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

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3. Vertical wave propagation

Before considering the general case of oblique wave propagation we first treat the special case of propagation along the vertical layers of a periodic atmospheric structure. This case serves to explain the approach and to introduce some definitions. In addition, in order to separate the diverse effects, we first investigate atmospheres having either temperature or velocity fluctuations.

3.1. Thermally structured atmosphere

A motionless atmosphere ([FORMULA]), structured only by fluctuations of temperature, hereafter will be called a thermally structured atmosphere. With [FORMULA] Eq. (19) reads

[EQUATION]

If the sound speed is a sinusoidal function of the horizontal coordinate,

[EQUATION]

Eq. (20) reduces to

[EQUATION]

which will be solved for arbitrary values of [FORMULA]. This equation is not the Mathieu equation, because the second and the first derivatives have periodic coefficients. According to the Floque theorem (Ince 1944) Eq. (22) has four different solutions

[EQUATION]

[EQUATION]

For [FORMULA] the solution has the period [FORMULA], which coincides with the period of the atmospheric structure defined by (21), while for [FORMULA] the period is [FORMULA], twice that period. An important difference between the even and odd solutions is the constant term [FORMULA]. Notice that the the individual terms of (23) and (24) are not solutions of (22). This is related to the reason why the perturbation method does not work for Eq. (15).

The even solution (23) of period [FORMULA] is considered first. Dispersion equation (A4) (see the derivation in the Appendix) has an infinite number of roots [FORMULA], which in the limit of a uniform atmosphere ([FORMULA]) are

[EQUATION]

while the eigenfunctions (23) are

[EQUATION]

The root with [FORMULA] corresponds to an acoustic plane wave propagating in the z -direction. For [FORMULA] there are acoustic waves propagating vertically, but standing horizontally. Discrete values of the horizontal wavenumber appear because here the uniform atmosphere is considered as a limiting case of the structured atmosphere. The periodicity of the eigenfunctions [FORMULA] is imposed by the periodic structure of the non-uniform case.

For [FORMULA] the first two roots of the dispersion equation (A4) are approximately

[EQUATION]

The vertically propagating wave with [FORMULA] has dispersion; its phase velocity is less than the mean sound speed, which is unity in our dimensionless variables. The slow-down of this acoustic wave increases with wavenumber [FORMULA] and with the amplitude [FORMULA] of the temperature fluctuation. The coefficients of the eigenfunction (23) are approximately equal to [FORMULA]. Relation (28) for [FORMULA] shows the dispersion of the waves. For this case the eigenfunction (23) contains all terms of the sum. The difference to the case [FORMULA] is that [FORMULA]. The wave mode (28) has a cut-off frequency and becomes evanescent, i.e. [FORMULA], for [FORMULA]. The occurrence of a cut-off frequency becomes clear if the structured atmosphere is considered as a multilayer wave guide, consisting of a set of layer waveguides of thickness [FORMULA]. The cut-off frequency is the same as in a uniform waveguide with sound speed [FORMULA] and thickness [FORMULA], and with [FORMULA] (Ingard 1988). The cut-off frequency corresponds the case where an integer number of half wavelengths fits into the thickness [FORMULA]. For [FORMULA] the wavenumber [FORMULA] is imaginary, and the wave is evanescent.

In the case [FORMULA] the cut-off frequencies [FORMULA] must be considered as eigenfrequencies of the structured atmosphere. This is similar to the vibrations of a one-dimensional lattice. But whereas in the latter case only propagation along the 1D lattice is possible we may consider two-dimensional wave propagation in the case of a structured atmosphere. We shall return to this point below in the context of oblique propagation. At present, the solutions of (A4) for [FORMULA] can be considered as waveguide modes of a multilayer waveguide for [FORMULA], and as eigenmodes of the structured atmosphere for [FORMULA]. Unfortunately the analogy with the waveguide is reasonable only for the 1D model, which we merely use in order to avoid the difficulties of the 3D treatment. To follow the analogy with lattice vibrations we shall denote the wave modes (25) vibrational waves. These modes were revealed for the first time in Paper 1, where they had been called high-frequency modes.

For the even solution with period [FORMULA] the dispersion relation is defined by Eq. (A6). In the limit [FORMULA] this dispersion equation has the solutions

[EQUATION]

For [FORMULA] we obtain vibrational waves with cut-off frequencies [FORMULA]. For example, the first root of the dispersion equation (A6) is

[EQUATION]

The treatment of the odd solutions (24) with spatial periods [FORMULA] and [FORMULA] is similar to the foregoing analysis of the solution (23). All odd modes are vibrational waves, corresponding to [FORMULA] (see (25) and (29)), because of [FORMULA].

3.2. Moving isothermal structured atmosphere

For an atmosphere with constant temperature, but with vertical flows

[EQUATION]

In this case Eq. (19) reduces to Eq. (A7). The first two roots of this dispersion equation (A11) are approximately

[EQUATION]

The dispersion equation (32) is valid for acoustic waves in an atmosphere consisting of layers with flows (31) of opposite direction. It differs essentially from the dispersion equation (28) for acoustic waves in a thermally structured atmosphere. In the low-frequency limit, [FORMULA], the phase velocity tends to [FORMULA] in the atmosphere with flows, i.e. to a limit that exceeds the sound speed; in the same limit the phase velocity of waves in the motionless thermally structured atmosphere is just the sound speed. In both cases the slow-down of the acoustic waves increases with increasing frequency.

3.3. Turbulent sound

In the moving atmosphere there exists another mode that is absent in the motionless atmosphere. The phase velocity of this mode is approximately given by

[EQUATION]

This mode was revealed for the first time in Paper 1. In order to explore its properties we determine the fluctuations of the pressure and of the vertical and horizontal velocity components for the solution (23),

[EQUATION]

[EQUATION]

[EQUATION]

If the coefficient [FORMULA] in (23) is zero or small in comparison with the other coefficients [FORMULA], the mean gas pressure fluctuation, calculated by integration of (35) over the space period, will to be small or zero. The average over [FORMULA] of the vertical velocity (36) does not vanish. Nor does the mean large-scale fluctuation [FORMULA] of the dynamic pressure vanish; it gets rather large if the phase velocity [FORMULA] as it is the case for this slow mode. It turns out that the gradient of the mean dynamical pressure indeed drives this mode, while the gradient of the mean gas pressure is absent. The mean horizontal velocity [FORMULA] is finite along with the mean vertical velocity [FORMULA]. Nevertheless, the mean [FORMULA] is zero. Since the slow mode is driven by the gradient of the mean dynamical pressure, we consider it as turbulent sound. The turbulent sound does not exist in an incompressible atmosphere, in spite of its solenoidal mean in the compressible case. Small-scale pressure and density fluctuations are a necessary condition for its existence.

The turbulent sound is not a unique mode. There is a number of similar modes, equal to the number of vibrational wave modes. They are distinguished in that one of the coefficients [FORMULA] in the solution (23) is much larger than all others. Moreover, there are turbulent sound modes corresponding to solutions (23) and (24) with period [FORMULA], which have exactly constant mean gas pressure because there is no [FORMULA] term for [FORMULA]. Turbulent sound modes require a special exploration, in particular because they produce strongly sheared flows. Even singularities of the vertical velocity (36) may appear, if [FORMULA] ; in this case the temperature and density fluctuations also become singular. We shall resume the discussion of the turbulent sound below after oblique propagation has been considered.

3.4. General case of a structured atmosphere

In the general case the structured atmosphere consists of hot upflows and cold downflows. The sound speed and the velocity of the flows are given by

[EQUATION]

Eq. (19) is reduced to Eq. (A12). The approximate solutions of the dispersion equation (A16) are

[EQUATION]

The joint action of the temperature and velocity fluctuations increases the dispersion of the acoustic and vibrational modes, (39) and (40). In the low-frequency limit, [FORMULA], the phase velocity of acoustic waves tends to [FORMULA] ; temperature fluctuation enhances the effect of flow on the phase velocity in this limit. The cut-off frequency of the vibrational modes decreases in comparison to the cases where only temperature or velocity fluctuations exist. The turbulent sound mode (41) depends on the temperature fluctuation only in higher orders of the expansion in terms of [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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