R A D I A T I V E D A M P I N G OF T R A P P E D G R A V I T Y W A V E S

I N THE S O L A R A T M O S P H E R E

P A T R I C I A ANDRI~ C L A R K

Univ. of Rochester, Rochester, N.Y. 14627, U.S.A.

and

A L F R E D CLARK, JR. Dept. of Mechanical and Aerospace Sciences and C.E.K. Mees Observatory,

Univ. of Rochester, Rochester, N. }7. 14627, U.S.A.

(Received 23 February, 1973)

Abstract. The radiative damping of trapped gravity waves in an optically thin atmosphere is studied for a stratified Boussinesq fluid. The character of the atmospheric eigenmodes depends on the distribution of the Brunt-V~iis~il/i frequency N and the radiative relaxation time r. The calculations for simple layer models show that if Nr is large over some finite fraction of the trapping region, then modes of long lifetime can exist. In order to suppress gravity waves entirely, it is necessary that Nr ~< 1 over the entire trapping region. Qualitative application of the results to the solar atmosphere leads to the conclusion that gravity wave eigenmodes of the solar atmosphere, although damped, are by no means eliminated by radiative effects.

1. Introduction

Uchida (1965, 1967) and Thomas et al. (1971) (called paper I hereafter) have suggested

that the five-min oscillations of the solar atmosphere are internal gravity waves t rapped in the low chromosphere. Such waves have periods comparable with 5 min

if the horizontal wavelength is of the order o f 1500 km - a circumstance consistent with the granule excitation concept developed by Meyer and Schmidt (1967) and

Stix (1970). As discussed in detail in paper I, the structure of the solar atmosphere is

such that internal gravity waves are t rapped in the first 10O0 or so km above the top

of the convection zone. These trapped waves are free eigenmodes o f the a tmosphere

and thus should be a prominent par t of the response to granule impact. The calculations o f Uchida, and of paper I, do not include the effects of radiative

transfer. As Souffrin (1966) has shown, internal gravity waves are highly susceptible

to radiative damping. The parameters which determine the damping in optically thin regions are the Brunt-V~iisglS. frequency N and the radiative relaxation time ~. Fo r

the case o f constant N and ~, there are no oscillatory solutions unless ~ > (2N) -1 (Souffrin (1966); Stix (1970)). In the solar atmosphere, ~ is a strong function of position. In the 1000 km region of gravity-wave trapping, the condit ion z < ( 2 N ) - 1 is satisfied in the 200 or 300 km just above the convection zone, but not in the remaining

part. In this situation, no clear-cut conclusion can be drawn f rom the results for a constant-property atmosphere. It is the purpose of this paper to analyze the damping of the gravity-wave eigenmodes for the case of variable z. Since the object is insight

Solar Physics 30 (1973) 319-325. All Rights Reserved Copyright �9 1973 by D. Reidel Publishing Company, Dordrecht-Holland

320 PATRICIA A. C L A R K A N D A L F R E D C L A R K , JR.

into mechanisms, and not simulation of the solar atmosphere, the simplest relevant model is used; namely, a stratified Boussinesq fluid with piecewise constant properties.

2. Basic Equations

Consider an equilibrium state in which the pressure, density and temperature are all functions of the vertical coordinate z. Since it is a Boussinesq fluid, the density varia- tions are small, and we can replace the density by its constant mean value 0o except in the buoyancy force. The linearized equations for small fluctuations about the equilib- rium state are well-known (Chandrasekhar, 1961), so we quote them without deriva- tion. The continuity and momentum equations are

divv = 0, (1) and

0o (~v/~t) = - Vp + 0g , (2)

where v is the velocity, p is the pressure fluctuation and 0 is the density fluctuation. The acceleration of gravity g is in the negative z-direction. The energy equation, for an optically thin gas, takes the form

~T/Ot + flw = - T / z , (3)

where Tis the temperature fluctuation, fl is the temperature gradient in the equilibrium state, w is the z-component o fv and z is the radiative relaxation time. Finally, we have the equation of state for a Boussinesq fluid,

0 = - Oo CzT, (4)

where c~ is the coefficient of thermal expansion. The quantities fi and z may depend on z, but not on the time t or the horizontal

position vector r. Thus, there are modes with frequency co and horizontal wave vector k proportional to e i (,ot+k.r). For these solutions, (1)-(4) are easily reduced to a single second-order equation:

d2w { N2 } dz z + k z ~o 2Ziog/z 1 w = 0 , (5)

where N = (9~zfl) 1/2 is the Brunt-V/iis[il~i frequency. The limiting cases of large and small ~ are instructive. For slow relaxation (o~r >> 1),

(5) reduces to

_ _ k 2 _ _ dz 2 + - 1 w = 0 , (6) (.O 2

which is the usual equation for adiabatic internal waves in a Boussinesq fluid. The other extreme of rapid radiative relaxation (roz~> 1) gives the limit

d2w dz~ 2 - k 2 w = O. (7)

RADIATIVE DAMPING OF TRAPPED GRAVITY WAVES IN THE SOLAR ATMOSPHERE 321

It is noteworthy that (7) can be obtained also by taking N 2 -~ 0 in (5). Thus, rapid radiative relaxation essentially unstratifies a Boussinesq fluid in the sense that the buoyancy force becomes unimportant in the dynamics of the fluctuations.

For the five-min oscillations in the solar atmosphere, the limit (7) is appropriate in the photosphere and very low chromosphere, whereas the limit (6) applies in the upper part of the region of gravity wave trapping. Thus, the behavior of the atmospheric eigenmodes cannot be determined by simple qualitative considerations, and it is necessary to adopt an explicit model and compute the results.

3. Analysis

We consider a piecewise constant model for the atmosphere. We take z positive out- ward, with z = 0 being the top of the convection zone (optical depth unity). The prin- cipal feature in the actual N2-distribution is a broad maximum in the low chromo- sphere. A crude model of this is N 2 = N 2 for 0 < z < H and N 2 =0 otherwise. Appro- priate numbers for the solar atmosphere are N 2 = 1 0 -3 s -2 and H = 1000 km (see paper I). In the region of gravity-wave trapping ,we take a layer model for the radiative relaxation time: z = z l for 0 < z < z 0 and z=z2 for Z o < z < H . The choice of numbers for z~ and ~2 will be discussed later. The general situation for the solar atmosphere corresponds to Nozl <~ 1 and Noz 2 > 1.

In the region z > H, we have taken N 2 = 0. The governing equation is (7) and z no longer appears. For z < 0 we may still use (7), but a little more explanation is required. In the solar atmosphere, the transition from optically thick to optically thin takes place in a layer much thinner than length scales associated with the five-min oscilla- tions. In the first approximation, we may take the atmosphere to be thin above z = 0 and thick below z = 0. Equation (5) does not hold in optically thick regions. By using the diffusion approximation in the optically thick part, it is not hard to show that the oscillations may be taken as adiabatic, and this leads to Equation (7) for w.

Thus, we have a four-layer model: Equation (7) holds for z < 0 and z > H , and Equation (5) obtains in the middle layers, with N 2 = N 2 throughout, and a jump in z (from -c~ to z2) at z o. The solutions in the four layers are connected by matching conditions at each interface. From the requirements of continuity for normal velocity and pressure, one can show that w and dw/dz must be continuous at each interface.

We put the equations in dimensionless form by using H as a length scale and N o ~ as a time scale. Then the solution for w has the following form:

w = Ae k~ for z < 0, (8)

w = B e x p ( i k 2 ~ z ) + C e x p ( - i k 2 ~ z ) for 0 < z < z 0, (9)

w = D e x p ( i k 2 z z ) + E e x p ( - i k 2 2 z ) for Z o < Z < l , (i0) and

w = F e -kz for z > l , (11) where

2 1 , 2 = { ( ( D 2 - - i(O/'Ci, 2 ) - 1 - - 1} 1/2 . (12)

322 PATRICIA A. CLARK AND ALFRED CLARK, JR.

In (8) and (11), the condit ions at z = _ oo already have been imposed. The matching condi t ions at the 3 interfaces lead to 6 homogeneous equat ions for the constants A - F .

The vanishing of the de terminant gives the eigenvalue equat ion

F ( k , co) = Q21 cos (k2azo) - R sin (k2,zo) = 0, where

Q = (1 + g22) 2 e ' { - (1 - i22) 2 e -/~,

R = (1 + i22) (22 - i22) e ir - (1 - i22) (22 + i22) e -i~,

with

-- k22 (1 - Zo).

The six constants are then given by

A = 1, B = �89 (1 - i / 2 , ) ,

where

and

c = �89 + g/21),

D = D (2,, 2 2 , Z 0 ) = G exp ( - i k22zo) ,

G = (42122) - ' { (2 , + 22) (21 - i) exp ( ik2,zo) +

-~ (22 - ,~1) (21 "}- i) exp ( - ik2,zo) } ,

E = E (2a, 22, Zo) = D (21, - 22, Zo),

(13)

(14)

(15)

(16)

F = e k {D exp (ik22) + E exp ( - ik22) } . (17)

Before discussing the numerical solution of (13), we consider some special cases, beginning with a one-layer model in which z, = T 2 = T. Then 21,2 = 2 and (13) reduces to

tan k2 = 22/(22 - 1). (18)

I t is easy to show that (18) has an infinity of real posi t ive roots 2j (k) . The resulting

eigenfrequencies are

i l [ 4 z z 1 '/2 a~s = - - + - - 2 1 , (19)

2 z - 2 z 1 + 2 j

which give damped oscillations for z > �89 (1 + 22) 1/2 and two aperiodic decaying modes for z < � 8 9 2 a/2 + 2 j ) .

As discussed earlier, the limit of rapid radiative relaxation (z ~ 0) is equivalent with the limit o f zero stratification (N 2 ~ 0). We may verify this for the present calculations by looking at the case % -+ 0. Then (13) becomes

tan [k22 (1 - Zo) ] = 222/(22 - 1), (20)

and this is just the eigenvalue equat ion for a one-layer model o f thickness 1 - z o. This shows tha t very rapid radiat ive relaxat ion in the lower layer shifts the eigen- functions upward.

Consider now the numerical solution of (13) for the complex frequency co. The technique used is a s t ra ightforward version of the a rgument principle. We select a

R A D I A T I V E D A M P I N G O F T R A P P E D G R A V I T Y W A V E S I N T H E S O L A R A T M O S P H E R E 323

square in the complex co plane within which a root is expected. The value o f F (k, co)

is determined at many points on the perimeter o f the square. I f the change in arg F

a round the square indicates a root, then the square is subdivided into four squares,

and the process is repeated. In this way, we nest down on the root to any desired

accuracy. Because o f the large number o f parameters (z~, z2, k, Zo, H, No) a complete charac-

terization o f the results is impractical, so we consider only selected parameter values.

We continue to work with dimensionless quantities, with occasional dimensional values being given in parentheses. We have the length scale H = 1000 km, and the

time-scale N o ~ --32 s. We take ~ = 0.5 (16 s), a value typical o f the layers immediately above the convection zone. We choose k= 4 .189 (4.189 x 10 -3 k m - a ) , corresponding

to a wavelength of 1500 km (twice a typical granule diameter). We take the thickness

o f the high-damping layer to be Zo=0.333 (333 km). For these choices, we have computed co for a number o f values of z 2, and the results for the lowest mode of

oscillation are shown in Table I. A convenient measure of the decay rate is the number

TABLE I Dimensionless complex eigenfrequencies co for trapped gravity waves. The radiative relaxation time in the lower third of the trapping region is zl = 0.5, and in the upper two-thirds is v2.The time scale is No -1 -- 32 s, where No is the Brunt-V~iis/il/~ frequency in the region of trapping. The number of oscillations in an e-folding decay time

is n = Re(og)/2~Im(co)

T$ (D H T2 (D H

0.5 1.46i, 0 . 5 3 9 i - 10.0 • 0.824 + 0.069i 1.9 1.0 • 0.672 • 0.517i 0.2 15.0 • 0.824 + 0.045i 2.9 3.0 • + 0.181i 0.7 20.0 • + 0.037i 3.5 5.0 • 0.820 + 0.113i 1.2 50.0 • 0.824 + 0.022i 6.0

o f periods in an e-folding decay time, n = Re (a~)/2rcIm(a0, and this quanti ty is shown

also in Table I. For small z2, the damping is great, as one would expect. For z2 ~> 5

(160 s), however, n > l and there are identifiable oscillations with a period T =

=2re/(0.82) =7 .7 (250 s). A choice o f '172 ) 5 is reasonable, since, according to Stix's tabulation, z exceeds 160 s for z > 400 km. Results for the second mode of oscillation are very similar in character, so we do not give them in detail.

We have carried out several tests to determine whether the general conclusions are sensitive to the details of the model, and we summarize them very briefly. First, we have computed the results for a 3-layer model in ~. The damping is still moderate.

A typical result is the following: co=0.69+0.11i , hence n = l . 0 , for H, No z and k the same as before, and for z = 0.5, 1.0 and 10.0 in successively higher thirds o f the t rapping region. For z = 1 0 0 rather than 10 in the third layer, the result is m = 0 . 6 9 + 0 . 0 6 i ,

giving n = 1.8. Second, we have computed the results for the two-layer model for other wave-lengths (in the range 500-5000 km), and again we find identifiable oscillations (n> 1) for Zz~>5.0. Finally, we have varied the thickness z o o f the high-damping

324 P A T R I C I A A. C L A R K A N D A L F R E D C L A R K , JR .

TABLE II Dimensionless complex eigenfrequencies co for trapped gravity waves. The radiative relaxation time is zl = 0.5 for 0 < z < zo and infinite for zo < z < 1. The time-scale is No -1 -- 32 s, where No is the Brunt-Vfiis/il/i

frequency in the trapping region

Zo CO ZO f.D

0.00 • 0.887 0.50 • 0.769 + 0.018i 0.10 • 0.873 + 0.005i 0.70 • 0.658 + 0.028i 0.33 i 0.824 + 0.012i 0.90 • 0.419 + 0.045i

layer. This has been done for T 1 =0.5, "c 2 = (30, and the previous values of H, k, and N o.

The results are given in Table II. The period changes appreciably with Zo, since the

effective thickness of the trapping region is essentially 1 - z o. The damping remains

small, however, even when z o is as large as 0.9.

4. Conclusions

The critical parameter in the damping of gravity waves is N% where N is the Brunt-

V/iisfilg frequency and z is the relaxation time. The damping is great for Nz ~ 1 and slight for Nz >> 1. In the case of variable N and ~, the criterion cannot be applied

locally; that is, it is not correct to say that oscillations will occur only where Nv >> 1. The existence of oscillatory modes depends on the entire distribution of N and z.

The present calculations show that there will be oscillatory modes unless Nz < 1 over the entire trapping regions. Thus, the buoyancy provided by a sublayer in which

Nv > 1 is sufficient to drive the whole layer in an oscillatory mode.

The model studied here is too idealized to allow quantitative conclusions about

waves in the solar atmosphere. Nevertheless, the mathematical results and the associated physical picture strongly support the qualitative conclusion that radiative damping will not expunge trapped gravity waves in the solar atmosphere. Schmidt and

Stix (1972) have reached the same conclusion on the basis of calculations (not presented) with a continuously varying z. (In the earlier comprehensive work by Stix (1970) for an isothermal, compressible atmosphere, detailed results were given for the radiative damping of acoustic waves, but not for internal gravity waves.)

As mentioned earlier, the eigenfunctions for damped waves accommodate to varia- tions in z by shifting somewhat to regions of larger z. In the solar atmosphere, the true eigenfunctions will be shifted to somewhat higher layers than the theoretical, adiabatic

eigenfunctions. At first, this suggests a difficulty in exciting such modes by penetrative convection. However, there is a compensating effect, also associated with small ~ in the lower levels. Penetrative convection is rendered much more effective in these layers since the small z essentially unstratifies the fluid. Thus, the small z (i) moves the eigenfunctions upward, and (ii) allows greater convective overshoot - a combination which, in effect, simply shifts the boundary of the convection zone upward.

RADIATIVE DAMPING OF TRAPPED GRAVITY WAVES IN THE SOLAR ATMOSPHERE 325

It is worth pointing out that very modest penetration is sufficient to excite waves.

A rising granule represents a coherent field of upward vertical momentum over a

horizontal scale of 750-1000 km. To excite gravity waves, it is only necessary that some of this momentum be transferred a few hundred kilometers upward. There is nothing to hinder this transfer in the solar atmosphere since the moderate stratification in the lower layers is rendered ineffective by the radiative relaxation. Observational evidence for significant penetration has been given by Frazier (1968a, b) and Sheeley and Bhatnagar (1971).

The observed properties of granules imply length and time scales which should excite internal gravity waves in the solar atmosphere. Radiative relaxation, while

causing some damping, is not sufficient to destroy the waves. It is difficult to see how such waves can be avoided in the solar atmosphere, although little can be said about

the amplitudes expected until the dynamics of wave excitation by penetrative convec- tion is better understood.

Acknowledgements

This work was supported in part by the National Aeronautics and Space Administra- tion ( N G R 33-019-126). Acknowledgement is made to the National Center for Atmospheric Research, which is sponsored by the National Science Foundation, for computer time in this research. We acknowledge helpful discussions of this work with Professor J. H. Thomas.

References

Chandrasekhar, S. : 1961, Hydrodynamic and Hydromagnetic Stability, Oxford. Frazier, E. N.: 1968a, Astrophys. J. 152, 557. Frazier, E. N. : 1968b, Z. Astrophys. 68, 345. Meyer, F. and Schmidt, H. U. : 1967, Z. Astrophys. 65, 274. Sheeley, N. R., Jr. and Bhatnagar, A. : 1971, Solar Phys. 18, 379. Souffrin, P. : 1966, Ann. Astrophys. 29, 55. Schmidt, H. U. and Stix, M. : 1972, preprint (Astron. Gesellschaft Meeting, Vienna, Sept. 1972). Stix, M. : 1970, Astron. Astrophys. 4, 189. Thomas, J. H., Clark, P. A., and Clark, A., Jr.: 1971, Solar Phys. 16, 51. Uchida, Y.: 1965, Astrophys. J. 142, 335. Uchida, Y.: 1967, Astrophys. J. 147, 181.