Erratum: "A Global Model for Circumgalactic and Cluster-core Precipitation" (2017, ApJ, 845, 80)

Section 3.3.3 of the published article attributed the saturation of thermally unstable internal gravity waves observed in numerical simulations of circumgalactic gas (e.g., McCourt et al. 2012; Meece et al. 2015) to nonlinear mode coupling. The published article presented a perturbation expansion of the fluid momentum equation in which the lowest order nonlinear terms represented interactions of wave triads that conserve wave momentum and energy. Those triad interactions can steadily transfer energy from a primary wave to other resonant pairs of waves that do not fully return the energy. Our calculation showed that the damping rate resulting from triad interactions was similar to the damping rate required to explain the observed saturation, and we called that process "buoyancy damping."

However, the published article incorrectly attributed buoyancy damping to resonant interactions of internal gravity waves with pairs of sound waves that are damped by radiative cooling. After publication, we learned that gravity waves cannot transfer their energy into resonant pairs of sound waves at the required rate. Wave triads consisting of one gravity mode and two acoustic modes can resonate, and the mode coupling constants are of the right order of magnitude to explain saturation of thermal instability through buoyancy damping, but there is no time-averaged net energy transfer from the gravity mode to the acoustic modes.

The most elegant proof that this energy-transfer channel is inefficient comes from Hasselmann (1967). Purely on the basis of symmetry arguments, Hasselmann showed that energy-conserving coupling of resonant wave triads cannot lead to time-averaged net energy flow from modes with lower frequencies to modes with greater frequencies. And internal gravity waves in a galactic atmosphere have lower frequencies than acoustic modes.

Nevertheless, the more general argument presented in Section 3.3.3 of the published article is still valid. Internal gravity waves do indeed decay through triad mode coupling at a rate that explains the observed saturation of thermal instability. However, they lose their energy to other gravity-wave modes, which ultimately dissipate into Kolmogorov turbulence by breaking. This damping mechanism is known as a "parametric subharmonic instability" and has been well studied in the contexts of terrestrial atmospheres and oceans (Staquet & Sommeria 2002).

In accordance with the analysis of Hasselmann (1967), the parametric subharmonic instability of internal gravity waves steadily channels gravity-wave energy into lower frequency modes. The shift in wave energy is therefore toward modes propagating at a greater angle away from the horizontal direction than the original wave does, and consequently they carry energy away from the atmospheric layer in which the original wave propagates. Those obliquely propagating modes also become more prone to dissipation through breaking as their wavevectors become more vertical. Most importantly, they do not return energy to the original internal gravity wave, which therefore damps. The same damping channel can cause saturation of thermal instability in galactic atmospheres in which the buoyancy timescale is shorter than the thermal timescale.