ti [ Ti

Then from equation(l1) the velocity disturbance,

1)

= &+I(2 = Zi),

i = 1,2, ‘.‘, m;

i = 1,2, ‘.‘) m; i=1,2

)..) m+l.

we can obtain

Tl =

the reflection

coefficient

r1 for the amplitude

of

7

[~~+l,ll2l/[~m+l,llll

where

N m+l,l

=

NlN2..Ni..Nm,

Ni = A,‘_,Azi. Hence the energy

transmittance

can be written T r,m+1

2.3

Two Kinds

of Special

1) For m = l,i.e., T,_2 =

1 -

(12)

.

model,

+ afP;Q;

where

For high frequency

InI

Cases

the two-layer

-$P:Q:

=

as

band, & < 1 , we can obtain

- 2a&:R:

+ &l-l,

(13)

LI Bo, WANG Shui / Chinese

Astronomy

and Astrophysics

Tr,2 = (1 ?;I)2 while for low frequency

T T,2

band,(i

that

51

+ -[(a1 79

-

cql)

log T + “;rlogcrl]z}-l.

1000

100 wave

Fig. 2 Energy transmittance

451

(14)

>> 1, it can be established

+ Ql-1)2

446-455

;

4 x 4{(w

25 (2001)

period

/s

versus wave period in the two-layer model

Fig.2 illustrates the profiles of energy transmittance of Alfven waves for three different values of temperature T2, while the intensity of the vertical magnetic field is 3 x 10W3 T and Tl = lo4 K. From Fig.2, it can be seen that the temperature jump serves as a high pass filter for AlfvCn waves. The energy transmittance decreases with the increase of the wave period. Furthermore, the transmittance approaches a constant Here equation [131is essentially the same to the results derived it appears in a somewhat different form. 2) For m = 2,i.e., the three-layer model,

Tr,3

=

~

128

at the high frequency in referencesf7y8],except _’ 1

[AB+CD+EF+-

,

(15)

+522<42

where

A = P;Q;

+ cxfP;tQ; - 2cqR;R,+,

B = P,‘Q3+ + ,;P,+Q,+ - 2azR;R,+, C = P,-Q;

+ a;P;Q;

D = -PzQ, E =

-Vdtz)Yo(G)Q:

F

~[JI

=

(<3)K

- 2alR;R,,

- a;P,-Q;

+ 2azR,R4+, + &dt2)K(S2)Pit

(<3)&+

+ &&3)Yo(t3)Q,+

limit. that

-

wR;t$l,

a2Rq+S3+1

.

452

Especially,

LI Bo, WANG Shui / Chinese

for high frequency

band,

Astronomy

and Astrophysics

25 (2001)

446-455

we have

Tr,s = l6cu~a~[(l+a~)2(lf~2)2+(l--~)2(l-~2)2+2(l-_(y~)(l--~)~~~2(~2-~s)]-1. Providing

the cosine terms

involved

T T It presents

the possible

and adopting

their extrema,

(16) we can find

ha2

r,3=

(1 + (YlCr2)2

4~~2 r,3 = (crl + a#

envelope

’

,cos2(<2

cos2(&

-

-

to the transmittance

E3) =

t3)

=

1,

-1.

curve at high frequency

band.

T,=lO'K T,=106K

Fig. 3

Effect of T3 on energy transmittance

in the three-layer

model

Fig.3 depicts the energy transmittance for the intensity of the vertical magnetic field equal to 3 x lo-“T, Tl = lO*K, T2 = 106K, and Ts is 106K, 1.5 x 106K, 2 x 106K, respectively. From Fig.3, it can be seen that at high frequency band, the energy transmittance curve indicates the oscillating form and its envelope can be roughly predicted by equation (16). This property shows the essential difference between the three-layer model and the two-layer one. With the parameters as set above, the higher the temperature Ts, the greater the temperature difference, and the greater the energy transmittance. Moreover, when the temperature difference increases, the first peak of the energy transmittance curve moves toward the low frequency band. It means that when the temperature difference increases, Alfven waves with lower frequencies can also transmit their energy through such a system. Fig.4 shows the energy transmittance for AlfvCn waves when the intensity of the vertical magnetic field is 3 x 10m3 T, Tl = lo* K, Ts = 2 x lo6 K, and T2 is lo5 K, 5 x lo5 K, lo6 K, respectively. From Fig.4, we can obtain that under the parameters as selected, the first peak of the energy transmittance also moves toward the low frequency band when Ts decreases, and the corresponding value of energy transmittance increases accordingly.

LI Bo, WANG Shui / Chinese

Astronomy

and Astrophysics

25 (2001)

T,=2x106K

I

10

100 wave

Fig. 4

1000

period

453

_

10000

/s

Effect of T2 on energy transmittance

3. ACTUAL

446-455

in the three-layer

ATMOSPHERIC

model

MODEL

We simplify the actual solar atmosphere as a 6-isothermal-layer system illustrated in Fig.5. By using equation (12), we calculate the energy transmittance of Alfven waves through such a system for different intensity Bs and inclination angle 8 of the magnetic field. Fig.6 illustrates the results in case of a vertical magnetic field with intensity Be being 3 x 10m3 T, 1 x 1O-2 T, 3 x 1O-2 T, while Fig.7 depicts the results in case of a magnetic field with a fixed intensity

Bs of 3 x 10e3 T, but with inclination

0

1

2 height

Fig. 5

angles 19being 0’,45’,75”,respectively.

3

4

5

/lO’km

Simplified lower parts of solar atmosphere

It can be observed from Fig.6 that as the intensity of the magnetic field increases,the energy transmittance decreases evidently ,while the oscillation behavior of the transmittance curve becomes less obvious. Fig.7 indicates that the transmittance depends to a significant

454

LI Bo, WANG

Shui / Chinese

1

Astronomy

10

100 wove

Fig. 6

Effect of magnetic

field strength

and Astrophysics

1000

25 (2001)

446-455

10000

period ,'s

Bo on energy transmittance

in multi-layer

models

extent on the inclination angle 8 of the magnetic field. When 0 is set to be 45” or 75”, energy transmittance exhibits an evident peak at frequency bands corresponding to the wave periods of several seconds, the position related to this peak moves toward the low frequency band as the slope of magnetic field increases. That is, when deviating from the vertical direction, the magnetic field leads to the increase of energy transmittance. This [lo]. The above results reveal that observation is consistent with the prediction by Hollweg such an idealized system is more suitable for the energy transfer of those Alfven waves with period of several seconds.

1

10

Fig. 7

1000

100 wave

period

10000

/s

Effect of inclination angle 6’ of magnetic field on energy transmittance in multi-layer models

LI Bo, WANG Shui / Chinese

Astronomy

and Astrophysics

25 (2001)

446-455

455

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