Heights to pressures (using the ICAO atmosphere)

Converts heights to pressures (in \(hPa\)) using the ICAO atmosphere.

The pressure (\(P\)) is retrieved as follow:

• For \(Z < 11000\) gpm

  \[P = 100.0- p_{b} \times P_{ICAO_{surface}}\]

  with

  \[p_{b} = (1.0 - p_{a})^{ZP_{11}}\]

  and

  \[p_{a} = L_{ICAO_{11}} \times Z \times \frac{1.0}{T_{ICAO_{surface}}}\]

• For \(11000 \leq Z < 20000.0\) gpm

  \[P = 100.0 \times 10^{p_{a}}\]

  with

  \[p_{a} = \log(P_{ICAO_{11}}) - p_{b}\]

and

\[p_{b} = \frac{g}{R_{d}} \times (Z - 11000) \times \frac{1.0}{T_{ICAO_{iso}}};\]

• For \(Z \geq 20000.0\) gpm

  \[P = 100.0 \times P_{ICAO_{22}} \times (1.0 - P_{a})^{ZP_{22}};\]

  with

  \[P_{a} = L_{ICAO_{22}} \times \frac{1.0}{T_{ICAO_{iso}}} \times (Z - 20000)\]

With:

• Temperature of isothermal layer: \(T_{ICAO_{iso}} = 216.65\) K

• The assume surface temperature: \(T_{ICAO_{surface}} = 288.15\) K

• The assume surface pressure: \(P_{ICAO_{surface}} = 1013.25\) hPa

• The assumed pressure at 11,000 gpm: \(P_{ICAO_{11}} = 226.32\) hPa

• The assumed pressure at 22,000 gpm: \(P_{ICAO_{22}} = 54.7487\) hPa

• The lapse rate for levels up to 11,000 gpm: \(L_{ICAO_{11}} = 6.5 \times 10^{-03}\) K/m

• The lapse rate from 11,000 gpm to 22,000 gpm: \(L_{ICAO_{22}} = -1.0 \times 10^{-03}\) K/m

• \(ZP_{11} = \frac{g}{R_{d} \times L_{ICAO_{11}}}\)

• \(ZP_{22} = \frac{g}{R_{d} \times L_{ICAO_{22}}}\)

• Specific gas constant for dry air: \(R_{d}\)

• Standard acceleration due to gravity: \(g\)

Saturated Vapor Pressure from T

The various formulations available to derive \(e_{sat}\) (Saturated Vapor Pressure) from \(T\) (temperature or dew point temperature) are:

• Rogers | default: Classical formula from Rogers and Yau (1989; Eq2.17)

  \[e_{sat} = 1000 \times 0.6112 \times \exp\left(17.67 \times \frac{T - 2.7315 \times 10^{2}}{T - 29.65}\right)\]

• Sonntag: Eqn 7, Sonntag, D., Advancements in the field of hygrometry, Meteorol. Zeitschrift, N. F., 3, 51-66, 1994. Most radiosonde manufacturers use Wexler, or Hyland and Wexler or Sonntag formulations, which are all very similar (Holger Vomel, pers. comm., 2011)

  \[ \begin{align}\begin{aligned}e_{sat} = \exp\left(\frac{-6096.9385}{T} + 21.2409642 - 2.711193 \times 10^{-2}\\ \times T + 1.673952 \times 10^{-5} \times T^{2} + 2.433502 \times \log(T)\right)\end{aligned}\end{align} \]

• Walko: Polynomial fit of Goff-Gratch (1946) formulation (Walko, 1991). The Walko formulation is computationally fastest of all methods, but becomes less accurate at extremely low temperatures, below roughly -70C.

  \[ \begin{align}\begin{aligned}e_{sat} = c_{0}+x \times (c_{1}+x \times (c_{2}+x \times (c_{3}+x \times (c_{4}+\\ x \times (c_{5}+x \times (c_{6}+x \times (c_{7}+x \times c_{8})))))))\end{aligned}\end{align} \]

  with:

  \[ \begin{align}\begin{aligned}c = [610.5851, 44.40316, 1.430341, 0.2641412 \times 10^{-1},\\ 0.2995057 \times 10^{-3}, 0.2031998 \times 10^{-5}, 0.6936113 \times 10^{-8},\\ 0.2564861 \times 10^{-11}, -0.3704404 \times 10^{-13}]\end{aligned}\end{align} \]

• Murphy: Murphy and Koop, Review of the vapour pressure of ice and supercooled water for atmospheric applications, Q. J. R. Meteorol. Soc (2005), 131, pp. 1539-1565.

  \[ \begin{align}\begin{aligned}e_{sat} = \exp\left(54.842763 - \frac{6763.22}{T} - 4.210 \times \log(T)\\ + 0.000367 \times T + tanh(0.0415 \times (T - 218.8))\\ \times (53.878 - \frac{1331.22}{T} - 9.44523 \times \log(T)\\ + 0.014025 \times T)\right)\end{aligned}\end{align} \]