Brian E. J. Rose, University at Albany
Planetary energy balance is the foundation for all climate modeling. So far we have expressed this through a globally averaged budget
$$C \frac{d T_s}{dt} = (1-\alpha) Q - OLR$$and we have written the OLR in terms of an emission temperature $T_e$ where by definition
$$ OLR = \sigma T_e^4 $$Using values from the observed planetary energy budget, we found that $T_e = 255$ K
The emission temperature of the planet is thus about 33 K colder than the mean surface temperature (288 K).
That's about -18ºC.
Let's plot global, annual average observed air temperature from NCEP reanalysis data.
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import xarray as xr
ncep_url = "http://www.esrl.noaa.gov/psd/thredds/dodsC/Datasets/ncep.reanalysis.derived/"
ncep_air = xr.open_dataset(ncep_url + "pressure/air.mon.1981-2010.ltm.nc",
decode_times=False)
print(ncep_air)
<xarray.Dataset> Dimensions: (lat: 73, level: 17, lon: 144, nbnds: 2, time: 12) Coordinates: * level (level) float32 1000.0 925.0 850.0 700.0 600.0 500.0 ... * lon (lon) float32 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 ... * time (time) float64 -6.571e+05 -6.57e+05 -6.57e+05 ... * lat (lat) float32 90.0 87.5 85.0 82.5 80.0 77.5 75.0 ... Dimensions without coordinates: nbnds Data variables: climatology_bounds (time, nbnds) float64 ... air (time, level, lat, lon) float64 ... valid_yr_count (time, level, lat, lon) float64 ... Attributes: description: Data from NCEP initialized reanalysis (4... platform: Model Conventions: COARDS not_missing_threshold_percent: minimum 3% values input to have non-missi... history: Created 2011/07/12 by doMonthLTM\nConvert... title: monthly ltm air from the NCEP Reanalysis References: http://www.esrl.noaa.gov/psd/data/gridded... dataset_title: NCEP-NCAR Reanalysis 1
# Take global, annual average and convert to Kelvin
from xarray.ufuncs import cos, deg2rad, log
coslat = cos(deg2rad(ncep_air.lat))
weight = coslat / coslat.mean(dim='lat')
Tglobal = (ncep_air.air * weight).mean(dim=('lat','lon','time'))
Tglobal.name
Tglobal
<xarray.DataArray (level: 17)> array([ 15.179082, 11.207002, 7.838327, 0.219941, -6.448343, -14.888844, -25.570467, -39.369685, -46.797908, -53.652235, -60.563551, -67.006048, -65.532927, -61.486637, -55.853581, -51.593945, -43.219982]) Coordinates: * level (level) float32 1000.0 925.0 850.0 700.0 600.0 500.0 400.0 ...
# a "quick and dirty" visualization of the data
Tglobal.plot()
[<matplotlib.lines.Line2D at 0x1246a3b00>]
Let's make a better plot.
Here we're going to use a package called metpy
to automate plotting this temperature profile in a way that's more familiar to meteorologists: a so-called skew-T plot.
from metpy.plots import SkewT
fig = plt.figure(figsize=(9, 9))
skew = SkewT(fig, rotation=30)
skew.plot(Tglobal.level, Tglobal, color='black', linestyle='-', linewidth=2, label='Observations')
skew.ax.set_ylim(1050, 10)
skew.ax.set_xlim(-75, 45)
# Add the relevant special lines
skew.plot_dry_adiabats(linewidth=0.5)
skew.plot_moist_adiabats(linewidth=0.5)
#skew.plot_mixing_lines()
skew.ax.legend()
skew.ax.set_title('Global, annual mean sounding from NCEP Reanalysis',
fontsize = 16)
Text(0.5,1,'Global, annual mean sounding from NCEP Reanalysis')
Note that surface temperature in global mean is indeed about 288 K or 15ºC as we keep saying.
So where do we find temperature $T_e=255$ K or -18ºC?
Actually in mid-troposphere, near 500 hPa or about 5 km height.
We can infer that much of the outgoing longwave radiation actually originates far above the surface.
Recall that our observed global energy budget diagram shows 217 out of 239 W m$^{-2}$ total OLR emitted by the atmosphere and clouds, only 22 W m$^{-2}$ directly from the surface.
This is due to the greenhouse effect.
So far we have dealt with the greenhouse in a very artificial way in our energy balance model by simply assuming
$$ \text{OLR} = \tau \sigma T_s^4 $$i.e., the OLR is reduced by a constant factor from the value it would have if the Earth emitted as a blackbody at the surface temperature.
Now it's time to start thinking a bit more about how the radiative transfer process actually occurs in the atmosphere, and how to model it.
# Using pre-defined code for the Planck function from the climlab package
from climlab.utils.thermo import Planck_wavelength
# approximate emission temperature of the sun in Kelvin
Tsun = 5780.
# boundaries of visible region in nanometers
UVbound = 390.
IRbound = 700.
# array of wavelengths
wavelength_nm = np.linspace(0., 3500, 400)
to_meters = 1E-9 # conversion factor
label_size=14
fig, ax = plt.subplots(figsize=(14,7))
ax.plot(wavelength_nm,
Planck_wavelength(wavelength_nm * to_meters, Tsun))
ax.grid()
ax.set_xlabel('Wavelength (nm)', fontsize=label_size)
ax.set_ylabel('Spectral radiance (W sr$^{-1}$ m$^{-3}$)', fontsize=label_size)
# Mask out points outside of this range
wavelength_vis = np.ma.masked_outside(wavelength_nm, UVbound, IRbound)
# Shade the visible region
ax.fill_between(wavelength_vis, Planck_wavelength(wavelength_vis * to_meters, Tsun))
title = 'Blackbody emission curve for the sun (T = {:.0f} K)'.format(Tsun)
ax.set_title(title, fontsize=label_size);
ax.text(280, 0.8E13, 'Ultraviolet', rotation='vertical', fontsize=12)
ax.text(500, 0.8E13, 'Visible', rotation='vertical', fontsize=16, color='w')
ax.text(800, 0.8E13, 'Infrared', rotation='vertical', fontsize=12);
/Users/br546577/anaconda3/lib/python3.6/site-packages/climlab/utils/thermo.py:159: RuntimeWarning: divide by zero encountered in true_divide u = h*c/l/k/T /Users/br546577/anaconda3/lib/python3.6/site-packages/climlab/utils/thermo.py:160: RuntimeWarning: invalid value encountered in true_divide return 2*k**5*T**5/h**4/c**3*u**5/(np.exp(u)-1)
The shape of the spectrum is a fundamental characteristic of radiative emissions (think about the color of burning coals in a fire – cooler = red, hotter = white)
Theory and experiments tell us that both the total flux of emitted radiation, and the wavelength of maximum emission, depend only on the temperature of the source!
The theoretical spectrum was worked out by Max Planck and is therefore known as the “Planck” spectrum (or simply blackbody spectrum).
Figure reproduced from Marshall and Plumb (2008): Atmosphere, Ocean, and Climate Dynamics
Going from cool to warm:
The integral of these curves over all wavelengths gives us our familiar $\sigma T^4$
Mathematically it turns out that
$$ λ_{max} T = \text{constant} $$(known as Wien’s displacement law).
By fitting the observed solar emission to a blackbody curve, we can deduce that the emission temperature of the sun is about 6000 K.
Knowing this, and knowing that the solar spectrum peaks at 0.6 micrometers, we can calculate the wavelength of maximum terrestrial radiation as
$$ λ_{max}^{Earth} = 0.6 ~ \mu m \frac{6000}{255} = 14 ~ \mu m $$This is in the far-infrared part of the spectrum.
Now let's look at normalized blackbody curves for Sun and Earth:
Figure reproduced from Marshall and Plumb (2008): Atmosphere, Ocean, and Climate Dynamics
There is essentially no overlap between the two spectra.
This is the fundamental reason we can discuss the solar “shortwave” and terrestrial “longwave” radiation as two distinct phenomena.
In reality all radiation exists on a continuum of different wavelengths. But in climate science we can get a long way by thinking in terms of a very simple “two-stream” approximation (short and longwave). We’ve already been doing this throughout the course so far!
Now look at the atmospheric absorption spectra.
(fraction of radiation at each wavelength that is absorbed on a single vertical path through the atmosphere)