Oscillatory convective modes in red giants: a possible explanation of the long secondary periods

We discuss properties of oscillatory convective modes in low-mass red giants, and compare them with observed properties of the long secondary periods (LSPs) of semi-regular red giant variables. Oscillatory convective modes are very nonadiabatic g$^{-}$ modes and they are present in luminous stars, such as red giants with $\log L/{\rm L}_\odot \ga 3$. Finite amplitudes for these modes are confined to the outermost nonadiabatic layers, where the radiative energy flux is more important than the convective energy flux. The periods of oscillatory convection modes increase with luminosity, and the growth times are comparable to the oscillation periods. The LSPs of red giants in the Large Magellanic Cloud (LMC) are observed to lie on a distinct period-luminosity sequence called sequence D. This sequence D period-luminosity relation is roughly consistent with the predictions for dipole oscillatory convective modes in AGB models if we adopt a mixing length of 1.2 pressure scale height ($\alpha = 1.2$). However, the effective temperature of the red-giant sequence of the LMC is consistent to models with $\alpha=1.9$, which predict periods too short by a factor of two.