## Appendix A: derivation of the perturbation equationsConsider a vertically stratified medium with unit vector , and similar dependences for the other pertinent physical quantities. The general Navier-Stokes equation for a fluid with constant coefficient of viscosity ( is the kinematic viscosity), in presence of chemical stratification and rotation reads where is the Levi-Civita pseudotensor. The heat equation, assuming a spatially homogeneous thermometric conductivity , has the standard form In Eq. A1 the Archimedian term accounting for
where . 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 , where
is the coefficient of thermal expansion, and
, the average density
and temperature of the surroundings. This condition implies a Mach
number . If We can eliminate the disturbing term by applying the operator to Eq. A3. If we define as the vorticity, Acting once more on this latter equality with and exploiting the well known identity one next finds Finally, project expressions A5 and A6 onto and , respectively, and profit of the special geometry of the problem to simplify the numerous terms and obtain where and 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 rateAssume that the field variables can be expanded in normal modes The operators appearing in the perturbation equations become and lead to the following system Eliminate in B1 by operating with
and using B3, eliminate After a few algebraic manipulations one eventually obtains the desired dispersion equation Setting , and rearranging, we recover the form (8) employed in our linear stability analysis. © European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |