## 2. Basic assumptions and the method## 2.1. Tube modelWe consider a thin, vertically oriented magnetic flux tube embedded in the solar atmosphere and excite nonlinear longitudinal tube waves by squeezing the tube symmetrically at different heights. We assume that the external pressure perturbations caused by the photospheric and subphotospheric turbulent motions are responsible for the squeezing and that these perturbations can be represented by a superposition of partial waves with random phases derived from the local turbulent flow field. The maximum rms velocity of the turbulent motions is taken from a range of observed velocities on the solar surface (e.g., Muller 1989; Nesis et al. 1992; Muller et al. 1994) as well as from numerical simulations of turbulent convection (e.g., Cattaneo et al. 1991; Steffen 1993). For the magnetic tube model, we assume that in the solar atmosphere described by model C of Vernazza et al. (1981) a vertically oriented flux tube is embedded and that the tube has a radius of and a field strength of at the height where externally the optical depth is . This field strength appears to be typical for magnetic flux tubes residing in the supergranular boundaries; for recent direct measurements of these fields see Solanki et al. (1996b). To show the dependence on the field strength we also consider tubes with and . The tubes are assumed to spread exponentially with height in accordance to horizontal pressure balance and magnetic flux conservation. As the maximum of the convective velocities both in mixing-length models and in numerical convection zone models occur deeper than , we select excitation heights at , 10 and 100 in optical depth measured outside the tube. Because the Vernazza et al. model goes only to , we extended that model by fitting a convection zone model obtained with the code of Bohn (1981, 1984). ## 2.2. Pressure balanceTo calculate wave energy spectra and fluxes carried by nonlinear
longitudinal tube waves, we must prescribe external pressure
fluctuations which lead to symmetrical squeezing of the tube. As
discussed earlier, the perturbations are caused by the turbulent
motions of the external medium. Therefore, in order to prescribe the
perturbations we have to know the physical properties of the
turbulence. Unfortunately, the properties of realistic turbulence
occuring on the Sun are presently unknown and currently no first
principle theory of turbulence exists. Therefore, the properties of
the turbulence occuring on and below the solar surface are usually
determined by specifying a turbulent energy spectrum. Many different
shapes of the turbulent energy spectrum in the solar atmosphere have
been proposed (e.g, Stein 1967; Bohn 1984; Musielak et al. 1989;
Goldreich & Kumar 1988, 1990). More recently, Musielak et al.
(1994) have combined some theoretical arguments about the turbulence
with the results of numerical simulations (e.g., Cattaneo et al. 1991)
as well as observational results (e.g., Zahn 1987; Muller et al. 1994)
and suggested that the spatial and temporal parts of the turbulent
energy spectrum can be described by an extended Kolmogorov spectrum
with a modified Gaussian frequency factor. Recently, Nordlund et al.
(1997) have argued that the Kolmogorov scaling with a power spectrum
slope of may not apply in regions of highly
non-isotropic motions found in convection zone simulations. They also
argue that the turbulence is reduced in rising bulk flows and enhanced
in downflows. This could increase our tube wave generation rates as
the magnetic tubes are indeed situated in the downflow regions. In
addition it has to be noted that bulk downflows along and outside the
magnetic flux tubes do not contribute to the wave generation. For the
superposed isotropic velocity fluctuations which we envision at our
shaking point, we expect on basis of the high Reynolds numbers, that
the Kolmogorov spectrum is well represented, particularly as the tube
excitation is at large optical depths (at the point of the maximum of
the convective velocity at to 100), where
radiation effects are still minor. For the range of this spectrum we
feel that it would be hard to deviate greatly from the typical length
scale in a gravitational atmosphere, the scale height Following Musielak et al. (1995) we write the pressure balance between the tube and the external medium as where is the time-dependent external turbulent pressure.
, ,
are the turbulent velocities in For the case of homogeneous isotropic turbulence there is no longer a dependence on due to the assumed homogeneity and for the three spatial components one has because of the assumed isotropy. Here is the rms velocity amplitude in one spatial direction, taken to be the same in the x, y and z-directions, it is independent of space and time. Note that our present definition of agrees with the notation in Paper I, and with that used in Eqs. (B6), (B10) and (18) of Musielak et al. (1995), while unfortunately the same notation was also used in a different sense in Eqs. (2), (6) and (8) of the latter paper to describe the total time-dependent 3-D turbulent velocity vector. From the above equations we find for the time-averaged external turbulent pressure After time-averaging of Eq. (1) and assuming homogeneous isotropic turbulence we obtain where is the average internal gas pressure and the average magnetic field strength. This equation shows that when one adds a turbulent flow outside the magnetic flux tube, then the mean external pressure is increased. This is due to the fact that in Eqs. (1), (2) the external turbulent pressure is a quantity which cannot be negative and thus must fluctuate around an average positive value. Typically the mean turbulent pressure leads to a small additional compression of the flux tube relative to the static case. With , and , we find . Note that because of this, has been neglected relative to in the paper of Musielak et al. (1995) in their analytical treatment of the generation of linear longitudinal tube waves. As we are interested in the fluctuating quantities we subtract Eq. (6) from (1) and upon linearization obtain where is the internal gas pressure perturbation and the magnetic field strength perturbation. ## 2.3. Turbulent velocity fluctuationsFollowing Paper I we assume that the horizontal turbulent velocity fluctuations in the x-direction can be represented by a linear combination of N = 100 partial waves where is the velocity amplitude of these waves, it will be determined by the turbulent energy spectrum (see below), is the wave frequency that ranges from 2.9 to 290 mHz with step mHz, and is an arbitrary but constant phase angle with being a random number in the interval [0, 1]. Note that these limits correspond to and , where is the acoustic cut-off frequency. Note also that the right hand side (RHS) of Eq. (8) cannot be formally treated as a Fourier decomposition of in time because of the random phases of the otherwise sinusoidal waves. To determine the velocity amplitudes , we follow the procedure outlined in Paper I. We begin with time averaging to find from which using Eqs. (3) and (4) we have As described in Paper I as well as by Musielak et al. (1995) the turbulent energy spectrum is normalized to where and are the spatial and temporal components of the turbulent energy spectrum, respectively. Here the factor results from the contributions to the kinetic energy from the three spatial directions. We now use Eq. (10) to write as from which we have with Comparing our present Eqs. (12), (13) with Eqs. (25), (26) of Paper I we notice an error in Paper I which leads to an underestimate of the transverse tube wave fluxes of that paper by a factor of 2. As mentioned in Paper I, the combined shaking by a horizontal velocity similar to could increase the transverse flux even more, but depends on correlation effects. As we assume that the shaking in x and y-directions are uncorrelated, the total transverse wave flux by shaking with and is a factor of two higher than that found when shaking with alone. Thus compared to the values given in Table 1 of Paper I, the true total transverse wave fluxes should be a factor of 2-4 higher. That is, the transverse wave flux should be of the order of . Note that contrary to the longitudinal wave case, the fluctuations do not contribute to the transverse wave flux. To specify the turbulent energy spectra appropriate for the solar convection zone we follow Musielak et al. (1994). These authors argue on the basis of observations and numerical simulations of the solar convection that a realistic turbulent energy spectrum should be reasonably well described by an extended Kolmogorov spectrum and a modified Gaussian frequency factor . The extended Kolmogorov spatial spectrum is given by where the factor is determined by the normalization condition and , where where is computed from Knowing and , we may now use Eqs. (13), (14) to calculate the velocity amplitude of the partial waves. The resulting amplitudes obtained in this way are shown in Fig. 1.
The results presented in Fig. 1 can now be used to compute the
turbulent velocity fluctuations at the
squeezing height using Eq. (8).
Fig. 2 shows (dashed) at the squeezing
height as a function of time and introduce the normalized power spectrum of the turbulent velocity by defining ; in the following we call simply the velocity spectrum. This spectrum has dimension of and is plotted in Fig. 3. The presented spectrum was obtained from temporal Fourier analysis of the last 2048 s of the velocity , where the velocity was sampled every sec. The lower limit of the spectrum is due to the total sampling time. As seen in Fig. 3, the velocity spectrum shows no discrete lines for any frequency but only broad noise which indicates completely aperiodic (chaotic) motions. This confirms that the velocity field described by Eq. (8) is indeed chaotic in nature as should be expected for the turbulent velocity fluctuations.
## 2.4. Fluctuating turbulent pressureIn the process of the longitudinal tube wave excitation the tube is compressed symmetrically by the external turbulent pressure. This turbulent pressure consists of a time averaged term which augments the external gas pressure and a fluctuating term which gives rise to longitudinal tube waves. Assuming isotropic homogeneous turbulence, the fluctuating turbulent pressure (see Eq. 7) is given by Here after Eq. (2) we should have used uncorrelated values of , and , but as the spectrum already consists of a large number of independent partial waves, the excitation with independent and will not bring anything new. Combining Eqs. (7) and (20), one sees that the external pressure fluctuations are translated into pressure and magnetic field fluctuations inside the tube. As our Lagrangian wave code uses a velocity boundary condition, we make use of the amplitude relations for longitudinal tube waves (Herbold et al. 1985, Eq. 59) to translate the gas pressure fluctuations into internal longitudinal velocity fluctuations where is the sound speed, the tube speed, is the Alfvén speed and is the ratio of specific heats. Note at this point that the amplitude relation of Eq. (21),
like relations for pure acoustic waves, is only valid for frequencies
large compared to the cut-off frequency for longitudinal tube waves;
this frequency was first introduced by Defouw (1976; see also Roberts
& Webb 1979). For wave frequencies close to this cut-off,
phase-shifts between and
enter Eq. (21), which approach
90 The remaining problem is now to connect in Eq. (7) the internal gas pressure fluctuation with the external turbulent pressure fluctuation . Here we follow Hasan (1997). Neglecting gravity and assuming that the squeezing of the tube occurs over a considerable height span (that is, taking his ) we find from his Eq. (35) from which, as is our , we derive Here is the plasma . Note that for , that is, when the magnetic field vanishes, this relation goes to the correct limit while for , when the tube becomes entirely empty of gas, (), the expected limit is reached. Note in passing that Eqs. (21) and (22) valid for longitudinal tube waves are very similar to the familiar amplitude relations of pure sound waves. Finally Eqs. (20), (23) allow to write Eq. (21) as Fig. 2 shows (solid) the internal longitudinal velocity which corresponds to the applied external velocity displayed in the same figure. The close relationship of the two velocities is apparent. While obviously oscillates around zero (see Fig. 2), is clearly very one-sided. However, as , the time-average of must also be zero. To check the oscillatory character of , we calculated = ( is shown solid in Fig. 2) and found that indeed oscillates around zero. The oscillations of are not symmetric because large absolute velocities produce large turbulent pressure fluctuations and thus large positive velocity spikes , while small velocities lead to small turbulent pressures and negative velocities . The negative velocity spikes are much smaller because, according to Eq. (24), they are at most proportional to . ## 2.5. Longitudinal wave energy fluxesFor the instantaneous longitudinal tube wave flux at height
Since we want to compare wave energy fluxes generated at different
heights in the tube it is appropriate to normalize the fluxes to the
solar surface, that is, to the height , for
which outside the tube we have . At this height
the tube radius has a cross-section , while at
other heights it has the cross-section where as above the overline indicates time averaging. The wave energy spectrum is calculated by taking the Fourier transform of by using the procedure outlined in Sect. 2.3. The expressions (25), (26) for the longitudinal wave energy flux
can be shown to be correct by introducing the wave solutions
(Eqs. (A.1) to (A.6)) of Herbold et al. (1985) into the energy
conservation equation for longitudinal tube waves (Hasan 1997,
Eqs. (30) to (32)) from which the correct dispersion relation for
these waves is recovered. We are very grateful to S. Hasan (1997,
private communication) and Y. Zhugzhda (1998, private communication)
for pointing out to us that the expression for
the longitudinal wave energy flux used in our previous papers
(Musielak et al. 1989, 1995) is inappropriate, as it mixes a wave flux
and a 'wave wind'. The total wave energy flux integrated over the tube
cross-section is where At this point we summarize the somewhat circuitous logic of the
procedure to obtain the generated longitudinal wave flux. The external
turbulent motions of the convection zone generate pressure
fluctuations (here is
the squeezing height and © European Southern Observatory (ESO) 1998 Online publication: September 8, 1998 |