On a priori bounding the growth of thermal instability waves

We have previously shown that the nonlinear growth of a finite-amplitude perturbation to a basic state given by a baroclinic zonal flow on the $\beta$-plane in a thermal quasigeostrophic reduced-gravity model can be a priori bounded. In this note we show that, unlike we stated earlier, Lyapunov stability can be proved even when buoyancy varies linearly with the meridional coordinate. In addition to rectifying our earlier results we expand them by deriving an instability saturation bound by making use of the existence of such a class of Lyapunov-stable basic states. This bound can be smaller than that one we estimated before, reinforcing our previous conclusions. We also present a numerical test of the accuracy of the derived bound.