T E M P E R A T U R E W A V E S IN THE S O L A R A T M O S P H E R E

Y. D. Z H U G Z D H A

Department of Cosmic Electrodynamics, IZMIRAN, Troitsk, Moscow Region, 142092, U.S.S.R.

(Received 20 March, 1989; in revised form 27 June, 1989)

Abstract. The properties of five-minute temperature waves in the photosphere are investigated. The phase and amplitude relations of temperature and acoustic waves are deduced. It is expected that the five-minute oscillations represent a mixture of acoustic and temperature waves. The temperature waves are generated due to linear interaction with acoustic waves.

It is well known that concurrent with the acoustic waves, temperature or heat waves can appear in the case of nonadiabatic disturbances (Landau and Lifshitz, 1959). The temperature waves are dissipative damped waves. Propagation of nonadiabatic hydrodynamic waves in a stratified medium have been con- sidered by Zhugzdha (1983). If stratification of heat exchange exists, a linear interaction of hydrodynamic and temperature waves arises. The temperature waves must be present in the solar atmosphere.

1. Introduction

We shall follow Landau and Lifshitz (1959) in consideration of temperature waves in a uniform medium and use the same notations. The following system of linearized

equations is used:

a(ap) _ p divv,

at

au p - - = - Y a p ,

at

&3T pc v ~ + p d i v v = 6 Q = - 7 B ,

(1)

a Q = zpe~ div(V6T).

We shall also use the equation for pressure disturbances

o@

3t - yp divv + (7 - 1)aQ. (2)

Following Landau and Lifshitz, consider oscillations with amplitudes which are pro- portional to exp i ( k x - cot). In this case the dispersion equation of nonadiabatic oscilla- tions (Landau and Lifshitz, 1959, p. 303) is

i(D3 k4 - k2 (co2 + ico + - - = 0 ,

\ 4 z zc, ~ (3)

Solar Physics 124: 205-209, 1989. �9 1989 KluwerAcademic Publishers. Printed in Belgium.

206 Y . D . Z H U G Z D H A

where c s and c r are adiabatic and isothermal sound velocities. This dispersion equation has two solutions which describe acoustic and temperature waves. In the case of high frequencies (r Z >> c 2) the approximate solutions of Equation (3) are equal to

ka o) C T = - - + i - - (c~ z - cZr) (acoustic wave), cr 2ZC) (4)

k T = ( l + i ) ~ Z (temperature wave),

and the acoustic wavelength is shorter than the temperature wavelength (k r ~ ka). In the case of low frequencies (co Z ~ c 2) the approximate solutions of Equation (3)

are

ka co (D2)~(~ z ~_) = _ + i c, 2c s s

_ _ + ( 0 2 -

z \ 4

(acoustic wave),

(temperature wave),

(5)

and the temperature wavelength is shorter than the acoustic wavelength (k r ~> ka). The disturbances of density, pressure, and temperature deduced from Equations (1)

and (2) are

b p = k V bp_ k ( i7co -k2z~ bT i (7- 1)k V. P co P co \ ~-__- ~ - / V , - (6) ' T i co- k2z

The disturbances of density, pressure, and temperature for acoustic and temperature waves are different. In the case of high frequencies (co X ~> c 2) the disturbances are approximately equal to

bp_ V @ _ V

p cr p Cr

3P-_(,/ icZr~ V '

p \ 'q 7oaZ/ c T

- - = ~)

T \'qYcoZI cr

bT i ( 7 - 1)c~ V

T co Z c T for acoustic waves,

@ - ( 7 + 1)( , / i~c~) v P k~lTcoZ/cr

(7)

for temperature waves .

The relative disturbances of pressure and density generated by high-frequency acoustic waves are greater than that generated by temperature waves at the same relative velocity amplitude VIe r. But the temperature disturbances for temperature waves are greater than for acoustic waves.

TEMPERATURE WAVES IN THE SOLAR ATMOSPHERE 207

In the case of low frequencies (oJ)~ ,~ c 2) the disturbances are approximately equal to

_ V ~ T V b p = V bp 7 - ( 7 - 1 ) - - for acoustic waves p c, p c, T c s

P \coZ/ cs P \ ~ Z / cs

b T _ ( ic~13/2V for temperature waves T \~oZ/ cs

When substituting the second relations from Equation (5) in Equation (6) the next term

in the expansion of co must be taken into account to avoid a singularity. The relative disturbances of density, pressure, and temperature for acoustic waves are the order of the relative velocity amplitude V/cs. The relative disturbances for low-frequency temperature waves are greater than those for acoustic waves. The disturbances of pressure and temperature for temperature waves are greater than the disturbances of

density.

2. Temperature Waves in the Solar Atmosphere

The theory of nonadiabatic hydrodynamic waves in an atmosphere with stratified heat exchange has been considered by Zhugzhda (1983). The equation of nonadiabatic oscillations in an isothermal atmosphere has the simple form:

+ - - + -k~ ( ~ Z ~ - ) T L - ~ z ~ + dz z H dz p

+ - - - - + + k~ b = 0 , (9) H dz o)2c~

where 8Q = pT/(d/dt)bs, k• is the horizontal wave number, H is the pressure scale. The main property of wave propagation in a nonuniform atmosphere is the existence

of a linear interaction between acoustic and temperature waves. A linear interaction arises in an atmospheric layer with kH R ~ 1, where k is the wave number of temperature waves (Equation (2)) and H R is the scale of a heat exchange stratification. In the solar photosphere H R is about 10-20 kin. The generation of temperature waves by acoustic and gravity waves is possibly due to their linear interaction. It should be emphasized that the generation of acoustic and gravity waves by temperature waves is also possible.

3. Temperature Waves in the Solar Photosphere

Temperature fluctuations related to solar granulation must generate temperature waves. Equation (7) can be used only for a crude estimation in the photosphere because a

208 Y . D . Z H U G Z D H A

diffuse approximation is not correct in this layer. The temperature conductivity of the solar photosphere is about 1011 cm 2 s-1 and the wavelength of temperature waves is around

2(km) = 10 x/fi (s), (10)

where P is the period of a wave. Consequently, the temperature waves have wavelengths around hundreds of kilome-

ters for a period of about a few minutes which fall in the range of granular dimensions. There is no mean size of a granule and the number of granules increases steeply down to the limit of resolution (Muller, 1988). Consequently, an interaction of temperature waves and granulation can occur.

Five-minute oscillations in the solar photosphere must generate temperature waves due to a linear interaction. In the layer of interaction, the temperature waves are not damped over short distances because of reinforcing by acoustic waves. In the photosphere the temperature disturbances of low-frequency (Equation (8)) temperature waves are connected with the velocity amplitude by

~_~ (e(s))3/2 v (11) Cs

The temperature fluctuations of the five-minute temperature waves are greater by a factor 103 than temperature fluctuations generated by pure acousticp-modes of the same velocity amplitude.

The maximum of temperature disturbances of low-frequency temperature waves 1 precedes the maximum of upward velocity by xz: for upgoing temperature waves and

delays by U: for downgoing waves - as follows from Equation (8). The maximum temperature of evanescent five-minute oscillations in the photosphere precedes the maximum of upward velocity by ire. It should be emphasized that Equations (4) and (5) were deduced for running acoustic waves and cannot be used for evanescent waves. The observable phase lag falls between 1re and - rc (Deubner and Fleck, 1989). The theory of linear interaction of nonadiabatic oscillations predicts the existence of temperature waves in the photosphere. We suppose that oscillations in the photosphere represent a mixture of acoustic and temperature waves and they can be separated by special investigation of the amplitude and phase relation between temperature and velocity oscillations.

The temperature waves can also generate acoustic waves. Consequently, it is possible to discuss the mechanism of the generation five-minute oscillations by temperature fluctuations which in turn are excited by turbulent convection. The condition of coupling with temperature waves is fulfilled for a wide frequency spectrum due to decreasing of Z with depth in the convective zone. It is obvious that temperature fluctuations produced by turbulent convection are higher than pressure fluctuations. We think that this means of wave generation must be taken into account also.

TEMPERATURE WAVES IN THE SOLAR ATMOSPHERE 209

4. Conclusion

In the solar photosphere five-minute temperature waves must occur. The acoustic and

temperature waves differ from one another by the amplitude and phase relations

between temperature and velocity. The observable phase lag is consistent with the

presence of temperature waves in the solar atmosphere. The five-minute oscillations in the photosphere represent a mixture of acoustic and temperature waves, which can be

separated by investigations of amplitude and phase relations between temperature and

velocity oscillations.

References

Deubner, F.-L. and Fleck, B.: 1989, Astron. Astrophys. 213, 423. Landau, L. and Lifshitz, E.: 1959, Fluid Mechanics, Pergamon Press, London, p. 303. Muller, R.: 1988, in R. Rutten and G. Severino (eds.), Solar and Stellar Granulation, Kluwer Academic

Publishers, The Netherlands, pp. 101-124. Zhugzhda, Y. D.: 1983, Astrophys. Space Sci. 95, 255.