This is a technical thread – not really “visualization” at all, just part of the series – looking for comment and verification.
Interested people can read a little about line shapes in Atmospheric Radiation and the “Greenhouse” Effect Part Twelve – The Curve of Growth - and of course there’s no problems with questions, but I haven’t aimed to explain the subject in this article.
I’ve been looking at the absorption line shapes in the upper troposphere/lower stratosphere.
Lorenz line shapes (collisional broadening) dominate in the lower atmosphere, Doppler broadening dominates in the upper atmosphere – and in some middle ground there is a “convolution” which is a Voigt line shape. This is bit of a pig because instead of being a nice well-formed function (like Lorenz or Doppler), each point on each line shape is calculated by an integral from -∞ to ∞. That’s one absorption line requiring many many integrals of a complex function between -∞ to ∞.
Here are a few pages explaining the topic from the excellent (very technical) Radiation & Climate, Vardavas & Taylor (2007).
The key point is the value of the mixing parameter, a, that determines whether the line needs to be modeled as a Voigt profile at all.
Note especially the last paragraph. If a > 1 for all lines of interest then we can use the simple Lorenz formula (pressure broadening).
So after playing around comparing line shapes for a while I realized that in the region of the atmosphere that we have been getting to in this series so far, perhaps the Voigt line shape wouldn’t be needed.
So let’s look at the likely values of a=bl0/γd.
My calculation, using eqns 4.5 & 4.11 and using v0 instead of ω0:
a = bl0.(p/p0).(T0/T)nair.(c/v0).(m/2RT)1/2
Also note that the formula for the Lorenz broadening (eqn 4.5) is an approximation, and the more accurate formula has an exponent which is measured (and can be retrieved from the HITRAN database) and here called nair.
nair varies from 0.41-0.78 for the CO2 lines between 200-2500 cm-1, and, just for interest, the measured line width at 1013 hPa and 296K (here called bl0) ranges from 0.055 – 0.095 cm-1.
Let’s re-arrange this formula to group things a bit better:
a = C . bl0/v0 . pT0nair/Tnair+0.5.
where the constant, C = c/p0 . (m/2R)1/2 = 152
p0 = 1013 hPa and T0 = 296K are the pressure and temperature at which the HITRAN measurements are made. Note that the formula requires SI units so p0=101300.
Just a note that the Matlab program seen in earlier articles in this series and described in Visualizing Atmospheric Radiation – Part Five – The Code uses gama for the line half width, nair for nair and v for v0.
So now we can plug some values for the lower stratosphere into the equation, take the whole HITRAN CO2 database and plot line strength, S vs a.
The reason for plotting line strength was expecting that there would be some values of a <1 and therefore requiring Voigt treatment and so wondering if they were so weak they could be ignored. Always want to skip the extra mile if possible..
So here is such a graph, with all 53,757 lines (in the range of interest) at 50 hPa and 210K:
The minimum value of the mixing parameter, a = 1.4. We are in the clear.
Comments please – mistakes in the formula re-arrangement?
Here is the MATLAB code for plotting the above graph:
% Voigt/Lorenz determination – value of mixing parameter, a,for CO2
% ref Vardavas & Taylor 2007, p91
% uses HITRAN database to determine where in atmosphere and line database
% a<1. When a>1 line profile is Lorenzian
p0=1.013e5; % std pressure for the HITRANS database in Pa
T0=296; % std temperature for HITRANS database in K
c=3e8; % speed of light
mco2=44.01e-3; % mass of mole of CO2
R=8.3143; % gas constant
% for now pick a temperature and pressure
p=5e3; % 50mbar
T=210; % temperature in K
% read in HITRAN for CO2 for the range considered in RTE program
[ v S iso gama nair ] = Hitran_read_0_2(2, 200, 2500, 1);
title(['p= ' num2str(p/100) ' hPa, T= ' num2str(T) ' K, Min a= ' num2str(min(a))]);
Here’s a histogram of values:
I was also interested in the breakdown of the above results in relevant bands. It’s clear that the mixing parameter will be lower when v is highest, and the 2500 cm-1 region is of less concern anyway as the atmospheric radiation is comparatively very low at this point (4 μm).
It does appear that the lines getting close to the Voigt threshold are those in the higher wavenumber (lower wavelength region, 4-5μm) that is anyway of less impact on the radiative balance in the atmosphere.
Part One - some background and basics
Part Two - some early results from a model with absorption and emission from basic physics and the HITRAN database
Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions
Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed
Part Five – The Code - code can be downloaded, includes some notes on each release
Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased
Part Eight – CO2 Under Pressure - how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases
Part Nine – Reaching Equilibrium - when we start from some arbitrary point, how the climate model brings us back to equilibrium (for that case), and how the energy moves through the system
Part Ten – “Back Radiation” - calculations and expectations for surface radiation as CO2 is increased
Part Eleven – Stratospheric Cooling - why the stratosphere is expected to cool as CO2 increases
Part Twelve – Heating Rates - heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.
Radiation and Climate, I.M. Vardavas & F.W. Taylor, Oxford Science Publications; International Series of Monographs on Physics 138 (2007)
The data used to create these graphs comes from the HITRAN database:
The HITRAN 2008 molecular spectroscopic database, by L.S. Rothman et al, Journal of Quantitative Spectroscopy & Radiative Transfer (2009)
The HITRAN 2004 molecular spectroscopic database, by L.S. Rothman et al., Journal of Quantitative Spectroscopy & Radiative Transfer (2005)