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

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2. Basic assumptions and the method

2.1. Tube model

We 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 [FORMULA] and a field strength of [FORMULA] at the height where externally the optical depth is [FORMULA]. 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 [FORMULA] and [FORMULA]. 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 [FORMULA], we select excitation heights at [FORMULA], 10 and 100 in optical depth measured outside the tube. Because the Vernazza et al. model goes only to [FORMULA], we extended that model by fitting a convection zone model obtained with the code of Bohn (1981, 1984).

2.2. Pressure balance

To 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 [FORMULA] 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 [FORMULA] 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 H, and the subsequent inertial breakup cascade to sizes as small as [FORMULA].

Following Musielak et al. (1995) we write the pressure balance between the tube and the external medium as

[EQUATION]

where p is the internal gas pressure, B the magnetic field strength in the tube, [FORMULA] the external gas pressure and

[EQUATION]

is the time-dependent external turbulent pressure. [FORMULA], [FORMULA], [FORMULA] are the turbulent velocities in x, y, z-directions which are functions of position [FORMULA] and time t. Upon time averaging one gets

[EQUATION]

For the case of homogeneous isotropic turbulence there is no longer a dependence on [FORMULA] due to the assumed homogeneity and for the three spatial components one has

[EQUATION]

because of the assumed isotropy. Here [FORMULA] 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 [FORMULA] 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 [FORMULA] 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

[EQUATION]

After time-averaging of Eq. (1) and assuming homogeneous isotropic turbulence we obtain

[EQUATION]

where [FORMULA] is the average internal gas pressure and [FORMULA] 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 [FORMULA], [FORMULA] and [FORMULA], we find [FORMULA]. Note that because of this, [FORMULA] has been neglected relative to [FORMULA] 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

[EQUATION]

where [FORMULA] is the internal gas pressure perturbation and [FORMULA] the magnetic field strength perturbation.

2.3. Turbulent velocity fluctuations

Following 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

[EQUATION]

where [FORMULA] is the velocity amplitude of these waves, it will be determined by the turbulent energy spectrum (see below), [FORMULA] is the wave frequency that ranges from 2.9 to 290 mHz with step [FORMULA] mHz, and [FORMULA] is an arbitrary but constant phase angle with [FORMULA] being a random number in the interval [0, 1]. Note that these limits correspond to [FORMULA] and [FORMULA], where [FORMULA] 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 [FORMULA] in time because of the random phases [FORMULA] of the otherwise sinusoidal waves.

To determine the velocity amplitudes [FORMULA], we follow the procedure outlined in Paper I. We begin with time averaging [FORMULA] to find

[EQUATION]

from which using Eqs. (3) and (4) we have

[EQUATION]

As described in Paper I as well as by Musielak et al. (1995) the turbulent energy spectrum is normalized to

[EQUATION]

where [FORMULA] and [FORMULA] are the spatial and temporal components of the turbulent energy spectrum, respectively. Here the factor [FORMULA] results from the contributions to the kinetic energy from the three spatial directions. We now use Eq. (10) to write [FORMULA] as

[EQUATION]

from which we have

[EQUATION]

with

[EQUATION]

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 [FORMULA] 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 [FORMULA] and [FORMULA] is a factor of two higher than that found when shaking with [FORMULA] 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 [FORMULA]. Note that contrary to the longitudinal wave case, the fluctuations [FORMULA] 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 [FORMULA] and a modified Gaussian frequency factor [FORMULA]. The extended Kolmogorov spatial spectrum is given by

[EQUATION]

where the factor [FORMULA] is determined by the normalization condition

[EQUATION]

and [FORMULA], where H is the scale height. [FORMULA] is the wave number where the turbulent cascade ends, for which Theurer et al. (1997) have estimated values as high as [FORMULA] where [FORMULA], for our purposes it is sufficient to take [FORMULA]. The modified Gaussian frequency factor is described by

[EQUATION]

where [FORMULA] is computed from

[EQUATION]

Knowing [FORMULA] and [FORMULA], we may now use Eqs. (13), (14) to calculate the velocity amplitude [FORMULA] of the partial waves. The resulting amplitudes [FORMULA] obtained in this way are shown in Fig. 1.

[FIGURE] Fig. 1. The velocity amplitude [FORMULA] for each partial wave of the frequency [FORMULA], computed using Eq. (13) for a prescribed rms turbulent velocity of [FORMULA].

The results presented in Fig. 1 can now be used to compute the turbulent velocity fluctuations [FORMULA] at the squeezing height [FORMULA] using Eq. (8). Fig. 2 shows [FORMULA] (dashed) at the squeezing height [FORMULA] as a function of time t up to [FORMULA]. This corresponds to [FORMULA]. It is seen that due to the many partial waves with random phases, [FORMULA] is very stochastic in nature. Since the RHS of Eq. (8) does not represent a Fourier decomposition of [FORMULA] in time, and since [FORMULA] can be treated as a non-periodic signal restricted in time to [FORMULA] s, we may now perform Fourier analysis of this signal by calculating its Fourier transform, [FORMULA], and the corresponding power spectrum, [FORMULA]. To normalize [FORMULA], we compute

[EQUATION]

and introduce the normalized power spectrum of the turbulent velocity [FORMULA] by defining [FORMULA]; in the following we call [FORMULA] simply the velocity spectrum. This spectrum has dimension of [FORMULA] and is plotted in Fig. 3. The presented spectrum was obtained from temporal Fourier analysis of the last 2048 s of the velocity [FORMULA], 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.

[FIGURE] Fig. 2. Horizontal fluctuating turbulent velocity [FORMULA] (dashed) computed from Eq. (8) as well as the internal longitudinal velocity [FORMULA] (solid) computed from Eq. (24) for the velocity amplitudes given in Fig. 1. Also shown are the zero velocity line (thin solid) and the time average [FORMULA] (solid) which from times [FORMULA] is essentially identical to the zero velocity line (which it should be).

[FIGURE] Fig. 3. Normalized power spectrum of the fluctuating turbulent velocity [FORMULA] shown in Fig. 2. This velocity spectrum is shown as a function of circular frequency [FORMULA].

2.4. Fluctuating turbulent pressure

In 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 [FORMULA] which augments the external gas pressure [FORMULA] 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

[EQUATION]

Here after Eq. (2) we should have used uncorrelated values of [FORMULA], [FORMULA] and [FORMULA], but as the spectrum already consists of a large number of independent partial waves, the excitation with independent [FORMULA] and [FORMULA] 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

[EQUATION]

where [FORMULA] is the sound speed, [FORMULA] the tube speed, [FORMULA] is the Alfvén speed and [FORMULA] 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 [FORMULA] and [FORMULA] enter Eq. (21), which approach 900. However, as the Lagrangian code permits to specify only one physical quantity (the velocity [FORMULA]) as boundary condition we are not affected by this phase-shift. Note that in an explicit time-dependent Lagrangian wave code the three unknowns (velocity and two thermodynamic variables) are exactly determined by the relations along the three characteristics (C+, C-, C0). At the top and bottom boundaries one characteristic is missing and is replaced by boundary conditions while the other two unknowns (e.g. [FORMULA] and S) are determined by relations along the remaining two characteristics (e.g. C- and the fluid path C0) by which the phase-shift is automatically computed and taken into account. This is particularly gratifying as for an acoustic wave spectrum (as opposed to a monochromatic wave) the influence of the phase-shift would be difficult to determine analytically.

The remaining problem is now to connect in Eq. (7) the internal gas pressure fluctuation [FORMULA] with the external turbulent pressure fluctuation [FORMULA]. 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 [FORMULA]) we find from his Eq. (35)

[EQUATION]

from which, as [FORMULA] is our [FORMULA], we derive

[EQUATION]

Here [FORMULA] is the plasma [FORMULA]. Note that for [FORMULA], that is, when the magnetic field vanishes, this relation goes to the correct limit [FORMULA] while for [FORMULA], when the tube becomes entirely empty of gas, ([FORMULA]), the expected limit [FORMULA] is reached. Note in passing that Eqs. (21) and (22) valid for longitudinal tube waves are very similar to the familiar amplitude relations [FORMULA] of pure sound waves. Finally Eqs. (20), (23) allow to write Eq. (21) as

[EQUATION]

Fig. 2 shows (solid) the internal longitudinal velocity [FORMULA] which corresponds to the applied external velocity [FORMULA] displayed in the same figure. The close relationship of the two velocities is apparent. While [FORMULA] obviously oscillates around zero (see Fig. 2), [FORMULA] is clearly very one-sided. However, as [FORMULA], the time-average of [FORMULA] must also be zero. To check the oscillatory character of [FORMULA], we calculated [FORMULA]=[FORMULA] ([FORMULA] is shown solid in Fig. 2) and found that [FORMULA] indeed oscillates around zero. The oscillations of [FORMULA] are not symmetric because large absolute velocities [FORMULA] produce large turbulent pressure fluctuations [FORMULA] and thus large positive velocity spikes [FORMULA], while small velocities [FORMULA] lead to small turbulent pressures and negative velocities [FORMULA]. The negative velocity spikes are much smaller because, according to Eq. (24), they are at most proportional to [FORMULA].

2.5. Longitudinal wave energy fluxes

For the instantaneous longitudinal tube wave flux at height z we take (Herbold et al. 1985, Eq. (58), Hasan 1997, Eq. (32))

[EQUATION]

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 [FORMULA], for which outside the tube we have [FORMULA]. At this height the tube radius has a cross-section [FORMULA], while at other heights it has the cross-section A. The normalized time-averaged flux with [FORMULA] can be written as

[EQUATION]

where as above the overline indicates time averaging. The wave energy spectrum is calculated by taking the Fourier transform of [FORMULA] 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 [FORMULA] 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 [FORMULA] where H is the enthalpy. By convention (Landau & Lifshitz 1959, Sect. 64) the first two terms arising from the linearized version of the mass flux [FORMULA] are usually not included in the wave flux. However, note that the additional term [FORMULA] used by Musielak et al. arises from the 'mass flux' term [FORMULA] and gives a contribution even when there is no net flow velocity [FORMULA], it thus will always be present in a wave. As in the Musielak et al. papers [FORMULA] was [FORMULA], the analytical longitudinal tube wave flux was overestimated by a factor of 1.38. Such a correction is small against the overall uncertainties in these type of calculations.

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 [FORMULA] (here [FORMULA] is the squeezing height and t the time) which can be translated into internal gas pressure fluctuations [FORMULA]. As our time-dependent longitudinal wave code requires velocity fluctuations [FORMULA] as boundary condition, we use the amplitude relations for longitudinal tube waves to replace [FORMULA] by [FORMULA]. One might think that, as we have [FORMULA] and [FORMULA], one could directly compute the longitudinal tube wave flux from Eq. (26) without using the wave code. This is wrong, as in such a product the phase-shift (time-delay) between the quantities would not be taken into account. For monochromatic waves this phase-shift is known, but not for a wave spectrum. In the wave code we take [FORMULA] as boundary condition, which is applied somewhat time-delayed but has the correct magnitude. The wave code then self-consistently generates new gas pressure fluctuations [FORMULA] which differ from the above mentioned [FORMULA], but now have the correct phase shift against the velocity fluctuations [FORMULA]. The wave code thus ensures that the quantities [FORMULA] and [FORMULA] have the correct phase shift required for the wave flux determination in Eqs. (25) and (26).

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Online publication: September 8, 1998
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