3. Results and discussion
For specified rms turbulent velocities in the range of to , squeezing at three different excitation heights (specified in terms of optical depths = 1, 10, 100), we have computed time-averaged longitudinal wave energy fluxes in magnetic flux tubes with field strengths and where is specified at . The tube was excited by specifying , using Eq. (24), where the fluctuating horizontal flow field with velocity is given by Eq. (8) and where . The partial wave amplitudes are computed from Eq. (13). This flow field is assumed to represent the isotropic turbulence acting as pressure fluctuations on the tube boundary.
The resulting normalized wave energy fluxes computed by using Eq. (26) for different cases of tube excitation are shown in Table 1. The results clearly demonstrate (see cases 10-13 or 14-17) that the fluxes depend strongly on the magnitude of the external turbulent velocity perturbation. Eq. (26) and cases 7 to 9 show that there is also a significant dependence on in the sense that with larger one gets smaller fluxes . This is due to the fact that a stronger tube becomes more rigid and thus less easily perturbed by outside pressure fluctuations. The numerically determined dependences on and (at ) can be roughly fitted by
with in (cm/s) and in (G). The approximate values from this formula are also shown in Table 1. A similar dependence of the longitudinal wave flux has been found in the analytical calculations by Musielak et al. (1989, 1995).
Table 1. Time-averaged longitudinal wave energy fluxes generated in a tube of magnetic field strength , squeezed by velocity fluctuations at optical depth , with an rms turbulent velocity . The fluxes are normalized to the solar surface at by using an area factor . Also shown are fitted flux values using Eq. (27).
Table 1 also shows that the wave fluxes do not depend much on the height of the excitation point (compare cases 1 to 3, 4 to 6, 10 to 17). For typical expected fluctuating velocities and field strengths the normalized longitudinal tube wave energy fluxes vary by roughly a factor of four around the value .
In all cases displayed in Table 1, the time dependent wave computations were carried out for tube sections starting at the selected squeezing height and extending a few grid points higher. As we are presently interested only in the wave generation and not in the wave propagation or the physics of the tube, the length of the tube section was chosen to be small in order to save computation time. All calculations were extended to 5000 s. In our computations we assumed adiabatic conditions and did not consider shock formation. We found that for larger rms turbulent velocities and for shallow squeezing heights the excitation produced rapid expansions of the computed tube section. This resulted in adiabatic cooling and eventually lead to unrealistically cold tubes with temperatures below 1000 K, where the wave computations broke down due to numerical difficulties. This occurred for km/s at , respectively. We attribute this behaviour to our adiabatic treatment. In addition these upper limits of the excitation strength points to the limit of validity of the thin tube approximation employed by us.
Fig. 4 shows for case 2 of Table 1 (displayed also in Figs. 1 and 2) the time-dependence of the longitudinal tube wave flux together with the averaged wave flux. It is seen that the instantaneous wave flux is very stochastic in nature, consisting of intense short duration bursts. Comparison of Figs. 4 and 2 shows that the spikes in longitudinal wave flux are directly related to spikes in and . We recall that this behaviour is also seen in the transverse wave generation computations of Paper I.
The spectrum corresponding to this instantaneous wave energy flux is presented in Fig. 5, which again shows the stochastic nature of longitudinal tube wave generation. The spectra obtained for all other cases of Table 1 are similar to the one shown in Fig. 5. Thus, in order to compare spectra generated at different heights, with different squeezing velocities and different magnetic field strengths, we have to smooth them as otherwise no systematic trends can be seen. A standard smoothing procedure has been used for all cases. As a result of this smoothing, the low-frequency shape of the spectrum is preserved, however, the high-frequency tail seen in Fig. 5 is removed because its contribution to the total wave energy flux is negligible (see Fig. 4).
Fig. 6 shows smoothed spectra of the longitudinal wave energy fluxes calculated for the tube and for the excitation at four different squeezing heights. It is seen that the smoothed wave energy spectra do not differ much with the excitation height for frequencies . However, for higher frequencies, there is a tendency that squeezing at greater heights produces more wave energy. Even so, the total wave energy fluxes do not differ much, which is consistent with the time-integrated total fluxes given in Table 1.
The results presented in Fig. 7 clearly demonstrate that the shapes of the generated spectra are practically independent of the rms turbulent velocity but its magnitude does strongly affect the total wave energy flux; the higher the velocity the more wave energy is generated. Moreover, the obtained results also show that when the turbulent velocity increases, more wave energy is generated in the high-frequency domain of the spectra.
Finally, Fig. 8 shows the dependence of the wave energy spectrum on the flux tube model for the excitation at a common depth . It is seen again that the shape of the spectra is practically independent of the strength of the magnetic field. However as noted above there is a significant dependence of the absolute wave flux on the field strength .
The present results are now compared with our previous ones. Analytical calculations of the generation of linear longitudinal tube waves were performed by Musielak et al. (1989, 1995) who found that typical wave energy fluxes carried by these waves were of the order of (for ). Comparing the value with our results presented in Table 1, one sees that the nonlinear excitation of longitudinal tube waves gives fluxes typically more than one order of magnitude higher. This indicates that the linear results give only lower bounds for realistic wave energy fluxes carried by these waves in the solar atmosphere. As already mentioned in Paper I, the comparison must be taken with some caution because there is an important difference between the analytical and numerical approaches. Namely, in the numerical approach the process of squeezing or shaking a tube takes place only at one specific height, whereas in the analytical approach a significant portion of the entire flux tube is affected. Thus, the main difference is that the numerical approach does not take into account any correlation effects while the analytical approach does include some of them. It is presently unclear whether these correlation effects would decrease or increase the total wave energy fluxes.
Another difference between analytical and numerical results is the dependence of the wave generation process on the magnetic field strength. According to Musielak et al. (1995), the efficiency of the generation of linear longitudinal tube waves is strongly affected by the strength of the tube magnetic field; the stronger the field the lower is the resulting wave energy flux. The effect can be explained by the fact that "stiffness" of a flux tube decreases (and the gas pressure inside the tube increases) when the field strength decreases and, as a result, it is easier to squeeze the tube by external turbulent motions. Thus, for the same external motions, higher wave energy fluxes should be generated for weaker magnetic fields. Both our numerical and analytical results show this effect. While our analytical results (Musielak et al. 1995, Table 1) give , for the case of , and , respectively, we presently have , , respectively for the same magnetic field strengths. This shows that for strong magnetic fields the difference between linear and nonlinear squeezing is particularly large.
Since the generated longitudinal tube waves are essentially acoustic waves guided by the magnetic field lines of a flux tube, it is also of interest to compare the tube wave fluxes with the acoustic wave fluxes outside the tube generated by the same turbulent motions in the solar convection zone. Recent calculations of the generation of linear acoustic waves performed by Musielak et al. (1994) showed that the total acoustic wave energy flux for the Sun is for the extended Kolmogorov energy spectrum with a modified Gaussian frequency factor, using a mixing length parameter of . That this choice of mixing-length parameter is realistic, has recently been discussed by Theurer et al. (1997). Note that the turbulence spectrum for the acoustic case is the same which we employ in the present paper. Surprisingly it turns out that the value of the acoustic wave energy flux is roughly equal to the typical longitudinal tube wave flux found in our present work.
One might think that finding the same wave flux inside and outside the tube implies that the magnetic flux tube does not play a major role in the outer solar atmosphere, a result clearly in contradiction with the observations. The surprise quickly fades when one realizes that the gas density inside the tube is roughly a factor of 6 smaller than the external density. For this low density the wave flux in the tube is large compared to the outside flux, leading to rapid shock formation, but also to severe NLTE effects which strongly modify the radiative losses. To consider all these effects it is necessary to perform detailed calculations of the propagation and dissipation of these waves within magnetic flux tubes. Previous calculations by Herbold et al. (1985) for monochromatic waves and for a chosen range of wave energy fluxes showed that the waves formed shocks in the lower part of the chromosphere and heat the tube effectively. These results have been confirmed recently by Fawzy et al. (1998) who investigated the dissipation rates for waves of various energy fluxes in tubes of different geometry. In addition, Cuntz et al. (1998) adopted the approach presented here to calculate longitudinal wave energy fluxes for K2V stars with different rotation rates and used these results to construct the first purely theoretical models of stellar chromospheres for rotating stars.
Finally, the longitudinal wave energy fluxes must be compared with the transverse fluxes of Paper I. From our above revised typical transverse wave fluxes we find a ratio . Estimates based directly on the observed velocities of the solar surface give similar magnitudes of the transverse wave energy fluxes (see Muller et al. 1994). These results have a simple physical explanation, namely, it is much easier to shake a magnetic flux tube by the external turbulent motions than to squeeze it. However, the dissipation rate for transverse waves is much lower than that for longitudinal waves. According to Ulmschneider et al. (1991), an efficient process of damping transverse waves is nonlinear coupling to longitudinal waves. This process may become active in the upper chromosphere after the propagating transverse waves reach large enough amplitudes. Thus, one may conclude that the upper chromospheric layers are heated by these waves, whereas longitudinal waves are responsible for the heating of the lower chromospheric layers. In the following, we show that this simplified picture has to be taken with caution.
Since the described calculations are limited to the thin flux tube approximation, a number of important physical processes that may strongly influence the energy propagation along the magnetic tubes are not considered. Aside of the appearance of compressive `sloshing modes' the dissipation properties of which have not been investigated, there is wave energy leakage from flux tubes to the external medium (Ziegler & Ulmschneider 1997a). Another process is the generation of wave energy from external acoustic waves. To get some idea about the efficiency of the latter processes, we refer to 2-D time-dependent MHD wave simulations performed by Huang (1996). The obtained results for magnetic slabs with showed that almost of the energy carried by transverse waves and of the energy carried by longitudinal waves can leak out to the external medium within two wave periods. In addition, Huang showed that the generation of wave energy in the magnetic slabs of the same by external acoustic waves is only about 15% for both wave modes. This makes the net leakage for transverse waves significant but practically allows to neglect the net leakage for longitudinal waves. The results of 3D time-dependent MHD simulations of magnetic flux tubes performed by Ziegler & Ulmschneider (1997b) showed that the leakage for transverse waves can be even higher. This means that these waves will contribute energy to the external acoustic wave field, but given the small filling factors of magnetic tubes outside the inner network, this transverse energy will not enhance much the outside acoustic flux.
© European Southern Observatory (ESO) 1998
Online publication: September 8, 1998