Gravity Modes in ZZ Ceti Stars. IV. Amplitude Saturation by Parametric Instability

ZZ Ceti stars (also known as DAV stars) exhibit small-amplitude photometric pulsations in multiple gravity modes. As the stars cool, their dominant modes shift to longer periods. We demonstrate that parametric instability limits overstable modes to amplitudes similar to those observed. In particular, it reproduces the trend that longer period modes have larger amplitudes. Parametric instability is a form of resonant three-mode coupling. It involves the destabilization of a pair of stable daughter modes by an overstable parent mode. The three modes must satisfy exact angular selection rules and approximate frequency resonance. The lowest instability threshold for each parent mode is provided by the daughter pair that minimizes (δω 2 + γ )/κ 2 , where κ is the nonlinear coupling constant, δω is the frequency mismatch, and γ d is the energy damping rate of the daughter modes. Parametric instability leads to a steady state if |δω| > γ d and to limit cycles if |δω| < γ d . The former behavior characterizes low radial order ( n ≤ 3) parent modes, and the latter those with n ≥ 5. In either case, the overstable mode's amplitude is maintained at close to the instability threshold value. Although parametric instability defines an upper envelope for the amplitudes of overstable modes in ZZ Ceti stars, other nonlinear mechanisms are required to account for the irregular distribution of amplitudes of similar modes and the nondetection of modes with periods longer than 1200 s. Resonant three-mode interactions involving more than one excited mode may account for the former and Kelvin-Helmholtz instability of the mode-driven shear layer below the convection zone for the latter.