Saturated Pseudoadiabats—A Noniterative Approximation

Abstract

Two noniterative approximations are presented for saturated pseudoadiabats (also known as moist adiabats). One approximation determines which moist adiabat passes through a point of known pressure and temperature, such as through the lifting condensation level on a skew T or tephigram. The other approximation determines the air temperature at any pressure along a known moist adiabat, such as the final temperature of a rising cloudy air parcel. The method used to create these statistical regressions is a relatively new variant of genetic programming called gene-expression programming. The correlation coefficient between the resulting noniterative approximations and the iterated data such as plotted on thermodynamic diagrams is over 99.97%. The mean absolute error is 0.28°C, and the root mean square error is 0.44 within a thermodynamic domain bounded by −30° < θw ≤ 40°C, P > 20 kPa, and −60° ≤ T ≤ 40°C, where θw, P, and T are wet-bulb potential temperature, pressure, and air temperature.

Technical note: A noniterative approach to modelling moist thermodynamics

“…However, such precision is unlikely to be necessary, as some of the thermodynamic relationships used in the truth iterative computations contain substantially larger errors than those introduced by the above optimization procedure (Davies-Jones, 2009;Koutsoyiannis, 2012). Moreover, conventional pseudoadiabatic diagrams, such as those used by US Air Force, Environment Canada and Air Transport Association of America, differ from each other by nearly 1 • C at the 20 kPa pressure level (Bakhshaii and Stull, 2013). The specific choice of thermodynamic constants and relationships undoubtedly affects the definition of truth used in this work; however, this has little effect on the overall validity of the approach.…”

“…Discontinuity and asymptomatic behaviour arising from error minimization for all transformed adiabats renders the parameter curves very difficult to model. A variety of functions would be necessary to capture the parameter behaviour, which in turn is likely to produce a complex and discontinuous error field, such as appeared in Bakhshaii and Stull (2013).…”

“…The moist adiabatic lapse rate is derived from conservation of moist entropy as a function of temperature, T , and saturated mixing ratio, r s , which itself is a nonlinear function of T and pressure P . To improve efficiency Davies-Jones (2008) proposed a different iterative method, based on inverting Bolton's formula (Bolton, 1980) for equivalent potential temperature valid for the pressure range 10 ≤ P ≤ 105 kPa and wet-bulb potential temperatures −20 < θ w < 40 • C. As a noniterative alternative, Bakhshaii and Stull (2013) offer an approximate solution devised using gene-expression programming. They provide two separate sets of equations for determining T (P , θ w ) and θ w (P , T ) for the domain bounded by −30 < θ w < 40 • C, P > 20 kPa and −60 < T < 40 • C. The complex nature of the problem required their splitting of the modelled region into subdomains, resulting in error discontinuity.…”

“…The method also produced fairly large errors (on the order of 1-2 • C) in the upper atmosphere. Despite the limitations, to our knowledge the only existing noniterative solution to approximate saturated pseudoadiabats is that by Bakhshaii and Stull (2013).…”

A noniteratiave approach to modelling moist thermodynamics

“…errors (on the order of 1-2°C) in the upper atmosphere. Despite the limitations, to our knowledge Bakhshaii and Stull (2013) is the only existing noniterative solution to approximate saturated pseudoadiabats.…”