An improved approximation for the moist-air entropy potential temperature $\theta_s$

TL;DR

This paper rigorously demonstrates that a previously proposed approximation of the moist-air entropy potential temperature is valid and introduces a second order approximation for more accurate computations in atmospheric science.

Contribution

It provides a rigorous validation of the leading order approximation and derives a new second order approximation for the moist-air entropy potential temperature.

Findings

The leading order approximation is confirmed as valid.

A second order approximation is derived for improved accuracy.

Impacts of the second order approximation are discussed.

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.