Thermodynamic exchange and mode stability in strongly nonadiabatic radial pulsations
We apply the quadratic balance relation for linear nonadiabatic pulsations to radial modes of a sequence of post-AGB envelope models covering $3.52\leq\log(T_{\rm eff})\leq4.6$. The relation separates the thermodynamic exchange power $\mathcal{P}_{\rm ex}$, response power $\mathcal{P}_{\rm rsp}$, and surface contribution $ΔB$, and introduces an effective norm through $\mathcal{P}_{\rm ex}+ΔB=2γχ_{\rm eff}\mathcal{E}_{\rm kin}$, where $γ$ is the mode growth rate. For nearly adiabatic modes $χ_{\rm eff}$ is close to unity, whereas in strongly nonadiabatic pulsations the compression and horizontal area deformation contributions can make it change magnitude or even reverse its sign. In the cooler models, a broad family of overtone modes has $χ_{\rm eff}<0$ and is damped despite positive thermodynamic exchange power. For these modes the response power supplies the opposing damping. Conversely, two low frequency modes in the hotter models are excited over a range of effective temperatures, even though their thermodynamic exchange power is negative. Their instability is produced by the positive response power. Strange modes generally have $χ_{\rm eff}>0$, apart from a narrow boundary-sensitive transition during which a cold strange mode transforms into an ordinary p-mode. The excited low frequency solution with $χ_{\rm eff}<0$ persists with a finer numerical mesh and such modes are found under the alternative outer boundary formulations examined, although their frequencies are boundary condition dependent. Thus, for strongly nonadiabatic pulsations, the sign of the thermodynamic exchange power alone does not determine mode stability.