moist_thermodynamics/saturation_vapor_pressures.py

+# -*- coding: utf-8 -*-
+"""
+Provides a collection of fits for saturation and sublimation vapor pressure
+
+Author: Bjorn Stevens (bjorn.stevens@mpimet.mpg.de)
+copygright, bjorn stevens Max Planck Institute for Meteorology, Hamburg
+
+License: BSD-3C
+"""
+#
+from . import constants
+import numpy as np
+
+
+def liq_wagner_pruss(T):
+    """Returns saturation vapor pressure (Pa) over planer liquid water
+
+    Encodes the empirical fits of Wagner and Pruss (2002), Eq 2.5a (page 399). Their formulation
+    is compared to other fits in the example scripts used in this package, and deemed to be the
+    best reference.
+
+    The fit has been verified for TvT <= T < = TvC.  For super cooled water (T<TvT) it deviates
+    from the results of Murphy and Koop where were developed for super-cooled water.  It is about
+    10% larger at 200 K, 25 % larter at 150 K, and then decreases again so it is 12% smaller at
+    the limit (123K) of the Murphy and Koop fit.  For accurate fits for super-cooled water the
+    function of Murphy and Koop should be used.
+
+    Args:
+        T: temperature in kelvin
+
+    Reference:
+        W. Wagner and A. Pruß , "The IAPWS Formulation 1995 for the Thermodynamic Properties
+    of Ordinary Water Substance for General and Scientific Use", Journal of Physical and Chemical
+    Reference Data 31, 387-535 (2002) https://doi.org/10.1063/1.1461829
+
+    >>> liq_wagner_pruss(np.asarray([273.16,305.]))
+    array([ 611.65706974, 4719.32683147])
+    """
+    TvC = constants.temperature_water_vapor_critical_point
+    PvC = constants.pressure_water_vapor_critical_point
+
+    vt = 1.0 - T / TvC
+    es = PvC * np.exp(
+        TvC
+        / T
+        * (
+            -7.85951783 * vt
+            + 1.84408259 * vt**1.5
+            - 11.7866497 * vt**3
+            + 22.6807411 * vt**3.5
+            - 15.9618719 * vt**4
+            + 1.80122502 * vt**7.5
+        )
+    )
+    return es
+
+
+def ice_wagner_etal(T):
+    """Returns sublimation vapor pressure (Pa) over simple (Ih) ice
+
+    Encodes the emperical fits of Wagner et al., (2011) which also define the IAPWS standard for
+    sublimation vapor pressure over ice-Ih
+
+    Args:
+        T: temperature in kelvin
+
+    Reference:
+        Wagner, W., Riethmann, T., Feistel, R. & Harvey, A. H. New Equations for the Sublimation
+        Pressure and Melting Pressure of H 2 O Ice Ih. Journal of Physical and Chemical Reference
+        Data 40, 043103 (2011).
+
+
+    >>> ice_wagner_etal(np.asarray([273.16,260.]))
+    array([611.655     , 195.80103377])
+    """
+    TvT = constants.temperature_water_vapor_triple_point
+    PvT = constants.pressure_water_vapor_triple_point
+
+    a1 = -0.212144006e2
+    a2 = 0.273203819e2
+    a3 = -0.610598130e1
+    b1 = 0.333333333e-2
+    b2 = 0.120666667e1
+    b3 = 0.170333333e1
+    theta = T / TvT
+    es = PvT * np.exp((a1 * theta**b1 + a2 * theta**b2 + a3 * theta**b3) / theta)
+    return es
+
+
+def liq_murphy_koop(T):
+    """Returns saturation vapor pressure (Pa) over liquid water
+
+    Encodes the empirical fit (Eq. 10) of Murphy and Koop (2011) which improves on the Wagner and
+    Pruß fits for supercooled conditions.
+
+    The fit has been verified for 123K <= T < = 332 K
+
+    Args:
+        T: temperature in kelvin
+
+    Reference:
+        Murphy, D. M. & Koop, T. Review of the vapour pressures of ice and supercooled water for
+        atmospheric applications. Q. J. R. Meteorol. Soc. 131, 1539–1565 (2005).
+
+    >>> liq_murphy_koop(np.asarray([273.16,140.]))
+    array([6.11657044e+02, 9.39696372e-07])
+    """
+
+    X = np.tanh(0.0415 * (T - 218.8)) * (
+        53.878 - 1331.22 / T - 9.44523 * np.log(T) + 0.014025 * T
+    )
+    return np.exp(54.842763 - 6763.22 / T - 4.210 * np.log(T) + 0.000367 * T + X)
+
+
+def liq_hardy(T):
+    """Returns satruation vapor pressure (Pa) over liquid water
+
+    Encodes the empirical fit (Eq. 10) of Hardy (1998) which is often used in the postprocessing
+    of radiosondes
+
+    Args:
+        T: temperature in kelvin
+
+    Reference:
+        Hardy, B., 1998, ITS-90 Formulations for Vapor Pressure, Frostpoint Temperature, Dewpoint
+        Temperature, and Enhancement Factors in the Range –100 to +100 °C, The Proceedings of the
+        Third International Symposium on Humidity & Moisture, London, England
+
+    >>> liq_hardy(np.asarray([273.16,260.]))
+    array([611.65715494, 222.65143353])
+    """
+    X = (
+        -2.8365744e3 / (T * T)
+        - 6.028076559e3 / T
+        + 19.54263612
+        - 2.737830188e-2 * T
+        + 1.6261698e-5 * T**2
+        + 7.0229056e-10 * T**3
+        - 1.8680009e-13 * T**4
+        + 2.7150305 * np.log(T)
+    )
+    return np.exp(X)
+
+
+def analytic(T, lx, cx):
+    """returns saturation vapor pressure over a given phase
+
+    Uses the rankine (constant specific heat, negligible condensate volume) approximations to
+    calculate the saturation vapor pressure over a phase with the specific heat cx, and phase
+    change enthalpy (from vapor) lx, at temperature T.
+
+    Args:
+        T: temperature in kelvin
+        lx: phase change enthalpy between vapor and given phase (liquid, ice)
+        cx: specific heat capacity of given phase (liquid, ice)
+
+    Returns:
+        value of saturation vapor pressure over liquid water in Pa
+
+    Reference:
+        Romps, D. M. Exact Expression for the Lifting Condensation Level. Journal of the Atmospheric
+        Sciences 74, 3891–3900 (2017).
+        Romps, D. M. Accurate expressions for the dew point and frost point derived from the Rankine-
+        Kirchhoff approximations. Journal of the Atmospheric Sciences (2021) doi:10.1175/JAS-D-20-0301.1.
+
+    >>> analytic(305.,constants.lvT,constants.cl)
+    4711.131611687174
+    """
+    TvT = constants.temperature_water_vapor_triple_point
+    PvT = constants.pressure_water_vapor_triple_point
+    Rv = constants.water_vapor_gas_constant
+
+    c1 = (constants.cpv - cx) / Rv
+    c2 = lx / (Rv * TvT) - c1
+    es = PvT * np.exp(c2 * (1.0 - TvT / T)) * (T / TvT) ** c1
+    return es
+
+
+def liq_analytic(T, lx=constants.lvT, cx=constants.cl):
+    """Analytic approximation for saturation vapor pressure over iquid
+
+    Uses the rankine (constant specific heat, negligible condensate volume) approximations to
+    calculate the saturation vapor pressure over liquid.  The procedure is described in Eq(4) of
+    Romps (2017) and best approximates the actual value for specific heats that differ slightly
+    from the best estimates of these quantities which are provided as default quantities.
+    Romps recommends cl = 4119 J/kg/K, and cpv = 1861 J/kg/K.
+
+    Args:
+        T: temperature in kelvin
+        lx: enthalpy of vaporization, at triple point, default constants.lvT
+        cl: specific heat capacity of liquid at triple point
+
+    Returns:
+        value of saturation vapor pressure over liquid water in Pa
+
+    Reference:
+        Romps, D. M. Exact Expression for the Lifting Condensation Level. Journal of the Atmospheric
+        Sciences 74, 3891–3900 (2017).
+        Romps, D. M. Accurate expressions for the dew point and frost point derived from the Rankine-
+        Kirchhoff approximations. Journal of the Atmospheric Sciences (2021) doi:10.1175/JAS-D-20-0301.1.
+
+    >>> liq_analytic(np.asarray([273.16,305.]))
+    array([ 611.655     , 4711.13161169])
+    """
+    return analytic(T, lx, cx)
+
+
+def ice_analytic(T, lx=constants.lsT, cx=constants.ci):
+    """Analytic approximation for saturation vapor pressure over ice
+
+    Uses the rankine (constant specific heat, negligible condensate volume) approximations to
+    calculate the saturation vapor pressure over ice.  The procedure is described in Eq(4) of
+    Romps (2017) and best approximates the actual value for specific heats that differ slightly
+    from the best estimates of these quantities which are provided as default quantities.
+    Romps recommends ci = 1861 J/kg/K, and cpv = 1879 J/kg/K.
+
+    Args:
+        T: temperature in kelvin
+        lx: enthalpy of sublimation, at triple point, default constants.lsT
+        ci: specific heat capacity of ice Ih, at triple point
+
+    Returns:
+        value of saturation vapor pressure over ice in Pa
+
+    Reference:
+        Romps, D. M. Exact Expression for the Lifting Condensation Level. Journal of the Atmospheric
+        Sciences 74, 3891–3900 (2017).
+        Romps, D. M. Accurate expressions for the dew point and frost point derived from the Rankine-
+        Kirchhoff approximations. Journal of the Atmospheric Sciences (2021) doi:10.1175/JAS-D-20-0301.1.
+
+
+    >>> ice_analytic(np.asarray([273.16,260.]))
+    array([611.655     , 195.99959431])
+    """
+    return analytic(T, lx, cx)
+
+
+def tetens(T, a, b):
+    """Returns saturation vapor pressure over liquid using the Magnus-Teten's formula
+
+    This equation is written in a general form, with the constants a and b determining the fit.  As
+    such it can be specified for either ice or water, or adapted as originally impelemented in ICON,
+    in which case PvT and TvT need to be substituted by Pv0 and T0.
+
+    Args:
+        T: temperature in kelvin
+
+    >>> tetens(285.,17.269,35.86)
+    1389.7114123472836
+    """
+
+    es = constants.PvT * np.exp(a * (T - constants.TvT) / (T - b))
+    return es
+
+
+def liq_tetens(T):
+    """Returns saturation vapor pressure over liquid using the Magnus-Teten's formula
+
+    This equation is what is used in the ICON code, hence its inclusion in this library.  The original
+    ICON implementation followed Murray's choice of constants (T0=273.15, Pv0=610.78, a=17.269, b=35.86).
+    This implementation is referenced to the triple point values of temperature and vapor and with
+    revised constants (a,b) chosen to better agree with the fits of Wagner and Pruss
+
+    Args:
+        T: temperature in kelvin
+
+    Reference:
+        Murray, F. W. On the Computation of Saturation Vapor Pressure. Journal of Applied Meteorology
+        and Climatology 6, 203–204 (1967).
+
+    >>> liq_tetens(np.asarray([273.16,305.]))
+    array([ 611.655     , 4719.73680592])
+    """
+    a = 17.41463775
+    b = 33.6393413
+
+    return tetens(T, a, b)
+
+
+def ice_tetens(T):
+    """Returns saturation vapor pressure over liquid using the Magnus-Teten's formula
+
+    This equation is what is used in the ICON code, hence its inclusion in this library.  The original
+    ICON implementation followed Murray's choice of constants (T0=273.15, Pv0=610.78, a=21.875, b=7.66).
+    This implementation is referenced to the triple point values of temperature and vapor and with
+    revised constants (a,b) chosen to better agree with the fits of Wagner and Pruss
+
+    Args:
+        T: temperature in kelvin
+
+    Reference:
+        Murray, F. W. On the Computation of Saturation Vapor Pressure. Journal of Applied Meteorology
+        and Climatology 6, 203–204 (1967).
+
+    >>> ice_tetens(np.asarray([273.16,260.]))
+    array([611.655     , 196.10072658])
+    """
+    a = 22.0419977
+    b = 5.0
+
+    return tetens(T, a, b)

setup.py

@@ -3,7 +3,7 @@ from setuptools import setup, find_packages
 
 setup(
     name="moist_thermodynamics",
-    version="0.3",
+    version="0.5",
     description="Constants and functions for the treatment of moist atmospheric thermodynamics",
     packages=find_packages(),
     install_requires=[