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Astron. Astrophys. 338, 311-321 (1998)

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1. Introduction

In this second of a series of papers we continue our study of the generation of different types of nonlinear magnetic tube waves in the solar atmosphere. In the first paper, see Huang et al. (1995, henceforth called Paper I), we have investigated the generation of nonlinear transverse tube waves by shaking a thin magnetic flux tube with velocity perturbations of an observed magnitude which vary according to an inferred spectrum of the solar turbulent convection. In the present paper, we apply external pressure perturbations to the tube and calculate the efficiency of generation of nonlinear longitudinal tube waves. Our approach is fully numerical and is, therefore, different from other previous (mostly analytical) treatments of the generation of magnetic tube waves (e.g., Musielak et al. 1989; Choudhuri et al. 1993a, 1993b; Musielak et al. 1995). In the work by Musielak et al. (1995), a general theory of interaction of magnetic flux tubes with the subsonic turbulent convection has been developed and used to investigate the generation of linear longitudinal and transverse tube waves. The obtained results show that the fluxes can be important for the heating observed in the solar chromospheric network and that they should be regarded only as lower bounds for realistic energy fluxes carried by these waves. Choudhuri et al. have investigated the generation of magnetic kink waves by rapid foot point motions of the magnetic flux tube. They argue that occasional rapid motions can account for the entire energy flux needed to heat the quiet corona. They find that pulses are much more efficient than continuous excitation for the transfer of wave energy to the solar corona and that the energy flux from pulses actually increases if there is a transition layer temperature jump in the atmosphere.

The approach presented by us in Paper I and in this paper can be justified by recent observations of the proper motions of footpoints of magnetic flux tubes at the photospheric level (Muller 1989; Nesis et al. 1992; Muller et al. 1994) and by time-dependent numerical simulations of the solar convection (e.g., Nordlund & Dravins 1990; Nordlund & Stein 1991; Cattaneo et al. 1991; Steffen 1993). Both the observations and numerical simulations clearly show that horizontal velocities as large as 3 [FORMULA] occur in the solar photosphere and at the top of the convection zone. Velocities of this magnitude and larger have also been seen by Title (1994, private communication) in his observational data. In addition, in some of these numerical simulations evidence has been found for the presence of horizontally propagating shock waves (Cattaneo et al. 1991; Steffen 1993, Solanki et al. 1996a, Steiner et al. 1996, 1998). The existence of granular transsonic flows has also been reported by Nesis et al. who interpreted some of their data as post-shock turbulence; this interpretation has been criticized by Solanki et al. who claim that there is no conclusive evidence for these flows in the Nesis et al. data but there is a signature of shocks in their data. Recently, Nordlund et al. (1997) have suggested that the granulation flow is seemingly laminar despite of the high Reynolds numbers involved and that the level of turbulence in this flow is much lower than previously assumed. Based on these contradictory results, it is clear that the problem of the existence of granular supersonic flows and post-shock turbulence is far from being solved despite the concentrated observational and theoretical efforts. Without going further into details of this discussion, we restrict our calculations to subsonic motions (see Sect. 2) and investigate numerically the interaction of these motions with magnetic flux tubes. The fact that this interaction may become an efficient source of magnetic tube waves which can propagate along the tubes and carry energy to the chromosphere and corona has already been recognized in the literature (see Narain & Ulmschneider 1996, and references therein). A rough estimate of the generated wave energy fluxes by Muller et al. (1994) clearly demonstrates that the amount of wave energy available for heating is sufficient to sustain the mean level of the observed radiative losses from both the solar chromosphere and corona.

The aim of the present paper is to calculate the wave energy fluxes generated by the interaction between a thin magnetic flux tube and the pressure fluctuations produced by turbulence in the solar photosphere and convection zone. We consider the continuous excitation of nonlinear longitudinal tube waves by these fluctuations and assume that the fluctuations are symmetric and are caused by the turbulent flow field. The fact that the imposed perturbations on a thin, vertically oriented magnetic flux tube are symmetric, results in a symmetric squeezing of the tube and in the excitation of purely longitudinal tube waves. Note that in this process the tube is not displaced from its vertical position.

This generation of longitudinal tube waves distinguishes the results of the present paper from those obtained in Paper I. Note that in Paper I, the excited nonlinear transverse tube waves were always coupled to longitudinal tube waves via the process of nonlinear mode coupling. As shown by Ulmschneider et al. (1991), the coupling can be an important way to dissipate the energy carried by the generated nonlinear transverse tube waves (Zhugzhda et al. 1995). In our present work, however, the excitation of nonlinear longitudinal tube waves is direct and a much more efficient shock dissipation will result in the chromospheric flux tube (Herbold et al. 1985).

In order to prescribe the external pressure fluctuations imposed on the tube, we specify the rms velocity amplitude of the external turbulent motions and use an extended Kolmogorov spectrum with a modified Gaussian frequency factor (Musielak et al. 1994). Note that the basic procedure for determining the fluctuations is similar to that presented in Paper I. The horizontal pressure balance translates the external pressure fluctuations into internal pressure and magnetic field fluctuations. Using the linear relations between longitudinal velocity-, pressure- and magnetic field perturbations, valid for small amplitude longitudinal tube waves, the fluctuations can be described by internal velocity perturbations alone which serve as piston boundary condition for the generated longitudinal tube wave.

To prescribe the external pressure perturbations by using a turbulent energy spectrum has another advantage. The representation of the perturbation spectrum by a large number of partial waves occasionally results in large wave amplitudes which are precisely those events which Choudhuri et al. envision to produce the bulk of the wave energy. We will show below (c.f. Fig. 4) that the longitudinal wave flux generation (like the transverse wave generation in Paper I) indeed displays this behaviour. Our approach to use realistic turbulent energy spectra thus permits to accurately describe the stochastic wave generation process.

The main numerical tool used in this paper is a one-dimensional, time-dependent, nonlinear MHD code originally developed by Herbold et al. (1985) to study the vertical propagation of longitudinal waves in thin magnetic flux tubes. A modified version of this code by Ulmschneider et al. (1991) has been used to study the propagation of longitudinal-transverse magnetic tube waves as well as the transverse wave generation in Paper I. Note that the use of the transverse equation of motion adopted from Spruit (1981) in the Ulmschneider et al. (1991) code has been criticized by a considerable number of authors (for details and the literature see Cheng 1992; Moreno-Insertis et al. 1997). These criticisms apply only to the back reaction force experienced by the tube upon swaying, but do not have any bearing on our present problem which does not allow for swaying. In addition, as shown by Osin et al. (1998), due to the lack of sizeable longitudinal flows, these criticisms do not appreciably influence the case of the transverse tube wave energy generation of Paper I or the wave propagation results of Ulmschneider et al. (1991). For our present calculation the transverse equation of motion was disregarded and the Herbold et al. code, which considers only the longitudinal equation of motion, was modified to perform our calculations of the excitation of nonlinear longitudinal tube waves.

We organize the paper as follows: Sect. 2 describes briefly the basic assumptions and the method; the results of our calculations are presented in Sect. 3, and finally Sect. 4 gives our conclusions.

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© European Southern Observatory (ESO) 1998

Online publication: September 8, 1998
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