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Astron. Astrophys. 318, 841-869 (1997)

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2. Convection in ATLAS9

We briefly summarize the mixing-length formulation and its modifications and indicate the values of the free parameters adopted for computing the grids of the ATLAS9 models (Kurucz, 1993a, 1994, 1995). The complete description of the mixing-length theory can be found in several papers and text-books. For our summary we refer mostly to Vitense (1953), Cox & Giuli (1968), and Mihalas (1970).

2.1. The mixing-length approach (ML) and the first ATLAS modification: [FORMULA]

The basic idea of the mixing-length theory is that the convective energy is carried out by upward and downward hot and cool bubbles, which travel for some distance L, the mixing-length, before dissolving. Convection may occur when the density of the bubble decreases at least as rapidly as the density of the surrounding medium. The Schwarzschild surface, where the condition is no longer fulfilled, represents the border of the convection zone in the atmosphere. Observations of the solar granulation have shown that this border does not correspond to the real situation in stars.

The mixing-length L is a free parameter having such values that the mixing-length to the pressure scale height ratio [FORMULA] is usually included between 0.5 and 2.0, for both models of internal structure and atmosphere. In addition to the value of the mixing-length there are other free parameters that have to be considered when the convective flux, the convective velocity, and the efficiency parameter are computed. We indicate which are the values adopted for these constants in the Kurucz (1993, 1994, 1995) ATLAS9 models.

In the framework of the mixing-length theory the convective flux can be expressed as (cf. Mihalas (1970), (6-274)):

4 [FORMULA] [FORMULA] =k [FORMULA] [FORMULA] [FORMULA] Tv ([FORMULA] - [FORMULA]), (1)

where the value of the free-parameter k is 0.5 if only upward or only downward bubbles are considered, while it is 1 if both upward and downward elements are taken into account.

The convective velocity is (cf. Mihalas (1970), (6-280)):

v= [FORMULA] L/2, (2)

where k1 is a free parameter which takes into account the work which is dissipated by the frictional forces. If half work is supposed to be dissipated k1 =0.5.

Finally, the efficiency parameter has the form (cf. Mihalas (1970), (6-286)):

[FORMULA] = [FORMULA] + [FORMULA] = [FORMULA] [FORMULA] v, (3)

where:

[FORMULA] = [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA], (4)

and

[FORMULA] = [FORMULA] v. (5)

The ratio V/A of the volume to the surface area of the convective element is a further arbitrary parameter which has to be fixed. For a sphere of diameter L, or a cube of side L, or a cylinder of diameter and height L, it is L/6. In these cases:

[FORMULA] = [FORMULA] v. (6)

With the notations:

[FORMULA] = [FORMULA] = [FORMULA] [FORMULA] L and y=1/2,

we recover the Henyey et al. (1965) formula for a linear temperature distribution inside the bubble:

[FORMULA] = [FORMULA] v [FORMULA] y. (7)

The grid of models from ATLAS9 were all computed with [FORMULA] =1.25. As it is discussed in Sect. 4, this value was fixed by Kurucz from the comparison of the observed and computed solar irradiance. The free parameters in the formulas (1), (2), and (4) have the values: k=0.5, k1 =0.5 and V/A=L/6. The efficiency factor is that given by formula (3), which includes [FORMULA]. The addition of [FORMULA] to [FORMULA] is the first modification of the mixing-length convection as was described by Kurucz (1970) for ATLAS5. A summary of the values assumed by the free parameters in the successive ATLAS versions is given in Castelli (1996).

2.2. The second modification of ML in ATLAS9: the horizontally averaged opacity (HAO)

In the convective layers, the Rosseland opacity [FORMULA] (j), corresponding to the temperature T(j) of the j layer is replaced by the opacity [FORMULA] (j), defined by

[FORMULA] = [FORMULA] + [FORMULA], (8)

namely [FORMULA] (j) is obtained by averaging the opacities of the hot and cold convective elements crossing the j layer. The numbers of these elements fix the value of the new free parameter f. ATLAS9 assumes f=1/2, which implies an equal number of hot and cold elements.

This modification of the standard ML was introduced in model atmospheres by Lester, Lane and Kurucz (1982) and it was taken from a work of Deupree (1979), who pointed out how, in this way, the convective flux in the upper part of a stellar envelope is very close to the flux obtained from a two-dimensional stellar model.

Because Lester et al. (1982) added also a variable mixing-length in their computations, the results from their paper can not be directly compared with the ATLAS9 results, when the third modification, namely the overshooting option, is switched off. We have tried to reproduce with ATLAS9 the convective fluxes computed by Deupree (1979) for an internal structure model with parameters L=50 [FORMULA], M=0.575 [FORMULA], [FORMULA] =6050 K, X=0.7, Z=0.001, and [FORMULA] =1.0. Fig. 1a shows the variation of [FORMULA] / [FORMULA] with temperature for both the ML (full line) and ML + HAO (dashed line) approaches in the case of an atmospheric model with parameters [FORMULA] =6050 K, [FORMULA] =2.5, [M/H]= -1, [FORMULA] =1.0, which correspond approximately to those used by Deupree (1979). Fig. 1b compares the corresponding curves as they are plotted in Fig. 1 of Deupree's (1979) paper. ATLAS9 does not yield the large difference between the two convective fluxes shown by Deupree (1979).

[FIGURE] Fig. 1. Comparison of convective fluxes computed with the standard mixing-length theory (ML)(full line) and the mixing-length theory modified for the horizontally averaged opacity (ML+HAO) (dashed line): a from ATLAS9 model; b Deupree's (1979) results based on an internal structure model

In ATLAS9, the horizontally average opacity acts in the sense of decreasing the convective flux. However, the values f=1/2 in formula (8) and k=1/2 in formula (1) are inconsistent, because the former implies the presence of both upward and downward convective elements, while the latter implies only upward (or only downward) moving bubbles.

Differences between the solar disk center intensity [FORMULA] computed with the standard ML and with ML+HAO occur in the 410.0-480.0 nm region. [FORMULA] from the model with ML+HAO convection is larger than [FORMULA] from the model with the only ML convection less than 1%. [FORMULA] from the model with ML+HAO convection and k=1 in formula (1) (both upward and downward moving bubbles) is lower than [FORMULA] from the model with the only ML convection and k=1/2 (only upward or only downward moving bubbles) less than 0.8%. The conclusion is that the effect of the HAO modification is nearly negligible, at least in the solar case.

2.3. The third modification of ML in ATLAS9: the "approximate overshooting"(AO)

The meaning of "approximate overshooting" was explained by Kurucz (1992). He assumes that "the center of a bubble stops at the top of the convection zone so that there is convective flux one bubble radius above the convection zone. That flux is found by computing the convective flux in the normal way and then smoothing it over a bubble diameter".

In practice, for an atmosphere with N layers, the "overshooting" flux [FORMULA] (j) at the j level is the ML convective flux [FORMULA] (j) averaged over a prefixed 2 [FORMULA] H(j) atmospheric thickness, where H(j) is the geometrical height of the j level in the atmosphere. If [FORMULA] is the pressure scale height, the thickness [FORMULA] H(j) is defined by:

[FORMULA] H(j)=min (W [FORMULA] (j)/2, H(N)-H(j), H(j)-H(1)), (9)

where W is a weight which controls the amount of smoothing and N indicates the last layer in the atmosphere (corresponding to [FORMULA] =100 for the models of the grids). In the ATLAS9 code stored on CD-ROM No.13 (Kurucz, 1993a), W was defined as:

W= [FORMULA], (10)

where [FORMULA] = [FORMULA] + [FORMULA]. If, owing to numerical instabilities, [FORMULA] becomes larger than [FORMULA], W is assumed to be equal 1.

This definition of W led to several discontinuities in the color indices from models with effective temperatures included between 7500 K and 6000 K. Color indices stored on CD-ROM No.13 (Kurucz, 1993a) and CD-ROM No.19 (Kurucz, 1994) are affected by such a shortcoming. For instance, North et al. (1994) discussed the discontinuities in the case of the Geneva photometric system. The discontinuities were caused by the different way how convection was computed in models having the convective zone entirely lying inside the atmosphere (as i.e. in a 7500,4.0 model) and in models having the convective zone extending beyond the bottom of the atmosphere (as i.e in a 7500,4.5 model). Owing to the definition of the weight W, in the first case, the convective flux was computed without "overshooting", while in the second case it was computed with "overshooting". We replaced the (10) with:

W= [FORMULA]. (11)

This new definition of W allowed to eliminate the disturbing discontinuities in the color indices and it has been adopted in a 1995 ATLAS9 version.

The "overshooting" convective flux is therefore:

[FORMULA] (j)= [FORMULA] [FORMULA]. (12)

Namely:

[FORMULA] (j)= [FORMULA] [FORMULA]

[FORMULA]. (13)

At each layer, the final convective flux is the maximum between [FORMULA] (j) and [FORMULA] (j). This last assumption makes the final convective flux greater than both the smoothed flux and the ML+HAO flux. Namely, in the convective atmosphere, the local flux [FORMULA] is replaced (when [FORMULA] [FORMULA] [FORMULA]) by the "non local" [FORMULA] flux, which is the average of the local flux [FORMULA] over a [FORMULA] (or shorter) height. As consequence, some positive convective flux is always present above the Schwarzschild layer.

The modification of the "approximate overshooting" is different from the physical overshooting as it is usually defined (i.e. Renzini, 1987; Kippenhahn & Weigert, 1990), because the "approximate overshooting" assumes a zero velocity of the mean convective element at the Schwarzschild surface and a positive convective flux above it. Physical overshooting implies a non zero velocity at the Schwarzschild surface and a negative convective flux above it.

The effect of the "overshooting" modification on the emerging radiation is larger than that yielded by a simply increasing of the [FORMULA] parameter in the ML and it will be discussed in the next sections.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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