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Astron. Astrophys. 354, 296-304 (2000)

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3. Analysis of the results and asymptotic solution

Before we solve these equations it is instructive to consider briefly the known cases of waves in one-dimensional media. For a plane wave in a stratified atmosphere the hydrodynamic equations admit only acoustic waves, but no internal gravity waves. The solution for impulsive excitation of a disturbance (see KRBM) gives an upward traveling pulse that amplifies exponentially with height; it is followed by a wake that oscillates at the acoustic cutoff period. By contrast, a plane acoustic wave due to an impulse in a homogeneous medium, for which the atmosphere of our every-day experience is a natural approximation, shows only the signal, but no wake. It is interesting to investigate the propagation of a disturbance in a 3D medium and to highlight the differences with the 1D cases. In order to gain some preliminary insight we study the asymptotic behavior of the solution in 3D.

3.1. Asymptotic analysis

Following Whitham (1974) we describe the solution in terms of a slowly varying wavetrain (i.e., with little variation in a typical wavelength and period), writing, for example, the pressure perturbation as


where the amplitude [FORMULA] and the phase [FORMULA] are slowly varying functions of position and time. We can then define a local wavenumber, [FORMULA], and a local frequency, [FORMULA], to obtain for the wavenumber [FORMULA] the equation of motion


(see Eq. 11.44 in Whitham, 1974) where [FORMULA] is the corresponding group velocity vector. The equation is valid in the limit [FORMULA].

According to Eq. (12), the wavenumber [FORMULA] is constant along group lines, which are defined by the equation, and each value of [FORMULA] propagates with the corresponding constant group velocity [FORMULA]. From the initial pulse, given by the superposition of modes encompassing the whole spectrum of wavenumbers, each wavenumber then propagates with its own, corresponding group velocity. A particular wavenumber found at position [FORMULA] at time t can be obtained by solving the pair of equations


for the wavenumber components.

The group velocity components in the vertical and horizontal directions are given by



where [FORMULA] is a root of Eq. (6) with the plus sign.

It is evident, from the dispersion relation (5) or from the system (7)-(10), that the complete solution is given by the superposition of acoustic and gravity waves. Here we confine our analysis to the acoustic waves.

The group velocity as a function of the moduli of the wavenumbers for the vertical and horizontal directions is plotted in Fig. 1, which shows that the behavior of the two curves is different at small and intermediate wavenumbers: in the horizontal direction the group velocity remains small up to [FORMULA] and then increases rapidly to the asymptotic value of unity, and in the vertical direction the growth begins steeply at [FORMULA] and then continues more gradually to the asymptotic value.

[FIGURE] Fig. 1. Group velocity vs vertical (solid line) and horizontal (dashed line) wavenumbers.

The evolution of the distribution of wavenumbers in space is described by Eqs. (13) and (14), with the group velocity [FORMULA] given by Eqs. (15) and (16). These equations can be solved for the components [FORMULA] and [FORMULA] in the vertical and horizontal directions as functions of [FORMULA] and [FORMULA]; the solutions are shown in Fig. 2. The two panels reflect the different behavior, described above, of the group velocity as a function of wavenumber; in both cases, small wavenumbers, which have low group velocity, are found close to the origin, while large wavenumbers, which have group velocity close to the speed of sound, are found near the pulse. At intermediate distances, the distribution of k values is much broader in the vertical than in the horizontal direction. This has immediate consequences for the nature of the solution: since the value of the wavenumber found at a particular position implies also the wavelength of the oscillation, one may expect (see Fig. 6 below) that in the horizontal direction the oscillations behind the front have almost constant wavelength, while in the vertical direction the wavelength gradually increases from the head of the wave towards the origin.

[FIGURE] Fig. 2. Vertical (upper panel) and horizontal (lower panel) distributions of wavenumbers as a function of [FORMULA] or [FORMULA], where t is fixed and [FORMULA].

Combining the phase velocity from the dispersion relation with the distribution of wavenumbers obtained from the evolution Eqs. (13) and (14) one can follow the propagation of points of constant phase, for example of a maximum. When a particular maximum is located close to the origin ([FORMULA]) it is characterized by low values of the wavenumber, and thus high phase velocity. As it moves away from the origin, its local wavenumber as well as its frequency increase (in the vertical direction, [FORMULA]), while its propagation velocity decreases (Fig. 2). Thus, as a particular maximum travels away from the origin and towards the head of the wave it is increasingly characterized by high-frequency components (which become important for shock formation in the nonlinear regime). Fig. 3 presents this behavior in more detail; it shows the phase velocity of three consecutive maxima as functions of distance from the source for the vertical (upper panel) and horizontal (lower panel) directions. In the horizontal direction the phase velocity is practically equal to the sound speed nearly throughout the whole region, and for all maxima. In the vertical direction the behavior is more complicated: Except for the head of the wave (the first maximum in Fig. 3, upper panel), which propagates at the sound speed, the later maxima have high phase velocity over increasingly extended height ranges. They therefore travel at increased phase velocity (Fig. 6, upper panel; see also KRBM Fig. 2), especially near the origin (where [FORMULA] as [FORMULA]), until they approach the head of the wave. As a consequence of this difference in behavior between the two directions, the surfaces of constant phase behind the wave front travel a longer distance in the vertical than in the horizontal direction and thus acquire oval shape (see Fig. 5 below).

[FIGURE] Fig. 3. Phase velocity for three consecutive maxima (solid: first maximum, dashed and dash-dotted: second and third maxima) in the vertical (upper panel) and horizontal (lower panel) directions.

3.2. Analysis of the results

We calculate the Fourier integral (7) numerically, taking the symmetry with respect to the z-axis into account, and interpret the results with the aid of the asymptotic analysis. Fig. 4 shows a representative solution for a pressure pulse at x[FORMULA]z[FORMULA]t[FORMULA]0, displaying the reduced pressure distribution, i.e., without the vertical amplification factor [FORMULA], in a vertical plane containing the origin, at a time t corresponding to about three times the acoustic cutoff period. As discussed above, the solution is given by the superposition of acoustic and internal gravity modes. The pressure variation due to internal gravity waves, which appears as radial stripes, remains confined to the vicinity of the origin since their group velocity is lower than that of acoustic waves. It is also highly anisotropic: internal gravity waves are excluded from the purely vertical direction, and in the horizontal direction their group velocity reaches only about 60% of the sound speed. Because of their higher group velocity acoustic waves become more prominent at greater distance from the origin. They are seen alone in Fig. 5, which shows that the head of the wave forms a spherical surface, with radius equal to [FORMULA].

[FIGURE] Fig. 4. Acoustic and internal gravity waves at [FORMULA] times the acoustic cutoff period, in a vertical plane through the origin. Wave crests and troughs are displayed by bright and dark shading, respectively.

[FIGURE] Fig. 5. Acoustic waves at [FORMULA] times the acoustic cutoff period in the vertical plane.

Recall that in the one-dimensional case the solution consists of a wave front followed by a wake that oscillates at a period approximately equal to the acoustic period ([FORMULA]). In the three-dimensional case this property is found again in the vertical direction (compare Fig. 6, upper panel, and KRBM, Fig. 2), but now also in the horizontal direction, albeit with much reduced amplitude.

[FIGURE] Fig. 6. Vertical (upper panel) and horizontal (lower panel) cuts of the pressure perturbation amplitude (in arbitrary units) at [FORMULA] times the acoustic cutoff period.

The oval shape of the pressure extrema is related to the distribution of [FORMULA]values in space (Fig. 2). As seen in Fig. 6 (lower panel), the pressure amplitude in the horizontal direction has wave crests that are nearly equidistant, consistent with the narrow distribution of [FORMULA]values and, consequently, wavelengths, through most of the range of [FORMULA]values. By contrast, the distance between pressure extrema in the vertical direction increases with distance from the head of the wave due to the corresponding increase in the phase velocity, from the sound speed at the head of the wave to larger values near the origin (Fig. 2), (and to infinitely large values in the asymptotic limit of [FORMULA]).

Since the disturbance is due to a point source, the wave starts out as a spherical wave. The pulse retains that shape but as can be seen in Fig. 5, the amplitude of the pulse increases from the vertical direction (both positive and negative) towards the horizontal direction.

The three-dimensional solution allows us to investigate the variation of the perturbation amplitude with direction relative to the vertical. We can look to this variation from two different perspectives: one possibility is that of considering the variation of the amplitude of a maximum at a fixed time and the other is that of considering it at a fixed height (see Fig. 7). In an astrophysical context this second case would correspond to looking at a fixed optical depth and therefore to a particular spectroscopic signature of the wave. In the first case (fixed time), the variation of the amplitude has two parts, the intrinsic variation of the reduced function, and the variation of the exponential factor, [FORMULA], that compensates for the exponential dependence of the background mass density on height and converts the reduced variables into the physical variables. But at a fixed height the distinction between reduced and physical variables is irrelevant.

[FIGURE] Fig. 7. Upper panel represents the positions of a wave crest at different times and their intersections with height of [FORMULA]. Lower panel shows a wave crest whose apex has reached the height of [FORMULA] at time [FORMULA].

We consider first the case of fixed height: Fig. 8 shows the variation of the amplitude of the pulse and of the first three maxima of the wave train as they cross the height of [FORMULA], as functions of the angle measured relative to the vertical direction. Note that, in the actual solar chromosphere, nonlinear effects would already be present at this height for typical initial perturbations; however, our motivation is to study the basic linear behavior of the wave propagation. Since the various points of the pulse, for example, reach the target height of [FORMULA] with position x and hence angle increasing with time (cf.. Fig. 7), the variation of the pressure amplitude reflects partly the spherical decay of the pulse with distance r from the origin ([FORMULA]), and partly the increase of the amplitude with angular distance from the vertical, as seen in Fig. 5. The net effect is a relatively slow decrease of the pressure amplitude, which reaches a factor of 2 at [FORMULA]. The decay is much faster for the wave crests in the wake, reaching a factor of 10 at [FORMULA] for the first and second maxima, and at [FORMULA] for the third maximum. The wave profile evidently narrows considerably with time.

[FIGURE] Fig. 8. Angle dependence of the (reduced) pressure amplitude relative to the vertical direction ([FORMULA]). The values are for the fixed height of [FORMULA] and refer to the pulse (-), and the 1st ([FORMULA]), 2nd ([FORMULA]) and 3rd ([FORMULA]) maxima of the wake.

A picture complementary to Fig. 8 is shown in Fig. 9, with the angle variation of the pressure perturbation at the times when the apices of the pulse and, subsequently, of the maxima in the wake reach the height of [FORMULA]. The upper panel gives the angle dependence of the reduced pressure (at the time when the apex reaches [FORMULA]), and the lower panel that of the physical pressure. The upper panel confirms the impression from Fig. 5 that the (reduced) pressure amplitude of the pulse grows with zenith angle. The crests of the wake weaken with separation from the vertical; they also have significantly lower amplitude. The influence of the exponential factor on the physical pressure is clearly seen in the lower panel, and it results in a much faster drop of the amplitudes with zenith angle. This is especially true for the maxima in the wake for which the oval shape causes a much faster decay than for the pulse and therefore a narrowing of the wave profile with the order of the maximum. For the pressure pulse (Fig. 9, lower panel), the decay in a half-angle of [FORMULA] is by nearly two orders of magnitude, and for the maxima in the wake, by nearly three orders of magnitude. Note that this decay reflects to a large extent the shape of the wave crests and is much less dramatic when the points along the crests have reached the target height, as described above.

[FIGURE] Fig. 9. Variation with zenith angle of the pulse (-), and the first three maxima of the wake ([FORMULA], resp.) at the instants when the apex for each feature has reached the target height of [FORMULA]; the upper panel shows the reduced pressure, the lower panel includes the exponential height factor.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000