# Copyright (c) 2008-2015 MetPy Developers.
# Distributed under the terms of the BSD 3-Clause License.
# SPDX-License-Identifier: BSD-3-Clause
"""Contains a collection of thermodynamic calculations."""
from __future__ import division
import numpy as np
import scipy.integrate as si
from .tools import find_intersections
from ..constants import Cp_d, epsilon, kappa, Lv, P0, Rd
from ..package_tools import Exporter
from ..units import atleast_1d, check_units, concatenate, units
exporter = Exporter(globals())
sat_pressure_0c = 6.112 * units.millibar
@exporter.export
@check_units('[pressure]', '[temperature]')
[docs]def potential_temperature(pressure, temperature):
r"""Calculate the potential temperature.
Uses the Poisson equation to calculation the potential temperature
given `pressure` and `temperature`.
Parameters
----------
pressure : `pint.Quantity`
The total atmospheric pressure
temperature : `pint.Quantity`
The temperature
Returns
-------
`pint.Quantity`
The potential temperature corresponding to the the temperature and
pressure.
See Also
--------
dry_lapse
Notes
-----
Formula:
.. math:: \Theta = T (P_0 / P)^\kappa
Examples
--------
>>> from metpy.units import units
>>> metpy.calc.potential_temperature(800. * units.mbar, 273. * units.kelvin)
<Quantity(290.96653180346203, 'kelvin')>
"""
return temperature * (P0 / pressure).to('dimensionless')**kappa
@exporter.export
@check_units('[pressure]', '[temperature]')
[docs]def dry_lapse(pressure, temperature):
r"""Calculate the temperature at a level assuming only dry processes.
This function lifts a parcel starting at `temperature`, conserving
potential temperature. The starting pressure should be the first item in
the `pressure` array.
Parameters
----------
pressure : `pint.Quantity`
The atmospheric pressure level(s) of interest
temperature : `pint.Quantity`
The starting temperature
Returns
-------
`pint.Quantity`
The resulting parcel temperature at levels given by `pressure`
See Also
--------
moist_lapse : Calculate parcel temperature assuming liquid saturation
processes
parcel_profile : Calculate complete parcel profile
potential_temperature
"""
return temperature * (pressure / pressure[0])**kappa
@exporter.export
@check_units('[pressure]', '[temperature]')
[docs]def moist_lapse(pressure, temperature):
r"""Calculate the temperature at a level assuming liquid saturation processes.
This function lifts a parcel starting at `temperature`. The starting
pressure should be the first item in the `pressure` array. Essentially,
this function is calculating moist pseudo-adiabats.
Parameters
----------
pressure : `pint.Quantity`
The atmospheric pressure level(s) of interest
temperature : `pint.Quantity`
The starting temperature
Returns
-------
`pint.Quantity`
The temperature corresponding to the the starting temperature and
pressure levels.
See Also
--------
dry_lapse : Calculate parcel temperature assuming dry adiabatic processes
parcel_profile : Calculate complete parcel profile
Notes
-----
This function is implemented by integrating the following differential
equation:
.. math:: \frac{dT}{dP} = \frac{1}{P} \frac{R_d T + L_v r_s}
{C_{pd} + \frac{L_v^2 r_s \epsilon}{R_d T^2}}
This equation comes from [Bakhshaii2013]_.
"""
def dt(t, p):
t = units.Quantity(t, temperature.units)
p = units.Quantity(p, pressure.units)
rs = saturation_mixing_ratio(p, t)
frac = ((Rd * t + Lv * rs) /
(Cp_d + (Lv * Lv * rs * epsilon / (Rd * t * t)))).to('kelvin')
return frac / p
return units.Quantity(si.odeint(dt, atleast_1d(temperature).squeeze(),
pressure.squeeze()).T.squeeze(), temperature.units)
@exporter.export
@check_units('[pressure]', '[temperature]', '[temperature]')
[docs]def lcl(pressure, temperature, dewpt, max_iters=50, eps=1e-2):
r"""Calculate the lifted condensation level (LCL) using from the starting point.
The starting state for the parcel is defined by `temperature`, `dewpt`,
and `pressure`.
Parameters
----------
pressure : `pint.Quantity`
The starting atmospheric pressure
temperature : `pint.Quantity`
The starting temperature
dewpt : `pint.Quantity`
The starting dew point
Returns
-------
`(pint.Quantity, pint.Quantity)`
The LCL pressure and temperature
Other Parameters
----------------
max_iters : int, optional
The maximum number of iterations to use in calculation, defaults to 50.
eps : float, optional
The desired absolute error in the calculated value, defaults to 1e-2.
See Also
--------
parcel_profile
Notes
-----
This function is implemented using an iterative approach to solve for the
LCL. The basic algorithm is:
1. Find the dew point from the LCL pressure and starting mixing ratio
2. Find the LCL pressure from the starting temperature and dewpoint
3. Iterate until convergence
The function is guaranteed to finish by virtue of the `max_iters` counter.
"""
w = mixing_ratio(saturation_vapor_pressure(dewpt), pressure)
new_p = p = pressure
eps = units.Quantity(eps, p.units)
while max_iters:
td = dewpoint(vapor_pressure(p, w))
new_p = pressure * (td / temperature) ** (1. / kappa)
if np.abs(new_p - p).max() < eps:
break
p = new_p
max_iters -= 1
else:
# We have not converged
raise RuntimeError('LCL calculation has not converged.')
return new_p, td
@exporter.export
@check_units('[pressure]', '[temperature]', '[temperature]')
[docs]def lfc(pressure, temperature, dewpt):
r"""Calculate the level of free convection (LFC).
This works by finding the first intersection of the ideal parcel path and
the measured parcel temperature.
Parameters
----------
pressure : `pint.Quantity`
The atmospheric pressure
temperature : `pint.Quantity`
The temperature at the levels given by `pressure`
dewpt : `pint.Quantity`
The dew point at the levels given by `pressure`
Returns
-------
`pint.Quantity`
The LFC pressure and temperature
See Also
--------
parcel_profile
"""
ideal_profile = parcel_profile(pressure, temperature[0], dewpt[0]).to('degC')
# The parcel profile and data have the same first data point, so we ignore
# that point to get the real first intersection for the LFC calculation.
x, y = find_intersections(pressure[1:], ideal_profile[1:], temperature[1:],
direction='increasing')
if len(x) == 0:
return np.nan * pressure.units, np.nan * temperature.units
else:
return x[0], y[0]
@exporter.export
@check_units('[pressure]', '[temperature]', '[temperature]')
[docs]def el(pressure, temperature, dewpt):
r"""Calculate the equilibrium level.
This works by finding the last intersection of the ideal parcel path and
the measured environmental temperature. If there is one or fewer intersections, there is
no equilibrium level.
Parameters
----------
pressure : `pint.Quantity`
The atmospheric pressure
temperature : `pint.Quantity`
The temperature at the levels given by `pressure`
dewpt : `pint.Quantity`
The dew point at the levels given by `pressure`
Returns
-------
`pint.Quantity, pint.Quantity`
The EL pressure and temperature
See Also
--------
parcel_profile
"""
ideal_profile = parcel_profile(pressure, temperature[0], dewpt[0]).to('degC')
x, y = find_intersections(pressure[1:], ideal_profile[1:], temperature[1:])
# If there is only one intersection, it's the LFC and we return None.
if len(x) <= 1:
return np.nan * pressure.units, np.nan * temperature.units
else:
return x[-1], y[-1]
@exporter.export
@check_units('[pressure]', '[temperature]', '[temperature]')
[docs]def parcel_profile(pressure, temperature, dewpt):
r"""Calculate the profile a parcel takes through the atmosphere.
The parcel starts at `temperature`, and `dewpt`, lifted up
dry adiabatically to the LCL, and then moist adiabatically from there.
`pressure` specifies the pressure levels for the profile.
Parameters
----------
pressure : `pint.Quantity`
The atmospheric pressure level(s) of interest. The first entry should be the starting
point pressure.
temperature : `pint.Quantity`
The starting temperature
dewpt : `pint.Quantity`
The starting dew point
Returns
-------
`pint.Quantity`
The parcel temperatures at the specified pressure levels.
See Also
--------
lcl, moist_lapse, dry_lapse
"""
# Find the LCL
l = lcl(pressure[0], temperature, dewpt)[0].to(pressure.units)
# Find the dry adiabatic profile, *including* the LCL. We need >= the LCL in case the
# LCL is included in the levels. It's slightly redundant in that case, but simplifies
# the logic for removing it later.
press_lower = concatenate((pressure[pressure >= l], l))
t1 = dry_lapse(press_lower, temperature)
# Find moist pseudo-adiabatic profile starting at the LCL
press_upper = concatenate((l, pressure[pressure < l]))
t2 = moist_lapse(press_upper, t1[-1]).to(t1.units)
# Return LCL *without* the LCL point
return concatenate((t1[:-1], t2[1:]))
@exporter.export
@check_units('[pressure]', '[dimensionless]')
[docs]def vapor_pressure(pressure, mixing):
r"""Calculate water vapor (partial) pressure.
Given total `pressure` and water vapor `mixing` ratio, calculates the
partial pressure of water vapor.
Parameters
----------
pressure : `pint.Quantity`
total atmospheric pressure
mixing : `pint.Quantity`
dimensionless mass mixing ratio
Returns
-------
`pint.Quantity`
The ambient water vapor (partial) pressure in the same units as
`pressure`.
Notes
-----
This function is a straightforward implementation of the equation given in many places,
such as [Hobbs1977]_ pg.71:
.. math:: e = p \frac{r}{r + \epsilon}
See Also
--------
saturation_vapor_pressure, dewpoint
"""
return pressure * mixing / (epsilon + mixing)
@exporter.export
@check_units('[temperature]')
[docs]def saturation_vapor_pressure(temperature):
r"""Calculate the saturation water vapor (partial) pressure.
Parameters
----------
temperature : `pint.Quantity`
The temperature
Returns
-------
`pint.Quantity`
The saturation water vapor (partial) pressure
See Also
--------
vapor_pressure, dewpoint
Notes
-----
Instead of temperature, dewpoint may be used in order to calculate
the actual (ambient) water vapor (partial) pressure.
The formula used is that from [Bolton1980]_ for T in degrees Celsius:
.. math:: 6.112 e^\frac{17.67T}{T + 243.5}
"""
# Converted from original in terms of C to use kelvin. Using raw absolute values of C in
# a formula plays havoc with units support.
return sat_pressure_0c * np.exp(17.67 * (temperature - 273.15 * units.kelvin) /
(temperature - 29.65 * units.kelvin))
@exporter.export
@check_units('[temperature]', '[dimensionless]')
[docs]def dewpoint_rh(temperature, rh):
r"""Calculate the ambient dewpoint given air temperature and relative humidity.
Parameters
----------
temperature : `pint.Quantity`
Air temperature
rh : `pint.Quantity`
Relative humidity expressed as a ratio in the range [0, 1]
Returns
-------
`pint.Quantity`
The dew point temperature
See Also
--------
dewpoint, saturation_vapor_pressure
"""
return dewpoint(rh * saturation_vapor_pressure(temperature))
@exporter.export
@check_units('[pressure]')
[docs]def dewpoint(e):
r"""Calculate the ambient dewpoint given the vapor pressure.
Parameters
----------
e : `pint.Quantity`
Water vapor partial pressure
Returns
-------
`pint.Quantity`
Dew point temperature
See Also
--------
dewpoint_rh, saturation_vapor_pressure, vapor_pressure
Notes
-----
This function inverts the [Bolton1980]_ formula for saturation vapor
pressure to instead calculate the temperature. This yield the following
formula for dewpoint in degrees Celsius:
.. math:: T = \frac{243.5 log(e / 6.112)}{17.67 - log(e / 6.112)}
"""
val = np.log(e / sat_pressure_0c)
return 0. * units.degC + 243.5 * units.delta_degC * val / (17.67 - val)
@exporter.export
@check_units('[pressure]', '[pressure]', '[dimensionless]')
[docs]def mixing_ratio(part_press, tot_press, molecular_weight_ratio=epsilon):
r"""Calculate the mixing ratio of a gas.
This calculates mixing ratio given its partial pressure and the total pressure of
the air. There are no required units for the input arrays, other than that
they have the same units.
Parameters
----------
part_press : `pint.Quantity`
Partial pressure of the constituent gas
tot_press : `pint.Quantity`
Total air pressure
molecular_weight_ratio : `pint.Quantity` or float, optional
The ratio of the molecular weight of the constituent gas to that assumed
for air. Defaults to the ratio for water vapor to dry air
(:math:`\epsilon\approx0.622`).
Returns
-------
`pint.Quantity`
The (mass) mixing ratio, dimensionless (e.g. Kg/Kg or g/g)
Notes
-----
This function is a straightforward implementation of the equation given in many places,
such as [Hobbs1977]_ pg.73:
.. math:: r = \epsilon \frac{e}{p - e}
See Also
--------
saturation_mixing_ratio, vapor_pressure
"""
return molecular_weight_ratio * part_press / (tot_press - part_press)
@exporter.export
@check_units('[pressure]', '[temperature]')
[docs]def saturation_mixing_ratio(tot_press, temperature):
r"""Calculate the saturation mixing ratio of water vapor.
This calculation is given total pressure and the temperature. The implementation
uses the formula outlined in [Hobbs1977]_ pg.73.
Parameters
----------
tot_press: `pint.Quantity`
Total atmospheric pressure
temperature: `pint.Quantity`
The temperature
Returns
-------
`pint.Quantity`
The saturation mixing ratio, dimensionless
"""
return mixing_ratio(saturation_vapor_pressure(temperature), tot_press)
@exporter.export
@check_units('[pressure]', '[temperature]')
[docs]def equivalent_potential_temperature(pressure, temperature):
r"""Calculate equivalent potential temperature.
This calculation must be given an air parcel's pressure and temperature.
The implementation uses the formula outlined in [Hobbs1977]_ pg.78-79.
Parameters
----------
pressure: `pint.Quantity`
Total atmospheric pressure
temperature: `pint.Quantity`
The temperature
Returns
-------
`pint.Quantity`
The corresponding equivalent potential temperature of the parcel
Notes
-----
.. math:: \Theta_e = \Theta e^\frac{L_v r_s}{C_{pd} T}
"""
pottemp = potential_temperature(pressure, temperature)
smixr = saturation_mixing_ratio(pressure, temperature)
return pottemp * np.exp(Lv * smixr / (Cp_d * temperature))
@exporter.export
@check_units('[temperature]', '[dimensionless]', '[dimensionless]')
[docs]def virtual_temperature(temperature, mixing, molecular_weight_ratio=epsilon):
r"""Calculate virtual temperature.
This calculation must be given an air parcel's temperature and mixing ratio.
The implementation uses the formula outlined in [Hobbs2006] pg.80.
Parameters
----------
temperature: `pint.Quantity`
The temperature
mixing : `pint.Quantity`
dimensionless mass mixing ratio
molecular_weight_ratio : `pint.Quantity` or float, optional
The ratio of the molecular weight of the constituent gas to that assumed
for air. Defaults to the ratio for water vapor to dry air
(:math:`\epsilon\approx0.622`).
Returns
-------
`pint.Quantity`
The corresponding virtual temperature of the parcel
Notes
-----
.. math:: T_v = T \frac{\text{w} + \epsilon}{\epsilon\,(1 + \text{w})}
"""
return temperature * ((mixing + molecular_weight_ratio) /
(molecular_weight_ratio * (1 + mixing)))
@exporter.export
@check_units('[pressure]', '[temperature]', '[dimensionless]', '[dimensionless]')
[docs]def virtual_potential_temperature(pressure, temperature, mixing,
molecular_weight_ratio=epsilon):
r"""Calculate virtual potential temperature.
This calculation must be given an air parcel's pressure, temperature, and mixing ratio.
The implementation uses the formula outlined in [Markowski2010] pg.13.
Parameters
----------
pressure: `pint.Quantity`
Total atmospheric pressure
temperature: `pint.Quantity`
The temperature
mixing : `pint.Quantity`
dimensionless mass mixing ratio
molecular_weight_ratio : `pint.Quantity` or float, optional
The ratio of the molecular weight of the constituent gas to that assumed
for air. Defaults to the ratio for water vapor to dry air
(:math:`\epsilon\approx0.622`).
Returns
-------
`pint.Quantity`
The corresponding virtual potential temperature of the parcel
Notes
-----
.. math:: \Theta_v = \Theta \frac{\text{w} + \epsilon}{\epsilon\,(1 + \text{w})}
"""
pottemp = potential_temperature(pressure, temperature)
return virtual_temperature(pottemp, mixing, molecular_weight_ratio)
@exporter.export
@check_units('[pressure]', '[temperature]', '[dimensionless]', '[dimensionless]')
[docs]def density(pressure, temperature, mixing, molecular_weight_ratio=epsilon):
r"""Calculate density.
This calculation must be given an air parcel's pressure, temperature, and mixing ratio.
The implementation uses the formula outlined in [Hobbs2006] pg.67.
Parameters
----------
temperature: `pint.Quantity`
The temperature
pressure: `pint.Quantity`
Total atmospheric pressure
mixing : `pint.Quantity`
dimensionless mass mixing ratio
molecular_weight_ratio : `pint.Quantity` or float, optional
The ratio of the molecular weight of the constituent gas to that assumed
for air. Defaults to the ratio for water vapor to dry air
(:math:`\epsilon\approx0.622`).
Returns
-------
`pint.Quantity`
The corresponding density of the parcel
Notes
-----
.. math:: \rho = \frac{p}{R_dT_v}
"""
virttemp = virtual_temperature(temperature, mixing, molecular_weight_ratio)
return (pressure / (Rd * virttemp)).to(units.kilogram / units.meter ** 3)
@exporter.export
@check_units('[temperature]', '[temperature]', '[pressure]')
[docs]def relative_humidity_wet_psychrometric(dry_bulb_temperature, web_bulb_temperature,
pressure, **kwargs):
r"""Calculate the relative humidity with wet bulb and dry bulb temperatures.
Parameters
----------
dry_bulb_temperature: `pint.Quantity`
Dry bulb temperature
web_bulb_temperature: `pint.Quantity`
Wet bulb temperature
pressure: `pint.Quantity`
Total atmospheric pressure
Returns
-------
`pint.Quantity`
Relative humidity
Notes
-----
.. math:: RH = 100 \frac{e}{e_s}
* :math:`RH` is relative humidity
* :math:`e` is vapor pressure from the wet psychrometric calculation
* :math:`e_s` is the saturation vapor pressure
References
----------
.. [10] WMO GUIDE TO METEOROLOGICAL INSTRUMENTS AND METHODS OF OBSERVATION WMO-No.8
(2008 edition, Updated in 2010) : PART 4
https://www.wmo.int/pages/prog/www/IMOP/CIMO-Guide.html
.. [11] Fan, Jinpeng. "Determination of the psychrometer coefficient A of the WMO
reference psychrometer by comparison with a standard gravimetric hygrometer."
Journal of Atmospheric and Oceanic Technology 4.1 (1987): 239-244.
See Also
--------
psychrometric_vapor_pressure_wet, saturation_vapor_pressure
"""
return (100 * units.percent * psychrometric_vapor_pressure_wet(dry_bulb_temperature,
web_bulb_temperature, pressure, **kwargs) /
saturation_vapor_pressure(dry_bulb_temperature))
@exporter.export
@check_units('[temperature]', '[temperature]', '[pressure]')
[docs]def psychrometric_vapor_pressure_wet(dry_bulb_temperature, wet_bulb_temperature, pressure,
psychrometer_coefficient=6.21e-4 / units.kelvin):
r"""Calculate the vapor pressure with wet bulb and dry bulb temperatures.
Parameters
----------
dry_bulb_temperature: `pint.Quantity`
Dry bulb temperature
wet_bulb_temperature: `pint.Quantity`
Wet bulb temperature
pressure: `pint.Quantity`
Total atmospheric pressure
psychrometer_coefficient: `pint.Quantity`
Psychrometer coefficient
Returns
-------
`pint.Quantity`
Vapor pressure
Notes
-----
.. math:: e' = e'_w(T_w) - A p (T - T_w)
* :math:`e'` is vapor pressure
* :math:`e'_w(T_w)` is the saturation vapor pressure with respect to water at temperature
:math:`T_w`
* :math:`p` is the pressure of the wet bulb
* :math:`T` is the temperature of the dry bulb
* :math:`T_w` is the temperature of the wet bulb
* :math:`A` is the psychrometer coefficient
Psychrometer coefficient depends on the specific instrument being used and the ventilation
of the instrument.
References
----------
.. [12] WMO GUIDE TO METEOROLOGICAL INSTRUMENTS AND METHODS OF OBSERVATION WMO-No.8
(2008 edition, Updated in 2010) : PART 4
https://www.wmo.int/pages/prog/www/IMOP/CIMO-Guide.html
.. [13] Fan, Jinpeng. "Determination of the psychrometer coefficient A of the WMO reference
psychrometer by comparison with a standard gravimetric hygrometer."
Journal of Atmospheric and Oceanic Technology 4.1 (1987): 239-244.
See Also
--------
saturation_vapor_pressure
"""
return (saturation_vapor_pressure(wet_bulb_temperature) - psychrometer_coefficient *
pressure * (dry_bulb_temperature - wet_bulb_temperature).to('kelvin'))