6. Applications to the Sun
It has long been known that the distribution of magnetic fields on the solar surface is highly inhomogeneous and that magnetic inhomogeneities outside sunspots form flux tube structures (e.g. Stenflo 1978; Zwaan 1989; Solanki 1993). Individual magnetic flux tubes are regions of intense magnetic fields that rapidly diverge in the solar chromosphere. The typical strength of the magnetic field inside these tubes at the base of the photosphere is about 1500 G but can also be less. There are turbulent motions in the solar convection zone where the tubes are rooted, but also in the solar photosphere. These motions may interact with the tubes leading to the generation of tube waves. High resolution white light observations of the Sun performed by Muller (1989), Nesis et al. (1992) and Muller et al. (1994) show that the velocity of turbulent motions can be as large as 2 .
It is important to find out how efficiently the observed turbulent motions generate linear and nonlinear magnetic tube waves. The problem of generation of these waves has been treated both analytically (Musielak et al. 1989, 1995; Choudhuri et al. 1993a, b) and numerically (Huang et al. 1995; Ulmschneider & Musielak 1998). The work by Huang et al. shows that the typical wave energy fluxes carried by nonlinear transverse tube waves are of the order of . However, the amount of energy which can be transferred to the chromosphere remains uncertain because the process of energy leakage to the external medium has not been taken into account. This is due to the use of the thin flux tube approximation by these authors. To estimate the efficiency of the leakage process, the slab structure is used to approximate a magnetic flux tube embedded in the solar photosphere (see Sect. 6.1).
Observations and numerical simulations show also evidence for the existence of large amplitude acoustic waves travelling horizontally in the upper layers of the solar convection zone (Nordlund & Dravis 1990; Cattaneo et al. 1991; Nordlund & Stein 1991; Steffen 1993; Steffen et al. 1994). It is of interest to investigate how efficiently these acoustic waves can supply energy into the magnetic flux tubes. The efficiency of this process is estimated here by using the model of two adjacent magnetic slabs (see Sect. 6.2).
6.1. Wave energy leakage
The results presented in Sect. 5 clearly show that in both cases of the longitudinal and transverse perturbation the energy carried by the body and surface waves leaks to the external medium. As a result of this process, external acoustic waves are generated. The wave energy leakage has been extensively discussed by Huang (1996) who obtained the results for magnetic slabs with and demonstrated that more than of the energy carried by transverse slab waves and almost of the energy carried by longitudinal slab waves can leak out to the external medium within two wave periods. In our present work the slab structure is used to simulate a flux tube embedded in the solar photosphere and to estimate the efficiency of wave energy leakage. It is well known that the solar photosphere may be approximated by a model consisting of two media: a magnetic region where the plasma is of order unity or smaller, and a weak field region where is very high. In the simulations performed here, the value of in the slab is chosen to be 1. The external medium is assumed to be field-free, which means that is infinite there. At the optical depth , the photospheric flux tubes have typical diameters of approximately 100 km and the sound speed is around 8 . For waves with periods of around 60 seconds, the diameter of the tube is normalized to 0.21 which is taken to be equal to the slab thickness. The velocity perturbation amplitude is chosen to be 0.25, which corresponds to a velocity perturbation of 2 and is in agreement with the observations (see above). Only transverse velocity perturbations are considered.
Snapshots of the resulting velocity field looks similarly as in Figs. 11 to 12. There are prominent acoustic waves in the external medium. Since the perturbations are imposed only on the slab, it is evident that a significant amount of wave energy must be leaking from the slab to the external medium. To estimate this amount of energy, we use the wave energy leakage ratio defined as
where and represent the energy carried by the external acoustic waves and the internal slab waves, respectively. They are given by
where is the total wave energy density, and and represent the part of the computational domain that is external and internal to the slab, respectively. According to Landau & Lifshitz (1959), the kinetic (), thermal () and magnetic () energy densities contribute to the total wave energy density as
where , , , and are perturbed quantities. The time-average of defined as
is used here as a quantitative measure of the amount of wave energy leaking to the external medium. The averaging time is chosen as 2.0 in these calculations. In general, it is desired to carry out the time average over a very long period of time. However, this is not feasible numerically as it would require an infinitely large computational domain to account for the wave spreading in time. Since the finite computational domain is used here, the simulations are stopped just before the fastest propagating wave reaches the boundary; the latter always happens when .
The calculated energy leakage ratio is 0.62, which means that of the energy carried by nonlinear slab waves leaks to the external medium within two wave periods. Now, applying this result to the wave energy fluxes calculated by Huang et al. (1995) for transverse tube waves, it is seen that more than half of the total wave energy flux (of ) carried by these waves will be transferred to the external medium and propagate there as acoustic waves. As shown recently by Ziegler & Ulmschneider (1997a,b), the efficiency of wave energy leakage can be even higher for more realistic magnetic flux tubes.
6.2. Generation of magnetic slab waves by external acoustic waves
To calculate the efficiency of the generation of magnetic slab waves excited by external acoustic waves, the model of two identical magnetic slabs is considered (see Sect. 2) and the external medium is field-free. The slabs are placed side by side in the computational domain and the distance between them is half an acoustic wavelength (see Figs. 15 and 16). The computational domain and the grid size are similar those chosen for our longitudinal and transverse wave calculations. The slab which is located closer to the center of the computational domain is called the first slab and the other the second slab. The distance between the first slab and the center of the computational domain, where the source of the external acoustic waves is located, is one acoustic wavelength. The first slab lies at , the second at . The wave source is aligned along the x-direction and has a width of of the acoustic wavelength. It must be noted that the values of these three parameters may have wide ranges in reality and that the particular values used here are chosen only for the purpose of simulation.
The numerical results obtained for are presented in Figs. 15 and 16, which show that at the time when the simulation was stopped, the external acoustic waves have already reached both slabs. It is seen that the propagating acoustic waves pass through both slabs and, as a result, MHD body and surface waves are excited in the slabs; the latter can be even better seen in Fig. 16. Obviously, the body and surface waves generated in the first slab are stronger than those observed in the second slab. The amount of energy that is lost by the acoustic waves due to the excitation of the magnetic slab waves on the first slab is and the corresponding value for the second slab is . Although these numbers are small, it is expected that the closer the slabs are to the wave source, the more energy will be transferred to them. The amount of energy transferred must also increase when the amplitude of the imposed perturbations is increased.
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998