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The growth rate of the compressible Rayleigh-Taylor instability is studied inthe presence of a background temperature gradient, $\Theta$, using a normalmode analysis. The effect of $\Theta$ variation is examined for three interfacetypes corresponding to combinations of the viscous properties of the fluids(inviscid-inviscid, viscous-viscous and viscous-inviscid) at different Atwoodnumbers, $At$, and, when at least one of the fluids' viscosity is non-zero, asa function of the Grashof number. For the general case, the resulting ordinarydifferential equations are solved numerically; however, dispersion relationsfor the growth rate are presented for several limiting cases. An analyticalsolution is found for the inviscid-inviscid interface and the correspondingdispersion equation for the growth rate is obtained in the limit of a large$\Theta$. For the viscous-inviscid case, a dispersion relation is derived inthe incompressible limit and $\Theta=0$. Compared to $\Theta=0$ case, the roleof $\Theta<0$ (hotter light fluid) is destabilizing and becomes stabilizingwhen $\Theta>0$ (colder light fluid). The most pronounced effect when$\Theta\neq 0$ is found at low $At$ and/or at large perturbation wavelengthsrelative to the domain size for all interface types. On the other hand, atsmall perturbation wavelengths relative to the domain size, the growth rate for$\Theta<0$ case exceeds the infinite domain incompressible constant densityresult. The results are applied to two practical examples, using sets ofparameters relevant to Inertial Confinement Fusion coasting stage and solarcorona plumes. The role of viscosity on the growth rate reduction is discussedtogether with highlighting the range of wavenumbers most affected by viscosity.The viscous effects further increase in the presence of background temperaturegradient, when the viscosity is temperature dependent.

Viscous effects on the Rayleigh-Taylor instability with background temperature gradient