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Astron. Astrophys. 323, 831-838 (1997)

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Appendix A: derivation of the perturbation equations

Consider a vertically stratified medium [FORMULA] with unit vector [FORMULA], and similar dependences for the other pertinent physical quantities. The general Navier-Stokes equation for a fluid with constant coefficient of viscosity [FORMULA] ([FORMULA] is the kinematic viscosity), in presence of chemical stratification and rotation reads

[EQUATION]

where [FORMULA] is the Levi-Civita pseudotensor. The heat equation, assuming a spatially homogeneous thermometric conductivity [FORMULA], has the standard form

[EQUATION]

In Eq. A1 the Archimedian term accounting for µ-gradients will be written as

[EQUATION]

where [FORMULA]. In the Boussinesq approximation, one demands that any density contrast only build up as the consequence of moderate temperature fluctuations, then enabling us to take an equation of state of the form [FORMULA], where [FORMULA] is the coefficient of thermal expansion, and [FORMULA], [FORMULA] the average density and temperature of the surroundings. This condition implies a Mach number [FORMULA]. If T, [FORMULA], p, [FORMULA] and [FORMULA] are the temperature, velocity components, pressure, molecular weight and angular frequency in the unperturbed configuration, and [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are the fluctuating quantities with [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] lower order corrections, the perturbation equations in the linear limit become

[EQUATION]

We can eliminate the disturbing term [FORMULA] by applying the operator [FORMULA] to Eq. A3. If we define [FORMULA] as the vorticity,

[EQUATION]

Acting once more on this latter equality with [FORMULA] and exploiting the well known identity [FORMULA] one next finds

[EQUATION]

Finally, project expressions A5 and A6 onto [FORMULA] and [FORMULA], respectively, and profit of the special geometry of the problem to simplify the numerous terms and obtain

[EQUATION]

where [FORMULA] and [FORMULA] are the z-components of the vorticity and the velocity. Eqs. A7, A8 and A4, supplemented by the closures 9 and 6, constitute the final set of perturbation equations.

Appendix B: the dispersion equation for the growth rate

Assume that the field variables can be expanded in normal modes

[EQUATION]

The operators appearing in the perturbation equations become

[EQUATION]

and lead to the following system

[EQUATION]

Eliminate [FORMULA] in B1 by operating with [FORMULA] and using B3, eliminate Z by operating with [FORMULA] and using B2, and finally eliminate both M and O by operating with n and using B4-B5. The resulting identity is further developed on the hypothesis that the vertical spatial variation of W is given by exp [FORMULA], which implies

[EQUATION]

After a few algebraic manipulations one eventually obtains the desired dispersion equation

[EQUATION]

Setting [FORMULA], [FORMULA] and rearranging, we recover the form (8) employed in our linear stability analysis.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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