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Astron. Astrophys. 325, 709-713 (1997)

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3. Results and discussion

3.1. Calculation of stochastic hydrodynamic models

For my study, I start with a monochromatic wave having a period of [FORMULA] s. In order to increase the stability of the hydrodynamic wave solutions, the radiation damping is switched on over a timespan of [FORMULA]  s. The wave computation is continued over 3500 time steps, corresponding to [FORMULA]  s. During that time 129 shocks are inserted into the atmosphere. The hydrodynamic atmosphere has then reached a dynamical steady state given by the balance of shock wave heating and radiative cooling.

Then I start to introduce shock waves with stochastically changing wave periods. I have selected two different wave period distributions. Both distributions are assumed to be Gaussian and centered at [FORMULA]  s. The first distribution (Spectrum 1) has a standard deviation of [FORMULA]  s, whereas the standard deviation of the second distribution (Spectrum 2) is [FORMULA]  s (see Fig. 1). The spectrum has been cut off to avoid "negative periods." Periods below [FORMULA]  s have also been excluded due to the limited hydrodynamic resolution in the wave models. The peak values of the wave period distributions have been chosen according to results from traditional acoustic energy generation models (Bohn 1984; Ulmschneider et al. 1996). As comparison to these stochastic wave models, I also calculate a monochromatic wave model with [FORMULA]  s. Contrary to waves considering period distributions, monochromatic waves rely on a fixed wave period only. The above-given wave period has been selected to ensure that shock overtaking events are avoided, which are found to occur even in monochromatic wave computations with sufficiently small wave periods (Theurer et al. 1997).

[FIGURE] Fig. 1. Distribution functions for the wave periods of the narrow (Spectrum 1) and broad frequency spectrum (Spectrum 2) used as inner atmospheric boundary condition of the models. The wave period used in the monochromatic wave model is indicated by an arrow.

Cuntz (1992a, b) already obtained some results concerning the behavior of stochastic wave models. He found that these models are characterized by shock-shock interaction due to the fact that a broad range of frequencies exists. The stochasticity of the wave field leads to episodic energy and momentum input to the atmosphere, which controls the temperature, the flow speed, and the thermodynamic quantities. The models are also characterized by a complicated hydrodynamic structure determined by a nonuniform distribution of shocks. The shock strengths and the shock speeds differ substantially and change non-monotonically with height. This result gives insight into the basic physics going on: after allowing the wave period to change stochastically, shocks with different strengths are introduced into the atmosphere. Different shock strengths cause different shock speeds leading to interacting, overtaking and merging of shocks ("shock-cannibalism"). Since the strength of an overtaking shock combines with the shocks it engulfs, its speed increases, so it overtakes more and more shocks in front of it and attains an even greater strength. Consequently, the amount of momentum and energy deposition that occurs varies drastically with time and atmospheric height. The direction of the flow alternates between outwardly and inwardly directed motions depending on the strengths of the shocks and the radiation-hydrodynamic history of the flow.

An example of that behavior is given in Fig. 2. It shows the temperature structure of a stochastic shock wave model at [FORMULA]  s and [FORMULA]  s after acoustic frequency spectra have been employed. It is found that the second and third shock (counted from lower to higher mass column densities) merge as the speed of the third shock is 11.5 km s-1 compared to 10.0 km s-1 of shock number two. Therefore, the post-shock temperature of the main shock increases from 7790 K to 8300 K while its strengths climbs from [FORMULA] = 4.23 to [FORMULA] = 4.84. The hydrogen ionization degree behind that shock also increases from 11.9% to 25%. The temperatures behind these shocks (which are extremely relevant for the formation of chromospheric line emission) clearly surpass those of the semiempirical chromosphere model of Basri et al. (1981), whereas the time-averaged atmospheric temperatures do not. They remain mostly between 3500 K and 4000 K, which is far beneath typical chromospheric values. This behavior is caused by quasi-adiabatic cooling preferably occurring in regions between the shocks. It is caused by the generation of considerable chromospheric velocity fields due to momentum transfer of strong shocks generated in the stochastic wave field.

[FIGURE] Fig. 2. Temperature distributions of two stochastic hydrodynamic models obtained [FORMULA]  s and [FORMULA]  s after acoustic frequency spectra have been employed (dotted and solid line, respectively). Also shown are the time-averaged temperature of the model (dashed line) and the temperature run of the semiempirical chromosphere model of Basri et al. (1981) (dashed-dotted line). The hydrodynamic model is calculated by using Spectrum 1.

3.2. Evaluation of chromospheric velocity fields

The main goal of this paper is to explore the consequences of shock-shock interaction for the generation of stochastic chromospheric velocity fields in [FORMULA]  Ori in a systematic manner. Therefore, I evaluate the behavior of the flow at different atmospheric heights corresponding to distinct values of the mass column density m. My check points are: log m = 0, -2, -4, -6 (cgs). By performing sufficiently long computer runs, I calculate the likelihood for the flow of being subsonic, supersonic, inflowing or outflowing. I will discuss the probability distribution for the flow velocity by using increments of 5 km s-1.

First of all, I evaluate the results for Spectrum 1. In order to minimize the impact of the starting model, which is the monochromatic wave model obtained at time step IT = 3500, I ignore the following [FORMULA]  s for the statistical analysis. The wave computation is then continued over a timespan of [FORMULA]  s. During that period of time, 51 shocks are inserted into the atmosphere. Due to overtaking and merging of shocks, strong atmospheric inflows and outflows are initiated. It is found that at relatively low mass column densities (i.e., farther out in the atmosphere) the likelihood for larger inflows and outflows increases (see Fig. 3). At log m = 0, the velocity of the flow stays within [FORMULA] 5 km s-1 range. At log m = -2, the distribution is slightly broader, but does not exceed [FORMULA] 10 km s-1. At log m = -4 and -6, however, the likelihood for the flow being larger than +10 km s-1 or smaller than -10 km s-1 is 8% and 21%, respectively. On the other hand, the absolute value of the flow speed barely exceeds 15 km s-1, except at log m = -6.

[FIGURE] Fig. 3. Histograms for the flow velocities at different atmospheric heights indicated by the mass column density m (g cm-2). Results are given for the monochromatic wave model and for Spectrum 1 and 2.

In the case of Spectrum 2, the results obtained are quite similar. It is found again that at log m = 0, the velocity of the flow velocity stays within [FORMULA] 5 km s-1. At log m = -2, the velocity distribution is again somewhat broader, but again does not exceed [FORMULA] 10 km s-1. At log m = -4 and -6, the likelihood for the absolute value of the flow for being larger than 10 km s-1 is now found to be 10% and 24%, respectively. The differences between Spectrum 1 and Spectrum 2 are insignificant, considering a margin of error of 2 percentage points. Huge differences however exist regarding the monochromatic wave calculation with [FORMULA]  s (see Fig. 3). I evaluated the behavior of the flow considering 20 wave periods. I found that the flow velocities almost never exceed the [FORMULA] 5 km s-1 range at log m = 0 and -2 and never ever exceed the [FORMULA] 10 km s-1 range. This result shows that acoustic frequency spectra are pivotal for generating chromospheric flow speeds outside the [FORMULA] 10 km s-1 range, which have been suggested by observations (Carpenter & Robinson 1997).

Now I investigate the flow velocity u in comparison with the local sound speed c, which is given by the local atmospheric temperature and the hydrogen ionization degree. The fraction [FORMULA] (= Mach number) is a further important characteristic of the atmospheric flow field. The Mach number of the flow is again evaluated at distinct values of the mass column density in the hydrodynamic models by using increments of 0.5. In case of an inflow, [FORMULA] is negative, in case of an outflow [FORMULA] is positive. The absolute value of [FORMULA] then decides whether the flow is subsonic or supersonic. The models show the following results: In case of the monochromatic wave model with [FORMULA]  s, the flow is essentially subsonic in the entire atmospheric domain. The likelihood for the flow to become supersonic increases slightly from 0% at log m = 0 to 5% at log [FORMULA] = -6. In case of the Spectrum 1, however, the likelihood for the flow for being supersonic increases continuously with decreasing mass column density. At log [FORMULA] = -2, -4, and -6, the likelihood for supersonic flows is 7%, 28%, and 43%, respectively. The margin of errors in this numbers is again about 2 percentage point indicating that the trend is real. In case of Spectrum 2, the likelihood for supersonic flows also increases with decreasing mass column density with the numbers now being higher than for Spectrum 1. At log [FORMULA] = -2, -4, and -6 the likelihood is now 10%, 35%, and 52%, respectively. It shows that at low mass column densities (i.e., relatively far outward in the chromosphere), the atmospheric flow is very often supersonic when acoustic frequency spectra are adopted. The differences concerning the Mach numbers between the two spectra appear to be significant. The fact that dependencies on the employed spectra occur for the Mach numbers, but not for velocities itself, is caused by the discrepancies in the chromospheric temperatures. Spectrum 2 contains a higher number of long-period shock waves leading to significantly stronger shocks in the atmosphere. These shocks initiate additional momentum transfer and quasi-adiabatic cooling, which counteracts chromospheric heating.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998