# -*- coding: utf-8 -*-
"""Functions related to water vapor and its thermodynamic effects
"""
from numbers import Number
import numpy as np
from typhon import constants
__all__ = [
'e_eq_ice_mk',
'e_eq_water_mk',
'e_eq_mixed_mk',
'density',
'mixing_ratio2specific_humidity',
'mixing_ratio2vmr',
'specific_humidity2mixing_ratio',
'specific_humidity2vmr',
'vmr2mixing_ratio',
'vmr2specific_humidity',
]
[docs]def e_eq_ice_mk(T):
r"""Calculate the equilibrium vapor pressure of water over ice.
.. math::
\ln(e_\mathrm{ice}) = 9.550426
- \frac{5723.265}{T}
+ 3.53068 \cdot \ln(T)
- 0.00728332 \cdot T
Parameters:
T (float or ndarray): Temperature [K].
Returns:
float or ndarray: Equilibrium vapor pressure [Pa].
See also:
:func:`~typhon.physics.e_eq_water_mk`
Calculate the equilibrium vapor pressure over liquid water.
:func:`~typhon.physics.e_eq_mixed_mk`
Calculate the vapor pressure of water over the mixed phase.
References:
Murphy, D. M. and Koop, T. (2005): Review of the vapour pressures of
ice and supercooled water for atmospheric applications,
Quarterly Journal of the Royal Meteorological Society 131(608):
1539â1565. doi:10.1256/qj.04.94
"""
if np.any(T <= 0):
raise ValueError('Temperatures must be larger than 0 Kelvin.')
# Give the natural log of saturation vapor pressure over ice in Pa
e = 9.550426 - 5723.265 / T + 3.53068 * np.log(T) - 0.00728332 * T
return np.exp(e)
[docs]def e_eq_water_mk(T):
r"""Calculate the equilibrium vapor pressure of water over liquid water.
.. math::
\ln(e_\mathrm{liq}) &=
54.842763 - \frac{6763.22}{T} - 4.21 \cdot \ln(T) \\
&+ 0.000367 \cdot T
+ \tanh \left(0.0415 \cdot (T - 218.8)\right) \\
&\cdot \left(53.878 - \frac{1331.22}{T}
- 9.44523 \cdot \ln(T)
+ 0.014025 \cdot T \right)
Parameters:
T (float or ndarray): Temperature [K].
Returns:
float or ndarray: Equilibrium vapor pressure [Pa].
See also:
:func:`~typhon.physics.e_eq_ice_mk`
Calculate the equilibrium vapor pressure of water over ice.
:func:`~typhon.physics.e_eq_mixed_mk`
Calculate the vapor pressure of water over the mixed phase.
References:
Murphy, D. M. and Koop, T. (2005): Review of the vapour pressures of
ice and supercooled water for atmospheric applications,
Quarterly Journal of the Royal Meteorological Society 131(608):
1539â1565. doi:10.1256/qj.04.94
"""
if np.any(T <= 0):
raise ValueError('Temperatures must be larger than 0 Kelvin.')
# Give the natural log of saturation vapor pressure over water in Pa
e = (54.842763
- 6763.22 / T
- 4.21 * np.log(T)
+ 0.000367 * T
+ np.tanh(0.0415 * (T - 218.8))
* (53.878 - 1331.22 / T - 9.44523 * np.log(T) + 0.014025 * T))
return np.exp(e)
[docs]def e_eq_mixed_mk(T):
r"""Return equilibrium pressure of water with respect to the mixed-phase.
The equilibrium pressure over water is taken for temperatures above the
triple point :math:`T_t` the value over ice is taken for temperatures
below :math:`T_tâ23\,\mathrm{K}`. For intermediate temperatures the
equilibrium pressure is computed as a combination
of the values over water and ice according to the IFS documentation:
.. math::
e_\mathrm{s} = \begin{cases}
T > T_t, & e_\mathrm{liq} \\
T < T_t - 23\,\mathrm{K}, & e_\mathrm{ice} \\
else, & e_\mathrm{ice}
+ (e_\mathrm{liq} - e_\mathrm{ice})
\cdot \left(\frac{T - T_t - 23}{23}\right)^2
\end{cases}
References:
IFS Documentation â Cy45r1,
Operational implementation 5 June 2018,
Part IV: Physical Processes, Chapter 12, Eq. 12.13,
https://www.ecmwf.int/node/18714
.. plot::
:include-source:
import numpy as np
import matplotlib.pyplot as plt
from typhon import physics
T = np.linspace(245, 285)
fig, ax = plt.subplots()
ax.semilogy(T, physics.e_eq_mixed_mk(T), lw=3, c='k', label='Mixed')
ax.semilogy(T, physics.e_eq_ice_mk(T), ls='dashed', label='Ice')
ax.semilogy(T, physics.e_eq_water_mk(T), ls='dashed', label='Water')
ax.set_ylabel('Vapor pressure [Pa]')
ax.set_xlabel('Temperature [K]')
ax.legend()
plt.show()
Parameters:
T (float or ndarray): Temperature [K].
See also:
:func:`~typhon.physics.e_eq_ice_mk`
Equilibrium pressure of water over ice.
:func:`~typhon.physics.e_eq_water_mk`
Equilibrium pressure of water over liquid water.
Returns:
float or ndarray: Equilibrium pressure [Pa].
"""
# Keep track of input type to match the return type.
is_float_input = isinstance(T, Number)
if is_float_input:
# Convert float input to ndarray to allow indexing.
T = np.asarray([T])
e_eq_water = e_eq_water_mk(T)
e_eq_ice = e_eq_ice_mk(T)
is_water = T > constants.triple_point_water
is_ice = T < (constants.triple_point_water - 23.)
e_eq = (e_eq_ice + (e_eq_water - e_eq_ice)
* ((T - constants.triple_point_water + 23) / 23)**2
)
e_eq[is_ice] = e_eq_ice[is_ice]
e_eq[is_water] = e_eq_water[is_water]
return float(e_eq) if is_float_input else e_eq
[docs]def density(p, T, R=constants.gas_constant_dry_air):
r"""Calculates gas density by ideal gas law.
.. math::
\rho = \frac{p}{R \cdot T}
Parameters:
p (float or ndarray): Pressure [Pa.]
T (float or ndarray): Temperature [K].
If type of T and p is ndarray, size must match p.
R (float): Gas constant [J K^-1 kg^-1].
Default is gas constant for dry air.
Returns:
float or ndarray: Density [kg/m**3].
See also:
:mod:`typhon.constants`
Module containing universal gas constant as well
as gas constants for dry air and water vapor.
Examples:
>>> density(1013e2, 300)
1.1763056653021122
"""
return p / (R * T)
[docs]def mixing_ratio2specific_humidity(w):
r"""Convert mass mixing ratio to specific humidity.
.. math::
q = \frac{w}{1 + w}
Parameters:
w (float or ndarray): Mass mixing ratio.
Returns:
float or ndarray: Specific humidity.
Examples:
>>> mixing_ratio2specific_humidity(0.02)
0.0196078431372549
"""
return w / (1 + w)
[docs]def mixing_ratio2vmr(w):
r"""Convert mass mixing ratio to volume mixing ratio.
.. math::
x = \frac{w}{w + \frac{M_w}{M_d}}
Parameters:
w (float or ndarray): Mass mixing ratio.
Returns:
float or ndarray: Volume mixing ratio.
Examples:
>>> mixing_ratio2vmr(0.02)
0.03115371853180794
"""
Md = constants.molar_mass_dry_air
Mw = constants.molar_mass_water
return w / (w + Mw / Md)
[docs]def specific_humidity2mixing_ratio(q):
r"""Convert specific humidity to mass mixing ratio.
.. math::
w = \frac{q}{1 - q}
Parameters:
q (float or ndarray): Specific humidity.
Returns:
float or ndarray: Mass mixing ratio.
Examples:
>>> specific_humidity2mixing_ratio(0.02)
0.020408163265306124
"""
return q / (1 - q)
[docs]def specific_humidity2vmr(q):
r"""Convert specific humidity to volume mixing ratio.
.. math::
x = \frac{q}{(1 - q) \frac{M_w}{M_d} + q}
Parameters:
q (float or ndarray): Specific humidity.
Returns:
float or ndarray: Volume mixing ratio.
Examples:
>>> specific_humidity2vmr(0.02)
0.03176931009073226
"""
Md = constants.molar_mass_dry_air
Mw = constants.molar_mass_water
return q / ((1 - q) * Mw / Md + q)
[docs]def vmr2mixing_ratio(x):
r"""Convert volume mixing ratio to mass mixing ratio.
.. math::
w = \frac{x}{1 - x} \frac{M_w}{M_d}
Parameters:
x (float or ndarray): Volume mixing ratio.
Returns:
float or ndarray: Mass mixing ratio.
Examples:
>>> vmr2mixing_ratio(0.04)
0.025915747437955664
"""
Md = constants.molar_mass_dry_air
Mw = constants.molar_mass_water
return x / (1 - x) * Mw / Md
[docs]def vmr2specific_humidity(x):
r"""Convert volume mixing ratio to specific humidity.
.. math::
q = \frac{x}{(1 - x) \frac{M_d}{M_w} + x}
Parameters:
x (float or ndarray): Volume mixing ratio.
Returns:
float or ndarray: Specific humidity.
Examples:
>>> vmr2specific_humidity(0.04)
0.025261087474946833
"""
Md = constants.molar_mass_dry_air
Mw = constants.molar_mass_water
return x / ((1 - x) * Md / Mw + x)