Gravity-Modes in ZZ Ceti Stars III. Eigenvalues and Eigenfuctions
  We report on numerical calculations of nonadiabatic eigenvalues and eigenfunctions for g-modes in ZZ Ceti variables. The spectrum of overstable $l=1$ modes delineates the instability strip. Its blue edge occurs where $\omega \tau_c \approx 1$ for the $n=1$ mode. Here $\omega$ is radian frequency and $\tau_c$ is about four times the thermal timescale at the bottom of the surface convection zone. As a ZZ Ceti cools, its convection zone deepens, longer period modes become overstable, but the critical value of $\omega\tau_c$ separating overstable and damped modes rises. The latter is a consequence of enhanced radiative damping for modes which propagate immediately below the convection zone. The critical value of $\omega\tau_c$ is of observational significance because modes with the smallest value of $\omega\tau_c$ are most observable photometrically. Maximum periods for overstable modes predicted for our cooler model envelopes are about a factor two longer than the observational upper limit of $1,200\s$. We assess a number of plausible resolutions for this discrepancy among which convective overshoot and nonlinear saturation look promising. The nonadiabatic eigenfunctions enable us to predict relative amplitudes and phases of photospheric variations of flux and velocity, quantities made accessible by recent observations. We also present asymptotic formula for damping rates of high order modes, a result of consequence for future investigations of nonlinear saturation of the amplidues of overstable modes. 
Yanqin Wu & Peter Goldreich