  The influence of stable and unstable stratification on the amplification of
coherent structures in turbulent channel flows is investigated by computing the
linear response to stochastic forcing. The responses to momentum and thermal
forcing are considered separately. Consistently with results of previous direct
numerical simulations, the influence of the mean flow stratification on
stochastic forcing amplifications is found to be non-negligible only for
streamwise-elongated large-scale structures. Unstable stratification is found
to enhance the peak variance of the response, except for the velocity response
to thermal forcing, and to increase the spanwise wavelength of the most
amplified structures. Stable stratification induces opposite effects. The
spanwise wavelengths maximizing the different types of variance amplifications,
all converge to approximately 6h when approaching the linear instability
threshold where large-scale coherent rolls become linearly unstable. In the
presence of even moderately unstable stratification, the profiles of turbulent
buoyancy and momentum fluxes and of rms vertical velocity of all types of most
amplified stochastic responses are nearly indistinguishable from those of the
critical mode becoming unstable at the critical Richardson number. The two most
energetic POD modes always contribute to more than 90% of the variance of the
response, except for the thermal response to thermal forcing. We conclude that
the same mechanism underlies the onset of the instability of coherent
large-scale rolls at the critical Richardson number and the amplification of
coherent large-scale structures at subcritical Richardson numbers.
