Meteorology

Meteorology#

Overview#

This section covers meteorology functions from NCL:

dewtemp_trh#

NCL’s dewtemp_trh calculates the dew point temperature given temperature and relative humidity using the equations from John Dutton’s “Ceaseless Wind” (pg. 273-274)[1] and returns a temperature in Kelvin.

Where, for the gas constant of water vapor (\(R_{v}\))of 461.5 \(\frac{J}{K*kg}\) (\(\frac{461.5}{1000 * 4.186} \frac{cal}{g*k}\)), the empirical value of the latent heat (pg. 273, Problem 8.3.1) is:

\(L_{lv} = 597.3 - 0.57(T - 273)\)

So, when \(h\) is the relative humidity, the dew point temperature (pg. 273, Equation 6, solved for as \(T_D\)) is:

\(T_D = \frac{T * L_{lv}}{L_{lv} - R_{v}Tlog(h)}\)

Important Note

To convert from Kelvin to Celsius -273.15 and to convert from Celsius to Kelvin +273.15

Grab and Go#

# Input: Single Value
from geocat.comp import dewtemp

temp_c = 18  # Celsius
relative_humidity = 46.5  # %

dewtemp(temp_c + 273.15, relative_humidity) - 273.15  # Returns in Celsius

np.float64(6.298141316024157)

# Input: List/Array
from geocat.comp import dewtemp

temp_kelvin = [291.15, 274.14, 360.3, 314]  # Kelvin
relative_humidity = [46.5, 5, 96.5, 1]  # %

dewtemp(temp_kelvin, relative_humidity) - 273.15  # Returns in Celsius

array([  6.29814132, -35.12955277,  86.22114845, -27.40981025])

daylight_fao56#

NCL’s daylight_fao56 calculates the maximum number of daylight hours as described in the Food and Agriculture Organization (FAO) Irrigation and Drainage Paper 56 (Chapter 3, Equation 34) [2].

Where the maximum number of daylight hours, \(N\), is:

\(N = \frac{24}{{\pi}} {\omega}_{s}\)

And \({\omega}_{s}\) is the sunset hour angle in radians (Chapter 3, Equation 25) [2] which is calculated from the latitude of the observer on Earth (\(\varphi\)) and the sun’s declination (\(\delta\)):

\({\omega}_{s} = arccos[-tan({\varphi})tan({\delta})]\)

Grab and Go#

# Input: Single Value
from geocat.comp import max_daylight

day_of_year = 246  # Sept. 3
latitude = -20  # 20 Degrees South

max_daylight(day_of_year, latitude)

array([[11.665592]], dtype=float32)

# Input: List/Array
from geocat.comp import max_daylight

# Spring Equinox (March 20), Summer Solstice (June 20), Autumn Equinox (Sept. 22), Winter Solstice (Dec. 21)
days_of_year = [79, 171, 265, 355]
latitudes = 40  # Boulder

max_daylight(days_of_year, latitudes)

array([[11.921149],
       [14.843202],
       [11.920901],
       [ 9.156431]], dtype=float32)

satvpr_temp_fao56#

NCL’s satvpr_temp_fao56 calculates saturation vapor pressure using temperature as described in the Food and Agriculture Organization (FAO) Irrigation and Drainage Paper 56 (Chapter 3, Equation 11) [2].

Where the saturation vapor pressure, \(e^°\) (kPa), at air temperature \(T\) (°C) is calculated as:

\(e^°(T) = 0.6108 {\exp}[\frac{17.27T}{T + 237.3}]\)

Grab and Go#

# Input: Single Value
from geocat.comp import saturation_vapor_pressure

temp = 50  # Fahrenheit

saturation_vapor_pressure(temp)

array(1.22796262)

# Input: List/Array
from geocat.comp import saturation_vapor_pressure

temp = [33, 50, 100, 212]  # Fahrenheit

saturation_vapor_pressure(temp)

array([  0.63594167,   1.22796262,   6.54556639, 102.21571649])

satvpr_tdew_fao56#

NCL’s satvpr_tdew_fao56 calculates the actual saturation vapor pressure using dewpoint temperature as described in the Food and Agriculture Organization (FAO) Irrigation and Drainage Paper 56 (Chapter 3, Equation 14) [2].

Where the actual vapor pressure, \(e_{a}\) (kPa), is saturation vapor pressure at a specific dewpoint temperature, \(T_{dew}\) (°C), which is calculated as:

\(e_{a} = e^°(T_{dew}) = 0.6108 {\exp}[\frac{17.27 T_{dew}}{T_{dew} + 237.3}]\)

# Input: Single Value
from geocat.comp import actual_saturation_vapor_pressure

temp = 35  # Fahrenheit

actual_saturation_vapor_pressure(temp)

array(0.68898447)

# Input: List/Array
from geocat.comp import actual_saturation_vapor_pressure

temp = [35, 60, 80, 200]  # Fahrenheit

actual_saturation_vapor_pressure(temp)

array([ 0.68898447,  1.76730647,  3.49620825, 80.00607017])

satvpr_slope_fao56#

NCL’s satvpr_slope_fao56 calculates the slope of the saturation vapor pressure curve as described in the Food and Agriculture Organization (FAO) Irrigation and Drainage Paper 56 (Chapter 3, Equation 13) [2].

Where the slope of saturation vapor pressure curve, \({\Delta}\) (kPa), at air temperature \(T\) (°C) is calculated as:

\({\Delta} = \frac{4098 (0.6108 {\exp}[\frac{17.27T}{T + 237.3}])}{(T + 237.3)^2}\)

# Input: Single Value
from geocat.comp import saturation_vapor_pressure_slope

temp = 60  # Fahrenheit

saturation_vapor_pressure_slope(temp)

array(0.11322096)

# Input: List/Array
from geocat.comp import saturation_vapor_pressure_slope

temp = [35, 60, 80, 200]  # Fahrenheit

saturation_vapor_pressure_slope(temp)

array([0.04941909, 0.11322096, 0.20552235, 2.99770999])

coriolis_param#

NCL’s coriolis_param calculates the Coriolis parameter at a given latitude

The Coriolis parameter (also known as the Coriolis frequency or the Coriolis coefficient) is calculated as twice the rotation rate (\({\Omega}\)) of the Earth times the sine of the latitude (\({\varphi}\))[3]

\(f = 2{\Omega}sin({\varphi})\)

The rotation rate depends on the length of the rotation period of the Earth (T) which is defined as one sidereal day (23 hours and 56 minutes):

\({\Omega} = \frac{2 * {\pi}}{T} = 7.292\text{e-5} \frac{rad}{s}\)

# Input: Single Value
from metpy.calc import coriolis_parameter
from metpy.units import units

latitude = 40  # degrees

coriolis_parameter(latitude * units.degree).magnitude

np.float64(9.374562340818716e-05)

# Input: List/Array
from metpy.calc import coriolis_parameter
from metpy.units import units

latitude = [-20, 40, 65]  # degrees

coriolis_parameter(latitude * units.degree).magnitude

array([-4.98810043e-05,  9.37456234e-05,  1.32178012e-04])

relhum#

NCL’s relhum calculates relative humidity given temperature, mixing ratio, and pressure. The percent of relative humidity (\({\Psi}\)) is based on the original NCL relhum code:

\({\Psi} = w (\frac{p - 0.378 * e_s}{0.622 * e_s}) * 100\)

Where \(w\) is the mass mixing ratio of water vapor and dry air, \(p\) is pressure, and \(e_s\) is the saturation vapor pressure for a given temperature.

The constant 0.622 represents the ratio of the molar mass of water vapor (\(M_w\)) in g/mol and the molar mass of dry air (\(M_d\)) in g/mol:

\(\frac{M_w}{M_d} = \frac{18.02}{28.9634} = 0.622\)

# Input: Single Value
from geocat.comp import relhum

temp = 303.15  # Kelvin
mixing_ratio = 0.018
pressure = 101325  # Pa

relhum(temp, mixing_ratio, pressure)

np.float64(68.05239617370448)

# Input: List/Array
from geocat.comp import relhum

temp = [375.15, 303.15, 315.15]  # Kelvin
mixing_ratio = [0.5, 0.018, 0.001]
pressure = [100325, 101325, 101400]  # Pa

relhum(temp, mixing_ratio, pressure)

array([43.78181802, 68.05239617,  1.92796564])

relhum_ice#

NCL’s relhum_ice calculates relative humidity with respect to ice, given temperature, mixing ratio, and pressure.

First,the approximation of vapor pressure is calculated from the Magnus Form.[4]

\({P_v} = c \exp^{\frac{a * T}{b + T}}\)

Where \(c\) is vapor pressure of water at 0 degrees Celsius (Pa), and \(a\) and \(b\) are saturation vapor pressure coefficient approximations over ice, as defined by the AEDki model.[5]

\(a = 22.571\), \(b = 273.71\), \(c = 6.1128\)

Then, the specific humidity (\(q_{st}\)) is calculated with \(p\) as pressure:

\(q_{st} = \frac{0.622 * P_v}{(p * 0.1) - 0.378 * P_v}\)

The constant 0.622 represents the ratio of the molar mass of vapor (\(M_w\)) and dry air (\(M_d\)) in g/mol:

\(\frac{M_w}{M_d} = \frac{18.02}{28.9634} = 0.622\)

And the 0.378 represents a correction constant where:

\(1 - \frac{M_w}{M_d} = 1 - 0.622 = 0.378\)

The percent of relative humidity (\({\Psi}\)) is calculated with \(w\) as the mixing ratio:

\({\Psi} = 100 * \frac{w}{q_{st}}\)

# Input: Single Value
from geocat.comp import relhum_ice

temp = 268.15  # Kelvin
mixing_ratio = 0.0037
pressure = 100000  # Pa

relhum_ice(temp, mixing_ratio, pressure)

np.float64(147.88012516016957)

# Input: List/Array
from geocat.comp import relhum_ice

temp = [268.15, 258.15, 250.15]  # Kelvin
mixing_ratio = [0.0037, 0.018, 0.001]
pressure = [100000, 100325, 101325]  # Pa

relhum_ice(temp, mixing_ratio, pressure)

array([ 147.88012516, 1756.81983351,  211.26793555])

relhum_water#

NCL’s relhum_water calculates relative humidity with respect to water, given temperature, mixing ratio, and pressure.

First,the approximation of vapor pressure is calculated from the Magnus Tetens form.[6]

\({P_v} = c \exp^{\frac{a (T- 273.16)}{T - b}}\)

Where \(c\) is vapor pressure of water at 0 degrees Celsius (Pa) as defined by the AEDki model[5], and \(a\) and \(b\) are saturation vapor pressure coefficient approximations, as defined by the Magnus Tetens model (Equation 6)[6]

\(a = 17.269\), \(b = 35.86\), \(c = 6.1128\)

Then, the specific humidity (\(q_{st}\)) is calculated with \(p\) as pressure:

\(q_{st} = \frac{0.622 * P_v}{(p * 0.1) - 0.378 * P_v}\)

The constant 0.622 represents the ratio of the molar mass of vapor (\(M_w\)) and dry air (\(M_d\)) in g/mol:

\(\frac{M_w}{M_d} = \frac{18.02}{28.9634} = 0.622\)

And the 0.378 represents a correction constant where:

\(1 - \frac{M_w}{M_d} = 1 - 0.622 = 0.378\)

The percent of relative humidity (\({\Psi}\)) is calculated with \(w\) as the mixing ratio:

\({\Psi} = 100 * \frac{w}{q_{st}}\)

# Input: Single Value
from geocat.comp import relhum_water

temp = 315.15  # Kelvin
mixing_ratio = 0.0037
pressure = 100000  # Pa

relhum_water(temp, mixing_ratio, pressure)

np.float64(7.025120800518423)

# Input: List/Array
from geocat.comp import relhum_ice

temp = [298.15, 315.15, 330.15]  # Kelvin
mixing_ratio = [0.0018, 0.0037, 0.001]
pressure = [100325, 100000, 101325]  # Pa

relhum_water(temp, mixing_ratio, pressure)

array([9.04879928, 7.0251208 , 0.87910297])

Python Resources#

Additional Reading#

References:#

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