A Two-Layer Model

This post is part of a primer on infrared spectroscopy and global warming. The previous post discusses the issue of saturation in the 14-micron band of carbon dioxide in a single-layer model.   The post before that discusses Beer's Law, and is a necessary prerequisite to understanding this post.  This post starts to look beyond the single-layer model, by discussing a two-layer model, and beginning a discussion of radiative transfer.

In the previous posts on Beer's Law, I started with the assumption that the radiation from the source was much hotter than the layer where the species of interest is located.  In such a case, the transmittance is equal to I_s/I₀ or

      t = I_s/I₀

in which  I_s is the signal seen at the detector and I₀ is the radiance from the source.  Let's eliminate that assumption and allow I₀ to be the radiance from a black body at some temperature T₀ I now introduce I₁, the radiance from a blackbody at the termperature of the gas in the cell, T₁.  The transmittance becomes:

      t = (I_s-I₁)/(I₀-I₁)

Note that if  I₀ much greater than I₁ and the absorption  is not saturated, so that  I_s is much greater than I₁, the equation approached the value of the single-layer model assumption.  Now, I rearrange this equation, solving for I_s  to show the signal that should be expected at the detector:

     I_s = I₁ + t(I₀-I₁)

If t = 1, all radiation is transmitted through the layer, then Is = I₀, as expected.  If t =0, and no radiation is transmitted through the layer, then I_s is equal to the radiation from a black body at T₁.

There is no restriction, that I₀ be greater than I₁. If the reverse is true, the signal shows the emission spectrum of the sample at T₁. In a previous post, I discussed some of the consequences of Kirchoff's Law, particularly that the black-body radiation curve acts as a natural limit to the amount of radiation that is absorbed or emitted by real matter.  Note that the radiance received at the detector in a two-layer model is constrained to be intermediate between the radiation from the two black-body temperatures!

Now I present an example.  Assume that the source black-body is at 288 K.  Assume the sample gas is at constant pressure of one atmosphere with a concentration of 380 ppm carbon dioxide.  Allow the cell to be one kilometer long at a constant temperature of 278 K.

The green curve shows the radiance at the detector in this example.  The black-body curves at 278 K and 288 K are shown for reference. Also, below I show the absorptivity spectrum of carbon dioxide for comparison:

In the low micron region, the green curve shows very little radiance as the black-body curves are not hot enough to have significant radiance at those wavelengths.  Up to about 8 microns, there is some slight absorbance that leads to a curve intermediate between the black-body curves.  Where there is no effective absorbance by carbon dioxide the curve approaches the the source curve at 288 K.  Near the 15-micron band, radiation is absorbed and the radiance approaches that of a black-body at the sample temperature. In this case, energy is being absorbed by the layer, and all things being equal, the layer will eventually get hotter.  When the layer gets hotter, it will radiate at a higher temperature!

Another useful way to look at the data is in terms of brightness temperature.  The brightness temperature is a way of converting a radiance spectrum into temperature units.  At a particular wavelength the spectrum is expressed in terms of the temperature of a black body emitting with the equivalent spectral radiance.

The brightness spectrum contains the same information as the radiance spectrum except that it is much easier to see the differences.  It is not necessary to plot the reference black-body curves: they would just be flat lines at their given temperatures.

Like the single-layer model  the two-layer model has limitations.  The most obvious limitation is that I have only treated two layers.  Temperature in the atmosphere is not restricted to two values, and a fuller atmospheric model must address that fact.  Additionally, I have not allowed the pressure to vary.  

The next post in this series expands the concept of a two-layer model to a three-layer model and discusses the implications of a multi-layer radiative transfer model.


Sources

• Alison K. Lazarevich, Christopher C. Carter, Michael E. Thomas, Isaac N. Bankman, Eric W. Rogala, and Richard J. Green,  Determination of CL Using Passive Remote Sensing of  Background Radiance, Sixth Joint Conference on Standoff Detection for Chemical and Biological Defense, 25-29 October 2004, Williamsburg, VA
• Chandrasekhar, S., 1950. Radiative Transfer, Oxford University Press, London (reprinted by Dover New York).
• Dry Adiabatic Lapse Rate
• NIST Webbook