Smithsonian Contributions to Astrophysics VOLUME 2, NUMBER 12 GRANULATION AND OSCILLATIONS OF THE SOLAR ATMOSPHERE by CHARLES WHITNEY SMITHSONIAN INSTITUTION Washington, D. C. 1958 Publications of the Astrophysical Observatory This series, Smithsonian Contributions to Astrophysics, was inaugurated in 1956 to provide a proper communication for the results of research con- ducted at the Astrophysical Observatory of the Smithsonian Institution. Its purpose is the "increase and diffusion of knowledge" in the field of astro- physics, with particular emphasis on problems of the sun, the earth, and the solar system. Its pages are open to a limited number of papers by other investigators with whom we have common interests. Another series is Annals of the Astrophysical Observatory. It was started in 1900 by the Observatory's first director, Samuel P. Langley, and has been published about every 10 years since that date. These quarto volumes, some of which are still available, record the history of the Observatory's researches and activities. Many technical papers and volumes emanating from the Astrophysical Observatory have appeared in the Smithsonian Miscellaneous Collections. Among these are Smithsonian Physical Tables, Smithsonian Meteorological Tables, and World Weather Records. Additional information concerning these publications may be secured from the Editorial and Publications Division, Smithsonian Institution, Washington, D. C. FKED L. WHIPPLE, Director, Astrophysical Observatory, Smithsonian Institution. Cambridge, Mass. For sale by the Superintendent of Documents, U. S. Government Printing Office Washington 25, D. C. ? Price 15 cents Granulation and Oscillations of the Solar Atmosphere By Charles Whitney1 The intensity of the continuous and line spectra of the solar disk and the velocities in- ferred from Doppler displacements show point- to-point fluctuations. The optical properties of the equipment and the earth's atmosphere greatly limit quantitative observations of these fluctuations. These essentially random fluctuations may be described by three parameters (although im- proved observing techniques may show these averages to be meaningless): (1) the mean size, or characteristic length; (2) the intensity con- trast; and (3) the mean lifetime or half-life. Summary of observations One-dimensional autocorrelation analyses indi- cate that the fluctuations in brightness are es- sentially random, although visual examination of photographs usually suggests the existence of characteristic lengths. In many cases the char- acteristic length thus derived (3000 to 5000 km) is undoubtedly a measure of the resolving power of the photograph. Recent studies by Rosch (1955, 1957), how- ever, support the view that a characteristic length of 1000 to 1500 km has a real significance. Indeed, he finds foreshortening toward the limb for granules of diameter 1000 km, which strongly suggests that these are not random dumpings of smaller granules. Data concerning lifetime and contrast have been summarized by Macris (1953). Correc- tions for scattered light have led to estimates of intensity contrasts as high as 30 to 40 percent, values corresponding to fluctuations of about 1 Smithsonian Astrophysical Observatory, and Harvard College Observatory. 500? K in brightness temperature. These esti- mates are extremely uncertain. The lifetimes of granules are of the order of minutes, judging from the changing appearance of the granulation pattern. Quantitative esti- mates give 2 to 5 minutes as the mean half-life. Evidence exists for a positive correlation be- tween continuum brightness and violet shift of the Fraunhofer lines. On the basis of plots of velocity and brightness, Richardson and Schwarz- schild (1950) suggest the presence of only a weak correlation. Plaskett's statistical analysis (1954) showed a correlation on two of the three plates he measured. However, the weakness of the correlations suggested to Plaskett that the absorption lines were formed above the granula- tion. As will become evident in the final section of this paper, it is not possible to base a physical model on such correlations measured on isolated plates. The analysis of a time sequence, yield- ing phase relations, is required before meaning- ful statements can be made concerning the physical connection between fluctuations in velocity and brightness. Visual examination of prints made from spec- tra obtained with the McMath-Hulbert vacuum spectrograph has convinced the present writer of a positive correlation between localized dark" ening of the photospheric continuum and the largest redward Doppler-displacements of me- tallic lines. A fact which may place restrictions on theo- ries of granulation is the striking similarity be- tween the patterns of Doppler displacements of strong (chromospheric) Fraunhofer lines and of weaker lines, presumably formed in the photosphere. This similarity also holds be- 365 366 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS tween neutral and ionized lines. McMath, Mohler, Pierce, and Goldberg (1956) have con- cluded that this similarity "implies either a chromospheric origin for the centers of all me- dium-strong Fraunhofer lines or the extension of the photospheric granules into the low chro- mosphere." Interpretations of the data There is little doubt of a physical connection between granulation and the instability of the deep photosphere (optical depth greater than unity) against convective motions. But, is the connection direct or indirect? Early inter- pretations held that the connection was direct and that granules were the rising convective cells themselves. But this interpretation no longer seems so reasonable as it once did, and it has yet to be investigated quantitatively in terms of variations of opacity, excitation, and the relative depths of line and continuum for- mation. Since the temperature gradient of the upper photosphere is less than the adiabatic gradient, this region is stable against convec- tive motions. Thus, it becomes difficult to explain the positive correlation between bright- ness and upward velocity or the extension of the motions into the low chromosphere. Further, Plaskett's (1956) calculations imply that the persistence of granulation to within 5" to 10" of the solar limb (see, e. g., Rosch, 1955, 1957) is not consistent with the hypothe- sis that brightness fluctuations are generated as deep as the top of the convection zone (opti- cal depth approximately unity). If we admit the extension of granulation into the upper photosphere or low chromosphere, and recognize the stability of this region against convective motions, it becomes necessary to assume that granulation is an effect of wave motions. That is, we must admit that the fluctuation energy is propagated into the critical region as wave energy, rather than being carried in as thermal energy by mass motions. The suggestion that sound waves are present in the solar atmosphere is not a new one (Bier- mann, 1946; Schwarzschild, 1948; Schatzman, 1953; Thomas, 1954), but previous investi- gators have limited themselves to the one- dimensional case of plane-waves propagated vertically. However, I treat the two dimen- sional equations of motion and am led to con- sider a family of solutions which differ physi- cally from those previously discussed. These solutions, which do not appear in the one- dimensional treatment, represent gravity waves or, more generally, mixtures of gravity waves and compressional waves. I adopt an approach suggested to me by Krook in which a "top" of the convection zone is postulated and is treated as a solid surface whose Z-coordinate fluctuates with X, Y, and time. These fluctuations of Z may be consid- ered as resulting from a combination of con- vective motions and sound waves. These motions of the "top" of the convective zone impress motions on the overlying atmosphere, and waves are generated. These waves will interfere constructively or destructively in a manner that depends on the characteristic fre- quencies and horizontal lengths of the fluctua- tions. Those that interfere constructively will be amplified, and can be expected to carry energy and momentum into the upper atmosphere. The present paper analyzes this situation by (a) considering the problem in two space- dimensions, (b) assuming temperature and gravity to be independent of height in the initial, undisturbed atmosphere, and (c) by treating only the steady-state of motions set up by the Fourier components of the fluctua- tions of height at the "top" of the convection zone. This restriction to steady-state solutions is a serious one, in view of the random nature of motions in the convection zone. We hope sub- sequently to remove this restriction and to treat the situation as an initial-value problem. The equations of motion 2 We shall use the Lagrangian formulation, and consider adiabatic oscillations in two space- dimensions. Capital letters and subscripts "0" shall denote unperturbed values, and lower- case letters shall denote deviations from equi- librium. The acceleration of gravity, g, is directed toward negative Z and is independent 1 In this and the following section we use the development given by Bjerknes et al. (1934, chapters 7, 8). NO. 12 of Z. Define a, B, y by dp a2 dp SOLAR ATMOSPHERE This equation has the solution C(Z) =< where a can be shown to equal the velocity of propagation of a small disturbance in a homo- geneous medium. The equations of motion in their linearized form become: d 2. 522 d2 d p . . p bX dZ p0 We assume solutions of the form x=A{Z) cos (A:X? at), z=C(Z) sin (ArX??<), Po Po sin (ikX- (1) (2) (3) (4) (5) (6) and introduce them into equations (1) to (3), obtaining the equations, -a 2A+kgC+kD=0, -u>2C+9C'+D'+g(a-B)D=0, (7) (8) (9) The primes denote differentiation with respect toZ. Before discussing the most general solution to this set of equations, we shall investigate four special cases. Case I: -4=0, &=0.?In this case, x vanishes, and the phases of z and p are independent of X. The solution represents plane waves propagated vertically. Equation (9) is now C'=-aD or D'=?-C". a Introducing equation (10) into (8) leads to the equation, C'-8gC'+a3aC=0. (11) When ij2>0, we have z=ef*zl\Clea. (17) ' ( i - J 4a,2 For the group velocity, we find from equation (15), (18) This is the velocity with which energy is trans- mitted through the medium. For an atmosphere which is initially iso- thermal, kT a3 P=Po?=Po?> where k and n are the Boltzmann constant and the mean molecular weight of the gas. Hence, since 8 is defined as B=yla*, 368 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS and, since a=l/a2 , we have /72<7* 7>l , the critical period is be- tween 3 and 5 minutes. Case II: A=0, k?*Q.?This is the case of purely transverse waves propagated horizon- tally, and we have from equations (7) to (9) the equation D'+g(a-fS)D=0, Hence for a non-vanishing solution, C'-agC=0, (19) D=-gC. (20) The solutions of equations (19)*and (20) are We note that the frequency of these oscilla- tions is independent of k, the wave number, and that it vanishes for an atmosphere in which a=/3. Two types of atmospheres for which a=/3 are a) an atmosphere in adiabatic equilibrium, and b), an atmosphere which is initially isothermal and which undergoes isothermal changes of state (7=1). For the solar conditions adopted above, we have c*-g ( 7 - 1 ) = ^ 1.84 10-3, Hence This solution represents plane longitudinal waves running horizontally. These are pure sound waves. The amplitude of the wave shows an exponential increase with height when /3>a, i. e., when dp0^ dp dp dp The case a=/3 is distinguished by an ampli- tude independent of height. Case IV: D=0.?The pressure variations now vanish and we have pure gravity waves. That is, the potential energy of wave motion appears in the form of gravitational potential alone. This is to be contrasted with the solu- tion in Case III , in which the gravitational potential of a particle is independent of time (since z=0), and the wave potential has the form of compression energy. Equations (7) to (9) give the expressions sec. x=A0ekZ cos (kX?ut), z=At*z sin (kX-o>t). The wave amplitude increases with height, and the particle paths are circular. Since a and /3 do not appear in this solution, and since the pressure fluctuations vanish everywhere, this solution is also valid for an infinitely deep, homogeneous, incompressible fluid with a free surface. NO. 12 SOLAR ATMOSPHERE 369 Summary of 'particular solutions.?We sum- marize the results of the above discussion by assembling the dispersion relations for the solutions considered. In the parenthetical com- ments, the words "vertical" and "horizontal" refer to the direction of phase propagation, or, equivalently, the direction of motion of the planes of constant phase. The words "trans- versal" and "longitudinal" refer to waves in which the particle motions are, respectively, perpendicular to and parallel to the direction of phase propagation. The absence of vertical- transverse waves is due to the atmosphere's inability to resist shearing motion, i. e., the neglect of viscosity. k Case I: Case II: u),= (y?l)g/a Case III: oi,=ka, Case IV: z ) , ( 3 3 ) ( 3 4 ) ( 3 5 ) Solutions (33) to (35) are completely specified by the parameters k and u or, equivalently, Jc and 7j. There is a double infinity of possible solutions corresponding to the infinite range of choices for o? and k. We now introduce a further condition on the solution, that the pressure variations vanish at a prescribed height, Zt. Such a condition is difficult to justify fully, but its reasonableness is suggested by the following. First, the recent models indicate that the chromosphere-corona transition consists in an abrupt rise in tempera- ture. For waves of sufficient length, this transition will appear as a discontinuity of temperature and density. The density above the discontinuity will be significantly less than that below the discontinuity, since the ratio of densities is essentially the reciprocal of the ratio of temperatures. Thus, for a sufficiently rapid and great increase of temperature with height, the discontinuity will behave essentially as a free (constant-pressure) boundary. The second consideration is that dissipation of wave energy by radiation (excluded from the present solution, since 7=1) may damp the wave and limit its amplitude at great heights. The condition that pressure variations shall vanish at height Zt gives the relation: or Hence ^-Q[~? (36) Thus, once the physical parameters Zt, g, and H are given, equation (36) becomes a "disper- sion" relation between OJ and k. When ri20).? We shall now evaluate these dispersion relations, commencing with the case ?72=O. From the assumed physical conditions, and taking Z r = 10 H, we derive ^=1600 km, ?=189 sec. CO These results are not sensitive to Z? as long as Zt> 10 H. When i?2>0, equation (36) becomes o>24-1.84 10-VI? 0.92 1Q-3 to2?1.84 10-37;H? 0.92 10~3' For each assumed value of co, solution of equation (47) gives the value of rjH. From the definition of 17, we then derive --i. (48) It is easily shown that equation (47) has no non-zero, real solution for 17 when w*< 1.10 10~3. Also, in the limit 10~ we have and the solution approaches that found above for 77=0. Therefore, we conclude that when ri^>0 the period of oscillation and the "wave- length" (X) in the horizontal direction must satisfy, respectively, the conditions 4x* T *-i. ioio-3 S 6 c 1 ' - 0.35 That is, the period must be less than 190 sec and the wavelength less than 1600 km. When we take Zt??>, the dispersion relation is changed only slightly, and the limiting period and wavelength are 208 and 1800 km, respec- tively. Tables 1 and 2 list the solutions for"Z,= 10H, and Z , = 00. When ?<140 sec, the dispersion relation becomes which is the relation for pure gravity waves. Consider, for example, a solution corresponding to Z,=10H, OJ2=2.00 10"3, ^ , = 10-5. The ver- tical displacements at the lower boundary have an amplitude AoH=15O cm. The horizontal and vertical displacements at Z=10H have an amplitude, according to table 1 of ^4=C=4.9 104 AoH=7A km. 372 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS TABLE 1.?Solutions of dispersion relation and corresponding amplitudes 1. 10X10-* 1. 20X10-* 2 00X10-* 3. 00X10-* 3.95X10-* 3. 95X10-* Period (sec) 2x 189 181 140 115 100 32 Wave- length(km) 2* k 1600 1450 868 580 440 44 Displacement amplitudes 4(10 H) C(10 H) 8.8X10* 1.3X10* 4.9X10* 7. 3X10? 1. 3X10? 4 X10" 8. 9X102 1.3X10* 4.9X10* 7. 3X104 1. 3X10? 4 X10w Relative density variation \P0/*-!H - 5 . 9 ?6. 3 -6.6X10-1 ?4. 4X10-2 - 3 . 4X10-* ? 5. 4X10-*5 Product of wave gravity (kg) 1. 07X10-* 1. 17X10-* 1. 98X10-* 3. 00X10-* 3. 95X10-* 3. 95X10-* TABLE 2.?Solution of dispersion relation and corresponding amplitudes 0. 92X10-* 1. 10X10-* 1. 20X10-* 2. 00X10-* 3. 00X10-* 3. 95X10-* 3. 95X10-* Period (sec) 2? to 208 189 181 140 115 100 32 Wave- length(km) 2x k 1800 1510 1380 830 550 420 4 Displacement amplitudes 4(10 H) C(l0H) 7.4X10* 1.0X10* 1.4X10* 4 7X10* 8.2X10* 1. 3X10? 4 X108* 9. 0X102 1. 1X10* 1.4X10* 4 7X10* 8. 2X108 1. 3X10? 4 X10<* Relative density variation \P0/x-iH - 1 . 2X101 - 7 . 6 - 5 . 7 ? 6. 6X10-1 ? 4. 4X10-2 ?3. 4X10-* ? 5. 4X10-*5 Product of wave gravity (kg) 0. 92X10-* 1. 10X10-* 1. 20 X10-* 2. 00X10-* 3. 00X10-3 3. 95X10-* 3.95X10-2 The density and pressure variations vanish at Z=10ff, but at the intermediate height Z=5H, table 1 gives an amplitude ???0.66 A<>=? 6.610~6. The pressure and density variations are, indeed, small. The horizontal and vertical components of particle velocity are x=A(a cos kX cos Q2 " 0 4 As frequency increases above ?Oi> the solu- tion follows Branch I, until the frequency ca^ is attained. Beyond this frequency, solutions on Branch II are also allowed. For still higher frequencies, solutions on the succeeding branches become allowed. For large frequencies the number of allowed solutions becomes proportional to the frequency. Figure 2 is a dispersion curve in which the logarithms of 2ir/k and 2T/? are the coordinates. Only the first four branches are included. The remaining branches would appear at equal intervals to the left of those depicted. Note that, for a given frequency, the interval be- tween allowed wavelengths increases with wave- length. That is, the shorter wavelengths are crowded together. Also, for periods less than, say, 102 seconds, the crowding is such that most of the allowed wavelengths lie near the straight asymptote. The equation of this asymptote is or a,? . K (52) That is, the relation between the period and the horizontal wavelength, \H, is just the rela- 374 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS FIGURE 2.?Solution of dispersion relation (ij l<0). Abscissa: wavelength of driving corrugation, 2w/k. Ordinate: period of oscillation, 2x/w. Dotted line shows the upper limit to the period. tion between period and wavelength of an acoustic wave. Evaluation of equations (40) to (42) shows that the amplitudes of oscillation increase exponentially upward with an e-folding distance of two scale-heights. These increases are modulated by a circular function of height, and thus show nodes. The first branch shows no nodes but each succeeding branch has one addi- tional node. Further, the horizontal and vertical amplitudes are comparable to each other, except for the cases &H??0 and kH-+ 0 as compressions! and gravitational oscil- lations, respectively. This dichotomy is used only to indicate that the potential energy associated with the oscillations appears pri- marily as compressional energy in one case, and as work done against gravity in the other. The difference in character of the compres- sional and gravitational solutions may be sum- marized as follows. The compressional solu- tions show an amplitude increasing as exp (Z/2H) and modulated by sine functions of height and horizontal distance. The density oscillations are virtually independent of per- iods. The gravitational solutions show a much more rapid increase with height, but they do not show a vertical modulation. The density oscillations decrease rapidly with period. Both types of solutions are restricted to per- iods less than about four minutes, and the gravitational solution is restricted to horizontal wavelengths less than about 2000 km. Thermal properties of the oscillations To evaluate amplitudes, we took 7=1, recog- nizing that radiative exchange of energy will tend to make the oscillations isothermal. We may obtain an estimate of the amplitude and phase of temperature changes in the following way (Whitney, 1955). Consider an optically thin element of matter whose equilibrium temperature and density are To and Po. Let the instantaneous values of these parameters be designated T=T0+8T, and use similar notation for the other variables. Then the conservation of energy requires Rpo vT 7?1 dt (53) where K, I and e are respectively the absorption coefficient, the mean intensity of incident radi- ation, and the rate of emission. If and it follows that RpnTn U 8T ST ? ?, d Sp > To=RpoTo Zt 70' ( 54 ) The assumption of sinusoidal variations of Sp and ST, i. e., Po NO. 12 SOLAR ATMOSPHERE 375 leads to the equation, E= Defining ^ 4KQ/w) (56) Hence, for vanishing opacity, # = 0, and the temperature and density variations are in phase with each other. Adopting the values of opacity and tempera- ture in the model solar atmosphere tabulated by Minnaert (1953) we derive the amplitudes and phases given in tables 3 and 4. These tables indicate that, for the optical depths contributing to continuum formation ( T > 0 . 1 ) and for periods greater than 10 seconds, the radiative exchange of energy is significant. For a given density oscillation, the temperature oscillation is reduced and advanced in phase. Thus, maximum temper- ature can occur nearly at the phase of maximum rate of compression rather than at maximum compression itself. TABLE 3.?Values of 2/?2a)~ I / 2 Optical depth (T) 0. 0001 . 001 . 129 . 294 . 672 Period of oscillation/ ? iin seconds 1 1. 00 1.00 1.00 1.00 1.00 10 1.00 1.00 0.99 . 97 .89 10? 1.00 0. 93 . 55 . 37 . 19 10? 0. 91 .24 .07 .04 .02 If we consider 100 seconds, or greater, as the order of magnitude of observed granule lifetimes, we see from table 3 that the tem- perature oscillations, for a given density oscilla- tion, increase rapidly with height in the atmos- phere. Further, since the wave solutions show a density oscillation increasing with height, the temperature oscillations associated with such a wave must increase with height. These arguments suggest that, if we asso- ciate the observed granulation with these waves, the mean height of granule formation must be greater than that of the mean continuum. Concerning the phase relations between velocity and temperature, we note that the wave solutions show density oscillation and displacement to be 180? out of phase. Hence, maximum downward velocity is in phase with maximum rate of compression. For adiabatic oscillations (#/?; 1) maximum temperature occurs at maximum downward displacement. For oscillations significantly affected by radia- tive exchange, (#/w5=l) the temperature phase- leads given in table 4 tend to shift maximum temperature toward the time of maximum downward velocity. TABLE 4.?Values of tan"1 (?/&>) in degrees. Optical depth (T) 0.0001 .011 . 129 .294 .672 Period of oscillation! ? lin seconds\? / 1 0.026 .23 .86 1. 4 2.9 10 0.26 2.3 8.6 14.3 26.9 10* 2.6 22.3 56. 1 6a 2 79. 1 10? 24 1 76.2 85.9 87. 6 88.8 In any physical situation the phase relations will be intermediate between these extremes, and the maximum temperature will occur between the times of maximum and zero velocity downward. Thus these wave solutions give a negative correlation between upward velocity and temperature perturbation. Spec- tra of the solar disk in which granulation is discernible seem to indicate a positive correla- tion (see p. 366). However, a positive correlation would arise from the present theory if the wave solution had the character of a running wave rather than the standing wave derived here. A running-wave solution would have resulted if we had not imposed the condition of vanishing pressure-oscillation at a specified height in the atmosphere. Whereas maximum density cor- responds in phase to maximum displacement for the standing wave, it corresponds to maxi- 376 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS mum upward velocity for the running wave. Thus, application of the above arguments to a running-wave solution suggests that the maxi- mum temperature should occur between zero velocity and maximum velocity upward, de- pending on the value of t?/?. Thus we may draw the following distinction: For a running wave, the maximum temperature should fall at or before maximum velocity upward; for the standing wave it should fall at or after maximum velocity downward. To use this distinction as a criterion for choosing between standing and running waves as the cause of the observed granulation re- quires the assumption that there is no change of wave phase with height in the relevant region of the solar atmosphere. This assumption is not inconsistent with results of high-resolution spectroscopy discussed on page 365. Admitting then, that the positive correlation between upward velocity and temperature rules out the standing-wave solution, we adopt the other alternative. But for the running solution, there is no unique dispersion-relation between frequency and horizontal wavelength? each value of one allows all values of the other. Thus we can say nothing about the spectrum of horizontal wavelengths. Tables 3 and 4, however, allow the comment that since optical depths less than 0.1 contribute only trivially to the continuous radiation, the lifetimes of significant temperature fluctuations must be less than 103 seconds. Also, the earlier arguments leading to a height of granule formation greater than that of the undisturbed continuum are still valid, since the running-wave solution also displays an amplification with height. It is a pleasure to acknowledge the discus- sions with Max Krook which led to this study. References BlEKMANN, L. 1946. Naturwissenschaften, vol. 33, p. 118. BJERKNES, V.; BJEKKNES, J.; SOLBEBG, H.; AND BERGERON, T. 1934. Hydrodynamique Physique. Lea Presses Universitarie de France, Paris. MACRIS, C. 1953. Ann. d'Astrophys., vol. 16, p. 19. MCMATH, R. R.; MOHLER, O. C ; PIERCE, A. K.; AND GOLDBERG, L. 1956. Astrophys. Journ., vol. 124, p. 1. MINNAERT, M. 1953. In Kuiper, ed., The sun, p. 127. University of Chicago Press. PLASKETT, H. H. 1954. Monthly Notices Roy. Astron. Soc. London, vol. 114, p. 251. 1956. In Beer, ed., Vistas in astronomy, vol. 1, p. 637. Pergamon Press, London. RICHARDSON, R. S., AND SCHWARZSCHILD, M. 1950. Astrophys. Journ., vol. I l l , p. 351. R5SCH, J. 1955. Comptes Rendus, Acad. Sci., Paris, vol. 240, p. 1630. 1957. L'ASTRONOMIE, April, p. 129. SCHATZMAN, E . 1953. Bull. Acad. Roy. Belgique (Class Sci.), vol. 39, p. 960. SCHWARZSCHILD, M. 1948. Astrophys. Journ., vol. 107, p. 1. THOMAS, R. N. 1954. Bull. Acad. Roy. Belgique (Class Sci.), vol. 40, p. 621. WHITNEY, C. 1955. Dissertation, Harvard University. Abstract The observations of solar granulation are briefly summarized and their interpretation is discussed. Steady- state solutions of the linearized equations of motion in two dimensions, subject to the boundary condition of an oscil- lating corrugation at the bottom of the solar atmosphere, are obtained and their observable properties outlined. Two points are emphasized. First, the motions of the solar atmosphere cannot be pure compression-waves, even in the region stable against convection. They should be considered as a mixture of compressional and gravita- tional waves. Second, a physical interpretation of granulation requires the determination, with time-resolved spectra, of the phase relations between brightness and velocity. U . S . G O V E R N M E N T P R I N