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Astron. Astrophys. 332, 314-324 (1998)

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5. Brillouin zones and vibrational waves

Even the simplest approximate dispersion equations for oblique modes are cumbersome due to the large number of modes: distorted acoustic modes, vibrational modes, and the turbulent sound mode. Therefore we present here numerical solutions of the dispersion equation (A19).The results are presented in the form [FORMULA] -diagrams, which are divided into so-called Brillouin zones separated by [FORMULA]. The reason of this separation is seen from the solution (42), which changes form one type to another when [FORMULA] is changed by 1. Therefore the first Brillouin zone already reveals all relevant physical effects.

5.1. Thermally structured atmosphere

A rather simple [FORMULA] -diagram is obtained in the case of a thermally structured atmosphere with [FORMULA]. It is well described by the approximate dispersion equation (43), and is presented in Fig. 1a. The solid lines correspond to horizontal propagation, [FORMULA]. The solid line passing through the origin is the acoustic mode that in the uniform atmosphere ([FORMULA]) is the straight line [FORMULA]. In the structured atmosphere, the behaviour of the acoustic branch changes drastically near the point where [FORMULA] and [FORMULA]. Beyond this point the phase velocity decreases with increasing [FORMULA], and the directions of the group and phase velocities are opposite. The second solution of the dispersion equation is a vibrational mode with cut-off frequency [FORMULA] ; it is shown by the solid curve passing through the point [FORMULA] on the ordinate axis. In the limit of a uniform atmosphere this curve is the straight line [FORMULA], which follows from (46) for the case of horizontal propagation, [FORMULA]. In the present case of a structured atmosphere ([FORMULA]) the curve changes its direction near the point [FORMULA], [FORMULA] ; its second part follows exactly the route taken by the acoustic branch in the case of a uniform atmosphere. Thus, the second parts of both the acoustic and vibrational modes are mutual extensions of the first parts of the respective other branch in the solution of the dispersion equation. There is a frequency gap between the two curves, where horizontal wave propagation is impossible. Waves with frequencies within this gap are evanescent. Such behaviour is inherent to waves in a periodic structure. The classical example is the behaviour of electrons in the periodic potential of a crystal lattice.

[FIGURE] Fig. 1. The ([FORMULA])-diagrams for acoustic and vibrational modes in a thermally structured atmosphere ([FORMULA]). Upper diagram: Modes propagating at [FORMULA] (solid), [FORMULA] (dotted), and [FORMULA] (dashed). Lower diagram: Acoustic modes with [FORMULA] (solid) and [FORMULA] (dashed); vibrational modes with [FORMULA] (solid) and [FORMULA] (dashed).

The [FORMULA] -diagram of Fig. 1a for the horizontal wave propagation is the conventional Brillouin zone diagram, and [FORMULA] marks the border between the first and second Brillouin zones; the frequency gap between the two solids curves corresponds to the energy barrier which can be passed only by means of the tunnel effect.

The analogy with the Brillouin zones is exact for the case of horizontal wave propagation. To explore the general case of oblique propagation let us consider waves with a fixed direction of propagation. The dotted and dashed lines in Fig. 1a correspond to waves with [FORMULA] equal to [FORMULA] and [FORMULA], respectively. The angle [FORMULA] is peculiar for the chosen value of [FORMULA], because the curves of acoustic and vibrational modes touch each other; both pass continuously through the point [FORMULA]. The frequency gap disappears, and acoustic waves pass through the border of the Brillouin zones without reflection.

For angles [FORMULA] it is possible to continuously prolong the acoustic and vibrational branches by choosing different angles of propagation for the two branches. The case where the vibrational mode propagates horizontally, while the acoustic wave is running at an angle [FORMULA], is shown by the solid curves of Fig. 1b. In this case the acoustic wave does not meet the potential barrier at the border between the Brillouin zones, but changes from oblique propagation in the first Brillouin zone to horizontal propagation in the second Brillouin zone. Partial reflection occurs due to refraction at the border of the Brillouin zones. If in the first Brillouin zone the propagation angle of the acoustic waves is smaller than [FORMULA] but larger than [FORMULA], then the frequency gap between the acoustic and vibrational modes is removed by the change in the direction of propagation. Acoustic waves that propagate at angles [FORMULA] tunnel through the border between the Brillouin zones and are transformed to horizontally propagating waves in the second Brillouin zone; in this way they meet the smallest possible potential barrier. The special angle [FORMULA] does not depend noticeably on the value of [FORMULA]. In the solar convection zone we should have [FORMULA].

The width of the potential barrier for the horizontally propagating waves is about [FORMULA] in terms of the dimensionless frequency [FORMULA]. Horizontally propagating acoustic waves passing through the Brillouin zone border undergo the strongest reflection, because the potential barrier is largest in this case. The reflection decreases with increasing angle of propagation as the potential barrier becomes narrower and disappears at [FORMULA]. However, reflection does not entirely cease at this angle, because the effect of changing the direction of propagation precludes the free passage of the Brillouin zone border. Only when the propagation angle reaches [FORMULA] the acoustic wave crosses the Brillouin zone without reflection. Thus, the effects of the potential barrier and the refraction at the Brillouin zone border favour waves with frequencies [FORMULA].

5.2. Vibrational waves

In order to explain the upper part, [FORMULA], of the [FORMULA] -diagram the analogy with solid state physics will be used. The microscopic theory of acoustic waves in a crystal lattice shows that along with acoustic waves a branch of vibrational waves exists. The vibrational phonons are elastic vibrations of the atoms around their equilibrium sites in the lattice. The propagation of these vibrations is considered as the vibrational waves. In the case of a continuous periodically structured atmosphere similar branches appear, as was shown in the preceding section. This means that acoustic energy can be transported in a structured atmosphere not only by acoustic waves but also by vibrational waves. In the lower part of the [FORMULA] -diagram ([FORMULA]) the vibrational waves are very different from the acoustic waves, because the directions of their phase and group velocities are opposite. Fig. 1a shows the branches of acoustic and vibrational waves for [FORMULA]. The behaviour of the two modes is typical for coupled oscillators. These curves are drawn for an imposed value of the propagation angle.

Fig. 1b shows the acoustic branch for [FORMULA] and the vibrational branch for [FORMULA]. The latter turns upwards after crossing the acoustic curve, so the behaviour of the two branches is similar after the crossing. For a prescribed point of the acoustic branch we may determine the angle for which the vibrational branch crosses. If the crossing happens on the ascending part of the vibrational branch, then coupling and exchange of energy between the two wave modes are possible. The boundary of the region where the coupling of acoustic and vibrational phonon branches occurs is defined by the points where the acoustic branches cross the minimum of the vibrational curve.

The border line connects the points [FORMULA] and [FORMULA] in the [FORMULA] -diagram of Fig. 1a. The coupling of acoustic and vibrational modes as well as the crossing of the Brillouin zone border needs a special treatment, because all modes are anisotropic and, consequently, the directions of the phase and group velocities do not coincide. But this is beyond the scope of this paper.

To give a comprehensive picture of the wave branches in a structured atmosphere the modes with space period [FORMULA] must be considered as well. This will be done for the moving atmosphere. One should not go too far with the analogy between pressure waves in the structured atmosphere and elastic oscillations of the crystal lattice, because the physics and the governing equations are rather different. But it is worth mentioning that the frequency [FORMULA] plays a similar role as the Debye frequency, and that [FORMULA] corresponds to the translation vector of the lattice.

5.3. Moving atmosphere

Fig. 2 shows the ([FORMULA])-diagram for the moving atmosphere with [FORMULA] and [FORMULA]. The diagrams of Fig. 1a and Fig. 2 show many similarities, but also a number of differences. For horizontal propagation there is no frequency gap between the vibrational and acoustic branches of Fig. 2. This is because flows that are perpendicular to the direction of wave propagation have no influence, and the coefficients in Eq. (15) do not depend on V. The frequency gap does exist for [FORMULA]. But this does not mean that the waves meet a potential barrier at the Brillouin zone border; they only undergo refraction in the same way as in a thermally structured atmosphere for [FORMULA] and [FORMULA]. The only difference is that in the moving atmosphere the refraction at the Brillouin zone border increases with the propagation angle [FORMULA], while in a thermally structured atmosphere it decreases and disappears for [FORMULA].

[FIGURE] Fig. 2. The ([FORMULA])-diagram for a moving atmosphere ([FORMULA]) for acoustic and vibrational modes with [FORMULA] (solid), [FORMULA] (dotted), [FORMULA] (dashed), and for the turbulent sound mode with [FORMULA] (dash-dotted).

The general case of a moving thermally structured atmosphere ([FORMULA], [FORMULA]) is shown in Fig. 3. The potential barrier for horizontally propagating waves appears due to the temperature structure, while the refraction for [FORMULA] results from the flows. The potential barrier and the refraction at the Brillouin zone border do not appear for a propagation angle [FORMULA]. All three diagrams of Figs. 1a, 2, and 3 show the same behaviour of the acoustic and vibrational branches for propagation angles [FORMULA], where linear coupling of these modes occurs. Fig. 4 shows a [FORMULA] - diagram for vibrational branches with period [FORMULA], defined by the dispersion equation (A20). The branches start at [FORMULA], where the central frequency of the first band gap, [FORMULA], is


There is no acoustic branch in this diagram, because acoustic waves, by definition, correspond to [FORMULA] in (42), that occur for solutions with space period [FORMULA] only. The vibrational waves with period [FORMULA] show coupling at the borders of the Brillouin zones.

[FIGURE] Fig. 3. The ([FORMULA])-diagram for a moving thermally structured atmosphere ([FORMULA], [FORMULA]) for acoustic and vibrational modes with [FORMULA] (solid), [FORMULA] (dash-dot-dotted), [FORMULA] (dotted), [FORMULA] (dashed), and for turbulent sound branches (dash-dotted).

[FIGURE] Fig. 4. The ([FORMULA])-diagram for vibrational modes period [FORMULA] with [FORMULA] (dotted), 3(dashed), and a turbulent sound mode (solid) with [FORMULA] in a moving thermally structured atmosphere ([FORMULA], [FORMULA]).

5.4. Turbulent sound

Turbulent sound branches appear in the [FORMULA] -diagrams of wave modes with space period [FORMULA] (Figs. 2 and 3) and [FORMULA] (Fig. 4). Eqs. (35)-(36) are valid for oblique propagation as well, but Eq. (37) should be replaced by


where [FORMULA] is still the component of the phase velocity along the direction of the flows: [FORMULA]. After averaging over the space period [FORMULA] Eq. (48) can be replaced by the approximate relation [FORMULA] ; this can be rewritten as a vector condition [FORMULA] that characterizes transversal waves. Thus, the turbulent sound is an almost transversal wave for [FORMULA]. We recall that the average of [FORMULA] is zero. The important issue for turbulent sound, as disclosed above, is whether the phase velocity along the flows is less than [FORMULA]. Figs. 5a, b show the dependence of the vertical phase velocity on the parameters [FORMULA] and [FORMULA], and on the propagation angle. The vertical phase velocity of the turbulent sound can be larger or smaller than [FORMULA], depending on the parameters. Singularities in the vertical and horizontal velocity components appear as a consequence of the inviscid treatment. If viscosity is taken into account, the frequencies become complex and the singularities are removed.

[FIGURE] Fig. 5. The vertical phase velocity [FORMULA] for turbulent sound in a moving thermally structured atmosphere with [FORMULA] and [FORMULA]. Upper diagram for even solutions, lower diagram for odd solutions; [FORMULA] (dotted), [FORMULA] (solid), [FORMULA] (dashed).

However, in the case of weak viscosity the wave energy is concentrated in thin layers, and small-scale shear flows appear near the locations where the singularities occur in the inviscid case. This means that even for a rather small mean wave amplitude the nonlinear effects should be taken into account. It seems that real turbulence has to generate turbulent sound among other modes. But the turbulent sound produces small-scale shear flows, which do not arise from other wave modes. This small-scale shear may in turn generate small-scale turbulence. Thus, turbulent sound is a likely mechanism of generating small-scale turbulence from large-scale turbulence.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998