An improved approximation for the moist-air entropy potential temperature ${\theta }_s$

Abstract

The moist-air entropy is defined in Marquet (QJRMS 2011, arXiv:1401.1097) by $\boxed{s=s_{ref}+c_{pd}\ln ({\theta }_s)}$ in terms of two constant values ($s_{ref}$, $c_{pd}$) and a potential entropy temperature denoted by ${\theta }_s$. It is shown in Marquet (2011) that a quantity denoted by $({\theta }_s{)}_1$ plays the role of a leading order approximation of ${\theta }_s$. The aim of this note is to demonstrate in a more rigorous way that $({\theta }_s{)}_1$ is indeed the leading order approximation of ${\theta }_s$, and to derive a second order approximation which may be used in computations of values, gradients or turbulent fluxes of moist-air entropy. Some impacts of this second order approximation are described in this brief version of a note to be submitted to the QJRMS.

2 pages, 3 figures, submitted for a WGNE "Blue Book" publication