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

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2. Basic equations

An equilibrium atmosphere with vertical hot upflows and cold downflows is considered. The equilibrium pressure [FORMULA] is constant in the entire atmosphere, while the equilibrium values of temperature [FORMULA], density [FORMULA] and vertical velocity [FORMULA] are arbitrary functions of x. All equilibrium variables are independent of the vertical coordinate z. The set of linearized hydrodynamical equations is

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The time derivative of the momentum equation is

[EQUATION]

The substitution of Eqs. (2) and (3) into the time derivative of the linearized equation of state (4) gives

[EQUATION]

After substitution of [FORMULA] from (1) into (6) and of (6) into (5) the set of Eqs. (1)-(4) is reduced to one vector equation for the velocity [FORMULA]

[EQUATION]

[EQUATION]

Now take the divergence of both sides of Eq. (7)

[EQUATION]

[EQUATION]

where the notation [FORMULA] is used. The substitution of (7) into (8) gives

[EQUATION]

[EQUATION]

[EQUATION]

Now a harmonic dependence of the variables on t and z is introduced

[EQUATION]

and the components of Eq. (7) and Eq. (9) are rewritten as

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where the tilde has been omitted. After elimination of [FORMULA] and [FORMULA] from Eqs. (10)-(12) a second-order equation for u is obtained

[EQUATION]

[EQUATION]

which in terms of the new dependent variable

[EQUATION]

reads

[EQUATION]

where [FORMULA] is a phase velocity. Eq. (15) defines the dependence on x of the amplitude of an acoustic wave with frequency [FORMULA] and vertical phase velocity [FORMULA]. In the limit of a uniform atmosphere, where [FORMULA] and [FORMULA], the equation is reduced to the dispersion relation [FORMULA], where [FORMULA] is the "horizontal" wavenumber. In the general case of a structured atmosphere the dispersion equation for acoustic waves could be defined by the condition that the wave amplitude is bounded at infinity. This condition corresponds to the requirement that [FORMULA] in the case of the uniform atmosphere.

In this paper I treat only the case where the equilibrium temperature, density and vertical velocity are periodic functions of x, with spatial period [FORMULA]:

[EQUATION]

Eq. (15) belongs to the class of equations with periodic coefficients. The treatment of such equations has been developed especially for the Mathieu and Hill equations, which often appear in physical problems (Ince 1944). However, Eq. (15) is more general than the Mathieu and Hill equations, and a more detailed treatment is necessary.

For convenience we introduce dimensionless variables

[EQUATION]

is the mean of the sound velocity over the space period [FORMULA]. The reason for the unusual definition of the spatial wavenumber [FORMULA] is that in this way Eq. (15) is reduced to the form used for the Mathieu and Hill equations. It is thus easier to refer to the properties of equations with periodic coefficients. In terms of the dimensionless variables Eq. (15) reads

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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