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

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2. The basic equations

The linearized hydrodynamic equations for adiabatic fluctuations, expressing the conservation of mass, momentum and internal energy for a gravitationally stratified, isothermal atmosphere can be written in the form:

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA] are, respectively, the perturbed gas pressure and density and the velocity; the subscript 0 denotes the unperturbed variables; [FORMULA] is the gravitational acceleration, [FORMULA] is the sound speed and [FORMULA] is the ratio of specific heats. The equilibrium values of unperturbed variables have the form:

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA] are, respectively, the values of pressure and density at the reference height [FORMULA]; and [FORMULA] is the pressure scale height.

In order to simplify the calculations it is convenient to assume the height dependence (see Lamb 1932)

[EQUATION]

and, as a consequence,

[EQUATION]

With these assumptions and measuring space in units of twice the pressure scale height, [FORMULA], and time in units of [FORMULA], where [FORMULA] is the cutoff period, Eqs. (1) become

[EQUATION]

where [FORMULA] is any of the variables (2) or (3) in non-dimensional form, i.e., [FORMULA], [FORMULA] and [FORMULA] i.e. the `reduced' perturbations, and [FORMULA] and [FORMULA] are Laplacian operators. Note that the reduced quantities p and [FORMULA] bear a vertical amplification as [FORMULA] even though the actual perturbations [FORMULA] and [FORMULA] decay with z. Note also that Eq. (4) reduces to the Klein-Gordon equation in the one-dimensional limit (see, e.g., KRBM).

We perform a plane-wave analysis assuming that all variables are proportional to [FORMULA]. Eq. (4) then yields the dispersion relation

[EQUATION]

(Bray & Loughhead 1974), where [FORMULA] is in units of [FORMULA] and [FORMULA] in units of [FORMULA]. Eq. (5) can be solved to obtain

[EQUATION]

where the plus sign is for the two roots for acoustic waves and the minus sign for the two roots for internal gravity waves . The solution of system (1), subject to given initial conditions at [FORMULA], can be written in the form of Fourier Integrals:

[EQUATION]

The summation goes over four modes corresponding to the four roots of the dispersion relation (5), i.e., both acoustic and internal gravity modes travelling in two directions.

In order to obtain the values of [FORMULA] of Eqs. 8-10, i.e., the relation between density, velocities and pressure in the plane-wave solutions, we have Fourier analyzed system (1), derived a set of algebraic equations (Appendix A), and determined the amplitudes [FORMULA] by matching the given initial conditions at [FORMULA] (Appendix B). As initial conditions we have considered a pressure pulse; the set of conditions becomes:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the amplitude of the initial perturbation and [FORMULA] is the spatial width of the Gaussian. Note that this four conditions completely determine the solution since we have fourth order equation. The analytical form for the amplitudes [FORMULA] is given in Appendix B.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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