RADIATIVE DAMPING OF GRAVITY WAVFS IH THE SOLAR ATMOSPHERE

Jerry D. Logan and Henry A. L1ill Department of Physics University of Arizona Tucson, Arizona 85721

ABSTRACT

The nonlocal character of the radiation f ie ld s iani f icant ly modi- fies the radiative damping of perturbations in the solar photosphere. Gravity waves are not usually considered to exist in the solar photo- sphere because the radiative damping time, when based on the Newtonian approximation, is too short. However, this restr ict ion does not abDlY to low order gravity waves. In fact, with the inclusion of nonlocal effects, the radiative damping for low order gravity waves becomes negative for some region in the photosDhere and thus acts as a drivina mechanism for gravity waves there.

I. INTRODUCTION

Several of the features of the observational results reported by Stebbins et al. (1980) have been interpreted by Hil l (1980) as evidence for the existence of gravity waves in the solar photosphere. Penetra- tive convection is an effective driving mechanism for gravity waves. This process should copiously produce gravity waves just above the con- vection zone in the solar atmosphere (Stein, 1967). However, the analyses of Souffrin (1966) and Stix (1970) using the Hewtonian approxi- mation for the radiative damping show that gravity waves cease to exist i f the radiative damping time, ~R, is less than the inverse of the Brunt-V~is~l~ frequency. The radiative damping time is quite small and less than the above cr i t ica l inverse frequency when computed for the lower photosphere usina the Newtonian approximation. As a result, i t has been argued on theoretical grounds that Fravity waves cannot exist in the lower photosphere, although an effective mechanism for their generation occurs there.

In examining the nonadiabatic term in the eneray equation describing ste l lar osci l lat ions, Hill and Logan (1980) discovered that the nonlocal character of the radiation f ie ld contributes s iani f icant ly to radiative damping of wave phenomena in ste l lar atmospheres. For gravity waves

Space Science Reviews 27 (1980) 301-306. 0038-6308/80/0273-301 $0.90. Copyright �9 1980 by D. ReideI Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

302 JERRY D. LOGAN AND HENRY A. HILL

th is might resolve the discrepancy between H i l l ' s (1980) in terpre ta t ion of the observational work of Stebbins et al . (198%)) and the previous theoret ical resu l t based on a local analysis.

2. THE BEHAVIOR OF RADIATIVE DAMPING FOR LOt,! ORDER GRAVITY HAVES

The nonadiabatic term in the energy equation contains a factor proport ional to the mean in tens i ty through which nonlocal ef fects are manifested. Let us consider the perturbat ion in the mean in tens i t y at some point in the op t i ca l l y th in atmosphere of the sun. The per- turbat ion in the mean in tens i ty w i l l be an integral of the Derturbation in the source funct ion over the ent i re atmosphere weighted by a trans- mission factor. I f the spat ial scale of the horizontal and ver t ica l var ia t ion of the disturbance is larger than the local opacity scale height, d is tant regions of the atmosphere can influence th is in tegra l . The value of the perturbat ion in the mean in tens i ty w i l l then deviate s i gn i f i can t l y from that represented only by a local value of the per- turbat ion in the source term.

The perturbat ion in the mean in tens i ty , J, for an atmosphere s t r a t i f i e d in the z d i rec t ion is aiven by

j , 1 dq~ do + S'e -#sec~ p~-tano dz = 4-~-~ o

(1)

where K is the mean gray absorption opacity, t is the opt ical depth de- fined with the mean gray opacity, to is the optical depth of the at- mosphere, S is the integrated (over frequency) source funct ion, and thepr ime ( ' ) indicates the Eulerian perturbat ion. I f we assume the horizontal var ia t ion in the source funct ion is of the form exD(ikhx), we can then express the var ia t ion in the source funct ion as

S' = q '~ khxe mz (2)

where m is a complex number that can be used to model the z dependence of S'. The horizontal var ia t ion can then be immediately intearated and we can wr i te

oo

J' = I q ' f ( s )ds - co

(3)

where

~/2 asH f (s ) = �89 I do(khHK s tan@)e-tSeC~ < pKH tanedo

0 K (4)

RADIATIVE DAMPING OF GRAVITY WAVES IN THE SOLAR ATMOSPHERE 303

and Jo is a zero order Bessel func t ion , H K is the local opac i ty scale height which is a func t ion of z, s is a dimensionless parameter def ined by

s = z/H K , (5)

and the po in t of evaluat ion fo r J' is at z = O. The r e l a t i o n s h i p be- tween the hor izonta l wavenumber fo r a wave in the solar atmosphere and the ~ value of the spher ical harmonic descr ib ing the o s c i l l a t i o n is :

kh = ,,/'~,(~, + 1)- (6) R

where R e is the solar radius. Fol lowing the seminal work of Spiegel (1957), we wr i t e the local

r ad ia t i ve damping time fo r a gray-absorbing atmosphere as

1 _ 4 ~ E -r c T '

R p (B' J ' ) (7)

where Cp is the spec i f i c heat at constant pressure. I f the hor izonta l and ve r t i ca l wavenumbers mu l t i p l i ed by the local opac i ty scale height are both less than one, the intearand in equation (3) is not l oca l i zed around the po in t of evaluat ion fo r J ' . The dimensionless temperature per tu rba t ion can be associated wi th a complex ve r t i ca l wavenumber B. To inves t iga te the e f f ec t of the nonlocal aspect of the rad ia t i on f i e l d , we set m equal to the real par t of B and expand J' as

el, r--0 �9 r/l l p:0

where t l is the op t ica l depth where the maximum con t r i bu t i on to the in tegra l of equation (3) occurs,

(s)

oo

= �89 f f(slWrds , (91 - c o (r)

and p is a binomial c o e f f i c i e n t . In order to i n t e g r a t e equat ion (9) one must, in gene ra l , have a r e l a t i o n s h i p between t and s. However, when z and c~ both equal zero, equat ion (9) can be i n t e g r a t e d d i r e c t l y . I-n p a r t i c u l a r , the i n t eg ra l for r equal to zero is

Ez(Y o) I~ = 1 (lO) 2 U

304 JERRY D. LOGAN AND HENRY A. HILL

where E 2 is the exponential integral given by

E2(t) = f e-Wtdw2 l w

( l l )

A f i n i t e optical depth for the atmosphere is easily seen from equation (lO) to decrease the magnitude of J'

Including only the f i r s t term in the series of equation (8), the damping time for a gray-absorbing atmosphere in L.T.E. in the low wave- number regime can be writ ten as

i = 16~B~- [ T-4 T-4 (B-m)zl ] TR CpTl3o t o - t l e I~163 (12)

where ~B is the Boltzmann constant and z I is the difference in heights between El and to. The typical value of the difference between E l and #o is 0.5 with E l being the greater of the two (cf. Hi l l and Logan, 1980). Using equation (12) and assuming a constant opacity scale height of 90 km with 5 - m eoual to zero, the damping times for the solar model of Bahcall et al. (1973) can be computed for d i f ferent s and m's. Figures la through I f show the computed damping times for mHK equal to O, 0.18, and 0.45 respectively, with s equal to l and lO00. The damping times computed in the ~lewtonian approximation are also shown for com- parison (represented by the dotted curves). In al l of the solid curves a prominent region of negative damping times appears. This corresponds to amplif ication of the disturbance.

3. IMPLICATIONS FOR GRAVITY WAVES IH THE SOLAR ATMOSPHERE

A gravity wave with a s value less than or of the order of several thousand and with IBHKI less than one is far from beinq overdamDed in the photosphere and in fact should be driven as shown above. The growth rate of an osc i l la t ion wi l l of course depend on i ts overall damping, but the wave wi l l not be damped out of existence a pr io r i . The change in the magnitude of the damping due to the inclusion of nonlocal effects not only allows gravity waves to overlan the region of penetra- t ive convection, which can mechanically generate the waves, but also permits a coupling of the waves through radiat ive transfer to the penetrative convection and to turbulent motion in the convection zone.

The Newtonian damping time is independent of scale when the non- local aspect of the radiat ion f i e ld is included. However, the radiat ive damping time for small wavenumbers d i f fers qua l i ta t ive ly from that for large wavenumbers. Letting the horizontal wavenumber go to i n f i n i t y to derive an effect ive Brunt-V~is~l~ frequency from the dispersion rela- t ion therefore automatically excludes the proper consideration of the low wavenumber behavior.

R A D I A T I V E DAMPING OF G R A V I T Y WAVES IN THE S O L A R ATMOSPHERE 305

4 x l O 2

2

E 0

E - 2 o 13

- 4

0.8

6 x [02

4

2

g E -~i 0

E

- 4

-6

, , \

= I 000

/~H K = 0.O

I I I I I ~ : 1 l x 102

,8Hx= O.O

2

= 0

E

~ - 2

- 4 f I I I I

0.0 0.8 -I,6 -2.4 -5,2 0.8 0.0

log T o

,SHK= 0 ] 8

2

- - I 0.0 -0.8 -I.6 -2 .4 -3.2 0.8 0.0

log To

0=1 BH x = 0.45

i / / /

-0,8 -I .6 -2,4

_to~ To _ I I Q=IIo00

~HK= 0.18

-5.2

I I I

0.8 -O.8 -I.6 -2.4 -5.2

log To

4xlO 2

E 0

E

- 4

I

0.8 0,0

i

Q = iooo

~HK= 0.45

I I I I I I I

-0.8 -I,6 -2.4 -5.2 0.8 OD -0 .8 - I .6 -2 .4 -5 .2

log ~ log To

Figure 1. Rad ia t i ve damping t imes computed from equat ion (12) as a func- t i on o f o p t i c a l depth f o r ~ = I and 1000 and f o r ~H~ = O, 0.18 and 0.45. The do t ted l i n e rep resen ts the Newtonian r a d i a t i v e damping t ime.

306 JERRY D. LOGAN AND HENRY A. HILL

An examination of Figures la through I f shows that the dr iv ing and therefore possibly the amplitude of an osc i l l a t i on both depend on i t s ver t ica l growth rate. This might explain the necessary "anomalous boundary condit ions" discussed by Hi l l (1978) in the comparison of the long period solar osc i l l a t ion observations and those of the 5 min modes.

Another consequence of dr iv ing is that the momentum and energy f lux of an amplif ied wave must increase. This must show uo in the momentum and energy budgets of the mean flow. Therefore, i f waves which have been amplif ied through the i r in teract ion with the radiat ion f i e ld are present, we would expect rad ia t ive equi l ibr ium to be established at a lower mean-field temperature than the local temperature for rad iat ive equi l ibr ium without the disturbances.

The above analysis shows that the ~!ewtonian approximation f a i l s to reproduce the qua l i ta t i ve behavior of rad ia t ive damping for low order grav i ty waves. One is led to conclude that a local analysis based on the ~Jewtonian approximation cannot be used to dismiss the existence of grav i ty waves in the solar photosphere nor used against the interpreta- t ion of observations in terms of grav i ty waves.

In order to understand the observational properties of pulsations in stars other than the sun where rad ia t ive damping in the envelope is

a l so important, i t may be necessary to consider the nonlocal aspect of the radiat ion f i e ld .

This work was supported in Dart by the National Science Foundation and the Air Force Off ice of Sc ien t i f i c Research.

REFERENCES

Bahcall, J. N., Huebner, W. F. Magee, N. H., Jr., Merts, A. L., and Ulrich, R. K.: 1973, Ap. J. 184, p. ].

Hill, H. A.: 1978, in The New Solar Physics, ed. J. Eddy (Boulder, Westview), p. 135.

Hill, H. A.: ]980, in these proceedings. Hill, H. A. and Logan, J. D.: 1980, submitted to Ap. J. Souffrin, P.: 1966, Ann. Astrophys. 29, p. 55. Spiegel, E. A.: 1957, Ap. J. 126, p. 202. Stebbins, R. T., Hill, H. A., Zanoni, R., and Davis, R. E..: 1980, in

"Nonradial and Nonlinear Stellar Pulsation," Lecture Notes in Physics, No. 125, ed. H. A. Hill and W. A. Dziembowski (Berlin. Springer-Verlag), p. 381.

Stein, R. F.: 1967, Solar Phys. 2, p. 385. Stix, M.: 1970, Astr. Ap. 4, p. ]89.

DISCUSSION

J. COX: How thick in kilometers is the driving region? LOGAN: I don't know how T and the depth in kilometers are related.

We go to log "c = - 0.8.