def equivalent_potential_temperature(pressure, temperature):
"""
Calculates equivalent potential temperature given an air parcel's
pressure and temperature.
The implementation uses the formula outlined in [5]
Parameters
----------
pressure: array_like
Total atmospheric pressure
temperature: array_like
The temperature
Returns
-------
array_like
The corresponding equivalent potential temperature of the parcel
Notes
-----
.. math:: \Theta_e = \Theta e^\frac{L_v r_s}{C_{pd} T}
References
----------
.. [5] Hobbs, Peter V. and Wallace, John M., 1977: Atmospheric Science, an Introductory
Survey. 78-79.
"""
pottemp = potential_temperature(pressure, temperature)
smixr = saturation_mixing_ratio(pressure, temperature)
return pottemp * np.exp(Lv * smixr / (Cp_d * temperature))
def evaluate(self, points):
"""Evaluate the estimated pdf on a set of points.
Parameters
----------
points : (# of dimensions, # of points)-array
Alternatively, a (# of dimensions,) vector can be passed in and
treated as a single point.
Returns
-------
values : (# of points,)-array
The values at each point.
Raises
------
ValueError : if the dimensionality of the input points is different
than the dimensionality of the KDE.
"""
points = np.atleast_2d(points)
dim, num_m = np.array(points).shape
if dim != self.dim:
raise ValueError("points have dimension {}, dataset has dimension "
"{}".format(dim, self.dim))
result = np.zeros(num_m)
if num_m >= self.num_dp:
# there are more points than data, so loop over data
for i in range(self.num_dp):
diff = self.dataset[:, i, np.newaxis] - points
tdiff = np.dot(self.inv_cov, diff)
energy = np.sum(diff * tdiff, axis=0) / 2.0
result = result + np.exp(-energy)
else:
# loop over points
for i in range(num_m):
diff = self.dataset - points[:, i, np.newaxis]
tdiff = np.dot(self.inv_cov, diff)
energy = np.sum(diff * tdiff, axis=0) / 2.0
result[i] = np.sum(np.exp(-energy), axis=0)
result = result / self.norm_factor
return result
def equivalent_potential_temperature(pressure, temperature, dewpoint):
r"""Calculate equivalent potential temperature.
This calculation must be given an air parcel's pressure, temperature, and dewpoint.
The implementation uses the formula outlined in [Bolton1980]_:
First, the LCL temperature is calculated:
.. math:: T_{L}=\frac{1}{\frac{1}{T_{D}-56}+\frac{ln(T_{K}/T_{D})}{800}}+56
Which is then used to calculate the potential temperature at the LCL:
.. math:: \theta_{DL}=T_{K}\left(\frac{1000}{p-e}\right)^k
\left(\frac{T_{K}}{T_{L}}\right)^{.28r}
Both of these are used to calculate the final equivalent potential temperature:
.. math:: \theta_{E}=\theta_{DL}\exp\left[\left(\frac{3036.}{T_{L}}
-1.78\right)*r(1+.448r)\right]
Parameters
----------
pressure: `pint.Quantity`
Total atmospheric pressure
temperature: `pint.Quantity`
Temperature of parcel
dewpoint: `pint.Quantity`
Dewpoint of parcel
Returns
-------
`pint.Quantity`
The equivalent potential temperature of the parcel
Notes
-----
[Bolton1980]_ formula for Theta-e is used, since according to
[DaviesJones2009]_ it is the most accurate non-iterative formulation
available.
"""
t = temperature
td = dewpoint
p = pressure
e = saturation_vapor_pressure(dewpoint)
r = saturation_mixing_ratio(pressure, dewpoint)
t_l = 56 + 1. / (1. / (td - 56) + np.log(t / td) / 800.)
th_l = t * (1000 / (p - e)) ** constants.kappa * (t / t_l) ** (0.28 * r)
th_e = th_l * np.exp((3036. / t_l - 1.78) * r * (1 + 0.448 * r))
return th_e
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 constants.sat_pressure_0c * np.exp(17.67 * (temperature - 273.15)
/ (temperature - 29.65))