A Brief Review of Radiation

6. A Brief Review of Radiation#

This notebook is part of The Climate Laboratory by Brian E. J. Rose, University at Albany.

2. Solar Radiation#

Let’s plot a spectrum of solar radiation.

For Python details, click to expand the code blocks (or see the code in the notebook).

#  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(10., 3500., 400)
to_meters = 1E-9  # conversion factor

label_size = 16
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);

../_images/994067a5f88fe0175ab10a270840710cc5e2333078dc5b2e24b9cc5219874a80.png

Spectrum peaks in the visible range
most energy at these wavelength.
No coincidence that our eyes are sensitive to this range of wavelengths!
Longer wavelengths called “infrared”, shorter wavelengths called “ultraviolet”.

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).

fig, ax = plt.subplots(figsize=(14,7))
wavelength_um = wavelength_nm / 1000
for T in [24000,12000,6000,3000]:
    ax.plot(wavelength_um, 
         (Planck_wavelength(wavelength_nm * to_meters, T) / T**4),
           label=str(T) + ' K')
ax.legend(fontsize=label_size)
ax.set_xlabel('Wavelength (um)', fontsize=label_size)
ax.set_ylabel('Normalized spectral radiance (W sr$^{-1}$ m$^{-3}$ K$^{-4}$)', fontsize=label_size)
ax.set_title("Normalized blackbody emission spectra $T^{-4} B_{\lambda}$ for different temperatures");

../_images/bae44d24e44a238517bb0ab937e3288df3605f36b6b424a51646d23024d45e11.png

Going from cool to warm:

total emission increases
maximum emission occurs at shorter wavelengths.

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.

3. Terrestrial Radiation and absorption spectra#

Terrestrial versus solar wavelengths#

Now let’s look at normalized blackbody curves for Sun and Earth:

fig, ax = plt.subplots(figsize=(14,7))
wavelength_um = np.linspace(0.1, 200, 10000)
wavelength_meters = wavelength_um / 1E6

for T in [6000, 255]:
    ax.semilogx(wavelength_um, 
         (Planck_wavelength(wavelength_meters, T) / T**4 * wavelength_meters),
           label=str(T) + ' K')
ax.legend(fontsize=label_size)
ax.set_xlabel('Wavelength (um)', fontsize=label_size)
ax.set_ylabel('Normalized spectral radiance (W sr$^{-1}$ m$^{-2}$ K$^{-4}$)', fontsize=label_size)
ax.set_title("Normalized blackbody emission spectra $T^{-4} \lambda B_{\lambda}$ for the sun ($T_e = 6000$ K) and Earth ($T_e = 255$ K)",
            fontsize=label_size);

../_images/4c5c9b1201799722722e229de69433b405f0e11567a328d252eb81808a65ea2e.png

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!

Atmospheric absorption spectra#

Now look at the atmospheric absorption spectra.

(fraction of radiation at each wavelength that is absorbed on a single vertical path through the atmosphere)

Marshall and Plumb Figure 2.5

Figure reproduced from Marshall and Plumb (2007): Atmosphere, Ocean, and Climate Dynamics [Marshall and Plumb, 2007]

Atmosphere is almost completely transparent in the visible range, right at the peak of the solar spectrum
Atmosphere is very opaque in the UV
Opacity across the IR spectrum is highly variable!
Look at the gases associated with various absorption features:
Main players include H\(_2\)O, CO\(_2\), N\(_2\)O, O\(_2\).
Compare to major constituents of atmosphere, in decreasing order:
- 78% N\(_2\)
- 21% O\(_2\)
- 1% Ar
- H\(_2\)O (variable)

The dominant constituent gases N\(_2\) and O\(_2\) are nearly completely transparent across the entire spectrum (there are O\(_2\) absorption features in far UV, but little energy at these wavelengths).
The greenhouse effect mostly involves trace constituents:
- O\(_3\) = 500 ppb
- N\(_2\)O = 310 ppb
- CO\(_2\) = 400 ppm (but rapidly increasing!)
- CH\(_4\) = 1.7 ppm
Note that most of these are tri-atomic molecules! There are fundamental reasons for this: these molecules have modes of rotational and vibration that are easily excited at IR wavelengths. See courses in radiative transfer!

Credits#

This notebook is part of The Climate Laboratory, an open-source textbook developed and maintained by Brian E. J. Rose, University at Albany.

It is licensed for free and open consumption under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.

Development of these notes and the climlab software is partially supported by the National Science Foundation under award AGS-1455071 to Brian Rose. Any opinions, findings, conclusions or recommendations expressed here are mine and do not necessarily reflect the views of the National Science Foundation.