Radiation hydrodynamics of the stratified solar plasma

C.-V. Meister

Astrophysical Institute Potsdam, Potsdam, Germany

Received 5 April 2001, published online 22 January 2002

Contents

Abstract

1. Introduction

Studying global solar oscillations, properties in the velocity and radiation intensity fluctuations of the acoustic-type modes (called p modes) were found, which provides a reason to conclude that the waves behave nonadiabatically [e.g., Deubner, 1991]. The most probable reasons for this nonadiabaticity are interactions between the acoustic waves and the processes of radiation and heat transport as well as turbulence. First theoretical investigations of the visibility of the luminosity variations of the Sun correlated with p modes were performed by Toutain and Gouttebroze [1988, 1993].

The interpretation of the wave observations was mostly limited by adiabatic models. In general, the nonadiabatic models are yet restricted to the case of gray, homogeneous atmospheres. Thus the further development of the theory of nongray and nonhomogeneous atmospheres stratified in the gravity field and structured by the magnetic field is needed to understand the more and more detailed experimental data.

Thus Ibánez and Plachco [1989] introduced in the basic radiation magnetohydrodynamic equations of ideal gases [Mihalas and Mihalas, 1984] the electron heat conductivity. Already Souffrin [1972] obtained a dispersion equation of nonadiabatic waves for an isothermal atmosphere using the Newton approximation for the radiation relaxation. Zhugzhda [1983] found an analytical solution for nonadiabatic waves in an isothermal atmosphere with density-dependent radiation transport. Further, in 1991, he presented a general solution for the fundamental mode of any atmosphere at all.

However, in all the publications mentioned, the magnetic field was neglected. The influence of the magnetic field in an isothermal stratified atmosphere was first considered by Babaev et al. [1995], who used the Newton approximation for the radiation loss term.

Staude et al. [1994] studied nonadiabatic hydrodynamic waves, for example, the 5-min oscillations of the solar photosphere and chromosphere, in a homogeneous, nongray, radiating and thermally conducting atmosphere under the condition of thermal equilibrium. It was found that the heat conductivity by particle collisions only (neglecting effective wave-particle collisions) seems to be less important than the heat conductivity due to the radiation transport.

In general, the numerical modeling of the nonadiabatic waves in radiating atmospheres showed that also the further development of analytical solutions is of importance, on the one side, to define the boundary conditions for the numerical calculations [Dzhalilov et al., 1992] and, on the other side, to guarantee a better analysis of the numerical results. Thus an analytical study of the radiation hydrodynamics of a stratified atmosphere is presented here.

2. Eddington Approximation of Radiation Magnetohydrodynamics

To describe low-frequency phenomena in a fluid plasma system with radiation transport being influenced by a gravity force, one can use radiation hydrodynamics. This means that the continuity equation

(1)

the momentum balance

(2)

and the energy balance of a nonadiabatic plasma system

(3)

should be considered: d/dt=(/ t+ u), where r is the mass density, p is the scalar pressure, r u is the momentum density, g is the gravitational acceleration, and g is the adiabatic coefficient of the plasma. S is the radiation source function, J represents the mean intensity of the scattered radiation field, L describes the radiation loss function, and k designates the opacity. Taking into account that the square of the sound velocity is equal to c²_s=gp/r, one can also present (5) in the form

(4)

Further, it is assumed that the thermodynamic equation of an ideal gas is applicable:

(5)

where m is the mean molecular weight. The radiation pressure P_R is given by Mihalas and Mihalas [1984]

(6)

The source function S of the electromagnetic radiation is assumed to be equal to the Planck function

(7)

where s = (2 p⁵ k_B⁴) / (15 h³ c²) = (5.6697 0.0029) 10^-8 W m ² K ^-4 is the Stefan-Boltzmann constant.

The momentum and energy equations for the radiation field are used in the Eddington approximation [Mihalas and Mihalas, 1984]

(8)

(9)

J is defined as a specific intensity averaged over all solid angles:

(10)

(11)

where f_R is the photon distribution function; that is f_R( r, t, n, p) d p is the number of photons, per volume, at place r and time t, with the momentum lying in the interval ( p, p+d p)=(hn/c)( n, n+d n). The n value points into the direction of the propagation of the radiation. H is the Eddington flux, which is defined by the first angular momentum of the specific intensity

(12)

This means that (8) and (9) describe isotropic media such as the spherical and the one-dimensional planar ones; n designates the photon frequency, h is the Planck constant, and c is the speed of the light. For the sake of simplicity, we do not distinguish between the flux-mean opacity (in (8)) and the intensity-mean opacities (equations (3) and (9)). Such an approximation was also introduced by Bogdan and Knölker [1989] for the magnetized nonstratified solar atmosphere. The system of equations (1)-(3), (8), and (9) forms a full system of equations for the three scalar parameters r, p, T, and the vector u.

3. Derivation of the Dispersion Equation

It is assumed that the plasma is gray and that it is slightly disturbed from its altitude-dependent equilibrium state (designated by the index "o")

(13)

(14)

The equilibrium value of the radiation intensity J_o is found equating I in (12) with the Kirchhoff-Planck function

(15)

and then averaging the result over the frequency n. The equilibrium values of the source function S_o and the radiation intensity J_o coincide.

The plasma and field parameters may be expressed by the sum of the equilibrium values and the disturbances, which are designated by the index "1,"

(16)

(17)

(18)

Further, an attempt is made to derive the dispersion equation of nonadiabatic waves in the nonhomogeneous solar substance and to find a way to obtain relations between the wave amplitudes of different plasma parameters. For the interpretation of experimental data on the solar atmosphere, it is especially important to study temperature and radiation intensity fluctuations. The method used here is analogous to the method applied for the case when the loss term L=0 by Roberts [1991].

Taking into account the radiation pressure (6), the equilibrium momentum balance (2) is written as (  g=-g n_z )

(19)

Further, expressing the mean radiation intensity J_o by (14) and T_o(z) by the effective solar temperature

(20)

(21)

T _eff is the effective temperature of a blackbody of solar radius r_S=(6.96260.0007) 10⁸ m, which radiates with the total solar luminosity L_S=(3.8450.006) 10²⁶ W. Value t is the optical depth of the solar atmosphere, which is assumed to be gray, which that means

(22)

and

(23)

Finally, one obtains the equilibrium condition

(24)

where A0.2105 N m ^-2 is known. For g the gravitational acceleration at the solar surface g=274 m s ^-2 will be used.

On the other side, T_o in the equilibrium formula J_o=sT⁴_o/p, can be expressed by the equation of state,

(25)

Forming the z derivative of J_o given by (25), and equating the result with J_o/ z of (24), one has

(26)

(27)

This relation gives a possibility to estimate k_o using tables of r_o and p_o found within the standard solar model. A comparison of the results with available opacity values allows an evaluation of the quality of the theory developed here of nonadiabatic solar waves. It should be also mentioned that deriving (26) and (27), the mean molecular weight m was taken to be altitude-independent. An improvement of (26) and (27) for altitude-dependent m(z) can easily be done.

In the case of low-temperature solar plasma with log T < 3.85, opacity tables with an accuracy of about 10^-3 are available (see, for example, the review by Gong and Däppen [1998]). In the solar atmosphere about 300 km thick ( t 1 ), the opacity averaged over the frequency is about 3 10^-8 cm ^-1. For the Sun in general, typical values of the opacity per unit of mass are 0.5 10⁴ cm ² g ^-1 [Weigert and Wendker, 1996].

Figure 1

Figure 2

Parameters of the photosphere and lower chromosphere of the Sun are presented in Table 1 and Figures 1 and 2 after Weigert and Wendker [1996]. On the basis of these parameters, opacities are calculated using the dependencies on T_o and p_o-r_o given by (26). The results for the opacities are compared with the continuum opacity vales at 500 nm presented by Kurucz [1979] for solar regions with T_o 11,000 K. It is clear that within the very simple model presented here with constant m, only the order of magnitude of the opacity in certain solar regions may be reproduced.

Considering the radiation equilibrium described by (8) and (9), one finds the relations

(28)

and

(29)

Equation (28) shows that in the case of nonuniform temperature the mean Eddington flux does not vanish. Considering only vertical gradients in the plasma, relation (29) results into a constant z component of the mean Eddington flux. Expressing J_o/ z by (23), one obtains

(30)

For the linearized momentum balance, one finds from (2)

(31)

(32)

(33)

The linearized continuity equation has the form

(34)

Here

(35)

The corresponding energy balance for the disturbances is (using dr/dt=-r v )

(36)

Expressing p_o/ z in (36) by (3), and T₁ by the linearized equation of state

(37)

one obtains

(38)

The equations (31)-(34) and (38) describe the disturbances of the plasma parameters r₁, v, and p₁, which are coupled to the radiation field by J₁. The latter value has to be found from (8) and (9).

For the disturbances of the radiation field, one finds from (8), (9), and (30)

(39)

(40)

(41)

Expressing J_o/ z in (41) by (23) and (28), and substituting H_1x by (39) and H_1z by (40), one obtains for the dependence of the radiation intensity J₁ on the plasma disturbances the following formula:

(42)

Now (31), (33)-(35), (38), and (42) have to be used to derive the dispersion equation of the nonadiabatic waves. These equations describe the disturbances v_x, v_z, p₁, J₁, r₁, and D_o.

As a next step, r₁ will be excluded from the system of equations. Therefore r₁ from (34),

(43)

is substituted into (33), (38), (42). One finds then

(44)

(45)

(46)

(47)

(48)

Further, expressing v_x by (31), D^o one can transform (35) into

(49)

and (44)-(46) and (49) describe p₁, J₁, v_z and D^o. Further, we try to exclude p₁ and v_z from the systems of equations, and to find an equation for D^o (according to the method of solution for the plasma without radiation) containing an additional J₁ contribution.

Thus first we exclude D^o from (44) to (46) and find p₁ and v_z as a function of J₁. Thus D^o from (49) will be substituted into (44)-(46) to obtain

(50)

(51)

(52)

Considering the limit of infinite Boltzmann number b (which is mathematically equivalent to s0, e.g., J_o0 ) in the energy balances (51) and (52), one can simplify the latter relations

(53)

(54)

Substituting p₁ from (45) (in the limit s0 )

(55)

into (49) for D^o, one finds

(56)

Then, forming the z derivative of (55) and substituting into the result the expression for v_z/ z found from (56), one has

(57)

Now p₁/ z from (57) is substituted into (44). Using the relation

(58)

one finds

(59)

(60)

(61)

The z derivative of (59) is written as

(62)

(63)

Multiplying (59) by

(64)

and subtracting (62) from the result, one obtains

(65)

In the case of a vanishing radiation transport J=0, A=0, (65) may be transformed into a simple relation:

(66)

(67)

(68)

where w_a and w_g are the acoustic cutoff frequency and the buoyancy frequency of a stratified atmosphere without radiation transport. For linear temperature profiles, a solution of (66) was first found by Lamb [e.g., Roberts, 1991]. This solution may be represented by confluent hypergeometric functions. In all other cases (66) and above all (65) for systems with radiation transport have to be solved numerically.

1. Therefore first the fluctuations of the intensity of the scattered radiation field have to be found. Thus D_o from (56) (with v_z from (54) and v_z/ z from the z derivative of (54)) and D^o* from the z derivative of (56) (with v_z, v_z/ z, ² v_z/ z² from (54) and its z derivatives) have to be substituted into (59). Consequently, a differential equation of the fourth order for J₁ with respect to z will be found, which has to be solved for given boundary conditions.

2. If one has obtained J₁, the solution for v_z may be obtained from (54).

3. Knowing J₁ and v_z, one may further determine p₁ using, for example, (53).

4. After that, it is possible to find v_x using (31).

5. Then, with the help of (42) (expressing D^o by (56)), r₁ may also be found.

6. From (37), one determines then temperature fluctuations especially important for interpretation of satellite experiments T₁.

7. Besides, from (39) and (40), one may obtain H_1x and H_1z, respectively.

Applications of this method to find the fluctuations of the parameters of the solar plasma and the radiation field will be presented in further papers.

3. Conclusions

In the present paper, former models of hydrodynamic adiabatic atmospheric waves are extended to the case of nonadiabatic acoustic-gravity waves in a system with radiation transport where the radiation relaxation is described in the Eddington approximation.

Effects of the stratification of the atmosphere are taken into account, but no influence of the mean magnetic field is considered. The temperature distribution is assumed to be nonuniform, and an altitudinal-dependent radiation pressure is taken into account in the momentum balances.

In analogy to the works for stratified plasmas neglecting nonadiabatic effects, a dispersion relation is found introducing the divergency of the plasma velocity as a new parameter. In the obtained dispersion relation, there appear new radiation-intensity-dependent coefficients and additional terms describing the wave damping by the radiation transport.

The momentum and energy balances for the particles and the radiation are studied here, contrary to other works. Thus a simple approximate relation between the opacity and the mean plasma pressure, the mean pressure gradient and the mass density is found. This relation may be used to estimate the quality of the model developed for the atmospheric acoustic-gravity waves. Since, for the sake of simplicity, the mean-molecular-weight gradients in the system are neglected here, one is able to estimate only the order of magnitude of the opacity.

A scheme is given, which indicates how to find successively the amplitudes of the plasma (velocity, mass density, pressure, temperature) and radiation field (radiation intensity, Eddington flux) parameters in a stratified atmosphere where hydrodynamic acoustic-gravity waves exist. To apply this scheme, boundary conditions for the radiation intensity fluctuations and their spatial derivaties up to the fourth order have to be taken into account.

Analytical expressions to calculate the amplitudes of the plasma and radiation field variations are presented for the case of a large Boltzmann number of the substance, i.e. for relatively cold atmospheres with high plasma pressure and rather low mass density, or for a substance with a polytropic coefficient of the order of unity. The obtained results may be used for the interpretation of experimental data obtained during the solar satellite experiments, e.g., CORONAS and SOHO.

Acknowledgments

References

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